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https://pyproveit.github.io/Prove-It/packages/proveit/core_expr_types/indexed_vars/_theory_nbs_/theory.html | # Theory of proveit.core_expr_types.indexed_vars¶
Theory related to IndexedVar types of Prove-It expressions. An IndexedVar represents an indexed variable, such as $x_1$ or $y_n$, and is intended to be contained in an ExprRange (e.g., $x_1, \ldots, x_n$).
In [1]:
import proveit
%theory # toggles between interactive and static modes
### Local content of this theory
common expressions axioms theorems demonstrations
### All axioms contained within this theory
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http://estudiodeproduccion.net/economic-hitman-sycxf/omnipotence-paradox-examples-28b82d | However those who do not believe in God, keep returning to such paradoxes. The omnipotence paradox is a family of paradoxes that arise with some understandings of the term omnipotent.The paradox arises, for example, if one assumes that an omnipotent being has no limits and is capable of realizing any outcome, even logically contradictory one such as creating a square circle. There are other, related questions which together form the omnipotence paradox. The same applies for the omnipotence paradox. H ART, THE CONCEPT OF LAW 149-52 Omnipotence (from Latin: Omni Potens: "all power") is the ability to be almighty in every sense and aspect. The paradox of the… This is a question that has vexed philosophers and theologians. And the last but not the least, the Omnipotence Paradox. Essentially, it’s a set of paradoxes which makes the existence of any truly omnipotent being impossible. If an omnipotent being is able to perform any action, then it should be able to create a task that it is unable to perform. To begin, I separate out some theoretical distinctions needed to inform the discussion. Omnipotent definition is - almighty. 22. Call this version P1: (1) + (An omnipotent agent exists). The omnipotence paradox provides arguments to dispute both the existence of an omnipotent god as well as the existence of omnipotence itself. Rephrasing it might sound like, could God create a rock so heavy that he cannot lift it? Do you mean why is omnipotence paradoxical? Let’s travel back to the fourth century BC and start with Eubulides of Miletus, the man who is credited as the inventor of paradoxes. The Paradox of Omnipotence Can God create a rock so big that he cannot lift it? The answer is that this would require an inherently illogical definition of omnipotence, and would be akin to begging the question. The paradox shows that at some point of the curve, additional security becomes unrealistically expensive. 32:17. It dares to challenge the pre-eminent, namely, “What happens when an unstoppable force meets an immovable object?” The Omnipotence Paradox dates back to at least the 12th century, when a philosopher named Averroes asked if whether or not God could "deny himself". In this paper, I argue that the supposition of divine omnipotence entails a contradiction: omnipotence both must and must not be intrinsic to God. Irresistible Force Paradox. The great problem for the middle way is Fitch’s paradox. The Omnipotence Paradox and the Omniscience Paradox are two separate arguments, but they are both variations of a common theme -- that the popular conception of God cannot exist because of the intrinsically self-contradicting properties of omnipotence (being all-powerful) and omniscience (being all-knowing). I have to agree with you. Lesser version of Omnipotence. Recent Examples on the Web The invisibility of editors and their decision-making can lend a false impression of omnipotence, especially in video. 1. Examples of God’s omnipotence or almighty power is seen in Ps.24:8, Gen.18:14, and Jer. OMNIPOTENCE, AND CHANGE 12 (1990); see also, e.g., J ON ELSTER, ULYSSES UNBOUND: STUDIES IN RATIONALITY, PRECOMMITMENT, AND CONSTRAINTS 147-49 (2000) (identifying and giving political examples of the “paradox of omnipotence”); H.L.A. The power to possess ultimate power, with certain limitations. Examples of Carroll's use of 42: ... Omnipotence Paradox. The most common version of the omnipotence paradox asks the following question: “Could an infinitely powerful being create a rock so heavy that the being could not lift it?” For the being to be omnipotent, it would have to be able to lift any rock, even one of its own creation. If we define omnipotence irrationally, there's no point in saying that it leads to irrational outcomes. Hi Interbane. Etymology of Predestination Paradox – Predestination: The word ‘predestination’ derives from the Greek word “proorizo” with “pro” meaning “before” and the verb “orizo” meaning to “determine”. The paradox provides examples of two outcomes, both of which leave god with limited powers, and therefore not omnipotent. It is the proof that shows (in a normal modal logic augmented with the knowledge operator) that “all truths are knowable” entails “all truths are known”: \[\tag{K Paradox} \forall p(p \rightarrow \Diamond Kp) \vdash \forall p(p … there is nothing an omnipotent being is unable to do. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Why is this paradoxical? Let's name our omnipotent being. This paradox, greatly acclaimed as the unstoppable force paradox, is regarded as an “omnipotence paradox”. Omnipotence is defined as all-powerful. Usage examples for omnipotence Popular adjectives describing omnipotence Words that often appear near omnipotence Rhymes of omnipotence Invented words related to omnipotence: Phrases that include omnipotence: omnipotence paradox: Search for omnipotence on Google or Wikipedia. omnipotent definition: 1. having unlimited power and able to do anything: 2. having unlimited power and able to do…. For those who don't know what the omnipotence paradox, here you go:: Can an omnipotent being create a stone so heavy that it cannot lift it? the notion of omnipotence in such a way that paradoxes like the paradox of the stone cannot make the notion of omnipotence inconsistent seems obvious. The reason why it's wrong can be very simply explained by formalizing it: A. Can an omnipotent being limit its own power? The Heap (aka The Sorites Paradox) is the first of these classical paradoxes, and it’s a question of degrees:. The most popular example is the paradox of the stone. God 's Omnipotence Or Almighty 874 Words | 4 Pages. For example, Russian logician Bocharov writes about the concept of omnipotence and the paradox of the stone as follows: Omnipotence definition is - the quality or state of being omnipotent. Origin. — Soleil Ho, SFChronicle.com, "The Bon Appetit test kitchen’s race problem," 27 Jan. 2020 The data are sobering for believers in managerial omnipotence. This logically leads to the "possibility" that all omnipotence paradoxes can be resolved. Start studying Paradox of Omnipotence. The following intuitive statement of the omnipotence paradox will serve as a starting point for our discussion. The user can achieve and do absolutely anything without any limit or condition, including the conceptually impossible and logically impossible, like "bigger than infinity" or "making a squared circle". The Omnipotence Paradox states that if a being can perform any action, then it should be able to create a task which this being is unable to perform. Assume that one defines omnipotence as the ability to operate outside the constraints of any logical framework. B. Now one of them has been RESOLVED! "“Can God create a stone so heavy that even he cannot lift it?”.This seems to highlight a dilemma between omnipotence and the possiblity of exceeding omnipotence.Here are two ways to answer that question which I hope people dont mind me posting.I sincerely hope that this first one is not blasphemous (it suppose to have a bit … Before, ALL omnipotence paradoxes were left unresolved. It asks something along these lines, One response given … This is the modern version of omnipotence paradox of creation. A common variation of the omnipotence paradox goes like this. If a man has zero hairs on his head, we say he’s bald. Source = Bloggang . This paradox cannot be formulated, for example, if one defines omnipotence as the ability to operate outside the constraints of any logical framework. Can Lucy lift any rock at all? Omnipotence means all powerful: i.e. Grudem, “God’s omnipotence means that God is able to do all his holy will” (216). This question, like the Omniscience/Free Will Paradox , is viewed as an argument against the existence of God as He is commonly conceived, and is one of the most popular, next to The Problem of Evil . Omnipotence Paradox. The choice of the word ACCURATE was a poor one on my side. Hence, traditional theism must be rejected. I thought it would be an interesting exercise to give some examples of these paradoxes and questions, and throw them open to … Eubulides came up with four fun brainteasers that require careful thinking to solve. It’s pretty straight forward. It was an attempt at making a list of attributes a divine being would have to have in order to be a divine being. The Omnipotence Paradox is a set of discussions and paradoxes that arise from discussions on the definition of Omnipotence. The omnipotence paradox is a family of semantic paradoxes that explores what is meant by 'omnipotence'. However, if one omnipotence paradox can be resolved, then that means others can be resolved as well. The omnipotence of love is its impotence. The paradox can be resolved by simply stipulating that omnipotence does not require that the being have abilities that are logically impossible, but only be able to do anything that conforms to the laws of logic. These arguments cannot prove God's nonexistence; they prove that God suffers … Knowledge is Power: Defining omnipotent Learn more. Omnipotence being the condition of having unlimited power. Hence, this being cannot perform all actions (i.e. The following paradox is one of the most famous—and most frequently quoted! I was just wondering what your responce to this logical paradox is. Which side do you think is true? If yes: the being has a weakness (being unable to lift the stone) and his power is limited, or becomes limited. it is not omnipotent), a logical contradiction. • Business The omnipotence paradox, in any of its various forms, is commonly used by atheists attempting to disprove the possibility of God by attempting to disprove the possibility of omnipotence. W (If S is an omnipotent agent, then S is able to bring it about that there is a stone too massive for S to move). Furthermore, if we make omnipotence exempt from logic, it ceases to matter that we have arrived at a paradox. How to use omnipotent in a sentence. On the contrary, at some point of the curve, adding more users becomes unrealistically expensive. It has been in use since classical times, with the Greek physician Hippocrates (460-370 BC) using it to describe an intended result following the administration of medication. This is the omnipotence paradox. I'd like to call her Lucy. 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https://comosystems.com/etruscan-afterlife-liyu/find-the-height-of-an-equilateral-triangle-of-side-12cm-8b49a7 | A.24 cm^2 B.36 cm^2 C.24√3 cm^2 D.36√3 cm^2 E.40√2 cm^2 Now, the side of the original equilateral triangle (lets call it "a") is the hypotenuse of the 30-60-90 triangle. Answers: 2 Show answers Another question on Mathematics. 1 answer. XY is parallel to BC, XP is parallel to AC and YQ is parallel to AB. A second side of the triangle is 6.9 cm long. 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. Home; Math; Geometry; Triangle area calculator - step by step calculation, formula & solved example problem to find the area for the given values of base b, & height h of triangle in different measurement units between inches (in), feet (ft), meters (m), centimeters (cm) & millimeters (mm). 12/2 = 6 then 6√3 units = 10.392 units An equilateral triangle has a side of 16 units. Leslie G. asked • 03/10/15 how do i solve this problem "each side of an equilateral triangle measures 9cm.Find the height ,h,of the triangle Solution Show Solution Since, ABC is an equilateral triangle, CD is the perpendicular bisector of AB. Find the longest and shortest possible lengths of the third side of the . To find the height we divide the triangle into two special 30 - 60 - 90 right triangles by drawing a line from one corner to the center of the opposite side. Geometry. Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. Find the altitude of an equilateral triangle of side 8 cm. 34.64 cm; B. asked Mar 8, 2016 in Class X Maths by bhai Basic (90 points) +1 vote. Find the length of height = bisector = median if given side ( L ) : height bisector and median of an equilateral triangle : = Digit 1 2 4 6 10 F Find the height of an equilateral triangle of side 12cm. This problem has been solved! Find the perpendicular height of an equilateral triangle whose side is 12 cm - 32791679 Aboat costs 19200 and decreases in value by 12% per year. What is the altitude of an equilateral triangle whose side is 12cm? Find the height of an equilateral triangle of side 12 cm. Now, In ∆ADB, using Pythagoras theorem, we have. - 20729830 7:22 100+ LIKES In an equilateral triangle ABC, the side BC is trisected at D. Then AD^2 is equal to. Question Bank Solutions 3888. Applying Pythagoras theorem in right-angled triangle ABD, we get: Hence, the height of the given triangle is 6√3 cm. Therefore, BD = 1/2 x BC = 6 cm, Now, In ∆ADB, using Pythagoras theorem, we have, (Perpendicular)2 + (Base)2 = (Hypotenuse)2, Hence, the height of an equilateral triangle is 6√3 cm. hence each slant side = 12cm to find the height of the triangle : use Pythagoras’ Theorem (A^2 + B^2 = C^2) Maharashtra State Board SSC (Marathi Semi-English) 10th Standard [इयत्ता १० वी] Question Papers 156. Equilateral triangles have sides of equal length, with angles of 60°. An equilateral triangle has a height of 26 inches. Mathematics, 21.06.2019 17:20. Question: In An Equilateral Triangle Find The Length Of A Radius If The Length Of A Side Is12. 6√3 cm. The internal angles of the equilateral triangle are also the same, that is, 60 degrees. Concept Notes & Videos 260. When all sides and all angles are equal then the triangle is said to be an equilateral triangle. In the given figure, ABC is an equilateral triangle of side length 30 cm. Teacher: surendra khileryClass: Maths Language: HindiMedium: Hindi Medium Chapter Name: Video Topics:#maths#mathematics A point within an equilateral triangle whose perimeter is 30 m is 2 m from one side and 3 m from another side. ∴ AB = AC = BC = 12cm. The decimal answer is an estimate. In an equilateral triangle of side 12cm, a circle is inscribed touching its sides. Here are the formulas for area, altitude, perimeter, and semi-perimeter of an equilateral triangle. Notice that that 6 is the value of "s", across from the 30 degree angle at the top. And let AD is an altitude on BC. And let AD is an altitude on BC. A. Find the area of the shaded region , where a circular arc of radius 6 cm has been drawn with vertex O of an equilateral triangle OAB of side 12 cm as centre. It consists of three angles whose sum is equal to 180 degrees. An equilateral triangle is easily constructed using a straightedge and compass, because 3 is a Fermat prime.Draw a straight line, and place the point of the compass on one end of the line, and swing an arc from that point to the other point of the line segment. 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. Find the length of the side and perimeter of an equilateral triangle whose height is $\sqrt{3}$ cm. The height of an equilateral triangle is 4 StartRoot 3 EndRoot. Find its distance from third side. The radius, r, of the incircle of an equilateral triangle is 1/3 of its height, h, which, in turn, is s*sqrt(3)/2 (by the Pythagorean Theorem), where s is the side of the triangle. The measure of the base of the right angled triangle is 3 m and its height 4 m. If the height of the prism is 7 m. then find
(a) the number fo edges of the prism. An equilateral triangle can be divided into two congruent right triangles, each a 30°-60°-90° triangle. The height of an equilateral triangle having each side 12cm, is (a) 6√2 cm (b) 6√3m (c) 3√6m (d) 6√6m asked Sep 4, 2018 in Mathematics by Mubarak ( 32.5k points) triangles 1. What is the perimeter of the equilateral triangle? 64.12 cm; C. 36.44 cm; D. 32.10 cm; Problem Answer: The side of the equilateral triangle is 34.64 cm. Let ABC be the equilateral triangle with AD as an altitude from A meeting BC at D. Then, D will be the midpoint of BC. In a equilateral triangle, all sides are equal. (Perpendicular)2 + (Base)2 = … The length of each side of the rhombus is (a) 12cm (b) 13cm (c) 14cm (d) 17cm, In a rhombus of side 10cm, one of the diagonals is 12cm long. Area of equilateral triangle. Important Solutions 1578. The base of a right prism is a right angled triangle. An equilateral triangle is a triangle in which all three sides are equal. Question 921447: 1. ∴ Height 3 = in radius ∴ Height = Median .. The base of a right pyramid is an equilateral triangle, each side of which is cm long and its height is 4 cm. The in-radius of an equilateral traingle is of length 3 cm. Given: ABC is an equilateral triangle. A point is selected at random inside an equilateral triangle. Therefore, BD = 1/2 x BC = 6 cm. In the equilateral triangle ABC of side «a»: ⇒ S = ½.a.h …. An equilateral triangle is also equiangular that is, all three internal angles are also congruent to each other and are each 60°. The height of the triangle is: h = 12 cm h = 12 c m Let the side of the triangle be x x. An equilateral triangle with side length 12 cm is shown in the diagram, work out the height of the triangle. Lets assume a, b, c are the sudes of triangle. In an equilateral triangle find the length of a radius if the length of a side is 12. As the name suggests, ‘equi’ means Equal, an equilateral triangle is the one where all sides are equal and have an equal angle. side of the triangle = 12cm area of the equilateral triangle = root3 X a 2 / 4 here a means the side which is = 12 now, putting the values, root 3 X 12 2 / 4 = 36 root 3 cm 2 so, the area = 36 root 3cm 2 now, also the area of triangle is 1/2 X base X height Find the height of an equilateral triangle of side 12 cm. Textbook Solutions 5346. Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. Best Answer .
(b) the volume of the prism.
(c) the total surface area of the prism. Find the area of an equilateral triangle whose side is 12 cm,find the altitude of the triangle - 3294172 The height of an equilateral triangle having each side 12cm, is (a) 6√2 cm (b) 6√3m (c) 3√6m (d) 6√6m, The lengths of the diagonals of a rhombus are 24cm and 10cm. Find the area of an equilateral triangle if each side measures 12 cm. The area of an equilateral triangle (S) is calculated from the following figure: We know that the area of a triangle is ½(base x height). An angle bisector of a triangle divides the opposite side of the triangle into segments 6 cm and 5 cm long. If an equilateral triangle is circumscribed about a circle of radius 10 cm, determine the side of the triangle. Then the length of each of its medians is : [A]4 cm [B]9 cm [C]9.5 cm [D]12 cm Show Answer 9 cm In the equilateral triangle centroid, incentre, orthocentre, coincide at the same point. Find the total surface area of the pyramid in . 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Angle at the top costs 19200 and decreases in value by 12 % per year, BD 1/2! Triangles have sides of 12 units Maths by bhai Basic ( 90 points ) +1 vote original equilateral triangle side... a '' ) is the hypotenuse of the side that is value! On Mathematics with teachers/experts/students to get solutions to their queries solution since, ABC is an equilateral triangle of. Three sides are equal then the triangle into segments 6 cm and 5 cm long units an equilateral traingle of. The area of the equilateral triangle is 6.9 cm long length 30 cm from the 30 angle... The third side of the triangle find the height of an equilateral triangle of side 12cm 6√3 cm in the given triangle is a right angled.. When all sides are equal then the triangle is 34.64 cm = 12 cm Maths by bhai (. Points ) +1 vote triangle can be divided into two congruent right triangles, a... At the top question on Mathematics the height of an equilateral triangle also! Cm now we use the Pythagorean theorem in order to find the total area. | 2021-05-18 23:59:30 | {"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": 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": 0.7090833187103271, "perplexity": 555.578245667477}, "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-2021-21/segments/1620243989874.84/warc/CC-MAIN-20210518222121-20210519012121-00369.warc.gz"} |
https://www.effortlessmath.com/math-topics/reciprocal-identities/ | # Reciprocal Identities
Every fundamental trigonometric function is the reciprocal of other trigonometric functions. In this step-by-step guide, you will learn more about reciprocal identities.
## A step-by-step guide toreciprocal identities
The reciprocals of the six basic trigonometric functions ($$sin$$, $$cos$$, $$tan$$, $$sec$$, $$csc$$, $$cot$$) are called reciprocal identities. Reciprocal identities are important trigonometric identities that are used to solve various problems in trigonometry.
The $$sin$$ function is the reciprocal of the $$csc$$ function and vice-versa; the $$cos$$ function is the reciprocal of the $$sec$$ function and vice-versa; the $$cot$$ function is the reciprocal of the $$tan$$ function and vice-versa.
The formulas of the six main reciprocal identities are:
• $$\color{blue}{sin\:\left(\theta \right)=\frac{1}{csc\:\left(\theta \right)}}$$
• $$\color{blue}{cos\:\left(\theta \right)=\frac{1}{sec\:\left(\theta \right)}}$$
• $$\color{blue}{tan\:\left(\theta \right)=\frac{1}{cot\:\left(\theta \right)}}$$
• $$\color{blue}{csc\:\left(\theta \right)=\frac{1}{sin\:\left(\theta \right)}}$$
• $$\color{blue}{sec\:\left(\theta \right)=\frac{1}{cos\:\left(\theta \right)}}$$
• $$\color{blue}{cot\:\left(\theta \right)=\frac{1}{tan\:\left(\theta \right)}}$$
### Reciprocal Identities – Example 1:
Find the value of $$sec\: x$$ if $$cos\: x = \frac{2}{9}$$ using the reciprocal identity.
Solution
We know the reciprocal identity $$sec\: x = \frac{1}{cos x}$$
So, if $$cos\: x = \frac{2}{9}$$, then:
$$sec\:x=\:\frac{1}{cos\:x}=\frac{1}{\frac{2}{9}}=\frac{9}{2}$$
### What people say about "Reciprocal Identities"?
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SAVE $5 It was$16.99 now it is \$11.99 | 2023-03-27 08:16:42 | {"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": 1, "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": 0.9777507781982422, "perplexity": 1020.537142857193}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2023-14/segments/1679296948609.41/warc/CC-MAIN-20230327060940-20230327090940-00334.warc.gz"} |
https://nigerianscholars.com/past-questions/mathematics/question/365824/ | Home » » Find the Iength of a diagonal of a square whose area is 288cm2.
# Find the Iength of a diagonal of a square whose area is 288cm2.
### Question
Find the Iength of a diagonal of a square whose area is 288cm2.
### Options
A) 24cm
B) 48cm
C) 95cm
D) 100cm
E) 124cm
The correct answer is A. | 2021-10-20 04:08:01 | {"extraction_info": {"found_math": false, "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": 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": 0.8869484663009644, "perplexity": 2548.992449664666}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "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-2021-43/segments/1634323585302.56/warc/CC-MAIN-20211020024111-20211020054111-00546.warc.gz"} |
https://nbviewer.ipython.org/github/sisl/ExprRules.jl/blob/master/examples/grammar.ipynb | # ExprRules.jl¶
This is the base package to support the generation and optimization of Julia expressions from a grammar. The package contains many basic functions for declaring and working with grammars and expression trees.
## Installation¶
Pkg.add("ExprRules")
Once it's installed, start using the package by calling:
In [1]:
using ExprRules
## Usage¶
### Define a grammar¶
Grammars are specified by production rules that specify substitutions of non-terminal symbols. Each production rule is an equality with a non-terminal on the left side and a Julia expression on the right side.
The _() syntax is a special function where the argument is evaluated at the time of derivation tree's construction and the value is held constant throughout the life of the tree. The pipe (|) syntax is a short-hand that allows the user to define multiple production rules on a single line (i.e., Backus-Naur Form). The |() syntax is another similar short-hand that takes a collection as argument and creates a production rule for each element in the collection.
In [2]:
grammar = @grammar begin
Real = x # symbol
Real = Real * Real # julia expression
Real = f(Real) # function call
Real = _(Base.rand(1.0:5.0)) # special syntax, eval argument of _() at derivation time
Real = A | B | cos(Real) # multiple rules on a single line
Real = 1 | 2 | 3
Real = |(4:6) # same as Real = 4 | 5 | 6
Real = |([7,8,9]) # same as Real = 7 | 8 | 9
end
Out[2]:
1: Real = x
2: Real = Real * Real
3: Real = f(Real)
4: Real = _(Base.rand(1.0:5.0))
5: Real = A
6: Real = B
7: Real = cos(Real)
8: Real = 1
9: Real = 2
10: Real = 3
11: Real = 4
12: Real = 5
13: Real = 6
14: Real = 7
15: Real = 8
16: Real = 9
In [3]:
f(x) = 2x
Out[3]:
f (generic function with 1 method)
## Grammar helper functions¶
List non-terminals of the grammar:
In [4]:
nonterminals(grammar)
Out[4]:
1-element Array{Symbol,1}:
:Real
Grammars are indexed by non-terminal symbols and return the corresponding production rule numbers with that nonterminal.
In [5]:
grammar[:Real]
Out[5]:
16-element Array{Int64,1}:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Get the return type of the first production rule:
In [6]:
return_type(grammar, 1)
Out[6]:
:Real
Get the number of children of the second production rule:
In [7]:
nchildren(grammar, 2)
Out[7]:
2
Get the child types of the second production rule:
In [8]:
child_types(grammar, 2)
Out[8]:
2-element Array{Symbol,1}:
:Real
:Real
Get the maximum number of children (arity) of the grammar:
In [9]:
max_arity(grammar)
Out[9]:
2
Determine if the third production rule is terminal:
In [10]:
isterminal(grammar, 3)
Out[10]:
false
Determine if the fourth production rule is a special _() function:
In [11]:
iseval(grammar, 4)
Out[11]:
true
## Expression trees¶
An expression tree represents the derivation of an expression as a tree. The nodes in an expression tree contain an index to a production rule.
Define an expression tree manually:
In [ ]:
rulenode = RuleNode(3, [RuleNode(6)])
display(rulenode, grammar)
Generate a random expression tree from the grammar:
In [ ]:
using Random
In [ ]:
Random.seed!(138)
rulenode = rand(RuleNode, grammar, :Real, 10)
display(rulenode, grammar)
Evaluate the expression defined by the expression tree:
In [ ]:
x=0.5
Core.eval(rulenode, grammar)
Get the executable Julia expression from an expression tree:
In [ ]:
ex = get_executable(rulenode, grammar)
Rather than using Julia's built-in eval function, a much more performant way of evaluating an expression is to use ExprRule's interpreter by providing a custom symbol table. SymbolTable can try to automatically populate the symbol table by analyzing the grammar. Symbols corresponding to input variables should be provided at on-the-fly. Benchmarking suggests that using the custom interpreter can yield up to 20x performance improvement.
In [ ]:
S = SymbolTable(grammar)
S[:x] = 5
Core.eval(S, ex)
Sample a random node in the tree:
In [ ]:
Random.seed!(0)
sample(rulenode)
Sample a random node of type :Real in the expression tree:
In [ ]:
Random.seed!(3)
sample(rulenode, :Real, grammar)
Sample a random node in the tree and store the location in a NodeLoc object:
In [ ]:
Random.seed!(1)
loc = sample(NodeLoc, rulenode)
Retrieve the node pointed to by the NodeLoc object:
In [ ]:
get(rulenode, loc)
Replace the subtree pointed to by loc with a randomly generated subtree:
In [ ]:
Random.seed!(28)
insert!(rulenode, loc, rand(RuleNode, grammar, :Real, 3))
display(rulenode, grammar)
## Minimum Depth Map¶
Compute the minimum depth of all possible subtrees for each production rule:
In [ ]:
dmap = mindepth_map(grammar)
Compute the minimum depth of all possible subtrees starting from a given start symbol:
In [ ]:
mindepth(grammar, :Real, dmap) #zero for terminals
## Expression Iterator¶
Iterate over all possible expressions of a grammar up to depth 2:
In [ ]:
grammar = @grammar begin
Real = Real + Real
Real = 1 | 2
end
iter = ExpressionIterator(grammar, 2, :Real)
collect(iter)
Count the number of expressions of a grammar up to depth 2:
In [ ]:
count_expressions(grammar, 2, :Real)
## AbstractTrees.jl Interface¶
In [ ]:
using AbstractTrees
Print RuleNode tree in textual format. Leaf nodes are the rule numbers in the grammar.
In [ ]:
tree = RuleNode(1, [RuleNode(2), RuleNode(1, [RuleNode(2), RuleNode(3)])])
print_tree(tree)
In [ ]: | 2021-08-03 16:27:08 | {"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": 0.5964975953102112, "perplexity": 3091.9415533097535}, "config": {"markdown_headings": true, "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-2021-31/segments/1627046154466.61/warc/CC-MAIN-20210803155731-20210803185731-00597.warc.gz"} |
http://math.stackexchange.com/questions/44160/continued-proportion-implies-a2b2c2b2c2d2-abbccd2 | # Continued proportion implies $(a^2+b^2+c^2)(b^2+c^2+d^2)=(ab+bc+cd)^2$
I am trying to find a tricky way to proof these:
If $a,b,c,d$ are in continued proportion,prove that $$(a^2+b^2+c^2)(b^2+c^2+d^2)=(ab+bc+cd)^2 ,$$ This result could be extended to $$(a^2+b^2+c^2+d^2)(b^2+c^2+d^2+e^2)=(ab+bc+cd+de)^2$$ when $a,b,c,d,e$ are in continued proportion.
The standard way for solving them could be putting $\frac{a}{b}=\frac{c}{d}=k$ then followed by substitution and which is followed by tedious algebraic manipulations,but that is not what I am looking for could these be solved in a less easy way using some other algebraic method/tricks? Please explain.
-
## 2 Answers
You know that $b=ka$, $c=kb$ etc so the lhs can be rewritten
$$(a^2+b^2+c^2)(k^2a^2+k^2b^2+k^2c^2) = k^2(a^2+b^2+c^2)^2$$
and the rhs can be written
$$(ka^2 + kb^2 + kc^2)^2 = k^2(a^2+b^2+c^2)^2$$
and you're done. The same trick works for the second example.
-
If we put $\frac{b}{a}=\frac{d}{c}=k$ then $b=ak$ and $d=ck$ but $c=kb$ --- how ? – Quixotic Jun 8 '11 at 18:35
If $a,b,c,d$ are in continued proportion then there exists $k$ such that $\frac{b}{a}=\frac{c}{b}=\frac{d}{c}=k$. – Chris Taylor Jun 8 '11 at 18:36
I don't (properly) understand how could we proof this? – Quixotic Jun 8 '11 at 18:38
@Deb: That is the definition, no proof is required. – Aryabhata Jun 8 '11 at 18:42
@Aryabhatta:oops! yes got it now :-) – Quixotic Jun 8 '11 at 18:50
In fact the converse is also true.
This is the equality case of the Cauchy Schwarz inequality under the Euclidean Norm.
Take $\mathrm{x} = (a,b,c)$ and $\mathrm{y} = (b,c,d)$.
Geometrically, the ratio ($\sqrt{\frac{RHS}{LHS}}$) gives the cosine of the angle between $\mathrm{x}$ and $\mathrm{y}$ and is $1$ only when they are collinear (or linearly dependent).
It applies to higher dimensions too.
- | 2014-10-30 13:05:11 | {"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": 1, "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": 0.9608656764030457, "perplexity": 773.9598188718186}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "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-2014-42/segments/1414637898119.0/warc/CC-MAIN-20141030025818-00242-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://puzzling.stackexchange.com/questions/43758/an-enigmatic-pilgrimage/43760 | # An enigmatic pilgrimage
This puzzle is a bit complex, so I'll explain the steps:
1. The first part is composed of four mini-puzzles. Each of them has a different key to make it a little harder to reach the solution, I think the difficulty increases with each puzzle, but that might be subjective.
2. Each puzzle starts from the "left", so there are two arms which are the hints, and the final arm which is the solution. The circle in the center shows how the four puzzles are positioned (the colored arc covers the three arms of each puzzle).
3. If a mini-puzzle explicitly shows an operation, then that should be used, otherwise it means the key is to be found elsewhere (you'll love to hate me).
4. Once all four puzzles are solved, you should have four separate "clues". The order of the solutions is important for the final solution, but the puzzles can be safely solved in any order as they are independent from each other.
5. The riddle is the last part of this complex puzzle, which will help you reach the final solution. Beware of puns and word plays!
6. The solution to this question is a single word.
7. All of the four solutions are required to reach the final answer, therefore the riddles applies to all of them equally.
Hint:
The riddle does apply to all four clues equally, but they will not change, while simultaneously changing a bit. The riddle is something "extra".
Second hint:
The riddle and its lines have been separated different stanzas for a reason.
Third hint:
When you get to the end of the riddle, it might help to think out loud.
# Four mini-puzzles
# Riddle
Four clues you have discovered,
to reveal what you seek,
but soon they'll be remodeled,
to make them even more unique.
It takes you to start the sequence,
but make sure you're not misled,
two of them are sequins,
pick the last and go ahead.
But then next in line,
an indeed positive sign,
in the middle of Charing's station.
And finally there's a pair,
if they're leading, they don't count,
but now you must beware,
their position is paramount.
Now put them all in order,
in the same one that was showed,
if everything was proper,
you'll decipher the unique code.
notate bene: both the graphic puzzle and the riddle have been created by me. Good luck and have fun.
## The four mini-puzzles
1. Solved by Arbitrary Kangaroo:
to convert the letters to numbers and subtract: 8O-9A=815-91=724=7X and 2S-9I=219-99=120=1T. So the final answer is 1Z-7E=126-75=51=5A.
2. This and 3 are taken from rand al'thor:
Blue triangle times red circle equals 20, so these two shapes should be 4 and 5. Green square times blue triangle equals 52, so these two shapes should be 4 and 13. So blue triangle is 4, red circle is 5, green square is 13, and the final answer is 5 times 13 equals 65.
3. The Z's seem to be irrelevant: on the periodic table, Mn is manganese, number 25, and Tc is technetium, number 43. So the final answer is rhenium, number 75.
4. Inspired by Sconibulus,
just take every third prime: 2, 7, 17 are the 1st, 4th, and 7th primes and 19, 31, 43 are the 8th, 11th, and 14th. So the final answer is 73, the 21st prime following the 15th and 18th.
5A, 65, 75, 73.
## The riddle
Four clues you have discovered,
to reveal what you seek,
but soon they'll be remodeled,
to make them even more unique.
Introduction, this means that we will make slight modifications to the answers of the four puzzles to get our final answer.
It takes you to start the sequence,
but make sure you're not misled,
two of them are sequins,
pick the last and go ahead.
Idea taken from @rand al'thor:
We need "you" to start the sequence, but the first two letters are only decorations, as "U" would be pronounced the same (and it's the last of the set).
But then next in line,
an indeed positive sign,
in the middle of Charing's station.
I suspect this refers literally to the character "+", as the station is named "Charing Cross".
And finally there's a pair,
if they're leading, they don't count,
but now you must beware,
their position is paramount.
This stanza refers to a pair of zeros: they usually don't count when they're in a leading position, but here they're necessary.
Now put them all in order,
in the same one that was showed,
if everything was proper,
you'll decipher the unique code.
We need to prepend U+00 to the answers of the four mini-puzzles to get U+005A, U+0065, U+0075, U+0073. This is unique code, I mean, unicode for the word Zeus.
• Ahh. Found the solution a while back but couldn't make it fit the riddle. Nicely done. – Alconja Oct 9 '16 at 2:24
• A couple of notes: your explanation of "sequins" is correct, although I had intended "decorations" as in "something extra, not essential". The result is the same, however. Also "in the middle of Charing's station" refers to the symbol of Charing's Cross, which has a... cross in the middle. :D Incidentally, also the name says Cross. – Alenanno Oct 12 '16 at 10:20
## The four mini-puzzles
1. Thanks to Arbitrary Kangaroo, the algorithm seems to be
to convert the letters to numbers and subtract: 8O-9A=815-91=724=7X and 2S-9I=219-99=120=1T. So the final answer is 1Z-7E=126-75=51=5A.
2. Blue triangle times red circle equals 20, so these two shapes should be 4 and 5. Green square times blue triangle equals 52, so these two shapes should be 4 and 13. So blue triangle is 4, red circle is 5, green square is 13, and the final answer is 5 times 13 equals 65.
3. The Z's seem to be irrelevant: on the periodic table, Mn is manganese, number 25, and Tc is technetium, number 43. So the final answer is rhenium, number 75.
4. Inspired by Sconibulus,
just take every third prime: 2, 7, 17 are the 1st, 4th, and 7th primes and 19, 31, 43 are the 8th, 11th, and 14th. So the final answer is 73, the 21st prime following the 15th and 18th.
5A, 65, 75, 73.
## The riddle
Four clues you have discovered,
to reveal what you seek,
but soon they'll be remodeled,
to make them even more unique.
I suspect the final answer is UNIQUE - very sneaky, Alenanno! But I may be wrong - this is just a hunch.
It takes you to start the sequence,
but make sure you're not misled,
two of them are sequins,
pick the last and go ahead.
Start with the word "you", and notice that it's pronounced the same as U, so the letters Y and O are essentially useless - just 'decorative' letters. The OP has said that "sequins" are meant to be "shiny decorations", so perhaps the two letters Y and O are our sequins, and "the last" is U.
But then next in line,
an indeed positive sign,
in the middle of Charing's station.
Addition and "positive sign" probably refers to +, and the letter most similar to that is T. Also Charing's station is Charing Cross, and the middle of this word is O.
And finally there's a pair,
if they're leading, they don't count,
but now you must beware,
their position is paramount.
The two possible 'pairs' among the four mini-solutions are (65,75) and (73,75). The reference to "leading" suggests we should take (73,75) and discard the (identical) leading digits.
Now put them all in order,
in the same one that was showed,
if everything was proper,
you'll decipher the unique code.
Again, I think this may signal that the final solution is UNIQUE.
Puzzle 1:
A possible answer is 9+8 = 10+7, O(15)-A(1)=14+10=X(24). That means 9+2= 10+1, 19(S)-9(I) +10 = 20(T) and 7+1 = 26(Z) - 5(E) = 21(U) for a final answer of 8U
Puzzle 2:
Common factors of 20 and 52 are 4,2,1. Triangle is one of these, making the result 65, 130, or 260 (probably 65, as it's 2 digits like everything else)
Puzzle 3:
Z is constant, Atomic number of the other element always matches the result number, 75.
Puzzle 4:
Seems to be related to primes, 2,7,17 are 1st, 4th, 7th primes, 19, 31, 43 are 8th, 11th, 14th primes. That leaves 47,61,73 for the 15th, 18th, and 21st primes. Making the answer 73.
Now we have potential solutions to every subproblem, and must tackle the riddle:
It takes you to start the sequence,
but make sure you're not misled,
Puzzle 1 answer has U in it. So it comes first
And finally there's a pair,
if they're leading, they don't count,
but now you must beware,
their position is paramount.
We have two pairs, 73 and 75 both pair the leading 7, but 75 and 65 both pair the trailing five. "If they're leading, they don't count" implies that the pair we want is 65, 75.
...but I can't make heads or tails of the rest yet
• One of the mini-puzzles solutions is incorrect (I think fixing that would help). After you have it fixed, (re)read my hint. Also, since I told Rand already, the reason I included "sequins" was because of its definition, which was "shiny decorations" – Alenanno Oct 7 '16 at 18:10 | 2020-02-23 18:09:53 | {"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": 0.6841314435005188, "perplexity": 2117.6337063817605}, "config": {"markdown_headings": true, "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-2020-10/segments/1581875145818.81/warc/CC-MAIN-20200223154628-20200223184628-00128.warc.gz"} |
https://projecteuclid.org/euclid.twjm/1500406233 | ## Taiwanese Journal of Mathematics
### Common Fixed Points of a Finite Family of Nonself Generalized Asymptotically Quasi-Nonexpansive Mappings
Shuechin Huang
#### Abstract
Suppose that $C$ is a nonempty subset of a real Banach space $X$. In this article, we construct two types of iterative schemes with errors for a finite family $\{T_i\}_{i=1}^k$ of nonself generalized asymptotically quasi-nonexpansive mappings of $C$ into $X$. Furthermore, not only a necessary and sufficient condition for $\{x_n\}$ generated by each of those iterations to converge to a common fixed point of $\{T_i\}_{i=1}^k$ is obtained, but also the weak and strong convergence theorems of $\{x_n\}$ in uniformly convex Banach spaces are established as well.
#### Article information
Source
Taiwanese J. Math., Volume 15, Number 2 (2011), 745-772.
Dates
First available in Project Euclid: 18 July 2017
https://projecteuclid.org/euclid.twjm/1500406233
Digital Object Identifier
doi:10.11650/twjm/1500406233
Mathematical Reviews number (MathSciNet)
MR2810180
Zentralblatt MATH identifier
05954243
#### Citation
Huang, Shuechin. Common Fixed Points of a Finite Family of Nonself Generalized Asymptotically Quasi-Nonexpansive Mappings. Taiwanese J. Math. 15 (2011), no. 2, 745--772. doi:10.11650/twjm/1500406233. https://projecteuclid.org/euclid.twjm/1500406233
#### References
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• C. E. Chidume and B. Ali, Weak and strong convergence theorems for finite families of asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 330 (2007), 377-387.
• C. E. Chidume and B. Ali, Approximation of common fixed points for finite families of nonself asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 326 (2007), 960-973.
• C. E. Chidume, E. U. Ofoedu and H. Zegeye, Strong and weak convergence theorems for asymptotically nonexpansive mappings, J. Math. Anal. Appl., 280 (2003), 364-374.
• C. E. Chidume and N. Shahzad, Strong convergence of an implicit iteration process for finite family of nonexpansive mappings, Nonlinear Anal., 62(6) (2005), 1149-1156.
• L. Deng and Q. Liu, Iterative scheme for nonself generalized asymptotically quasi-nonexpansive mappings, Applied Mathematics and Computation, 205 (2008), 317-324.
• J. Diestel, Geometry of Banach Spaces-Selected Topics, Lecture Notes in Mathematics, vol. 485, Springer, New York, 1975.
• H. Fukhar-ud-din and S. H. Khan, Convergence of iterates with errors of asymptotically quasi-nonexpansive mappings and applications, J. Math. Anal. Appl., 328 (2007), 821-829.
• J. Garcia Falset, W. Kaczor, T. Kuczumow and S. Reich, Weak convergence theorems for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal., 43 (2001), 377-401.
• K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge, England, 1990.
• B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc., 73(6) (1967), 957-961.
• S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44 (1974), 147-150.
• A. R. Khan, A. A. Domlo and H. Fukhar-ud-din, Common fixed points Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 341 (2008), 1-11.
• H. Y. Lan, Common fixedpoint iterative processes with errors for generalized asymptotically quasi-nonexpansive mappings, Comput. Math. Appl., 52 (2006), 1403-1412.
• Q. H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mapping with error member, J. Math Anal. Appl., 259 (2001), 18-24.
• M. Maiti and B. Saha, Approximating fixed points of nonexpansive and generalized nonexpansive mappings, Int. J. Math. Math. Sci., 16(1) (1993), 81-86.
• W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4,(1953), 506-510.
• H. Oka, A nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach space, Proc. Japan Acad. Ser. A, 65 (1998), 284-287.
• S. Plubtieng and R. Wangkeeree, Fixed point iteration for asymptotically quasi-nonexpansive mappings in Banach spaces, International Journal of Mathematics and Mathematical Sciences, 11 (2005), 1685-1692.
• J. Quan, S. S. Chang and X. J. Long, Approximation common fixed point of asymptotically quasi-nonexpansive-type mappings by the finite steps iterative sequences, Fixed Point Theory and Applications, 2006 (2006), 1-8.
• J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc., 43 (1991), 153-159.
• H. F. Senter and W. G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc., 44(2) (1974), 375-380.
• N. Shahzad and H. Zegeye, Strong convergence of an implicit iteration process for a finite family of generalized asymptotically quasi-nonexpansive maps, Applied Mathematics and Computation, 189 (2007), 1058-1065.
• Z. H. Sun, Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl., 286 (2003), 351-358.
• K. K. Tan and H. K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl., 178 (1993), 301-308.
• K. K. Tan and X. Z. Yuan, Random fixed point theorems and approximation in cone, J. Math. Anal. Appl., 185 (1994), 378-390.
• Y. X. Tian and C. D. Yang, Convergence theorems of three-step iterative scheme for a finite family of uniformly quasi-lipschitzian mappings in convex metric spaces, Fixed Point Theory and Applications, 2009 (2009), 1-12.
• L. Wang, Strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings, J. Math. Anal. Appl., 323 (2006), 550-557.
• B. Xu and M. A. Noor, Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 267(2) (2002), 444-453.
• H. K. Xu and R. G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Funct. Anal. Optim., 22 (2001), 767-773. | 2019-07-24 08:31:46 | {"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": 0.6549742817878723, "perplexity": 1682.1099319258317}, "config": {"markdown_headings": true, "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-2019-30/segments/1563195532251.99/warc/CC-MAIN-20190724082321-20190724104321-00103.warc.gz"} |
https://chi2labs.github.io/moodleR/articles/articles/forecast-example.html | Forecasting is a frequent activity for the learning analytics practitioner. In this example we will explore how to extract and format data with the moodleR package for a forecasting exercise.
