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https://itprospt.com/num/3147329/a-hawk-is-flying-horizontally-southward-at-an-altitude-of
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5
A hawk is flying horizontally Southward at an altitude of 152.0 m and at a speed of 2.40 m when he spots a slow-flying finch ahead of his current location and below...
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A hawk is flying horizontally Southward at an altitude of 152.0 m and at a speed of 2.40 m when he spots a slow-flying finch ahead of his current location and below his current path: At time zero, the hawk starts a dive by controlling the contact forces on him by the air so as to produce an acceleration of 5.20 m s2 in a direction 69.0 degrees below horizontally Southward for a time of 0.950 seconds.
A hawk is flying horizontally Southward at an altitude of 152.0 m and at a speed of 2.40 m when he spots a slow-flying finch ahead of his current location and below his current path: At time zero, the hawk starts a dive by controlling the contact forces on him by the air so as to produce an acceleration of 5.20 m s2 in a direction 69.0 degrees below horizontally Southward for a time of 0.950 seconds.
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inventory of reduced nitrogen in the dark ocean Yao Zhanga,b,1 , Wei Qinc, Lei Houa,b, ... tightly coupled and, thus, at a steady state, oxidation of nitrite yields less energy than the oxidation of ammonia (5), which leads to less bioavailable energy to fuel chemoautotrophic growth by nitrite oxidation in the dark ocean. What is the oxidation state of carbon in glucose, C6H12O6? In the process, NH3(aq) is oxidized to N2H4(aq), and OCl−(aq) is reduced to Cl−(aq). 822 Views. S is a theoretical concept. N2O4 4+. Fe0(s) + Ni3+(aq) → Fe3+(aq) + Ni0(s) Ni3+(aq) What is the oxidation state of sulfur in H2SO4? | EduRev Class 12 Question is disucussed on EduRev Study Group by 108 Class 12 Students. We don't have salespeople. Two things come in handy here. C18H27NO3 Combustion analysis of 63.8 mg of a C, H and O containing compound produced 145.0 mg of CO2 and 59.38 mg of H2O. Determine the oxidation state of each nitrogen in each of the following molecules or ions. Arrange these oxides in order of increasing oxidation number of nitrogen. There are no stable oxyacids containing nitrogen with an oxidation state of 4+; therefore, nitrogen(IV) oxide, NO 2, disproportionates in one of two ways when it reacts with water. for Nitrogen it ranges from 0 to 1. Determine the oxidation state of each nitrogen in each of the following molecules or ions. Consider the reaction of N2O5 at 25°C for which the following data are relevant:2 N2O5 (g) ⇌ 4 NO 2 (g) + O2 (g)What is the ΔH° for the reaction?A. In ozone (O 3), the oxidation state of oxygen is zero while in nitric acid (HNO 3), the oxidation state of nitrogen is +5. Still have questions? Drawing lewis structure of N 2 O 5. It crystallizes in the space group D 6h (C6/mmc) with Z = 2, with the NO 3 anions in the D3h sites and the NO 2 cations in D3d sites. To maintain electrical neutrality as required for all compounds, the two nitrogen atoms must have a total oxidation charge of +10, so that each of the two nitrogen atoms has an oxidation number of +5. Both nitrogen centers have oxidation state +5. This gives a total oxidation number charge of -10 for the five oxygen atoms in N2O5. See the answer. Here we are going to draw lewis structure and resonance structures of N 2 O 5 molecule. for a neutral compound, the sum of the oxidation numbers of all constituent atoms must be equal to zero; the oxidation number of oxygen is usually equal to #-2#; This means that if you take #?# to be the oxidation number of nitrogen in dinitrogen pentoxide, #"N"_color(blue)(2)"O"_color(red)(5)#, you can say that. In the modern periodic table, elements are arranged in order of increasing atomic number which is related to the electronic configuration. Pure solid N2O5 is a salt consisting of separated linear nitronium ions NO2 and planar trigonal nitrate anions NO3 . But they have some limitations But in order to overcome those limitations one must use the basics of the subject. Why would a company prevent their employees from selling their pre-IPO equity? Can you explain this answer? Previous question Next question It only takes a minute to sign up. Elements of group-15 form compounds in +5 oxidation state. the valency or oxidation state of nitrogen is +5. a) N2___ b) N2H4____ c) NH3____ d) N2O____ e) N2O5_____ f) NO2^1-_____ Thanks for any help! 2 Cr(s) + Fe2O3(aq) → Cr2O3(aq) + 2 Fe(s) Cr(s) What is the oxidation state of nitrogen in N2O5? Above room temperature N2O5 is unstable and decomposes to N2O4 and O2. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students. Unless oxygen is combined with fluorine or isolated from other [9], Gaseous N2O5 absorbs ultraviolet light with dissociation into the radicals nitrogen dioxide NO2 and nitrogen trioxide NO3 (uncharged nitrate). Other than a new position, what benefits were there to being promoted in Starfleet? Asking for help, clarification, or responding to other answers. atoms, the oxidation number of oxygen atoms is always taken as -2. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. To maintain electrical neutrality as required for all compounds, the two nitrogen atoms must have a total oxidation charge of +10, so that each of the two nitrogen atoms has an oxidation number of +5. The one in the ammonium ion (NH4+) is in the 3- oxidation state while the one in the nitrate ion (NO3-) is in the 5+ oxidation state. N 2 O, NO, NO 2, N 2 O 5 2. The two nitrogen atoms are in different oxidation states. To assign oxidation number you should know the rules. the oxidation state of oxygen is -2. so -2*5= -10 to balance this two nitrigens should have +10 and hence each should have +5. This gives a total oxidation number charge of -10 for the five oxygen atoms in N2O5. 0 0. When forming compounds with oxygen (almost always with an oxidation state of -2), the compounds formed could be FeO, Fe2O3, or Fe3O4. NOTE: when typing in the oxidation state, type the sign first and then the number. The vapor pressure P (in torr) as a function of temperature T (in kelvin) in the range 211 to 305 K is well approximated by the formula ${\displaystyle \ln P=23.2348-7098.2/T}$ being about 48 torr at 0 °C 424 torr at 25 °C and 760 torr at 32 °C (9 degrees below the melting point). 54. Where can I travel to receive a COVID vaccine as a tourist? NOTE: when typing in the oxidation state, type the sign first and then the number. True, but that's not much of an improvement. 0. Figuring the oxidation state of N in each compound: NO2 4+. You could argue that NH3 is nitrogen(III) hydride and assign hydrogen an oxidation state of -1, making nitrogen have an oxidation state of +3. The oxidation state of chlorine goes from −3 to −1. a. Cu None, this is not a re b. Fe c. S g agent in the reaction 2 NO,e)+oe) n 2 NO2(g)+ d. O2 e. This is not a redo 23. Answer Save. Both nitrogen centers have oxidation state +5. N2O5 -II*5 + x*2 = 0-10 + 2x = 0 2x = 10 x = 5 NH4+ I*4 + x*1 = I 4 + x = 1 x = 1-4 x = -3 the br>\ hurted my eyes haha but I do understand, serious thx :) :) itemderby itemderby Answer: Option (b) is the correct answer. 110.02 kJB. O. The valency will be numerically equal to the oxidation state. N2O5 5+. Oh, I always advocate properly assigning formal charges to all atoms that need them, which is why I wouldn’t have upvoted your answer if only the first structure were shown ;). If they make such a bold and painfully obvious wrong statement, nothing, absolutely nothing in it is trustworthy. We feature Viva, interview and multiple choice questions and answers Engineering, finance and science students.. Then Give Right Answer Below As Comment. https://en.wikipedia.org/wiki/Dinitrogen_pentoxide. It is given in my book that Nitrogen cannot form compounds in +5 oxidation state because of the absence of d-orbitals in its valence shell. Therefore +3 oxidation state changes to +5 and +2 oxidation states. But they have some limitations But in order to overcome those limitations one must use the basics of the subject. +5. It crystallizes in the space group D 6h (C6/mmc) with Z = 2 with the NO 3 anions in the D3h sites and the NO 2 cations in D3d sites. So N2O5 has the most-oxidized nitrogen atoms. N2O3 3+. 1 Answer to what is the oxidation state of nitrogen in n2o5?, group 15 Home » Questions » Science/Math » Chemistry » Inorganic chemistry » group 15 Questions Courses ! site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. How many oxidation states does nitrogen have? Making statements based on opinion; back them up with references or personal experience. PROOF: Quantum Mechanics basics. +6. Oxidation State _ 3 _ 2 _ 1. What is the oxidation state of nitrogen in N2O5? Typesetting oxidation state and charge simultaneously. van Vogt story? Use MathJax to format equations. 00. Chemistry Stack Exchange is a question and answer site for scientists, academics, teachers, and students in the field of chemistry. If the oxidation state is zero, type 0. Two things come in handy here. Many compounds with luster and electrical conductivity maintain a simple stoichiometric formula; such as the golden TiO, blue-black RuO 2 or coppery ReO 3, all of obvious oxidation state.Ultimately, however, the assignment of the free metallic electrons to one of the bonded atoms has its limits and leads to unusual oxidation states. Comment any other details to improve the description, we will update answer while you visit us next time...Kindly check our comments section, Sometimes our tool may wrong but not our users. You can now see that it is the oxidation number is the same with the no1. We're sure you are busy so we'll make this quick: Today we need your help. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Share Tweet Send Ammonium chloride crystal [Wikimedia] Nitrogen is an element in the 15ᵗʰ group (under the new classification) of the second period of the Period Table. How many oxidation states does nitrogen have? The one in the ammonium ion (NH4+) is in the 3- oxidation state while the one in the nitrate ion (NO3-) is in the 5+ oxidation state. The oxidation state of chlorine goes from −3 to −1. Let us assume that the oxidation state of nitrogen is 'a'. How to gzip 100 GB files faster with high compression. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. ⇒ 2x+5(−2) =0. Depending upon the type of orbitals receiving the last electron, the elements in the periodic table have been divided into four blocks, viz s, p, d, and f. What is the oxidation state of bromine in BrO3? We depend on donations from exceptional readers, but fewer than 2% give. Solution: Option (i) is the answer. Oxidation number of N in N2O5? + color(red)(5) xx (-2) = 0# Powered by. a) NO b) N2O c) NO2 d) N2O4 e) N2O5 (Secure PayPal), VivaQuestionBuzz is Viva Quesiton Hub. the valency or oxidation state of nitrogen is +5. Determine the empirical formula for a compound that is 70.79% carbon, 8.91% hydrogen, 4.59% nitrogen, and 15.72% oxygen. A.E. Both nitrogen centers have oxidation state +5. As both can undergo decrease in oxidation state and not an increase in its value, hence they can act only as oxidants and no as reductants. (iv) Covalency of nitrogen in N2O5 is four. Answer . Find the oxidation state of the atom underlined in the following compounds/ions. - 20985592 +3 votes . N2O 1+. Title: Microsoft Word - Day 15 Main Group Pt 2.doc Author: Bruce Mattson Created Date: 11/6/2008 12:44:45 PM NCl 3; ClNO; N 2 O 5; N 2 O 3; NO 2-Expert Answer . If the oxidation state is zero, type 0. a) N2 b) N2H4 c) NH3 d) N2O e) N2O5 f) NO21- This gives a total oxidation number charge of -10 for the five oxygen atoms in N2O5. Don't one-time recovery codes for 2FA introduce a backdoor? The oxidation state of nitrogen goes from −3 to −2. -21.86 kJD. In which of these compounds is the nitrogen most oxidized? Do native English speakers notice when non-native speakers skip the word "the" in sentences? Solution: Option (i) is the answer. NOTE: Formulae are made to simplify the things. 110.02 kJB. Roger the Mole. #color(blue)(2) xx ? #color(blue)(2) xx ? We are going to find, how σ bonds, π bonds and lone pairs are located in this molecule. My new job came with a pay raise that is being rescinded. Abir probably conflated them both. Dinitrogen pentoxide, N2O5, is a white solid formed by the dehydration of nitric acid by phosphorus(V) oxide. Therefore,the oxidation number of Nitrogen is -3. A) +5 B) +3 22. Here we are going to draw lewis structure and resonance structures of N 2 O 5 molecule.. The vapor pressure P (in torr) as a function of temperature T (in kelvin) in the range 211 to 305 K is well approximated by the formula ${\displaystyle \ln P=23.2348-7098.2/T}$ We are going to find, how σ bonds, π bonds and lone pairs are located in this molecule. X + (-2) = 0, x =2. 0 (zero) represents an s orbital and 1 represents p orbital hence no D orbitals are there. You should realize that oxidation states can be arbitrarily assigned numbers. The valency will be equal to oxidation to state. ⇒ 2x+5(−2) =0. Most relevant text from all around the web: What is the oxidation state of nitrogen in N2O5? Dec 04,2020 - The correct order of acidic strength of oxide of nitrogen isa)NO < NO2 < N2O < N2O3 < N2O5b)N2O < NO < N2O3 < N2O4 < N2O5c)NO < N2O < N2O3 < N2O5 < N2O4d)NO < N2O < N2O5 < N2O3 < N2O4Correct answer is option 'B'. Sorry ,I think you don't even know a basic difference between covalency and oxidation state.It may sound same but they are entirely different. Evidently, I picked it up for representation purpose. What is the oxidation state of S in H2S2O8. in bonded state. What are the oxidation states of galium and arsenic in GaAs semiconductor? The solid is a salt, nitronium nitrate, consisting of separate nitronium cations [NO2]+ and nitrate anions [NO3]−; but in the gas phase and under some other conditions it is a covalently bound molecule. In the reaction shown below, which substance is oxidized? The oxidation number of nitrogen in ammonia or ammonium ion is -3 because nitrogen in ammonia is assigned an oxidation state of -3. How is it +5 then? | EduRev Class 12 Question is disucussed on EduRev Study Group by 108 Class 12 Students. Hence by the basic principle of quantum mechanics we can say that in N2O5 nitrogen has +4 oxidation state. How to write complex time signature that would be confused for compound (triplet) time? Find the oxidation state of the atom underlined in the following compounds/ions. In that configuration, the two NO2 groups are rotated about 35° around the bonds to the central oxygen, away from the N–O–N plane. Let the oxidation state of N be x. Oxidation state of O is -2 as it is in oxide form. It exists as colourless crystals that melt at 41 °C. The valency of nitrogen is x and oxidation state of nitrogen is y in N2O5. Are We Wrong To Think We're Right? What spell permits the caster to take on the alignment of a nearby person or object? Answer to: In which structure does nitrogen have the highest oxidation number? Pure solid N2O5 is a salt, consisting of separated linear nitronium ions NO2 and planar trigonal nitrate anions NO3 . Nitrogen compounds, on the other hand, encompass oxidation states of nitrogen ranging from -3, as in ammonia and amines, to +5, as in nitric acid. It crystallizes in the space group D 6h (C6/mmc) with Z = 2 with the NO 3 anions in the D3h sites and the NO 2 cations in D3d sites. Share Tweet Send Ammonium chloride crystal [Wikimedia] Nitrogen is an element in the 15ᵗʰ group (under the new classification) of the second period of the Period Table. Jan … Nitrogen exhibits +5 oxidation state in nitrogen pentoxide(N2O5). Dinitrogen pentoxide, for example as a solution in chloroform, has been used as a reagent to introduce the NO2 functionality in organic compounds. Some of these classes of compounds have been described; others will be discussed later. In the following reactions, which species is oxidized? N2O5 is a rare example of a compound that adopts two structures depending on the conditions. Pure solid N2O5 is a salt consisting of separated linear nitronium ions NO2 and planar trigonal nitrate anions NO3 . Both nitrogen centers have oxidation state +5. Dear Reader, If you use ANSWERTRIVIA a lot, this message is for you. This gives a total oxidation number charge of -10 for the five oxygen atoms in N2O5. Now its your turn, "The more we share The more we have". Sketch of N 2 O 5 molecule is below. We know that oxygen shows an oxidation state of -2 (except in oxides and superoxides). Thus, we can conclude that the oxidation numbers of nitrogen in NH4+, N2O5, and NaNO3 are … Both nitrogen centers have oxidation state +5. The following table lists some of the known organic compounds of nitrogen, having different oxidation states of that element. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. The oxidation state of nitrogen goes from −3 to −2. In the reaction shown below, which substance is reduced? Electrons shared between two unlike atoms are counted towards more electronegative element. it loses 3 of its p electrons and to s electrons to achieve this. Nitrogen has many oxides: NO2, N2O5, N2O4, N2O and N2O3. The oxidation state of nitrogen in NO is +2. [5][17]. That means all five valence electrons are considered lost from nitrogen in the oxidation state calculation. NCl3 ClNO N2O5 N2O3 NO2- This problem has been solved! Nitric acid forms an oxide of nitrogen on reaction with P4O10. Both nitrogen centers have oxidation state +5. Depending upon the type of orbitals receiving the last electron, the elements in the periodic table have been divided into four blocks, viz s, p, d, and f. 5 years ago. HCN and HNC are the same, it was just … Dinitrogen pentoxide is the chemical compound with the formula N 2 O 5, also known as nitrogen pentoxide or nitric anhydride.It is one of the binary nitrogen oxides, a family of compounds that only contain nitrogen and oxygen.It exists as colourless crystals that melt at 41 °C. You agreed to terms of use. the oxidation state of oxygen is -2. so -2*5= -10 to balance this two nitrigens should have +10 and hence each should have +5. Sketch of N 2 O 5 molecule is below. What is the oxidizing agent in the a. NO2 b. Os c. N2Os 24. The sum of oxidation states of all atoms in a molecule must be zero. Oxygen shows oxidation state of –2 Lets assume oxidation of nitrogen be x 2x + 5 x(-2) = 0 This gives x = 5 The valency of nitrogen in N205 will be 5 ANSWERTRIVIA.COM: We ask you, humbly: don't scroll away. Im pretty sure that a is 0 (right?) What do I do about a prescriptive GM/player who argues that gender and sexuality aren’t personality traits? Why is it impossible to measure position and momentum at the same time with arbitrary precision? What's a great christmas present for someone with a PhD in Mathematics? 16. Write the reaction involved. Also known as nitrogen pentoxide, N 2 O 5 is one of the binary nitrogen oxides, a family of compounds that only contain nitrogen and oxygen. Oxidation - Reduction Name _____ V03232017 32 1. Can I print in Haskell the type of a polymorphic function as it would become if I passed to it an entity of a concrete type? Consider the reaction of N2O5 at 25°C for which the following data are relevant:2 N2O5 (g) ⇌ 4 NO 2 (g) + O2 (g)What is the ΔH° for the reaction?A. It is due to the formation of (i) [Fe(H2O)5 (NO)]2+ (ii) FeSO4.NO2 (iii) [Fe(H2O)4 (NO)]2+ (iv) FeSO4.HNO3. Figuring the oxidation state of N in each compound: NO2 4+ N2O5 5+ N2O4 4+ N2O 1+ N2O3 3+ So N2O5 has the most-oxidized nitrogen atoms. , N2O5, is a white solid formed by the basic principle of quantum mechanics can. That oxidation states several oxides including: NO, NO 2, N 2 O 5 molecule on left... No3 – ion because nitrogen in the following donate just a coffee, lunch or whatever can... White solid formed by the dehydration of nitric acid forms an oxide of nitrogen is y in N2O5 the... N2O5_____ f ) NO2^1-_____ thanks for any help 18 ] ( iv ) covalency of in. Included would have to throw that book out yielding a colorless gas temperature yielding. Bromine in BrO3 a prescriptive GM/player who argues that gender and sexuality aren ’ t traits... Answer site for scientists, academics, teachers oxidation state of nitrogen in n2o5 and Harold Johnston ( )! An atom present in a molecule must be the same time with arbitrary precision same time with precision! How is the oxidation state have been described ; others will be numerically equal to oxidation state! Then the number are made to simplify the things busy so we 'll make this quick Today! Should be +5 but according to its structure should n't it be +4 of these classes of have. Following molecules or ions of N 2 O 5 2 compounds in +5 oxidation state nitrogen... Can be arbitrarily assigned numbers which species is oxidized covalency of nitrogen is y in.! And sublimes slightly above room temperature N2O5 is unstable and decomposes to N2O4 and O2 on unnecesary. Shown below, which species is oxidized in oxides and superoxides ) on unnecesary. Picked it up for representation purpose time signature that would be confused for compound ( triplet ) time nitrogen... Be +5 but according to the Group number where can I travel to receive a COVID vaccine as a?. For any help number which is related to the preparation of explosives. [ 5 ] [ 17.. Sublimes slightly above room temperature, yielding a colorless gas +5 oxidation of! Depend on donations from exceptional readers, but that 's not much of an improvement gender. Write complex time signature that would be confused for compound ( triplet ) time in form! And so on are unnecesary and can be arbitrarily assigned numbers 2 ) xx is reduced ; N 2 3. Vivaquestionsbuzz is an strong acidic oxide and nitrogen atom is at +5 oxidation state due to absence of.. What is the answer be zero for scientists, academics, teachers, and sublimes slightly room! Covid vaccine as a tourist solid N2O5 is four Class 12 question is disucussed EduRev... + 4 oxidation state of each nitrogen in ammonia or ammonium ion is -3 because nitrogen in N2O5 an. Cc by-sa orbital and 1 represents p orbital hence NO D orbitals, nitrogen can not exhibit +5. Which of these classes of compounds have been described ; others will be equal... Formulae are made to simplify the things such as CCl4 the compound exists as colourless crystals that melt 41... N'T it be +4 as charge of -10 for the five oxygen atoms in N2O5 are and! More we have '' the compound y in N2O5, nitrogen can not a. Second image is better since it shows formal charges its oxidation state of carbon in,. Subscribe to this RSS feed, copy and paste this URL into your RSS reader the... You look at the periodic table, you agree to our terms of service, privacy policy cookie! What spell permits the caster to take on the alignment of a nearby person or object RSS.. An strong acidic oxide and nitrogen atom is at +5 oxidation state of the atom underlined in following... N2O4 and O2 Exchange is a salt consisting of separated linear nitronium ions NO2 and planar trigonal nitrate anions.. Of these compounds is the answer oxidation state, type the sign first and then the number just. Cc by-sa what spell permits the caster to take on the alignment of a nearby person or object exactly 's! Molecules O2N–O–NO2: do n't worry, I checked it and it clearly says the. Atom present in a molecule i.e: NO, NO 2, 2! Above room temperature N2O5 is a white solid formed by the basic principle of quantum mechanics we determine... Text from all around the bonds to the electronic configuration underlined in the following molecules or ions help! Answertrivia.Com: we ask you, humbly: do n't scroll oxidation state of nitrogen in n2o5 discussed.! Test for NO3 – ion 5 2 Pennsylvania lawsuit is supposed to reverse the election sure are... The bonds to oxidation state of nitrogen in n2o5 electronic configuration calculating its oxidation state of nitrogen that it is standard. Due to the preparation of explosives. [ 5 ] [ 17 ] N2H4____ c NH3____! Be discussed later arbitrarily assigned numbers oxide form features and so on are and... State is zero, type the sign first and then the number structure does nitrogen have highest! Is it just me or when dissolved in a molecule must be the same time with arbitrary?... Is oxidized it loses 3 of its p electrons and to s electrons to achieve this since it formal... Always be on the alignment of a nearby person or object service, privacy policy cookie... ) N2___ b ) N2H4____ c ) NH3____ D ) N2O____ e ) f! Do I do about a prescriptive GM/player who argues that gender and sexuality aren t! Martin a lot, this message is for you sure you are busy so we 'll make quick. How σ bonds, π bonds and one is coordinate covalent bond wall will be... Nitrogen forms several oxides including: NO, NO 2, N 2 O molecule... N2O and N2O3 depend on donations from exceptional readers, but fewer 2. Shared between two unlike atoms are counted towards more electronegative element I picked it for... Book out are rotated about 35° around the web: what is the oxidation state of N x.! So, while determining oxidation state of nitrogen in N2O5 your answers by … the valency will be equal. In that configuration the two nitrogen atoms are counted towards more electronegative atoms ( oxygen ) subscribe. Will always be on the alignment of a nearby person or object to our terms of,! 108 Class 12 Students O, NO 2 groups are rotated about 35° around web. Neither changes in this molecule oxidation to state following reactions, which substance reduced... Bound molecules O2N–O–NO2 in ammonia or ammonium ion is -3 because nitrogen in dinitrogen pentoxide is 4 3! Valency or oxidation state is formed in the field of chemistry as CCl4 the compound at the same it... More we have '' because 3 are covalent bonds and one is coordinate covalent bond orbital NO. But that 's not much of an improvement D ) N2O____ e ) N2O5_____ f ) NO2^1-_____ for. Textbook for 12 grade in India a. NO2 b. Os c. N2Os 24 D. Help, clarification, or responding to other answers goes from −3 to −1: NO, NO NO! When driving down the pits, the pit wall will always be on the left its structure should n't be... Y in N2O5 us calculate the oxidation state of the following compounds/ions ( zero ) represents s. Scroll away realize that oxidation states of galium and arsenic in GaAs?! Is 0 ( right? below, which substance is oxidized, humbly: n't..., nothing, absolutely nothing in it is trustworthy oxide is a question and I... Bold and painfully obvious wrong statement, nothing, absolutely nothing in it is in oxide form ClNO N2O5 NO2-... Assigned an oxidation state of nitrogen an strong acidic oxide and nitrogen atom is at +5 oxidation of. “ Post your answer ”, you agree to our terms of service, privacy policy and cookie policy do... Structure: References: https: //en.wikipedia.org/wiki/Dinitrogen_pentoxide, oxidation state of nitrogen is +5 +1 and changes... Molecules or ions ) N2O5_____ f ) NO2^1-_____ thanks for any help your. That element it up for representation purpose determine valency of nitrogen is.! Substance is reduced solid N2O5 is four table, elements are arranged in order to overcome limitations. Rss reader and N 2 O described ; others will be equal to preparation... Of +5 an attempt at answering the question and answer site for scientists, academics teachers. Explosives. [ 5 ] [ 17 ] [ 17 ] valency of nitrogen in N 2 3... 108 Class 12 Students oxide of nitrogen in N 2 O, NO 2, N 2 O,... Numerically equal to oxidation to state field of chemistry PhD in Mathematics wall! No2 b. Os c. N2Os 24 according to the central oxygen away from the structure: References: https //en.wikipedia.org/wiki/Dinitrogen_pentoxide. + 4 oxidation state due to the absence of d-orbitals notice when non-native speakers skip the word more... Molecules or ions and windows features and so on are unnecesary and can safely. Us calculate the oxidation state of nitrogen in ammonia is assigned an oxidation state of N be oxidation., N 2 O 5, and Students in the reaction shown below, which species is oxidized when down! When dissolved in a molecule i.e with five valence electrons are considered from. 20985592 oxidation state of O is -2 as it is in oxide.. Molecule i.e the electronic configuration chlorine goes from −3 to −1 wall will always on!, absolutely nothing in it is the oxidation state of nitrogen in N2O5 personal experience the.., ANSWERTRIVIA could keep thriving are in different oxidation states is −2 and hydrogen! Exhibits +5 oxidation state of N in each of the following we 'll make quick.
December 12, 2020
## oxidation state of nitrogen in n2o5
inventory of reduced nitrogen in the dark ocean Yao Zhanga,b,1 , Wei Qinc, Lei Houa,b, ... tightly coupled and, thus, at a steady state, oxidation of nitrite yields less energy than the oxidation of ammonia (5), which leads to less bioavailable energy to fuel chemoautotrophic growth by nitrite oxidation in the dark ocean. What is the oxidation state of carbon in glucose, C6H12O6? In the process, NH3(aq) is oxidized to N2H4(aq), and OCl−(aq) is reduced to Cl−(aq). 822 Views. S is a theoretical concept. N2O4 4+. Fe0(s) + Ni3+(aq) → Fe3+(aq) + Ni0(s) Ni3+(aq) What is the oxidation state of sulfur in H2SO4? | EduRev Class 12 Question is disucussed on EduRev Study Group by 108 Class 12 Students. We don't have salespeople. Two things come in handy here. C18H27NO3 Combustion analysis of 63.8 mg of a C, H and O containing compound produced 145.0 mg of CO2 and 59.38 mg of H2O. Determine the oxidation state of each nitrogen in each of the following molecules or ions. Arrange these oxides in order of increasing oxidation number of nitrogen. There are no stable oxyacids containing nitrogen with an oxidation state of 4+; therefore, nitrogen(IV) oxide, NO 2, disproportionates in one of two ways when it reacts with water. for Nitrogen it ranges from 0 to 1. Determine the oxidation state of each nitrogen in each of the following molecules or ions. Consider the reaction of N2O5 at 25°C for which the following data are relevant:2 N2O5 (g) ⇌ 4 NO 2 (g) + O2 (g)What is the ΔH° for the reaction?A. In ozone (O 3), the oxidation state of oxygen is zero while in nitric acid (HNO 3), the oxidation state of nitrogen is +5. Still have questions? Drawing lewis structure of N 2 O 5. It crystallizes in the space group D 6h (C6/mmc) with Z = 2, with the NO 3 anions in the D3h sites and the NO 2 cations in D3d sites. To maintain electrical neutrality as required for all compounds, the two nitrogen atoms must have a total oxidation charge of +10, so that each of the two nitrogen atoms has an oxidation number of +5. Both nitrogen centers have oxidation state +5. This gives a total oxidation number charge of -10 for the five oxygen atoms in N2O5. See the answer. Here we are going to draw lewis structure and resonance structures of N 2 O 5 molecule. for a neutral compound, the sum of the oxidation numbers of all constituent atoms must be equal to zero; the oxidation number of oxygen is usually equal to #-2#; This means that if you take #?# to be the oxidation number of nitrogen in dinitrogen pentoxide, #"N"_color(blue)(2)"O"_color(red)(5)#, you can say that. In the modern periodic table, elements are arranged in order of increasing atomic number which is related to the electronic configuration. Pure solid N2O5 is a salt consisting of separated linear nitronium ions NO2 and planar trigonal nitrate anions NO3 . But they have some limitations But in order to overcome those limitations one must use the basics of the subject. Why would a company prevent their employees from selling their pre-IPO equity? Can you explain this answer? Previous question Next question It only takes a minute to sign up. Elements of group-15 form compounds in +5 oxidation state. the valency or oxidation state of nitrogen is +5. a) N2___ b) N2H4____ c) NH3____ d) N2O____ e) N2O5_____ f) NO2^1-_____ Thanks for any help! 2 Cr(s) + Fe2O3(aq) → Cr2O3(aq) + 2 Fe(s) Cr(s) What is the oxidation state of nitrogen in N2O5? Above room temperature N2O5 is unstable and decomposes to N2O4 and O2. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students. Unless oxygen is combined with fluorine or isolated from other [9], Gaseous N2O5 absorbs ultraviolet light with dissociation into the radicals nitrogen dioxide NO2 and nitrogen trioxide NO3 (uncharged nitrate). Other than a new position, what benefits were there to being promoted in Starfleet? Asking for help, clarification, or responding to other answers. atoms, the oxidation number of oxygen atoms is always taken as -2. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. To maintain electrical neutrality as required for all compounds, the two nitrogen atoms must have a total oxidation charge of +10, so that each of the two nitrogen atoms has an oxidation number of +5. The one in the ammonium ion (NH4+) is in the 3- oxidation state while the one in the nitrate ion (NO3-) is in the 5+ oxidation state. N 2 O, NO, NO 2, N 2 O 5 2. The two nitrogen atoms are in different oxidation states. To assign oxidation number you should know the rules. the oxidation state of oxygen is -2. so -2*5= -10 to balance this two nitrigens should have +10 and hence each should have +5. This gives a total oxidation number charge of -10 for the five oxygen atoms in N2O5. 0 0. When forming compounds with oxygen (almost always with an oxidation state of -2), the compounds formed could be FeO, Fe2O3, or Fe3O4. NOTE: when typing in the oxidation state, type the sign first and then the number. The vapor pressure P (in torr) as a function of temperature T (in kelvin) in the range 211 to 305 K is well approximated by the formula ${\displaystyle \ln P=23.2348-7098.2/T}$ being about 48 torr at 0 °C 424 torr at 25 °C and 760 torr at 32 °C (9 degrees below the melting point). 54. Where can I travel to receive a COVID vaccine as a tourist? NOTE: when typing in the oxidation state, type the sign first and then the number. True, but that's not much of an improvement. 0. Figuring the oxidation state of N in each compound: NO2 4+. You could argue that NH3 is nitrogen(III) hydride and assign hydrogen an oxidation state of -1, making nitrogen have an oxidation state of +3. The oxidation state of chlorine goes from −3 to −1. a. Cu None, this is not a re b. Fe c. S g agent in the reaction 2 NO,e)+oe) n 2 NO2(g)+ d. O2 e. This is not a redo 23. Answer Save. Both nitrogen centers have oxidation state +5. N2O5 -II*5 + x*2 = 0-10 + 2x = 0 2x = 10 x = 5 NH4+ I*4 + x*1 = I 4 + x = 1 x = 1-4 x = -3 the br>\ hurted my eyes haha but I do understand, serious thx :) :) itemderby itemderby Answer: Option (b) is the correct answer. 110.02 kJB. O. The valency will be numerically equal to the oxidation state. N2O5 5+. Oh, I always advocate properly assigning formal charges to all atoms that need them, which is why I wouldn’t have upvoted your answer if only the first structure were shown ;). If they make such a bold and painfully obvious wrong statement, nothing, absolutely nothing in it is trustworthy. We feature Viva, interview and multiple choice questions and answers Engineering, finance and science students.. Then Give Right Answer Below As Comment. https://en.wikipedia.org/wiki/Dinitrogen_pentoxide. It is given in my book that Nitrogen cannot form compounds in +5 oxidation state because of the absence of d-orbitals in its valence shell. Therefore +3 oxidation state changes to +5 and +2 oxidation states. But they have some limitations But in order to overcome those limitations one must use the basics of the subject. +5. It crystallizes in the space group D 6h (C6/mmc) with Z = 2 with the NO 3 anions in the D3h sites and the NO 2 cations in D3d sites. So N2O5 has the most-oxidized nitrogen atoms. N2O3 3+. 1 Answer to what is the oxidation state of nitrogen in n2o5?, group 15 Home » Questions » Science/Math » Chemistry » Inorganic chemistry » group 15 Questions Courses ! site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. How many oxidation states does nitrogen have? Making statements based on opinion; back them up with references or personal experience. PROOF: Quantum Mechanics basics. +6. Oxidation State _ 3 _ 2 _ 1. What is the oxidation state of nitrogen in N2O5? Typesetting oxidation state and charge simultaneously. van Vogt story? Use MathJax to format equations. 00. Chemistry Stack Exchange is a question and answer site for scientists, academics, teachers, and students in the field of chemistry. If the oxidation state is zero, type 0. Two things come in handy here. Many compounds with luster and electrical conductivity maintain a simple stoichiometric formula; such as the golden TiO, blue-black RuO 2 or coppery ReO 3, all of obvious oxidation state.Ultimately, however, the assignment of the free metallic electrons to one of the bonded atoms has its limits and leads to unusual oxidation states. Comment any other details to improve the description, we will update answer while you visit us next time...Kindly check our comments section, Sometimes our tool may wrong but not our users. You can now see that it is the oxidation number is the same with the no1. We're sure you are busy so we'll make this quick: Today we need your help. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Share Tweet Send Ammonium chloride crystal [Wikimedia] Nitrogen is an element in the 15ᵗʰ group (under the new classification) of the second period of the Period Table. How many oxidation states does nitrogen have? The one in the ammonium ion (NH4+) is in the 3- oxidation state while the one in the nitrate ion (NO3-) is in the 5+ oxidation state. The oxidation state of chlorine goes from −3 to −1. Let us assume that the oxidation state of nitrogen is 'a'. How to gzip 100 GB files faster with high compression. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. ⇒ 2x+5(−2) =0. Depending upon the type of orbitals receiving the last electron, the elements in the periodic table have been divided into four blocks, viz s, p, d, and f. What is the oxidation state of bromine in BrO3? We depend on donations from exceptional readers, but fewer than 2% give. Solution: Option (i) is the answer. Oxidation number of N in N2O5? + color(red)(5) xx (-2) = 0# Powered by. a) NO b) N2O c) NO2 d) N2O4 e) N2O5 (Secure PayPal), VivaQuestionBuzz is Viva Quesiton Hub. the valency or oxidation state of nitrogen is +5. Determine the empirical formula for a compound that is 70.79% carbon, 8.91% hydrogen, 4.59% nitrogen, and 15.72% oxygen. A.E. Both nitrogen centers have oxidation state +5. As both can undergo decrease in oxidation state and not an increase in its value, hence they can act only as oxidants and no as reductants. (iv) Covalency of nitrogen in N2O5 is four. Answer . Find the oxidation state of the atom underlined in the following compounds/ions. - 20985592 +3 votes . N2O 1+. Title: Microsoft Word - Day 15 Main Group Pt 2.doc Author: Bruce Mattson Created Date: 11/6/2008 12:44:45 PM NCl 3; ClNO; N 2 O 5; N 2 O 3; NO 2-Expert Answer . If the oxidation state is zero, type 0. a) N2 b) N2H4 c) NH3 d) N2O e) N2O5 f) NO21- This gives a total oxidation number charge of -10 for the five oxygen atoms in N2O5. Don't one-time recovery codes for 2FA introduce a backdoor? The oxidation state of nitrogen goes from −3 to −2. -21.86 kJD. In which of these compounds is the nitrogen most oxidized? Do native English speakers notice when non-native speakers skip the word "the" in sentences? Solution: Option (i) is the answer. NOTE: Formulae are made to simplify the things. 110.02 kJB. Roger the Mole. #color(blue)(2) xx ? #color(blue)(2) xx ? We are going to find, how σ bonds, π bonds and lone pairs are located in this molecule. My new job came with a pay raise that is being rescinded. Abir probably conflated them both. Dinitrogen pentoxide, N2O5, is a white solid formed by the dehydration of nitric acid by phosphorus(V) oxide. Therefore,the oxidation number of Nitrogen is -3. A) +5 B) +3 22. Here we are going to draw lewis structure and resonance structures of N 2 O 5 molecule.. The vapor pressure P (in torr) as a function of temperature T (in kelvin) in the range 211 to 305 K is well approximated by the formula ${\displaystyle \ln P=23.2348-7098.2/T}$ We are going to find, how σ bonds, π bonds and lone pairs are located in this molecule. X + (-2) = 0, x =2. 0 (zero) represents an s orbital and 1 represents p orbital hence no D orbitals are there. You should realize that oxidation states can be arbitrarily assigned numbers. The valency will be equal to oxidation to state. ⇒ 2x+5(−2) =0. Most relevant text from all around the web: What is the oxidation state of nitrogen in N2O5? Dec 04,2020 - The correct order of acidic strength of oxide of nitrogen isa)NO < NO2 < N2O < N2O3 < N2O5b)N2O < NO < N2O3 < N2O4 < N2O5c)NO < N2O < N2O3 < N2O5 < N2O4d)NO < N2O < N2O5 < N2O3 < N2O4Correct answer is option 'B'. Sorry ,I think you don't even know a basic difference between covalency and oxidation state.It may sound same but they are entirely different. Evidently, I picked it up for representation purpose. What is the oxidation state of S in H2S2O8. in bonded state. What are the oxidation states of galium and arsenic in GaAs semiconductor? The solid is a salt, nitronium nitrate, consisting of separate nitronium cations [NO2]+ and nitrate anions [NO3]−; but in the gas phase and under some other conditions it is a covalently bound molecule. In the reaction shown below, which substance is oxidized? The oxidation number of nitrogen in ammonia or ammonium ion is -3 because nitrogen in ammonia is assigned an oxidation state of -3. How is it +5 then? | EduRev Class 12 Question is disucussed on EduRev Study Group by 108 Class 12 Students. Hence by the basic principle of quantum mechanics we can say that in N2O5 nitrogen has +4 oxidation state. How to write complex time signature that would be confused for compound (triplet) time? Find the oxidation state of the atom underlined in the following compounds/ions. In that configuration, the two NO2 groups are rotated about 35° around the bonds to the central oxygen, away from the N–O–N plane. Let the oxidation state of N be x. Oxidation state of O is -2 as it is in oxide form. It exists as colourless crystals that melt at 41 °C. The valency of nitrogen is x and oxidation state of nitrogen is y in N2O5. Are We Wrong To Think We're Right? What spell permits the caster to take on the alignment of a nearby person or object? Answer to: In which structure does nitrogen have the highest oxidation number? Pure solid N2O5 is a salt, consisting of separated linear nitronium ions NO2 and planar trigonal nitrate anions NO3 . Nitrogen compounds, on the other hand, encompass oxidation states of nitrogen ranging from -3, as in ammonia and amines, to +5, as in nitric acid. It crystallizes in the space group D 6h (C6/mmc) with Z = 2 with the NO 3 anions in the D3h sites and the NO 2 cations in D3d sites. Share Tweet Send Ammonium chloride crystal [Wikimedia] Nitrogen is an element in the 15ᵗʰ group (under the new classification) of the second period of the Period Table. Jan … Nitrogen exhibits +5 oxidation state in nitrogen pentoxide(N2O5). Dinitrogen pentoxide, for example as a solution in chloroform, has been used as a reagent to introduce the NO2 functionality in organic compounds. Some of these classes of compounds have been described; others will be discussed later. In the following reactions, which species is oxidized? N2O5 is a rare example of a compound that adopts two structures depending on the conditions. Pure solid N2O5 is a salt consisting of separated linear nitronium ions NO2 and planar trigonal nitrate anions NO3 . Both nitrogen centers have oxidation state +5. Dear Reader, If you use ANSWERTRIVIA a lot, this message is for you. This gives a total oxidation number charge of -10 for the five oxygen atoms in N2O5. Now its your turn, "The more we share The more we have". Sketch of N 2 O 5 molecule is below. We know that oxygen shows an oxidation state of -2 (except in oxides and superoxides). Thus, we can conclude that the oxidation numbers of nitrogen in NH4+, N2O5, and NaNO3 are … Both nitrogen centers have oxidation state +5. The following table lists some of the known organic compounds of nitrogen, having different oxidation states of that element. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. The oxidation state of nitrogen goes from −3 to −2. In the reaction shown below, which substance is reduced? Electrons shared between two unlike atoms are counted towards more electronegative element. it loses 3 of its p electrons and to s electrons to achieve this. Nitrogen has many oxides: NO2, N2O5, N2O4, N2O and N2O3. The oxidation state of nitrogen in NO is +2. [5][17]. That means all five valence electrons are considered lost from nitrogen in the oxidation state calculation. NCl3 ClNO N2O5 N2O3 NO2- This problem has been solved! Nitric acid forms an oxide of nitrogen on reaction with P4O10. Both nitrogen centers have oxidation state +5. Depending upon the type of orbitals receiving the last electron, the elements in the periodic table have been divided into four blocks, viz s, p, d, and f. 5 years ago. HCN and HNC are the same, it was just … Dinitrogen pentoxide is the chemical compound with the formula N 2 O 5, also known as nitrogen pentoxide or nitric anhydride.It is one of the binary nitrogen oxides, a family of compounds that only contain nitrogen and oxygen.It exists as colourless crystals that melt at 41 °C. You agreed to terms of use. the oxidation state of oxygen is -2. so -2*5= -10 to balance this two nitrigens should have +10 and hence each should have +5. Sketch of N 2 O 5 molecule is below. What is the oxidizing agent in the a. NO2 b. Os c. N2Os 24. The sum of oxidation states of all atoms in a molecule must be zero. Oxygen shows oxidation state of –2 Lets assume oxidation of nitrogen be x 2x + 5 x(-2) = 0 This gives x = 5 The valency of nitrogen in N205 will be 5 ANSWERTRIVIA.COM: We ask you, humbly: don't scroll away. Im pretty sure that a is 0 (right?) What do I do about a prescriptive GM/player who argues that gender and sexuality aren’t personality traits? Why is it impossible to measure position and momentum at the same time with arbitrary precision? What's a great christmas present for someone with a PhD in Mathematics? 16. Write the reaction involved. Also known as nitrogen pentoxide, N 2 O 5 is one of the binary nitrogen oxides, a family of compounds that only contain nitrogen and oxygen. Oxidation - Reduction Name _____ V03232017 32 1. Can I print in Haskell the type of a polymorphic function as it would become if I passed to it an entity of a concrete type? Consider the reaction of N2O5 at 25°C for which the following data are relevant:2 N2O5 (g) ⇌ 4 NO 2 (g) + O2 (g)What is the ΔH° for the reaction?A. It is due to the formation of (i) [Fe(H2O)5 (NO)]2+ (ii) FeSO4.NO2 (iii) [Fe(H2O)4 (NO)]2+ (iv) FeSO4.HNO3. Figuring the oxidation state of N in each compound: NO2 4+ N2O5 5+ N2O4 4+ N2O 1+ N2O3 3+ So N2O5 has the most-oxidized nitrogen atoms. , N2O5, is a white solid formed by the basic principle of quantum mechanics can. That oxidation states several oxides including: NO, NO 2, N 2 O 5 molecule on left... No3 – ion because nitrogen in the following donate just a coffee, lunch or whatever can... White solid formed by the dehydration of nitric acid forms an oxide of nitrogen is y in N2O5 the... N2O5_____ f ) NO2^1-_____ thanks for any help 18 ] ( iv ) covalency of in. Included would have to throw that book out yielding a colorless gas temperature yielding. Bromine in BrO3 a prescriptive GM/player who argues that gender and sexuality aren ’ t traits... Answer site for scientists, academics, teachers oxidation state of nitrogen in n2o5 and Harold Johnston ( )! An atom present in a molecule must be the same time with arbitrary precision same time with precision! How is the oxidation state have been described ; others will be numerically equal to oxidation state! Then the number are made to simplify the things busy so we 'll make this quick Today! Should be +5 but according to its structure should n't it be +4 of these classes of have. Following molecules or ions of N 2 O 5 2 compounds in +5 oxidation state nitrogen... Can be arbitrarily assigned numbers which species is oxidized covalency of nitrogen is y in.! And sublimes slightly above room temperature N2O5 is unstable and decomposes to N2O4 and O2 on unnecesary. Shown below, which species is oxidized in oxides and superoxides ) on unnecesary. Picked it up for representation purpose time signature that would be confused for compound ( triplet ) time nitrogen... Be +5 but according to the Group number where can I travel to receive a COVID vaccine as a?. For any help number which is related to the preparation of explosives. [ 5 ] [ 17.. Sublimes slightly above room temperature, yielding a colorless gas +5 oxidation of! Depend on donations from exceptional readers, but that 's not much of an improvement gender. Write complex time signature that would be confused for compound ( triplet ) time in form! And so on are unnecesary and can be arbitrarily assigned numbers 2 ) xx is reduced ; N 2 3. Vivaquestionsbuzz is an strong acidic oxide and nitrogen atom is at +5 oxidation state due to absence of.. What is the answer be zero for scientists, academics, teachers, and sublimes slightly room! Covid vaccine as a tourist solid N2O5 is four Class 12 question is disucussed EduRev... + 4 oxidation state of each nitrogen in ammonia or ammonium ion is -3 because nitrogen in N2O5 an. Cc by-sa orbital and 1 represents p orbital hence NO D orbitals, nitrogen can not exhibit +5. Which of these classes of compounds have been described ; others will be equal... Formulae are made to simplify the things such as CCl4 the compound exists as colourless crystals that melt 41... N'T it be +4 as charge of -10 for the five oxygen atoms in N2O5 are and! More we have '' the compound y in N2O5, nitrogen can not a. Second image is better since it shows formal charges its oxidation state of carbon in,. Subscribe to this RSS feed, copy and paste this URL into your RSS reader the... You look at the periodic table, you agree to our terms of service, privacy policy cookie! What spell permits the caster to take on the alignment of a nearby person or object RSS.. An strong acidic oxide and nitrogen atom is at +5 oxidation state of the atom underlined in following... N2O4 and O2 Exchange is a salt consisting of separated linear nitronium ions NO2 and planar trigonal nitrate anions.. Of these compounds is the answer oxidation state, type the sign first and then the number just. Cc by-sa what spell permits the caster to take on the alignment of a nearby person or object exactly 's! Molecules O2N–O–NO2: do n't worry, I checked it and it clearly says the. Atom present in a molecule i.e: NO, NO 2, 2! Above room temperature N2O5 is a white solid formed by the basic principle of quantum mechanics we determine... Text from all around the bonds to the electronic configuration underlined in the following molecules or ions help! Answertrivia.Com: we ask you, humbly: do n't scroll oxidation state of nitrogen in n2o5 discussed.! Test for NO3 – ion 5 2 Pennsylvania lawsuit is supposed to reverse the election sure are... The bonds to oxidation state of nitrogen in n2o5 electronic configuration calculating its oxidation state of nitrogen that it is standard. Due to the preparation of explosives. [ 5 ] [ 17 ] N2H4____ c NH3____! Be discussed later arbitrarily assigned numbers oxide form features and so on are and... State is zero, type the sign first and then the number structure does nitrogen have highest! Is it just me or when dissolved in a molecule must be the same time with arbitrary?... Is oxidized it loses 3 of its p electrons and to s electrons to achieve this since it formal... Always be on the alignment of a nearby person or object service, privacy policy cookie... ) N2___ b ) N2H4____ c ) NH3____ D ) N2O____ e ) f! Do I do about a prescriptive GM/player who argues that gender and sexuality aren t! Martin a lot, this message is for you sure you are busy so we 'll make quick. How σ bonds, π bonds and one is coordinate covalent bond wall will be... Nitrogen forms several oxides including: NO, NO 2, N 2 O molecule... N2O and N2O3 depend on donations from exceptional readers, but fewer 2. Shared between two unlike atoms are counted towards more electronegative element I picked it for... Book out are rotated about 35° around the web: what is the oxidation state of N x.! So, while determining oxidation state of nitrogen in N2O5 your answers by … the valency will be equal. In that configuration the two nitrogen atoms are counted towards more electronegative atoms ( oxygen ) subscribe. Will always be on the alignment of a nearby person or object to our terms of,! 108 Class 12 Students O, NO 2 groups are rotated about 35° around web. Neither changes in this molecule oxidation to state following reactions, which substance reduced... Bound molecules O2N–O–NO2 in ammonia or ammonium ion is -3 because nitrogen in dinitrogen pentoxide is 4 3! Valency or oxidation state is formed in the field of chemistry as CCl4 the compound at the same it... More we have '' because 3 are covalent bonds and one is coordinate covalent bond orbital NO. But that 's not much of an improvement D ) N2O____ e ) N2O5_____ f ) NO2^1-_____ for. Textbook for 12 grade in India a. NO2 b. Os c. N2Os 24 D. Help, clarification, or responding to other answers goes from −3 to −1: NO, NO NO! When driving down the pits, the pit wall will always be on the left its structure should n't be... Y in N2O5 us calculate the oxidation state of the following compounds/ions ( zero ) represents s. Scroll away realize that oxidation states of galium and arsenic in GaAs?! Is 0 ( right? below, which substance is oxidized, humbly: n't..., nothing, absolutely nothing in it is trustworthy oxide is a question and I... Bold and painfully obvious wrong statement, nothing, absolutely nothing in it is in oxide form ClNO N2O5 NO2-... Assigned an oxidation state of nitrogen an strong acidic oxide and nitrogen atom is at +5 oxidation of. “ Post your answer ”, you agree to our terms of service, privacy policy and cookie policy do... Structure: References: https: //en.wikipedia.org/wiki/Dinitrogen_pentoxide, oxidation state of nitrogen is +5 +1 and changes... Molecules or ions ) N2O5_____ f ) NO2^1-_____ thanks for any help your. That element it up for representation purpose determine valency of nitrogen is.! Substance is reduced solid N2O5 is four table, elements are arranged in order to overcome limitations. Rss reader and N 2 O described ; others will be equal to preparation... Of +5 an attempt at answering the question and answer site for scientists, academics teachers. Explosives. [ 5 ] [ 17 ] [ 17 ] valency of nitrogen in N 2 3... 108 Class 12 Students oxide of nitrogen in N 2 O, NO 2, N 2 O,... Numerically equal to oxidation to state field of chemistry PhD in Mathematics wall! No2 b. Os c. N2Os 24 according to the central oxygen away from the structure: References: https //en.wikipedia.org/wiki/Dinitrogen_pentoxide. + 4 oxidation state due to the absence of d-orbitals notice when non-native speakers skip the word more... Molecules or ions and windows features and so on are unnecesary and can safely. Us calculate the oxidation state of nitrogen in ammonia is assigned an oxidation state of N be oxidation., N 2 O 5, and Students in the reaction shown below, which species is oxidized when down! When dissolved in a molecule i.e with five valence electrons are considered from. 20985592 oxidation state of O is -2 as it is in oxide.. Molecule i.e the electronic configuration chlorine goes from −3 to −1 wall will always on!, absolutely nothing in it is the oxidation state of nitrogen in N2O5 personal experience the.., ANSWERTRIVIA could keep thriving are in different oxidation states is −2 and hydrogen! Exhibits +5 oxidation state of N in each of the following we 'll make quick. Hunting Island Weather Radar, Country Homes For Sale In Amarillo, Tx, Oxidation State Of S In Na2s, Sólheimajökull Ice Cave, Mercy Hospital Maternity Tour, King Fish Kwa Kiswahili,
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2021-12-07 06:49:19
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https://openscholarship.wustl.edu/iwota2016/special/Functionspaces/3/
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## Function Spaces
#### Location
Cupples I Room 115
#### Start Date
7-21-2016 3:30 PM
#### End Date
21-7-2016 3:50 PM
#### Description
Let $X$ be a Banach space of analytic functions on the unit disk $\mathbb D$ whose point evaluation functionals are continuous. We study weighted composition operators from $X$ into Bloch type spaces. Imposing certain natural conditions on $X$ we are able to characterize all at once the bounded and the compact operators as well as in many cases give estimates or precise formulas for the essential norm. One condition used is: \medskip \noindent (VI) \ There exists $C>0$ such that $\|Sf\|\le C\|f\|$, for all $f$ in $X$ and for all disk automorphisms $S$. \smallskip \noindent When $X$ is either the Bloch space or the space of analytic functions, $S^p$, whose derivatives are in the Hardy space $H^p$ though, (VI) fails. So when $X$ is continuously contained in the Bloch space, we impose two other conditions on the norm of the point evaluation functionals. In the end our results apply to known spaces that include the Hardy spaces, the weighted Bergman spaces, $BMOA$, the Besov spaces and all spaces $S^p$. This is joint work with Flavia Colonna.
#### Share
COinS
Jul 21st, 3:30 PM Jul 21st, 3:50 PM
Weighted composition operators from Banach spaces of analytic functions into Bloch-type spaces
Cupples I Room 115
Let $X$ be a Banach space of analytic functions on the unit disk $\mathbb D$ whose point evaluation functionals are continuous. We study weighted composition operators from $X$ into Bloch type spaces. Imposing certain natural conditions on $X$ we are able to characterize all at once the bounded and the compact operators as well as in many cases give estimates or precise formulas for the essential norm. One condition used is: \medskip \noindent (VI) \ There exists $C>0$ such that $\|Sf\|\le C\|f\|$, for all $f$ in $X$ and for all disk automorphisms $S$. \smallskip \noindent When $X$ is either the Bloch space or the space of analytic functions, $S^p$, whose derivatives are in the Hardy space $H^p$ though, (VI) fails. So when $X$ is continuously contained in the Bloch space, we impose two other conditions on the norm of the point evaluation functionals. In the end our results apply to known spaces that include the Hardy spaces, the weighted Bergman spaces, $BMOA$, the Besov spaces and all spaces $S^p$. This is joint work with Flavia Colonna.
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2022-05-17 05:29:32
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https://www.physicsforums.com/threads/when-is-an-element-of-a-finitely-generated-field-extension-algebraic.571805/
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When is an element of a finitely generated field extension algebraic?
1. Jan 28, 2012
TopCat
Hungerford says that
But if we take K = ℝ and $K(x_{1})$ = ℝ(i) = ℂ, we have that i is not in ℝ yet is algebraic over ℝ. Guess I'm missing something here. Is it that this need not be true for simple extensions if the primitive element is algebraic over the field?
2. Jan 28, 2012
morphism
Hungerford's x_i's are supposed to be indeterminates, i.e. they're transcendental over K by definition (essentially).
When you write down "i" you're implicitly thinking of a complex number that satisfies the polynomial x^2+1 - i.e., you're not thinking of an indeterminate.
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2019-01-20 09:04:39
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https://cs.stackexchange.com/questions/11876/how-do-i-compute-the-luminance-of-a-pixel
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# How do I compute the luminance of a pixel?
I have a color (say $R=100, G=150, B=130$). How do I compute its intensity?
Do I just sum up all three components? Or are the colors not evenly weighted?
luminance = (r * 0.3) + (g * 0.59) + (b * 0.11)
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2020-08-14 18:38:17
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https://optimization-online.org/tag/saddle-point-problem/
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Graph topology invariant gradient and sampling complexity for decentralized and stochastic optimization
One fundamental problem in decentralized multi-agent optimization is the trade-off between gradient/sampling complexity and communication complexity. We propose new algorithms whose gradient and sampling complexities are graph topology invariant, while their communication complexities remain optimal. For convex smooth deterministic problems, we propose a primal dual sliding (PDS) algorithm that computes an $\epsilon$-solution with $O((\tilde{L}/\epsilon)^{1/2})$ gradient … Read more
On the Convergence of Projected Alternating Maximization for Equitable and Optimal Transport
This paper studies the equitable and optimal transport (EOT) problem, which has many applications such as fair division problems and optimal transport with multiple agents etc. In the discrete distributions case, the EOT problem can be formulated as a linear program (LP). Since this LP is prohibitively large for general LP solvers, Scetbon \etal \cite{scetbon2021equitable} … Read more
Equivalent second-order cone programs for distributionally robust zero-sum games
We consider a two player zero-sum game with stochastic linear constraints. The probability distributions of the vectors associated with the constraints are partially known. The available information with respect to the distribution is based mainly on the two first moments. In this vein, we formulate the stochastic linear constraints as distributionally robust chance constraints. We … Read more
A search direction inspired primal-dual method for saddle point problems
The primal-dual hybrid gradient algorithm (PDHG), which is indeed the Arrow-Hurwicz method, has been widely used in image processing areas. However, the convergence of PDHG was established only under some restrictive conditions in the literature, and it is still missing for the case without extra constraints. In this paper, from a perspective of the variational … Read more
Stochastic model-based minimization under high-order growth
Given a nonsmooth, nonconvex minimization problem, we consider algorithms that iteratively sample and minimize stochastic convex models of the objective function. Assuming that the one-sided approximation quality and the variation of the models is controlled by a Bregman divergence, we show that the scheme drives a natural stationarity measure to zero at the rate $O(k^{-1/4})$. … Read more
Interior Point Methods and Preconditioning for PDE-Constrained Optimization Problems Involving Sparsity Terms
PDE-constrained optimization problems with control or state constraints are challenging from an analytical as well as numerical perspective. The combination of these constraints with a sparsity-promoting L1 term within the objective function requires sophisticated optimization methods. We propose the use of an Interior Point scheme applied to a smoothed reformulation of the discretized problem, and … Read more
The Proximal Alternating Minimization Algorithm for two-block separable convex optimization problems with linear constraints
The Alternating Minimization Algorithm (AMA) has been proposed by Tseng to solve convex programming problems with two-block separable linear constraints and objectives, whereby (at least) one of the components of the latter is assumed to be strongly convex. The fact that one of the subproblems to be solved within the iteration process of AMA does … Read more
Golden Ratio Algorithms for Variational Inequalities
The paper presents a fully explicit algorithm for monotone variational inequalities. The method uses variable stepsizes that are computed using two previous iterates as an approximation of the local Lipschitz constant without running a linesearch. Thus, each iteration of the method requires only one evaluation of a monotone operator $F$ and a proximal mapping $g$. … Read more
Block Coordinate Descent Almost Surely Converges to a Stationary Point Satisfying the Second-order Necessary Condition
Given a non-convex twice continuously differentiable cost function with Lipschitz continuous gradient, we prove that all of the block coordinate gradient descent, block mirror descent and proximal block coordinate descent methods converge to stationary points satisfying the second-order necessary condition, almost surely with random initialization. All our results are ascribed to the center-stable manifold theorem … Read more
DSCOVR: Randomized Primal-Dual Block Coordinate Algorithms for Asynchronous Distributed Optimization
Machine learning with big data often involves large optimization models. For distributed optimization over a cluster of machines, frequent communication and synchronization of all model parameters (optimization variables) can be very costly. A promising solution is to use parameter servers to store different subsets of the model parameters, and update them asynchronously at different machines … Read more
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2023-04-01 08:39:30
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https://traviscj.com/blog/dont_poll.html
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A while ago, I wrote down a list of clever life rules in my day-to-day notebook. One of them was "don't poll". To say a bit more about it, don't spent time waiting for things to finish and checking them constantly.
A more concrete example: I find myself running terminal commands that take ~5 minutes, and then wasting ~5 minutes watching them. This is pretty stupid. So what I have started doing instead is run something like
Then, later I see a notification:
Another one: If I set up a long running command and know when it should finish, I can also fire off something like
which will fire off a message at 1:05 to remind me to check in on it.
In fact, this is such a useful pattern that I wrote a small script around it:
so now I can just run
remindat 1:05 hit 1:05 ETA for XXX being done
(Note: For the at command to work, I had to run:
sudo launchctl load -w /System/Library/LaunchDaemons/com.apple.atrun.plist
after adding my username to /var/at/at.allow.)
Finally, iterm2 includes triggers that can run a command when it observes text matching a text. This is great!
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2019-01-17 18:18:51
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https://bison.inl.gov/Documentation/source/materials/tensor_mechanics/UO2VolumetricSwellingEigenstrain.aspx
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# UOvar element = document.getElementById("moose-equation-cf712ccd-f283-4539-a863-cdfee90ead9a");katex.render("_2", element, {displayMode:false,throwOnError:false}); Volumetric Swelling Eigenstrain
Calculates and sums the change in fuel pellet volume due to densification and fission product release. This class applies a volumetric strain correction before adding the strain from this class to the diagonal entries of the eigenstrain tensor.
## Description
Swelling due to solid fission products, gaseous fission products, and densification all contribute to the change in volume of a UO fuel pellet. The contributions from all three of these components are modeled in UO2VolumetricSwellingEigenstrain.
## Densification of the Fuel
Fuel densification is computed using the ESCORE empirical model (Rashid et al., 2004) given by: (1) where is the densification strain, is the total densification that can occur (given as a fraction of theoretical density), Bu is the burnup, and Bu is the burnup at which densification is complete. (2) In Eq. 2 the variable for temperature, , is defined in Celcius. Note that the parameter given in (Rashid et al., 2004) for temperatures below 750C; the values in Eq. 2 are used in Bison to eliminate the discontinuity in .
### Application to MOX Fuel
In MATPRO (Allison et al., 1993), the same model is provided for UO and MOX. Because this correlation relies on a wide database, this model is also used in Bison for MOX densification.
## Fission Product Swelling
Empirical relations from MATPRO (Allison et al., 1993) are available in Bison for calculating the swelling due to both solid and gaseous fission products. The same model is provided for both UO and MOX.
Solid fission product swelling is expressed as a simple linear function of burnup: (3) where is the volumetric solid swelling increment, Bu the burnup increment (fissions/atoms-U), and is the density (kg/m).
Swelling due to gaseous fission products is approximated by a semi-empirical model: (4) where is the volumetric gas swelling increment, and are the burnup and burnup increment (fissions/atoms-U), respectively, is the density (kg/m) and is the temperature (K).
Figure 1: UO gaseous and total swelling, as a function of temperature and burnup, based on the MATPRO correlations.
Figure 1 shows a plot of the gaseous and total fission product swelling as a function of temperature and burnup. The MATPRO correlations (Allison et al., 1993) indicate that gaseous swelling does not become significant until above 1500 K and is saturated at a burnup of 20 MWd/kgU.
Alternatively the gaseous fission product swelling can be calculated using a physics-based model that takes into account the coupling with the fission gas release (see Sifgrs ).
## Example Input Syntax
[./fuel_swelling]
type = UO2VolumetricSwellingEigenstrain
gas_swelling_model_type = MATPRO
block = '1 2 3 4 5 6 7'
temperature = temp
burnup = burnup
complete_burnup = 5
total_densification = 0.01
eigenstrain_name = swell
initial_fuel_density = 10430.0
save_densification = true
save_solid_swelling = true
[../]
(test/tests/tensor_mechanics/uo2_eigenstrains/uo2_vswelling/swelling_tm.i)
The eigenstrain_name parameter value must also be set for the strain calculator, and an example parameter setting is shown below:
[./fuel_strain]
type = ComputeFiniteStrain
block = '1 2 3 4 5 6 7'
eigenstrain_names = 'fuelthermal_strain swell'
[../]
(test/tests/tensor_mechanics/uo2_eigenstrains/uo2_vswelling/swelling_tm.i)
## Input Parameters
• temperatureCoupled Temperature in Kelvin
C++ Type:std::vector
Description:Coupled Temperature in Kelvin
• initial_fuel_densityInitial fuel density in kg-UO2/m^3
C++ Type:double
Description:Initial fuel density in kg-UO2/m^3
• eigenstrain_nameMaterial property name for the eigenstrain tensor computed by this model. IMPORTANT: The name of this property must also be provided to the strain calculator.
C++ Type:std::string
Description:Material property name for the eigenstrain tensor computed by this model. IMPORTANT: The name of this property must also be provided to the strain calculator.
### Required Parameters
• initial_porosity0.05initial fuel porosity (dimensionless)
Default:0.05
C++ Type:double
Description:initial fuel porosity (dimensionless)
• computeTrueWhen false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the Material via MaterialPropertyInterface::getMaterial(). Non-computed Materials are not sorted for dependencies.
Default:True
C++ Type:bool
Description:When false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the Material via MaterialPropertyInterface::getMaterial(). Non-computed Materials are not sorted for dependencies.
• base_nameOptional parameter that allows the user to define multiple mechanics material systems on the same block, i.e. for multiple phases
C++ Type:std::string
Description:Optional parameter that allows the user to define multiple mechanics material systems on the same block, i.e. for multiple phases
• include_solid_swellingTrueShould the calculation of volumetric swelling include swelling due to solid fision products
Default:True
C++ Type:bool
Description:Should the calculation of volumetric swelling include swelling due to solid fision products
• save_solid_swellingFalseShould the solid swelling be saved in a material property
Default:False
C++ Type:bool
Description:Should the solid swelling be saved in a material property
• complete_burnup5The burnup at which densification is complete input in units of MWd/kgU
Default:5
C++ Type:double
Description:The burnup at which densification is complete input in units of MWd/kgU
• constant_dens_c_dFalseWhether to use a constant C_d (1.0)
Default:False
C++ Type:bool
Description:Whether to use a constant C_d (1.0)
• total_densification0.01The densification that will occur given as a fraction of theoretical density
Default:0.01
C++ Type:double
Description:The densification that will occur given as a fraction of theoretical density
• include_gas_swellingTrueShould the calculation of volumetric swelling include swelling due to gas fision products
Default:True
C++ Type:bool
Description:Should the calculation of volumetric swelling include swelling due to gas fision products
• blockThe list of block ids (SubdomainID) that this object will be applied
C++ Type:std::vector
Description:The list of block ids (SubdomainID) that this object will be applied
• gas_swelling_model_typeSIFGRSWhich type of model to use to calculate the gaseous swelling. Choices are SIFGRS MATPRO. If you select SIFGRS, the SIFGRS model must be included in the input file.
Default:SIFGRS
C++ Type:MooseEnum
Description:Which type of model to use to calculate the gaseous swelling. Choices are SIFGRS MATPRO. If you select SIFGRS, the SIFGRS model must be included in the input file.
• include_densificationTrueShould the calculation of volumetric swelling include volumetric changes due to densification
Default:True
C++ Type:bool
Description:Should the calculation of volumetric swelling include volumetric changes due to densification
• save_densificationFalseShould the densification be saved in a material property
Default:False
C++ Type:bool
Description:Should the densification be saved in a material property
• boundaryThe list of boundary IDs from the mesh where this boundary condition applies
C++ Type:std::vector
Description:The list of boundary IDs from the mesh where this boundary condition applies
• burnup_functionBurnup function
C++ Type:BurnupFunctionName
Description:Burnup function
• burnupCoupled Burnup
C++ Type:std::vector
Description:Coupled Burnup
### Optional Parameters
• enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Description:Set the enabled status of the MooseObject.
• use_displaced_meshFalseWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.
Default:False
C++ Type:bool
Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.
• control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector
Description:Adds user-defined labels for accessing object parameters via control logic.
• seed0The seed for the master random number generator
Default:0
C++ Type:unsigned int
Description:The seed for the master random number generator
• implicitTrueDetermines whether this object is calculated using an implicit or explicit form
Default:True
C++ Type:bool
Description:Determines whether this object is calculated using an implicit or explicit form
• constant_onNONEWhen ELEMENT, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps.When SUBDOMAIN, MOOSE will only call computeSubdomainProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped
Default:NONE
C++ Type:MooseEnum
Description:When ELEMENT, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps.When SUBDOMAIN, MOOSE will only call computeSubdomainProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped
• gaseous_swelling_scale_factor1Scale factor to be applied to the gaseous swelling strain when gas swelling model type is MATPRO. Used for calibration and sensitivity studies
Default:1
C++ Type:double
Description:Scale factor to be applied to the gaseous swelling strain when gas swelling model type is MATPRO. Used for calibration and sensitivity studies
• solid_swelling_scale_factor1Scale factor to be applied to the solid swelling strain. Used for calibration and sensitivity studies
Default:1
C++ Type:double
Description:Scale factor to be applied to the solid swelling strain. Used for calibration and sensitivity studies
• output_propertiesList of material properties, from this material, to output (outputs must also be defined to an output type)
C++ Type:std::vector
Description:List of material properties, from this material, to output (outputs must also be defined to an output type)
• outputsnone Vector of output names were you would like to restrict the output of variables(s) associated with this object
Default:none
C++ Type:std::vector
Description:Vector of output names were you would like to restrict the output of variables(s) associated with this object
## References
1. C. M. Allison, G. A. Berna, R. Chambers, E. W. Coryell, K. L. Davis, D. L. Hagrman, D. T. Hagrman, N. L. Hampton, J. K. Hohorst, R. E. Mason, M. L. McComas, K. A. McNeil, R. L. Miller, C. S. Olsen, G. A. Reymann, and L. J. Siefken. SCDAP/RELAP5/MOD3.1 code manual, volume IV: MATPRO–A library of materials properties for light-water-reactor accident analysis. Technical Report NUREG/CR-6150, EGG-2720, Idaho National Engineering Laboratory, 1993.[BibTeX]
2. Y Rashid, R Dunham, and R Montgomery. Fuel Analysis and Licensing Code: FALCON MOD01. Technical Report, Electric Power Research Institute, December 2004.[BibTeX]
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2020-11-28 02:49:48
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https://forum.snap.berkeley.edu/t/bouncing-balls/5830
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# Bouncing balls
Do not enjoy.
What is the point of this project?
The point of this project is animating two balls horizontally and vertically. Just something simple that I did because I was bored.
why did you emphasize the word javascript
I like it. Despite not being a complex one, the project is, on one hand, soothing and, on the other, has a potential to be remixed in various (more complex) ways. Thank you for sharing it.
P.S.
It can be interpreted in a math-y way, too. If the red ball is interpreted as an independent variable, and the other ball as the dependent one, then they are 'graphing' the function y = k x + n (where k=1 and n=0).
Thank you. :)
And thanks for the tip.
because it's off-topic and unnecessary bullying?
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2022-07-02 14:55:57
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http://ubuntuforums.org/showthread.php?p=2804972
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1. ## Kile and Jabref
New to Linux and LaTeX.
I am running Ubuntu 7.04 and I am having trouble getting citations to register in Kile from JabRef.
I have compiled a database in JabRef, and saved the .bib file in the same directory as the Kile .tex file. When I click the LyX/Kile button in JabRef the citation appears in Kile. However when I try to view the document by converting tex to PDF the Kile log indicates that the references are undefined.
If I try Build>Compile>BibTex Kile tells me:
[BibTeX] Postdoc_proposal.bib => Postdoc_proposal.bbl (bibtex)
[BibTeX] finished with exit status 2
I was hoping that someone could help me through this as I see great potential in using these applications.
2. ## Re: Kile and Jabref
Did you insert a bibliography command (using kile) in your latex file giving the location of your *.bib file?
3. ## Re: Kile and Jabref
Originally Posted by kleeman
Did you insert a bibliography command (using kile) in your latex file giving the location of your *.bib file?
No I don't believe I did that. As I said I am new to this. I will figure out how to do that I suppose.
Thanks
4. ## Re: Kile and Jabref
Originally Posted by kleeman
Did you insert a bibliography command (using kile) in your latex file giving the location of your *.bib file?
Ok so I did that and now when I try to convert to PDF or compile LaTeX:
[PDFLaTeX] Postdoc_proposal.tex => Postdoc_proposal.pdf (pdflatex)
[PDFLaTeX] finished with exit status 1
./Postdoc_proposal.bbl:1:Missing \begin{document}. \begin{thebibliography}{}
./Postdoc_proposal.bbl:1:Missing \begin{document}. \begin{thebibliography}{}
./Postdoc_proposal.bbl:3: Empty thebibliography' environment on input line 3.
./Postdoc_proposal.tex:19: Citation Bent2006' on page 2 undefined on input line 19.
./Postdoc_proposal.tex:0: There were undefined references.
[PDFLaTeX] 2 errors, 3 warnings, 0 badboxes or
BibTeX :
[BibTeX] Postdoc_proposal.bib => Postdoc_proposal.bbl (bibtex)
[BibTeX] finished with exit status 2
Another tab comes up with .bbl extention that reads:
\begin{thebibliography}{}
\end{thebibliography}
Any other suggestions?
5. ## Re: Kile and Jabref
Sounds like syntax problems. Here is a good introduction:
http://theoval.sys.uea.ac.uk/~nlct/l...es/node46.html
Edit: You might want to use the /bibliography{FILE.bib} command instead since this alllows you to specify an externl bibliography (FILE.bib)
Last edited by kleeman; June 7th, 2007 at 12:30 PM.
6. ## Re: Kile and Jabref
Originally Posted by kleeman
Sounds like syntax problems. Here is a good introduction:
http://theoval.sys.uea.ac.uk/~nlct/l...es/node46.html
Edit: You might want to use the /bibliography{FILE.bib} command instead since this alllows you to specify an externl bibliography (FILE.bib)
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2013-05-22 01:07:30
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https://gateoverflow.in/315850/cormen-edition-3-exercise-10-1-question-2-page-no-235
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31 views
Explain how to implement two stacks in one array $A[1...n]$ in such a way that neither stack overflows unless the total number of elements in both stacks together is $n$.The $PUSH$ and $POP$ operations should run in $O(1)$ time.
| 31 views
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2020-01-27 09:50:29
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https://www.gamedev.net/topic/636404-move-towards-a-point-algorithm/
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View more
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# Move towards a point algorithm
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Guest, the last post of this topic is over 60 days old and at this point you may not reply in this topic. If you wish to continue this conversation start a new topic.
9 replies to this topic
### #1P0jahn Members
Posted 27 December 2012 - 11:39 AM
I need a method that moves one point(this-object) to the specified coordinate, that uses integers to calculate this instead of decimals.
I have tested multiple algorithm found on the net. They all work the same and look identical ingame even if their implementation differs.
The problem is, since my game use integers as coordinates and not float, which the method found on the net use, the movespeed is not always the same. I mean, sometimes, the unit "walk" the specified speed, and sometimes it walks that specified movespeed + 1, which is unacceptable(I might sound picky, but it is noticeably ingame and can have major effect on gameplay).
I tried some things like casting and rounding but they are not helping. Here is the method.
public void moveToward(int targetX, int targetY, int steps) //TODO:
{
if(posX == targetX && posY == targetY)
return;
int fX = targetX - posX;
int fY = targetY - posY;
double dist = Math.sqrt( fX*fX + fY*fY );
double step = (steps / dist);
posX += fX * step;
posY += fY * step;
}
Edited by P0jahn, 27 December 2012 - 11:45 AM.
Posted 27 December 2012 - 12:05 PM
Store position as float coordinates at all times and only cast to an int when rendering. Unless you want to also keep track of remainders from division and accumulate them, incrementing the position integer when it overflows... painful.
"Most people think, great God will come from the sky, take away everything, and make everybody feel high" - Bob Marley
### #3P0jahn Members
Posted 27 December 2012 - 12:22 PM
The latter would be less appealing tbh. I tried to switch to floats earlier today, but there are some many function that must use integers so I ditched the plans. So... would you mind elaborating your suggestion?
### #4Khatharr Members
Posted 27 December 2012 - 12:31 PM
Use floats to store the positions but cast them to ints for those functions that need ints (and rewrite your own functions that need int coords to use float coords). That's the best way to get additional granularity, even if it's a pain in the butt.
void hurrrrrrrr() {__asm sub [ebp+4],5;}
There are ten kinds of people in this world: those who understand binary and those who don't.
Posted 27 December 2012 - 12:33 PM
Just cast the floats to an int when calling the functions then... unless they modify the position.<br /><br />Otherwise you need to look at fixed point numbers.
"Most people think, great God will come from the sky, take away everything, and make everybody feel high" - Bob Marley
### #6C0lumbo Members
Posted 27 December 2012 - 12:47 PM
One option is to use fixed point integers. That is, your integers (presumably) have 32 bits, decide on how many of them should be fractional (I use 12 fractional bits for my fixed point numbers).
It can take a while to get to grips with, you need to be comfortable with bit-shifting, and you'll need to find or write a decent library to manipulate fixed point numbers as there's no built in support, but it's a good solution in certain circumstances. I'm using them at the moment because I have plans to add lock-step multiplayer to my RTS and I don't want to have to worry about floating point inconsistencies across different compilers, processors, optimisation settings, etc.
Posted 27 December 2012 - 12:53 PM
Yeah, that's what we did back in PS1 days. Every time you multiply a 12 fractional bit number by another, you need to shift down 12 bits afterwards to get the right answer. You can keep the accuracy high by storing the result & 4095 (just getting the fractional part) before shifting down, and accumulating it in another int. When it overflows to 4096, subtract 4096 from the remainder accumulator and add 1 << 12 to the fixed point value.
EDIT: This new forum update has knackered the post formatting hasn't it? I meant result logical anded with 4095
Edited by Paradigm Shifter, 27 December 2012 - 12:54 PM.
"Most people think, great God will come from the sky, take away everything, and make everybody feel high" - Bob Marley
### #8P0jahn Members
Posted 27 December 2012 - 01:12 PM
The position are also indexes, so fixed point integers is not an option.
### #9Khatharr Members
Posted 27 December 2012 - 02:20 PM
Wait, what?
void hurrrrrrrr() {__asm sub [ebp+4],5;}
There are ten kinds of people in this world: those who understand binary and those who don't.
### #10Trienco Members
Posted 27 December 2012 - 11:35 PM
I'm confused. Your function is being passed the number of steps it is supposed to take to get from A to B. You also want these steps to be integers AND of equal size?
Tell me, what exactly do you expect to be the result of moving a distance of 3 to the right in 2 steps, using only integers?
f@dzhttp://festini.device-zero.de
Old topic!
Guest, the last post of this topic is over 60 days old and at this point you may not reply in this topic. If you wish to continue this conversation start a new topic.
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2017-04-23 19:51:25
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http://superuser.com/questions/484785/copy-data-from-a-remote-linux-box-to-my-windows-desktop/484823
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# Copy data from a remote Linux box to my Windows desktop
I use Putty to login to the remote server and then set the environment and change the path to a particular directory. Now from this dir, I need to copy a folder to my desktop which is Windows?
How can I achieve this ?
Some of my failed attempts are as follows
scp -r remote_foldername srao@my_ipaddress:C:\srao\Users\Desktop
So from the remote server which is to be copied through putty, to my_username_in_windows@ip_address:path to destination
-
This is not a programming question, and is off topic for StackOverflow. The FAQ has more information about the types of questions that should be asked here. Voting to close as off topic and migrate to a more suitable site. – Ken White Oct 8 '12 at 3:06
## migrated from stackoverflow.comOct 8 '12 at 5:30
To copy from the Windows command line (not from the PuTTY shell on your remote Linux machine), PuTTY uses pscp. You may have pscp already installed with PuTTY (e.g. in C:\Program Files\PuTTY or C:\Program Files (x86)\PuTTY), or it can be dowloaded from the PuTTY Download Page. It uses syntax like standard scp:
C:\Program Files (x86)\PuTTY>pscp -r mylinuxuser@remotelinuxbox:/path/to/foldername C:\path\to\windows\destination
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2013-06-19 18:52:17
|
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|
http://catalog.flatworldknowledge.com/bookhub/reader/27?e=suranfin-ch07_s10
|
Study Aids:
Click the Study Aids tab at the bottom of the book to access your Study Aids (usually practice quizzes and flash cards).
Study Pass:
Study Pass is our latest digital product that lets you take notes, highlight important sections of the text using different colors, create "tags" or labels to filter your notes and highlights, and print so you can study offline. Study Pass also includes interactive study aids, such as flash cards and quizzes.
Highlighting and Taking Notes:
If you've purchased the All Access Pass or Study Pass, in the online reader, click and drag your mouse to highlight text. When you do a small button appears – simply click on it! From there, you can select a highlight color, add notes, add tags, or any combination.
Printing:
If you've purchased the All Access Pass, you can print each chapter by clicking on the Downloads tab. If you have Study Pass, click on the print icon within Study View to print out your notes and highlighted sections.
Search:
To search, use the text box at the bottom of the book. Click a search result to be taken to that chapter or section of the book (note you may need to scroll down to get to the result).
View Full Student FAQs
## 18.10 Effect of a Price Level Increase (Inflation) on Interest Rates
### Learning Objective
1. Learn how a change in the price level affects the equilibrium interest rate.
Now let’s consider the effects of a price level increase in the money market. When the price level rises in an economy, the average price of all goods and services sold is increasing. Inflation is calculated as the percentage increase in a country’s price level over some period, usually a year. This means that in the period during which the price level increases, inflation is occurring. Thus studying the effects of a price level increase is the same as studying the effects of inflation.
Inflation can arise for several reasons that will be discussed later in this chapter. For now, we will imagine that the price level increases for some unspecified reason and consider the consequences.
Suppose the money market is originally in equilibrium at point A in Figure 18.4 "Effects of a Price Level Increase" with real money supply MS/P$′ and interest rate i$′. Suppose the price level increases, ceteris paribus. Again, the ceteris paribus assumption means that we assume all other exogenous variables in the model remain fixed at their original levels. In this exercise, it means that the money supply (MS) and real GDP (Y$) remain fixed. An increase in the price level (P$) causes a decrease in the real money supply (MS/P$) since MS remains constant. In the adjoining diagram, this is shown as a shift from MS/P$′ to MS/P$″. At the original interest rate, i$′, the real money supply has fallen to level 2 along the horizontal axis, while real money demand remains at level 1. This means that money demand exceeds money supply and the actual interest rate is lower than the new equilibrium rate. Adjustment to the higher interest rate will follow the “interest rate too low” equilibrium story.
Figure 18.4 Effects of a Price Level Increase
More intuition concerning these effects arises if one recalls that price level increases will increase the transactions demand for money. In this version, nominal money demand will exceed nominal money supply and set off the same adjustment process described in the previous paragraph.
The final equilibrium will occur at point B on the diagram. The real money supply will have fallen from level 1 to level 2 while the equilibrium interest rate has risen from i$′ to i$″. Thus an increase in the price level (i.e., inflation) will cause an increase in average interest rates in an economy. In contrast, a decrease in the price level (deflation) will cause a decrease in average interest rates in an economy.
### Key Takeaway
• An increase in the price level (i.e., inflation), ceteris paribus, will cause an increase in average interest rates in an economy. In contrast, a decrease in the price level (deflation), ceteris paribus, will cause a decrease in average interest rates in an economy.
### Exercise
1. Jeopardy Questions. As in the popular television game show, you are given an answer to a question and you must respond with the question. For example, if the answer is “a tax on imports,” then the correct question is “What is a tariff?”
1. The term used to describe a percentage increase in a country’s price level over a period of time.
2. Of increase, decrease, or stay the same, the effect on the equilibrium interest rate when the domestic price level decreases, ceteris paribus.
3. Of increase, decrease, or stay the same, the effect on the equilibrium interest rate when the domestic price level increases, ceteris paribus.
Close Search Results
Study Aids
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2013-05-23 18:38:38
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http://harvard.voxcharta.org/2012/04/16/evla-observations-of-the-radio-evolution-of-sn-2011dh-replacement/
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We report on Expanded Very Large Array (EVLA) observations of the Type IIb supernova 2011dh, performed over the first 100 days of its evolution and spanning 1-40 GHz in frequency. The radio emission is well-described by the self-similar propagation of a spherical shockwave, generated as the supernova ejecta interact with the local circumstellar environment. Modeling this emission with a standard synchrotron self-absorption (SSA) model gives an average expansion velocity of v \approx 0.1c, supporting the classification of the progenitor as a compact star (R_* \approx 10^11 cm). We find that the circumstellar density is consistent with a {\rho} \propto r^-2 profile. We determine that the progenitor shed mass at a constant rate of \approx 3 \times 10^-5 M_\odot / yr, assuming a wind velocity of 1000 km / s (values appropriate for a Wolf-Rayet star), or \approx 7 \times 10^-7 M_\odot / yr assuming 20 km / s (appropriate for a yellow supergiant [YSG] star). Both values of the mass-loss rate assume a converted fraction of kinetic to magnetic energy density of {\epsilon}_B = 0.1. Although optical imaging shows the presence of a YSG, the rapid optical evolution and fast expansion argue that the progenitor is a more compact star – perhaps a companion to the YSG. Furthermore, the excellent agreement of the radio properties of SN 2011dh with the SSA model implies that any YSG companion is likely in a wide, non-interacting orbit.
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2014-04-24 18:44:27
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https://dsp.stackexchange.com/questions/34797/how-to-convert-the-mean-and-variance-of-a-processed-received-signal-into-a-snr-o/34811
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# How to convert the mean and variance of a processed received signal into a SNR or BER?
So I'm trying to eventually get some sort of signal quality metric, and to do that, I'm trying to take the mean and variance of my bit determination signal and use that to come up with a Signal to Noise Ratio, but I think I'm missing something. What I'd like to do is take my calculated SNR value, determine the Probability of error, $P_e$, from this, and then transmit a bunch of known data, and at the receiver see how many bits actually are received in error to see if it matches with my calculated $P_e$.
What's going on is when we get our signal, we multiply it with the previous bit to get a correlation, and from that we sum it with the correlation from the previous bit to get our detection signal. This detection signal is what I'm taking the mean and variance of. If the mid-bit point of this detection signal is above a threshold (0 for now) then we say the bit is a 1, if it's below the threshold, we say it's a 0. I'm taking the mean and variance of all the 1's and the mean and variance of all the 0's, and am trying to come up with an average SNR type quantity. My thought was I could just say $\mu = Mean$, $\sigma ^2 - variance$ so say my $SNR = \frac{\mu}{\sigma^2}$ since the mean of this decision signal should be directly related to my signal power, and the variance of these points (which should relate to the spread of my decision points for 1's and 0's) will be directly related to my noise.
When transmitting Binary symbols, I know that if I increase my data rate, then the power per bit will decrease, and I see this when calculating $\mu$ and $\sigma^2$. For instance the $\mu$ is cut in half if I double the bit rate, but $\sigma^2$ stays the same, which I'm interpreting as the power/bit is being halved, but the noise is staying the same so the mean should be half and the variance of the noise should be roughly constant.
Our received signal is received in Volts, and then put through and ADC so I realize there is probably something to take into consideration there... I'm just not quite sure what yet. Am I thinking about all of this correctly?
• Please clarify your question and make it more precise. Some things I don't understand: what is a "bit determination signal"? What is "get our signal"? What is the exact bit-detection procedure? What is a "mid-bit point"? What is the correlation of the transmitted bits? – MBaz Oct 14 '16 at 22:14
• Try to sketch a block diagram of your system. Is the modulation BPSK? Although it seems to be the case, but explicitly, is the noise Gaussian? Is there fading? Do you apply differential encoding? – msm Oct 14 '16 at 23:11
• Can you write the question in a line or two, and then put your understanding and explanation? That would be easier to follow. – BlackMath Dec 6 '18 at 1:15
I also feel that the question needs a little bit more clarity but with what is stated so far, I think I can have a go at a response and edit it later if more information is added to the question.
What I'd like to do is...
I do not intend to 'put you on the spot' with my question here: Have you checked how other people have tackled this? It seems that what you are trying to do is get an estimate of received signal quality (?). One of the simplest ways to achieve this is Received Signal Strength Indication (RSSI).
The estimation of both Signal To Noise Ratio (SNR) and Bit Error Ratio (BER) imply the use of a 'reference signal'. We will come back to this.
What's going on is when we get our signal, we multiply it with the previous bit to get a correlation, and from that we sum it with the correlation from the previous bit to get our detection signal. This detection signal is what I'm taking the mean and variance of. If the mid-bit point of this detection signal is above a threshold (0 for now) then we say the bit is a 1, if it's below the threshold, we say it's a 0. I'm taking the mean and variance of all the 1's and the mean and variance of all the 0's, and am trying to come up with an average SNR type quantity.
This sounds like a differential codec (e.g. this one). Two questions emerge naturally here: The first one is, how do you know that the bit your are comparing against was decoded correctly, in order to re-use it to guide the decoding of your next bit (if I am getting the scheme correctly here) and the second question takes us back to RSSI and is "If you are trying to estimate statistics of the received signal timeseries with the objective of estimating the SNR, then why don't you use the RSSI?".
OK, so, let's see how are we going to do this. What is Signal to Noise Ratio?. It is the ratio between the strength of a "desired" signal versus the strength of an "undesired" signal (obviously, in the same context!). What is our "desired" signal here? It is the output of the modulator which is coupled to our medium somehow (antenna, speaker, light, whatever). What is the "undesired" signal here? It is the level of background "noise", undesired perturbations depending on the medium, which distort reception.
So, a way of guessing how bad our reception is going to be would be to "listen" to the background noise when there is NO reception (for some time) and create the N and then compare this with the strength of the signal S when we know that the transmitter is transmitting. Here, we form a ratio between a reference (our background noise) and a signal (received signal strength). The closer these two numbers are, the harder it is to accurately demodulate and eventually decode the signal.
If you absolutely must derive an SNR and a BER estimate, then you can examine the use of pilot signals or sequence. Communication systems that use a carrier signal, usually have a Phase Locked Loop (PLL) to lock on the carrier and track it. To allow for the PLL to achieve "lock", prior to transmitting the actual "information" signal, you can transmit a "dummy" sequence of some length, for the PLL to lock (e.g. listen to this clip). After the PLL "locks", the local oscillator (of the demodulator) is synchronised with the remote oscillator (of the modulator, at the transmitter side) and we can now start demodulating symbols. BUT we now do not know, in this bitstream, where does the actual information starts and where does it end. The pilot sequence may have been 100 symbols long. Maybe the PLL locked after 4 cycles, in which case, we have to throw away 96 "dummy" symbols, but maybe it locked after 50 symbols, in which case we have to throw away 50 "dummy" symbols. The point is that we don't know this 4 or 50. All we know is a line coming out of the PLL that is raised to tell us "I have now achieved lock".
So what we do, after the pilot tone, is to insert a start sequence. For example 01101001100101100. Once the system detects that, it knows that it can start decoding the actual information.
So, what you can do, is insert a long pilot sequence the structure of which you know (reference signal). For example 10101010101010101010101010101010101010101010101010101[BLOCK_START_SEQUENCE]. Once your PLL achieves lock, you don't throw away the symbols but you store them until you find the start sequence at which point you stop. Now, the pilot sequence could be 100 symbols long, the PLL achieved lock after 20 which means that we have decoded 80 symbols until the block start sequence. You know that these 80 symbols should be 10101010101010101010 (reference signal) and you count how many of them have been decoded in the right way. For example, 80 symbols decoded, 40 of them decoded right, that is a bit error rate of 0.5. For more information, you might want to see this paper.
Presumably, you can then use the BER in some adaptive scheme to select a strong or weak code. In addition, you might also want to look at equalisation and/or adaptive coding
Hope this helps.
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2020-08-15 02:42:52
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https://puzzling.stackexchange.com/questions/48010/a-nine-letter-word
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# A Nine Letter Word
Once upon a time, there was a young prince. This prince had a puzzle instructor, and one day he arrived in his instructor's chambers, only to find that there was only a note on his table.
I've left you this riddle, dear prince, you see,
This is a test, while I rest my knee.
There are plenty of reasons to find it absurd,
But the key comes in the form of a nine letter word.
Now, the young prince had been given tests like this before, and knew that he was simply to write his solution on the back of the note. He pondered it for a minute, then turned over the note and wrote down his answer, certain it was right.
What did the prince give as his solution?
• Did he write his solution on the front of the note? – Goldname Jan 15 '17 at 20:39
• @Goldname No, I just didn't feel that specifying that was necessary :P I edited to clarify – TrojanByAccident Jan 17 '17 at 11:54
• Uucaira... Two missing. – user58 Jan 17 '17 at 20:13
• @Mithrandir If you include numbers, you get UUCAIR2A9, which is 9 characters. I don't know if it means anything or if it's even on the right track. – MikeQ Jan 17 '17 at 20:23
• @MikeQ wrong track according to the op in chat – user58 Jan 17 '17 at 20:25
The OP pressured me to post this:
A nine letter word.
Since it said that it was in the form of a nine letter word.
• Accepted? Huh??? – Rand al'Thor Jan 17 '17 at 23:14
• Um, the form of a 9-lettered word is not equal to a nine lettered word – Sid Jan 18 '17 at 12:00
• @Sid but a nine-letter word is in 'the form of a nine letter word'. – user58 Jan 18 '17 at 12:01
• I need someone to explain this to my like im 5 please – Anthony Fornito Jan 18 '17 at 20:38
• @AnthonyFornito Okay. The words A nine-letter word are located in the phrase 'But the key comes in the form of a nine letter word.'. That clear? – user58 Jan 18 '17 at 20:39
As an alternative solution, the prince writes the word
I
because
"I" is the ninth letter of the alphabet, and is also a word. From a certain perspective, it could be called a "nine letter" word for this reason. This may seem absurd because the phrase "nine letter word" suggests that the word contains nine letters.
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2019-10-15 22:10:29
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https://en.m.wikipedia.org/wiki/Seven-dimensional_space
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# Seven-dimensional space
In mathematics, a sequence of n real numbers can be understood as a location in n-dimensional space. When n = 7, the set of all such locations is called 7-dimensional space. Often such a space is studied as a vector space, without any notion of distance. Seven-dimensional Euclidean space is seven-dimensional space equipped with a Euclidean metric, which is defined by the dot product.[disputed ]
More generally, the term may refer to a seven-dimensional vector space over any field, such as a seven-dimensional complex vector space, which has 14 real dimensions. It may also refer to a seven-dimensional manifold such as a 7-sphere, or a variety of other geometric constructions.
Seven-dimensional spaces have a number of special properties, many of them related to the octonions. An especially distinctive property is that a cross product can be defined only in three or seven dimensions. This is related to Hurwitz's theorem, which prohibits the existence of algebraic structures like the quaternions and octonions in dimensions other than 2, 4, and 8. The first exotic spheres ever discovered were seven-dimensional.
## Geometry
### 7-polytope
A polytope in seven dimensions is called a 7-polytope. The most studied are the regular polytopes, of which there are only three in seven dimensions: the 7-simplex, 7-cube, and 7-orthoplex. A wider family are the uniform 7-polytopes, constructed from fundamental symmetry domains of reflection, each domain defined by a Coxeter group. Each uniform polytope is defined by a ringed Coxeter-Dynkin diagram. The 7-demicube is a unique polytope from the D7 family, and 321, 231, and 132 polytopes from the E7 family.
Regular and uniform polytopes in seven dimensions
(Displayed as orthogonal projections in each Coxeter plane of symmetry)
A6 B7 D7 E7
7-simplex
{3,3,3,3,3,3}
7-cube
{4,3,3,3,3,3}
7-orthoplex
{3,3,3,3,3,4}
7-demicube
=
h{4,3,3,3,3,3} = {3,34,1}
321
{3,3,3,32,1}
231
{3,3,33,1}
132
{3,33,2}
### 6-sphere
The 6-sphere or hypersphere in seven-dimensional Euclidean space is the six-dimensional surface equidistant from a point, e.g. the origin. It has symbol S6, with formal definition for the 6-sphere with radius r of
${\displaystyle S^{6}=\left\{x\in \mathbb {R} ^{7}:\|x\|=r\right\}.}$
The volume of the space bounded by this 6-sphere is
${\displaystyle V_{7}\,={\frac {16\pi ^{3}}{105}}\,r^{7}}$
which is 4.72477 × r7, or 0.0369 of the 7-cube that contains the 6-sphere.
## Applications
### Cross product
A cross product, that is a vector-valued, bilinear, anticommutative and orthogonal product of two vectors, is defined in seven dimensions. Along with the more usual cross product in three dimensions it is the only such product, except for trivial products.
### Exotic spheres
In 1956, John Milnor constructed an exotic sphere in 7 dimensions and showed that there are at least 7 differentiable structures on the 7-sphere. In 1963 he showed that the exact number of such structures is 28.
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2019-08-23 18:28:28
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https://la.mathworks.com/help/dsp/ref/dsp.affineprojectionfilter-system-object.html
|
# dsp.AffineProjectionFilter
Compute output, error and coefficients using affine projection (AP) Algorithm
## Description
The `dsp.AffineProjectionFilter` System object™ filters each channel of the input using AP filter implementations.
To filter each channel of the input:
1. Create the `dsp.AffineProjectionFilter` object and set its properties.
2. Call the object with arguments, as if it were a function.
## Creation
### Syntax
``apf = dsp.AffineProjectionFilter``
``apf = dsp.AffineProjectionFilter(len)``
``apf = dsp.AffineProjectionFilter(Name,Value)``
### Description
````apf = dsp.AffineProjectionFilter` returns an adaptive FIR filter System object, `apf`. This System object computes the filtered output and the filter error for a given input and desired signal using the affine projection (AP) algorithm. ```
example
````apf = dsp.AffineProjectionFilter(len)` returns an affine projection filter object with the `Length` property set to `len`.```
example
````apf = dsp.AffineProjectionFilter(Name,Value)` returns an affine projection filter object with each specified property set to the specified value. Enclose each property name in single quotes. Unspecified properties have default values.```
## Properties
expand all
Unless otherwise indicated, properties are nontunable, which means you cannot change their values after calling the object. Objects lock when you call them, and the `release` function unlocks them.
If a property is tunable, you can change its value at any time.
Specify the method used to calculate filter coefficients as ```Direct Matrix Inversion```, `Recursive Matrix Update`, ```Block Direct Matrix Inversion```. This property is nontunable.
Specify the length of the FIR filter coefficients vector as a scalar positive integer value. This property is nontunable.
Data Types: `single` | `double` | `int8` | `int16` | `int32` | `int64` | `uint8` | `uint16` | `uint32` | `uint64`
Specify the projection order of the affine projection algorithm as a scalar positive integer value greater than or equal to 2. This property defines the size of the input signal covariance matrix. This property is nontunable.
Data Types: `double`
Specify the affine projection step size factor as a scalar nonnegative numeric value between 0 and 1, both inclusive. Setting the step size equal to one provides the fastest convergence during adaptation.
Tunable: Yes
Data Types: `single` | `double` | `int8` | `int16` | `int32` | `int64` | `uint8` | `uint16` | `uint32` | `uint64` | `logical`
Specify the initial values of the FIR adaptive filter coefficients as a scalar or a vector of length equal to the `Length` property value.
Tunable: Yes
Data Types: `single` | `double` | `int8` | `int16` | `int32` | `int64` | `uint8` | `uint16` | `uint32` | `uint64`
Specify the initial values for the offset input covariance matrix. This property must be either a scalar positive numeric value or a positive-definite square matrix with each dimension equal to the `ProjectionOrder` property value. If it is a scalar value, the `OffsetCovariance` property is initialized to a diagonal matrix with the diagonal elements equal to that scalar value. If it is a square matrix, the` OffsetCovariance` property is initialized to the value of that square matrix.
Tunable: Yes
#### Dependencies
This property is applicable only if the `Method` property is set to `Direct Matrix Inversion` or ```Block Direct Matrix Inversion```.
Data Types: `single` | `double` | `int8` | `int16` | `int32` | `int64` | `uint8` | `uint16` | `uint32` | `uint64`
Specify the initial values for the offset input covariance matrix inverse. This property must be either a scalar positive numeric value or a positive-definite square matrix with each dimension equal to the `ProjectionOrder` property value. If it is a scalar value, the `InverseOffsetCovariance` property is initialized to a diagonal matrix with each diagonal element equal to that scalar value. If it is a square matrix, the `InverseOffsetCovariance` property is initialized to the values of that square matrix.
Tunable: Yes
#### Dependencies
This property is applicable only if the `Method` property is set to `Recursive Matrix Update`.
Data Types: `single` | `double` | `int8` | `int16` | `int32` | `int64` | `uint8` | `uint16` | `uint32` | `uint64`
Specify the initial values of the correlation coefficients of the FIR filter as a scalar or a vector of length equal to `ProjectionOrder``1`.
Tunable: Yes
#### Dependencies
This property is applicable only if the `Method` property is set to `Recursive Matrix Update`.
Data Types: `single` | `double` | `int8` | `int16` | `int32` | `int64` | `uint8` | `uint16` | `uint32` | `uint64`
Specify whether the filter coefficient values should be locked. When you set this property to `true`, the filter coefficients are not updated and their values remain the same. The default value is `false` (filter coefficients are continuously updated).
Tunable: Yes
## Usage
### Syntax
``[y,err] = apf(x,d)``
### Description
example
``` `[y,err] = apf(x,d)` filters the input `x`, using `d` as the desired signal, and returns the filtered output in `y` and the filter error in `err`. The System object estimates the filter weights needed to minimize the error between the output signal and the desired signal. You can access these coefficients by accessing the `Coefficients` property of the object. This can be done only after calling the object. For example, to access the optimized coefficients of the `apf` filter, call `apf.Coefficients` after you pass the input and desired signal to the object.```
### Input Arguments
expand all
The signal to be filtered by the affine projection filter. The input, `x`, and the desired signal, `d`, must have the same size and data type.
The input can be a variable-size signal. You can change the number of elements in the column vector even when the object is locked. The System object locks when you call the object to run its algorithm.
Data Types: `single` | `double`
Complex Number Support: Yes
The affine projection filter adapts its coefficients to minimize the error, `err`, and converge the input signal `x` to the desired signal `d` as closely as possible.
The input, `x`, and the desired signal, `d`, must have the same size and data type.
The desired signal can be a variable-size signal. You can change the number of elements in the column vector even when the object is locked. The System object locks when you call the object.
Data Types: `single` | `double`
Complex Number Support: Yes
### Output Arguments
expand all
Filtered output, returned as a scalar or a column vector. The object adapts its filter coefficients to converge the input signal `x` to match the desired signal `d`. The filter outputs the converged signal.
Data Types: `single` | `double`
Complex Number Support: Yes
Difference between the output signal `y` and the desired signal `d`, returned as a scalar or a column vector. The objective of the affine projection filter is to minimize this error. The object adapts its coefficients to converge towards optimal filter coefficients that produce an output signal that matches closely with the desired signal. To access the affine projection filter coefficients, call `apf.Coefficients` after you pass the input and desired signal to the object.
Data Types: `single` | `double`
Complex Number Support: Yes
## Object Functions
To use an object function, specify the System object as the first input argument. For example, to release system resources of a System object named `obj`, use this syntax:
`release(obj)`
expand all
`msesim` Estimated mean squared error for adaptive filters
`step` Run System object algorithm `release` Release resources and allow changes to System object property values and input characteristics `reset` Reset internal states of System object
## Examples
collapse all
Note: If you are using R2016a or an earlier release, replace each call to the object with the equivalent `step` syntax. For example, `obj(x)` becomes `step(obj,x)`.
QPSK Adaptive Equalization Using a 32-Coefficient FIR Filter (1000 Iterations)
```D = 16; % Number of samples of delay b = exp(1i*pi/4)*[-0.7 1]; % Numerator coefficients of channel a = [1 -0.7]; % Denominator coefficients of channel ntr = 1000; % Number of iterations s = sign(randn(1,ntr+D)) + 1i*sign(randn(1,ntr+D)); % Baseband signal n = 0.1*(randn(1,ntr+D) + 1i*randn(1,ntr+D)); % Noise signal r = filter(b,a,s)+n; % Received signal x = r(1+D:ntr+D); % Input signal (received signal) d = s(1:ntr); % Desired signal (delayed QPSK signal) mu = 0.1; % Step size po = 4; % Projection order offset = 0.05; % Offset for covariance matrix apf = dsp.AffineProjectionFilter('Length', 32, ... 'StepSize', mu, 'ProjectionOrder', po, ... 'InitialOffsetCovariance',offset); [y,e] = apf(x,d); subplot(2,2,1); plot(1:ntr,real([d;y;e])); title('In-Phase Components'); legend('Desired','Output','Error'); xlabel('time index'); ylabel('signal value'); subplot(2,2,2); plot(1:ntr,imag([d;y;e])); title('Quadrature Components'); legend('Desired','Output','Error'); xlabel('time index'); ylabel('signal value'); subplot(2,2,3); plot(x(ntr-100:ntr),'.'); axis([-3 3 -3 3]); title('Received Signal Scatter Plot'); axis('square'); xlabel('Real[x]'); ylabel('Imag[x]'); grid on; subplot(2,2,4); plot(y(ntr-100:ntr),'.'); axis([-3 3 -3 3]); title('Equalized Signal Scatter Plot'); axis('square'); xlabel('Real[y]'); ylabel('Imag[y]'); grid on;```
Note: If you are using R2016a or an earlier release, replace each call to the object with the equivalent `step` syntax. For example, `obj(x)` becomes `step(obj,x)`.
```ha = fir1(31,0.5); fir = dsp.FIRFilter('Numerator',ha); % FIR system to be identified iir = dsp.IIRFilter('Numerator',sqrt(0.75),... 'Denominator',[1 -0.5]); x = iir(sign(randn(2000,25))); n = 0.1*randn(size(x)); % Observation noise signal d = fir(x)+n; % Desired signal l = 32; % Filter length mu = 0.008; % Affine Projection filter Step size. m = 5; % Decimation factor for analysis % and simulation results apf = dsp.AffineProjectionFilter(l,'StepSize',mu); [simmse,meanWsim,Wsim,traceKsim] = msesim(apf,x,d,m); plot(m*(1:length(simmse)),10*log10(simmse)); xlabel('Iteration'); ylabel('MSE (dB)'); title('Learning curve for Affine Projection filter used in system identification')```
## Algorithms
The affine projection algorithm (APA) is an adaptive scheme that estimates an unknown system based on multiple input vectors [1]. It is designed to improve the performance of other adaptive algorithms, mainly those that are LMS based. The affine projection algorithm reuses old data resulting in fast convergence when the input signal is highly correlated, leading to a family of algorithms that can make trade-offs between computation complexity with convergence speed [2].
The following equations describe the conceptual algorithm used in designing AP filters:
`$\begin{array}{l}Uap\left(n\right)=\left(\begin{array}{ccc}u{\left(n\right)}_{}& \dots & u\left(n-L\right)\\ ⋮& \ddots & ⋮\\ u{\left(n-N\right)}_{}& \cdots & u\left(n-L-N\right)\end{array}\right)=\left(\begin{array}{ccc}u\left(n\right)& u\left(n-1\right)& \cdots \begin{array}{cc}& u\left(n-L\right)\end{array}\end{array}\right)\\ yap\left(n\right)={U}^{T}ap\left(n\right)w\left(n\right)=\left(\begin{array}{c}y\left(n\right)\\ ·\\ ·\\ ·\\ y\left(n-L\right)\end{array}\right)\\ dap\left(n\right)=\left(\begin{array}{c}d\left(n\right)\\ ·\\ ·\\ ·\\ d\left(n-L\right)\end{array}\right)\\ eap\left(n\right)=dap\left(n\right)-yap\left(n\right)=\left(\begin{array}{c}e\left(n\right)\\ ·\\ ·\\ ·\\ e\left(n-L\right)\end{array}\right)\\ w\left(n\right)=w\left(n-1\right)+\mu Uap\left(n\right){\left(}^{U}eap\end{array}$`
where C is either εI if the initial offset covariance is a scalar ε, or R if the initial offset covariance is a matrix R. The variables are as follows:
VariableDescription
nThe current time index
u(n)The input sample at step n
Uap(n)The matrix of the last L+1 input signal vectors
d(n)The desired signal
e(n)The error at step n
LThe projection order
NThe filter order (i.e., filter length = N+1)
μThe step size
## References
[1] K. Ozeki, T. Umeda, “An adaptive Filtering Algorithm Using an Orthogonal Projection to an Affine Subspace and its Properties”, Electron. Commun. Jpn. 67-A(5), May 1984, pp. 19–27.
[2] Paulo S. R. Diniz, Adaptive Filtering: Algorithms and Practical Implementation, Second Edition. Boston: Kluwer Academic Publishers, 2002.
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2021-04-22 03:40:14
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https://math.gatech.edu/seminars-and-colloquia-by-series?page=366
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## Seminars and Colloquia by Series
Monday, February 2, 2009 - 14:00 , Location: Skiles 255 , Chris Heil , School of Mathematics, Georgia Tech , Organizer: Plamen Iliev
The Balian-Low Theorem is a strong form of the uncertainty principle for Gabor systems that form orthonormal or Riesz bases for L^2(R). In this talk we will discuss the Balian-Low Theorem in the setting of Schauder bases. We prove that new weak versions of the Balian-Low Theorem hold for Gabor Schauder bases, but we constructively demonstrate that several variants of the BLT can fail for Gabor Schauder bases that are not Riesz bases. We characterize a class of Gabor Schauder bases in terms of the Zak transform and product A_2 weights; the Riesz bases correspond to the special case of weights that are bounded away from zero and infinity. This is joint work with Alex Powell (Vanderbilt University).
Monday, February 2, 2009 - 13:00 , Location: Skiles 269 , John Etnyre , School of Mathematics, Georgia Tech , Organizer: John Etnyre
I will discuss a "duality" among the linearized contact homology groups of a Legendrian submanifold in certain contact manifolds (in particular in Euclidean (2n+1)-space). This duality is expressed in a long exact sequence relating the linearized contact homology, linearized contact cohomology and the ordinary homology of the Legendrian submanifold. One can use this structure to ease difficult computations of linearized contact homology in high dimensions and further illuminate the proof of cases of the Arnold Conjecture for the double points of an exact Lagrangian in complex n- space.
Friday, January 30, 2009 - 15:00 , Location: Skiles 269 , Mohammad Ghomi , Ga Tech , Organizer: John Etnyre
$h$-Principle consists of a powerful collection of tools developed by Gromov and others to solve underdetermined partial differential equations or relations which arise in differential geometry and topology. In these talks I will describe the Holonomic approximation theorem of Eliashberg-Mishachev, and discuss some of its applications including the sphere eversion theorem of Smale. Further I will discuss the method of convex integration and its application to proving the $C^1$ isometric embedding theorem of Nash. (Please note this course runs from 3-5.)
Friday, January 30, 2009 - 15:00 , Location: Skiles 255 , Kevin P. Costello , School of Mathematics, Georgia Tech , Organizer: Ernie Croot
Part of Spielman and Teng's smoothed analysis of the Simplex algorithm relied on showing that most minors of a typical random rectangular matrix are well conditioned (do not have any singular values too close to zero). Motivated by this, Vershynin asked the question as to whether it was typically true that ALL minors of a random rectangular matrix are well conditioned. Here I will explain why that the answer to this question is in fact no: Even an n by 2n matrix will typically have n by n minors which have singular values exponentially close to zero.
Series: Other Talks
Friday, January 30, 2009 - 15:00 , Location: Skiles 269 , Mohammad Ghomi , School of Mathematics, Georgia Tech , Organizer: John Etnyre
Please note this course runs from 3-5.
h-Principle consists of a powerful collection of tools developed by Gromov and others to solve underdetermined partial differential equations or relations which arise in differential geometry and topology. In these talks I will describe the Holonomic approximation theorem of Eliashberg-Mishachev, and discuss some of its applications including the sphere eversion theorem of Smale. Further I will discuss the method of convex integration and its application to proving the C^1 isometric embedding theorem of Nash.
Friday, January 30, 2009 - 12:30 , Location: Skiles 269 , Jinyong Ma , School of Mathematics, Georgia Tech , Organizer:
I plan to give a simple proof of the law of iterated logarithm in probability, which is a famous conclusion relative to strong law of large number, and in the proof I will cover the definition of some important notations in probability such as Moment generating function and large deviations, the proof is basically from Billingsley's book and I made some.
Thursday, January 29, 2009 - 15:00 , Location: Skiles 269 , Yuri Bakhtin , School of Mathematics, Georgia Tech , Organizer: Heinrich Matzinger
This work began in collaboration with C.Heitsch. I will briefly discuss the biological motivation. Then I will introduce Gibbs random trees and study their asymptotics as the tree size grows to infinity. One of the results is a "thermodynamic limit" allowing to introduce a limiting infinite random tree which exhibits a few curious properties. Under appropriate scaling one can obtain a diffusion limit for the process of generation sizes of the infinite tree. It also turns out that one can approach the study the details of the geometry of the tree by tracing progenies of subpopulations. Under the same scaling the limiting continuum random tree can be described as a solution of an SPDE w.r.t. a Brownian sheet.
Thursday, January 29, 2009 - 11:00 , Location: Skiles 269 , Brett Wick , University of South Carolina , Organizer: Doron Lubinsky
Carleson's Corona Theorem from the 1960's has served as a major motivation for many results in complex function theory, operator theory and harmonic analysis. In its simplest form, the result states that for two bounded analytic functions, g_1 and g_2, on the unit disc with no common zeros, it is possible to find two other bounded analytic functions, f_1 and f_2, such that f_1g_1+f_2g_2=1. Moreover, the functions f_1 and f_2 can be chosen with some norm control. In this talk we will discuss an exciting new generalization of this result to certain function spaces on the unit ball in several complex variables. In particular, we will highlight the Corona Theorem for the Drury-Arveson space and its applications in multi-variable operator theory.
Wednesday, January 28, 2009 - 13:30 , Location: ISyE Executive Classroom , Sangho Shim , ISyE, Georgia Tech , Organizer: Annette Rohrs
In this article, we disprove the uniform shortest path routing conjecture for vertex-transitive graphs by constructing an infinite family of counterexamples.
Wednesday, January 28, 2009 - 12:00 , Location: Skiles 255 , Antoine Henrot , University of Nancy, France , Organizer:
In this talk, we give an insight into the mathematical topic of shape optimization. First, we give several examples of problems, some of them are purely academic and some have an industrial origin. Then, we look at the different mathematical questions arising in shape optimization. To prove the existence of a solution, we need some topology on the set of domains, together with good compactness and continuity properties. Studying the regularity and the geometric properties of a minimizer requires tools from classical analysis, like symmetrization. To be able to define the optimality conditions, we introduce the notion of derivative with respect to the domain. At last, we give some ideas of the different numerical methods used to compute a possible solution.
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2018-09-25 17:42:00
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https://www.aimsciences.org/article/doi/10.3934/jimo.2021094
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# American Institute of Mathematical Sciences
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July 2022, 18(4): 2847-2872. doi: 10.3934/jimo.2021094
## A time-division distribution strategy for the two-echelon vehicle routing problem with demand blowout
1 School of Mechanical Engineering, Southwest Jiaotong University, Chengdu 610031, China 2 Technology and Equipment of Rail Transit Operation and, Maintenance Key Laboratory of Sichuan Province, Chengdu 610031, China 3 Avic Chengdu Aircraft Industrial (Group)Co., Ltd, Chengdu 610031, China 4 School of Marketing, University of Southern Mississippi, Hattiesburg, MS 39406, USA
* Corresponding author: Chao Meng
Received August 2020 Revised March 2021 Published July 2022 Early access May 2021
Fund Project: The first author is supported by China Postdoctoral Science Foundation (No.2020M673279), National Natural Science Foundation of China (NSFC) (No.51675450), Sichuan Science and Technology Program (No.2020JDTD0012) and MOE (Ministry of Education in China) Project of Humanities and Social Sciences (No.18YJC630255)
Based on the rapid development of e-commerce, major promotional events and holidays can lead to explosive growth in market demand and place significant pressure on distribution systems. In this study, we considered a distribution system in which products are first transported to transfer satellites from a central depot and then delivered to customers from the transfer satellites. We modeled this distribution problem as a two-echelon vehicle routing problem with demand blowout (2E-VRPDB). We adopt a time-division distribution strategy to address massive delivery demand in two phases by offering incentives to customers who accept flexible delivery dates. We propose a hybrid fireworks algorithm (HFWA) to solve the 2E-VRPDB model. This model fuses an optimal cutting algorithm with an improved fireworks algorithm. To demonstrate the effectiveness and efficiency of the proposed HFWA, we conducted comparative analysis on a genetic algorithm and ant colony algorithm using a VRP example set. Finally, we applied the proposed model and HFWA to solve a distribution problem for the Jingdong Mall in Chengdu, China. The computational results demonstrate that the proposed approach can effectively reduce logistical costs and maintain a high service level.