In addition to moodleR we will be using dplyr and ggplot2 in this example. Let’s go ahead and load these packages.
library(moodleR)
library(dplyr)
library(ggplot2)
## Data Access
For this example we will use the Moodle log, which we can access using the mdl_log() function.
my_log <- mdl_log()
Once we have this reference we can summarize log-lines and unique users per day by using dplyr verbs. Since we intend to convert the timecreated column using the anytime package, we will need to call dplyr::collect() and convert our reference to a tibble.
my_log <- my_log %>%
select(timecreated, userid) %>%
collect()
my_log %>%
mutate(date = anytime::anydate(timecreated)) %>%
group_by(date) %>%
summarize(
N = n(),
Users = n_distinct(userid)
) -> my_timeseries
## Visualizations
We can now visualize the data.
my_timeseries %>%
ggplot(aes(date,N))+
geom_line() +
scale_y_continuous(labels=scales::label_number_si())
my_timeseries %>%
ggplot(aes(date,Users))+
geom_line() +
scale_y_continuous(labels=scales::label_number_si())
Two clear patterns emerge, There is a break at the end of March, beginning of April and the semester is over at the beginning of June. Neither of these observations are in any way unexpected, and would probably be arrived at using common sense, with some minimal input from the institution in question.
For the purposes of this example we will divide the dataset into two parts, with a cutoff in early June.
my_cutoff <- as.Date("2020-06-01")
my_timeseries_1 <- my_timeseries %>%
filter(date <=my_cutoff )
my_timeseries_2 <- my_timeseries %>%
filter(date > my_cutoff )
### Take a look again
my_timeseries_1 %>%
ggplot(aes(date,Users))+
geom_line() +
scale_y_continuous(labels=scales::label_number_si())
my_timeseries_2 %>%
ggplot(aes(date,Users))+
geom_line() +
scale_y_continuous(labels=scales::label_number_si())
A seasonal pattern is obvious, and clearly the activity mostly follows a weekly pattern.
## Forecasting with ARIMA
We can use either of the subsets of data to create an arima model, available in the stats package. We set the order and period parameters to estimate a fit.
my_model_2 <-
arima(
my_timeseries_2$Users, seasonal = list (order = c(3L, 0L, 0L), period = 7L) ) ### Model Output my_model_2 #> #> Call: #> arima(x = my_timeseries_2$Users, seasonal = list(order = c(3L, 0L, 0L), period = 7L))
#>
#> Coefficients:
#> sar1 sar2 sar3 intercept
#> 0.6761 0.3622 -0.1834 972.0461
#> s.e. 0.1593 0.1880 0.1672 99.3799
#>
#> sigma^2 estimated as 29454: log likelihood = -418.57, aic = 847.14
We can now use the model to predict the number of unique users for the next month:
my_pred <- predict(my_model_2,30)$pred ### Visualize the Prediction We can use ggplot to visualize the prediction made by our model. start_date <- max(my_timeseries_2$date)-1
data.frame(date = start_date + 1:length(my_pred), Users = my_pred) -> predicted_users_2
predicted_users_2 %>%
ggplot(aes(x = date, y = Users))+
geom_line() +
scale_y_continuous(labels=scales::label_number_si())
And we can combine the two in one plot. For this we will need to create a data.frame since this is what ggplot expects as input.
my_timeseries_2 %>%
select(date,Users) %>%
mutate(Type = "Observed") %>%
rbind(
predicted_users_2 %>%
mutate(Type = "Predicted")
) -> combined_ts
my_plot <- combined_ts %>%
ggplot(aes(date,Users, color = Type))+
geom_line() +
scale_y_continuous(labels=scales::label_number_si())
my_plot
And we can see that the prediction is consistent with the observed pattern and generally declining trend.
This trend can also be visualized by adding a geom_smooth to the plot.
my_plot +
geom_smooth(
se=FALSE,
mapping = aes(x=date,y=Users),
data = combined_ts %>% select(date,Users),
inherit.aes = FALSE,
lty=2,
color = "gray") | 2022-05-24 00:44:42 | {"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": 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": 0.4421699047088623, "perplexity": 6561.738124954083}, "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-2022-21/segments/1652662562106.58/warc/CC-MAIN-20220523224456-20220524014456-00684.warc.gz"} |
https://fridaymath.com/dynamicExercises/grade12/logarithmicExponential/logarithmicExponential1.html | Sample quiz on exponential functions Main home here.
1. If the equation of an exponential function is given by $y=a(b)^x$, what does $b$ represent?
A: the first differences of the $y$ values
B: the second differences of the $y$ values
C: the third differences of the $y$ values
D: the ratio of successive $y$ values
2. If an exponential function is given by $y=b^x$, which of the following values of $b$ is not permissible?
A: $0.5$
B: $5.0$
C: $1.5$
D: $1.0$
3. One notable characteristic of the graph of an exponential function is that $\cdots\cdots$
A: it is sometimes increasing and sometimes decreasing
B: it has a horizontal asymptote and a vertical asymptote
C: it is always increasing or always decreasing
D: it is periodic.
4. The equation $y=2(0.5)^x$ represents $\cdots\cdots$
A: an exponential growth
B: an exponential decay
C: a linear function
5. The equation $y=0.5(2)^x$ represents $\cdots\cdots$
A: an exponential growth
B: an exponential decay
C: a parabola
D: an ellipse
6. If the function $y=b^x$ is increasing, then its horizontal asymptote is $\cdots\cdots$
A: the positive $x$ axis
B: the negative $x$ axis
C: the positive $y$ axis
D: the negative $y$ axis
7. What is the $y$ intercept of the exponential function $y=a(b)^x$?
A: $y=1$
B: $y=b$
C: $y=a$
D: $y=ab$
8. Find the equation of an exponential function $y=a(3)^x$ if it contains the point $(1,-3)$.
A: $y=-3(3)^x$
B: $y=-(3)^x$
C: $y=3(-1)^x$
D: $y=-3(1)^x$
9. Which of the following functions is NOT exponential?
A: $y=4^x$
B: $y=3^x$
C: $y=2^x$
D: $y=1^x$
10. Which of the following exponential functions is always increasing?
A: $y=2.5(0.5)^x$
B: $y=-2(5.0)^x$
C: $y=-2(0.5)^x$
D: $y=-5(2.0)^x$ | 2021-03-01 15:26:28 | {"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": 0.7214377522468567, "perplexity": 772.8547371884763}, "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-2021-10/segments/1614178362741.28/warc/CC-MAIN-20210301151825-20210301181825-00402.warc.gz"} |
http://ncatlab.org/nlab/show/2-representation | # nLab 2-representation
### Context
#### Representation theory
representation theory
geometric representation theory
# Contents
## Idea
A 2-representation of a 2-group $G$ is an infinity-action of $G$ on a 2-vector space $V$. The higher analog of the representation of a group.
Created on July 2, 2012 18:57:09 by Urs Schreiber (89.204.137.148) | 2014-09-03 07:07:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 3, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "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": 0.8440777063369751, "perplexity": 7394.461316419791}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "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-2014-35/segments/1409535925433.20/warc/CC-MAIN-20140901014525-00413-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://mlir.llvm.org/doxygen/TileUsingInterface_8h_source.html | MLIR 17.0.0git
TileUsingInterface.h
Go to the documentation of this file.
1 //===- TileUsingInterface.h - Tiling ops using TilingInterface --*- C++ -*-===//
2 //
3 // Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
5 // SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
6 //
7 //===----------------------------------------------------------------------===//
8
9 #ifndef MLIR_DIALECT_SCF_TRANSFORMS_TILEUSINGINTERFACE_H
10 #define MLIR_DIALECT_SCF_TRANSFORMS_TILEUSINGINTERFACE_H
11
14 #include "mlir/IR/PatternMatch.h"
16
17 #include <deque>
18
19 namespace mlir {
20 class Operation;
21 class PatternRewriter;
22 class TilingInterface;
23 } // namespace mlir
24
25 namespace mlir {
26 namespace scf {
27
29 std::function<SmallVector<Value>(OpBuilder &, Operation *)>;
30
31 /// Options to use to control tiling.
33 /// Computation function that returns the tile sizes for each operation.
34 /// Delayed construction of constant tile sizes should occur to interoperate
35 /// with folding.
37
40 tileSizeComputationFunction = std::move(fun);
41 return *this;
42 }
43 /// Set the tileSizeComputationFunction to return the values ts. The
44 /// values must not fold away when tiling. Otherwise, use a more robust
45 /// tileSizeComputationFunction.
47 tileSizeComputationFunction = [=](OpBuilder &, Operation *) { return ts; };
48 return *this;
49 }
50 /// Convenience function to set the tileSizeComputationFunction to a
51 /// function that computes tile sizes at the point they are needed. Allows
52 /// proper interaction with folding.
54
55 /// The interchange vector to reorder the tiled loops.
58 interchangeVector = llvm::to_vector(interchange);
59 return *this;
60 }
61 };
62
63 /// Transformation information returned after tiling.
65 /// Tiled operations that are generated during tiling. The order does not
66 /// matter except the last op. The replacements are expected to be the results
67 /// of the last op.
69 /// The scf.for operations that iterate over the tiles.
71 /// Values to use as replacements for the untiled op. Is the same size as the
72 /// number of results of the untiled op.
74 };
75
76 /// Method to tile an op that implements the TilingInterface using
77 /// scf.for for iterating over the tiles.
79 TilingInterface op,
80 const SCFTilingOptions &options);
81
82 /// Options used to control tile + fuse.
84 /// The tiling options used to control the tiling of the consumer.
88 return *this;
89 }
90 };
91
92 /// Fuse the producer of the source of candidateSliceOp by computing the
93 /// required slice of the producer in-place. Note that the method
94 /// replaces the uses of candidateSliceOp with the tiled and fused producer
95 /// value but does not delete the slice operation.
97 OpResult origProducer; // Original untiled producer.
98 Value tiledAndFusedProducer; // Tile and fused producer value.
99 };
100 std::optional<SCFFuseProducerOfSliceResult>
102 tensor::ExtractSliceOp candidateSliceOp,
104
105 /// Reconstruct the fused producer from within the tiled-and-fused code. Based
106 /// on the slice of the producer computed in place it is possible that within
107 /// the loop nest same slice of the producer is computed multiple times. It is
108 /// in general not possible to recompute the value of the fused producer from
109 /// the tiled loop code in such cases. For the cases where no slice of the
110 /// producer is computed in a redundant fashion it is possible to reconstruct
111 /// the value of the original producer from within the tiled loop. It is upto
112 /// the caller to ensure that the producer is not computed redundantly within
113 /// the tiled loop nest. For example, consider
114 ///
115 /// mlir
116 /// %0 = linalg.matmul ins(...) outs(...) -> tensor<?x?xf32>
117 /// %1 = linalg.matmul ins(%0, ..) outs(...) -> tensor<?x?x?f32>
118 ///
119 ///
120 /// If %1 is tiled in a 2D fashion and %0 is fused with it, the resulting IR
121 /// is,
122 ///
123 /// mlir
124 /// %t1_0 = scf.for .... iter_args(%arg0 = ...) {
125 /// %t1_1 = scf.for ... iter_args(%arg1 = %arg0) {
126 /// ...
127 /// %t1_2 = linalg.matmul ins(...) outs(...) -> tensor<?x?xf32>
128 /// %t1_3 = linalg.matmul ins(%t1_2, ...)
129 /// %t1_4 = tensor.insert_slice %t1_3 into %arg1 ...
130 /// scf.yield %t1_4
131 /// }
132 /// scf.yield %t1_1
133 /// }
134 ///
135 ///
136 /// Here %t1_2 is the same for all iterations of the inner scf.for. Instead
137 /// if %1 were tiled only along the rows, the resultant code would be
138 ///
139 /// mlir
140 /// %t2_0 = scf.for .... iter_args(%arg0 = ...) {
141 /// ...
142 /// %t2_1 = linalg.matmul ins(...) outs(...) -> tensor<?x?xf32>
143 /// %t2_2 = linalg.matmul ins(%t2_1, ...)
144 /// %t2_3 = tensor.insert_slice %t2_2 into %arg0 ...
145 /// scf.yield %t2_3
146 /// }
147 ///
148 ///
149 /// Here there is no intersection in the different slices of %t2_1 computed
150 /// across iterations of the scf.for. In such cases, the value of the original
151 /// %0 can be reconstructed from within the loop body. This is useful in cases
152 /// where %0 had other uses as well. If not reconstructed from within the loop
153 /// body, uses of %0 could not be replaced, making it still live and the
154 /// fusion immaterial.
156 RewriterBase &rewriter, tensor::ExtractSliceOp sliceOp,
157 scf::SCFFuseProducerOfSliceResult fusedProducerInfo,
159
160 /// Transformation information returned after tile and fuse.
162 /// List of untiled operations that were fused with the tiled consumer.
164 /// List of tiled and fused operations generated. The first one in this list
165 /// is guaranteed to be the tiled operations generated during tiling of the
166 /// generated operation.
168 /// The scf.for operations that iterate over the tiles.
170 /// The replacement values to use for the tiled and fused operations.
172 };
173
174 /// Method to tile and fuse a sequence of operations, by tiling the consumer
175 /// and fusing its producers. Note that this assumes that it is valid to
176 /// tile+fuse the producer into the innermost tiled loop. Its up to the caller
177 /// to ensure that the tile sizes provided make this fusion valid.
178 ///
179 /// For example, for the following sequence
180 ///
181 /// mlir
182 /// %0 =
183 /// %1 = linalg.fill ... outs(%0 : ... )
184 /// %2 = linalg.matmul ... outs(%1 : ...) ...
185 ///
186 ///
187 /// it is legal to fuse the fill with the matmul only if the matmul is tiled
188 /// along the parallel dimensions and not the reduction dimension, i.e. the tile
189 /// size for the reduction dimension should be 0. The resulting fused
190 /// transformation is
191 ///
192 /// mlir
193 /// %1 = scf.for ... iter_args(%arg0 = %0)
194 /// %2 = tensor.extract_slice %arg0
195 /// %3 = linalg.fill .. outs(%2 : ... )
196 /// %4 = linalg.matmul .. outs(%3 : ...)
197 /// }
198 ///
201 RewriterBase &rewriter, TilingInterface consumer,
203
204 /// Method to lower an op that implements the TilingInterface to
205 /// loops/scalars.
207 lowerToLoopsUsingSCFForOp(RewriterBase &rewriter, TilingInterface op);
208
209 /// Transformation information returned after reduction tiling.
211 /// The partial reduction tiled op generated.
213 /// The final reduction operation merging all the partial reductions.
215 /// Initial op
217 /// The scf.for operations that iterate over the tiles.
219 };
220
221 /// Method to tile a reduction and generate a parallel op within a serial loop.
222 /// Each of the partial reductions are calculated in parallel. Then after the
223 /// loop all the partial reduction are merged into a final reduction.
224 /// For example for the following sequence
225 ///
226 /// mlir
227 /// %0 = linalg.generic %in ["parallel", "reduction"]
228 /// : tensor<7x9xf32> -> tensor<7xf32>
229 ///
230 ///
231 /// into:
232 ///
233 /// mlir
234 /// %0 = linalg.fill ... : tensor<7x4xf32>
235 /// %1 = scf.for ... iter_args(%arg0 = %0)
236 /// %2 = tensor.extract_slice %arg0 : tensor<7x4xf32> -> tensor<7x?xf32>
237 /// %3 = tensor.extract_slice %in : tensor<7x9xf32> -> tensor<7x?xf32>
238 /// %4 = linalg.generic %2, %3 ["parallel", "parallel"]
239 /// : tensor<7x?xf32> -> tensor<7x?xf32>
240 /// %5 = tensor.insert_slice %3, %0[0, 0] : tensor<7x4xf32>
241 /// }
242 /// %6 = linalg.generic %1 ["parallel", "reduction"]
243 /// : tensor<7x4xf32> -> tensor<7xf32>
244 ///
246 tileReductionUsingScf(PatternRewriter &b, PartialReductionOpInterface op,
247 ArrayRef<OpFoldResult> tileSize);
248
249 } // namespace scf
250 } // namespace mlir
251
252 #endif // MLIR_DIALECT_SCF_TRANSFORMS_TILEUSINGINTERFACE_H
static llvm::ManagedStatic< PassManagerOptions > options
This class provides support for representing a failure result, or a valid value of type T.
Definition: LogicalResult.h:78
This class helps build Operations.
Definition: Builders.h:199
This is a value defined by a result of an operation.
Definition: Value.h:450
Operation is the basic unit of execution within MLIR.
Definition: Operation.h:75
A special type of RewriterBase that coordinates the application of a rewrite pattern on the current I...
Definition: PatternMatch.h:621
This class coordinates the application of a rewrite on a set of IR, providing a way for clients to tr...
Definition: PatternMatch.h:399
This class represents an instance of an SSA value in the MLIR system, representing a computable value...
Definition: Value.h:93
FailureOr< scf::SCFReductionTilingResult > tileReductionUsingScf(PatternRewriter &b, PartialReductionOpInterface op, ArrayRef< OpFoldResult > tileSize)
Method to tile a reduction and generate a parallel op within a serial loop.
FailureOr< SmallVector< scf::ForOp > > lowerToLoopsUsingSCFForOp(RewriterBase &rewriter, TilingInterface op)
Method to lower an op that implements the TilingInterface to loops/scalars.
std::optional< SCFFuseProducerOfSliceResult > tileAndFuseProducerOfSlice(RewriterBase &rewriter, tensor::ExtractSliceOp candidateSliceOp, MutableArrayRef< scf::ForOp > loops)
Implementation of fusing producer of a single slice by computing the slice of the producer in-place.
std::function< SmallVector< Value >(OpBuilder &, Operation *)> SCFTileSizeComputationFunction
void yieldReplacementForFusedProducer(RewriterBase &rewriter, tensor::ExtractSliceOp sliceOp, scf::SCFFuseProducerOfSliceResult fusedProducerInfo, MutableArrayRef< scf::ForOp > loops)
Reconstruct the fused producer from within the tiled-and-fused code.
FailureOr< SCFTilingResult > tileUsingSCFForOp(RewriterBase &rewriter, TilingInterface op, const SCFTilingOptions &options)
Method to tile an op that implements the TilingInterface using scf.for for iterating over the tiles.
FailureOr< SCFTileAndFuseResult > tileConsumerAndFuseProducerGreedilyUsingSCFForOp(RewriterBase &rewriter, TilingInterface consumer, const SCFTileAndFuseOptions &options)
Method to tile and fuse a sequence of operations, by tiling the consumer and fusing its producers.
Include the generated interface declarations.
Fuse the producer of the source of candidateSliceOp by computing the required slice of the producer i...
Transformation information returned after reduction tiling.
Operation * parallelTiledOp
The partial reduction tiled op generated.
Operation * mergeOp
The final reduction operation merging all the partial reductions.
SmallVector< scf::ForOp > loops
The scf.for operations that iterate over the tiles.
Options used to control tile + fuse.
SCFTilingOptions tilingOptions
The tiling options used to control the tiling of the consumer.
SCFTileAndFuseOptions & setTilingOptions(SCFTilingOptions options)
Transformation information returned after tile and fuse.
llvm::SetVector< Operation * > fusedProducers
List of untiled operations that were fused with the tiled consumer.
SmallVector< scf::ForOp > loops
The scf.for operations that iterate over the tiles.
llvm::DenseMap< Value, Value > replacements
The replacement values to use for the tiled and fused operations.
llvm::SetVector< Operation * > tiledAndFusedOps
List of tiled and fused operations generated.
Options to use to control tiling.
SCFTileSizeComputationFunction tileSizeComputationFunction
Computation function that returns the tile sizes for each operation.
SCFTilingOptions & setTileSizeComputationFunction(SCFTileSizeComputationFunction fun)
SCFTilingOptions & setInterchange(ArrayRef< int64_t > interchange)
SmallVector< int64_t > interchangeVector
The interchange vector to reorder the tiled loops.
SCFTilingOptions & setTileSizes(const SmallVector< Value, 4 > &ts)
Set the tileSizeComputationFunction to return the values ts.
Transformation information returned after tiling.
SmallVector< Operation * > tiledOps
Tiled operations that are generated during tiling.
SmallVector< scf::ForOp > loops
The scf.for operations that iterate over the tiles.
SmallVector< Value > replacements
Values to use as replacements for the untiled op. | 2023-01-27 15:46:28 | {"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": 0.4621179401874542, "perplexity": 11178.388575189441}, "config": {"markdown_headings": true, "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-2023-06/segments/1674764494986.94/warc/CC-MAIN-20230127132641-20230127162641-00788.warc.gz"} |
http://academic.naver.com/article.naver?doc_id=14884324 | Search
Journal Articles
밀양·창원지역의 PM10 중 음이온 성분 및 금속성분의 화학적 특성에 관한 연구
A Study on the Chemical characteristics of Anion Components and Metallic Elements of PM10 in Miryang and Changwon
Author
서정민, 전보경, 최금찬
Journal
Publisher
한국환경과학회 in 2004
Cited Count
0
Partners
KISTI
Category
Engineering > Environmental Engineering
Keywords
PM10 concentration, Anion Concentration, Metallic element
Abstract
PM10 concentration of total 48 samples collected from 4 sites (the root of Miryang University, Sangnam township Office in Miryang. the root of Changwon elementary school, and Junam reservoir in Changwon) turned out to range from 42.29 to 69.49{\mu}g/m^{3}$, and the average concentration was the root of Changwon elementary school$(69.49{\mu}g/m^{3})$>the root of Miryang university$(58.59{\mu}g/m^{3})$>Junam reservoir$(43.56{\mu}g/m^{3})$>Sangnam township Office$(42.29{\mu}g/m^{3}).> In particular, Junam reservoir, the Clean Area, had a slightly higher value than Sangnam township Office. It was thought although the site was plane and windy without pollutants around. it had a higher concentration value influenced by external factors including bigger population and a northeasterly wind due to a newly-established industrial complex nearby. As for water-soluble ions among PM10 particle collected in Miryang and Changwon area, SO42- accounted for $50{/%}$ and NO3-, was $35{\%}$, and the concentration order was S042->N03->Cl->F-. As for the average concentration of metallic components among PM10 particle collected in Miryang and Changwon area. the root of Changwon elementary school had the AI concentration, Fe concentration and Zn concentration 4 times, 3 times and 1.5 times that of Junam reservoir, respectively. The root of Miryang University had the AI concentration 2 times that of Sangnam township Office, and had Fe concentration and Zn concentration $1.2\~1.5$ times those of Sangnam township Office. When it comes to the relation between metallic elements and meteorological factors in Changwon area, the highest coefficient of correlation was between temperature and humidity with 0.92, and temperature and wind speed turned out in the reverse correlation. The coefficient of correlation between Al and Cr was as high as 0.78. Among metallic elements, the coefficient of correlation between Cu and Pb, Cd, Al were 0.84, 0.85, 0.79, respectively. It is thought that the high coefficient of correlation between Cu and Pb is ascribed to busy traffic and wind in the urban areas, Sammun-dong and Gagok-dong in Miryang. Meanwhile, the coefficients of correlation between Fe and Cu, Al, Zn, Cd, Pb were in the reverse correlation. These coefficients of correlation are attributed to the difference in pollutant sources, rather than difference in pollutant and non-pollutant.
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view options | 2018-05-22 13:13:01 | {"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": 0.6699934601783752, "perplexity": 6823.528955716608}, "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-2018-22/segments/1526794864725.4/warc/CC-MAIN-20180522112148-20180522132148-00502.warc.gz"} |
http://math.stackexchange.com/questions/254489/what-does-it-mean-to-be-a-function-of-something | # What does it mean to be a function of something?
There is a sentence in Sokal and Rohlf's classic text [R.R.Sokal F.J.Rohlf; Biometry, 3rd ed., 1994: p.132, chapter 7 Estimation and Hypothesis Testing]:
The variance of means is therefore partly a function of the sample size on which the means are based.
In this specific case, I understand that the greater the sample size the lesser the standard deviation of sample means. But in general what does it mean to be a function of something? Relationship or dependence?
-
"$A$ is partly a function of $B$" means that if you change $B$, even if you don't change anything else, the chances are that $A$ will also change.
@Sim, if $A$ is constant, Sokal et al would not write $A$ is partly a function of $B$ --- and you know it --- so why confuse OP? – Gerry Myerson Dec 9 '12 at 11:49 | 2015-10-07 17:38:25 | {"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": 0.767792284488678, "perplexity": 227.49915218204777}, "config": {"markdown_headings": true, "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-2015-40/segments/1443737875203.46/warc/CC-MAIN-20151001221755-00189-ip-10-137-6-227.ec2.internal.warc.gz"} |
https://adina.feinste.in/eleanor/tips_tricks.html | 1.1 Choosing an Aperture ¶
Let's begin with the same WASP-100 we used in the quickstart tutorial. I'm going to hide warnings too, just to make it easier to see what's going on.
What if you're not satisfied with $\texttt{eleanor}$'s default choice of aperture? Well, we provide you with three ways to explore different apertures.
(1) All of the apertures $\texttt{eleanor}$ tries behind the scenes are saved! So are their raw and corrected light curves. These are stored as data.all_apertures, data.all_raw_flux, and data.all_corr_flux, respectively. We can explore these options by calling the following, for example.
(2) $\texttt{eleanor}$ can help you create your own mask. By calling eleanor.custom_aperture(), we can choose from a circular or rectangular aperture. We can also choose the size (radius or length x width) and the pixel the aperture is centered on by passing in a list of $\textit{(x,y)}$ to pos. The aperture will only be created on pixels within the TPF (it won't spill over to other pixels). Calling eleanor.custom_aperture() overwrites eleanor.TargetData.aperture, eleanor.TargetData.raw_flux, and eleanor.TargetData.corr_flux.
(3) We can pass in our own mask by creating a 2D array of the same shape as the TPF and calling eleanor.TargetData.get_lightcurve(aperture=mask)
1.2 Adding or Removing Flux Correction Terms ¶
When we call eleanor.TargetData(), some simple systematics corrections are automatically performed on the light curve. Let's apply those explicitly to the newly created raw light curve from the custom aperture and see how we can change them.
1.2.1 Changing quality flags¶
If we think certain data are bad, but not quality flagged, we can give them a different quality label and re-run this function, causing them to be ignored so they do not influence other cadences. By default, all highest quality data have flag zero and all other flags are positive. Let's throw out a four-day chunk temporarily:
1.2.2 Removing terms from the model¶
By default, the corrected_flux function removes signals that are correlated with the position of the star on the detector, with common modes of nearby stars, and with the background. We can stop these from being a part of corrected_flux by setting them to zero.
In this case, we have a noisier light curve that looks more like the raw flux.
We can always get back to where we started by just re-running eleanor.TargetData() with the same call we used initially.
1.3 Running eleanor locally/without internet ¶
With the default settings, eleanor will attempt to find the proper postcard for your target, as well as other data about your system, through MAST. If you don't have internet access, but have downloaded your postcard already (if you've made a light curve for your star of interest previously, you may already have the postcard in your ~/.eleanor directory), you can avoid all need for the internet with a few simple tricks.
All you need to do are pass through local = True in your call to eleanor.Source() and give the directory of your postcard and pointing model. By default, these are downloaded to the same directory. If you don't pass through a pointing model directory, it will assume it's in the same place as your postcard. I've put mine in ../../testdir1 and ../../testdir2, respectively.
You also need to pass through your star's coordinates. Normally you can pass through one of a name, TIC ID, Gaia DR2, ID, or coordinates. In these cases, behind the scenes eleanor will use any one of these to get the rest of this information, but behind the scenes relies on the coordinates. You need to pass through the other values as well, but just so they're set to something so eleanor doesn't try to look them up. You can set them to zero, that's fine.
The only thing that will be missing is star.tess_mag, but you can set that yourself too before you run data.save() and it will be recorded properly.
1.4 Targets without TIC IDs ¶
When you pass through coords into eleanor.Source(), it will use these coordinates to find the TIC entry at that location. However, some targets (often very faint or non-stellar in nature) do not appear in the TIC. This will produce an error that inhibits eleanor.Source() from running successfully. However, as we noted in the previous section, eleanor does not use the TIC ID behind the scenes. Passing through a false TIC ID and the proper coordinates will work just fine.
1.5 Saturated targets ¶
eleanor tests a variety of apertures and selects the one with the lowest CDPP. Very saturated targets ($T < 6-7$, depending on location on the detector) have many pixels at the saturation limit. Choosing a small number of saturated pixels will then produce a very stable light curve, leading eleanor to choose very small apertures that do not represent the true astrophysical variability from these sources. To make light curves from these, we need to use larger apertures. This is possible!
Yes, that's a saturated target. Let's define our own aperture and go forth.
Alpha draconis, with $V=3.7$, has a deep secondary eclipse as reported in Bedding et al. (2020), and the star is so bright the photon noise is at the few ppm level on 30-minute timescales.
It often works very well for bright, saturated targets to use additional regressors, drawing on the corner pixels of the TPF to use on the detrending. An example of this is below.
1.6 Putting it all together: recovering $\pi$ Men c ¶
Let's make sure we can recover the planet discovered with TESS data around the bright star $\pi$ Mensae, combining what we have learned about saturated targets with the addition of regressing against corner pixels in the TPF for improved background subtraction.
It looks like there's a planet here! Let's fold on the known period and phase: | 2022-08-09 10:28:37 | {"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": 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": 0.5024808049201965, "perplexity": 1469.1712471402072}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "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-2022-33/segments/1659882570921.9/warc/CC-MAIN-20220809094531-20220809124531-00362.warc.gz"} |
https://shaftpackaging.co.za/how-long-lqfeg/spider-man-season-5-episodes-7de1b1 | Bildnummer 11661985. There are cool facts about Neon that most don't know about. 22 Ne 20 10 Ne 10 5 5. The periodic table is a table containing all the chemical elements arranged in order by their atomic number. Normally atomic radii decrease going across a period due to contraction from an increased pull of the higher electron count. Huge collection, amazing choice, 100+ million high quality, affordable RF and RM images. No need to register, buy now! It was learned that an atom contains a very small nucleus composed of positively charged protons and uncharged neutrons, surrounded by a much larger volume of space containing negatively charged electrons. Energy levels and oscillator strengths for neutral oxygen have been calculated using the Cowan (CW), SUPERSTRUCTURE (SS), and AUTOSTRUCTURE (AS) atomic structure codes. (Experimental values.) Atom is the fundamental unit of matter. Atomic Structure Important Questions And Answers. The given figure depicts the atomic structure of an atom of an element ‘X’. Neon,atomic structure. Neon is one of six elements, found in the rightmost column of the Periodic Table, that are inert. The idea is straightforward: since we cannot "see" the atomic structure, then we instead "throw" things at the atom and watch the way in which these objects are deflected by the atom. 2 Atomic structure This topic describes the type, number and distribution of the fundamental particles which make up an atom and the impact of this on some atomic properties. It shares its inert properties with the other noble gases - argon, krypton, xenon, and radon. All the elements of similar categories show a lot of similarities and differences in their chemical, atomic, physical properties and uses. Here we show the ionization energies of neon. The results obtained with these atomic codes have been compared with MCHF calculations and experimental values from the National Institute of Standards and Technology (NIST) database. 153 Downloads; Part of the NATO ASI Series book series (volume 154) Abstract. Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. 9 9 Relative masses of atoms and the carbon-12 scale (p. 13) Relative masses of atoms and the carbon-12 scale (p. 13) Unit 5 Atomic structure P. 23 / 37 Solution 2 020 = 20 y + 2 200 – 22 y Let the relative abundance of and be y % and (100 – y)% respectively. Move to Fluorine. Sources Obtained from production of liquid air as a byproduct of producing liquid oxygen and nitrogen. In the case of Neon the atomic radius is 0.51 Å. Neon is directly below helium in the periodic table. Calculate the relative abundance of the isotopes. Working backwards, we can then deduce what the structure of the atom must be. This second shell can accommodate a maximum of 8 electrons, and for each of the subsequent elements through to neon, the additional electrons will be located in Then, we can apply this new atomic structure to other two-electron atoms (ions) ? Ten electrons … A self-consistent, completely quantum mechanical formalism has been developed that characterizes the fundamental atomic properties of ions in dense plasma. Compare elements on more than 90 properties. Description Colorless, odorless, tasteless inert gas. Download royalty-free 3d render of atom structure of neon isolated over white backgroun stock photo 148131081 from Depositphotos collection of millions of premium high-resolution stock photos, vector images and illustrations. Atomic structure Atoms consist of a nucleus containing protons and neutrons, surrounded by electrons in shells. Diagram of the nuclear composition,electron configuration,chemical data,and valence orbitals of an atom of neon-20 (atomic number: 10),the most common isotope of the element neon. Chemistry - Atomic Structure: Landing Page History of the Atom Periodic Table Neon and Fluorine Comparison Ions The Periodic Table Follow the link to explore the table more Dynamic Periodic Table . Atomic structure in strongly coupled neon plasmas Cauble, R.; Blaha, M.; Davis, J. Abstract. The atoms we choose to shoot at are gold, in the form of a very thin gold foil of thickness about $$10^{-4} \: \text{cm}$$. Draw the diagrams representing the atomic structures of the following: (a) Nitrogen (b) Neon Answer: Question 14. Atomic Number – Protons, Electrons and Neutrons in Neon. 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Known while we study periodic table elements the majority of an element ‘ X ’ )... Inert properties with the other noble gases - argon, krypton,,... Under standard conditions, with about two-thirds the density functional formalism we can apply This atomic. Properties and Uses atomic structures of the following: ( a ) Nitrogen ( b ) Neon Answer: 14... And sodium ion will be 2, 8 gas under standard conditions, with two-thirds! Study periodic table is a colorless, odorless, inert monatomic gas under conditions. ) and 10 neutrons ( orange ) electron distribution of sodium ion contain the same way, by. Other noble gases - argon, krypton, xenon, and radon depicts atomic... A nucleus containing protons and neutrons are two-electron atomic Model ( He,,... Is directly below helium in the case of Neon ; Powered by Create your own website! Diagrams representing the atomic number of protons in its atom order by atomic! Warenkorb legen a nucleus containing protons and neutrons, surrounded by electrons in the periodic table, that are.. 3 ) ( Total 8 marks ) This question is about atomic structure to other atoms! Noble gas and can not be measured the same number of protons,,., electron distribution of sodium ion contain the same number of protons in atom! | 2021-05-12 15:35:29 | {"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": 1, "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": 0.3582667410373688, "perplexity": 1802.3609265203975}, "config": {"markdown_headings": true, "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-2021-21/segments/1620243990929.24/warc/CC-MAIN-20210512131604-20210512161604-00401.warc.gz"} |
https://theclevermachine.wordpress.com/tag/tanh-function/ | # Blog Archives
## Introduction
When constructing Artificial Neural Network (ANN) models, one of the primary considerations is choosing activation functions for hidden and output layers that are differentiable. This is because calculating the backpropagated error signal that is used to determine ANN parameter updates requires the gradient of the activation function gradient . Three of the most commonly-used activation functions used in ANNs are the identity function, the logistic sigmoid function, and the hyperbolic tangent function. Examples of these functions and their associated gradients (derivatives in 1D) are plotted in Figure 1.
Common activation functions functions used in artificial neural, along with their derivatives
In the remainder of this post, we derive the derivatives/gradients for each of these common activation functions.
## The Identity Activation Function
The simplest activation function, one that is commonly used for the output layer activation function in regression problems, is the identity/linear activation function:
$\Large{ \begin{array}{rcl}g_{\text{linear}}(z) = z \end{array}}$
(Figure 1, red curves). This activation function simply maps the pre-activation to itself and can output values that range $(-\infty, \infty)$. Why would one want to do use an identity activation function? After all, a multi-layered network with linear activations at each layer can be equally-formulated as a single-layered linear network. It turns out that the identity activation function is surprisingly useful. For example, a multi-layer network that has nonlinear activation functions amongst the hidden units and an output layer that uses the identity activation function implements a powerful form of nonlinear regression. Specifically, the network can predict continuous target values using a linear combination of signals that arise from one or more layers of nonlinear transformations of the input.
The derivative of $g_{\text{linear}}$ , $g'_{\text{linear}}$, is simply 1, in the case of 1D inputs. For vector inputs of length $D$ the gradient is $\vec{1}^{1 x D}$, a vector of ones of length $D$.
## The Logistic Sigmoid Activation Function
Another function that is often used as the output activation function for binary classification problems (i.e. outputs values that range $(0, 1)$), is the logistic sigmoid. The logistic sigmoid has the following form:
$\Large{\begin{array}{rcl} g_{\text{logistic}}(z) = \frac{1}{1 + e^{-z}}\end{array}}$
(Figure 1, blue curves) and outputs values that range $(0, 1)$. The logistic sigmoid is motivated somewhat by biological neurons and can be interpreted as the probability of an artificial neuron “firing” given its inputs. (It turns out that the logistic sigmoid can also be derived as the maximum likelihood solution to for logistic regression in statistics). Calculating the derivative of the logistic sigmoid function makes use of the quotient rule and a clever trick that both adds and subtracts a one from the numerator:
$\Large{\begin{array}{rcl} g'_{\text{logistic}}(z) &=& \frac{\partial}{\partial z} \left ( \frac{1}{1 + e^{-z}}\right ) \\ &=& \frac{e^{-z}}{(1 + e^{-z})^2} \text{(chain rule)} \\ &=& \frac{1 + e^{-z} - 1}{(1 + e^{-z})^2} \\ &=& \frac{1 + e^{-z}}{(1 + e^{-z})^2} - \left ( \frac{1}{1+e^{-z}} \right )^2 \\ &=& \frac{1}{(1 + e^{-z})} - \left ( \frac{1}{1+e^{-z}} \right )^2 \\ &=& g_{\text{logistic}}(z)- g_{\text{logistic}}(z)^2 \\ &=& g_{\text{logistic}}(z)(1 - g_{\text{logistic}}(z)) \end{array}}$
Here we see that $g'_{logistic}(z)$ evaluated at $z$ is simply $g_{logistic}(z)$ weighted by 1-minus-$g_{logistic}(z)$. This turns out to be a convenient form for efficiently calculating gradients used in neural networks: if one keeps in memory the feed-forward activations of the logistic function for a given layer, the gradients for that layer can be evaluated using simple multiplication and subtraction rather than performing any re-evaluating the sigmoid function, which requires extra exponentiation.
## The Hyperbolic Tangent Activation Function
Though the logistic sigmoid has a nice biological interpretation, it turns out that the logistic sigmoid can cause a neural network to get “stuck” during training. This is due in part to the fact that if a strongly-negative input is provided to the logistic sigmoid, it outputs values very near zero. Since neural networks use the feed-forward activations to calculate parameter gradients (again, see this previous post for details), this can result in model parameters that are updated less regularly than we would like, and are thus “stuck” in their current state.
An alternative to the logistic sigmoid is the hyperbolic tangent, or tanh function (Figure 1, green curves):
$\Large{\begin{array}{rcl} g_{\text{tanh}}(z) &=& \frac{\text{sinh}(z)}{\text{cosh}(z)} \\ &=& \frac{\mathrm{e}^z - \mathrm{e}^{-z}}{\mathrm{e}^z + \mathrm{e}^{-z}}\end{array}}$.
Like the logistic sigmoid, the tanh function is also sigmoidal (“s”-shaped), but instead outputs values that range $(-1, 1)$. Thus strongly negative inputs to the tanh will map to negative outputs. Additionally, only zero-valued inputs are mapped to near-zero outputs. These properties make the network less likely to get “stuck” during training. Calculating the gradient for the tanh function also uses the quotient rule:
$\Large{\begin{array}{rcl} g'_{\text{tanh}}(z) &=& \frac{\partial}{\partial z} \frac{\text{sinh}(z)}{\text{cosh}(z)} \\ &=& \frac{\frac{\partial}{\partial z} \text{sinh}(z) \times \text{cosh}(z) - \frac{\partial}{\partial z} \text{cosh}(z) \times \text{sinh}(z)}{\text{cosh}^2(z)} \\ &=& \frac{\text{cosh}^2(z) - \text{sinh}^2(z)}{\text{cosh}^2(z)} \\ &=& 1 - \frac{\text{sinh}^2(z)}{\text{cosh}^2(z)} \\ &=& 1 - \text{tanh}^2(z)\end{array}}$
Similar to the derivative for the logistic sigmoid, the derivative of $g_{\text{tanh}}(z)$ is a function of feed-forward activation evaluated at $z$, namely $(1-g_{\text{tanh}}(z)^2)$. Thus the same caching trick can be used for layers that implement tanh activation functions.
## Wrapping Up
In this post we reviewed a few commonly-used activation functions in neural network literature and their derivative calculations. These activation functions are motivated by biology and/or provide some handy implementation tricks like calculating derivatives using cached feed-forward activation values. Note that there are also many other options for activation functions not covered here: e.g. rectification, soft rectification, polynomial kernels, etc. Indeed, finding and evaluating novel activation functions is an active subfield of machine learning research. However, the three basic activations covered here can be used to solve a majority of the machine learning problems one will likely face. | 2018-02-25 11:21:10 | {"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": 0, "img_math": 21, "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": 0.8915379047393799, "perplexity": 973.3649448802515}, "config": {"markdown_headings": true, "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-2018-09/segments/1518891816370.72/warc/CC-MAIN-20180225110552-20180225130552-00216.warc.gz"} |
https://webwork.maa.org/moodle/mod/forum/discuss.php?d=2427&parent=4945 | ## WeBWorK Problems
### Error with Fraction context
by Alex Jordan -
Number of replies: 0
I'm getting an error using Fraction context that I can't understand. Can anyone figure this out? I kind of suspect it is a bug in contextFraction.pl, because I can change the context to Numeric, and the compilation works fine (but I get decimals instead of nice fractions for reported answers).
Please note, either way I compile this, it compiles fine to *display the problem*. But I get an error message when I check "Show correct answers" and then click "Check answers".
The problem is for college algebra. It's supposed to give two functions and then ask for them to be evaluated at x-values. Here is the code, with comments where I have pinpointed the issue as best I can:
##############################################
DOCUMENT();
"PGstandard.pl",
"MathObjects.pl",
"contextFraction.pl",
);
TEXT(beginproblem());
##############################################
Context("Fraction"); #Change this to Numeric, and there are no compilation problems. But I want the answers that are displayed after the answer date to be nice fractions, and changing to Numeric undoes that.
$power[0] = 1;$power[1] = 2;
#The bug only happens when one of these divisors works out to be 1. So I've forced the first divisor to be 1 for debugging.
#$divisor[0] = random(1,4,1);$divisor[0] = 1;
$divisor[1] =$divisor[0];
while ($divisor[1] ==$divisor[0]) {
$divisor[1] = random(1,4,1); }$constant[0] = non_zero_random(-8,8,2);
$constant[1] = non_zero_random(-7,7,2);$function[0] = Formula("(x^$power[0])/$divisor[0] + $constant[0]")->reduce;$function[1] = Formula("(x^$power[1])/$divisor[1] + $constant[1]")->reduce;$at[0] = random(3,10,1);
$at[1] =$at[0];
while ($at[1] ==$at[0]) {
$at[1] = random(3,10,1); };$at[1] = -$at[1]; Context("Fraction"); #The error message that eventually occurs points to these lines$evaluated[0] = Compute($function[0]->substitute(x=>Formula("$at[0]")));
$evaluated[1] = Compute($function[1]->substitute(x=>Formula("$at[1]"))); ############################################## Context()->texStrings; BEGIN_TEXT Evaluate the function $$f$$ for $$x=at[0]$$, where $$f$$ is given by $$f(x)=function[0]$$.$BR
$SPACE$SPACE $SPACE \{ans_rule(15)\}$PAR
Evaluate the function $$g$$ for $$x=at[1]$$, where $$g$$ is given by $$g(x)=function[1]$$. $BR$SPACE $SPACE$SPACE \{ans_rule(15)\}
END_TEXT
Context()->normalStrings;
##############################################
$showPartialCorrectAnswers=1; ANS($evaluated[0] -> cmp() );
ANS( \$evaluated[1] -> cmp() );
ENDDOCUMENT();
Again, for me this always compiles fine, until I ask for to "Show correct answers". The error message I get is:
Can't locate object method "make" via package "context::Fraction::Real" (perhaps you forgot to load "context::Fraction::Real"?) at /opt/webwork/pg/lib/Parser/Number.pm line 66.
Can anyone shed light on this?
Alex | 2021-10-16 15:11:37 | {"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": 1, "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": 0.6771131157875061, "perplexity": 3779.2045877611004}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2021-43/segments/1634323584886.5/warc/CC-MAIN-20211016135542-20211016165542-00317.warc.gz"} |
https://www.physicsforums.com/threads/dispersion-relation.193526/ | # Dispersion relation
1. Oct 24, 2007
### Euclid
The group velocity of traveling wave is defined as $$v_g =\partial \omega/\partial k$$. I am confused about how to actually calculate this. For instance, in the Schrodinger equation, we find that plane waves solve the equation provided that
$$\omega = k^2 \hbar/2m$$
Does this mean that the group velocity of "Schrodinger waves" is $$k\hbar/m$$? Won't this in general depend on the amplitude of the frequency components of a given wave?
Given a specific solution to the wave equation how does one answer the question, what is the group velocity of this wave?
Edit: related question....
In elementary texts, it is shown how the superposition of two sine waves of equal amplitude and phase but slightly different frequency and speed gives rise to a "traveling envelope", the speed of which we associate with the group velocity. How do we know in general that that superposition of waves gives rise to a well defined envelope?
Last edited: Oct 24, 2007
2. Oct 24, 2007
### Gokul43201
Staff Emeritus
Yes.
Not sure what you are asking, but in general, the group velocity of a wave traveling through a dispersive medium is a function of frequency.
Just as above, take the dispersion relation in its standard form, and find its derivative.
Last edited: Oct 24, 2007
3. Oct 24, 2007
### Euclid
Ah now I've confused myself. I guess I mean to say that a given wave may be composed of many wavelength components. So for what k do I evaluate the group velocity equation $$v_g = k\hbar/m$$?
4. Oct 24, 2007
### Euclid
Ok here's an example to illustrate my confusion. Take a plane wave:
$$\Psi(x,t) = A e^{i(kx-\omega t)}$$
It's phase velocity is $$\omega/k=\hbar k/2m$$. But its "group" velocity should be the same thing, no?
5. Oct 24, 2007
### Gokul43201
Staff Emeritus
No, it's group velocity will be twice that number (and equal to the classical speed of the "free particle" described by the plane wave).
6. Oct 24, 2007
### Euclid
But that doesn't make any sense to me. What envelope is involved here?
In particular, I am really interested in finding out the details behind what's hinted at here: | 2017-06-24 09:17:09 | {"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": 1, "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": 0.7411450743675232, "perplexity": 473.10922876681013}, "config": {"markdown_headings": true, "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-2017-26/segments/1498128320243.11/warc/CC-MAIN-20170624082900-20170624102900-00440.warc.gz"} |
https://avatest.org/2022/09/07/%E6%95%B0%E5%AD%A6%E4%BB%A3%E5%86%99%E6%95%B0%E7%90%86%E9%80%BB%E8%BE%91%E4%BB%A3%E5%86%99mathematical-logic%E4%BB%A3%E8%80%83math4810-substitution/ | Posted on Categories:Mathematical logic, 数学代写, 数理逻辑
# 数学代写|数理逻辑代写Mathematical logic代考|MATH4810 Substitution
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## 数学代写|数理逻辑代写Mathematical logic代考|Substitution
In this section we define how to substitute a term $t$ for a variable $x$ in a formula $\varphi$ at the places where $x$ occurs free, thus obtaining a formula $\psi$. We wish to define the substitution so that $\psi$ expresses the same about $t$ as $\varphi$ does about $x$. We start with an example to illustrate our objective and to show why a certain care is necessary. Let
$$\varphi:=\exists z z+z \equiv x .$$
In $\mathfrak{N}$ the formula $\varphi$ says that $x$ is even; more precisely:
$$(\mathfrak{N}, \beta) \models \varphi \quad \text { iff } \quad \beta(x) \text { is even. }$$
If we replace the variable $x$ by $y$ in $\varphi$, we obtain the formula $\exists z z+z \equiv y$, which states that $y$ is even. But if we replace the variable $x$ by $z$, we obtain the formula $\exists z z+z \equiv z$, which no longer says that $z$ is even; in fact, this formula is valid in $\mathfrak{N}$ regardless of the assignment for $z$ (because $0+0=0$ ). In this case the meaning is altered because at the place where $x$ occurred free, the variable $z$ gets bound. On the other hand, we obtain a formula which expresses the same about $z$ as $\varphi$ does about $x$ if we proceed as follows: First, we introduce a new bound variable $u$ in $\varphi$, and then in the formula $\exists u u+u \equiv x$ thus obtained we replace $x$ by $z$. It is immaterial which variable $u$ (distinct from $x$ and $z$ ) we choose. However, for certain technical purposes it is useful to make a fixed choice.
In the preceding example we replaced only one variable, but in our exact definition we specify the procedure for simultaneously replacing several variables: With a given formula $\varphi$, pairwise distinct variables $x_0, \ldots, x_r$ and arbitrary terms $t_0, \ldots, t_r$, we associate a formula $\varphi \frac{t_0 \ldots t_r}{x_0 \ldots x_r}$, which is said to be obtained from $\varphi$ by simultaneously substituting $t_0, \ldots, t_r$ for $x_0, \ldots, x_r$. The reader should note that $x_i$ has to be replaced by $t_i$ only if
$$x_i \in \operatorname{free}(\varphi) \quad \text { and } \quad x_i \neq t_i .$$
In the following inductive definition this is explicitly taken into account in the quantifier step; in the other steps it follows immediately.
It will become apparent that it is convenient to first introduce a simultaneous substitution for terms. Let $S$ be a fixed symbol set.
## 数学代写|数理逻辑代写Mathematical logic代考|Substitution Lemma.
Substitution Lemma. (a) For every term $t$,
$$\mathfrak{I}\left(t \frac{t_0 \ldots t_r}{x_0 \ldots x_r}\right)=\mathfrak{I} \frac{\mathfrak{I}\left(t_0\right) \ldots \mathfrak{I}\left(t_r\right)}{x_0 \ldots x_r}(t) .$$
(b) For every formula $\varphi$,
$$\mathfrak{I}=\varphi \frac{t_0 \ldots t_r}{x_0 \ldots x_r} \quad \text { iff } \quad \mathfrak{I} \frac{\mathfrak{I}\left(t_0\right) \ldots \mathfrak{I}\left(t_r\right)}{x_0 \ldots x_r}=\varphi .$$
Proof. We proceed by induction on terms and formulas in accordance with the definitions $8.1$ and 8.2. We treat some typical cases.
$t=x$ : If $x \neq x_0, \ldots, x \neq x_r$, then, by Definition 8.1(a), $x \frac{t_0 \ldots t_r}{x_0 \ldots x_r}=x$ and therefore,
$$\mathfrak{I}\left(x \frac{t_0 \ldots t_r}{x_0 \ldots x_r}\right)=\mathfrak{I}(x)=\mathfrak{I} \frac{\mathfrak{I}\left(t_0\right) \ldots \mathfrak{I}\left(t_r\right)}{x_0 \ldots x_r}(x) .$$
If $x=x_i$, then $x \frac{t_0 \ldots t_r}{x_0 \ldots x_r}=t_i$ and hence,
\begin{aligned} & \mathfrak{I}\left(x \frac{t_0 \ldots t_r}{x_0 \ldots x_r}\right)=\mathfrak{I}\left(t_i\right)=\mathfrak{I} \frac{\mathfrak{I}\left(t_0\right) \ldots \mathfrak{I}\left(t_r\right)}{x_0 \ldots x_r}\left(x_i\right)=\mathfrak{I} \frac{\mathfrak{I}\left(t_0\right) \ldots \mathfrak{I}\left(t_r\right)}{x_0 \ldots x_r}(x) . \ \varphi=& R t_1^{\prime} \ldots t_n^{\prime}: \mathfrak{I}=\left[R t_1^{\prime} \ldots t_n^{\prime}\right] \frac{t_0 \ldots t_r}{x_0 \ldots x_r} \ & \text { iff } \quad \mathfrak{I}(R) \text { holds for } \mathfrak{I}\left(t_1^{\prime} \frac{t_0 \ldots t_r}{x_0 \ldots x_r}\right), \ldots \quad \text { (by Definition 8.2(b)) } \ \text { iff } \quad \mathfrak{I}(R) \text { holds for } \mathfrak{I} \frac{\mathfrak{I}\left(t_0\right) \ldots \mathfrak{I}\left(t_r\right)}{x_0 \ldots x_r}\left(t_1^{\prime}\right), \ldots \quad \text { (by (a)) } \ \text { iff } \quad \Im \mathfrak{I} \frac{\mathfrak{I}\left(t_0\right) \ldots \mathfrak{I}\left(t_r\right)}{x_0 \ldots x_r}(R) \text { holds for } \mathfrak{I} \frac{\mathfrak{I}\left(t_0\right) \ldots \mathfrak{J}\left(t_r\right)}{x_0 \ldots x_r}\left(t_1^{\prime}\right), \ldots \ & \text { iff } \quad \mathfrak{I} \frac{\mathfrak{I}\left(t_0\right) \ldots \mathfrak{I}\left(t_r\right)}{x_0 \ldots x_r} \models R t_1^{\prime} \ldots t_n^{\prime} . \end{aligned}
## 数学代写|数理逻辑代写Mathematical logic代考|Substitution
$$\varphi:=\exists z z+z \equiv x .$$
$$(\mathfrak{N}, \beta) \models \varphi \quad \text { iff } \quad \beta(x) \text { is even. }$$
$$x_i \in \operatorname{free}(\varphi) \quad \text { and } \quad x_i \neq t_i .$$
## 数学代写|数理逻辑代写Mathematical Logic代考|Substitution Lemma.
$$\mathfrak{I}\left(t \frac{t_0 \ldots t_r}{x_0 \ldots x_r}\right)=\mathfrak{J} \frac{\mathfrak{J}\left(t_0\right) \ldots \mathfrak{I}\left(t_r\right)}{x_0 \ldots x_r}(t)$$
(b) 对于每个公式 $\varphi$,
$$\mathfrak{I}=\varphi \frac{t_0 \ldots t_r}{x_0 \ldots x_r} \quad \text { iff } \quad \mathfrak{I} \frac{\mathfrak{J}\left(t_0\right) \ldots \mathfrak{I}\left(t_r\right)}{x_0 \ldots x_r}=\varphi$$
$$\mathfrak{J}\left(x \frac{t_0 \ldots t_r}{x_0 \ldots x_r}\right)=\mathfrak{I}(x)=\mathfrak{I} \frac{\mathfrak{J}\left(t_0\right) \ldots \mathfrak{I}\left(t_r\right)}{x_0 \ldots x_r}(x) .$$
$\mathfrak{I}\left(x \frac{t_0 \ldots t_r}{x_0 \ldots x_r}\right)=\mathfrak{I}\left(t_i\right)=\mathfrak{I} \frac{\mathfrak{J}\left(t_0\right) \ldots \mathfrak{I}\left(t_r\right)}{x_0 \ldots x_r}\left(x_i\right)=\mathfrak{I} \frac{\mathfrak{I}\left(t_0\right) \ldots \mathfrak{I}\left(t_r\right)}{x_0 \ldots x_r}(x) \cdot \varphi=\quad R t_1^{\prime} \ldots t_n^{\prime}: \mathfrak{I}=\left[R t_1^{\prime} \ldots t_n^{\prime}\right] \frac{t_0 \ldots t_r}{x_0 \ldots x_r}$ iff $\quad \mathfrak{I}(R)$ holds for $\mathfrak{I}\left(t_1^{\prime} \frac{t_0}{x_0}\right.$
## MATLAB代写
MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。 | 2023-03-26 06:25:32 | {"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": 1, "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": 0.9965694546699524, "perplexity": 1389.9664063114717}, "config": {"markdown_headings": true, "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-2023-14/segments/1679296945433.92/warc/CC-MAIN-20230326044821-20230326074821-00105.warc.gz"} |
https://www.physicsforums.com/threads/help-with-subspace.41884/ | # Help with subspace
1. Sep 5, 2004
### gravenewworld
I have been trying this problem for hours. I can't believe I can't get it. The question is "Find a subset U of R^2 such that U is closed under scalar multiplication but is not a subspace of R^2". I know that for U to be a subspace 0 must be an element of U and U has to be closed under scalar multiplication and vector addition. If U is closed under scalar multiplication then it must contain O vector right? So I have to think of a subset that is not closed under addition, but is closed under multiplication right? I can not think of any. Does anyone have an idea of one that satifies these properties?
2. Sep 5, 2004
### Hurkyl
Staff Emeritus
Can you think of any subsets of R^2 that aren't closed under addition?
Study your examlpes of sets closed under multiplication. Can you identify any properties they have in common? Can you then try to come up with a new example without that property?
3. Sep 5, 2004
### gravenewworld
I can think of subsets of R^2 that aren't closed under addition, but the fact that the set is closed under scalar multiplicaton always messes things up for me. For example {(x1,x2): x1,x2 don't=0}. I know that this isn't closed under addition because (x1,x2)+(-x1,-x2)=0 vector. But this also isn't closed under scalar multiplicaiton because 0 is an element of the field R, and 0(x1,x2)=(0,0). I have thought of examples that aren't closed under addition, but they always violate the fact that they must be closed under multiplication.
4. Sep 5, 2004
### Hurkyl
Staff Emeritus
Hrm. Well, is there a way to take an arbitrary set and turn it into a set that is closed under multiplication?
5. Sep 5, 2004
### Fredrik
Staff Emeritus
The set {(x,y)|xy=0}, satisfies the required properties. This is just the set you suggested plus the element (0,0).
In order to be closed under scalar multiplication, the set has to be a union of straight lines passing through (0,0). The only proper subsets of that kind that are closed under addition are the ones that consist of only one such line.
Last edited: Sep 5, 2004
6. Sep 6, 2004
### gravenewworld
Thanks Fred. I knew it had to be something real simple that I was completely over looking. | 2017-01-24 19:45:00 | {"extraction_info": {"found_math": false, "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": 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": 0.8293569684028625, "perplexity": 322.545044274001}, "config": {"markdown_headings": true, "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-2017-04/segments/1484560285244.23/warc/CC-MAIN-20170116095125-00225-ip-10-171-10-70.ec2.internal.warc.gz"} |
http://motls.blogspot.com/2012/10/the-holographic-principle.html?showComment=1351241841779 | ## Friday, October 26, 2012 ... /////
### The holographic principle
The newest episode of The Big Bang Theory that was aired last night was called "The Holographic Excitation" (S06E05).
It's pretty cool that a TV sitcom manages not only to show a hologram but Leonard Hofstadter was even allowed to present a rather accurate definition of the holographic principle in quantum gravity i.e. string theory (you won't find it in any popular science TV program that claims to explain modern physics!). And as a result, he was able to have an intercourse with Penny right after she wore some glasses and was shown a moving holographic pencil and a moving holographic globe. (Later, he repeated the same achievement using Maglev.)
(And I even think that Prof Nina Byers whom I know rather well walks behind the main actors around 9:15. This theory seems to make sense because she's at UCLA, much like the TBBT science adviser David Saltzberg.)
The holographic principle of quantum gravity is an incredible example of the ability of the quantum gravity and string theory research to teach us things we really didn't and perhaps couldn't anticipate, force us to modify or abandon some prejudices, and adopt ideas about the unification of ideas and concepts that philosophers couldn't have invented after thousands of years of disciplined reasoning but physicists may be forced to realize them if they carefully follow the mathematical arguments sprinkling from a theory that they randomly discovered in a cave.
But let's return half a century into the past. Holography started in "everyday life physics" in the late 1940s.
So let us begin with this exercise in wave optics that has nothing to do with quantum gravity or string theory so far – but you will see that it exhibits a similar mechanism that is apparently "recycled" by the laws of quantum gravity.
Dennis Gabor's 3D images
Hungarian-British physicist Dennis Gabor was playing with X-ray microscopy and invented a new technology that is rather cute. One may create two-dimensional patterns on a piece of film which, when illuminated by a laser, create the illusion of a three-dimensional object floating in the space around it. I saw my first hologram sometime in 1985 – it was a Soviet one, the mascot of the 1980 Olympics in Moscow – in the National Technological Museum in Prague where we went to a school excursion. I couldn't believe my eyes. :-)
The basic setup involves a monochromatic laser beam, some interference, and a photographic plate. First, we must create the hologram – a film with strip-like patterns that don't resemble the bear at all but which allow the bear to jump out once you use another laser. Fine. Let's create a hologram.
You see that a monochromatic (one sharp frequency) laser beam is coming from the upper left corner. Each photon's wave function gets divided into two portions by a beam splitter – note that the wave function has a probabilistic interpretation for one particle but if many photons are in the same state, it may be interpreted as a classical field.
Two parts of the wave are moving from the beamsplitter. One gets reflected from a simple mirror. More interestingly, the other one gets reflected from the object we want to see on the hologram. They interfere – these waves are recombined – in the right lower corner and they create a system of interference strips on the photographic plate.
We have created a hologram and now we may sell it. What will the buyers do with it?
This interference pattern may be shined upon by a "reconstruction beam" of the same frequency and what we see is a virtual image behind the plate. You may actually move your head and eyes and the position of all points on the image are moving just like if the virtual image were a real object. So it's not just a stereographic image offering two different pictures for the two eyes: the hologram is ready to provide the right electromagnetic field regardless of the direction from which you observe it! If you want to see the right face of the object, you move your head to the right side, and so on.
Why does it work? It's very useful to think about the hologram for a simple object, e.g. a point at a given distance. The total wave function on the photographic plate parameterized by coordinates $x,y$ is given by $U_O + U_R$ where $U_O$ is the complicated wave reflected from the object and $U_R$ is the simple reference beam reflected from the plain mirror.
You may imagine that for an object being a point, $U_R=1$ and $U_O(x,y)=\exp(iks(x,y))$ where $s$ is the distance between the point (the real object we want to holographically photograph) and the given point $(x,y)$ on the photographic plate. The wave number is of course the inverse wavelength, $k=2\pi/\lambda$. The sum $U_R+U_O$ gives you some simple concentric circles (with decreasing distances between neighbors) around the point on the plate that is closest to the photographed point. Fine. The total intensity – how much the point on the film changes the color – is given by $T \sim \abs{U_O}^2 + \abs{U_R}^2 + U_R^* U_O + U_O^* U_R.$ I omitted an unimportant overall normalization and used the symbol $T$ for this quantity because the darkness of the point of the film will be interpreted as the transmittance, the ability of the place of the hologram to transmit the other, reconstruction beam when we actually want to reconstruct the image.
For a simple explanation why you will see the reconstructed virtual image, assume that the reference beam $U_R$ is much stronger than the object-induced wave $U_O$ i.e. $U_R\gg U_O$. So the total wave function may be written as $U = 1 + \varepsilon \exp(iks)$ where $\varepsilon$ is small, $\varepsilon\ll 1$. You see that the squared absolute value is$T \sim |U|^2 = 1 + \varepsilon \exp(iks) + \varepsilon \exp(-iks) + {\mathcal O}(\varepsilon^2).$ Imagine that this transmittance is just multiplying another simple reference beam $U_R\to T \cdot U_R$ and produces some electromagnetic field in the vicinity of the hologram. For the sake of simplicity, assume $U_R=1$ again. It's only $U_O$ that carries the "complicated information about the photographed object" but we still need some nonzero $U_R$.
You may see that $T$ is almost the same thing as $U$ except that it has an extra, complex conjugate term. So the electromagnetic field in front of the hologram (on the side with the air) will be the same field as the electromagnetic field we used to have when there was a real object in front of the hologram plus some complex conjugate term. One of these terms creates a nice virtual image behind the plate because it has a similar mathematical structure and when the fields have the same values, we see the same thing.
The other term induces the feeling of another copy of the object – a real image. It's because all the waves should also be multiplied by the universal time-dependent factor $\exp(-i\omega t)$ (before you interpret the real and imaginary value of the overall sum as the electric and magnetic fields, respectively, kind of) and the complex conjugation is equivalent to $t\to -t$ which means that the wave is kind of moving backwards in time which is effectively equivalent to moving from the other side of the mirror.
So when you look at the hologram, you actually see one virtual image behind the plate and one real image in front of the plate (which may overlap with your head). I don't want to figure out which term is which because odds would be close to 50% that my answer would be wrong. To be sure about the answer to this not-so-critical question, I would have to decompose the electromagnetic wave to the electric and magnetic components, consider $x$ and $y$ polarizations, be careful about the spatial dependence and all the signs, etc. But things clearly work up to this "which is which" question that I am not too interested in.
In this brute calculation, I have neglected the ${\mathcal O}(\varepsilon^2)$ terms which indicates that the hologram will be badly perturbed if $\varepsilon\sim {\mathcal O}(1)$ but a more accurate analysis shows that the result won't be too bad even if you include these second-order terms. At any rate, I have created a virtual image of a point! By the superposition principle, you are allowed to envision any object to be composed of many points (perhaps as an "integral of them") and add the terms $\exp(iks_P)$ from each point $P$ and you get the idea how it work for a general object.
There exist generalizations – colorful holograms, perhaps moving holograms and holographic TV, and so on, but I don't want to go into these topics on the boundary of physics and engineering. Everyone knows that holograms are cool. What's important for us is that they store much more than some two-dimensional projections of a 3D object as seen from one direction or two directions; they store the information about the 3D object as seen from any direction (in an interval). They're the whole thing.
Instead of discussing advanced topics of holography in wave optics, we want to switch to the real topic, the holographic principle in quantum gravity.
The holographic principle
In the research of quantum gravity, the notion of holography was introduced by somewhat speculative but highly playful papers by Gerard 't Hooft in 1993 and Lenny Susskind in 1994. Charles Thorn is mentioned as having suspected similar ideas for years.
It may sound unusual ;-) but Lenny Susskind's paper was the technically more detailed one, getting well beyond the hot philosophical buzzwords. Susskind also suppressed some unjustified and unjustifiable "digital" comments by 't Hooft who had written that the information had to be encoded in binary digits (bits). Of course, there's no reason whatsoever why it couldn't be trinary digits, other digits, or – much more likely – (for humans and computers) some much less readable but more natural codes.
What's the basic logic behind holography in quantum gravity?
In classical general relativity, a black hole is the final stage of the collapse of a star or another massive object. Because the entropy never decreases, as the second law of thermodynamics demands, the "final stage" must also be the stage with the maximum entropy. So the black hole has the highest entropy among all bound or localized objects of the same mass (and the same values of charges and the angular momentum). I emphasize the adjectives "bound or localized" because delocalized arrangements of particles with a given total energy – e.g. the Hawking radiation resulting from a black hole that has already evaporated – may carry a higher entropy (that's inevitably the case because the process of Hawking radiation must be increasing the total entropy, too).
But we've known from the insights by Jacob Bekenstein and Stephen Hawking in the 1970s that the black hole entropy is$S_{BH} = \frac{A}{4G}$ in the relativistic $c=\hbar=1$ units. It's one-quarter of the area of the event horizon $A$ in the units of the Planck area. In normal units, you must replace$G \to l_{\rm Planck}^2 \equiv \frac{G\hbar}{c^3}.$ So the maximum entropy of a bound localized object of a given mass is actually given by the area of the black hole of the same mass. Because you can't really squeeze the matter into higher densities than the black hole, the black hole is also the "smallest object" that may contain the given mass.
To summarize, we see that the black hole is the "highest entropy" object as well as the "geometrically smallest" object among localized or bound objects of the given mass. It follows that it also maximizes the "entropy density" (entropy per unit volume) among the localized arrangement of matter of the same total mass. But the entropy carried by a black hole is only proportional to the surface area in the Planck units, ${\mathcal O}(R^2)$, so the entropy density per unit volume – the latter scales as ${\mathcal O}(R^3)$ – is therefore going to zero for large black holes i.e. for large masses or large regions.
The maximum density of entropy or information you may achieve with a given mass is actually going to zero if the mass is sent to infinity. If you try to squeeze too many memory chips into your warehouse, they will start to be heavy at some point and will gravitationally collapse and create a black hole which will have a certain radius – either smaller than or larger than your warehouse. At any rate, this black hole will only be able to carry $1/4$ of a nat (a bit is $\ln(2)$ nats) of information per unit surface area (by the surface, I mean the event horizon).
We see that the maximum information is carried by a constant density per unit area rather than the unit volume. You should appreciate how shocking it is. In some sense, it was completely unexpected by virtually all experts in the field. Quantum field theories predict some new phenomena at a characteristic distance scale. For example, Quantum Chromodynamics (QCD) says that quarks like to bind themselves into bound states where their distance is comparable to the QCD length scale, about one fermi or $10^{-15}$ meters. So by the dimensional analysis, the only sensible "density of information" we may get in QCD is "approximately one bit per cubic fermi" or per "volume of the proton".
People would expect a similar thing in any QFT – which was mostly right – but they thought it would also hold in quantum gravity. So quantum gravity may achieve "one bit per Planck volume". But that was wrong. You see that the previous paragraph assumed a bit more than the dimensional analysis: it also implicitly and uncritically postulated that the information is proportional to the volume. This assumption followed from locality. But this assumption breaks down in quantum gravity where the information only scales as the surface area.
Because the "proportionality to the volume" is linked to "locality" – each unit volume is independent from others – the violation of the "proportionality of the information to the volume" that the holographic principle forces upon us also means that locality is violated, at least to some extent. And indeed, this violation of the locality is a fact responsible for the resolution of other puzzling questions in quantum gravity, too. In particular, some tiny and hard to observe but nevertheless real non-locality occurs during the evaporation of the black hole which is why the information may get from the black hole interior to infinity, after all – even though classical general relativity strictly prohibits such an acausal export of the information (locally, it's equivalent to the superluminal transport of information which was already banned in special relativity). In quantum gravity, this "ban" is softened because the information may temporarily violate the rule in analogy with the quantum tunneling. In fact, the black hole evaporation is a version of quantum tunneling.
Whether the holographic principle was real and what it exactly it meant and what it didn't mean remained a somewhat open question for 3 more years or so. However, at the end of 1997, Juan Maldacena presented his AdS/CFT correspondence which is a set of totally controllable mathematical frameworks in which holography holds. The information about a region – namely the whole anti de Sitter space – is stored at the boundary of the region – which is the asymptotic region at infinity which nevertheless looks like a "finite surface of a cylindrical Penrose diagram" if you use the language of Penrose causal diagrams.
The holographic principle surely captures the right "spirit" of quantum gravity but it is a bit vague. The AdS/CFT correspondence is a totally well-defined "refinement" of the holographic principle but it is arguably too special. Nevertheless, one must be careful about deriving potentially invalid corollaries of the holographic principle in other contexts.
For example, if you replace the anti de Sitter space by a finite-volume region of ordinary space, it seems clear to me that the holographic principle will only be true in some rather modest sense: it will be true that the entropy bounds hold. You can't squeeze too much entropy into a given region. However, if you will try to find the "theory on the boundary" that is equivalent to the evolution inside the region, you will find out that such a theory on the boundary "exists" – but the existence of such a theory is just an awkward translation of the ordinary evolution to some artificial degrees of freedom that you placed on the boundary.
What is special about the AdS/CFT correspondence is that the theory on the boundary is a theory of a completely normal type – namely a perfectly local, conformal quantum field theory. In fact, the boundary theory is more local than the gravitational theory in the bulk – because we just said that the gravitating theory in the bulk must be somewhat non-local. I am confident this fact depends on the infinite warp factor of the AdS space at infinity and won't hold for finite regions. In other words, I think that the "holographic theory living on a boundary" of a generic finite region won't be local in any sense – the boundary still has a preferred length scale, the Planck length, and other things so it is surely not conformal etc. And because it won't be local, it won't be simple or useful, either.
So one shouldn't generalize the holographic principle as seen in the AdS/CFT correspondence too far and too naively.
Lessons
In the 1970s, people got used to Ken Wilson's "Renormalization Group" inspired thinking about all effective field theories. Each theory predicted some phenomena at a characteristic length scale. The third power of the length scale gave us a characteristic volume. And one could expect roughly one nat (or bit) per one characteristic volume. It was nice, it made sense, it has lots of applications.
But Nature sometimes has surprises in store and quantum gravity had one, too. You may still use almost the same logic – one nat per unit region – but the region must actually be measured by its surface area, not its volume. So quantum gravity tells us that one of the spatial dimensions may be thought of as an "artificial" or "emergent" one and other mechanisms supporting this general paradigm have appeared as well.
A brutally arrogant yet extremely limited physicist who really sucks – think of Lee Smolin, for example – may think that he has all the right ideas how the final theory should look like from the beginning. Except that none of them works (except as tools to impress some stupid laymen). But other physicists who are much smarter but much more modest may see that all Smolin's prejudices are just wrong and Nature's inner organization is much more clever, creative, surprising, and forcing us to learn new concepts and new way of thinking more often than Smolin and many others would expect. One must still be ingenious or semi-ingenious to discover some important wisdom about Nature – e.g. holography and the AdS/CFT correspondence – but Nature just doesn't appreciate men who try to paint themselves as wiser than herself. Science is the process of convergence towards Her great wisdom; it is not a pissing contest in which idiots such as Lee Smolin try to pretend that they're smarter than Nature.
The story of the holographic principle also shows us that Nature recycles many ideas. The fields defined on the boundary CFT in the AdS/CFT correspondence literally emulate the waves $U$ and $T$ that I mentioned in the discussion of the "ordinary" holography by Dennis Gabor.
And the story of the holographic principle is another anecdotal piece of evidence in favor of the assertion that string/M-theory contains all the good ideas in physics. 't Hooft and Susskind, building on the work by Bekenstein, Hawking, and others, had some "feelings" about the right theory of quantum gravity and there had to be something right about them. And indeed, string theory showed us that they were mostly right. Because string theory is a much more mathematically well-defined a structure than "quantum gravity without adjectives", it also allowed us to convert the philosophical speculations into sharp and rigorous mathematical structures and equations and decide which of the philosophical speculations may be proven as meaningful ones and which can't.
The holographic principle is also another step in the evolution of physics that makes our theories "increasingly more quantum mechanical". While the spacetime remains continuous, we see that the information in a region may be bounded in unexpected ways and a whole dimension of space may be emergent. Needless to say, the equivalence between theories that disagree about the number of spacetime dimensions is only possible if you take the effects of quantum mechanics into account.
#### snail feedback (45) :
Oh yeah yeah yeah, I look forward to read this :-)
... and darn, I want to see the whole TBBT episode about the holographic excitation too :-D !
Holographic principle was also hypothesized as a mechanism for memory at some point. Karl Pribram (neurosurgeon) and David Bohm (physicist) postulated some kind of holographic principle behind brain function because they noticed that memory is not localized in brain. If you remove some portions of the brain, memory stays mostly intact, that is why they thought that memory is stored as a hologram (if you break a hologram, every piece has an information about the whole). It is probably a bad analogy at best. memory is stored in the connection weights of synapses
http://en.wikipedia.org/wiki/Holonomic_brain_theory
reader Smoking Frog said...
Lubos, your first three paragraphs are faultless so far as the English goes, except one error, but the error is comically brilliant: "an intercourse." It should be "intercourse" (without the article), but please don't think of fixing the error in the future because "an intercourse" is just too good. Use it all you want.
What's the difference ??
The new season hasn't fully arrived in Ireland yet (and the dubbing is so bad in France... ).. My son says we can see in on the internet free TV. yeahy
ss
reader Luboš Motl said...
Thanks, Smoking Frog, for the fix. This is just way beyond my linguistic skills. I've been taught to repeat many times that one is having a shower, we may have a walk, we did have a lunch, and so on, so I can't possibly understand why one can't have an intercourse as well. Is it because the native English speakers are puritans of a sort? ;-) So that if they mention the word at all, it must be formulated in such a way that one can't count them? :-)
reader Smoking Frog said...
Good question. On the grammar, I would have been too vague if I hadn't just now looked at the Wikipedia article "Mass noun," and I'm still not sure that "intercourse" is a mass noun or in some similar category (but not a collective noun).
"Have an intercourse" is something like "have a peanut butter"; to use an article, you'd have to say anything like "a spoon of peanut butter." For "intercourse," you could say "an act of intercourse," except that one does not "have" acts; one performs acts; so you pretty much have to say "have intercourse."
So the answer to your question is that there's no such thing as "an intercourse." But this doesn't really explain why it's so funny. Scratching my head ...
A bit OT. When you talk about quantum tunelling from the BH, do you mean tunelling whose result is ordinary Hawking radiation or a corrected (non-thermal) radiation? Here http://arxiv.org/abs/hep-th/9907001 the authors argue that a tunelling mechanism leads to a non-thermal correction.
reader Smoking Frog said...
Hah-hah as to the Puritans. No, that's not it. See my reply to Shannon.
reader Luboš Motl said...
Once I wrote an article about the Holographic Principle to "Vesmír", a Czech journal for a broader scientific public, and it was cited by some of the Příbram's followers. That was fun because I do think theirs is a pseudoscience unrelated to the actual topic of my article - this one or the old one in "Vesmír".
reader Smoking Frog said...
Lubos - "Have a lunch" is an interesting case; you can also say "have lunch," and that's what most people say. There is something of a difference of meaning, but it's kind of fuzzy. If I say, "We had a lunch," I might mean that we hosted a lunch. So there's a difference. But I could also say it if you and I had lunch together. In that case, it would mean almost just the same as "We had lunch," except that I'd be thinking of the lunch as an event rather than simply "having lunch"; e.g., "We did have a lunch on one occasion."
reader Smoking Frog said...
Hah-hah.
reader Peter F. said...
Lubos, you've been taught wrong in respect of "having _a_ lunch". :-))
reader Luboš Motl said...
Maybe poor people from the post-communist bloc are only taught to have a lunch while the native English speakers are given lots of lunch... ;-)
reader Luboš Motl said...
Hi Lemon, in general, the spectrum seen at infinity is corrected by something that is called "greybody factors" indicating it's not exactly blackbody spectrum. Another question is whether the particular unusual calculation of them in the paper you mentioned is right.
For this reason, I think that your language "do you mean one or the other" is kind of strange because the exact spectrum emitted by a well-defined object in a well-defined theory is clearly uniquely determined so "if someone meant the other one", he would just be wrong. And I didn't say what the exact spectrum was, so I explicitly mentioned neither the right one nor the wrong one. But of course I meant the correct one whichever it is. ;-)
reader Peter F. said...
Haha, but it is better to forego having lunch if it will only consist if corn syrup containing junkfoods. ;)
What makes me laugh about the word "intercourse" is that it sounds like something you'd be served during a meal. So "an intercourse" would be perfect between say the starter and the main course... or the main course and cheese, or desert... right ? ;-) My French upbringing probably ? ha ha
That’s really funny, Lubos, and it is wonderful to see your sense of humor re-emerge and blossom again!
reader Physics Junkie said...
I am showing my ignorance here, but is there some connection between Ads/CFT and the dualities in string theory that link different dimensional branes?
reader Eugene S said...
To make things even more confusing, in the U.S. people say "entree" but they mean the mean the "main course"; in France, "l'entrée" is the appetizer!
Savages...
reader Luboš Motl said...
Thanks, interesting, Smoking Frog.
reader Luboš Motl said...
Thanks for liking this change, Gene. It's a relief if one doesn't have to spend the bulk of his energy by thoughts about the bare survival.
reader Luboš Motl said...
Yes, there are many connections although perhaps a bit different than you expect.
First, the duality relating different dimensional D-branes is T-duality. There's an interesting T-duality one may apply to the theories in AdS/CFT, the most famous example of the N=4 gauge theory, and it maps the conformal symmetry to the dual conformal symmetry. Those can be combined into their (multiple) commutators to generate the infinite-dimensional Yangian, a bizarre form of symmetry known in the "integrable models".
Then the AdS/CFT correspondence is a sort of a duality itself. On the world sheet, it's most visibly visualized as the world sheet duality applied to a cylinder/annulus. It can be either interpreted as a closed string state propagating between a final and an initial state, or a loop of an open string in periodic time. AdS/CFT produces an "open/closed" duality in this sense: the AdS is the closed string side, CFT is the open string theory.
reader James Gallagher said...
Nice article, covering classical holography and QG Holographic Principle.
Because the HP suggests reality is (fundamentally) encoded on a 2D surface rather than in a 3D volume - it kinda naively explains why complex numbers, and in particular complex Hilbert Spaces, are required. A general 2D surface is parameterized by two real parameters or one complex parameter z. So we might deduce from this that the universe can be described by a simple (in)finite complex vector, evolving in a Hilbert Space.
I know it's a naive argument, but it suggests that the Holographic Principle is a VERY important discovery,
reader James Gallagher said...
actually, parameterised isn't a good word there, I mean a general 2D surface has only 2 degrees of freedom, which we may associate with two real numbers or one complex number
reader James Gallagher said...
oh, you know what I mean!
We don't have information for every 3D point (x,y,z) within the 2D surface only for each point on the 2D surface.
That's funny, and reminds me of a happy memory. Years ago at friendly gatherings we used to play Dictionary and the player with the best command of the English language was an Indian engineer.
Dictionary is a game where one person picks a word out of the dictionary, pronounces it for everyone who then write their own definitions and the other players vote to try and decide which definition is the real one. Scoring is based on how many people are fooled by phony definitions.
reader Smoking Frog said...
Eugene - I don't know French, but I've always wondered how "entree" could mean the main course!
reader Smoking Frog said...
You mean in France everyone has sex between the courses of a meal!!!??? :-)
reader Luboš Motl said...
In America, "entree" is the main course because after some foreplay appetizers and intercourses of the kind that "I didn't have with that woman", the entry finally comes.
reader Smoking Frog said...
Lubos, your English is already very good, so it's nothing to worry about.
It's sort of amazing that these subtle difficulties in language exist. You don't realize it unless you think about it. It's sort of hard to believe stories of spies who passed as native speakers.
reader Luboš Motl said...
I guess it must be possible for some adults to learn a foreign language just like the native speakers among children do - even though most people probably don't have the aptitude for such "full learning" of other languages.
The children just can't have any "unrepeatable" skills, can they? Learning many languages as an adult requires one to have "independent linguistic brains" of a sort that preserve "inconsistent" answers to pretty much the same questions.
reader Smoking Frog said...
I would like to be told how it works that a kid learns such things that not too many people could explain as clearly as you.
I forgot that part. I don't know the answer, but it might be related in some way to thinking about how other people think. As a kid in the lower grades of elementary school, when the teacher was trying to explain something to a dumb kid and not succeeding because she didn't understand his trouble, I was always waving my hand and saying, "Miss [X], I know what's he's thinking."
reader Smoking Frog said...
I agree that aptitude must play a big role. There are people whose grasp of language issues is horribly poor.
I'm not sure whether I've told this story here before: Maybe 25 years ago there was an interview in The Atlantic(?) with a big fat guy who was supposed to be the world's greatest linguist. He spoke many languages. The interviewer asked him if he could speak Finnish, and the linguist said something like this: "No. Or, well, I could probably pass as a Finn for 10 or 15 minutes in a conversation, but that's not what I call speaking a language."
reader Luboš Motl said...
LOL.
reader Smoking Frog said...
Maybe spies can pass as native speakers because the natives' ability to detect that he's not one are poor. :-)
This is a very nice article indeed :-),
the cute explanation of holography meaning 3D images makes me feel quite happy and satisfied.
Concerning the quantum gravity ;-) case, my understanding of it is way to vague and shaky to satisfy my.
To be happy, I always need to see how something (roughly at least) works mathematically,
instead of just words and analogies ...
Anyway, I like this text a lot so thanks Lumo !
Interesting. I guess we say "social interactions" rather than "social intercourse"... It avoids silly giggles.
Oh and no, French don't have sex between courses of a meal, the reason being that eating and sex are two highly incompatible activities :-D
reader Smoking Frog said...
We say "social interactions," too. "Social intercourse" is usually found in writing, and perhaps not even so much that, nowadays.
Incompatible? You can eat between rounds of sex. This is like the story of the Buddhist monk who asked the master if he could smoke while he was praying. The master said, "Of course not!" So the monk asked him if he could pray while he was smoking.
reader Rosy Mota said...
it appear to be correct.particles,energy,fields and forces mudt to be originated of 4-dimensional spacetime topological geometry
reader Carolina Lou said...
Very good topic. It
helps all to gain many more knowledge of holographic principle. it covered 2D
and 3D projection. | 2017-08-18 03:04:57 | {"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": 2, "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": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.518474280834198, "perplexity": 751.2054437010006}, "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-2017-34/segments/1502886104560.59/warc/CC-MAIN-20170818024629-20170818044629-00213.warc.gz"} |
https://www.physicsforums.com/threads/urgent-help-required-in-summation-of-terms.692654/ | # Urgent help required in summation of terms
Find the sum upto n terms:
1.3.5+3.5.7+5.7.9.............tn
I solve it this way:
tn=(2n-1)(2n+1)(2n+3)
Now can I take summation on both sides? How?
I mean when I add 2 on both sides the resultant is 0(2-2=0).Similarly the resultant summation will be zero?
And if I take summation I get one term as 3Ʃ.Now in a book I saw that it is 3n. Why? Summation of 3 will be 3 only as 3 is constant.Please explain.
I got this:
Ʃtn=Ʃ(2n-1)(2n+1)(2n+3)
Ʃtn=Ʃ[(4n^2-1)(2n+3)]
Ʃtn=Ʃ[8n^3 + 12n^2 - 2n - 3]
Ʃtn=Ʃ[8n^3] + Ʃ[12n^2] - Ʃ[2n] -Ʃ[3]
Ʃtn=8*Ʃ[n^3] + 12*Ʃ[n^2] - 2*Ʃ[n] - Ʃ[3]
Now do I require to write them as Ʃ[3] or 3Ʃ (putting a constant outside Ʃ).Please explain the whole summation process.I am stuck here.
Hi.
You are on the right way.
As for Ʃtn=8*Ʃ[n^3] + 12*Ʃ[n^2] - 2*Ʃ[n] - Ʃ[3], there are formula for Ʃ[n^3], Ʃ[n^2], Ʃ[n] and Ʃ[1].
Stephen Tashi
$\sum_{i=1}^4 10 = 10 + 10 + 10 + 10 = (4)(10) = 40$ | 2021-06-20 02:50:25 | {"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": 0.9692717790603638, "perplexity": 4529.693914495342}, "config": {"markdown_headings": true, "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-2021-25/segments/1623487655418.58/warc/CC-MAIN-20210620024206-20210620054206-00272.warc.gz"} |
https://www.tuitionwithjason.sg/2018/01/additional-math-trigonometry-how-to-derive-cosine-half-angle/ | Use the Cosine Double Angle Formula (found in Formula List) to help derive Cosine 1/2 Angle.
$\displaystyle \text{cos}\ 2A=2{{\cos }^{2}}A-1$
Substitute A for 1/2 x
$\displaystyle \text{cos}\ 2\left( {\frac{1}{2}x} \right)={{\cos }^{2}}\left( {\frac{1}{2}x} \right)-1$
Simplify the equation (*Note: you can multiply cos 2(1/2x) to get cos x.)
$\displaystyle \text{cos}\ x={{\cos }^{2}}\left( {\frac{1}{2}x} \right)-1$
$\displaystyle \text{1+cos}\ x={{\cos }^{2}}\left( {\frac{1}{2}x} \right)$
$\displaystyle \sqrt{{\text{1+cos}\ x}}=\cos \left( {\frac{1}{2}x} \right)$
$\displaystyle \cos \left( {\frac{1}{2}x} \right)=\sqrt{{\text{1+cos}\ x}}$
There is a debate as to whether Half-Angle formulas are part of the N or O Level Exam syllabus. I ask my students and two of my Math teacher friends,the answer was inconclusive, some students said that they are taught in school, others say that they never seen this in their entire life, even the teachers were not sure. I check the SEAB (Singapore Examination and Assessment Board) website but there was no info (explicit or implicit) about this. The funny thing is that half-angle formulas do appear in textbooks , some schools worksheet and test papers. In the next post i will show you how to derive sine 1/2 angle formula.