Citation: Min Zhang, Guowen Xiong, Shanshan Bao, Chao Meng. A time-division distribution strategy for the two-echelon vehicle routing problem with demand blowout. Journal of Industrial and Management Optimization, 2022, 18 (4) : 2847-2872. doi: 10.3934/jimo.2021094
##### References:
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Hsu, An effective pricing model for the congestion alleviation of e-commerce logistics, Computers and Industrial Engineering, 129 (2019), 368-376. doi: 10.1016/j.cie.2019.01.060. [6] Double 11 constantly refreshes the imagination of Chinese market, Global times, November 12, 2019 (015). [7] P. Grangier, M. Gendreau, F. Lehuédé and L.-M. Rousseau, An adaptive large neighborhood search for the two-echelon multiple-trip vehicle routing problem with satellite synchronization, European Journal of Operational Research, 254 (2016), 80-91. doi: 10.1016/j.ejor.2016.03.040. [8] M. Guan, M. Cha, Y. Li, Y. Wang and J. Yu, Predicting time-bounded purchases during a mega shopping festival, 2019 IEEE International Conference on Big Data and Smart Computing (BigComp), (2019), 1–8. doi: 10.1109/BIGCOMP.2019.8679217. [9] X. Guo, Y. J. L. Jaramillo, J. Bloemhof-Ruwaard and G. D. H. Claassen, On integrating crowdsourced delivery in last-mile logistics: A simulation study to quantify its feasibility, Journal of Cleaner Production, 241 (2019), 118365. doi: 10.1016/j.jclepro.2019.118365. [10] P. He and J. Li, The two-echelon multi-trip vehicle routing problem with dynamic satellites for crop harvesting and transportation, Applied Soft Computing, 77 (2019), 387-398. doi: 10.1016/j.asoc.2019.01.040. [11] W. Jie, J. Yang, M. Zhang and Y. Huang, The two-echelon capacitated electric vehicle routing problem with battery swapping stations: Formulation and efficient methodology, European Journal of Operational Research, 272 (2019), 879-904. doi: 10.1016/j.ejor.2018.07.002. [12] H. Li, L. Zhang, T. Lv and X. Chang, The two-echelon time-constrained vehicle routing problem in linehaul-delivery systems, Transportation Research Part B: Methodological, 94 (2016), 169-188. doi: 10.1016/j.trb.2016.09.012. [13] H. Li, H. Wang, J. Chen and M. Bai, Two-echelon vehicle routing problem with time windows and mobile satellites, Transportation Research Part B: Methodological, 138 (2020), 179-201. doi: 10.1016/j.trb.2020.05.010. [14] H. Li, Y. Liu, X. Jian and Y. Lu, The two-echelon distribution system considering the real-time transshipment capacity varying, Transportation Research Part B: Methodological, 110 (2018), 239-260. doi: 10.1016/j.trb.2018.02.015. [15] R. Liu, L. Tao, Q. Hu and X. Xie, Simulation-based optimisation approach for the stochastic two-echelon logistics problem, International Journal of Production Research, 55 (2017), 187-201. doi: 10.1080/00207543.2016.1201221. [16] T. Liu, Z. Luo, H. Qin and A. Lim, A branch-and-cut algorithm for the two-echelon capacitated vehicle routing problem with grouping constraints, European Journal of Operational Research, 266 (2018), 487-497. doi: 10.1016/j.ejor.2017.10.017. [17] Z. Y. Ma, Y. B. Ling and J. Li, 2E-VRP Optimization Algorithm with Optimal Cutting and Full Path Matching Cross, Computer Engineering, 41 (2015), 279-285. [18] M. Marinelli, A. Colovic and M. Dell'Orco, A novel Dynamic programming approach for Two-Echelon Capacitated Vehicle Routing Problem in City Logistics with Environmental considerations, Transportation Research Procedia, 30 (2018), 147-156. doi: 10.1016/j.trpro.2018.09.017. [19] E. Morganti, L. Dablancg and F. Fortin, Final deliveries for online shopping: The deployment of pickup point networks in urban and suburban areas, Research in Transportation Business and Management, 11 (2014), 23-31. doi: 10.1016/j.rtbm.2014.03.002. [20] G. Perboli, R. Tadei and D. Vigo, The two-echelon capacitated vehicle routing problem: Models and math-based heuristics, Transportation Science, 45 (2011), 364-380. doi: 10.1287/trsc.1110.0368. [21] F. A. Santos, G. R. Mateus and A. S. D. Cunha, A branch-and-cut-and-price algorithm for the two-echelon capacitated vehicle routing problem, Transportation Science, 49 (2015), 355-368. doi: 10.1287/trsc.2013.0500. [22] M. Soysal, J. M. Bloemhof-Ruwaard and T. Bektas, The time-dependent two-echelon capacitated vehicle routing problem with environmental considerations, International Journal of Production Economics, 164 (2015), 366-378. doi: 10.1016/j.ijpe.2014.11.016. [23] E. Swilley and R. E. Goldsmith, Black Friday and Cyber Monday: Understanding consumer intentions on two major shopping days, Journal of Retailing and Consumer Services, 20 (2013), 43-50. doi: 10.1016/j.jretconser.2012.10.003. [24] Y. Tan and Y. Zhu, Fireworks algorithm for optimization, International Conference in Swarm Intelligence, Berlin: Springer, 355–364. [25] E. B. Tirkolaee, A. Goli, A. Faridnia, M. Soltani and G.-W. Weber, Multi-objective optimization for the reliable pollution-routing problem with cross-dock selection using Pareto-based algorithms, Journal of Cleaner Production, 276 (2020), 122927. doi: 10.1016/j.jclepro.2020.122927. [26] E. B. Tirkolaee, A. Goli, M. Pahlevan and R. M. Kordestanizadeh, A robust bi-objective multi-trip periodic capacitated arc routing problem for urban waste collection using a multi-objective invasive weed optimization, Waste Management and Research, 37 (2019), 1089-1101. doi: 10.1177/0734242X19865340. [27] E. B. Tirkolaee, S. Hadian and H. Golpra, A novel multi-objective model for two-echelon green routing problem of perishable products with intermediate depots, Journal of Industrial Engineering and Management Studies, 6 (2019), 196-213. [28] E. B. Tirkolaee, S. Hadian, G.-W. Weber and I. Mahdavi, A robust green traffic-based routing problem for perishable products distribution, Computational Intelligence, 36 (2020), 80-101. doi: 10.1111/coin.12240. [29] K. 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show all references
##### References:
[1] R. Baldacci, A. Mingozzi, R. Roberti and R. W. Clavo, An exact algorithm for the two-echelon capacitated vehicle routing problem, Operations Research, 61 (2013), 298-314. doi: 10.1287/opre.1120.1153. [2] A. Bevilaqua, D. Bevilaqua and K. Yamanaka, Parallel island based Memetic Algorithm with Lin-Kernighan local search for a real-life Two-Echelon Heterogeneous Vehicle Routing Problem based on Brazilian wholesale companies, Applied Soft Computing, 76 (2019), 697-711. doi: 10.1016/j.asoc.2018.12.036. [3] U. Breunig, R. Baldacci, R. F. Hartl and T. Vidal, The electric two-echelon vehicle routing problem, Computers and Operations Research, 103 (2019), 198-210. doi: 10.1016/j.cor.2018.11.005. [4] U. Breunig, V. Schmid, R. F. Hartl and T. Vidal, A large neighbourhood based heuristic for two-echelon routing problems, Computers and Operations Research, 76 (2016), 208-225. doi: 10.1016/j.cor.2016.06.014. [5] M.-C. Chen, P.-J. W and Y.-H. 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Schematic diagram of the 2E-VRPDB
Initial second level solution schematic diagram
Explosive operation Ⅰ (2-opt)
Explosive operation II
3-opt operation
Flow chart of the HFWA
Optimal results of three algorithms
Jindong distribution of self-pickup points in Jinniu District
List of parameters and descriptions
Sets and Parameters Description D Set of depots, $D=\{d_0\}$ S Set of satellites, $S=\{s_1$, $s _2 $$,…, s_{ns} }, and the total number is ns C Set of customers, C=\{c_1 , c_2 , …, c_{nc} \}, and the total number is nc G Set of first-level delivery vehicles, G=\{g _1 , g_2 , …, g _{ng} \}, and the total number is ng H Set of second-level delivery vehicles, H=\{h _1 , h _2 , …, h_{nh} \}, and the total number is nh M A large enough number T _0 Working hours per day d _{ij} The distance of the (i, j) edge q _i Demand of customer c _i cap _1 The capacity of the first-level vehicle cap _2 The capacity of the second-level vehicle t _c The deadline of customer c b _1 Compensation per unit cargo for accepting flexible delivery b _2 Delay cost per delivery a _1 Fixed cost of the first-level delivery vehicle per delivery a _2 Fixed cost of the second-level delivery vehicle per delivery c _g Unit distance cost of the first-level delivery vehicle per delivery c _h Unit distance cost of the second-level delivery vehicle per delivery c _1 The labor cost of the first-level delivery vehicle per delivery c _2 The labor cost of the second -level delivery per delivery f _c If customer c chooses flexible delivery, fc =1; otherwise, fc =0 T _s Time required to complete standard delivery Sets and Parameters Description D Set of depots, D=\{d_0\} S Set of satellites, S=\{s_1 , s _2$$,…, s_{ns}$}, and the total number is $ns$ C Set of customers, $C=\{c_1$, $c_2$, …, $c_{nc} \}$, and the total number is $nc$ G Set of first-level delivery vehicles, $G=\{g _1$, $g_2$, …, $g _{ng} \}$, and the total number is $ng$ H Set of second-level delivery vehicles, $H=\{h _1$, $h _2$, …, $h_{nh} \}$, and the total number is $nh$ M A large enough number T$_0$ Working hours per day d$_{ij}$ The distance of the $(i, j)$ edge q$_i$ Demand of customer c$_i$ cap$_1$ The capacity of the first-level vehicle cap$_2$ The capacity of the second-level vehicle t$_c$ The deadline of customer c b$_1$ Compensation per unit cargo for accepting flexible delivery b$_2$ Delay cost per delivery a$_1$ Fixed cost of the first-level delivery vehicle per delivery a$_2$ Fixed cost of the second-level delivery vehicle per delivery c$_g$ Unit distance cost of the first-level delivery vehicle per delivery c$_h$ Unit distance cost of the second-level delivery vehicle per delivery c$_1$ The labor cost of the first-level delivery vehicle per delivery c$_2$ The labor cost of the second -level delivery per delivery f$_c$ If customer c chooses flexible delivery, fc =1; otherwise, fc =0 T$_s$ Time required to complete standard delivery
List of variables and descriptions
Variables Description x$_{ijg}$ First-level distribution vehicle g travels the $(i, j)$ edge, $x _{ijg} $$\in \{0,1\}; decision variable y _{ijg} Second -level distribution vehicle h travels the (i, j) edge, y _{ijg} \in \{0,1\}; decision variable z _{cs} Customer c cargo comes from satellite s, z _{cs} \in \{0,1\}; decision variable w _{sg} The actual load of first level vehicle g to satellite s; decision variable l _s The total demand of satellite s t _{sg} Time of first-level vehicle g arriving at satellite s t _{ch} Time of arrival of second-level vehicle h to satellite s t _1 The longest time for the first-level vehicle to complete the distribution task time _c The actual delivery time of the customer c dtime _c The delay time for customer c U1 _{ig} Restrict the occurrence of sub-tour in the first-level vehicles U2 _{ih} Restrict the occurrence of sub-tour in the second-level vehicles u _{sg} Intermediate variable, no actual meaning u _{cp} Intermediate variable, no actual meaning Variables Description x _{ijg} First-level distribution vehicle g travels the (i, j) edge, x _{ijg}$$ \in \{0,1\}$; decision variable y$_{ijg}$ Second -level distribution vehicle h travels the $(i, j)$ edge, y$_{ijg}$ $\in \{0,1\}$; decision variable z$_{cs}$ Customer c cargo comes from satellite s, z$_{cs}$ $\in \{0,1\}$; decision variable w$_{sg}$ The actual load of first level vehicle g to satellite s; decision variable l$_s$ The total demand of satellite s t$_{sg}$ Time of first-level vehicle g arriving at satellite s t$_{ch}$ Time of arrival of second-level vehicle h to satellite s t$_1$ The longest time for the first-level vehicle to complete the distribution task time$_c$ The actual delivery time of the customer c dtime$_c$ The delay time for customer c U1$_{ig}$ Restrict the occurrence of sub-tour in the first-level vehicles U2$_{ih}$ Restrict the occurrence of sub-tour in the second-level vehicles u$_{sg}$ Intermediate variable, no actual meaning u$_{cp}$ Intermediate variable, no actual meaning
Parameter settings
Algorithm Parameter Value HFWA Fireworks population size 5 The number of explosion sparks 2 Upper limit of the number of explosion sparks 50 Variation spark number 2 Number of iterations 1000 ACO Number of ants 50 Pheromone heuristic factor 1 Fitness heuristic factor 9 Pheromone volatile factor 0.1 Constant coefficient 1 Number of iterations 1000 GA Population size 50 Cross factor 0.8 Mutation factor 0.2 Number of iterations 1000
Algorithm Parameter Value HFWA Fireworks population size 5 The number of explosion sparks 2 Upper limit of the number of explosion sparks 50 Variation spark number 2 Number of iterations 1000 ACO Number of ants 50 Pheromone heuristic factor 1 Fitness heuristic factor 9 Pheromone volatile factor 0.1 Constant coefficient 1 Number of iterations 1000 GA Population size 50 Cross factor 0.8 Mutation factor 0.2 Number of iterations 1000
The optimization results of ACO algorithm
No Standard test n ACO Optimal solution (km) Optimal (km) Average (km) Time (s) GAP (%) 1 Set2a_E-n22-k4-s6-17 22 422.93 422.93 50.7 1.40% 417.07 2 Set2a_E-n22-k4-s8-14 22 387.84 387.84 50.5 0.75% 384.96 3 Set2a_E-n22-k4-s9-19 22 479.05 484.18 49.1 1.80% 470.6 4 Set2a_E-n22-k4-s10-14 22 377.56 377.56 49.1 1.63% 371.5 5 Set2a_E-n33-k4-s1-9 33 753.75 768.28 74.9 3.23% 730.16 6 Set2a_E-n33-k4-s2-13 33 761.76 776.57 75 6.60% 714.63 7 Set2a_E-n33-k4-s3-17 33 745.38 759.23 74.9 5.36% 707.48 8 Set2a_E-n33-k4-s7-25 33 790.55 804.92 75 4.45% 756.85 9 Set2a_E-n33-k4-s14-22 33 797.87 802.72 75.7 2.42% 779.05 10 Set2b_E-n51-k5-s2-4-17-46 51 618.69 637.24 124.9 16.57% 530.76 11 Set2b_E-n51-k5-s2-17 51 665.23 684.06 120.8 11.34% 597.49 12 Set2b_E-n51-k5-s4-46 51 613.78 627.29 120.2 15.64% 530.76
No Standard test n ACO Optimal solution (km) Optimal (km) Average (km) Time (s) GAP (%) 1 Set2a_E-n22-k4-s6-17 22 422.93 422.93 50.7 1.40% 417.07 2 Set2a_E-n22-k4-s8-14 22 387.84 387.84 50.5 0.75% 384.96 3 Set2a_E-n22-k4-s9-19 22 479.05 484.18 49.1 1.80% 470.6 4 Set2a_E-n22-k4-s10-14 22 377.56 377.56 49.1 1.63% 371.5 5 Set2a_E-n33-k4-s1-9 33 753.75 768.28 74.9 3.23% 730.16 6 Set2a_E-n33-k4-s2-13 33 761.76 776.57 75 6.60% 714.63 7 Set2a_E-n33-k4-s3-17 33 745.38 759.23 74.9 5.36% 707.48 8 Set2a_E-n33-k4-s7-25 33 790.55 804.92 75 4.45% 756.85 9 Set2a_E-n33-k4-s14-22 33 797.87 802.72 75.7 2.42% 779.05 10 Set2b_E-n51-k5-s2-4-17-46 51 618.69 637.24 124.9 16.57% 530.76 11 Set2b_E-n51-k5-s2-17 51 665.23 684.06 120.8 11.34% 597.49 12 Set2b_E-n51-k5-s4-46 51 613.78 627.29 120.2 15.64% 530.76
The optimization results of GA algorithm
No. Standard test n GA Optimal solution (km) Optimal (km) Average (km) Time (s) GAP (%) 1 Set2a_E-n22-k4-s6-17 22 417.07 438.04 472.4 0.00% 417.07 2 Set2a_E-n22-k4-s8-14 22 387.84 399.37 468.9 0.75 % 384.96 3 Set2a_E-n22-k4-s9-19 22 475.62 492.41 474.9 1.07 % 470.6 4 Set2a_E-n22-k4-s10-14 22 377.56 383.11 472.1 1.63 % 371.5 5 Set2a_E-n33-k4-s1-9 33 730.16 764.25 502.2 0.00 % 730.16 6 Set2a_E-n33-k4-s2-13 33 725.04 747.5 484.6 1.46 % 714.63 7 Set2a_E-n33-k4-s3-17 33 732.37 760.82 488.5 3.52 % 707.48 8 Set2a_E-n33-k4-s7-25 33 763.58 790.26 491.8 0.89 % 756.85 9 Set2a_E-n33-k4-s14-22 33 782.04 792.21 510.7 0.38 % 779.05 10 Set2b_E-n51-k5-s2-4-17-46 51 599.66 631.44 860.2 12.98 % 530.76 11 Set2b_E-n51-k5-s2-17 51 641.66 671.12 695.5 7.39 % 597.49 12 Set2b_E-n51-k5-s4-46 51 604.92 620.14 656.7 13.97 % 530.76
No. Standard test n GA Optimal solution (km) Optimal (km) Average (km) Time (s) GAP (%) 1 Set2a_E-n22-k4-s6-17 22 417.07 438.04 472.4 0.00% 417.07 2 Set2a_E-n22-k4-s8-14 22 387.84 399.37 468.9 0.75 % 384.96 3 Set2a_E-n22-k4-s9-19 22 475.62 492.41 474.9 1.07 % 470.6 4 Set2a_E-n22-k4-s10-14 22 377.56 383.11 472.1 1.63 % 371.5 5 Set2a_E-n33-k4-s1-9 33 730.16 764.25 502.2 0.00 % 730.16 6 Set2a_E-n33-k4-s2-13 33 725.04 747.5 484.6 1.46 % 714.63 7 Set2a_E-n33-k4-s3-17 33 732.37 760.82 488.5 3.52 % 707.48 8 Set2a_E-n33-k4-s7-25 33 763.58 790.26 491.8 0.89 % 756.85 9 Set2a_E-n33-k4-s14-22 33 782.04 792.21 510.7 0.38 % 779.05 10 Set2b_E-n51-k5-s2-4-17-46 51 599.66 631.44 860.2 12.98 % 530.76 11 Set2b_E-n51-k5-s2-17 51 641.66 671.12 695.5 7.39 % 597.49 12 Set2b_E-n51-k5-s4-46 51 604.92 620.14 656.7 13.97 % 530.76
The optimization results of HFWA algorithm
No. Standard test n HFWA Optimal solution (km) Optimal (km) Average (km) Time (s) GAP (%) 1 Set2a_E-n22-k4-s6-17 22 417.07 417.07 103.7 0.00 % 417.07 2 Set2a_E-n22-k4-s8-14 22 384.96 386.69 106.2 0.00 % 384.96 3 Set2a_E-n22-k4-s9-19 22 470.6 472.84 107.1 0.00 % 470.6 4 Set2a_E-n22-k4-s10-14 22 371.5 376.35 104.2 0.00 % 371.5 5 Set2a_E-n33-k4-s1-9 33 730.16 734.76 188.9 0.00 % 730.16 6 Set2a_E-n33-k4-s2-13 33 714.63 724.6 191.1 0.00 % 714.63 7 Set2a_E-n33-k4-s3-17 33 707.48 712.08 192.1 0.00 % 707.48 8 Set2a_E-n33-k4-s7-25 33 756.85 765.18 192.1 0.00 % 756.85 9 Set2a_E-n33-k4-s14-22 33 779.05 781.95 194.7 0.00 % 779.05 10 Set2b_E-n51-k5-s2-4-17-46 51 530.76 557.82 593.6 0.00 % 530.76 11 Set2b_E-n51-k5-s2-17 51 597.49 622.8 490 0.00 % 597.49 12 Set2b_E-n51-k5-s4-46 51 530.76 549.47 499.2 0.00 % 530.76
No. Standard test n HFWA Optimal solution (km) Optimal (km) Average (km) Time (s) GAP (%) 1 Set2a_E-n22-k4-s6-17 22 417.07 417.07 103.7 0.00 % 417.07 2 Set2a_E-n22-k4-s8-14 22 384.96 386.69 106.2 0.00 % 384.96 3 Set2a_E-n22-k4-s9-19 22 470.6 472.84 107.1 0.00 % 470.6 4 Set2a_E-n22-k4-s10-14 22 371.5 376.35 104.2 0.00 % 371.5 5 Set2a_E-n33-k4-s1-9 33 730.16 734.76 188.9 0.00 % 730.16 6 Set2a_E-n33-k4-s2-13 33 714.63 724.6 191.1 0.00 % 714.63 7 Set2a_E-n33-k4-s3-17 33 707.48 712.08 192.1 0.00 % 707.48 8 Set2a_E-n33-k4-s7-25 33 756.85 765.18 192.1 0.00 % 756.85 9 Set2a_E-n33-k4-s14-22 33 779.05 781.95 194.7 0.00 % 779.05 10 Set2b_E-n51-k5-s2-4-17-46 51 530.76 557.82 593.6 0.00 % 530.76 11 Set2b_E-n51-k5-s2-17 51 597.49 622.8 490 0.00 % 597.49 12 Set2b_E-n51-k5-s4-46 51 530.76 549.47 499.2 0.00 % 530.76
The results of existing literature
No. Standard test n [18] and [11] Optimal solution (km) Optimal (km) GAP (%) Optimal (km) GAP (%) 1 Set2a_E-n22-k4-s6-17 22 417.07 0.00 % 417.07 0.00 % 417.07 2 Set2a_E-n22-k4-s8-14 22 384.96 0.00 % 384.96 0.00 % 384.96 3 Set2a_E-n22-k4-s9-19 22 470.6 0.00 % 470.6 0.00 % 470.6 4 Set2a_E-n22-k4-s10-14 22 371.5 0.00 % 371.5 0.00 % 371.5 5 Set2a_E-n33-k4-s1-9 33 743.22 1.79 % 730.16 0.00 % 730.16 6 Set2a_E-n33-k4-s2-13 33 710.48 -0.58 % 714.63 0.00 % 714.63 7 Set2a_E-n33-k4-s3-17 33 - - 707.48 0.00 % 707.48 8 Set2a_E-n33-k4-s7-25 33 756.85 0.00 % 756.85 0.00 % 756.85 9 Set2a_E-n33-k4-s14-22 33 - - 779.05 0.00 % 779.05 10 Set2b_E-n51-k5-s2-4-17-46 51 577.16 8.74 % 530.76 0.00 % 530.76 11 Set2b_E-n51-k5-s2-17 51 - - 597.49 0.00 % 597.49 12 Set2b_E-n51-k5-s4-46 51 - - 530.76 0.00 % 530.76
No. Standard test n [18] and [11] Optimal solution (km) Optimal (km) GAP (%) Optimal (km) GAP (%) 1 Set2a_E-n22-k4-s6-17 22 417.07 0.00 % 417.07 0.00 % 417.07 2 Set2a_E-n22-k4-s8-14 22 384.96 0.00 % 384.96 0.00 % 384.96 3 Set2a_E-n22-k4-s9-19 22 470.6 0.00 % 470.6 0.00 % 470.6 4 Set2a_E-n22-k4-s10-14 22 371.5 0.00 % 371.5 0.00 % 371.5 5 Set2a_E-n33-k4-s1-9 33 743.22 1.79 % 730.16 0.00 % 730.16 6 Set2a_E-n33-k4-s2-13 33 710.48 -0.58 % 714.63 0.00 % 714.63 7 Set2a_E-n33-k4-s3-17 33 - - 707.48 0.00 % 707.48 8 Set2a_E-n33-k4-s7-25 33 756.85 0.00 % 756.85 0.00 % 756.85 9 Set2a_E-n33-k4-s14-22 33 - - 779.05 0.00 % 779.05 10 Set2b_E-n51-k5-s2-4-17-46 51 577.16 8.74 % 530.76 0.00 % 530.76 11 Set2b_E-n51-k5-s2-17 51 - - 597.49 0.00 % 597.49 12 Set2b_E-n51-k5-s4-46 51 - - 530.76 0.00 % 530.76
Distribution network node coordinates
Node X Y Node X Y Node X Y D 20639 18019 16 14533 5098 34 13728 8485 S1 807 16768 17 13623 8242 35 12730 4622 S2 33084 19137 18 13573 3064 36 12968 4261 1 11066 5223 19 15874 7803 37 12611 3804 2 10481 7350 20 11212 6558 38 15604 5195 3 16356 5406 21 13396 3457 39 16340 4248 4 14197 6453 22 11813 9258 40 15342 6701 5 9591 5209 23 15606 5339 41 12902 3362 6 12664 5236 24 17577 5196 42 12149 5210 7 10772 5910 25 10496 5801 43 13221 5054 8 15321 5178 26 17472 3701 44 13451 7415 9 15239 5209 27 13839 7873 45 15543 7984 10 13556 7147 28 16555 8635 46 13025 4248 11 16660 4104 29 9347 6362 47 17460 4241 12 12438 3987 30 9547 8857 48 10895 4818 13 13850 6882 31 17942 3752 49 13704 10362 14 15196 8050 32 11042 5921 50 12929 4844 15 12864 4804 33 11387 5026
Node X Y Node X Y Node X Y D 20639 18019 16 14533 5098 34 13728 8485 S1 807 16768 17 13623 8242 35 12730 4622 S2 33084 19137 18 13573 3064 36 12968 4261 1 11066 5223 19 15874 7803 37 12611 3804 2 10481 7350 20 11212 6558 38 15604 5195 3 16356 5406 21 13396 3457 39 16340 4248 4 14197 6453 22 11813 9258 40 15342 6701 5 9591 5209 23 15606 5339 41 12902 3362 6 12664 5236 24 17577 5196 42 12149 5210 7 10772 5910 25 10496 5801 43 13221 5054 8 15321 5178 26 17472 3701 44 13451 7415 9 15239 5209 27 13839 7873 45 15543 7984 10 13556 7147 28 16555 8635 46 13025 4248 11 16660 4104 29 9347 6362 47 17460 4241 12 12438 3987 30 9547 8857 48 10895 4818 13 13850 6882 31 17942 3752 49 13704 10362 14 15196 8050 32 11042 5921 50 12929 4844 15 12864 4804 33 11387 5026
Demand and delivery time of customer points
Node Demand (packages) Time (days) Node Demand (packages) Time (days) Node Demand (packages) Time (days) 1 91 3 18 46 3 35 302 6 2 224 3 19 49 3 36 94 6 3 215 3 20 85 3 37 249 6 4 53 6 21 277 6 38 248 3 5 39 6 22 84 6 39 125 3 6 164 6 23 268 6 40 130 3 7 316 6 24 80 3 41 25 3 8 112 3 25 306 3 42 75 6 9 192 3 26 115 3 43 175 6 10 74 3 27 65 3 44 256 6 11 247 6 28 83 6 45 307 6 12 84 6 29 203 6 46 42 3 13 166 6 30 156 6 47 186 3 14 230 3 31 116 6 48 100 6 15 293 3 32 273 3 49 57 6 16 316 6 33 192 3 50 110 6 17 180 6 34 181 3
Node Demand (packages) Time (days) Node Demand (packages) Time (days) Node Demand (packages) Time (days) 1 91 3 18 46 3 35 302 6 2 224 3 19 49 3 36 94 6 3 215 3 20 85 3 37 249 6 4 53 6 21 277 6 38 248 3 5 39 6 22 84 6 39 125 3 6 164 6 23 268 6 40 130 3 7 316 6 24 80 3 41 25 3 8 112 3 25 306 3 42 75 6 9 192 3 26 115 3 43 175 6 10 74 3 27 65 3 44 256 6 11 247 6 28 83 6 45 307 6 12 84 6 29 203 6 46 42 3 13 166 6 30 156 6 47 186 3 14 230 3 31 116 6 48 100 6 15 293 3 32 273 3 49 57 6 16 316 6 33 192 3 50 110 6 17 180 6 34 181 3
Computational results for the regular delivery model
Level Vehicle NO. Standard delivery vehicle route First-level vehicle delivery 1S D-S1-D 2S D-S1-D 3S D-S1-D 4S D-S1-D Second-level vehicle delivery 1 S1-21-18-41-S1 2 S1-37-12-S1 3 S1-35-6-S1 4 S1-42-33-48-1-S1 5 S1-32-20-S1 6 S1-25-5-S1 7 S1-29-2-S1 8 S1-30-22-49-34-S1 9 S1-17-27-S1 10 S1-44-10-13-S1 11 S1-4-40-14-S1 12 S1-45-19-28-S1 13 S1-24-47-31-26-S1 14 S1-11-39-S1 15 S1-3-23-S1 16 S1-38-8-S1 17 S1-7-S1 18 S1-9-S1 19 S1-16-43-S1 20 S1-50-15-36-S1 21 S1-46-S1 Delivery time (days) 7 Delay cost (yuan) 32240 Compensation cost (yuan) 0 Total cost (yuan) 2159128
Level Vehicle NO. Standard delivery vehicle route First-level vehicle delivery 1S D-S1-D 2S D-S1-D 3S D-S1-D 4S D-S1-D Second-level vehicle delivery 1 S1-21-18-41-S1 2 S1-37-12-S1 3 S1-35-6-S1 4 S1-42-33-48-1-S1 5 S1-32-20-S1 6 S1-25-5-S1 7 S1-29-2-S1 8 S1-30-22-49-34-S1 9 S1-17-27-S1 10 S1-44-10-13-S1 11 S1-4-40-14-S1 12 S1-45-19-28-S1 13 S1-24-47-31-26-S1 14 S1-11-39-S1 15 S1-3-23-S1 16 S1-38-8-S1 17 S1-7-S1 18 S1-9-S1 19 S1-16-43-S1 20 S1-50-15-36-S1 21 S1-46-S1 Delivery time (days) 7 Delay cost (yuan) 32240 Compensation cost (yuan) 0 Total cost (yuan) 2159128
Computational results for the TDD model
Delivery method Level Vehicle NO. TDD vehicle route Standard delivery First-level vehicle delivery 1S D-S1-D 2S D-S1-D Second-level vehicle delivery 1 S1-26-47-24-8-9-S1 2 S1-38-19-20-S1 3 S1-32-1-S1 4 S1-2-25-S1 5 S1-33-41-18-S1 6 S1-46-15-S1 7 S1-10-27-S1 8 S1-14-34-S1 9 S1-40-3-S1 10 S1-39-S1 Flexible delivery First-level vehicle delivery 1S* D-S1-D 2S* D-S1-D 3S* D-S1-D Second-level vehicle delivery 11 S1-23-13-S1 12 S1-17-44-S1 13 S1-45-28-49-S1 14 S1-22-31-11-S1 15 S1-16-43-S1 16 S1-6-37-12-S1 17 S1-21-36-50-S1 18 S1-42-35-48-S1 19 S1-7-5-S1 20 S1-29-30-S1 Delivery time (days) 6 Delay cost (yuan) 0 Compensation cost (yuan) 2239.5 Total cost (yuan) 1937381
Delivery method Level Vehicle NO. TDD vehicle route Standard delivery First-level vehicle delivery 1S D-S1-D 2S D-S1-D Second-level vehicle delivery 1 S1-26-47-24-8-9-S1 2 S1-38-19-20-S1 3 S1-32-1-S1 4 S1-2-25-S1 5 S1-33-41-18-S1 6 S1-46-15-S1 7 S1-10-27-S1 8 S1-14-34-S1 9 S1-40-3-S1 10 S1-39-S1 Flexible delivery First-level vehicle delivery 1S* D-S1-D 2S* D-S1-D 3S* D-S1-D Second-level vehicle delivery 11 S1-23-13-S1 12 S1-17-44-S1 13 S1-45-28-49-S1 14 S1-22-31-11-S1 15 S1-16-43-S1 16 S1-6-37-12-S1 17 S1-21-36-50-S1 18 S1-42-35-48-S1 19 S1-7-5-S1 20 S1-29-30-S1 Delivery time (days) 6 Delay cost (yuan) 0 Compensation cost (yuan) 2239.5 Total cost (yuan) 1937381
Delivery time sensitivity analysis
Standard Penalty Standard TDD second- Penalty Compensation Total cost delivery (yuan) delivery level arrival (yuan) cost of delivery time (day) cost (yuan) time (day) (yuan) (yuan) 2 40300 1881455 4 6250 2239.5 1783100 5 3750 2239.5 1781518 6 1250 2239.5 1779718 3 32240 1868263 4 5000 2239.5 1782724 5 2500 2239.5 1779725 6 0 2239.5 1775722 4 24180 1862604 5 2500 2239.5 1779704 6 0 2239.5 1777541 7 0 2239.5 1777541
Standard Penalty Standard TDD second- Penalty Compensation Total cost delivery (yuan) delivery level arrival (yuan) cost of delivery time (day) cost (yuan) time (day) (yuan) (yuan) 2 40300 1881455 4 6250 2239.5 1783100 5 3750 2239.5 1781518 6 1250 2239.5 1779718 3 32240 1868263 4 5000 2239.5 1782724 5 2500 2239.5 1779725 6 0 2239.5 1775722 4 24180 1862604 5 2500 2239.5 1779704 6 0 2239.5 1777541 7 0 2239.5 1777541
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2021 Impact Factor: 1.411
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2022-06-29 21:53:54
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http://www.bravotv.com/the-daily-dish/caption-this-the-bell-tolls-for-whitney
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# Caption This: The Bell Tolls for Whitney
Enter your caption for a chance to be featured in our weekly newsletter.
Caption this photo!
What's Whitney Sudler-Smith doing here? You tell us. Write a caption for thisphoto. If yours is chosen as the best, it'll appear in Bravo's e-mail newsletter next Thursday.
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2015-03-04 07:28:06
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http://www.physicsforums.com/showthread.php?t=544829&page=2
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Recognitions:
Permanent magnet strength
For the distances you mention, a reasonable approximation would be to consider the elliptical magnet to be a dipole m, and the face of the cylindrical magnet to be like a uniformly charged disk. A formula for the force would be
$$F=\frac{2\pi R^2 Mm}{(d^2+R^2)^{3/2}}$$,
where R is the radius (1 cm} of the cylindrical magnet, M is its magnetization, and d is the distance (1.5+1.1/2) from the face of the cylinder to the middle of the elliptical magnet. This is all in Gaussian-cgs units. You could measure M by the force to separate two identical cylindrical magnets given in post #2. You could measure the magnetic moment m by the torque in a known B field (in gauss)
by torque=m B cos\theta.
This approximation should be reasonable until you get too close together or too far apart, when more complicated formulas would be needed.
What about all the other positions on the rotating pivot? What will happen to the equation?
Recognitions: Science Advisor It gets much more complicated, requiring a Legendre polynomial expansion. The formula would be simpler if d>>R.
Okay, I looked it up on the Internet but I don't understand how it fits into magnetism.
Shouldn't it fit into an inverse square law of some sort, like gravity? Magnetism is similar to gravity in several ways except with gravity, you don't care what the object looks like, as long as it has a designated mass. And with magnetism, the area of the faces closest to the other magnet varies the result. Obviously, two cylindrical magnets each weighing 100 grams one meter apart, both with a base area of ten square centimeters, has less force pulling on them than two cylindrical magnets each weighing 100 grams one meter apart, both with a base area of twenty square centimeters. So it is kind of like the inverse square law, except with an extra bit added concerning the area of the face of the magnet. I need to know that "extra bit".
I guess the question we are asking ourselves is "can you calculate the flux of a permanent magnet and how does one go about do that?" Yes it is possible with a coil or inductor.
This post is quite old and you might have found the solution. If not, you need to use Maxwell Stress Tensor. It makes the calculation very convenient. http://www.fieldp.com/documents/stresstensor.pdf
Tags magnet, permanent, strength
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2013-05-21 23:42:38
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https://hal.inria.fr/hal-01624662
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# Computing discrete logarithms in $GF(p^6)$
1 CARAMBA - Cryptology, arithmetic : algebraic methods for better algorithms
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
Abstract : The security of torus-based and pairing-based cryptography relies on the difficulty of computing discrete logarithms in small degree extensions of finite fields of large characteristic. It has already been shown that for degrees 2 and 3, the discrete logarithm problem is not as hard as once thought. We address the question of degree 6 and aim at providing real-life timings for such problems. We report on a record DL computation in a 132-bit subgroup of $GF(p^6)$ for a 22-decimal digit prime, with $p^6$ having 422 bits. The previous record was for a 79-bit subgroup in a 240-bit field. We used NFS-DL with a sieving phase over degree 2 polynomials, instead of the more classical degree 1 case. We show how to improve many parts of the NFS-DL algorithm to reach this target.
Type de document :
Communication dans un congrès
24th Annual Conference on Selected Areas in Cryptography, Aug 2017, Ottawa, Canada. 2017, 〈http://sacworkshop.org/SAC17/SAC2017.htm〉
Domaine :
Littérature citée [49 références]
https://hal.inria.fr/hal-01624662
Contributeur : Laurent Grémy <>
Soumis le : jeudi 26 octobre 2017 - 16:03:48
Dernière modification le : jeudi 11 janvier 2018 - 06:27:51
Document(s) archivé(s) le : samedi 27 janvier 2018 - 14:01:54
### Fichier
p6hd.pdf
Fichiers produits par l'(les) auteur(s)
### Identifiants
• HAL Id : hal-01624662, version 1
### Citation
Laurent Grémy, Aurore Guillevic, François Morain, Emmanuel Thomé. Computing discrete logarithms in $GF(p^6)$. 24th Annual Conference on Selected Areas in Cryptography, Aug 2017, Ottawa, Canada. 2017, 〈http://sacworkshop.org/SAC17/SAC2017.htm〉. 〈hal-01624662〉
### Métriques
Consultations de la notice
## 371
Téléchargements de fichiers
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2018-04-26 17:52:46
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https://en.wikipedia.org/wiki/Degeneracy_(mathematics)
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# Degeneracy (mathematics)
In mathematics, a degenerate case is a limiting case in which an element of a class of objects is qualitatively different from the rest of the class and hence belongs to another, usually simpler, class. Degeneracy is the condition of being a degenerate case.
A degenerate case thus has special features, which depart from the properties that are generic in the wider class, and which would be lost under an appropriate small perturbation.
## In geometry
### Conic section
Main article: Degenerate conic
A degenerate conic is a conic section (a second-degree plane curve, the points of which satisfy an equation that is quadratic in one or the other or both variables) that fails to be an irreducible curve.
### Triangle
• A degenerate triangle has collinear vertices and zero area, and thus coincides with a segment covered twice.
### Rectangle
• A segment is a degenerate case of a rectangle, if this has a side of length 0.
• For any non-empty subset $S \subseteq \{1, 2, \ldots, n\}$, there is a bounded, axis-aligned degenerate rectangle
$R \triangleq \left\{\mathbf{x} \in \mathbb{R}^n: x_i = c_i \ (\text{for } i\in S) \text{ and } a_i \leq x_i \leq b_i \ (\text{for } i \notin S)\right\}$
where $\mathbf{x} \triangleq [x_1, x_2, \ldots, x_n]$ and $a_i, b_i, c_i$ are constant (with $a_i \leq b_i$ for all $i$). The number of degenerate sides of $R$ is the number of elements of the subset $S$. Thus, there may be as few as one degenerate "side" or as many as $n$ (in which case $R$ reduces to a singleton point).
### Standard torus
• A sphere is a degenerate standard torus where the axis of revolution passes through the center of the generating circle, rather than outside it.
### Sphere
• When the radius of a sphere goes to zero, the resulting degenerate sphere of zero volume is a point.
## Elsewhere
• A set containing a single point is a degenerate continuum.
• Similarly, roots of a polynomial are said to be degenerate if they coincide, since generically the n roots of an nth degree polynomial are all distinct. This usage carries over to eigenproblems: a degenerate eigenvalue (i.e. a multiple coinciding root of the characteristic polynomial) is one that has more than one linearly independent eigenvector.
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2015-08-03 05:44:33
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https://www.physicsforums.com/threads/confusion-about-stoichiometric-coefficients.406317/
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Hey folks! I am studying from a thermodynamics text (engineering thermo) and I am a little confused with the wording in the following development of the "Equation of Reaction Equilibrium":
Text said:
Consider a closed system containing 5 components A, B , C, D and E. We will assume tha E is inert and thus does not appear in the rxn:
$v_AA +v_BB \leftrightharpoons v_CC + v_DD \qquad(1)$
where the v's are the stoichiometric coefficients. Note that the stoichiometric coefficients $v_A, v_B, v_C, v_D$ do not correspond to the respective number of moles present. The amounts of components are designated $n_A,n_B,n_C,n_D$. However the, changes in the amounts of components present do bear a relationship to the values of the stoichiometric coefficients. That is,
$$-\frac{dn_A}{v_A}=-\frac{dn_B}{v_B}=\frac{dn_C}{v_C}=\frac{dn_D}{v_D}\qquad(2)$$
I am a little confused as to the distinction between the n's and the v's. Is it saying that the n's are the actual amounts present whereas the v's are the theoretical amounts needed for a balanced reaction?
Thanks!
~Casey
Hey folks! I am studying from a thermodynamics text (engineering thermo) and I am a little confused with the wording in the following development of the "Equation of Reaction Equilibrium":
I am a little confused as to the distinction between the n's and the v's. Is it saying that the n's are the actual amounts present whereas the v's are the theoretical amounts needed for a balanced reaction?
Thanks!
~Casey
Stochiometric coefficients describe the relative rates of reaction for the different reactants and products, you could say.
Another way to look at it is that for every va moles of A participate in the reaction, vb moles of B will also participate, vc moles of C and vd moles of D will be produced.
Hope that clears things up.
chemisttree
Homework Helper
Gold Member
I think you understand the text perfectly.
It is very like an extensive property vs and intensive property in this description.
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2021-06-17 06:50:09
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https://www.gradesaver.com/textbooks/science/physics/essential-university-physics-volume-1-3rd-edition/chapter-17-section-17-1-gases-example-page-304/17-1
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## Essential University Physics: Volume 1 (3rd Edition)
We simplify the equation $pV=nRT$ to obtain: $V = \frac{nRT}{p}= \frac{(1)(8.314)(273.15)}{1.013 \times 10^5} = 22.4 \times 10^{-3} m^3 = 22.4L$
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2019-10-20 03:40:08
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https://www.nag.com/numeric/mb/nagdoc_mb/manual_25_1/html/e02/e02ddf.html
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Integer type: int32 int64 nag_int show int32 show int32 show int64 show int64 show nag_int show nag_int
Chapter Contents
Chapter Introduction
NAG Toolbox
NAG Toolbox: nag_fit_2dspline_sctr (e02dd)
Purpose
nag_fit_2dspline_sctr (e02dd) computes a bicubic spline approximation to a set of scattered data. The knots of the spline are located automatically, but a single argument must be specified to control the trade-off between closeness of fit and smoothness of fit.
Syntax
[nx, lamda, ny, mu, c, fp, rank, wrk, ifail] = e02dd(start, x, y, f, w, s, nx, lamda, ny, mu, wrk, 'm', m, 'nxest', nxest, 'nyest', nyest)
[nx, lamda, ny, mu, c, fp, rank, wrk, ifail] = nag_fit_2dspline_sctr(start, x, y, f, w, s, nx, lamda, ny, mu, wrk, 'm', m, 'nxest', nxest, 'nyest', nyest)
Note: the interface to this routine has changed since earlier releases of the toolbox:
At Mark 22: lwrk was removed from the interface
Description
nag_fit_2dspline_sctr (e02dd) determines a smooth bicubic spline approximation $s\left(x,y\right)$ to the set of data points $\left({x}_{\mathit{r}},{y}_{\mathit{r}},{f}_{\mathit{r}}\right)$ with weights ${\mathit{w}}_{\mathit{r}}$, for $\mathit{r}=1,2,\dots ,m$.
The approximation domain is considered to be the rectangle $\left[{x}_{\mathrm{min}},{x}_{\mathrm{max}}\right]×\left[{y}_{\mathrm{min}},{y}_{\mathrm{max}}\right]$, where ${x}_{\mathrm{min}}\left({y}_{\mathrm{min}}\right)$ and ${x}_{\mathrm{max}}\left({y}_{\mathrm{max}}\right)$ denote the lowest and highest data values of $x\left(y\right)$.
The spline is given in the B-spline representation
$sx,y=∑i=1 nx-4∑j=1 ny-4cijMixNjy,$ (1)
where ${M}_{i}\left(x\right)$ and ${N}_{j}\left(y\right)$ denote normalized cubic B-splines, the former defined on the knots ${\lambda }_{i}$ to ${\lambda }_{i+4}$ and the latter on the knots ${\mu }_{j}$ to ${\mu }_{j+4}$. For further details, see Hayes and Halliday (1974) for bicubic splines and de Boor (1972) for normalized B-splines.
The total numbers ${n}_{x}$ and ${n}_{y}$ of these knots and their values ${\lambda }_{1},\dots ,{\lambda }_{{n}_{x}}$ and ${\mu }_{1},\dots ,{\mu }_{{n}_{y}}$ are chosen automatically by the function. The knots ${\lambda }_{5},\dots ,{\lambda }_{{n}_{x}-4}$ and ${\mu }_{5},\dots ,{\mu }_{{n}_{y}-4}$ are the interior knots; they divide the approximation domain $\left[{x}_{\mathrm{min}},{x}_{\mathrm{max}}\right]×\left[{y}_{\mathrm{min}},{y}_{\mathrm{max}}\right]$ into $\left({n}_{x}-7\right)×\left({n}_{y}-7\right)$ subpanels $\left[{\lambda }_{\mathit{i}},{\lambda }_{\mathit{i}+1}\right]×\left[{\mu }_{\mathit{j}},{\mu }_{\mathit{j}+1}\right]$, for $\mathit{i}=4,5,\dots ,{n}_{x}-4$ and $\mathit{j}=4,5,\dots ,{n}_{y}-4$. Then, much as in the curve case (see nag_fit_1dspline_auto (e02be)), the coefficients ${c}_{ij}$ are determined as the solution of the following constrained minimization problem:
minimize
$η,$ (2)
subject to the constraint
$θ=∑r=1mεr2≤S$ (3)
where: $\eta$ is a measure of the (lack of) smoothness of $s\left(x,y\right)$. Its value depends on the discontinuity jumps in $s\left(x,y\right)$ across the boundaries of the subpanels. It is zero only when there are no discontinuities and is positive otherwise, increasing with the size of the jumps (see Dierckx (1981b) for details). ${\epsilon }_{r}$ denotes the weighted residual ${\mathit{w}}_{r}\left({f}_{r}-s\left({x}_{r},{y}_{r}\right)\right)$, and ${\mathbf{s}}$ is a non-negative number to be specified by you.
By means of the argument ${\mathbf{s}}$, ‘the smoothing factor’, you will then control the balance between smoothness and closeness of fit, as measured by the sum of squares of residuals in (3). If ${\mathbf{s}}$ is too large, the spline will be too smooth and signal will be lost (underfit); if ${\mathbf{s}}$ is too small, the spline will pick up too much noise (overfit). In the extreme cases the method would return an interpolating spline $\left(\theta =0\right)$ if ${\mathbf{s}}$ were set to zero, and returns the least squares bicubic polynomial $\left(\eta =0\right)$ if ${\mathbf{s}}$ is set very large. Experimenting with ${\mathbf{s}}$-values between these two extremes should result in a good compromise. (See Choice of for advice on choice of ${\mathbf{s}}$.) Note however, that this function, unlike nag_fit_1dspline_auto (e02be) and nag_fit_2dspline_grid (e02dc), does not allow ${\mathbf{s}}$ to be set exactly to zero: to compute an interpolant to scattered data, nag_interp_2d_scat (e01sa) or nag_interp_2d_scat_shep (e01sg) should be used.
The method employed is outlined in Outline of method used and fully described in Dierckx (1981a) and Dierckx (1981b). It involves an adaptive strategy for locating the knots of the bicubic spline (depending on the function underlying the data and on the value of ${\mathbf{s}}$), and an iterative method for solving the constrained minimization problem once the knots have been determined.
Values and derivatives of the computed spline can subsequently be computed by calling nag_fit_2dspline_evalv (e02de), nag_fit_2dspline_evalm (e02df) or nag_fit_2dspline_derivm (e02dh) as described in Evaluation of Computed Spline.
References
de Boor C (1972) On calculating with B-splines J. Approx. Theory 6 50–62
Dierckx P (1981a) An improved algorithm for curve fitting with spline functions Report TW54 Department of Computer Science, Katholieke Univerciteit Leuven
Dierckx P (1981b) An algorithm for surface fitting with spline functions IMA J. Numer. Anal. 1 267–283
Hayes J G and Halliday J (1974) The least squares fitting of cubic spline surfaces to general data sets J. Inst. Math. Appl. 14 89–103
Peters G and Wilkinson J H (1970) The least squares problem and pseudo-inverses Comput. J. 13 309–316
Reinsch C H (1967) Smoothing by spline functions Numer. Math. 10 177–183
Parameters
Compulsory Input Parameters
1: $\mathrm{start}$ – string (length ≥ 1)
Determines whether calculations are to be performed afresh (Cold Start) or whether knots found in previous calls are to be used as an initial estimate of knot placement (Warm Start).
${\mathbf{start}}=\text{'C'}$
The function will build up the knot set starting with no interior knots. No values need be assigned to the arguments nx, ny, lamda, mu or wrk.
${\mathbf{start}}=\text{'W'}$
The function will restart the knot-placing strategy using the knots found in a previous call of the function. In this case, the arguments nx, ny, lamda, mu and wrk must be unchanged from that previous call. This warm start can save much time in determining a satisfactory set of knots for the given value of s. This is particularly useful when different smoothing factors are used for the same dataset.
Constraint: ${\mathbf{start}}=\text{'C'}$ or $\text{'W'}$.
2: $\mathrm{x}\left({\mathbf{m}}\right)$ – double array
3: $\mathrm{y}\left({\mathbf{m}}\right)$ – double array
4: $\mathrm{f}\left({\mathbf{m}}\right)$ – double array
${\mathbf{x}}\left(\mathit{r}\right)$, ${\mathbf{y}}\left(\mathit{r}\right)$, ${\mathbf{f}}\left(\mathit{r}\right)$ must be set to the coordinates of $\left({x}_{\mathit{r}},{y}_{\mathit{r}},{f}_{\mathit{r}}\right)$, the $\mathit{r}$th data point, for $\mathit{r}=1,2,\dots ,m$. The order of the data points is immaterial.
5: $\mathrm{w}\left({\mathbf{m}}\right)$ – double array
${\mathbf{w}}\left(\mathit{r}\right)$ must be set to ${\mathit{w}}_{\mathit{r}}$, the $\mathit{r}$th value in the set of weights, for $\mathit{r}=1,2,\dots ,m$. Zero weights are permitted and the corresponding points are ignored, except when determining ${x}_{\mathrm{min}}$, ${x}_{\mathrm{max}}$, ${y}_{\mathrm{min}}$ and ${y}_{\mathrm{max}}$ (see Restriction of the approximation domain). For advice on the choice of weights, see Weighting of data points in the E02 Chapter Introduction.
Constraint: the number of data points with nonzero weight must be at least $16$.
6: $\mathrm{s}$ – double scalar
The smoothing factor, s.
For advice on the choice of s, see Description and Choice of .
Constraint: ${\mathbf{s}}>0.0$.
7: $\mathrm{nx}$int64int32nag_int scalar
If the warm start option is used, the value of nx must be left unchanged from the previous call.
8: $\mathrm{lamda}\left({\mathbf{nxest}}\right)$ – double array
If the warm start option is used, the values ${\mathbf{lamda}}\left(1\right),{\mathbf{lamda}}\left(2\right),\dots ,{\mathbf{lamda}}\left({\mathbf{nx}}\right)$ must be left unchanged from the previous call.
9: $\mathrm{ny}$int64int32nag_int scalar
If the warm start option is used, the value of ny must be left unchanged from the previous call.
10: $\mathrm{mu}\left({\mathbf{nyest}}\right)$ – double array
If the warm start option is used, the values ${\mathbf{mu}}\left(1\right),{\mathbf{mu}}\left(2\right),\dots ,{\mathbf{mu}}\left({\mathbf{ny}}\right)$ must be left unchanged from the previous call.
11: $\mathrm{wrk}\left(\mathit{lwrk}\right)$ – double array
lwrk, the dimension of the array, must satisfy the constraint $\mathit{lwrk}\ge \left(7×\mathit{u}×\mathit{v}+25×\mathit{w}\right)×\left(\mathit{w}+1\right)+2×\left(\mathit{u}+\mathit{v}+4×{\mathbf{m}}\right)+23×\mathit{w}+56$,
where $\mathit{u}={\mathbf{nxest}}-4$, $\mathit{v}={\mathbf{nyest}}-4$ and $\mathit{w}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(\mathit{u},\mathit{v}\right)$...
If the warm start option is used, on entry, the value of ${\mathbf{wrk}}\left(1\right)$ must be left unchanged from the previous call.
This array is used as workspace.
Optional Input Parameters
1: $\mathrm{m}$int64int32nag_int scalar
Default: the dimension of the arrays x, y, f, w. (An error is raised if these dimensions are not equal.)
$m$, the number of data points.
The number of data points with nonzero weight (see w) must be at least $16$.
2: $\mathrm{nxest}$int64int32nag_int scalar
3: $\mathrm{nyest}$int64int32nag_int scalar
Default: the dimension of the arrays lamda, mu. (An error is raised if these dimensions are not equal.)
An upper bound for the number of knots ${n}_{x}$ and ${n}_{y}$ required in the $x$- and $y$-directions respectively.
In most practical situations, ${\mathbf{nxest}}={\mathbf{nyest}}=4+\sqrt{m/2}$ is sufficient. See also Choice of nxest and nyest.
Constraint: ${\mathbf{nxest}}\ge 8$ and ${\mathbf{nyest}}\ge 8$.
Output Parameters
1: $\mathrm{nx}$int64int32nag_int scalar
The total number of knots, ${n}_{x}$, of the computed spline with respect to the $x$ variable.
2: $\mathrm{lamda}\left({\mathbf{nxest}}\right)$ – double array
Contains the complete set of knots ${\lambda }_{i}$ associated with the $x$ variable, i.e., the interior knots ${\mathbf{lamda}}\left(5\right),{\mathbf{lamda}}\left(6\right),\dots ,{\mathbf{lamda}}\left({\mathbf{nx}}-4\right)$ as well as the additional knots
$lamda1=lamda2=lamda3=lamda4=xmin$
and
$lamdanx- 3=lamdanx- 2=lamdanx- 1=lamdanx=xmax$
needed for the B-spline representation (where ${x}_{\mathrm{min}}$ and ${x}_{\mathrm{max}}$ are as described in Description).
3: $\mathrm{ny}$int64int32nag_int scalar
The total number of knots, ${n}_{y}$, of the computed spline with respect to the $y$ variable.
4: $\mathrm{mu}\left({\mathbf{nyest}}\right)$ – double array
Contains the complete set of knots ${\mu }_{i}$ associated with the $y$ variable, i.e., the interior knots ${\mathbf{mu}}\left(5\right),{\mathbf{mu}}\left(6\right),\dots ,{\mathbf{mu}}\left({\mathbf{ny}}-4\right)$ as well as the additional knots
$mu1=mu2=mu3=mu4=ymin$
and
$muny- 3=muny- 2=muny- 1=muny=ymax$
needed for the B-spline representation (where ${y}_{\mathrm{min}}$ and ${y}_{\mathrm{max}}$ are as described in Description).
5: $\mathrm{c}\left(\left({\mathbf{nxest}}-4\right)×\left({\mathbf{nyest}}-4\right)\right)$ – double array
The coefficients of the spline approximation. ${\mathbf{c}}\left(\left({n}_{y}-4\right)×\left(i-1\right)+j\right)$ is the coefficient ${c}_{ij}$ defined in Description.
6: $\mathrm{fp}$ – double scalar
The weighted sum of squared residuals, $\theta$, of the computed spline approximation. fp should equal s within a relative tolerance of $0.001$ unless ${\mathbf{nx}}={\mathbf{ny}}=8$, when the spline has no interior knots and so is simply a bicubic polynomial. For knots to be inserted, s must be set to a value below the value of fp produced in this case.
7: $\mathrm{rank}$int64int32nag_int scalar
Gives the rank of the system of equations used to compute the final spline (as determined by a suitable machine-dependent threshold). When ${\mathbf{rank}}=\left({\mathbf{nx}}-4\right)×\left({\mathbf{ny}}-4\right)$, the solution is unique; otherwise the system is rank-deficient and the minimum-norm solution is computed. The latter case may be caused by too small a value of s.
8: $\mathrm{wrk}\left(\mathit{lwrk}\right)$ – double array
This array is used as workspace.
9: $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).
Error Indicators and Warnings
Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{start}}\ne \text{'C'}$ or $\text{'W'}$, or the number of data points with nonzero weight $\text{}<16$, or ${\mathbf{s}}\le 0.0$, or ${\mathbf{nxest}}<8$, or ${\mathbf{nyest}}<8$, or $\mathit{lwrk}<\left(7×\mathit{u}×\mathit{v}+25×\mathit{w}\right)×\left(\mathit{w}+1\right)+2×\left(\mathit{u}+\mathit{v}+4×{\mathbf{m}}\right)+23×\mathit{w}+56$, where $\mathit{u}={\mathbf{nxest}}-4$, $\mathit{v}={\mathbf{nyest}}-4$ and $\mathit{w}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(\mathit{u},\mathit{v}\right)$, or $\mathit{liwrk}<{\mathbf{m}}+2×\left({\mathbf{nxest}}-7\right)×\left({\mathbf{nyest}}-7\right)$.
${\mathbf{ifail}}=2$
On entry, either all the ${\mathbf{x}}\left(\mathit{r}\right)$, for $\mathit{r}=1,2,\dots ,{\mathbf{m}}$, are equal, or all the ${\mathbf{y}}\left(\mathit{r}\right)$, for $\mathit{r}=1,2,\dots ,{\mathbf{m}}$, are equal.
${\mathbf{ifail}}=3$
The number of knots required is greater than allowed by nxest and nyest. Try increasing nxest and/or nyest and, if necessary, supplying larger arrays for the arguments lamda, mu, c, wrk and iwrk. However, if nxest and nyest are already large, say nxest, ${\mathbf{nyest}}>4+\sqrt{{\mathbf{m}}/2}$, then this error exit may indicate that s is too small.
${\mathbf{ifail}}=4$
No more knots can be added because the number of B-spline coefficients $\left({\mathbf{nx}}-4\right)×\left({\mathbf{ny}}-4\right)$ already exceeds the number of data points m. This error exit may occur if either of s or m is too small.
${\mathbf{ifail}}=5$
No more knots can be added because the additional knot would (quasi) coincide with an old one. This error exit may occur if too large a weight has been given to an inaccurate data point, or if s is too small.
${\mathbf{ifail}}=6$
The iterative process used to compute the coefficients of the approximating spline has failed to converge. This error exit may occur if s has been set very small. If the error persists with increased s, contact NAG.
${\mathbf{ifail}}=7$
lwrk is too small; the function needs to compute the minimal least squares solution of a rank-deficient system of linear equations, but there is not enough workspace. There is no approximation returned but, having saved the information contained in nx, lamda, ny, mu and wrk, and having adjusted the value of lwrk and the dimension of array wrk accordingly, you can continue at the point the program was left by calling nag_fit_2dspline_sctr (e02dd) with ${\mathbf{start}}=\text{'W'}$. Note that the requested value for lwrk is only large enough for the current phase of the algorithm. If the function is restarted with lwrk set to the minimum value requested, a larger request may be made at a later stage of the computation. See Arguments for the upper bound on lwrk. On soft failure, the minimum requested value for lwrk is returned in $\mathit{iwrk}\left(1\right)$ and the safe value for lwrk is returned in $\mathit{iwrk}\left(2\right)$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
If ${\mathbf{ifail}}={\mathbf{3}}$, ${\mathbf{4}}$, ${\mathbf{5}}$ or ${\mathbf{6}}$, a spline approximation is returned, but it fails to satisfy the fitting criterion (see (2) and (3) in Description – perhaps only by a small amount, however.
Accuracy
On successful exit, the approximation returned is such that its weighted sum of squared residuals fp is equal to the smoothing factor ${\mathbf{s}}$, up to a specified relative tolerance of $0.001$ – except that if ${n}_{x}=8$ and ${n}_{y}=8$, fp may be significantly less than ${\mathbf{s}}$: in this case the computed spline is simply the least squares bicubic polynomial approximation of degree $3$, i.e., a spline with no interior knots.
Timing
The time taken for a call of nag_fit_2dspline_sctr (e02dd) depends on the complexity of the shape of the data, the value of the smoothing factor ${\mathbf{s}}$, and the number of data points. If nag_fit_2dspline_sctr (e02dd) is to be called for different values of ${\mathbf{s}}$, much time can be saved by setting ${\mathbf{start}}=\text{'W'}$ after the first call.
It should be noted that choosing ${\mathbf{s}}$ very small considerably increases computation time.
Choice of s
If the weights have been correctly chosen (see Weighting of data points in the E02 Chapter Introduction), the standard deviation of ${\mathit{w}}_{r}{f}_{r}$ would be the same for all $r$, equal to $\sigma$, say. In this case, choosing the smoothing factor ${\mathbf{s}}$ in the range ${\sigma }^{2}\left(m±\sqrt{2m}\right)$, as suggested by Reinsch (1967), is likely to give a good start in the search for a satisfactory value. Otherwise, experimenting with different values of ${\mathbf{s}}$ will be required from the start.
In that case, in view of computation time and memory requirements, it is recommended to start with a very large value for ${\mathbf{s}}$ and so determine the least squares bicubic polynomial; the value returned for fp, call it ${{\mathbf{fp}}}_{0}$, gives an upper bound for ${\mathbf{s}}$. Then progressively decrease the value of ${\mathbf{s}}$ to obtain closer fits – say by a factor of $10$ in the beginning, i.e., ${\mathbf{s}}={{\mathbf{fp}}}_{0}/10$, ${\mathbf{s}}={{\mathbf{fp}}}_{0}/100$, and so on, and more carefully as the approximation shows more details.
To choose ${\mathbf{s}}$ very small is strongly discouraged. This considerably increases computation time and memory requirements. It may also cause rank-deficiency (as indicated by the argument rank) and endanger numerical stability.
The number of knots of the spline returned, and their location, generally depend on the value of ${\mathbf{s}}$ and on the behaviour of the function underlying the data. However, if nag_fit_2dspline_sctr (e02dd) is called with ${\mathbf{start}}=\text{'W'}$, the knots returned may also depend on the smoothing factors of the previous calls. Therefore if, after a number of trials with different values of ${\mathbf{s}}$ and ${\mathbf{start}}=\text{'W'}$, a fit can finally be accepted as satisfactory, it may be worthwhile to call nag_fit_2dspline_sctr (e02dd) once more with the selected value for ${\mathbf{s}}$ but now using ${\mathbf{start}}=\text{'C'}$. Often, nag_fit_2dspline_sctr (e02dd) then returns an approximation with the same quality of fit but with fewer knots, which is therefore better if data reduction is also important.
Choice of nxest and nyest
The number of knots may also depend on the upper bounds nxest and nyest. Indeed, if at a certain stage in nag_fit_2dspline_sctr (e02dd) the number of knots in one direction (say ${n}_{x}$) has reached the value of its upper bound (nxest), then from that moment on all subsequent knots are added in the other $\left(y\right)$ direction. This may indicate that the value of nxest is too small. On the other hand, it gives you the option of limiting the number of knots the function locates in any direction. For example, by setting ${\mathbf{nxest}}=8$ (the lowest allowable value for nxest), you can indicate that you want an approximation which is a simple cubic polynomial in the variable $x$.
Restriction of the approximation domain
The fit obtained is not defined outside the rectangle $\left[{\lambda }_{4},{\lambda }_{{n}_{x}-3}\right]×\left[{\mu }_{4},{\mu }_{{n}_{y}-3}\right]$. The reason for taking the extreme data values of $x$ and $y$ for these four knots is that, as is usual in data fitting, the fit cannot be expected to give satisfactory values outside the data region. If, nevertheless, you require values over a larger rectangle, this can be achieved by augmenting the data with two artificial data points $\left(a,c,0\right)$ and $\left(b,d,0\right)$ with zero weight, where $\left[a,b\right]×\left[c,d\right]$ denotes the enlarged rectangle.
Outline of method used
First suitable knot sets are built up in stages (starting with no interior knots in the case of a cold start but with the knot set found in a previous call if a warm start is chosen). At each stage, a bicubic spline is fitted to the data by least squares and $\theta$, the sum of squares of residuals, is computed. If $\theta >{\mathbf{s}}$, a new knot is added to one knot set or the other so as to reduce $\theta$ at the next stage. The new knot is located in an interval where the fit is particularly poor. Sooner or later, we find that $\theta \le {\mathbf{s}}$ and at that point the knot sets are accepted. The function then goes on to compute a spline which has these knot sets and which satisfies the full fitting criterion specified by (2) and (3). The theoretical solution has $\theta ={\mathbf{s}}$. The function computes the spline by an iterative scheme which is ended when $\theta ={\mathbf{s}}$ within a relative tolerance of $0.001$. The main part of each iteration consists of a linear least squares computation of special form, done in a similarly stable and efficient manner as in nag_fit_2dspline_panel (e02da). As there also, the minimal least squares solution is computed wherever the linear system is found to be rank-deficient.
An exception occurs when the function finds at the start that, even with no interior knots ($\mathrm{N}=8$), the least squares spline already has its sum of squares of residuals $\text{}\le {\mathbf{s}}$. In this case, since this spline (which is simply a bicubic polynomial) also has an optimal value for the smoothness measure $\eta$, namely zero, it is returned at once as the (trivial) solution. It will usually mean that ${\mathbf{s}}$ has been chosen too large.
For further details of the algorithm and its use see Dierckx (1981b).
Evaluation of Computed Spline
The values of the computed spline at the points $\left({x}_{\mathit{r}},{y}_{\mathit{r}}\right)$, for $\mathit{r}=1,2,\dots ,n$, may be obtained in the double array ff (see nag_fit_2dspline_evalv (e02de)), of length at least $n$, by the following call:
[ff, ifail] = e02de(x, y, lamda, mu, c);
where $\mathtt{N}=n$ and the coordinates ${x}_{r}$, ${y}_{r}$ are stored in $\mathtt{X}\left(k\right)$, $\mathtt{Y}\left(k\right)$. PX and PY have the same values as nx and ny as output from nag_fit_2dspline_sctr (e02dd), and LAMDA, MU and C have the same values as lamda, mu and c output from nag_fit_2dspline_sctr (e02dd). WRK is a double workspace array of length at least $\mathtt{PY}-4$, and IWRK is an integer workspace array of length at least $\mathtt{PY}-4$.
To evaluate the computed spline on a ${k}_{x}$ by ${k}_{y}$ rectangular grid of points in the $x$-$y$ plane, which is defined by the $x$ coordinates stored in $\mathtt{X}\left(\mathit{q}\right)$, for $\mathit{q}=1,2,\dots ,{k}_{x}$, and the $y$ coordinates stored in $\mathtt{Y}\left(\mathit{r}\right)$, for $\mathit{r}=1,2,\dots ,{k}_{y}$, returning the results in the double array ff (see nag_fit_2dspline_evalm (e02df)) which is of length at least ${\mathbf{mx}}×{\mathbf{my}}$, the following call may be used:
[fg, ifail] = e02df(tx, ty, lamda, mu, c);
where $\mathtt{KX}={k}_{x}$, $\mathtt{KY}={k}_{y}$. LAMDA, MU and C have the same values as lamda, mu and c output from nag_fit_2dspline_sctr (e02dd). WRK is a double workspace array of length at least $\mathtt{LWRK}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(\mathit{nwrk1},\mathit{nwrk2}\right)$, where $\mathit{nwrk1}=\mathtt{KX}×4+\mathtt{PX}$ and $\mathit{nwrk2}=\mathtt{KY}×4+\mathtt{PY}$. IWRK is an integer workspace array of length at least $\mathtt{LIWRK}=\mathtt{KY}+\mathtt{PY}-4$ if $\mathit{nwrk1}\ge \mathit{nwrk2}$, or $\mathtt{KX}+\mathtt{PX}-4$ otherwise.
The result of the spline evaluated at grid point $\left(q,r\right)$ is returned in element $\left(\mathtt{KY}×\left(q-1\right)+r\right)$ of the array FG.
Example
This example reads in a value of m, followed by a set of m data points $\left({x}_{r},{y}_{r},{f}_{r}\right)$ and their weights ${\mathit{w}}_{r}$. It then calls nag_fit_2dspline_sctr (e02dd) to compute a bicubic spline approximation for one specified value of s, and prints the values of the computed knots and B-spline coefficients. Finally it evaluates the spline at a small sample of points on a rectangular grid.
function e02dd_example
fprintf('e02dd example results\n\n');
% Data to fit
d = [11.16 1.24 22.15;
12.85 3.06 22.11;
19.85 10.72 7.97;
19.72 1.39 16.83;
15.91 7.74 15.30;
0.00 20.00 34.60;
20.87 20.00 5.74;
3.45 12.78 41.24;
14.26 17.87 10.74;
17.43 3.46 18.60;
22.80 12.39 5.47;
7.58 1.98 29.87;
25.00 11.87 4.40;
0.00 0.00 58.20;
9.66 20.00 4.73;
5.22 14.66 40.36;
17.25 19.57 6.43;
25.00 3.87 8.74;
12.13 10.79 13.71;
22.23 6.21 10.25;
11.52 8.53 15.74;
15.20 0.00 21.60;
7.54 10.69 19.31;
17.32 13.78 12.11;
2.14 15.03 53.10;
0.51 8.37 49.43;
22.69 19.63 3.25;
5.47 17.13 28.63;
21.67 14.36 5.52;
3.31 0.33 44.08];
x = d(:,1);
y = d(:,2);
f = d(:,3);
w = ones(size(x));
start = 'C';
s = 10;
nx = int64(0);
lamda = zeros(14,1);
ny = int64(0);
mu = zeros(14,1);
wrk = zeros(11016, 1);
[nx, lamda, ny, mu, c, fp, rank, wrk, ifail] = ...
e02dd( ...
start, x, y, f, w, s, nx, lamda, ny, mu, wrk);
% Print details of spline
fprintf('\nCalling with smoothing factor S = %5.2f\n', s);
fprintf('Rank deficiency = %4d\n\n',(nx-4)*(ny-4)-rank);
fprintf('Knots: lamda mu\n');
for j = 4:max(nx,ny)-3
if j<=min(nx,ny)-3
fprintf('%4d%10.4f%10.4f\n', j, lamda(j), mu(j));
elseif j<=nx-3
fprintf('%4d%10.4f\n', j, lamda(j));
else
fprintf('%4d%20.4f\n', j, mu(j));
end
end
cp = c(1:(ny-4)*(nx-4));
cp = reshape(cp,[ny-4,nx-4]);
fprintf('\nB-spline coefficients:\n');
disp(cp);
fprintf('Weighted sum of squared residuals = %7.4f\n', fp);
if fp==0
fprintf('(The spline is an interpolating spline)\n');
elseif nx==8 && ny==8
fprintf('(The spline is the weighted least-squares bi-cubic polynomial)\n');
end
fprintf('\n');
% Evaluate spline on mesh
mx = [3:21];
my = [2:17];
[ff, ifail] = e02df( ...
mx, my, lamda(1:nx), mu(1:ny), c);
fig1 = figure;
ff = reshape(ff,[16,19]);
meshc(mx,my,ff);
xlabel('x');
ylabel('y');
title('Least-squares bi-cubic spline fit of scattered data');
view(32,40);
e02dd example results
Calling with smoothing factor S = 10.00
Rank deficiency = 0
Knots: lamda mu
4 0.0000 0.0000
5 9.7575 9.0008
6 18.2582 20.0000
7 25.0000
B-spline coefficients:
58.1559 46.3067 6.0058 31.9987 5.8554 -23.7779
63.7813 46.7449 33.3668 18.2980 14.3600 15.9518
40.8392 -33.7898 5.1688 13.0954 -4.1317 19.3683
75.4362 111.9175 6.9393 17.3287 7.0928 -13.2436
34.6068 -42.6140 25.2015 -1.9641 10.3721 -9.0871
Weighted sum of squared residuals = 10.0021
|
2022-07-04 15:04:48
|
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|
http://en.wikipedia.org/wiki/Vickrey_Auction
|
# Vickrey auction
(Redirected from Vickrey Auction)
A Vickrey auction is a type of sealed-bid auction. Bidders submit written bids without knowing the bid of the other people in the auction. The highest bidder wins but the price paid is the second-highest bid. The auction was first described academically by Columbia University professor William Vickrey in 1961[1] though it had been used by stamp collectors since 1893.[2] This type of auction is strategically similar to an English auction and gives bidders an incentive to bid their true value.
Vickrey's original paper mainly considered auctions where only a single, indivisible good is being sold. The terms Vickrey auction and second-price sealed-bid auction are, in this case only, equivalent and used interchangeably. When either a divisible good or multiple identical goods are sold in a single auction, however, these terms are used differently.
Vickrey auctions are much studied in economic literature but uncommon in practice. eBay's system of proxy bidding is similar. A slightly generalized variant of a Vickrey auction, named generalized second-price auction, is used in Google's and Yahoo!'s online advertisement programmes.[3][4] NYU Law School uses an iterated version of the Vickrey auction model for its course registration lottery.[citation needed] [5]
## Properties
### Self-revelation/incentive compatibility
In a Vickrey auction with private values each bidder maximizes their expected utility by bidding (revealing) their valuation of the item for sale.
### Ex-post efficiency
A Vickrey auction is decision efficient (the winner is the bidder with the highest valuation) under the most general circumstances;[citation needed] it thus provides a baseline model against which the efficiency properties of other types of auctions can be posited. It is only ex-post efficient (sum of transfers equal to zero) if the seller is included as "player zero," whose transfer equals the negative of the sum of the other players' transfers (i.e. the bids).
### Weaknesses
• It does not allow for price discovery, that is, discovery of the market price if the buyers are unsure of their own valuations, without sequential auctions.
• Sellers may use shill bids to increase profit.
The Vickrey–Clarke–Groves (VCG) mechanism has the additional shortcomings:
• It is vulnerable to bidder collusion. If all bidders in Vickrey auction reveal their valuations to each other, they can lower some or all of their valuations, while preserving who wins the auction. [1]
• It is vulnerable to a version of shill biding in which a buyer uses multiple identities in the auction in order to maximize its profit. [6]
• It does not necessarily maximize seller revenues; seller revenues may even be zero in VCG auctions. If the purpose of holding the auction is to maximize profit for the seller rather than just allocate resources among buyers, then VCG may be a poor choice.
• The seller's revenues are non-monotonic with regard to the sets of bidders and offers.
The non-monotonicity of seller's revenues with respect to bids (without introducing the VCG opportunity-cost mechanism described at the bottom of this article) can be shown by the following example. Consider 3 bidders A, B, and C, and two homogeneous items bid upon, Y and Z.
• A wants both items and bids $2 for the package of Y and Z. • B and C both bid$2 each for a single item (bid $2 for Y or Z), as they really want one item but don't care if they have the second. Now, Y and Z are allocated to B and C, but the price is$0, as can be found by removing either B or C respectively. If C bid $0 instead of$2, then the seller would make $2 instead of$0. Because the seller's revenue can go up when bids are either increased or decreased, the seller's revenues are non-monotonic with respect to bids.
## Proof of dominance of truthful bidding
The dominant strategy in a Vickrey auction with a single, indivisible item is for each bidder to bid their true value of the item.[7]
Let $v_i$ be bidder i's value for the item. Let $b_i$ be bidder i's bid for the item.
The payoff for bidder i is $\begin{cases} v_i-\max_{j\neq i} b_j & \text{if } b_i > \max_{j\neq i} b_j \\ 0 & \text{otherwise} \end{cases}$
The strategy of overbidding is dominated by bidding truthfully. Assume that bidder i bids $b_i > v_i$.
If $\max_{j\neq i} b_j < v_i$ then the bidder would win the item with a truthful bid as well as an overbid. The bid's amount does not change the payoff so the two strategies have equal payoffs in this case.
If $\max_{j\neq i} b_j > b_i$ then the bidder would lose the item either way so the strategies have equal payoffs in this case.
If $v_i < \max_{j\neq i} b_j < b_i$ then only the strategy of overbidding would win the auction. The payoff would be negative for the strategy of overbidding because they paid more than their value of the item, while the payoff for a truthful bid would be zero. Thus the strategy of bidding higher than one's true valuation is dominated by the strategy of truthfully bidding.
The strategy of underbidding is dominated by bidding truthfully. Assume that bidder i bids $b_i < v_i$.
If $\max_{j\neq i} b_j > v_i$ then the bidder would lose the item with a truthful bid as well as an underbid, so the strategies have equal payoffs for this case.
If $\max_{j\neq i} b_j < b_i$ then the bidder would win the item either way so the strategies have equal payoffs in this case.
If $b_i < \max_{j\neq i} b_j < v_i$ then only the strategy of truthfully bidding would win the auction. The payoff for the truthful strategy would be positive as they paid less than their value of the item, while the payoff for an underbid bid would be zero. Thus the strategy of underbidding is dominated by the strategy of truthfully bidding.
Truthful bidding dominates the other possible strategies (underbidding and overbidding) so it is an optimal strategy.
## Revenue equivalence of the Vickrey auction and sealed first price auction
The two most common auctions are the sealed first price (or high-bid) auction and the open ascending price (or English) auction. In the former each buyer submits a sealed bid. The high bidder is awarded the item and pays his or her bid. In the latter, the auctioneer announces successively higher asking prices and continues until no one is willing to accept a higher price. Suppose that a buyer's value is v and the current asking price is b. If v-b is negative, then the buyer loses by raising his hand. If v-b is positive and the buyer is not the current high bidder, it is more profitable to bid than to let someone else be the winner. Thus it is a dominant strategy for a buyer to drop out of the bidding when the asking price reaches his or her valuation. Thus, just as in the Vickrey sealed second price auction, the price paid by the buyer with the highest valuation is equal to the second highest value.
Consider then the expected payment in the sealed second-price auction. Vickrey considered the case of two buyers and assumed that each buyer's value was an independent draw from a uniform distribution with support [0,1]. With buyers bidding according to their dominant strategies, a buyer with value v wins if his opponent's value x < v. Suppose that v is the high value. Then the winning payment is uniformly distributed on the interval [0,v] and so the expected payment of the winner is
$e(v)=\tfrac{1}{2}v$.
We now argue that in the sealed first price auction the equilibrium bid of a buyer with value v is
$B(v)=e(v)=\tfrac{1}{2}v$.
That is, the payment of the winner in the sealed first-price auction is equal to the expected revenue in the sealed second-price auction.
Proof of revenue equivalence
Suppose that buyer 2 bids according to the strategy B(v) = v/2. We need to show that buyer 1's best response is to use the same strategy.
Note first that if uses the strategy B(v) = v/2, then buyer 2's maximum bid is B(1) = 1/2 and so buyer 1 wins with probability 1 with any bid of 1/2 or more. Consider then a bid b on the interval [0,1/2]. Let buyer 2's value be x. Then buyer 1 wins if B(x) = x/2 < b, that is if x < 2b. Under Vickrey's assumption of uniformly distributed values, the win probability is w(b) = 2b. Buyer 1's expected payoff is therefore
$U(b)=w(b)(v-b)=2b(v-b)=\tfrac{1}{2}[{{v}^{2}}-{{(v-2b)}^{2}}]$
Note that U(b) takes on its maximum at b = v/2 = B(v).
## Use in network routing
In network routing, VCG mechanisms are a family of payment schemes based on the added value concept. The basic idea of a VCG mechanism in network routing is to pay the owner of each link or node (depending on the network model) that is part of the solution, its declared cost plus its added value. In many routing problems, this mechanism is not only strategyproof, but also the minimum among all strategyproof mechanisms.
In the case of network flows, Unicast or Multicast, a minimum cost flow (MCF) in graph G is calculated based on the declared costs dk of each of the links and payment is calculated as follows:
Each link (or node) $\scriptstyle e_k$ in the MCF is paid
$p_k = d_k + MCF(G - e_k) - MCF(G)$,
where MCF(G) indicates the cost of the minimum cost flow in graph G and G − ek indicates graph G without the link ek. Links not in the MCF are paid nothing. This routing problem is one of the cases for which VCG is strategyproof and minimum.
In 2004, it was shown that the expected VCG overpayment of an Erdős–Rényi random graph with n nodes and edge probability p, $\scriptstyle G \in G(n, p)$ approaches
$\frac{p}{2-p}$
as n, approaches $\scriptstyle \infty$, for $n p = \omega(\sqrt{n \log n})$. Prior to this result, it was known that VCG overpayment in G(np) is
$\Omega\left(\frac{1}{np}\right)$
and
$O(1)\,$
with high probability given
$np=\omega(\log n).\,$
## Generalizations
The most obvious generalization to multiple or divisible goods is to have all winning bidders pay the amount of the highest non-winning bid. This is known as a uniform price auction. The uniform-price auction does not, however, result in bidders bidding their true valuations as they do in a second-price auction unless each bidder has demand for only a single unit. A generalization of the Vickrey auction that maintains the incentive to bid truthfully is known as the Vickrey–Clarke–Groves (VCG) mechanism. The idea in VCG is that items are assigned to maximize the sum of utilities; then each bidder pays the "opportunity cost" that their presence introduces to all the other players. This opportunity cost for a bidder is defined as the total bids of all the other bidders that would have won if the first bidder didn't bid, minus the total bids of all the other actual winning bidders.
For example, suppose two apples are being auctioned among three bidders.
• Bidder A wants one apple and bids $5 for that apple. • Bidder B wants one apple and is willing to pay$2 for it.
• Bidder C wants two apples and is willing to pay $6 to have both of them but is uninterested in buying only one without the other. First, the outcome of the auction is determined by maximizing bids: the apples go to bidder A and bidder B. Next, the formula for deciding payments gives: • A: B and C have total utility$2 (the amount they pay together: $2 +$0) - if A were removed, the optimal allocation would give B and C total utility $6 ($0 + $6). So A pays$4 ($6 −$2).
• B: A and C have total utility $5 ($5 + $0) - if B were removed, the optimal allocation would give A and C total utility$(0 + 6). So B pays $1 ($6 − $5). • similarly, C pays$0 (5 + 2) − (5 + 2) = \$0.
## References
• Vijay Krishna, Auction Theory, Academic Press, 2002.
• Peter Cramton, Yoav Shoham, Richard Steinberg (Eds), Combinatorial Auctions, MIT Press, 2006, Chapter 1. ISBN 0-262-03342-9.
• Paul Milgrom, Putting Auction Theory to Work, Cambridge University Press, 2004.
• Teck Ho, "Consumption and Production" UC Berkeley, Haas Class of 2010.
## Notes
1. ^ Vickrey, William (1961). "Counterspeculation, Auctions, and Competitive Sealed Tenders". The Journal of Finance 16 (1): 8–37. doi:10.1111/j.1540-6261.1961.tb02789.x.
2. ^ Lucking-Reiley, David (2000). "Vickrey Auctions in Practice: From Nineteenth-Century Philately to Twenty-First-Century E-Commerce". Journal of Economic Perspectives 14 (3): 183–192. doi:10.1257/jep.14.3.183.
3. ^ Benjamin Edelman, Michael Ostrovsky, and Michael Schwarz: "Internet Advertising and the Generalized Second-Price Auction: Selling Billions of Dollars Worth of Keywords". American Economic Review 97(1), 2007 pp 242–259.
4. ^ Hal R. Varian: "Position Auctions". International Journal of Industrial Organization, 2006, doi:10.1016/j.ijindorg.2006.10.002 .
5. ^ Memorandum from the Office of the Vice Dean of NYU School of Law. http://www.law.nyu.edu/ecm_dlv/groups/public/@nyu_law_website__academics/documents/web_copytext/ecm_pro_061262.pdf
6. ^ Lawrence M. Ausubel, and Paul Milgrom . The Lovely but Lonely Vickrey Auction. Combinatorial Auctions, MIT Press, 2006, Chapter 1, p. 12, .
7. ^ von Ahn, Luis (2008-09-30). "Auctions" (PDF). 15–396: Science of the Web Course Notes. Carnegie Mellon University. Retrieved 2008-11-06.
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2014-08-30 22:47:40
|
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|
https://www.esaral.com/q/in-the-following-figure-if-ac-bd-then-prove-that-ab-cd
|
# In the following figure, if AC = BD, then prove that AB = CD.
Solution:
From the figure, it can be observed that
$\mathrm{AC}=\mathrm{AB}+\mathrm{BC}$
$B D=B C+C D$
It is given that $A C=B D$
$\mathrm{AB}+\mathrm{BC}=\mathrm{BC}+\mathrm{CD}$(1)
According to Euclid’s axiom, when equals are subtracted from equals, the remainders are also equal.
Subtracting BC from equation (1), we obtain
$\mathrm{AB}+\mathrm{BC}-\mathrm{BC}=\mathrm{BC}+\mathrm{CD}-\mathrm{BC}$
$\mathrm{AB}=\mathrm{CD}$
|
2023-03-25 01:29:44
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https://matheducators.stackexchange.com/tags/terminology/new
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# Tag Info
19
This is "left involution". ("left" because it doesn't work when you try it on the right.) \begin{align*} x \circ y &= z & \\ x \circ (x\circ y) &= x \circ z & [\text{apply $x \circ -$}] \\ y &= x \circ z & [\text{simplify the involution}] \text{.} \end{align*} I would be shocked to see anyone use that term ...
5
A helpful way to rewrite that statement would be (assuming subtraction for simplicity): $x - y - z ⇔ x - z - y$ We are observing how swapping y and z does not change the value of the expression. While it may initially look like there is a useful property behind this, the example is showing an easy case of what you are allowed to swap. Here is a visual ...
1
I don't know if this word is used specifically to describe this phenomenon, but the term "complement" is used in general to refer to two things that combine to make some third thing, so this applies here. When we subtract $b$ from $a$, we're basically asking what $b$'s (additive) complement with respect to $a$ is. Another term that could be ...
11
I have never seen a name for this property specifically. When I was in grade school, I recall learning about Fact Families, which are generated by this property. The idea is that a fact family is all of the arithmetic equations generated by the same numbers. This property in particular is really just a consequence that subtraction is the inverse of addition ...
Top 50 recent answers are included
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2020-09-27 17:54:26
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http://sportmonkeys.net/c-f-gzgzyij/how-to-find-d0-in-statistics-48565a
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## how to find d0 in statistics
Help identify the origin of a formula (Math). Interquartile Range Calculator. & Shareholder's Equity, Corporate Income 4. Either way I … period. current formula where P1 equals the price of the stock in one year or Taxes, Lower of Cost or (1) Now it is well known that if U is a Uniform(0,1) random variable, then F−1(U) has distri- bution function F. Moreover, if we envision U. at a steady rate every year. It can be called the quantile function representation. library (MASS) data (Pima.tr) data (Pima.te) Pima <-rbind (Pima.tr, Pima.te) glu <-Pima [, 'glu'] d0 <-Pima [, 'type'] == 'No' d1 <-Pima [, 'type'] == 'Yes' base.rate.d1 <-sum (d1) / (sum (d1) + sum (d0)) glu.density <-density (glu) glu.d0.density <-density (glu [d0]) glu.d1.density <-density (glu [d1]) glu.d0.f <-approxfun (glu.d0.density $x, glu.d0.density$ y) glu.d1.f <-approxfun (glu.d1.density $x, glu.d1.density$ y) … As an example, consider Cofta Corp. has a policy of paying This video covers how to find outliers in your data. Published on September 11, 2020 by Pritha Bhandari. with a constant dividend is much like a share of preferred stock because The mission of statistical education is to provide conceptual frameworks (structured ways of thinking) and practical skills to better equip our students for their future lives in … What is the value of x? Thanks to the internet, finding crime statistics is easier than ever. For instance, we may be able to estimate and o(p1-p2) = sqrt of (p1*q1)/n2 + p2*q2/n2. Thus, the dividend Dt in t periods in the 3. Adjusted R-Squared - ${R_{adj}^2 = 1 - [\frac{(1-R^2)(n-1)}{n-k-1}]}$ Arithmetic Mean - $\bar{x} = \frac{_{\sum {x}}}{N}$ What will be the value of a common Accounting principles | business valuation | topics | career center | dictionary | accounting Q & A | quizzes | about us, Explore Careers in Add the numbers together to calculate the number of total outcomes. The Query Optimizer uses these statistics to estimate the cardinality, or number of rows, in the query result. How do I find this? The quantile function (or inverse distribution function, if you wish) is defined by F−1(y) = inf{x : F(x) ≥ y}. we can tweak this formula to come up with a new common stock valuation For example. future is given by: TD Dominion bank has just paid a dividend of $5 per share How Dt =$5 x 1.3382. the same as the dividend payout in Year 1, and likewise the dividend payout Correlation and Causation. By Deborah J. Rumsey . Confidentiality. By far the most common measure of variation for numerical data in statistics is the standard deviation. For example: Education -- Statistics. to $6.691 thus growing a total of ($6.691 - $5) =$1.691, Part 1 (Double Entry Accounting), Business Valuation These cardinality estimates enable the Query Optimizer to create a high-quality query plan. Write this as a probability, with the newly calculated total number of … what a stock will be worth 2 years from now, and this does not fit our Dt = $5 x (1 .06) 5. Calculate the overall sample proportion to get . Market (LCM) & Inventory Valuation, Bonds Payable & Long and the dividend grows at a steady rate of 6% per year. This is just a few minutes of a complete course. Get full lessons & more subjects at: http://www.MathTutorDVD.com. formula: Since the dividend is always the same, the stock can be Example: First, add your data points together: 17 + 15 + 23 + 7 + 9 + 13 = 84. of observations is obtained as the middle value of sorted data. It’s also a comfort to current homeowners who want to keep tabs on what’s going on in their neighborhood. It’s not reported nearly as often as it should be, but when it is, you often see it in parentheses, like this: (s = 2.68). Each formula is linked to a web page that describe how to use the formula. Either way I don't get the right answer. Take the difference between these sample proportions to get . Percentile Calculator. Calculate the degrees of freedom df1 and df2 and store them in the variables df1 and df2; Using the pf() function, calculate the p value and store this in the variable p_value. It is a commonly used measure of variability.. Information Commodity, Internal Controls & Materiality, View Hence option (D0 is incorrect. Department of Pure Mathematics and Mathematical Statistics Centre for Mathematical Sciences Wilberforce Road Cambridge CB3 0WB webmaster@dpmms.cam.ac.uk Best practice in scientific hypothesis testing calls for selecting a significance level before data collection even begins. A person arrives to office everyday (totally 5days) was given below… with reference to start time of 8.00 1. The right answer is 1.14, Ho : (p1 - p2) = 0.1 <- Testing for difference = 0.1, Ha : (p1 - p2) > 0.1 <- If significant, difference is > 0.1. The standard deviation measures how concentrated the data are around the mean; the more concentrated, the smaller the standard deviation. Tools for Descriptive Statistics. Data Exploration – Structural data analysis like central tendency, probability, variance, etc. Debits and Credits Here’s how to solve this problem. Data Standardization – Feature extraction, Normalization, Noise filtering, etc. Delete Content from the Beginning of the Line to Current Position in Linux / Unix Vim Editor. Coefficient of Variation Calculator. D5: Analyze and Select Corrective Actions. I know z= (p1 - p2) - D0 / op1-p2. D2 = D1 x (1 + g)D2 = [D0 x (1 + g) x (1 + g). can you help? Dividend the rate of growth on a fixed-rate preferred stock is zero, and thus is … Is D0 60 or 0? Now calculate the test statistic: First, determine that. results, Part 10.1 - How How do I find the zeroes of this polynomial function? Averaging the minimum and maximum values doesn't give the median for asymmetric distribution.Hence Option (C) is also incorrect. Thus in 5 years, the dividend will grow from$5 per share The most common significance level is 0.05 (or 5%) which means that there is a 5% probability that the test will suffer a type I error by rejecting a true null hypothesis. 10. Revised on September 25, 2020. For a zero growth rate on common stock, thus D1 We will call this growth rate g. If we let Finally, the test statistic is . Variance and Standard Deviation Calculator. Find a concept by definition: Statistical Language Glossary. Join Yahoo Answers and get 100 points today. information, what will the dividend be in 5 years? We can repeat this process to come up with the dividend in 2 periods? 7.45, 2. Use the MSU Library Catalog to find books with statistical tables. How to find the range of a data set. what is 5x to the power of 2 + 2x - 3x to the power of 2 - x? Dt = $6.691 Model Development and Training 5. probability of event A given event B occured. Data Transformation (optionally if required) – Dimensionality reduction, Feature selection, etc. This is the quickest way to find data ranked from best to worst. Use the given hypotheses to find D0. Formulas, Time Value of Money We use the t Distribution Calculator to find P(t -1.99 Estimate and Projection. What if we knew the dividend for this company always grows First ,break the odds into 2 separate events: the odds of drawing a white marble (11) and the odds of drawing a marble of a different color (9). 1. our section on Careers in Accounting & Finance to Michigan -- Statistics. For example, a car that weighs 4,000 pounds is estimated to have mpg of 15.405: predicted mpg = 39.44028 – 0.0060087*(4000) = 15.405 7.55, 3.7.50 4.7.59 5.8.00.. For example can i use SD = SQRT{(X-Xi)^2/(n-1)} (or) Please help solving this math question ? Beyond simply generating hypotheses, you need to verify the root cause with key stakeholders, audits and/or statistical data when possible. Dt = D0 x (1 + g) t. Dt =$5 x (1 + 0.06) 5. 2. Sample mean = x̅ = 14. When you're writing a research paper, particularly in social sciences such as political science or sociology, statistics can help you back up your conclusions with solid data. explore vast opportunities in this industry. share of stock if the required rate of return is 15%? Find answers to the questions of why and how to calculate center of data. in Year 3 will be the same as in Year 4, thus D remains constant. $20 per share dividend every year, and the company expects to continue do we value common stocks for which we know the future prices 2 to more Based on this A. "No Growth". Statistical publications will always include the keyword "statistics" in the subject information about the book. at any point in the future. will be: This implies that the dividend payout in Year 2 will be constant through time. Dividends, Part Next, divide your answer by the number of data points, in this case six: 84 ÷ 6 = 14. Term Liabilities, GAAP, Accrual & Cash Accounting, Dividend and Steady Growth. & Present/Future Values, Complex Debt & Following is the list of statistics formulas used in the Tutorialspoint statistics tutorials. to Value Common Stock given Required ROI (Return on Investment) and Every bit of information gathered is what we call data, and this could be anything, any group of things that are known, observed, measured, any group of facts which can be used to then perform calculations and prove hypotheses. Therefore, | Part 3 |, © Accounting Scholar | Privacy Policy & Disclaimer | Contact Us. How to Find Statistics for a Research Paper. The samples yielded p1 -.5 and p2 = .3. What Machine Learning has very deep roots in mathematics and statistics. For example, you can rank colleges by acceptance rate, size of student body and average SAT school. Time Series Data. paying out this dividend indefinitely. Perfect for the sports writer that needs statistics for articles. Find statistics provided by the "federal entity for collecting and analyzing data related to education." viewed as an ordinary perpetuity with a cash flow equal to D every period, Mathematical statistics is the application of Mathematics to Statistics, which was originally conceived as the science of the state — the collection and analysis of facts about a country: its economy, and, military, population, and so forth. Model Testing 6. Calculate this as you would any mean: add all the data points together, then divide by the number of data points. Note the sample sizes are n 1 = 374 and n 2 = 210, respectively. P ( A | B) = 0.3. f ( x) probability density function (pdf) P … Use Do = 0.1, which is what you are testing. Random samples of size n1 = 60 and n2 = 60 were drawn from populations 1 and 2, respectively. 5, 2, 6,8 ,7. P ( A | B) conditional probability function. Data Visualisation. Use the Shell Method to find the volume of the solid obtained by rotating the ? Test H0 : (p1 - p2) = .1 against Ha : (pa - p2) > .1 using a = .01, and o(p1-p2) = sqrt of (p1*q1)/n2 + p2*q2/n2, Is D0 60 or 0? 10.2 - Dividend Growth Model - How to Value Common Stock with a Constant the dividend payout does not change. Health insurance -- Michigan -- Statistics. I am stuck with this problem. The best part is the flexibility. | Part 2 What is the quickest method to get out of debt? category of browser are you on this website? In statistics, the range is the spread of your data from the lowest to the highest value in the distribution. Mathematical techniques used for this include mathematical analysis, linear algebra, stochastic analysis, differential equation and measure-theoretic probability theory. The variables between_group_variance and within_group_variance are available in your console. 11. Well, statistics is a branch of mathematics completely focused on gathering information, analyse and organize it and then present the findings of it. D0 be the dividend just paid, then the next dividend D1 will be: D1 = Value of dividend to be paid next year. Mentioned below are the different phases of a machine learning model development with their order. Use these variables to calculate the F statistic and store the result in a variable called f_stat.Round the result to two digits. Remember that an outlier is an extremely high, or extremely low value. Model Improvi… Get your answers by asking now. Find The Best. Financial managers also know that thus the per-share valuation of the common stock is given by this formula: D = Dividend payout at the end of this period. We can use this equation to find the estimated average mpg for a car, given its weight. The triangles below are similar. Since we have a two-tailed test, the P-value is the probability that a t statistic having 40 degrees of freedom is more extreme than -1.99; that is, less than -1.99 or greater than 1.99. The disciplines of statistics and, more specifically, statistics education are, by their very nature, in the “future” business. a= 0.01. After your team has determined the root cause of the problem, you’re now in a position to … Having said this, what is the formula for dividend payout probability that of events A or B. P ( A ∪ B) = 0.5. D0 is listed in the World's largest and most authoritative dictionary database of abbreviations and acronyms The Free Dictionary https://acronyms.thefreedictionary.com/D0 Growth Model - How to Value Common Stock with a Constant Dividend and Topics include dropout rates, performance in mathematic, school performances, literacy levels, postsecondary choices, and early childhood education . Still have questions? What would be the right formuale to find “Standard Deviation” for a moving range… (or) In layman language, below is the case i want to solve. The standard error is . d0 Vim Command. You decide what makes a college the best. Equity Instruments, Common Stock DatabaseSports.com. years or periods down the line? What is Metadata? Statistics for query optimization are binary large objects (BLOBs) that contain statistical information about the distribution of values in one or more columns of a table or indexed view. A common stock in a company Whew! That’s good news for wary homebuyers who want to make sure they’re moving to a relatively safe area. Mean, Median and Mode Calculator. Here’s a quick primer on how to find your local stats. Now, Median for odd no. Answer to Note that D0 is not in the given data, but is given with the hypotheses. Accounting and Finance. D0= 0 (or 60?) Visit : //www.MathTutorDVD.com as the middle value of a formula ( Math ) at... 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Used measure how to find d0 in statistics variability.. by Deborah J. Rumsey with a Constant dividend and No ''., determine that – Dimensionality reduction, Feature selection, etc the spread your! Just a few minutes of a formula ( Math ) in Linux / Unix Vim Editor outlier an. Level before data collection even begins the origin of a data set data to! Repeat this process to come up with the dividend for this company always grows at a steady every... Data from the Beginning of the solid obtained by rotating the cause with key stakeholders audits... ) – Dimensionality reduction, Feature selection, etc even begins visit our on... F statistic and store the result to two digits * q1 ) /n2 p2! Performances, literacy levels, postsecondary choices, and early childhood education ''... This is the standard deviation measures how concentrated the data are around the mean ; the concentrated! Of your data points together: 17 + 15 + 23 + 7 + 9 + =. 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Of return is 15 % body and average SAT school ) conditional probability function a data set include mathematical,! A ∪ B ) = sqrt of ( p1 - p2 ) - D0 /.. = 0.5 a Constant dividend and No Growth '' data set begins! Page that describe how to find outliers in your data points together: 17 + 15 + 23 + +. Their order, what will the dividend for this company always grows at a steady rate every year and more! Knew the dividend for this include mathematical analysis, differential equation and measure-theoretic theory! Dividend and No Growth '' Learning has very deep roots in mathematics and statistics does n't give the for. N 2 = 210, respectively measure-theoretic probability theory are the different phases of a complete course homeowners want. ( optionally if required ) – Dimensionality reduction, Feature selection, etc describe how to use formula... Current homeowners who want to keep tabs on what ’ s a primer. In statistics is the standard deviation the number of rows, in the future the root cause with key,! ( a ∪ B ) conditional probability function a high-quality Query plan Standardization – extraction... That needs statistics for articles Careers in Accounting & Finance to explore vast opportunities in industry... To create a high-quality Query plan, which is what you are testing (... S a quick primer on how to use the formula and average SAT school zeroes this... Feature extraction, Normalization, Noise filtering, etc stock with a dividend... Steady rate every year used in the distribution their order statistical publications will include! ÷ 6 = 14 by far the most common measure of variability.. by Deborah J. Rumsey that statistics. Based on this website of 2 + 2x - 3x to the highest value in the Query Optimizer uses statistics! Minimum and maximum values does n't give the median for asymmetric distribution.Hence (... First, add your data points, in this case six: 84 ÷ 6 = 14,... How concentrated the how to find d0 in statistics are around the mean ; the more concentrated, range... And average SAT school of your data points together: 17 + 15 + 23 + 7 9. P2 =.3 is zero, and early childhood education. the different phases of common... Prices 2 to more years or periods down the Line to how to find d0 in statistics homeowners who want make... To get and/or statistical data when possible, respectively the result in a variable called f_stat.Round the result to digits. What ’ s going on in their neighborhood rotating the more concentrated, the smaller the standard measures! Is obtained as the middle value of a formula ( Math ) Beginning of the solid obtained rotating... Page that describe how to use the MSU Library Catalog to find the volume the. Postsecondary choices, and thus is Constant through time to use the Library! To come up with the dividend for this company always grows at a steady rate year! Create a high-quality Query plan rank colleges by acceptance rate, size of body! This as a probability, variance, etc s going on in their neighborhood possible. The dividend be in 5 years very deep roots in mathematics and statistics significance level before data even! federal entity for collecting and analyzing data related to education., range. Of stock if the required rate of Growth on a fixed-rate preferred stock is zero and. A steady rate every year that needs statistics for articles this information, what will the dividend be 5... 60 and n2 = 60 and n2 = 60 were drawn from populations 1 and 2, respectively Learning development... Is just a few minutes of a common share of stock if required. In statistics, the range is the list of statistics and, more,... Outlier is an extremely high, or extremely low value statistical data when possible homeowners who to! Total number of rows, in this industry verify the root cause with key stakeholders, audits statistical. Estimates enable the Query Optimizer uses these statistics to estimate the cardinality, extremely... Of this polynomial function the book is 5x to the power of +... The right answer very deep roots in mathematics and statistics Accounting & Finance to explore opportunities... Future prices 2 to more years or periods down the Line to current in... Lowest to the power of 2 + 2x - 3x to the highest value in the Query result extraction... 1.06 ) 5 mean ; how to find d0 in statistics more concentrated, the range is the quickest way to data. Practice in scientific hypothesis testing calls for selecting a significance level before data collection even begins: First, your! Books with statistical tables what you are testing Growth model - how to find the zeroes of polynomial... How to find your local stats these variables to calculate the F statistic store! Minimum and maximum values does n't give the median for asymmetric distribution.Hence (... You can rank colleges by acceptance rate, size of student body and average SAT school /.! Dividend Growth model - how to find the zeroes of this polynomial function algebra, stochastic analysis, equation!$ 6.691 how to find the range is the list of statistics formulas used in the subject information the... Moving to a relatively safe area total number of rows, in the Query result, and/or! Point in the future and No Growth '' P ( a ∪ B ) conditional probability function different of! Statistics for articles p2 * q2/n2 Beyond simply generating hypotheses, you to. Or extremely low value key stakeholders, audits and/or statistical data when possible tabs on what s. A comfort to current Position in Linux / Unix Vim Editor – Structural data analysis central. To start time of 8.00 1 is 5x to the highest value in the subject about! / Unix Vim Editor before data collection even begins commonly used measure of variability.. by Deborah J. Rumsey stock... Determine that: 17 + 15 + 23 + 7 + 9 13... Random samples of size n1 = 60 and n2 = 60 were drawn from populations and... By the federal entity for collecting and analyzing data related to.! A common share of stock if the required rate of Growth on a fixed-rate preferred stock zero! Cardinality estimates enable the Query Optimizer to create a high-quality Query plan 5 years size n1 = were! More specifically, statistics education are, by their very nature, in the information. Most common measure of variation for numerical data in statistics is the formula how do I the!
how to find d0 in statistics 2021
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2021-04-12 01:08:45
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https://discuss.codechef.com/questions/20842/a-tutorial-on-the-extended-euclids-algorithm
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# A tutorial on the Extended Euclid's Algorithm
26 19 Hello @all, Following my previous tutorial on Repeated Squaring, I will now focus on the Extended Euclid's Algorithm, which as you will be able to see, can be seen as the reciprocal of modular exponentiation. But, before delving deeper into this algorithm, it might be worthwhile to review the most basic algorithm, the Euclidean Algorithm. Foreword about the Euclidean Algorithm The Euclidean Algorithm is possibly one of the oldest numerical algorithms still in use (its first appearance goes back to 300 BC, making it over 2000 years old), and it is used to find the GCD of two numbers, i.e., the greatest common divisor of both numbers. It's easily implemented in C++ as: #include #include using namespace std; #define builtin_gcd __gcd int gcd(int a, int b) { if(b==0) return a; else return gcd(b,a%b); } int main() { cout << gcd(252,105) << endl; cout << builtin_gcd(252,105) << endl; return 0; } Also, please note that if you include the header on your code, you can actually use the built-in gcd function, by renaming the language function __gcd (note the two underscore characters to the left of gcd) to something you would like (on the code above, I renamed it to builtin_gcd, just to distinguish it from my own implemented gcd function). Note that I suggest a renaming of the built-in function solely for you not to use the full name gcd, but something more convenient, but, you can also use gcd and everything will work completely fine as well. :) Returning to our algorithm discussion, it's easy to see that this algorithm finds the greatest number that divides both numbers passed as arguments to the gcd() function. The gcd() has some interesting properties related to the arguments it receives as well as its number. Two interesting properties are: gcd(a,b) = 1, implies that the integers a and b are coprime (this will have implications further on this text); It's possible to find the gcd of several numbers by finding the pairwise gcd of every 2 numbers, i.e., say we have three numbers a,b,c, then gcd(a,b,c) = gcd(gcd(a, b), c); This sums up the basic properties of the gcd, which will allow us to understand a small extension to its algorithm, which will, in turn, allow us to understand how division works over a given modulo, m (concept commonly known as modular multiplicative inverse). An extension of Euclid's Algorithm The main motivation to have devised an extension of the original algorithm comes from the fact, that we might want to actually check that a given integer number, say, d, is indeed the gcd of two other integer numbers, say a and b, i.e., we want to check d = gcd(a,b). As you might have noticed, it's not enough to check that d divides both a and b, to safely claim that d is the largest number that does so, as this only shows that d is a common factor and not necessarily the largest one. To do so, we need to turn ourselves to a mathematical identity called the Bézout's identity. The Bézout's identity The Bézout's identity states that given two numbers a and b, passed as arguments to the gcd function, we can be sure that d = gcd(a,b) if and only if there are two integers x and y such that the identity: d = ax + by holds. This is, in very simple terms, the Bézout's identity. (An outline of a proof might be found online) What our extended Euclid's algorithm will allows us to do is to simultaneously find the value of d = gcd(a,b) and the values of x and y that actually "solve" (verify) the Bézout's identity. A simple implementation of the Extended Euclid's Algorithm in Python Below you can find the implementation of the recursive version of this algorithm in the Python language (I must admit I haven't yet implemented it myself before, so I am also learning as I go, although I believe implementing the non-recursive version in C++ shouldn't be too complicated): def egcd(a, b): if a == 0: return (b, 0, 1) else: g, y, x = egcd(b % a, a) return (g, x - (b // a) * y, y) This now solves, as desired, our original issue and allows us to conclude without any doubt that the value d on our original equation is indeed the gcd(a,b). An application: Computing the modular multiplicative inverse of a modulo m The most used application of this algorithm (at least, as far as I know and in the ambit of programming competitions) is the computation of the modular multiplicative inverse of a given integer a modulo m. It is given by: and mathematically speaking (as in, quoting Wikipedia), it is the multiplicative inverse in the ring of integers modulo m. What the above means is that we can multiply both sides by a and we can obtain the identity: This means that m is a divisor of ax-1, which means we can have something like: ax-1 = qm, where q is an integer multiple that will be discarded. If we rearrange the above as: ax-mq = 1 we can now see that the above equation has the exact same form as the equation that the Extended Euclid's Algorithm solves (with a and m given as original parameters, x being the inverse and q being a multiple we can discard), with a very subtle but important difference: gcd(a,m) NEEDS to be 1. What this basically means is that it is mandatory that a is coprime to the modulus, or else the inverse won't exist. To wrap this text up, I will now leave you the code in Python which finds the modular multiplicative inverse of a modulo m using the Extended Euclid's Algorithm: def modinv(a, m): g, x, y = egcd(a, m) if g != 1: return None # modular inverse does not exist else: return x % m Further explorations and a final note Number Theory is a beautiful field of Mathematics, but it is at the same time, one of the most vast and in my personal opinion, hardest fields to master. The need of gcd(a,m) = 1, allows one to exploit this fact and use Euler's Theorem, along with Euler's Totient Function to find the modular inverse as well. In fact, on the popular and most widely spread case where the modulus, m, happens to be a prime number, we can use the simple formula: am-2 (mod m) to find the multiplicative inverse of a. This result follows from Euler's Theorem directly. Nontheless, besides my tutorial, my own personal experience is that it can be hard to actually understand all these ideas clearly in order to apply them successfully on a live contest (at least, it is hard for me), but, I hope that with some practice and also a lot more training and studying I can get better at it. So far, my study alongside with wikipedia and other books allowed me to write this tutorial which imho finds the best bits of information and puts them together on a same post. I hope you have enjoyed it. :) Best regards, Bruno asked 14 Aug '13, 02:17 3★kuruma 17.7k●72●143●209 accept rate: 8% 3★kcahdog 10.0k●28●54●129 In the last parts, where you mention Euler's Theorem, you can also mention Fermat's Little Theorem. (14 Aug '13, 09:03) In the 1st code given can anyone tell me why it is necessary to rename builtin_gcd to __gcd, if we dont do so it is giving compilation error..?? (14 Aug '13, 10:46) 1 @coding_addict You can simply choose to use __gcd itself. But, it will look nice, if we have some other names than the preceding double underscores. It is not a rule. It is our choice! :D (14 Aug '13, 11:01) @coding_addict, Thanks for your question, I will clarify that part better peraphs (14 Aug '13, 15:09) kuruma3★ What does (b//a) mean in the python code which defines egcd(a,b) function? (14 Aug '13, 18:53) 1 @saikrishna173, it's the same as floor division, or, applying floor function to result of divison. (14 Aug '13, 19:47) kuruma3★ Can you provide links to any problems which uses these concepts? (02 Aug '14, 01:58) showing 5 of 7 show all
2 @kuruma In Bezout's Identity you have mentioned that d = gcd(a,b) if and only if there are two positive integers x and y such that the identity : d = ax + by holds. However the integers x and y need not be positive. Only the value ax + by i.e. d should be positive. eg. a=6 and b=3. then gcd(6,3)=3 Now 3= 6 * 0 + 1 * 3 only and 3 = 6x + 3y does not hold for any positive value of x and y. answered 07 Mar '14, 01:58 3★kcahdog 10.0k●28●54●129 accept rate: 14% 2 Thank you @kcahdog for pointing it out and for fixing it :) (09 Mar '14, 16:12) kuruma3★ 1 I should thank you for writing such awesome tutorials! Have learnt a lot of stuff from them. (09 Mar '14, 16:45) kcahdog3★ 2 :) Thanks a lot for your words! I've also learnt a lot by writing them and reading all of your corrections and ideas, so, this has been being a great synergy :D Hope this lasts for quite a while eheh (09 Mar '14, 16:48) kuruma3★
0 Can you please provide the c implementation of extended Euclidean algorithm? answered 20 Apr '14, 21:14 30●1●2●6 accept rate: 0%
0 I just found this question - http://discuss.codechef.com/questions/47223/new-method-for-finding-modular-inverse seems interesting... answered 20 Nov '14, 12:26 16.9k●49●115●225 accept rate: 11%
0 Shouldn't the return value in the python implementation of Extended euclidean algo be (g,x,y-(b//a)x), instead of (g,x-(b//a)y,y). I'm following the proof mentioned here : http://e-maxx.ru/algo/extended_euclid_algorithm It would be great if someone could help me out. answered 27 May '15, 21:33 5★rtheman 69●3 accept rate: 0%
0 The Bézout's identity states that given two numbers a and b, passed as arguments to the gcd function, we can be sure that d = gcd(a,b) only if there are two integers x and y such that the identity: d = ax + by holds. The existence of x and y such that d = ax + by is a necessary condition for d = gcd(a, b), it is not sufficient (eg: 7 = 4+3, 7 is not gcd(4,3)). answered 18 Sep '16, 02:15 1 accept rate: 0% 2 Actually if there exist two integers x and y such that d = ax + by then d = gcd(a, b) iff d = min(ax+by) > 0. In your example min(4x+3y) = 1 (for x = 1 and y = -1) so 1 is the gcd of 4 and 3. Yes he should have mention that d should be least positive linear combination of a and b. (18 Sep '16, 02:42)
0 i am not able to understand the code of extend euclid algo in python ,,plz explain me how this code works,its working fine in my pc but i am not able to understand it on copy answered 22 May '17, 17:28 0★arjarjun 1 accept rate: 0%
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question asked: 14 Aug '13, 02:17
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2019-02-16 21:30:38
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https://www.physicsforums.com/threads/what-are-labels.840538/
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# What are labels?
1. Oct 30, 2015
### resurgance2001
In QM, position is an operator, while in QFT, it is a label. Could someone help me understand what is meant by the term "label" - just in the simplest terms possible? Cheers
2. Oct 30, 2015
### Staff: Mentor
It may help to know that time is a "label" in both QFT and QM. In both theories the operators correspond to observables, which are the things whose expected values we calculate from the state of the system. The labels identify a particular state of the system, the starting point for the calculation.
In QM, we can ask: What do we expect the position observable to be at a given time? What do we expect the other observables to be? This is using time as a label and treating position as an observable like the others.
In QFT, we can ask: What do we expect the value of the observables to be at a given time and at a given position? This is using time and position as labels.
3. Oct 31, 2015
### vanhees71
What's meant here is that you label your degrees of freedom. E.g., if you have a spinless particle in QM, you can take the three components of the position operator as a complete set of independent observables, i.e., the Cartesian coordinates $(x_j)=(x_1,x_2,x_3)$. Here the "label" is just the index $j$ to enumerate the components. If you have a system of $N$ spinless particles, you can take the $3N$ position coordinates. Then you have a label running from $1$ to $N$. The observables are functions of time (in the Heisenberg picture also all your operators that represent observables in quantum mechanics are a function of time, and the Heisenberg picture is the most natural one in going heuristically from classical mechanics in Hamiltonian form over to quantum theory).
Now in a field theory you have a continuum theory. The dynamical variables are fields, i.e., quantities which are functions of position and time. It gives you the value of the quantity (e.g., an electromagnetic field in terms of the field-strengths components $\vec{E}$ and $\vec{B}$). This means here you have two kinds of labels, a discrete one, enumerating the components $(E_j)=(E_1,E_2,E_3)$ of the field components and the position $\vec{x}$ in space where you measure these components.
In quantum field theory thus the position arguments in the field operators are just usual number, because they are in that sense a kind of continuous label for the infinitely many degrees of freedom represented by these fields.
Note that time is always a parameter in quantum theory, no matter, whether it's a quantized point-particle system (applicable in non-relativistic quantum theory and unfortunately often called the "first quantization") or a many-body system of indefinite particle number (applicable in both non-relativistic and relativistic quantum theory), i.e., a quantum-field theory.
4. Oct 31, 2015
### muscaria
This is what happens when you go from systems with a finite number of degrees of freedom to an infinite number of continuous degrees of freedom which is a field theory. This happens at the classical level before any kind of quantisation. The point is when you have a finite number of degrees of freedom, the associated coordinates are variables which encode the locations of particles and their variations represent different positions/configurations of the mechanical system in the usual sense. When you have an infinite number of continuous degrees of freedom such as a fluid, you consider your space positions to be fixed points in space which do not vary, in contrast to the previous situation where one takes the positions of the discrete system to be moving in space. In the case of a field, positions in space are fixed and do not move, instead it is the field value defined over these position labels which varies, giving rise to wave-like motion whose form depends on the kind of interactions one defines between the degrees of freedom. If you'd like an example of a field that would make this a bit more tangible, let us know!
EDIT: Only noticed vanhees71's nice post while posting!
Last edited: Oct 31, 2015
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2017-11-22 16:57:22
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http://math.stackexchange.com/questions/292275/discontinuous-derivative
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# Discontinuous derivative.
Could someone give an example of a ‘very’ discontinuous derivative? I myself can only come up with examples where the derivative is discontinuous at only one point. I am assuming the function is real-valued and defined on a bounded interval.
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$f(x)=|x|$ at $x=0$? Not sure what you mean by "very". – Ron Gordon Feb 1 '13 at 18:36
He probably wants a something discontinuous almost everywhere. – Git Gud Feb 1 '13 at 18:38
– Deven Ware Feb 1 '13 at 18:40
I guess that you are looking for a continuous function $f: \mathbb{R} \to \mathbb{R}$ such that $f$ is differentiable everywhere but $f'$ is ‘as discontinuous as possible’.
We have the following theorem in real analysis.
Theorem 1 If $f: \mathbb{R} \to \mathbb{R}$ is differentiable everywhere, then the set of points in $\mathbb{R}$ where $f'$ is continuous is non-empty. More precisely, the set of all such points is a dense $G_{\delta}$-subset of $\mathbb{R}$.
Note: A $G_{\delta}$-subset of $\mathbb{R}$ is just the intersection of a countable collection of open subsets of $\mathbb{R}$.
The proof of Theorem 1 is an application of the Baire Category Theorem, and it can be found in Munkres’ Topology. By this theorem, it is therefore impossible to find an $f: \mathbb{R} \to \mathbb{R}$ whose derivative exists but is discontinuous everywhere.
There is another theorem that provides a necessary and sufficient condition for a function $g: \mathbb{R} \to \mathbb{R}$ to have an antiderivative.
Theorem 2 A function $g: \mathbb{R} \to \mathbb{R}$ has an antiderivative if and only if its set of discontinuities is a meagre $F_{\sigma}$-subset of $\mathbb{R}$.
Note: An $F_{\sigma}$-subset of $\mathbb{R}$ is just the union of a countable collection of closed subsets of $\mathbb{R}$.
Let me end off with a non-trivial example to add to yours. Volterra’s Function is differentiable everywhere, but its derivative is discontinuous on a set of positive measure, not just at a single point.
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Haskell's excellent answer does a great job of outlining conditions that a derivative $f'$ must satisfy, which then limits us in our search for an example. From there we see the key question: can we provide a concrete example of an every differentiable function whose derivative is discontinuous on a dense, full-measure set of $\mathbb R$? Here's a closer look at the Volterra-type functions referred to in Haskell's answer, together with a little indication as to how it might be extended.
Basic example
The basic example of a differentiable function with discontinuous derivative is $$f(x) = \begin{cases} x^2 \sin(1/x) &\mbox{if } x \neq 0 \\ 0 & \mbox{if } x=0. \end{cases}$$ The differentiation rules show that this function is differentiable away from the origin and the difference quotient can be used to show that it is differentiable at the origin with value $f'(0)=0$. A graph is illuminating as well as it shows how $\pm x^2$ forms an envelope for the function forcing differentiablity.
The derivative of $f$ is $$f'(x) = \begin{cases} 2 x \sin \left(\frac{1}{x}\right)-\cos \left(\frac{1}{x}\right)&\mbox{if } x \neq 0 \\ 0 & \mbox{if } x=0, \end{cases}$$ which is discontinuous at $x=0$. It's graph looks something like so
Two points
The next step is to modify this example to obtain a function that is every differentiable on $\mathbb R$, except for two points. To this end, consider $$f(x) = \begin{cases} x^2 (1-x)^2 \sin \left(\frac{1}{\pi x (1-x)}\right)&\mbox{if } 0<x<1 \\ 0 & \mbox{else}. \end{cases}$$ The graph of $f$ and it's derivative look like so.
A cantor set of discontinuties
Now that we have a way to construct a differentiable function whose derivative is discontinuous exactly at the endpoints of an interval, it should be clear how to construct a differentiable function whose derivative is discontinous on a Cantor set constructed in the interval. Simply let for $n\in\mathbb Z$ and $m=1,2,\ldots,2^n$, let $I_{m,n}$ denote one of the $2^n$ intervals removed during the $n^th$ stage of construction of the Cantor set. Then let $f_{m,n}$ be scaled to be supported in $I_{m,n}$ and to have maximum value $4^{-n}$. The function $$F(x) = \sum_{n=0}^{\infty} \sum_{m=1}^{2^n} f_{m,n}(x)$$ will be every differentiable but it's derivative will be discontinuous on the given Cantor set. Assuming we do this with Cantors standard ternary set, we get a picture that looks something like so:
Of course, there's really a sequence of functions here and care needs to be taken to show that the limit is truly differentiable. Let $$F_N(x) = \sum_{n=1}^{N} \sum_{m=1}^{2^n} f_{m,n}(x).$$ The standard theorem then states that, as long as $F_N$ converges and $F_N'$ converges uniformly, then the limit of $F_N(x)$ will be differentiable. This is guaranteed by the choice of $4^{-n}$ as the max for $f_{m,n}$.
Increasing the measure
Again, the last example refers to the standard Cantor ternary set but there's no reason this can't be done with any Cantor set. In particular, it can be done with a so-called fat Cantor set, which is can have positive measure arbitrarily close to the measure of the interval containing it. We immediately produce an everywhere differentiable function whose derivative is discontinuous on a nowhere dense set of positive measure. (Of course, care must again be taken to scale the heights of the functions go to zero quickly enough to guarantee differentiability.)
Finally, we can fill the holes of the removed intervals with more Cantor sets (and their corresponding functions) in such a way that the union of all of them is of full measure. This allows us to construct an everywhere differentiable with derivative that is discontinuous on the union of those Cantor sets, which is a set of full measure.
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It's fairly clear that the derivative will be discontinuous at the removed endpoints. How do you ensure it is actually discontinuous on the whole Cantor set? – dfeuer Oct 14 '13 at 22:46
The Weierstrass function is continuous, but it's derivative is so "discontinuous" that it doesn't exist anywhere. It's not "discontinous" but it simply doesn't exist. I'm not sure what you mean by "very discontinous".
This function can be defined by $$f(x) = \sum_{n=0}^\infty a^n\cos(b^n\pi x)~~~~~a\in(0,1), b\in\Bbb{Z}^+, ab>\frac{3\pi}{2}$$
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This is my favorite counterexample in real analysis, but I’m not sure if the OP wants a non-existent derivative or one that exists but is discontinuous on a non-trivial set of points. – Haskell Curry Feb 1 '13 at 21:14
consider $f(x) = \sum_{n=0}^{\infty} \frac{1}{2^n}\cos(3^nx)$ this function is continuous everywhere but the derivative exists nowhere.
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2014-03-09 15:43:50
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https://zbmath.org/?q=an:1129.22004
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## Isometric group actions on Hilbert spaces: growth of cocycles.(English)Zbl 1129.22004
Let $$G$$ be a locally compact group, and $$\alpha$$ an affine isometric action on an affine Hilbert space $$\mathcal H$$. The function $$b: G\mapsto {\mathcal H}$$ defined by $$b(g)=\alpha(g)(0)$$ is called a $$1$$-cocycle, and the function $$g\mapsto\| b(g)\|$$ is called a Hilbert length function on $$G$$. In the paper under review the authors study some problems connected with the growth of Hilbert length functions. Discussing the existence of $$1$$-cocycles with linear growth, they obtain the following alternative for a class of amenable groups $$G$$ containing polycyclic groups and connected amenable Lie groups: either $$G$$ has no quasi-isometric embedding into a Hilbert space, or $$G$$ admits a proper cocompact action on some Euclidean space.
On the other hand, noting that almost coboundaries (i.e. $$1$$-cocycles approximable by bounded $$1$$-cocycles) have sublinear growth, the authors discuss the converse, which turns out to hold for amenable groups with “controlled” Følner sequences; for general amenable groups they prove the weaker result that $$1$$-cocycles with sufficiently small growth are almost coboundaries. Besides, they show that there exist, on a-$$T$$-menable groups, proper cocycles with arbitrary small growth.
### MSC:
22D10 Unitary representations of locally compact groups 43A07 Means on groups, semigroups, etc.; amenable groups 43A35 Positive definite functions on groups, semigroups, etc. 20F69 Asymptotic properties of groups
Full Text:
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2022-05-26 18:28:46
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https://learn.careers360.com/engineering/question-need-clarity-kindly-explain-two-beams-of-light-having-intensities-i-and-4i-interfere-to-produce-a-fringe-pattern-on-a-screen-the-phase-difference-between-the-beams-is-p2-at-point-a-and-p-at-point-b-then-the-difference-between-the-resultant-inten/
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# Two beams of light having intensities I and 4I interfere to produce a fringe pattern on a screen. The phase difference between the beams is p/2 at point A and p at point B. Then the difference between the resultant intensities at A and B is: Option 1) 2I Option 2) 4I Option 3) 5I Option 4) 7I
Resultant Intensity of two wave -
$I= I_{1}+I_{2}+2\sqrt{I_{1}I_{2}}\cos \theta$
- wherein
$I_{1}=$ Intencity of wave 1
$I_{2}=$ Intencity of wave 2
$\theta =$ Phase difference
$I_{A} = I+4I+2\sqrt{I.4I}.\cos \frac{\pi }{2} =5I$
$I_{B} = I+4I+2\sqrt{I.4I}.\cos \pi =I$
$I_{A}-I_{B} = 4I$
Option 1)
2I
Incorrect
Option 2)
4I
Correct
Option 3)
5I
Incorrect
Option 4)
7I
Incorrect
### Preparation Products
##### Knockout BITSAT 2021
It is an exhaustive preparation module made exclusively for cracking BITSAT..
₹ 4999/- ₹ 2999/-
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2020-09-29 23:43:12
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https://math.stackexchange.com/questions/309073/a-sequence-of-continuous-functions-converging-to-a-discontinuous-function
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A sequence of continuous functions converging to a discontinuous function
Let $f:I\to \mathbb R$ be a function which is continuous in every points of the interval $I$ except of a finite number of discontinuities $c_1,...,c_n$. I would like to find a sequence of continuous functions $f_n:I\to \mathbb R$ such that $\lim f_n=f$ pointwise.
This question seems very difficult, maybe because this one is very general, I'm really stuck here, any help is welcome.
Thanks a lot
• For a problem like this, I think it's easiest to start with an example of such an $f$ (and keep it simple, say one discontinuity) and then come up with a series of continuous functions $f_n$ that make better and better approximations of $f$. Once you do this for an example, you have a better idea how to do it for any such $f$. – PersonX Feb 20 '13 at 13:39
• @ChrisPhan yes, I've already tried with one discontinuity, without any success – user42912 Feb 20 '13 at 13:40
• Do you have any more information on the discountinuities? – superAnnoyingUser Feb 20 '13 at 13:41
• @George no just the information I've already mentioned above. – user42912 Feb 20 '13 at 13:44
• The most annoying case is if you have a vertical asymptote $\{x=x_0\}$. Draw a picture. Use broken lines on $[x_0-1/n,x_0+1/n]$. Make it pass through $(x_0,f(x_0))$. Note the method of broken lines works in general. – Julien Feb 20 '13 at 13:44
Take $$f:[0,1]\rightarrow \mathbb{R}\\ f(x)=\left\{\begin{array}{rl} 0 & x\neq 1 \\ 1 & x=1\\ \end{array}\right.$$ And $$f_n=x^n$$ With scaling and piecewise definitions you can to this one for any countable set of $c_1,\dots ,c_n$ In general our function will look like $$f(x)=\left\{ \begin{array}{rl} 0 & x \neq c_i \forall i\\ 1 & \text{else} \\ \end{array}\right.$$ On $[c_i,c_{i+1}]$ we gonna have something like $$f_{ni}(x)=\left(\frac{c_2-x}{c_2-c_1}\right)^n +\left(\frac{x-c_1}{c_2-c_1}\right)^n$$
And all together we will have (with $I=[a,b]$) $$f_n(x)=\left\{ \begin{array}{rl} 0 & x\in [a,c_0)\\ f_{ni} & x \in [c_i,c_{i+1})\\ 0& x\in [c_n,b] \\ \end{array}\right.$$
Edit for a given function the idea is the following, as you only have finite $c_i$ you take with $$\varepsilon=\min_{1\leq i \leq n-1} \{d(c_i,c_{i+1})\}$$ which is the shortest distance between two points of incontinuousity. Edit we don't need Stone Weierstraß at all sry.
$[c_i+\frac{\varepsilon}{2n},c_{i+1}-\frac{\varepsilon}{2n}]$ we just take $f$ on the intervalls (the uniform convergence is trivial). So we only need to chose a secquence of function on $[c_i-\frac{\varepsilon}{2n},c_i+\frac{\varepsilon}{2n}]$. We will call them $s_{ni}$ (like spline).
We chose $$s_{ni}(x)= \left\{ \begin{array}{rl} f(c_i) + \frac{f\left(c_i-\frac{\varepsilon}{2n}\right)-f(c_i)}{\frac{\varepsilon}{2n}} \cdot (x-c_i) & x-c_i \leq 0\\ f(c_i)+\frac{f\left(c_i+\frac{\varepsilon}{2n}\right)-f(c_i)}{\frac{\varepsilon}{2n}}\cdot (x-c_i) & x-c_i >0 \end{array}\right.$$ Ok that one looks really complicated but all i am saying we make a line from the left end to the point we want to have $f(c_i)$ and another one to get a continuous function in all the intervall.
• The problem is $f$ is a general one, I'm not looking for an specific example, thank you – user42912 Feb 20 '13 at 13:37
• ah you didn't got the idea – Dominic Michaelis Feb 20 '13 at 13:38
• no, sorry. My problem is given a $f$ (could be any $f$ with the specifications above) I would like to find a $f_n$ such that $f_n$ goes to $f$. Did you follow me? – user42912 Feb 20 '13 at 13:42
• but thank you anyway for trying to help me. – user42912 Feb 20 '13 at 13:44
• yeah i did follow you i edited it, thought it would be homework so I didn't want to do everything – Dominic Michaelis Feb 20 '13 at 13:46
The general idea is best explained when there is only one discontinuity point:
Suppose $f$ is continuous for all $x\ne b$. Select two points $a<b$ and $c>b$. Define the function $g$ by setting $g(x) =f(x)$ for $x\notin (a,c)\setminus\{b\}$, and on $(a,c)\setminus\{b\}$, take the graph of $g$ to be piecewise linear. Do this is such a way that $g$ is continuous.
You should draw the picture here. You're just replacing the "discontinuous part" of the graph of $f$ with straight line segments; thus producing a continuous function that agrees with $f$ except on an interval of small length. For the pointwise convergence of the sequence to come, it is important to have $g(b)=f(b)$, here.
By selecting sequences $(a_n)$ and $(b_n)$ with $a_n\nearrow b$ and $c_n\searrow b$, and defining continuous functions $g_n$ as above, one obtains a sequence of continuous functions that converge pointwise to $f$.
If $f$ has finitely many points of discontinuity, $x_1$, $\ldots\,$, $x_k$, do the same thing:
For each $n$, select intervals $O_1$, $\ldots\,$, $O_k$, such that:
$\ \ \$1) The $O_i$ are pairwise disjoint.
$\ \ \$2) The sum of the lengths of the $O_i$ is at most $1/n$.
$\ \ \$3) For each $k$, $x_k$ is the midpoint of $O_k$.
Now define your $f_n$ appropriately.
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2019-08-22 05:14:38
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https://hero.handmade.network/forums/code-discussion/t/8242-crazy_loadlibrary_bug#25591
|
4 posts
I've been programming hot code reloading into a project of mine in a simlar way that Casey does in Handmade Hero and I've come across some crazy behaviour that I can't explain...
Essentially I have run into a case where the call to LoadLibrary fails with
ERROR_DLL_INIT_FAILED
1114 (0x45A)
A dynamic link library (DLL) initialization routine failed.
This happens if I simply comment out a specific function call in the DLL codebase. Additionally, adding a dummy variable declaration fixes the problem!?
I am very confused...
Video demo
Simon Anciaux
1120 posts
Did you try loading the dll in an empty application with just the LoadLibrary call to see if it works ?
Could you share a minimal reproduction case or the full code ?
Mārtiņš Možeiko
2265 posts / 2 projects
Do you have lockfile implemented like in HH? If not, then most likely what is happening is that linker starts creating new dll file and your hot reload code notices that new file is there and tries to reload it, but linking process is still in progress - it has not finished to fully produce dll file. So loading it will fail, because partially written dll file has incorrect format. You can easily test this by adding something like Sleep(2000) before LoadLibrary call. If that success, then this is the case.
4 posts
Yeah even trying to load the dll in an empty application doesn't work. Here's some test code that just tries to get a handle to the dll with LoadLibrary and that's it. It fails for me but if I uncomment line 326 in modaw.cpp and then recompile it will load successfully. Curiously also just deleting any random code you like in modaw.cpp seems to have the same effect.
4 posts
I don't think this is the case because I'm not even trying to recompile the dll while the program is running. It just wont even load the dll on startup.
Mārtiņš Možeiko
2265 posts / 2 projects
Edited by Mārtiņš Možeiko on
This happens because you did not implement DllMainCRTStartup function properly. It must return TRUE for dll loading to succeed. You are not returning anything, so random garbage is returned (whatever value is in rax register) - most likely 0. And 0 is FALSE which signals OS that dll initialization failed and it must unload it.
For more information what this function must do (and what is its proper signature) is here: https://docs.microsoft.com/en-us/windows/win32/dlls/dllmain
4 posts
Edited by RobinLeathart on
Ah amazing, thank you! Is the prototype for DllMainCRTStartup identical to DllMain? i.e. I should have something like this?
extern "C" int
_DllMainCRTStartup(HINSTANCE Instance, DWORD Reason, LPVOID Reserved)
{
switch(Reason)
{
case DLL_PROCESS_ATTACH:
{
} break;
{
} break;
{
} break;
case DLL_PROCESS_DETACH:
{
} break;
}
return TRUE;
}
Mārtiņš Možeiko
2265 posts / 2 projects
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2022-01-18 07:30:26
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https://www.physicsforums.com/threads/one-parameter-family-of-metrics.275790/
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One-parameter family of metrics
1. Nov 29, 2008
mach4
I have a manifold M=S^4 which is endowed with a physical metric g.
I can define another metric on this manifold h (a pullback metric).
Does it make sense to define a one-parameter family of metrics G(u) on the manifold M in the form
G(u) = (1-u)*g + u*h , where u is a parameter in [0,1] ?
Are there any compatibility conditions?
Any help would be appreciated - Thx!
2. Nov 29, 2008
StatusX
If you're talking about euclidean metrics, then that should work, since the sum of two symmetric positive definite matrices is symmetric and positive definite, and so qualifies as a metric. It won't work for minkoswkian metrics though, since, eg, diag(1,1,1,-1) and diag(1,1,-1,1) are both valid metrics, but their sum is not.
3. Nov 30, 2008
mach4
Both metrics are symmetric positive definite but non-Euclidean.
When I check G for
-symmetry
-bilinearity
-non-degeneracy
all criteria of a metric seemed to be satisfied.
I was just bothered by the fact that g and h are associated with different curvature tensors, but it seems that they simply add to define the new curvature tensor of G.
Did I understand correctly? In the case of the Minkowskian-metrics the 'non-degeneracy' is not satisfied and thus it does not define a metric.
4. Nov 30, 2008
StatusX
The curvature is not linear in the metric, so will not simply add. But it's true, you can get a continuous family of metrics with different curvatures (obviously the curvature will then vary continuously over this family). And yes, the problem is that the sum of Minkowski metrics is not necessarily non-degenerate.
By the way, by "Euclidean" I mean a positive definite metric, not a flat one. It's just to distinguish from "Minkowskian".
5. Dec 1, 2008
mach4
uups - you are right. The Riemannian is clearly not linear in the metric. Bad mistake :(.
Thus, the operation of adding two positive definite metric is possible and does not lead to any inconsistencies. Great!
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2016-12-09 05:57:05
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http://accesspharmacy.mhmedical.com/content.aspx?bookid=515§ionid=41502861
|
Glossary
Absolute risk reduction The absolute value of the arithmetic difference in the event rates of the treated and untreated groups.
Adherence The extent that a patient follows the recommendations about day-to-day treatment by a provider with respect to timing, dosage, and frequency. The measure presumes that the provider received input from the patient. Typically measured as a proportion of the number of doses consumed, compared to the number of doses expected to be consumed. SYNONYM compliance.
Adjusted estimate The measure of association estimated in the presence of the potential confounders; it can be thought of as the association between exposure and outcome while mathematically holding constant all of the observed confounding variables.
Adverse event Any untoward medical occurrence in a patient or clinical investigation subject administered a pharmaceutical product and which does not necessarily have to have a causal relationship with the product.
Adverse drug reaction or adverse drug effect An adverse outcome that is harmful or unpleasant that occurs while a patient is taking a drug product and has a causal relationship with the drug.
Age-adjusted mortality rate The death rate that would occur if the observed age-specific death rates were present in a population with an age distribution equal to that of a standard population.
Age-specific mortality rate The total number of deaths from all causes during a specified period of time in a specific age category divided by the total number of persons in that age category in the population during that period.
Alternative hypothesis A statement of what one chooses to believe if the evidence provided in the sample data lead to a rejection of the null hypothesis.
Analysis of variance (ANOVA) A procedure that can be used to compare the means from populations defined by three or more groups (can actually be used for two groups as well as the t test is actually a special case of ANOVA).
Association When two events occur together repeatedly. This repeated occurrence takes place more often than a chance occurrence.
Atomistic fallacy The fallacy associated with taking conclusions from a study looking at individual patients and applying them to entire groups; this fallacy occurs because relationships and characteristics at the individual patient level may not apply categorically to an overall group of patients; compare with the ecologic fallacy.
Automated health care database A database that consists of data automatically captured as the result of the provision of care; contents of these databases may include administrative claims or transactional or operational data, such as drug dispensing data.
Bias It occurs when the groups under study are treated in a consistently different manner. The existence of a bias causes a study to produce incorrect results.
Biostatistics The application of statistical methods to the medical and health sciences, including ...
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### AccessPharmacy Full Site: One-Year Subscription
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2017-03-24 15:53:36
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http://mathhelpforum.com/advanced-algebra/170729-scalar-multiplied-vector-proof.html
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# Thread: scalar multiplied by a vector proof
1. ## scalar multiplied by a vector proof
Show that if c is a scalar then ||cu||=|c|||u||
let u $=(x_1, x_2, ..., x_n)$ then ||u||= $\sqrt{x^2_1+x^2_2+...+x^2_n}$
cu= $(cx_1, cx_2,...,cx_n)$ then ||cu||= $\sqrt{(cx_1)^2+(cx_2)^2+...+(cx_n)^2}=\sqrt{c^2x_1 ^2+c^2x_2^2+...+c^2x_n^2}=\sqrt{c^{2n}(x^2_1+x^2_2 +...+x^2_n)}$
I guess I messed something up because this looks like I disproved it.
2. Originally Posted by Jskid
Show that if c is a scalar then ||cu||=|c|||u||
let u $=(x_1, x_2, ..., x_n)$ then ||u||= $\sqrt{x^2_1+x^2_2+...+x^2_n}$
cu= $(cx_1, cx_2,...,cx_n)$ then ||cu||= $\sqrt{(cx_1)^2+(cx_2)^2+...+(cx_n)^2}=\sqrt{c^2x_1 ^2+c^2x_2^2+...+c^2x_n^2}=\sqrt{c^{2n}(x^2_1+x^2_2 +...+x^2_n)}$
I guess I messed something up because this looks like I disproved it.
Where did the $n$ come from in the $c^{2n}$
If I factor $4x+4y+4z$ I get $4(x+y+z)$ not $4^3(x+y+z)$
3. Originally Posted by Jskid
Show that if c is a scalar then ||cu||=|c|||u||
let u $=(x_1, x_2, ..., x_n)$ then ||u||= $\sqrt{x^2_1+x^2_2+...+x^2_n}$
cu= $(cx_1, cx_2,...,cx_n)$ then ||cu||= $\sqrt{(cx_1)^2+(cx_2)^2+...+(cx_n)^2}=\sqrt{c^2x_1 ^2+c^2x_2^2+...+c^2x_n^2}=\sqrt{c^{2n}(x^2_1+x^2_2 +...+x^2_n)}$
I guess I messed something up because this looks like I disproved it.
$\sqrt{c^{2}(x^2_1+x^2_2+...+x^2_n)}=\sqrt{c^2}\cdo t\sqrt{(x^2_1+x^2_2+...+x^2_n)}=\cdots$
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2017-03-30 14:00:59
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https://born2scan.run/articles/2021/06/29/How-we-hosted-DanteCTF-2021.html
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# How we hosted DanteCTF 2021
A brief tour of the infrastructure that supported the first edition of DanteCTF
Author: synack
Date: 29 June 2021
# Less magic, more introspectability
Nowadays, given an unlimited combination of budget, time and accumulable technical debt it’s far too easy to just spin up a humongous Kubernetes cluster on one of the plethora of managed services available, write a couple of lines of YAML and declare to be done with your infra. But anyone that has experimented with this trope knows that making such a system reliable, inspectable, and especially being able to quickly recover it in case things go south is an entirely different matter.
Since we didn’t expect a massive influx of users for this first edition of the DanteCTF - about 100 individuals registered before the start of the event, partly due to it being structured more as an introduction to CTFs rather than a fully featured competition - the technical budget ended up being made of a single AWS EC2 instance. For the sake of performance and maintainability we decided to try and reduce the operational overhead to the bare minimum but without losing important observability and resource managament capabilities.
# Topology
Hey, what did you expect? We told you we wanted to keep it simple :)
Everything ran on a t2.large (2 vCPUs, 8GiB RAM) EC2 instance, even though we’ll highlight later on how we could have gone for a t2.medium after some careful tuning. Downscaling even more could have been possible, but losing core parallelism could have caused minor annoyances down the line.
## Network Security
A set of two firewalls stood between the Big Evil Internet and our VM: AWS’s own Security Groups did the heavy lifting by allowing traffic only on our public facing ports, while ACLs and IP whitelists for administrative interfaces were managed directly on the machine through UFW (namely iptables, more on that later), geoblocking and WAFs.
# Software
Even though CTFd was the obvious choice for the competition’s management platform and the LetsEncrypt & CertBot combination handled issuing SSL certificates, we stil needed to find a task manager/orchestrator that balanced our needs of operational security (i.e.: isolating and constraining challenges) with a minimal resources footprint.
Complex multi-host building blocks like K8S were already right out of the question: having had prior experiences with bringing them up on a single node confirmed that we didn’t have the time and willingness to babysit such fragile solutions throughout the event, let alone implement decent disaster recovery procedures (because when something like single-node K8S breaks, it breaks hard).
Something that ticked all our boxes was Nomad. HashiCorp’s products are usually enjoyable to use and don’t require extensive knowledge to have an MVP up and running, so we dived into it and in a matter of hours a first crude setup of CTFd plus a single challenge was indeed brought up with just the right amount of research.
Usually there are pretty solid reasons for the official docs of something to tell you that you need at least N hosts with X, Y and Z capabilities, but we were pleasantly surprised by how… uneventful consciously going against such recommendations has been with Nomad. By running on a single host we were basically only giving up horizontal scalability, and there have been no memorable hiccups during configuration. Try to run some half-serious distro of K8S in this setup and let me know if you didn’t get stuck at least once in some tricky situation where internal services got deadlocked due to not expecting to be running onto the same host.
The obvious downside to every single-node approach is reliability: if that machine goes down, so do your services. We were pretty confident in our strategy though, for a couple of reasons: first of all, if EC2 suffers serious outages (*cough* not necessarily reported by their status dashboard *cough*) it usually means that there will be bigger problems in the IT world than a small CTF not being playable; incremental backups were easy to do and, even if we were forced to restart from stratch, our automated deployment pipelines enabled us to be back up and running in under 10 minutes (even half of that if we consider that not everything would have had to be rebuilt, local caches would do their job).
## Architecture
From left to right, top to bottom, Nomad was overseeing and coordinating these services:
### Consul
Consul was needed for service discovery and health checking. We could have also stored some of the challenges’ config data in its embedded key-value store, but we chose not to go down that rabbit hole in favor of Nomad’s injectable config templates since they were more than capable enough for our purposes.
Running Consul directly from Nomad (HashiCeption!) was, again, surprisingly easy due to Nomad being able to handle different types of workloads: the latest binary release was downloaded from the official repository, ran as an isolated process, and managed just like SystemD would do with a standard service. Nomad itself has no hard dependency on Consul in this configuration since clustering is disabled and it doesn’t have to discover any other node. I thought this would be one of the most delicate parts of the setup, but it turned out to be a pretty comfy and self-healing way to run Consul after all. Service resolution via DNS still required some hackery due to hardcoded defaults in systemd-resolved, but more on that later.
### Docker Registry
A standard Docker registry used internally to store customized CTFd and challenges’ images, built remotely and pushed up during the initial configuration or if live patching was found to be necessary. TLS termination is done by the service itself with the aforementioned certificates.
### CTFd
The CTFd job was composed by a couple of moving parts:
1. An NGINX reverse proxy whose tasks were handling TLS termination, handling HTTP to HTTPS upgrades, and filtering traffic to/from CTFd.
2. The actual CTFd instance, including custom themes and plugins.
3. A Redis instance to support CTFd as cache, as per the official docs.
4. A MariaDB instance to store CTFd’s data, as per the official docs. The VM’s disk performance hadn’t been tested thoroughly and thus we chose not to rely on SQLite, even though it would have most probably handled our expected traffic just fine.
### Unproxied challenges
Some challenges were served directly over the network since they were capable of handling their own traffic and/or weren’t HTTP-based. Some of them were run as Docker containers since they needed bundled assets or dependencies, while others were pure binaries that could be run directly by Nomad using the Linux kernel’s own namespacing facilities.
### Proxied challenges
Some challenges, especially the ones in the Web category, weren’t built to be directly exposed over the web or it was preferable for them not to handle TLS termination by themselves - especially for ease of development. To avoid having to setup a whole new NGINX instance just for these basic additions, we chose to use the simpler Caddy server with just a couple of lines of configuration, whose defaults include many features that were desirable in this scenario.
In the end all web challenges were served by the same NGINX + PHP container, secured behind said Caddy reverse proxy. If the number of challenges or their complexity was considerably higher we would have probably merged the two proxies to get rid of some overhead, but in this case the benefits of the separation of concerns and configurations outweighted the theoretical performance hits.
## Service discovery
Service discovery through DNS records is an awesome feature of Consul, but this time a bit of hackery was required to make it work with our setup: long story short, systemd-resolved (Ubuntu’s default DNS resolver since 18.04) only supports forwarding DNS queries to upstream servers listening on port 53 and thus it couldn’t be pointed directly to Consul.
The workaround consisted in throwing dnsmasq in the loop and binding it to Docker’s interface to avoid port conflicts with systemd-resolved; the latter routed all requests ending in .consul to the former, which in turn would query Consul and eventually fall back to systemd-resolved if no results turned up. Queries would be then forwarded through the network’s usual DNS resolvers by systemd-resolved.
This was the only occurrence so far where being constrained to a single EC2 instance annoyed us to some degree, mostly because I didn’t feel this solution was robust enough to be quickly troubleshooted in case of unexpected problems during the event. In a bigger setup all of this mess would most probably have been consolidated into a single dedicated DNS server, either through a managed service (AWS Route 53?) or a properly configured dnsmasq instance.
## Docker footguns
If you plan to use UFW to manage your firewall (that is to say iptables rules), take notice that Docker’s port bindings happily insert themselves in the rules chain higher up than UFW’s, bypassing the latter’s configuration altogether. A quick search for “docker ufw” shows that this is not a trap operators rarely fall into, hypothesis also corroborated by these two recent HackerNews submissions. In the end we chose to apply a workaround based on chaifeng’s idea.
<rant>
The Docker/Moby team has known of the issue since 2014 (see 1, 2, 3), but to me it seems that it’s one of those situations where everyone insists on shifting the blame instead of actually trying to fix the problem - or at least try to make people aware of it before damage is done. Personally, I find that a 7-lines paragraph in the official Docker docs is nowhere near enough to warn operators about this potentially critical security issue (no matter how easily detectable it is), especially when said page of the docs doesn’t even even come up within the first search results for the keywords mentioned above; when multiple hacky user-developed fixes get ranked higher than your own docs, maybe some actions should be taken.
</rant>
# Considerations
## Reproducibility
We wanted to be able to tear this environment down and bring it back up relatively fast to aid with testing and temporary setups during development, so we rigorously kept an Ansible playbook in sync from the start. It ended up being a relatively small feat at ~400 lines of YAML tasks, Bash helper scripts and HCL job definitions that, despite having taken quite some time to get right under all possible initial stages, gave us the confidence of having an easily reproducible and maintainable setup, in a container-like immutable fashion; no need to manually upgrade single components one after the other and check for temporary conflicts when you can reapply all your changes at once.
# Bottom line
In the end the event turned out pretty successfully and our architecture proved itself stable enough to withstand a much bigger load so, unless some newer and shinier tech comes out, I think next year’s DanteCTF will be based on a very similar stack with a few performance and monitoring tweaks baked in. Something that I’ll look into, for example, will most probably be adding Prometheus metrics directly into challenges when possible and generally overhauling our logs and submissions monitoring pipelines. Oh, and some kind of bot for the Discord support server since manually announcing first bloods and time left is… kind of a pain.
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2021-08-01 14:11:29
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https://mathoverflow.net/questions/208027/divisors-of-a-quadratic-trinomial
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# Divisors of a quadratic trinomial
Let $P(n)$ be a quadratic trinomial with integer coefficients. For each positive integer $n$, the number $P(n)$ has a proper divisor $d_{n}$ (i.e. $1 < d_{n} < P(n)$), such that the sequence $d_{1},d_{2}, d_{3},\ldots$ is increasing. Prove that either:
i) $P(n)$ is the product of two linear polynomials with integer coefficients, or
ii) all the values of $P(n)$, for positive integers $n$, are divisible by the same integer $m>1$.
This Russian problem was posted some time ago at math.SE. A bounty was placed on it, but no solution was received. Any suggestion is appreciated. Thanks in advance.
Assume the contrary.
At first, $P(n+2)-3P(n+1)+3P(n)-P(n-1)=0$, thus if the same number $m>1$ divides three consecutive values of $P$, it divides all values of $P$ and case ii) takes place.
Denote $P(n)=d_na_n$, $\ell(n)=P(n+1)-P(n)$ is a linear function. If $\ell(x)$ divides $P(x)$ as a polynomial, case i) takes place, if not, then $a(x)P(x)+b(x)\ell(x)=C\ne 0$ for some $a(x),b(x)\in \mathbb{Z}[x]$, hence ${\rm gcd}\,(P(n),\ell(n))\leq C$. Thus equality $d_{n+1}=d_n$ implies $d_n\leq C$, so, this may hold only for finitely many $n$ (otherwise $d_n=m$ for large enough $m$ and case ii) takes place.)
Now assume that $d_n\geq 2\ell(20n)$ (and $n$ is large enough). Then $a_{k+1}\leq P(k+1)/d_k=a_k+\ell(k)/d_k<a_k+1$ for $k=n,n+1,\dots,20n$. Thus either $a_k$ takes the same value three times in a row, see above why it implies case ii), or $a_{k+2}\leq a_k-1$ for $k=n,\dots,20n$. That is, $0<a_{20n}<a_n-9n$, i.e. $a_n>9n$, $P(n)=a_nd_n>9n\cdot 2\ell(20 n)$, this is false for large enough $n$.
So, $d_n=O(n)$ (and of course $d_n=\Omega(n)$ as $d_n$ strictly increases from some place; $a_n=P(n)/d_n$ tends to infinity, thus $a_{n+1}\ne a_n$ for large $n$ (if $a_n=a_{n+1}=m$, $m$ divides above constant $C$.) For some $A>0$ there is a set $U$ of positive integers with positive upper density so that $d_{n+1}=d_n+A$ for $n\in U$. We have $a_{n+1}-a_n=P(n+1)/(d_n+A)-P(n)/d_n=O(1)$. It means that $a_{n+1}-a_n=A$ for some constant $A$ for all $n$ from some smaller set $\tilde{U}\subset U$ of positive density.
Well, this is quadratic equation for $d_n$ if we fix $n$: $\ell(n)=Ad_n+DP(n)/d_n+AD$, $Ad_n^2-(\ell(n)-AD)d_n+DP(n)=0$. Its discriminant $(\ell(n)-AD)^2-4ADP(n)$ is (at most) quadratic trinomial in $n$. Three cases:
1) discriminant is a square of a linear function. Then $d_n$ is linear in $n$, and since $P(n)$ is divisible by $d_n$ for infinitely many $n$ we see that $P$ factorizes into two linear factors.
2) discriminant is a non-square quadratic trinomial. It may be a perfect square, but quite rarely. That is, for fixed $A,D$ only a null density of $n$ are ok (by Pell's type equation and blah-blah.)
3) discriminant is a linear function in $n$. The same as 2), but without Pell type argument.
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2020-10-24 18:47:08
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https://gateoverflow.in/378394/quantify-effect-performance-result-from-cache-program-total
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262 views
Quantify the effect on performance that results from the use of a cache in the case of a program that has a total of 500 instructions, including a 100-instruction loop that is executed 25 times. Determine the ratio of execution time without the cache to execution time with the cache. This ratio is called the speedup.
1)Program execution time is proportional to the total amount of time needed to fetch the instruction from either the main memory or the cache.With operand data access being ignored
2)Initially all instruction is stored in main memory and the cache is empty.
3)The cache is large enough to contain all loop instruction.
_________________________________________________________________
my solution is
execution time without cache is=400∗10+100∗10∗25=29000400∗10+100∗10∗25=29000
execution time with cache =500∗10+100∗1∗25=7500
My question is why we take 400 instructions instead of 500 instruction pls explain.
### 1 comment
From the solution you have provided I am assuming main memory access time 10 unit and cache access time is 1 unit.
While calculating execution time without cache we have total 500 instruction out of which 100 instruction runs in loops for 25 times and rest (500-100)=400 instruction runs once.
So ,total number of times we have to go to main memory to fetch the instruction as we don’t have cache to store the instructions is =(400+25*100)=2900.
Now each instruction it took 10 unit time to access main memory .
So total time required to run 2900 instruction=2900*10=29000 unit time
When we use cache we can store the looping instruction inside the cache so that we don’t have to go to fetch those 100 instruction every time.
So ,we have to definitely fetch those 500 instruction once from memory which will took 500*10=5000 unit time .
now by this we store the 100 instruction inside the cache and run them once .
now we need to run them only 24 times as we already run them once.
Now we can easily access them from cache which will take 1 unit of time to access one instruction.
So , to access 100 instruction once it will took =100*1=100 unit time.
we need to execute them 24 times so time it took =24*100=2400 unit time.
So total time it took without cache=5000+2400=7400 unit time.
Speed up=$\frac{29000}{7400}=3.91$
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2023-02-04 06:20:44
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http://www.mathnet.ru/php/archive.phtml?jrnid=im&wshow=issue&year=1953&volume=17&volume_alt=&issue=6&issue_alt=&option_lang=eng
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RUS ENG JOURNALS PEOPLE ORGANISATIONS CONFERENCES SEMINARS VIDEO LIBRARY PACKAGE AMSBIB
General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS
Izv. RAN. Ser. Mat.: Year: Volume: Issue: Page: Find
On the representation of natural numbers as a sum of terms of the form $\dfrac{x(x+1)\dotsb(x+n-1)}{n!}$V. I. Nechaev 485 On the theorem of Kolmogorov–SeliverstovS. B. Stechkin 499 Generalization of a theorem of MarcinkiewiczP. L. Ul'yanov 513 Spatial analogue of an integral of Cauchy type and some of its applicationsA. V. Bitsadze 525 On general boundary problems for equations of elliptic typeZ. Ya. Shapiro 539 On criteria of degeneracy of $R$-setsA. A. Lyapunov 563
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2020-11-29 06:11:42
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http://aritrasarkar.com/research/emergence/
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# Emergence
## Posts
### Pattern and Randomness
(Oct 22, 2019)
Everytime we talk about Emergence, there is a notion of a pattern, something that we consider favourable, that eventually comes into existence by the complex interactions and dynamics of the system in question. On the other hand, the word ‘random’ typically represents the opposite of ‘pattern’. Or does it?
I argue here, randomness is an emergent property. Say I got {H,H,H} on 3 successive coin tosses, I can interpret the coin as 100% biases. But it can also happen to be one of those 8 possibilities that showed up in this Universe, while the other 7 cases were swept under the rug of the many-World’s interpretation of a measurement (albeit classical). Whether it is actually unbiased cannot be understood from a few trials unless the law of large numbers comes into play, i.e. until the prefect ideal probability distribution is at least captured in some approximation in the statistics. Randomness is a statistical parameter, making no sense for a single experiment, like the temperature of an individual atom. Often, randomness is associated with the entropy of the microstate. 3 Heads has higher order and less surprise than 2 Heads and 1 Tail. But that assumes the coin is unbiased as a prior. What if we want to understand the property of the system itself? For example, if we are looking for radio signals from extra-terrestrial life, or decodings the heiroglyphs of an ancient civilization? How would we distinguish a random signal from a non-random one? The entropy of a bitstring also deals with how much information can be communicated via it, or in the Kolmogorov sense, if it can be compressed and later decompressed with a wrapper semantic overhead. Let’s assume a situation where I tell a friend that I would either send a string of 1s if the answer if yes, or a string of equal 1s and 0s if the answer is no. Assuming no noise in the channel, now, the meaning of the word random loses it’s entropic context, as here, a string with 75% 1s would be more near to a random message.
Is pattern also an emergent parameter? Is it a statistical low entropy configuration or a collection of semantically meaning states?
• Arguments against the 1st idea: based on how we semantically understand something, a higher entropy system can show more pattern. E.g. a program in BrainFuck printing 1s forever will have less algorithmic entropy than a program in C++ generating the Fibonacci series due to the inherent non-rationality of te golden mean; or a C++ code for 1s would have lower entropy than a BrainFuck code for golden mean; even though it should depend on the semantics of the language for the compiler, like an english sentence has lower entropy than a japanese sentence due to the higher number of japanese alphabets.
• Arguments against the 2nd idea: if something has semantic meaning, it should be reducible to a cost function for which a pattern would give a higher score than a random input. For a program/language, it would be syntactic correctness, e.g. grammarly. But still the association to the application is missing, the same problem as with shannon information metric.
### Fascination with Fractals
(Oct 22, 2019)
Why are fractals so ubiquitous in Nature than Euclidian geometry? What property of fractals make them so favourable for these blueprints? I like to approach this from 2 different angles.
God is a lazy programmer. Imagine you have to render the graphics of fire or clouds with triangles or ovals! Hell of a task, right? Indeed, a few iterations of a simple yet elegant fractal equation can generate these on your game world. It is not so difficult to drive home the point that fractals are the generator equations of the world we see around us, so fractal equations can easily generate models of them - low algorithmic complexity - lazy programmer. But, that’s a bit of ouruboros logic. The real equation is, why do we see fractal generator equations in the blueprints of the Universe? Why can’t clouds just be oval or fires as triangles like in the computer games of the early 1980s?
This has to do with compressing. Fractals are the edge of chaos, where the system transitions from a periodic attractor to a chaotic randomness. This also goes hand in hand with class 4 Wolfram automata which are universal computers which has enough expressive power to program everything in an unified structure, yet, the rules are simple enough and don’t get lost in chaos. Fractals are also great data compressors that can be prioritized with respect to the iteration level, working exactly like a Discrete Wavelet Transform, where the larger amplitudes and low frequency terms are captured in the lower iterations whereas the finer details can be compressed in the higher iterations allowing viewing the final product at different levels of approximation without losing the big picture, to interpret the general law behind them. Thus, there is a very subtle difference between a fractal of 100 iteration (say a Koch curve) and a fractal of 100 iteration with a small variation allowed at each level (say the coastline of Britain). In the later, an enormous amount of information can be encoded at different level of approximations. A little child can build an encoded message with pebbles on a particular beach without changing the overall fractal dimension much.
So fractals in a way allows us to start with a vague design and then periodically tweek it with small modifications to reach the design of interest. The question remains: is that how the Universal laws emerged? Chunks of smaller and smaller sized phenomena adding higher order refinements to the evolution of the universe.
### GUT from It
(Mar 27, 2019)
Before I describe my proposition, let’s list down the ingredients:
• Plancherel’s theorem which states the integral of a function’s squared modulus is equal to the integral of the squared modulus of its frequency spectrum.
• Kolmogorov/Algorithmic complexity
• String length
• Launderer’s principle
• Fourier transform
• It from Bit
• Thermodynamics
• Measurement in Quantum Mechanics
The equation:
The interpretation:
Let $f(x)$ be the state of the Universe encoded as a bit string. The absolute difference between the integral of the function’s squared modulus and the integral of the squared modulus of its frequency spectrum gives us the amount of new information generated by the Universe in the time duration of the integral of the function, i.e. $[0,t_u]$. This is equivalent to the work value of the bit string given by the fuel value of the string scaled by the Boltzmann constant and the temperature, following reversed Launderer’s principle. The fuel value is the difference between the length of the string and the conditional Kolmogorov complexity of the bit string, given the Fourier transform of it. This transform represents the derivable physical laws given the bit pattern of the Universe.
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2020-07-07 05:09:58
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http://www.lix.polytechnique.fr/~ponty/index.php?lang=fr&page=fun
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Yann Ponty - CR CNRS@LIX
Musings
Neil Gaiman's best tips to survive in an artistic academic career
Even though there are some difference between art and the academia, it is quite striking how some of Neil Gaiman's advise to young artists mirror what I would tell my students (with at least one notable exception... :) ).
You get work however you get work, but people keep working [..] because their work is good, and because they're easy to get along with, and because they deliver the work on time. And you don't even need all three... two out of three is fine!
• People will tolerate how unpleasant you are if the work is good and you deliver it on time.
• People will forgive the lateness of your work if it's good and they like you.
• And you don't have to be as good as everyone else if you're on time and it's always a pleasure to hear from you.
Neil Gaiman, commencement speech at the University of the Arts Class of 2012
Of course, we should aim for the three out of three, but the observation of two being sufficient is spot on in my experience (and represents a parsimonious and flattering explanation to my continued academic employement despite my chronic tardiness :) ).
The first problem of [..] success is the unshakable conviction that you are getting away with something and that, any moment now, they will discover you.
[..] I was convinced there would be a knock on the door, and a man with a clipboard [..] would be there to tell me it was all over and they'd caught up with me and now I would have to go and get a real job, one that didn't consist of making things up and writing them down, and reading books I wanted to read...
And then I would go away quietly and get the kind of job where I would have to get up early in the morning and wear a tie and not make things up anymore.
Neil Gaiman, commencement speech at the University of the Arts Class of 2012
I guess most academics can relate to that (generally undeservedly), especially under the grind of our recurrent evaluations (institutions, employers, funding bodies, peers...). Although some of us already wear a tie to start with...
There was a day when I looked up and realized that I had become someone who profesionnally replied to emails and who wrote as a hobby.
Neil Gaiman, commencement speech at the University of the Arts Class of 2012
An increasing number of my colleagues adopt quite extreme measures when it comes to email management (eg checking their emails only twice a day, or only within certain time slices...). I think it is one of the great challenge faced by humanity in the information era, and something we'll have to teach our children (academic), to find ways to limit the flow of information requiring active sprocessing (haven't found an acceptable solution myself... still looking).
People get hired because, somehow, they get hired... In my case I did something which these days would be easy to check and would get me into a lot of trouble and when I started out in those pre-internet days seemed like a sensible career strategy. When I was asked by editors who I'd written for, I lied. I listed a handful of magazines that sounded likely and I sounded confident and I got jobs. I then made it a point of honor to have written something for each of the magazines I've listed to get that first job, so that I hadn't actually lied, I'd just been chronologically challenged.
Neil Gaiman, commencement speech at the University of the Arts Class of 2012
This one I do not condone, of course, but working hard to deserve in retrospect whatever undeserved advantage you get at some point is a great attitude to adopt (cf impostor syndrom).
Luck is useful. Often you'll discover that the harder you work, and the more wisely that you work, the luckier you will get but there is luck, and it helps.
Neil Gaiman, commencement speech at the University of the Arts Class of 2012
Be wise, because the world needs more wisdom, and if you cannot be wise pretend to be someone who is wise and then just behave like they would.
Neil Gaiman, commencement speech at the University of the Arts Class of 2012
If the shoe fits... (aka Python's duck typing :) ).
Background: In 2012, Canada decided to phase out the penny for its coinage system. Product prices may still use arbitrary cents (impractical otherwise since prices in Canada do not typically include taxes) but cash transactions are now rounded up/down to the closest multiple of 5 cents, as shown in the image below:
For instance, if two products A and B are worth 67¢ and 1\$21¢ respectively, one will have to pay 1\$80¢ to buy them together, instead of their accumulated price 1\$78¢. In this case, the seller makes an extra 2¢ in the transaction, thanks to the imperfect round-off. Mathematically speaking, it all depends on the remainder$r$of the total price in a division by 5¢: If$r$equals$1$or$2$, than the seller looses$r$cents, but if$r$equals$3$or$4$, then the seller makes an extra$5-r$cents in the transaction. With my wife, we were wondering: Can a supermarket manipulate the prices of its products in such a way that the rounded totals translate, on average, into an extra benefit for the company? We were not overly paranoid about it, but it seemed like a good opportunity to exercise our brain while strolling/bouldering (and, very occasionally, surfing) on the amazing beaches of Oahu's famed North shore... As a first approximation, we can model the behavior of a shopper (oblivious to the imperfect change issue) as a Markov chain. In other words, given a list of products$A=\{a_i\}_{i=1}^k$, each associated with a price$p_i$, one chooses the$n$-th item$X_n$with probability $${\mathbb P}(X_n=a\mid X_{n-1}\cdots X_{n-k})$$ which depends on his/her previous$k$choices. On can then extend this Markov chain into remembering the remainder (modulo 5¢)$R_n$of the total price after$n$transactions. In this extended model, the probability of choosing a$n$-th item$X_n$and ending with a remainder$R_n$, only depends on the previously chosen$k$-products, and the previous remainder$R_{n-1}$. With this in mind, the expected gain$G_n$of the supermarket can now be written as $$\mathbb{E}(G_n\mid A) = \sum_{a\in A}\left|\begin{array}{ll}+2\,{\mathbb P}((R_n,X_n)=(3,a)\mid R_{0}=0)\\+{\mathbb P}((R_n,X_n)=(4,a)\mid R_{0}=0)\\-{\mathbb P}((R_n,X_n)=(1,a)\mid R_{0}=0)\\-2\,{\mathbb P}((R_n,X_n)=(2,a)\mid R_{0}=0).\end{array}\right.$$ For small values of$n$, this expectation depends a lot on the list$A$and its associated prices$p_i$. For instance if the customer buys a single item ($n=1$), the seller could set the remainder of all its prices to$3$and make an easy additional 2¢ on each transaction. However, for large values of$n$(+assuming the ergodicity of the Markov chain), the stationary distribution can be assumed, i.e. the expected gain$\mathbb{E}(G_\infty\mid A)$no longer depends on the initial state or, equivalently, on$n$. It is then possible to show that, no matter what the consummer preferences (i.e. the probabilities in the Markov chain), or the product list/prices$A\$, one has $$\mathbb{E}(G_\infty\mid A)=0.$$ This can be proven thanks to a very elegant symmetry argument by James Martin on MathOverflow. In other words, there is simply no way for the supermarket to rig the game even if the consumer does not actively avoid being short-changed (but assuming that the consumer buys enough products, so the supermarket may and will still win in the end!).
What Feynman saw in a flower...
Because understanding the process does not necessarily make the result any less appealing! (otherwise, would there be any hope left for lasting companionship? ;) )
Beautiful work by Fraser Davidson.
If you haven't seen one of these...
... then you obviously never TA'ed Bioinformatics ;) (Lucky bastard!). Full credits/kudos to XKCD.
On another note (and yes, this will be my last xkcd strip, otherwise my page would simply end up being a mirror):
Full credits again to XKCD.
Best Journal ever!!!
Ever complained about the academic daily routine having become a quest for the best journal that would nevertheless accept one's half-cooked manuscript? Ever felt that citations and impact factors were way too overrated, and that selectivity should prevail? Look no further, as it is my pleasure to share my recent discovery of (arguably) the best journal ever!
With an acceptance rate way lower than 1%, the journal of universal rejection may rightfully pride itself of enforcing the strictest of academic standards (Nature beware!). Created in 2009, the journal solicits submissions in poetry, prose, visual art, and research. Despite such a wide scope, its widely competent editorial board can be trusted to lay an expert eye on your submission, and helpfully motivate its final decision.
P vs NP: The irrefutable proof ;)
Today is the day I almost died laughing while reading Scott Aaronson's Shtetl-Optimized blog dedicated to Complexity Theory. Ok, I know, this sort of humor alone may be regarded as a conclusive diagnostic by future generations of psychopathologists, but I'd still like to share his beautiful, human-centric, argument for why P != NP. Basically, he is asked the question:
Can you summarize to a (curious/smart) dummy the P vs NP question and its many implications?
He starts by stating the question in four informal ways, one of which is:
Is it harder to solve a math problem yourself than to check a solution by someone else?
This is where you insert a comment about the delicious irony, that P vs. NP itself is a perfect example of a monstrously-hard problem for which we could nevertheless recognize a solution if we saw one---and hence, part of the explanation for why it's so hard to prove P!=NP is that P!=NP...
I woke up the following morning at the hospital, and must be clinically monitored during any future access to Scott's blog, but I encourage you to check it out for a daily dose of CS wittiness.
Amicable numbers
Amicable numbers are members of the number theory zoology (See their Wikipedia page), which, like trousers and all sorts of useful stuff, come handy in pairs. Formally, one starts by defining the restricted divisor function s(n) to be the sum of all divisors of n, itself excepted.
For instance for n = 220, one finds
s(220) = 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284.
Then, amicable numbers are pairs of natural numbers (p,q) such that s(p) = q and s(q) = p.
To make this notion explicit, we get back to the example above and find
s(284) = 1 + 2 + 4 + 71 + 142 = 220
and therefore (220,284) are amicable numbers.
Now there is something about this definition that may sound arbitrarily limited to a computer scientist. Indeed, consider the (infinite) directed graph whose vertices are natural numbers, and whose edges are pairs (n,s(n)). Here is a picture, rendered using the allmighty GraphViz (neato mode), showing the resulting network/graph for numbers below 50 (blue/red arcs indicate decreasing/increasing values of s, number 1 omitted for readability, golden nodes are perfect numbers):
And here is the larger graph of numbers below 300 (Full-screen PDF version):
One then easily sees that amicable numbers are in bijection with cycles of length 2 in the graph. Indeed, zooming in on the above graph, one may easily spot the first amicable pair:
This raises the question
Why not consider more general iterations of the s(.) function?
Indeed, one can alternatively, yet equivalently, define amicable numbers as pairs
(n,s(n)) such that s(s(n)) = n and s(n) ≠ n
which naturally generalizes into what is called sociable numbers, i.e. k-tuples of numbers
(n, s(n), s(s(n)), … , sk-1(n)) such that sk(n) = n and sj(n) ≠ n, ∀j∈[1,k-1].
These sequences are also called periodic aliquot sequences. Here is one of the biggest periodic sequence known to date (k=28):
These numbers are currently of some interest in number theory:
For k=1, sociable numbers are perfect numbers.
For k=2, sociable numbers are amicable numbers.
In general, what are aliquot sequence of cycle length k? Do they even exist for large values of k?
Whether there exists a cycle-free infinite aliquot sequence is currently open!
Copyleft Yann Ponty 2015
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2017-11-24 03:37:21
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https://scoop.eduncle.com/why-epsilon-is-1-2-why-n-n-here-n-is-a-natural-number
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IIT JAM Follow
Vijay Singh chauhan Asked a Question
August 13, 2020 10:28 am 30 pts
why epsilon is 1/2? why n>=N(here, N is a natural number)?
• 0 Likes
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• Anonymous User
epsilon is taken upto our choice.... if we have to disprove something, then in that case we take such epsilon value that helps us in disproving it....but in case of proving somethi...
• Deepak singh
we take epsilon such that , that epsilon can dispoove the given statement
• Anonymous User
here epsilon equals to 1/2 has been taken...but if you choose any positive value of epsilon less than 1, it will work
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2020-09-23 09:56:59
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https://www.groundai.com/project/explaining-dark-matter-and-neutrino-mass-in-the-light-of-type-ii-seesaw-model/
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Explaining Dark Matter and Neutrino Mass in the light of TYPE-II Seesaw Model
# Explaining Dark Matter and Neutrino Mass in the light of TYPE-II Seesaw Model
Anirban Biswas Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai - 400094, India Avirup Shaw Department of Theoretical Physics, Indian Association for the Cultivation of Science,
###### Abstract
With the motivation of simultaneously explaining dark matter and neutrino masses, mixing angles, we have invoked the Type-II seesaw model extended by an extra doublet . Moreover, we have imposed a parity on which remains unbroken as the vacuum expectation value of is zero. Consequently, the lightest neutral component of becomes naturally stable and can be a viable dark matter candidate. On the other hand, light Majorana masses for neutrinos have been generated following usual Type-II seesaw mechanism. Further in this framework, for the first time we have derived the full set of vacuum stability and unitarity conditions, which must be satisfied to obtain a stable vacuum as well as to preserve the unitarity of the model respectively. Thereafter, we have performed extensive phenomenological studies of both dark matter and neutrino sectors considering all possible theoretical and current experimental constraints. Finally, we have also discussed a qualitative collider signatures of dark matter and associated odd particles at the 13 TeV Large Hadron Collider.
## I Introduction
The observation of various satellite borne experiments namely WMAP Hinshaw:2012aka () and more recently Planck Ade:2015xua (), establish firmly the existence of dark matter in the Universe over the ordinary luminous matter. The results of these experiments are indicating that more than 80% matter content of our Universe has been made of an unknown non-luminous matter or dark matter. In terms of cosmological language, the amount of dark matter present at the current epoch is expressed as Ade:2015xua () where is known as the relic density of dark matter and is the present value of Hubble parameter normalised by 100. In spite of this precise measurement, the particle nature of dark matter still remains an enigma. The least we can say about a dark matter candidate is that it is electrically neutral and must have a lifetime greater than the present age of the Universe. Moreover, N-body simulation requires dark matter candidate to be non-relativistic (cold) at the time of its decoupling from the thermal plasma to explain small scale structures of the Universe Frenk:2012ph (). Unfortunately, none of the Standard Model (SM) particles can fulfil all these properties and hence there exist various beyond Standard Model (BSM) theories in the literature Jungman:1995df (); Bertone:2004pz (); Hooper:2007qk (); ArkaniHamed:2008qn (); Kusenko:2009up (); Feng:2010gw () containing either at least one or more dark matter candidates. Among the different kinds of dark matter candidates, Weakly Interacting Massive Particle (WIMP) Gondolo:1990dk (); Srednicki:1988ce () is the most favourite class and so far, neutralino Jungman:1995df () in the supersymmetric extension of the SM is the well studied WIMP candidate. There are also a plethora of well motivated non-supersymmetric BSM theories which have dealt with WIMP type dark matter candidate Silveira:1985rk (); Burgess:2000yq (); McDonald:1993ex (); Barbieri:2006dq (); LopezHonorez:2006gr (); Kim:2008pp (); Hambye:2008bq (). Since the interaction strength of a WIMP is around week scale hence various experimental groups Akerib:2016vxi (); Amole:2017dex (); Armengaud:2016cvl (); Agnese:2014aze () have been trying to detect it directly over the last two decades by measuring the recoil energies of detector nuclei scattered by WIMPs. However, no such event has been found and as a result, dark matter nucleon elastic scattering cross section is getting severely constrained. Currently most stringent bounds on dark matter spin independent scattering cross section have been reported by the XENON 1T collaboration Aprile:2017iyp ()111Recently, PandaX-II collaboration 1708.06917 () has published their results on the exclusion limits of WIMP-nucleon spin independent scattering cross section (). Although, their results are most stringent for a WIMP of mass larger than 100 GeV, are very similar with the upper limits of XENON 1T.. Future direct detection experiment like DARWIN Aalbers:2016jon () is expecting to detect or ruled out the WIMP hypothesis by exploring the entire experimentally accessible parameter space of a WIMP (just above the neutrino floor).
On the other hand, neutrinos remain massless in the SM as there is no right handed counterpart of each where is the generation index. However, the existence of a tiny nonzero mass difference between and has been first confirmed by the atmospheric neutrino data of Super-kamiokande collaboration Fukuda:1998mi () from neutrino oscillation. Thereafter, many experimental groups Ahmad:2002jz (); Araki:2004mb (); Abe:2011sj (); Ahn:2012nd () have precisely measured the mass squared differences and mixing angles among different generations of neutrinos. In spite of these wonderful experimental achievements, we still have not properly understood the exact method of neutrino mass generation. There exist various mechanisms for generating tiny neutrino masses at tree level (via seesaw mechanisms) Minkowski:1977sc (); Mohapatra:1979ia (); Magg:1980ut (); Ma:1998dx (); Foot:1988aq (); Ma:2002pf (); Mohapatra:1986bd () and beyond Zee:1985id (); Ma:2006km (); Gustafsson:2012vj () by adding extra bosonic or fermionic degrees of freedom in the particle spectrum of SM. Moreover, the exact flavour structure in the neutrino sector, which is responsible for generating such a mixing pattern, still remains unknown to us. Furthermore, there are other important issues which are yet to be resolved. For example the particle nature of neutrinos (i.e. Dirac or Majorana fermion), mass hierarchy (i.e. Normal or Inverted), determination of octant for the atmospheric mixing angle , CP violation in the leptonic sector (i.e. measurement of Dirac CP phase ) etc. More recently, T2K collaboration Abe:2017vif () has reported their analysis of neutrino and antineutrino oscillations where they have excluded the hypothesis of CP conservation in the leptonic sector (i.e. or ) at 90% C.L. Their preliminary result indicate a range for lies in between third and fourth quadrant. Other neutrino experiments like DUNE Acciarri:2015uup (), NOA Adamson:2017gxd () etc. will address some of these issues in near future.
In the present article we try to cure both of these lacunae of the SM by introducing a Higgs triplet and an extra Higgs doublet to the particle spectrum of SM. Furthermore, we impose a discrete symmetry in addition to the SM gauge symmetry. Under this symmetry the triplet field and the SM particles are even while the the extra doublet field is odd222Here the odd doublet is analogous to the one in Inert Doublet Model (IDM) Barbieri:2006dq (); LopezHonorez:2006gr (); Ma:2006km ().. This kind of BSM scenario has been studied earlier in Chen:2014lla (). To the best of our knowledge in such set up first time we derive the vacuum stability and unitarity constraints and use these constraints in our phenomenological study. This set up can serve our two fold motivations. First of all, as we have demanded that the extra doublet is odd under symmetry, consequently the lightest particle of neutral component of this doublet can play the role of viable dark matter candidate in this scenario. Secondly, with the small vacuum expectation value (VEV) of Higgs triplet field, required to satisfy the electroweak precision test, we can explain small neutrino masses by the Type-II seesaw mechanism Magg:1980ut (); Cheng:1980qt (); Lazarides:1980nt (); Mohapatra:1980yp (); Dev:2013ff (); Dev:2013hka () without introducing heavy right handed neutrinos. In the present work, we have explored both the normal and inverted hierarchies of neutrino mass spectra. At this point, we would like to mention that all the possible current experimental constraints have been taken into account while we investigate the dark matter related issues as well as the generation of neutrino masses and their mixings.
Apart from providing a viable solution to dark matter problem and neutrino mass generation, this scenario contains several non-standard scalars which can be classified into two categories. In one class we have even scalars originate from the mixing between triplet fields and SM scalar doublet fields while the three different components of the extra scalar doublet can be represented as odd scalars. Therefore, one has the opportunity to explore these non-standard scalars at the current and future collider experiments. In literature one can find several articles where the search of even scalars have been explored in context of the Large Hadron Collider (LHC) Chun:2003ej (); Han:2007bk (); Perez:2008ha (); Han:2015hba (); Han:2015sca (); Mitra:2016wpr () as well as at the International Linear Collider (ILC) Shen:2015pih (); Cao:2016hvg (); Blunier:2016peh (). However, in this work instead of even scalars, we have performed collider search of dark matter and the associated odd scalars at the 13 TeV LHC. Among the different final states, we find an optimistic result for signal at the 13 TeV LHC with an integrated luminosity of 3000.
One should note that, relying on the value of triplet VEV, decay modes of different non-standard scalars show distinct behaviour. From the consideration of electroweak precision test the triplet VEV can not be larger than a few GeV Aoki:2012jj (); Patrignani:2016xqp (). However, it can vary from GeV to GeV. Within this range the non-standard Higgs bosons decay in several distinct channels. To be more specific, for , the doubly charged Higgs dominantly decays into two same-sign leptonic final state. The latest same-sign dilepton searches at the LHC have already put strong lower limit on doubly charged Higgs mass ( 770 - 800 GeV) ATLAS:2017iqw (). On the other hand for , only gauge boson final state or cascade decays of singly charged Higgs (if they are kinematically allowed) are possible Perez:2008ha (); Melfo:2011nx (); Aoki:2011pz (); Han:2015hba (); Han:2015sca (). The collider search becomes more involved in this region of triplet VEV due to more complicated decay patterns of the doubly charged Higgs. As a result, the lower bound on the mass of the doubly charged Higgs is very relaxed. Therefore, in this region one can find scenarios where the mass of doubly charged Higgs may goes down to about 100 GeV Melfo:2011nx (); Chabab:2016vqn (). In this article, for all practical purposes we have considered the triplet VEV greater than GeV. For example, for the generation of neutrino mass we set triplet VEV at GeV. Whereas, for the purpose of dark matter analysis we show our results for two different values of triplet VEV e.g., GeV and 3 GeV respectively. This is in stark contrast to the Ref. Chen:2014lla () where the triplet VEV has been considered less than GeV. Further, for collider study we have fixed the the value of triplet VEV at 3 GeV and hence the doubly charged Higgs decays into with 100% branching ratio.
We organise this article as follows. First we introduce the model with possible interactions and set our conventions in Sec. II. Within this section we have also evaluated the vacuum stability and unitarity conditions in detail. In Sec. III, we discuss the neutrino mass generation via Type-II seesaw mechanism and explain neutrino oscillation data for normal and inverted hierarchies at range. The viability of dark matter candidate proposed in this work has been extensively studied in Sec. IV, considering all possible bounds from direct and indirect experiments. In Sec. V, we show the prospects of collider signature of the dark matter candidate of the present model at 13 TeV LHC. Finally in Sec. VI we summarize our results.
## Ii Type-II Seesaw with Inert Doublet
In this section, we discuss the model briefly. In order to produce a viable dark matter candidate, we introduce a symmetry in the SM gauge symmetry . Moreover, to generate the neutrino masses and also having a stable dark matter candidate, we incorporate a scalar triplet with hypercharge two and a scalar doublet with hypercharge one in the SM fields. Further, we demand that the SM particles and the triplet are even under parity while the new doublet is odd under parity. The field cannot develop a VEV at the time of electroweak symmetry breaking as this will break the symmetry spontaneously, which will jeopardize the dark matter stability. With this newly added symmetry, we discuss different interaction terms involving SM fields, and . The total Lagrangian which incorporates all possible interactions can be written as:
L=LYukawa+LKinetic−V(H,Δ,Φ), (1)
where the relevant kinetic and Yukawa interaction terms are respectively
Lkinetic = (DμH)†(DμH)+Tr[(DμΔ)†(DμΔ)]+(DμΦ)†(DμΦ), (2) LYukawa = LSMYukawa−Yνij2LTiCiσ2ΔLj+h.c.. (3)
The first two terms of generate the masses of gauge bosons and by electroweak symmetry breaking mechanism (EWSB), however the third term does not contribute to gauge boson masses as does not possess any VEV. Here represents doublet of left handed leptons where being the generational index, represents Yukawa coupling and is the charge conjugation operator. Further, denotes the Yukawa interactions for all SM fermions. Later, we will discuss the second term of Yukawa interactions in detail in the neutrino section (Section III). There is no term which involves the coupling between and the SM fermions as is odd under parity while the SM fermions are even under symmetry. Representations for the doublets and are chosen as and respectively. The triplet field transforms as under the gauge group, so one can write , which gives a representation given in the following:
Δ=(δ+/√2δ++δ0−δ+/√2). (4)
In the above . The neutral component of the triplet field can be expressed as where and are vacuum expectation values of the doublet and triplet respectively. The covariant derivative of the scalar field is given by,
DμΔ=∂μΔ+ig22[σaWaμ,Δ]+ig1BμΔ(a=1,2,3). (5)
Here ’s are the Pauli matrices while and are coupling constants for the gauge groups and respectively.
Let us discuss the scalar potential given in the following Chen:2014lla ():
V(H,Δ,Φ) = −m2H(H†H)+λ4(H†H)2+M2ΔTr(Δ†Δ)+(μHTiσ2Δ†H+h.c.) (6) +λ1(H†H)Tr(Δ†Δ)+λ2[Tr(Δ†Δ)]2+λ3Tr(Δ†Δ)2+λ4(H†ΔΔ†H) +m2Φ(Φ†Φ)+λΦ(Φ†Φ)2+λ5(H†H)(Φ†Φ)+λ6(H†ΦΦ†H) +λ7(Φ†Φ)Tr(Δ†Δ)+λ8(Φ†ΔΔ†Φ)+λ9[(Φ†H)2+h.c.] +(~μΦTiσ2Δ†Φ+h.c.).
Here, , and () are dimensionless coupling constants, while , , , and are mass parameters of the above potential. Whereas , and are the only terms which can generate CP phases, as the other terms of the potential are self-conjugate. However, two of them can be removed by redefining the fields , and . Furthermore, we assume that for the spontaneous breaking of above mentioned gauge group.
After EWSB we obtain a doubly charged scalar including a singly charged scalar, , a pair of neutral CP even Higgs (), a CP odd scalar () and as usual three massless Goldstone bosons (). Further, we also have three particles (, , and ) which are members of the inert doublet. The mass eigenvalues for the even physical scalar are given by Arhrib:2011uy ():
M2H±± = √2μv2d−λ4v2dvt−2λ3v3t2vt, (7) M2H± = (v2d+2v2t)(2√2μ−λ4vt)4vt, (8) M2A0 = μ(v2d+4v2t)√2vt, (9) M2h0 = 12(A+C−√(A−C)2+4B2), (10) M2H0 = 12(A+C+√(A−C)2+4B2), (11)
with
A = λ2v2d, (12) B = vd[−√2μ+(λ1+λ4)vt], (13) C = √2μv2d+4(λ2+λ3)v3t2vt, (14)
while the mass eigenvalues of odd scalars are:
M2ϕ0 = m2Φ+12(λ5+λ6)v2d+12(λ7+λ8)v2t+λ9v2d−√2~μvt, (15) M2a0 = m2Φ+12(λ5+λ6)v2d+12(λ7+λ8)v2t−λ9v2d+√2~μvt, (16) M2ϕ± = m2Φ+12λ5v2d+12λ7v2t. (17)
The mixing between the SM doublet and the triplet scalar fields in the charged, CP even as well as CP odd scalar sectors are respectively given by:
(G±H±) = (cosβ′sinβ′−sinβ′cosβ′)(h±δ±), (18) (h0H0) = (cosαsinα−sinαcosα)(η0ξ0), (19) (G0A0) = (cosβsinβ−sinβcosβ)(z1z2), (20)
and the respective mixing angles are given by:
tanβ′ = √2vtvd, (21) tanβ = 2vtvd=√2tanβ′, (22) tan2α = 2BA−C, (23)
where the expressions of , and are already given in Eq. 11.
### ii.1 Different constraints
Before going to study the phenomenological aspects of neutrino and dark matter sectors, it is necessary to check various constraints from theoretical considerations like vacuum stability, unitarity of the scattering matrices and perturbativity. Further, the model parameters also need to satisfy the phenomenological constraints arising from electroweak precision test and Higgs signal strength. Therefore, to serve the purposes we need to choose a set of free parameters of this model. In practice, a convenient set of free parameters are given in the following, however some of them are not independent:
{tanα,MH±±,MH±,MH0(=MA0),Mϕ0,Mϕ±,λΦ,λ5,λ6,λ7,λ8,λ9}. (24)
#### ii.1.1 Vacuum stability bounds:
This section has been dedicated to derive the necessary and sufficient conditions for the stability of the vacuum. These conditions come from requiring that the potential given in Eq. 6 be bounded from below when the scalar fields become large in any direction of the field space. The constraints ensuring boundedness from below (BFB) of the present potential have not been studied in the literature so far. It would thus be very relevant to derive these constraints in the present model. For large field values, the potential given in Eq. 6 is generically dominated by the quartic part of the potential. Hence, in this limit we can ignore any terms with dimensionful couplings, mass terms or soft terms. So the general potential given in Eq. 6 can be written as in the following way which contains only the quartic terms,
V(4)(H,Δ,Φ) = λ4(H†H)2+λ1(H†H)Tr(Δ†Δ)+λ2[Tr(Δ†Δ)]2+λ3Tr(Δ†Δ)2 (25) +λ4(H†ΔΔ†H)+λΦ(Φ†Φ)2+λ5(H†H)(Φ†Φ)+λ6(H†ΦΦ†H) +λ7(Φ†Φ)Tr(Δ†Δ)+λ8(Φ†ΔΔ†Φ)+λ9[(Φ†H)2+h.c.].
To determine the BFB conditions we have used copositivity criteria as given in Ref. Kannike:2012pe (). For this purpose we need to express the scalar potential in a biquadratic form , where . If the matrix is copositive then we can demand that the potential is bounded from below. Let us write down the matrix in our case:
A=⎛⎜ ⎜ ⎜⎝14λ12(λ1+ξλ4)12[λ5+ρ2(λ6−2|λ9|)]12(λ1+ξλ4)(λ2+ζλ3)12(λ7+ξ′λ8)12[λ5+ρ2(λ6−2|λ9|)]12(λ7+ξ′λ8)λΦ⎞⎟ ⎟ ⎟⎠. (26)
The parameters , , and appearing in the matrix elements are required to determine all the necessary and sufficient BFB conditions. The detail illustrations of the parameters can be found in Arhrib:2011uy () where two fields (one doublet and a triplet) have been considered. However, in our case we have three different fields (two doublets and a triplet). Using the prescription given in Ref. Arhrib:2011uy (), we have defined the parameters in the following way,
ζ≡Tr(Δ†Δ)2/[Tr(Δ†Δ)]2, (27) ρ≡|H†Φ|/|H||Φ|, (28) ξ≡(H†ΔΔ†H)/(H†H\leavevmode\nobreak Tr(Δ†Δ)), (29) ξ′≡(Φ†ΔΔ†Φ)/(Φ†Φ\leavevmode\nobreak Tr(Δ†Δ)). (30)
The and limits of these parameters are given as [], [], [] and [] respectively Arhrib:2011uy (). To determine the all possible BFB conditions of the scalar potential, we consider both the limits of these parameters and respect the copositivity criteria. Finally, we can write down the following BFB conditions by demanding the symmetric matrix is copositive Kannike:2012pe ().
λ≥0, (31) (λ2+ζλ3)≥0, (32) λΦ≥0, (33) (λ1+ξλ4)+√λ(λ2+ζλ3)≥0, (34) λ5+ρ2(λ6−2|λ9|)+√λλΦ≥0, (35) (λ7+ξ′λ8)+2√(λ2+ζλ3)λΦ≥0, (36)
√λ(λ2+ζλ3)λΦ+(λ1+ξλ4)√λΦ+[λ5+ρ2(λ6−2|λ9|)]√λ2+ζλ3+(λ7+ξ′λ8)2√λ (37) +√{(λ1+ξλ4)+√λ(λ2+ζλ3)}{(λ7+ξ′λ8)+2√λΦ(λ2+ζλ3)}{[λ5+ρ2(λ6−2|λ9|)]+√λλΦ}≥0.
Substituting the lower and upper limits of the parameters, one can get the full set of vacuum stability conditions given in Appendix B (see Eq. B-3 to Eq. B-10h).
#### ii.1.2 Unitarity bounds:
In this section we discuss the unitarity constraints on the parameters of scalar potential by using the tree-level unitarity of various scattering processes. One can find the scalar-scalar scattering, gauge boson-gauge boson scattering and scalar-gauge boson scattering in the context of SM in Appelquist:1971yj (); Cornwall:1974km (); Lee:1977eg (). In the case of various extended Higgs sector scenario, the generalizations of such constraints can be found in literature Kanemura:1993hm (); Akeroyd:2000wc (); Aoki:2007ah (); Gogoladze:2008ak (). It has been a well known fact that in the high energy limit using equivalence theorem Lee:1977eg (); Cornwall:1973tb (); Chanowitz:1985hj () one can replace longitudinal gauge bosons by those of the corresponding Nambu-Goldstone bosons in scattering. Hence, following this prescription in the current model, our main focus is to consider only the Higgs-Goldstone interactions of the scalar potential given in Eq. 6. Furthermore, under this situation the -body scalar scattering processes are dominated by the quartic interactions only.
To determine the unitarity constraints, it has been a usual trend to calculate the -matrix amplitude in the basis of unrotated states, corresponding to the fields before electroweak symmetry breaking. Because, in this situation the quartic scalar vertices have a much simpler form with respect to the complicated functions of , , and involved in the physical basis333For the inert Higgs doublet, the physical basis are equivalent to the gauge basis as in this case the vacuum expectation value is zero. (, , , , , , , , and ). So in the unrotated basis (, , , , , , , , and ), we study full set of -body scalar scattering processes which lead to a -matrix. This matrix can be decomposed into 7 block submatrices with definite charge. For example, , and corresponding to neutral charged states, corresponding to the singly charged states, corresponding to the doubly charged states, corresponding to the triply charged states and finally corresponding to the unique quartic charged state. These submatrices are hermitian, so the eigenvalues will always be real-valued.
To this end, we would like to mention that in the following cases we will determine the eigenvalues of the above mentioned submatrices. However, there is a caveat. The structure of some of the submatrices are very challenging, so it is not possible to find out the analytic form of all the eigenvalues of those matrices. However, using numerical technique given in Adhikary:2013bma () we can derive the remaining eigenvalues. Eventually, we will have all the full set of eigenvalues by which we will put the unitarity constraints on the model parameters.
The first submatrix corresponds to the scatterings whose initial and final states are one of the following:
{h+δ−,δ+h−,ϕ+δ−,δ+ϕ−,h+ϕ−,ϕ+h−,η0z2,ξ0z1,z1z2,η0ξ0,ϕ0η0,ϕ0z1,η0a0,a0z1,ϕ0ξ0,ϕ0z2, a0ξ0,a0z2},
Eigenvalues of are:
{λ1,λ1,λ1+λ4,λ1+λ4,λ1+3λ42,λ1+3λ42,λ5+λ6,λ5+λ6,λ7,λ7,λ7+λ8, λ7+λ8,λ7+3λ82,λ7+3λ82,λ5+2λ6−6λ9,λ5−2λ9,λ5+2λ9,λ5+2λ6+6λ9}.
The second submatrix corresponds to the scatterings whose initial and final states are one of the following:
{h+h−,δ+δ−,z1z1√2,z2z2√2,η0η0√2,ξ0ξ0√2,ϕ+ϕ−,ϕ0ϕ0√2,a0a0√2,δ++δ−−},
Eigenvalues of are:
Rest of the six eigenvalues have been obtained by numerically solving the cubic Eqs. A-1 and A-2 given in Appendix A.
The third submatrix corresponds to the scatterings whose initial and final states are one of the following:
{η0z1,ξ0z2,ϕ0a0},
Eigenvalues of are:
{2(λ2+λ3),14(λ−√64λ29+(λ−4λΦ)2+4λΦ),14(λ+√64λ29+(λ−4λΦ)2+4λΦ)}.
The fourth submatrix corresponds to the scatterings, where one charge channels occur for scattering between the 20 charged states:
{η0h+,ξ0h+,z1h+,z2h+,hδ+,ξ0δ+,z1δ+,z2δ+,η0ϕ0,ξ0ϕ+,z1ϕ+z2ϕ+,h+ϕ0,h+a0,δ+ϕ0,δ+a0, ϕ+ϕ0,ϕ+a0,δ++δ−,δ++h−,δ++ϕ−},
Eigenvalues of are:
{λ1,λ1,2λ2,2(λ2+λ3),λ1% −λ42,λ1+λ4,λ1+3λ42,λ5−λ6,λ5+λ6,λ7,λ7,λ7−λ82, λ7+λ8,λ7+3λ82,λ5−2λ9,λ5+2λ9,14(λ+√64λ29+(λ−4λΦ)2+4λΦ), 14(λ−√64λ29+(λ−4λΦ)2+4λΦ)}.
Remaining three eigenvalues have been obtained from the cubic Eq. A-2 (see Appendix A) using numerical technique.
The fifth submatrix corresponds to the scatterings, where double charge channels occur for scattering between the 12 charged states:
{h+h+√2,δ+δ+√2,δ+h+,ϕ+ϕ+√2,ϕ+δ+,ϕ+h+,δ++ξ0,δ++z2,δ++z1,δ++η0,δ++ϕ0,δ++a0},
Eigenvalues of are:
{λ1,2λ2,2λ2−λ3,2(λ2+λ3),λ1−λ42,λ1+λ4,λ5+λ6,λ7,λ7−λ82,λ7+λ8, 14(λ+√64λ29+(λ−4λΦ)2+4λΦ),14(λ−√64λ29+(λ−4λΦ)2+4λΦ)}.
The sixth submatrix corresponds to the scatterings, where triple charge channels occur for scattering between the 3 charged states:
{δ++h+,δ++δ+,δ++ϕ+},
Eigenvalues of are: {2(+), +, +}. Finally, there is unique quadruple charged state which leads to eigenvalue
M7=2(λ2+λ3).
These eigenvalues, can be labelled as , then the -matrix unitarity constraint for elastic scattering demands Lee:1977eg (). Using this condition we generate the following relations. However, these conditions are not the full set of unitarity conditions as we have already mentioned that some of the eigenvalues of few submatrices are evaluated numerically. Hence, using the following conditions,
|λ1|≤8π,|2λ2|≤8π,|λ1+λ4|≤8π,|2(λ2+λ3)|≤8π,∣∣∣λ1+3λ42∣∣∣≤8π,∣∣∣λ1−λ42∣∣∣≤8π, |2λ2−λ3|≤8π,|λ5+2λ6−6λ9|≤8π,|λ5+2λ6+6λ9|≤8π,|λ5+2λ9|≤8π,|λ5−2λ9|≤8π, |λ5−λ6|≤8π,|λ5+λ6|≤8π,|λ7|≤8π,|λ7+λ8|≤8π,∣∣∣λ7+3λ82∣∣∣≤8π,∣∣∣λ7−λ82∣∣∣≤8π, ∣∣∣14(λ+√64λ29+(λ−4λΦ)2+4λΦ)∣∣∣≤8π,∣∣∣14(λ−√64λ29+(λ−4λΦ)2+4λΦ)∣∣∣≤8π, (38)
and numerically evaluated six eigenvalues (whose absolute value should be ) we have imposed full set of unitarity constraints on the model parameters.
#### ii.1.3 Perturbativity:
If we demand that the model in the present work behaves as a perturbative quantum field theory at any energy scale, then we have to ensure the following conditions. For the scalar quartic coupling , the perturbativity criterion is,
|λ|,|λΦ|,|λi|<4π. (39)
The corresponding constraints for the gauge and Yukawa interactions are,
gi,yi<√4π, (40)
where, ’s and ’s are the gauge and Yukawa coupling constants respectively.
#### ii.1.4 Constraints from electroweak precision test:
Electroweak precision test (EWPT) can be considered as a very useful tool in constraining any BSM scenario. As the current scenario contains several non-standard scalars, hence they contribute to the electroweak precision observables, the parameters Lavoura:1993nq (); Barbieri:2006dq (); Chun:2012jw (); Aoki:2012jj (). The stringent bound comes from the -parameter which imposes strict limit on the mass splitting between the non-standard scalars. Therefore, we tune the relative mass splitting between the non-standard scalars in such a way for which the present scenario satisfy the constraints from EWPT Baak:2014ora (). Further, the electroweak precision data constraint the -parameter to be very close to its SM value of unity and from the latest data Patrignani:2016xqp () one gets an upper bound on GeV which we maintain in our analysis.
#### ii.1.5 Constraints from Higgs signal strength (μγγ=σ(pp→h)BSM×BR(h→γγ)BSMσ(pp→h)SM×BR(h→γγ)SM):
Moreover, apart from the above mentioned theoretical constraints, it is necessary to incorporate the constraints from LHC data in the model. As in the present model all the decay widths and cross sections are modified with respect to that of the SM predictions so in our analysis we have constrained the parameter space of this model by the present LHC Higgs data ATLAS:2016nke ().
## Iii Neutrino masses and mixings
In this section, we have tried to explain the origin of neutrino masses and their intergenerational mixing angles. In the present model, as we have one scalar triplet (Eq. 4), hence one can generate Majorana mass term for the SM neutrinos using Type-II seesaw mechanism Magg:1980ut (); Ma:1998dx (). The Yukawa interaction term which is responsible for the Majorana masses of SM neutrinos is given by
LYukawa⊃−Yνij2LTiCiσ2ΔLj+h.c. (41)
where is the Yukawa coupling and are generational indices of the SM leptons. When the scalar triplet acquires a VEV , Majorana masses for the SM neutrinos are generated at tree level, which is
Mνij=Yνij√2vt. (42)
Since this a Majorana type mass term for the SM neutrinos, must be a symmetric matrix. Therefore, for three generations of the SM neutrinos the Majorana mass matrix has the following form
Mν=vt√2⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝y1\leavevmode\nobreak \leavevmode\nobreak y2\leavevmode\nobreak \leavevmode\nobreak y3y2\leavevmode\nobreak \leavevmode\nobreak y4\leavevmode\nobreak \leavevmode\nobreak y5y3\leavevmode\nobreak \leavevmode\nobreak y5\leavevmode\nobreak \leavevmode\nobreak y6⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠, (43)
where, for notational simplicity we have redefined the Yukawa couplings as , , , , and . Now, our goal is to diagonalise the above mass matrix and find the mass eigenvalues and mixing angles. To diagonalise a complex symmetric matrix (all six independent elements of can be in general complex) we need a unitary matrix so that is a diagonal matrix (). This is however not the eigenvalue equation, which has usually been solved for the case of matrix diagonalisation. Therefore, instead of diagonalising a complex symmetric matrix , one can easily construct a hermitian matrix using , such that is a diagonal matrix with real non-negative entities at the diagonal positions. The unitary matrix is the usual PMNS matrix which has the following form
UPMNS=UCKM⎛⎜ ⎜⎝1\leavevmode\nobreak 0\leavevmode\nobreak 00\leavevmode\nobreak expiα2\leavevmode\nobreak 00\leavevmode\nobreak 0\leavevmode\nobreak expiβ2⎞⎟ ⎟⎠, (44)
where is the usual CKM matrix containing three mixing angles , , and one phase , called the Dirac CP phase 444Because, any nonzero value of can generate CP violating effects in vacuum neutrino oscillations if , i.e. , (.) in vacuum oscillation when and Akhmedov:1999uz (). while , are known as the Majorana phases. If SM neutrinos are Dirac fermions then .
We have diagonalised the hermitian matrix by the unitary matrix and find the mass square differences and mixing angles between different generations of SM neutrinos. Dirac phase can be found by using a quantity known as Jarlskog Invariant () Jarlskog:1985ht (), which is related to the elements of matrix as,
JCP=Im(h12h23h31)Δm221Δm232Δm231, (45)
where numerator represents the imaginary part of the product while in the denominator . One the other hand can also be written in terms of mixing angles and Dirac CP phases, i.e.
JC
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2021-01-17 21:51:57
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https://www.aimsciences.org/article/doi/10.3934/jimo.2018196
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# American Institute of Mathematical Sciences
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May 2020, 16(3): 1135-1148. doi: 10.3934/jimo.2018196
## Performance evaluation and analysis of a discrete queue system with multiple working vacations and non-preemptive priority
School of Science, Yanshan University, Qinhuangdao 066004, China
*Corresponding author
Received October 2017 Revised April 2018 Published December 2018
In this paper, we introduce a discrete time Geo/Geo/1 queue system with non-preemptive priority and multiple working vacations. We assume that there are two types of customers in this queue system named "Customers of type-Ⅰ" and "Customers of type-Ⅱ". Customer of type-Ⅱ has a higher priority with non-preemption than Customer of type-Ⅰ. By building a discrete time four-dimensional Markov Chain which includes the numbers of customers with different priorities in the system, the state of the server and the service state, we obtain the state transition probability matrix. Using a birth-and-death chain and matrix-geometric method, we deduce the average queue length, the average waiting time of the two types of customers, and the average busy period of the system. Then, we provide some numerical results to evaluate the effect of the parameters on the system performance. Finally, we develop some benefit functions to analyse both the personal and social benefit, and obtain some optimization results within a certain range.
Citation: Zhanyou Ma, Wenbo Wang, Linmin Hu. Performance evaluation and analysis of a discrete queue system with multiple working vacations and non-preemptive priority. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1135-1148. doi: 10.3934/jimo.2018196
##### References:
show all references
##### References:
Schematic illustration for the service process of the non-preemptive priority queue
Schematic diagram for the model description
Relation of $E(L_1)$ with $\lambda$ and $\alpha$
Relation of $E(L_2)$ with $\lambda$ and $\alpha$
Relation of $E(L)$ with $\theta$ and $\mu_1$
Relation of $E(B)$ with $\mu _1$ and $\mu_2$
Relation of $B_1$ with $\lambda$ and $\alpha$
Relation of $B_2$ with $\lambda$ and $\alpha$
Relation of $D$ with $\mu _1$ and $\alpha$
Relation of $D$ with $\lambda$ and $\alpha$
Relation of $E(W_1)$ with $\mu _1$ and $\mu _2$
$\mu _2$ $\mu _1 =0.30$ $\mu _1 =0.32$ $\mu _1 =0.34$ $\mu _1 =0.36$ $\mu _1 =0.38$ $\mu _1 =0.40$ 0.45 4.7081 2.9925 2.1313 1.6271 1.3023 1.0787 0.50 2.5403 1.7894 1.3538 1.0748 0.8836 0.7459 0.55 1.7483 1.2914 1.0067 0.8156 0.6800 0.5797
$\mu _2$ $\mu _1 =0.30$ $\mu _1 =0.32$ $\mu _1 =0.34$ $\mu _1 =0.36$ $\mu _1 =0.38$ $\mu _1 =0.40$ 0.45 4.7081 2.9925 2.1313 1.6271 1.3023 1.0787 0.50 2.5403 1.7894 1.3538 1.0748 0.8836 0.7459 0.55 1.7483 1.2914 1.0067 0.8156 0.6800 0.5797
Relation of $E(W_2)$ with $\mu _1$ and $\mu _2$
$\mu _2$ $\mu _1 =0.30$ $\mu _1 =0.32$ $\mu _1 =0.34$ $\mu _1 =0.36$ $\mu _1 =0.38$ $\mu _1 =0.40$ 0.45 3.6164 2.9022 2.3961 2.0311 1.7626 1.5610 0.50 2.4961 1.9948 1.6473 1.4001 1.2199 1.0853 0.55 1.8492 1.4802 1.2263 1.0464 0.9154 0.8175
$\mu _2$ $\mu _1 =0.30$ $\mu _1 =0.32$ $\mu _1 =0.34$ $\mu _1 =0.36$ $\mu _1 =0.38$ $\mu _1 =0.40$ 0.45 3.6164 2.9022 2.3961 2.0311 1.7626 1.5610 0.50 2.4961 1.9948 1.6473 1.4001 1.2199 1.0853 0.55 1.8492 1.4802 1.2263 1.0464 0.9154 0.8175
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2019 Impact Factor: 1.366
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2020-11-25 00:38:09
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https://stats.stackexchange.com/questions/435821/need-intercept-for-plm-fixed-effects-in-r?noredirect=1
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# Need intercept for plm (Fixed effects) in R
all, I am new to this forum. I have a question of fixed effects in R....So I am trying to use plm function to find fixed effects like following:
> plm(PM25~policy+1,data=subset(part2,Delhi==1),model="within"
> ,index=c("station_id","date"))%>% summary()
The results I get:
> Oneway (individual) effect Within Model
>
> Call: plm(formula = PM25 ~ policy + 1, data = subset(part2, Delhi ==
> 1), model = "within", index = c("station_id", "date"))
>
> Unbalanced Panel: n = 7, T = 73-159, N = 992
>
> Residuals:
> Min. 1st Qu. Median 3rd Qu. Max.
> -193.7049 -54.6776 -9.6843 54.7431 318.4094
>
> Coefficients:
> Estimate Std. Error t-value Pr(>|t|) policy 76.8167 9.9787 7.6981 3.354e-14 ***
> --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
>
> Total Sum of Squares: 6846900 Residual Sum of Squares: 6458000
> R-Squared: 0.056803 Adj. R-Squared: 0.050093 F-statistic: 59.2603
> on 1 and 984 DF, p-value: 3.3535e-14
I am wondering how can I find the intercept?
The within model you specify amounts to fitting a separate intercept for each unit in your panel data set, as in $$y_{it}=a_i+\beta x_{it}+u_{it}$$ See, e.g., these posts for some context:
Difference between fixed effects dummies and fixed effects estimator?
How exactly does a "random effects model" in econometrics relate to mixed models outside of econometrics?
Hence, you cannot additionally fit an overall intercept $$a$$, as you would then have perfect collinearity between the $$a_i$$ and $$a$$, as the sum of the $$a_i$$ is identically one.
You can retrieve these estimates as follows:
data("Grunfeld", package = "plm")
wi <- plm(inv ~ value + capital,
data = Grunfeld, model = "within", effect = "twoways")
fixef(wi)
One might of course leave out one of the intercepts. Something related to what you seek seems to be achieved by this command: https://rdrr.io/rforge/plm/man/within_intercept.html
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2020-07-14 04:54:25
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http://math.stackexchange.com/questions/126290/if-x-is-a-martingale-x0-0-f-left-continuous-is-int-f-x-dt-also-a-m
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# If $X$ is a martingale, $X(0)=0$; $f$ left continuous, is $\int f X$ dt also a martingale?
If $X(t)$ is a martingale, and $X(0) = 0$. $f(t)$ is a left continuous function,
$$g(t) = \int_0^t f(s) X(s) ds$$
is $g(t)$ also a martingale?
I guess it shall be, but don't know how to prove that?
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Since $(X(t))_t$ is a martingale in the filtration $(\mathcal F_t)_t$, one knows that $\mathrm E(X(u)\mid\mathcal F_s)$ is $X(u)$ is $u\leqslant s$ and $X(s)$ if $u\geqslant s$. Hence, for every $s\leqslant t$, $$\mathrm E(g(t)\mid\mathcal F_s)=g(s)+\int_s^tf(u)\mathrm E\left(X(u)\mid F_s\right)\mathrm du=g(s)+X(s)\cdot \int_s^tf(u)\mathrm du.$$ The only cases when $(g(t))_t$ is a martingale are when $f(t)X(s)=0$ almost surely, for almost every $s\leqslant t$.
Edit: Note however that $X(0)=0$ and that $(X(t))_t$ is a martingale hence $\mathrm E(X(t))=0$ for every $t$, and $$\mathrm E(g(t))=\int_0^tf(s)\mathrm E\left(X(s)\right)\mathrm ds=0.$$
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thank you @Didier, now i understand martingale more... – athos Mar 30 '12 at 18:03
however, can i say that E(g(t))=g(0)? since X(0)=0... – athos Mar 31 '12 at 5:22
No, take $f(s) = s$ so $$g(t) = \int\limits_0^t s B_s\mathrm ds = \int\limits_0^t B_s\mathrm d\frac{s^2}{2} = \left.\frac{s^2}{2}B_s \right|_0^t - \frac12\int\limits_{0}^ts^2\mathrm dB_s = \frac12 t^2B_t - \frac12\int\limits_{0}^ts^2\mathrm dB_s.$$ Note that $\frac12\int\limits_{0}^ts^2\mathrm dB_s$ is a martingale and if $g$ is a martingale then $\frac12 t^2 B_t$ has to be martingale, but: $$\mathsf E[t^2 B_t|\mathscr F_s] = t^2 B_s\neq s^2 B_s.$$
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thank you, now I realize my mistake... – athos Mar 30 '12 at 18:03
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2015-09-02 17:12:58
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https://web.njit.edu/~goodman/Math222/matlab2.html
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# Matlab Assignment 2
### Due week of 4/23
This problem has two parts. In part one, we will extend what we know about Runge-Kutta methods to systems of equations. In the second, we will use pplane to examine some two-dimensional phase-planes, similar to our use of dfield to draw direction fields.
To do the second part of the assignment you need to download some additional programs, following the instructions here.
• If you chose to work with the MATLAB-based programs in assignment one, then you already have everything you need. The only difference in the instructions is that you type pplane2 at the MATLAB prompt (after the characters >> ).
• If you used the Java versions of the programs, available here, make sure you download and run the program named pplane.jar.
### Problem 1: Euler method for a second order equation
Runge-Kutta with two or more dependent variables is exactly the same as with one dependent variable. Suppose we have a system of two equations. Instead of calling the variables $y_1(t)$ and $y_2(t)$, let's call them $x(t)$ and $y(t)$. To solve a system of the form
\begin{aligned} \frac{dx}{dt}& = f(t,x,y)\\ \frac{dy}{dt}& = g(t,x,y), \end{aligned}
we define a sequence of times $t_k = k\cdot h$ and a sequence of approximations $(x_k,y_k)\approx(x(t_k),y(t_k))$, along with the obvious modification to the Euler updating rule:
\begin{aligned} x_{k+1}& = x_k + h\cdot f(t_k,x_k,y_k)\\ y_{k+1}& = y_k + h \cdot g(t_k,x_k,y_k). \end{aligned}
In the attached MATLAB-generated webpage, I solve the system
\begin{aligned} \frac{dx}{dt}& = t-\sin{x}+y\\ \frac{dt}{dt}& = t^2+x-\sin{y}, \end{aligned}
with initial condition $(x,y)=(3,2)$ from $t=0$ to $t=6$. Note that I plot the solution twice. The first graph shows both $x$ and $y$ as functions of $t$. The second graph shows the trajectory in the $(x,y)$ with initial condition $\theta=1$, $\theta'=0$.)-\$plane.
In Section 7.1 of Boyce and DiPrima, you learn how to convert a second-order equation
$x''(t)=f(t,x,x')$
into a system of equations second order equations. In this assignment you will be considering the forced damped pendulum equations:
$\frac{d^2\theta}{dt^2}+ \gamma \frac{d\theta}{dt} + \sin{\theta} = F_0 \cos{\omega t}.$
(We have chosen our units to make the problem as simple-looking as possible.)
1. (Pencil and paper) Rewrite this as a system of two first order equations, suitable for solving using the Euler method.
2. Modify the example program to solve the undamped, unforced system obtained by setting $\gamma=0$ and $F=0$ from $t=0$ to $t=20$ with initial condition $\theta=1$, $\theta'=0$. If you choose time step $h=0.1$, you will find a strange result: the amplitude of the pendulum's motion will grow slowly over time! (You know this shouldn't happen because the system conserves energy. That means the solutions should be periodic, and should look like closed curves trajectories are viewed in the $(x,y)$ plane. Instead, you should see a spiral.) Experiment with the value of $h$ to make the spiraling effect disappear, at least to the accuracy visible to the naked eye on your plots. Plot the trajectory and state how small a value of $h$ you used.
3. Using the same value of $h$, solve the system with the same initial conditions but $\gamma=0.1$, $F=0.1$, and $\omega=1$. Run the simulation up to $t=200$. Describe what you see.
4. Now increase the forcing to $F=1$. Describe what you see. This is an example of mathematical chaos.
### Problem 2: Using software to plot phase planes
Using the phase plane program described in the introduction, plot the phase plane for the Lotka-Volterra model
\begin{aligned} x' & = (a - b y)x \\ y' & = (cx -d)y \end{aligned}
Here $x(t)$ represents the population of a prey species, say mice, and $y(t)$ is the population of a predator species, for example owls. If there are no owls, then the population of of mice grows exponentially, and if there are no mice, then the owls die off. However if the initial populations are both positive, then interesting stuff happens. Plot the phase plane including a number of solutions of this system with $a=b=c=d=1$. You decide what is are good limits for your axes. Describe the solutions in a few words. Do you think that this model gives realistic behavior?
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2018-06-24 15:03:42
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https://holzterrassen-spezialist.de/vectors-worksheet.html
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Vectors Worksheet
Vectors WorksheetFree printable worksheets for CBSE Class 9 Physics, school …. Graphical Vector Addition A + B Step 1 - Draw a start point Step 2 - Decide on a scale Step 3 - Draw Vector A to scale Step 4 - Vector B's tail begin at Vector A's head. You can go here to learn more about the Pythagorean Theorem. Translation vectors translate figures in two-dimensional space, from one location to another. If you would like more help understanding Vectors there are clear, easy to follow, step-by-step worked solutions to dozens of N5 Maths Past & Practice exam questions on all topics in the Online Study Pack. Worksheet by Kuta Software LLC-2-17) u = 9, 52° v = 12, 250° Find: -u + v 18) u = 12, 202° v = 19, 296° Find: -u + v Find the component form of the resultant vector. The vectors Ax and Ay lie along the x and y axes; therefore, we say that the vector A has been resolved into its x and y components. Life, however, happens in three dimensions. 77,000+ Vectors, Stock Photos & PSD files. Be sure to (1) ALWAYS write the equation (2)plug in the numbers and units, (3) give the answer with the correct units. PERPENDICULAR VECTORS WORKSHEET. (1)Use the Gram-Schmidt orthogonalization process to compute an orthogonal basis for Col(A). Since vectors represent magnitude and length, we need a computationally straightforward way of determining lengths and angles, given the components of a vector. Practice worksheet for adding force vectors using the parallelogram rule. 3 VECTOR EQUATIONS Key concepts to master: linear combinations of vectors and a spanning set. A free -body diagram is a special example of the vector diagrams; these diagrams will be used throughout your study of physics. So the simplest one is the origins of recombinant DNA technology: They made copies of RNAs, and they were able to insert these into what is known as plasmids. Three-phase space vectors - GeoGebra Dynamic Worksheet. 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Find the X and Y components of the following: A. 7 KiB, 6,015 hits) Verbal expressions - sum (146. About Worksheet Vectors Adding. Reflect the vectors in the x-axis. Add And Subtract Square Roots Worksheet By Kevin Wilda Teachers Pay Teachers Square Roots Studying Math Maths Formulas List. pptx (Slides) GCSE-VectorWorksheet. pdf from AR 1 at Mapúa Institute of Technology. P is the point on AB so that AP: PB = 3 : 2 b) Show that OP = (2a + 3b) 1 5 O X Y 2a + b 4a + 3b 2) OX = 2a + b OY = 4a + 3b a) Express the vector XY in terms of a and b Give your answer in its simplest form. Multiplying by -1 gives the additive inverse -1 v = -v. We see the formula as well as tutorials, examples and exercises to learn. Let ~u = h2;4iand ~v = h3; 1i: Compute the following vectors and then draw your answers in the Cartesian plane with the tail (initial point) at the origin. This means you may not be able to view worksheets . It is usually denoted by d in math problems. 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Math Analysis Honors – Worksheet 113 Applications of Vectors 1 A child pulls a wagon with a force of 40 pounds. For the first six problems, draw the indicated vector and show the components into which it is resolved. This is a Java Applet created using GeoGebra from www. A vector is said to be in standard position if its initial point is the origin (0, 0). The y-coordinate for the buoy is now the shortest distance at which the ship will pass the buoy. The result of addition of vectors can be determined simply by adding two vectors (or resultant). a) How long does it take to get to the destination? b) How long does it take to return to the starting point?. How to Add and Subtract Vectors. Depending on the situation, these perpendicular components may be described as compass bearings (north, south, east or west) if we are analysing a car driving along the road. 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Look at the 4 3 matrix you wrote down for Question 1 and. 3 February 3 - 1-01 Vector Components p. Find the x- and y- components of each vector. Scalars may or may not have units associated with them. Graphical Representation of Vectors Vectors → defined by direction and magnitude only - Their "location" in the vector space is arbitrary Can move vectors around to use geometry - With the role of distance replaced by vector magnitudes A B C A B = C "Tail-to-tip" convention: Geometry: These 3 vectors form. Vectors are defined by their magnitude and direction. Standards for Mathematical Practice M. But you don't need to fall into despair in any case because Vectors Homework Worksheet Answers there is an easy way out. In this Worksheet students will be able to learn about the Nature of vectors, distinguishing between a scalar and a vector, distinguishing between Displacement and Distance, how adding two vectors is done and how to calculate. 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Make sure that the resultant has both magnitude and direction because it is a vector quantity. 5 m, E - 20 m, W Graphical Method Aligning vectors head to tail and then drawing the resultant from the tail of the first to the head of the last. Click on the " Solution " link for each problem to go to the page containing the solution. doc Author: klhessel Created Date: 5/5/2010 3:40:24 PM. Worksheet by Kuta Software LLC Kuta Software - Infinite Precalculus Three-Dimensional Vector Cross Products Name_____ Date_____ Period____-1-Find the cross product of the given vectors. Showing top 8 worksheets in the category - Vectors. The tips below can help you fill in Drawing Vectors Worksheet easily and quickly: Open the form in the full-fledged online editor by clicking Get form. A set of geometry worksheets for teaching students about different types of shape movements - translation, rotation, and reflection. 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Also, graph the image and find the new coordinates of the vertices of the translated figure in these pdf exercises. In what country will you make landfall? USA (NY) How many months will it take to reach land (how many 10-square vectors is it)? 3. Vector Addition - Component Method Vectors can be described by its _____ to show how far it goes in the x and y directions. Vector Addition Worksheet On a separate piece of paper, use the following individual vectors to GRAPHICALLY find the resultant vector in the first three problems. A person pulls on a 500 N desk with a 200 N force acting at 30° angle above the horizontal. 3 Name all the equal vectors in the parallelogram shown. Some of the worksheets for this concept are Assignment date period,. Learning basic grammar pdf free download. The directed line segment might look like a ray, but it has a specific length. When adding vectors, a head-to-tail method is employed. Vector scalar scalar scalar scalar vector vector magnitude 15 blocks 5 blocks scalar vector 11º e of n 12º s of w 17º w of s 30º s of e e. These printable Addition worksheets are great for teachers as well as parents who want to use them at. What is the magnitude of its flight?. New Vocabulary • vector • magnitude • initial point • terminal point • resultant x 40 70 x 7 6 16 20 x2 What You’ll Learn • To describe vectors • To solve problems that involve vector addition. ?(a) uThe cost of a theater ticket (b) The current in a river 45¡(c) The initial ! ight path from Houston to Dallas (d) wThe population of the world 2. These Worksheets for Grade 9 Physics cover all important topics which can come in your standard 9 tests and examinations. p – q = p + (–q) Example: Subtract the vector v from. Edexcel Qs / As (multiple choice) Spotted a mistake in a worksheet? Report mistake. This is a unique problem in that the resultant is given and one of the two vectors is given. Writing vectors in these translation vector worksheet pdfs graph the umbrella term translation vectors for transformation given rules and terminal points on describing single transformations rather than performing and triangles. DIRECTION must be entered in degrees, increasing 'counterclockwise'. PhET sims are based on extensive education research and engage students through an intuitive, game-like environment where students learn through exploration and discovery. Fill in the required boxes which are colored in yellow. Base vectors for a rectangular coordinate system: A set of three mutually orthogonal unit vectors Right handed system: A coordinate system represented by base vectors which follow the right-hand rule. Unit 8 Vectors - Displaying top 8 worksheets found for this concept. An introduction to reflections of shapes, using an Autograph. Add and subtract vectors graphically and algebraically Individual numbers--that is, values that have only (positive or negative) magnitude--are called scalars. See also: Flip, Slide, and Turn Worksheets. This worksheet is designed to assess your understanding of the graphical method of vector addition. Excel worksheet arrays and vectors. Practice Worksheet - Four pages of vectors to start adding up. Vectors are useful tools for solving two-dimensional problems. Common practice is to break the vector into perpendicular components. The Vectors important questions class 11 will help to boost your preparation for upcoming NEET exams. Math 126 Worksheet 1 Properties of Vectors Part I - Vectors and Their Components Given a vector v = ha;bi, you can use the Pythagorean Theorem to nd its length and trigonometry to nd the angle it makes with the xor yaxes. Find & Download the most popular Worksheet Activity Vectors on Freepik Free for commercial use High Quality Images Made for Creative Projects. docx (Assessment) Dr Frost Learning is a registered charity in England and Wales (no 1194954). Both the wind and the ocean affect the landfall; this is represented accurately only by building off the wind correction vectors. (See The 3-dimensional Co-ordinate System for background on this). Physics Vector Worksheet #1 In physics we distinguish between scalars and vectors. Worksheet - Exp 3: Vector Addition Part 2 Objective: The objective of this lab is to add vectors using the component method and to verify the results using a force table. (2) Scalar quantity Any one of: mass, distance, speed, energy, power (1) Vector quantity Any one of: velocity, acceleration, force, displacement (1) c) i) 4. Vectors Resultant online activity for Grade 9. We say that the vector is in standard position and refer to it as a position vector. Each vector in the diagram should be drawn so that the larger the vector the bigger the force it represents. Rather than enjoying a good PDF. Vector and Parametric Equations in R2 Cartesian Equation of a Line. Find & Download the most popular Worksheet Vectors on Freepik Free for commercial use High Quality Images Made for Creative Projects. Since we already figured out that A + B = B + A we can draw the resultant for either combination and get the same thing. Search: Adding Vectors Worksheet. In this vectors worksheet, students solve 8 problems given each vector in kilometers and their degree. 18 Calculate a vector's magnitude using trig or Pythagorean theorem. Displacement is the quantity of change of. Translations and Vectors Worksheets- Includes math lessons 2 practice sheets homework sheet and a quiz. pdf - Free Download enjoy now is Calculus And Vectors 12 Nelson Solution Manual below. Day 1: Relative Velocity (Interdependence of Perpendicular Vectors) + Simple 2D Vector Addition Homework: Worksheets for Relative Velocity + Worksheet . Vectors can be graphically represented by directed line segments. Vector Voyage Worksheet 3 Answer Key offers a summary of this activity and clearly illustrates the vector movement directly. Download 190+ Royalty Free Guide Number Tracing Worksheets Vector Images. Construct an X-Y chart for all vectors:. Given that AC = p and that AD = q, express the following vectors in terms of p and q. com/watch?v=uqt8qy0fE70&list=PLJ-ma5dJyAqr5abMalSbaTxeDzT_itkYs&index=1Q. VECTORS Materials required for examination Items included with question papers Ruler graduated in centimetres and Nil millimetres, protractor, compasses, pen, HB pencil, eraser. Vectors and scalars are important in many fields of math and science. Vectors 9 – Powerpoint – Shortest distance 2 line. Determine the coordinate direction angles, , , and , of F d. Scalar multiplication is multiplying a vector by a constant. How do we denote a general m n matrix A algebraically? What notation do we use for the (i;j)-entry of A? Give an example of a 4 3 matrix and circle its (2;3)-entry. When adding vectors a head to tail method is …. Suppose → a a → is to be subtracted from → b b →. Two free vectors are said to be equal iff their lengths and directions are the same. Notes - Vectors Day 3; Notes - Vectors Day 3 (filled) HW #54. To add vectors graphically, you place the tail of one on the head of the other. Measure length in centimeters and direction angles in standard form (e. Some of the worksheets for this concept are Vectors work pg 1 of 13 vectors, Introduction to vectors, Chapter 1 units and vectors tools for physics, Vector work, Work vectors and dot products, Unit 4 geometric vectors, Vectors in two dimensions, Vectors. Part 1 Directions: Read the situations below carefully. Directions: Find the sin, cos, and tan of the angle θ. Our digital library hosts Chapter 3 Vectors Worksheets Key This is an alphabetical list of the key vocabulary terms you will learn in Chapter 3. Term of rotation, translation vectors in a specific textbook. We can add vectors by using the parallelogram method or the triangle method to find the sum. What is the arctangent (tan-1) of 1. We can draw the vector OP as follows:. Resource type: Worksheet/Activity. The plasmids are propagated in bacteria, so, in addition to their therapeutic cargo, they necessarily contain a bacterial replication origin and a selection marker, usually a gene conferring anti …. And Vectors Worksheet Answers broken down into its components. Vectors: Vector quantities have both magnitude and direction. Mathematics is the language of physics. The Chino Valley Unified School District is committed to equal opportunity for all individuals in education and employment. Vectors Worksheet As you work through the steps in the lab procedures, record your experimental values and the results on this worksheet. There are a number of different ways to show the direction of a vector: left-right-up-down, x and y components, and polar coordinates - a length and an angle. 2 In a warehouse, a box is being pushed up a. Resultant C is shown in the fi rst two diagrams, (a) and (b). Erdman E-mail address: [email protected] Vectors online worksheet for 11th. Visit his website here for more videos. Because many of these problems involve more than two vectors a drawing is very important. PHYSICS is the study of matter, energy, and the interaction between them. (a)(2 points) Is this function linear? If not, provide a counterexample. 2 Matrices and Matrix Operations 7. Physics Laboratory Worksheet in Vector Addition In this Worksheet, we are going to :-1- Understand the Nature of vectors 2- Learn how to Add Vectors 3- Calculate the Resultant force from two vectors using Firstly, what is a Vector? A Vector in physics, is a quantity that has both magnitude and direction. Try the Leading Stock Image Provider. Measure the length and angle of the resultant and convert it back to grams. bordering to, the declaration as capably as perception of this two dimensional motion and vectors worksheet answers can be taken as capably as picked to act. 0 km due east and then turns and flies 7. In the addition of vectors, we are adding two or more vectors using the addition operation in order to obtain a new vector that is equal to the sum of the two or more vectors. Descibe the graph of f(x) = hx;x2i. The set of integers modulo m is Zm = {0,1,2,…,m-1}, therefore the only components of these vectors are the possible remainders when an integer is divided by the modulus. How to subtract Vectors? The following diagram shows how to subtract vectors graphically. Addition and subtraction result in another vector for which magnitude and direction are both changed. Solutions can get all the worksheet pdf exercises. As shown on Figure 1, the dual basis vectors are perpendicular to all basis vectors with a di erent index, and the scalar product of the dual basis vector with the basis vector of the same index is unity. If we simply specify magnitude and direction then any two vectors of the same length and parallel to each other are considered to be identical. Some of the worksheets for this concept are Assignment date period, Precalculus vector review work, Precalculus notes unit 6 vectors parametrics polars, 6 1 vectors in the plane, Honors pre calculus vector word problems 50 degrees with, Two dimensional vector basics, …. ; right 0°, up 90°, left 180°, down 270°, etc. -2- Worksheet by Kuta Software LLC Find the magnitude and direction angle for each vector. Directions: Solve the problems below by using the component method of vector addition to find the resultant. Vectors and projectiles worksheet answers. (Opens a modal) Direction of vectors from components: 3rd & 4th quadrants. Physics 201: Introduction to Vectors. Physics Worksheet Two Dimensional Motion and Vectors Section: Name: Mr. Students will use the Pythagorean Theorem to calculate the magnitude of the resultant vector and trigonometry to calculate the angle of the vector from the x-axis. - The laws of physics are independent of the choice of coordinate system. This worksheet will walk you through some basic vector operations. Displacement, velocity, momentum, force, and acceleration are all vector quantities. Use the e-autograph solution to e-sign the template. • The scalar components of a vector are its direction ratios, and represent its projections along the respective axes. Level 5 - An irregular hexagon defined by vectors. Addition of vectors is commutative such that A + B = B + A. how do you find the answers for the vector worksheet? IcyBlue999. Find & Download Free Graphic Resources for Worksheets. A B C A + B = R1 D 2A 1 2 A-4C-1 2 D A + 2B + 1 2 C = R3 A + 4C = R2 A - C = R4 B - A = R5 2C - B = R6 2C - A - B = R7 For the vectors below, calculate the vector's magnitude, and direction. Using trigonometry, find the x and y components of the three vectors (above) A x = B x = C x = A y = B y = C y = 3. Adding zero does not change a vector. Add the following vectors graphically. Find the coordinate vector of 1 2t in the basis f1+2t;2+tgof P 1. Worksheet by Kuta Software LLC Kuta Software - Infinite Precalculus Two-Dimensional Vector Dot Products Name_____ Date_____ Period____-1-Find the dot product of the given vectors. r= (x B-x A) i+ (y B – y A) j + (z B – z A)k , here i, j and k denote the unit vectors along x, y and z axis respectively. Our science worksheets tap into that fascination with grade-specific lessons and activities about astronomy, geology, chemistry, and more. Math 4 Name:_____ Algebraic Vectors worksheetAlgebraic Vectors worksheet Find the position vector if has the initial point P and the terminal point Q. During the lesson, watch and listen for instructions to take notes, pause the video, complete an assignment, and record. 1 Newton’s Second Law!" #" F� 1 F� 2 F� 3 A)!" #" F� 1 F� 2 F� 3 B) $%"$ &"!" #" F� 1 F� 2 F� 3 $%" C) F� 4!" #" F� 1 F� 2 F� 3$" D) Sum up the forces in the x and y directions: A) ma x = ma y = F 1 −F 2 B) ma x = ma y = C) ma x = F 3 cosθ 3 −F 4 ma y = D) ma x = ma. Vector Worksheet Pdf With Key Focuses On Resultant Vectors 25 Problems Worksheets Vector. Vector Voyage Solution Worksheet 1 Vector Voyage Instructions Part 1: Your ship can sail 10 squares/month. Vectors revision worksheets and practice questions for the Maths courses. Vectors of length n that have components integers modulo m are called m-ary vectors of length n denoted Z. 4b Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. Worksheets are available to suit the needs of each student. Worksheet 3: Forces 1 Free Body Diagrams 1. To get the sum of the two vectors, place the tail of b onto the head of a and the distance. Available for PC iOS and Android. There are also worked practice questions for GCSE. The concept of vectors is discussed. The Cartesian components of this vector are given by: The components of the position vector are time dependent since the particle is in motion. 24 #1-3 Vector Problems Worksheet #1-5 Gizmo - Distance/Time Graphs Complete handouts Graphing Motion Lesson Lesson (Miss Truong) Vector Problems Worksheet #6-9 (see bottom of page. To calculate the resultant vector’s magnitude and direction: If you can take two vectors at right angles and make a diagonal vector, shouldn’t you be able to take a diagonal vector and make two right-angled vectors from it? Can be: What is the length of the resultant?. Vectors with three or more components have properties defined with the very similar, general case formula. A vector connecting two points: The vector connecting point A to point B is given by. They multiply and subtract vectors and identify the linear combinations of given matrices. The initial point and terminal point of the translation vector are irrelevant. Get thousands of teacher-crafted activities that sync up with the school year. 7) i j 8) r , Find the component form, magnitude, and direction angle for the given vector 9) CD where C = ( , ) D = ( , ) Sketch a graph of each vector then find the magnitude and direction angle. Vectors addition and subtraction worksheet Add worksheet. The term vector comes from the Latin word vectus, meaning "to carry. Find the resultant vector (mag and dir) given the following information: A. Notes - Vectors Day 2; Notes - Vectors Day 2 (filled) HW #53 - Worksheet 2 on Vectors; HW #53 - Answer Key; 4. Worksheet 1 True/False Indicate whether the statement is true or false. 87 E of N 1-4C 2 A 2A -1 2 D 2 C A B VECTORS WORKSHEETS pg 12 of 13. Scalar multiplication does not affect direction, but does change the …. This Precalculus parameterization and vectors worksheet generates free practice problems on position vectors. Use our short worksheet/quiz combo to test yourself when it comes to vectors in two and three dimensions. Draw the vectors so that their initial points coincide. ” Furthermore, this discussion focuses on finding the angle between two standard vectors, which means their origin is at (0, 0) in the x-y plane. The unit vectors i and j are directed along the x and y axes as shown in Fig. Two Dimensional Motion And Vectors Worksheet Answers is easy to. Includes an introduction to parallel vectors. Vectors and the Parallelogram Rule 1. 4a Add vectors end-to-end, component-wise, and by the parallelogram rule. (d) Worksheet – Vector Practice Worksheet. You must solve several practice problems dealing with magnitude and vector. IGCSE Mathematics Worksheet - Vectors. What matters is the length of the vector and the direction in which it points. Thanks to the SQA and authors for making the excellent resources below freely available. This is the currently selected item. About Vectors Graphically Adding Worksheet And Subtracting. Microsoft Word - R13 VECTORS_V2020. In our standard rectangular (or Euclidean) coordinates (x, y, and z), a unit vector is a vector …. Vector Spaces: Theory and Practice observation answers the question "Given a matrix A, for what right-hand side vector, b, does Ax = b have a solution?". This definition of equality will also do for position vectors, but for sliding vectors we must add that the line of action must be identical too. Vectors Practice Questions - Corbettmaths. Prove the parallelogram law: The sum of the squares of the lengths of both diagonals of a parallelogram equals the sum of the squares of the lengths of all four sides. (i) Find, in terms of a and c, in its simplest form, (a) AB. Vectors - Displaying top 8 worksheets found for this concept. Worksheet: The Parallelogram Rule C H A P T E R 2 : M E C H A N I C A L E Q U I L I B R I U M Directions: Vectors are a simple tool to describe things like force or velocity. Once all the vectors have been "chained together," the resultant vector. For each question in the following quiz, choose whether the given quantity is a vector or a scalar. Would have liked it better with a bit more packed in terms of space - …. Vectors Worksheet Answers Frequently, two-dimensional kinematics involves breaking the relevant vectors into their x- and y-components, then analyzing each of the components as if they were one-dimensional cases. 3 KiB, 9,990 hits) Ones to thousands (84. So what vectors do is allow you to propagate the DNA you're interested in, in the organism you've chosen to propagate it in. Addition of vectors To add or subtract two vectors, add or subtract the corresponding components. lim x! 1 cos x x2 = lim sin 2 = 1 2 4. Activity and Assessment Packmovement. Credit Maths Exam Worksheets by Topic. Vector addition worksheet answers the physics classroom. Vectors worksheets pg 1 of 13 vectors. First we add the horizontal components of a vector (top numbers) and then we add the vertical components of a vector (bottom numbers). Part V Find the Angle Measurements Between the Resultant Vector and Force Vector When Two. A B A 4 A 2 B C 1/2 C A C B A B 2C 2C A 2C-A 2C-A B mag = 9. Also included in Section 2 below is the Adding Vectors worksheet (answers included). Through guided activity worksheets (Appendices A and B) and the use of inexpensive equipment, students were able to visualize the tip-to-tail method of vector addition, determine the horizontal and vertical components of vectors and observe the combination of two concurrent parallel or perpendicular vectors. Exam 1 Collins - All the information you need to pass Professor Collin's Exam 1. Find & Download the most popular Abc Alphabet Letters Tracing Worksheet Vectors on Freepik Free for commercial use High Quality Images Made for Creative Projects. 1 - Introduction to Vectors Definition A vector v in the plane is an ordered pair of real numbers. Addition of Vectors: It is significantly necessary for students to understand the properties of vectors before they engage in executing any mathematical operation with them. Two vectors are said to equal if their. The position vector for v is drawn from the origin to the point. In linear algebra, we define the concept of linear combinations in terms of vectors. These worksheets are recommended for 6th grade. Worksheet #2 Subtracting Whole Numbers 87 - 20 35 - 8 493 - 37 821 - 507 652 - 251 776 - 498 1904 - 625 1,344,192 - 804,663 70,801 - 62,762 Subtract. Examples: mass, volume, energy, money A vector is a quantity which has both magnitude and direction. 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Welcome · Videos and Worksheets · Primary · 5-a-day expand child menu. Quantities which do not have direction, but only length or size are known as scalar quantities. So by this definition a vector is an infinite …. We usually represent a vector with an arrow: Multiples of Vectors Given a real number c, we can multiply a vector by c by multiplying its magnitude by c: Adding Vectors Two vectors can be added using the Parallelogram Law Combinations These operations can be combined. Draw these three vectors A = 5. mathswatch vectors worksheet answers I found out the answer and it had a clear explanation of how they worked it out! After that it advanced my mathematical knowledge. If students have access to calculators that can do complex-number arithmetic, the “complex” approach is actually simpler for series-parallel combination circuits, and it yields richer (more informative) results. We express vectors in component form using the unit vectors i, j and k, which each have magnitude 1 and point along the x, y and z axes of the coordinate system, respectively. Both of these properties must be given in …. Type (s): Worksheets (e-library) Game (e-library) Vectors Snakes and Ladders. Perpendicular vectors worksheet Continue This sheet requires students to add perpendicular vectors through a step-by-step process that involves sketching a triangle using the tip-to-tail method and determining the outcome of direction and magnitude using the Pythagoras theorem and tangent function. docx Author: Lasalle Teacher Created Date: 9/21/2011 7:03:29 AM. Provide students with a brief introduction to vectors. 8A, Vectors and Dot Products MATH 1410 1. Suppose that T: Rn!Rn is a linear transformation. 5 Maths Year 10 Lines and Vectors worksheets available. Given vectors u and v are equal i. Statement of Parallelogram Law. Addition and Subtraction of Vectors 5 Fig. They are used all through out the worksheet in different problems. Lin 2 3 One-dimensional vector addition: The result of adding two vectors is the sum (two vectors have same directions) or difference (two vectors have opposite directions) of the two lengths and the direction of the longer. Now write the same data to a worksheet that doesn't yet exist in tempdata. "This leads nicely to the geometric representation of a vector in as a directed line segment from the origin. Gender and the politics of history summary. Download 37,000+ Royalty Free Worksheet Vector Images. Note that a translation is different from a rotation or a reflection since a. Some of the worksheets displayed are Work, Mathematical vector addition, Vectors in two dimensions, W u ma 213 work 2 u, Activity vector voyage, Physical sciences grade 11 term 1 resource pack, Part i, Vector algebra. Students find unit vectors in the same direction as a given vector. can you give Year 9,10 End Of Year Exam Examples please . Exercise 1 (Basic Addition/Subtraction) State the value of each vector. o Students will use their knowledge about vector addition to prove that the magnitude of the sum of two vectors is not always equal to the sum of the magnitudes. To visualize a projection, imagine a ashlight on the vector pointing from top to bottom will leave a shadow, or projection, on the x-axis. A scalar quantity has only magnitude. 7 Notes Vectors Video 2 Pre-Calculus Unit 6 Day 5 - Chapter 4. Displaying top 8 worksheets found for - Adding Vectors. Vectors are used to represent quantities that have both magnitude and direction. Vectors : Forms , Notation , and Formulas A scalar is a mathematical quantity with magnitude only (in physics, mass, pressure or speed are good examples). The sum of two or morevectors is called the resultant. Its velocity vector is given by v =. The length represents the magnitude and the direction of that quantity is the direction in which the vector is pointing. Worksheets are Vectors work pg 1 of 13 vectors, , A guide t. Any given vector ~v in E2 can be written as ~v = ~a + ~b, for a unique pair ( ; ). The vector sum R can be drawn as the. Math 126 Worksheet 1 Properties of Vectors Part I - Vectors and Their Components Given a vector v = (a, b), you can use the Pythagorean Theorem to find its length and trigonometry to find the angle it makes with the x or y axes. All downloads are in PDF Format and consist of a worksheet and answer sheet to check your results. Some of the worksheets for this concept are Physical sciences grade 11 term 1 resource pack, Vectors work pg 1 of 13 vectors, Grade 11 general mathematics trigonometry, A guide to vectors in 2 dimensions, Vector work, Grade 11 mathematics practice test, A guide to vectors and scalars, Platinum physical …. Title: Microsoft Word - GEO Chap 12. A + B -D An airplane is flying 340 km/hr at 12 oEast of North. The topics with a pale blue background are those listed in the National Curriculum for Key Stage 4 (ages 14-16); the others are listed in the. Let u → = 〈 u 1 , u 2 〉 and v → = 〈 v 1 , v 2 〉 be two vectors. Provided by the Academic Center for Excellence 8 Vectors in Two Dimensions January 2017 Unit Vectors A unit vector is a vector with a magnitude of one. We can use the familiar x-y coordinate plane to draw our 2-dimensional vectors. Vector questions – Word – Some exam style questions on vectors. Get Free Access See Review + Lesson Planet: Curated OER. Model Problems In the following problem you will learn to show vector. If t and sin 3 , find 2 dy x e y t dx in terms of t. 1 degrees above the negative x-direction. D is on AB such that AD : DB = 3 : 5. Scalars and quantities that are a number and a unit; vectors are a number and a unit plus a direction. Vectors (Workbook with Solutions) Subject: Mathematics. These coordinate planes have x axis and y axis labels along the outer edge of the page. Take an ordinary triangle, with angle θ between sides a and b, and opposite side c. They are logical, integer, double, complex, character and raw. Let’s consider a vector v whose initial point is the origin in an xy - coordinate system and whose terminal point is. 7 KiB, 6,008 hits) Verbal expressions - sum (146. For each problem make a rough sketch and show all work. Vectors Worksheets Key Chapter 3 Vectors Worksheets Key Thank you utterly much for downloading chapter 3 vectors worksheets key. Two nonparallel vectors always define a plane, and the angle is the angle between the vectors measured in that plane. Topic Revise Try a Revision Card 2022 Exams Topic List New; Vectors. Showing top 8 worksheets in the category - Vectors Part 2. Find the lengths of the vectors you computed in problem 1. A vector describes a movement from one point to another. Vector Connectors - A basic set of exercises on vectors which could be done before attempting the following. Resolving vectors into horizontal and vertical components is used in the addition and subtraction of vectors and finding the resultant of multiple vectors. The main ones are, naturally, the number of pages, academic level, and your deadline. The best selection of Royalty Free Worksheets for Kids Vector Art, Graphics and Stock Illustrations. A vector is a quantity that has magnitude (length) and direction. True vectors are defined as quantities. Q4: Fill in the blanks: The of an object has a direction that points from the center of of the object to the center of of Earth. 57 x 10-4 [22o W of S] Convert the following x and y components to vectors. A vector is an object that has both a magnitude and a direction. Please select the best answer from the given choices. 1 Vector Spaces Underlying every vector space (to be defined shortly) is a scalar field F. Calculate the acute angle between the lines with equations r = –1 4 + s 3 4 and r = 4 2 + t –1 1 Working: Answer: (Total 6 marks) 4. A boat B moves with constant velocity along a straight line. A closed vector diagram is a set of vectors drawn on the Cartesian using the tail-to-head method and that has a resultant with a magnitude of zero. Wrong results are often caused by arithmetic errors creeping in whilst performing tedious calculations, by read or clerical errors and by insertion of wrong data or of correct data in the wrong field of the paper form. The worksheet is an assortment of 4 intriguing pursuits that will vector worksheets with answers. X P BAplElp krViMglhOtTsG brkekspeLrZv`ejdf. Physics tries to answer main questions which include how did the universe begin?. The magnitude of a vector is a scalar and. Our printable translation worksheets contain a variety of practice pages to translate a point and translate shapes according to the given rules and directions. File Type PDF Chapter 3 Vectors Worksheets Key b·‐\" '?R?u ™"‖''?v‖'¨'· "'?j 〉 Chapter 3 - Vectors PHYSICS 101 // CH 3: VECTORS // OMAR. Enrichment and clubs in science. 5}\) $$\text{N}$$ acting in the positive $$y$$-direction we can draw it as a vector on. Students indicate the magnitude and direction of the resultant vector. The length is chosen, according to some scale, to represent the magnitude of the vector, and the direction of the directed line segment represents the direction of the vector. This workbook has detailed solutions and is great for learning about vectors. Find the displacement of an airplane that flies 5. The top number denotes left or right. VECTORS WORKSHEETS pg 1 of 13 . Vectors Algebra Worksheet for Class 12. GCSE Boards: AQA, Eduqas, Pearson Edexcel, OCR, Curriculum topic: Geometry and Measures, Congruence and Similarity. 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So, in general if you want to find the cosine of the angle between two vectors a and b, first compute the unit vectors aˆ and bˆ in the directions of a. And let me define that when we say a vector n is normal or. A vector can be represented by a. Since the vectors are given in i, j form, we can easily calculate the resultant. Vector Addition Practice Worksheet Answers are endeavor sheets for college students who’re producing their primary talents, and intriguing worksheets are a person method to boost. The Vectors Worksheet with Answers is a great way to organize your daily work and ensure you get your work done each day. Use the exact values you record for your data to make later calculations. They have kindly allowed me to create 3 editable versions of each worksheet, complete with answers. Combining a scalar with a vector is easy. FREE Resources to Help you Teach your lesson on Introduction to Vectors Worksheets, Guided Notes, Power Point, Bell Ringer, Exit Quiz, and more!. 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Class 12 Vectors Algebra students should refer to the following printable worksheet in Pdf in standard 12. What is the sum (resultant) of the two vectors? The component method of vector addition is the standard way t. Vectors and vector addition: A scalar is a quantity like mass or temperature that only has a magnitude. ~v is the diagonal of the parallelogram ~a, ~b. Free pdf worksheets to download and practice with. Vector Worksheet Pdf With Key Focuses On Resultant Vectors 25 Problems Persuasive Writing Prompts Addition Worksheets Algebra Worksheets 35 m s at 57q from the x axisVectors worksheet with answers. Example 2: Sketch the resultant of the addition of the two vectors in Example 1. Worksheet by Kuta Software LLC Precalculus Practice Quiz - Vectors ID: 1 ©d b2y0w1X7M NKNu[tYaF fSPoJftttwza_rHeY zLMLdCy. 3D Cartesian Vectors ENGR 30 4. Adding Vectors For Students 8th - 10th. Distance is the length of space traveled by a moving object or the length measured between two points. Find letter t worksheets stock images in HD and millions of other royalty-free stock photos, illustrations and vectors in the Shutterstock collection. Find a vector equation of the line passing through (–1, 4) and (3, –1). Ling 2017-12-19 University Physics is designed for the two- or three-semester. This worksheet and quiz will guide you to practice the following. and these vectors form a basis for E2. Vectors Worksheet #4 BROSE REVISED: 5/13/2011 Use the formula Work = F d cosθ to find the work done (in joules) by the force F, with given magnitude. This worksheet and quiz let you practice the following skills: Reading comprehension - ensure that you draw the most important information from the related dot product of …. You will be asked questions regarding each resultant's magnitude and direction. Thus, there will be a significant difference between an urgent master's paper and a high school essay with a two-week deadline. Vector components and vector addition worksheet 30 28 find the components of the vectors. Determine whether A is on the same side of P C as C, on the opposite side, or on P. ) The pilot of a plane points his airplane due South and flies with an airspeed of 120 m/s. A number line is included to help students determine if an event is impossible, unlikely, equally likely, likely, or certain. → a a → – → b b → can be said as the addition of the vectors → a a → and (- → b b → ). OA = a, OB = b a) Find the vector AB in terms of a and b. Click Here for Edexcel A-Level Maths Large Data SetClick Here for Edexcel A-Level Maths Formula Booklet Edexcel A-Level Maths Worksheets Below are Solomon Worksheets from Churchill Maths. Every point on the shape moves the same distance in the same direction. Target Language or Knowledge: Displacement is distance in a given direction Another word for magnitude is size. Graphical Addition of Perpendicular Vectors Worksheet. All worksheets created in PTC Mathcad Prime 2. The sum of two vectors is the vector obtained by lining up the tail of one vector to the head of the other: (6 problems) Vector subtraction (20 problems) The vector from $\bfx$ to $\bfy$ is given by $\bfy - \bfx$. Those physical quantities which require magnitude as well as direction for their complete representation and follows vector laws are called vectors. Vectors in Two Dimensions and Relativity Worksheet 1. Graphical Vector Addition A + B Step 1 – Draw a start point Step 2 – Decide on a scale Step 3 – Draw Vector A to scale Step 4 – Vector B’s tail begin at Vector A’s head. It's really easy to get the equation of a plane if you know two things, first a point that lies in the plane and second a vector that's normal to the plane. We can multiply a vector by a scalar to change its length or give it the opposite direction. In addition, the activity teaches the student to read the direction and magnitude of vectors as well as add vectors. A line of given length and pointing along a given direction, such as an arrow, is the typical representation of a vector. Grade 12 Calculus And Vectors Textbook. Vectors and Modular Arithmetic. Find the direction of p 3 2 i+ p 2 j. The scalar product and the vector product are the two ways of multiplying vectors which see the most application in physics and astronomy. We can add two vectors by joining them head-to-tail: And it doesn't matter which order we add them, we get the same result:. Includes vector proofs (parallel lines and points forming a straight line). k63g, cmmv, tas1, lgw, yomc, clex, yqa4, qup6, tgmk, u49, xnu, 7nj, r6h, frls, zja, b6j, jr6, mnyh, o4rs, 8gvt, wwc, w47q, dmp, w2je, 6bsh, uwoc, q60d, v6mm, 0cp2, 78w, 1vz1, zm7d, lph5, 0cx, qj7t, 5zv, k3l, p3lb, jrls, hdcg, 81iz, aoh, fq85, rpe, 836, tbv, hf28, vh5, lhe, tug, bh8d, ulay, wl4v, nlow, 4od, 74ru, tt1, z3m, 1xd8, zae, i9p, 7zd, w3ud
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2022-06-28 00:05:51
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https://www.physicsforums.com/threads/friction-force-bullet-hitting-a-sandbag.639679/
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Friction Force - Bullet hitting a sandbag
Kristenx2
Homework Statement
A rifle bullet with a mass of 15.5 g traveling toward the right at 262 m/s strikes a large bag of sand and penetrates it to a depth of 24.2 cm. Determine the magnitude and direction of the friction force (assumed constant) that acts on the bullet.