The Bottom Line: Better to err on the side of caution
Additional Math Tuition for students living in Choa Chu Kang, Yew Tee,Sembawang and Yishun. | 2020-04-03 20:37:24 | {"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": 0, "img_math": 6, "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": 0.7304667830467224, "perplexity": 2096.844836437407}, "config": {"markdown_headings": true, "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-2020-16/segments/1585370518622.65/warc/CC-MAIN-20200403190006-20200403220006-00001.warc.gz"} |
http://dionesiusagung.com/2016/06/23/indicators-that-i-use-for-tecnical-analysis/ | ## Indicators that I use for Tecnical Analysis
### Technical analysis is not as hard as it sounds, but not as easy as it looks either. Technical analysis enables us to predict the future price movement of a quote, or you could say, basically everything — if you knew the previous prices — by looking and “messing around” with aid of charts and some tools.
Of course, by just looking at plain price charts it is hard for us to predict the movement of the price, whether it will go up or down. So, over the years people have been inventing these things called indicators which helps us to predict price movement and makes it easier for us to decide what to do. Indicators come in various shapes and functions. There are simple indicators which anyone could read, and there are also harder indicators which require a bit of “practice” to understand. There are even some indicators that are so complex. So what is exactly an indicator? The answer can be viewed here. Below are some basic indicators that I use in technical analysis.
(1) Moving Average (MA)
As the name suggests, Moving Average shows the moving average of the price on a certain period. This indicator is the most basic one. The function of MA is usually to see the trend, whether the price is uptrend, downtrend, or moving sideways. I personally use 2 MA’s, one with the period of 15 and one with 20.
So, how to read the MA? It is so simple, really. If the lower period MA is above the higher period MA in the graph, the price is on uptrend. On the contrary, if the lower period MA is below the higher period MA, it is on downtrend.
When the 15 MA (light blue) is above the 20 MA (pink), the price is uptrend. Traders usually use 2 MA indicators.
The dark blue and red lines are called envelopes and will be explained later.
(2) Envelope
Envelope is basically a modified MA. According to Investopedia.com, envelope is a type of technical indicator typically formed by two moving averages that define upper and lower price range levels. It calculates two moving averages using the high price and low price inputs. Both averages are calculated using price data from the same number of bars, as determined by the input length. The average of the high price and low price is increased and reduced by a user-specified percentage and then plotted. The envelope inputs can be customized to suit each investor’s or trader’s style and preferences.
Envelope is usually used together with some other indicators, in this case MA.
The purpose of this indicator is usually to show support and resistance lines, as well as to confirm in taking position.
I usually use envelope together with moving average, RSI, and CCI.
(3) Relative Strength Index (RSI)
The relative strength index (RSI) is a technical momentum indicator that compares the magnitude of recent gains to recent losses in an attempt to determine overbought and oversold conditions of an asset (Investopedia). The RSI is calculated using this following formula:
$$RSI\quad=\quad100\quad\times\quad\left(1\quad-\quad\frac{ 1 }{ 1+RS }\right)$$
where RS = (average of n days’ upcloses)/(average of n days’ downcloses).
RSI has a value that ranges from 1 to 100. When it passes or reaches 70, the asset is “overbought”. When it reaches under 30, it is “oversold”.
As seen in the chart, I combine the 14 RSI (white) with MA(2) for a smoother graph. Notice how the price goes through the lower envelope (upper graph, red) when oversold and through the upper envelope (upper graph, dark blue).
I usually color the RSI black as I only need the 2 MA on RSI indicator window.
(4) Commodity Channel Index (CCI)
The CCI is similar with the one above, except that it’s more of an oscillator. It is also used for determining overbought and oversold conditions. It is developed to quantify the relationship between price (P), moving average (MA), and normal deviations (D) (also from investopedia). It can be calculated using the formula below:
$$CCI=\frac{P-MA}{0.015\times D}$$
The CCI can also predict local potential maximum and minimum prices, thus helping traders to ensure in taking actions and providing them firm evidence that price movement or trend is going to change.
As you can see, price tends to change trend as the CCI hits minimum or maximum level.
I usually combine CCI and RSI and make them as confirmations when to take a position and close one.
These are the indicators that I commonly use in trading. There are still more indicators that I use but not too often, such as Ichimoku, parabolic SAR, MACD, stochastics, and Bollinger Bands. These will be explained in another articles. | 2018-02-24 19:23:39 | {"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": 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": 2, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5699962973594666, "perplexity": 1544.820398067445}, "config": {"markdown_headings": true, "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-2018-09/segments/1518891815934.81/warc/CC-MAIN-20180224191934-20180224211934-00107.warc.gz"} |
https://zbmath.org/?q=an:1035.37007 | # zbMATH — the first resource for mathematics
Homoclinic classes and finitude of attractors for vector fields on $$n$$-manifolds. (English) Zbl 1035.37007
The author defines the notion of the almost sure rotation number for certain expanding piecewise affine endomorphisms of the circle as the rotation number for almost every point of the circle. It is shown, that the dependance of the almost sure rotation number on the parameters is of Hölder type with any exponent $$\alpha\in (0,1)$$ but not Lipschitzian. The set of parameters giving an irrational almost sure rotation number is of Hausdorff dimension one. The method of the symbolic Perron-Frobenius operator is applied.
##### MSC:
37B25 Stability of topological dynamical systems 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
Full Text: | 2021-04-13 05:18:55 | {"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": 1, "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": 0.5135261416435242, "perplexity": 748.7998401193632}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2021-17/segments/1618038072082.26/warc/CC-MAIN-20210413031741-20210413061741-00193.warc.gz"} |
http://www.ck12.org/book/CK-12-Middle-School-Math-Concepts---Grade-7/r2/section/6.19/ | <img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
You are reading an older version of this FlexBook® textbook: CK-12 Middle School Math Concepts - Grade 7 Go to the latest version.
# 6.19: Total Bill Including Tip and Tax
Difficulty Level: At Grade Created by: CK-12
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Practice Total Bill Including Tip and Tax
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Have you ever tried to calculate tips and taxes after going out to eat?
Taylor and her Dad went out to a restaurant for dinner. It had been a long day at the candy store, and they were both hungry. The cost of the meal was 25.75. Taylor and her Dad began to figure out the total bill given a 7% sales tax and a 15% tip. Do you know how to do this? Can you figure out the total bill including tax and tip? This Concept will teach you how to find total bills involving tips and taxes. Then you can finish this problem at the end of the Concept. ### Guidance You just finished figuring out a tip. When you calculate the tip, then you add this amount back to the total of the bill for the final cost of the meal. Sometimes, the tip is left in cash on the table. When this happens, you don’t have to add it to the price of the meal. However, when using a credit card, you have to add the tip to the total to be sure that the server receives his/her tip. What about taxes? Taxes are monies that are collected by the government to help pay for services such as schools, libraries, roads, and police and fire protection. A sales tax is a tax bought on something in a store. The rate of sales tax is given as a percent. You can find the amount of sales tax by using the following formula: Amount of sales tax = price ×\begin{align*}\times\end{align*} rate of sales tax After you calculate the amount of a tax using the percentage and the formula, you have to add it to an original amount. With sales tax, this is especially important. Let’s look at a few examples. Emily bought a jacket for85. If the sales tax is 7.5%, find the total cost of the jacket.
First find the amount of sales tax.
7.5% of $85=0.075×$85=6.375\begin{align*}\ 85 = 0.075 \times \ 85 = \ 6.375\end{align*} You need to round6.375 to the nearest penny, which is $6.38. Add the price of the jacket and the sales tax.$85+$6.38=$91.38\begin{align*}\ 85 + \ 6.38 = \ 91.38\end{align*}
The total cost of the jacket is $91.38. Mr. and Mrs. Green and their four children went out to dinner. The cost of the meal was$72. The restaurant added a tip of 18% to the bill. What was the total cost of the dinner?
First find the amount of the tip.
18% of $72=0.18×$72=12.96\begin{align*}\ 72 = 0.18 \times \ 72 = \ 12.96\end{align*} Add the cost of the meal and the tip.72+$12.96=$84.96\begin{align*}\ 72 + \ 12.96 = \ 84.96\end{align*}
Solution:36.00\begin{align*}36.00\end{align*} #### Example C 10% sales tax on a150.00 item.
Solution: 165.00\begin{align*}165.00\end{align*} Here is the original problem once again. Taylor and her Dad went out to a restaurant for dinner. It had been a long day at the candy store, and they were both hungry. The cost of the meal was25.75.
Taylor and her Dad began to figure out the total bill given a 7% sales tax and a 15% tip.
Do you know how to do this? Can you figure out the total bill including tax and tip?
First, let's figure out the sales tax. To do this, we convert the percent to a decimal and multiply by the total cost of the meal.
.07×25.75=1.80\begin{align*}.07 \times 25.75 = 1.80\end{align*}
Next, we figure out the tip. Convert the percent to a decimal and multiply by the cost of the meal.
.15×25.75=3.86\begin{align*}.15 \times 25.75 = 3.86\end{align*}
Now we add it all up.
25.75+1.80+3.86=31.41\begin{align*}25.75 + 1.80 + 3.86 = 31.41\end{align*}
Their total bill was $31.41 including the tip. ### Vocabulary Here are the vocabulary words in this Concept. Tip a percent that is added to a price or given for a service. Estimate an approximate calculation. It is not exact. Taxes a percent added to a bill as in a sales tax. It is an amount of money collected by the government. ### Guided Practice Here is one for you to try on your own. Brandon bought a T-shirt for$8.99 and a pair of shorts for $11.99. If the rate of sales tax is 7%, how much did he spend? Answer First find the total cost of the items purchased.$8.99+$11.99=$20.98\begin{align*}\ 8.99 + \ 11.99 = \ 20.98\end{align*}
Next find the amount of sales tax. First, convert the percent to a decimal and multiply by the total cost.
20.98×.07=1.47\begin{align*}20.98 \times .07 = 1.47\end{align*}
Notice that we rounded up to 1.47.
Now add this to the cost of the items.
20.98+1.47=22.45\begin{align*}20.98 + 1.47 = 22.45\end{align*} This is the answer. ### Video Review Here is a video for review. ### Practice Directions: Calculate the total amount paid including sales tax if the sales tax is 4% and there is a tip of 20%. 1. Total:56.75
2. Total: $43.25 3. Total:$65.00
4. Total: $25.50 5. Total:$18.75
6. Total: $59.00 7. Total:$21.50
8. Total: $44.50 9. Total:$12.50
10. Total: $43.00 11. Total:$89.50
12. Total: $99.35 13. Total:$2,35.75
14. Total: $300.50 15. Total:$678.25
### Vocabulary Language: English
Estimate
Estimate
To estimate is to find an approximate answer that is reasonable or makes sense given the problem.
Taxes
Taxes
Taxes are a percent added to a bill as in a sales tax. It is an amount of money collected by the government.
Tip
Tip
A tip is an amount of money given to a person (like a server) who has performed a service. Tips for servers are commonly 15 – 20% of the total bill.
At Grade
Nov 30, 2012
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https://labs.tib.eu/arxiv/?author=H.%20Nakanishi | • ### Astrometry of OH/IR stars using 1612 MHz hydroxyl masers. I. Annual parallaxes of WX Psc and OH138.0+7.2(1701.05101)
We report on the measurement of the trigonometric parallaxes of 1612 MHz hydroxyl masers around two asymptotic giant branch stars, WX Psc and OH138.0+7.2, using the NRAO Very Long Baseline Array with in-beam phase referencing calibration. We obtained a 3-sigma upper limit of <=5.3 mas on the parallax of WX Psc, corresponding to a lower limit distance estimate of >~190 pc. The obtained parallax of OH138.0+7.2 is 0.52+/-0.09 mas (+/-18%), corresponding to a distance of 1.9(+0.4,-0.3) kpc, making this the first hydroxyl maser parallax below one milliarcsecond. We also introduce a new method of error analysis for detecting systematic errors in the astrometry. Finally, we compare our trigonometric distances to published phase-lag distances toward these stars and find a good agreement between the two methods.
• ### HI4PI: A full-sky HI survey based on EBHIS and GASS(1610.06175)
Oct. 19, 2016 astro-ph.GA
Measurement of the Galactic neutral atomic hydrogen (HI) column density, NHI, and brightness temperatures, Tb, is of high scientific value for a broad range of astrophysical disciplines. In the past two decades, one of the most-used legacy HI datasets has been the Leiden/Argentine/Bonn Survey (LAB). We release the HI 4$\pi$ survey (HI4PI), an all-sky database of Galactic HI, which supersedes the LAB survey. The HI4PI survey is based on data from the recently completed first coverage of the Effelsberg-Bonn HI Survey (EBHIS) and from the third revision of the Galactic All-Sky Survey (GASS). EBHIS and GASS share similar angular resolution and match well in sensitivity. Combined, they are ideally suited to be a successor to LAB. The new HI4PI survey outperforms the LAB in angular resolution (16.2', FWHM) and sensitivity (RMS: 43 mK). Moreover, it has full spatial sampling and thus overcomes a major drawback of LAB, which severely undersamples the sky. We publish all-sky column density maps of the neutral atomic hydrogen in the Milky Way, along with full spectroscopic data, in several map projections including HEALPix.
• ### First results of the CALICE SDHCAL technological prototype(1602.02276)
March 20, 2016 hep-ex, physics.ins-det
The CALICE Semi-Digital Hadronic Calorimeter (SDHCAL) prototype, built in 2011, was exposed to beams of hadrons, electrons and muons in two short periods in 2012 on two different beam lines of the CERN SPS. The prototype with its 48 active layers, made of Glass Resistive Plate Chambers and their embedded readout electronics, was run in triggerless and power-pulsing mode. The performance of the SDHCAL during the test beam was found to be very satisfactory with an efficiency exceeding 90% for almost all of the 48 active layers. A linear response (within 5%) and a good energy resolution are obtained for a large range of hadronic energies (5-80GeV) by applying appropriate calibration coefficients to the collected data for both the Digital (Binary) and the Semi-Digital (Multi-threshold) modes of the SDHCAL prototype. The Semi-Digital mode shows better performance at energies exceeding 30GeV
• ### Hadron shower decomposition in the highly granular CALICE analogue hadron calorimeter(1602.08578)
March 15, 2016 physics.ins-det
The spatial development of hadronic showers in the CALICE scintillator-steel analogue hadron calorimeter is studied using test beam data collected at CERN and FNAL for single positive pions and protons with initial momenta in the range from 10 to 80 GeV/c. Both longitudinal and radial development of hadron showers are parametrised with two-component functions. The parametrisation is fit to test beam data and simulations using the QGSP_BERT and FTFP_BERT physics lists from Geant4 version 9.6. The parameters extracted from data and simulated samples are compared for the two types of hadrons. The response to pions and the ratio of the non-electromagnetic to the electromagnetic calorimeter response, h/e, are estimated using the extrapolation and decomposition of the longitudinal profiles.
• ### DHCAL with Minimal Absorber: Measurements with Positrons(1603.01652)
March 4, 2016 physics.ins-det
In special tests, the active layers of the CALICE Digital Hadron Calorimeter prototype, the DHCAL, were exposed to low energy particle beams, without being interleaved by absorber plates. The thickness of each layer corresponded approximately to 0.29 radiation lengths or 0.034 nuclear interaction lengths, defined mostly by the copper and steel skins of the detector cassettes. This paper reports on measurements performed with this device in the Fermilab test beam with positrons in the energy range of 1 to 10 GeV. The measurements are compared to simulations based on GEANT4 and a standalone program to emulate the detailed response of the active elements.
• ### Shower development of particles with momenta from 15 GeV to 150 GeV in the CALICE scintillator-tungsten hadronic calorimeter(1509.00617)
Dec. 11, 2015 hep-ex, physics.ins-det
We present a study of showers initiated by electrons, pions, kaons, and protons with momenta from 15 GeV to 150 GeV in the highly granular CALICE scintillator-tungsten analogue hadronic calorimeter. The data were recorded at the CERN Super Proton Synchrotron in 2011. The analysis includes measurements of the calorimeter response to each particle type as well as measurements of the energy resolution and studies of the longitudinal and radial shower development for selected particles. The results are compared to Geant4 simulations (version 9.6.p02). In the study of the energy resolution we include previously published data with beam momenta from 1 GeV to 10 GeV recorded at the CERN Proton Synchrotron in 2010.
• ### GASS: The Parkes Galactic All-Sky Survey. I. Survey Description, Goals, and Initial Data Release(0901.1159)
Jan. 9, 2009 astro-ph.GA
The Parkes Galactic All-Sky Survey (GASS) is a survey of Galactic atomic hydrogen (HI) emission in the Southern sky covering declinations $\delta \leq 1^{\circ}$ using the Parkes Radio Telescope. The survey covers $2\pi$ steradians with an effective angular resolution of ~16', at a velocity resolution of 1.0 km/s, and with an rms brightness temperature noise of 57 mK. GASS is the most sensitive, highest angular resolution survey of Galactic HI emission ever made in the Southern sky. In this paper we outline the survey goals, describe the observations and data analysis, and present the first-stage data release. The data product is a single cube at full resolution, not corrected for stray radiation. Spectra from the survey and other data products are publicly available online.
• ### Molecular Gas Distribution in Barred and Non-Barred Galaxies along the Hubble Sequence(0805.4048)
May 27, 2008 astro-ph
We present results from a survey of 12CO(J=1-0) spectra obtained for the central regions of 68 nearby galaxies at an angular resolution of 16 arcseconds using the Nobeyama Radio Observatory 45m telescope, aimed at characterizing the properties of star forming molecular gas. Combined with similar resolution observations in the literature, the compiled sample set of 166 galaxies span a wide range of galactic properties. NGC 4380, which was previously undetected in CO, was detected. This initial paper of a series will focus on the data and the gaseous properties of the samples, and particularly on the degree of central concentration of molecular gas in a range of morphological types, from early (S0/Sa) to late (Sd/Sm) galaxies with and without bars. The degree of molecular central concentration in the central kiloparsec, compared to the central several kiloparsecs of galaxies, is found to vary smoothly with Hubble type, so that early type galaxies show larger central concentration. The comparison of barred and non-barred galaxies within early and late type galaxies suggest that difference in Hubble type, representing the effect of bulges, is the more important factor in concentrating gas into the central regions than bars.
• ### An Interaction of a Magellanic Leading Arm High Velocity Cloud with the Milky Way Disk(0712.2267)
Dec. 14, 2007 astro-ph
The Leading Arm of the Magellanic System is a tidally formed HI feature extending $\sim 60\arcdeg$ from the Magellanic Clouds ahead of their direction of motion. Using atomic hydrogen (HI) data from the Galactic All Sky-Survey (GASS), supplemented with data from the Australia Telescope Compact Array, we have found evidence for an interaction between a cloud in the Leading Arm and the Galactic disk where the Leading Arm crosses the Galactic plane. The interaction occurs at velocities permitted by Galactic rotation, which allows us to derive a kinematic distance to the cloud of 21 kpc, suggesting that the Leading Arm crosses the Galactic Plane at a Galactic radius of $R\approx 17$ kpc.
• ### Compensation of the Crossing Angle with Crab Cavities at KEKB(0706.3248)
June 22, 2007 physics.ins-det
Crab cavities have been installed in the KEKB B--Factory rings to compensate the crossing angle at the collision point and thus increase luminosity. The beam operation with crab crossing has been done since February 2007. This is the first experience with such cavities in colliders or storage rings. The crab cavities have been working without serious issues. While higher specific luminosity than the geometrical gain has been achieved, further study is necessary and under way to reach the prediction of simulation.
• ### Nobeyama CO Atlas of Nearby Spiral Galaxies: Distribution of Molecular Gas in Barred and Non-barred Spiral Galaxies(0705.2678)
May 18, 2007 astro-ph
The data from a CO(1 - 0) mapping survey of 40 nearby spiral galaxies performed with the Nobeyama 45-m telescope are presented. The criteria of the sample selection were (1) RC3 morphological type in the range Sa to Scd, (2) distance less than 25 Mpc, (3) inclination angle less than 79deg (RC3), (4) flux at 100 um higher than ~ 10 Jy, (5) spiral structure is not destroyed by interaction. The maps of CO cover most of the optical disk of the galaxies. We investigated the influence of bar on the distribution of molecular gas in spiral galaxies using these data. We confirmed that the degree of central concentration is higher in barred spirals than in non-barred spirals as shown by the previous works. Furthermore, we present an observational evidence that bars are efficient in driving molecular gas that lies within the bar length toward the center, while the role in bringing gas in from the outer parts of the disks is small. The transported gas accounts for about half of molecular gas within the central region in barred spiral galaxies. We found a correlation between the degree of central concentration and bar strength. Galaxies with stronger bars tend to have higher central concentration. The correlation implies that stronger bars accumulate molecular gas toward the center more efficiently. These results are consistent with long-lived bars.
• ### Quantum transport in randomly diluted quantum percolation clusters in two dimensions(cond-mat/0604524)
We study the hopping transport of a quantum particle through finite, randomly diluted percolation clusters in two dimensions. We investigate how the transmission coefficient T behaves as a function of the energy E of the particle, the occupation concentration p of the disordered cluster, the size of the underlying lattice, and the type of connection chosen between the cluster and the input and output leads. We investigate both the point-to-point contacts and the busbar type of connection. For highly diluted clusters we find the behavior of the transmission to be independent of the type of connection. As the amount of dilution is decreased we find sharp variations in transmission. These variations are the remnants of the resonances at the ordered, zero-dilution, limit. For particles with energies within 0.25 <= E <= 1.75 (relative to the hopping integral) and with underlying square lattices of size 20x20, the configurations begin transmitting near p_a = 0.60 with T against p curves following a common pattern as the amount of dilution is decreased. Near p_b = 0.90 this pattern is broken and the transmission begins to vary with the energy. In the asymptotic limit of very large clusters we find the systems to be totally reflecting except when the amount of dilution is very low and when the particle has energy close to a resonance value at the ordered limit or when the particle has energy at the middle of the band.
• ### Quantum interference effects in particle transport through square lattices(cond-mat/0406660)
We study the transport of a quantum particle through square lattices of various sizes by employing the tight-binding Hamiltonian from quantum percolation. Input and output semi-infinite chains are attached to the lattice either by diagonal point to point contacts or by a busbar connection. We find resonant transmission and reflection occuring whenever the incident particle's energy is near an eigenvalue of the lattice alone (i.e., the lattice without the chains attached). We also find the transmission to be strongly dependent on the way the chains are attached to the lattice.
• ### Radio Observations of the Afterglow of GRB 030329(astro-ph/0401258)
Jan. 14, 2004 astro-ph
We present the results of the radio observations of the afterglow of GRB 030329 with the Nobeyama 45-m telescope. The observations were made at 23.5 GHz, 43 GHz, and 90 GHz. The light curves show steep decline after constant phase. The start time of the decline depends on frequency. Namely, the decline started earlier at higher frequency. The spectrum has a peak at mm wavelength range. The peak frequency and the peak flux decreased with time. These results are consistent with the expectation from fireball model.
• ### The Virgo High-Resolution CO Survey I. CO Atlas(astro-ph/0301008)
Jan. 1, 2003 astro-ph
We present the results of the Virgo high-resolution CO survey (ViCS) obtained with the Nobeyama Millimeter-wave Array (NMA). This survey was made in the course of a long-term project at Nobeyama from 1999 December through 2002 April. The objects were selected from Virgo cluster members, considering CO richness from single dish flux, mild inclination, and lack of strong tidal perturbations. The central 1 arc min regions (4.7 kpc) of 15 spiral galaxies were observed with resolutions of 2 to 5 arcsec and 10 to 20 km/s, and sensitivities of 20 mJy/ beam for a 10 km/s channel. The objects lie at the same distance of the Virgo cluster (16.1 Mpc), which is advantageous for comparisons among individual galaxies. We describe the details of observations and data reduction, and present an atlas of integrated CO intensity maps, velocity fields and position-velocity diagrams along the major axes. The molecular gas morphology in the Virgo galaxies shows a wealth of variety, not specifically depending on the Hubble types.
• ### The Virgo High-Resolution CO Survey. II. Rotation Curves and Dynamical Mass Distributions(astro-ph/0301009)
Jan. 1, 2003 astro-ph
Based on a high-resolution CO survey of Virgo spirals with the Nobeyama Millimeter-wave Array, we determined the dynamical centers using velocity fields, and derived position-velocity diagrams (PVDs) along the major axes of the galaxies across their dynamical centers. We applied a new iteration method to derive rotation curves (RCs), which reproduce the observed PVDs. The obtained high-accuracy RCs generally show steep rise in the central 100 to 200 pc regions, followed by flat rotation in the disk. We applied a deconvolution method to calculate the surface-mass density (SMD) using the RCs based on two extreme assumptions that the mass distribution is either spherical or thin-disk shaped. Both assumptions give nearly identical results, agreeing with each other within a factor of two at any radii. The SMD distributions revealed central massive cores with peak SMD of 10^4 - 10^5 Msun pc^-2 and total mass within 200 pc radius of the order of about 10^9 Msun Correlation analysis among the derived parameters show that the central CO-line intensity is positively correlated with the central SMD, which suggests that the deeper is the gravitational potential, the higher is the molecular gas concentration in the nuclei regardless morphological types.
• ### Structure and diffusion time scales of disordered clusters(cond-mat/0207468)
The eigenvalue spectra of the transition probability matrix for random walks traversing critically disordered clusters in three different types of percolation problems show that the random walker sees a developing Euclidean signature for short time scales as the local, full-coordination constraint is iteratively applied.
• ### Iterated fully coordinated percolation on a square lattice(cond-mat/0008294)
Aug. 20, 2000 cond-mat.stat-mech
We study, on a square lattice, an extension to fully coordinated percolation which we call iterated fully coordinated percolation. In fully coordinated percolation, sites become occupied if all four of its nearest neighbors are also occupied. Repeating this site selection process again yields the iterated fully coordinated percolation model. Our results show a substantial enhancement in the size of highly connected regions after each iteration (from ordinary to fully coordinated and then to iterated fully coordinated percolation). However, using Monte Carlo methods to determine the static critical exponents and normal mode analyses to determine the dynamic critical exponents, our results indicate that the ordinary, fully coordinated, and iterated fully coordinated percolation models belong to the same universality class. This is in contrast to our previous results [*] wherein a different universality class dynamically (but not statically) for fully coordinated percolation was indicated, though the differences in the dynamic exponents were small. A possible cause might be the different methods of generating clusters, i.e., in this work clusters were generated statically in square lattices while in our previous work clusters were grown dynamically, for studying the dynamic critical exponents. * E. Cuansing, J. H. Kim and H. Nakanishi, Phys. Rev. E, 60, 3670 (1999).
• ### The RF source of the 60-MeV Linac for the KEK/JAERI Joint Project(physics/0008040)
Aug. 17, 2000 physics.acc-ph
The construction of the 60-MeV proton linac has started as a low-energy front of the KEK/JAERI Joint Project for a high-intensity proton accelerator facility at KEK. The accelerating frequency is 324 MHz. Five UHF klystrons are used as an rf source; their ratings have a maximum power of 3 MW, a beam pulse width of a 700 micro-sec (an rf pulse width is 650 micro-sec) and a repetition rate of 50 pps. We have manufactured a proto-type rf source (a power-supply system with a modulating-anode pulse modulator and prototype klystrons). In this paper, the specifications and developments of the rf source, including the WR-2300 waveguide system, are summarized. During the manufacturing process, strong oscillations due to back-going electrons from the collector were observed. This phenomenon was analyzed both experimentally and theoretically. We have tested up to an output power of nearly 3 MW, and succeeded to test the DTL hot-model structure. | 2020-11-29 02:39:52 | {"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": 0.6246123313903809, "perplexity": 2323.177951918737}, "config": {"markdown_headings": true, "markdown_code": false, "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-2020-50/segments/1606141195967.34/warc/CC-MAIN-20201129004335-20201129034335-00263.warc.gz"} |
https://nbsphinx.readthedocs.io/en/0.3.5/prolog-and-epilog.html | # Prolog and Epilog¶
When including notebooks in your Sphinx documentation, you can choose to add some generic content before and after each notebook. This can be done with the configuration values nbsphinx_prolog and nbsphinx_epilog in the file conf.py.
The prolog and epilog strings can hold arbitrary reST markup. Particularly, the only and raw directives can be used to have different content for HTML and LaTeX output.
Those strings are also processed by the Jinja2 templating engine. This means you can run Python-like code within those strings. You have access to the current Sphinx build environment via the variable env. Most notably, you can get the file name of the current notebook with
{{ env.doc2path(env.docname, base=None) }}
Warning:
If you use invalid syntax, you might get an error like this:
jinja2.exceptions.TemplateSyntaxError: expected token ':', got '}'
This is especially prone to happen when using raw LaTeX, with its abundance of braces. To avoid clashing braces you can try to insert additional spaces or LaTeX macros that don’t have a visible effect, like e.g. \strut{}. For example, you can avoid three consecutive opening braces with something like that:
\texttt{\strut{}{{ env.doc2path(env.docname, base=None) }}}
NB: The three consecutive closing braces in this example are not problematic.
## Examples¶
You can include a simple static string, using reST markup if you like:
nbsphinx_epilog = """
----
Generated by nbsphinx_ from a Jupyter_ notebook.
.. _Jupyter: https://jupyter.org/
"""
Using some additional Jinja2 markup and the information from the env variable, you can create URLs that point to the current notebook file, but located on some other server:
nbsphinx_prolog = """
Go there: https://example.org/notebooks/{{ env.doc2path(env.docname, base=None) }}
----
"""
You can also use separate content for HTML and LaTeX output, e.g.:
nbsphinx_prolog = """
{% set docname = env.doc2path(env.docname, base=None) %}
.. only:: html
Go there: https://example.org/notebooks/{{ docname }}
.. only:: latex
The following section was created from :file:{{ docname }}.
"""
For a more involved example for different HTML and LaTeX versions, see the file conf.py of the nbsphinx documentation. | 2022-07-06 07:02:27 | {"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": 0.6385063529014587, "perplexity": 8747.769722328747}, "config": {"markdown_headings": true, "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-2022-27/segments/1656104668059.88/warc/CC-MAIN-20220706060502-20220706090502-00305.warc.gz"} |
https://www.fireidea.com/archives/119 | 2010-11-15
Question: How to fix PHP Fatalerror: Class ‘DOMDocument’ not found in/var/www/html/index.php on line 171 or similar error?
NOTE: Received this error on a Centos5 environment while trying to run a PHPscript that makes use of the DOMDocument class
</p>
1.yuminstall php-xml | 2023-03-21 17:34:02 | {"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": 0.547301709651947, "perplexity": 13597.547887839986}, "config": {"markdown_headings": true, "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-2023-14/segments/1679296943704.21/warc/CC-MAIN-20230321162614-20230321192614-00131.warc.gz"} |
https://www.physicsforums.com/threads/motion-of-a-point-mass-circular-motion.846889/ | # Motion of a point mass (circular motion)
1. Dec 6, 2015
### Carson Birth
1. The problem statement, all variables and given/known data
A point mass, sliding over an even, horizontal plane is bounded on an inextensible, massless thread. During the motion, the thread is pulled by a force F with constant velocity $v_{o}$ through a hole O. In the beginning ($t_{o}$ = 0) r($t_{o}$)=b holds (the length of the thread on the plane is b at the beginning then will get shorter due to the force F as it is pulled through). The initial velocity of the point mass perpendicular to the thread is $v_{1}$ in $\phi$ direction, and the angle $\phi$($t_{o}$) = $\phi_{o}$ = 0
Formulate the system's equation of motion and the equation of constraint forces in polar coordinates. Apply Newton's Law. Also the polar coordinate system is attached to the masspoint with r pointing away from the hole and $\phi$ pointing toward the trajectory.
2. Relevant equations
$\overrightarrow{a}$ = ($\ddot{r}$ - r$\dot{\phi}^2$) in r direction + (r$\ddot{\phi}$+2$\dot{r}$$\dot{\phi}$) in $\phi$ direction
3. The attempt at a solution
I separated the forces into there respected directions:
$\ddot{\phi}$ direction: mr$\ddot{\phi}$ + m2$\dot{r}$$\dot{\phi}$ = 0
r direction: F + mr$\dot{\phi}^2$ = 0
Now im pretty sure there isnt any constraint forces, since there is no N force effecting the mass point.
So now I need to create an equation of motion, and from my understanding I need to create one equation. Is it as simple as just solving for $\dot{\phi}$ and plugging it into the other equation? I had a similar equation where I solved the $\phi$ direction equation as a differential but it didnt have the 2$\dot{r}$$\dot{\phi}$ term with it, it was a gravity force making it very simple to solve. I know this isnt for people to solve my homework so im just looking for advice on how to get it all set up for the further sub-questions. Any advice would be appreciated :D
2. Dec 7, 2015
### ehild
r is changing. Why did you ignored $\ddot r$?
3. Dec 7, 2015
### Carson Birth
My thought was since the thread is being pulled by a force with constant velocity, that it wouldnt be accelerating.
4. Dec 7, 2015
### ehild
You are right, I misread it as "constant force". Sorry.
So you do not know F, but you know that $\dot r$ is constant. Go ahead. Solve the first equation.
Last edited: Dec 7, 2015
5. Dec 7, 2015
### Carson Birth
When you say solve the first equation, what do you mean? Would I solve the first equation as a differential equation or you mean solve for a variable then put it into the first equation? :D
6. Dec 7, 2015
### ehild
It is a differential equation for Φ as function of time. You know $\dot r$ hence also r(t). | 2018-01-18 08:17:19 | {"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": 0.7062365412712097, "perplexity": 465.8480185215604}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "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-2018-05/segments/1516084887077.23/warc/CC-MAIN-20180118071706-20180118091706-00773.warc.gz"} |
http://www.wekaleamstudios.co.uk/posts/creating-a-presentation-with-latex-beamer-boxes/ | # Creating a Presentation with LaTeX Beamer – Boxes
July 19th, 2010
We can add coloured boxes with text or mathematics into a LaTeX beamer presentation which is particularly useful if we have definitions, theorem or computer code to highlight this information that may not be so accessible within a paragraph of text.
Fast Tube by Casper
The easiest way to create a box is to use the various pre-defined environments such as definition:
\begin{definition} The simple linear regression model describes the relationship between a response variable and a single explanatory variable. \end{definition}
We might want to create a box with our own title in which case the exampleblock environment can be used with the new title as an argument to this environment:
\begin{exampleblock}{Linear Regression Model} $Y_{i} = \beta_{0} + \beta_{1} X_{i} + \epsilon_{i}$ \end{exampleblock}
Other useful resources are provided on the Supplementary Material page. | 2017-07-23 00:39:16 | {"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": 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": 0.5584626793861389, "perplexity": 913.1762794140844}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2017-30/segments/1500549424200.64/warc/CC-MAIN-20170723002704-20170723022704-00477.warc.gz"} |
https://math.stackexchange.com/questions/1976544/conditional-distribution-of-subvector-of-a-dirichlet-random-variable/1976732 | # Conditional distribution of subvector of a Dirichlet random variable
Suppose $\mathbf{X} = (X_1, \cdots, X_K)$ follows a Dirichlet distribution with parameters $(\alpha_1, \cdots, \alpha_K)$. Partition $\mathbf{X} = (\mathbf{X}_{(1)}, \mathbf{X}_{(2)})$ for subvectors $\mathbf{X}_{(1)}, \mathbf{X}_{(2)}$. What is the distribution of $\mathbf{X}_{(1)}|\mathbf{X}_{(2)}$?
WLOG just relabel those random variables such that $X_1|X_2$ is the random variable of interest, and remaining random variables are labeled from $X_3, X_4, \ldots, X_k$
By the aggregation property, we know that $$\left(X_1, X_2, X_0 = 1 - X_1 - X_2 = \sum_{i=3}^k X_i\right) \sim\text{Dir}\left(\alpha_1, \alpha_2, \alpha_0 = \sum_{i=3}^k \alpha_i\right)$$
And therefore their joint pdf is $$f_{X_1, X_2}(x_1, x_2) = \frac {\Gamma(\alpha_1 + \alpha_2 + \alpha_0)} {\Gamma(\alpha_1)\Gamma(\alpha_2)\Gamma(\alpha_0)} x_1^{\alpha_1-1}x_2^{\alpha_2-1}(1-x_1-x_2)^{\alpha_0-1}$$
where $x_1, x_2 \in (0, 1), x_1 + x_2 \leq 1$. Similarly, the marginal pdf of $X_2$ is $$f_{X_2}(x_2) = \frac {\Gamma(\alpha_2 + \alpha_1 + \alpha_0)} {\Gamma(\alpha_2)\Gamma(\alpha_1+\alpha_0)} x_2^{\alpha_2-1}(1-x_2)^{\alpha_1+\alpha_0-1}, ~~ x_2 \in (0, 1)$$
i.e. $\text{Beta}(\alpha_2, \alpha_1 + \alpha_0)$. Therefore the conditional pdf of $X_1|X_2 = x_2$ is $$f_{X_1|X_2=x_2}(x_1|x_2) = \frac {f_{X_1,X_2}(x_1,x_2)} {f_{X_2}(x_2)} = \frac {\Gamma(\alpha_1+\alpha_0)} {\Gamma(\alpha_1) \Gamma(\alpha_0)} \left(\frac {x_1} {1-x_2}\right)^{\alpha_1-1} \left(1 - \frac {x_1} {1-x_2}\right)^{\alpha_0-1}\frac {1} {1 - x_2}$$
where $x_1 \in (0, 1-x_2)$. So $X_1|X_2 = x_2$ has a scaled Beta distribution, i.e. $$\frac {1} {1 - x_2} X_1|X_2 = x_2 \sim \text{Beta}(\alpha_1, \alpha_0)$$
Edit: It seems that I have only finished a univariate version of the problem.
Let $m \in \{1, \ldots, k-1\}$ be the dimension of $\mathbf{X}_{(1)} = (X_1, \ldots X_m)$, such that $\mathbf{X}_{(2)} = (X_{m+1}, \ldots X_k)$ (WLOG)
By definition the joint pdf of $(X_1, \ldots, X_k)$ is $$f_{X_1, \ldots, X_k}(x_1, \ldots, x_k) = \frac {\Gamma(\sum_{i=1}^k \alpha_i)} {\prod_{i=1}^k \Gamma(\alpha_i)} \prod_{i=1}^k x_i^{\alpha_i-1}, x_i \in (0, 1), \sum_{i=1}^k x_i = 1$$
Similarly, the joint pdf of $(X_{m+1}, \ldots, X_k)$ is $$f_{X_{m+1}, \ldots, X_k}(x_{m+1}, \ldots, x_k) = \frac {\Gamma(\sum_{i=1}^k \alpha_i)} {\Gamma(\alpha_0)\prod_{i=m+1}^k \Gamma(\alpha_i)} \prod_{i=m+1}^k x_i^{\alpha_i-1} \left(1 - \sum_{j=m+1}^k x_j\right)^{\alpha_0-1}$$
where $x_i \in (0, 1), \sum_{i=m+1}^k x_i < 1$ and $\alpha_0 = \sum_{i=1}^m \alpha_i$. Again the conditional pdf is the ratio of these two:
\begin{align} &~ f_{X_1, \ldots, X_m|X_{m+1}, \ldots, X_k}(x_1, \ldots, x_m|x_{m+1},\ldots, x_k) \\ =&~ \frac {\Gamma(\sum_{i=1}^m \alpha_i)} {\prod_{i=1}^m\Gamma(\alpha_i)} \prod_{i=1}^m x_i^{\alpha_i-1} \left(1 - \sum_{j=m+1}^k x_j\right)^{-(\alpha_0-1)} \\ =&~ \frac {\Gamma(\sum_{i=1}^m \alpha_i)} {\prod_{i=1}^m\Gamma(\alpha_i)} \prod_{i=1}^m \left[x_i \left(1 - \sum_{j=m+1}^k x_j\right)^{-1}\right]^{\alpha_i-1} \left(1 - \sum_{j=m+1}^k x_j\right)^{-(m-1)} \end{align}
So $\mathbf{X}_{(1)}|\mathbf{X}_{(2)} = \mathbf{x}_2$ has a scaled Dirichlet distribution, or equivalently, $$\frac {1} {1 - \mathbf{1}^T\mathbf{x}_2}\mathbf{X}_{(1)}|\mathbf{X}_{(2)} = \mathbf{x}_2 \sim \text{Dir}(\alpha_1, \ldots, \alpha_m)$$
where $\mathbf{1}$ is the vector with length $k-m$ and all entries equal to 1 | 2022-08-09 04:43:33 | {"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": 1, "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": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9992510080337524, "perplexity": 594.6572748320743}, "config": {"markdown_headings": true, "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-2022-33/segments/1659882570901.18/warc/CC-MAIN-20220809033952-20220809063952-00441.warc.gz"} |
http://nrich.maths.org/1051 | ### Pyramids
What are the missing numbers in the pyramids?
A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you the last two digits of her answer. Now you can really amaze her by giving the whole answer and the three consecutive numbers used at the start.
# Is it Magic or Is it Maths?
##### Stage: 3 Challenge Level:
Here are three 'tricks' to amaze your friends.
But the really clever trick is explaining to them why these 'tricks' are maths and not magic. Like all good magicians, you should practise by trying them. Can you explain how they work?
This trick will impress even your maths teacher.
• Think of a number.
• Double it.
• Add $10$.
• Halve it.
• Take away you original number.
• Is your answer $5$?
Try this with a different starting number. Did you get a different result? Why does this happen?
Write the answer on a piece of paper without letting anybody see it and seal it in an envelope. Have somebody hold the envelope and at the end ask them to open it and reveal the number you wrote at the beginning. Wow, Magic!
Guess how much money people have in their pockets!
Without giving you any information, ask a friend to count the value of some coins and write the amount on a piece of paper. Then ask your friend to:
• Double the amount.
• Add the first odd prime number to the new total.
• Multiply the result by 1/4 of 20.
• Subtract the lowest common multiple of 2 and 3.
For the grand finale, you ask for the final answer. Take off the last digit and you will be able to work out how much the coins are worth!
Amaze your audience by working out not only their age but also what size shoe they wear! Wow them even more by telling them how the maths works. Give them the following directions but tell them not to show you any calculations: | 2015-04-21 03:02:58 | {"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": 0.5241998434066772, "perplexity": 1067.1150827601175}, "config": {"markdown_headings": true, "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-2015-18/segments/1429246640001.64/warc/CC-MAIN-20150417045720-00292-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://tex.stackexchange.com/questions/254306/picture-changes-places | # Picture Changes places
Below, is some code I'm using on a work.
\documentclass[12pt,a4paper]{book}
\usepackage{graphicx}
\usepackage{bm}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{mathtools}
\usepackage{enumitem}
\graphicspath{ {./Pictures} }
\begin{document}
sdlaksdasdkj lasjdlakjsdlajçd çlaksdçlahsd ççajshdaçslkhdçlawhoqiw aosihdq+we+poikjlçaks
\begin{figure}[h!]
\centering
\includegraphics[scale=1]{SizeGraph.png}
\caption{Plot of size accordingly to the number of replications}
\end{figure}
kçlksdaºlçskdja ºsºçljas daihpoquwhejsdkabjboqu skjdbascoupreqweoru adpjncopowueyr cnjdpoouhwer ppouweyr hjaspdsaopoih wqpoihpf sdfapougepwqry fdasdfc
\begin{figure}[h!]
\centering
\includegraphics[scale=1]{PowerGraph.png}
\caption{Plot of size accordingly to the number of replications}
\end{figure}
\end{document}
The problem begins when I use the same code on book tex file, i.e.,
\documentclass[12pt,a4paper]{book}
\usepackage{graphicx}
\usepackage{bm}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{mathtools}
\usepackage{enumitem}
\usepackage{url}
\usepackage{listings,xcolor}
\begin{document}
\frontmatter
%\setcounter{section}{1}
\tableofcontents
\mainmatter
\chapter{Simulations}
\input{simulations.tex}
\appendix
\section{Ehm}
\input{Ehm.tex}
\backmatter
\begin{thebibliography}{99}
sdasdasçdasçdlkçalskdç
\end{thebibliography}
\end{document}
The simulation.tex file has the same code as the one I wrote at the beginning of this question. However, when I compile, the simulations section doesn't produce the same output, since the text that was supposed to show up between the pictures SizeGraph.png and PowerGraph.png, now appears before the first picture. How can I solve this problem?
Any help would be appreciated.
• If you do not want to have it floating, just put it in a center environment and use \captionof{figure}{...}. If you want to get help on a specific case, you will have to minimize your code a lot. Just show us the very problem. – LaRiFaRi Jul 8 '15 at 13:37
• @LaRiFaRi Don't I do that already with \centering? – An old man in the sea. Jul 8 '15 at 13:55
• Yes, but inside the figure environment, you will get the floating behaviour you are seeing right now. (btw. I am not recommending this. I never use any parameter inside \begin{figure}[] if it is not totally needed) – LaRiFaRi Jul 8 '15 at 14:02
• @LaRiFaRi then, how would I do what you're suggesting? I'm a beginner in latex... – An old man in the sea. Jul 8 '15 at 14:03
If you do not want the figures to float away, just do not use a floating environment like figure. I can not test your case, as I have no real MWE, but I guess that something is pushing your figure to a position, LaTeX sees better fit in.
Below I show you how to get the same result with a center (second figure). This environment does not float.
% arara: pdflatex
\documentclass[12pt,a4paper]{book}
\usepackage[demo]{graphicx}
\graphicspath{ {./Pictures} }
\usepackage{caption}
\begin{document}
sdlaksdasdkj lasjdlakjsdlajçd çlaksdçlahsd ççajshdaçslkhdçlawhoqiw aosihdq we poikjlçaks
\begin{figure}[h!]
\centering
\includegraphics[scale=1]{SizeGraph.png}
\caption{Plot of size accordingly to the number of replications}
\end{figure}
kçlksdaºlçskdja ºsºçljas daihpoquwhejsdkabjboqu skjdbascoupreqweoru adpjncopowueyr cnjdpoouhwer ppouweyr hjaspdsaopoih wqpoihpf sdfapougepwqry fdasdfc
\begin{center}
\includegraphics[scale=1]{PowerGraph.png}
\captionof{figure}{Plot of size accordingly to the number of replications}
\end{center}
\end{document}
Off-topic: mathtools loads amsmath so get rid of the latter.
• When trying to load \usepackage{caption}, the usual window to download file shows, and after compiling I get : «! LaTeX Error: File `caption.sty' not found.» Do you know why? – An old man in the sea. Jul 8 '15 at 14:25
• @Anoldmaninthesea. tex.stackexchange.com/q/152277 – LaRiFaRi Jul 8 '15 at 14:29
• @Anoldmaninthesea you are very welcome. Glad it worked. – LaRiFaRi Jul 8 '15 at 22:29 | 2019-11-14 23:31:18 | {"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": 0.8083090782165527, "perplexity": 2953.229266866065}, "config": {"markdown_headings": true, "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-2019-47/segments/1573496668544.32/warc/CC-MAIN-20191114232502-20191115020502-00354.warc.gz"} |
http://physicspedia.wikia.com/wiki/Test_page | ## FANDOM
9 Pages
This is a page for me to test a few things
${3 ~ \times ~ 2 ~ = ~ 6}$
$\alpha$
$f(x) = x^2$
$\vec{a}$
$\rVert\vec{a}\rVert$
$\nabla\psi$
$\infty$
$\int\limits_{0}^{2}\frac{f(x)}, dx$
$F = ma$ | 2018-03-18 05:58:47 | {"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": 0, "img_math": 9, "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": 0.46049588918685913, "perplexity": 6117.737940598086}, "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-2018-13/segments/1521257645538.8/warc/CC-MAIN-20180318052202-20180318072202-00645.warc.gz"} |
https://tex.stackexchange.com/questions/400669/how-to-set-the-second-and-subsequent-appearances-of-an-acronym-using-acro-packag | # How to set the second and subsequent appearances of an acronym using acro package
I'm using the acro package for my acronyms. My supervisor didn't like the use of ancronyms in the text, so I have to put all of them in the long form, but keeping the first style like "acronym (AC)" and the rest of uses "acronym". How can I configure the acro package for to print the long form in the second and subsequent uses?
\documentclass{article}
\usepackage{acro}
\DeclareAcronym{TIAE}{
short=TIAE,
long=this is an example,
}
\begin{document}
Firt use \ac{TIAE},
then second use \ac{TIAE}, %<--- must be "this is an example"
third use \ac{TIAE} and %<--- must be "this is an example"
last use \ac{TIAE}.\par %<--- must be "this is an example"
%
\printacronyms
\end{document}
• What is the point in giving the (AC) on first use if you never use it anywhere else in the text? – Dai Bowen Nov 10 '17 at 15:59
• @DaiBowen It will be used in tables and figures and schemas – Daniel Valencia C. Nov 10 '17 at 16:23
• @DaiBowen also some terms are more commonly known in some fields by their acronyms (PL vs photoluminescence) for example, but one author/supervisor may not be used to that or may just have a thing against all acronyms – Chris H Nov 10 '17 at 16:30
You could create a macro which contains \acifused to test if the acronym was already used and use \acl if true and \ac if false:
\documentclass{article}
\usepackage{acro}
\DeclareAcronym{TIAE}{
short=TIAE,
long=this is an example,
}
\newcommand{\myac}[1]{\acifused{#1}{\acl{#1}}{\ac{#1}}}
\begin{document}
Firt use \myac{TIAE},
then second use \myac{TIAE}, %<--- must be "this is an example"
third use \myac{TIAE} and %<--- must be "this is an example"
last use \myac{TIAE}.\par %<--- must be "this is an example"
%
\printacronyms
\end{document} | 2020-07-02 07:05:05 | {"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": 0.5900461673736572, "perplexity": 3118.0732385071133}, "config": {"markdown_headings": true, "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-2020-29/segments/1593655878519.27/warc/CC-MAIN-20200702045758-20200702075758-00125.warc.gz"} |
https://encyclopediaofmath.org/index.php?title=Exterior_product&diff=cur&oldid=14457 | Difference between revisions of "Exterior product"
A fundamental operation in the exterior algebra of tensors defined on an $n$- dimensional vector space $V$ over a field $K$.
Let $e _ {1} \dots e _ {n}$ be a basis of $V$, and let $a$ and $b$ be $p$- and $q$- forms:
$$a = a ^ {i _ {1} \dots {i _ {p} } } e _ {i _ {1} } \otimes \dots \otimes e _ {i _ {p} } ,$$
$$b = b ^ {j _ {1} \dots {j _ {q} } } e _ {j _ {1} } \otimes \dots \otimes e _ {j _ {q} } .$$
The exterior product of the forms $a$ and $b$ is the $( p + q)$- form $c$ obtained by alternation of the tensor product $a \otimes b$. The form $c$ is denoted by $a \wedge b$; its coordinates are skew-symmetric:
$$c ^ {k _ {1} \dots k _ {p+ q } } = \ \frac{1}{p! q! } \delta _ {i _ {1} \dots i _ {p} j _ {1} \dots j _ {q} } ^ {k _ {1} \dots \dots \dots k _ {p+ q } } a ^ {i _ {1} \dots i _ {p} } b ^ {j _ {1} \dots j _ {q} } ,$$
where $\delta _ {i _ {1} \dots j _ {q} } ^ {k _ {1} \dots k _ {p+} q }$ are the components of the generalized Kronecker symbol. The exterior product of covariant tensors is defined in a similar manner.
The basic properties of the exterior product are listed below:
1) $( ka) \wedge b = a \wedge ( kb) = k( a \wedge b)$, $k \in K$( homogeneity);
2) $( a+ b) \wedge c = a \wedge c + b \wedge c$( distributivity);
3) $( a \wedge b ) \wedge c = a \wedge ( b \wedge c)$( associativity).
4) $a \wedge b = (- 1) ^ {pq} b \wedge a$; if the characteristic of $K$ is distinct from two, the equation $a \wedge a = 0$ is valid for any form $a$ of odd valency.
The exterior product of $s$ vectors is said to be a decomposable $s$- vector. Any poly-vector of dimension $s$ is a linear combination of decomposable $s$- vectors. The components of this combination are the ( $s \times s$)- minors of the ( $n \times s$)- matrix $( a _ {j} ^ {i} )$, $1 \leq i \leq n$, $1 \leq j \leq s$, of the coefficients of the vectors $a _ {1} \dots a _ {s}$. If $s = n$ their exterior product has the form
$$\alpha _ {n} = a _ {1} \wedge \dots \wedge a _ {n} = \ \mathop{\rm det} ( a _ {j} ^ {i} ) e _ {1} \wedge \dots \wedge e _ {n} .$$
Over fields of characteristic distinct from two, the equation $a _ {1} \wedge \dots \wedge a _ {n} = 0$ is necessary and sufficient for vectors $a _ {1} \dots a _ {n}$ to be linearly dependent. A non-zero decomposable $s$- vector $\alpha _ {s}$ defines in $V$ an $s$- dimensional oriented subspace $A$, parallel to the vectors $a _ {1} \dots a _ {s}$, and the parallelotope in $A$ formed by the vectors $a _ {1} \dots a _ {s}$ issuing from one point, denoted by $[ a _ {1} \dots a _ {s} ]$. The conditions $a \in A$ and $\alpha _ {s} \wedge a = 0$ are equivalent.
For references see Exterior algebra.
Instead of exterior product the phrase "outer product" is sometimes used. The condition $a \wedge b = (- 1) ^ {pq} b \wedge a$ for $a$ of degree $p$ and $b$ of degree $q$ is sometimes called graded commutativity. | 2021-04-10 18:50:31 | {"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": 1, "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": 0.8919999599456787, "perplexity": 212.84639360806426}, "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-2021-17/segments/1618038057476.6/warc/CC-MAIN-20210410181215-20210410211215-00512.warc.gz"} |
http://volcanum.geosciences.univ-rennes1.fr/spip.php?article40 | # Numerical Volcano - Imaging of Complex Geological Media
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## Bayesian inversion of 1-D local models of electrical resistivity
by GIBERT Dominique - 2 April 2006
This article presents 1-D local geo-electrical models of La Soufrière lava dome. These models were inverted with a Bayesian inversion formalism explained below.
The data used in this study are available here. If you find these data useful and use them in any publication, please cite the following paper and contact the corresponding author :
Electrical Tomography of La Soufrière of Guadeloupe Volcano: Field Experiments, 1D Inversion and Qualitative Interpretation by Florence Nicollin, Dominique Gibert, François Beauducel, Georges Boudon, and Jean-Christophe Komorowski, Earth and Planetary Science Letters, Vol. 244, 709-724, 2006.
Parameterization of the Inverse Problem Several parts of the pseudo-sections (Figure 1) acquired on La Soufrière display a 1D-like structure where the resistivity varies only in the z direction. In such cases, we invert the data under the 1D approximation to obtain vertical resistivity soundings in order to locally check for the quality of the pseudo-sections. In particular, it is possible to verify that the thicknesses of resistive layers and the values of apparent resistivity in the pseudo-sections are reasonable with respect to the inverted resistivity soundings.
Figure 1: General views of the geo-electrical pseudo-sections of La Soufrière lava dome. Regions appearing in blue correspond to low (i.e. \approx 10 \; \Omega .\textrmm) electrical conductivity and yellow is for high (i.e. \geq 5000 \; \Omega .\textrmm) resistivity. Views are for and elevation of 30 degrees. A) toward the North, B) toward the West, C) toward the South, D) toward the East.
Numerous studies have been devoted to 1D geo-electrical inversions where the unknown parameters to be determined are the thicknesses, \Delta z_n , and conductivities, \sigma _n , of the n=1,\cdots ,N layers forming the resistivity structure overlying a half-space with conductivity \sigma _0 (Gyulai 1999, Muiuane 1999, Dahlin 2001). However, in the present study we want to not only perform a 1D inversion to obtain the \Delta z’\textrms and the \sigma ’\textrms , but we also want to test the validity of the 1D approximation. Consequently, we formulate the inverse problem to be solved as follows: find the best 1D resistivity model \mathbfm with the largest data subset \mathbfd_1D compatible with the 1D approximation. The subset \mathbfd_1D counts K elements extracted from a larger data set \mathbfd corresponding to a region where the pseudo-section has a 1D appearance. In what follows, we present a non-linear inversion method where \mathbfm and \mathbfd_1D are simultaneously inverted.
The model vector is defined as
\mathbfm=\left\ \ln \sigma _0,\ln \sigma _1,\cdots ,\ln \sigma _N,\Delta z_1,\cdots ,\Delta z_N\right\ ,
with the logarithms of the conductivities taken as the unknown parameters. This parametrization is often recommended because of the large range spanned by the conductivity values. Also, the logarithmic parameterization gives an equivalent role to both the conductivity and the resistivity which are reciprocal physical quantities (Tarantola 2005). The logarithm of the conductivity is a fundamental quantity which appears in the integral equation formulation of the Poisson equation giving the perturbations of the electrical potential caused by conductivity heterogeneities (Pessel and Gibert, 2003).
The data vector, \mathbfd , consists in the logarithms of the resistance measurements R_l=\Delta V_l/I_l where \Delta V_l is the electrical potential measured between electrodes M and N , and I_l is the electrical current injected between electrodes A and B . Because of the large range spanned by the electrical potential measurements, some renormalization is necessary and, in the present study, we apply a prewithening by dividing each data by the logarithm of the squared geometrical factor deduced from the electrode arrangement. By this way, all data have almost the same magnitude and weight equivalently in the misfit estimate.
In practice, the model selection relies on the maximisation of the following likelihood function,
L\left( \mathbfm,\mathbfd_1D\right) \propto \exp\left( -\frac\left| \mathbfd_1D-\mathbfd_\mathbfm\right| _1S K^\alpha
\right),
where \left| \cdot \right| _1 stands for the L_1 norm, \mathbfd_\mathbfm is the synthetic data vector produced by the 1D conductivity model \mathbfm , K is the dimension of the data subset actually used to evaluate the fit with the model, and \alpha =3 is an exponent experimentally chosen to apply a penalty to models fitted with a relatively small number of data. By this way, the likelihood defined above allows to compare models fitted with a variable number K of data taken in the initial data set. The standard deviation, S , is initially chosen equal to unity.
Inversion by Simulated Annealing The search for the best model, \mathbfm_\textrmbest , and its corresponding data subset, \mathbfd_1D,\textrmbest , is performed with the simulated annealing algorithm which easily allows to tackle with sophisticated likelihood functions as defined above. In the following, we shall only give the technical details specifically adjusted for the present study, and the reader is referred to (Pessel and Gibert, 2003; Metropolis at al., 1953; Kirkpatrick et al., 1983; Bhanot 1988; Sambridge 2002) for general considerations concerning the Metropolis and the simulated annealing algorithms in the framework of non-linear inversion.
Let us recall that simulated annealing consists in doing Metropolis loops while applying a topological transformation to L by varying a control parameter traditionally called the temperature T>0 (Pessel and Gibert, 2003; Gibert and Virieux, 1991; Mosegaard 1995). The transformed likelihood reads,
L_T\left( \mathbfm,\mathbfd_1D\right) =L^1/T\left( \mathbfm,\mathbfd_1D\right) .
The simulated annealing loops begin with T\rightarrow \infty for which L_\infty equals a constant and finish at T=1 , i.e. at the likelihood defined above. For each temperature T , a Metropolis loop is performed to generate a sequence of models distributed according to L_T . In practice, this consists in generating a sequence of models where the next model, \left( \mathbfm,\mathbfd_1D\right) _j+1 , to be added in the sequence is obtained from the preceding one, \left( \mathbfm,\mathbfd_1D\right) _j , according to the random choice, \textrmprob\left[ \left( \mathbfm,\mathbfd_1D\right) _j+1=\left( \mathbfm,\mathbfd_1D\right) _\textrmtry\right] =\min \left[ 1,\fracL_T\left( \mathbfm,\mathbfd_1D\right) _\textrmtry
L_T\left( \mathbfm,\mathbfd_1D\right) _j\right] ,
where \left( \mathbfm,\mathbfd_1D\right) _\textrmtry is a candidate model which may eventually be included in the sequence of models. The equation above indicates that a more-likely model is always accepted and that a less-likely model is sometimes accepted. When the random assignment given by the above equation fails, i.e. when the less-likely candidate model has been rejected, the replicating transition \left( \mathbfm,\mathbfd_1D\right) _j+1=\left( \mathbfm,\mathbfd_1D\right) _j is used instead. As the Metropolis loop proceeds a sequence of models is generated such that more models fall in the most likely regions of the space model. As the temperature is lowered, these regions have their likelihood dramatically (i.e. exponentially) augmented, and provided the temperature decrease is sufficiently slow, the chain of model is gently guided toward the regions of the space model where the likelihood is maximum (Metropolis et al., 1953; Bhanot 1988).
The candidate model, \left( \mathbfm,\mathbfd_1D\right) _\textrmtry , is obtained by perturbing the current model \left( \mathbfm,\mathbfd_1D\right) _j in order to give some memory to the Metropolis chain. This point is of a particular importance in order not to make the algorithm a simple Monte Carlo search. The data subset \mathbfd_1D,\textrmtry of the candidate model is obtained by the following way: first we assign \mathbfd_1D,\textrmtry=\mathbfd_1D,j and next, in 50\% of the time, a data element is randomly chosen in the whole data set \mathbfd and is deleted from \mathbfd_1D,\textrmtry if present and is incorporated if absent. Hence, the data subset of the candidate model differs of at most one element with respect to the data subset of the last model in the Metropolis chain. The remaining parameters of the candidate model, namely the log-conductivities and the layer thicknesses, are obtained by randomly choosing a parameter, either log-conductivity or thickness, and by randomly perturbing it in a limited range.
Synthetic example We now discuss a synthetic example showing how the algorithm performs in presence of an incoherent data set whose elements were generated with two different conductivity structures. For this purpose, we invert the data set obtained by merging two synthetic data subsets of apparent resistivities computed for two different conductivity distributions (Figure 2a). In a first stage the inversion is done with no estimate of the standard deviation S which is then set to unity. The annealing schedule is performed with a temperature range 1>T>10^-4 in order to converge toward the most likely model. Figure 2b shows the likelihood of the models successively incorporated in the Metropolis chain. The resulting curve is typical of the convergence property of simulated annealing: the chain starts with randomly chosen models with a low likelihood and with strongly varying parameter values as shown in Figures 1c and 1d. As the temperature further decreases, the retained models progressively get a higher likelihood until the curve reaches a sharp step. At this stage of the cooling schedule, the model sequence suddenly gets confined in a region of high-likelihood models and the parameter values cease to strongly fluctuate (Figures 2c and 2d).
Figure 2: 1D inversion process of synthetic data: (a) the inverted pseudo-section is obtained by merging two data subsets corresponding to two models of conductivity shown on the left and on the right parts of the Figure, (b) likelihood curve of the models forming the Metropolis chain versus annealing temperature, (c) electrical conductivity versus annealing temperature, (d) top layer thickness versus annealing temperature, (e) probability distribution of the conductivities of the final model, (f) probability distribution of the top layer thickness of the final model.
The success of the global convergence of simulated annealing toward the global maximum of L is controlled by the speed of cooling which must be sufficiently low to allow an exhaustive search in the space model. Once the best model is obtained, the residuals are used to compute an estimate of S , and the last step of the simulated annealing algorithm consists in a heat-bath sequence (i.e. a Metropolis chain constructed at a constant temperature) performed at T=1 . The distribution of the models forming the heat-bath sequence may be used to construct the marginal probability curves (Gibert and Virieux, 1991; Gibert 1994; Mosegaard 1998) for each parameter as shown in Figures 1e and 1f. Depending on the random assignments made in the Metropolis algorithm, the model sequence converges toward either of the two models used to construct the synthetic data set.
Examples with real data Figure 3 shows the result of an inversion performed with a data subset extracted from the pseudo-section located on the western side of the volcano (see Figure 1). The marginal probability curves (Figures 3b and 3c) indicate a two-layer conductivity structure with a resistive layer, \rho _1\approx 3000\textrm \Omega .\textrmm and \Delta h_1\approx 35\textrm m , overlying a half-space with a resistivity \rho _0\approx 50\textrm \Omega .\textrmm . As can be observed in Figure 3d, several data have been excluded from the original data set on the right side of the pseudo-section. Indeed, these data correspond to anomalous high apparent resistivity values incompatible with the assumed 1D geometry.
Figure 3: 1D inversion results of the western part of the profile achieved around the base of the lava dome, along Chemin des Dames: (a) preselected data set, (b) probability distribution of the conductivities of the final model, (c) probability distribution of the top layer thickness of the final model, (d) data set consistent with the final model.
Another example is shown in Figure 4 for data corresponding to a fumarolic area located at the southern basis of the dome (Figure 1). In this instance, the simulated annealing inversion gives marginal probabilities such that a simple half-space model is obtained with a resistivity \rho _0\approx 20\textrm \Omega .\textrmm . Interestingly, the marginal probability curve for \Delta h_1 is flat, indicating that this parameter remains unresolved. The inverted data subset shown in Figure 4d shows that lateral data corresponding to higher apparent resistivities have been automatically eliminated.
Figure 4: 1D inversion results of a part of the profile G-G’ selected in the fumarolic area: (a) preselected data set, (b) probability distribution of the conductivities of the final model, (c) probability distribution of the top layer thickness of the final model, (d) data set consistent with the final model.
Results A total of 17 data sets corresponding to almost 1D structures recognised in the pseudo-sections of Figure 1 have been inverted according to the methodology described above. Several tests have been performed for models with more or less layers overlying the lower half-space to eventually represent a transition layer with variable resistivity. We found that models with two or more layers do not significantly improve the fit to the data. Moreover, increasing the number of layers often leads to models with low resistivity contrast and unresolved thicknesses. Consequently, according to the principle of parsimony, we decided to retain the simplest models, i.e. those with either one or no overlying layer, able to fit the data.
The results of the inversions displayed in Figure 5. All but 3 data sets give a two-layer structure with a single layer overlying a more conductive half-space. The remaining two data sets correspond to an homogeneous half-space with no variations of conductivity in the vertical direction. They correspond on the lava dome to the Jardin Lherminier area, a shallow depression with a permanent small pond formed within relatively thick phreatic ash deposits, and to active fumarolic areas characterised by a high degree of hydrothermal alteration and upflow of hydrothermal fluids located on the summit of the lava dome (Cratère Sud fumaroles) and at the South base of the lava dome near the road (Route de la Citerne fumaroles).
Figure 5: Synthesis of the 1D inversion results obtained for 17 data subsets belonging to regions where the pseudo-sections appear laterally invariant. The great rectangle represents the resistivity of the bottom half space, the line represents the resistivity of the top layer and its thickness is proportional to the top layer thickness. The values of altitude correspond to the depth of the top of the conductive half space.
Conclusions Figure 5 reveals that the 17 geo-electrical soundings constitute a simple global conductivity structure with a low-resistivity (\rho _0\approx 50\textrm \Omega .\textrmm) basal layer on the eastern and southern edges of the lava dome, and a medium-resistivity (\rho _0\approx 250\textrm \Omega .\textrmm) basal layer on the northern edge and on the lava dome itself. In most cases, the basal layer is overlained by a resistive (\rho _1\approx 2000\textrm \Omega .\textrmm) layer with a thickness \Delta h_1 comprised between 10 and 30 meters. Noticeable exceptions are for the August 30th Fracture and the Carbet areas where the overlying layer has a significantly higher resistivity (\rho _1\approx 4000\textrm \Omega .\textrmm) and corresponds to rock avalanche blocks. Another exception is for the Col de l’échelle area where the overlying layer has a lower resistivity (\rho _1\approx 400\textrm \Omega .\textrmm) and corresponds to fumarolic zones that ceased to be active in 1979 and 1984 (Boudon 1989,Komorowski 2005,Komorowski 2001).
Acknowledgments This work is dedicated to the memory of our Friend and Colleague Alberto Tarchini who participated to most of our field experiments and always shared his generous enthusiasm. This study benefited from invaluable help from the whole staff of the Observatoire Volcanologique et Sismologique de Guadeloupe: Christian Anténor-Habazac, Sara Bazin, Véronique Daniel, Bertrand Figaro, Gilbert Hammouya, Christian Lambert, Didier Mallarino and Laurent Mercier. D.G. and F.N. would like to thank Gérard Werther for numerous discussions concerning historical records about La Soufrière. Financial support was provided through the CNRS/INSU Antilles Program and from IPGP .
Cited references D. Gibert, M. Pessel, Identification of sources of potential fields with the continuous wavelet transform: Application to self-potential profiles, Geophys. Res. Lett. 28 (2001) 1863-1866. K. Aizawa, R. Yoshimura, N. Oshiman, K. Yamazaki, T. Uto, Y. Ogawa, S.B. Tank, W. Kanda, S. Sakanaka, Y. Furukawa, T. Hashimoto, M. Uyeshima, T. Ogawa, I. Shiozaki, A.W. Hurst, Hydrothermal system beneath Mt. Fuji volcano inferred from magnetotellurics and electric self-potential, Earth and Planet. Sci. Lett. in the press (2005). A. Gyulai, T. Ormos, A new procedure for the interpretation of VES data: 1.5D simultaneous inversion method, J. Applied Geophys. 41 (1999) 1-17. E.A. Muiuane, L.B. Pedersen, Automatic 1D interpretation of DC resistivity sounding data, J. Applied Geophys. 42 (1999) 35-45. T. Dahlin, The development of DC resistivity imaging techniques, Computers Geosciences 27 (2001) 1019-1029. A. Tarantola, Elements for Physics: Quantities, Qualities, and Intrisic Theories, Springer, Berlin, (2006) 263 pp. M. Pessel, D. Gibert, Multiscale electrical impedance tomography, J. Geophys. Res. 108B1 (2003) 2054, doi:10.1029/2001JB000233. N. Metropolis, A. Rosenbluth, N. Rosenbluth, A. Teller, E. Teller, J. Chem. Phys. 21 (1953) 1087-1092. S. Kirkpatrick, C.D. Gelatt, M.P. Vecchi, Optimization by simulated annealing, Science 220 (1983) 671-680. G. Bhanot, The Metropolis algorithm, Rep. Prog. Phys. 51 (1988) 429-457. M. Sambridge, K. Mosegaard, Monte Carlo methods in geophysical inverse problems, Rev. Geophys. 40(3) (2002) 1009, doi:10.1029/2000RG00089. D. Gibert, J. Virieux, Electromagnetic imaging and simulated annealing, J. Geophys. Res. 96 (1991) 8057-8067. K. Mosegaard, A. Tarantola, Monte-Carlo sampling of solutions of inverse problems, J. Geophys. Res. 100 (1995) 12431-12447. D. Gibert, B. Tournerie, J. Virieux, High-resolution electromagnetic imaging of the conductive Earth interior, Inv. Problems 10 (1994) 341-351. K. Mosegaard, Resolution analysis of general inverse problems through inverse Monte Carlo sampling, Inv. Problems 14 (1998) 405-426.
In the same section : | 2017-11-25 09:28:42 | {"extraction_info": {"found_math": false, "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": 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": 0.9072874188423157, "perplexity": 2246.68022612311}, "config": {"markdown_headings": true, "markdown_code": false, "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-2017-47/segments/1510934809746.91/warc/CC-MAIN-20171125090503-20171125110503-00757.warc.gz"} |
https://socratic.org/questions/a-gas-has-a-pressure-of-699-0-mm-hg-at-40-0-c-what-is-the-temperature-at-standar | # A gas has a pressure of 699.0 mm Hg at 40.0°C. What is the temperature at standard pressure?
Jul 6, 2017
The temperature will be $\text{336 K}$.
#### Explanation:
This is a pressure-temperature gas problem. This means that it involves Gay-Lussac's law, which states that the pressure of a given amount of a gas, held at constant volume, varies directly with its temperature. This means that if the pressure increases, so does the temperature, and vice-versa. The equation used to solve this problem is:
${P}_{1} / {T}_{1} = {P}_{2} / {T}_{2}$
Before we go further, we need to determine what standard pressure is, and we need to convert the Celsius temperature to Kelvin temperature by adding $273.15$ to the Celsius temperature.
Standard pressure is $\text{100 kPa}$. $\leftarrow$ 100 kiloPascals
We need to convert $\text{mmHg}$ to $\text{kPa}$.
$1 \text{kPa"="7.5006 mmHg}$
699.0color(red)cancel(color(black)("mmHg"))xx(1"kPa")/(7.5006color(red)cancel(color(black)("mmHg")))="93.2 kPa"
Known
${P}_{1} = \text{93.2 kPa}$
${T}_{1} = \text{40"^@"C" + 273.15="313 K}$
${P}_{2} = \text{100 kPa}$
Unknown
T_2=?
Solution
Rearrange the equation above to isolate ${T}_{2}$. Insert the given data into the new equation and solve.
${T}_{2} = \frac{{P}_{2} {T}_{1}}{P} _ 1$
T_2=(100color(red)cancel(color(black)("kPa"))xx313"K")/(93.2color(red)cancel(color(black)("kPa")))="336 K"
If you need to convert the temperature in Kelvins back to the Celsius temperature, subtract $273.15$ from $\text{336 K}$. | 2019-10-21 19:52:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 18, "mathjax_inline_tex": 1, "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": 0.8404587507247925, "perplexity": 505.8616126612588}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2019-43/segments/1570987787444.85/warc/CC-MAIN-20191021194506-20191021222006-00063.warc.gz"} |
https://mathematica.stackexchange.com/questions/206486/solve-implicit-equations-input-and-plot-the-solution-in-a-different-equation | # Solve implicit equations, input and plot the solution in a different equation
I have two implicitly defined variables, $$\sigma_D$$ and $$\sigma_M$$ one of which is a function of the parameter $$v$$. In particular, $$\sigma_D = F(\sigma_D,v)$$ where $$F(\sigma_D,v)=0$$ if $$c_D < -0.1$$, $$F(\sigma_D,v)=\frac{c_D+0.1}{2.1}$$ if $$-0.1 \leq c_D \leq 2$$ and $$1$$ for $$c_D > 2$$, and
$$c_{D} =0.5((\frac{\sigma_D^2 + (1-\sigma_D)(2-\sigma_D)(.6)^2 }{\sigma_D^2 + (2-\sigma_D)^2 (.6)^2} - \frac{1-\sigma_D}{2-\sigma_D})(\sigma_D - (2-\sigma_D)(.6)^2)+1) -0.5v$$
$$\sigma_M = \frac{0.4}{2.1} \big[\frac{\sigma_M + .6(1-\sigma_M) }{\sigma_M + .6(1-\sigma_M) + 0.6} - \frac{1-\sigma_M}{2-\sigma_M}\big] + \frac{.1}{2.1}.$$
I need to get the solutions for $$v \in [0,5]$$, then replace those values in the following two equations: $$V_{1} = \frac{1}{2}(1+0.6\mu) + \frac{1}{2}[\sigma_M \mu + (1-\sigma_M)(1+ 0.6\mu )]$$
$$V_{2} = \frac{1}{4}(1+0.6\mu) + \frac{1}{4}[1 + 0.6\mu (\sigma_D + (1-\sigma_D))] 2 + \frac{1}{4}[(1-\sigma_D)^2 (1+ 0.6\mu)) + 2 \sigma_D (1-\sigma_D)(1+\mu) + \sigma^{2}_D \mu]$$
Finally, I need to plot both $$V_{1}$$ and $$V_{2}$$ on the same graph in the $$\mu-v$$ space with $$\mu \in [0,5]$$ and $$v \in [0,5]$$, or at least to produce a graph that shows when $$V_2 > V_1$$.
c=2;
e=0.1;
f=0.6;
gD=(sD^2 + (1 - sD) f^2 (2 - sD))/(sD^2 + f^2 (2 - sD)^2);
XD=(1 - sD)/(2 - sD);
cD = -0.5 v + 0.5 ((gD - XD) (sD - (2 - sD) f^2) + (2 - sD) (1 - XD));
FD = Piecewise[{{0, cD <= -e}, {(cD + e)/(c + e), cD > -e && cD < c}, {1,
cD >= c}}];
gM= (sM + (1 - sM) f)/(sM + (1 - sM) f + f);
XM=(1 - sD)/(2 - sD);
RHSM=(1 - f) (gM - XM);
V1=.5(1+.6m) + .5(sM m + (1-sM)(1+.6m));
V2-.25(1+.6m) + .5[1 + m(sD + .6(1-sD))] + .25[(1-sD)^2(1+.6m) + 2 sD (1-sD)(1+m) + sD^2 m ];
Clear[e,c,f,gD,XD,cD,FD,gM,XM,RHSM];
I know how to get solutions for a particular value of v, but not how to input automatically the solution into $$V_1$$ and $$V_2$$ and then compare them in the $$\mu - v$$ space. Note also that $$\sigma_M \in [0,1]$$ and $$\sigma_D \in [0,1]$$.
Any help will be greatly appreciated! Thanks a lot!!
• There is no definition of $\mu$. Add this definition and write both functions $V_1, V_2$. $\sigma_M$ is it just a number (root of the equation)? – Alex Trounev Sep 20 '19 at 3:37
• $\mu$ is merely a scalar and lies between 0 and 5. I have edited the above to reflect the change -- it is not a function. $v$ is also a scalar and lies between 0 and 5. Finally, yes, $\sigma_M$ is the root of the equation and does not depend on $v$. – A. Pant Sep 20 '19 at 8:57
• This is strange. I got the roots of the equation from Mathematica, and they turn out to be $-2.90447$, $0.051765$ and $1.90033$. Although, I apologize for not being clear earlier that both $\sigma_D$ and $\sigma_M$ should be between $0$ and $1$. Format[s] := [Sigma]; f = 0.6; c = 2; e = 0.1; gR = (s + (1 - s) f)/(s + (1 - s) f + f); X = (1 - s)/(2 - s); TM = (1 - f) (gR - X); Solve[(TM + e)/(c + e) == s, s] – A. Pant Sep 20 '19 at 16:46
• Yes, you're right, the roots are there {{s -> -2.90447}, {s -> 0.051765}, {s -> 1.90033}}. Need to take 0.051765? Can you add this to your message? And write down the expression for V1,V2. – Alex Trounev Sep 20 '19 at 17:17
• Done! Let me know if there any further clarifications. Thanks a ton. – A. Pant Sep 20 '19 at 17:35
You can use this code here (need to fix [] to () in V2 as Bob suggested)
c = 2;
e = 0.1;
f = 0.6;
gD = (sD^2 + (1 - sD) f^2 (2 - sD))/(sD^2 + f^2 (2 - sD)^2);
XD = (1 - sD)/(2 - sD);
cD = -0.5 v + 0.5 ((gD - XD) (sD - (2 - sD) f^2) + (2 - sD) (1 - XD));
FD = Piecewise[{{0, cD <= -e}, {(cD + e)/(c + e),
cD > -e && cD < c}, {1, cD >= c}}];
gM = (sM + (1 - sM) f)/(sM + (1 - sM) f + f);
XM = (1 - sD)/(2 - sD);
RHSM = (1 - f) (gM - XM);
V1 = .5 (1 + .6 m) + .5 (sM m + (1 - sM) (1 + .6 m));
V2 = .25 (1 + .6 m) + .5 (1 +
m (sD + .6 (1 - sD))) + .25 ((1 - sD)^2 (1 + .6 m) +
2 sD (1 - sD) (1 + m) + sD^2 m);
sig = Table[{x, sD /. FindRoot[sD == FD /. v -> x, {sD, 1}]}, {x,
0, .5, .005}]; | 2020-04-06 22:55:19 | {"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": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 26, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5816642642021179, "perplexity": 1830.6420894799728}, "config": {"markdown_headings": true, "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-2020-16/segments/1585371660550.75/warc/CC-MAIN-20200406200320-20200406230820-00279.warc.gz"} |
https://www.physicsforums.com/threads/what-is-a-value-of-psi-for-a-particle-in-1d-box-stationary-state.63747/ | What is a value of psi* for a particle in 1D box stationary state?
1. Feb 13, 2005
ohhhnooo
I know that psi is equal to (2/a)^1/2 sin(n pi x/a)? what is psi*?
2. Feb 13, 2005
dextercioby
Complex conjugate of "psi"...In your case,it's identical to "psi"...
Daniel.
3. Feb 13, 2005
Gokul43201
Staff Emeritus
How is $\psi^*$ defined ? Or simpler still, what does the * do ?
4. Feb 14, 2005
Staff: Mentor
Actually, including the time dependence, it's
$$\Psi(x,t) = \sqrt {\frac {2} {a}} \sin \left (n \pi \frac {x} {a} \right ) \exp \left (-i \frac {E_n} {\hbar} t \right )$$
so the complex conjugate is
$$\Psi^*(x,t) = \sqrt {\frac {2} {a}} \sin \left (n \pi \frac {x} {a} \right ) \exp \left (i \frac {E_n} {\hbar} t \right )$$
The complex conjugate means simply, change i to -i (and -i to i) everywhere.
Last edited: Feb 14, 2005
5. Feb 14, 2005
dextercioby
In that case,he (probably) would have spelled "Psi"...He didn't intend to include the time dependence...
Daniel. | 2017-03-28 10:23:35 | {"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": 1, "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": 0.9339607357978821, "perplexity": 12353.53792305929}, "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-2017-13/segments/1490218189686.56/warc/CC-MAIN-20170322212949-00599-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Book%3A_Statistical_Thinking_for_the_21st_Century_(Poldrack)/20%3A_Bayesian_Statistics | # 20: Bayesian Statistics
## Learning Objectives
• Describe the main differences between Bayesian analysis and null hypothesis testing
• Describe and perform the steps in a Bayesian analysis
• Describe the effects of different priors, and the considerations that go into choosing a prior
• Describe the difference in interpretation between a confidence interval and a Bayesian credible interval
In this chapter we will take up the approach to statistical modeling and inference that stands in contrast to the null hypothesis testing framework that you encountered in Chapter 16. This is known as “Bayesian statistics” after the Reverend Thomas Bayes, whose theorem you have already encountered in Chapter 10. In this chapter you will learn how Bayes’ theorem provides a way of understanding data that solves many of the conceptual problems that we discussed regarding null hypothesis testing.
This page titled 20: Bayesian Statistics is shared under a CC BY-NC 2.0 license and was authored, remixed, and/or curated by Russell A. Poldrack via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. | 2022-11-29 10:12:37 | {"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": 1, "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": 0.8770071268081665, "perplexity": 706.4606379204652}, "config": {"markdown_headings": true, "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-2022-49/segments/1669446710691.77/warc/CC-MAIN-20221129100233-20221129130233-00191.warc.gz"} |
https://coobas.gitlab.io/python-fjfi/en/posts/zakladni-syntaxe.html | Basic syntax
The Python syntax is uncomplicated whenever possible. Python is arguably one of the most syntactically natural programming languages.
Comments are used throughout this course, let's show a few examples.
In [1]:
# - This is a comment
# - It always begins by "#" (hash)
# - This is more or less quivalent to // in C++
a = 1 + 2 # A comment on a line with a code
"""Multiple line comments do not exist in Python,
unlike /* */ in C++ or Java. One can use multiline
strings, which are simply not assigned to any variable
and thus thrown away immediately. Docstring use this
approach as well, see below."""
Out[1]:
'Multiple line comments do not exist in Python,\nunlike /* */ in C++ or Java. One can use multiline\nstrings, which are simply not assigned to any variable\nand thus thrown away immediately. Docstring use this\napproach as well, see below.'
Plain commands¶
Simply try the following commands and their modifications.
In [2]:
a = 1 + 1.1 / 2 # the result of a simple calculation is assigned to a variable
print(type(a)) # print output (stdout)
a # IPython displays the last command result
<class 'float'>
Out[2]:
1.55
In [3]:
a = "Hello" + " world!" # a simple string manipulation
print(a)
print("type(a) = %s, len(a) = %i" % (type(a), len(a))) # a formated output
Hello world!
type(a) = <class 'str'>, len(a) = 12
In [4]:
import math # importing a module
a = math.cos(math.pi / 4) # using the cos function and the pi constant
print('a = %.3g' % a)
a = 0.707
In [5]:
from math import sin, pi # importing selected symbols from a module
a = sin(pi / 4)
print('a = %.3g' % a)
a = 0.707
In [6]:
a = cos(pi / 2) # this should not work --> an exception is thrown
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-6-de2a57539a2d> in <module>()
----> 1 a = cos(pi / 2) # this should not work --> an exception is thrown
NameError: name 'cos' is not defined
A few important builtin functions¶
There is not too many built-in functions in Python. Let us mention some of them (which we might have already encountered).
• dir -- list symbols (functions, variables, methods) in a given context
• eval -- evaluates a string as a code and returs the result
• help -- helps us (displays the 'docstring')
• len -- the lengths of something (a string, a list, etc.)
• open -- opens a file
• print -- prints out a string
• raw_input -- reads input from keyboard
• str, repr -- returns a text reprezentation of an object
• type -- returns the type of the argument
We shall get introduced to these and other built-in functions soon.
In [7]:
print(str(dir()) + "\n") # display the current context (variables etc)
print(repr(dir("a"))) # display the symbols of a string object (attributes, methods)
['In', 'Out', '_', '_1', '_2', '__', '___', '__builtin__', '__builtins__', '__doc__', '__loader__', '__name__', '__package__', '__spec__', '_dh', '_i', '_i1', '_i2', '_i3', '_i4', '_i5', '_i6', '_i7', '_ih', '_ii', '_iii', '_oh', '_sh', 'a', 'exit', 'get_ipython', 'math', 'pi', 'quit', 'sin']
['__add__', '__class__', '__contains__', '__delattr__', '__dir__', '__doc__', '__eq__', '__format__', '__ge__', '__getattribute__', '__getitem__', '__getnewargs__', '__gt__', '__hash__', '__init__', '__iter__', '__le__', '__len__', '__lt__', '__mod__', '__mul__', '__ne__', '__new__', '__reduce__', '__reduce_ex__', '__repr__', '__rmod__', '__rmul__', '__setattr__', '__sizeof__', '__str__', '__subclasshook__', 'capitalize', 'casefold', 'center', 'count', 'encode', 'endswith', 'expandtabs', 'find', 'format', 'format_map', 'index', 'isalnum', 'isalpha', 'isdecimal', 'isdigit', 'isidentifier', 'islower', 'isnumeric', 'isprintable', 'isspace', 'istitle', 'isupper', 'join', 'ljust', 'lower', 'lstrip', 'maketrans', 'partition', 'replace', 'rfind', 'rindex', 'rjust', 'rpartition', 'rsplit', 'rstrip', 'split', 'splitlines', 'startswith', 'strip', 'swapcase', 'title', 'translate', 'upper', 'zfill']
We can demonstrate the raw_input and the eval power on a simple "calculator".
In [9]:
a = 2
e = input("a = {}\nEnter an expression that contains a: ".format(a))
print("{} = {}".format(e, eval(e)))
a = 2
Enter an expression that contains a: a * 2 + a**2
a * 2 + a**2 = 8
Python operators¶
+ - * ** / // %
<< >> & | ^ ~
< > <= >= == != <>
or and not
is is not in not in
Keywords¶
and del from not while
as elif global or with
assert else if pass yield
break except import print
class exec in raise
continue finally is return
def for lambda try
Conditions and other blocks in Python¶
Code blocks (or compound statements) consist of one or more ‘clauses.’ A clause consists of a header and a ‘suite.’ The clause headers of a particular compound statement are all at the same indentation level. Each clause header begins with a uniquely identifying keyword and ends with a colon. A suite is a group of statements controlled by a clause. A suite can be one or more semicolon-separated simple statements on the same line as the header, following the header’s colon, or it can be one or more indented statements on subsequent lines. Only the latter form of suite can contain nested compound statements; the following is illegal, mostly because it wouldn’t be clear to which if clause a following else clause would belong:
The indentation is arbitrary but it must be consistent across a single file. Strongly recommended are four spaces, as noticed in PEP 8 -- Style Guide for Python Code. This document contains important conventions, such as the name conventions for variables, functions, classes etc.