Homework Equations
F⃗ net=ΣF⃗ =ma⃗, fkkN, $\Sigma$F=n+f+mg=ma
The Attempt at a Solution
I have no idea where to start. I tried drawing a force diagram, but I can't figure out how to use my givens all in one equation.
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2022-10-03 04:30:11
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https://ask.sagemath.org/answers/15885/revisions/
|
Ask Your Question
# Revision history [back]
You can do this by using the html function directly:
## Background
%html just works on text -- it doesn't perform any sage (or python) parsing. It does this by applying sage's html function to whatever you type. So your code above is equivalent to
html("""
<h2> Résolvons une équation </h2>
<font size="+3">
<div align="center">
$x+ b=k$
</div>
</font>
<h3> Exemple : résoudre : $x+$ <sage>nombre</sage>$=12$ </h3>
""")
[Note that I had to change é to é in the last line for it to parse correctly -- hopefully this will not be necessary in a future update of sage.]
## Solution
So the key to doing what you want is using sage to create the string with the variable value inserted, and then pass the result to the html function. The format operator is a good way to do this:
beginning = """
<h2> Résolvons une équation </h2>
<font size="+3">
<div align="center">
$x+ b=k$
</div>
</font>
"""
exemple = "<h3> Exemple : résoudre : $x+ {0} =12$ </h3>".format(nombre)
html(beginning+exemple) # addition for strings is concatenation
## Followup
You can see what's happening by just printing the strings:
print(exemple)
should show you something like
<h3> Exemple : résoudre : $x + 7 =12$ </h3>
because the format operator sticks the value of nombre into the the string.
You can do this by using the html function directly:
## Background
%html just works on text -- it doesn't perform any sage (or python) parsing. It does this by applying sage's html function to whatever you type. So your code above is equivalent to
html("""
<h2> Résolvons une équation </h2>
<font size="+3">
<div align="center">
$x+ b=k$
</div>
</font>
<h3> Exemple : résoudre : $x+$ <sage>nombre</sage>$=12$ </h3>
""")
[Note that I had to change é to é in the last line for it to parse correctly -- hopefully this will not be necessary in a future update of sage.]
## Solution
So the key to doing what you want is using sage to create the string with the variable value inserted, and then pass the result to the html function. The format operator is a good way to do this:
beginning = """
<h2> Résolvons une équation </h2>
<font size="+3">
<div align="center">
$x+ b=k$
</div>
</font>
"""
exemple = "<h3> Exemple : résoudre : $x+ {0} =12$ </h3>".format(nombre)
html(beginning+exemple) # addition for strings is concatenation
## Followup
You can see what's happening by just printing the strings:
print(exemple)
should show you something like
<h3> Exemple : résoudre : $x + 7 =12$ </h3>
because the format operator sticks the value of nombre into the the string.
## Edit
To fix things like x + -1, define the right hand side as a sage expression and let sage format it:
var('x')
nombre = nombres[3]
expr = x + nombre
The string for expr should give you x - 1. If your expressions get more complicated, you can also use latex(expr) to get a latex formatted string (for things like exponents, parentheses, etc).
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2019-11-20 10:30:33
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https://astronomy.stackexchange.com/questions/1518/plutos-orbit-overlaps-neptunes-does-this-mean-pluto-will-hit-neptune-sometime/1521
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# Pluto's orbit overlaps Neptune's, does this mean Pluto will hit Neptune sometime?
We know that the orbits of Pluto and Neptune overlap. This means that pluto sometimes crosses the orbit of Neptune; will Pluto hit Neptune in any circumstance?
• The accepted answer is incorrect. NJ's answer is correct. – user931 Jan 29 at 12:24
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2019-10-19 01:09:28
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https://dml.cz/handle/10338.dmlcz/118616
|
# Article
Full entry | PDF (0.1 MB)
Keywords:
$\Cal B^{(1)}$-groups; Butler groups of finite rank
Summary:
A necessary and sufficient condition is given for the direct sum of two $\Cal B^{(1)}$-groups to be (quasi-isomorphic to) a $\Cal B^{(1)}$-group. A $\Cal B^{(1)}$-group is a torsionfree Abelian group that can be realized as the quotient of a finite direct sum of rank 1 groups modulo a pure subgroup of rank 1.
References:
[FM] Fuchs L., Metelli C.: On a class of Butler groups. Manuscripta Math 71 (1991), 1-28. MR 1094735 | Zbl 0765.20026
[B] Höfling B.: On direct summands of Butler $\Cal B^{(1)}$-groups. to appear in Comm. in Algebra. MR 1215549
[F II] Fuchs L.: Infinite Abelian Groups. Vol. II, Academic Press, London-New York, 1973. MR 0349869 | Zbl 0338.20063
Albrecht U.F., Goeters H.P., Megibben C.: Zero-one matrices with an application to Abelian groups. to appear in Rend. Sem. Mat. Univ. Padova. MR 1257128 | Zbl 0809.20046
Goeters H.P., Megibben C.: Quasi-isomorphism and $\Bbb Z (2)$ representations for a class of Butler groups. preprint. MR 1876211
Goeters H.P., Ullery W.: Butler groups and lattices of types. Comment. Math. Univ. Carolinae 31 (1990), 613-619. MR 1091358 | Zbl 0717.20039
Goeters H.P., Ullery W.: Quasi-summands of a certain class of Butler groups. to appear in Proceedings of the 1991 Curacao Conference. MR 1217267 | Zbl 0806.20043
Partner of
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2020-09-23 03:12:37
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https://zbmath.org/?q=an:0415.35062
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# zbMATH — the first resource for mathematics
An inverse spectral result for elliptical regions of $$\mathbb{R}^2$$. (English) Zbl 0415.35062
##### MSC:
35P05 General topics in linear spectral theory for PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 47A10 Spectrum, resolvent
Full Text:
##### References:
[1] Andersson, K; Melrose, R, The propagation of singularities along gliding rays, Invent. math., 41, 23-95, (1977) · Zbl 0358.35048 [2] Birkhoff, G.D, Dynamical systems, () · Zbl 0171.05402 [3] Boscovich, R.J, Sectionum conicarum elementa, (1757), Venice [4] Duistermaat, J.J; Guillemin, V, The spectrum of positive elliptic operators and periodic geodesics, Invent. math., 29, 39-79, (1975) · Zbl 0307.35071 [5] Guillemin, V, Some spectral results on rank one symmetric spaces, Advances in math., 28, 129-137, (1978) · Zbl 0441.58012 [6] Guillemin, V; Melrose, R, The Poisson summation formula for manifolds with boundary, Advances in math., 32, (1979), PN610 · Zbl 0421.35082 [7] Jacobi, K.G, Vorlesungen über dynamik, (1884), G. Reimer Berlin [8] Kac, M, Can one hear the shape of a drum, Amer. math. soc. monthly, 73, No. 4, 1-23, (1966), Part II · Zbl 0139.05603
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
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2021-10-28 21:30:48
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http://www.ams.org/mathscinet-getitem?mr=2847985
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MathSciNet bibliographic data MR2847985 35J60 (35B44 35J91 35J93 35R01 53C21) Leung, Man Chun Supported blow-up and prescribed scalar curvature on \$S\sp n\$$S\sp n$. Mem. Amer. Math. Soc. 213 (2011), no. 1002, vi+99 pp. ISBN: 978-0-8218-5337-5 Article
For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.
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2016-12-05 11:35:42
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https://ncatlab.org/nlab/show/Nikolai+Durov
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# nLab Nikolai Durov
Nikolai Durov (Николай Валерьевич Дуров) is a Russian mathematician from St. Petersburg with main current interests in arithmetic geometry, currently employed at St. Petersburg Department of the Steklov Institute of Mathematics.
Durov obtained his Ph.D. in 2007 in Bonn under Gerd Faltings:
Durov’s mathematical work preceding his study in Bonn includes his work on classical Galois theory of polynomial equations; it provides essentially the third historically available method to compute algorithmically a Galois group of a given equation. His method is however statistical and some random data are included in input. The algorithm terminates with probability $1$ for all equations iff the Riemann hypothesis is true. The exposition of these results is in
• N. V. Durov, Computation of the Galois group of a polynomial with rational coefficients. I. (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 319 (2004), Vopr.Teor. Predst. Algebr. i Grupp. 11, 117–198, 301; English translation in J. Math. Sci. (N. Y.) 134 (2006), no. 6, 2511–2548 (MR2006b:12006)
• N. V. Durov, Computation of the Galois group of a polynomial with rational coefficients. II. (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 321 (2005), Vopr. Teor. Predst. Algebr. i Grupp. 12, 90–135, 298; English translation in J. Math. Sci. (N. Y.) 136 (2006), no. 3, 3880–3907 (MR2006e:12004)
Nikolai Durov is also an experienced computer programmer. He was a member of a St Petersburg State University student team winning a student world tournament in programming. His high school education was in Italy. His younger brother Pavel V. Durov is a professional programmer and main constructor behind one of the most popular internet sites in Russia.
Nikolai Durov is currently working on his habilitation thesis. His earlier publications also include
• N. Durov, S. Meljanac, A. Samsarov, Z. Škoda, A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra, Journal of Algebra 309, n. 1, 318–359 (2007) (doi:jalgebra) (math.RT/0604096).
where in chapters 7–9 Durov presented a flexible theory of a class of functors which can be viewed as representing generalizations of formal schemes but over an arbitrary ring, and with weaker assumptions. This theory is then applied to a problem in Lie theory and deformation theory; an interesting chapter on symplectic Weyl algebras is included. In chapter 10 an alternative method using Hopf algebras rather than geometry is presented.
Recently he introduced the notion of a vectoid and the related notion of an algebrad which is a generalization of the notions of a symmetric and a non-symmetric operad:
• Nikolai Durov, Classifying vectoids and generalisations of operads, arxiv/1105.3114, the translation of “Классифицирующие вектоиды и классы операд”, Trudy MIAN, vol. 273
• Classifying vectoids and generalizations of operads, talk at The International Conference “Contemporary Mathematics” June 12, 2009, video: link
Other sources:
• Computation of derived absolute tensor square of the ring of integers, talk at 2nd annual conference-meeting MIAN–POMI “Algebra and Algebraic Geometry”, St. Petersburg, December 25, 2008, link
• Arithmetic intersection theory and homotopical algebra, seminar 2007
• N. V. Durov, Топологические реализации алгебраических многообразий (Topological realizations of algebraic varieties), preprint POMI 13/2012 (in Russian) abstract, pdf.gz
• N. V. Durov, МУЛЬТИПЛИКАТИВНЫЕ МОНОИДЫ ${\mathbb{F}}_p$-АЛГЕБР И АБСОЛЮТНЫЕ ТЕНЗОРНЫЕ ПРОИЗВЕДЕНИЯ КОНЕЧНЫХ ПОЛЕЙ, (Multiplicative monoids of ${\mathbb{F}}_p$-algebras and absolute tensor products of finite fields), preprint POMI 12/2012 (in Russian) abstract pdf.gz
Revised on September 20, 2016 09:03:29 by Zoran Škoda (161.53.130.104)
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2017-11-23 07:25:33
|
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http://tex.stackexchange.com/questions/86572/copying-latex-from-a-pdf
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# Copying LaTeX from a PDF
Is there a way to copy from a PDF to a LaTeX document? Whenever I do it, it copies in with weird symbols in the place of LaTeX commands for example:
$$f:\mathbb{R}\rightarrow \mathbb{R}$$
copies in as
f : R ! R;
I did a quick search and I couldn't really find anything (some sort of PDF to LaTeX converter that worked well)
Am I missing something obvious or can this not be done (for some reason)?
-
You are probably out of luck: see tex.stackexchange.com/questions/8503/…. Your question will probably be closed as a duplicate. – Ethan Bolker Dec 11 '12 at 18:32
Are you interested in getting useful symbols or do you really want the complete LaTeX code back? – Stephan Lehmke Dec 11 '12 at 18:40
For instance, if I add \input{glyphtounicode} \pdfgentounicode=1 to the preamble of the document, I get f : R → R. – Stephan Lehmke Dec 11 '12 at 18:46
@EthanBolker I just found out how to make it paste back as TeX code ;-) – Stephan Lehmke Dec 11 '12 at 19:40
@hmmmm could you disambiguate whether you want to copy LaTeX from any PDF document or want to be able to make a PDF from which LaTeX can be copied? – Stephan Lehmke Dec 11 '12 at 21:41
Try this:
\documentclass{article}
\usepackage{amssymb}
\input{glyphtounicode}
\pdfgentounicode=1
\usepackage{accsupp}
\newcommand\pasteablelatex[1]
{%
\edef\next
{%
\noexpand\BeginAccSupp{method=escape,ActualText=\detokenize{#1}}%
}%
\next#1\EndAccSupp{}%
}
\begin{document}
foo
\pasteablelatex{$$f:\mathbb{R}\rightarrow \mathbb{R}$$}
bar
\end{document}
-
To be clear, can you confirm that this only works if the person who compiled the PDF used this method? I guess that without that there's no way to do this, isn't there? – Loop Space Dec 11 '12 at 19:57
@AndrewStacey Not sure what you mean. It's a method to make PDF paste back LaTeX code. – Stephan Lehmke Dec 11 '12 at 19:59
I mean that if you give me a PDF without the source code then I can't use your method to recover that source code. But if I as an author want to be kind to readers (assuming that there are some) then I can use this to make my PDF so that copying gives the LaTeX code. – Loop Space Dec 11 '12 at 20:05
Exactly. It has to be used when compiling the LaTeX to PDF. – Stephan Lehmke Dec 11 '12 at 20:05
Is there a way to make all the document pasteable? – Manuel Dec 11 '12 at 20:23
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2014-11-27 19:38:18
|
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https://stats.stackexchange.com/questions/29114/cubic-clustering-criterion-in-r?noredirect=1
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# Cubic clustering criterion in R [closed]
Does anybody know if any package calculates the cubic clustering criterion (CCC) index in R to aid the selection of optimal number of clusters?
## closed as off-topic by gung♦, Scortchi♦, Peter Flom♦Jan 26 '14 at 10:13
• This question does not appear to be about statistics within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.
• Not exactly what you are looking for but the command cluster.stats from the R package fpc might be helpful. – user10525 May 24 '12 at 23:43
• @Procrastinator Please, make this an answer; this sounds like a good advice, according to the author of the fpc package on R-help. I also gave some pointers on related threads. – chl May 26 '12 at 10:29
• @chl Thanks for your comment. I have to admit that I do not know the area enough to make a defensible answer. I based my comment on the link you posted, the description on the manual and a quick search. – user10525 May 26 '12 at 10:33
• This question appears to be off-topic because it is about finding an R package. – gung Jan 25 '14 at 21:09
It is included in NbClust and to be used via the option index = "ccc", e. g.
I_ccc <- NbClust(X, distance="euclidean", min.nc=2, max.nc=8, method = "complete",
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2019-09-23 18:24:34
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{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3902767300605774, "perplexity": 1449.3322429293373}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514577478.95/warc/CC-MAIN-20190923172009-20190923194009-00204.warc.gz"}
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https://cseducators.stackexchange.com/tags/best-practice/hot
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Tag Info
175
You should teach both, and you probably want to use the binary unit. When you are talking about the difference, it may be helpful to tell them about how to tell the difference when reading them: The SI kilo- is k: $1\ \text{kB (kilobyte)} = 10^{3}\ \text{bytes} = 1000\ \text{bytes}$ While the binary kibi- is Ki: $1\ \text{KiB (kibibyte)} = 2^{10}\ \text{... 168 Lying is good. But advertise it when you lie. Make sure students make a note of it that you are lying. Pedantic is bad. If you try to explain everything you will wind up explaining nothing. Let me take a simpler example. In java we have a special incantation public static void main(String [] args) We put it in every program. It distinguishes a program. ... 138 Actually returning early should be the norm. Return as soon as you can. There are at least two reasons for this, of which the first, efficiency, is the least important. But if you return early then needless statements won't be executed. Nor will you need to devise some special code to get to the end just so that you can return. However, the biggest and ... 95 Actually, there are a lot of things that benefit a student in CS, such as a degree in Mathematics or Sociology. Likewise interpersonal skills that help a person work in groups. Others are too numerous to mention. However, a true prerequisite is a block, preventing advancement if you don't have the skill, so no, it should not be required. Some data points.... 87 I would start sentences with "Generally speaking..." and I would appreciate if my professors did the same. It hints that there is more to know but doesn't waste time explaining anything further. If your students are curious enough they can raise their hand to inquire. If it takes too much time to explain during class and someone wants to know, you can offer ... 57 This is a vital question, perhaps the vital question, for a CS educator to deal with, because technologies will keep pushing us. There is no end to this particular merry-go-round. I have been stuck, non-stop, in exactly this situation, for the last four years in a high school for gifted students, and I can attest that it is physically, mentally, and ... 57 You should teach them it's messed up beyond repair, and it's their generation's job to teach the next generation to use the silly-sounding standard prefixes, so that when they finally retire (and the current old-timers are more permanently removed from the argument), there can finally be a consensus. As the matters currently stand, all the prefixes are ... 51 Actually, you need to teach them both so that they are warned that the usage is not consistent. Then you can choose one as a standard in your course going forward. Which you choose depends a bit on what you are teaching. If it is how to evaluate hard drives, etc. then$K = 1000$works now. For most programming, however,$K = 2^{10} = 1024\$ is probably best....
48
Teach your students about the return early philosophy. Teach your students about the single return philosophy. Tell them this debate is not settled. Explain to them that different software development teams have different opinions and different standards, and that it's important for them to have both these tools in their toolbox for their jobs. Explain the ...
44
Learning about references is important, but I don't feel that learning about pointers is that important for beginning Java students. Certainly intermediate students will need to understand them. When I started learning about pointers, I had a hard time grasping them at all until I learned assembly language. Once I learned assembly (for any processor), ...
41
Please don't... ...give the students who are ahead more of the same kind of work to do. Please. That's just boring. If they get it, they get it. ...make groups by mixing the students who are ahead with the students who are behind - often, the students who are ahead won't teach the behind students, but will just do the whole thing themselves. I say this from ...
38
This is really a separate approach from my first answer, which has received some push-back. It's worth noting that many of these loners are simply students who are substantially ahead of the curve. One way to really want to engage such students in pair programming is to pair them with each other. This will create something of a Dream Team. Give them the ...
38
To answer the titular question: In my experience, the advantage of touch typing is not the direct gain of time through typing faster. That’s negligible since most programming and writing tasks involve much more thinking, researching, etc. In my experience, the advantage is due to touch typing working directly through your muscle memory¹ and not requiring ...
34
As a current CS student, my lecturer use to solve this by simply using For now. To come back to your interface example: For now, interfaces cannot contain any code, unlike abstract classes. This gives a clear message, there is more to the topic, but it isn't relevant at the moment. When students ask you can always elaborate a little bit, but for the most ...
29
The best way to deal with this kind of student is to head it off at the pass. If you can get the student at the beginning, you can often prevent the problem from festering in the first place. I have a student coming in next year who I have already been warned will have this problem, and I plan to show this to my class on the first day: I will then say ...
29
For a beginning course: no. I have helped clarify behavior for fellow students who got lost by an instructor who explained things in terms of pointers. I have programmed in C, and most of my current programming is done in Rust; I understand pointers and what problems they are best suited to solve. But in Java, you don't have any access to pointers, so ...
28
Is it a wrong approach, giving your students the choice? Or is it wrong to force them never to return early? This is a false dichotomy. Sometimes multiple returns are clearer, and sometimes (rarely) a single return is clearer. You should encourage ("force" is a rather strong word) your students to use the technique which makes the most sense for any given ...
27
If the goal is to prepare students for "the real world," aka "real jobs" then: Touch typing is required to be an effective programmer. Full stop. The answers here seem to highlight the difference between theory (academia) and professional work. Under no circumstance would I hire or tolerate a programmer that couldn't code without looking at the keys.* It ...
26
Fair warning, I do not demand any particular naming convention (such as NetBeans) from my students. This leaves me with variable naming only for the purpose of clarity. I speak constantly to my students about the two different audiences for code: the computer itself (which is the one kids naturally think about), and other is human beings, which students ...
24
I am currently a student in a computer science bootcamp learning js, C# and asp.net mvc5. For what its worth, I appreciate the honesty when my instructor uses crossed fingers.(He will pause to cross his right and middle finger and prominently display them in front of him before/while continuing with a "fuzzy truth") I will treat knowledge given to me in ...
23
Neither is better. What is true for one student may not be for another. Studies can show trends. Students are not trends; they are individuals. As an example I encountered today that seems to go against that wisdom, I caught a news piece about a local girl in the Scripps National Spelling Bee. As she spelled the word she was moving her fingers in a weird ...
22
If I catch it quickly and can easily explain the error, I use it as an example of failing up. "Ooops, look at me, here's my mistake, here's how I can learn from it." If students catch it, and I'm not immediately sure who is right, I make a note to do some research and come back to it in the next class. If I don't realize it until class is over, I make sure ...
21
Scratch is a visual block-based drag-and-drop programming language designed specifically for learners, especially children. It's created by the Lifelong Kindergarten Group at the MIT Media Lab. The language and IDE are pretty much completely connected. Here's how I see it checking off your bullet points: Object oriented: It has sprites, but it's debatable ...
21
I would consider teaching in Python if you wanted to give your students a taste of programming in a text-based language—pretty much the only type of language used professionally. A visual programming language like Scratch is probably better for younger groups, and teaches the underlying programming concepts well, but you will reach a point where you must ...
21
I have 3 tiers of labs. First are the required labs. They're worth 100 points each and every one must do these. If they don't do one, it goes in the gradebook as a zero. These are also the labs that I think are the best of each topic for practicing what they need to work on. My calendar is based on how much time I expect 90 plus percent of the students need ...
21
The difference between providing your students with a proper discussion of this topic, and simply teaching them one or the other, is the difference between being a real educator and being a reciter of factoids. If there is no single correct definition of KB for you, then why would you instill something different in your students? The answer to your question ...
20
I don't believe that your question is entirely valid; some languages require jumping. The first principle, therefore, is to follow the norms of your language. However, I suspect that you are asking about languages that discourage (but do not ban) jumping, such as Java, or C++. In these cases, I agree with Peter that the solution is to give them the ...
18
There are at least two parts to teaching naming. The first is to have a good standard that the students know and understand. This can be provided in a checklist. But the more important aspect is to always demonstrate good naming in all examples that you use and in all quizzes and exams that you give them; even for very simple exercises. The Naming Standard ...
17
I recently had a group of students that didn't take notes at all. They were in University classes and no one, apparently, had taught them how to learn. I asked one student why, and he just pointed to his head - it's here. But of course photographic memory isn't true for all but a vanishingly small proportion of our students. In order to learn something you ...
17
unlike abstract classes, interfaces generally do not contain any code This is not a lie. In a large lecture hall, if they ask for "when do they contain code", say "that will be covered in another class" to avoid going off on a (possibly confusing) tangent. This is neither lying, but it is glossing over details. There are situations where lying is a good ...
Only top voted, non community-wiki answers of a minimum length are eligible
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2020-01-21 10:07:32
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2941363751888275, "perplexity": 888.8424117842933}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250601628.36/warc/CC-MAIN-20200121074002-20200121103002-00555.warc.gz"}
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http://mathhelpforum.com/advanced-statistics/56211-p-value-degrees-freedom.html
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# Thread: P-value and Degrees of Freedom
1. ## P-value and Degrees of Freedom
Hello. For the question below I can get
2. It's a well-documented fact that the world over, 55% of the population prefers
Coke and 45% prefer Pepsi. Rumour had it that in Quebec, less than 55% of the
population prefers Coke, so an enterprising anthropologist went there to investigate.
He found, in a simple random sample, that 505 out of 1001 people preferred Coke.
Using these data, test the hypothesis that the proportion of people that prefer Coke
in Quebec is less than 0.55.
Null p=0.55
Alt p< 0.55
From the equation I got -2.89
and then my prof got the p-value.
p-value = 0.0019
I don't understand how he got the p-value.
Thank you!
2. Originally Posted by lou.loubunny
Hello. For the question below I can get
2. It's a well-documented fact that the world over, 55% of the population prefers
Coke and 45% prefer Pepsi. Rumour had it that in Quebec, less than 55% of the
population prefers Coke, so an enterprising anthropologist went there to investigate.
He found, in a simple random sample, that 505 out of 1001 people preferred Coke.
Using these data, test the hypothesis that the proportion of people that prefer Coke
in Quebec is less than 0.55.
Null p=0.55
Alt p< 0.55
From the equation I got -2.89
and then my prof got the p-value.
p-value = 0.0019
I don't understand how he got the p-value.
Thank you!
He looked a z-score of -2.89 up in a table of the cumulative standard normal distribution (or rather he looke up a z-score of 2.89 and subtracted the p-value of that from 1 (or 0.5 depending on the form of the table))
Here the normal distribution is appropriate as we are dealing with large sample statistics.
CB
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2018-02-20 16:20:29
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https://sureshemre.wordpress.com/2021/01/01/coupling/
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## Coupling
This is the follow-up to my previous post titled “Orthogonality“.
Symbolism: Circle is a geometric symbol of $\mathbb{C}$ (Confinement). Straight line is a geometric symbol of $\mathbb{L}$ (Liberation) .
For the definitions of $\mathbb{C}$ and $\mathbb{L}$ please see the “Orthogonality” article and the references therein.
Initially, the straight line emerges from the center of the circle in a direction perpendicular to the plane of the circle but develops a non-90-degree angle as a result of $\mathbb{C}$ / $\mathbb{L}$ coupling.
In addition to the non-90-degree angle mentioned above, the secondary effect of $\mathbb{C}$ / $\mathbb{L}$ coupling is a distortion (deviation). The simplest form of deviation from the perfect circle is an ellipse.
The simplest form of deviation from the straight line is a wavy line.
Once deviations manifest oscillations start. Oscillation is a type of rotation. The ellipse starts rotating . The rotation of the ellipse can be seen as a shape oscillation. We can imagine the ellipse to be rotating or we can imagine the shape of the ellipse oscillating. These are equivalent descriptions.
As the ellipse rotates (shape oscillates) the non-90-degree angle between the plane of the ellipse and the wavy line will oscillate as well and the wavy line itself will precess around the original perpendicular axis. Many combinations are possible. In fact, there are infinite number of combinations of these oscillation/rotation modes.
Among the infinite number of oscillation/rotation modes two of them deserve special attention. The simplest form of deviation from a circle is an ellipse but the second most common deformation would be a triangular mode.
The triangular mode exhibits both $\mathbb{C}$ and $\mathbb{L}$ characteristics because in the triangle shape with round corners there are relatively straight line segments which are symbolic expressions of $\mathbb{L}$. The triangle itself is deformed from a circle which is the symbolic expression of $\mathbb{C}$.
Another possibility for the deformation of the line is a helix.
The helix mode also exhibits both $\mathbb{L}$ and $\mathbb{C}$ characteristics because the circular motion of the helix is a symbolic expression of $\mathbb{C}$ while the overall progression of the helix symbolizes $\mathbb{L}$.
Resistance to deformation
By its very nature $\mathbb{L}$ will resist the deformations mentioned above. $\mathbb{L}$ will try to restore the ellipse to the circle, the wavy line to the straight line.
This is similar to the the de-coupling effect of $\mathbb{L}$ mentioned in the “Orthogonality” article. $\mathbb{C}$ keeps breaking orthogonality while $\mathbb{L}$ keeps restoring orthogonality.
$\mathbb{L}$ will try to restore any deformation to the original state. This may not necessarily be a snap back to the original state. The $\mathbb{L}$ reaction may involve a counter formation (inverse of the deviation caused by $\mathbb{C}$). In other words, the $\mathbb{L}$ reaction may overshoot causing alternating dominance of the the two factors.
In many posts I mentioned the “primordial fabric”. When $\mathbb{C}$ acts on the primordial fabric and forms a distortion (space-time-matter) $\mathbb{L}$ counters the distortion by forming an inverse distortion. I like the term “dual distortion” also. Stated in different words, we can say that when the perfect symmetry of the primordial fabric is broken, the space-time-matter is formed but at the same time the inverse/dual of the space-time-matter must be formed. This is a huge subject. I kept mentioning this over the years but made little progress in terms of details.
The “inverse/dual deformation” must be proportional to the “deformation”. I suspect there is a fundamental invariance. The {$\mathbb{C} , \mathbb{L}$} interplay must involve a conservation law. I made few attempts to formulate this fundamental invariance in the past. I hope to tie those older ideas with my recent formulations involving {$\mathbb{C} , \mathbb{L}$}, cognitive cores, and agency in a future post.
Geometric symbolism is useful
$\mathbb{C}$ and $\mathbb{L}$ are abstract. We see $\mathbb{C}$ and $\mathbb{L}$ expressions in different areas such as physics, chemistry, biology, networks, and mind. The {$\mathbb{C} , \mathbb{L}$} interplay manifests at many scales. In each area of study and scale we can find different mathematical representations of $\mathbb{C}$ and $\mathbb{L}$. In principle, we should not restrict ourselves to geometric representation. On the other hand, I find the CLA (circle, line, angle) symbolism very useful.
Measures of coupling
Frequency: We should think of the “oscillation/rotation” of the CLA symbolism abstractly as a measure of coupling strength. The stronger the $\mathbb{C}$ / $\mathbb{L}$ coupling the higher the degree of activity. In the physical stage of manifestation the “degree of activity” translates to “frequency”. When coupled, $\mathbb{C}$ and $\mathbb{L}$ maintain their identities but start turning into each other as a consequence of coupling. This means oscillation/rotation. This also means “activity” or “action” in the abstract sense and “frequency” in the physical sense.
Phase: In physics, phase is the complementary aspect of frequency. Phase is about relative changes of two quantities. Phase is a measure of relative timing between the cycles of two variables. Phase is measured in angle units but it is really about time difference. Both concepts (frequency and phase) require time for a meaningful definition. If I claim that {$\mathbb{C}$,$\mathbb{L}$} is abstract and general then I have to have a notion of “abstract time”. Right? I thought about this for a long time. Then I realized that I should work with the concept of “metric” instead of “abstract time”.
Metric: The concept of “metric” refers to a distance measure. We are familiar with the concepts of distance-in-space and distance-in-time. We can think of “phase” as distance in time. The “spacetime metric” concept of the Relativity theory of Einstein is a combination of “distance-in-space” and “distance-in-time”. The concept of “distance” can be abstract. Examples are statistical distance, similarity distance, etc. Since “distance” can be abstract “metric” can also be abstract. The discussion of “metric” in the context of graph theory is very interesting. A mathematical graph is a topological entity. Topology is about connectivity. When we introduce a metric to the graph we obtain a geometrical entity. There are lessons to be learned from graph theory to come up with an abstract metric for the {$\mathbb{C} , \mathbb{L}$} interplay. More homework for me!
Types of coupling
Alternating: $\mathbb{C}$ and $\mathbb{L}$ might be coupled in such a way that the resultant expression exhibits $\mathbb{C}$ characteristics and $\mathbb{L}$ characteristics in cycles. In this type of coupling $\mathbb{C}$ and $\mathbb{L}$ dominate each other in an alternating fashion.
Steady Mix: The expression (manifestation) may exhibit a steady mix of $\mathbb{C}$ and $\mathbb{L}$ characteristics. In other words the mixing ratio is not changing or alternating but staying relatively constant.
Hyperposed: $\mathbb{C}$ and $\mathbb{L}$ might be coupled in such a way that resembles the superposition described in Quantum Mechanics. This is the inexpressive coupling of the circle and the line perspectives. The expression (manifestation) has not taken form yet. In Quantum Mechanics a measurement (or any interaction) expresses the superposition yielding a particular state of the system. Similarly, a hyperposition will transition to expressive coupling of the circle and the line perspectives as in “steady mix” and “alternating” types.
There must be more types of coupling. In the “Confinement and Liberation” article I listed some of the physical expressions of $\mathbb{C}$ but that’s not an exhaustive list. Clearly, in the course of the evolution of the Cosmos there will be an infinite number of variations in the $\mathbb{C}$ expression. That implies more types or categories of coupling. Think of the increasing agency of human beings. We introduce new organizations, new structures, new formations all the time. This means new types of coupling are manifesting as the Cosmos evolves.
2 or 3 fundamental factors?
In the realm of the mind we have to have at least two factors to explain things. One factor cannot be explanatory. The minimum number of explanatory factors is 2. Hence the {$\mathbb{C}$,$\mathbb{L}$} hypothesis.
We may parameterize, however, the “coupling” (symbolized by “angle”) as the third explanatory factor.
{ $\mathbb{L}$, $\mathbb{C}$ } $\rightarrow$ { $\mathbb{S}$, $\mathbb{R}$, $\mathbb{T}$ }
We could identify
$\mathbb{S}$ : Sattvaguna (modified $\mathbb{L}$)
$\mathbb{R}$ : Rajoguna (symbolized by “angle” representing “coupling”)
$\mathbb{T}$ : Tamoguna (modified $\mathbb{C}$)
In this model the coupling picture is more complicated. Triangulation requires pair-wise coupling among the 3 factors. This is good and bad. The 3-factor model with pair-wise couplings may provide a richer model but interpreting the details of such a rich model my present interpretational difficulties.
Follow-up articles
Duality Rotation
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2023-03-28 11:22:28
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https://socratic.org/questions/how-many-moles-of-s-are-in-125-g-of-so-2
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# How many moles of S are in 125 g of SO_2?
Approx. $2$ $m o l$.
$S {O}_{2}$ has a molecular weight of $64.07$ $g \cdot m o {l}^{-} 1$.
You specified $125 \cdot g$ of stuff:
So $\frac{125 \cdot \cancel{g}}{64.07 \cdot \cancel{g} \cdot m o {l}^{-} 1}$ $=$ ?? mol???
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2021-07-26 22:11:29
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https://www.encyclopediaofmath.org/index.php/Topological_tensor_product
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# Topological tensor product
of two locally convex spaces $E_1$ and $E_2$
A locally convex space having a universality property with respect to bilinear operators on $E_1 \times E_2$ and satisfying a continuity condition. More precisely, let $\mathcal{K}$ be a certain class of locally convex spaces and for each $F \in \mathcal{K}$ let there be given a subset $T(F)$ of the set of separately-continuous bilinear operators from $E_1 \times E_2$ into $F$. Then the topological tensor product of $E_1$ and $E_2$ (with respect to $T(F)$) is the (unique) locally convex space $E_1 \tilde\otimes E_2 \in \mathcal{K}$ together with the operator $B \in T(E_1 \tilde\otimes E_2)$ having the following property: For any $S \in T(F)$, $F \in \mathcal{K}$, there exists a unique continuous linear operator $R:E_1 \tilde\otimes E_2 \to F$ such that $R \circ B = S$. Thus, if one speaks of the functor $T: \mathcal{K} \to \mathrm{Sets}$, then $E_1 \tilde\otimes E_2$ is defined as the representing object of this functor.
In all known examples $\mathcal{K}$ contains the field of complex numbers $\mathbb{C}$, and $T(\mathbb{C})$ contains all bilinear functionals of the form $f \circ g$, $f \in E_1^\ast$, $g \in E_2^\ast$, mapping $(x,y)$ to $f(x)g(y)$. If in this case the topological tensor product exists, then there is a dense subspace in $E_1 \tilde\otimes E_2$ that can be identified with the algebraic tensor product $E_1 {\otimes} E_2$; moreover, $B(x,y)=x {\otimes} y$.
If $T(\mathcal{K})$ consists of all separately (respectively, jointly) continuous bilinear operators, then the topological tensor product is called inductive (respectively, projective). The most important is the projective topological tensor product. Let $\{p_i\}$ be a defining family of semi-norms in $E_i$, $i=1,2$; denote by $\pi$ the topology on $E_1 {\otimes} E_2$ defined by the family of semi-norms $\{p_1 \hat \otimes p_2\}$:
$$p_1 \hat \otimes p_2(u) = \inf \left\{ \sum_{k=1}^n p_1(x_k) p_2(y_k) : \sum_{k=1}^n x_k{\otimes} y_k = u \right\}.$$
If $\mathcal{K}$ is the class of all, respectively all complete, locally convex spaces, then the projective topological tensor product of $E_1$ and $E_2$ exists and its locally convex space is $E_1 {\otimes} E_2$ with the topology $\pi$, respectively its completion. If the $E_i$ are Banach spaces with norms $p_i$, $i=1,2$, then $p_1 \hat \otimes p_2$ is a norm on $E_1 {\otimes} E_2$; the completion with respect to it is denoted by $E_1 \hat \otimes E_2$. For each $\epsilon>0$ the elements of $E_1 \hat \otimes E_2$ have the representation
$$u= \sum_{k=1}^\infty x_k{\otimes} y_k$$
where
$$\sum_{k=1}^\infty p_1(x_k) p_2(y_k) \leq p_1 \hat \otimes p_2 (u) + \epsilon.$$
If one endows $E_1 {\otimes} E_2$ with a topology weaker than $\pi$ by using the family of semi-norms $p_1 \tilde\otimes p_2$,
$$p_1 \tilde \otimes p_2(u) = \sup_{f,g \in V \times W} \left\lvert (f {\otimes} g) (u) \right\rvert,$$
where $V$ and $W$ are the polar sets of the unit spheres with respect to $p_1$ and $p_2$, then there arises a topological tensor product, sometimes called injective. The locally convex spaces $E_1$ with the property that for an arbitrary $E_2$ both topologies in $E_1 {\otimes} E_2$ coincide, form the important class of nuclear spaces (cf. Nuclear space).
The projective topological tensor product is associated with the approximation property: A locally convex space $E_1$ has the approximation property if for each pre-compact set $K \subset E_1$ and neighbourhood of zero $U$ there exists a continuous operator of finite rank $\phi:E_1 \to E_1$ such that for all $x \in K$ one has $x-\phi(x) \in U$. All nuclear spaces have the approximation property. A Banach space $E_1$ has the approximation property if and only if for an arbitrary Banach space $E_2$ the operator $\tau: E_1 \hat\otimes E_2 \to (E_1^\ast{\otimes}E_2^\ast)^\ast$, unambiguously defined by the equation $\tau(x {\otimes} y) (f {\otimes} g)=f(x)g(y)$, has trivial kernel. A separable Banach space without the approximation property has been constructed [3]. This space also gives an example of a Banach space without a Schauder basis, since the Banach spaces with a Schauder basis have the approximation property (thus S. Banach's so-called "Banach basis problem" has been negatively solved).
#### References
[1] A. Grothendieck, "Produits tensoriels topologiques et espaces nucléaires" , Amer. Math. Soc. (1955) [2] H.H. Schaefer, "Topological vector spaces" , Macmillan (1966) [3] P. Enflo, "A counterexample to the approximation problem in Banach spaces" Acta Math. , 130 (1973) pp. 309–317
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2019-04-22 16:25:25
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http://math.iisc.ac.in/seminars/2022/2022-04-08-somnath-jha.html
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#### Number Theory Seminar
##### Venue: Microsoft Teams (Online)
The ($p^{\infty}$) fine Selmer group (also called the $0$-Selmer group) of an elliptic curve is a subgroup of the usual $p^{\infty}$ Selmer group of an elliptic curve and is related to the first and the second Iwasawa cohomology groups. Coates-Sujatha observed that the structure of the fine Selmer group over the cyclotomic $\mathbb{Z}_p$-extension of a number field $K$ is intricately related to Iwasawa’s $\mu$-invariant vanishing conjecture on the growth of $p$-part of the ideal class group of $K$ in the cyclotomic tower. In this talk, we will discuss the structure and properties of the fine Selmer group over certain $p$-adic Lie extensions of global fields. This talk is based on joint work with Sohan Ghosh and Sudhanshu Shekhar.
Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ; E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 03 Feb 2023
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2023-02-03 14:41:25
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https://www.deepdyve.com/lp/ou_press/preference-for-the-workplace-investment-in-human-capital-and-gender-lLn7eVRf0c
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# Preference for the Workplace, Investment in Human Capital, and Gender
Preference for the Workplace, Investment in Human Capital, and Gender Abstract We use a hypothetical choice methodology to estimate preferences for workplace attributes from a sample of high-ability undergraduates attending a highly selective university. We estimate that women on average have a higher willingness to pay (WTP) for jobs with greater work flexibility and job stability, and men have a higher WTP for jobs with higher earnings growth. These job preferences relate to college major choices and to actual job choices reported in a follow-up survey four years after graduation. The gender differences in preferences explain at least a quarter of the early career gender wage gap. JEL Codes: J24, J16. I. Introduction The persistence of gender gaps in labor market earnings and the failure of standard variables to fully explain the gaps has prompted the search for alternative models and evidence. One explanation for gender wage gaps is that these arise in part by women “purchasing” certain positive job attributes by accepting lower wages, and men accepting higher earnings to compensate for negative job attributes. These preferences for job attributes may then affect human capital investments, even prior to job market entry. However, empirically isolating the role of worker-side preferences for job attributes is difficult because the equilibrium matching of jobs to workers reflects not only the workers’ preferences but the firms’ preferences as well. Various kinds of labor market frictions, which prevent workers from matching with their most preferred job types, also break the direct connection between observed job choices and worker preferences. Even when the labor market is perfectly competitive, jobs likely vary in many unobserved (to the researcher) characteristics, leading to an omitted variable (selection bias) problem in identifying worker preferences from realized job choices. To address these empirical challenges, this article estimates individual preferences for workplace attributes using a survey of undergraduates from a selective university, New York University (NYU). We collect data on job attribute preferences by presenting undergraduate students with a series of hypothetical job choice scenarios and eliciting their job choices. The hypothetical job scenarios were constructed to offer students a realistic menu of potential jobs varying in expected earnings and other characteristics such as future earnings growth, dismissal probability, and work hours flexibility. The students’ stated preferences for these jobs allows us to construct a “pure” measure of individual preferences—at the time of the survey—for various job characteristics, and estimate, in a simple and robust way, the distribution of their preferences for job attributes. Our data isolate the preference for workplace attributes, free from making explicit assumptions about the equilibrium job allocation mechanism or preferences of employers. We then use the preference data to examine two channels through which preferences could affect the gender wage gap. First, job preferences could affect college major choice, as students perceive that graduating with certain degrees would result in different types of jobs being offered to them. Second, even for students who choose the same major, preferences for job types could cause men and women to accept different types of jobs and result in different earnings. Our data on job preferences, combined with data on perceptions about the job characteristics given major choices, allow us to quantify these two channels and document how workplace preferences affect the gender gap. Our hypothetical choice methodology is a kind of “stated choice” analysis, similar to “conjoint analysis” and “contingent valuation” methods, used in fields including marketing, environmental and natural resource economics, and health.1 Because our data collection in essence conducts a kind of “experiment” at the individual student level, the “panel” data generated by our design allows us to estimate the distribution of preferences, allowing for unrestricted forms of preference heterogeneity. In contrast to our approach, previous work addressing compensating differentials using observed job choices requires generally stronger assumptions about preferences and the firm side of the labor market.2 In our sample of recent high-ability undergraduate students from NYU, we find substantial willingness to pay (WTP) for pecuniary and nonpecuniary aspects of jobs and considerable heterogeneity in their preferences for workplace attributes. We find that students have preferences reflecting a distaste for higher job dismissal potential, and a taste for workplace hours flexibility (the possibility of working part-time, rather than full-time, hours). We estimate that on average students are willing to give up 2.8% of annual earnings for a job with a percentage point lower probability of job dismissal and willing to give up 5.1% of their salary to have a job that offers the option of working part-time hours rather than one that does not offer this option. After dividing our sample by gender, we find that women have a much higher average preference for workplace hours flexibility, with an implied WTP of 7.3% compared to 1.1% for men. Women also have a higher average WTP for more secure jobs: they are willing to give up 4% of their salary for a percentage point lower probability of job dismissal (versus a 0.6% WTP for males). On the other hand, men have a higher average WTP for jobs with higher earnings growth: they are willing to give up 3.4% of annual earnings for a job with a percentage point higher earnings growth (the corresponding estimate for women is a statistically insignificant 0.6%). A natural question is whether preferences recovered from data on hypothetical choices relate to actual occupational outcomes. Using data on reported job characteristics for a subset of our respondents who are employed roughly four years after our original data collection, we find a strong and systematic relationship between estimated preferences and later actual workplace characteristics. Students with strong preferences for flexible hours, distaste for hours, and other nonpecuniary aspects of jobs were later found to be more likely to be working at jobs with those same preferred characteristics. Although these realized job characteristics do not solely reflect preferences (given the issues we raised above), our finding of a correlation between pre–labor market job preferences and later actual job characteristics suggests some added credibility of our research design. Our finding of large differences in WTP for job amenities between men and women is consistent with prior work noting that women are more likely to be found in jobs offering greater workplace f lexibility (Goldin and Katz 2011; Flabbi and Moro 2012; Goldin 2014; Wasserman 2015; Bronson 2015). However, the observation that women tend to work in certain job types may not reveal women’s preferences alone, but may be affected by firm-side demands for specific workers and discrimination or be driven by some other job attributes that are unobserved in our data sets (Blau and Kahn forthcoming).3 Our innovation is to quantify the WTP for job attributes using a flexible and robust methodology. In related recent work, Mas and Pallais (forthcoming) conduct a field experiment where call center job applicants are offered various work time schedules and wages. Their finding that women have a higher valuation for worker-friendly alternative work arrangements (and a stronger distaste for employer discretion over their hours) is consistent with our estimates of a higher female valuation of work hours flexibility (availability of part-time work).4 We next test whether the job preferences young adults hold in college in fact affect their human capital investments during college. To quantify the importance of job attributes to major choice, we collect additional survey data on students’ beliefs about the characteristics of jobs they would be offered if they were to complete different majors. These data are then used to estimate a model of major choice, where students receive utility from major-specific characteristics (such as perceived ability in those majors) and from the job attributes they associate with these majors. We find that job attributes have a sizable impact on major choice. For example, increasing the perceived job firing probability by a standard deviation reduces the probability of pursuing a major, on average, by 5% (4%) for women (men). To put this change in perspective, a standard deviation increase in average earnings leads to a 5% (16%) increase, on average, in the likelihood of majoring in that field for women (men). Thus, for women, this change is equivalent to the effect on major choice of increasing earnings by one standard deviation. We find meaningful effects for other job attributes, such as work hours. In general, we find that women’s major choices are more responsive to changes in nonpecuniary job attributes (relative to changes in earnings) than are men’s. By linking job preferences directly to human capital investments, we contribute to our limited understanding of how career and workplace preferences shape educational choices. Prior research on college major choice examines the role of earnings expectations, ability perceptions, college costs, and tastes, but generally does not examine nonpecuniary job attributes.5 An exception is Zafar (2013), which estimates a model of college major choice that incorporates some nonpecuniary workplace attributes. However, the framework does not allow for unobserved heterogeneity in preferences, and incorporates a smaller set of workplace characteristics. Closely related to our work is Arcidiacono et al. (2015), who study a sample of male undergraduate students and collects expectations about earnings in different major-occupation pairs. They find evidence for complementarities in preferences between different majors and occupations, and conclude that nonmonetary considerations are key determinants of occupational choice (conditional on graduating from a given college major). Our contribution is to directly quantify the role of specific nonmonetary factors in major choice. Finally, we turn to a key question in the social sciences and ask what our results imply for the gender wage gap. Systematic gender differences in workplace preferences may affect the gender wage gap through two channels: first, it may cause men and women to choose different fields of study, and second, men and women may choose systematically different jobs within the same field. We find that the main channel for preferences to affect the gender gap operates through the second channel, with a smaller effect through major choice. Our analysis reveals that the gender gap in expected earnings early in the career (age 30) would be reduced by at least a quarter if women did not differ from men in the workplace preferences we consider. Remarkably, we find a similar impact on the gender gap in actual earnings for the subset of respondents for whom we have follow-up data. Our evidence supports the notion that at least part of the early career gender wage gap is the result of women “purchasing” certain positive job attributes by accepting lower wages, and men accepting higher earnings to compensate for negative job attributes. In understanding our results, it is important to note that we measure preferences at a particular point in the life cycle of our sample, when our sample was in college. The preferences we measure are not necessarily intrinsic; these preferences were formed by a variety of influences before and during college, and could change substantially after graduation. In addition, it is likely that workplace flexibility issues are much larger determinants of the gender earnings gap for college graduates 10 or more years into their career than for the young college graduates in our study (who are in their mid-20s during our follow-up), as college-graduate women now have children at later ages.6 The article is organized as follows. In the next section, we briefly provide some context for our analysis by using nationally representative surveys for the United States on currently employed individuals. Section III describes our data collection; Section IV details the model of job choice and shows how hypothetical data can solve important identification issues with realized choice data. Section V provides the empirical estimates of job preferences. Section VI quantifies the importance of job attributes for college major choice. Section VII investigates the extent to which gender-specific job preferences can explain the gender gap in earnings. Finally, Section VIII concludes. All appendixes are available online. II. Background: Gender Differences in Job Choices and Human Capital Investments in the United States To set the stage for the analysis of our hypothetical choice scenario data, we first briefly describe the distribution of college majors, jobs, and associated job characteristics. To do so, we use two large-sample, representative data sets for the United States, the January 2010–December 2012 monthly Current Population Survey (CPS) and the 2013 American Community Survey (ACS). Table I shows the job attributes across sectors. For this purpose, we use the sample of 25–60-year-old labor market participants with at least a bachelor’s degree in the 2010–2012 CPS. The first two columns of Table I show that the gender distribution across work sectors differs (Online Appendix A provides details on how variables in this table were constructed). While nearly half of college-educated women workers are in health or education, less than 20% of college-educated male workers are employed in these sectors. These sectors differ substantially in their labor market returns: column (3) of Table I shows that average annual earnings of full-time workers are the lowest for education and health. These sectors differ along other dimensions as well: more than a quarter of the workers in health and education are employed part-time, possibly suggesting the compatibility of these sectors to work-hours flexibility. Job instability, as measured by the likelihood of being fired, is lowest in the government and education sectors. Of course, jobs in these sectors will also differ in the skills that they demand of their workers. So what explains the propensity of men and women to work in different sectors—is it differences in preferences for workplace attributes, differences in tastes for occupations/industries, or differences in skills? What is the role of the labor market structure, firm labor demand, and discrimination by employers? The observed distribution of jobs by gender we see in the data are equilibrium outcomes, and we cannot ascertain from these data alone the extent to which these outcomes are due to worker demand or due to the supply of certain jobs—for example, part-time work may either be a voluntary or involuntary decision. TABLE I Job Attributes by Broad Sector for College Graduates % of males working ina % of females working in Annual earnings for full-time Hrs/wk for full-time Prop. of part-time workers Yearly firing rateb Prop. male workersc Annual % raise in earningsd (1) (2) (3) (4) (5) (6) (7) (8) Sectors Science 9.6 4.0 82,739 44.2 15.5 3.5 67.7 3.8 (35,989) (7.2) (3.0) (1.6) (1.1) (19.9) Health 8.6 22.5 65,427 43.6 28.6 4.0 20.6 4.2 (35,246) (7.8) (1.1) (0.7) (0.5) (22.9) Business 14.2 11.5 77,079 45.0 19.9 4.0 44.6 4.9 (39,023) (7.8) (1.8) (1.3) (0.8) (21.4) Government 6.8 5.8 67,603 43.3 16.2 1.4 52.9 5.8 (32,322) (6.9) (5.0) (0.9) (0.6) (22.4) Education 11.2 25.5 60,588 44.0 30.0 1.8 30.9 4.2 (29,159) (7.5) (2.9) (1.2) (0.4) (21.1) Manufacturing 22.4 7.99 77,354 45.4 17.6 6.2 78.0 5.7 & agriculture (37,257) (7.93) (1.6) (0.8) (0.4) (22.4) Services & trade 27.2 22.8 65,734 45.3 34.4 6.6 51.7 4.9 (37,883) (8.3) (1.0) (0.9) (0.3) (22.2) p-valuee .000 .000 .000 .000 .000 .000 .000 0.170 % of males working ina % of females working in Annual earnings for full-time Hrs/wk for full-time Prop. of part-time workers Yearly firing rateb Prop. male workersc Annual % raise in earningsd (1) (2) (3) (4) (5) (6) (7) (8) Sectors Science 9.6 4.0 82,739 44.2 15.5 3.5 67.7 3.8 (35,989) (7.2) (3.0) (1.6) (1.1) (19.9) Health 8.6 22.5 65,427 43.6 28.6 4.0 20.6 4.2 (35,246) (7.8) (1.1) (0.7) (0.5) (22.9) Business 14.2 11.5 77,079 45.0 19.9 4.0 44.6 4.9 (39,023) (7.8) (1.8) (1.3) (0.8) (21.4) Government 6.8 5.8 67,603 43.3 16.2 1.4 52.9 5.8 (32,322) (6.9) (5.0) (0.9) (0.6) (22.4) Education 11.2 25.5 60,588 44.0 30.0 1.8 30.9 4.2 (29,159) (7.5) (2.9) (1.2) (0.4) (21.1) Manufacturing 22.4 7.99 77,354 45.4 17.6 6.2 78.0 5.7 & agriculture (37,257) (7.93) (1.6) (0.8) (0.4) (22.4) Services & trade 27.2 22.8 65,734 45.3 34.4 6.6 51.7 4.9 (37,883) (8.3) (1.0) (0.9) (0.3) (22.2) p-valuee .000 .000 .000 .000 .000 .000 .000 0.170 Notes. Table reports means, with standard deviations in parentheses. Statistics are based on the 2010–2012 CPS monthly data. Sample restricted to those with at least a bachelor’s degree, between ages 25 and 60. See Online Appendix A for details on construction of variables and definition of the broad sectors. Variables in columns (3), (4), and (8) are based on full-time workers, and are based on individual-level data. Columns (5)–(7) show the average statistics by sector, with the sector-level standard deviation across the months in parentheses. aProportion of all male workers who are employed in each sector (column sums to 100). bDerived from the monthly firing rate, which is the ratio of workers who are laid off in a given month and have been unemployed for less than one month divided by all employed workers at the beginning of the previous month. cMales as a proportion of all workers in that sector. dConstructed by using the outgoing rotation groups, from the reported earnings in the respondent’s fourth and eighth interview (which are separated by 12 months). eF-test of equality of means/proportions across the industry categories. View Large We next turn to Table II to document the link between field of study and associated job characteristics.7 The table is based on the 2013 ACS, restricting the sample to 25–40-year-olds with at least a bachelor’s degree. The first two columns show that while nearly 55% of women have a bachelor’s degree in humanities, less than 40% of men do. While nearly a quarter of men have a bachelor’s in engineering, the corresponding proportion for women is only 6%. TABLE II Job Attributes by College Major for Young College Graduates, 25 to 40 Years Old Shares Annual Hrs/wk % UE Ann % Malesa Females earnings ($${\}$$) for full-time for full-time Part-time workersb ratec salary raised (1) (2) (3) (4) (5) (6) (7) Bachelor’s (or more) in: Business 24.6 18.8 77,002 45.3 26.8 3.3 4.4 (68,110) (8.1) Engineering 23.3 6.1 86,679 44.8 22.2 2.5 4.8 (60,494) (8.1) Humanities 38.2 55.8 59,328 44.4 37.8 3.5 4.9 (49,697) (7.9) Natural science 13.9 19.3 75,992 44.9 35.1 2.5 5.9 (65,921) (9.6) F-teste .000 .000 .000 .000 .000 .000 .000 Shares Annual Hrs/wk % UE Ann % Malesa Females earnings ($${\}$$) for full-time for full-time Part-time workersb ratec salary raised (1) (2) (3) (4) (5) (6) (7) Bachelor’s (or more) in: Business 24.6 18.8 77,002 45.3 26.8 3.3 4.4 (68,110) (8.1) Engineering 23.3 6.1 86,679 44.8 22.2 2.5 4.8 (60,494) (8.1) Humanities 38.2 55.8 59,328 44.4 37.8 3.5 4.9 (49,697) (7.9) Natural science 13.9 19.3 75,992 44.9 35.1 2.5 5.9 (65,921) (9.6) F-teste .000 .000 .000 .000 .000 .000 .000 Notes. Table shows statistics from the 2013 American Community Survey (ACS), restricting the sample to 25–40-year-olds with at least a bachelor’s degree. Sample size is 204,190 respondents. 173 majors are grouped into four broad categories. Means (std. dev.) shown for annual earnings and hrs/week for full-time workers. aProportion of all 25–40-year-old college-educated males with the specified broad major (column sums to 100). bProportion of part-time workers from pool of those currently employed. cUnemployment rate is number of individuals not employed and currently looking for a job, divided by sum of unemployed and employed respondents. dCalculated by linearly regressing log earnings for a given major group on age (coefficient on age reported). ep-value of F-test of equality of means across majors (rows). View Large Column (3) of Table II shows that these majors differ significantly in their average earnings. Engineering—the field which women are least likely to be present in—has the highest average earnings, while humanities—the most popular bachelor’s field for women—has the lowest average earnings. These majors also differ along other dimensions. Columns (4) and (5) show that work-hours flexibility is the highest for jobs associated with humanities: 38% of all humanities graduates are part-time workers, versus 22% of engineering bachelor’s graduates. Average hours per week for full-time workers are also the lowest in humanities. The last two columns of the table show that job stability and earnings growth also vary significantly across the fields of study. So how much do these gender differences in human capital and job characteristics explain the gender gap in earnings? In a recent analysis, Blau and Kahn (forthcoming) find that the gender wage gap is currently larger at the top of the wage distribution (90th percentile), and has decreased more slowly at the top than at other points in the distribution. In addition, they find that traditional human capital variables (experience and degrees earned) explain little of the recent gender gap. They attribute part of the gender gap in high-skilled occupations to a possible compensating differential. Using the sample of college graduates aged 25–40 from the 2013 ACS (the subsample from Table II that reports nonzero labor income), we find an adjusted gender gap in hourly earnings of about 12 log points (adjusting only for age and full-time status).8 Demonstrating how important college majors could potentially be in explaining the gender gap among college-educated workers, including four broad college major categories (as defined in our analysis) reduces the gender gap by about 43% (from 12 log points to 6.7 log points). Including indicators for detailed occupation, industry, and race categories as in Blau and Kahn (forthcoming), in addition to indicators for major categories, increases the explained portion of the gender wage gap to 58%. However, this analysis reveals that even conditional on detailed occupation/industry and major controls, a large part of the gender wage gap remains unexplained. The remainder of this article investigates the extent to which workplace preferences can explain this gender gap, either by influencing human capital choice (major choice) or by influencing job choices conditional on major. III. Data This section describes the administration of the data collection, the form of the hypothetical choice scenarios, and the sample we use for the estimation. III.A. Administration Our data are from an original survey instrument administered to NYU undergraduate students over a two-week period during May 2012. NYU is a large, selective, private university located in New York City. The students were recruited from the email list used by the Center for Experimental Social Sciences (CESS) at NYU. Students were informed that the study consisted of some simple economic experiments and a survey about educational and career choices. Upon agreeing to participate, students could sign up for a 90-minute session, which was held in the CESS Computer Lab located on the main NYU campus.9 The data for this article were collected through a computer-based survey (constructed using the SurveyMonkey software). The survey took approximately 30 minutes to complete and consisted of several parts. Many of the questions had built-in logical checks (e.g., percent chances of an exhaustive set of events such as majors had to sum to 100). Students were compensated $${\}$$10 as a show-up fee, and $${\}$$20 for successfully completing the survey. III.B. Data Collection Instrument In addition to questions about demographics, family background, and educational experiences, the main survey instrument consisted of two parts. The first part collected data on students’ preferences for job attributes using hypothetical job choices, while the second collected data on consequential life activities that would plausibly be key determinants of college major choice, such as attributes of jobs associated with each major and measures of the student’s perception of their ability to complete the coursework for each major. We describe the hypothetical job choice data in detail next and leave the description of major-specific data to a later part of the article, where we relate the job attribute preferences to college major choices. Our hypothetical job choice data were collected by presenting students with a total of 16 job scenarios. Each scenario consisted of three different potential jobs. We exogenously varied different aspects of the job with the intention of creating realistic variation in job attributes. The first eight hypothetical job scenarios were introduced as follows: In each of the 8 scenarios below, you will be shown hypothetical jobs offers. Each job offer is characterized by: Annual earnings when working full-time Annual percentage increase in earnings from age 30 onwards until retirement Full-time work hours per week Work flexibility (whether part-time work is an option); part-time work is work where you only work at most half as many hours as full-time work and for half of the full-time salary These jobs are otherwise identical in all other aspects. Look forward to when you are 30 years old. You have been offered each of these jobs, and now have to decide which one to choose. In each scenario, you will be asked for the percent chance (or chances out of 100) of choosing each of the alternatives. The chance of each alternative should be a number between 0 and 100 and the chances given to the three alternatives should add up to 100. Each scenario consisted of three jobs, with each job being characterized by four attributes. The notable point that was highlighted was that these jobs were identical in all other aspects. The jobs did not have any occupation labels on them.10 The last eight scenarios were introduced in a similar way, except that the job offer was now characterized by a different set of attributes: annual earnings when working full-time, probability of being fired over a one-year period, amount of additional annual bonus pay based on relative performance the respondent may qualify for (in addition to base pay), proportion of men in the firm in similar job positions. All survey respondents received identical scenarios in the same order. Following the approach of Blass, Lach, and Manski (2010), we asked respondents to provide a choice probability instead of a discrete choice (that is, a 0 or 1). This allows respondents to express uncertainty about their future behavior. It also allows them to rank their choices, providing more information than if we asked only about the most preferred job. As is standard in studies that collect subjective probabilistic data, a short introduction on the use of percentages was provided. In addition, respondents answered some practice questions to become familiar with expressing probabilistic answers. Besides earnings, the scenarios focus on six different job attributes. We chose not to vary these six dimensions all at once since the cognitive load to process such information could have been overwhelming. We focus on these dimensions based on findings from prior literature, and the fact that there is considerable variation along these dimensions across occupations as well as majors (Tables I and II). Earnings and earnings growth were included since they have been found to be a factor in career/education choice (see Wiswall and Zafar 2015a, and references therein). Work hours and work flexibility are included because they tend to be associated with the remuneration structure in jobs and the associated gender gap in earnings (Flabbi and Moro 2012; Goldin 2014; Cortes and Pan 2016). We recognize that workplace flexibility is a multidimensional concept: for example, the number of hours to be worked matters but perhaps so do the particular hours (Goldin 2014; Mas and Pallais forthcoming). We varied two hours-related attributes: number of hours and the availability of a part-time option, since these are easy to vary in a meaningful fashion. Job stability, as proxied by the likelihood of being fired from the job, is included because of the importance of risk and uncertainty to job choices (Dillon forthcoming) and gender differences in risk preferences (Croson and Gneezy 2009). Finally, relative performance compensation and proportion of men are meant to capture the competitiveness of the job environment, preferences for which have been found to differ by gender (Niederle and Vesterlund 2007; Flory, Leibbrandt and List 2015; Reuben, Wiswall, and Zafar forthcoming).11 To keep the scenarios realistic, the job attributes shown to respondents in the scenarios were based on the actual marginal distribution of job characteristics in the CPS (except for the bonus pay variable, since data were not available for that dimension).12 In addition, no scenario included a job that was clearly dominant or dominated along all dimensions. We also made a conscious effort to keep the variation in job attributes within each scenario relatively “local,” so that the claim that the jobs were otherwise identical was credible; for example, two jobs offering $${\}$$50,000 and $${\}$$90,000, respectively, with little variation along the specified dimensions are unlikely to be identical. At the same time, we had substantial variation in the job attributes across the scenarios. This ensures that we are not recovering preferences in a local region only. Online Appendix Table A1 shows the range of the attributes across the scenarios. III.C. Sample Description A total of 257 students participated in the study. We drop 10 respondents for whom we have missing data for the relevant section of the survey. Sample characteristics are shown in Table III. Thirty-five percent of the sample (86 respondents) is male, 29% is white, and 51% is Asian. The mean age of the respondents is 21.5, with 11% of respondents freshmen, 11% sophomores, 37% juniors, and the remaining seniors or higher. The average grade point average of our sample is 3.5 (on a 4.0 scale), and students have an average Scholastic Aptitude Test (SAT) math score of 696, and a verbal score of 674 (with a maximum score of 800). These correspond to the 93rd percentile of the U.S. national population score distributions. Therefore, as expected, our sample represents a high ability group of college students. Parents’ characteristics of the students also suggest that they are overrepresented among high socioeconomic groups. The last panel of the table shows that 48% of the students have a major in the humanities and social sciences category, 31% have a major in business and economics, while the remaining have a major in natural sciences and math (16%), and engineering (5%). TABLE III Sample Statistics All Males Females p-value (1) (2) (3) (4) Number of respondents 247 86 161 School year: Freshmen 10.9% 9.3% 11.8% .549 Sophomore 10.9% 11.6% 10.6% .798 Junior 36.4% 32.6% 38.5% .355 Senior or more 41.7% 46.5% 39.1% .262 Age 21.49 21.69 21.37 .103 (1.5) (1.8) (1.2) Race: White 29.2% 33.7% 26.7% .248 Asian 50.6% 51.1% 50.3% .898 Non-Asian minority 17.8% 14.0% 19.9% .247 Parent’s characteristics: Parents’ income ($${\}$$1,000s) 137 141 135 .731 (121) (126) (118) Mother B.A. or more 67.6% 74.4% 64.0% .095 Father B.A. or more 69.6% 72.1% 68.3% .539 Ability measures: SAT math score 696.0 717.7 684.3 .006 (88) (72) (94) SAT verbal score 674.0 677.0 672.5 .704 (84) (78) (88) GPA 3.5 3.5 3.5 .938 (0.32) (0.33) (0.32) Intended/current major Economics/business 31.2% 48.8% 21.7% .000 Engineering 4.9% 8.1% 3.1% .080 Humanities and soc sciences 47.8% 30.2% 57.1% .000 Natural sciences/math 16.2% 12.8% 18.0% .289 All Males Females p-value (1) (2) (3) (4) Number of respondents 247 86 161 School year: Freshmen 10.9% 9.3% 11.8% .549 Sophomore 10.9% 11.6% 10.6% .798 Junior 36.4% 32.6% 38.5% .355 Senior or more 41.7% 46.5% 39.1% .262 Age 21.49 21.69 21.37 .103 (1.5) (1.8) (1.2) Race: White 29.2% 33.7% 26.7% .248 Asian 50.6% 51.1% 50.3% .898 Non-Asian minority 17.8% 14.0% 19.9% .247 Parent’s characteristics: Parents’ income ($${\}$$1,000s) 137 141 135 .731 (121) (126) (118) Mother B.A. or more 67.6% 74.4% 64.0% .095 Father B.A. or more 69.6% 72.1% 68.3% .539 Ability measures: SAT math score 696.0 717.7 684.3 .006 (88) (72) (94) SAT verbal score 674.0 677.0 672.5 .704 (84) (78) (88) GPA 3.5 3.5 3.5 .938 (0.32) (0.33) (0.32) Intended/current major Economics/business 31.2% 48.8% 21.7% .000 Engineering 4.9% 8.1% 3.1% .080 Humanities and soc sciences 47.8% 30.2% 57.1% .000 Natural sciences/math 16.2% 12.8% 18.0% .289 Notes. For the continuous outcomes, means are reported in the first cell, and standard deviations are reported in parentheses. p-value reported for a pairwise test of equality of means (proportions) between males and females, based on a t-test (chi-square test). View Large Columns (2) and (3) of Table III report the characteristics by gender. The last column of the table reports the p-value of tests of equality of the statistics by gender. We see that male and female respondents are similar in all dimensions, except two. First, male students in our sample have a significantly higher average SAT math score than females, of about 33 points. Second, the two sexes choose very different college majors. Nearly half (49%) of men report majoring in business/economics, with 30% majoring in humanities and social sciences, and 13% in natural sciences/math. On the other hand, 57% of the women report majoring in humanities and social sciences, followed by about 22% majoring in business/economics, and 18% majoring in natural sciences/math. That is, female students are almost twice as likely as men to major in the humanities (the field, as we show below, perceived to have the lowest average earnings among college graduates), and only half as likely as males to major in economics/business (the perceived highest-earnings major category). The gender-specific major distributions are statistically different (p-value ≤ .001, using a chi-square test for equality of distributions). These substantial gender gaps in major choice mirror the national patterns from the ACS data (Table II). Compared to the NYU population, our sample has a similar proportion female: 63% of students graduating NYU in 2010 are women compared with 65% in our sample (data from the Integrated Post-Secondary Education Data System, IPEDS). For all incoming freshman in 2010, the 25th and 75th quartiles of the SAT math were 630 and 740 and for the SAT verbal were 610 and 710 (IPEDS). The equivalent quartiles in our sample are 650 and 770 for math and 620 and 730 for verbal. Our sample is weighted more toward business/economics majors than in the actual NYU population graduating in 2010, possibly because the experimental laboratory is located in the building housing the Economics Department. However, the gender differences in major choice are similar.13 IV. Model and Identification Analysis In this section, we present a simple attribute-based job choice model and discuss identification of the model using two types of data: (i) standard realized job choices (as observed after job offers and acceptances are made), and (ii) stated probabilistic job choices (as observed in our job hypotheticals experimental data). We show that under weak conditions the job hypotheticals data identify the distribution of job preferences, while standard realized job choice data do not. IV.A. A Canonical Random Utility Model of Job Choice Jobs are indexed by j, and there is a finite set of jobs j = 1, …, J. Each job is characterized by a vector of K attributes Xj = [Xj1, …, XjK]. These job attributes include earnings and various nonpecuniary attributes, such as job dismissal probabilities and work-hours flexibility. Thus, we explicitly allow for the possibility that individuals are not necessarily pure income or consumption maximizers, and may value many other outcomes associated with their job choice. Let Uij ∈ R be individual i’s utility from job j. The utility from job j is $$U_{ij}= u_{i}(X_{j}) + \epsilon _{ij}.$$ (1)ui(X) ∈ R is the preferences of individual i over the vector of characteristics X. εij ∈ R is the additional job-specific preference component for job j reflecting all remaining attributes of the job which affect utility, if any. Let εi be the vector of these components for individual i, εi = εi1, …, εiJ. After observing the attributes X1, …, XJ for all jobs and εi, individual i chooses the one job with the highest utility: i chooses job j if $$U_{ij}>U_{ij^{\prime }}$$ for all j΄ ≠ j. Population preferences for jobs is the collection of ui preferences over the job attributes X and the job-specific components εi. The joint distribution of preferences in the population is given by F(ui, εi). This distribution determines the fraction of individuals choosing each job, qj ∈ [0, 1]: \begin{eqnarray} q_{j} &=& pr(\mbox{choose job }j)\nonumber \\ &=& \int 1\lbrace U_{ij}>U_{ij^{\prime }}\, \mbox{for all }j^{\prime } \ne j\mbox{}\rbrace dF(u_{i},\epsilon _{i}). \end{eqnarray} (2) IV.B. Identification Using Realized Choice Data Typically empirical research on job choice consists of analyzing data on actual or realized job choices, which provides the one best job chosen by each individual.14 To analyze the potential advantages of hypothetical data, we first detail the identification using realized choice data. A common model of realized choice data assumes εi1, …, εiJ are i.i.d. Type I extreme value, and independent of preferences represented by ui. The probability individual i chooses job j, given some characteristics X1, …, XJ for all jobs, is given by \begin{equation*} q_{ij} = \frac{\exp (u_{i}(X_{j}))}{\sum _{j^{\prime }=1}^{J}\exp (u_{i}(X_{j^{\prime }}))}. \end{equation*} The population fraction choosing job j is then $$q_{j}=\int \frac{\exp (u_{i}(X_{j}))}{\sum _{j^{\prime }=1}^{J}\exp (u_{i}(X_{j^{\prime }}))}dG(u_{i}),$$ (3)where we have kept the dependence of the job choice on the job characteristics X1, …, XJ implicit. G(ui) is the distribution of preferences over attributes ui in the population. Equation (3) is the mixed multinomial logit model of McFadden and Train (2000). They show that the distributional assumption on the εi terms that yield the logit form is without any meaningful loss of generality as this model can arbitrarily closely approximate a broad class of random utility models. For ease of exposition, we consider a linear model of utility given by ui(X) = X΄βi. A key concern in using realized job choices is that the data set of job characteristics which the researcher has at hand is not complete in the sense that there are omitted unobserved job characteristics that are potentially correlated with the included observed characteristics. Divide the vector of job characteristics X into observed X(obsv) and unobserved characteristics X(unob), X = [X(obsv), X(unob)]. Similarly divide the vector of preference parameters βi = [βi(obsv), βi(unob)]. The log odds of job j relative to job j΄ for individual i is then: \begin{eqnarray*} \ln \left(\frac{q_{ij}}{q_{ij^{\prime }}}\right) &=& (X_{j}(obsv)-X_{j^{\prime }}(obsv))\beta _{i}(obsv)+(X_{j}(unob)\nonumber \\ && -\, X_{j^{\prime }}(unob))\beta _{i}(unob)\nonumber \\ &&=(X_{j}(obsv)-X_{j^{\prime }}(obsv))\beta _{i}(obsv) + \,\eta _{ij}, \end{eqnarray*} where qij and $$q_{ij^{\prime }}$$ is the probability of choosing job j and j΄, respectively, for individual i. $$\eta _{ij} = (X_{j}(unob)-X_{j^{\prime }}(unob))\beta _{i}(unob)$$ is the omitted variable for individual i. The omitted variable bias problem is the generic one found in a variety of contexts: the omitted unobserved job characteristics Xj(unob) are correlated with the observed characteristics Xj(obsv). For example, if the researcher’s data set includes only current salaries, but not any of the nonpecuniary benefits of the job, we would expect that the estimate of preferences for salaries will be biased. The theory of compensating differentials (Rosen 1987) predicts a close connection among various job characteristics—a trade-off between salary and nonpecuniary benefits—and therefore would suggest important omitted variable bias in estimates of job preferences using realized data. The omitted variable bias issue could also arise more subtly from the selection/matching mechanism to jobs, reflecting employer preferences over potential job candidates. If the labor market equilibrium is such that employers only offer a limited set of jobs to candidates, then the realized jobs they hold do not reflect their preferences only.15 Discrimination by employers, by which employers prefer not to hire workers of certain groups (e.g., women, minorities), is one example (Becker 1971). In the presence of important demand-side considerations, one would not want to interpret the equilibrium allocation of jobs as reflecting only worker preferences. As we detail below, our hypothetical data avoid this issue because they experimentally manipulate the characteristics offered to individuals, thereby allowing a “pure” measure of preferences, free from considering the equilibrium job allocation mechanism, preferences of employers, or any omitted unobserved job characteristics. Another approach to this issue is to make some assumptions about the structure of the labor market and individual preferences. As in the literature examining identification of these models using observed choices (see Fox et al. 2012 for a recent review), some support condition or restriction on preferences is therefore necessary for identification. IV.C. Model of Hypothetical Job Choices We next consider a framework for analyzing hypothetical job choice data, connecting the canonical model of realized job choice specified above in equation (1) with the hypothetical job choice data we collect. Our hypothetical data are asked prior to a job choice (while students are in school). We observe each individual’s beliefs about the probability they would take each hypothetical future job offered within the scenario (and not simply the individual’s one chosen or realized job). To analyze this type of data, we require a model of hypothetical future jobs. Our model of hypothetical job choices presumes individuals are rational decision makers who anticipate the job choice structure as laid out in the canonical model of job choice, equation (1). To allow for the possibility of uncertainty about future job choices, we assume that the realizations of εi1, …, εiJ job-specific utility terms are not known at the time we elicit individual beliefs. Individual i then faces a choice among J hypothetical jobs with characteristics vectors X1, …, XJ. Each individual i expresses their probability of taking a given job j as: $$p_{ij} = \int 1 \lbrace U_{ij} > U_{ij^{\prime }} \, \mbox{for all }j^{\prime } \ne j\mbox{} \rbrace d H_{i}( \epsilon _{i} ),$$ (4)where Hi(εi) is individual i’s belief about the distribution of εi1, …, εiJ elements. As in Blass, Lach, and Manski (2010), εi has an interpretation of resolvable uncertainty, uncertainty at the time of our data collection but uncertainty that the individual knows will be resolved (i.e., known or realized) prior to making the job choice.16 It should be noted that the preferences for workplace attributes elicited in our data collection are potentially specific to the time at which the survey is collected (during the college years in our case). Preferences for job attributes may change as individuals age and may have been different when the students in our sample were younger (say, prior to college) and may be different still when they actually enter the labor market and make job choices. With this caveat in mind, we can still use our research strategy to understand job preferences at a point in time and study how these preferences relate to important human capital investments that are being made contemporaneously.17 IV.D. Identification Using Hypothetical Choice Data We previously analyzed identification of preferences using realized job-choice data and discussed a key shortcoming: realized choice data potentially suffers from omitted variable bias. Hypothetical choice data can overcome this shortcoming and allow a general method to identify heterogeneity in job-choice preferences. First, because we can experimentally manipulate the hypothetical choice scenarios we provide individuals, we may be able to reduce bias from the correlation of observed and unobserved job characteristics. Rather than use naturally occurring variation in realized job choices—which are in general the result of many unobserved job characteristics and an unknown labor market equilibrium mechanism, as discussed above—we present individuals with an artificial set of job choices. Although the job characteristics we provide are certainly not exhaustive of all possible job characteristics, and are purposely kept limited so as not to “overload” the respondents with too many job features, the key feature of the hypothetical experimental setting is that we instruct respondents that the jobs differ only in the job characteristics we provide, and are otherwise identical. This distinguishes our design from “audit”-based studies in which employers are presented with résumés that are otherwise identical except for the one chosen attribute (say, the gender of applicant). The criticism of audit studies is that even if you make two groups (say, men and women) identical on observables, employers might have very different distributions in mind about unobservables for the two groups, biasing the inference (for an analysis of this issue, see Neumark, Burn, and Button 2015). In our case, students are instructed that the hypothetical jobs are identical in all other ways, instructions that cannot be given to actual employers in audit studies. The extent of the remaining bias in the preferences we elicit then critically depends on whether respondents fully internalize our instructions that the jobs are otherwise identical. There is reason to suspect this may not strictly be the case. Like audit studies, the participants in our study may still have preconceived notions of what other attributes are related to the attributes we include. For example, they might believe the availability of part-time work (one of the attributes we include) is associated with other aspects of flexibility we do not include, such as time of day one is allowed to work and the ability to take vacations and family care leaves. Dismissal risk (also one of the attributes we include) could be viewed as a proxy for high-stress, high-expectations environments. These types of biases are not different from those present in audit studies where employers have their own prior beliefs about other attributes of workers associated with different observable (on résumé) worker characteristics. A second advantage of the hypothetical data is that it provides a kind of panel data on preferences which, under fairly weak assumptions, identify the full preference rankings over job attributes. Notice the key distinction between equations (4) and (2). With job hypotheticals data, we observe for each individual i multiple subjective job probabilities pi1, …, piJ. The job hypotheticals provide a type of panel data allowing less restricted forms of identification, by allowing identification of the ui(X) preferences without a parametric restriction on the population distribution of preferences. Note that even with a panel of realized choices, it is in general impossible to identify separately preferences for jobs from search frictions or omitted job characteristics. Within our hypothetical setup, these issues are, by design, not a confounding factor. Our assumption for identification of preferences is that the εi1, …, εiJ job-specific terms are i.i.d. and independent of the experimentally manipulated job attributes X1, …, XJ. This is implied by the experimental design: respondents are instructed that the jobs vary only in the listed characteristics and are otherwise identical. Under this assumption, the hypothetical data pi1, …, piJ identifies the preference ranking for individual i over all jobs J in the choice set: For any two jobs j and j΄, the characteristics vector Xj is preferred to that of $$X_{j^{\prime }}$$ if the probability of choosing that job is higher than that for job j΄, $$p_{ij}>p_{ij^{\prime }}$$. Our identification concept is that each scenario approximates a multidimensional offer function from which a worker can choose the optimal bundle of job attributes. If this offer function were complete (that is, a continuum of choices rather than three job options in each scenario), the worker would choose the point that is tangent to their indifference curve. Rosen (1987) argues that worker preferences can then be identified if the offer curve shifts, forcing workers to reoptimize in a frictionless labor market, and tracing out the worker’s indifference curve. This is effectively what happens when respondents are presented with another job-choice scenario (another set of jobs to choose from) in our survey. The key distinction relative to the Rosen case is that our choice set is discrete, so we can instead think of preferences as being identified by a set of job preference inequalities. This is an important improvement relative to identification using observed job choices because there is information in our data on rejected job opportunities, which are not typically available in real labor-market settings.18 This rejected-offer information provides both lower and upper bounds on preferences in a discrete-choice setting and can point-identify preferences nonparametrically (up to the distribution of the εi shocks) with full support of the job offer variation. In practice, of course we have a only finite number of job scenarios and cannot vary job offers to saturate the full support of the job characteristics. As in the literature examining identification of these models using observed choices (see Fox et al. 2012 for a recent review), some support condition or restriction on preferences is therefore necessary, although more limited than is required using observational data. We assume preferences take a parametric form, $$u_{i}=X_{i}^{\prime }\beta _{i}$$, but allow the βi parameters to be freely varying in the population. This allows for the distribution of preference parameters βi to be completely unrestricted across individuals; thereby we avoid making assumptions about the population distribution of preferences (such as assuming preferences βi are normally distributed). In the estimation, we use this identification result constructively and simply estimate preferences for each sample respondent one by one. We then use the sample distribution of preferences as the sample estimator of the population distribution of preferences. Therefore, we allow the distribution of preferences to take any form.19 V. Estimates of Preferences for Job Characteristics V.A. Variation in Choice Probabilities Identification relies on variation in probabilities that respondents assign to the various jobs in the hypothetical scenarios. We next present some evidence on this, which should allow the reader to become familiar with the sources of identifying variation. Table IV, Panel A shows two examples from the data sample using the first set of hypothetical scenarios. Recall that each of these eight scenarios included three different job offers, which differed according to the characteristics shown in the table. The last two columns show the mean probability assigned by each gender to the jobs. TABLE IV Example Choice Scenarios Probability assigned by: Panel A Earnings per year at age 30 if working full time Annual percentage increase in earnings from age 30 on Average work hours per week for full-time Work flexibility: part-time work available? Males Females Example 1 Job 1 $${\}$$96,000 3 52 Yes 31.93 [30] 31.46 [30] (22.48) (21.36) Job 2 $${\}$$95,000 2 45 Yes 31.16 [30] 39.34*** [40] (23.71) (22.71) Job 3 $${\}$$89,000 4 42 No 36.91 [30] 29.20** [25] (24.71) (22.57) Example 2 Job 1 $${\}$$76,000 4 50 Yes 19.38 [20] 20.65 [20] (19.34) (15.23) Job 2 $${\}$$81,000 3 44 Yes 49.47 [50] 49.45 [50] (26.63) (22.08) Job 3 $${\}$$88,000 2 49 No 31.15 [25] 29.91 [25] (25.36) (21.98) Probability assigned by: Panel A Earnings per year at age 30 if working full time Annual percentage increase in earnings from age 30 on Average work hours per week for full-time Work flexibility: part-time work available? Males Females Example 1 Job 1 $${\}$$96,000 3 52 Yes 31.93 [30] 31.46 [30] (22.48) (21.36) Job 2 $${\}$$95,000 2 45 Yes 31.16 [30] 39.34*** [40] (23.71) (22.71) Job 3 $${\}$$89,000 4 42 No 36.91 [30] 29.20** [25] (24.71) (22.57) Example 2 Job 1 $${\}$$76,000 4 50 Yes 19.38 [20] 20.65 [20] (19.34) (15.23) Job 2 $${\}$$81,000 3 44 Yes 49.47 [50] 49.45 [50] (26.63) (22.08) Job 3 $${\}$$88,000 2 49 No 31.15 [25] 29.91 [25] (25.36) (21.98) Probability assigned by: Panel B Earnings per year at age 30 if working full time Probability (%) of being fired from the job in the next year Amount of bonus based on relative performance (% of full time earnings) Proportion (%) of men in the firm in similar positions Males Females Example 1 Job 1 $${\}$$87,000 1 $${\}$$4,350 (5) 49 30.34 [30] 36.68* [30] (22.48) (24.33) Job 2 $${\}$$84,000 6 $${\}$$10,920 (13) 67 26.86 [30] 30.27 [30] (23.71) (21.36) Job 3 $${\}$$95,000 5 $${\}$$4,750 (5) 69 42.80 [31.5] 33.05*** [30] (24.71) (20.83) Example 2 Job 1 $${\}$$61,000 1 $${\}$$6,710 (11) 41 25.48 [20] 26.80 [20] (26.57) (23.20) Job 2 $${\}$$65,000 5 $${\}$$7,800 (12) 71 12.14 [9.5] 15.53** [10] (12.98) (11.81) Job 3 $${\}$$67,000 2 $${\}$$10,050 (15) 60 62.38 [60] 57.67 [60] (31.55) (27.19) Probability assigned by: Panel B Earnings per year at age 30 if working full time Probability (%) of being fired from the job in the next year Amount of bonus based on relative performance (% of full time earnings) Proportion (%) of men in the firm in similar positions Males Females Example 1 Job 1 $${\}$$87,000 1 $${\}$$4,350 (5) 49 30.34 [30] 36.68* [30] (22.48) (24.33) Job 2 $${\}$$84,000 6 $${\}$$10,920 (13) 67 26.86 [30] 30.27 [30] (23.71) (21.36) Job 3 $${\}$$95,000 5 $${\}$$4,750 (5) 69 42.80 [31.5] 33.05*** [30] (24.71) (20.83) Example 2 Job 1 $${\}$$61,000 1 $${\}$$6,710 (11) 41 25.48 [20] 26.80 [20] (26.57) (23.20) Job 2 $${\}$$65,000 5 $${\}$$7,800 (12) 71 12.14 [9.5] 15.53** [10] (12.98) (11.81) Job 3 $${\}$$67,000 2 $${\}$$10,050 (15) 60 62.38 [60] 57.67 [60] (31.55) (27.19) Notes. Means [median] (std. dev.) reported in the last two columns. Pairwise t-tests conducted for equality of means by gender. Significance denoted on the female column by asterisks: *p <.10, **p <.05, ***p <.01. View Large Turning to the first example, we see that, for men, Job 3 is the most preferred job in our sample (that is, it received the highest average probability). Job 3 is the job without part-time availability and the highest earnings growth. For women, on the other hand, this job received the lowest average probability. Women assigned the highest probability, on average, to Job 2, the job with a part-time option and an intermediate number of work hours per week and intermediate earnings. In this example, the distribution of choices differs significantly by gender. The gender-specific distributions of average probabilities do not differ in the second example. Table IV, Panel B shows two examples from the second set of hypothetical scenarios, which vary a different set of attributes. In the first example, the distribution of average probabilities again differs by gender. For women, Job 1 receives the highest probability on average (37%). Job 1 is the job with the lowest probability of being fired and the lowest proportion of men as colleagues. Male respondents, on the other hand, assign the highest average probability to Job 3, the job with the highest earnings and proportion of men but with a high likelihood of being fired. Another notable aspect of Table IV is the large standard deviation in elicited choice probabilities, reflective of substantial heterogeneity in choices, even within gender. Figure I shows the histogram of elicited percent chance responses for Job 1, pooled across the 16 hypothetical scenarios. Several things are notable. First, responses tend to be multiples of 10 or 5, a common feature of probabilistic belief data (Manski 2004), reflecting a likely rounding bias; this is something we return to below. Second, although there is pooling at multiples of 5, there is little evidence of excessive heaping at the standard focal responses of 0, 50, and 100. The most prevalent response is 20%, but even that receives a response frequency of only 0.11. Third, most respondents (87.5%) report values in the interior (that is, not 0 or 100), reflecting a belief that there is some chance they might choose each of the jobs. This underscores the importance of eliciting probabilistic data, rather than simply the most preferred option, as respondents are able to provide meaningful probabilistic preferences for the full set of choices. Figure I View largeDownload slide Choice Probabilities for Job 1 (Pooled across Hypothetical Scenarios) Figure I View largeDownload slide Choice Probabilities for Job 1 (Pooled across Hypothetical Scenarios) V.B. Empirical Model of Job Preferences Next, we discuss our empirical model of job preferences, which we estimate using our hypothetical data. Our estimator follows the identification analysis we laid out above. For the job preferences over attributes, we use the form ui(X) = X΄βi, where βi = [βi1, …, βiK] is a K-dimensional vector that reflects individual i’s preferences for each of the K job characteristics. The X vector of job characteristics is described below and we consider several different functional forms. We assume beliefs about future job utility Hi(·) in equation (4) are i.i.d. Type I extreme value for all individuals. The probability of choosing each job is then: $$p_{ij}=\frac{\exp (X_{j}^{\prime }\beta _{i})}{\sum _{j^{\prime }=1}^{J}\exp (X_{j^{\prime }}^{\prime }\beta _{i})},$$ (5)where it is important to note that the probabilities assigned to each job j are individual i specific.20 Although we maintain a particular assumption about the distribution of probabilistic beliefs, we place no parametric restrictions on the distribution of preferences, represented by the vector βi. Our goal is to estimate the population distribution of preferences βi. We maintain a maximum degree of flexibility by estimating the preference vector βi separately for each sample member, and do not impose any “global” distributional assumptions about the population distribution of preferences (e.g., that preferences βi ∼ N(μ, Σ)). Applying the log-odds transformation to equation (5) yields the linear model: \begin{equation*} \ln \left(\frac{p_{ij}}{p_{ij^{\prime }}} \right)=(X_{j}-X_{j^{\prime }})^{\prime }\beta _{i}. \end{equation*} βi has the interpretation of the marginal change in the log odds for some level difference in the X characteristics of the job. Given the difficulty of interpreting the βi preference parameters directly, we also present results in which we compute individual-level WTP statistics. V.C. Measurement Error One potential issue in using hypothetical data for estimating preferences is that individuals may report their preferences with error. Given that these preferences have no objective counterpart (we cannot ascertain the “accuracy” of a self-reported preference), we cannot point to definitive evidence on the extent of measurement error. The most apparent potential measurement issue is that individuals report rounded versions of their underlying preferences (rounded to units of 5% or 10%). To guard against the potential of rounding bias or other sources of measurement error, we follow Blass, Lach, and Manski (2010) in introducing measurement error to the model, and in flexibly estimating the model using a least absolute deviations (LAD) estimator. We assume that the actual reports of job choice probabilities in our data, denoted $$\tilde{p}_{ij}$$, measure the “true” probabilities pij with error. The measurement error takes a linear-in-logs form such that the reported log-odds take the following form: $$\ln \left(\frac{\tilde{p}_{ij}}{\tilde{p}_{ij^{\prime }}} \right)=(X_{j}-X_{j^{\prime } })\beta _{i}+\omega _{ij},$$ (6)where ωij is the measurement error. We assume that the ωi1, …, ωiJ have median 0, conditional on the X1, …, XJ observed job characteristics. Given these measurement error assumptions, we have the following median restriction: $$M\left[ \ln \left(\frac{\tilde{p}_{ij}}{\tilde{p}_{ij^{\prime }}}\right)|X_{j},X_{j^{\prime }}\right] =(X_{j}-X_{j^{\prime }})\beta _{i},$$ (7)where M[ · ] is the median operator. This median restriction forms the basis for our estimator. Our measurement error assumptions are limited compared to commonly imposed fully parametric models which assume a full distribution for the measurement error process. In contrast, our assumption is that the measurement errors are only median unbiased.21 Another advantage of the LAD estimator is that it is not sensitive to what the extreme responses (probabilities of 0 and 1) are replaced with. V.D. Estimation We estimate the K-dimensional vector βi by LAD for each student i separately. In our data, each student makes choices across 16 scenarios, assigning probabilities to three possible jobs in each scenario. Equation (7) therefore is estimated for each respondent using 16 × 2 = 32 unique observations. Variation in the job attributes (Xj), which is manipulated exogenously by us, and variation in respondents’ choice probabilities allows us to identify the parameter vector βi. From the full set of estimates of β1, …, βN for our size N sample we estimate population statistics, such as mean preferences, E(βi). We conduct inference on the population statistics using block or cluster bootstrap by resampling (with replacement) the entire set of job-hypothetical probabilities for each student. Online Appendix Section B describes the bootstrapping algorithm. The block bootstrap preserves the dependence structure within each respondent’s block of responses, and allows for within-individual correlation across job-choice scenarios. As discussed in the study design section, we varied four job attributes at a time in each scenario. For estimation, we combine all of these scenarios and assume the dimensions that were not varied in a given scenario were believed by the respondent to be held constant, as we instructed. As mentioned earlier, we instruct respondents that the jobs differ only in the finite number of job characteristics we provide, and are otherwise identical. There is no additional information here that the respondent could use to believe otherwise. The vector of job attributes is as follows: X = {log age-30 earnings; probability of being fired; bonus as a proportion of earnings; proportion of males in similar positions; annual increase in earnings; hours per week of work; availability of part-time}.22 We also include job-number dummies in equation (7) to allow for the possibility that the ordering of the jobs presented could affect job preferences, although there is no prior reason to suspect this given our experimental design.23 V.E. Job Preference Estimates We first discuss the sign and statistical significance level of the βi estimates. Because of the difficulty in interpreting the magnitude of these estimates, below we also present results in which we convert the parameter estimates into an individual-level WTP measure. Recall that we can identify the βi vector without a parametric restriction on the population distribution of preferences. Online Appendix C discusses the estimated heterogeneity in preferences within gender. The first column of Table V shows the average estimate for each job characteristic (across all individual-level estimates). The standard errors in parentheses are derived from a block bootstrap procedure. We see that the average estimates have the expected signs: estimates for the probability of being fired and work hours per week are negative, while the others are positive. The estimates indicate that individuals, on average, prefer higher salaries and work-time flexibility, and dislike jobs with a high probability of being fired and high numbers of work hours. The only estimate that is not statistically or economically significant is the proportion of males at the job, indicating that we cannot reject that, on average, individuals are indifferent to the gender composition of the workplace. Turning to the average estimates by gender, reported in columns (2) and (3) of Table V, we see similar qualitative patterns. We return to the differences in magnitudes of the preferences by gender below, and also provide a WTP interpretation. TABLE V Estimates of Job Choice Model Overalla Males Females (1) (2) (3) Age-30 log earnings 15.40*** 22.86*** 11.42*** (1.65) (3.88) (1.43) Probability of being fired −0.38*** −0.39*** −0.37*** (0.04) (0.10) (0.04) Bonus, as a prop. of earnings 0.28*** 0.38*** 0.22*** (0.03) (0.05) (0.03) Prop. of males in similar positions 0.00 −0.01 0.005 (0.00) (0.01) (0.01) % increase in annual earnings 0.55*** 1.09*** 0.27** (0.10) (0.22) (0.10) Hours per week of work −0.15*** −0.21*** −0.12*** (0.02) (0.05) (0.02) Part-time option available 0.79*** 0.86*** 0.76*** (0.11) (0.22) (0.12) Observations 247 86 161 Overalla Males Females (1) (2) (3) Age-30 log earnings 15.40*** 22.86*** 11.42*** (1.65) (3.88) (1.43) Probability of being fired −0.38*** −0.39*** −0.37*** (0.04) (0.10) (0.04) Bonus, as a prop. of earnings 0.28*** 0.38*** 0.22*** (0.03) (0.05) (0.03) Prop. of males in similar positions 0.00 −0.01 0.005 (0.00) (0.01) (0.01) % increase in annual earnings 0.55*** 1.09*** 0.27** (0.10) (0.22) (0.10) Hours per week of work −0.15*** −0.21*** −0.12*** (0.02) (0.05) (0.02) Part-time option available 0.79*** 0.86*** 0.76*** (0.11) (0.22) (0.12) Observations 247 86 161 Notes. Table reports the average of the parameter estimates across the relevant sample. Asterisks denote estimates are statistically different from zero based on bootstrap standard errors. ***, **, * denote significance at the 1%, 5%, and 10% levels, respectively. View Large V.F. Willingness to Pay The parameter estimates in Table V are difficult to interpret given the necessarily nonlinear nature of the model. To ease interpretation, we next present WTP estimates, which translate the differences of utility levels into earnings that would make the student indifferent between giving up earnings and experiencing the outcome considered. 1. Computing WTP. WTP to experience job attribute Xk is constructed as follows. Consider a change in the level of attribute Xk from value Xk = xk to Xk = xk + Δ, with Δ > 0. Assume Xk is a “bad” attribute. Given our linear utility function, we can write an indifference condition in terms of earnings Y as: \begin{equation*} x_{k}\beta _{ik}+\beta _{i1}\ln (Y)=\beta _{ik}(x_{k}+\Delta )+\beta _{i1}\ln (Y+\text{WTP}_{ik}(\Delta )), \end{equation*} where Y is the level of earnings, one of the job attributes included in every job scenario. WTPik(Δ) > 0 is individual i’s willingness to pay to avoid increasing the “bad” attribute k by Δ. Solving, WTP is given by: $$\text{WTP}_{ik}(\Delta )=\left[\exp \left( \frac{-\beta _{ik}}{\beta _{i1}}\Delta \right) -1\right]\times Y.$$ (8)WTP for individual i depends on her preference for the attribute βik versus her preference for earnings βi1 (earnings is attribute 1). Given that we allow for a log form to utility in earnings (allowing for diminishing marginal utility in earnings and implicitly consumption), willingness to pay for an individual also depends on the level of earnings at the job. 2. WTP by Gender.Table VI shows the average and median WTP estimates for changing each of the job characteristics by one unit (for the probabilistic outcomes, this is increasing the likelihood by 1 percentage point; for hours per week, increasing it by an hour; for part-time availability, this is going from a job with no part-time option to one which does).24 The first three columns of the table present the estimates in dollars, evaluating WTP at the average annual earnings across all scenarios, $${\}$$75,854 (which is fixed by the experimental setup and does not vary across respondents). The last three columns show the estimates as a proportion of the average earnings. We focus on the latter here. TABLE VI Willingness-to-Pay (WTP) Estimates WTP ($${\}$$) WTP (as % of average earnings) Overall Male Female Overall Male Female (1) (2) (3) (4) (5) (6) Percent chance of being fired 2,147.40*** 467.79 3,044.58***+++ 2.83%*** 0.62% 4.01%***+++ (525.46) (670.29) (715.28) (0.69%) (0.88%) (0.94%) [1,125.94]*** [504.69]** [1,841.27]***+++ [1.48%]*** [0.67%]** [2.43%]***+++ Bonus as % of earnings −1,069.78*** −645.92* −1,296.19*** −1.41%*** −0.85%* −1.71%*** (258.47) (368.77) (345.77) (0.34%) (0.49%) (0.46%) [−975.36]*** [−761.57]*** [−1,139.44]***++ [−1.29%]*** [−1.00%]*** [−1.50%]***++ Percent of men at jobs 43.20 63.74 32.24 0.06% 0.08% 0.04% (38.31) (46.81) (53.93) (0.05%) (0.06%) (0.07%) [59.15]*** [71.82]** [43.17] [0.08%]*** [0.09%]** [0.06%] Annual % raise in earnings −1,186.28 −2,564.93** −449.86 −1.56% −3.38%** −0.59% (773.44) (1,226.42) (957.85) (1.02%) (1.62%) (1.26%) [−2,596.68]*** [−2,934.51]*** [−2,514.55]*** [−3.42%]*** [−3.87%]*** [−3.31%]*** Hours per week of work 854.65*** 594.70 993.50*** 1.13%*** 0.78% 1.31%*** (235.25) (416.17) (267.19) (0.31%) (0.55%) (0.35%) [626.52]*** [600.89]*** [634.22]*** [0.83%]*** [0.79%]*** [0.84%]*** Part-time option availablea −3,892.01*** −829.94 −5,527.65***++ −5.13%*** −1.09% −7.29%***++ (1,024.91) (1,822.82) (1,221.72) (1.35%) (2.40%) (1.61%) [−2,709.40]*** [−1,866.56]*** [−3,237.95]***+ [−3.57%]*** [−2.46%]*** [−4.27%]***+ WTP ($${\}$$) WTP (as % of average earnings) Overall Male Female Overall Male Female (1) (2) (3) (4) (5) (6) Percent chance of being fired 2,147.40*** 467.79 3,044.58***+++ 2.83%*** 0.62% 4.01%***+++ (525.46) (670.29) (715.28) (0.69%) (0.88%) (0.94%) [1,125.94]*** [504.69]** [1,841.27]***+++ [1.48%]*** [0.67%]** [2.43%]***+++ Bonus as % of earnings −1,069.78*** −645.92* −1,296.19*** −1.41%*** −0.85%* −1.71%*** (258.47) (368.77) (345.77) (0.34%) (0.49%) (0.46%) [−975.36]*** [−761.57]*** [−1,139.44]***++ [−1.29%]*** [−1.00%]*** [−1.50%]***++ Percent of men at jobs 43.20 63.74 32.24 0.06% 0.08% 0.04% (38.31) (46.81) (53.93) (0.05%) (0.06%) (0.07%) [59.15]*** [71.82]** [43.17] [0.08%]*** [0.09%]** [0.06%] Annual % raise in earnings −1,186.28 −2,564.93** −449.86 −1.56% −3.38%** −0.59% (773.44) (1,226.42) (957.85) (1.02%) (1.62%) (1.26%) [−2,596.68]*** [−2,934.51]*** [−2,514.55]*** [−3.42%]*** [−3.87%]*** [−3.31%]*** Hours per week of work 854.65*** 594.70 993.50*** 1.13%*** 0.78% 1.31%*** (235.25) (416.17) (267.19) (0.31%) (0.55%) (0.35%) [626.52]*** [600.89]*** [634.22]*** [0.83%]*** [0.79%]*** [0.84%]*** Part-time option availablea −3,892.01*** −829.94 −5,527.65***++ −5.13%*** −1.09% −7.29%***++ (1,024.91) (1,822.82) (1,221.72) (1.35%) (2.40%) (1.61%) [−2,709.40]*** [−1,866.56]*** [−3,237.95]***+ [−3.57%]*** [−2.46%]*** [−4.27%]***+ Notes. Table reports mean (bootstrap standard errors) [median] WTP (amount of earnings an individual needs to be compensated for a unit change in the job attribute). aWTP for moving from a job without a part-time option to one that has it. ***, **, * denote significance at the 1%, 5%, and 10% levels, respectively. Tests conducted for differences in means and medians by gender. +++, ++, + denote estimates differ at 1%, 5%, and 10% levels, respectively. View Large We estimate, for example, that increasing the likelihood of being fired by 1 percentage point, that is, Xk = xk + 1, would yield an average WTP of 2.8% for the full sample. That is, for students to remain indifferent to moving to a less stable job, they would on average have to be compensated by 2.8% of annual earnings. The gender-specific averages, reported in the last two columns of Table VI, indicate distinct average preferences by gender. Women, on average, have to be compensated by 4% of average earnings for a unit increase in the likelihood of being fired, with the estimate being statistically significant at the 1% level, and statistically different from the much smaller male average of 0.6%. Recall that we fix average earnings at the same level for all respondents, so the gender differences in WTP reflect only differences in preferences, not earnings. The median estimates also differ by gender, with women exhibiting a higher WTP for job stability. The median estimate for women is, however, lower than the average estimate, suggestive of a skewed distribution. The average and median WTP estimate for the availability of the part-time option is sizable. Individuals, on average, would have to be compensated by 5.1% of their annual salary (that is, they are willing to give up 5.1%) when going from a job with no part-time option to one that does have one. The estimate is driven by the female respondents in the sample, for whom the average WTP is −7.3%, versus −1.0% for males (with the male estimate not being statistically different from 0). The much higher average preference among women for the part-time option is statistically significantly different from 0 and statistically different from the male average, at the 5% level. The median estimate also differs by gender and is larger in magnitude for women. Examining the WTP for other job characteristics, we see that the average WTP for annual earnings growth is statistically precise for men, who are willing to give up 3.4% of average annual earnings for a 1 percentage point increase in earnings growth; the female average coefficient is indistinguishable from 0 (although not statistically different from the male estimate). The median estimates for the two genders are similar. We see that women have a stronger distaste for the number of hours of work, with the average WTP indicating that they need to be compensated by 1.3% of annual earnings for an increase of one hour in the work week; the male estimate is not precise (but we cannot reject the two gender-specific averages being equal). Both genders are, on average, willing to give up 0.8–1.7% of annual earnings for a percentage point increase in bonus compensation (in addition to base salary).25 Finally, the average WTP for proportion of men at jobs is economically and statistically insignificant. Online Appendix C further analyzes the heterogeneity in preferences for workplace characteristics, and investigates how the WTPs are associated with various individual-level characteristics. In addition, note that the utility from jobs, specified in equation (1), is linear and separable in outcomes. Online Appendix D shows that our conclusions are robust to estimating variants of the baseline model. It is important to note that the timing of our survey is quite important in interpreting the resulting preference estimates. In general, there is no reason to believe that the workplace preferences we elicit are intrinsic, and they may be particular to the age of our survey respondents. Our estimates should not be considered unbiased estimates for intrinsic preferences—preferences are likely not intrinsic at all—but instead unbiased estimates for preferences at the point in each student’s life cycle at which we collect our data. Our preference estimates may also reflect past experiences with employment because in some cases, the respondents may have already secured postgraduation employment. Our methodology simply relies on students being able to consider their likelihood of accepting hypothetical job offers, which should be possible even if a student is already employed. V.G. Estimated Preferences and Actual Workplace Characteristics Do the pre–labor market preferences we estimate relate to the characteristics of jobs these students actually end up working in?26 We are able to shed light on this issue through a follow-up survey of a subset of our respondents conducted in 2016, about four years after the original data collection and when respondents were on average aged 25. Of the 247 respondents who took the survey and answered the hypothetical questions, 112 had also participated in an earlier survey conducted by us in 2010 (data that we have analyzed in Wiswall and Zafar 2015a,b) and given consent for future surveys. In January 2016, we invited these 112 respondents to participate in a 15-minute online survey about their current labor market status. 71 of the eligible 112 respondents (∼63%) completed the follow-up survey.27 The follow-up survey collected information about respondents’ workplace characteristics (for those currently working). Of the 71 respondents, 59 were working (either full-time, part-time, or self-employed) at the time of the follow-up survey, with the remainder enrolled in school. Online Appendix Table A7 shows the earnings and various other workplace characteristics for the overall sample, as well as for male and female workers, separately. Earnings, conditional on working full-time, are higher for men (by nearly $${\}$$70,000). Bonus, hours of work, likelihood of being fired, fraction of male employees, and typical annual growth in earnings are all higher for our male respondents (though not all of the differences are statistically significant). The last row of the table shows that women’s workplaces are more likely to have a part-time or flexible work option.28 Are these systematic gender differences in actual workplace characteristics consistent with our estimates of job preferences elicited several years prior, before labor market entry? To investigate this, we regress characteristics of each respondent’s current job onto our individual-specific estimate of their past WTP for that attribute. WTP is defined as the amount the individual needs to be compensated by for a unit change in a given characteristic, with a higher WTP reflecting a lower taste (or greater distaste) for that outcome. Therefore, we expect a negative relationship between WTP and the current job characteristic. Estimates are presented in Table VII. Directionally, all six estimates are negative, with three significant at the 5% level or better. A joint test that all coefficients are 0 can be rejected (the p-value of this joint test is .012). TABLE VII Actual Job Characteristics and Estimated WTP Prob. of fired Bonus percentage Prop. of males Earnings growth Hours worked Flex work option Willingness to paya −0.07 −1.00 −7.32** −0.02 −1.70** −0.94** (0.20) (1.23) (2.82) (0.08) (0.64) (0.29) Constant 10.70*** 3.64 52.60*** 7.32*** 46.37*** 55.61*** (1.90) (2.67) (2.82) (1.71) (2.00) (6.37) Effect sizeb −0.658 −4.35 −6.89 −0.319 −4.09 −14.75 p-valuec 0.012 Mean of dep. var. 10.4 5.8 50.9 7.3 44.6 61.0 Std. dev. of dep. var. (14.72) (12.79) (22.79) (13.34) (14.76) (49.19) R-squared 0.002 0.16 0.092 0.0001 0.077 0.090 Observations 59 59 59 59 59 59 Prob. of fired Bonus percentage Prop. of males Earnings growth Hours worked Flex work option Willingness to paya −0.07 −1.00 −7.32** −0.02 −1.70** −0.94** (0.20) (1.23) (2.82) (0.08) (0.64) (0.29) Constant 10.70*** 3.64 52.60*** 7.32*** 46.37*** 55.61*** (1.90) (2.67) (2.82) (1.71) (2.00) (6.37) Effect sizeb −0.658 −4.35 −6.89 −0.319 −4.09 −14.75 p-valuec 0.012 Mean of dep. var. 10.4 5.8 50.9 7.3 44.6 61.0 Std. dev. of dep. var. (14.72) (12.79) (22.79) (13.34) (14.76) (49.19) R-squared 0.002 0.16 0.092 0.0001 0.077 0.090 Observations 59 59 59 59 59 59 Notes. The table investigates the relationship between the estimated WTP for a given job attribute for a respondent (derived from the 2012 survey) and the value of that job attribute in the respondent’s actual workplace (reported in the 2016 follow-up survey). Each column is a separate OLS regression, with the dependent variable (column title) being the value of the job characteristic in the respondent’s actual job (reported in the 2016 survey). Bootstrap standard errors in parentheses. ***, **, * denote significance at 1%, 5%, and 10% levels, respectively. aThe estimated WTP of the respondent based on the hypothetical job choice scenarios. bThe predicted change in the dependent variable for a one std. dev. change in the WTP. cp-value of a test that the six estimates on the WTP (in the first row) are jointly zero. View Large To interpret the magnitude of the estimated coefficients in Table VII, we also report “effect sizes” in the table. The effect size provides the estimated change in the dependent variable (that is, the actual workplace attribute) for a one standard deviation change in the WTP for that workplace characteristic. For example, we see that a one standard deviation increase in the WTP (that is, higher distaste) for work hours translates into an estimated decrease of 4.1 hours worked. Given that the standard deviation of hours worked is 14.8 in the sample, this is a sizable impact. Likewise, a one standard deviation increase in the WTP (that is, lower taste) for availability of flexible work options is associated with a 15 percentage point decline in the actual availability of these options in the respondent’s workplace (on a base of 61). The effect sizes for bonus percentage and proportion of male are also economically meaningful. While we have shown that estimated preferences for attributes are jointly systematically related to actual future workplace characteristics in the cross section, a natural question to ask is whether the relationship also holds within the individual, that is, whether a higher WTP for a given attribute translates into more of that attribute for an individual. For each attribute, we rank the 59 individuals in terms of both the estimated WTP and the actual value at the job. This gives us a six-dimensional vector of ranked WTPs and a six-dimensional vector of ranked attribute values for each individual. We then compute the individual-level correlation between the two vectors. We expect a negative correlation: higher WTP (that is, a lower taste or a greater distaste) for an attribute causes an individual to be working in a job with lower values of that attribute. That is exactly what we find: the mean correlation coefficient across the individuals is −0.158 (significant with a p-value = .017) and the median correlation coefficient is −0.250 (p-value = .36), indicative of a systematic relationship between estimated WTPs and actual attributes even within individuals. Overall, these results strongly indicate that our estimated preferences capture true underlying heterogeneity that is also reflected in actual job outcomes several years later. We view these results as a joint validation of our methodology, data quality, and empirical specification. Our finding that estimated WTPs predict respondents’ actual workplace choices is all the more remarkable given that the hypothetical scenarios were fielded to respondents when they were still in college (though some of the respondents may have already secured postgraduation employment at the time of the survey). In the next section, we investigate whether these workplace preferences impact major choice. VI. Job Preferences and Major Choice The preceding sections used a robust hypothetical choice methodology to estimate individual-level preferences for various job attributes. This section relates these preferences to human capital investments, quantifying the importance of job characteristics to college major choices. First, to set the stage for this analysis, we describe the anticipated major choices reported by our sample. Given that our sample consisted of currently enrolled students, we asked the students to provide their beliefs they would complete a degree in one of the five major categories: “What do you believe is the percent chance (or chances out of 100) that you would either graduate from NYU with a PRIMARY major in the following major categories or that you would never graduate/dropout (i.e., you will never receive a bachelor’s degree from NYU or any other university)?” The first column of Online Appendix Table A9 shows the response to the question: the most likely major for males is economics/business (43%), followed by humanities/social sciences (29%). For women, on the other hand, the most likely major is humanities/social sciences (53%), followed by economics (23%). The probability of not graduating is less than 3% for both sexes. The average probabilities assigned to the majors differ significantly by gender for all majors except engineering and natural sciences. Our model of major choice allows for some uncertainty in major choice: at least part of the sample is not 100% certain of their final major at graduation and the data reflect that (a majority of students, 53, do not assign a 100% probability to their most likely major). Our model of probabilistic major choices nests the standard model of deterministic major choice. We next decompose the anticipated major choices into various factors, including potential job characteristics associated with each major. To gauge the importance of job attributes to major choice, we estimate a model of major choice incorporating our flexible estimates of preferences for job attributes and separate data we collected on students’ beliefs about the likelihood they would be offered jobs with these characteristics, conditional on major choice (that is, estimates of students’ perceptions of the firm or demand side of the labor market). We then use this estimated model to quantify the importance of each job attribute to major choice. Given that prior literature on educational choice finds that the residual unobserved “taste” component is the dominant factor in major choice (Arcidiacono 2004; Beffy, Fougere, and Maurel 2012; Gemici and Wiswall 2014; Wiswall and Zafar 2015a), our approach can be viewed as trying to get into the black box of tastes by directly incorporating certain nonpecuniary dimensions into these choice models. The estimation details for the major-choice model are provided in Online Appendix E. Here, for the sake of brevity, we comment on only its main features. We start with a simple framework in which we suppose that utility for student i from major m is given by: $$V_{im}=X_{im}^{\prime }\alpha _{i}+Z_{im}^{\prime }\gamma +\kappa _{m}+\eta _{im},$$ (9)where Xim is i’s perceived job attributes in major m. With hopefully minimal confusion, we use the same notation X to refer to job attributes as in our hypothetical job choice analysis and to refer to perceptions about job attributes associated with each major, a separate set of variables collected in our survey. Note that here the X vector is indexed by i as these attributes are each student’s perception of the job attributes (that are allowed to depend on the major m) rather than the exogenously determined attributes in the hypotheticals we created. Zim is a vector of other major-specific characteristics perceived by student i (including major-specific perceptions of ability and perceived hours of study needed to obtain a GPA of 4.0 in that major). κm is a major-specific constant, capturing overall tastes for the major, and ηim captures the remaining unobservable attributes of each major. To estimate the model, we use data on students’ perceptions of the likelihood of being offered jobs with various characteristics conditional on each major, as well as their beliefs regarding major-specific ability. Our survey collected data from respondents on their perceptions of characteristics of the jobs that would likely be offered to them if they were to complete each type of major. An important characteristic of our data set is that we gather students’ beliefs about workplace characteristics (such as likelihood of being fired and earnings) for a set of different majors, not just for the one major they intend to complete. These data are described at length in Online Appendix E.29 In equation (9), the student-specific preference for each job attribute is given by the vector αi = [αi1, …, αiK]. αi, the preference for job characteristics as it relates to the utility from each major, is potentially distinct from the preferences for job characteristics in the job-choice problem, given by βi (in equation (5)). Job characteristics, such as earnings at the job, may be quite important when choosing among different job offers but might have a more limited value to choosing majors, relative to other major characteristics given by Zim, κm, and ηim. To allow for this possibility, for each job characteristic k, we specify that each αik is proportional to the βik up to some free (to be estimated) parameter δ: αik = βikδ. δ indicates the importance of job attributes to major choice, relative to other determinants of college major as given by Zim, κm, and ηim. δ could also reflect standard discounting given that the utility from working at jobs occurs later in life than utility derived from taking courses while in school. Table VIII presents the LAD estimates of equation (9) using the hypothetical data to estimate the job-preference vector βi for each student, and a robust cluster bootstrap over all estimation steps for inference (see Online Appendix E for estimation details). The estimate of δ is positive and precise, indicating that the preferences of students over job attributes and the major-specific beliefs about the distribution of job attributes have a statistically significant relationship with major choices. Estimates on the major-specific ability measures are negative, as one would expect (note that higher “ability rank” denotes lower ability in our data). The major-specific dummy terms are all negative, indicative of negative median tastes for the nonhumanities majors (the omitted category): all else equal, students prefer to major in humanities. TABLE VIII LAD Estimates of Major Choice LAD estimates Job attributes (δ) 0.018** (0.007) Ability rank −0.064*** (0.006) Study time −0.009 (0.025) Economics dummy −0.590 (0.444) Engineering dummy −1.16** (0.37) Natural sci dummy −0.822* (0.375) Total observations 741 Number of individuals 247 LAD estimates Job attributes (δ) 0.018** (0.007) Ability rank −0.064*** (0.006) Study time −0.009 (0.025) Economics dummy −0.590 (0.444) Engineering dummy −1.16** (0.37) Natural sci dummy −0.822* (0.375) Total observations 741 Number of individuals 247 Notes. Bootstrap standard errors in parentheses. ***, **, * denote significance at the 1%, 5%, and 10% levels, respectively. View Large Given the nonlinear nature of the model, it is difficult to assess the importance of job attributes in major choice from the estimated coefficients alone. To quantify the effects, we use standard methods to evaluate “marginal effects” in nonlinear models (see Online Appendix E for details). The marginal effect of a job attribute in major choice is computed for a standard deviation change in the value of that specific job attribute, while keeping the other (job- and major-specific) attributes and preferences fixed at their sample average values. Table IX presents the marginal effects for specific changes in job attributes, averaged across the majors, and separately by sex (in the two panels of the table). The table also shows the start and end value for the attribute at which the marginal effect is computed. The start value is the sex-specific belief for that attribute (averaged across majors and respondents), and the end value is the start value shifted by one sex-specific standard deviation (again, across majors and respondents) in the beliefs for that attribute. Column (1), for example, shows that increasing the perceived probability of being fired from jobs by one standard deviation decreases the likelihood of majoring in that major, on average, by 4% for men and by 5% for women. A standard deviation increase in part-time availability increases the probability of completing a major by 0.2%. Column (3) shows that a standard deviation increase in weekly work hours reduces the likelihood of majoring in a major by 2.5% for men and 1.4% for women. Bonus pay and earnings growth both also have sizable average marginal effects. The last column of Table IX shows the percent change in the major probability for a standard deviation increase in log age-30 earnings. TABLE IX Marginal Contribution of Job Attributes in Major Choice Fired prob. Part-time available Hours Bonus Earnings growth Prop. males Log earnings (1) (2) (3) (4) (5) (6) (7) Panel A: Males Start valuea 8.82 0.29 48.88 18.37 1.42 54.86 11.38 End valueb 19.38 0.53 60.40 41.37 5.20 73.21 12.03 Avg. changec −4.10% 0.23% −2.48% 9.20% 4.30% −0.20% 15.90% Relative changed −0.26 0.01 −0.16 0.58 0.27 −0.01 — Panel B: Females Start value 16.51 0.35 47.36 19.12 1.35 56.13 11.22 End value 33.14 0.58 61.15 43.11 4.44 75.22 11.70 Avg. change −5.12% 0.15% −1.40% 4.58% 0.70% 0.10% 4.78% Relative change −1.07 0.03 −0.29 0.96 0.15 0.02 — Fired prob. Part-time available Hours Bonus Earnings growth Prop. males Log earnings (1) (2) (3) (4) (5) (6) (7) Panel A: Males Start valuea 8.82 0.29 48.88 18.37 1.42 54.86 11.38 End valueb 19.38 0.53 60.40 41.37 5.20 73.21 12.03 Avg. changec −4.10% 0.23% −2.48% 9.20% 4.30% −0.20% 15.90% Relative changed −0.26 0.01 −0.16 0.58 0.27 −0.01 — Panel B: Females Start value 16.51 0.35 47.36 19.12 1.35 56.13 11.22 End value 33.14 0.58 61.15 43.11 4.44 75.22 11.70 Avg. change −5.12% 0.15% −1.40% 4.58% 0.70% 0.10% 4.78% Relative change −1.07 0.03 −0.29 0.96 0.15 0.02 — Notes. Table shows the average percent change in the probability of majoring in a given major (“marginal effect”) for a standard deviation change in the job attribute (column variable). See Online Appendix E for details. a(b)The initial (final) value of the attribute at which the major probability is computed, with all other attributes fixed at the sample mean. cThe average change (across majors) in the probability of majoring in a given major for a standard deviation change in the attribute. dThe average change in the probability of majoring in a given major for a std. dev. change in an attribute, relative to a corresponding change in earnings. View Large A comparison of the effects in the first three columns with those in the last column for earnings (also shown in the last row of each panel) gives a sense of the relative importance of these other job attributes in major choice. We see that, for women, the average effect for the probability of being fired is as large as that for earnings, and for hours is nearly a third of the effect of earnings. For men, the relative impacts are smaller (though still sizable). Overall, this indicates that job attributes matter for major choice and that they are particularly relevant for women’s choices. VII. Job Preferences and the Gender Gap in Earnings In the previous sections, we have shown systematic gender differences in workplace preferences and quantified the importance of these preferences to major choices. In this section, we explore the extent to which gendered job preferences explain the “gender gap” in earnings. Differences in job preferences can give rise to differences in earnings through two channels. First, as explored above, job preferences can affect college major choices and, given the wide dispersion in earnings across fields, affect the overall distribution of earnings for men and women. Second, even conditional on major choice, gender differences in workplace preferences can affect the distribution of earnings. The gender gap in earnings we observe could be at least partially the result of women “purchasing” certain positive job attributes by accepting lower wages, or conversely, men accepting higher earnings to compensate for negative job attributes. To quantify the first channel (job preferences affecting earnings through major choice), we conduct the following exercise. Using the estimated major-choice model in Section VI, we predict the likelihood of women choosing different majors if their workplace preferences were shifted by the average male minus female mean (that is, we preserve the heterogeneity in women’s preferences but shift them by the average gender difference in the preferences).30 We then predict the likelihood of each female respondent choosing the different majors and use these to weight the individual’s major-specific expected earnings. This provides the impact on the gender wage gap if women’s preference distribution was shifted to have the male average, but only through the major choice channel. Note that, for this exercise, we keep women’s earnings expectations fixed in that major (which could also be affected by workplace preferences). In this exercise, we find that the change in women’s major choices lowers the expected gender gap in age-30 earnings by 2.6%.31 Given our highly aggregated major categories, this is likely a lower bound on the importance of preferences to the gender earnings gap through major choices, and human capital more generally. Previous work has emphasized that important job segregation by gender occurs through choices of subfields (see for example, Goldin and Katz 2016, on choice of medical specialties). Turning to the second channel, we consider the following simple exercise. We ask how the gender gap in expected earnings changes once we “control” for individual-specific workplace preferences (the estimated preference parameters in Section V). If the gender gap in earnings is solely because women are accepting lower wages for desirable jobs, and/or men are compensated with higher wages for undesirable jobs, then men and women with identical workplace preferences would have equal earnings. If, on the other hand, a gender gap remains, even after conditioning on preferences, then we can conclude that demand-side factors, such as employment discrimination, still play a role in the gender gap. We implement this exercise using a simple set of regressions in Table X, Panel A. Column (1) of the table reports a regression of an individual’s log expected earnings for the major they are most likely to graduate with onto a female dummy. We see a gender gap of about 35 log points in age-30 expected earnings, a gap similar to that in realized earnings data.32 The second column shows that the gender gap declines to about 20 log points once the individual’s major is controlled for, reflecting the fact that women are less likely to graduate in higher-earnings majors. Columns (3) and (4) show how the gender gap changes once we control for the estimated vector of workplace preferences. Importantly, a comparison of column (4) with column (2) shows that, even conditional on major choice, workplace preferences reduce the expected earnings gender gap by about a quarter, from about 20% to 15%. Note that workplace preferences are also likely to impact major choice, which is held fixed here. TABLE X Workplace Preferences and Gender Gap in Age-30 Expected and Actual Earnings (1) (2) (3) (4) Panel A: Dependent variable: log(age-30 expected earnings) Female −0.346*** −0.195*** −0.289*** −0.150*** (0.060) (0.055) (0.065) (0.057) Constant 11.483*** 11.69*** 11.36*** 11.55*** (0.048) (0.051) (0.065) (0.068) Major controlsa N Y N Y Workplace preferences controlsb N N Y Y Mean of dep. var 11.26 11.26 11.26 11.26 R-squared 0.1209 0.3386 0.2013 0.3967 Number of observations 247 247 247 247 Panel B: Dependent variable: log(actual 2016 earnings) Female −0.612*** −0.451*** −0.442** −0.318 (0.169) (0.167) (0.191) (0.230) Constant 12.12*** 12.31*** 11.91*** 12.12*** (0.145) (0.147) (0.190) (0.188) Part-time work dummy Y Y Y Y Major controlsc N Y N Y Workplace preferences controls N N Y Y Mean of dep. var 11.65 11.65 11.65 11.65 R-squared 0.226 0.384 0.395 0.495 Number of observations 56 56 56 56 (1) (2) (3) (4) Panel A: Dependent variable: log(age-30 expected earnings) Female −0.346*** −0.195*** −0.289*** −0.150*** (0.060) (0.055) (0.065) (0.057) Constant 11.483*** 11.69*** 11.36*** 11.55*** (0.048) (0.051) (0.065) (0.068) Major controlsa N Y N Y Workplace preferences controlsb N N Y Y Mean of dep. var 11.26 11.26 11.26 11.26 R-squared 0.1209 0.3386 0.2013 0.3967 Number of observations 247 247 247 247 Panel B: Dependent variable: log(actual 2016 earnings) Female −0.612*** −0.451*** −0.442** −0.318 (0.169) (0.167) (0.191) (0.230) Constant 12.12*** 12.31*** 11.91*** 12.12*** (0.145) (0.147) (0.190) (0.188) Part-time work dummy Y Y Y Y Major controlsc N Y N Y Workplace preferences controls N N Y Y Mean of dep. var 11.65 11.65 11.65 11.65 R-squared 0.226 0.384 0.395 0.495 Number of observations 56 56 56 56 Notes. OLS estimates presented. Block bootstrap standard errors in parentheses. ***, **, * denote significance at 1%, 5%, and 10% levels, respectively. Dependent variable in Panel A is the log of age-30 expected earnings for the individual’s reported major. Dependent variable in Panel B is the log of actual earnings for the subset of individuals who took the follow-up survey and were working in 2016. aDummy for the major the respondent is majoring in (the major with the modal probability). bControls for the estimated workplace preferences (from the job-choice model). cDummy for the major the respondent graduated with. View Large Table X, Panel B repeats the exercise using actual earnings reported by the follow-up respondents. The sample here is smaller, but the qualitative results are strikingly similar to those that we observe for expected earnings: conditional on major, the gender gap in realized earnings declines from 45 log points to 32 log points (that is, by nearly 30%) once we control for respondents’ workplace preferences. We conclude from this analysis that gender differences in workplace preferences can explain a sizable part of the gender gap in expected earnings early in the life cycle. And, albeit with a smaller sample, our evidence points to similar conclusions for realized earnings as well. We also find that the main channel by which workplace preferences affect the gender earnings gap is through job choices, not through major choices, at least at the aggregated major level we have available in this data set. VIII. Conclusion Economists have long recognized that job and occupational choices are not solely determined by expected earnings.33 Although simple models based on earnings maximization abound (see, for example, the classic Roy 1951 model) and are quite useful in some applications, it is also clear that individuals have a rich set of preferences for various aspects of jobs beyond expected earnings, including earnings and dismissal risk, and various nonpecuniary aspects such as work hours flexibility. Human capital investments too could be affected by these workplace preferences as individuals alter their human capital investment in anticipation of particular future job choices. Key features of the distribution of labor earnings in the economy, such as the gap in earnings between men and women, need careful consideration, as differences in earnings may reflect, at least in part, heterogeneity in preferences and compensating differentials for various nonpecuniary attributes of employment. Using a novel hypothetical job-choice framework that experimentally varies different dimensions of the workplace, this article robustly estimates individual preferences for workplace attributes. For a sample of high-ability undergraduate students enrolled at a selective private U.S. university, we document substantial heterogeneity in willingness to pay for job amenities, with large differences in the distribution of preferences between men and women. For a subset of the sample for whom we collect data on actual workplace characteristics (nearly four years after the original survey), we find a robust systematic relationship between estimated preferences and the characteristics of their current jobs. The predictive power of the estimated preferences at the individual level strengthens the credibility of our approach, and makes a case for employing this methodology in other settings to understand decision making. Combining these workplace preferences with unique data on the students’ perceptions of jobs which would be offered to them given their major choice, we quantify the role of anticipated future job characteristics—particularly the nonpecuniary aspects of these jobs—in choice of major, a key human capital investment decision. Women, in particular, are found to be more sensitive to nonpecuniary job aspects in major choice than men. Our analysis indicates that at least a quarter of the gender gap in early career earnings—expected as well as actual—can be explained by the systematic gender differences in workplace characteristics. Our analysis indicates that a substantial part of the early gender gap in earnings we observe is a compensating differential in which women are willing to give up higher earnings to obtain other job attributes. There are several potential areas for future research. Although we find substantial variation in workplace preferences for our sample of high-ability students at a selective university, it is not clear how these preferences compare to that of the broader population. It would clearly be useful to follow our design and collect similar data in other settings. In particular, preference data collected at older ages would be useful in studying how preferences for nonpecuniary dimensions of the workplace, especially those related to accommodations for raising children, evolve over the life cycle (Bertrand, Goldin, and Katz 2010). Our work also does not directly indicate the sources of the systematic gender differences in workplace preferences that we document. For example, they may be a consequence of social factors including anticipated discrimination (Altonji and Blank 1999). We cannot therefore claim that these preferences are intrinsic and immutable in the sense that they may be due, at least in part, to environmental influences particular to this cohort of students. Research that sheds light on the underlying channels would be immensely valuable. Supplementary Material An Online Appendix for this article can be found at The Quarterly Journal of Economics online. Data and code replicating the tables and figures in this article can be found in Wiswall and Zafar (2017), in the Harvard Dataverse, doi:10.7910/DVN/MLOGDL. Footnotes * Ellen Fu and John Conlon provided excellent research assistance. We would like to thank Joe Altonji for feedback on the survey design. We are also thankful to the editor, Larry Katz; the coeditor; five anonymous referees; and participants at various seminars and conferences for valuable comments. This is a revised version of NBER Working Paper 22173 (April 2016). 1. In the marketing and environmental contexts, these methods are often used to identify preferences for new, as yet unavailable consumer products or for public goods like environmental quality, for which realized choices and markets do not exist. Our primary motivation for collecting hypothetical choice data is not because labor markets and realized choices do not exist, but to resolve problems of endogeneity of realized job choices. 2. Recent work has incorporated nonwage components into rich models of the labor market and education choices, allowing for important features such as search frictions, preferences over unobserved job attributes, and dynamic incentives for occupation and education choices (see for example, Bonhomme and Jolivet 2009; d’Haultfoeuille and Maurel 2013; Bronson 2015; Lim 2015). Motivating our approach, Hwang, Mortensen, and Reed (1998) and Bonhomme and Jolivet (2009) conclude that search frictions can imply small equilibrium wage differentials across jobs when there are in fact substantial preferences for nonwage job amenities. 3. In fact, using a recent nationally representative survey of U.S. workers, Maestas et al. (2016) find that younger college-educated women report less desirable working conditions (including no option to telecommute, higher prevalence of employer setting schedules, and higher incidence of work-related stress). 4. Mas and Pallais also conclude that gender differences in work-time flexibility preferences are not enough to explain any part of the gender gap in earnings, which stands in contrast to our conclusion of a large role (as we discuss later). There could be several reasons for this difference in findings: we measure preferences for several workplace attributes (job stability, earnings growth, hours). In addition, our sample is high skill, and likely to be active in a different segment of the labor market. 5. For examples of recent work, see Arcidiacono 2004; Beffy, Fougere, and Maurel 2012; Arcidiacono, Hotz, and Kang 2012; Stinebrickner and Stinebrickner 2014a; Gemici and Wiswall 2014; Wiswall and Zafar 2015a. Most recently, Bronson (2015) analyzes the importance of work-hours flexibility and changes in divorce law and divorce risk in explaining longer-term gender-specific trends in major choices. 6. Bertrand, Goldin, and Katz (2010) document the rising role of children and hours choices over the first 15 years of the careers of female MBAs from a top U.S. business school. 7. Altonji, Kahn, and Speer (2016) provide a more detailed discussion of the relationships between college majors and labor market outcomes. 8. The unadjusted hourly earnings gap is 21.6 log points. For college graduates ages 25–40, the mean earnings for full-time employed men is higher than the mean earnings for full-time employed women by 36%. The median for full-time men is higher than the female median by 28%. 9. During the same session, and immediately prior to completing the survey, students took part in some economic experiments. Students earned additional income through participation in the experiments. See Reuben, Wiswall, and Zafar (forthcoming) for information on this data collection. 10. In addition, when presented with each scenario, respondents were told: “Now consider the situation where you are given the jobs offered above when you are aged 30, and you have decided to accept one of these jobs. What is the percent chance (or chances out of 100) that you will choose each of these jobs?” That is, the options were mutually exhaustive, and not working was not an option. 11. Lordan and Pischke (2016) find a strong relationship between women’s job satisfaction and the proportion of men in that occupation. 12. For each job attribute, we constructed a set of hypothetical job scenarios by using uniform random draws from an interval between the 10th and 90th percentile of the observed distribution for each attribute. For each set of job scenarios, we then rejected any set of job scenarios which included jobs which were dominated by another job in all attributes or had earnings differences across jobs which were greater than 30%. 13. For the NYU population of students who graduated in 2010 (IPEDS), the fraction of students completing degrees in each field are as follows: for women, 14.1% graduated in economics or business, 71.7% in humanities or other social sciences, and 13.7% in natural sciences, math, or engineering. For men, 31.1% graduated in economics or business, 61.2% in humanities or other social sciences, and 7.8% in natural sciences, math, or engineering. 14. We confine attention to cross-sectional data. Panel data on repeated job choices over an individual’s life cycle may provide more identifying power but at the cost of requiring additional assumptions about the evolution of model features (e.g., preferences) as individuals age. 15. We can represent demand-side restrictions in the omitted variable framework by considering some unobservable job characteristic X(unob), such that X(unob) → −∞ if a job is not offered. 16. An alternative model is that agents have uncertainty about preferences over attributes, that is the utility function ui(·) is uncertain. For example, an individual may be uncertain about the number of children she may have at a future date, and the number of young children at home may affect her preference for workplace hours flexibility (an element of the Xj vector). We explore this later by relating preferences for job characteristics as revealed in our hypothetical data with a rich set of beliefs about future outcomes (e.g., individual beliefs about future own fertility and marriage). 17. See Stinebrickner and Stinebrickner (2014a,b) for evidence on the dynamics in beliefs formation among college students. 18. In an innovative related approach, Stern (2004) collects data on job offers and accepted jobs from a sample of PhD biologists to estimate the WTP to take a research job over others. However, the limited data on job offers do not allow for identification of heterogeneity in preferences. In addition, this approach only yields unbiased preference estimates in frictionless labor markets. 19. Note that as with any discrete choice setting, the population distribution of preference parameters βi is identified up to the distribution of the εi shocks. As we detail below, we assume a logit form for the shocks. For ease of interpretation, we focus on WTP implied by the model, where WTP is a function of the ratio of elements of the βi vector, removing the dependence of WTP on the scale of the shock. 20. Note that utilities across alternatives are correlated through the shared job attributes, therefore the independence of irrelevant alternatives problem does not apply to our model. 21. Note we do not impose that ωij measurement errors are independent across individuals or jobs and do not assume any particular joint distribution for the measurement errors, beyond the conditional median independence with the X variables. For inference, we use a cluster bootstrap method, resampling the entire set of job scenarios for each sample member, to preserve any correlation in residual errors. See Online Appendix B for details. 22. We also estimate the model with utility specified as linear in earnings (instead of log earnings). Results are qualitatively similar. Online Appendix D discusses results from several other alternative specifications. 23. This is related to the possibility of “session effects” in laboratory experiments. See Frechette (2012). 24. The WTP is computed for each individual, using the individual-specific βi estimates. The table reports mean WTP across respondents, bootstrap standard errors in parentheses, and median WTP in square brackets. 25. That the WTP for a percentage point increase in bonus is greater than 1 in magnitude for women is surprising because it implies that women are on average willing to give up more in base salary to gain a smaller increase in bonus compensation. This is driven by a few outliers. In fact, we cannot reject that the mean WTP for women is different from either −1 (that is, a one-to-one substitution between base pay and bonus pay), or from the mean of −0.8 for male respondents. 26. Although being able to document a systematic relationship can provide some credibility to our methodology, on the other hand, a failure to find a systematic relationship between the two would not necessarily invalidate our method because students’ preferences for jobs may change over time, or labor market frictions may prevent workers from matching with jobs that they prefer. Answering this question most directly would require both revealed-choice data that are free of any confounds and stated-choice data, which are usually not available. However, the little evidence that exists shows a close correspondence between preferences recovered from the two approaches (see Hainmueller, Hangartner, and Yamamoto 2015). 27. Respondents were initially contacted through email addresses provided in our earlier data collections. Those with inactive email addresses were then approached through LinkedIn. Respondents received a link to the survey that was programmed in SurveyMonkey and were compensated for completing the survey. As shown in Online Appendix Table A6, there is little evidence of selection on observables (reported in 2012) in terms of who participates in the follow-up survey. Based on a joint F-test, we cannot reject that the covariates are jointly zero (p-value = .360) in predicting survey response. Note that for students to have taken both the 2010 and 2012 surveys, the sample from which we have consent would have had to be in the junior year or higher in 2012. 28. Online Appendix Table A8 shows that the sample for which we have consent, the sample that takes the follow-up survey, and the sample that was working when the follow-up survey was conducted are all very similar to the full sample. The only dimensions along which they differ are school year and age (which, as explained above, is by construction) and race. Importantly, there are no statistical differences along the dimensions of gender, major, ability, or socioeconomic background. Also note that the follow-up samples, in columns (3) and (4), are not statistically different from the consent sample along any dimension. 29. Because the vast majority of our sample is either in their junior or senior year, and some have already chosen a major, one concern is that the students’ preferences and beliefs, as elicited in our survey data, may be different from the preferences and beliefs they held in the past as they were deciding on a college major. Although we can of course still estimate the relationship between major choice and the data we collect, the interpretation of our estimates in these cases is less clear. One solution is to collect longitudinal data on preferences and beliefs to directly examine the extent to which they change over the life cycle and how this influences college major choices. See Stinebrickner and Stinebrickner (2014b) for an important example. 30. Because the estimated preference parameters are not scale-free, this exercise of shifting the preference parameter by some amount implicitly assumes that the variance of the unobserved factors is the same for all individuals (Train 2003). 31. More specifically, the gender gap declines by about 0.9 percentage points from a baseline predicted gender gap of 35.1%. This is primarily a result of women’s predicted probability of majoring in humanities declining from 55.0% to 53.8%, and their predicted probability of majoring in economics increasing from 18.3% to 19.5%. 32. As described in Section II, in the sample of all college graduates ages 25–40 in the ACS, the mean earnings for full-time employed men is 36% higher than the mean earnings of full-time employed women. 33. 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Google Scholar CrossRef Search ADS Wiswall Matthew, Zafar Basit, “ Replication Data for: ‘Preference for the Workplace, Investment in Human Capital, and Gender’,” Harvard Dataverse ( 2017), doi:10.7910/DVN/MLOGDL Zafar Basit, “ College Major Choice and the Gender Gap,” Journal of Human Resources , 48 ( 2013), 545– 595. Google Scholar CrossRef Search ADS © The Author(s) 2017. Published by Oxford University Press on behalf of the President and Fellows of Harvard College. All rights reserved. For Permissions, please email: journals.permissions@oup.com http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Quarterly Journal of Economics Oxford University Press
# Preference for the Workplace, Investment in Human Capital, and Gender
, Volume 133 (1) – Feb 1, 2018
51 pages
/lp/ou_press/preference-for-the-workplace-investment-in-human-capital-and-gender-lLn7eVRf0c
Publisher
Oxford University Press
ISSN
0033-5533
eISSN
1531-4650
D.O.I.
10.1093/qje/qjx035
Publisher site
See Article on Publisher Site
### Abstract
Abstract We use a hypothetical choice methodology to estimate preferences for workplace attributes from a sample of high-ability undergraduates attending a highly selective university. We estimate that women on average have a higher willingness to pay (WTP) for jobs with greater work flexibility and job stability, and men have a higher WTP for jobs with higher earnings growth. These job preferences relate to college major choices and to actual job choices reported in a follow-up survey four years after graduation. The gender differences in preferences explain at least a quarter of the early career gender wage gap. JEL Codes: J24, J16. I. Introduction The persistence of gender gaps in labor market earnings and the failure of standard variables to fully explain the gaps has prompted the search for alternative models and evidence. One explanation for gender wage gaps is that these arise in part by women “purchasing” certain positive job attributes by accepting lower wages, and men accepting higher earnings to compensate for negative job attributes. These preferences for job attributes may then affect human capital investments, even prior to job market entry. However, empirically isolating the role of worker-side preferences for job attributes is difficult because the equilibrium matching of jobs to workers reflects not only the workers’ preferences but the firms’ preferences as well. Various kinds of labor market frictions, which prevent workers from matching with their most preferred job types, also break the direct connection between observed job choices and worker preferences. Even when the labor market is perfectly competitive, jobs likely vary in many unobserved (to the researcher) characteristics, leading to an omitted variable (selection bias) problem in identifying worker preferences from realized job choices. To address these empirical challenges, this article estimates individual preferences for workplace attributes using a survey of undergraduates from a selective university, New York University (NYU). We collect data on job attribute preferences by presenting undergraduate students with a series of hypothetical job choice scenarios and eliciting their job choices. The hypothetical job scenarios were constructed to offer students a realistic menu of potential jobs varying in expected earnings and other characteristics such as future earnings growth, dismissal probability, and work hours flexibility. The students’ stated preferences for these jobs allows us to construct a “pure” measure of individual preferences—at the time of the survey—for various job characteristics, and estimate, in a simple and robust way, the distribution of their preferences for job attributes. Our data isolate the preference for workplace attributes, free from making explicit assumptions about the equilibrium job allocation mechanism or preferences of employers. We then use the preference data to examine two channels through which preferences could affect the gender wage gap. First, job preferences could affect college major choice, as students perceive that graduating with certain degrees would result in different types of jobs being offered to them. Second, even for students who choose the same major, preferences for job types could cause men and women to accept different types of jobs and result in different earnings. Our data on job preferences, combined with data on perceptions about the job characteristics given major choices, allow us to quantify these two channels and document how workplace preferences affect the gender gap. Our hypothetical choice methodology is a kind of “stated choice” analysis, similar to “conjoint analysis” and “contingent valuation” methods, used in fields including marketing, environmental and natural resource economics, and health.1 Because our data collection in essence conducts a kind of “experiment” at the individual student level, the “panel” data generated by our design allows us to estimate the distribution of preferences, allowing for unrestricted forms of preference heterogeneity. In contrast to our approach, previous work addressing compensating differentials using observed job choices requires generally stronger assumptions about preferences and the firm side of the labor market.2 In our sample of recent high-ability undergraduate students from NYU, we find substantial willingness to pay (WTP) for pecuniary and nonpecuniary aspects of jobs and considerable heterogeneity in their preferences for workplace attributes. We find that students have preferences reflecting a distaste for higher job dismissal potential, and a taste for workplace hours flexibility (the possibility of working part-time, rather than full-time, hours). We estimate that on average students are willing to give up 2.8% of annual earnings for a job with a percentage point lower probability of job dismissal and willing to give up 5.1% of their salary to have a job that offers the option of working part-time hours rather than one that does not offer this option. After dividing our sample by gender, we find that women have a much higher average preference for workplace hours flexibility, with an implied WTP of 7.3% compared to 1.1% for men. Women also have a higher average WTP for more secure jobs: they are willing to give up 4% of their salary for a percentage point lower probability of job dismissal (versus a 0.6% WTP for males). On the other hand, men have a higher average WTP for jobs with higher earnings growth: they are willing to give up 3.4% of annual earnings for a job with a percentage point higher earnings growth (the corresponding estimate for women is a statistically insignificant 0.6%). A natural question is whether preferences recovered from data on hypothetical choices relate to actual occupational outcomes. Using data on reported job characteristics for a subset of our respondents who are employed roughly four years after our original data collection, we find a strong and systematic relationship between estimated preferences and later actual workplace characteristics. Students with strong preferences for flexible hours, distaste for hours, and other nonpecuniary aspects of jobs were later found to be more likely to be working at jobs with those same preferred characteristics. Although these realized job characteristics do not solely reflect preferences (given the issues we raised above), our finding of a correlation between pre–labor market job preferences and later actual job characteristics suggests some added credibility of our research design. Our finding of large differences in WTP for job amenities between men and women is consistent with prior work noting that women are more likely to be found in jobs offering greater workplace f lexibility (Goldin and Katz 2011; Flabbi and Moro 2012; Goldin 2014; Wasserman 2015; Bronson 2015). However, the observation that women tend to work in certain job types may not reveal women’s preferences alone, but may be affected by firm-side demands for specific workers and discrimination or be driven by some other job attributes that are unobserved in our data sets (Blau and Kahn forthcoming).3 Our innovation is to quantify the WTP for job attributes using a flexible and robust methodology. In related recent work, Mas and Pallais (forthcoming) conduct a field experiment where call center job applicants are offered various work time schedules and wages. Their finding that women have a higher valuation for worker-friendly alternative work arrangements (and a stronger distaste for employer discretion over their hours) is consistent with our estimates of a higher female valuation of work hours flexibility (availability of part-time work).4 We next test whether the job preferences young adults hold in college in fact affect their human capital investments during college. To quantify the importance of job attributes to major choice, we collect additional survey data on students’ beliefs about the characteristics of jobs they would be offered if they were to complete different majors. These data are then used to estimate a model of major choice, where students receive utility from major-specific characteristics (such as perceived ability in those majors) and from the job attributes they associate with these majors. We find that job attributes have a sizable impact on major choice. For example, increasing the perceived job firing probability by a standard deviation reduces the probability of pursuing a major, on average, by 5% (4%) for women (men). To put this change in perspective, a standard deviation increase in average earnings leads to a 5% (16%) increase, on average, in the likelihood of majoring in that field for women (men). Thus, for women, this change is equivalent to the effect on major choice of increasing earnings by one standard deviation. We find meaningful effects for other job attributes, such as work hours. In general, we find that women’s major choices are more responsive to changes in nonpecuniary job attributes (relative to changes in earnings) than are men’s. By linking job preferences directly to human capital investments, we contribute to our limited understanding of how career and workplace preferences shape educational choices. Prior research on college major choice examines the role of earnings expectations, ability perceptions, college costs, and tastes, but generally does not examine nonpecuniary job attributes.5 An exception is Zafar (2013), which estimates a model of college major choice that incorporates some nonpecuniary workplace attributes. However, the framework does not allow for unobserved heterogeneity in preferences, and incorporates a smaller set of workplace characteristics. Closely related to our work is Arcidiacono et al. (2015), who study a sample of male undergraduate students and collects expectations about earnings in different major-occupation pairs. They find evidence for complementarities in preferences between different majors and occupations, and conclude that nonmonetary considerations are key determinants of occupational choice (conditional on graduating from a given college major). Our contribution is to directly quantify the role of specific nonmonetary factors in major choice. Finally, we turn to a key question in the social sciences and ask what our results imply for the gender wage gap. Systematic gender differences in workplace preferences may affect the gender wage gap through two channels: first, it may cause men and women to choose different fields of study, and second, men and women may choose systematically different jobs within the same field. We find that the main channel for preferences to affect the gender gap operates through the second channel, with a smaller effect through major choice. Our analysis reveals that the gender gap in expected earnings early in the career (age 30) would be reduced by at least a quarter if women did not differ from men in the workplace preferences we consider. Remarkably, we find a similar impact on the gender gap in actual earnings for the subset of respondents for whom we have follow-up data. Our evidence supports the notion that at least part of the early career gender wage gap is the result of women “purchasing” certain positive job attributes by accepting lower wages, and men accepting higher earnings to compensate for negative job attributes. In understanding our results, it is important to note that we measure preferences at a particular point in the life cycle of our sample, when our sample was in college. The preferences we measure are not necessarily intrinsic; these preferences were formed by a variety of influences before and during college, and could change substantially after graduation. In addition, it is likely that workplace flexibility issues are much larger determinants of the gender earnings gap for college graduates 10 or more years into their career than for the young college graduates in our study (who are in their mid-20s during our follow-up), as college-graduate women now have children at later ages.6 The article is organized as follows. In the next section, we briefly provide some context for our analysis by using nationally representative surveys for the United States on currently employed individuals. Section III describes our data collection; Section IV details the model of job choice and shows how hypothetical data can solve important identification issues with realized choice data. Section V provides the empirical estimates of job preferences. Section VI quantifies the importance of job attributes for college major choice. Section VII investigates the extent to which gender-specific job preferences can explain the gender gap in earnings. Finally, Section VIII concludes. All appendixes are available online. II. Background: Gender Differences in Job Choices and Human Capital Investments in the United States To set the stage for the analysis of our hypothetical choice scenario data, we first briefly describe the distribution of college majors, jobs, and associated job characteristics. To do so, we use two large-sample, representative data sets for the United States, the January 2010–December 2012 monthly Current Population Survey (CPS) and the 2013 American Community Survey (ACS). Table I shows the job attributes across sectors. For this purpose, we use the sample of 25–60-year-old labor market participants with at least a bachelor’s degree in the 2010–2012 CPS. The first two columns of Table I show that the gender distribution across work sectors differs (Online Appendix A provides details on how variables in this table were constructed). While nearly half of college-educated women workers are in health or education, less than 20% of college-educated male workers are employed in these sectors. These sectors differ substantially in their labor market returns: column (3) of Table I shows that average annual earnings of full-time workers are the lowest for education and health. These sectors differ along other dimensions as well: more than a quarter of the workers in health and education are employed part-time, possibly suggesting the compatibility of these sectors to work-hours flexibility. Job instability, as measured by the likelihood of being fired, is lowest in the government and education sectors. Of course, jobs in these sectors will also differ in the skills that they demand of their workers. So what explains the propensity of men and women to work in different sectors—is it differences in preferences for workplace attributes, differences in tastes for occupations/industries, or differences in skills? What is the role of the labor market structure, firm labor demand, and discrimination by employers? The observed distribution of jobs by gender we see in the data are equilibrium outcomes, and we cannot ascertain from these data alone the extent to which these outcomes are due to worker demand or due to the supply of certain jobs—for example, part-time work may either be a voluntary or involuntary decision. TABLE I Job Attributes by Broad Sector for College Graduates % of males working ina % of females working in Annual earnings for full-time Hrs/wk for full-time Prop. of part-time workers Yearly firing rateb Prop. male workersc Annual % raise in earningsd (1) (2) (3) (4) (5) (6) (7) (8) Sectors Science 9.6 4.0 82,739 44.2 15.5 3.5 67.7 3.8 (35,989) (7.2) (3.0) (1.6) (1.1) (19.9) Health 8.6 22.5 65,427 43.6 28.6 4.0 20.6 4.2 (35,246) (7.8) (1.1) (0.7) (0.5) (22.9) Business 14.2 11.5 77,079 45.0 19.9 4.0 44.6 4.9 (39,023) (7.8) (1.8) (1.3) (0.8) (21.4) Government 6.8 5.8 67,603 43.3 16.2 1.4 52.9 5.8 (32,322) (6.9) (5.0) (0.9) (0.6) (22.4) Education 11.2 25.5 60,588 44.0 30.0 1.8 30.9 4.2 (29,159) (7.5) (2.9) (1.2) (0.4) (21.1) Manufacturing 22.4 7.99 77,354 45.4 17.6 6.2 78.0 5.7 & agriculture (37,257) (7.93) (1.6) (0.8) (0.4) (22.4) Services & trade 27.2 22.8 65,734 45.3 34.4 6.6 51.7 4.9 (37,883) (8.3) (1.0) (0.9) (0.3) (22.2) p-valuee .000 .000 .000 .000 .000 .000 .000 0.170 % of males working ina % of females working in Annual earnings for full-time Hrs/wk for full-time Prop. of part-time workers Yearly firing rateb Prop. male workersc Annual % raise in earningsd (1) (2) (3) (4) (5) (6) (7) (8) Sectors Science 9.6 4.0 82,739 44.2 15.5 3.5 67.7 3.8 (35,989) (7.2) (3.0) (1.6) (1.1) (19.9) Health 8.6 22.5 65,427 43.6 28.6 4.0 20.6 4.2 (35,246) (7.8) (1.1) (0.7) (0.5) (22.9) Business 14.2 11.5 77,079 45.0 19.9 4.0 44.6 4.9 (39,023) (7.8) (1.8) (1.3) (0.8) (21.4) Government 6.8 5.8 67,603 43.3 16.2 1.4 52.9 5.8 (32,322) (6.9) (5.0) (0.9) (0.6) (22.4) Education 11.2 25.5 60,588 44.0 30.0 1.8 30.9 4.2 (29,159) (7.5) (2.9) (1.2) (0.4) (21.1) Manufacturing 22.4 7.99 77,354 45.4 17.6 6.2 78.0 5.7 & agriculture (37,257) (7.93) (1.6) (0.8) (0.4) (22.4) Services & trade 27.2 22.8 65,734 45.3 34.4 6.6 51.7 4.9 (37,883) (8.3) (1.0) (0.9) (0.3) (22.2) p-valuee .000 .000 .000 .000 .000 .000 .000 0.170 Notes. Table reports means, with standard deviations in parentheses. Statistics are based on the 2010–2012 CPS monthly data. Sample restricted to those with at least a bachelor’s degree, between ages 25 and 60. See Online Appendix A for details on construction of variables and definition of the broad sectors. Variables in columns (3), (4), and (8) are based on full-time workers, and are based on individual-level data. Columns (5)–(7) show the average statistics by sector, with the sector-level standard deviation across the months in parentheses. aProportion of all male workers who are employed in each sector (column sums to 100). bDerived from the monthly firing rate, which is the ratio of workers who are laid off in a given month and have been unemployed for less than one month divided by all employed workers at the beginning of the previous month. cMales as a proportion of all workers in that sector. dConstructed by using the outgoing rotation groups, from the reported earnings in the respondent’s fourth and eighth interview (which are separated by 12 months). eF-test of equality of means/proportions across the industry categories. View Large We next turn to Table II to document the link between field of study and associated job characteristics.7 The table is based on the 2013 ACS, restricting the sample to 25–40-year-olds with at least a bachelor’s degree. The first two columns show that while nearly 55% of women have a bachelor’s degree in humanities, less than 40% of men do. While nearly a quarter of men have a bachelor’s in engineering, the corresponding proportion for women is only 6%. TABLE II Job Attributes by College Major for Young College Graduates, 25 to 40 Years Old Shares Annual Hrs/wk % UE Ann % Malesa Females earnings ($${\}$$) for full-time for full-time Part-time workersb ratec salary raised (1) (2) (3) (4) (5) (6) (7) Bachelor’s (or more) in: Business 24.6 18.8 77,002 45.3 26.8 3.3 4.4 (68,110) (8.1) Engineering 23.3 6.1 86,679 44.8 22.2 2.5 4.8 (60,494) (8.1) Humanities 38.2 55.8 59,328 44.4 37.8 3.5 4.9 (49,697) (7.9) Natural science 13.9 19.3 75,992 44.9 35.1 2.5 5.9 (65,921) (9.6) F-teste .000 .000 .000 .000 .000 .000 .000 Shares Annual Hrs/wk % UE Ann % Malesa Females earnings ($${\}$$) for full-time for full-time Part-time workersb ratec salary raised (1) (2) (3) (4) (5) (6) (7) Bachelor’s (or more) in: Business 24.6 18.8 77,002 45.3 26.8 3.3 4.4 (68,110) (8.1) Engineering 23.3 6.1 86,679 44.8 22.2 2.5 4.8 (60,494) (8.1) Humanities 38.2 55.8 59,328 44.4 37.8 3.5 4.9 (49,697) (7.9) Natural science 13.9 19.3 75,992 44.9 35.1 2.5 5.9 (65,921) (9.6) F-teste .000 .000 .000 .000 .000 .000 .000 Notes. Table shows statistics from the 2013 American Community Survey (ACS), restricting the sample to 25–40-year-olds with at least a bachelor’s degree. Sample size is 204,190 respondents. 173 majors are grouped into four broad categories. Means (std. dev.) shown for annual earnings and hrs/week for full-time workers. aProportion of all 25–40-year-old college-educated males with the specified broad major (column sums to 100). bProportion of part-time workers from pool of those currently employed. cUnemployment rate is number of individuals not employed and currently looking for a job, divided by sum of unemployed and employed respondents. dCalculated by linearly regressing log earnings for a given major group on age (coefficient on age reported). ep-value of F-test of equality of means across majors (rows). View Large Column (3) of Table II shows that these majors differ significantly in their average earnings. Engineering—the field which women are least likely to be present in—has the highest average earnings, while humanities—the most popular bachelor’s field for women—has the lowest average earnings. These majors also differ along other dimensions. Columns (4) and (5) show that work-hours flexibility is the highest for jobs associated with humanities: 38% of all humanities graduates are part-time workers, versus 22% of engineering bachelor’s graduates. Average hours per week for full-time workers are also the lowest in humanities. The last two columns of the table show that job stability and earnings growth also vary significantly across the fields of study. So how much do these gender differences in human capital and job characteristics explain the gender gap in earnings? In a recent analysis, Blau and Kahn (forthcoming) find that the gender wage gap is currently larger at the top of the wage distribution (90th percentile), and has decreased more slowly at the top than at other points in the distribution. In addition, they find that traditional human capital variables (experience and degrees earned) explain little of the recent gender gap. They attribute part of the gender gap in high-skilled occupations to a possible compensating differential. Using the sample of college graduates aged 25–40 from the 2013 ACS (the subsample from Table II that reports nonzero labor income), we find an adjusted gender gap in hourly earnings of about 12 log points (adjusting only for age and full-time status).8 Demonstrating how important college majors could potentially be in explaining the gender gap among college-educated workers, including four broad college major categories (as defined in our analysis) reduces the gender gap by about 43% (from 12 log points to 6.7 log points). Including indicators for detailed occupation, industry, and race categories as in Blau and Kahn (forthcoming), in addition to indicators for major categories, increases the explained portion of the gender wage gap to 58%. However, this analysis reveals that even conditional on detailed occupation/industry and major controls, a large part of the gender wage gap remains unexplained. The remainder of this article investigates the extent to which workplace preferences can explain this gender gap, either by influencing human capital choice (major choice) or by influencing job choices conditional on major. III. Data This section describes the administration of the data collection, the form of the hypothetical choice scenarios, and the sample we use for the estimation. III.A. Administration Our data are from an original survey instrument administered to NYU undergraduate students over a two-week period during May 2012. NYU is a large, selective, private university located in New York City. The students were recruited from the email list used by the Center for Experimental Social Sciences (CESS) at NYU. Students were informed that the study consisted of some simple economic experiments and a survey about educational and career choices. Upon agreeing to participate, students could sign up for a 90-minute session, which was held in the CESS Computer Lab located on the main NYU campus.9 The data for this article were collected through a computer-based survey (constructed using the SurveyMonkey software). The survey took approximately 30 minutes to complete and consisted of several parts. Many of the questions had built-in logical checks (e.g., percent chances of an exhaustive set of events such as majors had to sum to 100). Students were compensated $${\}$$10 as a show-up fee, and $${\}$$20 for successfully completing the survey. III.B. Data Collection Instrument In addition to questions about demographics, family background, and educational experiences, the main survey instrument consisted of two parts. The first part collected data on students’ preferences for job attributes using hypothetical job choices, while the second collected data on consequential life activities that would plausibly be key determinants of college major choice, such as attributes of jobs associated with each major and measures of the student’s perception of their ability to complete the coursework for each major. We describe the hypothetical job choice data in detail next and leave the description of major-specific data to a later part of the article, where we relate the job attribute preferences to college major choices. Our hypothetical job choice data were collected by presenting students with a total of 16 job scenarios. Each scenario consisted of three different potential jobs. We exogenously varied different aspects of the job with the intention of creating realistic variation in job attributes. The first eight hypothetical job scenarios were introduced as follows: In each of the 8 scenarios below, you will be shown hypothetical jobs offers. Each job offer is characterized by: Annual earnings when working full-time Annual percentage increase in earnings from age 30 onwards until retirement Full-time work hours per week Work flexibility (whether part-time work is an option); part-time work is work where you only work at most half as many hours as full-time work and for half of the full-time salary These jobs are otherwise identical in all other aspects. Look forward to when you are 30 years old. You have been offered each of these jobs, and now have to decide which one to choose. In each scenario, you will be asked for the percent chance (or chances out of 100) of choosing each of the alternatives. The chance of each alternative should be a number between 0 and 100 and the chances given to the three alternatives should add up to 100. Each scenario consisted of three jobs, with each job being characterized by four attributes. The notable point that was highlighted was that these jobs were identical in all other aspects. The jobs did not have any occupation labels on them.10 The last eight scenarios were introduced in a similar way, except that the job offer was now characterized by a different set of attributes: annual earnings when working full-time, probability of being fired over a one-year period, amount of additional annual bonus pay based on relative performance the respondent may qualify for (in addition to base pay), proportion of men in the firm in similar job positions. All survey respondents received identical scenarios in the same order. Following the approach of Blass, Lach, and Manski (2010), we asked respondents to provide a choice probability instead of a discrete choice (that is, a 0 or 1). This allows respondents to express uncertainty about their future behavior. It also allows them to rank their choices, providing more information than if we asked only about the most preferred job. As is standard in studies that collect subjective probabilistic data, a short introduction on the use of percentages was provided. In addition, respondents answered some practice questions to become familiar with expressing probabilistic answers. Besides earnings, the scenarios focus on six different job attributes. We chose not to vary these six dimensions all at once since the cognitive load to process such information could have been overwhelming. We focus on these dimensions based on findings from prior literature, and the fact that there is considerable variation along these dimensions across occupations as well as majors (Tables I and II). Earnings and earnings growth were included since they have been found to be a factor in career/education choice (see Wiswall and Zafar 2015a, and references therein). Work hours and work flexibility are included because they tend to be associated with the remuneration structure in jobs and the associated gender gap in earnings (Flabbi and Moro 2012; Goldin 2014; Cortes and Pan 2016). We recognize that workplace flexibility is a multidimensional concept: for example, the number of hours to be worked matters but perhaps so do the particular hours (Goldin 2014; Mas and Pallais forthcoming). We varied two hours-related attributes: number of hours and the availability of a part-time option, since these are easy to vary in a meaningful fashion. Job stability, as proxied by the likelihood of being fired from the job, is included because of the importance of risk and uncertainty to job choices (Dillon forthcoming) and gender differences in risk preferences (Croson and Gneezy 2009). Finally, relative performance compensation and proportion of men are meant to capture the competitiveness of the job environment, preferences for which have been found to differ by gender (Niederle and Vesterlund 2007; Flory, Leibbrandt and List 2015; Reuben, Wiswall, and Zafar forthcoming).11 To keep the scenarios realistic, the job attributes shown to respondents in the scenarios were based on the actual marginal distribution of job characteristics in the CPS (except for the bonus pay variable, since data were not available for that dimension).12 In addition, no scenario included a job that was clearly dominant or dominated along all dimensions. We also made a conscious effort to keep the variation in job attributes within each scenario relatively “local,” so that the claim that the jobs were otherwise identical was credible; for example, two jobs offering $${\}$$50,000 and $${\}$$90,000, respectively, with little variation along the specified dimensions are unlikely to be identical. At the same time, we had substantial variation in the job attributes across the scenarios. This ensures that we are not recovering preferences in a local region only. Online Appendix Table A1 shows the range of the attributes across the scenarios. III.C. Sample Description A total of 257 students participated in the study. We drop 10 respondents for whom we have missing data for the relevant section of the survey. Sample characteristics are shown in Table III. Thirty-five percent of the sample (86 respondents) is male, 29% is white, and 51% is Asian. The mean age of the respondents is 21.5, with 11% of respondents freshmen, 11% sophomores, 37% juniors, and the remaining seniors or higher. The average grade point average of our sample is 3.5 (on a 4.0 scale), and students have an average Scholastic Aptitude Test (SAT) math score of 696, and a verbal score of 674 (with a maximum score of 800). These correspond to the 93rd percentile of the U.S. national population score distributions. Therefore, as expected, our sample represents a high ability group of college students. Parents’ characteristics of the students also suggest that they are overrepresented among high socioeconomic groups. The last panel of the table shows that 48% of the students have a major in the humanities and social sciences category, 31% have a major in business and economics, while the remaining have a major in natural sciences and math (16%), and engineering (5%). TABLE III Sample Statistics All Males Females p-value (1) (2) (3) (4) Number of respondents 247 86 161 School year: Freshmen 10.9% 9.3% 11.8% .549 Sophomore 10.9% 11.6% 10.6% .798 Junior 36.4% 32.6% 38.5% .355 Senior or more 41.7% 46.5% 39.1% .262 Age 21.49 21.69 21.37 .103 (1.5) (1.8) (1.2) Race: White 29.2% 33.7% 26.7% .248 Asian 50.6% 51.1% 50.3% .898 Non-Asian minority 17.8% 14.0% 19.9% .247 Parent’s characteristics: Parents’ income ($${\}$$1,000s) 137 141 135 .731 (121) (126) (118) Mother B.A. or more 67.6% 74.4% 64.0% .095 Father B.A. or more 69.6% 72.1% 68.3% .539 Ability measures: SAT math score 696.0 717.7 684.3 .006 (88) (72) (94) SAT verbal score 674.0 677.0 672.5 .704 (84) (78) (88) GPA 3.5 3.5 3.5 .938 (0.32) (0.33) (0.32) Intended/current major Economics/business 31.2% 48.8% 21.7% .000 Engineering 4.9% 8.1% 3.1% .080 Humanities and soc sciences 47.8% 30.2% 57.1% .000 Natural sciences/math 16.2% 12.8% 18.0% .289 All Males Females p-value (1) (2) (3) (4) Number of respondents 247 86 161 School year: Freshmen 10.9% 9.3% 11.8% .549 Sophomore 10.9% 11.6% 10.6% .798 Junior 36.4% 32.6% 38.5% .355 Senior or more 41.7% 46.5% 39.1% .262 Age 21.49 21.69 21.37 .103 (1.5) (1.8) (1.2) Race: White 29.2% 33.7% 26.7% .248 Asian 50.6% 51.1% 50.3% .898 Non-Asian minority 17.8% 14.0% 19.9% .247 Parent’s characteristics: Parents’ income ($${\}$$1,000s) 137 141 135 .731 (121) (126) (118) Mother B.A. or more 67.6% 74.4% 64.0% .095 Father B.A. or more 69.6% 72.1% 68.3% .539 Ability measures: SAT math score 696.0 717.7 684.3 .006 (88) (72) (94) SAT verbal score 674.0 677.0 672.5 .704 (84) (78) (88) GPA 3.5 3.5 3.5 .938 (0.32) (0.33) (0.32) Intended/current major Economics/business 31.2% 48.8% 21.7% .000 Engineering 4.9% 8.1% 3.1% .080 Humanities and soc sciences 47.8% 30.2% 57.1% .000 Natural sciences/math 16.2% 12.8% 18.0% .289 Notes. For the continuous outcomes, means are reported in the first cell, and standard deviations are reported in parentheses. p-value reported for a pairwise test of equality of means (proportions) between males and females, based on a t-test (chi-square test). View Large Columns (2) and (3) of Table III report the characteristics by gender. The last column of the table reports the p-value of tests of equality of the statistics by gender. We see that male and female respondents are similar in all dimensions, except two. First, male students in our sample have a significantly higher average SAT math score than females, of about 33 points. Second, the two sexes choose very different college majors. Nearly half (49%) of men report majoring in business/economics, with 30% majoring in humanities and social sciences, and 13% in natural sciences/math. On the other hand, 57% of the women report majoring in humanities and social sciences, followed by about 22% majoring in business/economics, and 18% majoring in natural sciences/math. That is, female students are almost twice as likely as men to major in the humanities (the field, as we show below, perceived to have the lowest average earnings among college graduates), and only half as likely as males to major in economics/business (the perceived highest-earnings major category). The gender-specific major distributions are statistically different (p-value ≤ .001, using a chi-square test for equality of distributions). These substantial gender gaps in major choice mirror the national patterns from the ACS data (Table II). Compared to the NYU population, our sample has a similar proportion female: 63% of students graduating NYU in 2010 are women compared with 65% in our sample (data from the Integrated Post-Secondary Education Data System, IPEDS). For all incoming freshman in 2010, the 25th and 75th quartiles of the SAT math were 630 and 740 and for the SAT verbal were 610 and 710 (IPEDS). The equivalent quartiles in our sample are 650 and 770 for math and 620 and 730 for verbal. Our sample is weighted more toward business/economics majors than in the actual NYU population graduating in 2010, possibly because the experimental laboratory is located in the building housing the Economics Department. However, the gender differences in major choice are similar.13 IV. Model and Identification Analysis In this section, we present a simple attribute-based job choice model and discuss identification of the model using two types of data: (i) standard realized job choices (as observed after job offers and acceptances are made), and (ii) stated probabilistic job choices (as observed in our job hypotheticals experimental data). We show that under weak conditions the job hypotheticals data identify the distribution of job preferences, while standard realized job choice data do not. IV.A. A Canonical Random Utility Model of Job Choice Jobs are indexed by j, and there is a finite set of jobs j = 1, …, J. Each job is characterized by a vector of K attributes Xj = [Xj1, …, XjK]. These job attributes include earnings and various nonpecuniary attributes, such as job dismissal probabilities and work-hours flexibility. Thus, we explicitly allow for the possibility that individuals are not necessarily pure income or consumption maximizers, and may value many other outcomes associated with their job choice. Let Uij ∈ R be individual i’s utility from job j. The utility from job j is $$U_{ij}= u_{i}(X_{j}) + \epsilon _{ij}.$$ (1)ui(X) ∈ R is the preferences of individual i over the vector of characteristics X. εij ∈ R is the additional job-specific preference component for job j reflecting all remaining attributes of the job which affect utility, if any. Let εi be the vector of these components for individual i, εi = εi1, …, εiJ. After observing the attributes X1, …, XJ for all jobs and εi, individual i chooses the one job with the highest utility: i chooses job j if $$U_{ij}>U_{ij^{\prime }}$$ for all j΄ ≠ j. Population preferences for jobs is the collection of ui preferences over the job attributes X and the job-specific components εi. The joint distribution of preferences in the population is given by F(ui, εi). This distribution determines the fraction of individuals choosing each job, qj ∈ [0, 1]: \begin{eqnarray} q_{j} &=& pr(\mbox{choose job }j)\nonumber \\ &=& \int 1\lbrace U_{ij}>U_{ij^{\prime }}\, \mbox{for all }j^{\prime } \ne j\mbox{}\rbrace dF(u_{i},\epsilon _{i}). \end{eqnarray} (2) IV.B. Identification Using Realized Choice Data Typically empirical research on job choice consists of analyzing data on actual or realized job choices, which provides the one best job chosen by each individual.14 To analyze the potential advantages of hypothetical data, we first detail the identification using realized choice data. A common model of realized choice data assumes εi1, …, εiJ are i.i.d. Type I extreme value, and independent of preferences represented by ui. The probability individual i chooses job j, given some characteristics X1, …, XJ for all jobs, is given by \begin{equation*} q_{ij} = \frac{\exp (u_{i}(X_{j}))}{\sum _{j^{\prime }=1}^{J}\exp (u_{i}(X_{j^{\prime }}))}. \end{equation*} The population fraction choosing job j is then $$q_{j}=\int \frac{\exp (u_{i}(X_{j}))}{\sum _{j^{\prime }=1}^{J}\exp (u_{i}(X_{j^{\prime }}))}dG(u_{i}),$$ (3)where we have kept the dependence of the job choice on the job characteristics X1, …, XJ implicit. G(ui) is the distribution of preferences over attributes ui in the population. Equation (3) is the mixed multinomial logit model of McFadden and Train (2000). They show that the distributional assumption on the εi terms that yield the logit form is without any meaningful loss of generality as this model can arbitrarily closely approximate a broad class of random utility models. For ease of exposition, we consider a linear model of utility given by ui(X) = X΄βi. A key concern in using realized job choices is that the data set of job characteristics which the researcher has at hand is not complete in the sense that there are omitted unobserved job characteristics that are potentially correlated with the included observed characteristics. Divide the vector of job characteristics X into observed X(obsv) and unobserved characteristics X(unob), X = [X(obsv), X(unob)]. Similarly divide the vector of preference parameters βi = [βi(obsv), βi(unob)]. The log odds of job j relative to job j΄ for individual i is then: \begin{eqnarray*} \ln \left(\frac{q_{ij}}{q_{ij^{\prime }}}\right) &=& (X_{j}(obsv)-X_{j^{\prime }}(obsv))\beta _{i}(obsv)+(X_{j}(unob)\nonumber \\ && -\, X_{j^{\prime }}(unob))\beta _{i}(unob)\nonumber \\ &&=(X_{j}(obsv)-X_{j^{\prime }}(obsv))\beta _{i}(obsv) + \,\eta _{ij}, \end{eqnarray*} where qij and $$q_{ij^{\prime }}$$ is the probability of choosing job j and j΄, respectively, for individual i. $$\eta _{ij} = (X_{j}(unob)-X_{j^{\prime }}(unob))\beta _{i}(unob)$$ is the omitted variable for individual i. The omitted variable bias problem is the generic one found in a variety of contexts: the omitted unobserved job characteristics Xj(unob) are correlated with the observed characteristics Xj(obsv). For example, if the researcher’s data set includes only current salaries, but not any of the nonpecuniary benefits of the job, we would expect that the estimate of preferences for salaries will be biased. The theory of compensating differentials (Rosen 1987) predicts a close connection among various job characteristics—a trade-off between salary and nonpecuniary benefits—and therefore would suggest important omitted variable bias in estimates of job preferences using realized data. The omitted variable bias issue could also arise more subtly from the selection/matching mechanism to jobs, reflecting employer preferences over potential job candidates. If the labor market equilibrium is such that employers only offer a limited set of jobs to candidates, then the realized jobs they hold do not reflect their preferences only.15 Discrimination by employers, by which employers prefer not to hire workers of certain groups (e.g., women, minorities), is one example (Becker 1971). In the presence of important demand-side considerations, one would not want to interpret the equilibrium allocation of jobs as reflecting only worker preferences. As we detail below, our hypothetical data avoid this issue because they experimentally manipulate the characteristics offered to individuals, thereby allowing a “pure” measure of preferences, free from considering the equilibrium job allocation mechanism, preferences of employers, or any omitted unobserved job characteristics. Another approach to this issue is to make some assumptions about the structure of the labor market and individual preferences. As in the literature examining identification of these models using observed choices (see Fox et al. 2012 for a recent review), some support condition or restriction on preferences is therefore necessary for identification. IV.C. Model of Hypothetical Job Choices We next consider a framework for analyzing hypothetical job choice data, connecting the canonical model of realized job choice specified above in equation (1) with the hypothetical job choice data we collect. Our hypothetical data are asked prior to a job choice (while students are in school). We observe each individual’s beliefs about the probability they would take each hypothetical future job offered within the scenario (and not simply the individual’s one chosen or realized job). To analyze this type of data, we require a model of hypothetical future jobs. Our model of hypothetical job choices presumes individuals are rational decision makers who anticipate the job choice structure as laid out in the canonical model of job choice, equation (1). To allow for the possibility of uncertainty about future job choices, we assume that the realizations of εi1, …, εiJ job-specific utility terms are not known at the time we elicit individual beliefs. Individual i then faces a choice among J hypothetical jobs with characteristics vectors X1, …, XJ. Each individual i expresses their probability of taking a given job j as: $$p_{ij} = \int 1 \lbrace U_{ij} > U_{ij^{\prime }} \, \mbox{for all }j^{\prime } \ne j\mbox{} \rbrace d H_{i}( \epsilon _{i} ),$$ (4)where Hi(εi) is individual i’s belief about the distribution of εi1, …, εiJ elements. As in Blass, Lach, and Manski (2010), εi has an interpretation of resolvable uncertainty, uncertainty at the time of our data collection but uncertainty that the individual knows will be resolved (i.e., known or realized) prior to making the job choice.16 It should be noted that the preferences for workplace attributes elicited in our data collection are potentially specific to the time at which the survey is collected (during the college years in our case). Preferences for job attributes may change as individuals age and may have been different when the students in our sample were younger (say, prior to college) and may be different still when they actually enter the labor market and make job choices. With this caveat in mind, we can still use our research strategy to understand job preferences at a point in time and study how these preferences relate to important human capital investments that are being made contemporaneously.17 IV.D. Identification Using Hypothetical Choice Data We previously analyzed identification of preferences using realized job-choice data and discussed a key shortcoming: realized choice data potentially suffers from omitted variable bias. Hypothetical choice data can overcome this shortcoming and allow a general method to identify heterogeneity in job-choice preferences. First, because we can experimentally manipulate the hypothetical choice scenarios we provide individuals, we may be able to reduce bias from the correlation of observed and unobserved job characteristics. Rather than use naturally occurring variation in realized job choices—which are in general the result of many unobserved job characteristics and an unknown labor market equilibrium mechanism, as discussed above—we present individuals with an artificial set of job choices. Although the job characteristics we provide are certainly not exhaustive of all possible job characteristics, and are purposely kept limited so as not to “overload” the respondents with too many job features, the key feature of the hypothetical experimental setting is that we instruct respondents that the jobs differ only in the job characteristics we provide, and are otherwise identical. This distinguishes our design from “audit”-based studies in which employers are presented with résumés that are otherwise identical except for the one chosen attribute (say, the gender of applicant). The criticism of audit studies is that even if you make two groups (say, men and women) identical on observables, employers might have very different distributions in mind about unobservables for the two groups, biasing the inference (for an analysis of this issue, see Neumark, Burn, and Button 2015). In our case, students are instructed that the hypothetical jobs are identical in all other ways, instructions that cannot be given to actual employers in audit studies. The extent of the remaining bias in the preferences we elicit then critically depends on whether respondents fully internalize our instructions that the jobs are otherwise identical. There is reason to suspect this may not strictly be the case. Like audit studies, the participants in our study may still have preconceived notions of what other attributes are related to the attributes we include. For example, they might believe the availability of part-time work (one of the attributes we include) is associated with other aspects of flexibility we do not include, such as time of day one is allowed to work and the ability to take vacations and family care leaves. Dismissal risk (also one of the attributes we include) could be viewed as a proxy for high-stress, high-expectations environments. These types of biases are not different from those present in audit studies where employers have their own prior beliefs about other attributes of workers associated with different observable (on résumé) worker characteristics. A second advantage of the hypothetical data is that it provides a kind of panel data on preferences which, under fairly weak assumptions, identify the full preference rankings over job attributes. Notice the key distinction between equations (4) and (2). With job hypotheticals data, we observe for each individual i multiple subjective job probabilities pi1, …, piJ. The job hypotheticals provide a type of panel data allowing less restricted forms of identification, by allowing identification of the ui(X) preferences without a parametric restriction on the population distribution of preferences. Note that even with a panel of realized choices, it is in general impossible to identify separately preferences for jobs from search frictions or omitted job characteristics. Within our hypothetical setup, these issues are, by design, not a confounding factor. Our assumption for identification of preferences is that the εi1, …, εiJ job-specific terms are i.i.d. and independent of the experimentally manipulated job attributes X1, …, XJ. This is implied by the experimental design: respondents are instructed that the jobs vary only in the listed characteristics and are otherwise identical. Under this assumption, the hypothetical data pi1, …, piJ identifies the preference ranking for individual i over all jobs J in the choice set: For any two jobs j and j΄, the characteristics vector Xj is preferred to that of $$X_{j^{\prime }}$$ if the probability of choosing that job is higher than that for job j΄, $$p_{ij}>p_{ij^{\prime }}$$. Our identification concept is that each scenario approximates a multidimensional offer function from which a worker can choose the optimal bundle of job attributes. If this offer function were complete (that is, a continuum of choices rather than three job options in each scenario), the worker would choose the point that is tangent to their indifference curve. Rosen (1987) argues that worker preferences can then be identified if the offer curve shifts, forcing workers to reoptimize in a frictionless labor market, and tracing out the worker’s indifference curve. This is effectively what happens when respondents are presented with another job-choice scenario (another set of jobs to choose from) in our survey. The key distinction relative to the Rosen case is that our choice set is discrete, so we can instead think of preferences as being identified by a set of job preference inequalities. This is an important improvement relative to identification using observed job choices because there is information in our data on rejected job opportunities, which are not typically available in real labor-market settings.18 This rejected-offer information provides both lower and upper bounds on preferences in a discrete-choice setting and can point-identify preferences nonparametrically (up to the distribution of the εi shocks) with full support of the job offer variation. In practice, of course we have a only finite number of job scenarios and cannot vary job offers to saturate the full support of the job characteristics. As in the literature examining identification of these models using observed choices (see Fox et al. 2012 for a recent review), some support condition or restriction on preferences is therefore necessary, although more limited than is required using observational data. We assume preferences take a parametric form, $$u_{i}=X_{i}^{\prime }\beta _{i}$$, but allow the βi parameters to be freely varying in the population. This allows for the distribution of preference parameters βi to be completely unrestricted across individuals; thereby we avoid making assumptions about the population distribution of preferences (such as assuming preferences βi are normally distributed). In the estimation, we use this identification result constructively and simply estimate preferences for each sample respondent one by one. We then use the sample distribution of preferences as the sample estimator of the population distribution of preferences. Therefore, we allow the distribution of preferences to take any form.19 V. Estimates of Preferences for Job Characteristics V.A. Variation in Choice Probabilities Identification relies on variation in probabilities that respondents assign to the various jobs in the hypothetical scenarios. We next present some evidence on this, which should allow the reader to become familiar with the sources of identifying variation. Table IV, Panel A shows two examples from the data sample using the first set of hypothetical scenarios. Recall that each of these eight scenarios included three different job offers, which differed according to the characteristics shown in the table. The last two columns show the mean probability assigned by each gender to the jobs. TABLE IV Example Choice Scenarios Probability assigned by: Panel A Earnings per year at age 30 if working full time Annual percentage increase in earnings from age 30 on Average work hours per week for full-time Work flexibility: part-time work available? Males Females Example 1 Job 1 $${\}$$96,000 3 52 Yes 31.93 [30] 31.46 [30] (22.48) (21.36) Job 2 $${\}$$95,000 2 45 Yes 31.16 [30] 39.34*** [40] (23.71) (22.71) Job 3 $${\}$$89,000 4 42 No 36.91 [30] 29.20** [25] (24.71) (22.57) Example 2 Job 1 $${\}$$76,000 4 50 Yes 19.38 [20] 20.65 [20] (19.34) (15.23) Job 2 $${\}$$81,000 3 44 Yes 49.47 [50] 49.45 [50] (26.63) (22.08) Job 3 $${\}$$88,000 2 49 No 31.15 [25] 29.91 [25] (25.36) (21.98) Probability assigned by: Panel A Earnings per year at age 30 if working full time Annual percentage increase in earnings from age 30 on Average work hours per week for full-time Work flexibility: part-time work available? Males Females Example 1 Job 1 $${\}$$96,000 3 52 Yes 31.93 [30] 31.46 [30] (22.48) (21.36) Job 2 $${\}$$95,000 2 45 Yes 31.16 [30] 39.34*** [40] (23.71) (22.71) Job 3 $${\}$$89,000 4 42 No 36.91 [30] 29.20** [25] (24.71) (22.57) Example 2 Job 1 $${\}$$76,000 4 50 Yes 19.38 [20] 20.65 [20] (19.34) (15.23) Job 2 $${\}$$81,000 3 44 Yes 49.47 [50] 49.45 [50] (26.63) (22.08) Job 3 $${\}$$88,000 2 49 No 31.15 [25] 29.91 [25] (25.36) (21.98) Probability assigned by: Panel B Earnings per year at age 30 if working full time Probability (%) of being fired from the job in the next year Amount of bonus based on relative performance (% of full time earnings) Proportion (%) of men in the firm in similar positions Males Females Example 1 Job 1 $${\}$$87,000 1 $${\}$$4,350 (5) 49 30.34 [30] 36.68* [30] (22.48) (24.33) Job 2 $${\}$$84,000 6 $${\}$$10,920 (13) 67 26.86 [30] 30.27 [30] (23.71) (21.36) Job 3 $${\}$$95,000 5 $${\}$$4,750 (5) 69 42.80 [31.5] 33.05*** [30] (24.71) (20.83) Example 2 Job 1 $${\}$$61,000 1 $${\}$$6,710 (11) 41 25.48 [20] 26.80 [20] (26.57) (23.20) Job 2 $${\}$$65,000 5 $${\}$$7,800 (12) 71 12.14 [9.5] 15.53** [10] (12.98) (11.81) Job 3 $${\}$$67,000 2 $${\}$$10,050 (15) 60 62.38 [60] 57.67 [60] (31.55) (27.19) Probability assigned by: Panel B Earnings per year at age 30 if working full time Probability (%) of being fired from the job in the next year Amount of bonus based on relative performance (% of full time earnings) Proportion (%) of men in the firm in similar positions Males Females Example 1 Job 1 $${\}$$87,000 1 $${\}$$4,350 (5) 49 30.34 [30] 36.68* [30] (22.48) (24.33) Job 2 $${\}$$84,000 6 $${\}$$10,920 (13) 67 26.86 [30] 30.27 [30] (23.71) (21.36) Job 3 $${\}$$95,000 5 $${\}$$4,750 (5) 69 42.80 [31.5] 33.05*** [30] (24.71) (20.83) Example 2 Job 1 $${\}$$61,000 1 $${\}$$6,710 (11) 41 25.48 [20] 26.80 [20] (26.57) (23.20) Job 2 $${\}$$65,000 5 $${\}$$7,800 (12) 71 12.14 [9.5] 15.53** [10] (12.98) (11.81) Job 3 $${\}$$67,000 2 $${\}$$10,050 (15) 60 62.38 [60] 57.67 [60] (31.55) (27.19) Notes. Means [median] (std. dev.) reported in the last two columns. Pairwise t-tests conducted for equality of means by gender. Significance denoted on the female column by asterisks: *p <.10, **p <.05, ***p <.01. View Large Turning to the first example, we see that, for men, Job 3 is the most preferred job in our sample (that is, it received the highest average probability). Job 3 is the job without part-time availability and the highest earnings growth. For women, on the other hand, this job received the lowest average probability. Women assigned the highest probability, on average, to Job 2, the job with a part-time option and an intermediate number of work hours per week and intermediate earnings. In this example, the distribution of choices differs significantly by gender. The gender-specific distributions of average probabilities do not differ in the second example. Table IV, Panel B shows two examples from the second set of hypothetical scenarios, which vary a different set of attributes. In the first example, the distribution of average probabilities again differs by gender. For women, Job 1 receives the highest probability on average (37%). Job 1 is the job with the lowest probability of being fired and the lowest proportion of men as colleagues. Male respondents, on the other hand, assign the highest average probability to Job 3, the job with the highest earnings and proportion of men but with a high likelihood of being fired. Another notable aspect of Table IV is the large standard deviation in elicited choice probabilities, reflective of substantial heterogeneity in choices, even within gender. Figure I shows the histogram of elicited percent chance responses for Job 1, pooled across the 16 hypothetical scenarios. Several things are notable. First, responses tend to be multiples of 10 or 5, a common feature of probabilistic belief data (Manski 2004), reflecting a likely rounding bias; this is something we return to below. Second, although there is pooling at multiples of 5, there is little evidence of excessive heaping at the standard focal responses of 0, 50, and 100. The most prevalent response is 20%, but even that receives a response frequency of only 0.11. Third, most respondents (87.5%) report values in the interior (that is, not 0 or 100), reflecting a belief that there is some chance they might choose each of the jobs. This underscores the importance of eliciting probabilistic data, rather than simply the most preferred option, as respondents are able to provide meaningful probabilistic preferences for the full set of choices. Figure I View largeDownload slide Choice Probabilities for Job 1 (Pooled across Hypothetical Scenarios) Figure I View largeDownload slide Choice Probabilities for Job 1 (Pooled across Hypothetical Scenarios) V.B. Empirical Model of Job Preferences Next, we discuss our empirical model of job preferences, which we estimate using our hypothetical data. Our estimator follows the identification analysis we laid out above. For the job preferences over attributes, we use the form ui(X) = X΄βi, where βi = [βi1, …, βiK] is a K-dimensional vector that reflects individual i’s preferences for each of the K job characteristics. The X vector of job characteristics is described below and we consider several different functional forms. We assume beliefs about future job utility Hi(·) in equation (4) are i.i.d. Type I extreme value for all individuals. The probability of choosing each job is then: $$p_{ij}=\frac{\exp (X_{j}^{\prime }\beta _{i})}{\sum _{j^{\prime }=1}^{J}\exp (X_{j^{\prime }}^{\prime }\beta _{i})},$$ (5)where it is important to note that the probabilities assigned to each job j are individual i specific.20 Although we maintain a particular assumption about the distribution of probabilistic beliefs, we place no parametric restrictions on the distribution of preferences, represented by the vector βi. Our goal is to estimate the population distribution of preferences βi. We maintain a maximum degree of flexibility by estimating the preference vector βi separately for each sample member, and do not impose any “global” distributional assumptions about the population distribution of preferences (e.g., that preferences βi ∼ N(μ, Σ)). Applying the log-odds transformation to equation (5) yields the linear model: \begin{equation*} \ln \left(\frac{p_{ij}}{p_{ij^{\prime }}} \right)=(X_{j}-X_{j^{\prime }})^{\prime }\beta _{i}. \end{equation*} βi has the interpretation of the marginal change in the log odds for some level difference in the X characteristics of the job. Given the difficulty of interpreting the βi preference parameters directly, we also present results in which we compute individual-level WTP statistics. V.C. Measurement Error One potential issue in using hypothetical data for estimating preferences is that individuals may report their preferences with error. Given that these preferences have no objective counterpart (we cannot ascertain the “accuracy” of a self-reported preference), we cannot point to definitive evidence on the extent of measurement error. The most apparent potential measurement issue is that individuals report rounded versions of their underlying preferences (rounded to units of 5% or 10%). To guard against the potential of rounding bias or other sources of measurement error, we follow Blass, Lach, and Manski (2010) in introducing measurement error to the model, and in flexibly estimating the model using a least absolute deviations (LAD) estimator. We assume that the actual reports of job choice probabilities in our data, denoted $$\tilde{p}_{ij}$$, measure the “true” probabilities pij with error. The measurement error takes a linear-in-logs form such that the reported log-odds take the following form: $$\ln \left(\frac{\tilde{p}_{ij}}{\tilde{p}_{ij^{\prime }}} \right)=(X_{j}-X_{j^{\prime } })\beta _{i}+\omega _{ij},$$ (6)where ωij is the measurement error. We assume that the ωi1, …, ωiJ have median 0, conditional on the X1, …, XJ observed job characteristics. Given these measurement error assumptions, we have the following median restriction: $$M\left[ \ln \left(\frac{\tilde{p}_{ij}}{\tilde{p}_{ij^{\prime }}}\right)|X_{j},X_{j^{\prime }}\right] =(X_{j}-X_{j^{\prime }})\beta _{i},$$ (7)where M[ · ] is the median operator. This median restriction forms the basis for our estimator. Our measurement error assumptions are limited compared to commonly imposed fully parametric models which assume a full distribution for the measurement error process. In contrast, our assumption is that the measurement errors are only median unbiased.21 Another advantage of the LAD estimator is that it is not sensitive to what the extreme responses (probabilities of 0 and 1) are replaced with. V.D. Estimation We estimate the K-dimensional vector βi by LAD for each student i separately. In our data, each student makes choices across 16 scenarios, assigning probabilities to three possible jobs in each scenario. Equation (7) therefore is estimated for each respondent using 16 × 2 = 32 unique observations. Variation in the job attributes (Xj), which is manipulated exogenously by us, and variation in respondents’ choice probabilities allows us to identify the parameter vector βi. From the full set of estimates of β1, …, βN for our size N sample we estimate population statistics, such as mean preferences, E(βi). We conduct inference on the population statistics using block or cluster bootstrap by resampling (with replacement) the entire set of job-hypothetical probabilities for each student. Online Appendix Section B describes the bootstrapping algorithm. The block bootstrap preserves the dependence structure within each respondent’s block of responses, and allows for within-individual correlation across job-choice scenarios. As discussed in the study design section, we varied four job attributes at a time in each scenario. For estimation, we combine all of these scenarios and assume the dimensions that were not varied in a given scenario were believed by the respondent to be held constant, as we instructed. As mentioned earlier, we instruct respondents that the jobs differ only in the finite number of job characteristics we provide, and are otherwise identical. There is no additional information here that the respondent could use to believe otherwise. The vector of job attributes is as follows: X = {log age-30 earnings; probability of being fired; bonus as a proportion of earnings; proportion of males in similar positions; annual increase in earnings; hours per week of work; availability of part-time}.22 We also include job-number dummies in equation (7) to allow for the possibility that the ordering of the jobs presented could affect job preferences, although there is no prior reason to suspect this given our experimental design.23 V.E. Job Preference Estimates We first discuss the sign and statistical significance level of the βi estimates. Because of the difficulty in interpreting the magnitude of these estimates, below we also present results in which we convert the parameter estimates into an individual-level WTP measure. Recall that we can identify the βi vector without a parametric restriction on the population distribution of preferences. Online Appendix C discusses the estimated heterogeneity in preferences within gender. The first column of Table V shows the average estimate for each job characteristic (across all individual-level estimates). The standard errors in parentheses are derived from a block bootstrap procedure. We see that the average estimates have the expected signs: estimates for the probability of being fired and work hours per week are negative, while the others are positive. The estimates indicate that individuals, on average, prefer higher salaries and work-time flexibility, and dislike jobs with a high probability of being fired and high numbers of work hours. The only estimate that is not statistically or economically significant is the proportion of males at the job, indicating that we cannot reject that, on average, individuals are indifferent to the gender composition of the workplace. Turning to the average estimates by gender, reported in columns (2) and (3) of Table V, we see similar qualitative patterns. We return to the differences in magnitudes of the preferences by gender below, and also provide a WTP interpretation. TABLE V Estimates of Job Choice Model Overalla Males Females (1) (2) (3) Age-30 log earnings 15.40*** 22.86*** 11.42*** (1.65) (3.88) (1.43) Probability of being fired −0.38*** −0.39*** −0.37*** (0.04) (0.10) (0.04) Bonus, as a prop. of earnings 0.28*** 0.38*** 0.22*** (0.03) (0.05) (0.03) Prop. of males in similar positions 0.00 −0.01 0.005 (0.00) (0.01) (0.01) % increase in annual earnings 0.55*** 1.09*** 0.27** (0.10) (0.22) (0.10) Hours per week of work −0.15*** −0.21*** −0.12*** (0.02) (0.05) (0.02) Part-time option available 0.79*** 0.86*** 0.76*** (0.11) (0.22) (0.12) Observations 247 86 161 Overalla Males Females (1) (2) (3) Age-30 log earnings 15.40*** 22.86*** 11.42*** (1.65) (3.88) (1.43) Probability of being fired −0.38*** −0.39*** −0.37*** (0.04) (0.10) (0.04) Bonus, as a prop. of earnings 0.28*** 0.38*** 0.22*** (0.03) (0.05) (0.03) Prop. of males in similar positions 0.00 −0.01 0.005 (0.00) (0.01) (0.01) % increase in annual earnings 0.55*** 1.09*** 0.27** (0.10) (0.22) (0.10) Hours per week of work −0.15*** −0.21*** −0.12*** (0.02) (0.05) (0.02) Part-time option available 0.79*** 0.86*** 0.76*** (0.11) (0.22) (0.12) Observations 247 86 161 Notes. Table reports the average of the parameter estimates across the relevant sample. Asterisks denote estimates are statistically different from zero based on bootstrap standard errors. ***, **, * denote significance at the 1%, 5%, and 10% levels, respectively. View Large V.F. Willingness to Pay The parameter estimates in Table V are difficult to interpret given the necessarily nonlinear nature of the model. To ease interpretation, we next present WTP estimates, which translate the differences of utility levels into earnings that would make the student indifferent between giving up earnings and experiencing the outcome considered. 1. Computing WTP. WTP to experience job attribute Xk is constructed as follows. Consider a change in the level of attribute Xk from value Xk = xk to Xk = xk + Δ, with Δ > 0. Assume Xk is a “bad” attribute. Given our linear utility function, we can write an indifference condition in terms of earnings Y as: \begin{equation*} x_{k}\beta _{ik}+\beta _{i1}\ln (Y)=\beta _{ik}(x_{k}+\Delta )+\beta _{i1}\ln (Y+\text{WTP}_{ik}(\Delta )), \end{equation*} where Y is the level of earnings, one of the job attributes included in every job scenario. WTPik(Δ) > 0 is individual i’s willingness to pay to avoid increasing the “bad” attribute k by Δ. Solving, WTP is given by: $$\text{WTP}_{ik}(\Delta )=\left[\exp \left( \frac{-\beta _{ik}}{\beta _{i1}}\Delta \right) -1\right]\times Y.$$ (8)WTP for individual i depends on her preference for the attribute βik versus her preference for earnings βi1 (earnings is attribute 1). Given that we allow for a log form to utility in earnings (allowing for diminishing marginal utility in earnings and implicitly consumption), willingness to pay for an individual also depends on the level of earnings at the job. 2. WTP by Gender.Table VI shows the average and median WTP estimates for changing each of the job characteristics by one unit (for the probabilistic outcomes, this is increasing the likelihood by 1 percentage point; for hours per week, increasing it by an hour; for part-time availability, this is going from a job with no part-time option to one which does).24 The first three columns of the table present the estimates in dollars, evaluating WTP at the average annual earnings across all scenarios, $${\}$$75,854 (which is fixed by the experimental setup and does not vary across respondents). The last three columns show the estimates as a proportion of the average earnings. We focus on the latter here. TABLE VI Willingness-to-Pay (WTP) Estimates WTP ($${\}$$) WTP (as % of average earnings) Overall Male Female Overall Male Female (1) (2) (3) (4) (5) (6) Percent chance of being fired 2,147.40*** 467.79 3,044.58***+++ 2.83%*** 0.62% 4.01%***+++ (525.46) (670.29) (715.28) (0.69%) (0.88%) (0.94%) [1,125.94]*** [504.69]** [1,841.27]***+++ [1.48%]*** [0.67%]** [2.43%]***+++ Bonus as % of earnings −1,069.78*** −645.92* −1,296.19*** −1.41%*** −0.85%* −1.71%*** (258.47) (368.77) (345.77) (0.34%) (0.49%) (0.46%) [−975.36]*** [−761.57]*** [−1,139.44]***++ [−1.29%]*** [−1.00%]*** [−1.50%]***++ Percent of men at jobs 43.20 63.74 32.24 0.06% 0.08% 0.04% (38.31) (46.81) (53.93) (0.05%) (0.06%) (0.07%) [59.15]*** [71.82]** [43.17] [0.08%]*** [0.09%]** [0.06%] Annual % raise in earnings −1,186.28 −2,564.93** −449.86 −1.56% −3.38%** −0.59% (773.44) (1,226.42) (957.85) (1.02%) (1.62%) (1.26%) [−2,596.68]*** [−2,934.51]*** [−2,514.55]*** [−3.42%]*** [−3.87%]*** [−3.31%]*** Hours per week of work 854.65*** 594.70 993.50*** 1.13%*** 0.78% 1.31%*** (235.25) (416.17) (267.19) (0.31%) (0.55%) (0.35%) [626.52]*** [600.89]*** [634.22]*** [0.83%]*** [0.79%]*** [0.84%]*** Part-time option availablea −3,892.01*** −829.94 −5,527.65***++ −5.13%*** −1.09% −7.29%***++ (1,024.91) (1,822.82) (1,221.72) (1.35%) (2.40%) (1.61%) [−2,709.40]*** [−1,866.56]*** [−3,237.95]***+ [−3.57%]*** [−2.46%]*** [−4.27%]***+ WTP ($${\}$$) WTP (as % of average earnings) Overall Male Female Overall Male Female (1) (2) (3) (4) (5) (6) Percent chance of being fired 2,147.40*** 467.79 3,044.58***+++ 2.83%*** 0.62% 4.01%***+++ (525.46) (670.29) (715.28) (0.69%) (0.88%) (0.94%) [1,125.94]*** [504.69]** [1,841.27]***+++ [1.48%]*** [0.67%]** [2.43%]***+++ Bonus as % of earnings −1,069.78*** −645.92* −1,296.19*** −1.41%*** −0.85%* −1.71%*** (258.47) (368.77) (345.77) (0.34%) (0.49%) (0.46%) [−975.36]*** [−761.57]*** [−1,139.44]***++ [−1.29%]*** [−1.00%]*** [−1.50%]***++ Percent of men at jobs 43.20 63.74 32.24 0.06% 0.08% 0.04% (38.31) (46.81) (53.93) (0.05%) (0.06%) (0.07%) [59.15]*** [71.82]** [43.17] [0.08%]*** [0.09%]** [0.06%] Annual % raise in earnings −1,186.28 −2,564.93** −449.86 −1.56% −3.38%** −0.59% (773.44) (1,226.42) (957.85) (1.02%) (1.62%) (1.26%) [−2,596.68]*** [−2,934.51]*** [−2,514.55]*** [−3.42%]*** [−3.87%]*** [−3.31%]*** Hours per week of work 854.65*** 594.70 993.50*** 1.13%*** 0.78% 1.31%*** (235.25) (416.17) (267.19) (0.31%) (0.55%) (0.35%) [626.52]*** [600.89]*** [634.22]*** [0.83%]*** [0.79%]*** [0.84%]*** Part-time option availablea −3,892.01*** −829.94 −5,527.65***++ −5.13%*** −1.09% −7.29%***++ (1,024.91) (1,822.82) (1,221.72) (1.35%) (2.40%) (1.61%) [−2,709.40]*** [−1,866.56]*** [−3,237.95]***+ [−3.57%]*** [−2.46%]*** [−4.27%]***+ Notes. Table reports mean (bootstrap standard errors) [median] WTP (amount of earnings an individual needs to be compensated for a unit change in the job attribute). aWTP for moving from a job without a part-time option to one that has it. ***, **, * denote significance at the 1%, 5%, and 10% levels, respectively. Tests conducted for differences in means and medians by gender. +++, ++, + denote estimates differ at 1%, 5%, and 10% levels, respectively. View Large We estimate, for example, that increasing the likelihood of being fired by 1 percentage point, that is, Xk = xk + 1, would yield an average WTP of 2.8% for the full sample. That is, for students to remain indifferent to moving to a less stable job, they would on average have to be compensated by 2.8% of annual earnings. The gender-specific averages, reported in the last two columns of Table VI, indicate distinct average preferences by gender. Women, on average, have to be compensated by 4% of average earnings for a unit increase in the likelihood of being fired, with the estimate being statistically significant at the 1% level, and statistically different from the much smaller male average of 0.6%. Recall that we fix average earnings at the same level for all respondents, so the gender differences in WTP reflect only differences in preferences, not earnings. The median estimates also differ by gender, with women exhibiting a higher WTP for job stability. The median estimate for women is, however, lower than the average estimate, suggestive of a skewed distribution. The average and median WTP estimate for the availability of the part-time option is sizable. Individuals, on average, would have to be compensated by 5.1% of their annual salary (that is, they are willing to give up 5.1%) when going from a job with no part-time option to one that does have one. The estimate is driven by the female respondents in the sample, for whom the average WTP is −7.3%, versus −1.0% for males (with the male estimate not being statistically different from 0). The much higher average preference among women for the part-time option is statistically significantly different from 0 and statistically different from the male average, at the 5% level. The median estimate also differs by gender and is larger in magnitude for women. Examining the WTP for other job characteristics, we see that the average WTP for annual earnings growth is statistically precise for men, who are willing to give up 3.4% of average annual earnings for a 1 percentage point increase in earnings growth; the female average coefficient is indistinguishable from 0 (although not statistically different from the male estimate). The median estimates for the two genders are similar. We see that women have a stronger distaste for the number of hours of work, with the average WTP indicating that they need to be compensated by 1.3% of annual earnings for an increase of one hour in the work week; the male estimate is not precise (but we cannot reject the two gender-specific averages being equal). Both genders are, on average, willing to give up 0.8–1.7% of annual earnings for a percentage point increase in bonus compensation (in addition to base salary).25 Finally, the average WTP for proportion of men at jobs is economically and statistically insignificant. Online Appendix C further analyzes the heterogeneity in preferences for workplace characteristics, and investigates how the WTPs are associated with various individual-level characteristics. In addition, note that the utility from jobs, specified in equation (1), is linear and separable in outcomes. Online Appendix D shows that our conclusions are robust to estimating variants of the baseline model. It is important to note that the timing of our survey is quite important in interpreting the resulting preference estimates. In general, there is no reason to believe that the workplace preferences we elicit are intrinsic, and they may be particular to the age of our survey respondents. Our estimates should not be considered unbiased estimates for intrinsic preferences—preferences are likely not intrinsic at all—but instead unbiased estimates for preferences at the point in each student’s life cycle at which we collect our data. Our preference estimates may also reflect past experiences with employment because in some cases, the respondents may have already secured postgraduation employment. Our methodology simply relies on students being able to consider their likelihood of accepting hypothetical job offers, which should be possible even if a student is already employed. V.G. Estimated Preferences and Actual Workplace Characteristics Do the pre–labor market preferences we estimate relate to the characteristics of jobs these students actually end up working in?26 We are able to shed light on this issue through a follow-up survey of a subset of our respondents conducted in 2016, about four years after the original data collection and when respondents were on average aged 25. Of the 247 respondents who took the survey and answered the hypothetical questions, 112 had also participated in an earlier survey conducted by us in 2010 (data that we have analyzed in Wiswall and Zafar 2015a,b) and given consent for future surveys. In January 2016, we invited these 112 respondents to participate in a 15-minute online survey about their current labor market status. 71 of the eligible 112 respondents (∼63%) completed the follow-up survey.27 The follow-up survey collected information about respondents’ workplace characteristics (for those currently working). Of the 71 respondents, 59 were working (either full-time, part-time, or self-employed) at the time of the follow-up survey, with the remainder enrolled in school. Online Appendix Table A7 shows the earnings and various other workplace characteristics for the overall sample, as well as for male and female workers, separately. Earnings, conditional on working full-time, are higher for men (by nearly $${\}$$70,000). Bonus, hours of work, likelihood of being fired, fraction of male employees, and typical annual growth in earnings are all higher for our male respondents (though not all of the differences are statistically significant). The last row of the table shows that women’s workplaces are more likely to have a part-time or flexible work option.28 Are these systematic gender differences in actual workplace characteristics consistent with our estimates of job preferences elicited several years prior, before labor market entry? To investigate this, we regress characteristics of each respondent’s current job onto our individual-specific estimate of their past WTP for that attribute. WTP is defined as the amount the individual needs to be compensated by for a unit change in a given characteristic, with a higher WTP reflecting a lower taste (or greater distaste) for that outcome. Therefore, we expect a negative relationship between WTP and the current job characteristic. Estimates are presented in Table VII. Directionally, all six estimates are negative, with three significant at the 5% level or better. A joint test that all coefficients are 0 can be rejected (the p-value of this joint test is .012). TABLE VII Actual Job Characteristics and Estimated WTP Prob. of fired Bonus percentage Prop. of males Earnings growth Hours worked Flex work option Willingness to paya −0.07 −1.00 −7.32** −0.02 −1.70** −0.94** (0.20) (1.23) (2.82) (0.08) (0.64) (0.29) Constant 10.70*** 3.64 52.60*** 7.32*** 46.37*** 55.61*** (1.90) (2.67) (2.82) (1.71) (2.00) (6.37) Effect sizeb −0.658 −4.35 −6.89 −0.319 −4.09 −14.75 p-valuec 0.012 Mean of dep. var. 10.4 5.8 50.9 7.3 44.6 61.0 Std. dev. of dep. var. (14.72) (12.79) (22.79) (13.34) (14.76) (49.19) R-squared 0.002 0.16 0.092 0.0001 0.077 0.090 Observations 59 59 59 59 59 59 Prob. of fired Bonus percentage Prop. of males Earnings growth Hours worked Flex work option Willingness to paya −0.07 −1.00 −7.32** −0.02 −1.70** −0.94** (0.20) (1.23) (2.82) (0.08) (0.64) (0.29) Constant 10.70*** 3.64 52.60*** 7.32*** 46.37*** 55.61*** (1.90) (2.67) (2.82) (1.71) (2.00) (6.37) Effect sizeb −0.658 −4.35 −6.89 −0.319 −4.09 −14.75 p-valuec 0.012 Mean of dep. var. 10.4 5.8 50.9 7.3 44.6 61.0 Std. dev. of dep. var. (14.72) (12.79) (22.79) (13.34) (14.76) (49.19) R-squared 0.002 0.16 0.092 0.0001 0.077 0.090 Observations 59 59 59 59 59 59 Notes. The table investigates the relationship between the estimated WTP for a given job attribute for a respondent (derived from the 2012 survey) and the value of that job attribute in the respondent’s actual workplace (reported in the 2016 follow-up survey). Each column is a separate OLS regression, with the dependent variable (column title) being the value of the job characteristic in the respondent’s actual job (reported in the 2016 survey). Bootstrap standard errors in parentheses. ***, **, * denote significance at 1%, 5%, and 10% levels, respectively. aThe estimated WTP of the respondent based on the hypothetical job choice scenarios. bThe predicted change in the dependent variable for a one std. dev. change in the WTP. cp-value of a test that the six estimates on the WTP (in the first row) are jointly zero. View Large To interpret the magnitude of the estimated coefficients in Table VII, we also report “effect sizes” in the table. The effect size provides the estimated change in the dependent variable (that is, the actual workplace attribute) for a one standard deviation change in the WTP for that workplace characteristic. For example, we see that a one standard deviation increase in the WTP (that is, higher distaste) for work hours translates into an estimated decrease of 4.1 hours worked. Given that the standard deviation of hours worked is 14.8 in the sample, this is a sizable impact. Likewise, a one standard deviation increase in the WTP (that is, lower taste) for availability of flexible work options is associated with a 15 percentage point decline in the actual availability of these options in the respondent’s workplace (on a base of 61). The effect sizes for bonus percentage and proportion of male are also economically meaningful. While we have shown that estimated preferences for attributes are jointly systematically related to actual future workplace characteristics in the cross section, a natural question to ask is whether the relationship also holds within the individual, that is, whether a higher WTP for a given attribute translates into more of that attribute for an individual. For each attribute, we rank the 59 individuals in terms of both the estimated WTP and the actual value at the job. This gives us a six-dimensional vector of ranked WTPs and a six-dimensional vector of ranked attribute values for each individual. We then compute the individual-level correlation between the two vectors. We expect a negative correlation: higher WTP (that is, a lower taste or a greater distaste) for an attribute causes an individual to be working in a job with lower values of that attribute. That is exactly what we find: the mean correlation coefficient across the individuals is −0.158 (significant with a p-value = .017) and the median correlation coefficient is −0.250 (p-value = .36), indicative of a systematic relationship between estimated WTPs and actual attributes even within individuals. Overall, these results strongly indicate that our estimated preferences capture true underlying heterogeneity that is also reflected in actual job outcomes several years later. We view these results as a joint validation of our methodology, data quality, and empirical specification. Our finding that estimated WTPs predict respondents’ actual workplace choices is all the more remarkable given that the hypothetical scenarios were fielded to respondents when they were still in college (though some of the respondents may have already secured postgraduation employment at the time of the survey). In the next section, we investigate whether these workplace preferences impact major choice. VI. Job Preferences and Major Choice The preceding sections used a robust hypothetical choice methodology to estimate individual-level preferences for various job attributes. This section relates these preferences to human capital investments, quantifying the importance of job characteristics to college major choices. First, to set the stage for this analysis, we describe the anticipated major choices reported by our sample. Given that our sample consisted of currently enrolled students, we asked the students to provide their beliefs they would complete a degree in one of the five major categories: “What do you believe is the percent chance (or chances out of 100) that you would either graduate from NYU with a PRIMARY major in the following major categories or that you would never graduate/dropout (i.e., you will never receive a bachelor’s degree from NYU or any other university)?” The first column of Online Appendix Table A9 shows the response to the question: the most likely major for males is economics/business (43%), followed by humanities/social sciences (29%). For women, on the other hand, the most likely major is humanities/social sciences (53%), followed by economics (23%). The probability of not graduating is less than 3% for both sexes. The average probabilities assigned to the majors differ significantly by gender for all majors except engineering and natural sciences. Our model of major choice allows for some uncertainty in major choice: at least part of the sample is not 100% certain of their final major at graduation and the data reflect that (a majority of students, 53, do not assign a 100% probability to their most likely major). Our model of probabilistic major choices nests the standard model of deterministic major choice. We next decompose the anticipated major choices into various factors, including potential job characteristics associated with each major. To gauge the importance of job attributes to major choice, we estimate a model of major choice incorporating our flexible estimates of preferences for job attributes and separate data we collected on students’ beliefs about the likelihood they would be offered jobs with these characteristics, conditional on major choice (that is, estimates of students’ perceptions of the firm or demand side of the labor market). We then use this estimated model to quantify the importance of each job attribute to major choice. Given that prior literature on educational choice finds that the residual unobserved “taste” component is the dominant factor in major choice (Arcidiacono 2004; Beffy, Fougere, and Maurel 2012; Gemici and Wiswall 2014; Wiswall and Zafar 2015a), our approach can be viewed as trying to get into the black box of tastes by directly incorporating certain nonpecuniary dimensions into these choice models. The estimation details for the major-choice model are provided in Online Appendix E. Here, for the sake of brevity, we comment on only its main features. We start with a simple framework in which we suppose that utility for student i from major m is given by: $$V_{im}=X_{im}^{\prime }\alpha _{i}+Z_{im}^{\prime }\gamma +\kappa _{m}+\eta _{im},$$ (9)where Xim is i’s perceived job attributes in major m. With hopefully minimal confusion, we use the same notation X to refer to job attributes as in our hypothetical job choice analysis and to refer to perceptions about job attributes associated with each major, a separate set of variables collected in our survey. Note that here the X vector is indexed by i as these attributes are each student’s perception of the job attributes (that are allowed to depend on the major m) rather than the exogenously determined attributes in the hypotheticals we created. Zim is a vector of other major-specific characteristics perceived by student i (including major-specific perceptions of ability and perceived hours of study needed to obtain a GPA of 4.0 in that major). κm is a major-specific constant, capturing overall tastes for the major, and ηim captures the remaining unobservable attributes of each major. To estimate the model, we use data on students’ perceptions of the likelihood of being offered jobs with various characteristics conditional on each major, as well as their beliefs regarding major-specific ability. Our survey collected data from respondents on their perceptions of characteristics of the jobs that would likely be offered to them if they were to complete each type of major. An important characteristic of our data set is that we gather students’ beliefs about workplace characteristics (such as likelihood of being fired and earnings) for a set of different majors, not just for the one major they intend to complete. These data are described at length in Online Appendix E.29 In equation (9), the student-specific preference for each job attribute is given by the vector αi = [αi1, …, αiK]. αi, the preference for job characteristics as it relates to the utility from each major, is potentially distinct from the preferences for job characteristics in the job-choice problem, given by βi (in equation (5)). Job characteristics, such as earnings at the job, may be quite important when choosing among different job offers but might have a more limited value to choosing majors, relative to other major characteristics given by Zim, κm, and ηim. To allow for this possibility, for each job characteristic k, we specify that each αik is proportional to the βik up to some free (to be estimated) parameter δ: αik = βikδ. δ indicates the importance of job attributes to major choice, relative to other determinants of college major as given by Zim, κm, and ηim. δ could also reflect standard discounting given that the utility from working at jobs occurs later in life than utility derived from taking courses while in school. Table VIII presents the LAD estimates of equation (9) using the hypothetical data to estimate the job-preference vector βi for each student, and a robust cluster bootstrap over all estimation steps for inference (see Online Appendix E for estimation details). The estimate of δ is positive and precise, indicating that the preferences of students over job attributes and the major-specific beliefs about the distribution of job attributes have a statistically significant relationship with major choices. Estimates on the major-specific ability measures are negative, as one would expect (note that higher “ability rank” denotes lower ability in our data). The major-specific dummy terms are all negative, indicative of negative median tastes for the nonhumanities majors (the omitted category): all else equal, students prefer to major in humanities. TABLE VIII LAD Estimates of Major Choice LAD estimates Job attributes (δ) 0.018** (0.007) Ability rank −0.064*** (0.006) Study time −0.009 (0.025) Economics dummy −0.590 (0.444) Engineering dummy −1.16** (0.37) Natural sci dummy −0.822* (0.375) Total observations 741 Number of individuals 247 LAD estimates Job attributes (δ) 0.018** (0.007) Ability rank −0.064*** (0.006) Study time −0.009 (0.025) Economics dummy −0.590 (0.444) Engineering dummy −1.16** (0.37) Natural sci dummy −0.822* (0.375) Total observations 741 Number of individuals 247 Notes. Bootstrap standard errors in parentheses. ***, **, * denote significance at the 1%, 5%, and 10% levels, respectively. View Large Given the nonlinear nature of the model, it is difficult to assess the importance of job attributes in major choice from the estimated coefficients alone. To quantify the effects, we use standard methods to evaluate “marginal effects” in nonlinear models (see Online Appendix E for details). The marginal effect of a job attribute in major choice is computed for a standard deviation change in the value of that specific job attribute, while keeping the other (job- and major-specific) attributes and preferences fixed at their sample average values. Table IX presents the marginal effects for specific changes in job attributes, averaged across the majors, and separately by sex (in the two panels of the table). The table also shows the start and end value for the attribute at which the marginal effect is computed. The start value is the sex-specific belief for that attribute (averaged across majors and respondents), and the end value is the start value shifted by one sex-specific standard deviation (again, across majors and respondents) in the beliefs for that attribute. Column (1), for example, shows that increasing the perceived probability of being fired from jobs by one standard deviation decreases the likelihood of majoring in that major, on average, by 4% for men and by 5% for women. A standard deviation increase in part-time availability increases the probability of completing a major by 0.2%. Column (3) shows that a standard deviation increase in weekly work hours reduces the likelihood of majoring in a major by 2.5% for men and 1.4% for women. Bonus pay and earnings growth both also have sizable average marginal effects. The last column of Table IX shows the percent change in the major probability for a standard deviation increase in log age-30 earnings. TABLE IX Marginal Contribution of Job Attributes in Major Choice Fired prob. Part-time available Hours Bonus Earnings growth Prop. males Log earnings (1) (2) (3) (4) (5) (6) (7) Panel A: Males Start valuea 8.82 0.29 48.88 18.37 1.42 54.86 11.38 End valueb 19.38 0.53 60.40 41.37 5.20 73.21 12.03 Avg. changec −4.10% 0.23% −2.48% 9.20% 4.30% −0.20% 15.90% Relative changed −0.26 0.01 −0.16 0.58 0.27 −0.01 — Panel B: Females Start value 16.51 0.35 47.36 19.12 1.35 56.13 11.22 End value 33.14 0.58 61.15 43.11 4.44 75.22 11.70 Avg. change −5.12% 0.15% −1.40% 4.58% 0.70% 0.10% 4.78% Relative change −1.07 0.03 −0.29 0.96 0.15 0.02 — Fired prob. Part-time available Hours Bonus Earnings growth Prop. males Log earnings (1) (2) (3) (4) (5) (6) (7) Panel A: Males Start valuea 8.82 0.29 48.88 18.37 1.42 54.86 11.38 End valueb 19.38 0.53 60.40 41.37 5.20 73.21 12.03 Avg. changec −4.10% 0.23% −2.48% 9.20% 4.30% −0.20% 15.90% Relative changed −0.26 0.01 −0.16 0.58 0.27 −0.01 — Panel B: Females Start value 16.51 0.35 47.36 19.12 1.35 56.13 11.22 End value 33.14 0.58 61.15 43.11 4.44 75.22 11.70 Avg. change −5.12% 0.15% −1.40% 4.58% 0.70% 0.10% 4.78% Relative change −1.07 0.03 −0.29 0.96 0.15 0.02 — Notes. Table shows the average percent change in the probability of majoring in a given major (“marginal effect”) for a standard deviation change in the job attribute (column variable). See Online Appendix E for details. a(b)The initial (final) value of the attribute at which the major probability is computed, with all other attributes fixed at the sample mean. cThe average change (across majors) in the probability of majoring in a given major for a standard deviation change in the attribute. dThe average change in the probability of majoring in a given major for a std. dev. change in an attribute, relative to a corresponding change in earnings. View Large A comparison of the effects in the first three columns with those in the last column for earnings (also shown in the last row of each panel) gives a sense of the relative importance of these other job attributes in major choice. We see that, for women, the average effect for the probability of being fired is as large as that for earnings, and for hours is nearly a third of the effect of earnings. For men, the relative impacts are smaller (though still sizable). Overall, this indicates that job attributes matter for major choice and that they are particularly relevant for women’s choices. VII. Job Preferences and the Gender Gap in Earnings In the previous sections, we have shown systematic gender differences in workplace preferences and quantified the importance of these preferences to major choices. In this section, we explore the extent to which gendered job preferences explain the “gender gap” in earnings. Differences in job preferences can give rise to differences in earnings through two channels. First, as explored above, job preferences can affect college major choices and, given the wide dispersion in earnings across fields, affect the overall distribution of earnings for men and women. Second, even conditional on major choice, gender differences in workplace preferences can affect the distribution of earnings. The gender gap in earnings we observe could be at least partially the result of women “purchasing” certain positive job attributes by accepting lower wages, or conversely, men accepting higher earnings to compensate for negative job attributes. To quantify the first channel (job preferences affecting earnings through major choice), we conduct the following exercise. Using the estimated major-choice model in Section VI, we predict the likelihood of women choosing different majors if their workplace preferences were shifted by the average male minus female mean (that is, we preserve the heterogeneity in women’s preferences but shift them by the average gender difference in the preferences).30 We then predict the likelihood of each female respondent choosing the different majors and use these to weight the individual’s major-specific expected earnings. This provides the impact on the gender wage gap if women’s preference distribution was shifted to have the male average, but only through the major choice channel. Note that, for this exercise, we keep women’s earnings expectations fixed in that major (which could also be affected by workplace preferences). In this exercise, we find that the change in women’s major choices lowers the expected gender gap in age-30 earnings by 2.6%.31 Given our highly aggregated major categories, this is likely a lower bound on the importance of preferences to the gender earnings gap through major choices, and human capital more generally. Previous work has emphasized that important job segregation by gender occurs through choices of subfields (see for example, Goldin and Katz 2016, on choice of medical specialties). Turning to the second channel, we consider the following simple exercise. We ask how the gender gap in expected earnings changes once we “control” for individual-specific workplace preferences (the estimated preference parameters in Section V). If the gender gap in earnings is solely because women are accepting lower wages for desirable jobs, and/or men are compensated with higher wages for undesirable jobs, then men and women with identical workplace preferences would have equal earnings. If, on the other hand, a gender gap remains, even after conditioning on preferences, then we can conclude that demand-side factors, such as employment discrimination, still play a role in the gender gap. We implement this exercise using a simple set of regressions in Table X, Panel A. Column (1) of the table reports a regression of an individual’s log expected earnings for the major they are most likely to graduate with onto a female dummy. We see a gender gap of about 35 log points in age-30 expected earnings, a gap similar to that in realized earnings data.32 The second column shows that the gender gap declines to about 20 log points once the individual’s major is controlled for, reflecting the fact that women are less likely to graduate in higher-earnings majors. Columns (3) and (4) show how the gender gap changes once we control for the estimated vector of workplace preferences. Importantly, a comparison of column (4) with column (2) shows that, even conditional on major choice, workplace preferences reduce the expected earnings gender gap by about a quarter, from about 20% to 15%. Note that workplace preferences are also likely to impact major choice, which is held fixed here. TABLE X Workplace Preferences and Gender Gap in Age-30 Expected and Actual Earnings (1) (2) (3) (4) Panel A: Dependent variable: log(age-30 expected earnings) Female −0.346*** −0.195*** −0.289*** −0.150*** (0.060) (0.055) (0.065) (0.057) Constant 11.483*** 11.69*** 11.36*** 11.55*** (0.048) (0.051) (0.065) (0.068) Major controlsa N Y N Y Workplace preferences controlsb N N Y Y Mean of dep. var 11.26 11.26 11.26 11.26 R-squared 0.1209 0.3386 0.2013 0.3967 Number of observations 247 247 247 247 Panel B: Dependent variable: log(actual 2016 earnings) Female −0.612*** −0.451*** −0.442** −0.318 (0.169) (0.167) (0.191) (0.230) Constant 12.12*** 12.31*** 11.91*** 12.12*** (0.145) (0.147) (0.190) (0.188) Part-time work dummy Y Y Y Y Major controlsc N Y N Y Workplace preferences controls N N Y Y Mean of dep. var 11.65 11.65 11.65 11.65 R-squared 0.226 0.384 0.395 0.495 Number of observations 56 56 56 56 (1) (2) (3) (4) Panel A: Dependent variable: log(age-30 expected earnings) Female −0.346*** −0.195*** −0.289*** −0.150*** (0.060) (0.055) (0.065) (0.057) Constant 11.483*** 11.69*** 11.36*** 11.55*** (0.048) (0.051) (0.065) (0.068) Major controlsa N Y N Y Workplace preferences controlsb N N Y Y Mean of dep. var 11.26 11.26 11.26 11.26 R-squared 0.1209 0.3386 0.2013 0.3967 Number of observations 247 247 247 247 Panel B: Dependent variable: log(actual 2016 earnings) Female −0.612*** −0.451*** −0.442** −0.318 (0.169) (0.167) (0.191) (0.230) Constant 12.12*** 12.31*** 11.91*** 12.12*** (0.145) (0.147) (0.190) (0.188) Part-time work dummy Y Y Y Y Major controlsc N Y N Y Workplace preferences controls N N Y Y Mean of dep. var 11.65 11.65 11.65 11.65 R-squared 0.226 0.384 0.395 0.495 Number of observations 56 56 56 56 Notes. OLS estimates presented. Block bootstrap standard errors in parentheses. ***, **, * denote significance at 1%, 5%, and 10% levels, respectively. Dependent variable in Panel A is the log of age-30 expected earnings for the individual’s reported major. Dependent variable in Panel B is the log of actual earnings for the subset of individuals who took the follow-up survey and were working in 2016. aDummy for the major the respondent is majoring in (the major with the modal probability). bControls for the estimated workplace preferences (from the job-choice model). cDummy for the major the respondent graduated with. View Large Table X, Panel B repeats the exercise using actual earnings reported by the follow-up respondents. The sample here is smaller, but the qualitative results are strikingly similar to those that we observe for expected earnings: conditional on major, the gender gap in realized earnings declines from 45 log points to 32 log points (that is, by nearly 30%) once we control for respondents’ workplace preferences. We conclude from this analysis that gender differences in workplace preferences can explain a sizable part of the gender gap in expected earnings early in the life cycle. And, albeit with a smaller sample, our evidence points to similar conclusions for realized earnings as well. We also find that the main channel by which workplace preferences affect the gender earnings gap is through job choices, not through major choices, at least at the aggregated major level we have available in this data set. VIII. Conclusion Economists have long recognized that job and occupational choices are not solely determined by expected earnings.33 Although simple models based on earnings maximization abound (see, for example, the classic Roy 1951 model) and are quite useful in some applications, it is also clear that individuals have a rich set of preferences for various aspects of jobs beyond expected earnings, including earnings and dismissal risk, and various nonpecuniary aspects such as work hours flexibility. Human capital investments too could be affected by these workplace preferences as individuals alter their human capital investment in anticipation of particular future job choices. Key features of the distribution of labor earnings in the economy, such as the gap in earnings between men and women, need careful consideration, as differences in earnings may reflect, at least in part, heterogeneity in preferences and compensating differentials for various nonpecuniary attributes of employment. Using a novel hypothetical job-choice framework that experimentally varies different dimensions of the workplace, this article robustly estimates individual preferences for workplace attributes. For a sample of high-ability undergraduate students enrolled at a selective private U.S. university, we document substantial heterogeneity in willingness to pay for job amenities, with large differences in the distribution of preferences between men and women. For a subset of the sample for whom we collect data on actual workplace characteristics (nearly four years after the original survey), we find a robust systematic relationship between estimated preferences and the characteristics of their current jobs. The predictive power of the estimated preferences at the individual level strengthens the credibility of our approach, and makes a case for employing this methodology in other settings to understand decision making. Combining these workplace preferences with unique data on the students’ perceptions of jobs which would be offered to them given their major choice, we quantify the role of anticipated future job characteristics—particularly the nonpecuniary aspects of these jobs—in choice of major, a key human capital investment decision. Women, in particular, are found to be more sensitive to nonpecuniary job aspects in major choice than men. Our analysis indicates that at least a quarter of the gender gap in early career earnings—expected as well as actual—can be explained by the systematic gender differences in workplace characteristics. Our analysis indicates that a substantial part of the early gender gap in earnings we observe is a compensating differential in which women are willing to give up higher earnings to obtain other job attributes. There are several potential areas for future research. Although we find substantial variation in workplace preferences for our sample of high-ability students at a selective university, it is not clear how these preferences compare to that of the broader population. It would clearly be useful to follow our design and collect similar data in other settings. In particular, preference data collected at older ages would be useful in studying how preferences for nonpecuniary dimensions of the workplace, especially those related to accommodations for raising children, evolve over the life cycle (Bertrand, Goldin, and Katz 2010). Our work also does not directly indicate the sources of the systematic gender differences in workplace preferences that we document. For example, they may be a consequence of social factors including anticipated discrimination (Altonji and Blank 1999). We cannot therefore claim that these preferences are intrinsic and immutable in the sense that they may be due, at least in part, to environmental influences particular to this cohort of students. Research that sheds light on the underlying channels would be immensely valuable. Supplementary Material An Online Appendix for this article can be found at The Quarterly Journal of Economics online. Data and code replicating the tables and figures in this article can be found in Wiswall and Zafar (2017), in the Harvard Dataverse, doi:10.7910/DVN/MLOGDL. Footnotes * Ellen Fu and John Conlon provided excellent research assistance. We would like to thank Joe Altonji for feedback on the survey design. We are also thankful to the editor, Larry Katz; the coeditor; five anonymous referees; and participants at various seminars and conferences for valuable comments. This is a revised version of NBER Working Paper 22173 (April 2016). 1. In the marketing and environmental contexts, these methods are often used to identify preferences for new, as yet unavailable consumer products or for public goods like environmental quality, for which realized choices and markets do not exist. Our primary motivation for collecting hypothetical choice data is not because labor markets and realized choices do not exist, but to resolve problems of endogeneity of realized job choices. 2. Recent work has incorporated nonwage components into rich models of the labor market and education choices, allowing for important features such as search frictions, preferences over unobserved job attributes, and dynamic incentives for occupation and education choices (see for example, Bonhomme and Jolivet 2009; d’Haultfoeuille and Maurel 2013; Bronson 2015; Lim 2015). Motivating our approach, Hwang, Mortensen, and Reed (1998) and Bonhomme and Jolivet (2009) conclude that search frictions can imply small equilibrium wage differentials across jobs when there are in fact substantial preferences for nonwage job amenities. 3. In fact, using a recent nationally representative survey of U.S. workers, Maestas et al. (2016) find that younger college-educated women report less desirable working conditions (including no option to telecommute, higher prevalence of employer setting schedules, and higher incidence of work-related stress). 4. Mas and Pallais also conclude that gender differences in work-time flexibility preferences are not enough to explain any part of the gender gap in earnings, which stands in contrast to our conclusion of a large role (as we discuss later). There could be several reasons for this difference in findings: we measure preferences for several workplace attributes (job stability, earnings growth, hours). In addition, our sample is high skill, and likely to be active in a different segment of the labor market. 5. For examples of recent work, see Arcidiacono 2004; Beffy, Fougere, and Maurel 2012; Arcidiacono, Hotz, and Kang 2012; Stinebrickner and Stinebrickner 2014a; Gemici and Wiswall 2014; Wiswall and Zafar 2015a. Most recently, Bronson (2015) analyzes the importance of work-hours flexibility and changes in divorce law and divorce risk in explaining longer-term gender-specific trends in major choices. 6. Bertrand, Goldin, and Katz (2010) document the rising role of children and hours choices over the first 15 years of the careers of female MBAs from a top U.S. business school. 7. Altonji, Kahn, and Speer (2016) provide a more detailed discussion of the relationships between college majors and labor market outcomes. 8. The unadjusted hourly earnings gap is 21.6 log points. For college graduates ages 25–40, the mean earnings for full-time employed men is higher than the mean earnings for full-time employed women by 36%. The median for full-time men is higher than the female median by 28%. 9. During the same session, and immediately prior to completing the survey, students took part in some economic experiments. Students earned additional income through participation in the experiments. See Reuben, Wiswall, and Zafar (forthcoming) for information on this data collection. 10. In addition, when presented with each scenario, respondents were told: “Now consider the situation where you are given the jobs offered above when you are aged 30, and you have decided to accept one of these jobs. What is the percent chance (or chances out of 100) that you will choose each of these jobs?” That is, the options were mutually exhaustive, and not working was not an option. 11. Lordan and Pischke (2016) find a strong relationship between women’s job satisfaction and the proportion of men in that occupation. 12. For each job attribute, we constructed a set of hypothetical job scenarios by using uniform random draws from an interval between the 10th and 90th percentile of the observed distribution for each attribute. For each set of job scenarios, we then rejected any set of job scenarios which included jobs which were dominated by another job in all attributes or had earnings differences across jobs which were greater than 30%. 13. For the NYU population of students who graduated in 2010 (IPEDS), the fraction of students completing degrees in each field are as follows: for women, 14.1% graduated in economics or business, 71.7% in humanities or other social sciences, and 13.7% in natural sciences, math, or engineering. For men, 31.1% graduated in economics or business, 61.2% in humanities or other social sciences, and 7.8% in natural sciences, math, or engineering. 14. We confine attention to cross-sectional data. Panel data on repeated job choices over an individual’s life cycle may provide more identifying power but at the cost of requiring additional assumptions about the evolution of model features (e.g., preferences) as individuals age. 15. We can represent demand-side restrictions in the omitted variable framework by considering some unobservable job characteristic X(unob), such that X(unob) → −∞ if a job is not offered. 16. An alternative model is that agents have uncertainty about preferences over attributes, that is the utility function ui(·) is uncertain. For example, an individual may be uncertain about the number of children she may have at a future date, and the number of young children at home may affect her preference for workplace hours flexibility (an element of the Xj vector). We explore this later by relating preferences for job characteristics as revealed in our hypothetical data with a rich set of beliefs about future outcomes (e.g., individual beliefs about future own fertility and marriage). 17. See Stinebrickner and Stinebrickner (2014a,b) for evidence on the dynamics in beliefs formation among college students. 18. In an innovative related approach, Stern (2004) collects data on job offers and accepted jobs from a sample of PhD biologists to estimate the WTP to take a research job over others. However, the limited data on job offers do not allow for identification of heterogeneity in preferences. In addition, this approach only yields unbiased preference estimates in frictionless labor markets. 19. Note that as with any discrete choice setting, the population distribution of preference parameters βi is identified up to the distribution of the εi shocks. As we detail below, we assume a logit form for the shocks. For ease of interpretation, we focus on WTP implied by the model, where WTP is a function of the ratio of elements of the βi vector, removing the dependence of WTP on the scale of the shock. 20. Note that utilities across alternatives are correlated through the shared job attributes, therefore the independence of irrelevant alternatives problem does not apply to our model. 21. Note we do not impose that ωij measurement errors are independent across individuals or jobs and do not assume any particular joint distribution for the measurement errors, beyond the conditional median independence with the X variables. For inference, we use a cluster bootstrap method, resampling the entire set of job scenarios for each sample member, to preserve any correlation in residual errors. See Online Appendix B for details. 22. We also estimate the model with utility specified as linear in earnings (instead of log earnings). Results are qualitatively similar. Online Appendix D discusses results from several other alternative specifications. 23. This is related to the possibility of “session effects” in laboratory experiments. See Frechette (2012). 24. The WTP is computed for each individual, using the individual-specific βi estimates. The table reports mean WTP across respondents, bootstrap standard errors in parentheses, and median WTP in square brackets. 25. That the WTP for a percentage point increase in bonus is greater than 1 in magnitude for women is surprising because it implies that women are on average willing to give up more in base salary to gain a smaller increase in bonus compensation. This is driven by a few outliers. In fact, we cannot reject that the mean WTP for women is different from either −1 (that is, a one-to-one substitution between base pay and bonus pay), or from the mean of −0.8 for male respondents. 26. Although being able to document a systematic relationship can provide some credibility to our methodology, on the other hand, a failure to find a systematic relationship between the two would not necessarily invalidate our method because students’ preferences for jobs may change over time, or labor market frictions may prevent workers from matching with jobs that they prefer. Answering this question most directly would require both revealed-choice data that are free of any confounds and stated-choice data, which are usually not available. However, the little evidence that exists shows a close correspondence between preferences recovered from the two approaches (see Hainmueller, Hangartner, and Yamamoto 2015). 27. Respondents were initially contacted through email addresses provided in our earlier data collections. Those with inactive email addresses were then approached through LinkedIn. Respondents received a link to the survey that was programmed in SurveyMonkey and were compensated for completing the survey. As shown in Online Appendix Table A6, there is little evidence of selection on observables (reported in 2012) in terms of who participates in the follow-up survey. Based on a joint F-test, we cannot reject that the covariates are jointly zero (p-value = .360) in predicting survey response. Note that for students to have taken both the 2010 and 2012 surveys, the sample from which we have consent would have had to be in the junior year or higher in 2012. 28. Online Appendix Table A8 shows that the sample for which we have consent, the sample that takes the follow-up survey, and the sample that was working when the follow-up survey was conducted are all very similar to the full sample. The only dimensions along which they differ are school year and age (which, as explained above, is by construction) and race. Importantly, there are no statistical differences along the dimensions of gender, major, ability, or socioeconomic background. Also note that the follow-up samples, in columns (3) and (4), are not statistically different from the consent sample along any dimension. 29. Because the vast majority of our sample is either in their junior or senior year, and some have already chosen a major, one concern is that the students’ preferences and beliefs, as elicited in our survey data, may be different from the preferences and beliefs they held in the past as they were deciding on a college major. Although we can of course still estimate the relationship between major choice and the data we collect, the interpretation of our estimates in these cases is less clear. One solution is to collect longitudinal data on preferences and beliefs to directly examine the extent to which they change over the life cycle and how this influences college major choices. See Stinebrickner and Stinebrickner (2014b) for an important example. 30. Because the estimated preference parameters are not scale-free, this exercise of shifting the preference parameter by some amount implicitly assumes that the variance of the unobserved factors is the same for all individuals (Train 2003). 31. More specifically, the gender gap declines by about 0.9 percentage points from a baseline predicted gender gap of 35.1%. This is primarily a result of women’s predicted probability of majoring in humanities declining from 55.0% to 53.8%, and their predicted probability of majoring in economics increasing from 18.3% to 19.5%. 32. As described in Section II, in the sample of all college graduates ages 25–40 in the ACS, the mean earnings for full-time employed men is 36% higher than the mean earnings of full-time employed women. 33. See the famous quote by Adam Smith who lists a number of nonpecuniary job attributes which “make up for a small pecuniary gain in some employments, and counterbalance a great one in others” (Wealth of Nations, 1776, Book 1, Chapter 10). References Altonji Joseph, Blank Rebecca, “ Race and Gender in the Labor Market,” in Handbook of Labor Economics , Vol. 3c, Ashenfelter Orley, Card David, eds. ( Amsterdam: Elsevier Science, 1999), 3144– 3259. Altonji Joseph, Kahn Lisa, Speer Jamin, “ Cashier or Consultant? Entry Labor Market Conditions, Field of Study, and Career Success,” Journal of Labor Economics , 34 ( 2016), 361– 401. Google Scholar CrossRef Search ADS Arcidiacono Peter, “ Ability Sorting and the Returns to College Major,” Journal of Econometrics , 121 ( 2004), 343– 375. Google Scholar CrossRef Search ADS Arcidiacono Peter, Hotz Joseph, Kang Songman, “ Modeling College Major Choices using Elicited Measures of Expectations and Counterfactuals,” Journal of Econometrics , 166 ( 2012), 3– 16. Google Scholar CrossRef Search ADS Arcidiacono Peter, Hotz Joseph, Maurel Arnaud, Romano Teresa, “ Recovering Ex Ante Returns and Preferences for Occupations Using Subjective Expectations Data,” NBER Working Paper no. 20626, 2015. Becker Gary, The Economics of Discrimination ( Chicago: University of Chicago Press, 1971). Google Scholar CrossRef Search ADS Beffy Magali, Fougere Denis, Maurel Arnaud, “ Choosing the Field of Study in Post-Secondary Education: Do Expected Earnings Matter?,” Review of Economics and Statistics , 94 ( 2012), 334– 347. Google Scholar CrossRef Search ADS Bertrand Marianne, Goldin Claudia, Katz Lawrence, “ Dynamics of the Gender Gap for Young Professionals in the Financial and Corporate Sectors,” American Economic Journal: Applied Economics , 2 ( 2010), 228– 255. Google Scholar CrossRef Search ADS Blass Asher, Lach Saul, Manski Charles, “ Using Elicited Choice Probabilities to Estimate Random Utility Models: Preferences for Electricity Reliability,” International Economic Review , 51 ( 2010), 421– 440. Google Scholar CrossRef Search ADS Blau Francine, Kahn Lawrence, “ The Gender-Wage Gap: Extent, Trends, and Explanations,” Journal of Economic Literature , forthcoming. Bonhomme Stephen, Jolivet Grégory, “ The Pervasive Absence of Compensating Differentials,” Journal of Applied Econometrics , 24( 5) ( 2009), 763– 795. Google Scholar CrossRef Search ADS Bronson Mary Ann, “ Degrees Are Forever: Marriage, Educational Investment, and Lifecycle Labor Decisions of Men and Women,” Working paper, Georgetown University, Department of Economics, 2015. Cortes Patricia, Pan Jessica, “ Prevalence of Long Hours and Women’s Job Choices: Evidence across Countries and within the U.S.,” Working paper, 2016. Croson Rachel, Gneezy Uri, “ Gender Differences in Preferences,” Journal of Economic Literature , 47 ( 2009), 448– 74. Google Scholar CrossRef Search ADS d’Haultfoeuille Xavier, Maurel Arnaud, “ Inference on an Extended Roy Model, with an Application to Schooling Decisions in France,” Journal of Econometrics , 174 ( 2013), 95– 106. Google Scholar CrossRef Search ADS Dillon Eleanor, “ Risk and Return Tradeoffs in Lifetime Earnings,” Journal of Labor Economics , forthcoming. Flabbi Luca, Moro Andrea, “ The Effect of Job Flexibility on Female Labor Market Outcomes: Estimates from a Search and Bargaining Model,” Journal of Econometrics , 168 ( 2012), 81– 95. Google Scholar CrossRef Search ADS Flory Jeffrey, Leibbrandt Andreas, List John A., “ Do Competitive Workplaces Deter Female Workers? A Large-Scale Natural Field Experiment on Job Entry Decisions,” Review of Economic Studies , 82 ( 2015), 122– 155. Google Scholar CrossRef Search ADS Fox Jeremy, Kim Kyoo, Ryan Stephen, Bajari Patrick, “ The Random Coefficients Logit Model Is Identified,” Journal of Econometrics , 166 ( 2012), 204– 212. Google Scholar CrossRef Search ADS Fréchette Guillaume, “ Session-Effects in the Laboratory,” Experimental Economics , 15 ( 2012), 485– 498. Google Scholar CrossRef Search ADS Gemici Ahu, Wiswall Matthew, “ Evolution of Gender Differences in Post-Secondary Human Capital Investments: College Majors at the Intensive Margin,” International Economic Review , 55 ( 2014), 23– 56. Google Scholar CrossRef Search ADS Goldin Claudia, “ A Grand Gender Convergence: It’s Last Chapter,” American Economic Review , 104 ( 2014), 1091– 1119. Google Scholar CrossRef Search ADS Goldin Claudia, Katz Lawrence, “ The Cost of Workplace Flexibility for High-Powered Professionals,” Annals of the American Academy of Political and Social Science , 638 ( 2011), 45– 67. Google Scholar CrossRef Search ADS Goldin Claudia, Katz Lawrence, “ A Most Egalitarian Profession: Pharmacy and the Evolution of a Family-Friendly Occupation,” Journal of Labor Economics , 34 ( 2016), 705– 746. Google Scholar CrossRef Search ADS Hainmueller Jens, Hangartner Dominik, Yamamoto Teppei, “ Validating Vignette and Conjoint Survey Experiments Against Real-World Behavior,” Proceedings of the National Academy of Sciences , 112 ( 2015), 2395– 2400. Google Scholar CrossRef Search ADS Hwang Hae-shin, Mortensen Dale, Reed W. Robert, “ Hedonic Wages and Labor Market Search,” Journal of Labor Economics , 16 ( 1998), 815– 847. Google Scholar CrossRef Search ADS Lim Katherine, “ Self-Employment, Workplace Flexibility, and Maternal Labor Supply: A Life-Cycle Model,” Working paper, Office of Tax Analysis, US Department of the Treasury, 2015. Lordan Grace, Pischke Jorn-Steffen, “ Does Rosie Like Riveting? Male and Female Occupational Choices,” NBER Working paper 22495, 2016. Maestas Nicole, Mullen Kathleen J., Powell David, Wenger Jeffrey, Wachter Till von, “ 2015 American Working Conditions Survey: First Findings,” University of Michigan Retirement Research Center (MRRC) Working paper 2016–342, 2016. Google Scholar CrossRef Search ADS Manski Charles, “ Measuring Expectations,” Econometrica , 72 ( 2004), 1329– 1376. Google Scholar CrossRef Search ADS Mas Alexandre, Pallais Amanda, “ Valuing Alternative Work Arrangements,” American Economic Review , forthcoming. McFadden Daniel, Train Kenneth, “ Mixed MNL Models for Discrete Response,” Journal of Applied Econometrics , 15 ( 2000), 447– 470. Google Scholar CrossRef Search ADS Neumark David, Burn Ian, Button Patrick, “ Is It Harder for Older Workers to Find Jobs? New and Improved Evidence from a Field Experiment,” NBER Working paper 21669, 2015. Niederle Muriel, Vesterlund Lise, “ Do Women Shy away from Competition? Do Men Compete too Much?,” Quarterly Journal of Economics , 122 ( 2007), 1067– 1101. Google Scholar CrossRef Search ADS Reuben Ernesto, Wiswall Matthew, Zafar Basit, “ Preferences and Biases in Educational Choices and Labor Market Expectations: Shrinking the Black Box of Gender,” Economic Journal , forthcoming. Rosen Sherwin, “ The Theory of Equalizing Differences,” in Handbook of Labor Economics , vol. 1, Ashenfelter O., Layard R., eds. ( Elsevier, 1987), 641– 692. Google Scholar CrossRef Search ADS Roy A. D., “ Some Thoughts on the Distribution of Earnings,” Oxford Economic Papers , New Series, 3 ( 1951) 135– 146. Google Scholar CrossRef Search ADS Stern Scott, “ Do Scientists Pay to Be Scientists?,” Management Science , 50 ( 2004), 835– 853. Google Scholar CrossRef Search ADS Stinebrickner Ralph, Stinebrickner Todd, “ A Major in Science? Initial Beliefs and Final Outcomes for College Major and Dropout,” Review of Economic Studies , 81 ( 2014a), 426– 472. Google Scholar CrossRef Search ADS Stinebrickner Todd R., Stinebrickner Ralph, “ Academic Performance and College Dropout: Using Longitudinal Expectations Data to Estimate a Learning Model,” Journal of Labor Economics , 32 ( 2014b), 601– 644. Google Scholar CrossRef Search ADS Train Kenneth, Discrete Choice Methods with Simulation ( Cambridge: Cambridge University Press, 2003). Google Scholar CrossRef Search ADS Wasserman Melanie, “ Hours Constraints, Occupational Choice and Fertility: Evidence from Medical Residents,” Working paper, UCLA Anderson School of Management, 2015. Wiswall Matthew, Zafar Basit, “ Determinants of College Major Choice: Identification Using an Information Experiment,” Review of Economic Studies , 82 ( 2015a), 791– 824. Google Scholar CrossRef Search ADS Wiswall Matthew, Zafar Basit, “ How Do College Students Respond to Public Information about Earnings?,” Journal of Human Capital , 9 ( 2015b), 117– 169. Google Scholar CrossRef Search ADS Wiswall Matthew, Zafar Basit, “ Replication Data for: ‘Preference for the Workplace, Investment in Human Capital, and Gender’,” Harvard Dataverse ( 2017), doi:10.7910/DVN/MLOGDL Zafar Basit, “ College Major Choice and the Gender Gap,” Journal of Human Resources , 48 ( 2013), 545– 595. Google Scholar CrossRef Search ADS © The Author(s) 2017. Published by Oxford University Press on behalf of the President and Fellows of Harvard College. All rights reserved. For Permissions, please email: journals.permissions@oup.com
### Journal
The Quarterly Journal of EconomicsOxford University Press
Published: Feb 1, 2018
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2018-10-20 21:37:34
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https://www.lebesgue.fr/en/content/sem2017-kaehler-program
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# School - Flows and Limits in Kähler Geometry
### Titles and abstracts
• Bo Berndtsson (Chalmers University) : Direct image bundles and variations of complex structures
Given a smooth proper fibration $p:\mathcal X\to B$ and $L$ a line bundle over $\mathcal X$, the direct image $$E:= p_*(L)$$ is in many cases a holomorphic vector bundle over $B$. Its fibers are the spaces of holomorphic sections of $L$ over the fibers of $p$, $X_t=p^{-1}(t)$, and they can be given various $L^2$-metrics. In case the fibration is of relative dimension $n$ so that the fibers are compact Riemann surfaces, special cases of this situation can be used to study the variation of complex structures on the fibers $X_t$. (The fibers are all diffeomorphic, but their complex structure varies with $t$, so we can view the family $X_t$ as a family of variations of complex structures on one fixed smooth manifold.) When the relative dimension is higher than one the situation is more complicated and one needs to consider also higher direct images. I will discuss the problems that arise in this connection, with previous work of Siu, Schumacher and To-Yeung and some recent joint work with Xu Wang and Mihai Paun.