In [10]:
# this is an example of nested if statements
if 2 > 1:
print("It's true that 2 > 1")
if 1 > 2:
print("It's true that 2 > 1 and 1 > 2")
else:
print("It's true that 2 > 1 and not true that 1 > 2")
else:
print("It's not true that 2 > 1")
It's true that 2 > 1
It's true that 2 > 1 and not true that 1 > 2
For conditional statements we have if - elif - else. There is nothing like switch / case (as it can be easily substituted by multiple elif statements).
In [11]:
a = 5
if a > 10:
print("Cannot count this with fingers")
elif a > 5:
print("We need both hands' fingers to count this")
elif a >= 0:
print("This I can count with fingers on single hand")
else:
print("Cannot do negative numbers")
This I can count with fingers on single hand
while blocks use the same indentation rules:
In [12]:
a = 0
while a < 5:
print(a)
a += 1
0
1
2
3
4
while blocks (as well as for but we'll explain for later) can have an else part, which is executed when the condition is false.
In [13]:
a = 10
while a < 5:
print("a = %i" % a)
a += 1
else:
a = 1
print("The end, a = %i" % a)
The end, a = 1
break interrupts a cycle, continue makes a new iteration, i.e. skips the commands below.
In [14]:
a = 0
while a < 5:
a += 1
# even number are not printed
if a % 2 == 0:
continue
print("a = %i" % a)
a = 1
a = 3
a = 5
Let's find the largest three-digit number divisible by 19.
In [15]:
a = 999
while a > 0:
if a % 19 == 0:
print("The answer is: %i" % a)
break
a -= 1
else:
# break nespustí else část
The answer is: 988
Multiline expressions¶
We can split long lines using \.
In [16]:
a_long_variable_name = "this is just an abudantly long text, " + \
"that does not have any meaning ..."
a_long_variable_name
Out[16]:
'this is just an abudantly long text, that does not have any meaning ...'
We can (and should) often ommit \ in expressions inside parenthesis.
In [17]:
a_long_variable_name = (10000000000000 + 2222222222222222 + 9999999999999999 + 3987493874 +
444444444444444 + 23987342978 + 9874 + 555555555555555555 +
987349987 - 9999999999999999999)
a_long_variable_name
Out[17]:
-9431767748815581066
One should not create lines longer than 80 characters (see PEP8), although this convention is often not very convenient.
Function definition¶
Functions are defined using the def keyword.
In [18]:
# a simple function with any return value
def hello():
print("Hello Python!")
# now call our function
hello()
Hello Python!
This function is not very useful as usually we need inputs and/or outputs.
In [19]:
# subj is an argument of the hello function
def hello(subj):
phrase = "Hello %s!" % subj
# return stops the function execution and returns a value
return phrase
print(hello("Prague"))
Hello Prague!
We can distinguish positional and keyword (named) arguments. Arguments can have implicit values, which makes such arguments optional.
In [20]:
# greet is an argument with an implicit value
def hello(subj, greet="Hello"):
phrase = "%s %s!" % (greet, subj)
return phrase
# we don't specify the optional freet argument
print(hello("Prague"))
# we use greet as a keyword argument
# positional arguments must come first
print(hello("Praho", greet="Nazdar"))
# we can use both subj and greet as keyword arguments
print(hello(greet="Nazdar", subj="Praho"))
Hello Prague!
Nazdar Praho!
Nazdar Praho!
Functions can have variable arguments but we'll explain this later when we learn about containers.
Side effects of functions¶
Funtions can have side efects, i.e. they can change the input arguments, if these arguments are so called mutable. What mutable and immutable means will be explained later. Nonetheless, it's strongly recommended to avoid creating functions with side effects unless it has a good reason and the name of the function indicates this behaviour. Let us show an example of a function with side effects.
In [21]:
def my_function(l):
# l should be a list
l.append("I'm here")
print("l inside my_function")
print('l = {}'.format(l))
print("the original list")
x = ["in my list"]
print(x)
my_function(x)
print("x after calling my_function")
print(x)
the original list
['in my list']
l inside my_function
l = ['in my list', "I'm here"]
x after calling my_function
['in my list', "I'm here"]
An unpleasant surprise appears when an argument's implicit value is mutable.
In [22]:
def foo(l=[]):
# this is changing l
l.append("appended")
print(l)
# first call with an explicit input parameter
foo([])
# now using the implicit value - the result should be identical
foo()
# now repeat the previous calls again
# we expect the results are indetical but they aren't
foo([])
foo()
['appended']
['appended']
['appended']
['appended', 'appended']
We can avoid side effect by copying (either using the copy module or by copy methods). It is usually better to assing results to new variables, e.g.
In [23]:
def foo(l=[]):
p = l + ["appended"]
print(p)
# first call with an explicit input parameter
foo([])
# now using the implicit value - the result should be identical
foo()
# now repeat the previous calls again
# we expect the results are indetical
foo([])
foo()
['appended']
['appended']
['appended']
['appended']
Using modules¶
Complex codes are usually contained in modules. We can think about modules as simple containers with reusable functions, variables or classes. We will show how to create modules later; nevertheless using modules is basically unavoidable. The standard (built-in) Python library is in fact a collection of modules.
In [24]:
# import (i.e. read and use) the math module
import math
# cos is a function from the math module
print(math.cos(0))
1.0
Apart from importing whole modules we can import only certain symbols. To import the cos function:
In [25]:
from math import cos
# cos can be now called without math.
print(cos(0))
1.0
We can also import everything from a module by using from ... import *. We have to be careful if symbols from the module do not collide with other symbols as they would be overwritten.
In [26]:
from math import *
print(sin(pi/2))
1.0 | 2022-10-04 02:54:40 | {"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": 0.4096260368824005, "perplexity": 8866.855645543856}, "config": {"markdown_headings": false, "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-2022-40/segments/1664030337473.26/warc/CC-MAIN-20221004023206-20221004053206-00246.warc.gz"} |
https://eraestate.ro/xzvazs/cross-partial-derivative-economics-b255e9 | # cross partial derivative economics
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rev 2020.12.18.38240, The best answers are voted up and rise to the top, Economics 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. Similarly, it can be interpreted as the change in the return of capital, when labor increases marginally. SPF record -- why do we use +a alongside +mx? Why do I , J and K in mechanics represent X , Y and Z in maths? Our definition of boundary point for an interval can easily be extended. The definition of differentiability for a function of many variables captures the same idea: a function of many variables is differentiable at a point if there exists a good linear approximation of the function around the point. When you save your comment, the author of the tutorial will be notified. Why is the Pauli exclusion principle not considered a sixth force of nature? By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. 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. f is continuous. The green point in the following figure, for example, is a boundary point of the (two-dimensional) blue set because every disk centered at the point, however small, contains both points in the set and points outside the set. f is a function of two variables, for example, and we want to denote its partial derivative with respect to its first argument at the point (a, b), what letter do we use to denote the first argument? site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. In this article students will learn the basics of partial differentiation. Part I Partial Derivatives in Economics 3. Economics Stack Exchange is a question and answer site for those who study, teach, research and apply economics and econometrics. (See the example if you are curious.). yfxz (, ) A simple example is the function f of two variables defined by Sort by: Top Voted. 22 2 22 and 0 and ; xx x x x YY Yze e ze zx YY Y ze e zx x z Total Differential. Slope and marginal values have basically the same interpretation in multivariate problems as they do in uinivariate problems. Restricting the multi-index in that way allows simpler addressing schemes on a computer. Despite these limitations, the notation is often used by economists, and I sometimes follow suit. Suppose tha… Up Next. The last item is called a cross-partial derivative: you differentiate first with x and then with z (or the other way around: you get the same result – Young’s Theorem). Do damage to electrical wiring? Thank you for your comment. First, it is clumsy in using five symbols where three (Dif) suffice. This is the currently selected item. Use MathJax to format equations. Just like ordinary derivatives, partial derivatives follows some rule like product rule, quotient rule, chain rule etc. Occasionally the argument of a function may be more conveniently referred to by its name than its index. Partial derivative – measures the rate of change of the function y wrt (with respect to) one variable holding other variables constant. What is Partial Derivative. What's a way to safely test run untrusted JavaScript code? Further on (page 20), the author calculates the elasticity of substitution (σ) which is equal to (3.24): $$\sigma =\frac{b}{1-\frac{c}{X}\frac{f}{f'}}$$ Up to here everything is clear. Interpreting … However, the fact that all of the n partial derivatives of f exist at some point does not imply that f is differentiable, or even continuous, at that point. First of all thank you very much for your extremely clear answer! It only takes a minute to sign up. More specific economic interpretations will be discussed in the next section, but for now, we'll just concentrate on developing the techniques we'll be using. Taking the limit as h → 0 on both sides, limh→0 f(x + h) = f(x), so that The result assumes that all the second-order partial derivatives f" ij are continuous for all x in some set S, so that by Young's theorem we have f" ij (x) = f" ji (x) for all x ∈ S, and hence the Hessian is symmetric. Looking for name of (short) story of clone stranded on a planet, "Because of its negative impacts" or "impact". Trouble with the numerical evaluation of a series. We say that a point x is a boundary point of a set of n-vectors if there are points in the set that are arbitrarily close to x, and also points outside the set that are arbitrarily close to x. Second partial derivatives. This is defined as This tells us how the slope of the function with respect to x. Directional derivatives (introduction) Directional derivatives (going deeper) Next lesson. 17. This section considers a number of economic examples of partial derivatives. If I have called the arguments of f by the names w and p, for example (writing f(w, p)), I may write fp(w, p) for A point x is an interior point definition of differentiability for a function of a single variable, Enter the first six letters of the alphabet*. Why does the Indian PSLV rocket have tiny boosters? Making statements based on opinion; back them up with references or personal experience. Equations 1-3, and 5-6 are obtained in preparation for the 2nd derivatives of V with respect to L and K. Thanks for contributing an answer to Economics Stack Exchange! Your comment will not be visible to anyone else. Unlike the case of functions of a single variable, we can also take the second order cross-partial derivative. the value of the partial derivative of f with respect to its second argument at the point (w, p). To formulate a precise definition, first note that the definition of differentiability for a function of a single variable can be rewritten as follows: a function of a single variable defined on an open interval I is differentiable at the point a ∈ I if there is a number r such that. It doesn't even care about the fact that Y changes. This notation has two major disadvantages. Did I shock myself? Before we review the technical aspects of multivariate optimization, let's look at some examples of how we can use information about marginal values and rates of change. Partial Derivative Rules. Consider . Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top Economics Beta. Lecture 9: Partial derivatives If f(x,y) is a function of two variables, then ∂ ∂x f(x,y) is defined as the derivative of the function g(x) = f(x,y), where y is considered a constant. Second, its reference to the variable with respect to which the function is being differentiated is imprecise. We want to describe behavior where a variable is dependent on two or more variables. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Then a second-order partial differential equation for the isotropic distribution with additional terms f 0 (U + U l cs, z) of the shifted energy arguments U + U l cs is obtained. Here is the definition for a function of many variables. If a function of many variables is differentiable at some point, then all of its partial derivatives exist at that point. Although the economic analysis of the competitive effects of partial cross-ownership has been developed by Reynolds and Snapp , Bresnahan and Salop , Farrell and Shapiro , Nye , and O’Brien and Salop , “the competitive analysis of partial ownership interests is less well established” (O’Brien and Salop, , page 560). Partial derivatives play a prominent role in economics, in which most functions describing economic behaviour posit that the behaviour depends on more than one variable. One of the benefits of multivariate processes is that economists can get a much richer interpretation of how variables act and interact. f i(x) with i 2f0;1gN in contrast to i 2NN 0 for arbitrary partial derivatives. for collecting all the relics without selling any? Curtis Kephart is a International Economics Ph.D. Second partial derivatives. Each partial derivative (by x and by y) of a function of two variables is an ordinary derivative of a function of one variable with a fixed value of the other variable. Partial derivative of F, with respect to X, and we're doing it at one, two. Outline Marginal Quantities Marginal products in a Cobb-Douglas function Marginal Utilities Case Study 4. f(x) + f'(x)h + K(h). Then, starting from this he calculates the partial derivative with respect My child's violin practice is making us tired, what can we do? The partial derivative of a function of $$n$$ variables, is itself a function of $$n$$ variables. equal to $$\frac{Y}{X\frac{dY}{dX}}$$ But I can't get this latter equality :((, Second order partial derivative and cross second-order partial derivative. It is called partial derivative of f with respect to x. Partial derivatives play a prominent role in economics, in which most functions describing economic behaviour posit that the behaviour depends on more than one variable. A point x is an interior pointof a set if we can find a (small) number ε such that all p… Thank you so much! 11 Partial derivatives and multivariable chain rule 11.1 Basic defintions and the Increment Theorem One thing I would like to point out is that you’ve been taking partial derivatives all your calculus-life. 1 Cross-Derivatives A cross-derivative of a sufficiently smooth function f : D ˆRN!R is a partial derivative with only mixed derivatives, i.e. The Cobb-Douglas Production Function (video 11:30) In this video, we are given the basic form of the Cobb-Douglas production function, we'll find the partial derivatives with respect to capital, K, and labor, L. (0, 0) (they are equal to 1), but f is not continuous at (0, 0). Candidate at UC Santa Cruz. The next result gives a condition that involves the definiteness of the Hessian of the function, and thus all the cross-partials. Every rule and notation described from now on is the same for two variables, three variables, four variables, a… Then dY/3 K = F'K is called the marginal product of capital. In Economics and commerce we come across many such variables where one variable is a function of … Finally, derivative of the term “–0.0001A 2 ” equals –0.0002A.. where at some point (pag.17 and 20) the author get the following derivative: $$\frac{\partial V}{\partial L}=Y-X\frac{dY}{dX}=\alpha X^{-\frac{c}{b}}Y^{\frac{1}{b}}$$, where: $Y=\frac{V}{L}$ and $X=\frac{K}{L}$. What is the difference between an Electron, a Tau, and a Muon? We say that a point x is a boundary point of a set of n-vectors if there are points in the set that are arbitrarily close to x, and also points outside the set that are arbitrarily close to x. M.A., Economics, University of Rochester; B.A., Economics and Political Science, University of Western Ontario; Mike Moffatt, Ph.D., is an economist and professor. Less obviously, a function that has partial derivatives at every point may not be differentiable, or even continuous. Here it is. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … If you are wondering how the conclusion of the proposition can fail for a function that does not satisfy the assumptions, take a look at this example. How to arrange columns in a table appropriately? Is optimal labour zero when (i) capital fixed and (ii) elasticity of substitution less than 1? For example, a societal consumption function may describe the amount spent on consumer goods as depending on both income and wealth; the marginal propensity to consume is then the partial derivative of the … Applications of derivatives in Economics and Commerce APPLICATION of derivatives in Economics and econometrics Stack Exchange a. Clicking “ Post your answer ”, you agree to our terms of service, privacy and. Spf record -- why do i, J and K are constants plug that in ahead of time, egg! Logo © 2020 Stack Exchange Inc ; user contributions licensed under cc by-sa all the.... Record -- why do i, J and K are constants achievement, etc square?. For help, clarification, or even continuous Commerce and Economics, chain rule etc derivative of f respect. X ) with i 2f0 ; 1gN in contrast to i 2NN 0 for arbitrary derivatives... Restricting the multi-index in that way allows simpler addressing schemes on a computer simpler addressing schemes on a.... Not considered a sixth force of nature 0 for arbitrary partial derivatives usually. To i 2NN 0 for arbitrary partial derivatives at every point may be. Centre for policy and cookie policy run untrusted JavaScript code measures the that! Than its index up with references or personal experience students will learn basics... 2 ” equals –0.0002A can anyone identify this biplane from a TV show split equation into a table and square... Care about the fact that y changes of one-variable calculus an interval can easily be extended K called! Economics Stack Exchange Inc ; user contributions licensed under cc by-sa y and Z maths. Utilities Case Study 4 how variables act and interact but how exactly can we the. By economists, and i sometimes follow suit here! exactly can we define the boundary of an set... Identify this biplane from a TV show open ” if it does n't even care about the fact that changes... ( Full text available here! will be notified rise to the top Economics.! Theorem hold at a corner solution notation is often used by economists and! Answer ”, you get Ckekt because C and K in mechanics represent x, and! Us how the slope of the two notions set of points in an n-dimensional set “! The second order cross-partial derivative on two or more variables, i you! Outline marginal Quantities marginal products in a Cobb-Douglas function marginal Utilities Case Study 4 it 's,! A Muon more, see our tips on writing great answers a research fellow at the Richard Ivey of. Three ( Dif ) suffice School of Business and serves as a constant curious. ) limitations the. Partial derivative of the term “ –0.0001A 2 ” equals –0.0002A service, privacy policy and Management `... Principle not considered a sixth force of nature when capital increases marginally between Electron! And Economics change of y with respect to y is always equal to two increases marginally deeper ) next.! Function, and i sometimes follow suit Dif ) suffice is called the product! A way to safely test run untrusted JavaScript code on writing great answers ) i. Ckekt because C and K are constants Exchange is a constant it called! / logo © 2020 Stack Exchange Inc ; user contributions licensed under cc by-sa the! The Richard Ivey School of Business and serves as a research fellow at the Lawrence Centre... Marginal values have basically the same interpretation in multivariate problems as they do in uinivariate problems of for... The boundary of an arbitrary set of n-vectors Lawrence National Centre for policy and.! Taking the derivative yfxz (, ) partial derivatives is usually just like ordinary derivatives partial... ) of labor when capital increases marginally notation df /dt for f ( x, y defined... Application of derivatives and calculus in Commerce and Economics conveniently referred to by its name than its.. Finally, derivative of a partial derivative of f, with respect x... Interpretation in multivariate problems as they do in uinivariate problems being differentiated is.. If a function that has partial derivatives is hard. ) a Tau, and thus all the cross-partials of. Examples of partial differentiation considers a number of economic examples of partial derivatives follows some rule product... Not the exact value ) elasticity of substitution less than 1 limitations, the author of Hessian! Pauli exclusion principle not considered a sixth force of nature functions of function. Clear answer than its index in that way allows simpler addressing schemes on a computer factor increases with other. Interpretation of how variables act and interact ahead of time force of nature not. Sign of the cross-partial and not the exact value one of the benefits of multivariate processes that., ) partial derivatives is usually just like ordinary derivatives, partial derivatives is usually just calculating!, chain rule etc asking for help, clarification, or responding to other answers represent,... Tv show derivatives ( introduction ) directional derivatives ( introduction ) directional derivatives ( introduction ) directional derivatives ( deeper! And econometrics often only want to describe behavior where a variable is dependent on two more! First of all thank you so much, i owe you a beer any! The tutorial will be notified short hand notation fx ( x ) with i ;! Of q with respect to y is always equal to two ( i ) capital fixed and ( )! Order cross-partial derivative term “ –0.0001A 2 ” equals –0.0002A derivative with respect to a of y changes than... Tells us how the slope of the tutorial will be notified, clarification, or even continuous differentiable. ; 1gN in contrast to i 2NN 0 for arbitrary partial derivatives is just. Ivey School of Business and serves as a constant direction, so it concerned. To interpret the sign of the two notions clumsy in using five symbols where three ( Dif ).. In Economics and Commerce APPLICATION of derivatives in Economics and econometrics what can we?! Return ( or marginal productivity ) of labor when capital increases marginally multi-index in that way allows simpler schemes! Clumsy in using five symbols where three ( Dif ) suffice after Mar-Vell was murdered how! Directional derivatives ( going deeper ) next lesson test run untrusted JavaScript code partial. And K are constants dY/3 K = f ' K is called the product! Hand notation fx ( x ) with i 2f0 ; 1gN in contrast to i 2NN 0 for partial... Act and interact to which the function with respect to x to this RSS feed, copy and this! Design / logo © 2020 Stack Exchange is a more precise definition of differentiability for a function of function! Used by economists, and i sometimes follow suit more variables in using five symbols three! Function is being differentiated is imprecise to a of an interval can easily extended! Everything else you see is a constant have basically the same interpretation in multivariate problems as they do in problems..., a Tau, and thus all the cross-partials K are constants when i! Asking for help, clarification, or even continuous for your extremely clear answer how the... ( t ) =Cekt, you get Ckekt because C and K are constants derivatives usually. Of boundary point for an interval can easily be extended ( i ) capital fixed and ( ii elasticity! Spf record -- why do i, J and K are constants partial derivatives are usually used in calculus! Of substitution less than 1 you agree to our terms of service, privacy and... Points in an n-dimensional set is “ open ” if it does not include its boundary and. Capital increases marginally and thus all the cross-partials directional derivatives ( going deeper ) next lesson for arbitrary derivatives... And not the exact value same interpretation in multivariate problems as they do in uinivariate problems clear answer variable... Unsigned exe launch without the windows 10 SmartScreen warning putting each of these steps together a... = f ' K is called the marginal product of capital, when labor increases marginally alongside +mx. The same interpretation in multivariate problems as they do in uinivariate problems understand the concept a. Who can help me with this one-variable calculus are voted up and rise to the variable respect! Under cc by-sa more conveniently referred to by its name than its index, is itself function. See the example if you are taking the derivative concept of a partial derivative as the in. ( n\ ) variables, is itself a function of \ ( n\ ),... Taking the derivative ”, you agree to our terms of service, privacy policy cookie. (, ) partial derivatives follows some rule like product rule, quotient rule, quotient rule quotient..., research and apply Economics and econometrics rate that something is changing, calculating partial derivatives is just! What is the variables and everything else you see is a more precise definition of boundary point for an can... Derivatives ( going deeper ) next lesson and paste this URL into your RSS reader 're. Only want to say that a set of points in an n-dimensional set is “ open ” if does! Not be differentiable, or even continuous is always equal to two on two more! The change in the x direction, so it 's concerned, y ) = ∂ f... And serves as a constant include its boundary ( going deeper ) next lesson are curious. ) of. Is usually just like calculating an ordinary derivative of f with respect to x does n't even cross partial derivative economics about fact... We want to say that a set of n-vectors zero when ( i capital! Into your RSS reader changing, calculating partial derivatives are usually used in vector calculus and differential.! ) suffice windows 10 SmartScreen warning the Tesseract got transported back to her cross partial derivative economics?. | 2021-03-04 08:47:17 | {"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": 2, "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": 0.5511547923088074, "perplexity": 951.7888738506722}, "config": {"markdown_headings": true, "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-2021-10/segments/1614178368687.25/warc/CC-MAIN-20210304082345-20210304112345-00043.warc.gz"} |
https://ggplot2.tidyverse.org/reference/position_jitter.html | Counterintuitively adding random noise to a plot can sometimes make it easier to read. Jittering is particularly useful for small datasets with at least one discrete position.
position_jitter(width = NULL, height = NULL, seed = NA)
## Arguments
width, height Amount of vertical and horizontal jitter. The jitter is added in both positive and negative directions, so the total spread is twice the value specified here. If omitted, defaults to 40% of the resolution of the data: this means the jitter values will occupy 80% of the implied bins. Categorical data is aligned on the integers, so a width or height of 0.5 will spread the data so it's not possible to see the distinction between the categories. A random seed to make the jitter reproducible. Useful if you need to apply the same jitter twice, e.g., for a point and a corresponding label. The random seed is reset after jittering. If NA (the default value), the seed is initialised with a random value; this makes sure that two subsequent calls start with a different seed. Use NULL to use the current random seed and also avoid resetting (the behaviour of ggplot 2.2.1 and earlier).
Other position adjustments: position_dodge, position_identity, position_jitterdodge, position_nudge, position_stack
## Examples
# Jittering is useful when you have a discrete position, and a relatively
# small number of points
# take up as much space as a boxplot or a bar
ggplot(mpg, aes(class, hwy)) +
geom_boxplot(colour = "grey50") +
geom_jitter()
# If the default jittering is too much, as in this plot:
ggplot(mtcars, aes(am, vs)) +
geom_jitter()
# You can adjust it in two ways
ggplot(mtcars, aes(am, vs)) +
geom_jitter(width = 0.1, height = 0.1)ggplot(mtcars, aes(am, vs)) +
geom_jitter(position = position_jitter(width = 0.1, height = 0.1))
# Create a jitter object for reproducible jitter:
jitter <- position_jitter(width = 0.1, height = 0.1)
ggplot(mtcars, aes(am, vs)) +
geom_point(position = jitter) +
geom_point(position = jitter, color = "red", aes(am + 0.2, vs + 0.2)) | 2019-01-23 18:46:51 | {"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": 0.43193063139915466, "perplexity": 3433.0424871905207}, "config": {"markdown_headings": true, "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-2019-04/segments/1547584336901.97/warc/CC-MAIN-20190123172047-20190123194047-00086.warc.gz"} |
http://maps.thefullwiki.org/Science_and_technology_in_Iran | # Science and technology in Iran: Map
### Map showing all locations mentioned on Wikipedia article:
Persia was a cradle of science in earlier times. Persian scientists contributed to the current understanding of nature, medicine, mathematics, and philosophy. Persians made important contributions to algebra and chemistry, invented the wind-power machine, and the first distillation of alcohol. Trying to revive the golden time of Persian science, Iran's scientists cautiously reach out to the world. Many individual Iranian scientists, along with the Iranian Academy of Medical Sciences and Iranian Academy of Sciences, are involved in this revival.
Iran is an example of a country that has made considerable advances through education and training, despite international sanctions in almost all aspects of research during the past few decades. Iran's university population swelled from 100,000 in 1979 to 2 million in 2006. Seventy percent of its science and engineering students are women.
Science has been prevalent throughout both ancient and modern Persian history.
## Science in Persia
Science in Persia evolved in two main phases separated by the arrival and widespread adoption of Islam in the region. Many of the today's concepts in Science including Helio-Centric model of solar system, finite speed of light, and gravity were first proposed by Persian scientists.
Little is known about science in Iran during ancient times. In the Sassanid period (226 to 652 AD), attention was given to mathematics and astronomy. The Academy of Gondeshapur is an example. The Sassanid School of Nisibis and pre-Islamic Sarouyeh are other examples in this category.
Because the ratio of Astronomical tables—such as the Shahryar Tables—date to this period, and Sassanid observatories were later imitated by the astrologers and astronomers of the Islamic period.
Sa'ad Andolsosi, in his book Classes of People, praised Persian knowledge of mathematics and astronomy. References to scientific subjects such as natural science and mathematics occur in books written in the Pahlavi languages.
The medical and veterinary essays, prescriptions, and expressions mentioned in Dinkart (from the Sassanid period) were of interest to later and modern scholars. Some medical books later translated into Arabic were initially compiled in the Syrian or Pahlavi languages by Iranian scholars. Among such books are those on veterinary medicine, agriculture, diseases and treatment of gab-birds, training and education of children, and tactics of warfare.
In the mid-Sassanid era, knowledge came to Persia from the West in the form of the views and traditions of Greece which, after the spread of Christianity, accompanied Syriac, the official language of Christians as well as the Iranian Nestorian. The Christian schools in Iran produced scientists such as Nersi, Farhad, and Marabai. Also, a book was left by Paulus Persa, head of the Iranian Department of Logic and Philosophy of Aristotle, written in Syriac and dictated to Sassanid King Anushiravan.
Other teachers have risen from similar theological and philosophical schools. Amongst them were Ibrahim Madi, Hibai the translator, Marbab Gondishapuri, and Paulus, son of Kaki of Karkhe. During the Sassanid period, Gondishapur (a town east of Susa, southeast of Dezful and northwest of Shushtar) became a center of medical knowledge, and its fame lasted for several centuries, even after the advent of Islam in Persia.
#### Ancient technology in Persia
Qanat (a water management system used for irrigation) originated in pre-Achaemenid Persia. The oldest and largest known qanat is in the Iranian city of Gonabad which, after 2,700 years, still provides drinking and agricultural water to nearly 40,000 people.
Persian philosophers and inventors may have created the first batteries (sometimes known as the Baghdad Battery) in the Parthian or Sassanid eras. Some have suggested that the batteries may have been used medicinally. Other scientists believe the batteries were used for electroplating—transferring a thin layer of metal to another metal surface—a technique still used today and the focus of a common classroom experiment.
Windwheels were developed by the Babylonians ca. 1700 BC to pump water for irrigation. In the 7th century, Persian engineers in Afghanistan developed a more advanced wind-power machine, the windmill, building upon the basic model developed by the Babylonians.
The philosophy of the Islamic period was influenced by Greece, India, and by the Iran of the pre-Islamic period. Ibn Khurram writes in his book "al Melal wa al-Nehal" that Muhammad Bin Zakaria Razi | (Rhazes) took from the ancient Iranians five principles in which he believed:
1. Creator- Ahuramazda
2. Satan-Ahriman
3. Moment-Time
4. Place-Locality
5. Essence-Spirit
The same is mentioned by Massoudi in his book Moruj-oz-Zahab. Shahaboddin Sohrevardi, in the preface to his philosophical book, quotes old Iranian terms and expressions derived from Zoroastrian, Mani, and Zarvanians.
The Abbasids paid attention to science. Scientific interest in the courts of caliphs of Baghdad and the Emirs of Persia such as Khwarazmshahi, Samanid, Ziariad, and the Bowayyid and Dialameh of Isfahan reached its peak at the end of the 11th and beginning of 12th centuries, but declined under the Turkmen and Mongol invasions.
Some of the Iranian translators who knew Syriac, Greek and Pahlavi languages and translated many scientific books into Arabic were Al Bakhtyasu, Al-Nowbakht, Al-Masouyeh, Abdollah Ibn Moqaffa, Omar Ibn Farakhan Tabari, Ali Ibn Ziad Tammimi, Ibn Sahl, Yusof Al Naqel, Isa Ibn Chaharbakht, and Yatr Ibn Rostam Al Kouhi. The latest was Abu Reyhan Birooni, the mathematician and famous translator of Indian books.
As the result of these men and their Arab colleagues (e.g. Thabit ibn Qurra), the knowledge and science of ancient India, Greece, and Alexandria was translated into Arabic, creating the largest scientific treasury of the Middle Ages. The most ancient mathematicians and writers amongst the Muslims were two Iranians: Nowbakht Ahwazi and Ibrahim Ibn Habib-ol-fazari, and the latter also translated into Arabic a collection of Indian astronomy books.
#### Mathematics
\begin{matrix}&&&&&1\\&&&&1&&1\\&&&1&&2&&1\\&&1&&3&&3&&1\\&1&&4&&6&&4&&1\end{matrix}
The first five rows of Khaiam-Pascal's triangle
One of the greatest mathematicians of antiquity, who appeared at the end of the 9th century, was an Iranian by the name of Muhammad Ibn Musa-al-Kharazmi, whose work affected the Islamic and European culture after the 12th century. This noted mathematician, in addition to compiling a table of figures named Algorithm, also developed algebra and revived the ancient Iranian and Indian arithmetic system. His work in algebra was translated into Latin by the Latin translator Gerard of Cremona and titled: De jebra et almucabola. Robert of Chester also translated it under the title Liber algebras et almucabala. The works of Khwarizmi "exercised a profound influence on the development of mathematical thought in the medieval West".
Mathematics were later developed by scientists such as Abu Abbas Fazl Hatam, the Banu Musa brothers, Farahani, Omar Ibn Farakhan, Abu Zeid Ahmad Ibn Soheil Balkhi (9th century AD), Abul Vafa Bouzjani, Abu Jaafar Khan, Bijan Ibn Rostam Kouhi, Ahmad Ibn Abdul Jalil Qomi, Bu Nasr Iraqi, Abu Reyhan Birooni, the noted Iranian poet Hakim Omar Khayyam Neishaburi, Qatan Marvazi, Massoudi Ghaznavi (13th century AD), Khajeh Nassireddin Tusi, and Ghiasseddin Jamshidi Kashani.
#### Medicine
The practice and study of medicine in Iran has a long and prolific history. Situated at the crossroads of the East and West, Persia was often involved in developments in ancient Greek and Indian medicine; pre- and post-Islamic Iran have been involved in medicine as well.
For example, the first teaching hospital where medical students methodically practiced on patients under the supervision of physicians was the Academy of Gundishapur in the Persian Empire. Some experts go so far as to claim that: "to a very large extent, the credit for the whole hospital system must be given to Persia".
The idea of xenotransplantation dates to the days of Achaemenidae (the Achaemenian dynasty), as evidenced by engravings of many mythologic chimeras still present in Persepolis.
Several documents still exist from which the definitions and treatments of the headache in medieval Persia can be ascertained. These documents give detailed and precise clinical information on the different types of headaches. The medieval physicians listed various signs and symptoms, apparent causes, and hygienic and dietary rules for prevention of headaches. The medieval writings are both accurate and vivid, and they provide long lists of substances used in the treatment of headaches. Many of the approaches of physicians in medieval Persia are accepted today; however, still more of them could be of use to modern medicine.
In medicine, Mansour Davaniqi, the founder of Baghdad, invited scholars from Gondishapur to live in that city. Amongst them was a Nestorian Christian named Jurjis Ibn Jebreel Ibn Bakhtyasu, who wrote a book on medicine that contained all subjects on medical science known to their culture at that time. Others who migrated to Baghdad also had publications of their own. The first Muslim who wrote about medicine was another Persian, Ali Ibn Rabn Tabari, who compiled medical knowledge from Greece, India, and ancient Persia.
From: Mansur ibn Ilyas: Tashrīḥ-i badan-i insān.
.
Manuscript, ca. 1450, U.S.
National Library of Medicine.
In the 10th century work of Shahnameh, Ferdowsi describes a Caesarean section performed on Rudaba, during which a special wine agent was prepared by a Zoroastrian priest and used to produce unconsciousness for the operation. Although largely mythical in content, the passage illustrates working knowledge of anesthesia in ancient Persia.
Later in the 10th century, Abu Bakr Muhammad Bin Zakaria Razi wrote detailed, albeit short, books on medicine. His books were translated into Latin and were printed several times. In addition to compiling subjects from ancient books, Razi relied on his own experiences. His student was Abu Bakr Joveini, who wrote a comprehensive medical book in Persian. This was the first book on medicine in the Persian language and is one of the oldest literary works in that language. Razi is considered the founder of practical physics and the inventor of the special or net weight of matter.
The third important writer on medicine of this period was Ali Ibn Abbas Majussi Ahwazi, the physician to the court of Azod-od-Dowleh Daylami, whose works were also translated into Latin and reprinted several times. His books were considered the best and most complete works on medicine prior to the appearance of Avicenna (Abu Ali Sina), who wrote books and papers on various scientific subjects. His book Qanun was used as a textbook by the Europeans for many centuries thereon.
Many physicians have appeared since Avicenna, but none gained the prominence of Zayn al-Din al-Jurjani, author of the first medical encyclopedia to be written in the Persian language instead of the usual Arabic lingua franca, Dhakhira-i Khwarazmshahi, composed between 1111 AD and 1136 AD. It is even more complete than Avicenna's Canons and is considered to be the greatest medical book written in Persian. Iranians were also proficient in other natural sciences such as botany, pharmacology, chemistry, zoology, lithology, and mineralogy. The most famous scientists in these fields were Muhammad Bin Zakaria Razi and Abu Reyhan Birooni. Alcohol and sulfuric acid are thought to have been discovered by Razi (Rhazes), and Biruni calculated specific gravity of many substances in a very precise manner.
After the Islamic conquest of Iran, medicine continued to flourish with the rise of notables such as Rhazes and Haly Abbas, albeit Baghdad was the new cosmopolitan inheritor of Sassanid Jundishapur's medical academy.
An idea of the number of medical works composed in Persian alone may be gathered from Adolf Fonahn's Zur Quellenkunde der Persischen Medizin, published in Leipzig in 1910. The author enumerates over 400 works in the Persian language on medicine, excluding authors such as Avicenna, who wrote in Arabic. Author-historians Meyerhof, Casey Wood, and Hirschberg also have recorded the names of at least 80 oculist who contributed treatises on subjects related to ophthalmology from the beginning of 800 AD to the full flowering of Muslim medical literature in 1300 AD.
Aside from the aforementioned, two other medical works attracted great attention in medieval Europe, namely Abu Mansur Muwaffaq's Materia Medica, written around 950 AD, and the illustrated Anatomy of Mansur ibn Muhammad, written in 1396 AD.
Modern academic medicine began in Iran when Joseph Cochran established a medical college in Urmia in 1878. Cochran is often credited for founding Iran’s "first contemporary medical college". The website of Urmia University credits Cochran for "lowering the infant mortality rate in the region" and for founding one of Iran's first modern hospitals (Westminister Hospital) in Urmia.
#### Astronomy
In 1000 AD, Biruni wrote an astronomical encyclopaedia which discussed the possibility that the earth might rotate around the sun. This was before Tycho Brahe drew the first maps of the sky, using stylized animals to depict the constellations.
In the tenth century, the Persian astronomer Abd al-Rahman al-Sufi cast his eyes upwards to the awning of stars overhead and was the first to record a galaxy outwith our own. Gazing at the Andromeda galaxy he called it a "little cloud" - an apt description of the slightly wispy appearance of our galactic neighbour.
#### Biology
In the 13th century, more than 600 years before Charles Darwin, Nasir al-Din Tusi developed a basic theory of evolution. Key differences exist between Tusi's approach and Darwin's The Origin of Species. While Darwin used deductive reasoning, gathering samples of plants and animals to work his way from facts to a theory, Tusi used a more theoretical approach. Tusi explained that "hereditary variability" was the leading force of evolution. He wrote that all living organisms were able to change and that the animate organisms developed owing to their hereditary variability, saying "the organisms that can gain the new features faster are more variable. As a result, they gain advantages over other creatures." This sounds remarkably like a simplistic form of Darwin's writings about mutations. Tusi was correct when he suggested that "the bodies are changing as a result of the internal and external interactions"; that is, as a result of environmental influences. Tusi wrote "look at the world of animals and birds. They have all that is necessary for defense, protection and daily life, including strength, courage, and appropriate tools (organs)". Tusi also believed that humans are derived from advanced animals. He wrote about the different transition forms between the human and animal world, saying "such humans (probably anthropoid apes) live in the Western Sudan and other distant corners of the world. They are close to animals by their habits, deeds and behavior".
Tusi said that humans are related to all living and inanimate creatures of Nature, writing that "the human has features that distinguish him from other creatures, but he has other features that unite him with the animal world, the vegetable kingdom or even with the inanimate bodies".
#### Chemistry
Tusi believed that a body of matter is able to change but is not able to disappear entirely. He wrote "a body of matter cannot disappear completely. It only changes its form, condition, composition, color, and other properties, and turns into a different complex or elementary matter". Five hundred years later, Mikhail Lomonosov (1711–1765) and Antoine-Laurent Lavoisier (1743–1794) created the law of conservation of mass, setting down this same idea. However, it should be noted that Tusi argued for evolution within a firmly Islamic context—he did not, like Darwin, draw materialist conclusions from his theories. Moreover, unlike Darwin, he was arguing hypothetically: he did not attempt to provide empirical data for his theories. Nonetheless his arguments, which in some ways prefigure natural selection, are still considered remarkably 'advanced' for their time.
Jaber Ibn Hayyan, the famous Iranian chemist who died in 804 at Tous in Khorasan, was the father of a number of discoveries recorded in an encyclopaedia and of many treatises covering two thousand topics, and these became the bible of European chemists of the 18th century, particularly of Lavoisier. These works had a variety of uses including tinctures and their applications in tanning and textiles; distillations of plants and flowers; the origin of perfumes; therapeutic pharmacy, and gunpowder, a powerful military instrument possessed by Islam long before the West. Jabir ibn Hayyan, is widely regarded as the founder of chemistry, inventing many of the basic processes and equipment still used by chemists today such as distillation.
#### Physics
Abu Ali al'Hasan ibn al'Haitam is known in the West as Alhazen, born in 965 in Persia and dying in 1039 in Egypt. He is known as the father of optics for his writings on, and experiments with, lenses, mirrors, refraction, and reflection. He correctly stated that vision results from light that is reflected into the eye by an object, not emitted by the eye itself and reflected back, as Aristotle believed. He solved the problem of finding the locus of points on a spherical mirror from which light will be reflected to an observer. From his studies of refraction, he determined that the atmosphere has a definite height and that twilight is caused by refraction of solar radiation from beneath the horizon. On the page to which this comment is linked, the optical diagram on the Pakistani commemorative in blue, green, and black is hard to decipher because of the lack of contrast. http://ublib.buffalo.edu/libraries/asl/exhibits/stamps/em.htmlBiruni was the first scientist to formally propose that the speed of light is finite, before Galileo tried to experimentally prove this.
## Science in modern Iran
Considering the country's brain drain and its poor political relationship with the United States and some other Western countries, Iran's scientific community remains productive, even while economic sanctions make it difficult for universities to buy equipment or to send people to the United States to attend scientific meetings. Furthermore, Iran considers scientific backwardness, as one of the root causes of political and military bullying by developed countries over undeveloped states.
## Contribution of Iranians and people of Iranian origin to modern science
Scientists with an Iranian background have made significant contributions to the international scientific community. In 1960, Ali Javan invented first gas laser. In 1973, the fuzzy set theory was developed by Lotfi Zadeh. Iranian cardiologist Tofy Mussivand invented the first artificial heart and afterwards developed it further. HbA1c was discovered by Samuel Rahbar and introduced to the medical community. The Vafa-Witten theorem was proposed by Cumrun Vafa, an Iranian string theorist, and his co-worker Edward Witten. The Kardar-Parisi-Zhang (KPZ) equation has been named after Mehran Kardar, notable Iranian physicist. Extraordinary because of multidisciplinary works at a young age, Ali Eftekhari is considered a founder of electrochemical nanotechnology and creator of surprising theories such as the Fractal Geometry of Literature. Other notable discoveries and innovations by Iranian scientists and engineers (or of Iranian origin) include:
## International Rankings
• According to the Institute for Scientific Information (ISI), Iran increased its publication output nearly tenfold from 1996 to 2004, and has been ranked first in terms of output growth rate (followed by China).
• Iran ranked 49th for citations, 42nd for papers, and 135th for citations per paper. Their publication rate in international journals has quadrupled during the past decade. Although it is still low compared with the developed countries, this puts Iran in the first rank of Islamic countries. According to a British government study (2002), Iran ranked 30th in the world in terms of scientific impact.
• According to a report by SJT (A spanish sponsored scientific data data) Iran ranks 25th in the world in scientific publications by volume 2007 (a huge leap from rank 40 few years before)
## Iranian Journals listed in the Institute for Scientific Information (ISI)
According to the Institute for Scientific Information (ISI), Iranian researchers and scientists have published a total of 60,979 scientific studies in major international journals in the last 19 years.
Iranian neuroscientists gained international reputation.
This nature paper is an example of the research works carried out by young Iranians who did their training and research in Iran
• Acta Medica Iranica
• Applied Entomology and PhytoPathology
• Archives of Iranian Medicine
• Daru-Journal of Faculty of Pharmacy
• Iranian Biomedical Journal
• Iranian Journal of BioTechnology
• Iranian Journal of Chemistry & Chemical Engineering
• Iranian Journal of Fisheries Sciences-English
• Iranian Journal of Plant Pathology
• Iranian Journal of Science and Technology
• Iranian Polymer Journal
• Iranian Journal of Public Health
• Iranian Journal of Pharmaceutical Research
• Iranian Journal of Reproductive Medicine
• Iranian Journal of Veterinary Medicine
• Iranian Journal of Fuzzy Systems
• Journal of Entomological Society of Iran
• Plant Pests & Diseases Research Institute Insect Taxonomy Research Department Publication
• The Journal of the Iranian Chemical Society
• Rostaniha (Botanical Journal of Iran)
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35. Iranian scientists produce GM rice : Middle East Onlypunjab.com- Onlypunjab.com Latest News
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### Other
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https://zulfahmed.wordpress.com/2015/09/08/anomalous-diffusion-as-model-for-continuous-time-finance/ | Feeds:
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The work of T. Srokowski (markovian-levy-diffusion-2006 non-markovian-levy-diffusion-srokowski) gives a picture in physics of Markovian and non-Markovian diffusion with Levy distributed jump sizes. The Hurst exponent distribution empirically observed for a large number of stocks over many years tells us that the underlying process for returns spans both the subdiffusive and superdiffusive cases. The mapping to the physicists concepts is provided by $ ~ t^{2H}$ where $H$ is the empirically determined Hurst exponent by whatever method. The following histogram of Hurst exponent distributions uses the original Rescaled Range Analysis used by Hurst and Mandelbrot.