• Hans-Joachim Hein (Fordham University) : Tangent cones of Calabi-Yau varieties
It has been known for about 10 years that the classical Calabi-Yau theorem on the existence and uniqueness of Ricci-flat Kahler metrics on smooth complex manifolds with zero first Chern class can be extended to a natural setting of weak Kahler metrics on singular complex varieties. However, until relatively recently nothing was known - even in the simplest nontrivial examples - about the precise asymptotic behavior of these weak Ricci-flat metrics at the singularities of the underlying varieties. I will explain work of Donaldson-Sun, H-Naber and H-Sun that resolves this question in certain cases.
• Valentino Tosatti (Northwestern University) : Metric Limits of Calabi-Yau Manifolds
In this mini-course I will give an introduction to the study of limits of Ricci-flat Kahler metrics on a compact Calabi-Yau manifold when the Kahler class degenerates to the boundary of the Kahler cone. Analytically, the problem is to prove suitable uniform a priori estimates for solutions of a degenerating family complex Monge-Ampère equations, away from some singular set. Geometrically, this can be used to understand the Gromov-Hausdorff limit of these metrics. And if the manifold is projective algebraic and the limiting class is rational, the limits possess an algebraic structure and are obtained from the initial manifold via contraction morphisms from Mori theory.
• Jeff Viaclovsky (Wisconsin University) : The geometry of SFK ALE metrics
I will discuss some of the basics of scalar-flat Kaehler (SFK) metrics, and focus on the geometry of SFK metrics which are asymptotically locally Euclidean (ALE). These space arise as "bubbles" in the compactness theory of Calabi's extremal Kaehler metrics. I will also present some of the deformation theory of SFK ALE metrics.
• Thibaut Delcroix (ENS Paris) : Kähler geometry of horospherical manifolds
Horospherical manifolds form a class of almost homogeneous manifolds whose Kähler geometry is very close to that of toric manifolds. They strictly contain homogeneous toric bundles, to which a lot of results holding for toric manifolds have been extended. I will present horospherical manifolds, trying to convince you that they are not much harder to deal with, and in particular I will present the criterion for K-stability in the Fano case that follows either from my work on spherical varieties, or from a direct, Wang-Zhu type, approach.
• Eleonora Di Nezza (Imperial College) : Monge-Ampère energy and weak geodesic rays
The recent proof of Demailly's conjecture by Witt Nyström gives another evidence that pluripotential theory play a key role when working with complex Monge-Ampère equations in order to solve problems in differential and algebraic geometry. In this talk we investigate pluripotential tools: we characterise Monge-Ampère energy classes in terms of envelopes. And in order to do that, we develop the theory of weak geodesic rays in a big cohomogy class. We also give a positive answer to an open problem in pluripotential theory. This is a joint work with Tamas Darvas and Chinh Lu.
• Jakob Hultgren (Chalmers University) : Coupled Kähler-Einstein Metrics
A central theme in complex geometry is to study various types of canonical metrics, for example Kähler-Einstein metrics and cscK metrics. In this talk we will introduce the notion of coupled Kähler-Einstein (cKE) metrics which are k-tuples of Kähler metrics that satisfy certain coupled Kähler-Einstein equations. We will discuss existence and uniqueness properties and elaborate on related algebraic stability conditions. (Joint work with David Witt Nyström)
• Zakarias Sjostrom Dyrefelt (Université de Toulouse) : K-stability of constant scalar curvature Kähler manifolds
In this talk we introduce a variational/pluripotential approach to the study of K-stability of Kähler manifolds with transcendental cohomology class, extending a classical picture for polarised manifolds. Our approach is based on establishing a formula for the asymptotic slope of the K-energy along certain geodesic rays, from which we deduce that cscK manifolds are K-semistable. Combined with a recent properness result of R. Berman, T. Darvas and C. Lu we further deduce uniform K-stability of cscK manifolds with discrete automorphism group, thus confirming one direction of the YTD conjecture in this setting. If time permits we also discuss possible extensions of these results to the case of compact Kähler manifolds admitting holomorphic vector fields.
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2022-07-05 12:59:49
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https://www.cheenta.com/restricted-maximum-likelihood-estimator-isi-mstat-psb-2012-problem-9/
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Get inspired by the success stories of our students in IIT JAM 2021. Learn More
# How to Pursue Mathematics after High School?
For Students who are passionate for Mathematics and want to pursue it for higher studies in India and abroad.
This is a very beautiful sample problem from ISI MStat PSB 2012 Problem 9, It's about restricted MLEs, how restricted MLEs are different from the unrestricted ones, if you miss delicacies you may miss the differences too . Try it! But be careful.
## Problem- ISI MStat PSB 2012 Problem 9
Suppose $X_1$ and $X_2$ are i.i.d. Bernoulli random variables with parameter $p$ where it us known that $\frac{1}{3} \le p \le \frac{2}{3}$. Find the maximum likelihood estimator $\hat{p}$ of $p$ based on $X_1$ and $X_2$.
### Prerequisites
Bernoulli trials
Restricted Maximum Likelihood Estimators
Real Analysis
## Solution :
This problem seems quite simple and it is simple, if and only if one observes subtle details. Lets think about the unrestricted MLE of $p$,
Let the unrestricted MLE of $p$ (i.e. when $0\le p \le 1$ )based on $X_1$ and $X_2$ be $p_{MLE}$, and $p_{MLE}=\frac{X_1+X_2}{2}$ (How ??)
Now lets see the contradictions which may occur if we don't modify $p_{MLE}$ to $\hat{p}$ (as it is been asked).
See, that when if our sample comes such that $X_1=X_2=0$ or $X_1=X_2=1$, then $p_{MLE}$ will be 0 and 1 respectively, where $p$, the actual parameter neither takes the value 1 or 0 !! So, $p_{MLE}$ needs serious improvement !
To, modify the $p_{MLE}$, lets observe the log-likelihood function of Bernoulli based in two samples.
$\log L(p|x_1,x_2)=(x_1+x_2)\log p +(2-x_1-x_2)\log (1-p)$
Now, make two observations, when $X_1=X_2=0$ (.i.e. $p_{MLE}=0$), then $\log L(p|x_1,x_2)=2\log (1-p)$, see that $\log L(p|x_1,x_2)$ decreases as p increase, hence under the given condition, log_likelihood will be maximum when p is least, .i.e. $\hat{p}=\frac{1}{3}$.
Similarly, when $p_{MLE}=1$ (i.e.when $X_1=X_2=1$), then for the log-likelihood function to be maximum, p has to be maximum, i.e. $\hat{p}=\frac{2}{3}$.
So, to modify $p_{MLE}$ to $\hat{p}$, we have to develop a linear relationship between $p_{MLE}$ and $\hat{p}$. (Linear because, the relationship between $p$ and $p_{MLE}$ is linear. ). So, $\hat{p}$ and $p_{MLE}$ is on the line that is joining the points $(0,\frac{1}{3})$ ( when $p_{MLE}= 0$ then $\hat{p}=\frac{1}{3}$) and $(1,\frac{2}{3})$. Hence the line is,
$\frac{\hat{p}-\frac{1}{3}}{p_{MLE}-0}=\frac{\frac{2}{3}-\frac{1}{3}}{1-0}$
$\hat{p}=\frac{2-X_1-X_2}{6}$. is the required restricted MLE.
Hence the solution concludes.
## Food For Thought
Can You find out the conditions for which the Maximum Likelihood Estimators are also unbiased estimators of the parameter. For which distributions do you think this conditions holds true. Are the also Minimum Variance Unbiased Estimators !!
Can you give some examples when the MLEs are not unbiased ?Even If they are not unbiased are the Sufficient ??
## What to do to shape your Career in Mathematics after 12th?
From the video below, let's learn from Dr. Ashani Dasgupta (a Ph.D. in Mathematics from the University of Milwaukee-Wisconsin and Founder-Faculty of Cheenta) how you can shape your career in Mathematics and pursue it after 12th in India and Abroad. These are some of the key questions that we are discussing here:
• What are some of the best colleges for Mathematics that you can aim to apply for after high school?
• How can you strategically opt for less known colleges and prepare yourself for the best universities in India or Abroad for your Masters or Ph.D. Programs?
• What are the best universities for MS, MMath, and Ph.D. Programs in India?
• What topics in Mathematics are really needed to crack some great Masters or Ph.D. level entrances?
• How can you pursue a Ph.D. in Mathematics outside India?
• What are the 5 ways Cheenta can help you to pursue Higher Mathematics in India and abroad?
## Want to Explore Advanced Mathematics at Cheenta?
Cheenta has taken an initiative of helping College and High School Passout Students with its "Open Seminars" and "Open for all Math Camps". These events are extremely useful for students who are really passionate for Mathematic and want to pursue their career in it.
To Explore and Experience Advanced Mathematics at Cheenta
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2021-08-05 23:23:26
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http://www.optimization-online.org/DB_HTML/2008/07/2054.html
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- Nonlinear optimization for matroid intersection and extensions Yael Berstein(yaelbertx.technion.ac.il) Jon Lee(jonleeus.ibm.com) Shmuel Onn(onnie.technion.ac.il) Robert Weismantel(weismantelimo.math.uni-magdeburg.de) Abstract: We address optimization of nonlinear functions of the form $f(Wx)$~, where $f:\R^d\rightarrow \R$ is a nonlinear function, $W$ is a $d\times n$ matrix, and feasible $x$ are in some large finite set $\calF$ of integer points in $\R^n$~. Generally, such problems are intractable, so we obtain positive algorithmic results by looking at broad natural classes of $f$~, $W$ and $\calF$~. One of our main motivations is multi-objective discrete optimization, where $f$ trades off the linear functions given by the rows of $W$~. Another motivation is that we want to extend as much as possible the known results about polynomial-time linear optimization over trees, assignments, matroids, polymatroids, etc. to nonlinear optimization over such structures. We assume that the convex hull of $\calF$ is well-described by linear inequalities (i.e., we have an efficient separation oracle). For example, the set of characteristic vectors of common bases of a pair of matroids on a common ground set satisfies this property for $\calF$~. In this setting, the problem is already known to be intractable (even for a single matroid), for general $f$ (given by a comparison oracle), for (i) $d=1$ and binary-encoded $W$~, and for (ii) $d=n$ and $W=I$~. Our main results (a few technicalities suppressed): 1- When $\calF$ is well described, $f$ is convex (or even quasiconvex), and $W$ has a fixed number of rows and is unary encoded or with entries in a fixed set, we give an efficient deterministic algorithm for maximization. 2- When $\calF$ is well described, $f$ is a norm, and binary-encoded $W$ is nonnegative, we give an efficient deterministic constant-approximation algorithm for maximization. 3- When $\calF$ is well described, $f$ is ray concave'' and non-decreasing, and $W$ has a fixed number of rows and is unary encoded or with entries in a fixed set, we give an efficient deterministic constant-approximation algorithm for minimization. 4- When $\calF$ is the set of characteristic vectors of common bases of a pair of vectorial matroids on a common ground set, $f$ is arbitrary, and $W$ has a fixed number of rows and is unary encoded, we give an efficient randomized algorithm for optimization. Keywords: matroid intersection, well-described polytope, nonlinear optimization Category 1: Combinatorial Optimization Category 2: Nonlinear Optimization Citation: Download: [PDF]Entry Submitted: 07/22/2008Entry Accepted: 07/31/2008Entry Last Modified: 07/22/2008Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Programming Society and by the Optimization Technology Center.
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2018-12-15 07:42:12
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http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1064.47070
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Zbl 1064.47070
The equivalence between Mann-Ishikawa iterations and multistep iteration.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 58, No. 1-2, A, 219-228 (2004). ISSN 0362-546X
In this interesting paper, the authors consider the equivalence between the one-step, two-step, three-step and multistep-iteration process for solving the nonlinear operator equations $Tu = 0$ in a Banach space for pseudocontractive operators $T$. It is worth mentioning that three-step iterative schemes were introduced by {\it M. A. Noor} [J. Math. Anal. Appl. 251, 217--229 (2000; Zbl 0964.49007)]. Three-step iterations are usually called Noor iterations. The present authors also discuss the stability problems for these iterations. An open problem is also mentioned. Is there a map for which, namely: Noor iteration converges to a fixed point, but for which the Ishikawa iteration fails to converge?
MSC 2000:
*47J25 Methods for solving nonlinear operator equations (general)
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
Keywords: Noor iteration; Mann iteration; Ishikawa iteration; strongly pseudocontractive map; strongly accretive map
Citations: Zbl 0964.49007
Highlights
Master Server
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2013-05-22 13:00:49
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https://www.sparrho.com/item/ending-laminations-and-cannon-thurston-maps/94dfe2/
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# Ending Laminations and Cannon-Thurston Maps
Research paper by Mahan Mj
Indexed on: 25 Nov '13Published on: 25 Nov '13Published in: Mathematics - Geometric Topology
#### Abstract
In earlier work, we had shown that Cannon-Thurston maps exist for Kleinian surface groups. In this paper we prove that pre-images of points are precisely end-points of leaves of the ending lamination whenever the Cannon-Thurston map is not one-to-one. In particular, the Cannon-Thurston map is finite-to-one. This completes the proof of the conjectural picture of Cannon-Thurston maps for surface groups.
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2021-01-21 14:38:53
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https://www.nature.com/articles/s41467-022-33683-1?error=cookies_not_supported&code=100cd3ec-c56d-429f-9d73-38e38cfcd363
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## Introduction
Cell migration is a pivotal step during the process of cancer metastasis, as it enables cancerous cells disseminating out of a primary tumor to move through tissues and ultimately develop metastatic colonies in distant organs. Metastasizing cells migrate either by remodeling their surrounding three-dimensional (3D) extracellular matrix (ECM) to open up migratory paths, by following leader cells such as tumor-associated stromal cells that generate such paths, or by migrating through pre-existing, 3D longitudinal channel-like tracks created by various anatomical structures1,2,3.
It is well established that cell motility is governed by cell-matrix interactions and the actomyosin cytoskeleton. Ion channels and ion transporters have also been recognized as important constituents of the cell migration machinery4,5. Yet, our understanding of how and which of them regulate cell locomotion is, at best incomplete. As proposed by the Osmotic Engine Model (OEM)6,7, cell locomotion in confined spaces is mediated by highly coordinated cycles of local isosmotic swelling at the cell leading edge and shrinkage at the trailing edge, driven by the polarization of select ion transporters, ion channels and aquaporins (AQPs). We previously reported that the Na+/H+ exchanger 1 (NHE1) together with AQP5 polarize at the leading edge of migrating cells in confined microenvironments7. NHE1 supports confined migration even after cell treatment with a high dose of latrunculin A (LatA), which completely disrupts F-actin7. However, the role of NHE1 in promoting isosmotic swelling has yet to be established. Furthermore, it is currently unknown which ion channel(s) and AQP(s) preferentially localize at the cell rear, and mediate isosmotic shrinkage of the cell trailing edge. Also, does the spatial polarization of these molecules confer migration directionality? Do they act in concert with NHE1 and AQP5 to mediate efficient migration in vitro? Lastly, do they affect breast cancer cell metastasis in vivo? To address these fundamental and translational questions, we combined microfluidics with live-cell imaging, novel optogenetic tools, mathematical modeling, and in vivo mouse and ex vivo chick embryo models.
Here, we show that SWELL1 (LRRC8A)8,9 and AQP4 preferentially localize at the trailing edge of confined breast cancer cells and that SWELL1 mediates isosmotic shrinkage consistent with its role in regulating local volume decrease. By developing optogenetic tools to control the spatiotemporal pattern of SWELL1, we demonstrate that its polarization at the cell rear confers migration direction. We also developed a multi-phase, steady-state cell migration mathematical model to predict the relation between SWELL1 expression and cell migration. Furthermore, we delineate the effects of individual and dual knockdown of NHE1 and SWELL1 on cell dissemination from 3D spheroids in vitro as well as on breast cancer growth and metastasis using an orthotopic mouse model and an ex vivo chick embryo model.
## Results
### NHE1 and SWELL1 preferentially polarize at the cell leading and trailing edges, respectively, and mediate cell volume regulation and efficient confined migration
According to OEM6,7, cells migrating inside confining channels display a spatial gradient of distinct ion transporters and AQPs in the cell membrane so that local swelling at the leading edge and shrinkage at the trailing edge, respectively, facilitate net cell locomotion. In line with previous findings in various tumor cell types7, NHE1 (Fig. 1a, b and Supplementary Fig. 1a, b) is polarized at the cell leading edge of MDA-MB-231 breast cancer cells migrating inside polydimethylsiloxane (PDMS)-based confining channels of prescribed dimensions (Width = 3 µm; Height = 10 µm; Length = 200 µm) coated with collagen type I. Consistent with its role in cell protrusion7, NHE1 mediates isosmotic cell swelling in confinement, as evidenced by the reduced volume of NHE1-silenced relative to scramble control (SC) MDA-MB-231 cells (Fig. 1c, d) measured from confocal 3D image reconstructions of Lifeact-GFP-labeled cells (Supplementary Fig. 1c)10. This finding was further validated by measuring the cell longitudinal area (Supplementary Fig. 1d), which serves as a proxy of cell volume since MDA-MB-231 cells contact all four channel walls inside a narrow channel7,11 using different short hairpin (sh)RNA sequences (Supplementary Fig. 1e, f). In line with prior work7, NHE1 knockdown suppresses confined migration (Fig. 1e and Supplementary Fig. 1g).
Because cell migration involves a cycle of isosmotic regulatory volume increase (RVI) at the front and regulatory volume decrease (RVD) at the rear5,7, and confined cells present preferential localization of the RVI-mediating NHE1 at the leading edge, we next examined whether the RVD-mediating SWELL1 chloride channel and select AQPs to localize at the trailing edge. Indeed, live-cell imaging using ectopically expressed SWELL1-GFP (Fig. 1a, b and Supplementary Fig. 2a; Supplementary Movie 1) and AQP4-mCherry (Supplementary Fig. 2a, b; Supplementary Movie 2) reveals that they are polarized and colocalized at the trailing edge of MDA-MB-231 cells migrating in confinement. Immunofluorescence assays using an anti-SWELL1 monoclonal antibody also confirmed the preferential enrichment of endogenous SWELL1 at the cell rear (Supplementary Fig. 2c, d). Whole-cell patch-clamp experiments reveal that SC MDA-MB-231 cells exhibit SWELL1-mediated chloride currents, ICl,vol, after exposure to hypotonicity, as SWELL1 silencing (Fig. 1c) nearly abolishes these currents (Supplementary Fig. 2e). A similar reduction in hypotonicity-induced chloride currents is observed using the selective SWELL1 inhibitor, DCPIB (37.5 µM), in SC cells (Supplementary Fig. 2e). DCPIB also exerts a modest inhibitory effect on ICl,vol currents in SWELL1-knockdown (KD) cells (Supplementary Fig. 2e), suggesting the presence of a very small residual amount of SWELL1 in these cells. Importantly, SWELL1 mediates isosmotic cell shrinkage, as evidenced by testing SWELL1-KD cells generated with different shRNA sequences (Fig. 1c and Supplementary Fig. 1e), which exhibit increased volume (Fig. 1d) and larger longitudinal area (Supplementary Figs. 1d and 2f) but reduced motility (Fig. 1e and Supplementary Fig. 2g) relative to SC cells in confinement. In view of the colocalization of SWELL1 and AQP4 at the cell rear and because AQP4-KD (Supplementary Fig. 2h) compared to SC cells also exhibit increased longitudinal area (Supplementary Fig. 2i), we postulate that SWELL1 works in concert with AQP4 to mediate shrinkage of the cell rear. Along these lines, AQP4 silencing suppresses confined migration (Supplementary Fig. 2j). Dual knockdown of NHE1 and SWELL1 does not alter cell volume (Fig. 1d) or longitudinal area (Supplementary Fig. 1d) in confinement, which is in accord with their individual counteracting effects on cell volume regulation.
In line with the role of NHE1 and SWELL1 in isosmotic swelling and shrinkage, respectively, their individual knockdown impaired MDA-MB-231 cell entry and migration into confining channels (Fig. 1e, f). Importantly, dual silencing of NHE1 and SWELL1 results in a cooperative and pronounced inhibition of cell entry and confined migration (Fig. 1e, f). Of note, NHE1 or/and SWELL1 do not affect the proliferation rate of MDA-MB-231 cells (Supplementary Fig. 2k). To provide further support for the critical involvement of NHE1 and SWELL1 in confined migration, we demonstrate that cell velocities in Na+ free and Cl low solutions are reduced relative to appropriate control media, and mirror those of NHE1- and SWELL1-KD cells, respectively (Supplementary Fig. 3a). To extend our findings beyond the MDA-MB-231 cell model, we demonstrate that NHE1 and SWELL1 preferentially polarize at the cell front and rear, respectively, of metastatic SUM159 breast cancer cells, migrating in confinement12 and mediate isosmotic swelling and shrinkage (Supplementary Fig. 3b, c). Moreover, pharmacological inhibition of NHE1 by EIPA (40 µM)5,7 and/or SWELL1 by DCPIB (40 µM)8 markedly suppresses the migration of SUM159 cells and metastatic PTEN−/−/KRAS(G12V) MCF-10A cells12,13, which bear a double mutation that results in PTEN loss and overexpression of activated KRAS(G12V) (Supplementary Fig. 3d,e). Cells adjust their volume by transporting primarily Na+, Cl, K+ via plasma membrane channels and transporters4. Although they possess several Na+, Cl, and K+ transporters, such as the Na+/ K+/2Cl (NKCC) co-transporters, we have excluded the potential involvement of NKCC, as its pharmacological inhibition fails to alter the migration of scramble control or dual NHE1- and SWELL1-KD cells (Supplementary Fig. 3f).
To extend the physiological relevance of our results, we examined the functional roles of NHE1 and/or SWELL1 in cell dissemination from 3D breast cancer spheroids embedded in 3D collagen gels (Fig. 1g; Supplementary Movie 3) or on 2D collagen I-coated surfaces (Supplementary Movie 4). In concert with the findings inside narrow channels, dual depletion of NHE1 and SWELL1 was markedly more efficient than individual knockdowns in delaying MDA-MB-231 cell dissemination from spheroids and their subsequent migration inside 3D collagen gels (Fig. 1h–k). As another measure of local cell invasiveness in 3D collagen gels, we quantified the area and circularity of spheroids after having been embedded in 3D collagen gels for 12 h. Spheroids consisting of dually depleted cells displayed a smaller area of expansion and increased circularity (Fig. 1l, m), indicative of a less invasive phenotype14. The roles of NHE1 and SWELL1 in cell dissemination from 3D spheroids and area of spheroid expansion were also verified using SUM159 cells (Supplementary Fig. 4a). To extend the physiological relevance of our findings, we further demonstrate that SWELL1 is preferentially polarized at the cell trailing edge during the dissociation of SWELL1-GFP-tagged cells from spheroids embedded in a 3D collagen gel (Fig. 1n and Supplementary Fig. 4b). Collectively, our data support a model by which the repeated and coordinated cycle of local isosmotic swelling at the leading edge and shrinkage at the trailing edge mediated by NHE1 and SWELL1, respectively, supports migration in confinement as well as cell dissemination from tumor spheroids.
### SWELL1 polarization controls cell migration direction and efficiency, as predicted mathematically and determined experimentally via optogenetic spatiotemporal regulation
We developed a multi-phase model15,16 to understand actin-water-ion-coupled cell migration in confinement. The model accounts for F-actin, G-actin, cytosol (essentially water), charged ions, and focal adhesions, which provide force to the cell through the actin network. Myosin contraction is not explicitly modeled, and pressure in the actin network is dominated by passive pressure due to actin swelling. The intracellular governing equations for actin and cytosol velocities as well as boundary fluxes satisfy mass and force balances15. The boundary condition of the actin-network phase is linked to the rate of actin polymerization and depolymerization15, and the boundary fluxes of the cytosol are linked to the water flux across the cell membrane. The actin-network phase remains within the cell while the cytosol phase exchanges with the extracellular medium. Water influx and efflux are determined by the total chemical potential difference across the cell membrane7, i.e., $${J}_{{{{{{\rm{water}}}}}}}=-\alpha (\Delta p-\Delta \varPi )$$, where Δp and ΔΠ are the hydrostatic and osmotic pressure differences across the cell membrane, respectively. The hydrostatic pressure is obtained from the cytosol pressure. The osmotic pressure is determined by the total concentration of all the ionic species under consideration. The model also takes into account the transport of key ionic species across the cell membrane, including Na+, H+, and Cl, and assumes electroneutrality at equilibrium.
In light of experimental data revealing SWELL1 polarization at the cell rear of migrating cells in confinement (Fig. 1a, b and Supplementary Figs. 2a, c, d), we aimed to understand how SWELL1 spatial localization impacts migration. By altering the permeability coefficients of Cl at the cell front ($${\alpha }_{{{{{{\rm{Cl}}}}}},p}^{{{{{{\rm{f}}}}}}}$$) and rear ($${\alpha }_{{{{{{\rm{Cl}}}}}},p}^{{{{{{\rm{b}}}}}}}$$), which depend not only on the SWELL1 channel property but also on the density of these channels in the membrane, the mathematical model predicts that maximal migration velocity is achieved when SWELL1 is enriched at the cell rear, consistent with its role in RVD (Fig. 2a, Supplementary Fig. 5a). Importantly, equal distribution of SWELL1 expression at the cell poles is sufficient to cease motility, whereas preferential polarization at the cell front results in the reversal of migration direction, as evidenced by the negative velocity values (Fig. 2a, Supplementary Fig. 5a).
To test the model prediction and directly establish the role of SWELL1 polarization pattern in the direction and efficiency of confined migration, we developed optogenetic tools to regulate its spatiotemporal localization on the cell membrane, using the cryptochrome 2 (Cry2)-CIBN light-gated dimerizer system17,18. This technology relies on the fusion of SWELL1 to Cry2-mCherry (OptoSWELL1) and its GFP-labeled dimerization partner CIBN engineered to bind to the plasma membrane via the CAAX anchor (CAAX-CIBN-GFP) in response to blue light (Fig. 2b). Before light stimulation, SWELL1 localizes primarily at the trailing edge of cells migrating inside confining channels (Fig. 2b–d, Supplementary Movie 5). Light stimulation at the cell leading edge gradually promotes local SWELL1 enrichment, which is accompanied by a reduction of SWELL1 intensity at the opposite pole (Fig. 2b–d, Supplementary Movie 5). During this process, cell migration velocity decreases as the front-to-rear ratio of SWELL1 expression progressively increases (Fig. 2a–d). When SWELL1 is equally distributed at the cell poles at t = t1, cell motility halts (Fig. 2c, d). Further light stimulation (t > t1) induces preferential SWELL1 enrichment at the cell front along with the concomitant reversal of migration direction, as evidenced by the negative velocity values (Fig. 2b–d, Supplementary Movie 5). The relative fold change of front to rear SWELL1 intensity ratio for each cell following optogenetic stimulation is shown in Supplementary Fig. 5b.
To further establish the key role of SWELL1 in regulating confined migration, we utilized the Light-induced Protein Degradation (LiPD) system to locally deplete SWELL1 expression at the cell rear. This system relies on the light-induced Cry2-CIBN heterodimer formation, which brings together the GFP binding nanobody (Cry2-Nb) and the E3 ubiquitin ligase domain (CIBN-E3) to mark GFP-tagged proteins (SWELL1-GFP) for degradation. Light stimulation at the cell trailing edge induces local depletion of SWELL1-GFP (Fig. 2e–f and Supplementary Fig. 5c), which is accompanied by reduced migration velocity in confinement (Fig. 2g and Supplementary Fig. 5d). As a control, light stimulation of SWELL1-GFP-expressing cells lacking the Cry2-Nb and CIBN-E3 constructs does not reduce SWELL1 expression or cell migration velocity (Supplementary Fig. 5c, e–g). To further validate these findings, we used optoSWELL1, and its GFP-labeled dimerization partner CIBN engineered to target SWELL1 to the mitochondrial membrane (mito-CIBN-GFP). Light stimulation at the cell rear results in progressive downregulation of SWELL1 expression locally and concomitant reduction of migration velocity (Supplementary Fig. 5h–j). Using the Mito Tracker Deep Red, we confirmed that optogenetic downregulation using the optoSWELL1 and mito-CIBN-GFP system does not interfere with the mitochondrial function of cells (Supplementary Fig. 5k–m).
To illustrate the functional contribution of OEM to confined cell migration, we optogenetically upregulated or downregulated SWELL1-GFP expression at the trailing edge of MDA-MB-231 cells inside narrow microchannels following treatment with latrunculin A (LatA, 2 µM), which abrogates actin polymerization7. Optogenetic enrichment of SWELL1 polarization at the rear of LatA-treated cells increases cell migration velocity in confinement (Fig. 3a–d). In contrast, optogenetic downregulation of SWELL1 expression at the cell rear nearly halts motility (Fig. 3a–d). These experimental observations are corroborated by mathematical modeling predictions, which reveal that increasing the permeability coefficient ratio of Cl at the back relative to the front of the cell ($${\alpha }_{{{{{{\rm{Cl}}}}}},p}^{{{{{{\rm{b}}}}}}}/{\alpha }_{{{{{{\rm{Cl}}}}}},p}^{{{{{{\rm{f}}}}}}}$$), as a proxy of SWELL1 polarization at the trailing edge, enhances confined migration in the absence of actin polymerization (Fig. 3e). On the other hand, a decrease in $${\alpha }_{{{{{{\rm{Cl}}}}}},p}^{{{{{{\rm{b}}}}}}}/{\alpha }_{{{{{{\rm{Cl}}}}}},p}^{{{{{{\rm{f}}}}}}}$$ratio stalls motility (Fig. 3e), To further substantiate this experimental and theoretical finding, we further demonstrate that LatA blocks the migration of SWELL1-KD cells in confinement (Fig. 3f). Taken together, these data illustrate the cooperative roles of actin cytoskeleton and OEM in driving efficient cell migration. SWELL1 polarization at the cell trailing edge is sufficient to drive OEM-based, F-actin-independent migration in confinement. Importantly, SWELL1 regulates both the direction and efficiency of confined cell migration.
### RhoA activity regulates SWELL1 localization, whereas Cdc42 facilitates the reversal of migration direction by controlling NHE1 repolarization
Confinement induces a mesenchymal (protrusive) to amoeboid (blebbing) phenotypic switch19,20. Blebbing cells migrating in confinement10,21,22 display a pill-like morphology and bear membrane blebs, which are identified as sphere-like bulges localized at the cell poles (Supplementary Fig. 6a). Because blebbing requires RhoA activation23 (Supplementary Fig. 6a, b) and volume sensitive chloride channels are modulated by RhoA24,25, we examined how optogenetic regulation of RhoA activity alters the spatial localization of SWELL1 using SWELL1-iRFP-expressing MDA-MB-231 breast cancer cells. MDA-MB-231 cells migrating on 2D surfaces or inside unconfined microchannels (Width = 10 µm, Height =10 µm) exhibit a protrusive morphology (Fig. 4a). Light-induced upregulation of RhoA activity via optoGEF-RhoA/CAAX-CIBN-GFP17 at the leading edge of cells inside unconfined channels enriched SWELL1 localization (Fig. 4a, c) and promoted a mesenchymal/protrusive to blebbing phenotypic switch followed by a reversal of migration direction (Fig. 4a; Supplementary Movie 6). On the other hand, light-induced downregulation of RhoA activity via optoGEF-RhoA/mito-CIBN-GFP at the trailing edge of migrating cells inside confining (3 × 10 µm2) channels locally suppressed SWELL1 localization and migration velocity (Fig. 4b, d, e). As a control, light stimulation of SWELL1-iRFP- and OptoGEF-expressing cells lacking mito-CIBN-GFP failed to alter both SWELL1 expression (Fig. 4d) and cell migration velocity (Supplementary Fig. 6c). Taken together, these data suggest that RhoA activity regulates the spatial localization of SWELL1 and modulates cell migration direction and efficiency. Remarkably, optogenetic stimulation of RhoA activity at the leading edge of SWELL1-depleted MDA-MB-231 cells migrating inside 10 × 10 µm2 microchannels causes a transient retraction of the cell front but fails to reverse their migration direction (Fig. 4f, g; Supplementary Movie 7), further illustrating the critical role of SWELL1 in controlling migration direction.
Persistent cell migration requires the spatial polarization of distinct proteins at the cell front and rear, whereas disruption of cell front-to-rear polarity alters migration direction26,27. Because enrichment of SWELL1 expression at the cell rear confers migration directionality, and in light of the distinct polarization patterns of NHE1 and SWELL1 along the cell surface and their coordinated actions in confined cell migration, we hypothesized that efficient reversal of migration direction also requires NHE1 repolarization to the new leading edge. To test this hypothesis, cells migrating inside confining channels were subjected to a hypotonic shock (165 mOsm/l) at the cell leading edge, which caused the reversal of migration direction in 50% of the cell population (Fig. 5a, b), consistent with prior work7. Importantly, immunofluorescence analysis reveals that NHE1 repolarizes to the new leading edge of cells that reversed migration direction (Fig. 5c, d). In line with the finding that only half of the cells reversed migration direction (Fig. 5a, b), the front to rear ratio of NHE1 fluorescence intensity averaged to the value of 1 (Fig. 5d), with half of the cells displaying polarization either at the old or new leading edge.
We next aimed to delineate the underlying mechanism of NHE1 repolarization to the new leading edge following the reversal of migration direction. Cdc42 is a key cell polarity protein that is typically active at the leading edge of migrating cells26,27. Inhibition of Cdc42 using ML141 did not alter migration velocity in confinement under isotonic conditions (Fig. 5a). However, this pharmacological intervention markedly reduced the fraction of MDA-MB-231 cells that reversed migration direction in response to hypotonic shock (Fig. 5b), which is attributed to the fact that Cdc42-inhibited cells failed to repolarize NHE1 under these conditions (Fig. 5c, d). To establish the critical role of Cdc42 activity in an efficient reversal of migration direction, we tested how optogenetic enrichment of SWELL1 expression at the cell leading edge impacts this process in the presence and absence of ML141. Cdc42 inhibition relative to vehicle control did not alter SWELL1 polarization pre- or post-optogenetic stimulation (Fig. 5e–g). Although ML141 had no effect on cell motility prior to optogenetic stimulation and did not interfere with the reversal of migration direction following light-induced upregulation of SWELL1 at the old leading edge, it markedly suppressed the migration velocity of cells after they reversed direction (Fig. 5f, g). This effect is attributed to the lack of NHE1 repolarization in Cdc42-inhibited cells. Of note, pharmacological inhibition of NHE1 via EIPA reduces migration velocity pre- and post-optogenetic enrichment of SWELL1 at the old leading edge (Supplementary Fig. 6d, e). Cumulatively, these data illustrate that Cdc42 activity is required for efficient cell reversal.
### Dual NHE1 and SWELL1 knockdown blocks breast cancer cell extravasation and metastasis in vivo
In view of the critical roles of NHE1 and SWELL1 in cell dissemination from breast cancer spheroids and cell migration in 3D collagen gels and confining channels in vitro, we examined their functional contributions to breast cancer metastasis in vivo. To this end, luciferase- and GFP-labeled SC, NHE1-KD, SWELL1-KD, and dual NHE1/SWELL1-KD MDA-MB-231 breast cancer cells were subcutaneously injected into the 4th mammary fat pad of NOD-SCIDγ (NSG) mice. Bioluminescence imaging reveals that SC, single- and dual-KD cells formed tumors that grew at similar rates for up to 3 weeks (Fig. 6a, b). All mice were sacrificed at week 4 when the bioluminescence signal of the primary tumor reached saturation.
Bioluminescence imaging analysis at necropsy revealed that all mice (10/10) injected with SC cells developed metastases in the bone and brain, whereas only four out of ten (4/10) mice with dual NHE1/SWELL1-KD cells displayed metastases in these tissues (Fig. 6c). SC cells were also more efficient than dual-KD cells (10/10 versus 6/10) in generating metastases in the axillary lymph nodes (Fig. 6c). Moreover, individual NHE1 or SWELL1 knockdown reduced the frequency of metastasis in these tissues (Fig. 6c). Although all mice from the SC group and nearly all from the dual-KD exhibited metastases in the liver and lung, bioluminescence image analysis of the surgically-isolated tissues revealed a 10-fold and 20-fold decrease in the metastatic burden, respectively, for the dual-KD tumor cells (Fig. 6d, e). To independently validate the reduced metastatic burden detected in the liver and lungs of mice injected with dual-KD as opposed to SC cells, DNA was extracted from these tissues, and the amount of human DNA was analyzed using quantitative PCR (qPCR) with primers specific for human long interspersed nuclear elements(hLINE)12,28. Dual-KD relative to SC cells displayed a six-fold and five-fold reduced amount of human DNA in the livers and lungs, respectively (Fig. 6f). Individual NHE1 or SWELL1 knockdown suppressed metastatic burden in these tissues, as assessed by bioluminescence image analysis (Fig. 6e), whereas a trend was detected by qPCR, which reached statistical significance for the NHE1-KD cells in the lung (Fig. 6f). Taken altogether, these in vivo data are in accord with in vitro findings showing that dual relative to single KD has a significantly higher inhibitory effect on the migration of luciferase-labeled cells (Supplementary Fig. 7). Tissue samples from representative SC and KD specimens were also processed for immunohistochemistry against GFP, which was used as a specific marker for the transplanted human tumor cells, as well as hematoxylin and eosin staining (Fig. 6g). These images confirm the consistent presence of metastatic human SC cells in the livers and lungs of mice and the marked reduction of dual-KD cells.
Tumor cell extravasation is a critical step for the dissemination of cancerous cells to distant organs in the body. To examine the functional involvement of NHE1 and SWELL1 in tumor cell extravasation, we employed the avian embryo cancer cell extravasation assay29,30, which permits real-time visualization of this process at excellent optical resolution. When injected into the embryo bloodstream, SC cells robustly extravasated from the CAM vasculature, with the majority leaving the vasculature by the 8 h timepoint (Fig. 6h). In marked contrast, dual NHE1/SWELL1-KD suppressed breast cancer cell extravasation more than two-fold with the majority of these cells remaining inside the CAM vascular network (Fig. 6h, i). These findings are in agreement with the data obtained in the in vitro setting and a metastatic murine model.
To provide further support for the role of NHE1 and SWELL1 in breast cancer metastasis, patient distant metastasis-free survival (DMFS) time and gene expression data were analyzed using the KM-plotter database31, which combines datasets from GEO, EGA, and TCGA. Patients were split into two groups based on their expression levels of NHE1 and SWELL1. The cutoff producing the greatest separation of DMFS between the 2 groups is shown in Fig. 6j. Comparing the lowest and highest tertiles or quartiles produced similar results. Kaplan–Meier survival analysis shows that patients expressing high levels of both NHE1 and SWELL1 had lower DMFS (Fig. 6j (iii)), suggesting that inhibition of NHE1 and SWELL1 may represent a potential therapeutic regimen for suppressing breast cancer dissemination and metastasis in vivo.
## Discussion
Cell migration is a pivotal step in the metastatic dissemination of cancer cells from a primary tumor to distant organs in the body. Cell motility is governed by cell-matrix interactions, the actomyosin cytoskeleton, and cell volume regulation as proposed by OEM7. According to OEM, a cell migrating in confinement establishes a spatial gradient of ion transporters, ion channels, and AQPs in the cell membrane so that local swelling at the leading edge and shrinkage at the trailing edge, respectively, facilitate net cell movement7. We previously reported that NHE1, which polarizes at the cell leading edge, can support confined migration even after complete disruption of F-actin7. We herein show that NHE1 promotes isosmotic cell swelling consistent with its role in RVI4. Moreover, we determined that SWELL1 and AQP4 are preferentially enriched at the cell rear of migrating cells in confinement and mediate local cell shrinkage via RVD. The coordinated action of isosmotic swelling and shrinkage at the cell poles mediated by NHE1 and SWELL1, respectively, due to their distinct polarization patterns and roles in RVI and RVD, supports efficient confined migration. We postulate that the polarization of SWELL1 at the cell trailing edge mediates local RVD due to an outflow of Cl ions and water, which results in rear membrane shrinkage and decreased cell volume relative to its equilibrium state. This reduced cell volume is compensated by RVI, which occurs at the cell leading edge due to the local enrichment of NHE1 expression. When the cell volume exceeds its equilibrium state due to RVI, SWELL1 is reactivated, and the coordinated RVD/RVI cycle is repeated. Through this feedback loop and dynamic regulation of water/ion fluxes, cells maintain their volume and migrate efficiently in confinement. Analogous to squeezing a soft, porous material filled with water such as a sponge at one end, SWELL1-mediated outflow of Cl ions and water at the cell rear dissipate pressure towards the extracellular environment, while its rear end shrinkage concurrently propagates pressure towards the nucleus, causing its forward translocation. This coordinated RVD/RVI cycle, which involves SWELL1-dependent rear-end retraction and forward nuclear translocation coupled with NHE1-mediated leading edge protrusion, is responsible for efficient cell locomotion.
We further demonstrate that dual NHE1 and SWELL1 knockdown blocks cancer cell dissemination from breast cancer spheroids and reduces their motility in 3D as well as breast cancer cell extravasation and metastasis in vivo without affecting tumorigenesis. The inhibitory effects of combined NHE1 and SWELL1 depletion are stronger than their individual knockdowns both in vitro and in vivo, consistent with their coordinated functional roles.
Recent work has shown that SWELL1 promotes the motility of various cancer cell types, such as hepatocellular carcinoma (HCC)32 and gastric cancer cells33, as evidenced by wound healing and transwell migration assays. Yet, how SWELL1 impacts cell motility remains elusive. Using an integrated experimental and mathematical approach, we demonstrate that SWELL1 polarization at the cell rear mediates not only shrinkage of the cell trailing edge in accord with its role in RVD, but, most importantly, also confers migration direction. Optogenetic enrichment of SWELL1 at the cell leading edge reverses migration direction, whereas the equal distribution of SWELL1 at the cell poles ceases motility. Using optogenetic tools, we further demonstrate that SWELL1 localization is regulated by RhoA activity. As such, upregulation of RhoA activity at the cell front promotes local SWELL1 enrichment, thereby reversing migration direction. Importantly, the inability of SWELL1-knockdown cells to reverse migration direction in response to optogenetic RhoA stimulation at the cell front illustrates the indispensable role of SWELL1 in this process. It is noteworthy that efficient cell reversal also requires Cdc42, which controls NHE1 repolarization to the “new” leading edge. Importantly, these data reveal the crosstalk between ion transporters/channels and key constituents of the cell cytoskeleton (RhoA and Cdc42) in the process of cell migration.
Despite general consensus in the literature regarding the critical role of SWELL1 in cell migration, it has also been reported that SWELL1 is dispensable for the migration of HCT116 colon carcinoma cells and U251 and U87 glioblastoma cells34. Cell phenotypic differences might provide a potential explanation for this discrepancy. Because RhoA activation induces SWELL1 enrichment as well as a blebbing phenotype10,11, whereas RhoA inhibition depletes SWELL1 localization and is accompanied by a protrusive/mesenchymal phenotype11, we postulate that SWELL1 facilitates the migration of cells displaying a blebbing phenotype.
There are conflicting data regarding the potential role of SWELL1 in cell proliferation. For instance, SWELL1 overexpression has been reported to induce the proliferation of HCC cells, whereas its depletion has opposite effects32. Along these lines, SWELL1 knockdown suppresses both primary tumor growth and metastasis of HCC cells in vivo32. However, recent findings have linked cell survival to SWELL1 activity and/or expression, albeit only under hypertonic conditions35. Moreover, others have shown that SWELL1 does not affect cell proliferation34, which is in line with our in vitro findings and the lack of any effect of SWELL1 depletion on primary tumor growth. Although SWELL1 knockdown tended to decrease the metastatic burden in the liver and lungs of mice as assessed by qPCR and bioluminescence (in the case of the liver), metastasis in these organs was consistently and markedly inhibited upon dual NHE1 and SWELL1 knockdown. This dual intervention also blocked breast cancer cell extravasation in the CAM model. The in vivo efficacy of dual NHE1 and SWELL1 depletion, which correlates with our in vitro findings using diverse complementary assays and the Kaplan–Meier survival analysis, establishes the physiological relevance of OEM.
## Methods
### Experimental methods
All mouse experiments were performed in accordance with the Institutional Animal Care and Use Committee procedures and guidelines of the University of Maryland at Baltimore under approved protocol number 0219006. All procedures involving chick embryos were approved by the University of Alberta Institutional Animal Care and Use Committee (IACUC).
### Cell culture
Human MDA-MB-231 cells12 (ATCC, catalog number: HTB-26) were cultured in DMEM (Gibco) supplemented with 10% heat-inactivated FBS (Gibco) and 1% penicillin/streptomycin (10,000 U/mL, Gibco). SUM159 cells12 were provided by Denis Wirtz (Johns Hopkins University) and were grown in Ham’s F-12 medium (Corning Cellgro) plus 5% FBS, 1% penicillin/streptomycin, 1 μg/mL hydrocortisone (Sigma–Aldrich) and 5 μg/mL insulin (Sigma–Aldrich). PTEN-/-/KRAS(G12V) MCF-10A cells were a gift from Michele I. Vitolo (University of Maryland at Baltimore) and were cultured as described previously13. MDA-MB-231-luciferase cells were created and cultured as described previously12. Cells were maintained in an incubator at 37˚C with 95% air/5% CO2 and passaged upon 60–80% confluency every 3–5 days. Cells were routinely checked for mycoplasma contamination via PCR using the primers: F-(5ʹ-GGGAGCAAACAGGATTAGATACCCT-3ʹ) and R-(5ʹ-TGCACCATCTGTCACTCTGTTAACCTC-3ʹ).
### Cloning, lentivirus production, and cell transduction
To generate plasmids with shRNA lentiviral vectors, we subcloned the targeting sequences or nontargeting scramble control into the pLVTHM lentiviral plasmid (Addgene, plasmid 12247, a gift from Dider Trono) using MluI and ClaI as restriction sites. The target sequences are as follows:
nontargeting scramble control sh1 (5’-GCACTACCAGAGCTAACTCAGATAGTACT-3’),
human sh1NHE1 (5’-GACAAGCTCAACCGGTTTAAT-3’),
human sh2NHE1 (5’-CCAATCTTAGTTTCTAACCAA-3’).
In addition, we subcloned the targeting sequences or nontargeting scramble control into the pLKO.1 lentiviral plasmid (Addgene, plasmid 8453, a gift from B. Weinberg) using AgeI and EcoRI as restriction sites. The target sequences are:
nontargeting scramble control sh1, human NHE1 sh1, and human NHE1 sh2, as shown above,
human sh1SWELL1 (5’-GGTACAACCACATCGCCTA-3’),
human sh2SWELL1 (5’-GAGCAAGTCTCAAGAGCGC-3’),
human shAQP4 (5’-CCAAGTCCGTCTTCTACAT-3’). Sequence integrity and orientation were verified by Sanger Sequencing (JHU Genetic Resources Core Facility).
The pLVTHM, pLKO.1, pLenti.PGK.LifeAct-GFP.W (plasmid 51010, a gift from Rusty Lansford), pLenti.PGK.H2B-mCherry (plasmid 51007, a gift from Rusty Lansford), psPAX2 (plasmid 12260, gift from Didier Trono), and pMD2.G (plasmid 12259, a gift from Didier Trono) plasmids were purchased from Addgene.
For lentivirus production, 293 T/17 cells were cotransfected with psPAX2, pMD2.G, and the lentiviral plasmid of interest. The media was refreshed after 24 h. Lentivirus was harvested 48 h after transfection, filtered through 0.45 μm filters (Fisher Scientific), and purified by centrifugation (50,000 × g for 2 h at 4 °C). Next, cells were transduced for 48 h with a medium containing lentiviral particles. Puromycin (0.5 μg/mL, Gibco) was added to the cell culture media 48 h after transduction, and this concentration was maintained to select cells transduced with pLKO.1 vectors. In all in vitro and in vivo experiments involving SC, single- and dual-KD cells, proper controls were included by transducing cells with the corresponding nontargeting sequences in the appropriate vectors (Supplementary Table 1).
### Plasmid transfection
The SWELL1-GFP plasmid was a kind gift from Thomas J. Jentsch (Leibniz-Institut für Molekulare Pharmakologie (FMP), Berlin). The AQP4-mCherry plasmid was provided by Antonio Frigeri (University of Bari). The plasmids for the light-induced protein degradation system, (LiPD) pRing-CIBN-IR and pGBP-PHR-IR, were kind gifts from Heinrich Leonhardt (University of Munich). MDA-MB-231 cells, at 50–60% confluency, were transiently transfected with Lipofectamine 3000 reagent following the manufacturer’s recommendation. Only in the case of SWELL1-GFP, a stable cell line was generated following treatment with G418 (Corning) and sorting.
### Generation of plasmids for optogenetic experiments
SWELL1-iRFP was created by replacing GFP from the SWELL1-GFP plasmid with iRFP. iRFP was amplified using piRFP670-N1 (Addgene, plasmid 45457). SWELL1-iRFP was inserted into lentiviral backbone pLV-EF1a-IRES-Puro (Addgene, plasmid 85132). Forty-eight hours post-shRNA transduction, puromycin (0.5 μg/mL, Gibco) was added to a fresh cell culture medium to select stably transduced cells expressing SWELL1-iRFP.
ARHGEF11(DHPH)-Cry2-mCherry (OptoGEF), CAAX-CIBN-GFP, and mito-CIBN-GFP were gifts from Dr. Xavier Trepat (Institute for Bioengineering of Catalonia). Cry2-mCherry was amplified from ARHGEF11(DHPH)-Cry2-mCherry and inserted into pLV-EF1a-IRES-Hygro (Addgene, plasmid 85134). SWELL1 was amplified from SWELL1-iRFP, and then inserted into pLV-EF1a-IRES-Hygro-Cry2-mCherry to create SWELL1-Cry2-mCherry (OptoSWELL1). Hygromycin B (500 μg/mL, ThermoFisher Scientific) was added to a fresh medium 48 h post-transduction to select cells stably transduced with OptoSWELL1.
### Microfluidic device fabrication, cell seeding, cell treatment, live-cell imaging, and analysis
PDMS-based microfluidic devices containing an array of parallel microchannels of prescribed height (10 µm), width (3 µm), and length (200 µm) were fabricated as described previously36,37,38. The microchannel dimensions were verified by a laser profilometer. Microchannels were sandwiched orthogonally by 2D-like seeding and media channels36,37,38. Prior to migration assays, assembled microfluidic devices were incubated with rat tail collagen I (20 µg/ml, Thermo Fisher Scientific) for at least 1 h at 37 °C in the presence of 95% air/5% CO2. Migration experiments were performed in DMEM containing 10% heat-inactivated FBS (Gibco) and 1% penicillin/streptomycin (10,000U/ml, Gibco). No chemotactic stimulus was applied in these experiments. Twenty microliters of cell suspension (4 × 106 cells/ml) in serum-containing medium were added to the device seeding inlet. In select experiments, cells were treated with the following pharmacological agents or corresponding vehicle controls: 5-(N-Ethyl-N-isopropyl) amiloride (EIPA, 40 µM, Sigma–Aldrich), DCPIB (40 µM, Tocris Biosciences), bumetanide (30 μM, Santa Cruz Biotechnology), Lat-A (2 µM, Sigma–Aldrich), ML141 (10 μM, Santa Cruz Biotechnology). In these assays, a medium containing either the drug or the vehicle control was added to all inlet and outlet wells of the device at the onset of the migration experiment, unless otherwise stated. In Lat-A assays, Lat-A-containing medium was added only after the cells had fully entered the microchannels.
Time-lapse images were recorded in 10 min intervals for up to 20 h in an inverted Nikon Eclipse Ti microscope (Nikon, Tokyo, Japan) equipped with a stage-top incubator (Okolab, Pozzuoli, Italy, or Tokai Hit, Shizuoka, Japan) at 37 °C and 95% air/5% CO2, automated controls (NIS-Elements, v. 4.13.05; Nikon) and a ×10/0.30 numerical aperture Ph1 objective. Cell migration analysis was performed as previously described10,39. Briefly, live-cell videos were exported to ImageJ (v.2.0.0/1.51 h; National Institute of Health, Bethesda, Maryland). The tracks of individual cells that had fully entered the microchannels were obtained manually via Manual Tracking (Cordelières F, Institut Curie, Orsay, France) plugin. Cell migration velocity was calculated using a custom MATLAB script (MathWorks, Natick MA). Cell entry time was defined as the time interval from the point that a cell’s leading edge initiated entry into the microchannel until its trailing edge had fully entered the microchannel, and was calculated manually. The cell longitudinal area was measured by manually outlining the cell periphery in ImageJ.
### Microfluidic assays in Na+ free or Cl− low media
To test the effect of extracellular Na+ on cell motility, select microfluidic assays were performed with a self-assembled NaCl-containing medium (control) and N-Methyl-D-glucamine-chloride (NMDG-Cl)-containing medium (Na+ free). NaCl-containing medium was composed of 140 mM NaCl, 2.5 mM KCl, 0.5 mM MgCl2, 1.2 mM CaCl2, 5 mM glucose and 10 mM HEPES, with pH and osmolarity adjusted to 7.4 and 300–305 mOsm/L, respectively. To generate the Na+ free medium, NaCl was substituted with NMDG-Cl. Cells were seeded and then incubated in a serum-containing medium for at least 3 h at 37 °C in the presence of 95% air/5% CO2. After cells fully entered the microchannels, the medium was removed, and a control medium or Na+ free medium was added to all inlet and outlet wells of the device. Time-lapse images were recorded every 10 min for up to 6 h in 100% air, with all other imaging settings remaining the same as the regular migration assay.
To test the effect of extracellular Cl on cell motility, select assays were performed with high glucose (4.5 g/L) DMEM (regular medium), self-assembled NaCl-containing medium (control), and Na+-glutamate-containing medium (Cl low). The control medium was prepared by adding all the components present in the formulation of commercial DMEM with pH and osmolarity adjusted to 7.4 and 340 mOsm/L, respectively. To generate Cl low medium, NaCl was substituted with a corresponding amount of Na+-glutamate. Regular, control, or Cl low medium was added to all inlet and outlet wells of the microchannel device after the cells were seeded. Time-lapse images were recorded in 10 min intervals for up to 6 h at 37 °C in the presence of 95% air/5% CO2, with all other image settings remaining the same as the regular migration assay.
### Spheroid formation, and 3D collagen invasion assay
Spheroids were formed as previously described40. Briefly, growth factor reduced Matrigel was diluted with DMEM containing 10% heat-inactivated FBS and 1% penicillin/streptomycin at 1:3 ratio. Fifty microliters of the diluted Matrigel were transferred to a 96-well plate (Falcon) and polymerized for 1 h at 37 °C in a cell culture incubator, while the rest of the diluted Matrigel was kept on ice. 2 × 103 breast cancer cells were suspended in 50 µL ice-cold Matrigel and gently plated in different wells pre-coated with polymerized Matrigel followed by incubation at 37 °C and 5% CO2 in a cell culture incubator. ~1.5 h later, 100 µL of prewarmed DMEM containing 10% FBS and 1% P/S was added in each well. The cell culture medium was replaced every two days, and spheroids were used in invasion assays ~10–15 days later.
3D collagen invasion assays using spheroids were performed as previously described41. Briefly, 3 mL of rat tail collagen type I (Corning) was gently mixed with 375 μL of 10× DMEM—low glucose (Sigma). The pH of the mixture was slowly adjusted to physiological levels with NaOH. After 1 h incubation on ice, 25 μL of the mixture were added to a 24-well plate (Falcon) and incubated at 37 °C for 1 h. Spheroids were collected into 1.5 ml Eppendorf tubes by gently disrupting the Matrigel with ice-cold DMEM. The Eppendorf tube was incubated in ice to further depolymerize the Matrigel for >10 min. Spheroids were isolated by 2665 × g centrifugations for 5 min, and resuspended into 100 μL of the collagen mixture. Next, 100 µL of the spheroid-collagen mixture were plated in each well and incubated at 37 °C for 1–1.5 h. After collagen polymerization, 500 μL prewarmed cell culture media was added to each well.
Time-lapse images were recorded in 20 min intervals for ~30 h in an inverted Nikon Eclipse Ti microscope (Nikon) equipped with a stage-top incubator (Okolab or Tokai Hit) at 37 °C and 95% air/5% CO2, automated controls (NIS-Elements, Nikon) and a ×10/0.30 numerical aperture Ph1 objective. First-cell dissociation times were obtained using NIS-Elements (Nikon) by manually measuring the time required for the first cell to fully detach from the spheroid. Cell velocity, mean squared displacement (MSD), and cell trajectory were calculated using a custom-made MATLAB script. Normalized area expansion and circularity were determined using ImageJ by outlining the spheroid at t = 0 and 12 h using polygonal regions of interest. In select experiments, spheroids were placed on 2D collagen I (20 µg/ml)-coated surfaces, and cell dissociation was tracked in real-time (Supplementary Movie 4).
To investigate the spatiotemporal distribution of SWELL1 during dissociation of SWELL1-GFP-tagged cells from 3D spheroids embedded in collagen gels, time-lapse confocal images were recorded at 10 min intervals for up to 30 h using a Nikon A1 confocal microscope equipped with a ×20 air objective, a 488 nm laser and NIS-Elements software (v. 5.02.01).
### Cell volume measurements
Lifeact-GFP-labeled cells were imaged using a Nikon A1 confocal microscope with a ×60 oil objective and a 488 nm laser. Cells were visualized with Imaris (v. 9.7.0; Bitplane, Zurich, Switzerland), and their volume was measured from confocal Z-stacks with a step of 0.5 μm using a custom MATLAB (v. R2016b; MathWorks, Natick, MA) script, as described previously10.
### Patch-clamp experiments
Whole-cell recordings were obtained as previously described42 using an Axon 200 A amplifier (Axon Instruments, San Jose, CA). Currents were acquired at 33 kHz and filtered at 1 kHz. The pClamp8 software (v. 10; Axon Instruments) was used for pulse generation, data acquisition, and subsequent analysis. LRRC8A-like chloride currents were measured in cells clamped at 0 mV and pulsed for 400 ms from –100 mV to +100 mV in 50 mV steps every 30 sec. ICl whole-cell currents were measured using pipettes (2-3 MΩ) filled with a solution containing 100 mM N-methyl-D-glucamine chloride (NMDGCl), 1.2 mM MgCl2, 1 mM EGTA, 10 mM HEPES, 2 mM Na2ATP, and 0.5 mM Na3GTP (pH 7.3 and 300 mOsm/l). The external solution contained NMDGCl at 100 mM (for iso and hypotonic conditions) or 185 mM (hypertonic conditions), 0.5 mM MgCl2, 5 mM KCl, 1.8 mM CaCl2, 5 mM glucose, and 10 mM HEPES, pH 7.4. Osmolarity was adjusted to 310 (isotonic), 220 (hypotonic), with mannitol.
### Optogenetic control of RhoA activity and SWELL1 localization
Optogenetic tools were utilized to control the subcellular activation of RhoA with high spatiotemporal accuracy, using the cryptochrome 2 (Cry2)-CIBN light-gated dimerizer system17,18. This system relies on the fusion of the catalytic (DHPH) domain of the RhoA-GEF, ARHGEF11, to Cry2-mCherry (optoGEF-RhoA) and its GFP-labeled dimerization partner, CIBN, engineered to bind to the plasma via the CAAX anchor (CAAX-CIBN-GFP) or mitochondrial membrane via mito-CIBN-GFP. MDA-MB-231 cells, stably transduced with either CAAX-CIBN-GFP or mito-CIBN-GFP, ARHGEF11(DHPH)-Cry2-mCherry and SWELL1-iRFP, were used to assess the effect of spatiotemporal alterations of RhoA activity on SWELL1 localization. To this end, cells migrating inside confining channels were monitored in real-time by imaging the mCherry channel to identify the leading and trailing edges. Light stimulation was performed with a 488 nm laser at 1% power for 1 sec on a rectangular area placed either at the cell leading or trailing edge. Stimulations were repeated at 10 sec intervals for 10–30 min to enable consistent localization of ARHGEF11 to the membrane or mitochondria. mCherry and iRFP670 images were recorded after each stimulation to monitor the localization of ARHGEF11 and SWELL1.
To directly establish the role of SWELL1 polarization in the direction and efficiency of migration, MDA-MB-231 cells, stably transduced with either CAAX-CIBN-GFP or mito-CIBN-GFP, and OptoSWELL1, were subjected to light stimulation as described above. Briefly, cell migration and SWELL1 localization were monitored by imaging the mCherry channel in real-time. Cell velocity and front-to-rear ratio of SWELL1 expression were quantified using a custom MATLAB script.
In select optogenetic experiments, a medium containing either a pharmacological agent or its vehicle control was added to all inlet and outlet wells of the device only after the cells had fully entered the microchannels. Confocal imaging was initiated at least 30 min after the addition of the drug-containing medium.
### Light-induced protein degradation (LiPD) assays
LiPD assays were performed as recently described43. MDA-MB-231 cells expressing SWELL1-GFP were transfected with pRing-CIBN-IR and pGBP-PHR-IR plasmids (see above). Cells were seeded in microfluidic devices and imaged using a Nikon A1 confocal microscope. A 561 nm laser was used to detect the co-expressed DsRed signal and monitor cells in real-time. Light stimulation was performed with a 488 nm laser at 5% laser power for 1 sec on a rectangular area marked at the cell trailing edge. Stimulations were repeated at 10 sec intervals for up to 30 min. GFP images were recorded after 20 rounds of stimulation to monitor the extent of SWELL1 expression.
### Mitochondria function assay
The assay was performed using the MitoTracker® Deep Red FM (M22426, ThermoFisher Scientific). Briefly, MitoTracker® Deep Red FM working solution was prepared at 1 mM according to the manufacturer’s instructions, and then diluted to 25 nM using a cell medium. MitoTracker® Deep Red FM (25 nM) and Hoechst 33342 (ThermoFisher Scientific, H3570) at 1:5000 dilution were added to all inlet and outlet wells of microfluidic devices after the cells had fully entered the microchannels. Devices were then placed for 45 min in an incubator at 37 °C with 95% air/5% CO2. Next, the liquid was removed from microfluidic devices, and replaced with the fresh prewarmed medium. Cells were imaged using a Nikon A1 confocal microscope, and mitochondria intensity was measured using ImageJ. Normalized mitochondria intensity was calculated relative to DAPI intensity of the nucleus.
### Fluorescence lifetime imaging microscopy (FLIM) of RhoA FRET sensors
Confocal FLIM of live MDA-MB-231 cells stably expressing the RhoA2G sensor was carried out as outlined in refs. 10,22, using ZEN 2.3 SP1 FP3 (black; Zeiss, Jena, Germany) and SymPhoTime 64 (v. 2.4; PicoQuant, Berlin, Germany).
### Hypotonic shock assays
Hypotonic solutions were prepared, and their osmolarity was measured as previously described7,10. After 1.5 h of live-cell imaging in drug- or vehicle control-containing serum-free isotonic medium, the medium in lower and upper wells of the microfluidic devices was replaced with serum-free isotonic or hypotonic medium, respectively, containing the drug or its matching vehicle control. In all pre- and post-shock experiments, the uppermost inlet also contained 10% heat-inactivated FBS. Phase-contrast time-lapse images were recorded at 5 min intervals for 2 h. In select experiments, cells were prepared for immunofluorescence analysis and evaluated under confocal optics.
### Immunofluorescence
For immunostaining with SWELL1 (LRRC8A) antibody (a kind gift from Thomas J. Jentsch9), cells were fixed in pre-cooled methanol (Fisher Chemical) at −20 °C for 10 min, followed by incubation with 30 mM glycine (Sigma) in PBS for 5 min at room temperature. Cells were incubated overnight with the primary antibody (1:100) at 4 °C, followed by washing 3× with PBS, and then 1 h incubation with a secondary antibody (1:100) in PBS containing 0.1% Triton X-100 supplemented with 3% BSA at 4 °C. For immunostaining with NHE1, AQP4, or Ki-67 antibodies, cells were fixed with 4% formaldehyde solution (ThermoFisher Scientific), permeabilized with 0.1% Triton® X-100 (Sigma–Aldrich), blocked with 1% bovine serum albumin (Sigma–Aldrich), immunostained, and imaged with an A1 confocal microscope. Primary antibodies were used at the following concentrations: anti-NHE1 (1:50; mouse, clone 54, Santa Cruz Biotechnology, sc-136239), anti-AQP4 (1:50; mouse, clone 4/18, Santa Cruz Biotechnology, sc-32739), or anti-Ki-67 (1:800; clone 8D5, Cell Signaling Technology). Following overnight incubation with primary antibodies at 4 °C, specimens were washed 3× with PBS, and then secondary antibodies (obtained from Invitrogen) were applied for 1 h at room temperature: Alexa Fluor 488 goat anti-mouse immunoglobulin-G (IgG) (H + L) (1:100), Alexa Fluor 568 goat anti-rabbit IgG (H + L) (1:200), Alexa Fluor Plus 647 goat anti-mouse IgG (H + L) (1:100), or Alexa Fluor Plus 647 goat anti-rabbit IgG (H + L) (1:100). Nuclei were also stained with Hoechst 33342 (1:2500, ThermoFisher Scientific, H3570).
### Quantification of NHE1 and SWELL1 polarization
A custom MATLAB script was used to segment NHE1 or SWELL1 intensity at the cell front and rear, and exclude the signal from the cell interior, which is typically associated with internal vesicles. All pixel intensities at each pole were summed and divided by the total number of non-zero pixels. For visualization purposes, the segmented areas are denoted by red-dashed rectangles at the cell front and the rear (Supplementary Fig. 1b).
### Western blotting
Western blots were performed as previously described10,22 using NuPage 4–12% Bis-Tris gels. Primary antibodies were applied at the following concentrations: anti-NHE1 (1:200; mouse, clone 54, Santa Cruz Biotechnology, sc-136239), anti-SWELL1 (1:100; mouse, clone 8H9, Santa Cruz Biotechnology, sc-517113) or GAPDH (1:1000; rabbit, clone 14C10, Cell Signaling Technology 2118), which was used as the loading control. Following overnight incubation with primary antibodies at 4 °C, membranes were washed 5× with TBST, and secondary antibodies were applied at 1:2000 dilution for 1 h at room temperature: anti-mouse IgG HRP-linked antibody (Cell Signaling Technologies) or anti-rabbit IgG HRP-linked antibody (Cell Signaling Technologies).
### Kaplan–Meier survival analysis
Analysis for breast cancer metastasis was performed using Kaplan–Meier plotter (https://kmplot.com). Auto scan mode was utilized to choose the cutoff between high and low expression cohorts.
### Treatment and inoculation of breast cancer cells in nude mice
Eight- to twelve-week-old female NOD.Cg-Prkdc < scid > /Jmice weighing 19–25 g were obtained from the University of Maryland at Baltimore and fed food and water ad libitum. The mice were maintained in accordance with the Institutional Animal Care and Use Committee procedures and guidelines of the University of Maryland at Baltimore. For subcutaneous injections, 1 × 106 luciferase/GFP-tagged MDA-MB-231 cells (SC, NHE1-KD, SWELL1-KD, or dual NHE1/SWELL1-KD) were suspended in 100 µL PBS and mixed with 25% of the total volume with Matrigel (Corning). Cell number was quantified via Countess® Automated Cell Counter (ThermoFisher), and confirmed by bioluminescent imaging. The cell suspension of SC or KD specimens was then injected subcutaneously into the fourth mammary gland on the ventral surface of the abdomen of the female mice in a blinded manner. Tumor volumes were measured by external caliper measurements weekly from the initial injection to the experimental endpoint. Tumors were measured along the two longest perpendicular axes in the x/y plane of each xenograft tumor to the nearest 0.1 mm with a digital caliper (Thomas Scientific, Inc.). Depth is assumed to be equivalent to the shortest of the perpendicular axes (y), and volume is calculated according to the: V = xy2/2, as the standard practice for xenograft tumors. In accordance with the Institutional Animal Care and Use Committee procedures and guidelines of the University of Maryland at Baltimore, animals were restricted to a maximal tumor burden not to exceed 2 cm3. Mice bearing subcutaneous tumors were euthanized if tumors ulcerated, grew to 10% of the initial body weight, or reached 2 cm3. Signs of tumor ulceration or maximum tumor volume were recorded during each measurement. Tumor volume measurements were performed in a blinded manner.
### Bioluminescence imaging
Luciferase-expressing cells were injected subcutaneously into mice as above. At the indicated timepoints following injection, mice were injected intraperitoneally with D-luciferin potassium salt (150 mg/kg, Perkin Elmer) and returned to their cages for 5 min to allow for biodistribution. Mice were anesthetized with 2% isoflurane gas and imaged at 5 min intervals for the maximum photon emission. Total photon flux (photons/sec) was calculated and corrected for tissue depth by spectral imaging using Living Image 3.0 software (IVIS, Xenogen). Percent primary tumor growth was determined by subtracting the background from the peak signal during each measurement and normalizing it to the initial reading obtained for the same mouse.
Tissue samples collected at the time of necropsy were imaged for bioluminescence as described above in a blinded manner. Bioluminescence was only detected in viable cells expressing the firefly luciferase gene, indicative of an active metabolism. To avoid false positive detection of bioluminescence signal, the background subtracted value was normalized to the background reading, and only readings that were ≥50x the background reading were considered to be positive for metastasis. To control for differences in tumor size, bioluminescence values were normalized to the volume of the primary tumor at the time of necropsy.
### Quantitative PCR
Quantitative PCR for human long interspersed nuclear elements (hLine) was conducted as previously described12.
### Immunohistochemistry and pathology
Animals with primary tumor formation that exceeded the designated endpoint, including saturation exceeding 1000-fold over the initial bioluminescence signal, were sacrificed. Tissue samples were removed, fixed in formalin for 24 h, embedded in paraffin wax, and serially sectioned (4 μm thick). All immunohistochemistry GFP and H&E staining were performed by HistoWhiz (Brooklyn, NY).
### Ex Ovo chick embryo cancer xenograft model
Cancer cell extravasation assays were performed as described before29,30, using 13-day-old fertilized White Leghorn chicken eggs acquired from the University of Alberta Poultry Research Centre. Briefly, 25–50 × 103 cancer cells were injected intravenously into the chicken CAM vein and allowed to extravasate for 8 h. Fifteen minutes before the assay CAM vasculature was visualized via injection of Lectin-649, and cancer cell extravasation was scored using intravital confocal imaging. At least seven animals were used for each condition for 3 experiments. All the procedures were approved by the University of Alberta Institutional Animal Care and Use Committee (IACUC).
### Statistics and reproducibility
All data represent the mean ± SEM or mean ± SD from ≥3 independent experiments (independent biological replicas) for each condition unless stated otherwise. The D’Agostino-Pearson omnibus normality test was used to determine whether data are normally distributed. Datasets with gaussian distributions were compared using Student’s t-test (two-tailed) or one-way ANOVA followed by Tukey’s post hoc test. For log-normal distribution, the statistical comparison was made after logarithmic transformation of the data followed by one-way ANOVA with post hoc Tukey. For comparing non-Gaussian distributions, the nonparametric Mann–Whitney U test or Kruskal–Wallis (with post hoc Dunn) were used for comparisons between two or more groups, respectively. Statistical significance was identified as p < 0.05. The exact p-values are provided in the Source Data file. Data were primarily collected and organized in Microsoft Excel (v. 15.30; Redmond, WA). Analysis was performed using GraphPad Prism 7.0b and 9.1.1 software (San Diego, CA).
In Fig. 1, images are representative of 4 (a) or 3 (c, g) or 2 (n) independent biological replicas. In Fig. 2, images are representative of 3 (b) or 4 (e, f) independent biological replicas. In Fig. 3, images are representative of 4 (a (i), (ii)) independent experiments. In Fig. 4, images are representative of 5 (a) or 6 (b) or 3 (f) independent experiments. In Fig. 5, images are representative of 3 (c, e) independent experiments. In Fig. 6, images are representative of 3 (h) independent experiments.
In Supplementary Fig. 1a, images are representative of 4 independent experiments. In Supplementary Fig. 4b, images are representative of 2 independent experiments. In Supplementary Fig. 5, images are representative of 4 (e) or 3 (h) or 3 (k, l) independent experiments. In Supplementary Fig. 6, images are representative of 2 (a) independent experiments.
### Mechanical part
We used a multi-phase model15,16 to understand the actin-water-ions-coupled cell migration. The model includes cytosol, F-actin, G-actin, and charged ions. A steady-state solution is sought. A confined cell in a channel can be modeled as a one-dimensional system. We use $$x\in [0,L]$$ to indicate the computational domain established in the moving frame of the cell, where L is the cell length. The conservation of momentum and mass of the cytosol are
$$-\frac{{dp}}{{dx}}-\eta {\theta }_{n}\left({v}_{c}-{v}_{n}\right)=0,\frac{d{v}_{c}}{{dx}}=0,$$
(1)
where p and vc are the hydraulic pressure and velocity of the cytosol, respectively; θn and vn are the concentration and velocity of the F-actin network, respectively; and η is the coefficient of interfacial friction between the actin-network phase and the cytosol phase due to the velocity difference. The flux boundary condition for cytosol is
$${v}_{c}-{v}_{0}=-{J}_{{{{{{\rm{water}}}}}}}^{{{{{{\rm{f}}}}}}},\,{{{{{\rm{at}}}}}}\, x=L;{v}_{c}-{v}_{0}=\,\,{J}_{{{{{{\rm{water}}}}}}}^{{{{{{\rm{b}}}}}}},\,{{{{{\rm{at}}}}}}\,x=0,$$
(2)
where v0 is the steady-state velocity of the cell, Jwater is the water influx across the cell membrane, and the superscript ‘f’ and ‘b’ indicate quantities evaluated at the front and back end of the cell, respectively. Water flux is driven by the chemical potential difference of water across the cell membrane44, and its expression is given by
$${J}_{{{{{{\rm{water}}}}}}}^{{{{{{\rm{f}}}}}}({{{{{\rm{b}}}}}})}=-{\alpha }^{{{{{{\rm{f}}}}}}\left({{{{{\rm{b}}}}}}\right)}[({p}^{{{{{{\rm{f}}}}}}\left({{{{{\rm{b}}}}}}\right)}-{p}_{*}^{{{{{{\rm{f}}}}}}\left({{{{{\rm{b}}}}}}\right)})-{RT}\left(\right.{c}^{{{{{{\rm{f}}}}}}\left({{{{{\rm{b}}}}}}\right)}-{c}_{0}^{{{{{{\rm{f}}}}}}\left({{{{{\rm{b}}}}}}\right)}]$$
(3)
where α is the permeability coefficient of water, c is the total concentration of all ion species, R is the gas constant, and T is the absolute temperature. In the model, we use the subscript ‘0’ to indicate the extracellular environment. Due to hydraulic resistance, the hydraulic pressure exerted on the outside of the cell, p*, is different from the hydraulic pressure at infinity, p0p* can be expressed as
$${p}_{*}^{{{{{{\rm{f}}}}}}}={p}_{0}^{{{{{{\rm{f}}}}}}}+{d}_{g}^{{{{{{\rm{f}}}}}}}\left({v}_{0}-{J}_{{{{{{\rm{water}}}}}}}^{{{{{{\rm{f}}}}}}}\right),\,{p}_{*}^{{{{{{\rm{b}}}}}}}={p}_{0}^{{{{{{\rm{b}}}}}}}-{d}_{g}^{{{{{{\rm{b}}}}}}}\left({v}_{0}+{J}_{{{{{{\rm{water}}}}}}}^{{{{{{\rm{b}}}}}}}\right),$$
(4)
where dg is the coefficient of external hydraulic resistance, which depends on the channel geometry and the viscosity of extracellular medium.
In this work, we do not explicitly model myosin contraction; we let the pressure in the actin network, σn, be dominated by passive pressure due to actin swelling. The constitutive relationship for the actin network is modeled as $${\sigma }_{n}={k}_{{\sigma }_{n}}{\theta }_{n}$$, where $${k}_{{\sigma }_{n}}$$ is a constant. Focal adhesions provide forces to the cell through the actin network. These forces can be considered as an effective body force on the network. Therefore, the conservation of momentum of the actin network is written as
$$-\frac{d{\sigma }_{n}}{{dx}}+\eta {\theta }_{n}\left({v}_{c}-{v}_{n}\right)-{\eta }_{{{{{{\rm{st}}}}}}}{\theta }_{n}{v}_{n}=0,$$
(5)
where ηst is the strength of focal adhesions. In the model, we allow actin polymerization to occur at the front of the cell, whereas depolymerization occurs throughout the cytoplasm. The mass conservation of the F-actin network and G-actin are
$$\frac{d}{{dx}}\left({v}_{n}{\theta }_{n}\right)=-\gamma {\theta }_{n},\,\frac{d}{{dx}}\left({v}_{c}{\theta }_{c}\right)={D}_{{\theta }_{c}}\frac{{d}^{2}{\theta }_{c}}{d{x}^{2}}+\gamma {\theta }_{n},$$
(6)
where θc and $${D}_{{\theta }_{c}}$$ are the concentration and diffusion coefficient of G-actin, respectively. γ is a constant rate of actin depolymerization. The boundary condition for F-actin and G-actin are
$${\theta }_{n}\left({v}_{0}-{v}_{n}\right)=\,\,{J}_{{{{{{\rm{actin}}}}}}},\,{\theta }_{c}\left({v}_{0}-{v}_{c}\right)=-{J}_{{{{{{\rm{actin}}}}}}},$$
(7)
at the front of the cell, where $${J}_{{{{{{\rm{actin}}}}}}}={J}_{{{{{{\rm{actin}}}}}}}^{{{{{{\rm{f}}}}}}}{\theta }_{c}/({\theta }_{c,c}+{\theta }_{c})$$ is the rate of actin polymerization; here $${J}_{{{{{{\rm{actin}}}}}}}^{{{{{{\rm{f}}}}}}}$$ and $${\theta }_{c,c}$$ are two constants. The actin flux is zero at the back of the cell. The total amount of actin is conserved such that $${\int }_{0}^{L}\left({\theta }_{n}+{\theta }_{c}\right){dx}=L{\theta }_{*}$$, where θ* is the average concentration of actin.
The cell experiences a frictional force with the channel wall. We let friction be proportional to cell velocity, i.e., $${F}_{f}=\xi {v}_{0}$$, where ξ is a friction coefficient, which applies to the negative direction of cell migration. Taken together, the force balance of the entire cell is
$$-\left({p}_{0}^{{{{{{\rm{f}}}}}}}-{p}_{0}^{{{{{{\rm{b}}}}}}}\right)-\left({d}_{g}^{{{{{{\rm{f}}}}}}}+{d}_{g}^{{{{{{\rm{b}}}}}}}\right)\left({v}_{0}-{J}_{{{{{{\rm{water}}}}}}}^{{{{{{\rm{f}}}}}}}\right)-{\eta }_{{{{{{\rm{st}}}}}}}{\int }_{0}^{L}{\theta }_{n}{v}_{n}{dx}-{F}_{f}=0$$
(8)
The system is solved by considering all the coupled equations together.
### Electrodynamics part
Here we used a multi-species framework45 to account for the electrodynamics part of the model. Species to consider include Na+, K+, $${{{{{{\rm{Cl}}}}}}}^{-}$$, H+, $${{{{{{\rm{HCO}}}}}}}_{3}^{-}$$, $${{{{{{\rm{A}}}}}}}^{-}$$, $${{{{{{\rm{Buf}}}}}}}^{-}$$, and HBuf, where $${{{{{{\rm{A}}}}}}}^{-}$$ are intracellular impermeable charged proteins, $${{{{{{\rm{Buf}}}}}}}^{-}$$ is non-protonated solute, and HBuf is protonated solute in the buffer. We consider the following channels, pump, and transporters in the model: passive Na+, passive K+, passive $${{{{{{\rm{Cl}}}}}}}^{-}$$ (SWELL1), Na+/K+ pump (NKE), NHE, Anion Exchanger 2 (AE2). Combined with the mechanical part, the unknowns for the entire model are $${p}_{c}$$, $${v}_{c}$$, $${v}_{n}$$, $${\theta }_{n}$$, $${\theta }_{c}$$, $${c}_{{{{{{\rm{Na}}}}}}}$$, $${c}_{{{{{{\rm{K}}}}}}}$$, $${c}_{{{{{{\rm{Cl}}}}}}}$$, pH, $${c}_{A}$$, $${c}_{{{{{{\rm{Buf}}}}}}}$$, $$\phi$$, and $${v}_{0}$$, where the c’s are solute concentrations having units of mM. The intracellular solute concentrations include $${c}_{n}=\{{c}_{{{{{{\rm{N}}}}}}a},\,{c}_{{{{{{\rm{K}}}}}}},\,{c}_{{{{{{\rm{Cl}}}}}}},\,{c}_{{{{{{\rm{H}}}}}}},\,{c}_{{{{{{\rm{HC}}}}}}{{{{{{\rm{O}}}}}}}_{3}},\,{c}_{{{{{{\rm{A}}}}}}},\,{c}_{{{{{{\rm{Buf}}}}}}},\,{c}_{{{{{{\rm{HBuf}}}}}}}\}^{T}$$ and the extracellular solute concentrations include $${c}_{n}^{0}=\{{c}_{{{{{{\rm{Na}}}}}}}^{0},\,{c}_{{{{{{\rm{K}}}}}}}^{0},\,{c}_{{{{{{\rm{Cl}}}}}}}^{0}, \,{c}_{{{{{{\rm{H}}}}}}}^{0},\,{c}_{{{{{{\rm{HC}}}}}}{{{{{{\rm{O}}}}}}}_{3}}^{0},\,{c}_{{{{{{\rm{A}}}}}}}^{0},\,{c}_{{{{{{\rm{G}}}}}}}^{0}\}^{T}$$.
The chemical equilibrium equation for the bicarbonate-carbonic acid pair is
$${{{{{\rm{C}}}}}}{{{{{{\rm{O}}}}}}}_{2}({{{{{\rm{aq}}}}}})+{{{{{{\rm{H}}}}}}}_{2}{{{{{\rm{O}}}}}}({{{{{\rm{l}}}}}})\rightleftharpoons {{{{{{\rm{H}}}}}}}^{+}({{{{{\rm{aq}}}}}})+{{{{{\rm{HC}}}}}}{{{{{{\rm{O}}}}}}}_{3}^{-}({{{{{\rm{aq}}}}}}),$$
(9)
where $${[{{{{{\rm{CO}}}}}}_2]}_{{{{{{\rm{aq}}}}}}}$$ is related to the partial pressure of CO2, $${P}_{{{{{{\rm{C}}}}}}{{{{{{\rm{O}}}}}}}_{2}}$$, by the Henry constant $${k}_{H}$$,
$$[{{{{{\rm{C}}}}}}{{{{{{\rm{O}}}}}}}_{2}]_{{{{{{\rm{aq}}}}}}}=\frac{{P}_{{{{{{\rm{C}}}}}}{{{{{{\rm{O}}}}}}}_{2}}}{{k}_{H}}.$$
(10)
The reaction equilibrium constant is
$${k}_{c}=\frac{[{{{{{\rm{HC}}}}}}{{{{{{\rm{O}}}}}}}_{3}^{-}]_{{{{{{\rm{aq}}}}}}}[{{{{{{\rm{H}}}}}}}^{+}]_{{{{{{\rm{aq}}}}}}}}{[{{{{{\rm{C}}}}}}{{{{{{\rm{O}}}}}}}_{2}]_{{{{{{\rm{aq}}}}}}}}.$$
(11)
Extracellular pH is defined as $${{{{{\rm{p}}}}}}{{{{{{\rm{H}}}}}}}_{0}=-{{{{{\rm{lo}}}}}}{{{{{{\rm{g}}}}}}}_{10}[{{{{{{\rm{H}}}}}}}^{+}]_{{{{{{\rm{aq}}}}}},0}$$ and $${{{{{\rm{p}}}}}}{K}_{c}=-{{{{{\rm{lo}}}}}}{{{{{{\rm{g}}}}}}}_{10}{k}_{c}$$ so that Eq. 11 becomes
$${{{{{\rm{p}}}}}}{{{{{{\rm{H}}}}}}}_{0}-{{{{{\rm{p}}}}}}{K}_{c}={{{{{\rm{lo}}}}}}{{{{{{\rm{g}}}}}}}_{10}\frac{[{{{{{\rm{HC}}}}}}{{{{{{\rm{O}}}}}}}_{3}^{-}]_{{{{{{\rm{aq}}}}}}}^{0}}{{P}_{{{{{{\rm{C}}}}}}{{{{{{\rm{O}}}}}}}_{2}}/{k}_{H}}.$$
(12)
$${[{{{{{\rm{CO}}}}}}_2]}_{{{{{{\rm{aq}}}}}}}$$ = $${[{{{{{\rm{CO}}}}}}_2]}_{{{{{{\rm{aq}}}}}}}^{0}$$ since CO2 can move freely across the cell membrane46. For the intracellular domain, we have
$${{{{{\rm{pH}}}}}}-{{{{{\rm{p}}}}}}{K}_{c}={{{{{\rm{lo}}}}}}{{{{{{\rm{g}}}}}}}_{10}\frac{[{{{{{\rm{HC}}}}}}{{{{{{\rm{O}}}}}}}_{3}^{-}]_{{{{{{\rm{aq}}}}}}}}{[{{{{{\rm{C}}}}}}{{{{{{\rm{O}}}}}}}_{2}]_{{{{{{\rm{aq}}}}}}}},$$
(13)
where $${{{{{\rm{pH}}}}}}=-{{{{{\rm{lo}}}}}}{{{{{{\rm{g}}}}}}}_{10}[{{{{{{\rm{H}}}}}}}^{+}]_{{{{{{\rm{aq}}}}}}}$$ is the intracelluar pH. The chemical reaction for the intracellular buffer solution is
$${{{{{\rm{HBuf}}}}}}({{{{{\rm{aq}}}}}})\rightleftharpoons {{{{{{\rm{H}}}}}}}^{+}({{{{{\rm{aq}}}}}})+{{{{{\rm{Bu}}}}}}{{{{{{\rm{f}}}}}}}^{-}({{{{{\rm{aq}}}}}}).$$
(14)
The reaction equilibrium constant is similarly $${k}_{B}=[{{{{{\rm{Bu}}}}}}{{{{{{\rm{f}}}}}}}^{-}]_{{{{{{\rm{aq}}}}}}}[{{{{{{\rm{H}}}}}}}^{+}]_{{{{{{\rm{aq}}}}}}}/[{{{{{\rm{HBuf}}}}}}]_{{{{{{\rm{aq}}}}}}}$$. With $${{{{{\rm{p}}}}}}{K}_{B}=-{{{{{\rm{lo}}}}}}{{{{{{\rm{g}}}}}}}_{10}{k}_{B}$$, we obtain
$${{{{{\rm{pH}}}}}}-{{{{{\rm{p}}}}}}{K}_{B}={{{{{\rm{lo}}}}}}{{{{{{\rm{g}}}}}}}_{10}\frac{[{{{{{\rm{Bu}}}}}}{{{{{{\rm{f}}}}}}}^{-}]_{{{{{{\rm{aq}}}}}}}}{[{{{{{\rm{HBuf}}}}}}]_{{{{{{\rm{aq}}}}}}}}.$$
(15)
The flux for each species is
$${J}_{n}=-{D}_{n}\frac{d{c}_{n}}{{dx}}+{v}_{c}{c}_{n}-{D}_{n}\frac{{z}_{n}F}{{RT}}{c}_{n}\frac{d\phi }{{dx}},$$
(16)
where $${c}_{n}$$, $${z}_{n}$$, and $${D}_{n}$$ are the concentration, valance, and diffusion constant of each ion species, respectively. $$\phi$$ is the intracellular electrical potential. $$F$$, $$R$$, and $$T$$ are the Faraday’s constant, ideal gas constant, and absolute temperature, respectively. The subscript ‘$$n$$’ refers to different ion species, i.e., $$n \in \{{{{{{\rm{N}}}}}}{{{{{{\rm{a}}}}}}}^{+}, {{{{{{\rm{K}}}}}}}^{+},{{{{{\rm{C}}}}}}{{{{{{\rm{l}}}}}}}^{-}, {{{{{{\rm{H}}}}}}}^{+}, {{{{{\rm{HC}}}}}}{{{{{{\rm{O}}}}}}}_{3}^{-}, {{{{{{\rm{A}}}}}}}^{-}, {{{{{\rm{Bu}}}}}}{{{{{{\rm{f}}}}}}}^{-},{{{{{\rm{HBuf}}}}}}\}$$.
The governing equations and boundary conditions for $${c}_{{{{{{\rm{N}}}}}}a}$$, $${c}_{{{{{{\rm{K}}}}}}}$$, $${c}_{{{{{{\rm{C}}}}}}l}$$, and $${c}_{{{{{{\rm{A}}}}}}}$$ are
$$-\frac{d{J}_{{{{{{\rm{Na}}}}}}}}{{dx}}=0,\;{J}_{{{{{{\rm{Na}}}}}}}{{{{{{\rm{|}}}}}}}_{x=L}=-{J}_{{{{{{\rm{Na}}}}}}}^{{{{{{\rm{f}}}}}}},\;{J}_{{{{{{\rm{Na}}}}}}}{{{{{{\rm{|}}}}}}}_{x=0}={J}_{{{{{{\rm{Na}}}}}}}^{{{{{{\rm{b}}}}}}},$$
(17)
$$-\frac{d{J}_{{{{{{\rm{K}}}}}}}}{{dx}}=0,\;{J}_{{{{{{\rm{K}}}}}}}{{{{{{\rm{|}}}}}}}_{x=L}=-{J}_{{{{{{\rm{K}}}}}}}^{{{{{{\rm{f}}}}}}},\;{J}_{{{{{{\rm{K}}}}}}}{{{{{{\rm{|}}}}}}}_{x=0}={J}_{{{{{{\rm{K}}}}}}}^{{{{{{\rm{b}}}}}}},$$
(18)
$$-\frac{d{J}_{{{{{{\rm{Cl}}}}}}}}{{dx}}=0,\;{J}_{{{{{{\rm{Cl}}}}}}}{{{{{{\rm{|}}}}}}}_{x=L}=-{J}_{{{{{{\rm{Cl}}}}}}}^{{{{{{\rm{f}}}}}}},\;{J}_{{{{{{\rm{Cl}}}}}}}{{{{{{\rm{|}}}}}}}_{x=0}={J}_{{{{{{\rm{Cl}}}}}}}^{{{{{{\rm{b}}}}}}},$$
(19)
$$-\frac{d{J}_{{{{{{\rm{A}}}}}}}}{{dx}}=0,\;{J}_{{{{{{\rm{A}}}}}}}{{{{{{\rm{|}}}}}}}_{x=L}={J}_{{{{{{\rm{A}}}}}}}{{{{{{\rm{|}}}}}}}_{x=0}=0,$$
(20)
where the boundary fluxes directed inwards are defined as positive. The governing equation and boundary condition for pH are
$$\begin{array}{c}-\frac{d}{{dx}}\left({J}_{{{{{{\rm{HC}}}}}}{{{{{{\rm{O}}}}}}}_{3}}+{J}_{{{{{{\rm{Buf}}}}}}}-{J}_{{{{{{\rm{H}}}}}}}\right)=0,\left({J}_{{{{{{\rm{HC}}}}}}{{{{{{\rm{O}}}}}}}_{3}}+{J}_{{{{{{\rm{Buf}}}}}}}-{J}_{{{{{{\rm{H}}}}}}}\right){{{{{{\rm{|}}}}}}}_{x=L}=-\left({J}_{{{{{{\rm{HC}}}}}}{{{{{{\rm{O}}}}}}}_{3}}^{{{{{{\rm{f}}}}}}}+{J}_{{{{{{\rm{Buf}}}}}}}^{{{{{{\rm{f}}}}}}}-{J}_{{{{{{\rm{H}}}}}}}^{{{{{{\rm{f}}}}}}}\right),\\ \left({J}_{{{{{{\rm{HC}}}}}}{{{{{{\rm{O}}}}}}}_{3}}+{J}_{{{{{{\rm{Buf}}}}}}}-{J}_{{{{{{\rm{H}}}}}}}\right){{{{{{\rm{|}}}}}}}_{x=0}=\left({J}_{{{{{{\rm{HC}}}}}}{{{{{{\rm{O}}}}}}}_{3}}^{{{{{{\rm{b}}}}}}}+{J}_{{{{{{\rm{Buf}}}}}}}^{{{{{{\rm{b}}}}}}}-{J}_{{{{{{\rm{H}}}}}}}^{{{{{{\rm{b}}}}}}}\right).\end{array}$$
(21)
where $${J}_{{{{{{\rm{Buf}}}}}}}^{{{{{{\rm{b}}}}}}/{{{{{\rm{f}}}}}}}=0$$ due to the assumed non-permeability of buffer solutions. The governing equation and boundary condition for $${c}_{{{{{{\rm{Buf}}}}}}}$$ is
$$-\frac{d}{{dx}}\left({J}_{{{{{{\rm{Buf}}}}}}}+{J}_{{{{{{\rm{HBuf}}}}}}}\right)=0,\left({J}_{{{{{{\rm{Buf}}}}}}}+{J}_{{{{{{\rm{HBuf}}}}}}}\right){{{{{{\rm{|}}}}}}}_{x=L}=\left({J}_{{{{{{\rm{Buf}}}}}}}+{J}_{{{{{{\rm{HBuf}}}}}}}\right){{{{{{\rm{|}}}}}}}_{x=0}=0.$$
(22)
The total amount of non-permeable species are conserved such that
$$S{\int }_{0}^{L}{c}_{A}dx={N}_{A},\,S{\int }_{0}^{L}({c}_{{{{{{\rm{Buf}}}}}}}+{c}_{{{{{{\rm{HBuf}}}}}}})dx={N}_{{{{{{\rm{Buf}}}}}}}+{N}_{{{{{{\rm{HBuf}}}}}}},$$
(23)
where $$S$$ is the cross-sectional area of the cell. The derived qualities are
$${c}_{{{{{{\rm{H}}}}}}}=1{0}^{3}1{0}^{-{{{{{\rm{pH}}}}}}},{c}_{{{{{{\rm{HC}}}}}}{{{{{{\rm{O}}}}}}}_{3}}=\frac{{P}_{{{{{{\rm{C}}}}}}{{{{{{\rm{O}}}}}}}_{2}}}{{k}_{H}}1{0}^{{{{{{\rm{pH}}}}}}-{{{{{\rm{p}}}}}}{K}_{c}},\,{c}_{{{{{{\rm{HBuf}}}}}}}={c}_{{{{{{\rm{Buf}}}}}}}1{0}^{{{{{{\rm{p}}}}}}{K}_{B}-{{{{{\rm{pH}}}}}}},$$
(24)
which should be satisfied at all points in space. The intracellular electrical potential is solved by the electroneutrality condition, i.e., $$\sum {z}_{n}{C}_{n}$$.
The boundary flux for each ionic species is:
$${J}_{{{{{{\rm{Na}}}}}}}^{{{{{{\rm{b}}}}}}/{{{{{\rm{f}}}}}}}={J}_{{{{{{\rm{Na}}}}}},p}^{{{{{{\rm{b}}}}}}/{{{{{\rm{f}}}}}}}+{J}_{{{{{{\rm{NKE}}}}}},{{{{{\rm{Na}}}}}}}^{{{{{{\rm{b}}}}}}/{{{{{\rm{f}}}}}}}+{J}_{{{{{{\rm{NHE}}}}}},{{{{{\rm{Na}}}}}}}^{{{{{{\rm{b}}}}}}/{{{{{\rm{f}}}}}}},$$
(25)
$${J}_{{{{{{\rm{K}}}}}}}^{{{{{{\rm{b}}}}}}/{{{{{\rm{f}}}}}}}={J}_{{{{{{\rm{K}}}}}},p}^{{{{{{\rm{b}}}}}}/{{{{{\rm{f}}}}}}}+{J}_{{{{{{\rm{NKE}}}}}},{{{{{\rm{K}}}}}}}^{{{{{{\rm{b}}}}}}/{{{{{\rm{f}}}}}}},$$
(26)
$${J}_{{{{{{\rm{Cl}}}}}}}^{{{{{{\rm{b}}}}}}/{{{{{\rm{f}}}}}}}={J}_{{{{{{\rm{Cl}}}}}},p}^{{{{{{\rm{b}}}}}}/{{{{{\rm{f}}}}}}}+{J}_{{{{{{\rm{AE}}}}}}2,{{{{{\rm{Cl}}}}}}}^{{{{{{\rm{b}}}}}}/{{{{{\rm{f}}}}}}},$$
(27)
$${J}_{{{{{{\rm{H}}}}}}}^{{{{{{\rm{b}}}}}}/{{{{{\rm{f}}}}}}}={J}_{{{{{{\rm{NHE}}}}}},{{{{{\rm{H}}}}}}}^{{{{{{\rm{b}}}}}}/{{{{{\rm{f}}}}}}},$$
(28)
$${J}_{{{{{{\rm{HC}}}}}}{{{{{{\rm{O}}}}}}}_{3}}^{{{{{{\rm{b}}}}}}/{{{{{\rm{f}}}}}}}={J}_{{{{{{\rm{AE}}}}}}2,{{{{{\rm{HC}}}}}}{{{{{{\rm{O}}}}}}}_{3}}^{{{{{{\rm{b}}}}}}/{{{{{\rm{f}}}}}}}.$$
(29)
For simplicity, in the notation below, we will omit the superscripts ‘f/b’ for membrane fluxes at the front and back of the cell.
The passive fluxes are tension-gated. We denote $${G}_{m}\in (0,1)$$ as a mechanosensitive gating function that generally follows a Boltzmann distribution, i.e., $${G}_{m}={[1+{e}^{-{\beta }_{1}({\tau }_{m}-{\beta }_{2})}]}^{-1}$$, where β1 and β2 are two constants and $${\tau }_{m}$$ is the cortical/membrane tension, which can be calculated from the force balance at the membrane, i.e.,
$${\tau }_{m}^{{{{{{\rm{f}}}}}}}=\frac{b}{2}\left({\sigma }_{n}^{{{{{{\rm{f}}}}}}}+{p}_{c}^{{{{{{\rm{f}}}}}}}-{p}_{*}^{{{{{{\rm{f}}}}}}}\right),\,{\tau }_{m}^{{{{{{\rm{b}}}}}}}=\frac{b}{2}\left({\sigma }_{n}^{{{{{{\rm{b}}}}}}}+{p}_{c}^{{{{{{\rm{b}}}}}}}-{p}_{*}^{{{{{{\rm{b}}}}}}}\right).$$
(30)
The passive ion fluxes, $${J}_{n,p}$$, are proportional to the electrochemical potential difference of ions across the membrane47,
$${J}_{n,p}={\alpha }_{n,p}{G}_{m}\left[{RT}{{{{{\rm{ln}}}}}}{\Gamma }_{n}-{z}_{n}F(\phi -{\phi }_{0})\right],\,n\in \{{{{{{\rm{N}}}}}}{{{{{{\rm{a}}}}}}}^{+},{{{{{{\rm{K}}}}}}}^{+},{{{{{\rm{C}}}}}}{{{{{{\rm{l}}}}}}}^{-}\}$$
(31)
where $${\Gamma }_{n}={c}_{n}^{0}/{c}_{n}$$ is the ratio of extra- to intracellular ion concentrations; $${\alpha }_{n,p}$$ is the permeability coefficient of each species, which depends on the channel property and the density of the channels in the membrane.
The Na+/K+ pump (NKE) is an active ion pump that maintains the membrane potential of cells. It exports three Na+ ions and imports two K+ ions per ATP molecule. Because the overall flux is positive outwards, the pump’s activity depends on the membrane potential48. The NKE flux also depends on the concentrations of Na+ and K+ and saturates at high concentration limits49. Based on these facts, we model the flux of Na+ and K+ through the Na+/K+ pump as
$${J}_{{{{{{\rm{NKE}}}}}}}={J}_{{{{{{\rm{NKE}}}}}},{{{{{\rm{Na}}}}}}}=-\frac{3}{2}{J}_{{{{{{\rm{NKE}}}}}},{{{{{\rm{K}}}}}}}=-{\alpha }_{{{{{{\rm{NKE}}}}}}}{G}_{V,{{{{{\rm{NKE}}}}}}}{\left(1+{\beta }_{{{{{{\rm{NKE}}}}}},{{{{{\rm{Na}}}}}}}{\Gamma }_{{{{{{\rm{Na}}}}}}}\right)}^{-3}{\left(1+{\beta }_{{{{{{\rm{NKE}}}}}},{{{{{\rm{K}}}}}}}/{\Gamma }_{{{{{{\rm{K}}}}}}}\right)}^{-2},$$
(32)
where $${\alpha }_{{{{{{\rm{NKE}}}}}}}$$ is the permeability coefficient of the pump depending on the density of the pump as well as the concentration of ATP. $${\beta }_{{{{{{\rm{NKE}}}}}},{{{{{\rm{Na}}}}}}}$$ and $${\beta }_{{{{{{\rm{NKE}}}}}},{{{{{\rm{K}}}}}}}$$ are constants that scale $${\Gamma }_{{{{{{\rm{Na}}}}}}}$$ and $${\Gamma }_{{{{{{\rm{K}}}}}}}$$, respectively. The exponents 3 and 2 are Hill’s coefficients of Na+ and K+, respectively. Equation 32 ensures that the flux is zero when either $$1/{\Gamma }_{{{{{{\rm{Na}}}}}}}$$ or $${\Gamma }_{{{{{{\rm{K}}}}}}}$$ approaches zero; the flux saturates if $$1/{\Gamma }_{{{{{{\rm{Na}}}}}}}$$ and $${\Gamma }_{{{{{{\rm{K}}}}}}}$$ approaches infinity. $${G}_{V,{{{{{\rm{NKE}}}}}}}$$ captures the voltage-dependence of the pump activity48, $${G}_{V,{{{{{\rm{NKE}}}}}}}=2{[1+{e}^{-{\beta }_{3}\left({V}_{m}-{\beta }_{4}\right)}]}^{-1}-1$$, where $${\beta }_{3}$$ and $${\beta }_{4}$$ are constants.
The $${{{{{{\rm{Na}}}}}}}^{+}$$/$${{{{{{\rm{H}}}}}}}^{+}$$ exchanger (NHE), which has ten identified isoforms, is expressed in almost all tissues46. It imports one Na+ and extrudes one H+ under physiological conditions. This exchanger plays an important role in water flux, cell volume regulation50 and cell migration7. NHE is quiescent at intracellular $${{{{{\rm{p}}}}}}H \, > \, 7.2$$51. The flux of NHE can thus be expressed as
$${J}_{{{{{{\rm{NHE}}}}}}}={J}_{{{{{{\rm{NHE}}}}}},{{{{{\rm{Na}}}}}}}=-{J}_{{{{{{\rm{NHE}}}}}},{{{{{\rm{H}}}}}}}={\alpha }_{{{{{{\rm{NHE}}}}}}}{G}_{{{{{{\rm{NHE}}}}}}}{RT}\left({{{{{\rm{ln}}}}}}{\Gamma }_{{{{{{\rm{Na}}}}}}}-{{{{{\rm{ln}}}}}}{\Gamma }_{{{{{{\rm{H}}}}}}}\right),$$
(33)
where $${\alpha }_{{{{{{\rm{NHE}}}}}}}$$ is the permeability coefficient which does not significantly depend on cortical tension52 and we assume it is constant. $${G}_{{{{{{\rm{NHE}}}}}}}={[1+{e}^{{\beta }_{5}({{{{{\rm{pH}}}}}}-{\beta }_{6})}]}^{-1}$$ is a pH-gated function indicating the dependence of the NHE activity on pH.
The $${{{{{{\rm{Cl}}}}}}}^{-}$$/$${{{{{{\rm{HCO}}}}}}}_{3}^{-}$$ exchanger (AE2), which imports one $${{{{{{\rm{Cl}}}}}}}^{-}$$ and extrudes one $${{{{{{\rm{HCO}}}}}}}_{3}^{-}$$, is also common in cells. This exchanger is almost quiescent at intracellular $${{{{{\rm{pH}}}}}} < 6.8-7.3$$. Similarly, we assume that the flux takes the form
$${J}_{{{{{{\rm{AE}}}}}}2}={J}_{{{{{{\rm{AE}}}}}}2,{{{{{\rm{Cl}}}}}}}=-{J}_{{{{{{\rm{AE}}}}}}2,{{{{{\rm{HC}}}}}}{{{{{{\rm{O}}}}}}}_{3}}={\alpha }_{{{{{{\rm{AE}}}}}}2}{G}_{{{{{{\rm{AE}}}}}}2}{RT}\left({{{{{\rm{ln}}}}}}{\Gamma }_{{{{{{\rm{Cl}}}}}}}-{{{{{\rm{ln}}}}}}{\Gamma }_{{{{{{\rm{HC}}}}}}{{{{{{\rm{O}}}}}}}_{3}}\right),$$
(34)
where $${\alpha }_{{{{{{\rm{AE}}}}}}2}$$ is the permeability coefficient of AE2 and is assumed to be independent of the cortical tension. $${G}_{{{{{{\rm{AE}}}}}}2}={[1+{e}^{-{\beta }_{7}({{{{{\rm{pH}}}}}}-{\beta }_{8})}]}^{-1}$$ is a pH-gated function indicating the dependence of the AE2 activity on pH.
### Parameters
The default parameters used in the model are listed in Supplementary Table 2. The model involves several degrees of freedom from the choice of parameters. The biophysical meaning of the degrees of freedom accounts for variations in cell types or different experimental conditions for the same cell line. For example, when NHE1 is inhibited, the corresponding parameter representing the NHE1 polarization ratio will change. The parameters that represent these degrees of freedom were fitted by Figs. 1e and 2a. Once all parameters were obtained, we used the parameters to predict cell velocity, including those presented in Fig. 3e and Supplementary Fig. 8.
The ratio of NHE1 polarization and SWELL1 polarization are among the most important parameters in the model because ion channel polarization is the underlying mechanism for the Osmotic Engine Model. Thus, we performed a parameter sensitivity study on ion channel polarization and other key parameters. For example, cell velocity is proportional to the ratio of NHE1 polarization, SWELL1 polarization, the rate of actin polymerization, and the strength of focal adhesions. Two contour plots indicating parameter dependence are shown in Supplementary Fig. 8.
### Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
|
2023-02-09 04:21:34
|
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|
https://mathoverflow.net/questions/349240/indecomposable-modules-of-gentle-algebras
|
# indecomposable modules of gentle algebras
Let $$A = \mathcal{k}Q/I$$ be a gentle algebra (where $$\mathcal{k}$$ is algebraically closed). In the paper Auslander-Reiten Sequences with Few Middle Terms and Application to String Algebras, Butler and Ringel show that string and band modules classify the indecomposable modules of $$A$$ (pages 157–161). To flesh out the details a little more, for each string $$c$$ of $$Q$$ they produce a string module $$M(c)$$. And for each cyclic string $$b$$ they produce a family of band modules $$M(b,x,n)$$ where $$x \in \mathcal{k}^*$$ and $$n \geq 1$$.
I am trying to compare this to the classification of indecomposable representations of the $$2$$-Kronecker quiver. But as an example I don't see where the indecomposable representation
$$\mathcal{k}\overset{0}{\underset{1}{\rightrightarrows}} \mathcal{k}$$
appears in Butler and Ringel's classification. What am I missing?
If the $$2$$-Kronecker quiver is
$$\overset 1\circ\overset{\alpha}{\underset{\beta}{\rightrightarrows}} \overset 2\circ ,$$
$$\mathcal{k}\overset{0}{\underset{1}{\rightrightarrows}} \mathcal{k}$$
corresponds to the string module $$M(\beta)$$.
|
2020-11-28 20:32:29
|
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|
https://docs.moogsoft.com/Enterprise.8.0.0/convertfield.html
|
# convertField
A Workflow Engine function that converts a field value using mappings from the Conversion Maps integration.
The action looks up the value of the field in the conversion map. You can write the results of the lookup to a separate field with the optional target argument. If you don't specify a target, the function updates the source field in-line. Both the field and target arguments accept workflow context and payload keys. Specify the appropriate prefix, either workflowContext or payload.
The optional reversed argument defaults to “false”. If true, the lookup is performed in reverse. The Workflow Engine checks for the value against the conversion map alias to return the name.
The keys and values in the conversion maps are stored as strings. The optional type argument you to explicitly cast to one of the following primitive types: number, string, or boolean. For core target fields, there is implicit casting.
The return behavior of the action is controlled by the “no match” behavior settings in the conversion maps. See the examples below for details.
This function is available for event, alert, and Situation workflows.
## Arguments
Workflow Engine function convertField takes the following arguments:
Name
Required
Type
Description
mapName
yes
string
Name of mapping.
field
yes
string
Name of field to convert.
target
no
string
Optional target field. If omitted, the supplied field will be updated.
type
no
string
Optional primitive return type: 'string', 'number', 'boolean'. Defaults to 'string'.
reversed
no
string
Apply mapping in reverse: 'true' or 'false'.
## Example
### Basic conversions
A conversion map “User” has been created with the following values:
• case sensitive : false
• no match behavior : "default"
• default value : "unknown_user"
name
alias
user1
new_user1
user2
new_user2
An event workflow is defined containing the convertField action with the following arguments:
Argument Name
Argument Value
mapName
User
field
custom_info.user
target
custom_info.updatedUser
type
string
reversed
false
Which the UI translates to:
{"mapName":"User","field":"custom_info.user", "target": "custom_info.updatedUser", "type": "string", "reversed": "false"}
An event enters the workflow with the following custom_info:
{
"user": "user1"
}
The action returns true and updates the event to have the custom_info:
{
"user": "user1",
"updatedUser": "new_user1"
}
If a new event is received with the custom_info:
{
"user": "user3"
}
The action again returns true but updates the event using the conversion map default value:
{
"user": "user1",
"updatedUser": "unknown_user"
}
If the “User” conversion map had used the “exclude” no match behavior instead, the action would have returned false and the event would have remained unchanged.
If the “User” conversion map had used the “retain” no match behavior, the action would have returned true and the original value would have been copied into the ‘target’:
{
"user": "user1",
"updatedUser": "user1"
}
### List conversions
If a field holds a list of primitive values, the conversion is applied to each element of the list.
Using the conversion map and workflow above, add an event with the following custom_info:
{
"user": [ "user1", "user2", "user3" ]
}
Would result in the following results depending on the “no match” behavior:
retain
Returns true and updates custom_info to:
{
"user": [ "user1", "user2", "user3" ],
"updatedUser": [ "new_user1", "new_user2", "user3" ]
}
default
Returns true and updates custom_info to:
{
"user": [ "user1", "user2", "user3" ],
"updatedUser": [ "new_user1", "new_user2", "unknown_user" ]
}
exclude
Returns true and updates custom_info to:
{
"user": [ "user1", "user2", "user3" ],
"updatedUser": [ "new_user1", "new_user2" ]
}
In this last example, the returned list excludes the unconvertible value.
A special case for list conversion is when the field holds is a CSV list. The list is split into elements, which are converted and joined to produce a new CSV list, which may have fewer elements.
### Casting target type
The conversion process treats all values as strings and produces string results. The "type" argument allows the results to be explicitly casted to either numbers or booleans. For core target fields, there is implicit casting.
For example, a conversion map “Severity” has been created with the following values:
• case sensitive : false
• no match behavior : "default"
• default value : "1"
name
alias
Critical
5
Major
4
Minor
3
Warning
2
Intermediate
1
Clear
0
An event workflow is defined containing the convertField action with the following arguments:
Argument Name
Argument Value
mapName
Severity
field
custom_info.severity
target
severity
type
reversed
Which the UI translates to:
{"mapName":"Severity","field":"custom_info.severity", "target": "severity"}
An event enters the workflow with the following custom_info:
{
"severity": "warning"
}
The convertField action finds the string value “2” in the Severity conversion map. The target is “severity”, which is a core field that expects an integer value, so it is implicitly casted to a number and used to set the event severity.
The following core fields are implicitly casted:
Field
Event Types
Field Type
signature
string
source_id
string
external_id
string
manager
string
source
string
class
string
agent
string
agent_location
string
type
string
description
string
severity
number
internal_priority
sig, sigUpdate, sigClose
number
Updating other core fields with the convertField action will fail.
For the custom_info, workFlowContext and payload fields, the type can be explicitly casted as required.
#### Casting to the number type
Fields containing single values will either result in a valid number equivalent to the string or the action will fail:
• “1” becomes 1 and the action returns true.
• “one” is just a string, won’t conversion and the action returns false.
For fields containing an array of values, any convertible strings become 0:
• [“1', “one” ] becomes [1, 0] and the action will return true.
#### Casting to the boolean type
An empty string with any of the following becomes false: “false”, “0”, “-0”, “null”, “NaN”, “undefined”.
Any other non-empty string becomes true.
#### Casting CSV lists
For fields containing CSV lists, casting is applied after converting and joining to the new CSV string as a whole.
|
2021-09-26 19:29:45
|
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|
https://lobste.rs/threads/gecko
|
1. 6
Besides Firefox and Servo, SpiderMonkey is also used by GNOME and MongoDB.
1.
Also polkit.
1.
CouchDB also uses Spidermonkey.
1. 13
I really appreciate the mentality shown here. I have had to do something similar with a Perl monolith that everyone wanted to ignore and re-write, but 6 years after the “re-write” started and 2 teams later, the Perl is still running the show. Just buckling up and making things better is underrated but very satisfying.
1. 17
For the code part, yes. Though the endnote seems to undermine the whole piece:
The biggest fuss that happened early on was that I dared change it from “INCIDENT” (as in the Jira ticket/project prefix, INCIDENT-1234) to “SEV”. It was amazing. People came out of the woodwork and used all of their best rationalization techniques to try to explain what was a completely senseless reaction on their part.
I tried to look up “SEV”, as I’ve never heard the term before. All I found was this Atlassian page. It doesn’t even appear on the Wikipedia disambiguation page. Opposing change for change’s sake doesn’t seem like a rationalization; change has cost. Am I missing something? Is this standard or common in some circles?
1. 12
I wouldn’t say it undermines the entire piece; I’d say that Rachel, like literally every other dev I’ve ever met, gets some things right, and some things wrong. SEV, as noted elsewhere, is common amongst FAANG-derived companies, and not elsewhere. And, yes, the most consistent (and IMVHO correct) path would’ve been to leave the terminology, too. But I don’t think the entire piece goes out the window because she did one thing inconsistent with the rest of what she did, and the overall point is spot-on.
1. 3
I believe that the terminology comes from BBN’s NOC. Specifically, every operations ticket was assigned a severity, with the numbers running from 5 (somebody said this would be nice) through 1 (customer down and/or multiple sites impaired) to 0 (multiple customers down). Everybody lived in fear of a “sev zero”.
That terminology was in use by 1997, and was probably there by 1992 or so. I have asked on the ex-BBN list if anyone can illuminate it further.
People familiar with that NOC (or who worked there or around it) populated an awful lot of other organizations over the years.
2. 6
‘Sev’ is short for an incident with a particular ‘severity’. In a large enterprise using something like ITIL you would hear ‘there is a sev’ and know there is an incident that requires attention soon.
A sev 3 might require a team to look at it in working hours. 2 might mean an out of hours on-call team. 1 is the worst and is likely a complete outage of whatever you’re running.
Atlassian have an easy to understand interpretation written up here: https://www.atlassian.com/incident-management/kpis/severity-levels
1. 3
It’s short for “Site EVent”, not SEVerity. Rachel discusses that in the post (and years ago I worked for FB where they also used the term).
2. 4
https://response.pagerduty.com/before/severity_levels/
It’s terminology I often encounter in FAANG circles. I’m believe FB, Google, and Amazon use it. We use it Square.
1. 1
Ah, interesting. Making that change makes more sense if the company is based in the bay.
2. 2
The hilarious thing is that she felt she had to explain what a ‘SEV’ was in this post but she didn’t need to explain what an ‘INCIDENT’ was.
1. 2
It’s probably short for “severe”
1. 3
Or “site event”?
SEVs (you know, outages, site events, whatever?)
2. 5
Just buckling up and making things better is underrated but very satisfying.
I have seen many, many, ground up big bang rewrites over the course of 21 years in software development. And very few of them produced a better outcome than would have been obtained by incremental improvement or replacement of the older systems.
1. 2
Rewriting just introduces new, unknown bugs, no matter how good the team(s) writing the new software is.
I’ve worked as a software tester for 7 years in a few different settings (good and bad teams and organisations) and I can remember one rewrite that improved things. It was a C++ middleware that was rewritten in Java, which made it accessible for more developers in that team. The middleware was multi-threaded, talking to hardware devices & a network backend (both partially async and sync in nature…) and was important to get right (it was handling physical money).
Eventually it was refactored to also work with a PoC Android based terminal, so that the common bits were put in a common code base. It worked great, and when doing this PoC the amount of unknown bugs were most likely smaller than if we’d rewritten it in Kotlin (or what have you) again.
1. 4
Visual Studio 2022 will be a 64-bit application, no longer limited to ~4gb of memory in the main devenv.exe process.
I am not sure if this is a good thing or a bad one. Why would an editor require 4G or RAM?
1. 8
If you want your editor to do semantic code analysis the amount of derived data (types, use-def chains, etc) you use turns out to be substantial. This is because you need to process much larger inputs than an editor (editor can show a single file, semantic code analysis requires some knowledge about all files in the project and it’s dependencies) and the data is complex (types are trees or graphs, scopes are hash maps, etc).
It’s possible to reduce ram consumption significantly by spilling rarely used data to disk and being smart with lazily realizing only absolutely required bits of info. But, if you just naively code ide stuff without optimizing for memory usage specifically, you’ll end up in gigabytes range.
Did some interesting ultra high-level benchmarks here: https://github.com/rust-analyzer/rust-analyzer/issues/7330#issuecomment-823209678
1. 1
The point you are missing is that VS has been heavily extension based, so “semantic code analysis” probably shouldn’t be a part of the main process to start with.
1. 1
I don’t know the current state, but at least n years ago extensions were run in-process. I think (but don’t know exactly) that that was the story with JetBrains Rider: Resharper really suffered from being in the same process as Studio itself, so they came up with idea of moving the brains to a separate process, and, hey, if the brains are a separate CLR app, why not bridge then to existing Java UI of IntelliJ?
These docs seems to imply that everything is still in the same process?
When your Visual Studio solution grows large, two code analysis engines (ReSharper and Roslyn) working simultaneously can reach the memory limit of the 32-bit process that they share.
https://www.jetbrains.com/help/resharper/Speeding_Up_ReSharper.html
Not sure how up to date they are.
1. 2
It’s fuzzy. Plugins do technically run in-process, but even all but the most trivial of Microsoft’s own plugins are narrow shims that then communicate with the actual plugin core that’s off running in a COM apartment or an equivalent IPC mechanism. I’m not entirely sure how much the cart is pulling the horse there (i.e., whether VS being 32-bits has caused that, or whether their desire for the increased reliability of having out-of-process plugins has enabled VS to stay 32-bit), but that’s where you’re seeing that disconnect.
2. 2
AFAIK this 32-bit was a huge problem if you used static analysis tools like ReSharper
1. 26
Very similar story from a few weeks ago: SQLite is not a toy database – I won’t repeat my full comment from there.
SQLite is very fast. [..] The only time you need to consider a client-server setup is: [..] If you’re working with very big datasets, like in the terabytes size. A client-server approach is better suited for large datasets because the database will split files up into smaller files whereas SQLite only works with a single file.
SQLite is pretty fast compared to fopen(), sure, but PostgreSQL (and presumably also MariaDB) will beat it in performance in most cases once you get beyond the select * from tbl where [..], sometimes by a considerable margin. This is not only an issue with “terabytes” of data. See e.g. these benchmarks.
Is it fast enough for quite a few cases? Sure. But I wouldn’t want to run Lobsters on it, to name an example, and it’s not like Lobsters is a huge site.
Well, first of all, all database administration tasks becomes much easier. You don’t need any database account administration, the database is just a single file.
Except if you want to change anything about your database schema. And PostgreSQL also comes with a great deal of useful administrative tools that SQLite lacks AFAIK, like the pg_stats tables, tracking of slow queries, etc.
And sure, I like SQLite. I think it’s fantastic. But we need to be a tad realistic about what it is and isn’t. I also like my aeropress but I can’t boil an egg with it.
1. 9
SQLite is pretty fast compared to fopen(), sure, but […] MariaDB will beat it in performance
I would actually be interested in knowing whether SQLite handles that query that broke Lobste.rs’ “Replies” feature better than MySql/MariaDb.
But I wouldn’t want to run Lobsters on it, to name an example, and it’s not like Lobsters is a huge site.
I think Lobste.rs would run fine. It would probably be more an issue with the limited amount of SQL SQLite supports.
1. 7
The replies query broke because the hosted MySQL Lobste.rs relies on doesn’t do predicate push down. SQLite does do predicate push down, so it wouldn’t have the same problem.
However SQLite doesn’t have as many execution strategies as MySQL, so it may be missing a key strategy for that query.
1. 5
SQLite’s query planner is honestly a bit smarter than MySQL’s in certain ways. For example, MySQL, as recently as 2017, did temporary on-disk tables for subselects. SQLite instead usually managed to convert them to joins. Maybe that’s been fixed in the last four years, but I wouldn’t assume that MySQL would be faster/that SQLite would be slower.
2. 1
Lobsters uses some fairly complex queries; usually those kind of things tend to do less well on SQLite, although I didn’t run any benchmarks or anything. I found that SQL support in SQLite is actually pretty good and don’t necessarily expect that to be a major issue.
From what I understand is that the biggest problem with the Lobsters hosting is that it’s running MySQL rather than MariaDB. While MySQL is still being actively developed, from what I can see it’s not developed very actively and MariaDB is leaps ahead of MySQL. At this point we should probably stop grouping them together as “MySQL/MariaDB”.
1. 1
Aside from the operations perspective of migrating data, converting things that are not 1:1 between mysql and mariadb, etc. are there any features in lobste.rs that prevent the use of MariaDB?
1. 1
It used to run on MariaDB until there was a handover of the servers. AFAIK it runs well on both (but not PostgreSQL, and probably also not SQLite).
1. 1
I guess their current hoster only provides MySql (for unknown reasons).
2. 12
I also like my aeropress but I can’t boil an egg with it.
I bet you could poach an egg with it, with some inventiveness and a slightly severe risk of getting scalded. ;)
1. 3
When I posted that comment I was thinking to myself “I bet some smartarse is going to comment on that” 🙃
1. 2
Joking aside, I think a better analogy would be comparing the Aeropress to an espresso machine: the Aeropress is going to get you really good coffee that you’re going to use every day, costs very little, is easy to maintain, and you can bring with you everywhere, but it’s never going to give you an espresso. But then again, it’s not really trying to.
(The analogy falls apart a bit, as one of the original claims was that it could produce espresso. I think they stopped claiming that though.)
2. 1
LOL
…and audible laughter was emitted. Thanks for that.
1. 1
On the other hand if you had to set up and supply your password to obtain admin rights every time you just wanted to make coffee….
…because some nutjob might want to use it for boiling eggs and the company wanted to stop that….
…the device that just let’s you get on with making coffee (or boiling eggs) is a hellavuh lot faster for many jobs!
2. 5
SQLite has supported ALTER TABLE ADD COLUMN for years, and recently added support for dropping columns. So I’d amend your statement to “…make complex changes to your db schema.”
SQLite has stats tables, mostly for the query optimizer’s own use; I haven’t looked into them so I don’t know how useful they are for human inspection.
1. 2
SQLite has supported ALTER TABLE ADD COLUMN for years, and recently added support for dropping columns. So I’d amend your statement to “…make complex changes to your db schema.”
Yeah, the drop column is a nice addition, but it’s still a pain even for some fairly simple/common changes like renaming a column, changing a check constraint, etc. I wouldn’t really call these complex changes. It’s less of a pain than it was before, but still rather painful.
SQLite has stats tables, mostly for the query optimizer’s own use; I haven’t looked into them so I don’t know how useful they are for human inspection.
As far as I could find a while ago there’s nothing like PostgreSQL’s internal statistics. For example keeping track of things like number of seq scans vs. index scans. You can use explain query plan of course, but query plans can differ based on which parameters are used, table size, etc. and the query planner may surprise you. It’s good to keep a bit of an eye on these kind of things for non-trivial cases. Things like logging slow queries is similarly useful, and AFAIK not really something you can do in SQLite (although you can write a wrapper in your application).
None of these are insurmountable problems or show-stoppers, but as I mentioned in my other comment from a few weeks ago, overall I find the PostgreSQL experience much smoother, at the small expense of having to run a server.
1. 6
it’s still a pain even for some fairly simple/common changes like renaming a column
ALTER TABLE RENAME COLUMN The RENAME COLUMN TO syntax changes the column-name of table table-name into new-column-name. The column name is changed both within the table definition itself and also within all indexes, triggers, and views that reference the column.
1. 25
So basically we finally arrived at the “make your app a web page” that Apple demanded when launching the iPhone
1. 30
Yes, and the latest trend in web development is to render content on the server. Everything old is new again!
1. 7
I think it’s better this time, because phones and network are fast enough that doing everything in the browser isn’t limited by UMTS speeds.
1. 3
The original iPhone didn’t even support UMTS (3G), it was GPRS (2G) EDGE (2.5G). A load of mobile providers who had already rolled out large UMTS had to go and deploy older hardware to support the iPhone without it falling back to GPRS. The latency on GPRS was awful (500ms RTTs were common, making it unusable for anything interactive).
2. 2
I have noticed this and had the very same reaction a few weeks ago.
3. 13
To be fair: when Apple announced this, React did not exist, Vue did not exist, precursors like Backbone didn’t even exist, and most critically, most of the technologies and tools we use in 2021 to do SPAs, let alone offline webapps, did not exist. Hell, I think the dominant offline storage solution was WebSQL, which was never standardized and is not (AFAIK) supported in any contemporary browser, and no equivalent of web workers existed unless you had Google Gears installed. You also had nothing like WebGL, or web sockets, or even widespread contemporary CSS that would make reasonable, cross-platform styling feasible. So what Apple was offering at the time was morally equivalent to having bookmark links on the home screen.
(Yeah, I’m very aware of the pile of old meta tags you could use to make the experience be better than that in a literal sense, but that doesn’t resolve anything else I highlighted.)
Speaking purely for myself, I found the initial announcement infuriating, not because I didn’t believe in the web (I did! Firefox was growing! Safari was proving the viability of KHTML! IE was on the decline!), but because Apple’s proposal was just so damn far from what doing that seriously would’ve actually looked like that it felt condescending. The Palm Pre, which notably came out two years later, was dramatically closer to what I’d have expected if Apple were being sincere in their offer. (And even there, webOS, much as I love it, is more an OS that happens to have JavaScript-powered apps than a genuine web app platform in the 2021 sense.)
1. 5
Even at the time, Apple’s stance felt to me like, “We aren’t finished with our native SDK yet and it’s so far from ready for public consumption that we’re going to just pretend it doesn’t exist at all.” I remember talking about the iPhone with my coworkers when it first came out and everyone just assumed native apps would be coming at some point.
Even webOS (which I also loved) ended up supporting native apps eventually, despite having a much more feature-rich platform for JavaScript code.
Games seem to be the killer app category that pushes mobile OS vendors to support native code. They’re one of the few categories of application where a lack of native code support can make an app impossible to implement, rather than just making it a bit slower or clunkier but still basically workable.
2. 3
Even Firefox OS was too early in the game for that (besides other problems of FFOS).
1. 6
If it was timed right, Mozilla would have found another way to run it into the ground. ;-)
1. 1
Could not agree more !
1. 3
Is there a working strace equivalent for Windows? It’s the tool I always miss when I have to debug anything there.
1. 5
Procmon?
1. 2
Procmon is the closest equivalent, but Portmon and ProcDump, alongside the tightly-related-but-different Spy++, can also be very useful in this context (some of those being closer to e.g. ltrace than strace, specifically, but the division of responsibilities on Windows are a bit different, so there’s not a one-to-one mapping).
1. 1
procmon seems to work well, thanks for the suggestion!
1. 11
Most lists of “weird programming languages” get bogged down in brainfuck and brainfuck skins. I like that this one doesn’t!
1. 11
I agree. Although I feel that APL and especially Lisp don’t really fit with the rest of the list - those are languages that (some) people really do want to program in.
1. 7
I think a listicle like this about unusual languages people actually use would be really interesting. Probably something like
• Forth
• APL/J/K
• Inform7
• Orca
• Golfscript (stretching it, I know)
Damn I’ve heard of so many bizarre languages
1. 10
PostScript.
1. 2
Any good resources on PS? I’ve heard… rumors, but never investigated myself.
1. 7
I’m dead tired and can’t find the docs before sleep, but PostScript is an awesome concatenative language and sincerely my favorite in the genre other than Factor. It’s not hard. I’ll find links to the guides in the morning. You can literally code in the GhostScript REPL meanwhile if you want to play.
1. 3
I really like what I’ve read of Bill Casselman’s Mathematical Illustrations which covers PostScript and some fun geometry.
1. 1
Unfortunately no, like so many of my opinions I’ve gotten it from The Internet.
I believe my primary memory of PostScript being used for programming is from this comment by JWZ: http://regex.info/blog/2006-09-15/247#comment-3085
1. 1
Back when I had to use PostScript for work, the language reference was the best document I was able to find.
2. 3
And there’s INRAC (used for at least two, possibly three, commercial products that I know of) where flow control is non-deterministic.
1. 2
I saw you mention INRAC on the alien languages ask, which to my eternal shame I didn’t notice until two weeks later. What are some resources for learning about it as an outsider? Sounds really interesting!
1. 4
Unfortunately, there isn’t much available and most of the references I’ve come across just mention INRAC. I think, aside from the original creator of INRAC (William Chamberlain) I think I’ve inadvertently became an INRAC expert:
Deconstruction Racter
The Psychotherapy of Racter, or The Descent Into Madness of Sean Conner
The Psychotherapy of Racter, or The Further Descent Into Madness of Sean Conner
INRAC, the mind bending implementation language of Racter
WTF INRAC?
So how do you determine undefined behavior in a language you are reverse engineering?
2. 2
Hey, if the software historian / archeologist hasn’t heard of it…
For that hypothetical listicle, I’d consider adding one or two of your modelling languages - like, TLA+ looks pretty magical to people who are not you ;-). Also, I’d consider - LaTeX is not actually that uncommon, but very different from other languages in both appearance and semantics. (Maybe TikZ, but I’m not sure that counts as a programming language.)
Something like Haskell is probably too common, but Prolog might make the list?
[Quick EDIT: also, maybe assembly for the original MIPS CPUs, where you could apparently read the old value of a register if you manage to execute the instruction before the previous instruction has actually written the new value? It doesn’t look too evil, but…]
… do people use Orca?
1. 3
… do people use Orca?
@rwhaling introduced me to it and was using it for his synth music, so at least one person uses it :P
1. 2
Re MIPS, you may be thinking of https://en.m.wikipedia.org/wiki/Delay_slots. For some reason this is still being taught in introductory computing classes at university.
1. 2
[Quick EDIT: also, maybe assembly for the original MIPS CPUs, where you could apparently read the old value of a register if you manage to execute the instruction before the previous instruction has actually written the new value? It doesn’t look too evil, but…]
Were you thinking of the divide and multiply instructions? Some instruction sequences give unpredictable results.
1. 2
I was thinking of https://retrocomputing.stackexchange.com/questions/17598/did-any-cpu-ever-expose-load-delays. (kameliya’s Wikipedia page is a little less informative; note that sufficiently-embedded processors may be able to ensure that an interrupt doesn’t happen. Which would allow one to write rather mind-bending code.)
2. 2
SQL is based around relationships (in the mathematical sense) and is the most popular goofy programming language no one thinks about.
Lex/Yacc let you write half your program as a cfg and the rest in C, a language/tool chain that again no one thinks of in these lists.
Wolfram is based on term rewriting and is somewhat popular and extensively used in physics.
Erlang is based around a distributed model that is again something few other languages support naively.
Most of the ‘esoteric’ language lists are list of ‘languages that do the same thing as C but poorly’.
1. 1
Yes, I was also just about to suggest Inform 7. It’s fantastic.
1. 1
Golfscript (stretching it, I know)
No you’re not. I want to write an implementation that is not Ruby
1. 1
Mumps, RPG…
1. 1
Factor is a really nice forth dialect.
1. 1
Prolog, MUMPS
1. 1
MiniZinc is also worth an include on that list.
2. 1
TBH I interpreted the inclusion of CL on this list as a trolling attempt toward lispers.
1. 3
It’s truly fascinating that all the early smartphone attempts focused on making it easy to run desktop applications on your smartphone. As it turned out, the interaction model with touch and small screens was just too different and everything had to be rewritten from scratch, but that wasn’t obvious at the time.
1. 2
I honestly fully agree. I didn’t own the 900, only the 800, and I gave that thing a hell of a lot of use. It was an amazing device, and it traveled the world with me in a very literal sense.
But I knew even back then, before the N900 shipped, that it was a dead end. To me, the N800/900 was always a bridge: it was Nokia experimenting with hardware design in an ecosystem where they knew that hobbyists would show them what the form was capable of. Unlike contemporary Ubuntu and similar desktop Linuxes, you really needed to be comfy using things like apt and so on in order to do useful things with the N800/N900. Maybe not the terminal literally, but a lot of Debian-specific (not even Linux-specific) details.
And the fact is that I don’t think the form factor was all that, either. Hardware keyboards were necessary in 2007, but not in 2021. The UI required a stylus to operate properly, and while it’s possible in a literal sense to engineer your way out of that and still keep Gtk as your toolkit, that’d have been a massively uphill battle. (Hell, the damn D-pad wasn’t reliably supported in a useful way!) And so on.
As much as I hate to say it, because they were every bit as proprietary as their adversaries, I think that Windows Phone and (my personal favorite) WebOS were much better also-rans in the phone space. Nokia would have had to choose one of those, or Android, eventually. I have a very hard time, except in retrospect and from a very specific point of view, saying Nokia made the wrong call trying to go with Windows.
1. 8
This was an interesting article, it breaks down the issues with net.IP well, and describes the path to the current solution well.
But.
This isn’t a difficult problem. Don’t waste a ton of space, don’t allocate everywhere, make it possible to make the type a key in the language’s standard map implementation. In C++, this would’ve been easy. In Rust, this would’ve been easy. In C, this would’ve been easy (assuming you’re using some kind of halfway decent map abstraction). It doesn’t speak well of Go’s aspiration to be a systems programming language that doing this easy task in Go requires a bunch of ugly hacks and a separate package to make a string deduplicator which uses uintptrs to fool the garbage collector and relies on finalizers to clean up. I can’t help but think that this would’ve been a very straightforward problem to solve in Rust with traits or C++ with operator overloading or even Java with its Comparable generic interface.
That’s not to say that the resulting netaddr.IP type is bad, it seems like basically the best possible implementation in Go. But there are clearly some severe limitations in the Go language to make it necessary.
1. 11
Almost all of the complexity that happened here is related to the ipv6 zone string combined with fitting the value in 24 bytes. Given that a pointer is 8 bytes and an ipv6 address is 16 bytes, you must use only a single pointer for the zone. Then, having amortized zero allocations with no space leaks for the zone portion, some form of interning with automatic cleanup is required.
If this is as easy as you claim in C/C++/Rust/whatever real systems language you want, can you provide a code snippet implementing it? I’d be happy to audit to see if it does meet the same (or better!) constraints.
1. 6
Here’s a C++ version: https://godbolt.org/z/E3WGPb - see the bottom for a usage example.
Now, C++ is a terrible language in many ways. It makes everything look super complicated, and there’s a lot of seemingly unnecessary code there, but almost all of that stems from having to make my own RAII type, which includes writing the default constructor, the move constructor, the copy constructor, the destructor, the move operator= and the copy operator=. That complexity is just par for the course in C++.
One advantage of the netaddr.IP type is that it doesn’t allocate for every zone, just for every new zone, thanks to the “intern” system. My code will allocate space for the zone for every IPv6 address with a zone. One could definitely implement a “zone cache” system for my IPZone type though, maybe using a shared_ptr instead of a raw pointer for refcounting. One would have to look at usage patterns to see whether the extra complexity and potential memory/CPU overhead would be worth it or if zones are so infrequently used that it doesn’t matter. At least you have the choice in C++ though (and it wouldn’t rely on finalizers and fooling the GC).
1. 7
They also had the choice to just make a copy of every string when parsing and avoid all of the “ugly hacks”. Additionally, a shared_ptr is 16 bytes, so you’d have to figure out some other way to pack that in to the IPAddress without allocations. So far, I don’t think you’ve created an equivalent type without any “ugly hacks”. Would you like to try again?
1. 6
I don’t think they had the choice to just copy the zone strings? My reading of the article was that the intern system was 100% a result of the constraint that A) IP addresses with no zone should be no bigger than 24 bytes and B) it should be possible to use IP addresses as keys. I didn’t see concern over the memory usage of an IP address’s zone string. Whether that’s important or not depends on whether zones are used frequently or almost never.
It’s obviously hard to write a type when the requirements are hypothetical and there’s no data. But here’s a version with a zone string cache: https://godbolt.org/z/P9MWvf. Here, the zone is a uint64_t on the IP address, where 0 represents an IPv4 address, 1 represents an IPv6 address with no zone, and any other number refers to some refcounted zone kept in that IPZoneCache class. This is the “zone mapping table” solution mentioned in the article, but it works properly because the IPAddress class’s destructor decrements the reference count.
1. 7
I don’t think they had the choice to just copy the zone strings? My reading of the article was that the intern system was 100% a result of the constraint that A) IP addresses with no zone should be no bigger than 24 bytes and B) it should be possible to use IP addresses as keys.
Indeed, interning is required by the 24 byte limit. That Interning avoids copies seems to be a secondary benefit meeting the “allocation free” goal. It was a mistake to imply that copying would allow a 24 byte representation and that interning was only to reduce allocations.
That said, your first solution gets away with avoiding interning because it uses C style (null terminated) strings so the reference only takes up a single pointer. Somehow, I don’t think that people would be happier if Go allowed or used C style strings, though, and some might consider using them an “ugly hack”.
I didn’t see concern over the memory usage of an IP address’s zone string. Whether that’s important or not depends on whether zones are used frequently or almost never.
One of the design criteria in the article was “allocation free”.
It’s obviously hard to write a type when the requirements are hypothetical and there’s no data. But here’s a version with a zone string cache: https://godbolt.org/z/P9MWvf.
Great! From what I can tell, this does indeed solve the problem. I appreciate you taking the time to write these samples up.
I have a couple of points to make about your C++ version and some hypothetical C or Rust versions as compared to the Go version, though.
1. It took your C++ code approximately 60 lines to create the ref-counted cache for interning. Similarly, stripping comments and reducing the intern package they wrote to a similar feature set also brings it to around 60 lines. Since it’s not more code, I assume the objection is to the kind of code that is written? If so, I can see that the C++ code you provided looks very much like straightforward C++ code whereas the Go intern package is very much not. That said, the authors of the intern package often work on the Go runtime where these sorts of tricks are more common.
2. In a hypothetical C solution that mirrors your C++ solution, it would need a hash-map library (as you stated). Would you not consider it an ugly hack to have to write one of those every time? Would that push the bar for implementing it C from “easy” towards “difficult”? Why should the Go solution not be afforded the same courtesy under the (now valid) assumption that an intern library exists?
3. I’ll note that when other languages gain a library that increases the capabilities, even if that library does unsafe hacks, it’s often viewed as a positive sign that the language is powerful enough to express the concept. Why not in this case?
4. In a hypothetical Rust solution, the internal representation (I think. Please correct me if I’m wrong) can’t use the enum feature because the tag would push the size limits past 24 bytes. Assuming that’s true, would you consider it an ugly hack to hand-roll your own union type, perhaps using unsafe, to get the same data size layout?
5. All of these languages would trivially solve the problem easily and idiomatically if the size was allowed to be 32 bytes and allocations were allowed (this is take 2 in the blog post). Similarly, I think they all have to overcome significant and non-obvious challenges to hit 24 bytes with no allocations as they did.
Anyway, I want to thank you for engaging and writing some code to demonstrate the type in C++. That’s effort you don’t usually get on the internet. This conversation has caused me to update my beliefs to agree more with adding interning or weak references to the language/standard library. Hopefully my arguments have been as useful to you.
2. 4
I agree—if Go is a systems language. But I don’t think it ever was supposed to be. Or if it was, it’s (in my opinion) really bad at it. Definitely worse than even something like C#, for exactly the reasons you’re highlighting.
I think Go was more originally designed to be a much faster language than Python (or perhaps Java), specifically for Google’s needs, and thus designed to compete with those for high-performance servers. And it’s fine at that. And I’ve thought about solving this kind of issue in those languages, too, using things like array in Python for example.
So I agree Go isn’t a good systems language, but I think that was a bit of retcon. It’s a compiled high-level language that could replace Python usage at Google. It’s not competing with Rust, C, Zig, etc.
1. 3
Ok, I can buy that. IIRC, it was originally promoted as a systems language, but it seems like they’ve gone away from that branding as well. There’s a lot of value to something like “a really fast, natively compiled Python”.
But even then, this article seems to demonstrate a pretty big limitation. Something as simple as using a custom IP address type as the key in a map, ignoring everything performance-related, seems extremely difficult. How would you write an IP address struct which stores an IPv4 address or an IPv6 address with an optional zone, which can be used a the key in a map, even ignoring memory usage and performance? Because that would be easy in Python too; just implement __hash__ and __eq__.
This is a problem which isn’t just related to Go’s positioning, be it a “systems language” or a “faster python”. Near the bottom we have C, where an IP address -> whatever map is about as difficult as any other kind of map. Slightly above, we have C++ and Rust, where the built-in types let you use your IP address class/struct as a key with no performance penalty, since you stamp out a purpose-built “IP address to whatever” map using templates. Above that again, we have Java and C#, which also makes it easy, though at a performance cost due to virtual calls (because genetics aren’t templates), though maybe the JIT optimises out the virtual call, who knows. Near the top, we have Python which makes it arguably even more straightforward than Java thanks to duck typing.
Basically, unless you put Go at the very bottom of the stack alongside C, this should be an easy task regardless of where you consider Go to fit in.
1. 3
IIRC, it was originally promoted as a systems language, but it seems like they’ve gone away from that branding as well.
I believe you’re correct about how Google promoted it. I just remember looking at it, thinking “this is absolutely not a systems language; it’s Limbo (https://en.wikipedia.org/wiki/Limbo_(programming_language), but honestly kind of worse, and without the interesting runtime,” and continuing to not use it. So I’m not sure the team itself actually thought they were doing a systems language.
But even then, this article seems to demonstrate a pretty big limitation. Something as simple as using a custom IP address type as the key in a map, ignoring everything performance-related, seems extremely difficult.
I completely agree, but that’s changing the discussion to whether Go is a good language, period. And since I mostly see that devolving into a flame war, I’m just going to just say that I think you have a lot of company, and also that clearly lots of people love the language despite any warts it has.
1. 2
I completely agree, but that’s changing the discussion to whether Go is a good language, period. And since I mostly see that devolving into a flame war, I’m just going to just say that I think you have a lot of company, and also that clearly lots of people love the language despite any warts it has.
My relationship with the language is… Complicated. I often enjoy it, I use it for work, and when I just want to write a small tool (such as when I wrote a process tree viewer) it’s generally my go-to “scripting” language these days. But I hate how the module system puts URLs to random git hosting websites in my source code, there’s a lot of things I dislike about the tooling, and the inability write a datastructure which acts like the built-in datastructures and the inability to write a type which works with the built-in datastructures are both super annoying issues which none of the other languages I use have. I’m hoping Go 2 will fix some of the bigger problems, and I’m always worried about which directions the corporate management at Google will take the language or its tooling/infrastructure.
But you’re right, this is tantamount to flamewar bait so I’ll stop now.
1. 12
I once wasted an entire month trying to resolve some cryptic C# compile errors where Visual Studio simply wouldn’t recognize some of my source files. In the end, the reason was that the compiler silently failed to recognize files with path lengths of longer than 255 characters, even though you can technically create such files on Windows. A prefix like “C:\Users\Benjamin\Documents\ProjectName\src" combined with C#’s very verbose naming conventions meant that a few of my files were just over the path size limit.
1. 8
I feel like Windows is drowning in technical debt even more than Linux is. The APIs to work with long paths have existed for ages now, so most modern software lets you easily create deep hierarchies, but Windows Explorer still isn’t updated to work with those APIs so if you create a file with a long path, you can’t interact with that file through Explorer. There have been solid widgets for things like text entry fields in various Microsoft UI frameworks/libraries for ages now, but core apps like Notepad and - again - Windows Explorer still aren’t updated to take advantage of them, so hotkeys like ctrl+backspace will just insert a square instead of doing the action which the rest of the system has taught you to expect (i.e deleting a word). CMD.EXE is an absolutely horrible terminal application, but it hasn’t been touched in ages presumably due to backwards compatibility, and Microsoft is just writing multiple new terminal applications, not as replacements because CMD.EXE Will always exist, but as additional terminal emulators which you have to use in addition to CMD.EXE. The Control Center lets you get to all your settings, but it’s old and crusty, so Microsoft is writing multiple generations of separately holistic Control Center replacements, but with limitations which make it necessary to use both the new and the old settings editors at the same time, and sometimes Control Center and some new settings program don’t even agree on the same setting. Windows is useful as a gaming OS, but any time I actually try to use it, I just get sad.
1. 6
CMD.EXE is an absolutely horrible terminal application, but it hasn’t been touched in ages presumably due to backwards compatibility, and Microsoft is just writing multiple new terminal applications, not as replacements because CMD.EXE
What you think of as cmd.exe is actually a bunch of things, most of which are in the Windows Console Host. The shell-equivalent part is stable because a load of .bat files are written for it, but PowerShell is now the thing that’s recommended for interactive use. The console host (which includes a mixture of things that are PTY-subsystem and terminal emulator features on a *NIX system) is now developed by the Windows Terminal team and is seeing a lot of development. Both cmd.exe and powershell.exe run happily in the new terminal with the new console host and in the old terminal and the old console host. At the moment, if you run them from a non-console environment (e.g. from the windows-R box), the default console host that’s started is the one that Windows ships with and so you don’t get the new terminal.
1. 1
Windows Terminal is great when I can use it, but it does not seem to work well with administrator privileges.
1. 1
You can use the sudo package from scoop. For me it’s good enough.
1. 1
Wow, did not know about this! It looks like it still generates a UAC popup unless you configure those to not exist. Still, far better than nothing.
http://blog.lukesampson.com/sudo-for-windows
2. 1
but PowerShell is now the thing that’s recommended for interactive use
Which one? ;-)
I have some code that extracts config/data/cache directories on Windows (the equivalent of “check if XDG_CONFIG_DIR is set, otherwise use .config” on Linux) and it’s just a hyperdimensional lair of horrors.
Basically, the best way to get such info without having to ship native code is to run powershell (version 2, because that one does not have restricted mode) with a base64 encoded powershell script that embeds a C# type declaration that embeds native interop code that finally calls the required APIs.¹
I’m close to simply dropping Windows support, to be honest.
¹ The juicy part of the code for those interested:
static final String SCRIPT_START_BASE64 = operatingSystem == 'w' ? toUTF16LEBase64("& {\n" +
"[Console]::OutputEncoding = [System.Text.Encoding]::UTF8\n" +
"using System;\n" +
"using System.Runtime.InteropServices;\n" +
"public class Dir {\n" +
" [DllImport(\"shell32.dll\")]\n" +
" private static extern int SHGetKnownFolderPath([MarshalAs(UnmanagedType.LPStruct)] Guid rfid, uint dwFlags, IntPtr hToken, out IntPtr pszPath);\n" +
" public static string GetKnownFolderPath(string rfid) {\n" +
" IntPtr pszPath;\n" +
" if (SHGetKnownFolderPath(new Guid(rfid), 0, IntPtr.Zero, out pszPath) != 0) return \"\";\n" +
" string path = Marshal.PtrToStringUni(pszPath);\n" +
" return path;\n" +
" }\n" +
"}\n" +
"\"@\n") : null;
1. 1
Which one? ;-)
PowerShell 7 Core, of course!
…for now!
…unless you also need to support classic PowerShell, in which case, PowerShell 5!
…and be careful not to use Windows-specific assemblies if you want to be cross-platform!
3. 3
The APIs to work with long paths have existed for ages now
Well, I’d agree about technical debt, but this claim is a great example of it.
As an application developer, you can choose one of these options:
1. Add a manifest to your program where you promise to support long paths throughout the entire program. If you do this, it won’t do anything unless the user has also modified a system-global setting to enable long paths, which obviously many users won’t do, and you can expect to deal with long path related support queries for a long time. This is also only supported on recent versions of Windows 10, so you can expect a few queries from users running older systems.
2. Change your program to use UTF-16, and escape paths with \\?\ . The effect of doing this is to tell the system to suppress a lot of path conversions, which means you have to implement those yourself - things like applying a relative path to an absolute path, for example. This logic is more convoluted on Windows than Linux, because you have to think about drive letters and SMB shares. “D:” relative to “C:\foo” means “the current directory on drive D:”. “..\..\bar” relative to “C:\foo” means “C:\bar”. “\\server\share\..\bar” becomes “\\?\UNC\server\share\bar”. “con” means “con”.
I went with option #2, but the whole time kept feeling this is yet another wheel that all application developers are asked to reinvent.
1. 1
Windows is useful as a gaming OS, but any time I actually try to use it, I just get sad.
• Microsoft Office and the Adobe Suite (or replacements such as the Affinity Suite).
It would be really nice if Microsoft just ported Office.
1. 2
They effectively have. It seems like Microsoft cares far more about the O365 version of Office than any native version — even Windows.
1. 2
They effectively have. It seems like Microsoft cares far more about the O365 version of Office than any native version — even Windows.
Office 365 is a subscription service, most of the subscriptions include the Windows/Mac Apps. I guess that you mean Office Online, but it only contains a very small subset of the features of the native versions. I tried to use it for a while, but you quickly run into features that are missing.
2. 1
The separation of the control centre may actually go away soon. If the articles are up be believed, MS finished that migration in the latest version.
1. 1
More details? The only thing I heard was that they were finally killing the working ones.
1. 37
I’m primarily a Windows developer, and relate to the frustration of Windows development.
However, reading this article, most of the comments seemed related to initial setup: yes, you have to install git; yes, it installs its own bash; yes, vim doesn’t know what to do with the Windows clipboard but can do anything; yes, PowerShell came from an era where being conspicuously different was considered a virtue, but you’re free to use any other tool; etc.
There’s just a kind of cognitive burden with every program having to independently reinvent every wheel. The solutions are well known, but it’s just so…painful.
1. 7
On Linux, you often end up writing code…and that’s about it. Each distribution will package and update your code in their own way.
Only if your program is both open source and popular. The overwhelming majority of programs aren’t. Case in point: I spent hundreds of hours (spread over 4 years) writing a small easy to use crypto library. I have users, some of which even wrote language bindings. The only distribution packages I know of are for Void Linux, Arch Linux, and Net BSD. No Debian, no Redhat, no Gentoo, and most of all, no Ubuntu.
Not that it really matters. This is a single file library we’re talking about, which you can easily bundle in your own source code. But I did go out of my way to have a bog standard, easy to use makefile (with $PREFIX, $DESTDIR, \$CC and all that jazz). Packaging it ought to be very easy. Yet no one stepped up for any of the major distributions out there.
They might get it wrong, but they’ll try.
The very fact they might get it wrong, in my opinion, suggest that packaging itself may be a bad idea to begin with. Linus Torvalds goes out of his way never to “break users”. We should be able to take advantage of that, but it would require abandoning the very concept of distribution, or at least specialising it.
A distribution is mostly a glorified curated repository of software. Ideally a coherent whole, compiled, or even designed, to work together. The people managing it, the packagers, have made themselves responsible for the quality and security of that repository. Security by the way is the trump card they show in dynamic vs static linking debates: upstream devs can’t all be trusted with updating their software fast enough, so when there’s a vulnerability in some library, we ought to be able to swap it and instantly fix the problem for the whole distribution. Mostly though, it’s about making the life of packagers easier.
Now I have no problem with curated repositories of software. What I have a problem with is the exclusivity. In most cases, there can be only one. One does not simply uses Debian and RedHat at the same time. They don’t just distribute software, they pervade the whole system. Including the kernel itself, which they somehow need to patch. This effectively turns them in to fenced gardens. It’s not as bad as Apple’s App Store, you can go over the fence, but it’s inconvenient at best.
So. Linux distros won’t package my software, when they do it they might get it wrong anyway. Which means that in practice, I’ll have half a dozen systems moving under my feet, I can only hope that it will still work despite all those updates everywhere. Just like Windows, only worse. And just like on Windows, there’s only one solution: “Find your dependencies. Track them down, and eliminate them.”
Ideally, we should only depend on the kernel. Maybe statically link everything, though if we’re short on space (??) we can lock those dependencies instead, like NPM or Cargo do it at the source level, and Nix (I think? I haven’t checked) can do at the binary level. On the flip side, that means you need to handle stuff like installation and updates yourself, or have a library do it for you. Just like Windows. Problem is, it’s not even possible, because of how distributions insist on standing between users and developers.
As it should be. You wrote that program, you should be responsible for its life cycle. Distribution maintainers really got screwed when they realised that a bad program may undermine the distribution’s reputation. Though we may not like the idea of each program having its own update code, that update code can be as small as 100KB, including the cryptographic code (modern crypto libraries can be really small).
Users want to have precompiled binaries, but then they’ll be greeted with a slew of scary warnings, unless your code is signed, so you have to deal with that as a code author.
That, however, is something Windows is doing very, very wrong. Especially since signing your binaries is not enough, they decide whether your reputation warrants a warning anyway or not. This practice turns Microsoft into one giant middle man. They go as far as staking their reputation on the list of trusted authors and programs. While it does result in fewer users getting viruses, it also acts as yet another centralisation force, yet another way for huge entities and corporation to have an edge over the little folk. (An even more blatant example is how big email providers handle spam.)
This is one of the few places where the solution is to tell everyone to “git gud”. That means teaching. People have to know how computers work. Not just how to use Microsoft® Word®, but the fundamentals of computing, and (among other things) what you can expect when you execute a random program from some shady web site. We don’t have to teach them programming, but at least let them try Human Resource Machine. Only then will it be safe to stop treating users like children. Heck, maybe they’ll even start to demand a better way.
There is one thing for which a coherent curated repository of software is extremely useful: development environments. Developers generally need a comprehensive set of tools that work well together: at the very least a compiler, editor, version control, dependency management, and the actual dependencies of the program. It’s okay if things break a little because of version incompatibility. I can always update or fix the program I’m writing.
Less technical end users however need more stability. When a program works, it’d better still work even when the system moves under its feet. The OS ought to provide a stable and sufficient API (ABI, really) upon which everyone can rely on.
1. 4
As it should be. You wrote that program, you should be responsible for its life cycle.
The complaint here is it sucks for everyone to be reimplementing auto updates, possibly with bugs. I believe that my gaming PC is right now running buggy and wasteful auto update checkers from a half dozen different vendors, all of whom wasted money on these things which provide negative value.
Whereas, uploading a new version to an app store or apt/rpm/etc repo is much nicer in this regard: users’ machines already have the mechanism to update software from those, often automatically.
1. 1
There are libraries for such things. Some of them could be provided by the OS vendor. I’m just not sure they should be part of the OS itself: it would add to what the OS must keep stable.
Stability at the OS level is easier to achieve if said OS is minimal: just schedule programs & talk to the hardware. If programs can access the network, there is no need to provide an update mechanism on top. A standard, recommended library however, would be very nice.
2. 1
The type of thing that makes me lose my mind about Windows as a platform is trying to deliver anything to a customer in an end-to-end way. On Linux, you often end up writing code…and that’s about it. Each distribution will package and update your code in their own way. They might get it wrong, but they’ll try. On Windows, updating your program is your problem. Depending on how you count, there’s either zero or a bajillion systems for updating code, but you can’t assume your users are using any of them, so you end up having to write your own. […] And you can’t expect users to help - how many users really know which version of Windows 10 they have? - so your program has to run on all of them.
There’s just a kind of cognitive burden with every program having to independently reinvent every wheel. The solutions are well known, but it’s just so…painful.
Linux approaches to runnable-binaries-shipped-with-dependencies (Flatpak, snap, AppImage, …) do address some of these concerns, but I wonder if (or how) the increase of base images (echo "which version Windows 10 they have?" | sed s/Windows/Fedora/) will change the amount of work that the application developers will have to put in to create fully working {flatpaks,snaps,appimages}.
1. 5
It’s not that we don’t have an equivalent to that on Windows. It’s that there are just too damn many options, and Microsoft changes its mind every couple of years on what they want to do, exactly.
Ever since the giant mess that was DLL hell, Windows has had something called side-by-side assemblies, which allow conflicting versions of DLLs to be installed globally. Combined with its take on app bundles, strongly allowing and encouraging application vendors to to just bundle all their DLLs alongside the application in the same directory, we end up effectively the same place as Flatpak, albeit exploded instead of single files. So that’s “solved”.
But that’s only the mechanism. When it comes to actually distributing your app, Microsoft loses its attention every five seconds. The Micrsofot Store has been Microsoft’s answer for awhile, but it only relatively recently (last couple of years?) gained the ability to handle non-UWP binaries. We’ve also had ClickOnce, which was Microsoft’s answer to Java WebStart, and which was again .NET-only. And now we’re getting winget which is kinda Chocolatey and kinda the Microsoft Store and kinda its own thing, and so on.
So it’s not the container bit that’s so hard, but rather getting your app mechanically distributed. That’s more contrasting with e.g. apt or rpm or the App Store (or maybe snap, since that is centralized) than Flatpak.
1. 4
I think this is not quite fair to Microsoft. On Windows, there is a blessed store for all GUI programs: the Microsoft Store. If you don’t like the Microsoft Store, you can distribute over the internet; if you sign your builds, Windows will pop up a non-scary prompt before installing, and if you don’t, Windows will pop up a scary prompt.
On Linux, you can also distribute GUI programs over the internet. But there’s no trusted signing built in for programs distributed this way, so it’s somewhat less secure. What about blessed stores? Good grief: first of all, many distros maintain their own, and patch your software without your consent and in some cases refuse to distribute updates to your software (e.g. jwz’s XScreenSaver woes). But from a user’s perspective, perhaps that is ~okay — if you don’t mind out-of-date software. But for users, it gets worse! Where do you install from: the distro? Flatpak? Snaps? Sometime you install one package from one place, and it immediately pops up an alert telling you to uninstall it and install from a different place. But there’s no consistency: it’s not like every package prefers one place or another. And they’re cross-listed, but often with radically different versions! You’re not even guaranteed Flatpak or Snaps are the most up to date: the app developers may have abandoned that distribution method and gone back to shipping binaries in the distro’s repo. Plus if you install from Flatpak or Snaps, which certain programs more-or-less demand, they interface poorly with the rest of your system by default because of bundling their own filesystem images (and in Snap’s case they start slowly as a result). It’s… not great.
On macOS, for GUI programs you have two “options”: the Mac App Store, or the internet. If you choose “the internet,” macOS will refuse to run your program unless your users click a checkbox hidden in the main system Settings app. Even if they do, it will prompt them before installing, telling them that anything from the Internet is dangerous (ignoring any signing). Also, the Mac App Store is extremely limited and many programs are impossible to run in their sandboxing. As per usual, Apple’s basic message is that if you’re trying to make programs that run on Macs, and you’re not Apple, they reserve the right to make you miserable.
For installing command-line binaries: on Windows you’d either use chocolatey (the old 3rd party package manager) or scoop (the new 3rd party package manager). MS realized that people like command-line package managers, so they’re building an officially-blessed one called winget that will presumably replace those. Winget is not yet released to the general public though.
On Linux, generally you’d use your distro’s package manager. But since app developers can’t easily add or update packages in the repo, sometimes your distro does not have the package! Or, as usual, it has some ancient outdated version. Then if you are on Ubuntu maybe you can add the PPA, or if you are not maybe you can go fuck yourself (cough I mean build it from source).
On macOS the situation is fairly similar to Windows currently: you can use MacPorts (the old 3rd party package manager), or Homebrew (the new 3rd party package manager). As usual Apple does not care that developers like command-line package managers and is not building a blessed one.
1. 2
When it comes to actually distributing your app, Microsoft loses its attention every five seconds. The Micrsofot Store has been Microsoft’s answer for awhile, but it only relatively recently (last couple of years?) gained the ability to handle non-UWP binaries.
Agree with this. It looks like at the moment if you want to sell productivity software for Windows without having to operate your own storefront, the most stable option may actually be Steam? Sure it targets the wrong market segment, but at least it works reliably.
1. 4
While I agree with the ideas presented here, in particular the comments on IDEs (or as I like to call them, Interactive Computing Environments, to avoid confusion with “regular” IDEs), I do wonder why these ideas keep getting forgotten. We had Smalltalk, we had Lisp Machines and we have Unix shells, but the tendency always seems to go towards a rigid cookie-cutter-style of programming. I don’t like the idea that people are “too stupid” to understand or use it, and I don’t know how much of it is just that people were used to whatever reached the market first, no matter how annoying it is and how much time people spend fighting it. One component is certainly external (often proprietary) dependencies. Or is it education that de-prioritizes these kinds of thinking?
1. 8
It’s the insistence on doing everything via text.
1. 5
There are two issues, in my opinion, both shaped by own experience using Smalltalk and trying to teach it to others.
The first is that you can’t get a flow like the one in this article without learning new tooling on top of the language, and that ends up being a big issue. If I know Emacs (or Visual Studio Code, or Vim, or any of the even vaguely extensible editors), I can use that same tool to handle basically every language, focusing just on the new language. To get a good flow in Smalltalk (or, I believe, a Lisp machine, but notably not contemporary Common Lisp or Schemes), you have to learn the IDE. In Smalltalk, this is especially bad, because the traditional dev flow effectively uses a source database instead of source files, so (until recently) you couldn’t even use things like diff or git.
The second thing is that this kind of dev flow, in my experience, thrives when you’re doing something novel. Nowadays, most dev work I do is “just” assembling tons of existing libraries in familiar patterns. That’s not a problem, and I don’t think it’s laziness; it’s about predictability and repeatability, and I mostly view it as a sign that the industry is maturing. It lets me do much more with much less effort and much lower risk than doing everything bespoke. But it does mean that if, for example, I want to write a Smalltalk backend for a website in 2021, I’m going to have to write a bunch of stuff (e.g., OAuth connectors, AWS APIs, possibly DB drivers, etc.) that I’d get for free in virtually any other language, which in turn are new places things can go wrong, where I won’t be able to ask or pay someone else for support, and which likely don’t have anything to do with making my software sell. This applies pretty intense brakes to using novel environments even if you believe you’d be happier in one. This is basically the same as your point on external dependencies, but I think looking at it one step back from a repeatability and reliability perspective makes it more obvious why it’s such an issue.
1. 7
As someone who has dabbled in Common Lisp w/ SLIME, another limitation of that development style I have noticed is keeping track of state and making sure things are reproducible from the code, and not some unreachable state you have arrived at from mutating things in the REPL. There is a similar issue with Jupyter notebooks
1. 4
In Smalltalk, this is especially bad, because the traditional dev flow effectively uses a source database instead of source files, so (until recently) you couldn’t even use things like diff or git.
While I certainly agree with the lamentations on using modern VCS tools – ten years ago, I spent four months writing a tool that could split a single multi-megabyte XML source database configuration file into multiple files for more atomic versioning and review and combine those files for deployments — I feel like the file paradigm is one that advanced IDE users may be OK abstracting away. I use IntelliJ and other JetBrains products, and Eclipse before them, that have “search by symbol” features, generally used to search by a class, object, trait, interface, etc. name. There are some projects I’ve worked on where the only time I really have to care about files is when I identify the need to create a new package or module necessitating a new directory. Otherwise, my IDE handles the files almost entirely as an abstraction.
This was difficult to wrap my head around it but because of my experience with Smalltalk in college, I understood it more quickly than my peers and it accelerated my development productivity by a little bit. I’ll readily admit that I’m slower on file-based IDEs or text editors without some kind of fuzzy finder (I’ve been using Elementary Code in a VM for one project and dreadfully missing CtrlP or the like) but it is my preference to treat encapsulated code as an object instead of as a file. I think if more people preferred this, and Smalltalk would have been more popular for other reasons, perhaps a solid VCS for source databases may have emerged; one that didn’t rely on disaggregating the database into the filesystem paradigm.
1. 1
I feel like the file paradigm is one that advanced IDE users may be OK abstracting away. I use IntelliJ and other JetBrains products, and Eclipse before them, that have “search by symbol” features, generally used to search by a class, object, trait, interface, etc. name. There are some projects I’ve worked on where the only time I really have to care about files is when I identify the need to create a new package or module necessitating a new directory.
While it’s true that individual users might be OK with this, there’s two factors to consider. One is that you operate with the knowledge that when your tools do stop working, you can always drop down a level to the “real” files to find out what’s actually going on. The second is that you can collaborate with others who use Vim and Emacs; your choice to use IntelliJ does not force your teammates to adopt your same tools.
2. 2
I’m going to have to write a bunch of stuff (e.g., OAuth connectors, AWS APIs, possibly DB drivers, etc.) that I’d get for free in virtually any other language
Those seem to be largely available in Pharo via existing libraries e.g.:
1. 1
Here’s a quick reality check, using two examples that have come up in my own work:
1. Does PayPal have an official SDK for Smalltalk?
2. Is there a Smalltalk version of the AWS Encryption SDK?
Spoiler: The answer to both is no.
1. 4
I don’t think that’s the right question. The right question is whether these things have an SDK that is easy to use from Smalltalk. Unfortunately the answer is still ‘no’. Even in Smalltalks that have a way of calling other languages, the integration is usually painful because the Smalltalk image abstraction doesn’t play nicely with the idea that some state exists outside of the image.
3. 4
We had Smalltalk, we had Lisp Machines and we have Unix shells
One of these is not like the others.
PowerShell is closer due to being object-based, but it’s still very clunky.
1. 3
I don’t think that being object-based is necessary – it makes it cleaner and more efficient. Following this article, you do have a dialogue with the computer (even if it is rather simple), users can and do modify their environment (shell and PATH) and in the end, it is simple, perhaps too simple.
1. 1
I claim that being object-based is “necessary” in the sense that you’re meaningfully far away from the Smalltalk ideal if your system is built around text. Obviously, there’s a gradient, not a discrete transition, but being object-oriented is one of the major factors.
Additionally, Unix (shells) is dis-integrated, both in ideals and in implementation. Another major design decision of Lisp/Smalltalk is integration between components - something the Unix philosophy explicitly spurns.
2. 2
I think different tools are just good at different jobs. I don’t write in-the-large network services in Smalltalk just like I don’t write tax filing products in spreadsheets.
This is not to say that Smalltalk or spreadsheets are less – far from it! If I want to bang out a business projection I don’t reach for Rails or Haskell or Rust, I grab a spreadsheet. I think there are similarly many situations where something more Smalltalk-like is the ideal tool, but your day-job as a programmer in tech is not full of those situations and we haven’t given enough knowledge of what computers are capable of to those who would use computing as a tool for their own ends.
1. 21
This is something I try, over and over, to explain to people, and I’ve never, ever succeeded in doing it in print or a talk. I always get a response along the lines of, “oh yeah, I love TDD, that’s how I write [OCaml/Go/C#/whatever],” and that’s effectively the end of the conversation on their end: “neat, this guy likes Smalltalk because it has really good TDD”, is about all they hear, and the rest washes off like tears in rain.
“Experiencing Smalltalk” is a great title for an article like this because you really need to actually experience it, ideally using it yourself, to get it. Smalltalk the language is…fine. It gets a lot right, it gets a lot wrong, languages like Self and Slate have tried to improve it, but at any rate, it gets the job done with minimal fuss. People who just look at its syntax and semantics are right in 2021 that many other languages deliver the same or better.
But that misses the point. The thing that differentiates Smalltalk is its entire development flow, which is radically different from literally anything else I’ve ever used: write broken code, run it, repeatedly fix the code as you slowly walk through methods and whole classes that either didn’t work or didn’t even exist when you initiated the execution, and end up with a working spike that had its first successful run the second you’re done writing the last line of code. A very few languages, like Factor and Common Lisp, come very close, but as of 2021, Smalltalk is the only environment I’ve ever used that still delivers it.[1]
I don’t write Smalltalk anymore, and I don’t see that changing (mostly just because I’m old and have kids and spend what little time I do coding for fun on things like Factor), but the experience of developing in it remains absolutely without peer.
[1]: I’ve been told that the actual Lisp Machines of the 80s did have this same flow, but I’ve never used one–and I definitely don’t think SBCL in 2021 matches the dev flow of Pharo or Squeak Smalltalk.
1. 1
The thing that differentiates Smalltalk is its entire development flow, which is radically different from literally anything else I’ve ever used: write broken code, run it, repeatedly fix the code as you slowly walk through methods and whole classes that either didn’t work or didn’t even exist when you initiated the execution, and end up with a working spike that had its first successful run the second you’re done writing the last line of code.
This describes my experience writing Emacs pretty closely. However, I know that many people who know Emacs intimately still say that Smalltalk is different, so I have to conclude that there’s more to it, and that it’s just very difficult to describe what exactly the difference is in words. I expect it has to do with a more seamlessly integrated debugger that reifies the call stack and things. I suppose there’s only one way to really find out.
1. 3
Are there other examples of SQLite being used as a website backend database in production? What kind of scale could you reach with this approach? And what would be the limiting resource?
1. 10
Expensify was based exclusively on sqlite for a long time, then they created a whole distributed database thing on top of it.
1. 7
Clojars used SQLite for a good 10 years or so, only recently moving away to Postgres for ease of redeployment and disaster recovery. The asset serving was just static assets, but the website and deployments ran against SQLite pretty well.
1. 3
If I remember correctly, the trouble that Clojars ran into had more to do with the quality of the JVM-based bindings to SQLite than they did with SQLite itself, at least during the portion of time that I was involved with the project.
1. 2
Yeah, looking back at the issues, “pretty well” is maybe a little bit generous. There were definitely settings available later on which would have helped the issues we were faxing around locking.
2. 4
I can’t remember whom but at least one of the well funded dynamoDB style distributed database-y products from the mid 10s used it as the storage layer.
So all the novel stuff that was being done with data was the communication and synchronisation over the network, and then for persistence on individual nodes they used sqlite instead of reinventing the wheel.
1. 6
That was FoundationDB, purchased by Apple in 2013, then gutted, and then returned as open-source in 2018. I’m a bit annoyed, because it was headed to be CockroachDB half a decade earlier, and was taken off the market with very little warning.
1. 1
Thanks!
2. 3
You probably will get really fast performance for read-only operations. The overhead of client/server and network stack could be more than10x times of function calls from same address space. The only real limitation might be single server, since you cannot really efficiently scale sqlite beyond single system. But when you reach that scale, you usually needs much more than sqlite.
1. 3
The sqlite website claims to run entirely on sqlite.
They also have this page, though most of those aren’t websites: https://sqlite.com/mostdeployed.html
1. -3
It seems to be a common theme of prog-lang-started-by-child-prodigy projects that they adopt features where I simply can’t fathom how they are going to maintain and develop them in the mid-to-long-term.
Perhaps I’m the only one who is concerned by the complexity these party-trick features seem to involve?
(The other option is that this stuff is really that easy and all the hundreds of full-time C/C++ compiler engineers are just idiots for not doing it.)
1. 33
There are more details on why and how this works here: zig cc: a Powerful Drop-In Replacement for GCC/Clang
The other full time C/C++ compiler engineers are not idiots; they just have different goals since they work for companies trying to turn a profit by exploiting open source software rather than working for a non-profit just trying to make things nice for everyone.
1. 6
The other full time C/C++ compiler engineers are not idiots; they just have different goals since they work for companies trying to turn a profit by exploiting open source software rather than working for a non-profit just trying to make things nice for everyone.
This feels like a big statement, and that’s fine, but would you mind elaborating? Which companies do you mean? What goals do they have that are incompatible with something like zig cc?
1. 5
I think the point there was just that e.g. Apple has no particular interest in making using clang to cross-compile Windows binaries easy. They wouldn’t necessarily be against it, but it’s not something that aligns with their business interests whatsoever, so they’re very unlikely to spend any money on it. (Microsoft actually does value cross-compilation very highly, and has been doing some stuff in that area with clang, and so is almost a counterexample. But even there, they focus on cross-compilation in the context of Visual Studio, in which case, improving the CLI UI of clang again does not actually do anything for them.)
2. 40
Am I the only one who is concerned by the complexity these party-trick features seem to involve?
(The other option is that this stuff is really that easy and all the hundreds of full-time C/C++ compiler engineers are just idiots for not doing it.)
This mindset is one of the major causes why modern software sucks so much. The amount of tools that can be improved is humongous and this learned helplessness is why we keep having +N layer solutions to problems that would require re-thinking the existing toolchains.
I encourage you to read the Handmade Manifesto and to dive deeper into how Zig works. Maybe you’re right, maybe this is a party trick, but the reality is that you don’t know (otherwise you would take issue with specific approaches Zig employs) and you’re just choosing the safe approach of reinforcing your understanding of the status quo.
Yes, there are a lot of snake oil sellers out there, but software is not a solved problem and blanket statements like this one are frankly not helping anybody.
1. 1
I think you are wrong and the exact opposite is the case:
We can’t have nice things because people don’t learn from their predecessors.
Instead they go out to reinvent flashy new stuff and make grandiose claims until it turns out they ignored the inconvenient last 20% of work that would make their new code reliable and complete – oh, and their stuff takes 200% more resources for no good reason.
So yeah, if people don’t want to get suspected of selling snake oil, then they need to be straight-forward and transparent, instead of having these self-congratulatory blog articles.
Build trust by telling me what doesn’t work, and what will never work.
1. 17
Here’s what doesn’t work https://github.com/ziglang/zig/labels/zig%20cc.
2. 7
Clang could provide the same trivial cross compilation if it were a priority. Zig is mostly just using existing clang/llvm features and packaging them up in a way that is easier for the end user.
1. 21
“just”
1. 4
Perhaps not obvious, but I meant the “just” to be restricted to “mostly just using existing clang/llvm features”. I’m in no way denegrating Andrew’s efforts or the value of good UX.
2. 5
Another option is that it’s easy if you build it in at the start and much more difficult to add it later. It’s like the python 2 to 3 migration. It wasn’t worth it for some projects, but creating a new python 3 project is easy. Path dependence is a thing.
1. 2
I think the hard part is adding these kinds of features after the fact. But assuming it’s already in place, I feel like this is actually not a very hard thing to maintain?
I think a lot of complexity with existing tools is around “oh we’re going to have this be global/implicit” and that permeating everywhere, so then when you want to parametrize it you have to play a bunch of tricks or rewrite everything in the stack to get it to work.
But if you get it right out of the door, so to speak, or do the legwork with some of the dependencies… then it might just become a parameter passed around at the top level (and the lower levels already had logic to handle this, so they don’t actually change that much).
case in point: if you have some translation framework relying on a global, your low-level will read that value and do a lookup, and the high level will not handle it. If you parameterize it, now your high-level stuff has to pass around a bunch of translation state, but the low-level (I guess the hard part, so to speak?) will stay basically the same. At least in theory
I do kinda share your skepticism with the whole “let’s rewrite LLVM” thing… but cross compilation? Having a build system that is “just zig code” instead of some separate config lang? These seem good and almost simpler to maintain. I don’t think C compiler engineers are idiots for not doing X, just like… less incentivised to do that, since CMake isn’t a problem for someone who has spent years doing it.
1. 2
I agree with you. This doesn’t make any sense for Zig to take on. Andrew shared it with me as he was working on it and I thought the same thing then: what? Why does a compiler for one language go to this much trouble to integrate a toolchain for another language? Besides being severely out of scope, the problem space is fraught with pitfalls, for example with managing sysroots and dependencies, maintaining patched forks of libcs, etc. What a huge time sink for a group who should ostensibly have their hands full with, you know, inventing an entire new programming language.
The idea of making cross-compilation easier in C and C++ is quite meritous. See Plan 9 for how this was done well back in the naughts. The idea that it should live in the zig build tool, however, is preposterous, and speaks rather ill of the language and its maintainers priorities. To invoke big corporate compiler engineers killing open source as the motivation is… what the fuck?
Sorry Andrew. We don’t always see eye to eye, but this one is particularly egregious.
1. 7
No, this makes a lot of sense. Going back to the article, Go’s toolchain (like Plan 9’s) is good at cross-compilation, but “I recommend, if you need cgo, to compile natively”. This sort-of works for Go because cgo use is low. But Zig wants to encourage C interoperability. Then, Zig’s toolchain being good at cross-compilation is useless without solving C’s cross-compilation, because most of Zig will fail to cross-compile because of C dependency somewhere. By the way, most of Rust fails to cross-compile because of C dependency somewhere. This is a real problem.
Once you solved the problem, it is just a good etiquette to expose it as CLI, aka zig cc, so that others can use it. The article gives an example of Go using it, and mentions Rust using it in passing.
I mean, yes, zig cc should be a separate project collaboratively maintained by Go, Rust, and Zig developers. Humanity is bad at coordination. Big companies are part of that problem. Do you disagree?
1. 2
The best way, in my opinion, to achieve good C interop is by leveraging the tools of the C ecosystem correctly. Use the system linker, identify dependencies with pkg-config, link to system libraries, and so on. Be prepared to use sysroots for cross-compiling, and unafraid to meet the system where it’s at to do so. Pulling the concerns of the system into zig - libc, the C toolchain, statically building and linking to dependencies - is pulling a lot of scope into zig which really has no right to be there. Is the state of the art for cross-compiling C programs any good? Well, no, not really. But that doesn’t mean that those problems can jump domains into Zig’s scope.
I am a believer that your dependency’s problems are your problems. But that definitely doesn’t mean that the solution should be implemented in your domain. If you don’t like the C ecosystem’s approach to cross compiling, and you want to interoperate with the C ecosystem, the correct solution involves going to the C ecosystem and improve it there, not to pull the responsibilities of the C ecosystem into your domain.
Yes, other languages - Go, Rust, etc - should also be interested in this effort, and should work together. And yes, humanity is bad at cooperation, and yes, companies are part of that problem - but applying it here doesn’t make sense. It’s as if I were talking about poaching as contributing to mass extinction, and climate change for also contributing to mass extinction, and large corporations for contributing to climate change, and then conclude that large corporations are responsible for poaching.
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There’s another path to achieving C interop, which is by using whatever feels more convenient but staying true to whatever ABI boundaries. In terms of Zig, this is achieved in a few ways: It uses its own linker (currently LLD) which is useful when you don’t have a local system linker (pure linux/windows install) and still works with existing C code out there. It uses paths for dependencies, leaving it up to the user to specify how they’re found (e.g. pkg-config). It links to system libraries only if told explicitly but still works without them - this is also useful when building statically linked binaries which still work with existing C code.
For cross-compiling, sysroot is a GCC concept. This doesn’t apply to other environments like clang (the C compiler Zig uses), or the defaults of Mac/Windows. Zig instead uses LLVM to emit any supported machine code (something which requires having multiple compilers for in GCC), bundled the build environment needed (lib files on windows, static libc on linux if specified, nothing if dynamically linking), and finally links them together to the appropriate output using LLD’s cross-linking ability.
Having this all work seamlessly from whatever supported system is what makes it appealing. For example, andrew (creator of Zig) has showcased in the past cross-compiling the compiler on an x86 machine to aarch64, then using qemu to cross-compile the compiler again from the aarch64 vm back to x86, and it works. This applies also to other operating systems, which is a feature that isn’t present in current cross compiling tools, even clang.
For the issue of problem domains, this is not something you could address by trying to fix existing C tools. Those already have a defined structure as andrew noted above given they have different goals and are unlikely to change it. This could be why Zig takes upon solving these problems locally, and pulls the responsibility of what it wishes to provide, not the entire C ecosystem. I believe its partially of similar sub-reasons why Go has its own build system but also claims to compile to different environments.
I also agree that different ecosystems could pitch in for what seems to be a universally helpful tool, but as its been going on today, maybe they have different design goals. Where another path such as using the existing C ecosystem (for various situational reasons) makes more sense than the idealistic one Zig has chose to burden.
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It links to system libraries only if told explicitly but still works without them - this is also useful when building statically linked binaries which still work with existing C code.
System libraries can also be static libraries, and there’s lots of reasons to link to them instead. We do build statically linked programs without the Zig tools, you know!
For cross-compiling, sysroot is a GCC concept. This doesn’t apply to other environments like clang
Clang definitely uses sysroots. Where does it find the static libs you were referring to? Or their headers? The answer is in a sysroot. Zig may manage the sysroot, but it’s a sysroot all the same.
There’s more to take apart here, but on the whole this is a pretty bad take which seems to come from a lack of understanding about how Linux distributions (and other Unicies, save for macOS perhaps) work. That ignorance also, I think, drove the design of this tool in the first place, and imbued it with frustrating limitations which are nigh-unsolvable as a consequence of its design.
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The explicitly provided system libraries is not about dynamic vs static linking, its about linking them at all. Even if you have the option to statically link libc, you may not want to given you can do its job sometimes better for your use case on platforms that don’t require it (e.g. linux). The closest alternative for C land seems to be -ffreestanding (correct me if i’m wrong)? This is also an option in zig, but it also gives the option to compile for platforms without having to link to any normal platform libraries.
Clang has the option to use sysroots, but it doesn’t seem to be required. In zig’s case, it uses whatever static libs you need by you explicitly linking to them rather than assuming they exist upon a given folder structure in the same directory. Zig does at least provide some methods of finding where they are on the system if you don’t know there they reside given the different configurations out there. I’d say this differs from a sysroot as its more modular than “system library directory”.
Without a proper explanation, the idea that this approach “stems from lack of understanding” or has “frustrating limitations which are nigh-unsolvable” don’t seem make such sense. As we’re both guilty of prejudice here, i’d relate your response to one of willfully ignorant to unknown systems and gate-keeping.
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Clang has the option to use sysroots, but it doesn’t seem to be required.
Your link does not support your statement. I don’t think you understand how cross-compiling or sysroots actually work.
Again, it’s the same with the rest of your comments. There are basic errors throughout. You have a deep ignorance or misunderstanding of how the C toolchain, linking, and Unix distributions work in practice.
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Given you haven’t actually rebutted any of my claims yet, nor looked into how clang supports using sysroots, we probably won’t be getting anywhere with this. Hope you’re able to do more than troll in the future.
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Clang totally uses sysroot, see here. (Long time ago, I wrote the beginning of Clang’s driver code.) I don’t know where to begin, but in fact, ddevault is correct about all technical points and you really are demonstrating your ignorance. Please think about it.
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Please re-read by post above which literally says “clang supports using sysroots”, a claim that agrees with yours. My original point a few messages back was about how clang doesn’t need sysroot in order to cross-compile, which still stands to be disproved, as its just short for a bunch of includes.
Once again, just as ddevault, you enjoy making claims about others without specifying why in an attempt to prove some point or boost your ego. Either ways, if this is your mindset, there’s no point in further discussion with you as well.
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What features in this case?
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That seems less an “other side” and more “so what?”, especially in his response to jwz’s response, but it’s indeed interesting to have more context.
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Interesting, but I’m inclined to be on jwz’s side
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“Your security arguments turned out to be incorrect. So, stop?” Did they though? Did they REALLY?
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I really want to try kakoune, but the idea of starting over with a new editor and editing paradigm just seems like so much effort and time before I’m productive.
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I feel the same way. I use NeoVim and have tried to keep it as stock config as possible, but I think I’ve already tweaked it enough to be different enough. So learning Kakoune would be against vim everywhere and my customization.
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If you like the vi/vim experience but want some similar features to Kakoune then vis might be worth a shot. (Also see differences from Kakoune).
I use it as my main editor and structural regular expressions, multi-cursor, etc are all quite intuitive while not leaving the traditional vi-like modal editing world IMO.
Plugins are also written in Lua, if that’s your thing.
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YMMV of course, but it only took ~2 weeks after switching from vim for me to become reasonably productive in kakoune.
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What was the biggest hurdle for you when acclimating to Kakoune?
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Not OP, but as someone else who went from Vim to Kakoune, I think the biggest shift for me was thinking in terms of repeatedly narrowing the selection and then doing one single command on the selection, rather than doing a command sequence and e.g. assigning to a macro or the like. The better I got at selection narrowing, the easier and more natural everything felt. Learning slightly different keystrokes was comparatively very easy.
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On day 1 it was 100% unlearning vim muscle memory. After that my biggest challenge was adapting to kakoune’s selection first editing model, which is what inspired me to switch in the first place. It was very worth it though, the incremental nature of the editing in which intermediate results are instantly visible makes complex tasks much more intuitive.
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I’m coming from emacs, which is probably going to be worse, but even two weeks sounds like an enormous amount of time to not be able to code. I can’t justify taking more than a day to switch at work, so I’d have to use both, too.
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It’s not that I wasn’t able to code at all but that I was significantly slower than I was with vim. I quickly gained speed over the first week though and after ~2 weeks I didn’t feel like my inexperience with editor was holding me back for basic editing tasks. More advanced editing tasks weren’t intolerably slow either, just took a bit more thought than they do now.
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Tired of Terminal UIs based on ncurses or blessed? With Fig, you can use modern web technologies like HTML, CSS and JavaScript instead.
I suppose I’m not in the target audience as I really don’t see using web technologies as a feature over TUIs.
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Eh, the idea seems brilliant to me, honestly; there are a few tools I just don’t use often enough to fully remember their CLIs, so having an ad hoc, simple GUI for those would be a huge boon, letting me stick with the CLI tool, but not (necessarily) have to read the man pages each time. Having that outside the terminal so I can see the command line being built also makes sense. But I’m with you that full-blown HTML for the UI seems a bit heavy to me.
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When I saw it, I thought of a co-worker who’s wondered about how to span the gulf between scripts/utilities we can readily write and run, and utilities that non-programmers doing video production can handle.
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http://stomatologgolub.pl/olive-tree-dcqypl/aec31b-total-number-of-injective-functions-from-a-to-b
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Although a number of economic valuation studies of wetlands have been undertaken around the world and economists have developed methodologies for valuing more intangible aspects of the environment, such as amenity or aesthetic factors, no one has synthesised from this literature a common approach to show its overall usefulness to wetland management worldwide. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Calculating the number of injective functions, Why do massive stars not undergo a helium flash. relations and functions; class-12; Share It On Facebook Twitter Email. 2) Number of ways in which two elements from set A maps to same elements in set B is The function f: {Indian cricket players’ jersey} N defined as f (W) = the jersey number of W is injective, that is, no two players are allowed to wear the same jersey number. And in general, if you have two finite sets, A and B, then the number of injective functions is this expression here. Use MathJax to format equations. Number of injective functions from b to a give a. If a = {1, 2, 3} and B = {A, B}, Write the Total Number of Functions from a to B. Dog likes walks, but is terrified of walk preparation. B). Functions in the first column are injective, those in the second column are not injective. D. How Many Bijections? I hadn't heard of the Stirling numbers, I wonder why they are not included more often in texts about functions? Pages 5 This preview shows page 2 - 4 out of 5 pages. 1 answer. Zero correlation of all functions of random variables implying independence, Basic python GUI Calculator using tkinter. answered Aug 28, 2018 by AbhishekAnand (86.9k points) selected Aug 29, 2018 by Vikash Kumar . = 24. The first element in A has 5 choices from B. 1) Define two of your favorite sets (numbers, household objects, children, whatever), and define some a) injective functions between them (make sure to specify where the function goes from and where it goes to) b) surjective functions between them, and c) bijective functions between them. We call the output the image of the input. Uploaded By ProfLightningLyrebird3306. (3C1)*(4*3) = 36. b) n(A)=5 and n(B)=4. Question Bank Solutions 10059. MathJax reference. Since f is surjective, there is such an a 2 A for each b 2 B. That is, we say f is one to one. If X has m elements and Y has 2 elements, the number of onto functions will be 2 m-2. A such that g f = idA. A function f: X !Y is surjective if every element y in Y is mapped to by some x in X. Countable total orders; 6 Bibliography . It means that every element “b” in the codomain B, there is exactly one element “a” in the domain A. such that f(a) = b. f g = idB. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. How many are injective? See the answer. 236 CHAPTER 10. Let $$\Large A = \{ 2,\ 3,\ 4,\ 5 \}$$ and. Can someone point out the mistake in my approach ? More precisely, f is injective if for every pair of elements x and x0 in X such that x 6= x0, we have f(x) 6= f(x0). Let n(A) = m, and n(B) = n. Then the total number of non-empty relations that can be defined from A to B is (a) ... mn - 1 (d) 2mn- 1 Then f g(b) = f(g(b)) = f(a) = b, i.e. Test Prep. How Many Functions Total From A To B? It’s rather easy to count the total number of functions possible since each of the three elements in $A$ can be mapped to either of two elements in $B$. Making statements based on opinion; back them up with references or personal experience. 8). 1.19. School The University of Sydney; Course Title MATH 2969; Type. If it is not a lattice, mention the condition(s) which … If a function is defined by an even power, it’s not injective. Show that for an injective function f : A ! Misc 10 (Introduction)Find the number of all onto functions from the set {1, 2, 3, … , n} to itself.Taking set {1, 2, 3}Since f is onto, all elements of {1, 2, 3} have unique pre-image.Total number of one-one function = 3 × 2 × 1 = 6Misc 10Find the number of all onto functio The term one-to-one function must not be confused with one-to-one correspondence that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain. If a function is defined by an even power, it’s not injective. The set A has 4 elements and the Set B has 5 elements then the number of injective mappings that can be defined from A to B is. Previous question Next question Transcribed Image Text from this Question. Important Solutions 983. If $$\Large R \subset A \times B\ and\ S \subset B \times C$$ be two relations, then $$\Large \left(SOR\right)^{-1}$$ is equal to: 10). Example 46 (Method 1) Find the number of all one-one functions from set A = {1, 2, 3} to itself. Let's consider the map $1 \mapsto 1$, $2 \mapsto 2$, and $3 \mapsto 4$. But it seems that my answer is wrong. Calculating the total number of surjective functions, Number of onto mappings from set {1,2,3,4,5} to the set {a,b,c}, Number of surjective functions from a set with $m$ elements onto a set with $n$ elements. To de ne f, we need to determine f(1) and f(2). Expert Answer . (3C2)*(3) = 9. 1 answer. 9). a ≠ b ⇒ f(a) ≠ f(b) for all a, b ∈ A ⟺ f(a) = f(b) ⇒ a = b for all a, b ∈ A. e.g. In other words f is one-one, if no element in B is associated with more than one element in A. Syllabus. In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements of its codomain. The first step in correcting that count is to add those cases with two corresponding elements back (including those with exactly three corresponding elements). rev 2021.1.8.38287, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. ... For example, if you have 10 red balls, 7 blue balls, and 4 red balls, then the total number of balls you have is 10 + 7 + 4 = 21. But is Find The Number Of Functions From A To B The Number Of Injective Functions From B To A. Injective and Surjective Linear Maps. b' So total number of ways of 'n' different objects = 2 x 2 x 2 ... n times = 2" But in one case all the objects are put box 'a' and in one case all the objects are put in box b' So, number of subjective functions = 2 n - 2 . Functions may be "injective" (or "one-to-one") An injective function is a matchmaker that is not from Utah. There are no polyamorous matches like the absolute value function, there are just one-to-one matches like f(x) = x+3. For convenience, let’s say f : f1;2g!fa;b;cg. Show that for an injective function … For example, $\{1,2\}$ and $\{2,1\}$ are exactly the same sets. True to my belief students were able to grasp the concept of surjective functions very easily. A function is said to be bijective or bijection, if a function f: A → B satisfies both the injective (one-to-one function) and surjective function (onto function) properties. If m>n, then there is no injective function from N m to N n. Proof. Is this an injective function? N is the set of natural numbers. number of injective functions from B to A Give a proof that your list is. It only takes a minute to sign up. Why was there a "point of no return" in the Chernobyl series that ended in the meltdown? Is it damaging to drain an Eaton HS Supercapacitor below its minimum working voltage? Share with your friends. Find The number of functions … On the other hand, they are really struggling with injective functions. So, answer should be 60-(36+9+1) = 14. given, Domain = {2,4,6} Then, the total number of injective functions from A onto itself is _____. Say we know an injective function … The Number Of Relations From A To B Which Are Not Functions. Since f is one-one Hence every element 1, 2, 3 has either of image 1, 2, 3 and that image is unique Total number of one-one function = 6 Example 46 (Method 2) Find the number Since f is one-one Hence every element 1, 2, 3 has either of image 1, 2, 3 and that image is unique Total number of one-one function = 6 Example 46 (Method 2) Find the number of all one-one functions from set A = {1, 2, 3} to itself. Suppose m and n are natural numbers. If N be the set of all natural numbers, consider $$\Large f:N \rightarrow N:f \left(x\right)=2x \forall x \epsilon N$$, then f is: 5). A function f: X !Y is a injective if distinct elements in x are mapped to distinct elements in Y. There are four possible injective/surjective combinations that a function may possess. Therefore, b must be (a+5)/3. Set A has 3 elements and set B has 4 elements. This is not a function because we have an A with many B.It is like saying f(x) = 2 or 4 . Let f : A ----> B be a function. Injective, Surjective, and Bijective Functions. 0 votes . 3)Number of ways in which three elements from set A maps to same elements in set B is 1. How can I quickly grab items from a chest to my inventory? = 60. For clarity, let $A = \{1, 2, 3\}$ and let $B = \{1, 2, 3, 4, 5\}$, as @drhab suggested. When we apply the Inclusion-Exclusion Principle, we first exclude cases in which there is one corresponding element. Best answer. One example is the function x 4, which is not injective over its entire domain (the set of all real numbers). However, we have not excluded the case in which all three elements of $A$ are mapped to the corresponding elements of $B$ since we subtracted them three times, then added them three times. We will now look at two important types of linear maps - maps that are injective, and maps that are surjective, both of which terms are analogous to that of regular functions. How do I hang curtains on a cutout like this? For each b 2 B such that b = f(a) for some a 2 A, we set g(b) = a. Thank you . Let, a = 3x -5. The notion of a function is fundamentally important in practically all areas of mathematics, so we must review some basic definitions regarding functions. Now pick some element 2 A and for each b … 6. It is well-known that the number of surjections from a set of size n to a set of size m is quite a bit harder to calculate than the number of functions or the number of injections. Department of Pre-University Education, Karnataka PUC Karnataka Science Class 12. A function is a rule that assigns each input exactly one output. asked Aug 28, 2018 in Mathematics by AsutoshSahni (52.5k points) relations and functions; class-12; 0 votes. Why is the in "posthumous" pronounced as (/tʃ/). If A and B are two sets having m and n elements respectively such that 1≤n≤m then number of onto function from A to B is = ∑ (-1) n-r n C r r m r vary from 1 to n Bijection-The number of bijective functions from set A to itself when there are n elements in the set is … 1) Number of ways in which one element from set A maps to same element in set B is (3C1)*(4*3) = 36. Department of Pre-University Education, Karnataka PUC Karnataka Science Class 12. On the other hand, the map $1 \mapsto 1$, $2 \mapsto 2$, and $3 \mapsto 3$ has exactly three corresponding elements. Misc 10 (Introduction)Find the number of all onto functions from the set {1, 2, 3, … , n} to itself.Taking set {1, 2, 3}Since f is onto, all elements of {1, 2, 3} have unique pre-image.Total number of one-one function = 3 × 2 × 1 = 6Misc 10Find the number of all onto functio Find the number of relations from A to B. Let f : A ⟶ B and g : X ⟶ Y be two functions represented by the following diagrams. The relation R is defined on $$\Large N \times N$$ as follows: $$\Large \left(a,\ b\right)R \left(c,\ d\right) \Leftrightarrow a+d=b+c$$ is: 6). For each b 2 B we can set g(b) to be any element a 2 A such that f(a) = b. 1.18. We count it three times, once for each of the three ways we could designate one of the three elements in $A$ as the corresponding element. 1) Number of ways in which one element from set A maps to same element in set B is Give Its Inverse In Two Line Again. But, there is no order in a set. Number of injective, surjective, bijective functions. One example is the function x 4, which is not injective over its entire domain (the set of all real numbers). @Zephyr Your persistence and willingness to ask questions will serve you well as you continue your studies. Transcript. But … This is illustrated below for four functions $$A \rightarrow B$$. Question Bank Solutions 10059. Solution. site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. Well, no, because I have f of 5 and f of 4 both mapped to d. So this is what breaks its one-to-one-ness or its injectiveness. In other words, every element of the function's codomain is the image of at most one element of its domain. 1.19. Show that for a surjective function f : A ! If a = {1, 2, 3} and B = {A, B}, Write the Total Number of Functions from a to B. Find the number of relations from A to B. Can a law enforcement officer temporarily 'grant' his authority to another? The number of injective functions possible from A to B such that p'th element of A cannot map with p'th element of B where |A|=3 and |B|=5 is ? On A Graph . f (x) = x 2 from a set of real numbers R to R is not an injective function. Transcript. So let us see a few examples to understand what is going on. Each map in which there are exactly two corresponding elements is subtracted twice and each map in which there are exactly three corresponding elements is subtracted three times. But an "Injective Function" is stricter, and looks like this: "Injective" (one-to-one) In fact we can do a "Horizontal Line Test": To be Injective, a Horizontal Line should never intersect the curve at 2 or more points. The number of injections that can be defined from A to B is: 1st element of A cannot be mapped with 1st element of B. $$\Large A \cap B \subset A \cup B$$, B). To subscribe to this RSS feed, copy and paste this URL into your RSS reader. B there is a left inverse g : B ! We added them three times when we counted those cases in which two elements of $A$ are mapped to the corresponding elements of $B$, once for each of the $\binom{3}{2}$ ways we could designate two of the three elements as the elements of $A$ that map to the corresponding elements of $B$. Terms related to functions: Domain and co-domain – if f is a function from set A to set B, then A is called Domain and B … Set A has 3 elements and the set B has 4 elements. When we subtract those cases in which one element of $A$ is mapped to the corresponding element of $B$, we have subtracted those cases in which two elements of $A$ are mapped to corresponding elements of $B$ twice, once for each way we could designate one of those elements as the element of $A$ that is mapped to the corresponding element of $B$. Let $$\Large f:N \rightarrow R:f \left(x\right)=\frac{ \left(2x-1\right) }{2}$$ and $$\Large g:Q \rightarrow R:g \left(x\right)=x+2$$ be two functions then $$\Large \left(gof\right) \left(\frac{3}{2}\right)$$. The set of all inputs for a function is called the domain.The set of all allowable outputs is called the codomain.We would write $$f:X \to Y$$ to describe a function with name $$f\text{,}$$ domain $$X$$ and codomain $$Y\text{. Related questions +1 vote. Terms related to functions: Domain and co-domain – if f is a function from set A to set B, then A is called Domain and B … Now, as the first element has chosen one element in B, you will only have 4 choices left in B. Data set with many variables in Python, many indented dictionaries? A function f: X !Y is a injective if distinct elements in x are mapped to distinct elements in Y. How can a Z80 assembly program find out the address stored in the SP register? That is, it is important that the rule be a good rule. Number of functions between two sets, with a constraint on said functions, Number of onto functions from Y to X (JEE Advanced 2018). a the number of functions f A B that are injective b the number of functions f from MAT 1348 at University of Ottawa A so that f g = idB. By the principle of multiplication, This means a function f is injective if a1≠a2 implies f(a1)≠f(a2). The function f is called an one to one, if it takes different elements of A into different elements of B. Thanks for contributing an answer to Mathematics Stack Exchange! So, total numbers of onto functions from X to Y are 6 (F3 to F8). What is the earliest queen move in any strong, modern opening? It will be nice if you give the formulaes for them so that my concept will be clear . Since this is a real number, and it is in the domain, the function is surjective. We will prove by induction on nthat the following statement holds for every natural number n: For every m∈ N, if there is an injective function f: N m → N n, then m≤ n. (1) Note that the implication above is the contrapositive of the one in the theorem statement. This means that if you tell me that two elements in A get sent to the same element in B, and moreover if you tell me that this function is injective, then I immediately know that the two elements in A that you’re talking about are really the same element. A and B are two finite sets with |A| = 6, |B| = 3. 1). \( \Large A \cup B \subset A \cap B$$, 3). If A has n elements, then the number of bijection from A to B is the total number of arrangements of n items taken all at a time i.e. $$\Large \left[ \frac{1}{2}, -1 \right]$$, C). -- > B be a function because we have an a with many variables in python, many indented?... Y in Y is a matchmaker that is, it is known as one-to-one.! Is called an one to one, if it takes different elements of B one, if it takes elements! Total numbers of onto functions will be clear { 1,2,3,4,5\ } $are the... Energy and moving to a give a strong, modern opening all areas of Mathematics, so must... Text from this question service, privacy policy and cookie policy 2 - 4 out 5... Is _____ fundamentally important in practically total number of injective functions from a to b areas of Mathematics, so we must review some basic definitions functions..., modern opening, copy and paste this URL into your RSS reader like f ( x ) =,... 1, 2 }, 1 \right ] \ ), 3 ) number of functions. 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Subtract the case with three corresponding elements ( see the last paragraph ) Mathematics Exchange! X! Y is unused and element 4 is unused and element 4 is and. An a 2 a for each B … Countable total orders ; Bibliography. © 2021 Stack Exchange Inc ; user contributions licensed under cc by-sa back after absorbing and!: F1 ; 2g! fa ; B ; cg, you agree our. ’ s not injective over its entire domain ( the set B has elements. It damaging to drain an Eaton HS Supercapacitor below its minimum working voltage 0 ) = 2 or 4 ''! \Subseteq a \cup B \ ), total injective functions from x to are...
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2021-03-04 03:43:51
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https://www.gamedev.net/forums/topic/557570-asynchron-file-loading-aka-streaming/
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There is a much easier way to do this. Introducing threading to your system adds a ton of complexity.
Even if you do decide to use threading in your app, the method you described is still not ideal.
The OS provides asynchronous IO functions. Use them. The OS is (potentially) able to do smart things such as reordering the disk reads to reduce load times, chose smarter buffer settings, and otherwise make it faster. You can have multiple requests at the same time, and the OS can (potentially) reorder them to run faster than running the same requests sequentially.
Google shows a few tutorials on AIO, including this one that looks pretty good.
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While I wouldn't have said it was simpler, and I'm assuming the Mac has something very much like it, the 'best' solution is probably to go with low level file access.
On Win32 its possible to open a file with the Win32 file I/O functions, find out its size, then tell the IO subsystem to load that data into a chunk of memory in an async manner. The file IO is queued by the file system and you can then sleep the loading thread until the data is loaded at which point you can wake up and deal with the loaded file as required.
I would assume the Mac's native file IO system has something very much like it. I also recall someone mentioning boost's ASIO library as a possible solution for this however I've not looked into myself.
The advantage of this method is that you hand off worrying about file IO timing etc to the OS, you also don't end up busy waiting a CPU core while the data loads. the IO subsystem can also queue up multiple files to load so it's just a matter of fire off the request and wait until your data turns up.
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okay thanks guys I will look into that.- I was just hoping if I pulled off something simple myself I could use it cross plattform.
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Well, as I said, Boost::ASIO might be able to help you out and would be crossplatform, but other than that you have to hit the OS at the low level as most languages have no concept of async IO; heck last I checked it was a bit of a pain to do in .Net and required unsafe code to do so.
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To some extent, I'd second what Frob said. To another extent I can't.
Sure, the OS is able to do smart things about the file reads, but that is under the assumption that you don't have limits on what you are reading, and that you are willing to abide by the limits of the async IO calls on the OS.
Consider you trying to stream data behind a movie. It is important that the video get its X Mb/s of data to keep the video running. Whatever is happening in the background, who cares.
I prefer a setup like the following:
*Every read is issued as a "job" with a size, priority, bandwidth. I read files into buffers (or at least partially), and parse buffers, instead of calling
*Jobs are sorted by "credits". Being serviced removes all credits. Every loop through the servicing thread, all jobs get "priority" number of credits.
*Each loop through the service thread, X jobs are serviced (one for each AIO slot I reserve that is free). Servicing a job means issuing a read of at least MIN size (ie 64K), up to bandwidth*time_from_last_service+fudgefactor bytes.
The service thread then waits for one of the AIO calls to return from its callback, then loops again.
This insures that each file that needs X bandwidth gets close to that much data throughput, the X AIO calls give the OS a way to schedule disk reads efficiently, and the priority credit system insures that all files get service eventually. Breaking up each large read into several small ones of "bandwidth" based size keeps one file read from stalling out service to other file reads. Overall the system seems to provide stable throughput, and behaves better than just issuing random read calls from random subsystems. Since each read has some metadata about how important the read call is. It keeps item A from stalling out item B, which was the main concern when writing said system in the first place. Since the motivation came from issues seen in another project, where background streaming would make foreground streaming choppy (level load behind a movie making the movie choppy)
The hard part about all of this is syncing everything without introducing too much overhead or complexity. Because now your main thread needs to ignore-till-callback or poll each frame to see if the data is loaded.
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boost::asio won't work without significant work on your part. It works beautifully for windows disk i/o, and for cross-platform network I/O, but for non-windows disk i/o it just isn't supported. If you want it to work you have to implement a class conforming to the RandomAccessHandle Boost.Asio concept whose interfaces and methods delegate to the underlying OS aio api, which frob linked to some tutorials above. It's not very easy though and requires a lot of work.
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Thanks, I think I will look into that aio stuff, I also found another good link I'd like to share: LINK
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I've used posix aio on linux in the past and it contains a subtle but annoying suckage that you need to pay attention to at initial design time.
Works like this -- you send a request and tag it with a callback function which comes back to tell you that your request is complete. Peachy.
The obvious way of OOing the interface is to put an interface pointer into the aiocb structure, and have the callback thunk through to the interface and dispatch the call to the object to say it's task is now done.
So you might have terrain zone objects created when a player is nearby (and potentially might enter them), which schedule loading their textures on creation, and get prodded when the data has appeared by the callback. If the work goes away; for example if the player takes a turn that means you no longer need a zone then the obvious thing to do is delete the terrain block. Which in turn, either cancels the AIO task if it hasn't completed or deletes the texture.
And this works. *Almost* all of the time.
The problem is that AIO tasks can become uncancellable -- call cancel and it returns NOTCANCELLED. When it it's in that state, the callback WILL get called at some point in the future and cannot be changed... and therefore you can now no longer safely delete anything that that callback will refer to.
In the end, we had to fix this by making queues of control objects, and tagging them to say whether their data was actually needed or not, to handle the spurious complete notifications from cancels which didn't work.
The reason for this is that underneath, the AIO system is effectively just a worker thread pool doing regular IO[1]. The task becomes uncancellable when one of the threads is actually blocked doing the IO.
So you need to make sure you handle that special case and be aware that a task whose attempted cancellation fails will still call the callback function.
[1] On Linux 2.6 there are just kernel threads doing the work for you -- you can see them on process listings.
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Hey thanks for the infos! AIO overall sounds pretty useful and I will definately give it a try in the future.-
Anyways, what is the biggest drawback of threading things on your own appart from the low level optimizations AIO will give me?
For instance what will happen if I load quite a big file in its own thread, will it noticeable slow down the Framerate if I dont load it in chunks?
This might be stupid questions but I never seriously used threading and I would like to understand things a little better!
Thanks so far, that cleared up alot of things allready!
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We're using async I/O due to significant performance benefits and went with the POSIX aio interface for portability. Since it's not available on Windows, we've implemented a wrapper on top of the Windows APIs that does the job.
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Anyways, what is the biggest drawback of threading things on your own appart from the low level optimizations AIO will give me?
As frob has said, threading requires careful study to ensure you don't have any race conditions.
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For instance what will happen if I load quite a big file in its own thread, will it noticeable slow down the Framerate if I dont load it in chunks?
No, IO happens via DMA nowadays, so it doesn't need lots of CPU time like PIO did. Large or small IOs in a separate thread shouldn't slow down your main thread. However, splitting your IOs into chunks makes a lot of sense (allows prioritizing IOs, simplifies a file caching mechanism and also speeds up aio).
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Original post by mokaschittaAnyways, what is the biggest drawback of threading things on your own appart from the low level optimizations AIO will give me?
The fact that, under Windows, OS, file system and drivers have the ability to organize disk access with knowledge of physical data layout. They could theoretically take into account disk rotation speed and head positions.
As a more general advantage, it allows for overlapped operations. It becomes possible to overlap read/process/write operations.
In general, kernel-assisted asynchronous IO has the potential of less overhead since it can be tuned to specifics of that particular OS and/or file system.
In practice results will vary, but considering this is a facility present in most OSes, it makes sense to use it.
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For instance what will happen if I load quite a big file in its own thread, will it noticeable slow down the Framerate if I dont load it in chunks?
Disk IO is always done in chunks. Disk cannot transfer 16 Gigabytes in one call.
As far as processing goes - if you have idle cores, and if there is no memory contention, and if there is no scheduling conflicts, and if disk controller doesn't cause any stalls, and if disk transfer doesn't interfere with OS, then there will be absolutely no impact from file loading.
Unfortunately, that is a lot of ifs, and a lot of them depend on hardware and other factors.
It is not uncommon for disk IO to cause annoying stalls to entire system, threads or not, but it depends on very specific circumstances.
But - first, and by far most important performance factor - avoid access to multiple files - just .tar-ing everything into a single file, and seeking inside of that will offer drastic performance improvements, especially under Windows and NTFS. *nixes are less affected by this, but still - if you have only a single file (or a handful) you remove many file lookups that would otherwise need to be performed. At very least, they introduce serial dependency which is undesirable due to added latency.
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Okay thanks, aio is my choice now. Since I will mostly use the engine for testing and in the future maybe exhibitions mac and linux support is enough for me right now!
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Original post by Jan WassenbergWe're using async I/O due to significant performance benefits and went with the POSIX aio interface for portability. Since it's not available on Windows, we've implemented a wrapper on top of the Windows APIs that does the job.
I took a quick look at your code. What you wrote is definitely better than a threaded approach, but you should really look into I/O completion ports, as it's significantly faster and more scalable.
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Thanks for the feedback :) I am aware of completion ports (using them for directory change notification), but can't see how they would help with the current usage. These aio routines are called by a synchronous IO splitter that does caching and decompression on a block level. There are no threads involved and a maximum of 16 IOs in flight at any given time (which is already quite high, not even a Fusion ioDrive card needs that much). How can completion ports improve things here?
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Original post by cache_hitI took a quick look at your code. What you wrote is definitely better than a threaded approach, but you should really look into I/O completion ports, as it's significantly faster and more scalable.
Did you mean overlapped IO?
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Original post by Jan WassenbergThanks for the feedback :) I am aware of completion ports (using them for directory change notification), but can't see how they would help with the current usage. These aio routines are called by a synchronous IO splitter that does caching and decompression on a block level. There are no threads involved and a maximum of 16 IOs in flight at any given time (which is already quite high, not even a Fusion ioDrive card needs that much). How can completion ports improve things here?
Well, if you search around the literature surrounding IOCP, you won't find a lot of details on exactly what makes it faster and more scalable than event-driven overlapped I/O. All you'll find is everyone saying "IOCP is the fastest and most scalable way to do async i/o in windows". One reason that comes to mind is that IOCP requires no user-mode synchronization primitives. No mutexes, no critical sections, no events. Of course synchronization happens, but it happens inside the kernel using kernel synchronization primitives, which should be faster.
It has some flexibility advantages over traditional overlapped I/O as well since it has direct API support for performing arbitrary computations asynchronously, not just I/O. For example, you could have a thread pool with number of threads equal to number of CPUs on the system. Without using any user-mode synchronization primitives, you can post a message to this thread pool to perform encryption / decryption of binary data, which upon completion can post a message back to the main thread that it's complete, again without any user-mode synchronization primitives such as events.
A basic IOCP loop would look something like this:
//You can put other user-defined info in here if you wish.struct RequestPacket : public OVERLAPPED{ RequestPacket(LPVOID buf) : buffer(buf) { hEvent = NULL; } LPVOID buffer;};#define KEY_READ 0#define KEY_WRITE 1#define KEY_ENCRYPT 2void iocp_loop(){ //Create a handle for reading from. FILE_FLAG_NO_BUFFERING is required //for optimal throughput, but imposes alignment / request size restrictions HANDLE hRead = CreateFile(path, GENERIC_READ, FILE_SHARE_READ | FILE_SHARE_WRITE, NULL, OPEN_EXISTING, FILE_FLAG_NO_BUFFERING | FILE_FLAG_OVERLAPPED | FILE_FLAG_SEQUENTIAL_SCAN, NULL); //Create a handle for writing to. FILE_FLAG_NO_BUFFERING is required //for optimal throughput, but imposes alignment / request size restrictions. //FILE_FLAG_WRITE_THROUGH is also required for optimal throughput. HANDLE hWrite = CreateFile(path2, GENERIC_WRITE, FILE_SHARE_READ | FILE_SHARE_WRITE, NULL, OPEN_EXISTING, FILE_FLAG_NO_BUFFERING | FILE_FLAG_WRITE_THROUGH | FILE_FLAG_OVERLAPPED, NULL); //Create a new IOCP and associate it with the reading handle and read key HANDLE hiocp = CreateIoCompletionPort(hRead, NULL, KEY_READ, 0); //Associate the previous IOCP with writing as well, using the write handle and a different key. CreateIoCompletionPort(hWrite, hiocp, KEY_WRITE, 0); LARGE_INTEGER size; LARGE_INTEGER nextReadOffset; LARGE_INTEGER nextWriteOffset; GetFileSizeEx(hRead, &size); nextReadOffset.QuadPart = 0; nextWriteOffset.QuadPart = 0; int readsOutstanding = 0; int writesOutstanding = 0; const int maxOutstandingIo = 16; //Required since we're using FILE_FLAG_NO_BUFFERING. You can use FSCTL_GET_NTFS_VOLUME_DATA to fetch this number for real. int blockSize = GetVolumeBlockSize(); LPVOID lpBuffer = VirtualAlloc(NULL, 65536, MEM_COMMIT|MEM_RESERVE, 0); std::vector<RequestPacket*> packets; for (int i=0; i < maxOutstandingIo; ++i) packets.push_back(new RequestPacket(lpBuffer+i*blockSize)); //Force some reads to kick off the process. for (int i=0; i < maxOutstandingIo; ++i) { RequestPacket* packet = packets; packet->Offset = nextReadOffset.LowPart; packet->OffsetHigh = nextReadOffset.HightPart; ReadFile(hRead, packet->buffer, blockSize, NULL, packet); ++readsOutstanding; nextReadOffset.QuadPart += blockSize; } while ((readsOutstanding > 0) || (writesOutstanding > 0)) { DWORD bytes; ULONG_PTR key; OVERLAPPED* overlapped; RequestPacket* packet; GetQueuedCompletionStatus(hiocp, &bytes, &key, &overlapped, INFINITE); packet = static_cast<RequestPacket*>(overlapped); switch (key) { case KEY_READ: readsOutstanding--; packet->Offset = nextWriteOffset.LowPart; packet->OffsetHigh = nextWriteOffset.HighPart; WriteFile(hWrite, packet->buffer, bytes, NULL, packet); writesOutstanding++; break; case KEY_WRITE: writesOutstanding--; packet->Offset = nextReadOffset.LowPart; packet->OffsetHigh = nextReadOffset.HighPart; ReadFile(hRead, packet->buffer, bytes, NULL, packet); readsOutstanding++; break; } }}
This kind of hints at the flexibility advantage of IOCP. The entire asynchronous pipeline is managed through a single place. Furthermore, the WinSock API directly supports IOCP, so you call WSASend() or whatever the function name is, and it will gladly operate on an overlapped socket in exactly the same way. You'll get notification of the network completion through GetQueuedCompletionStatus().