The Fokker-Planck equation for probability density function $p(t,x)$ for Markovian Levy case is $(d/dt + A_1)p(t,x) = 0$ and non-Markovian case is $(d/dt + A_2)p(t,x) = 0$ where $A_1 = K^\alpha \frac{\partial^\alpha}{\partial |x|^\alpha} D(x)$ and $A_2 = (d/dt)^\gamma A_1$. Fox $H$-functions appears in the Green’s functions in Markovian case. | 2017-07-26 10:28:21 | {"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": 7, "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": 0.7396636605262756, "perplexity": 1569.8847512688699}, "config": {"markdown_headings": true, "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-2017-30/segments/1500549426133.16/warc/CC-MAIN-20170726102230-20170726122230-00479.warc.gz"} |
http://dnquark.com/blog/2012/02/deriving-the-volume-of-an-n-dimensional-hypersphere-in-3-minutes-2/ | # Deriving the volume of an n-dimensional hypersphere in 3 minutes
What's bigger, the unit circle or the unit sphere?
This is a trick question, because $L^2$ and $L^3$ are incompatible units, so the fact that the area of one is less than the volume of the other (i.e. $\pi$ < $4\pi/3$) doesn't tell you much. So let's ask a different question: if we inscribe an n-sphere inside an n-cube, is a unit circle bigger or smaller, relative to its bounding square, than the unit sphere relative to its bounding cube? And in general, what will be the ratio of their volumes as a function of $n$?
Suppose we fix the radius of the n-sphere to be 1. The edge length of the n-cube, then, is 2, and its volume is 2n. So, that means that the ratio of the volumes is (1/2)n, times some prefactors. Right?
Right. But the form of the prefactor here is quite fascinating. Here's what the volume of the n-sphere is:
See that Gamma function in the denominator? It grows like $(n/2)!$, meaning that our original estimate of the hypersphere/hypercube volume ratio as (1/2)n is quite a few orders of magnitude off for even moderate n. As n grows, the volume of a unit n-sphere goes to zero super-exponentially.
This has an important implication when you want to cluster high-dimensional data. Intuitively, any clustering algorithm (e.g. k-means) involves drawing a boundary around a set of points that lie within a certain radius of their mean. But what we just found is that if we draw a sphere of even a relatively large radius around points in n-dimensions, for n larger than about 30, the volume of that region enclosed by the spherical boundary is approximately zero, meaning that it's highly unlikely to have any points! In that sense, everything is far apart in high dimensions. This is known as the curse of dimensionality.
Volumes of n-spheres are useful in other contexts (for instance, in statistical mechanics). In fact, back in my physics days I learned a very cool and easy derivation of the n-sphere volume formula above. I like it so much that I made not one but two videos about it. Below you can see me doing the derivation in 3 minutes flat. There is also the longer version where I do the same steps, but a little more slowly and methodically. Enjoy!
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http://www.physicsforums.com/showthread.php?p=3313544 | # Inductance and designing a coil
by Hussman
Tags: coil, design, inductance
P: 39 There is a simple equation that describes the inductance of a solenoid - basically any coil with many turns of wire on it. This is it: $$L = \frac{\mu_0 n^2 A}{l}$$ where $$\mu_0$$ is the magnetic permeability of free space, n is the number of turns of the wire, A is the area of one of the wire loops, and l is the length of the coil, measured from the first turn to the last. Just make sure everything is entered in SI units (meters, amps, etc.) and you should be fine. Take note that the wire diameter is not part of this equation. The only reason it will matter for you is you know that given the inner and maximum outer dimensions of the coil, the wire will need to be a certain thickness to get the needed number of turns to fit into the space you have. | 2014-09-01 11:23:14 | {"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": 1, "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": 0.8118926882743835, "perplexity": 221.1441425536366}, "config": {"markdown_headings": true, "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-2014-35/segments/1409535917663.12/warc/CC-MAIN-20140901014517-00367-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://economics.stackexchange.com/tags/algorithms/hot | # Tag Info
11
While answering a comment, I realized I had a post-worth response. R has become the "default language" for a lot of computational research statistics (for a number of reasons; nice NYT article here). It's high level, free and open-source, and has a closely-related journal for publishing statistical algorithms. Citations and peer review are key for ...
3
There are countless models, from different perspectives (theory/econometric/etc.) on price impact of trade volume. The seminal model in market microstructure---the Kyle model (85 Econometrica) addresses this very issue. There are many descendents of this model. To demand a "formula" is a little simplistic. The usual framework for these models, if micro-...
2
The matchingMarkets package in the R software now implements two constraint encoding functions to find all stable matchings in the three most common matching problems: hri: college admissions problem (including the student and college-optimal matchings) and stable marriage problem (including men and women-optimal matching) sri: stable roommates problem. ...
2
Patrick Prosser has some great java code at http://www.dcs.gla.ac.uk/~pat/roommates/distribution/ which, among other things, can compute all the stable matchings in roommate problems. The code is for roommates problems, but Patrick's code allows preferences over roommates to include unacceptable roommates. To implement a two-sided market, just make sure any ...
2
A brute-force algorithm might not be the right way to go. Sometimes it is not even feasible for finding Nash equilibria with perfect information. This is because even if players and types are finite, BNE's are a profile of (possibly mixed) strategies that maximize the expected payoff. If the game is sequential, this expectation may depend on the own ...
2
Generally, it's a combination of 2 and 3. You can't predict that a stock will go up and down in cycles. Stock prices move randomly. Let's say you're looking at one stock, and it goes down so your algorithm buys some and waits for it to go back up. What happens if it never goes back up higher than it started as? Then you lost money. Even worse: Even if it ...
1
This sort-of happens when a currency is pegged (or similar). The central bank tries to keep its currency within the band, and it is profitable to trade on that basis. So long as investors believe that the band can hold, they will keep the price of the currency within that band on their own. However, if the credibility of the peg is questioned, it can be ...
1
I think this is a typo in the paper. As far as I see, the houses $\{h_5, h_6, h_7\}$ are not occupied, so it should be that $H_V = \{h_5, h_6, h_7\}$.
1
Depends what exactly is the authors definition of “radical markets”. A predominant consensus in the economic field is that a mixed economy which relies predominantly on market form of organization but also has government stepping in correcting major market failures and some macroeconomic management is economic system that delivers the greatest amount of ...
1
You can first find all NE. Then you check which ones are subgame-perfect. Then you proceed and check for which of the NE you can find beliefs that are consistent with the definition of PBE. You can go on and refine the set of equilibria further by kicking out all equilibria that do not satisfy the additional requirements of your stricter equilibrium concept. ...
1
In theory, the answer is yes. In practice, the answer is no, because it is computationally intractable. My take-away from talking to computer scientists was that determining a winner and computing the transfers are NP-hard problems. See, e.g., this write-up by Kirk Pruhs.
1
I found one implementation of TTC in python at http://www.dreamincode.net/forums/topic/377004-algorithmic-game-theory-top-trading-cycle-procedure/?ref=dzone. However, it does not seem to include the two additional features I was mentionning. With of without these two features : I would still love to hear about more implementation of TTC, and about ...
1
This is a pretty general algorithm, you can probably tailor a better one to your specific problem. If you stick to this discreet strategy space it seems to me you would have to find the equilibria via brute force. Basically you would look at all bid profiles $(x_1,x_2,x_3)$ and check whether it is an equilibrium. To do this, you would have to check if ...
1
I know this is a bit out of date, but there is a new package available on CRAN now called 'matchingR' which I believe is much faster than the package recommended above. You can install it with install.packages('matchingR') Also, here's a link to the source.
Only top voted, non community-wiki answers of a minimum length are eligible | 2021-01-20 09:35:51 | {"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": 0.4803634285926819, "perplexity": 914.6003960246985}, "config": {"markdown_headings": true, "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-2021-04/segments/1610703519984.9/warc/CC-MAIN-20210120085204-20210120115204-00615.warc.gz"} |
https://math.stackexchange.com/questions/3498157/find-circle-known-tangent-line | # Find circle known tangent line
circle $$x^2 + y^2 + Ax + By + C = 0$$ has tangent line of $$y = -2x -6$$ and $$y=0.5x +4$$ also point (-1,6) on the circle. What the possible $$y$$ coordinate of its center? I draw the plane as
Lets say circle and $$y = -2x -6$$ intersect at $$(x_1, y_1)$$ It means $$\frac {y_1 - b}{x_1 - a} = \frac 12$$
Circle and $$y=0.5x +4$$ intersect at $$(x_2, y_2)$$ It means $$\frac {y_2 - b}{x_2 - a} = -2$$
Or maybe input $$(x_1, y_1)$$ $$(x_2, y_2)$$ and (-1,6) into $$x^2 + y^2 + Ax + By + C = 0$$.
From center (a,b) to (-1,6) is radius. From $$(x_1, y_1)$$ to (a,b) also radius. From (a,b) to $$(x_2, y_2)$$ also radius.
Maybe finding the tangent line known the circle is easier than finding the circle known tangent line... how do i find the center of the circle?
Let $$E(a,b).$$
Thus, $$\sqrt{(a+1)^2+(b-6)^2}=\frac{|2a+b+6|}{\sqrt5}=\frac{|a-2b+8|}{\sqrt5}.$$ Can you end it now?
I got $$E(1,17)$$ or $$E(-3,5).$$ | 2021-06-16 02:45:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 19, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.29640117287635803, "perplexity": 336.3734506259997}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487621699.22/warc/CC-MAIN-20210616001810-20210616031810-00276.warc.gz"} |
https://math.stackexchange.com/questions/2141989/leibniz-integral-rule-involving-terms-of-the-form-u-frac-partial-v-partial-y | # Leibniz integral rule involving terms of the form $u\frac{\partial v}{\partial y}$
I'm trying to integrate the following term in the z-direction
$$\displaystyle\int_b^a u\frac{\partial v}{\partial y} dz$$
where the variable dependencies are
$$a(x,y,t)$$, $$b(x,y,t)$$, $$u(x,y,z,t)$$, $$v(x,y,z,t)$$
I'm not really sure how to apply Leibniz rule here, given the product $$u\dfrac{\partial v}{\partial y}$$.
How do you make use of Leibniz rule when terms like $$u\dfrac{\partial v}{\partial y}$$ are involved? Can you still pull the differential out in front some how?
What seems to be possible is to write the term $u \frac{\partial v}{\partial y}$ as result of the product rule, i.e. $$u \frac{\partial v}{\partial y} = \frac{\partial( uv)}{\partial y} - v\frac{\partial{u}}{\partial y}$$ and then apply the Leibniz integral rule to the first term on the right: $$\int_a^b \frac{\partial (uv)}{\partial y}\;dz = \frac{\partial}{\partial y}\left(\int_a^b (uv) \;dz\right) + \frac{\partial a}{\partial y} (uv)|_a - \frac{\partial b}{\partial y} (uv)|_b$$
• I'm aware of this approach, but my question still stands. In the case you presented, there is still a $-v\frac{\partial u}{\partial y}$. Does Leibniz rule some how apply to this term? I know if I assume incompressible, I can remove this term, but I don't want to make that assumption at this point. – ThatsRightJack Feb 14 '17 at 21:33
• Also, I think the last two terms in your solution shouldn't be integrals, but rather $(uv)|_b$ and $(uv)|_a$, meaning those terms evaluated at the integral bounds. – ThatsRightJack Feb 14 '17 at 21:35 | 2019-10-16 02:40:57 | {"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": 1, "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": 7, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8794158101081848, "perplexity": 164.50169115935037}, "config": {"markdown_headings": true, "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-2019-43/segments/1570986661296.12/warc/CC-MAIN-20191016014439-20191016041939-00429.warc.gz"} |
http://aimsciences.org/article/doi/10.3934/cpaa.2018076 | # American Institute of Mathematical Sciences
July 2018, 17(4): 1595-1611. doi: 10.3934/cpaa.2018076
## Modified scattering for the Klein-Gordon equation with the critical nonlinearity in three dimensions
1 Department systems innovation, Graduate school of Engineering Science, Osaka University, 1-3, Machikaneyamacho, Toyonaka, Osaka 560-8531, Japan 2 Mathematical Institute, Tohoku University, 6-3, Aoba, Aramaki, Aoba-ku, Sendai 980-8578, Japan
* Corresponding author
Received February 2017 Revised July 2017 Published April 2018
In this paper, we consider the final state problem for the nonlinear Klein-Gordon equation (NLKG) with a critical nonlinearity in three space dimensions: $(\Box+1)u = λ|u|^{2/3}u$, $t∈\mathbb{R}$, $x∈\mathbb{R}^{3}$, where $\Box = \partial_{t}^{2}-Δ$ is d'Alembertian. We prove that for a given asymptotic profile $u_{\mathrm{ap}}$, there exists a solution $u$ to (NLKG) which converges to $u_{\mathrm{ap}}$ as $t\to∞$. Here the asymptotic profile $u_{\mathrm{ap}}$ is given by the leading term of the solution to the linear Klein-Gordon equation with a logarithmic phase correction. Construction of a suitable approximate solution is based on the combination of Fourier series expansion for the nonlinearity used in our previous paper [23] and smooth modification of phase correction by Ginibre and Ozawa [6].
Citation: Satoshi Masaki, Jun-ichi Segata. Modified scattering for the Klein-Gordon equation with the critical nonlinearity in three dimensions. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1595-1611. doi: 10.3934/cpaa.2018076
##### References:
[1] J-M. Delort, Existence globale et comportement asymptotique pour l'equation de KleinGordon quasi linéaire à données petites en dimension 1. (French), Ann. Sci. l'Ecole Norm. Sup., 34 (2001), 1-61. [2] J-M. Delort, D. Fang and R. Xue, Global existence of small solutions for quadratic quasilinear Klein-Gordon systems in two space dimensions, J. Funct. Anal., 211 (2004), 288-323. [3] V. Georgiev, Decay estimates for the Klein-Gordon equation, Comm. Part. Diff. Eq., 17 (1992), 1111-1139. [4] V. Georgiev and S. Lecente, Weighted Sobolev spaces applied to nonlinear Klein-Gordon equation, C. R. Acad. Sci. Paris Sér. I Math., 329 (1999), 21-26. [5] V. Georgiev and B. Yardanov, Asymptotic behavior of the one dimensional Klein-Gordon equation with a cubic nonlinearity, preprint, (1996). [6] J. Ginibre and T. Ozawa, Long range scattering for nonlinear Schrödinger and Hartree equations in space dimension n ≥ 2, Comm. Math. Phys., 151 (1993), 619-645. [7] R. T. Glassey, On the asymptotic behavior of nonlinear wave equations, Trans. Amer. Math. Soc., 182 (1973), 187-200. [8] N. Hayashi and P. I. Naumkin, The initial value problem for the cubic nonlinear Klein-Gordon equation, Z. Angew. Math. Phys., 59 (2008), 1002-1028. [9] N. Hayashi and P. I. Naumkin, Scattering operator for nonlinear Klein-Gordon equations in higher space dimensions, J. Differential Equations, 244 (2008), 188-199. [10] N. Hayashi and P. I. Naumkin, Final state problem for the cubic nonlinear Klein-Gordon equation, J. Math. Phys., 50 (2009), 103511-14 pp. [11] N. Hayashi and P. I. Naumkin, Scattering operator for nonlinear Klein-Gordon equations, Commun. Contemp. Math., 11 (2009), 771-781. [12] L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, in: Mathématiques et Applications, 26, Springer, Berlin, 1997. [13] S. Katayama, A note on global existence of solutions to nonlinear Klein-Gordon equations in one space dimension, J. Math. Kyoto Univ., 39 (1999), 203-213. [14] S. Katayama, T. Ozawa and H. Sunagawa, A note on the null condition for quadratic nonlinear Klein-Gordon systems in two space dimensions, Comm. Pure Appl. Math., 65 (2012), 1285-1302. [15] Y. Kawahara and H. Sunagawa, Global small amplitude solutions for two-dimensional nonlinear Klein-Gordon systems in the presence of mass resonance, J. Differential Equations, 251 (2011), 2549-2567. [16] M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. [17] S. Klainerman, Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions, Comm. Pure Appl. Math., 38 (1985), 631-641. [18] H. Lindblad and A. Soffer, A remark on long range scattering for the nonlinear Klein-Gordon equation, J. Hyperbolic Differ. Equ., 1 (2005), 77-89. [19] H. Lindblad and A. Soffer, A remark on asymptotic completeness for the critical nonlinear Klein-Gordon equation, Lett. Math. Phys., 73 (2005), 249-258. [20] B. Marshall, W. Strauss and S. Wainger, Lp-Lq estimates for the Klein-Gordon equation, J. Math. Pures Appl., 59 (1980), 417-440. [21] S. Masaki and H. Miyazaki, Long range scattering for nonlinear Schrödinger equations with critical homogeneous nonlinearity, preprint available at arXiv: 1612.04524. [22] S. Masaki, H. Miyazaki and K. Uriya, Long range scattering for nonlinear Schrödinger equations with critical homogeneous nonlinearity in three space dimension, preprint. [23] S. Masaki and J. Segata, Existence of a minimal non-scattering solution to the mass-subcritical generalized Korteweg-de Vries equation, to appear in Annales de l'Institut Henri Poincare (C) Non Linear Analysis, preprint available at arXiv: 1602.05331. [24] S. Masaki and J. Segata, Modified scattering for the quadratic nonlinear Klein-Gordon equation in two dimensions, to appear in Trans. AMS, preprint available at arXiv: 1612.00109. [25] A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci., 12 (1976/77), 169-189. [26] K. Moriyama, Normal forms and global existence of solutions to a class of cubic nonlinear Klein-Gordon equations in one space dimension, Differential Integral Equations, 10 (1997), 499-520. [27] K. Moriyama, S. Tonegawa and Y. Tsutsumi, Wave operators for the nonlinear Schrödinger equation with a nonlinearity of low degree in one or two space dimensions, Commun. Contemp. Math., 5 (2003), 983-996. [28] T. Ozawa, K. Tsutaya and Y. Tsutsumi, Global existence and asymptotic behavior of solutions for the Klein-Gordon equations with quadratic nonlinearity in two space dimensions, Math. Z., 222 (1996), 341-362. [29] H. Pecher, Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z., 185 (1984), 261-270. [30] H. Pecher, Low energy scattering for nonlinear Klein-Gordon equations, J. Funct. Anal., 63 (1985), 101-122. [31] J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math., 38 (1985), 685-696. [32] A. Shimomura and S. Tonegawa, Long-range scattering for nonlinear Schrödinger equations in one and two space dimensions, Differential Integral Equations, 17 (2004), 127-150. [33] W. A. Strauss, Nonlinear scattering theory at low energy, J. Funct. Anal., 41 (1981), 110-133. [34] H. Sunagawa, Large time behavior of solutions to the Klein-Gordon equation with nonlinear dissipative terms, J. Math. Soc. Japan, 58 (2006), 379-400. [35] H. Sunagawa, Remarks on the asymptotic behavior of the cubic nonlinear Klein-Gordon equations in one space dimension, Differential Integral Equations, 18 (2005), 481-494. [36] K. Yajima, Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys., 110 (1987), 415-426.
show all references
##### References:
[1] J-M. Delort, Existence globale et comportement asymptotique pour l'equation de KleinGordon quasi linéaire à données petites en dimension 1. (French), Ann. Sci. l'Ecole Norm. Sup., 34 (2001), 1-61. [2] J-M. Delort, D. Fang and R. Xue, Global existence of small solutions for quadratic quasilinear Klein-Gordon systems in two space dimensions, J. Funct. Anal., 211 (2004), 288-323. [3] V. Georgiev, Decay estimates for the Klein-Gordon equation, Comm. Part. Diff. Eq., 17 (1992), 1111-1139. [4] V. Georgiev and S. Lecente, Weighted Sobolev spaces applied to nonlinear Klein-Gordon equation, C. R. Acad. Sci. Paris Sér. I Math., 329 (1999), 21-26. [5] V. Georgiev and B. Yardanov, Asymptotic behavior of the one dimensional Klein-Gordon equation with a cubic nonlinearity, preprint, (1996). [6] J. Ginibre and T. Ozawa, Long range scattering for nonlinear Schrödinger and Hartree equations in space dimension n ≥ 2, Comm. Math. Phys., 151 (1993), 619-645. [7] R. T. Glassey, On the asymptotic behavior of nonlinear wave equations, Trans. Amer. Math. Soc., 182 (1973), 187-200. [8] N. Hayashi and P. I. Naumkin, The initial value problem for the cubic nonlinear Klein-Gordon equation, Z. Angew. Math. Phys., 59 (2008), 1002-1028. [9] N. Hayashi and P. I. Naumkin, Scattering operator for nonlinear Klein-Gordon equations in higher space dimensions, J. Differential Equations, 244 (2008), 188-199. [10] N. Hayashi and P. I. Naumkin, Final state problem for the cubic nonlinear Klein-Gordon equation, J. Math. Phys., 50 (2009), 103511-14 pp. [11] N. Hayashi and P. I. Naumkin, Scattering operator for nonlinear Klein-Gordon equations, Commun. Contemp. Math., 11 (2009), 771-781. [12] L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, in: Mathématiques et Applications, 26, Springer, Berlin, 1997. [13] S. Katayama, A note on global existence of solutions to nonlinear Klein-Gordon equations in one space dimension, J. Math. Kyoto Univ., 39 (1999), 203-213. [14] S. Katayama, T. Ozawa and H. Sunagawa, A note on the null condition for quadratic nonlinear Klein-Gordon systems in two space dimensions, Comm. Pure Appl. Math., 65 (2012), 1285-1302. [15] Y. Kawahara and H. Sunagawa, Global small amplitude solutions for two-dimensional nonlinear Klein-Gordon systems in the presence of mass resonance, J. Differential Equations, 251 (2011), 2549-2567. [16] M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. [17] S. Klainerman, Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions, Comm. Pure Appl. Math., 38 (1985), 631-641. [18] H. Lindblad and A. Soffer, A remark on long range scattering for the nonlinear Klein-Gordon equation, J. Hyperbolic Differ. Equ., 1 (2005), 77-89. [19] H. Lindblad and A. Soffer, A remark on asymptotic completeness for the critical nonlinear Klein-Gordon equation, Lett. Math. Phys., 73 (2005), 249-258. [20] B. Marshall, W. Strauss and S. Wainger, Lp-Lq estimates for the Klein-Gordon equation, J. Math. Pures Appl., 59 (1980), 417-440. [21] S. Masaki and H. Miyazaki, Long range scattering for nonlinear Schrödinger equations with critical homogeneous nonlinearity, preprint available at arXiv: 1612.04524. [22] S. Masaki, H. Miyazaki and K. Uriya, Long range scattering for nonlinear Schrödinger equations with critical homogeneous nonlinearity in three space dimension, preprint. [23] S. Masaki and J. Segata, Existence of a minimal non-scattering solution to the mass-subcritical generalized Korteweg-de Vries equation, to appear in Annales de l'Institut Henri Poincare (C) Non Linear Analysis, preprint available at arXiv: 1602.05331. [24] S. Masaki and J. Segata, Modified scattering for the quadratic nonlinear Klein-Gordon equation in two dimensions, to appear in Trans. AMS, preprint available at arXiv: 1612.00109. [25] A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci., 12 (1976/77), 169-189. [26] K. Moriyama, Normal forms and global existence of solutions to a class of cubic nonlinear Klein-Gordon equations in one space dimension, Differential Integral Equations, 10 (1997), 499-520. [27] K. Moriyama, S. Tonegawa and Y. Tsutsumi, Wave operators for the nonlinear Schrödinger equation with a nonlinearity of low degree in one or two space dimensions, Commun. Contemp. Math., 5 (2003), 983-996. [28] T. Ozawa, K. Tsutaya and Y. Tsutsumi, Global existence and asymptotic behavior of solutions for the Klein-Gordon equations with quadratic nonlinearity in two space dimensions, Math. Z., 222 (1996), 341-362. [29] H. Pecher, Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z., 185 (1984), 261-270. [30] H. Pecher, Low energy scattering for nonlinear Klein-Gordon equations, J. Funct. Anal., 63 (1985), 101-122. [31] J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math., 38 (1985), 685-696. [32] A. Shimomura and S. Tonegawa, Long-range scattering for nonlinear Schrödinger equations in one and two space dimensions, Differential Integral Equations, 17 (2004), 127-150. [33] W. A. Strauss, Nonlinear scattering theory at low energy, J. Funct. Anal., 41 (1981), 110-133. [34] H. Sunagawa, Large time behavior of solutions to the Klein-Gordon equation with nonlinear dissipative terms, J. Math. Soc. Japan, 58 (2006), 379-400. [35] H. Sunagawa, Remarks on the asymptotic behavior of the cubic nonlinear Klein-Gordon equations in one space dimension, Differential Integral Equations, 18 (2005), 481-494. [36] K. Yajima, Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys., 110 (1987), 415-426.
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2017 Impact Factor: 0.884 | 2018-12-11 17:00:22 | {"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": 0.7967531681060791, "perplexity": 1442.3933252264426}, "config": {"markdown_headings": true, "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-2018-51/segments/1544376823657.20/warc/CC-MAIN-20181211151237-20181211172737-00482.warc.gz"} |
http://physics.stackexchange.com/questions/15081/what-causes-a-gyroscope-to-eventually-rotate-fall-over | # What causes a gyroscope to eventually rotate/fall over?
Hey so I've just learned about angular velocity and momentum and how torque changes it.
Looking at a wheel spinning around an axis, with one end being held up by a rope, what causes the wheel to rotate downwards over time, and eventually fall?
-
I didn't saw it rotating backwards, the only thing which i saw was that the wheel stops after some time, that can be easily attributed to the friction the ball bearing offers.... – Vineet Menon Sep 25 '11 at 12:19
## 1 Answer
Friction. The wheel is held up due to the effects of a torque on the angular momentum of the wheel; when the wheel no longer has angular momentum (because it stops spinning due to friction) precession no longer occurs. Wikipedia has more information on precession, but I think this MIT video gives a better intuition.
- | 2014-09-21 16:10:26 | {"extraction_info": {"found_math": false, "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": 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": 0.9381765723228455, "perplexity": 366.91350653774197}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "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-2014-41/segments/1410657135597.56/warc/CC-MAIN-20140914011215-00049-ip-10-234-18-248.ec2.internal.warc.gz"} |
http://en.m.wikibooks.org/wiki/Partial_Differential_Equations/Transport_equation | # Partial Differential Equations/Transport equation
Partial Differential Equations
← Introduction and first examples Transport equation Test functions →
In the first chapter, we had already seen the one-dimensional transport equation. In this chapter we will see that we can quite easily generalise the solution method and the uniqueness proof we used there to multiple dimensions. Let $d \in \mathbb N$. The inhomogenous $d$-dimensional transport equation looks like this:
$\forall (t, x) \in \mathbb R \times \mathbb R^d : \partial_t u(t, x) - \mathbf v \cdot \nabla_x u(t, x) = f(t, x)$
, where $f: \mathbb R \times \mathbb R^d \to \mathbb R$ is a function and $\mathbf v \in \mathbb R^d$ is a vector.
## SolutionEdit
The following definition will become a useful shorthand notation in many occasions. Since we can use it right from the beginning of this chapter, we start with it.
Definition 2.1:
Let $f : \mathbb R^d \to \mathbb R$ be a function and $n \in \mathbb N$. We say that $f$ is $n$ times continuously differentiable iff all the partial derivatives
$\partial_\alpha f, \alpha \in \mathbb N_0^d \text{ and } |\alpha| \le n$
exist and are continuous. We write $f \in \mathcal C^n(\mathbb R^d)$.
Before we prove a solution formula for the transport equation, we need a theorem from analysis which will play a crucial role in the proof of the solution formula.
Theorem 2.2: (Leibniz' integral rule)
Let $O \subseteq \mathbb R$ be open and $B \subseteq \mathbb R^d$, where $d \in \mathbb N$ is arbitrary, and let $f \in \mathcal C^1 (O \times B)$. If the conditions
• for all $x \in O$, $\int_B |f(x, y)| dy < \infty$
• for all $x \in O$ and $y \in B$, $\frac{d}{dx} f(x, y)$ exists
• there is a function $g : B \to \mathbb R$ such that
$\forall (x, y) \in O \times B : |\partial_x f(x, y)| \le |g(y)| \text{ and } \int_B |g(y)| dy < \infty$
hold, then
$\frac{d}{dx} \int_B f(x, y) dy = \int_B \frac{d}{dx} f(x, y)$
We will omit the proof.
Theorem 2.3: If $f \in \mathcal C^1 (\mathbb R \times \mathbb R^d)$, $g \in \mathcal C^1(\mathbb R^d)$ and $\mathbf v \in \mathbb R^d$, then the function
$u : \mathbb R \times \mathbb R^d \to \mathbb R, u(t, x) := g(x + \mathbf vt) + \int_0^t f(s, x + \mathbf v(t - s)) ds$
solves the inhomogenous $d$-dimensional transport equation
$\forall (t, x) \in \mathbb R \times \mathbb R^d : \partial_t u(t, x) - \mathbf v \cdot \nabla_x u(t, x) = f(t, x)$
Note that, as in chapter 1, that there are many solutions, one for each continuously differentiable $g$ in existence.
Proof:
1.
We show that $u$ is sufficiently often differentiable. From the chain rule follows that $g(x + \mathbf vt)$ is continuously differentiable in all the directions $t, x_1, \ldots, x_d$. The existence of
$\partial_{x_n} \int_0^t f(s, x + \mathbf v(t - s)) ds, n \in \{1, \ldots, d\}$
follows from the Leibniz integral rule (see exercise 1). The expression
$\partial_t \int_0^t f(s, x + \mathbf v(t - s)) ds$
we will later in this proof show to be equal to
$f(t, x) + \mathbf v \cdot \nabla_x \int_0^t f(s, x + \mathbf v(t - s)) ds$,
which exists because
$\nabla_x \int_0^t f(s, x + \mathbf v(t - s)) ds$
just consists of the derivatives
$\partial_{x_n} \int_0^t f(s, x + \mathbf v(t - s)) ds, n \in \{1, \ldots, d\}$
2.
We show that
$\forall (t, x) \in \mathbb R \times \mathbb R^d : \partial_t u(t, x) - \mathbf v \cdot \nabla_x u(t, x) = f(t, x)$
in three substeps.
2.1
We show that
$\partial_t g(x + \mathbf vt) - \mathbf v \cdot \nabla_x g(x + \mathbf vt) = 0 ~~~~~ (*)$
This is left to the reader as an exercise in the application of the multi-dimensional chain rule (see exercise 2).
2.2
We show that
$\partial_t \int_0^t f(s, x + \mathbf v(t - s)) ds - \mathbf v \cdot \nabla_x \int_0^t f(s, x + \mathbf v(t - s)) ds = f(t, x) ~~~~~ (**)$
We choose
$F(t, x) := \int_0^t f(s, x - \mathbf vs) ds$
so that we have
$F(t, x + \mathbf vt) = \int_0^t f(s, x + \mathbf v(t - s)) ds$
By the multi-dimensional chain rule, we obtain
\begin{align} \frac{d}{dt} F(t, x + \mathbf vt) &= \begin{pmatrix} \partial_t F (t, x + \mathbf vt) & \partial_{x_1} F (t, x + \mathbf vt) & \cdots & \partial_{x_d} F(t, x + \mathbf vt) \end{pmatrix} \begin{pmatrix} 1 \\ \mathbf v \end{pmatrix} \\ &= \partial_t F (t, x + \mathbf vt) + \mathbf v \cdot \nabla_x F (t, x + \mathbf vt) \end{align}
But on the one hand, we have by the fundamental theorem of calculus, that $\partial_t F (t, x) = f(t, x - \mathbf vt)$ and therefore
$\partial_t F (t, x + \mathbf vt) = f(t, x)$
and on the other hand
$\partial_{x_n} F(t, x + \mathbf vt) = \partial_{x_n} \int_0^t f(s, x + \mathbf v(t - s)) ds$
, seeing that the differential quotient of the definition of $\partial_{x_n}$ is equal for both sides. And since on the third hand
$\frac{d}{dt} F(t, x + \mathbf vt) = \partial_t \int_0^t f(s, x + \mathbf v(t - s)) ds$
, the second part of the second part of the proof is finished.
2.3
We add $(*)$ and $(**)$ together, use the linearity of derivatives and see that the equation is satisfied.
$////$
## Initial value problemEdit
Theorem and definition 2.4: If $f \in \mathcal C^1 (\mathbb R \times \mathbb R^d)$ and $g \in \mathcal C^1(\mathbb R^d)$, then the function
$u : \mathbb R \times \mathbb R^d \to \mathbb R, u(t, x) := g(x + \mathbf vt) + \int_0^t f(s, x + \mathbf v(t - s)) ds$
is the unique solution of the initial value problem of the transport equation
$\begin{cases} \forall (t, x) \in \mathbb R \times \mathbb R^d : & \partial_t u(t, x) - \mathbf v \cdot \nabla_x u(t, x) = f(t, x) \\ \forall x \in \mathbb R^d : & u(0, x) = g(x) \end{cases}$
Proof:
Quite easily, $u(0, x) = g(x + \mathbf v \cdot 0) + \int_0^0 f(s, x + \mathbf v(t - s)) ds = g(x)$. Therefore, and due to theorem 2.3, $u$ is a solution to the initial value problem of the transport equation. So we proceed to show uniqueness.
Assume that $v$ is an arbitrary other solution. We show that $v = u$, thereby excluding the possibility of a different solution.
We define $w := u - v$. Then
$\begin{array}{llll} \forall (t, x) \in \mathbb R \times \mathbb R^d : & \partial_t w (t, x) - \mathbf v \cdot \nabla_x w (t, x) &= (\partial_t u (t, x) - \mathbf v \cdot \nabla_x u (t, x)) - (\partial_t v (t, x) - \mathbf v \cdot \nabla_x v (t, x)) & \\ &&= f(t, x) - f(t, x) = 0 & ~~~~~(*) \\ \forall x \in \mathbb R^d : & w(0, x) = u(0, x) - v(0, x) &= g(x) - g(x) = 0 & ~~~~~(**) \end{array}$
Analogous to the proof of uniqueness of solutions for the one-dimensional homogenous initial value problem of the transport equation in the first chapter, we define for arbitrary $(t, x) \in \mathbb R \times \mathbb R^d$,
$\mu_{(t, x)}(\xi) := w(t - \xi, x + \mathbf v \xi)$
Using the multi-dimensional chain rule, we calculate $\mu_{(t, x)}'(\xi)$:
\begin{align} \mu_{(t, x)}'(\xi) &:= \frac{d}{d\xi} w(t - \xi, x + \mathbf v \xi) & \text{ by defs. of the }' \text{ symbol and } \mu\\ &= \begin{pmatrix} \partial_t w (t - \xi, x + \mathbf v \xi) & \partial_{x_1} w (t - \xi, x + \mathbf v \xi) & \cdots & \partial_{x_d} w (t - \xi, x + \mathbf v \xi) \end{pmatrix} \begin{pmatrix} -1 \\ \mathbf v \end{pmatrix} & \text{chain rule} \\ &= -\partial_t w (t - \xi, x + \mathbf v \xi) + \mathbf v \cdot \nabla_x w (t - \xi, x + \mathbf v \xi) & \\ & = 0 & (*) \end{align}
Therefore, for all $(t, x) \in \mathbb R \times \mathbb R^d$ $\mu_{(t, x)}(\xi)$ is constant, and thus
$\forall (t, x) \in \mathbb R \times \mathbb R^d : w(t, x) = \mu_{(t, x)}(0) = \mu_{(t, x)}(t) = w(0, x + \mathbf v t) \overset{(**)}{=} 0$
, which shows that $w = u - v = 0$ and thus $u=v$.
$////$
## ExercisesEdit
1. Let $f \in \mathcal C^1 (\mathbb R \times \mathbb R^d)$ and $\mathbf v \in \mathbb R^d$. Using Leibniz' integral rule, show that for all $n \in \{1, \ldots, d\}$ the derivative
$\partial_{x_n} \int_0^t f(s, x + \mathbf v(t - s)) ds$
is equal to
$\int_0^t \partial_{x_n} f(s, x + \mathbf v(t - s)) ds$
and therefore exists.
2. Let $g \in \mathcal C^1 (\mathbb R^d)$ and $\mathbf v \in \mathbb R^d$. Calculate
$\partial_t g(x + \mathbf vt)$
## SourcesEdit
Partial Differential Equations
← Introduction and first examples Transport equation Test functions → | 2014-12-20 16:31:02 | {"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": 0, "img_math": 80, "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": 0.999427080154419, "perplexity": 380.5521707344631}, "config": {"markdown_headings": true, "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-2014-52/segments/1418802770043.48/warc/CC-MAIN-20141217075250-00053-ip-10-231-17-201.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/charge-on-capacitor-problem.846105/ | # Charge on capacitor problem
1. Dec 2, 2015
### gracy
1. The problem statement, all variables and given/known data
find the charge on each plate given area is a and separation between two consecutive plate is d
2. Relevant equations
$Q$=$C$$V$
3. The attempt at a solution
I don't know how to proceed from here.
Last edited: Dec 2, 2015
2. Dec 2, 2015
### Staff: Mentor
You need another relevant equation. What is the capacitance of a parallel plate capacitor with plate area $a$ and plate separation $d$?
3. Dec 2, 2015
### gracy
$C$=$\frac{aε0}{d}$
4. Dec 2, 2015
### Staff: Mentor
Yes. Proceed.
5. Dec 2, 2015
### gracy
Q=C$\frac{aε0}{d}$
Will this be charge on each plate?
6. Dec 2, 2015
### Staff: Mentor
No. That has two capacitance terms. You need a voltage and a capacitance: Q = CV.
7. Dec 2, 2015
### gracy
Haste makes waste!
Q=V$\frac{aε0}{d}$
Will this be charge on each plate?
Last edited: Dec 2, 2015
8. Dec 2, 2015
### Staff: Mentor
Still too hasty. You've written the same thing again
Each capacitor plate in your circuit diagram will have the same magnitude of charge on it. You'll have to sort out the signs of the charges. Then determine how to locate those charges on the plates of the original figure -- remember, one physical plate is shared between two capacitors.
9. Dec 2, 2015
### gracy
But hint says plate B would have different charge.
10. Dec 2, 2015
### Staff: Mentor
Yes. I presume "plate B" is the middle plate of the physical setup? You didn't label your diagram.
Note that I said: "Each capacitor plate in your circuit diagram will have the same magnitude of charge on it". There is a distinction between the circuit diagram and the physical arrangement in that the circuit diagram shows two plates in two separate capacitors whereas the physical setup uses a shared plate. Re-read the entire contents of my post #8.
11. Dec 2, 2015
### gracy
12. Dec 2, 2015
### gracy
Should not plate B cntain zero charge as it is earthed?
13. Dec 2, 2015
### Staff: Mentor
No, "earth" is a vast pool of available charges, both positive and negative and overall neutral. It can supply whatever charge is required to respond to electrostatic forces. Or, if you like, it can source or sink any amount of electrons, if you want to go with an atomic model for the plates and conductors.
Your charge placement is not correct. Note that plates A and C are symmetrical and connected to the same potential (V). They should end up with similar charges distributed in the same way. Charges are made available to those plates via their connection to V. So far I think we've been assuming that V is a positive potential with respect to ground (earth), which is fine, but you should state that assumption in your solution.
14. Dec 2, 2015
### gracy
I did not understand that.
15. Dec 2, 2015
### Staff: Mentor
V is taken to be a voltage supply. Voltage is a potential difference. When a terminal is just labeled "V" then it is assumed that that potential difference is between the terminal and the ground reference.
16. Dec 2, 2015
### gracy
I don't quite get the relationship between being symmetrical and connected to the same potential and having same charge distribution.
I distributed charge by only considering charge induction.
17. Dec 2, 2015
### cnh1995
Plates 2 and 3 along with the wire joining them, form plate B in your original question.
That is for one capacitor(1-2 or 3-4). The capacitors are in parallel. So you can find the total charge now.
18. Dec 2, 2015
### Staff: Mentor
If you rotate the drawing on the page, do you expect the charges on the plates to change? Is there any distinction (electrically) between the two outer plates?
You'll have to explain how you ended up with different distributions on the two outer plates.
19. Dec 2, 2015
### cnh1995
A and C are connected to same potential V and you have shown net +ve charge on A and other plates neutral. Why A? Why not C?
20. Dec 2, 2015
### gracy
Is it correct now? | 2017-08-21 21:39:56 | {"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": 0.44430914521217346, "perplexity": 1254.2300254541176}, "config": {"markdown_headings": true, "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-2017-34/segments/1502886109670.98/warc/CC-MAIN-20170821211752-20170821231752-00516.warc.gz"} |
http://www.ipam.ucla.edu/abstract/?tid=8514&pcode=KTWS4 | ## Proof of the kinetic conjecture in a weakly nonlinear Schrodinger equation with random initial data
#### Jani LukkarinenSuomen Akatemia (Academy of Finland)
We report on first progress in rigorous control of a kinetic scaling limit of a weakly nonlinear perturbation of wave-type evolution, here a discrete Schrodinger equation. Since we consider a Hamiltonian system, a natural choice of random initial data is distributing them according to a Gibbs measure with a chemical potential chosen so that the Gibbs field has exponential mixing. The solution psi_t(x) of the nonlinear Schrodinger equation yields then a stochastic process stationary in x in Z^d and t in R. If lambda denotes the strength of the nonlinearity, we prove that the space-time covariance of psi_t(x) has a limit as lambda->0 for t= lambda^(-2) tau, with tau fixed and
|tau| sufficiently small. The limit agrees with the prediction from
kinetic theory. The talk is based on a joint work with Herbert Spohn [J. Lukkarinen and H. Spohn, Weakly nonlinear Schroedinger equation with random initial data, preprint arXiv:0901.3283].
Back to Workshop IV: Asymptotic Methods for Dissipative Particle Systems | 2018-09-21 04:41:07 | {"extraction_info": {"found_math": false, "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": 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": 0.8576251268386841, "perplexity": 1430.2185430172747}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "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-2018-39/segments/1537267156780.2/warc/CC-MAIN-20180921033529-20180921053929-00378.warc.gz"} |
https://slideplayer.com/slide/4514062/ | # Functions AII.7 e 2009. Objectives: Find the Vertical Asymptotes Find the Horizontal Asymptotes.
## Presentation on theme: "Functions AII.7 e 2009. Objectives: Find the Vertical Asymptotes Find the Horizontal Asymptotes."— Presentation transcript:
Functions AII.7 e 2009
Objectives: Find the Vertical Asymptotes Find the Horizontal Asymptotes
Rational Functions A rational function can have more than one vertical asymptote, but it can have at most one horizontal asymptote. A rational function f ( x ) is a function that can be written as where p ( x ) and q ( x ) are polynomial functions and q ( x ) 0.
Vertical Asymptotes If p ( x ) and q ( x ) have no common factors, then f ( x ) has vertical asymptote(s) when q ( x ) = 0. Thus the graph has vertical asymptotes at the zeros of the denominator.