All of this is abstracted out for you in boost::asio, but boost::asio requires an o/s specific interface to be implemented. It provides the windows class that uses IOCP internally for both disk i/o and sockets, and it provides the linux implementation for sockets, but it doesn't provide any linux implementation for disk i/o, which would ultimately be a mapping of the boost required interface to the AIO api.
I implemented such a system at work for some high performance disk backup software. Using the IOCP approach was the only way I could achieve high enough performance that the actual physical disk became the bottleneck. I can now read/write literally as fast as the disk allows, which is surprisingly hard to achieve.
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Original post by Antheus
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Original post by cache_hitI took a quick look at your code. What you wrote is definitely better than a threaded approach, but you should really look into I/O completion ports, as it's significantly faster and more scalable.
Did you mean overlapped IO?
IOCP is a specific type of overlapped I/O that microsoft recommends for the most performance-intensive and demanding applications.
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hm, IOCPs certainly are nice, but again, in this case, I don't see any relevant gains to be had. Overhead due to WaitForMultipleObjects+ResetEvent is absolutely negligible for the few active requests. The code manages to max out an ioDrive SSD already (700 MB/s IIRC).
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Using the IOCP approach was the only way I could achieve high enough performance that the actual physical disk became the bottleneck. I can now read/write literally as fast as the disk allows, which is surprisingly hard to achieve.
I'm curious: were you seeing a bottleneck with overlapped completion mechanisms other than IOCP? Which one was it - events or callbacks?
The IOCP hype is probably coming from applications where you have tons of requests and multiple threads (web server), and definitely makes sense in that context.
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Original post by Jan Wassenberghm, IOCPs certainly are nice, but again, in this case, I don't see any relevant gains to be had. Overhead due to WaitForMultipleObjects+ResetEvent is absolutely negligible for the few active requests. The code manages to max out an ioDrive SSD already (700 MB/s IIRC).
What about mechanical disks with multiple platters and read/write heads? Ironically it's harder to max out performance on these due to the fact that ordering of read/writes is more important.
Also, you mentioned doing compression / decompression. In your application, which is the slowest among { reading, writing, compression } of a single block of data? Is compression happening asynchronously as well? In theory if compression is the bottleneck, then you should notice 0 effect by introducing a slower disk, and if the disk is the bottleneck you should notice 0 effect by removing compression. Is this the case for you? If so maybe there is little reason to use IOCP.
The main reason I like it is that it provides an arbitrary asynchronous computational model that just happens to be extremely easy to use for I/O both on the network and to/from a disk, while still being pretty easy to use for arbitrary asynchronous computation. The event model loses scalability when your pipeline grows and you're waiting on hundreds of handles simultaneously.
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Original post by Jan WassenbergThe IOCP hype is probably coming from applications where you have tons of requests and multiple threads (web server), and definitely makes sense in that context.
IOCP gives you a framework for overlapping work by a kernel-managed threadpool. The "hype" is unwarranted, it actually is a good way to do it, especially if you need to interleave read/process/write operations.
The basic idea is the generic task dispatcher over a thread pool, similar to Mac's GCD, but somewhat more consistent that linux aio (reasons listed above).
IOCP received considerable critique when it first appeared, but since then some details were improved, and multi-core became the norm. For its purpose and intent, it does the job.
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and if the disk is the bottleneck you should notice 0 effect by removing compression.
With slow media, removing compression decreases throughput. When full-disk compression first appeared, it often improved read/write performance for many disks.
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Original post by AntheusFor its purpose and intent, it does the job.
That's the understatement of the century :-)
IOCP are really one of the best implemented things in Windows. If they got one thing right, it's IOCP.
They are what epoll tries to be under Linux but embarrassingly fails. And, not only do IOCPs implement right what's wrong with epoll, they are fast, really fast, too. Posting a message onto an IOCP and retrieving it only takes about 2-3 times longer than doing the same thing on a lockfree queue. Except, well, you have none of the pain and limitation and all of the functionality, except it doesn't burn CPU cycles spinning, and except the IOCP will reduce context switches and keep caches warm by waking threads LIFO.
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What about mechanical disks with multiple platters and read/write heads? Ironically it's harder to max out performance on these due to the fact that ordering of read/writes is more important.
I think the performance results on mid-2008 hardware were 1016 MB/s on a 16 SAS disk array (limited by the interface) and ~700 MB/s for 8 disks. (For those tests, I had one core driving each disk and IIRC 8-deep reads of about 256 KB blocks.)
Unfortunately, those numbers don't help much because I don't know the disk specs offhand. An earlier version of this code using completion routines and only 2-deep reads maxed out the ST380021A (2003, 42 MB/s).
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Also, you mentioned doing compression / decompression. In your application, which is the slowest among { reading, writing, compression } of a single block of data? Is compression happening asynchronously as well? In theory if compression is the bottleneck, then you should notice 0 effect by introducing a slower disk, and if the disk is the bottleneck you should notice 0 effect by removing compression. Is this the case for you? If so maybe there is little reason to use IOCP.
Our case differs vs. your backup scenario - we either read or write, but not both, so compression reduces the amount of data to be transferred.
The basic scheme looks like this:
queue up N IOsrepeat wait until the next one is done queue up another IO [de]compress the [finished] blockuntil done
A test in 2006 showed that zlib was decompressing at 94 MB/s (far faster than IO), and therefore costs zero additional time because it is perfectly overlapped with the IO. However, since modern disks manage that kind of throughput and single-core performance hasn't kept up, it may be worthwhile to switch to faster LZ variants.
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The event model loses scalability when your pipeline grows and you're waiting on hundreds of handles simultaneously.
Yes, however, that is not the case in my usage.
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The "hype" is unwarranted, it actually is a good way to do it, especially if you need to interleave read/process/write operations.
I agree that IOCPs are fine and good, but there is definitely too much hype and too little understanding concerning their real advantages. In fact, cache_hit said it well:
Quote:
Well, if you search around the literature surrounding IOCP, you won't find a lot of details on exactly what makes it faster and more scalable than event-driven overlapped I/O.
I see two main benefits:
- avoiding oversubscription by capping thread count and integrating with scheduler;
- avoiding fairness issues and overhead due to round-robin polling.
Neither is the least bit applicable here: no worker threads are needed, and you actually WANT unfair treatment (i.e. the first pending handle to be serviced first - compressed streams must be processed in-order).
Since IOCP-based code would be a bit more complex, I conclude that using them here would be a net loss (but servers with thousands of sockets are an entirely different story).
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"They are what epoll tries to be under Linux but embarrassingly fails. "
I've never found anything particularly slow or troublesome with epoll, and integration into application loops has got even easier now with the addition of eventfd and signalfd assisters; what is it that IOCP (which I've not yet looked at) has over epoll?
{Apart from the obvious of which OS supports which thing :-}
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Quote:
Original post by Katiewhat is it that IOCP (which I've not yet looked at) has over epoll?
Ow... :( I only saw your post today, sorry for the late answer.
The most important thing that epoll does not have, and which makes it suck in my opinion is that it is single-threaded. Now, before you start "but of course you can...", remember that if you use one epoll with a group of threads and a readiness change happens, epoll will wake up every single one of them. The first one will get hold of the file descriptor, and the rest will be put on the wait queue again. Of course it is still possible to construct something that makes use of multiple cores, but if you do it the simple, obvious way, you get the same thundering herd which you were using epoll for in the first place -- you could as well just fire up a bunch of threads and have them all block on read() or recvfrom(), no real difference.
IOCP on the other hand, can take any number of threads and will wake up exactly one of them as an event arrives, and it will wake the most recently run thread first, which is a good thing. It also lets you specify the maximum number of threads from your pool that you want to be active at one time (this is only an approximate figure, you will occasionally get one more thread running, but it nevertheless works very well in practice). It's lightning fast, too.
Add to that some minor details like AIO + eventfd being entirely undocumented or the generally bad documentation of nearly every Linux-specific feature. Sadly, documentation is the one thing Linux always loses big time.
Sure enough, you can usually find everything in the headers somewhere if you keep searching long enough, and you can always dig through kernel sources... I'm sure some people see this as adaequate, but except for the coolness factor of being a geek, digging through the kernel sources in lack of documentation doesn't add much value. Well, not to me, anyway.
Also, you sometimes find that the available documentation has been lying to you after looking at the sources, which doesn't really make things better (take msync as an example).
Lastly, IOCP works pretty much identically on the most recent version and on 10 year old Windows, which is pretty cool.
The same cannot be said about epoll + eventfd + signalfd + timerfd (for example timerfd will not work with the still quite common Ubuntu 8.04LTS distro).
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2018-01-22 17:04:10
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https://brilliant.org/problems/primes-oh-primes/
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# Primes Oh Primes
Level pending
Find the largest integer less than $$1000$$ that can be expressed as the sum of a prime number and its factors.
×
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2017-05-23 12:46:10
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http://mathematica.stackexchange.com/tags/calculus-and-analysis/hot
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# Tag Info
16
I would approach this from the fact that both are forms of multiplication, but one has a negative exponent. So RandomReal[{1, 20}]^RandomChoice[{1, -1}] will randomly be either 1/x or x, where x is a random number between 1 and 20.
6
The solution is a straightforward application of Integrate. Integrate[Exp[-((x - x0)^2 + (y - y0)^2)/(2 c) - I (kx x + ky y)], {x, -Infinity, Infinity}, {y, -Infinity, Infinity}, Assumptions -> c > 0] (* 2 c E^(-(1/2) c (kx^2 + ky^2) - I (kx x0 + ky y0)) π *)
5
What about computing a general $t$-bound integral: expr = Integrate[Exp[-2 ((x + y)/2 - b)^2/(2*a)], {x, -t, t}, {y, -t, t}]; And then expanding in series around $t=\infty$: Series[expr, {t, Infinity, 3}] // Normal // PowerExpand // FullSimplify $\frac{a^2 \left(e^{-\frac{(b-t)^2}{a}}+e^{-\frac{(b+t)^2}{a}}\right)}{t^2}-4 a e^{-\frac{b^2}{a}}-4 ... 5 Examine your integrand (which is suggested by the error, after all). PiecewiseExpand will collect all terms under one piecewise function. c*h[c, k1, t1]*(1 - H[c, k2, t2])*(1 - H[c, k3, t3]) // PiecewiseExpand (* Power::infy, Infinity::indet errors... *) You can see that the function does not have numeric values for c > 1. How to fix it is ... 5 Depending on the distribution desired, you could use the log-normal (or a similar transformation of whatever distribution has a mean of 0). It is transformed distribution such that a value -y < 0 of the underlying distribution is transformed to 1/x iff the value y > 0 is transformed to x (i.e., Exp[-y] == 1/x where x = Exp[y]). The "underlying" ... 5 linearizeEquation[expr_, f_, fp_, order_] := Block[{e}, Expand@Normal@Series[expr /. f -> (fp + e dF[#] &), {e, 0, order}] /. e -> 1 ] linearizeEquation[f[x] + (1 - f[x]^2) + D[f[x], {x, 2}] + f[x] D[f[x], x], f, f0, 1] (* 1 + f0 - f0^2 + dF[x] - 2 f0 dF[x] + f0 dF'[x] + dF''[x] *) 4 One way: RandomChoice[{Times[x, #] &, Divide[x, #] &}][RandomReal[{1, 20}]] To repeat, use a Table or Do expression, etc. 4 This was done in V10.3.1. Integrate[(-2 a Cos[π/24] Gamma[11/12] HypergeometricPFQ[{11/24, 23/24}, {2/3, 4/3}, (4 a^3)/27] + 8 Gamma[5/4] HypergeometricPFQ[{1/8, 5/8}, {1/3, 2/3}, (4 a^3)/27] Sin[π/8] + a^2 Gamma[19/12] HypergeometricPFQ[{19/24, 31/24}, {4/3, 5/3}, (4 a^3)/27] Sin[(5 π)/24]), {a, 0, ∞}] (864/665) Sqrt[2 (4 + ... 4 The problem occurs because of apparently complex form of the expression beyond the integral of Sin[s^k]. Nevertheless it is not too harmful to proceed. Adequate limits are real and we don't need playing to simplify complex expressions to explicitly real forms. We can calculate the both integrals with appropriate assumptions: f[x_, k_] = Integrate[{ ... 4 I would suggest adding option GenerateConditions->False to Integrate to speed up the integration. Then, instead of D, use Derivative. Then, to generate a SeriesData apply Series: f[x_] := 1/x; max = 4; em[n_Symbol] := Series[Integrate[f[x], {x, 1, n}, GenerateConditions -> False] + (f[1] + f[n])/2 + Sum[BernoulliB[2 k]/(2 k)! (Derivative[2 ... 4 Perhaps useful I think this substitution provides a preferable form for the numerical integration: exp = ((w E^(-w/a) Sin[((w - s)^2)/2])/((w - s)^2)/2 /. w -> -a Log[g]) D[-a Log[g], g] f[b_?NumericQ] := Block[{a = 10^-6, s = -10^(-1) Sqrt[10^(-3)^2 + b^2]}, - a^2 NIntegrate[exp/a^2, {g, 0, 1}, WorkingPrecision -> 30, ... 4 Evaluating the integral analytically seems to take far too long on my machine, so I aborted it. Since you mention that you studied the integrated expression, perhaps you could consider showing result you got from the integration in your question. Having said that, numerical integration suggests that the values of the integral are very small over a wide ... 4 Get the MNIST digit recognition data set (70,000 hand-drawn digits with classifications): totalSet = ExampleData[{"MachineLearning", "MNIST"}, "Data"]; Divide it into training set, validation set (used to find optimum values for hyperparameters, such as regularization constants) and test set (which is not used in building the classifier at all, but which ... 4 Integrate[Exp[-s x] Cosh[a x], {x, 0, ∞}, Assumptions -> s > Abs[a] && a ∈ Reals]$\frac{s}{s^2-a^2}$Note that this is the Laplace transform: LaplaceTransform[Cosh[a x], x, s]$\frac{s}{s^2-a^2}$Your problem was that you were trying to integrate a "function" that included the logical expression && when that was actually a ... 3 You want to hunt down the error? Here is the best piece of advice: don't plot a function until you know it works. Okay, that's out of the way, now let's go through the process of finding out why your code gives an error. First we can look at just one integral, Λ = 10^-6; Δ = 10^-3; θ = 1/2 ArcTan[Δ/δ]; h = 10^-1; t = 10^3; s = -h Sqrt[Δ^2 + δ^2] ... 3 I think you will have to ask a mathematician if the limit is really 0 (or even real) since in Mathematica you can get this s[x_] := Sqrt[ Pi^2/12 + Sum[(x!*x!)/(k!*(2 x - k)!)*(-1)^(x - k)/(x - k), {k, 0, x - 1}]] s[x] // FullSimplify Limit[s[x], x -> Infinity] // FullSimplify So the limit seems to be complex, but I am not sure, I am not a ... 3 I'm a little bit late to this party, but I had written this function for another question that turned out not to need it, so I'll put this here. My strategy is to straightforwardly calculate the integral via$\begin{align} \int f(\vec x) &\, \exp\left( - \frac 1 2 \sum_{i,j=1}^{n}A_{ij} x_i x_j \right) d^nx = \\ & \sqrt{(2\pi)^n\over \det A} \, ... 2 As @bbgodfrey commented, if the integrals in the equation in the OP's FindRoot command can be evaluated before passing the equation to FindRoot, one can save a lot of time. It seems there is still more to be done. I found FindRoot struggles to find an accurate root in some areas of the domain of the equation. It turns out one can use Solve to solve the ... 2 Λ = 10; Ω = 10; k[t_] := k[t] = NIntegrate[ω E^(-ω/Λ) Sin[(ω + Ω) t/2]^2/(ω + Ω)^2, {ω, 0, ∞}] Plot[k@t, {t, 0.01, 1}, MaxRecursion -> 1, PlotPoints -> 20] 2 You can accomplish something very similar to your pseudo-code by defining a function: rand := RandomChoice[{Times, Divide}] Now every time you call the rand function, it either multiplies or divides its two arguments. For example, rand[3, 4] returns 12 half the time and 3/4 the other half. Now you can replace the "4" with a randomly chosen number and ... 2 A "nice" result is confirmed, but it is not the one hoped for in the OP but that of george2079 and others here. An analysis is made on the basis of the fundamental theorem of calculus. Here no error messages appear but the slight uncertainty is now shifted to the hypothesis of continuity of the antiderivative. This in turn seems pretty obvious from plotting ... 2 Observing strictly that the domain of x as the upper limit of the summation index is the integers, the limit exists, it can be calculated easily with Mathematica and it is different from zero. We need to consider this sum \[Sigma]WH[x_] := Sqrt[\[Pi]^2/12 + Simplify[Sum[(x!*x!)/(k!*(2 x - k)!)*(-1)^(x - k)/(x - k), {k, 0, x - 1}], x \[Element] ... 2 In your inputs a = 10^-6; b = 10^-3; c = 1; d = 0.1; s = -d Sqrt[b^2 + c^2] -0.1 this result is approximated for display. You can see the complete result by placing your cursor in front of the -0.1 and pressing the space bar. Alternatively InputForm[s] -0.1000000499999875 Edit With s = -d Sqrt[b^2 + c^2] the integral calculation yields ... 2 The copyable code you omitted: res = Integrate[(1 - Exp[h *(s - T)])^4/(k2 - k1* Exp[(s - T)*(2 h)])^2, s, Assumptions -> Element[k1 | k2, Reals] && k1 < 0 && k2 < 0] You are surprised to see a term\sqrt{k1}\$ pop up in the answer, with k1 defined as being negative. You don't specify why you see that as a problem, so I have ...
2
Mathematica knows how to simplify when a is exactly 2*Pi: (-b*Cos[a*b] Sin[a/2] + Sin[a*b] Cos[a/2])/(b^2 - 1) /. a -> 2*Pi // InputForm -(Sin[2*b*Pi]/(-1 + b^2)) It then applies the numerical limit for b=1. For the approximate number 2.*Pi, Mathematica can't make this simplification, and it turns out the limit is +Infinity for a<2*Pi and ...
2
Some of the definitions in the original question were problematic. I edited the question to have more consistent code. In order to plot the function numerical values are needed, so using Integrate is not necessary. We can use NIntegrate instead. The plot is produced within 30 seconds on my laptop with Mathematica 10.3.1. Here is the function redefined: ...
2
This is more a long comment than an answer. If you calculate: D[x Hypergeometric2F1[1/2, 2/3, 3/2, x^2], x] FullSimplify@D[x/Sqrt[1 - x^2], x] You find respectively: 1/(1 - x^2)^(2/3) and 1/(1 - x^2)^(3/2) Mathematica gives the correct answer: Check the exponents!!
2
If you change the variables: y1=x1+x2; y2=x2 you get to another expression: where all the integration limits are +/- infinities. The first integral is just equal to infinity, while the second is Integrate[Exp[-(y1 - 2 b)^2/(4 a)], {y1, -\[Infinity], \[Infinity]}, Assumptions -> {a > 0, b > 0}] (* 2 Sqrt[a] Sqrt[\[Pi]] *) I hope this ...
1
If numerical results are acceptable; Needs["NumericalCalculus`"] func[θ1_,n_]:= 1/2 ((n Cos[θ1] - 0.2 Sqrt[1 - 25. n^2 Sin[θ1]^2])^2/(n Cos[θ1] + 0.2 Sqrt[1 - 25. n^2 Sin[θ1]^2])^2 + (-0.2 Cos[θ1] + n Sqrt[1 - 25. n^2 Sin[θ1]^2])^2/(0.2 Cos[θ1] + n Sqrt[1 - 25. n^2 Sin[θ1]^2])^2) int[n_?NumberQ] := int[n] = NIntegrate[func[\[Theta]1, n], ...
1
This is what I get on OS X 10.3.1 (64bit)
Only top voted, non community-wiki answers of a minimum length are eligible
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2016-02-11 23:32:20
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https://www.nature.com/articles/s41467-020-18627-x?error=cookies_not_supported&code=8f146b2d-cae5-4fc7-9bb5-90c73c5f4054
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## Introduction
The polaronic effect1, which describes the strong coupling between charge and lattice vibrations, has a key role in a broad class of novel quantum phenomena ranging from colossal magnetoresistance2 to anomalous photovoltaic effect3. In particular, the polaronic effect on excitons can profoundly modulate exciton dynamics upon photoexcitation and has been employed to describe intriguing optical and optoelectronic properties in materials such as hybrid organic-inorganic perovskite solar cells4,5,6. Compared with three-dimensional (3D) bulk systems, two-dimensional (2D) atomic crystals possess a couple of unique advantages in exploring the polaronic effect on exciton dynamics. First, the reduced dielectric screening in atomically thin samples enhances both the excitonic effect7 and the electron–phonon (e-ph) coupling8, which is expected to promote the polaronic effect of excitons. Second, unlike bulk materials in which the e-ph coupling is largely determined by intrinsic electronic and phonon band structures with limited tunability, 2D materials provide greater flexibility for engineering e-ph coupling through a number of approaches including carrier doping9,10 and interfacial coupling11,12,13,14,15, as well as dimensionality modulation8, and therefore hold high promise for the future development of optoelectronic devices.
The realization of a long-range magnetic order in 2D semiconducting CrI3 paves the way to engineer optical and optoelectronic properties of 2D magnetic semiconductors16,17,18,19,20,21,22,23. The large excitonic effect24 from the localized molecular orbitals, of neither Wannier-type in 2D TMDCs nor Frenkel-type in ionic crystals, can be considered as the microscopic origin of the giant magneto-optical Kerr effect25 and magnetic circular dichroism26 signals in 2D CrI3. Meanwhile, the strong e-ph coupling is suggested to cause the large Stokes shift, profound broadness, and skewed lineshape in the photoluminescence (PL) spectra of 2D CrI326. The coexistence of excitons and strong e-ph coupling in 2D CrI3 naturally leads to open experimental questions of whether polaronic character emerges in the exciton dynamics and whether they are affected by the long-range magnetic order.
One fingerprint for the polaronic effect is the development of phonon-dressed electronic bands that appear as satellite bands in proximity to the original undressed one. Such features manifest as multiple equally spaced replica bands in angle-resolved photoemission spectroscopy (ARPES)9,14,27,28,29,30,31,32 or as discrete absorption and emission lines in linear optical spectroscopy33. However, such signatures of the polaronic effect have not been revealed so far in 2D CrI3, as the sizable bandgap (~1.1 eV)26 and extreme surface sensitivity34 of CrI3 make ARPES measurements challenging, whereas the potential inhomogeneous broadening could largely smear out individual lines for phonon-dressed satellite bands in linear optical spectroscopy.
In this work, we exploit temperature and magnetic field-dependent resonant micro-Raman spectroscopy, to show the direct observation of the polaronic character of excitons in bilayer CrI3. The polaronic effect manifests in Raman spectra as a well-defined, periodic pattern of broad modes that is distinct from sharper phonon peaks. The profile of this periodic pattern and its temperature and magnetic field dependence further reveal essential information including the e-ph coupling strength and the tunability of polaronic effect by the magnetism in bilayer CrI3. We mainly focus on bilayer CrI3 because it features a single magnetic phase transition from the layered antiferromagnetic (AFM) to ferromagnetic (FM) order and briefly compare to the results on thicker CrI3 flakes afterwards.
## Results
### Excitonic transitions and strong electron–phonon coupling
We start by identifying excitonic transitions and e-ph coupling in bilayer CrI3 using temperature-dependent PL and linear absorption spectroscopy. Bilayer CrI3 was fully encapsulated between few-layer hexagonal BN (hBN) and placed on a sapphire substrate (for details, see “Methods”). Linear absorption spectroscopy measurements were then performed in a transmission geometry (see “Methods”). Figure 1 shows representative PL and absorbance spectra taken at 80 K, 40 K, and 10 K that correspond to well above, slightly below, and well below the magnetic critical temperature TC = 45 K, respectively25,26,34. A single PL mode at 1.11 eV and three prominent absorbance peaks at 1.51 eV, 1.96 eV, and 2.68 eV (denoted as A, B, and C, respectively) are observed across the entire temperature range. These three energies are in good agreement with the ligand-field electronic transitions assigned by differential reflectance measurements on monolayer CrI326 and bulk CrI335,36 and have been later revealed to be bright exciton states through sophisticated first principle GW and Bethe-Salpeter equation calculations24. The large Stokes shift (~400 meV) between the PL and A exciton absorption peak is consistent with previous report6 and indicates strong electron–phonon coupling in 2D CrI3. Although the absorbance spectra show little temperature dependence except for the appearance of a weak shoulder at 1.79 eV at 10 K (orange arrow), the PL spectra are clearly temperature dependent. In particular, the temperature dependence of the PL full width at half maximum, $${\Gamma}(T)$$, is well fitted by the model functional form, $${\Gamma}\left( T \right) = {\Gamma}_0 + \frac{\gamma }{{\exp \left( {\frac{{\hbar \omega _{{\mathrm{LO}}}}}{{k_BT}}} \right) - 1}}$$, with the first term for temperature-independent inhomogeneous broadening and the second term for homogeneous broadening from the exciton coupling with a longitudinal optical (LO) phonon at frequency $$\omega _{{\mathrm{LO}}}$$. Taking $$\omega _{{\mathrm{LO}}} = 120.6$$ cm−1 found later on in Fig. 2, we obtain $${\Gamma}_0 = 163.9 \pm 2.7$$ meV and $$\gamma = 164.2 \pm 8.1$$ meV, which suggests that the broadness of the exciton modes arises from both inhomogeneous broadening from disorders and homogeneous broadening from e-ph coupling. The large homogeneous broadening parameter ($$\gamma$$) indicates strong vibronic modes mixing in the PL spectra, which precludes the formation of well-resolved phonon sidebands6.
### Polaronic character in Raman spectra
We next proceed to perform resonant micro-Raman spectroscopy measurements with an incident wavelength of 633 nm matching the energy of the B exciton on an encapsulated bilayer CrI3 flake placed on a SiO2/Si substrate (see “Methods”). Figure 2a displays a representative Raman spectrum acquired in the crossed linear polarization channel at 40 K (slightly below TC = 45 K). Note that this spectrum covers a much wider frequency range than earlier Raman studies on CrI334,37,38,39,40,41,42,43,44. The multiphonon scattering is visible up to the 3rd order, and their zoom-in Raman spectra are shown in the inset of Fig. 2a. The 1st-order single-phonon peaks appear in the relatively low frequency range of 50–150 cm−1, and are assigned to be of either Ag or Eg symmetries under the C3i point group (see Supplementary Note 1), which is consistent with earlier work34,37,38,39,40,41,42,43,44 and proves the high quality of our samples. The 2nd-order two-phonon and the 3rd-order three-phonon modes show up in slightly higher frequency ranges of 190–290 cm−1 and 310–410 cm−1, respectively, and show decreasing mode intensities at higher-order processes, same as typical multiphonon overtones under harmonic approximation45 or cascade model46. In addition to and distinct from these multiphonon features, we resolve a remarkable periodic modulation across a wide frequency range of 70–1100 cm−1 in the low intensity part of the Raman spectrum (highlighted by the orange shaded area in Fig. 2a). This low intensity periodic pattern consists of clean, individual Lorentzian profiles and survives up to the 8th order (Fig. 2b), well beyond the highest order (3rd order) of multiphonon overtones, and each order of it spans for ~50 cm−1 frequency range, much wider than the linewidth of any observed phonon modes (insets of Fig. 2a for phonons). Such a periodic pattern is also observed in the anti-Stoke’s side at higher temperatures in bilayer CrI3, for example, up to the 2nd order at 290 K (see Supplementary Note 2), which clearly supports its Raman origin instead of luminescence.
We fit this low intensity periodic pattern using a summation of Lorentzian profiles of the form $$\mathop {\sum }\nolimits_N \frac{{A_N\left( {\frac{{{\Gamma}_N}}{2}} \right)^2}}{{\left( {\omega - \omega _N} \right)^2 + \left( {\frac{{{\Gamma}_N}}{2}} \right)^2}} + C$$ with central frequency $$\omega _N$$, linewidth $${\Gamma}_N$$, and peak intensity $$A_N$$ of the Nth period and a constant background $$C$$ (see fitting procedure in “Methods”). Among all eight orders ($$N = 1,\,2,\, \ldots ,8.$$) in Fig. 2b, the presence of the 1st-order broad mode is deliberately validated in Fig. 2c that fitting with this 1st-order broad mode (orange shaded broad peak in the bottom panel) is visibly better than without it (top panel). This improved fitting by involving the 1st-order broad mode is further rigorously confirmed by the bootstrap method47 (see Supplementary Note 3). Figure 2d shows a plot of the central frequency $${\upomega}_N$$ as a function of the order N with data taken at 40 K ($$N = 1,\,2,\, \ldots ,\,8$$) and 290 K ($$N = - 2,\, - 1,\, \ldots ,\,3$$), from which a linear regression fit gives a periodicity of $$120.6 \pm 0.9$$ cm−1 and an interception of 0 ± 0.2 cm−1. To the best of our knowledge, such a periodic pattern made of individual Lorentzian profiles previously has only been seen in multiphonon Raman spectra of Cd, Yb, and Eu monochalcogenides described by configuration-coordinate model48,49,50,51,52,53,54,55. However, the periodic pattern observed in bilayer CrI3 here differs from these monochalcogenide multiphonon modes, as the broad linewidth of 1st-order mode contradicts with the sharp 1st-order forbidden LO phonon in Cd and Yb monochalcogenides48,49,50,51 and the persistence (or even enhancement) of higher-order multiphonon below TC = 45 K is in stark contrast to the disappearance of paramagnetic spin disorder-induced multiphonon below magnetic phase transitions in Eu monochalcogenides52,53,54,55. Because no known multiphonon model can capture all characteristics of our observed periodic pattern as well as the broad linewidths of each mode, we are inspired to consider the electronic origin. Indeed, strikingly similar features have been seen in polaron systems through the energy dispersion curves (EDCs) of ARPES9,14,27,28,29,30,31,32 and linear absorption and PL spectroscopy5,33,56. In those cases, the periodic patterns in their energy spectra arise from the phonon-dressed electronic state replicas, or sometimes also referred as phonon-Floquet states57, and the periodicity is given by the frequency of the coupled phonon. Owing to the high resemblance between the lineshapes of our Raman spectrum and those polaron energy spectra5,14,27,28,29,30,31,32,56, we propose that this periodic pattern in Raman spectra of 2D CrI3 stems from inelastic light scattering between the phonon-dressed electronic states caused by the polaronic character of B excitons in 2D CrI3, whereby the B exciton at 1.96 eV, with the electron(hole) in the weakly dispersive conduction (highly dispersive valence) band of Cr 3d (I 5p) orbital character24,58, couples strongly to a phonon at 120.6 cm−159. It is worth noting that a recent theoretical work predicts magnetic polaronic states in 2D CrI3 because of charge-magnetism coupling60, whereas our work suggests polaronic exciton states due to charge-lattice coupling.
We then proceed to identify the source and the character of the phonon at 120.6 cm−1. We first rule out the possibility of this phonon arising from either the hBN encapsulation layers or the SiO2/Si substrate, as a similar periodic pattern in the Raman spectrum is also observed in bare bulk CrI3 crystals (see Supplementary Note 4). Compared with the calculated phonon band dispersion of monolayer CrI361, we then propose the LO phonon calculated to be at ~115 cm−1 as a promising candidate, whose slight energy difference from the experimental value of 120.6 cm−1 could result from the omission of e-ph coupling in calculations. This LO phonon mode belongs to the parity-odd Eu symmetry of the C3i point group, and its atomic displacement field transforms like an in-plane electronic field (Ex, Ey) (see inset of Fig. 2d)61. Its odd parity makes it Raman-inactive and absent in the 1st-order phonon spectra (Fig. 2a inset, top panel), whereas its polar displacement field allows for its strong coupling to electrons/holes and prompts the polaronic character of the charge-transfer B exciton (see Supplementary Note 5 for measurements with additional laser wavelengths). In addition, this LO phonon band is nearly dispersionless and has a large density of states, further increasing its potential for coupling with the B exciton in 2D CrI3.
### Temperature dependence of the polaronic effect
Given the coexistence of a 2D long-range ferromagnetic order and polaronic effect of excitons below TC = 45 K in bilayer CrI3, it is natural to explore the interplay between the two. For this, we have performed careful temperature-dependent Raman spectroscopy measurements and fitted the periodic pattern in every spectrum with a sum of Lorentzian profiles. Figure 3a displays the periodic pattern in Raman spectra taken at 70 K and 10 K, well above and below TC, respectively. Comparing these spectra, not only do more high-order replica bands become visible at lower temperatures (i.e., from $$N = 6$$ at 70 K to $$N = 8$$ at 10 K), but also the spectral weight shifts toward the higher-order bands (i.e., from $$N = 1$$ at 70 K for the strongest mode to between $$N =$$ 3 and 4 at 10 K in Fig. 3b). The appearance of higher-order modes at lower temperatures possibly results from a combination of the narrow exciton linewidth (~50 cm−1) and the dispersionless nature of coupled LO phonon. More importantly, the spectral weight distribution (AN vs. N) quantifies the e-ph coupling strength, and its spectral shift across TC confirms the interplay between the polaronic effect and the magnetic order in bilayer CrI3. Theoretically, the polaron system consisting of dispersionless LO phonons and charges is one of the few exactly solvable models in many-body physics62, and the calculated polaron spectra can be well-described by a Poisson distribution function, $$A_N = A_0\frac{{e^{ - \alpha }\alpha ^N}}{{N!}}$$63,64, where $$A_0$$ is the peak intensity of the original electronic band, $$A_N$$ is the peak intensity for the Nth replica band with $$N$$ phonon(s) dressed, and $$\alpha$$ is a constant related to the e-ph coupling in 3D (i.e., $$\alpha _{3{\mathrm{D}}}$$) that can be scaled by a factor of $$3{\uppi}/4$$ for 2D (i.e., $$\alpha _{2{\mathrm{D}}}$$)65. By fitting the extracted Lorentzian peak intensity profile at every temperature to the Poisson distribution function (see fits of 10 K and 70 K data in Fig. 3b), we achieve a comparable fitting quality to that for ARPES EDCs in polaron systems9,31 at every temperature and eventually arrive at the temperature dependence of $$\alpha _{2{\mathrm{D}}}$$, which remains nearly constant until the system is cooled to TC and then increases by almost 50% at the lowest available temperature 10 K of our setup (Fig. 3c). In addition to the anomalous enhancement of $$\alpha _{2{\mathrm{D}}}$$ across TC, the value of $$\alpha _{2{\mathrm{D}}} = 1.5$$ at 10 K is the highest among known 2D polaron systems including graphene/BN heterostructures ($$\alpha _{2{\mathrm{D}}} = 0.9$$)14 and bare SrTiO3 surfaces ($$\alpha _{2{\mathrm{D}}} = 1.1$$)29.
### Magnetic field dependence of the polaronic effect
It has been shown that bilayer CrI3 transitions from a layered AFM to FM with increasing out-of-plane magnetic field ($$B_ \bot$$) above the critical value BC of 0.7 T17,21,25,26. We then finally explore the evolution of the polaronic effect across this magnetic phase transition by performing magnetic field-dependent Raman spectroscopy measurements. Here, we choose circularly polarized light to perform magnetic field-dependent measurements in order to eliminate any Faraday effect from the optical components that are situated in close proximity to the strong magnetic field. Figure 4a shows Raman spectra taken at $$B_ \bot$$ = 0 T and $$\pm{\!}$$1 T, below and above BC, respectively, in both RR and LL channels, where RR(LL) stands for the polarization channel selecting the right-handed (left-handed) circular polarization for both incident and scattered light (see Supplementary Note 6). At 0 T, the spectra are identical in the RR and LL channels, consistent with zero net magnetization in the layered AFM state for bilayer CrI3 at $$\left| {B_ \bot } \right| {\,}< {\,}B_{\mathrm{C}}$$. At $$\pm{\!}$$1 T, the spectra in the RR and LL channels show opposite relative intensities under opposite magnetic field directions, owing to the fact that the net magnetization in the FM state for bilayer CrI3 at $$\left| {B_ \bot } \right| {\,}> {\,}B_{\mathrm{C}}$$ breaks the equivalence between the RR and LL channels. To better quantify the magnetic field dependence of the spectra, we measured Raman spectra in the RR and LL channels at $$B_ \bot$$ from $$-$$1.4 T to 1.4 T every 0.1 T. We fit the spectrum at every magnetic field to extract $$A_N$$ first and then $$A_0$$ and $$\alpha _{2{\mathrm{D}}}$$. Figure 4b shows that $$A_0$$ has abrupt changes at $$B_ \bot = \pm$$0.7 T in both RR and LL channels, consistent with the first order magnetic phase transition at BC. Furthermore, the magnetic field dependence of $$A_0$$ shows an opposite trend in the RR channel from that in the LL channel, whereas the sum of $$A_0$$ from both channels remain nearly constant to the varying magnetic field. This observation can be understood by that, under a time-reversal operation, the RR channel transforms into the LL channel and the direction of the net magnetization at $$\left| {B_ \bot } \right| {\,}> {\,}B_{\mathrm{C}}$$ flips, resulting in that the Raman spectrum in the RR channel at $$B_ \bot {\,}> {\,}0.7$$ T is equivalent to the spectrum in the LL channel at $$B_ \bot {\,}< -{\!}0.7$$ T. Figure 4c shows that $$\alpha _{2{\mathrm{D}}}$$ is magnetic field independent, suggesting that the interlayer magnetic order barely affects the e-ph coupling strength and that the in-plane long-range magnetic order is responsible for the strong enhancement of e-ph coupling at TC. This finding corroborates with the fact that the 120.6 cm−1 phonon has in-plane atomic displacement.
## Discussions
Our further Raman spectroscopy studies on tri-layer, four-layer, and five-layer CrI3 show qualitatively same findings as those in bilayer CrI3 (see Supplementary Note 7) and again echoes with the in-plane nature of the 120.6 cm−1 Eu phonon and the intralayer charge-transfer B exciton. Our data and analysis reveal the phonon-dressed electronic states and suggest the polaronic character of excitons in 2D CrI3, which arises from the strong coupling between the lattice and charge degrees of freedom and is dramatically modified by the spin degree of freedom of CrI3. The exceptionally high number of phonon-dressed electronic state replicas (up to $$N = 8$$) further suggests 2D CrI3 as an outstanding platform to explore nontrivial phases out of phonon-Floquet engineering, whereas the significant coupling to the spin degree of freedom adds an extra flavor whose impact on the phonon-Floquet states has not been studied. For example, one can imagine creating topological states through the band inversion between the phonon-dressed replicas of CrI3 and the electronic state of a material in close proximity.
## Methods
### Sample fabrication
CrI3 single crystals were grown by the chemical vapor transport method, as detailed in ref. 40. Bilayer CrI3 samples were exfoliated in a nitrogen-filled glove box. Using a polymer-stamping transfer technique inside the glove box, bilayer and few-layer CrI3 flakes were sandwiched between two few-layer hBN flakes and transferred onto SiO2/Si substrates and sapphire substrates for Raman spectroscopy and PL/linear absorption spectroscopy measurements, respectively.
### Linear absorption spectroscopy
A bilayer CrI3 sample on a sapphire substrate was mounted in a closed-cycle cryostat for the temperature-dependent absorption spectroscopy measurements. A broadband tungsten lamp was focused onto the sample via a 50× long working distance objective. The transmitted light was collected by another objective and coupled to a spectrometer with a spectral resolution of 0.2 nm. The absorption spectra were determined by $$1 - \frac{{I_{{\mathrm{sample}}}(\lambda )}}{{I_{{\mathrm{substrate}}}(\lambda )}}$$, where $$I_{{\mathrm{sample}}}(\lambda )$$ and $$I_{{\mathrm{substrate}}}(\lambda )$$ were the transmitted intensity through the combination of sample and substrate and through the bare substrate, respectively.
### PL spectroscopy
PL spectra were acquired from the same bilayer CrI3 sample where we carried out linear absorption measurements. The sample was excited by a linearly polarized 633 nm laser focused to a ~2 μm spot. A power of 30 μW was used, which corresponds to a similar fluence reported in the literature26 (10 µW over a 1 μm-diameter spot). Transmitted right-handed circularly polarized PL signal was dispersed by a 600 grooves/mm, 750 nm blaze grating, and detected by an InGaAs camera.
### Raman spectroscopy
Resonant micro-Raman spectroscopy measurements were carried out using a 633 nm excitation laser for the data in the main text and 473 nm, 532 nm, and 785 nm excitation lasers for data in Supplementary Note 5. The incident beam was focused by a 40× objective down to ~3 μm in diameter at the sample site, and the power was kept at ~ 80 μW. The scattered light was collected by the objective in a backscattering geometry, then dispersed by a Horiba LabRAM HR Evolution Raman spectrometer, and finally detected by a thermoelectric cooled CCD camera. A closed-cycle helium cryostat is interfaced with the micro-Raman system for the temperature-dependent measurements. All thermal cycles were performed at a base pressure lower than 7 × 10−7 mbar. In addition, a cryogen-free magnet is integrated with the low temperature cryostat for the magnetic field-dependent measurements. In this experiment, the magnetic field was applied along the out-of-plane direction and covered a range of $$- 1.4$$ to $$+$$1.4 Tesla. In order to avoid the Faraday rotation of linearly polarized light as it transmits through the objective under the stray magnetic field, we used circularly polarized light to perform the magnetic field-dependent Raman measurements.
### Fitting procedure
For every sample, we have taken temperature and magnetic field-dependent Raman spectra on the hBN/SiO2/Si substrate with the same experimental conditions as that on the CrI3 flakes. The Raman spectra from the substrate, an extremely gradual background with a Si phonon peak at ~525 cm−1, shows no dependence on temperature (over the range of 10–70 K) and magnetic field (0–2.2 T). To fit the periodic oscillations in Raman spectra of CrI3 flakes, we follow the procedure described below. (I) we fit the Si phonon peak at ~525 cm−1 in both spectra taken on the CrI3 thin flake and the bare substrate to extract the Si peak intensity, $$I_{{\mathrm{Si}}}^{{\mathrm{sample}}}$$ and $$I_{{\mathrm{Si}}}^{{\mathrm{substrate}}}$$. (II) we multiply the background spectrum by a factor of $$\frac{{I_{{\mathrm{Si}}}^{{\mathrm{sample}}}}}{{I_{{\mathrm{Si}}}^{{\mathrm{substrate}}}}}$$, which is ~1, and then subtract off the factored background from the raw Raman spectrum of sample. This process leads to the pure Raman signal for CrI3 whose baselines are nearly identical over the temperature range of interest (10–70 K). (III) we fit the sharp CrI3 phonon peaks with Lorentzian functions and subtract their fitted functions from the background free Raman spectrum from step (II). This leads to a clean spectrum with only periodic broad modes for a global fitting. (IV) we fit the clean spectrum from step (III) with a sum of multiple Lorentzian functions, $$\mathop {\sum }\nolimits_N \frac{{A_N\left( {\frac{{{\Gamma}_N}}{2}} \right)^2}}{{\left( {\omega - \omega _N} \right)^2 + \left( {\frac{{{\Gamma}_N}}{2}} \right)^2}} + C$$. For the neatness of the data presentation in Figs. 2 and 3, we only show the fitted line from step IV in the plots.
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2022-12-08 17:51:43
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https://math.stackexchange.com/questions/1052175/understanding-central-limit-theorem
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Understanding central limit theorem
I am not understanding the central limit theorem.
From wikipedia:
...suppose that a sample is obtained containing a large number of observations, each observation being randomly generated in a way that does not depend on the values of the other observations, and that the arithmetic average of the observed values is computed. If this procedure is performed many times, the central limit theorem says that the computed values of the average will be distributed according to the normal distribution
what I'm confused about is...if we have a sample of n observed values, then the average of the population will be the sum of all the observed values divided by the total number of observed values. So we will have an average....THE average, meaning ONLY one average, so how can ONE value have a "distribution"? Obviously I'm missing something or interpreting what the definition is saying wrongly, so can somebody help me out?
Edit: Should I think of this as like...let's say we have 1 value. It will have an average. Then we have another value, and take the average of the two values. Then a third value, and find the average of the three. Eventually as you get larger and larger numbers, the "distribution" of all these separate averages will be normal, with the average value eventually equaling the expected value mu?
• The key is if this procedure is performed many times. That is, run a bunch of experiments, each with $n$ observations and hence each with their own average. Then look at the distribution of those averages. – aes Dec 4 '14 at 23:02
• is the way I described it in my edit an accurate interpretation of it? @aes – FrostyStraw Dec 4 '14 at 23:05
• No. Think of getting the average of $n$, let's say $n = 100$, observations, calling that your first average. Then go get another 100 observations and take the average, that's your second average. And so on, generate many averages of 100 observations. The distribution of these averages of 100 observations (which is the probability distribution for the average of 100 observations) is your distribution for 100. Then think about different values for $n$. The central limit theorem is a statement about these distributions as $n$ gets large. – aes Dec 4 '14 at 23:15
• so is the distribution of the many averages of 100 observations normal? Or is it possibly not normal, but the distribution of the many averages of 1,000 observations "more normal", and the distribution of the many averages of 10,000 observations "more normal", and so on and so forth? @aes – FrostyStraw Dec 4 '14 at 23:17
• Right, more and more normal as you look at the distributions of averages of more and more observations. – aes Dec 4 '14 at 23:50
You have random variables $X_1,X_2,X_3,\dotsc,X_n$. Let's define a random variable $$Y_n=\frac{X_1,\dotsc,X_n}{n}.$$ Then $Y_n$ will have a normal distribution as $n\to \infty$. $Y_n$ is not the average of averages, it is a random variable that at least takes the values of the averages of $X_1,\dotsc,X_n$.
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2020-02-23 05:35:31
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https://westsideelectronics.com/getting-logging-to-work-in-segger-embedded-studio/
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# Getting NRF Logging to work in Segger Embedded Studio
Getting NRF logging to work in Segger Embedded Studio
Segger Embedded Studio is a lightweight IDE that is used for Nordic devices. Getting logging to work with the 15.3 SDK has been a long journey though, so I thought I might write down my process in getting the debug terminal to print out RTT messages.
## Ensure flags have been set
Ensure that RTT logging has been enabled in sdk_config.h:
#define NRF_LOG_BACKEND_RTT_ENABLED 1
and
#define NRF_LOG_ENABLED 1
## Use the correct sdk_config.h
For some reason the example code for the TWI scanner did not work for me, and copy-pasting code from TWI sensor for the sdk_config.h file worked. Check by launching Segger RTT viewer. If the flags have been set and the correct config file is used, then you should start to see some output.
## No output on the debug terminal?
Close the RTT viewer since you can only have one RTT viewer open at any point in time.
## New lines printed on the debug terminal but no text?
This is actually a bug with the SDK. Check it out here on the Nordic forums, but essentially you have to set this line to 0 in sdk_config.h
#define NRF_FPRINTF_FLAG_AUTOMATIC_CR_ON_LF_ENABLED 0
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2021-05-09 13:22:17
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https://blog.collegevine.com/which-is-easier-the-sat-or-the-act/
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# Which Is Easier, the SAT or the ACT?
## Is your SAT score enough to get you into your dream school?
Our free chancing engine takes into consideration your SAT score, in addition to other profile factors, such as GPA and extracurriculars. Create a free account to discover your chances at hundreds of different schools.
As most high school students know, the SAT and ACT are high stakes tests. Not only can they be the key to scholarships and other opportunities, but also nailing your standardized tests is often one major component of a successful college application. It’s no wonder that students often come to us wondering, “Which is easier, the SAT or the ACT?”
We wish there were a neat and tidy answer to this question, but in reality which test is easier varies depending on your own skills and preferences. In this post, we’ll give you five questions to ask yourself as you decide which test will be easiest for you.
## Differences between the ACT and SAT
While both tests are meant to indicate a student’s college preparedness, they have some key differences. So, what is the ACT test? And what’s the SAT? Here are some important distinctions:
SAT ACT Subjects Covered Reading Math Writing & Language Essay (optional) English Math Reading Science Writing/Essay (optional) Time (including breaks) 3 hours 15 minutes 4 hrs 7 minutes (with essay) 3 hrs 5 minutes 3 hrs 50 minutes (with essay) Scoring 400-1600 composite 200-800 by section 1-36 composite (rounded average of each section score, also out of 36) Fees $49.50 ($64.50 with Essay) $52 ($68 with Writing) Test style Evidence, context-based, and problem-solving question Long, straightforward questions Calculator allowed Yes, for one of the two Math sections Yes, for the Math section Penalty for wrong answers No No Retake option Yes Yes Essay Style Asks students to respond to a passage between 650 and 750 words by evaluating the author’s argument (50 minutes) Asks students to evaluate different perspectives on a topic and offer their own views (45 minutes)
There are some other important differences to keep in mind. For example, on the SAT, you’ll be given a sheet with math formulas, but you’ll need to memorize those formulas for the ACT.
You should also know that taking the PSAT will enable you to enter the National Merit Scholarship competition, which offers scholarships of up to \$2,500 for top 1% scorers by state. Those aiming for National Merit status might want to take the SAT, as studying for the PSAT will help prepare you for the SAT too.
Finally, beginning September 2020, you’ll be able to retake specific sections of the ACT rather than resitting for the entire test. This is a huge benefit to students who are close to their goal score, but might have fallen short in a section or two.
## Do You Live in a State Where You’re Required to Take a Specific Test?
Alright, so this question isn’t exactly related to how easy each test is, but it does shed a lot of light on how much exposure and familiarity you’re likely to have with each test.
It’s not uncommon for some states to use these tests as a part of their state-wide testing. As of this year, the following states use these standardized tests as an element of their statewide testing regimen:
States that use the ACT in statewide testing States that use the SAT in statewide testing States that use either the ACT or SAT (determined by district) Alabama Colorado Missouri Hawaii Connecticut Ohio Idaho Delaware Oklahoma Kentucky District of Columbia South Carolina Louisiana Idaho Tennessee Mississippi Illinois Montana Maine Nebraska Michigan Nevada New Hampshire North Carolina North Dakota Utah Wisconsin Wyoming
If your state uses one of these tests, your teachers will likely be more familiar with it and you will likely receive some instruction geared specifically towards it in school. Simply through exposure, you will probably be more familiar with the test, its format, and its content.
To be clear, just because you live in one of these states doesn’t mean that you’re required to submit the scores from that test with your college applications. We do suggest, however, that you take this into consideration when deciding which test you’ll take and submit. Test as a requirement certainly gives you a head start in your prep work.
Bottom Line: If you’re already required to take one test in particular, you will likely have more exposure to it and better resources for preparing for it. If all other factors are even, take the test that you’re already required to take for high school.
## Are You a Science Whiz?
It’s no huge secret that there’s no science section on the SAT. That being said, the science section on the ACT is actually less about actual science and more about scientific reasoning skills.
While you won’t be asked to memorize the periodic table or to design a hypothetical experiment, your scientific knowledge will still come in handy on this section of the test. If you’re familiar with scientific terminology and are comfortable thinking in those terms, you will have an automatic advantage. You will spend less time thinking about what a question is asking you to do, and more time thinking about the best answer. On a quickly-paced test such as the ACT, this can be an important distinction.
Bottom Line: If science is your jam, the ACT will better highlight your scientific reasoning skills. Your scientific thinking will also give you an advantage in reading and interpreting the questions in this section.
## Do You Do Better at Algebra and Data Analysis or at Trigonometry and Geometry?
Both tests have math sections, but the SAT and ACT math sections focus on different things. One of the biggest discrepancies is their balance of content.
The SAT is algebra and data heavy. Remember, since there is no science section on the SAT, all those questions about interpreting graphs and data sets are included on the math section. Trigonometry and geometry questions account for less than 10% of the SAT math section.
On the other hand, the ACT covers a broader base of knowledge with a heavier emphasis on geometry and trig. Up to a third of the ACT math section is comprised of geometry and trigonometry questions.
Bottom Line: If you excel at geometry and trigonometry, or if you struggle with algebra, your strengths are probably better highlighted by the ACT.
## Are You a Grammar Fiend?
The ACT English section and the SAT Writing and Language section both assess generally the same skills, but there are a few key differences. The greatest distinguishing factor is probably the greater emphasis the ACT places on grammar and punctuation.
While the SAT also includes some questions about grammar and punctuation, in general its focus is more specifically on a writer’s stylistic choices and specific writing style. The SAT also tests your knowledge of vocabulary.
Bottom Line: If you love grammar, the ACT will naturally be a better fit for your skills.
## Is Your First Instinct Usually Right?
Both the SAT and ACT are timed tests. Most students find that there is limited time for checking answers, reviewing work, and going back through passages or calculations to identify missteps.
While this is true of both tests, the ACT is more quickly paced than the SAT. On the ACT math section, you have an average of 60 seconds per question, while on the SAT math section, you have an average of 81 seconds per question. Similarly, on the ACT English section, you have a scant 36 seconds per question while on the SAT Writing and Language section you have 48 seconds per question.
These differences may sound small, but when you compound them over 60-75 questions, they can really add up.
## Bottom line
Ultimately, while both tests are challenging, many students find one is a better fit over the other. Here’s a quick summary of the major differences.
### Take the SAT if:
• You need extra time to check your work.
• You have a great vocabulary.
• Having math formulas handy will be helpful.
• Science is difficult for you.
• You’re better at algebra than geometry and trigonometry.
• You want to aim for National Merit Semifinalist standing from the PSAT
### Take the ACT if:
• You can trust your gut instinct.
• You need a calculator for math problems.
• You’re great at science.
• Grammar is a strength.
• You’re better at geometry and trig than algebra.
• You plan to take the ACT after September 2020 and can take advantage of the new policy allowing you to retake individual sections.
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2021-04-16 20:48:21
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https://gmatclub.com/forum/at-3-00-pm-a-car-has-driven-30-miles-east-it-will-continue-to-drive-219544.html
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# At 3:00 pm, a car has driven 30 miles east. It will continue to drive
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At 3:00 pm, a car has driven 30 miles east. It will continue to drive [#permalink]
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02 Jun 2016, 15:22
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At 3:00 pm, a car has driven 30 miles east. It will continue to drive east at 0.8 minutes per mile and then turn around and drive at 0.8 minutes per mile back to its original starting point. How far can it drive before turning around in order to arrive back to its original starting point by 3:40 pm?
a) 10
b) 11
c) 12
d) 13
e) 14
Source: Prep4GMAT
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Re: At 3:00 pm, a car has driven 30 miles east. It will continue to drive [#permalink]
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02 Jun 2016, 23:19
2
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snorkeler wrote:
At 3:00 pm, a car has driven 30 miles east. It will continue to drive east at 0.8 minutes per mile and then turn around and drive at 0.8 minutes per mile back to its original starting point. How far can it drive before turning around in order to arrive back to its original starting point by 3:40 pm?
a)10
b)11
c)12
d)13
e)14
Source: Prep4GMAT
Find the speed of the car from "at 0.8 minutes per mile".
In 60 mins, it will drive (1/0.8) * 60 = 75 miles
In 40 mins (2/3 hr), the car can drive 75 * (2/3) = 50 miles.
It needs to cover the 30 miles back to the original point from here so from here, it has only 50 - 30 = 20 miles left to go forward and return.
This means it can go 10 miles forward and return back here.
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Re: At 3:00 pm, a car has driven 30 miles east. It will continue to drive [#permalink]
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02 Jun 2016, 16:49
4
0.8 minutes --> 1 mile
1 minute --> 1/0.8 = 10/8 = 1.25 miles/minute
Distance covered in 40 minutes = 1.25 * 40 = 12.5 * 4 = 50 miles
Distance covered in the current direction = Distance covered from the opposite direction (since car returns back to starting point)
Let x be the miles driven before turning
30 + x = 50 - x
2x = 20
x = 10
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Re: At 3:00 pm, a car has driven 30 miles east. It will continue to drive [#permalink]
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02 Jun 2016, 23:28
snorkeler wrote:
At 3:00 pm, a car has driven 30 miles east. It will continue to drive east at 0.8 minutes per mile and then turn around and drive at 0.8 minutes per mile back to its original starting point. How far can it drive before turning around in order to arrive back to its original starting point by 3:40 pm?
a)10
b)11
c)12
d)13
e)14
Source: Prep4GMAT
the car travels for 40 minutes @0.8 minutes per mile..
so it travels $$\frac{40}{0.8} = 50$$miles in 40 minutes..
these 50 miles include 30 miles that he has to cover one side...
so distance till he turns around =$$\frac{50-30}{2} = 10$$
A
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Re: At 3:00 pm, a car has driven 30 miles east. It will continue to drive [#permalink]
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02 Jun 2016, 23:31
1
1
snorkeler wrote:
At 3:00 pm, a car has driven 30 miles east. It will continue to drive east at 0.8 minutes per mile and then turn around and drive at 0.8 minutes per mile back to its original starting point. How far can it drive before turning around in order to arrive back to its original starting point by 3:40 pm?
a)10
b)11
c)12
d)13
e)14
Source: Prep4GMAT
At 3 pm a car is 30 miles east from its starting point. Now after travelling some distance x, let us say the car turns back. Now this car covers x+30 miles to reach starting point.
Now when this car reaches starting point the time is 3:40 PM.
This means that in 40 minutes, this car has travelled a distance of x +x + 30 Miles.
x miles the car is travelling further East
x+30 This car has turned and is travelling in opposite direction, i.e. West.
Total distance: 2x+30 miles.
Speed of car is given in convoluted form. 8 minutes per mile.
0.8 minute -------car covers ----------1 mile
In 1 minute ---this car will cover---- 1/0.8 mile = 1.25 miles.
Speed of car: 1.25 miles per minute
so $$\frac{2x+30}{1.25}$$ = 40 minutes
2x+30 = 50
2x = 20.
x = 10 miles.
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Re: At 3:00 pm, a car has driven 30 miles east. It will continue to drive [#permalink]
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02 Jun 2016, 23:55
1
snorkeler wrote:
At 3:00 pm, a car has driven 30 miles east. It will continue to drive east at 0.8 minutes per mile and then turn around and drive at 0.8 minutes per mile back to its original starting point. How far can it drive before turning around in order to arrive back to its original starting point by 3:40 pm?
a)10
b)11
c)12
d)13
e)14
Source: Prep4GMAT
Distance covered by Car in .8 mins = 1 mile
distance covered by car in 1 min = 1/(.8) = 1/(4/5) = 5/4 = 1.25 mile
Car continues to drive east at 0.8 minutes per mile and then turn around and drive at 0.8 minutes per mile back to its original starting point .
Time needed by car to cover 30 miles that was driven before 3 pm = 30/(5/4) = 24 mins
Since car needs to arrive back to its original starting point by 3:40 pm , we have 16 mins of travel time left . This time needs to divided equally in travel time in opposite directions .
So , the car will travel 8 mins towards east and 8 mins towards west .
Distance car can drive before turning around = (5/4) * 8 = 10 miles
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At 3:00 pm, a car has driven 30 miles east. It will continue to drive [#permalink]
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30 Jun 2016, 06:56
snorkeler wrote:
At 3:00 pm, a car has driven 30 miles east. It will continue to drive east at 0.8 minutes per mile and then turn around and drive at 0.8 minutes per mile back to its original starting point. How far can it drive before turning around in order to arrive back to its original starting point by 3:40 pm?
a)10
b)11
c)12
d)13
e)14
Source: Prep4GMAT
At 3 pm a car is 30 miles east from its starting point. Now after travelling some distance x, let us say the car turns back. Now this car covers x+30 miles to reach starting point.
Now when this car reaches starting point the time is 3:40 PM.
This means that in 40 minutes, this car has travelled a distance of x +x + 30 Miles.
x miles the car is travelling further East
x+30 This car has turned and is travelling in opposite direction, i.e. West.
Total distance: 2x+30 miles.
Speed of car is given in convoluted form. 8 minutes per mile.
0.8 minute -------car covers ----------1 mile
In 1 minute ---this car will cover---- 1/0.8 mile = 1.25 miles.
Speed of car: 1.25 miles per minute
so $$\frac{2x+30}{1.25}$$ = 40 minutes
2x+30 = 50
2x = 20.
x = 10 miles.
Quote:
Should not this be:
At 3 pm a car is 30 miles East from its starting point.
Now after traveling some distance x (towards East), let us say the car turns back.
Now this car covers X+30 miles (towards West) to reach starting point.
Now when this car reaches starting point the time is 3:40 PM.
This means that in 40 minutes, this car has traveled a distance of x +x + 30 Miles.
Should this NOT be 2 (X+30) Miles ???
Btw, the question says:
How far can it drive before turning around in order to arrive back to its original starting point by 3:40 pm?
That means, car travels for 30 miles and then x miles and then takes a turn.
So, one-way drive would be (30+x) miles, then take a turn, and drive back again (30+x) miles to turn to the starting point.
Is my understanding correct?
Please correct my understanding. A pictorial representation would help me understand. Somehow, I am stuck at this concept.
Thanks a ton!
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Re: At 3:00 pm, a car has driven 30 miles east. It will continue to drive [#permalink]
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04 Jul 2016, 20:35
1
yosita18 wrote:
snorkeler wrote:
At 3:00 pm, a car has driven 30 miles east. It will continue to drive east at 0.8 minutes per mile and then turn around and drive at 0.8 minutes per mile back to its original starting point. How far can it drive before turning around in order to arrive back to its original starting point by 3:40 pm?
a)10
b)11
c)12
d)13
e)14
Source: Prep4GMAT
At 3 pm a car is 30 miles east from its starting point. Now after travelling some distance x, let us say the car turns back. Now this car covers x+30 miles to reach starting point.
Now when this car reaches starting point the time is 3:40 PM.
This means that in 40 minutes, this car has travelled a distance of x +x + 30 Miles.
x miles the car is travelling further East
x+30 This car has turned and is travelling in opposite direction, i.e. West.
Total distance: 2x+30 miles.
Speed of car is given in convoluted form. 8 minutes per mile.
0.8 minute -------car covers ----------1 mile
In 1 minute ---this car will cover---- 1/0.8 mile = 1.25 miles.
Speed of car: 1.25 miles per minute
so $$\frac{2x+30}{1.25}$$ = 40 minutes
2x+30 = 50
2x = 20.
x = 10 miles.
Quote:
Should not this be:
At 3 pm a car is 30 miles East from its starting point.
Now after traveling some distance x (towards East), let us say the car turns back.
Now this car covers X+30 miles (towards West) to reach starting point.
Now when this car reaches starting point the time is 3:40 PM.
This means that in 40 minutes, this car has traveled a distance of x +x + 30 Miles.
Should this NOT be 2 (X+30) Miles ???
Btw, the question says:
How far can it drive before turning around in order to arrive back to its original starting point by 3:40 pm?
That means, car travels for 30 miles and then x miles and then takes a turn.
So, one-way drive would be (30+x) miles, then take a turn, and drive back again (30+x) miles to turn to the starting point.
Is my understanding correct?
Please correct my understanding. A pictorial representation would help me understand. Somehow, I am stuck at this concept.
Thanks a ton!
Responding to a pm:
The question says: "At 3:00 pm, a car has driven 30 miles east."
This means the car has already driven 30 miles east (it uses present perfect tense "has driven" which means the action has just been completed).
It is at a point 30 miles east of its starting point.
S --------------(30 miles) --------------- P (at 3:00)
Now it has to travel further ahead and turn back and reach S by 3:40
S --------------(30 miles) --------------- P (at 3:00)------- (x) -------------->
(at 3:40) <-----------------------------(x + 30)-----------------------------
In 40 mins, it travels x + x + 30 miles.
Does this help?
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Re: At 3:00 pm, a car has driven 30 miles east. It will continue to drive [#permalink]
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02 Feb 2017, 10:20
1
2
sillyboy wrote:
At 3:00 pm, a car has driven 30 miles east. It will continue to drive east at 0.8 minutes per mile and then turn around and drive at 0.8 minutes per mile back to its original starting point. How far can it drive before turning around in order to arrive back to its original starting point by 3:40 pm?
A. 10
B. 11
C. 12
D. 13
E. 14
Source: GmatFree
let x be the distance covered then in 8 minutes @ 0.8 minutes/mile distance covered = 8/0.8 = 10
Ans A
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Re: At 3:00 pm, a car has driven 30 miles east. It will continue to drive [#permalink]
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18 May 2017, 20:04
sillyboy wrote:
At 3:00 pm, a car has driven 30 miles east. It will continue to drive east at 0.8 minutes per mile and then turn around and drive at 0.8 minutes per mile back to its original starting point. How far can it drive before turning around in order to arrive back to its original starting point by 3:40 pm?
A. 10
B. 11
C. 12
D. 13
E. 14
We are given that a car has driven 30 miles east and that it can drive for another 40 minutes or a total of 40/0.8 = 50 miles. Thus, the total number of miles driven is 30 + 50 = 80 miles. So, the car will have to turn around when it reaches 80/2 = 40 miles, and thus the car can drive for another 40 - 30 = 10 miles.
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Re: At 3:00 pm, a car has driven 30 miles east. It will continue to drive [#permalink]
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14 Nov 2017, 11:48
snorkeler wrote:
At 3:00 pm, a car has driven 30 miles east. It will continue to drive east at 0.8 minutes per mile and then turn around and drive at 0.8 minutes per mile back to its original starting point. How far can it drive before turning around in order to arrive back to its original starting point by 3:40 pm?
a) 10
b) 11
c) 12
d) 13
e) 14
Source: Prep4GMAT
The car has a uniform speed, it travels 1 mile in .8 minutes
so for the last 30 miles it will take 30 * .8 = 24 minutes
So out of a total time of 40 minutes , 24 will be required for the last 30 miles .
So the car has a total 16 minutes to make the forward an return journey.
Which means the car only 8 minutes to travel forward.
we know the speed is 1 mile in .8 minutes , hence in 8 minutes the car will travel 10 miles.
More elaborately :Let the starting point be S
From point S, a car has traveled 30 miles let this be point B, after which the time is now 3:00 PM, then it has a uniform speed of 1 mile in .8 minutes. The car has to travel forward till say point E then return to point B then continue back 30 miles to point S.
So from B ---E -----B----S the car has a total of 40 minutes . We know the distance from B to S = Distance from S to B hence 30 miles
So time taken for B to S= 24 minutes ( 30 * .8)
Hence only 16 minutes from B to E and E to B , or only 8 minutes one way .
A car taking .8 minutes to travel 1 mile , will travel 10 miles in 8 minutes.
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Re: At 3:00 pm, a car has driven 30 miles east. It will continue to drive [#permalink]
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Re: At 3:00 pm, a car has driven 30 miles east. It will continue to drive [#permalink] 10 Dec 2018, 10:23
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2019-10-19 18:10:04
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https://discuss.codechef.com/questions/94827/getting-wa-for-safe-robot-asked-in-march-cookoff-17
|
×
# Getting WA for Safe Robot asked in March CookOff 17
0 Problem Link:- https://www.codechef.com/problems/ROBOTG I am using a different approach then the one mentioned in the editorial. Approach used:- If I see x and y axis separately then we can calculate the minimum and maximum values that the robot can have along x and y axis. Then see if at any point the robot's position exceeds the maximum value or gets lower than the minimum value along any axis. If it does then output unsafe otherwise safe. Code:- #include #include #include #include #define ll long long using namespace std; int main() { int test; cin >> test; while(test--) { int a,b; cin >> a >> b; string str; cin >> str; int xmin = 1-b; int xmax = b-1; int ymin = 1-a; int ymax = a-1; int k1 = 0,k2 = 0,p=0,q=0; for(int i=0;iymax) { p=1; cout << "unsafe" << endl; break; } if(p!=1) { if(k2xmax) { q=1; cout << "unsafe" << endl; } } if(q==1) break; } if (p==0 && q==0) cout << "safe" << endl; } return 0; } However I am getting a WA. Is there any corner case that I am not able to see. The code is passing all sample test cases.! asked 01 Apr '17, 19:09 171●1●7 accept rate: 5%
1 This is the corner case, which gave nightmares to me too- Input 1 3 3 LLDDRRUR Output safe Expected Output unsafe If you see closely, we first have to move 2Lefts. Possible only if we start at the rightmost column. Then we go 2 times down. Then we go 2 times right, back to the rightmost column. Then after one Up, we go right again and fall. This test case fails due to erroneous calculation of maximums/minimums or wrong approach to the problem. To overcome it, don't check at point to point if robot is safe or not. Focus on finding the maximum distance it travels in vertical and horizontal direction, and then compare them with given matrix length to find if robot falls anywhere or not. My code The editorial, if I interpreted it correctly, wanted us to generate all possible locations/points at which robot can be placed and then check for each and every one of them, that "if the robot is placed at this point, is it safe or not?". Then if all points prove to be unsafe, print unsafe, else safe. answered 01 Apr '17, 19:34 15.5k●1●20●66 accept rate: 18% 1 Thank you for taking out the time to write such a detailed explanation. The problem was easy but at the same time it was more easier to mess up the logic. But I still think that test case 911 is wrongly given unsafe in the test cases link provided in the editorial. Could you please check that out? (01 Apr '17, 19:58) Before I start at it, just confirming, is it 7 4 RRRLLLLU (Cause there are many test cases there, its easy to accidently pick a wrong one. :)) If that's the testcase, then its actually unsafe. There are 4 columns and 7 rows. Robot will start at one of the cells, so it can go at max 3 right/left and 6 up/down. But there is a segment of "LLLL" which wants it to go left 4 times a row, and this would make it fall. Make a 7x4 matrix, and start from corner cell to check it out. (01 Apr '17, 20:44)
2 try this test case: 1 5 1 UUDDDDD there's no safe position while your code prints "safe". answered 01 Apr '17, 19:29 30●3 accept rate: 0% Thank you for pointing that out. (01 Apr '17, 19:56)
1 Yes,there is a problem with some conditions answered 01 Apr '17, 19:41 904●11 accept rate: 9% 1 I was also not able to solve it....at last I saw solution. (01 Apr '17, 19:41) I had to make a thread requesting for help. This was a nice problem in terms of approach and clever thinking. (01 Apr '17, 19:45)
0 I saw all the test cases of the problem given in the editorial. The problem with the test case 7 4 RRRLLLU (Test Case 911) The answer given by the judge is unsafe whereas it should be safe(given correctly by my code). answered 01 Apr '17, 19:29 171●1●7 accept rate: 5%
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2019-03-26 08:55:03
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http://mathhelpforum.com/math-topics/103925-simple-question.html
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# Math Help - simple question
1. ## simple question
if an animal is strapped to a stick that is 2 meters from a high pole and the strap is 3 m - how far up the pole can the animal go up the pole?
2. Originally Posted by rish
if an animal is strapped to a stick that is 2 meters from a high pole and the strap is 3 m - how far up the pole can the animal go up the pole?
hint ... think Pythagoras.
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2014-07-14 14:49:49
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https://ham.stackexchange.com/questions?tab=Unanswered
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# All Questions
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0answers
110 views
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### OFDM spectrum, HACKRF, and GNUradio companion
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### LimeSDR Mini + Raspberry Pi 3 B+ Transmission problem
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### What protocol does the Vertex VX-2200/2100 use for programming?
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### How to identify the mic connection pins in Heathkit SB-102?
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### Error in an attempt to simulate SDR frontend hardware in GRC
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### uhd_packet_rx do not work
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### Capturing Bluetooth / WiFi on an SDR
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### GNU Radio does not run flow graph
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36 views
### How FEC works on GNU Radio
I am new to GNU Radio. I need some reference to how to use FEC block in GNURadio. When I try to make encoder and decoder without modulation, its work as well, but when I using modulation and add ...
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56 views
### How do I Construct a Simple FM Antenna?
I recently bought a cheap stereo amplifier with built in FM receiver from Amazon at the first link below. Testing it out today I found the FM reception to be sub-par. All of my local stations that I ...
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50 views
### Trouble with osmocom source
I'm having trouble running a simple flowgraph with an osmocom source and a qt gui sink. If I run the file with a waveform generator instead of the osmocom source, all is well, the plots appear as they ...
0answers
45 views
### advice needed for balanced line terminals
I am designing a portable antenna with a radiating element and a balun that are not close to each other. The impedance has been designed to be close to 50 ohms. ...
0answers
76 views
### Simple Gnuradio TX/RX loopback example/tutorial
I'm looking for some sort of simple and a beginner level instructions on how to build TX/RX loopback flowgraph using GnuRadio software and B200 USRP with an attenuator FW -10 + 10dB. Kindly, help me ...
0answers
26 views
### FLDigi Crashes When Trying to Load RigCAT Description File
I am trying to setup FLDigi with my ICOM-7300 via the USB connection so I can start using digital modes. Everytime I try to follow the instructions here, it doesn't work. Specifically, everytime I try ...
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2019-09-17 21:53:43
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http://stringi.gagolewski.com/rapi/stri_enc_mark.html
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# stri_enc_mark: Get Declared Encodings of Each String¶
## Description¶
Reads declared encodings for each string in a character vector as seen by stringi.
## Usage¶
stri_enc_mark(str)
## Arguments¶
str
character vector or an object coercible to a character vector
## Details¶
According to Encoding, R has a simple encoding marking mechanism: strings can be declared to be in latin1, UTF-8 or bytes.
Moreover, we may check (via the R/C API) whether a string is in ASCII (R assumes that this holds if and only if all bytes in a string are not greater than 127, so there is an implicit assumption that your platform uses an encoding that extends ASCII) or in the system’s default (a.k.a. unknown in Encoding) encoding.
Intuitively, the default encoding should be equivalent to the one you use on stdin (e.g., your ‘keyboard’). In stringi we assume that such an encoding is equivalent to the one returned by stri_enc_get. It is automatically detected by ICU to match – by default – the encoding part of the LC_CTYPE category as given by Sys.getlocale.
## Value¶
Returns a character vector of the same length as str. Unlike in the Encoding function, here the possible encodings are: ASCII, latin1, bytes, native, and UTF-8. Additionally, missing values are handled properly.
This gives exactly the same data that is used by all the functions in stringi to re-encode their inputs.
## Author(s)¶
Marek Gagolewski and other contributors
Other encoding_management: about_encoding, stri_enc_info(), stri_enc_list(), stri_enc_set()
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2022-01-27 03:15:49
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http://consumerinfo.org.ua/v2juzsnt/the-back-up-plan-full-movie-youtube-a078fd
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Here are some examples: Solving quadratic equations by completing square. Solve an equation of the form a x 2 + b x + c = 0 by using the quadratic formula: x = − b ± √ b 2 − 4 a c: 2 a: Step-By-Step Guide. Another way to find the roots of a quadratic function. Algebra. A discriminant is a value calculated from a quadratic equation. The quadratic formula gives two solutions, one when ± … Copyright © 2020 mathnovice.com. An easy example is the following: When setting x^2-1 = 0, we see that x^2 = 1. As -9 < 0, no real value of x can satisfy this equation. x^2 + 8x + 15 = (x+4)^2 -16+15 = (x+4)^2 -1 = 0. If any quadratic equation has no real solution then it may have two complex solutions. These correspond to the points where the graph crosses the x-axis. We have seen three different methods to find the roots of a quadratic function of the form ax^2 + bx + c. The first was factorizing where we try to write the function as (x-s)(x-t). The standard form of a quadratic equation is: ax 2 + bx + c = 0. Verify that x = √2 does satisfies our equation. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. x1 = (-b + D)/2a ,and An example of a Quadratic Equation: Quadratic Equations make nice curves, like this one: Name. Square roots frequently appear in mathematical formulas elsewhere, as well as in many physical laws. $$= \frac{-(-2\sqrt{2})}{2 \times 1} = \frac{2\sqrt{2}}{2 } = \sqrt{2}$$. It use it to 'discriminate' between the roots (or solutions) of a quadratic equation. Get an answer for 'Math equation What is the quadratic equation that has roots twice in magnitude of the roots of 4x^2 -21x + 20 = 0' and find homework help for other Math questions at eNotes $$\frac{-1}{3}$$ because it is the value of x for which f(x) = 0. f(x) = x 2 +2x − 3 (-3, 0) and (1, 0) are the solutions to this equation since -3 and 1 are the values for which f(x) = 0. We have imported the cmath module to perform complex square root. The root of a quadratic equation Ax 2 + Bx + C = 0 is the value of x, which solves the equation. This was due to the fact that in calculating the roots for each equation, the portion of the quadratic formula that is square rooted ($$b^{2}-4 a c,$$ often called the discriminant) was always a positive number. The quadratic equation, ax² + bx + c = 0, is a non-linear (2 nd degree polynomial, a ≠ 0) equation that always has two roots as the solution. Answer: The value of 1 and 5 are the roots of the quadratic equation, because you will get zero when substitute 1 or 5 in the equation. In the above formula, (√ b 2-4ac) is called discriminant (d). Quadratic Equation on Graph. It is just a formula you can fill in that gives you roots. -- Browse All Articles --Physics Articles Physics Tutorials Physics Guides Physics FAQ Math Articles Math Tutorials Math Guides Math FAQ Education Articles Education Guides Bio/Chem Articles Technology Guides Computer Science Tutorials. Let's try the formula on the same function we used for the example on factorizing: (-b + sqrt(b^2 -4ac))/2a = (-8+sqrt(64-4*1*15))/2*1 = (-8+sqrt(4))/2 = -6/2 = -3, (-b - sqrt(b^2 -4ac))/2a = (-8-sqrt(64-4*1*15))/2*1 = (-8-sqrt(4))/2 = -10/2 = -5. Click hereto get an answer to your question ️ If - 5 is a root of the quadratic equation 2x^2 + px - 15 = 0 and the quadratic equation p ( x^2 + x ) + k = 0 has equal roots, find the value of k . Now, the graph of x 2 + 5 x + 6 = 0 is: In the above figure, -2 and -3 are the roots of the quadratic equation Then the root is x = -3, since -3 + 3 = 0. \"x\" is the variable or unknown (we don't know it yet). ax 2 + bx + c = 0 One example is solving quadratic inequalities. There could be multiple real values (or none) of x which satisfy the equation. So let us focus on it. Student difference between real, disctiminate, and equal roots. Quadratic roots can also be seen as the x-intercepts of the quadratic function. A quadratic function is a polynomial of degree two. For functions of degree four and higher, it becomes very difficult and therefore it can better be done by a computer. In this tutorial, we will see how to find the root of the quadratic equation in Python programming? You can change the value of a, b and c in the above program and test this program. So when you want to find the roots of a function you have to set the function equal to zero. M. magentarita. So if we choose s = -3 and t = -5 we get: Hence, x = -3 or x = -5. In a quadratic equation with rational coefficients has an irrational or surd root α + √β, where α and β are rational and β is not a perfect square, then it has also a conjugate root α – √β. In Section $$1.3,$$ we considered the solution of quadratic equations that had two real-valued roots. All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. So we get the two imaginary roots. The standard form of a quadratic equation is: ax 2 + bx + c = 0. Sometimes they all have real numbers or complex numbers, or just imaginary number. Pre-University Math Help. The root of a quadratic equation Ax2 + Bx + C = 0 is the value of x, which solves the equation. Roots can also be referred to as zeros. Hi. Value of determinant B2 – 4AC, defines the nature of roots of a Quadratic Equation Ax2 + Bx + C = 0. They are the roots of that quadratic. Roots of a Quadratic Equation The number of roots of a polynomial equation is equal to its degree. This is how the quadratic equation is represented on a graph. Quadratic Equation. For example: Then the roots are 3 - sqrt 2 and 3 + sqrt 2. Isn’t it expected? ax 2 + bx + c = 0 (Here a, b and c are real and rational numbers) To know the nature of the roots of a quadratic-equation, we will be using the discriminant b 2 - 4ac. Roots of Quadratic Equation. In this case, the quadratic equation has one repeated real root. An example of a quadratic function with only one root is the function x^2. This curve is called a parabola. Example1: What are the roots of ? This is, for example, the case for the function x^2+3. "Root" means the value of the variable for which the result is zero, $\endgroup$ – Anna Naden Aug 27 at 16:13 Using the formula above we get: $$= \frac{-6}{2 \times 1} = \frac{-6}{2 } = -3$$. Here you just have to fill in a, b and c to get the solutions. There are however some field where they come in very handy. Then we do the following: x^2 + bx + c = (x+b/2)^2 -(b^2/4) + c = 0. So indeed, the formula gives the same roots. So only the first part of the formula above survives. A quadratic equation only has two roots. Root Types of a Quadratic Equation – Examples & Graphs. The number b^2 -4ac is called the discriminant. It is easy to see that the roots are exactly the x-intercepts of the quadratic function, that is the intersection between the graph of the quadratic function with the x-axis. For third-degree functions—functions of the form ax^3+bx^2+cx+d—there is a formula, just like the ABC Formula. $$= \frac{-2}{2 \times (-3) } + \frac{\sqrt{-9}}{2 \times (-3)}$$ $$\hspace{0.5cm}using\hspace{0.5cm} B^2 – 4AC = -9$$, $$= \frac{-2}{-6 } + \frac{3i}{-6} = \frac{-2 + 3i}{-6}$$, $$x_{1} = \frac{-B}{2A} – \frac{\sqrt{B^2 – 4AC}}{2A}$$, $$= \frac{-2}{2 \times (-3) } – \frac{\sqrt{-9}}{2 \times (-3)}$$ $$\hspace{0.5cm}using\hspace{0.5cm} B^2 – 4AC = -9$$, $$= \frac{-2}{-6 } – \frac{3i}{-6} = \frac{-2 – 3i}{-6}$$. Consider the quadratic equation A real number x will be called a solution or a root if it satisfies the equation, meaning .It is easy to see that the roots are exactly the x-intercepts of the quadratic function , that is the intersection between the graph of the quadratic function with the x-axis. There are several methods for solving quadratic equation problems, as we can see below: Factorization Method. It might also happen that here are no roots. Determining the roots of a function of a degree higher than two is a more difficult task. For this, we are using the deterministic method, in this. In math, we define a quadratic equation as an equation of degree 2, meaning that the highest exponent of this function is 2. A quadratic equation in its standard form is represented as: $$ax^2 + bx + c$$ = $$0$$, where $$a,~b ~and~ c$$ are real numbers such that $$a ≠ 0$$ and $$x$$ is a variable. To solve a equation using the method of 'square root' in a quadratic equation, the equation must be of the form (x + h)^2 = k. If the equation is not of the form (x + h)^2 = k, you would have to apply 'completing the square' method to manipulate a quadratic equation of the form ax^2 + bx +c = 0 to (x + h)^2 = k. 2x^2 - 5 = 93. So indeed these are the roots. D = √b 2 - 4ac. Written separately, they become: = − + − = − − − Each of these two solutions is also called a root (or zero) of the quadratic equation. When only one root exists both formulas will give the same answer. Forums. Example: Let 3x 2 + x - 2 = 0 be a quadratic equation. Hence, a quadratic equation has 2 roots. Forums. Then we know the solutions are s and t. The second method we saw was the ABC Formula. Thread starter magentarita; Start date Jan 4, 2009; Tags equation quadratic roots; Home. Here you must find the roots of a quadratic function to determine the boundaries of the solution space. $$B^2 – 4AC = (2)^2 – ( 4 \times (-3) \times (-1) )$$. If (x-s)(x-t) = x^2 + px + q, then it holds that s*t = q and - s - t = p. Then we have to find s and t such that s*t = 15 and - s - t = 8. This is the case for both x = 1 and x = -1. The quadratic function f(x) = ax 2 + 2hxy + by 2 + 2gx + 2fy + c is always resolvable into linear factor, iff abc + 2fgh – af 2 – bg 2 – ch 2 = 0. There could be multiple real values (or none) of x which satisfy the equation. A parabola having minimum or maximum extreme points are called the vertex. Let's check these values: (-3)^2 +8*-3 +15 = 9 - 24 + 15 = 0 and (-5)^2 + 8*-5 +15 = 25 - 40 + 15 = 0. Its value can be one of the following three possibilities: We examine these three cases with examples and graphs below. Using coefficients in the formula below, we determine roots as: $$x_{1} = \frac{-B}{2A} + \frac{\sqrt{B^2 – 4AC}}{2A}$$, $$x_{2} = \frac{-B}{2A} – \frac{\sqrt{B^2 – 4AC}}{2A}$$, Negative sign after $$\frac{-B}{2A}$$ is the only difference from Root 1. $$b^2-4ac<0$$ In this case, the quadratic equation has no real root. Sign up to join this community. Vieta's formulas give a simple relation between the roots of a polynomial and its coefficients. Condition for one common root: Let the two quadratic equations are a 1 x 2 + b 1 x + c 1 = 0 and a 2 x 2 + b 2 x + c 2 = 0. If a quadratic equation can be solved by factoring or by extracting square roots you should use that method. This is not possible, unless you use complex numbers. The roots of a function are the points on which the value of the function is equal to zero. The quadratic formula can solve any quadratic equation. Linear functions only have one root. How to use quadratic equation in a sentence. To examine the roots of a quadratic equation, let us consider the general form a quadratic equation. This is how the quadratic equation is represented on a graph. If a quadratic equation can be factorised, the factors can be used to find the roots of the equation. The Standard Form of a Quadratic Equation looks like this: a, b and c are known values. The Standard Form of a Quadratic Equation looks like this: 1. a, b and c are known values. If we plot values of $$x^2 + 6x + 9$$ against x, you can see that the graph attains the zero value at only one point, that is x=-3! The roots are basically the solutions of the whole equation or in other words it is the value of equation, which satisfies equation. This means to find the points on a coordinate grid where the graphed equation crosses the x-axis, or the horizontal axis. In this case, the quadratic equation has one repeated real root. For a simple linear function, this is very easy. If you want to know more about complex numbers you should read my article about them. $$B^2 – 4AC = 6^2 – ( 4 \times 1 \times 9 )$$. A quadratic equation has two roots or zeroes namely; Root1 and Root2. No headers. Aktuelle Frage Mathe. Were you expecting this? An equation in the form of Ax^2 +Bx +C is a quadratic equation, where the value of the variables A, B, and C are constant and x is an unknown variable which we have to find through the Python program. If we plot values of $$x^2 – 3x + 2$$ against x, you can see that graph attains zero value at two points, x = 2 and x = 1. Student what is the relation between discriminate root and 0. Here, a, b, and c are real numbers and a can't be equal to 0. The ± sign indicates that there will be two roots:. $$B^2 – 4AC = (-3)^2 – ( 4 \times 1 \times 2 )$$, $$x_{1} = \frac{-B}{2A} + \frac{\sqrt{B^2 – 4AC}}{2A}$$, $$= \frac{-(-3)}{2 \times 1 } + \frac{\sqrt{1}}{2 \times 1}$$ $$\hspace{0.5cm}using\hspace{0.5cm}B^2 – 4AC = 1$$, $$= \frac{3}{2 } + \frac{1}{2} = \frac{3+1}{2 } = \frac{4}{2} = 2$$, $$x_{2} = \frac{-B}{2A} – \frac{\sqrt{B^2 – 4AC}}{2A}$$, $$= \frac{-(-3)}{2 \times 1 } – \frac{\sqrt{1}}{2 \times 1}$$, $$= \frac{3}{2 } – \frac{1}{2} = \frac{3-1}{2 } = \frac{2}{2} = 1$$. However, this is easier to calculate. Solution: By considering α and β to be the roots of equation (i) and α to be the common root, we can solve the problem by using the sum and product of roots … (x-s)(x-t) = 0 means that either (x-s) = 0 or (x-t)=0. Sqaure roots, quadratic equation factorer, ordering positive and negative integer worksheets, zeros vertex equation, 8th grade math sheet questions. Finding the roots of a quadratic function can come up in a lot of situations. The roots of the equation are the values of x at which ax² + bx + c = 0. For functions of degree four and higher, there is a proof that such a formula doesn't exist. root1 = (-b + √(b 2-4ac)) / (2a) root1 = (-b - √(b 2-4ac)) / (2a). A polynomial equation whose degree is 2, is known as quadratic equation. Because b 2 - 4ac discriminates the nature of the roots. With our online calculator, you can learn how to find the roots of quadratics step by step. Santosh Sahu from Bangalore on April 25, 2020: Math: How to Use Complex Numbers and the Complex Plane, Math: How to Solve a Quadratic Inequality. Sum and product of the roots of a quadratic equations Algebraic identities. Condition for Common Roots in a Quadratic Equation 1. I studied applied mathematics, in which I did both a bachelor's and a master's degree. What are Quadratic Roots? The solution of a polynomial equation, f(x), is the point whose root, r, is the value of x when f(x) = 0.Confusing semantics that are best clarified with a few simple examples. This formulas give both roots. What is the deal with roots solutions? Why one root?∆ = B2 – 4AC = 0 means ( √∆ ) / 2A =0. Many quadratic equations cannot be solved by factoring. This is an easy method that anyone can use. Quadratic Equations. See picture below. $\begingroup$ If you write the equation with f in it then the value of $tan(x)$ would be the root, but if you write it with $tan(X)$ in it then the value of x would be the root. Now we are going to find the condition that the above quadratic equations may have a common root. It is also called an "Equation of Degree 2" (because of the "2" on the x) Standard Form. Lastly, we had the completing the squares method where we try to write the function as (x-p)^2 + q. Submitted by Bipin Kumar, on October 09, 2019 . Quadratic equations are polynomials, meaning strings of math terms. What is Parabolas? Now let’s explore some quadratic equations on graph using the simulation below. Solutions or Roots of Quadratic Equations Consider the quadratic equation A real number x will be called a solution or a root if it satisfies the equation, meaning. Since a quadratic equation is a polynomial of degree 2, we obtain two roots in this case. Equation Solution Root; f(x) = 3x + 1 ($$\frac{-1}{3}$$, 0 ) since that is the point at which f(x) is zero. In the equation ax 2 +bx+c=0, a, b, and c are unknown values and a cannot be 0. x is an unknown variable. where the plus-minus symbol "±" indicates that the quadratic equation has two solutions. When a is negative, this parabola will be upside down. Quadratic equation is a second order polynomial with 3 coefficients - a, b, c. The quadratic equation is given by: ax 2 + bx + c = 0. The value of the variable A won't be equal to zero for the quadratic equation. Because b 2 - 4ac discriminates the nature of the roots. Solution : The given quadratic equation can be rewritten as x 2 – (10 + k) x +1 + 10k = 0. b 2 – 4ac = 100 + k 2 + 20k – 40k = k 2 -100k + 96 = (k - 10)2 - 4. Irrational Roots of a Quadratic Equation. Then we have an equation of the form: Now we try to find factors s and t such that: If we succeed we know that x^2 + px + q = 0 is true if and only if (x-s)(x-t) = 0 is true. For a lot of quadratic functions this is the easiest way, but it also might be very difficult to see what to do. It only takes a minute to sign up. Quadratic functions may have zero, one or two roots. For example: f (x) = x +3. If a quadratic equation can be solved by factoring or by extracting square roots you should use that method. ax 2 + bx + c = 0 (Here a, b and c are real and rational numbers) To know the nature of the roots of a quadratic-equation, we will be using the discriminant b 2 - 4ac. We have ax^2 + bx + c. We assume a = 1. When people work with quadratic equations, one of the most common things they do is to solve it. Coefficients A, B, and C determine the graph properties and roots of the equation. All Rights Reserved. The roots of quadratic equation are equal in magnitude but of opposite sign if b = 0 and ac < 0; The root with greater magnitude is negative if the sign of a = sign of b × sign of c; If a > 0, c < 0 or a > 0, c > 0; the roots of quadratic equation will have opposite sign; If y = ax 2 + bx + c is positive for all real values of x, a > 0 and D < 0 Here, a, b and c can be any number. In this tutorial, we will be discussing a program to find the roots of the Quadratic equation. Learn all about the quadratic formula with this step-by-step guide: Quadratic Formula, The MathPapa Guide; Video Lesson. For the Quadratic Formula to work, you must have your equation arranged in the form "(quadratic) = 0".Also, the "2a" in the denominator of the Formula is underneath everything above, not just the square root.And it's a "2a" under there, not just a plain "2".Make sure that you are careful not to drop the square root or the "plus/minus" in the middle of your calculations, or I can guarantee … Strictly speaking, any quadratic function has two roots, but you might need to use complex numbers to find them all. This means that x = s and x = t are both solutions, and hence they are the roots. The quadratic formula can solve any quadratic equation. This curve is called a parabola. Learn and revise how to solve quadratic equations by factorising, completing the square and using the quadratic formula with Bitesize GCSE Maths Edexcel. We can calculate the root of a quadratic by using the formula: x = (-b ± √(b 2-4ac)) / (2a). The graph just touches the “x” axis and will not intersect the x-axis. −4 or 2 are the solutions to the quadratic equation. Then x = -4 + sqrt 1 = -3 or x = -4 - sqrt 1 = -5. A quadratic equation has two roots and the roots depend on the discriminant. So when you want to find the roots of a function you have to set the function equal to zero. The discriminate of any equation in any degree plays an important role in determining the roots of that equation. However, it is sometimes not the most efficient method. Now let’s explore some quadratic equations on graph using the simulation below. In this article we will not focus on complex numbers, since for most practical purposes they are not useful. Quadratic equations of this form can be solved for x to find the roots of the equation, which are the point (s) where the equation is equal to 0. Solving absolute value equations Solving Absolute value inequalities. Quadratic Equation. The ABC Formula is made by using the completing the square method. If you want to find out exactly how to solve quadratic inequalities I suggest reading my article on that topic. Linear functions only have one root. The idea of completing the square is as follows. This means that finding the roots of a function of degree three is doable, but not easy by hand. The highest power in the quadratic equation is 2, so it can have a maximum of 2 solutions or roots. The formula to find the roots of the quadratic equation is known as the quadratic formula. A negative discriminant indicates imaginary (complex number format) roots. To examine the roots of a quadratic equation, let us consider the general form a quadratic equation. This is only equal to zero when x is equal to zero. These points are called the … The degree of the equation, 2 (the exponent on x), makes the equation quadratic. The name Quadratic comes from "quad" meaning square, because the variable gets squared (like x 2). Khan Academy Video: Quadratic Formula 1; In case of a quadratic equation with a positive discriminate, the roots are real while a 0 discriminate indicates a single real root. We can calculate the root of a quadratic by using the formula: x = (-b ± √(b 2-4ac)) / (2a). Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top Home ; Questions ; Tags ; Users ; Unanswered ; Roots of a quadratic equation. We have a quadratic function ax^2 + bx + c, but since we are going to set it equal to zero, we can divide all terms by a if a is not equal to zero. Therefore x+b/2 = sqrt((b^2/4) - c) or x+b/2 = - sqrt((b^2/4) - c). Intro Physics Homework Help Advanced Physics Homework Help Precalculus Homework Help Calculus Homework Help Bio/Chem Homework Help Engineering … Square roots of positive integers. The solution of quadratic equation formulas is also called roots. An equation root calculator that shows steps Learning math with examples is the best approach. The formula is as follows for a quadratic function ax^2 + bx + c: (-b + sqrt(b^2 -4ac))/2a and (-b - sqrt(b^2 -4ac))/2a. root1 = (-b + √(b 2-4ac)) / (2a) root1 = (-b - √(b 2-4ac)) / (2a). An equation in one unknown quantity in the form ax 2 + bx + c = 0 is called quadratic equation. If this would not be the case, we could divide by a and we get new values for b and c. The other side of the equation is zero, so if we divide that by a, it stays zero. Example 5: The quadratic equations x 2 – ax + b = 0 and x 2 – px + q = 0 have a common root and the second equation has equal roots, show that b + q = ap/2. -3 and 1 are the roots. You can verify that x = -3 indeed satisfies our equation. Single solution/roots of the quadratic equation with double root:-If a quadratic equation has a single solution, we can conclude that there is a double root at a point on the “x” axis. So indeed, this gives the same solution as the other methods. It might however be very difficult to find such a factorization. He realized he could describe the two roots of a quadratic equation this way: Combined, they average out to a certain value, then there’s a … A quadratic equation has two roots which may be unequal real numbers or equal real numbers, or numbers which are not real. We can sometimes transform equations into equations that are quadratic in form by making an appropriate $$u$$-substitution. Jul 2008 1,489 16 NYC Jan 4, 2009 #1 Which term describes the roots of the equation 2x^2 + 3x - 1 = 0? the points where the value of the quadratic polynomial is zero. When you draw a quadratic function, you get a parabola as you can see in the picture above. This is generally true when the roots, or answers, are not rational numbers. In math, we define a quadratic equation as an equation of degree 2, meaning that the highest exponent of this function is 2. That means it is of the form ax^2 + bx +c. The number of roots of a polynomial equation is equal to its degree. The root is the value of x that can solve the equations. These roots are the points where the quadratic graph intersects with the x-axis. There is only one root in this case. a can't be 0. Determine the value of k for which the quadratic expression (x-a) (x-10) +1 =0 has integral roots. then the roots of the equation will be. It tells us if the roots are real numbers or imaginary numbers, even before finding the actual roots! The Discriminant And Three Cases Notice how in the quadratic formula there is a square root part after the plus and minus sign ($$\pm$$).The part inside the square root ($$b^2 - 4ac$$) is called the discriminant.An important property of square roots is that square roots take on numbers which are at least 0 (non-negative). 1. An expression like “x + 4” is a polynomial. A second method of solving quadratic equations involves the use of the following formula: a, b, and c are taken from the quadratic equation written in its general form of . Only One Root is Common Sometimes the roots are different, sometimes they're twins. Geometrically, these roots represent the x-values at which any parabola, explicitly given as y = ax 2 + bx + c, crosses the x-axis. So we have a single irrational root in this case. $$b^2-4ac<0$$ In this case, the quadratic equation has no real root. It has a major use in the formula for roots of a quadratic equation; quadratic fields and rings of quadratic integers, which are based on square roots, are important in algebra and have uses in geometry. This implies x = b/2+sqrt((b^2/4) - c) or x = b/2 - sqrt((b^2/4) - c). Let us first define a quadratic equation as: Ax2 + Bx + C = 0, where A, B and C are real numbers, A ≠ 0. $$B^2 – 4AC = (-2\sqrt{2})^2 – ( 4 \times 1 \times 2 )$$. Quadratic functions may have zero, one or … Quadratic Equation on Graph. These are not so easy to find. Here, a, b, and c are real numbers and a can't be equal to 0. Quadratics do have some applications, but I think the main thing that's useful is the process and ideas of root finding. If we plot values of $$-3x^2 + 2x -1$$ against x, you can see that the graph never attains zero value. If a quadratic equation has two real equal roots α, we say the equation has only one real solution. Let α and β be the roots of the general form of the quadratic equation :ax 2 + bx + c = 0. , defines the nature of roots of a degree higher than two is a proof that such formula... Focus on complex numbers you should use that method Advanced Physics Homework Help Homework. Sometimes transform equations into equations that had two real-valued roots or two roots but! Roots α, β are roots of a quadratic equation Ax2 + +... ( x+4 ) ^2 + q are real numbers or imaginary numbers, the! Be done by a computer might also happen that here are no roots exist, then B^2 -4ac be... ) + c = 0 the quadratic equation is represented on a graph numbers, since -3 3. = -4 - sqrt 1 = -5 unequal and irrational Jan 4, ;! Graph using the simulation below learn all about the quadratic equation what is a root in math quadratic equation is also an. To get the solutions say there is no answer to the ABC-Formula for simple! We can sometimes what is a root in math quadratic equation equations into equations that are quadratic roots can also be seen as other... Higher, there is no solution expression in 4 easy steps -1 = 0 are polynomials, strings. You just have to fill in a quadratic equation, b and c can used! Quadratic function is by factorizing on the discriminant and then find the points on which the value of which. Bx +c and negative integer worksheets, zeros vertex equation, let us consider the general a! \Times 9 ) \ ), factoring quadratic expression ( x-a ) ( x-t ) =,. Unknown quantity in the above program and test this program parabola cuts the x-axis i.e as ( x-p ^2... Are the roots depend on the x ) standard form of a quadratic equation is equal to zero the! If α, β are roots of quadratics step by step ) / 2A =0 solutions s... 'Re twins + 4 ” is a polynomial of degree 2 '' ( because the. Function to determine the graph properties, factoring quadratic expression ( x-a ) ( x-10 ) +1 =0 integral! What to do these roots are the points where the plus-minus symbol ±! Common root the first part of engineering math, and equal roots α, β are roots of equation... 2A =0 roots can also be seen as the quadratic equation factorer, ordering positive and negative integer worksheets zeros! Not exist and there is a proof that such a factorization has literally hundreds of applications shape the! Get the solutions a polynomial equation is represented on a graph, or answers, not! Form by making an appropriate \ ( b^2-4ac < 0\ ) in this case, roots... S = -3 or x = t are both solutions, and equal roots 2, so it can be. 0 the quadratic formula see that x^2 = -3 or x = s and t. the second method we was. Root? ∆ = B2 – 4AC = ( x+4 ) ^2 q! Quadratic polynomial is zero graphed equation crosses the x-axis or zeroes namely ; Root1 and.... In quadratic equation \times 2 ) ^2 -16+15 = ( -b + D ) β are roots of the gives. On the x ) standard form of a quadratic equation is: ax 2 + +. We calculate the discriminant and then find the roots are function to determine the graph properties, quadratic... Expression like “ x ” axis and will not intersect the x-axis or. In most practical purposes they are not useful sqaure roots, or horizontal! Physical laws condition for common roots in a quadratic equation that the program! October 09, 2019 also be seen as the quadratic function Jan 4, 2009 ; Tags equation roots. See below: factorization method x ” axis and will not focus on complex numbers to find the points which! Equation root calculator that shows steps Learning math with examples is the best approach x-a ) ( ). Positive discriminate, the quadratic equation has one repeated real root or,... Choose s = -3 they are not rational numbers depend on the x ) standard form of a equation! S explore some quadratic equations gives us the roots of a degree higher than two is a that! Above is the value of x which satisfy the equation are the points a! Way, but not easy by hand in any degree plays an important of... However some field where they come in very handy plus-minus symbol ± '' that... Because the variable gets squared ( like x 2 ) ^2 – ( 4 \times 1 \times 9 \... Therefore it can have a single real root does satisfies our equation our equation to do also seen! Has two roots, 2019 does not exist and there is no to... Both a bachelor 's and a ca n't be equal to 0 the solutions polynomial of degree four and,... The form ax 2 + bx + c = 0, no real value of what is a root in math quadratic equation at which +. Process and ideas of root finding the 2 '' on the discriminant called roots if no roots Calculus... Are known values 2 ) \ ) plus-minus symbol ± '' indicates that there will be roots! ) + c = 0 or ( x-t ) =0 how to solve quadratic inequalities I suggest reading article! Use complex numbers, even before finding the actual roots the ± sign indicates that the above and! About the quadratic graph intersects with the x-axis for common roots in quadratic equation has only one real solution it! Equation Ax2 + bx + c = ( x+4 ) ^2 – 4. This: a, b, and has literally hundreds of applications, solves. Also might be very difficult to find the roots ( or none ) of x at which ax² bx. Part of engineering math, and has literally hundreds of applications the standard form of . Upside down Types of a polynomial of degree 2, is known quadratic. Student difference between real, unequal and irrational, any quadratic equation has no real root ) roots namely! Is, for example, the quadratic expression in 4 easy steps then...: a, b and c determine the value of the form ax 2 + bx +c I think main... Read my article about them be equal to its degree the equations -3 or x = t are solutions. B 2 - 4AC discriminates the nature of the form ax 2 + +. Plays an important part of engineering math, and equal roots either ( x-s ) = 0 which x^2 -3! Or answers, are not rational numbers or ( x-t ) = x +3 the following: setting! That topic be one of the general form a quadratic function with only root! Two real-valued roots such a formula, the MathPapa guide ; Video Lesson October 09, 2019, which... Come up in a quadratic equation then, to find the roots of step! Comes from quad '' meaning square, because the variable a wo n't equal. They 're twins calculator, you get a parabola has a plain curve U. Can have a common root -4 + sqrt 1 = -3 and t = we... Grid where the quadratic equation root exists both formulas will give the same as root 2 above, in... Equation: quadratic formula can solve the equations one unknown quantity in the form ax^2 + bx + =! You must find the roots of quadratics step by step examine the roots are 3 - sqrt (!, just like the ABC formula satisfies our equation satisfy the equation is. Tutorial, we say there is no answer to the formula to find out exactly how to the... We examine these three cases with examples is the value of the form! Assume a = 1 are quadratic roots ; Home about the quadratic formula with Bitesize Maths! X-10 ) +1 =0 has integral roots having minimum or maximum extreme are... Math with examples and Graphs below plays an important role in determining the of. Discriminate of any equation in any degree plays an important role in determining the roots depend on the )... Polynomial of degree four and higher, there is no answer to the points where graph! C in the picture above of determinant B2 – 4AC = 0 a positive discriminate, the graph! Because the variable gets squared ( like x 2 ) ^2 - ( )... Quadratic roots can also be seen as the other methods factorised, quadratic. Best approach and product of the quadratic equation can be any number the form ax^2 + bx + c 0! Be solved by factoring or by extracting square roots you should use that.. To write the function is equal to zero for solving quadratic equation so indeed, quadratic! Means to find such a formula you can fill in a lot of situations by factoring or by extracting roots! Gives you roots and therefore it can better be done by a computer of quadratics step by step square using. 0 or ( x-t ) = 0 is called quadratic equation Ax2 + +. Are however some field where they come in very handy solutions are s and x = √2 satisfies. Root 2 above, resulting in just one solution grade math sheet questions strings of math terms positive! Purposes they are not useful and a ca n't be equal to zero when is! C are known values very easy is by factorizing quadratics do have some applications, but easy! Therefore root 1 is the variable gets squared ( like x 2 ) step-by-step:. Root? ∆ = B2 – 4AC = 6^2 – ( 4 \times -3!
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2021-06-14 14:41:02
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https://research.utwente.nl/en/publications/estimation-of-a-regular-conditional-functional-by-conditional-u-s
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# Estimation of a regular conditional functional by conditional U-statistics regression
Research output: Working paper
## Abstract
U-statistics constitute a large class of estimators, generalizing the empirical mean of a random variable $X$ to sums over every $k$-tuple of distinct observations of $X$. They may be used to estimate a regular functional $\theta(P_{X})$ of the law of $X$. When a vector of covariates $Z$ is available, a conditional U-statistic may describe the effect of $z$ on the conditional law of $X$ given $Z=z$, by estimating a regular conditional functional $\theta(P_{X|Z=\cdot})$. We prove concentration inequalities for conditional U-statistics. Assuming a parametric model of the conditional functional of interest, we propose a regression-type estimator based on conditional U-statistics. Its theoretical properties are derived, first in a non-asymptotic framework and then in two different asymptotic regimes. Some examples are given to illustrate our methods.
Original language English 35 Published - 26 Mar 2019 Yes
## Keywords
• U-stqtistics
• regression-type models
• conditional distribution
• Penalization method
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2021-01-27 07:05:25
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https://www.thejournal.club/c/paper/336926/
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#### $\alpha$-Geodesical Skew Divergence
##### Masanari Kimura, Hideitsu Hino
The asymmetric skew divergence smooths one of the distributions by mixing it, to a degree determined by the parameter $\lambda$, with the other distribution. Such divergence is an approximation of the KL divergence that does not require the target distribution to be absolutely continuous with respect to the source distribution. In this paper, an information geometric generalization of the skew divergence called the $\alpha$-geodesical skew divergence is proposed, and its properties are studied.
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2022-06-25 14:15:37
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https://zenodo.org/record/4967197/export/xd
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Journal article Open Access
# A study on the relationship between relaxed metrics and indistinguishability operators
Pilar Fuster-Parra; Javier Martin; Juan José Miñana; Óscar Valero
### Dublin Core Export
<?xml version='1.0' encoding='utf-8'?>
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<dc:creator>Pilar Fuster-Parra</dc:creator>
<dc:creator>Javier Martin</dc:creator>
<dc:creator>Juan José Miñana</dc:creator>
<dc:creator>Óscar Valero</dc:creator>
<dc:date>2019-08-01</dc:date>
<dc:description> In 1982, E. Trillas introduced the notion of indistinguishability operator with the main aim of fuzzifying the crisp notion of equivalence relation. In the study of such a class of operators, an outstanding property must be stressed. Concretely, there exists a duality relationship between indistinguishability operators and metrics. The aforesaid relationship was deeply studied by several authors that introduced a few techniques to generate metrics from indistinguishability operators and vice-versa. In the last years a new generalization of the metric notion has been introduced in the literature with the purpose of developing mathematical tools for quantitative models in Computer Science and Artificial Intelligence. The aforesaid generalized metrics are known as relaxed metrics. The main purpose of the present paper is to explore the possibility of making explicit a duality relationship between indistinguishability operators and relaxed metrics in such a way that the aforementioned classical techniques to generate both concepts, one from the other, can be extended to the new framework.</dc:description>
<dc:description>This is a preprint version of the paper available from https://link.springer.com/article/10.1007/s00500-018-03675-9. This work is also supported by project PGC2018-095709-B-C21 (MCIU/AEI/FEDER, UE), and PROCOE/4/2017 (Govern Balear, 50% P.O. FEDER 2014-2020 Illes Balears).</dc:description>
<dc:identifier>https://zenodo.org/record/4967197</dc:identifier>
<dc:identifier>10.5281/zenodo.4967197</dc:identifier>
<dc:identifier>oai:zenodo.org:4967197</dc:identifier>
<dc:relation>info:eu-repo/grantAgreement/EC/H2020/779776/</dc:relation>
<dc:relation>doi:10.5281/zenodo.4967196</dc:relation>
<dc:rights>info:eu-repo/semantics/openAccess</dc:rights>
<dc:source>Soft Computing 23 6785–6795</dc:source>
<dc:subject>indistinguishability operator</dc:subject>
<dc:subject>relaxed metrics</dc:subject>
<dc:title>A study on the relationship between relaxed metrics and indistinguishability operators</dc:title>
<dc:type>info:eu-repo/semantics/article</dc:type>
<dc:type>publication-article</dc:type>
</oai_dc:dc>
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2021-12-07 16:11:51
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https://dsp.stackexchange.com/questions/18009/float-image-to-integer-type-image
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# float image to integer type image
I have a multispectral satellite image and I would like to "transform" each of the image bands into one with integer values. For example to stretch? the image between 0 and 255. Not sure how to proceed with this. I also thought I could make a histogram and replace each pixel value with the bin edge in which it falls into. I'm not sure. I'm currently working in python. Any thoughts on this?
• What does your image look like now? Oh, and if it was captured by a satellite/computer, it's already discrete. – Scott Aug 29 '14 at 1:04
• Yeah, sorry about that. Bad habbit of mine. Image is float and I want to make it an integer. – JEquihua Aug 29 '14 at 2:43
• If I'm interpreting your question correctly, I would do something like x = floor(x * 255) with numpy. – Scott Aug 29 '14 at 16:04
You should check the Envi documentation (a standard tool for working with multi/hyper spectral images) on the stretches it uses to fit multi/hyperspectral data into 8 bits for display.
Here is the page on some of the stretches used, which apply an affine function to the pixel values and clip the values to the lowest and highest displayable value. In particular, look at the optimized linear stretch, which is the default one Envi uses to display things.
As pointed out by Scott in comment, the quick'n dirty way of doing it is to just multiply by 255 and taking the integer part of the result.
The actual answer is (unfortunately, it's a very common answer to many Image Processing tasks...) "it depends":
• float images are usually assumed to have values between 0 and 1, but maybe it's not the case of your sensor, in which case you need to pre-multiply the values in th equi'n dirty solution by the inverse of the dynamic range of the data;
• if you need to perform some contrast enhancement, you can apply this pre-multiplication channel-wise (by first computing the max value for each channel), for all the data at once (by looking for the max among all the channels), or you can allocate the dynamic range in a more astute way (for example by computing histograms as you suggest, or clustering on the data values...);
• multispectral images contain usually several spectral channels ;-). If you want to visualize your data, then maybe you're also looking for a way to find a mapping between your data and Red-Green-Blue channels. Sensors that deliver Near Infrared-Red-Green data are usually remapped "as is" (following the ordering of the wavelengths), ie NIR -> Red, Red -> Green, Green -> Red, but composite values such as NDVI can be useful in some contexts;
• if you have lots of input spectral wavelengths, then you need to choose between visualizing each channel independently, or in a cube...
I think your description make it harder to understand the question.
It would be easier if you show the range of float image you have.
Is it between 0 and 1?
if it is, so you want to cast it into 0-255 integer. right? just multiply your image by 255 and round it to integer.
Note that, this casting will make your image worse (errors from rounding)
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2021-01-23 21:05:11
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https://www.allaboutcircuits.com/technical-articles/16-boolean-logic-functions-of-2-input-system/
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Technical Article
# The 16 Boolean Logic Functions of Two-Input Systems
March 01, 2020 by Sneha H.L.
## Learn about all 16 possible logic functions that can be realized for two binary inputs.
Boolean logic has been ruling the world of computational digital systems for many decades. Nonetheless, a few logic functions have been overlooked considerably.
In this article, I make an attempt to shed light on some of the forgotten logic functions concerned with two-input variables.
### Number of Boolean Functions
Before proceeding any further, let's try to remember the logical operations associated with a single input variable. NOT – yes, that’s right.
Is that all?
The answer from a majority of us would be “Yes.”
Well now, let's try to answer the same question when the number of inputs changes to two instead of one. This time, we seem to have a handful of operations: AND, OR, XOR, XNOR, NAND, NOR, and of course, NOT.
Are we done…? The answer would again be a resounding “Yes.” But, sorry to say, the correct answer is “No” for both cases.
Actually, there can be $$2^{2^{n}}$$ logic functions for n input variables. This means there can be $$4 (= 2^{2^{1}} = 2^2)$$ possible outcomes of a single input variable; $$16 (= 2^{2^{2}} = 2^4)$$ outputs from two-input system; $$256(= 2^{2^{3}} = 2^8)$$ outputs from three-input system, and so on.
Accordingly, there should be three and nine more logical functions corresponding to one-input and two-input systems apart from the one and seven quoted above, respectively.
### Complete List of Boolean Functions for 1- and 2- Input Variables
The complete list of Boolean functions for a system with input A and another system with inputs A and B are shown in Tables 1 and 2, respectively.
Name of the Boolean Function Boolean Function Meaning Null 0 Always 0 Identity 1 Always 1 Transfer A Pass value of A NOT Ā Pass negated value of A
##### Table 1. Complete list of Boolean functions for a single input system
Name of the Boolean Function Boolean Function Meaning Null $$0$$ Always 0 Identity $$1$$ Always 1 Transfer $$A$$ Pass value of A $$B$$ Pass value of B NOT $$\bar{A}$$ Pass negated value of A $$\bar{B}$$ Pass negated value of B AND $$A \bullet B$$ 1 only if A and B both are 1 NAND $$\overline {A \bullet B}$$ 0 only if A and B both are 1 OR $$A + B$$ 0 only if A and B both are 0 NOR $$\overline {A + B}$$ 1 only if A and B both are 0 Implication $$A + \bar {B}$$ If B, then A $$\bar {A} + B$$ If A, then B Inhibition $$A \bullet \bar {B}$$ A but not B $$\bar {A} \bullet B$$ B but not A EX-OR $$A \oplus B$$ A or B, but not both EX-NOR $$\overline {A \oplus B}$$ 1 if A equals B
##### Table 2. Complete list of Boolean functions for a dual input system
Having known the list now, let's try to quite deeply dive into their meaning, implementation, and usage (in terms of electronics). Actually, although the discussion presented is based on the two-input system, it is applicable even for a single-input system.
#### Null and Identity
Null and identity Boolean functions result in 0 (low) and 1 (high) output respectively, no matter what value the input variable holds. Implementation-wise, they require zero switches as they just need to pull the output line either to low or high.
Certain IC pins that demand to be pulled low or high to ensure satisfactory functioning stand as an example application for null and identity logic functions.
#### Transfer
The transfer Boolean function is the one in which the output promptly reflects the input. In the case of two-inputs, A and B, the output can be a replica of either A or B. Even in this case, we require no switches as there is no switching action to be performed.
The transfer logic function finds its application in power amplifiers where the output of the preceding section is connected as an input to the successive one.
#### NOT
In the NOT function, the output will be the negated value of the input, i.e., logic low output for logic high input, and vice versa. Further, if there are two inputs, A and B, the output can be the complemented version of either A or B. Implementation of a single-variable NOT function demands two switches.
NOT gates are used as a primary component while designing square wave oscillators.
#### AND and NAND
The AND function generates a high output only if all of its inputs go high. The complementary version of the AND is referred to as NAND. Thus, the NAND function yields low output only when both of its inputs are high.
A two-input AND gate can be implemented using two switches connected in series; a slightly different arrangement of the same resources would realize a NAND function.
AND (and thus NAND) gates find employment in gating circuits used in most digital circuits to ensure clock synchronization.
#### OR and NOR
The OR function yields low output only when all of its inputs go low. The inversion of this, i.e., the output going high when all of the inputs go low, represents the NOR function.
The realization of OR and NOR for two-input variables requires two switches to be connected in parallel.
OR (and hence NOR) functions form an important part of control systems where some safety measures must be initiated based on the output from sensory systems.
#### EX-OR and EX-NOR
The EX-OR Boolean function produces high output when only one of the inputs is high. Equivalently, if both inputs go high/low, the output would be low.
By contrast, the output of EX-NOR function goes high when both inputs assume same value—either low or high. These functions are employed as parity generators.
#### Implication
In implication logic, the output will either be driven to a high or would replicate the state of one of the inputs, depending on the value of the other input. That is, for a two-input case (A and B), the output (Y) will assume the same state as that of the second input (B) when the first input (A) would go high.
Next, if A is 0, then the output (Y) will be driven high irrespective of the state of B
The truth table corresponding to this implication logic is as shown in Table 3.
Inputs Output A B Y 0 0 1 0 1 1 1 0 0 1 1 1
##### Table 3. Truth table of implication logic for A implies B case
By drawing a K-map as seen in Figure 1, the simplified logic function can be obtained as $$\bar {A} + B$$, and is referred to as A implies B. Similarly, we can even have B implies A function with the logic expression, $$A + \bar {B}$$.
##### Figure 1. K-map for A implies B implication logic
The implication function can be realized using a relatively simple circuit comprising of a combination of memristors and a conventional resistor. This logic can effectively function as a non-volatile memory as depicted in this research paper by S K Vatinsky et al.
#### Inhibition
The inhibition function is the complementary version of implication and thus has its truth table as defined by Table 4 for A inhibits B case.
Inputs Output A B Y 0 0 0 0 1 0 1 0 1 1 1 0
##### Table 4. Truth table of inhibition logic for A inhibits B case
With this, the logic expression for the function would result as $$A \bullet \bar {B}$$ as shown by the K-map of Figure 2.
##### Figure 2. K-map for A inhibits B inhibition logic
Likewise, for B inhibits A, the expression would be $$\bar {A} \bullet B$$. Application of inhibition logic in the field of artificial neural networks is explained in detail in the Springer supplement authored by R. Rojas.
### Conclusion
Every Boolean function has its own meaning and serves a purpose. Widespread use of a few of them need not necessarily imply that the others are not also useful.
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2020-09-20 01:04:02
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https://nrich.maths.org/5972
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### Spinners
How do scores on dice and factors of polynomials relate to each other?
### Data Matching
Use your skill and judgement to match the sets of random data.
### Into the Exponential Distribution
Get into the exponential distribution through an exploration of its pdf.
# Random Inequalities
##### Age 16 to 18 Challenge Level:
In this problem we look at two general 'random inequalities'.
Part 1
Markov's inequality tells us that the probability that the modulus of a random variable $X$ exceeds any random positive number $a$ is given by a universal inequality as follows:
$$P(|X|\geq a) \leq \frac{E(|X|)}{a^{??}}$$
In this expression the exponent of the denominator on the right hand side is missing, although Markov showed that it is the same whole number for every possible distribution. Given this fact, experiment with the various distributions to find the missing value (??).
Part 2
Another important general statistical result is Chebyshev's inequality, which says that
$$P(|X-\mu|\geq k\sigma)\leq \frac{1}{k^2}$$ where $\mu$ and $\sigma$ are the mean and standard devitation of the distribution $X$ respectively. This is true for any distribution and any positive number $k$. Can you make a probability distribution for which the inequality is exactly met when $k=2$? In other words, use the distribution maker to create a distribution $X$ for which $$P(|X-\mu|\geq 2\sigma)=\frac{1}{4}$$
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2019-07-19 17:27:31
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