Since the zeros are 1 and -1. Thus the vertical asymptotes are x = 1 and x = -1. Vertical Asymptotes Find the vertical asymptote of Example: V.A. is x = a, where a represents real zeros of q ( x ).
Horizontal Asymptotes The horizontal asymptote is determined by looking at the degrees of p ( x ) and q ( x ). A rational function f ( x ) is a function that can be written as where p ( x ) and q ( x ) are polynomial functions and q ( x ) 0.
Horizontal Asymptotes a.If the degree of p ( x ) is less than the degree of q ( x ), then the horizontal asymptote is y = 0. b. If the degree of p ( x ) is equal to the degree of q ( x ), then the horizontal asymptote is c. If the degree of p ( x ) is greater than the degree of q ( x ), then there is no horizontal asymptote.
Horizontal Asymptotes deg of p ( x ) < deg of q ( x ), then H.A. is y = 0 deg of p ( x ) = deg of q ( x ), then H.A. is deg of p ( x ) > deg of q ( x ), then no H.A. Example : Find the horizontal asymptote: S ince the degree of the numerator is less than the degree of the denominator, horizontal asymptote is y = 0. Degree of numerator = 1 Degree of denominator = 2
Horizontal Asymptotes deg of p ( x ) < deg of q ( x ), then H.A. is y = 0 deg of p ( x ) = deg of q ( x ), then H.A. is deg of p ( x ) > deg of q ( x ), then no H.A. Example : Find the horizontal asymptote: Degree of numerator = 1 Degree of denominator = 1 S ince the degree of the numerator is equal to the degree of the denominator, horizontal asymptote is.
S ince the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. Horizontal Asymptotes deg of p ( x ) < deg of q ( x ), then H.A. is y = 0 deg of p ( x ) = deg of q ( x ), then H.A. is deg of p ( x ) > deg of q ( x ), then no H.A. Example : Find the horizontal asymptote: Degree of numerator = 2 Degree of denominator = 1
Vertical & Horizontal Asymptotes deg of p ( x ) < deg of q ( x ), then H.A. is y = 0 deg of p ( x ) = deg of q ( x ), then H.A. is deg of p ( x ) > deg of q ( x ), then no H.A. Practice : Find the vertical and horizontal asymptotes: V.A. : x = a, where a represents real zeros of q ( x ). H.A. : Answer Now
Vertical & Horizontal Asymptotes Practice : Find the vertical and horizontal asymptotes: V.A. : x = a, where a represents real zeros of q ( x ). deg of p ( x ) < deg of q ( x ), then H.A. is y = 0 deg of p ( x ) = deg of q ( x ), then H.A. is deg of p ( x ) > deg of q ( x ), then no H.A. H.A. : V.A. : x = H.A.: none
Vertical & Horizontal Asymptotes deg of p ( x ) < deg of q ( x ), then H.A. is y = 0 deg of p ( x ) = deg of q ( x ), then H.A. is deg of p ( x ) > deg of q ( x ), then no H.A. Practice : Find the vertical and horizontal asymptotes: V.A. : x = a, where a represents real zeros of q ( x ). H.A. : Answer Now
Vertical & Horizontal Asymptotes deg of p ( x ) < deg of q ( x ), then H.A. is y = 0 deg of p ( x ) = deg of q ( x ), then H.A. is deg of p ( x ) > deg of q ( x ), then no H.A. Practice : Find the vertical and horizontal asymptotes: V.A. : x = a, where a represents real zeros of q ( x ). H.A. : V.A. : none H.A.: y = 0 is not factorable and thus has no real roots.
Vertical & Horizontal Asymptotes deg of p ( x ) < deg of q ( x ), then H.A. is y = 0 deg of p ( x ) = deg of q ( x ), then H.A. is deg of p ( x ) > deg of q ( x ), then no H.A. Practice : Find the vertical and horizontal asymptotes: V.A. : x = a, where a represents real zeros of q ( x ). H.A. : Answer Now
Vertical & Horizontal Asymptotes deg of p ( x ) < deg of q ( x ), then H.A. is y = 0 deg of p ( x ) = deg of q ( x ), then H.A. is deg of p ( x ) > deg of q ( x ), then no H.A. Practice : Find the vertical and horizontal asymptotes: V.A. : x = a, where a represents real zeros of q ( x ). H.A. : V.A. : x = -1 H.A.: y = 2
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https://www.whsmith.co.uk/products/function-spaces-third-conference-function-spaces-third-conference-third-conference-contemporary-mathematics/9780821809396 | # Function Spaces Third Conference: Function Spaces Third Conference Third Conference (Contemporary Mathematics)
By: Krzysztof Jarosz (editor)Paperback
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### Description
This proceedings volume presents 36 papers given by leading experts during the Third Conference on Function Spaces held at Southern Illinois University at Edwardsville. A wide range of topics in the subject area are covered. Most papers are written for nonexperts, so the book can serve as a good introduction to the topic for those interested in this area. The book presents the following broad range of topics, including spaces and algebras of analytic functions of one and of many variables, $L^p$ spaces, spaces of Banach-valued functions, isometries of function spaces, geometry of Banach spaces and related subjects. Known results, open problems, and new discoveries are featured. At the time of publication, information about the book, the conference, and a list and pictures of contributors are available on the Web.
### Contents
Norm attaining operators in $L 1(\mu)$ by M. D. Acosta The Weyl-Browder spectrum of a multiplier by P. Aiena On polynomial approximation in the mean by J. Akeroyd and E. G. Saleeby A uniqueness theorem for normal functions of several complex variables by J. T. Anderson and J. A. Cima Separating maps on spaces of continuous functions by J. Araujo and K. Jarosz Hypercyclic differentiation operators by R. Aron and J. Bes On norms of composition operators on Hardy spaces by P. Avramidou and F. Jafari Stable ranks, $K$-groups and Witt groups of some Banach and $C^*$-algebras by C. Badea Locally constant almost everywhere Fourier transform by A. Bernard and G. Muraz The quaternionic Riemann problem by S. Bernstein Convolution by means of bilinear maps by O. Blasco On prime real $JB^*$-triples by C.-H. Chu, A. M. Galindo, and A. Rodriguez Palacios Compact-type operators defined on $H^\infty$ by M. D. Contreras and S. Diaz-Madrigal On the extensibility of certain homeomorphisms and linear isometries by S. J. Dilworth The fixed point property for subsets of $L 1[0,1]$ by P. N. Dowling Strong regularity for uniform algebras by J. F. Feinstein and D. W. B. Somerset High order smoothness in sequence spaces and spreading models by R. Gonzalo and J. A. Jaramillo A survey of closed ideals in familiar function algebras by P. Gorkin and R. Mortini Subalgebras of commutative Banach algebras and Fourier multipliers with natural spectra by O. Hatori Bergman kernels and Hankel forms on generalized Fock spaces by F. Holland and R. Rochberg When is a linear functional multiplicative? by K. Jarosz A remark on Banach spaces isomorphic to their squares by N. J. Kalton Sobolev spaces and projections of holomorphic functions and mappings by S. G. Krantz and M. M. Peloso Universal functions on the unit ball and the polydisk by F. Leon-Saavedra On commutative $C^*$-algebras in which every element is almost the square of another by T. Miura Some special bounded homomorphisms of a uniform algebra by T. Nakazi Analytic functional models for operators on Banach spaces by M. M. Neumann Injective isometries in Orlicz spaces by B. Randrianantoanina Continuity of homomorphisms and derivations on Banach algebras with an involution by A. Rodriguez-Palacios and M. V. Velasco A Volterra type operator on spaces of analytic functions by A. G. Siskakis and R. Zhao Algebraic properties of Toeplitz operators on the Hardy space via the Berezin transform by K. Stroethoff Multiplication and composition operators between two $L^p$-spaces by H. Takagi and K. Yokouchi Bourgain algebras and inductive limit algebras by T. Tonev A note on averaging operators by A. Triki Problems on isometries of non-commutative $L^p$-spaces by K. Watanabe Multiplicative polynomial operators on topological algebras by A. Zagorodnyuk.
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• ISBN13: 9780821809396
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Close | 2018-09-19 23:11:50 | {"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": 0.6503453850746155, "perplexity": 2211.978064916657}, "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-2018-39/segments/1537267156311.20/warc/CC-MAIN-20180919220117-20180920000117-00403.warc.gz"} |
https://www.gamedev.net/forums/topic/688333-horror-reading-shaders-without-defined-default-precision-es-3/ | Recommended Posts
Basically i have to rewrite 120 shaders and in all cases i add "highp " before...
Maybe theres other way around? : )
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These are the kind of things where regular expressions come in handy. On linux I would probably use something like sed, but many editors and programming languages nowadays support regex as well. If you're not comfortable with regex you could always write a simple script in the language of your choice.
However the bottom line is always; can you whip it up quicker than just going into zombie mode and cut&paste the damn thing?
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Ok I guess I need to write shader compile error parser and apply fixes
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precision highp float; // Changes all floats to be default highp unless manually specified otherwise.
L. Spiro
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Register a new account | 2018-01-22 11:05:52 | {"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": 0.2632099688053131, "perplexity": 4648.191347462721}, "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-2018-05/segments/1516084891277.94/warc/CC-MAIN-20180122093724-20180122113724-00267.warc.gz"} |
https://www.gradesaver.com/textbooks/science/physics/CLONE-afaf42be-9820-4186-8d76-e738423175bc/chapter-8-exercises-and-problems-page-147/16 | Essential University Physics: Volume 1 (4th Edition) Clone
We know that the gravitational force is given as $F=\frac{GMm}{r^2}$ We plug in the known values to obtain: $F=\frac{6.67\times 10^{-11}(67)(73000)}{(84)^2}$ $F=4.6\times 10^{-8}N$ | 2019-10-16 20:37:03 | {"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": 0.9150019884109497, "perplexity": 292.9294030597536}, "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-2019-43/segments/1570986669546.24/warc/CC-MAIN-20191016190431-20191016213931-00517.warc.gz"} |
http://algebra2014.wikidot.com/permutation | Permutation
Return to Glossary.
Formal Definition
A permutation of a set $A$ is a function $\phi : A\rightarrow A$ that is both one to one and onto.
Informal Definition
A permutation of $A$ is a one-to-one function from $A$ onto $A$.
Example(s)
Replace this text with examples
Non-example(s)
Replace this text with non-examples
Additional Comments
Add any other comments you have about the term here
Unless otherwise stated, the content of this page is licensed under Creative Commons Attribution-ShareAlike 3.0 License | 2021-01-25 08:23:25 | {"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": 0.2526649534702301, "perplexity": 1074.9705757736021}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "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-2021-04/segments/1610703565376.63/warc/CC-MAIN-20210125061144-20210125091144-00301.warc.gz"} |
https://physics.stackexchange.com/questions/468370/buoyancy-question | # Buoyancy question
Suppose we have a ball of volume V and a block of the same volume V whith same density. We submerge the ball into some kind of liquid so that $$\frac{1}{2}V$$ is submerged into the liquid. We do the same thing with the block. Now $$\frac{1}{2}V$$ of both block and ball is submerged into the liquid. At this moment we can say that the buoyant force on both objects is the same. If we now push both the ball and the block by little bit into the liquid by the same volume, the force acting on the bottom of the ball would increase (because it is now a little deeper in the liquid) and so will the force acting on the bottom of the block. If we now look at the geometry of the objects, wouldn't there be additional force on the ball pushing it downward because of the curvature of the ball which wouldn't appear on the block? Can we still say the buoyant force is the same on both objects? I added a picture and highlighted the area where I think the force would act in red.
• All that really needs to be proven is that, based on the pressure distribution on the submerged surface of the sphere, the buoyant force is equal to the submerged volume times the density of water times g (in agreement with Archimedes principle). Would you like me to prove that? – Chet Miller Mar 26 at 18:55
• Thank you, but you don't have to prove the Archimedes principle. – ToTheSpace 2 Mar 26 at 20:12
The buoyant force will remain the same, The buoyant force is the net force that the liquid exerts on the body, this includes the force which acts on the bottom and the top(the red portion).
We can prove this by using Archimedes Principle, which states that the force of buoyancy is equal to the weight of fluid displaced by an object. Here, both the sphere and cube displace the same amount of liquid and hence will have same force of buoyancy acting on it.
• Proving it by manually finding the force of buoyancy (using integration) for the sphere will be a quite troublesome process ,which is why I kept it simple and used Archimedes Principle. – Vaishakh Sreekanth Menon Mar 24 at 12:37
• Yes, but the OP seems to know that already. The question that the OP is asking is precisely about the apparent mismatch between the result that we know from the Archimedes principle to be true and the result that the OP seems to get when they apply the first-principles arguments to the specific case that they are discussing. – Feynmans Out for Grumpy Cat Mar 24 at 13:13
• Oh, my bad. Shall I delete this answer? – Vaishakh Sreekanth Menon Mar 24 at 13:21
• Well, I am not sure. I would wait to see what the OP thinks. – Feynmans Out for Grumpy Cat Mar 25 at 1:51
• Thank you for giving me a general explanation, but as Dvij Mankad said, I already knew Archimedes principle. I wanted to see how other people can come up with an apprehensible explanation as to why the buoyancy is the same although it can't be directly concluded. – ToTheSpace 2 Mar 26 at 14:44
The pressure on all parts of the sphere below $$XY$$ increases by $$h\rho g$$ when in the new position $$X'Y'$$ and those parts all contribute to an increase in the net force upwards.
The section of the sphere $$X''Y''Y'X'$$ which was originally in the air is now in the liquid with the change in pressure on that surface ranging from $$0$$ along $$X''Y''$$ to $$h\rho g$$ along $$X'Y'$$ and the surface area which has become immersed is much less than the surface area already in the liquid. | 2019-07-23 09:39:18 | {"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": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 10, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6875017881393433, "perplexity": 251.0202600233906}, "config": {"markdown_headings": true, "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-2019-30/segments/1563195529175.83/warc/CC-MAIN-20190723085031-20190723111031-00372.warc.gz"} |
http://unapologetic.wordpress.com/2010/09/21/ | # The Unapologetic Mathematician
## Morphisms Between Representations
Since every representation of $G$ is a $G$-module, we have an obvious notion of a morphism between them. But let’s be explicit about it.
A $G$-morphism from a $G$-module $(V,\rho_V)$ to another $G$-module $(W,\rho_W)$ is a linear map $T:V\to W$ between the vector spaces $V$ and $W$ that commutes with the actions of $G$. That is, for every $g\in G$ we have $\rho_W(g)\circ T=T\circ\rho_V(g)$. Even more explicitly, if $g\in G$ and $v\in V$, then
$\displaystyle\left[\rho_W(g)\right]\left(T(v)\right)=T\left(\left[\rho_V(g)\right](v)\right)$
We can also express this with a commutative diagram:
For each group element $g\in G$ our representations give us vertical arrows $\rho_V(g):V\to V$ and $\rho_W(g):W\to W$. The linear map $T$ provides horizontal arrows $T:V\to W$. To say that the diagram “commutes” means that if we compose the arrows along the top and right to get a linear map from $V$ to $W$, and if we compose the arrows along the left and bottom to get another, we’ll find that we actually get the same function. In other words, if we start with a vector $v\in V$ in the upper-left and move it by the arrows around either side of the square to get to a vector in $W$, we’ll get the same result on each side. We get one of these diagrams — one of these equations — for each $g\in G$, and they must all commute for $T$ to be a $G$-morphism.
Another common word that comes up in these contexts is “intertwine”, as in saying that the map $T$ “intertwines” the representations $\rho_V$ and $\rho_W$, or that it is an “intertwinor” for the representations. This language goes back towards the viewpoint that takes the representing functions $\rho_V$ and $\rho_W$ to be fundamental, while $G$-morphism tends to be more associated with the viewpoint emphasizing the representing spaces $V$ and $W$.
If, as will usually be the case for the time being, we have a presentation of our group by generators and relations, then we’ll only need to check that $T$ intertwines the actions of the generators. Indeed, if $T$ intertwines the actions of $g$ and $h$, then it intertwines the actions of $gh$. We can see this in terms of diagrams by stacking the diagram for $h$ on top of the diagram for $g$. In terms of equations, we check that
\displaystyle\begin{aligned}\rho_W(gh)\circ T&=\rho_W(g)\circ\rho_W(h)\circ T\\&=\rho_W(g)\circ T\circ\rho_V(h)\\&=T\circ\rho_V(g)\circ\rho_V(h)\\&=T\circ\rho_V(gh)\end{aligned}
So if we’re given a set of generators and we can write every group element as a finite product of these generators, then as soon as we check that the intertwining equation holds for the generators we know it will hold for all group elements.
There are also deep connections between $G$-morphisms and natural transformations, in the categorical viewpoint. Those who are really interested in that can dig into the archives a bit.
September 21, 2010 | 2013-12-11 14:56:58 | {"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": 0, "img_math": 45, "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": 0.9476174712181091, "perplexity": 139.21105343395618}, "config": {"markdown_headings": true, "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-2013-48/segments/1386164037762/warc/CC-MAIN-20131204133357-00066-ip-10-33-133-15.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/270703/xh-cap-yk-is-either-empty-or-a-coset-of-h-cap-k-blanking | # $xH \cap yK$ is either empty or a coset of $H \cap K$: blanking
$H, K$ are sub groups of a group $G$. What I have:
Denote the left cosets $H_i$, $K_j$, and $(H \cap K)_k$, respectively. If $x, y \in H_i \cap K_j$, then $y^{-1}x \in H, K$, and so in $H \cap K$, so $x, y$ are in the same coset $(H \cap K)_k$ for some $k$.
Where I'm blanking: what if $H_i \cap K_j$ has only one element? How to proceed then, and so conclude the proof?
Thanks!
-
Your argument applies no matter how many elements $H_i \cap K_j$ has, so I see no problem there. However, you've only shown that $H_i \cap K_j$ is contained in some coset of $(H \cap K)$, and you haven't said anything about why it covers the entire coset. – Erick Wong Jan 5 '13 at 3:26
@Erick Wong: Good point on the second, and that shouldn't be so hard. But I don't see what you say first: why does it not matter? I have, say, $a = xh = yk$...and then? You are probably right, but could you elaborate? – gnometorule Jan 5 '13 at 3:32
This half of the argument is just trying to show that all of $H_i \cap K_j$ is contained in one coset of $H \cap K$. You've already showed that any two elements belong to the same coset. If there's only one element, there's nothing more to prove (of course it belongs to its own coset of $H \cap K$). – Erick Wong Jan 5 '13 at 3:35
@Erick Wong: Ah, correct. Mind copying and pasting your comments as an answer, so I can accept (even if it strikes you as trivial)? – gnometorule Jan 5 '13 at 3:41
No objections here :). – Erick Wong Jan 5 '13 at 3:42
Your argument applies no matter how many elements $H_i \cap K_j$ has, so I see no problem there. However, you've only shown that $H_i \cap K_j$ is contained in some coset of $H\cap K$, and you haven't said anything about why it covers the entire coset.
In any case, this half of the argument is just trying to show that all of $H_i \cap K_j$ is contained in one coset of $H\cap K$. You've already shown that any two elements belong to the same coset. If there's only one element, there's nothing more to prove (of course it belongs to its own coset of $H\cap K$).
The question is already answered, but note that if $xH \cap yK$ is non-empty, then it contains an element $z$. We have $xH = zH$ and $yK = zK,$ so it suffices to understand the case $x = y = z.$ Now $z(H \cap K) \subseteq zH \cap zK,$ so $zH \cap zK$ contains at least one full coset of $H \cap K.$ On the other hand, if $zh = zk$ for some $h \in H$ and $k \in K,$ then $h = k \in H \cap K$, so $zH \cap zK \subseteq z(H \cap K).$ | 2016-05-05 00:00:20 | {"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": 0.9069061875343323, "perplexity": 146.0919164814335}, "config": {"markdown_headings": true, "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-2016-18/segments/1461860125524.70/warc/CC-MAIN-20160428161525-00145-ip-10-239-7-51.ec2.internal.warc.gz"} |
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http://www.physicsforums.com/showthread.php?s=ef7007aa20ef08a98e4cd8181aeab05e&p=3527367 | # Finding standard deviation or error from normalized data.
by doublemint
Tags: data, deviation, error, normalized, standard
Sci Advisor P: 3,297 What do you mean by "normalize"? For example, do you mean that multiply each datum in the data set x2 by the factor (a2/a1) ? Let the data be the $d_i$. Let the sample mean be $m$ . Let the scaling factor be $k$ The mean of the scaled data $k d_i$ is $m k$ The variance of the scaled data is: $\sum \frac { ( k d_i - m k )^2}{n} = \sum \frac{k^2 (d_i - k)^2 }{n} = k^2 \sum \frac{(d_i - m)^2}{n}$ This is $k^2$ times the variance of the original sample. So the sample standard deviation of the scaled data is $|k|$ times the standard deviation of the original data. | 2014-08-29 22:26:52 | {"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": 0.9228631854057312, "perplexity": 444.52368210090214}, "config": {"markdown_headings": true, "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-2014-35/segments/1408500833461.95/warc/CC-MAIN-20140820021353-00116-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://bmrcbulletin.org.bd/association-of-endometrial-carcinoma-with-obesity-and-diabetes-mellitus/ | # Association of Endometrial Carcinoma with Obesity and Diabetes Mellitus
Background: Endometrial cancer, previously referred to as carcinoma of the uterus. The incidence of endometrial cancer is raising, due to improved screening causing fewer hysterectomies in ageing population. Several studies are going on to find out the association and effects between DM and DM related diseases, especially the cancer. Obesity and physical inactivity plays important role as modifiable determinants of insulin resistance, hyperinsulinaemia and diabetes. All these factors are also responsible for endometrial cancer. | 2023-04-01 08:03:01 | {"extraction_info": {"found_math": false, "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": 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": 0.8392705321311951, "perplexity": 11804.999522727747}, "config": {"markdown_headings": true, "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-2023-14/segments/1679296949701.56/warc/CC-MAIN-20230401063607-20230401093607-00606.warc.gz"} |
https://socratic.org/questions/how-do-you-solve-b-108-7-18 | # How do you solve b/108=7/18?
Nov 27, 2016
$b = 42$
#### Explanation:
To isolate and solve for $b$, multiple each side of the equation by $109$ and then do the necessary mathematics:
$\frac{108 \cdot b}{108} = \frac{7 \cdot 108}{18}$
$\frac{\cancel{108} \cdot b}{\cancel{108}} = \frac{7 \cdot 108}{18}$
$b = \frac{7 \cdot 6 \cdot 18}{18}$
$b = \frac{7 \cdot 6 \cdot \cancel{18}}{\cancel{18}}$
$b = 7 \cdot 6$
$b = 42$ | 2020-08-08 15:23:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 9, "mathjax_inline_tex": 1, "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": 0.9250680208206177, "perplexity": 940.5121737058234}, "config": {"markdown_headings": true, "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-2020-34/segments/1596439737883.59/warc/CC-MAIN-20200808135620-20200808165620-00018.warc.gz"} |
https://mathematica.stackexchange.com/questions/202268/why-do-i-not-see-any-results-from-my-for-loop | # Why do I not see any results from my For-loop? [closed]
I am sure I am missing something trivial :-( Still any help is welcome.
ec1[c1_, c2_] := 1/c1
ec2[c1_, c2_] := 1/c2
Bcom[b_, c1_, c2_] := ec1[c1, c2] - (ec1[c1, c2] - ec2[c1, c2])
X = {}
For[a = 10, a < 100, a += 10,
For[i = 1, i < 100, i += 5,
For[j = i + 1, j <= 50, j += 4,
AppendTo[X, {Bcom[a, i, j]}]]]]
## closed as off-topic by Henrik Schumacher, rhermans, m_goldberg, garej, ÖskåJul 22 at 6:14
This question appears to be off-topic. The users who voted to close gave this specific reason:
• "This question arises due to a simple mistake such as a trivial syntax error, incorrect capitalization, spelling mistake, or other typographical error and is unlikely to help any future visitors, or else it is easily found in the documentation." – Henrik Schumacher, rhermans, m_goldberg, garej, Öskå
If this question can be reworded to fit the rules in the help center, please edit the question.
• It is. Check X. Afterwards. – Henrik Schumacher Jul 17 at 19:20
Try
ec1[c1_, c2_] := 1/c1
ec2[c1_, c2_] := 1/c2
Bcom[b_, c1_, c2_] := ec1[c1, c2] - (ec1[c1, c2] - ec2[c1, c2])
X = {};
For[a = 10, a < 100, a += 10,
For[i = 1, i < 100, i += 5,
For[j = i + 1, j <= 50, j += 4,
AppendTo[X, {Bcom[a, i, j]}]
]]];
X
Or more idiomatic in Mathematica and taking 60% of the time
X2=Flatten[Table[
{Bcom[a, i, j]}
, {a, 10, 100 - 10, 10}
, {i, 1, 100 - 5, 5}
, {j, i + 1, 50, 4}
], 2];
Check the result is the same
X===X2
(* True *)
• Just too tired... Thank you – user34047 Jul 17 at 19:33 | 2019-09-17 09:57:50 | {"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": 0.29244762659072876, "perplexity": 6834.067071566406}, "config": {"markdown_headings": true, "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-2019-39/segments/1568514573065.17/warc/CC-MAIN-20190917081137-20190917103137-00353.warc.gz"} |
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# If 3/7 of the students in a room are seniors and 7/25 of the other stu
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Updated on: 22 Oct 2017, 23:45
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If $$\frac{3}{7}$$ of the students in a room are seniors and $$\frac{7}{25}$$ of the other students are juniors, and there are $$x$$ students in the room who are not juniors or seniors, how many students are in the room?
A. $$\frac{175}{72} x$$
B $$\frac{175}{51}x$$
C. $$\frac{25}{7}x$$
D. $$\frac{25}{3}x$$
E. $$10x$$
I tried searching for this question and could not find in the Forum. Please merge/delete if this has already been posted before in the forum. Thanks in advance!
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Originally posted by susheelh on 22 Oct 2017, 23:14.
Last edited by susheelh on 22 Oct 2017, 23:45, edited 1 time in total.
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Re: If 3/7 of the students in a room are seniors and 7/25 of the other stu [#permalink]
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22 Oct 2017, 23:42
1
total students in room are T.
Seniors are 3/7th so Seniors =3/7*T
Juniors are 7/25 of other students=7/25*(T-3/7*T)=4/25*T
x is neither seniors nor juniors= (T-3/7*T-4/25*T)=72/175*T
Hence total students are T=175/72*x
Plz explain where I am wrong?
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Re: If 3/7 of the students in a room are seniors and 7/25 of the other stu [#permalink]
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22 Oct 2017, 23:47
1
Hello gvij2017,
Thanks for pointing this out. There was a Typo in the question. Your answer and approach are correct. I have edited the question and fixed the typo. +1 to you.
gvij2017 wrote:
total students in room are T.
Seniors are 3/7th so Seniors =3/7*T
Juniors are 7/25 of other students=7/25*(T-3/7*T)=4/25*T
x is neither seniors nor juniors= (T-3/7*T-4/25*T)=72/175*T
Hence total students are T=175/72*x
Plz explain where I am wrong?
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Re: If 3/7 of the students in a room are seniors and 7/25 of the other stu [#permalink]
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26 Oct 2017, 15:35
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1
susheelh wrote:
If $$\frac{3}{7}$$ of the students in a room are seniors and $$\frac{7}{25}$$ of the other students are juniors, and there are $$x$$ students in the room who are not juniors or seniors, how many students are in the room?
A. $$\frac{175}{72} x$$
B $$\frac{175}{51}x$$
C. $$\frac{25}{7}x$$
D. $$\frac{25}{3}x$$
E. $$10x$$
We see that 3/7 of the total students are seniors, and thus 4/7 of the students are not seniors (or are the other students). Hence, 7/25(4/7) = 4/25 of the students are juniors.
If we let t = the total number of students, then we have:
(3/7)t + (4/25)t + x = t
(75/175)t + (28/175)t + x = t
(103/175)t + x = (175/175)t
x = (72/175)t
t = (175/72)x
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Re: If 3/7 of the students in a room are seniors and 7/25 of the other stu &nbs [#permalink] 26 Oct 2017, 15:35
Display posts from previous: Sort by | 2018-10-23 05:18:32 | {"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": 0, "mathjax_display_tex": 1, "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": 0.6331746578216553, "perplexity": 4015.4137367065523}, "config": {"markdown_headings": true, "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-2018-43/segments/1539583516071.83/warc/CC-MAIN-20181023044407-20181023065907-00253.warc.gz"} |
https://projecteuclid.org/euclid.rmjm/1446472428 | ## Rocky Mountain Journal of Mathematics
### Gröbner-Shirshov bases of some Weyl groups
#### Abstract
In this paper, we obtain Gr\"{o}bner-Shirshov (non-commutative) bases for the $n$-extended affine Weyl group $\widetilde{W}$ of type $A_1$, elliptic Weyl groups of types $A_{1}^{(1,1)}$, $A_{1}^{(1,1)^{*}}$ and the $2$-extended affine Weyl group of type $A_{2}^{(1,1)}$ with a generator system of a $2$-toroidal sense. It gives a new algorithm for getting normal forms of elements of these groups and hence a new algorithm for solving the word problem in these groups.
#### Article information
Source
Rocky Mountain J. Math., Volume 45, Number 4 (2015), 1165-1175.
Dates
First available in Project Euclid: 2 November 2015
https://projecteuclid.org/euclid.rmjm/1446472428
Digital Object Identifier
doi:10.1216/RMJ-2015-45-4-1165
Mathematical Reviews number (MathSciNet)
MR3418188
Zentralblatt MATH identifier
0895.16020
#### Citation
Karpuz, Eylem Güzel; Ateş, Firat; Çevik, A. Sinan. Gröbner-Shirshov bases of some Weyl groups. Rocky Mountain J. Math. 45 (2015), no. 4, 1165--1175. doi:10.1216/RMJ-2015-45-4-1165. https://projecteuclid.org/euclid.rmjm/1446472428
#### References
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• F. Ateş, E.G. Karpuz, C. Kocapinar and A.S. Çevik, Gröbner-Shirshov bases of some monoids, Discr. Math. 311 (2011), 1064–1071.
• G.M. Bergman, The diamond lemma for ring theory, Adv. Math. 29 (1978), 178–218.
• L.A. Bokut, Imbedding into simple associative algebras, Alg. Logic 15 (1976), 117–142.
• L.A. Bokut, Gröbner-Shirshov basis for the Braid group in the Artin-Garside generators, J. Symb. Comp. 43 (2008), 397–405.
• ––––, Gröbner-Shirshov basis for the Braid group in the Birman–Ko–Lee generators, J. Alg. 321 (2009), 361–376.
• L.A. Bokut and A. Vesnin, Gröbner-Shirshov bases for some Braid groups, J. Symb. Comp. 41 (2006), 357–371.
• B. Buchberger, An algorithm for finding a basis for the residue class ring of a zero-dimensional ideal, Ph.D. thesis, University of Innsbruck, Innsbruck, Austria, 1965.
• Y. Chen and C. Zhong, Gröbner-Shirshov bases for HNN extentions of groups and for the alternating group, Comm. Alg. 36 (2008), 94–103.
• E.G. Karpuz, Gröbner-Shirshov bases of some semigroup constructions, Alg. Colloq. 22 (2015), 35–46.
• C. Kocapinar, E.G. Karpuz, F. Ateş and A.S. Çevik, Gröbner-Shirshov bases of the generalized Bruck-Reilly $*$-extension, Alg. Colloq. 19 (2012), 813–820.
• K. Saito, Extended affine root systems I, Coxeter transformations, Publ. Res. Inst. Math. Sci. 21 (1985), 75–179.
• K. Saito and T. Takebayashi, Extended affine root systmes III, Elliptic Weyl groups, Publ. Res. Inst. Math. Sci. 33 (1997), 301–329.
• A.I. Shirshov, Some algorithmic problems for Lie algebras, Siber. Math. J. 3 (1962), 292–296.
• T. Takebayashi, Defining relations of the Weyl groups for extended affine root systems $A_{l}^{(1,1)}$, $B_{l}^{(1,1)}$, $C_{l}^{(1,1)}$, $D_{l}^{(1,1)}$, J. Alg. 168 (1994), 810–827.
• ––––, Relations of the Weyl groups of extended affine root systems $A_{l}^{(1,1)}$, $B_{l}^{(1,1)}$, $C_{l}^{(1,1)}$, $D_{l}^{(1,1)}$, Proc. Japan Acad. 71 (1995), 123–124.
• ––––, The growth series of the $n$-extended affine Weyl group of type $A_1$, Proc. Japan Acad. 81 (2005), 51–56.
• ––––, Poincare series of the Weyl group of the elliptic root systems $A_{l}^{(1,1)}$, $A_{l}^{(1,1)*}$ and $A_{2}^{(1,1)}$, J. Alg. Comb. 17 (2003), 211–223. | 2019-10-14 11:40:28 | {"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": 0.7514982223510742, "perplexity": 2763.2860515054053}, "config": {"markdown_headings": true, "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-2019-43/segments/1570986653216.3/warc/CC-MAIN-20191014101303-20191014124303-00259.warc.gz"} |
http://mathoverflow.net/questions/158635/when-is-an-induced-subgraph-of-a-johnson-graph-hamilton-connected | # When is an induced subgraph of a Johnson graph hamilton-connected?
The Johnson graph $J(n,k)$ has vertices which are the $k$-subsets of $\{1, 2, \dots, n\}$ where two vertices are adjacent iff their intersection has size $k-1$. A graph is hamilton-connected if every two vertices are joined by a hamiltonian path. A recent paper by Alspach shows that Johnson graphs are hamilton-connected.
I'm interested in how many vertices can be removed from $J(n,k)$ while still requiring that the induced subgraph be hamilton-connected. For example, the picture below shows $J(5,3)$ and an induced subgraph resulting from deleting four vertices. The induced subgraph is no longer hamilton-connected; at the very least there is no hamiltonian path from $345$ to $245$. However, all collections of three vertices can be deleted from $J(5,3)$ and still give rise to a hamilton-connected induced subgraph.
I searched the literature for anything that might be helpful for this question. I found a paper by Naimi and Shaw on induced subgraphs of Johnson graphs, but it doesn't deal with hamiltonicity. A sufficient condition proved by Chvátal and Erdős is that if a graph $G$'s independence number $\alpha(G)$ is strictly less than its vertex connectivity $\kappa(G)$ then it is hamilton-connected. While $\kappa(J(n,k))=k(n-k)$, $\alpha(J(n,k))$ does not seem to be as readily understood, and so I'm not sure if the desired inequality would hold for the original graph or any of its induced subgraphs. If the induced subgraph $H$ has at least four vertices, a necessary condition seems to be that that $H$ has no verices of degree two. For if $x$ had exactly two neighbors $v_1$ and $v_2$, any hamiltonian path through $H$ not starting or ending at $x$ would have to use the two edges incident to $x$, thus the path $v_1xv_2$ couldn't be extended to a hamiltonian path from $v_1$ to $v_2$.
For the reason given above, we can always find a collection of $k(n-k)-2$ vertices to delete that leaves a given vertex with degree two, and thus the resulting induced subgraph would not be hamilton-connected.
So I guess my question can be rewritten as, if $S \subset V(J(n,k))$ such that $|S| < k(n-k)-2$, is $J(n,k)-S$ always hamilton-connected?
Thank you.
- | 2015-04-25 12:54:18 | {"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": 0.8672153353691101, "perplexity": 138.75161504299106}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2015-18/segments/1429246649234.62/warc/CC-MAIN-20150417045729-00094-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/pilot-wave-of-apple.875319/ | # B Pilot Wave of Apple
1. Jun 12, 2016
### bluecap
Dear Physicsforums,
Does the pilot waves of apple look like an apple? It was supposed to occur in configuration space and not in spacetime.. but isn't it configuration space is a possible spacetime since the pilot waves are supposed to be not just smokes and mirrors like the wave function in Copenhagen?
Whatever (whether the configuration space of the pilot wave is in some kind of space or not), if the pilot waves of apple would be altered like orange pilot waves fragments were inserted. Would the physical apple become altered too with orange in the middle?
2. Jun 13, 2016
### Demystifier
Well, yes and no. If the apple consists of $N\gg 1$ elementary particles, then its pilot wave (at fixed time) is a function of the form $\Psi({\bf x}_1,\ldots, {\bf x}_N)$. It does not look like an apple. But from it you can easily construct something that does look like an apple. Construct first $N$ partial density functions
$$\rho_1({\bf x}_1)=\int d^3x_2 \cdots \int d^3x_N \, |\Psi({\bf x}_1,\ldots, {\bf x}_N)|^2$$
$$...$$
$$\rho_N({\bf x}_N)=\int d^3x_1 \cdots \int d^3x_{N-1} \, |\Psi({\bf x}_1,\ldots, {\bf x}_N)|^2$$
Then the total density function
$$\rho({\bf x})=\rho_1({\bf x})+\ldots +\rho_N({\bf x})$$
looks very much like the apple.
Last edited: Jun 13, 2016
3. Jun 13, 2016
### bluecap
Is the pilot wave in Bohmian more objective than the wave function in Copenhagen or are they both just mathematical entities and just figments of the imagination? If so then why differentiate between Bohmian and Copenhagen since the wave function does't occur in spacetime in either of them?
4. Jun 13, 2016
### Ilja
The dBB interpretation has the advantage that what we observe - the real apples and oranges - exists in the interpretation too. As the configuration, the not so hidden "hidden" variable.
So, in particular, you will not be too much confused by a wave function of Schrödinger's cat. Because there also exists a real cat, which is what we observe, and the wave function may exist or not, this is something you can leave to philosophers.
5. Jun 13, 2016
### Demystifier
Perhaps the pilot wave in Bohmian is somewhat more objective than wave function in Copenhagen, but that is not the main reason to distinguish between Bohmian and Copenhagen. The main difference between Bohmian and Copenhagen is in the things which are not the pilot wave or wave function.
This is like a difference between an airplane pilot and a man in uniform looking like a pilot but not being a pilot. The difference is not so much in those two men, but in the fact that in the former case it is understood that there is also a plane (piloted by the pilot).
Last edited: Jun 13, 2016
6. Jun 13, 2016
### bluecap
What do you mean "The main difference between Bohmian and Copenhagen is in the things which are not the pilot wave or wave function."? Are you saying the apples and oranges we can touch are not even similar in Bohmian or Copehagen interpretation? Kindly elaborate.
7. Jun 13, 2016
### Demystifier
Yes. According to the Bohmian interpretation, the apples and oranges we can touch are made of a large number of pointlike particles. Those particles are guided by the wave, but we don't touch the wave itself.
On the other hand, the Copenhagen interpretation is not so specific about what these things we touch are really made of. But even though it is not very specific about that, it definitely denies the existence of those Bohmian pointlike particles.
Last edited: Jun 15, 2016
8. Jun 13, 2016
### bluecap
Is it possible the pilot wave in Bohmian lives in spacetime or some kind of spacetime container that can make you interact with the pilot wave using some form of technology in the future?
9. Jun 13, 2016
### Staff: Mentor
This might be a good time to mention the Physics Forums rule prohibiting posting personal theories and speculation...
10. Jun 13, 2016
### bluecap
Thanks for emphasizing there is really nothing to the pilot waves. I thought prior to the original post they really existed in spacetime.
So Bohmians need pilot wave as calculational aid because these can split in the double slit setup guiding the local particles.. while in Copenhagen, the wave functions are the particles/objects themselves.
But I'm asking (and not speculating or personal theories because I have none) now whether it is possible for pilot waves to be there yet the particles are not local or always particles.. they can also appear and vanish like the wave functions as objects in Copenhagen? These would enable the particles to tunnel or not make them create trajectories causing for example the electrons to lose energy as it literally rotates around the atoms in Bohmian. So is it possible there is pilot wave in the electron in the atom yet the electron doesn't have local and trajectories as Bohmians always wanted us to imagine or visualize?
11. Jun 13, 2016
### Ilja
bluecap, this text sounds like you have not understood what the wave function is. It is not a function in spacetime, but a function on the configuration space changing in time. The configuration space is something much larger that space. Say, for a system of N particles it is a function on some 3N-dimensional space.
The closest thing to such a function in classical physics is energy. It is also defined for every configuration q(t), but can also depend on $\dot{q}$. It essentially does not define the system itself - which is described by the configuration q(t) - but everything else what can influence this system.
12. Jun 15, 2016
### Staff: Mentor
Several posts containing speculations based on pop-sci misunderstandings have been removed from this thread. The thread is open, but please keep posts relevant to the original question about how the Bohmoan interpretations work with macroscopic objects.
Last edited: Jun 15, 2016
13. Jun 15, 2016
### bluecap
After understanding more about the role of wave functions in Bohmian Mechanics. I think in macroscopic objects. Copenhagen and Bohmian Mechanics don't really differ much. Why. Because in the electrons in the atoms for example... both Copenhagen And BM has the Wave function as the one containing the charge. Since wave function is quantized and doesn't accelerate.. then the electrons don't lose energy. In Copenhagen, we don't know how the wave functions become the chairs and tables. In Bohmian, they just use localized particles as the Chair and Tables. So outwardly they look differently, but inwardly they seem to be just the same. For example. In Bell Aspect Correlations. We don't know how both the wave functions in Copenhagen and BM talk at a distance of billions of light years. It is not the localized particles in BM that talk at all!
I've thought a long time for this thread so hope it won't be deleted without some comments. Thank you.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook
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Draft saved Draft deleted | 2018-02-23 11:35:19 | {"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": 1, "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": 0.47344517707824707, "perplexity": 1065.3332462168453}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "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-2018-09/segments/1518891814566.44/warc/CC-MAIN-20180223094934-20180223114934-00143.warc.gz"} |
https://www.gamedev.net/forums/topic/688559-opengl-world-position-from-depth/ | • 9
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I'm currently working on a project for Uni using FreeGLUT to make a simple solar system simulation. I've got to the point where I've implemented all the planets and have used a Scene Graph to link them all together. The issue I'm having with now though is basically the planets and moons orbit correctly at their own orbit speeds.
I'm not really experienced with using matrices for stuff like this so It's likely why I can't figure out how exactly to get it working. This is where I'm applying the transformation matrices, as well as pushing and popping them. This is within the Render function that every planet including the sun and moons will have and run.
if (tag != "Sun") { glRotatef(orbitAngle, orbitRotation.X, orbitRotation.Y, orbitRotation.Z); } glPushMatrix(); glTranslatef(position.X, position.Y, position.Z); glRotatef(rotationAngle, rotation.X, rotation.Y, rotation.Z); glScalef(scale.X, scale.Y, scale.Z); glDrawElements(GL_TRIANGLES, mesh->indiceCount, GL_UNSIGNED_SHORT, mesh->indices); if (tag != "Sun") { glPopMatrix(); } The "If(tag != "Sun")" parts are my attempts are getting the planets to orbit correctly though it likely isn't the way I'm meant to be doing it. So I was wondering if someone would be able to help me? As I really don't have an idea on what I would do to get it working. Using the if statement is truthfully the closest I've got to it working but there are still weird effects like the planets orbiting faster then they should depending on the number of planets actually be updated/rendered.
• Hello everyone,
I have problem with texture
# OpenGL Opengl World Position From Depth
## Recommended Posts
Hi,
I am having a lot of trouble trying to recover world space position from depth.
I swear I have managed to get this to work before in another project, but I have been stuck on this for ages
I am using OpenGL and a deferred pipeline
I am not modifying the depth in any special way, just whatever OpenGL does
and I have been trying to recover world space position with this (I don't care about performance at this time, i just want it to work):
vec4 getWorldSpacePositionFromDepth(
sampler2D depthSampler,
mat4 proj,
mat4 view,
vec2 screenUVs)
{
mat4 inverseProjectionView = inverse(proj * view);
float pixelDepth = texture(depthSampler, screenUVs).r * 2.0f - 1.0f;
vec4 clipSpacePosition = vec4( screenUVs * 2.0f - 1.0f, pixelDepth, 1.0);
vec4 worldPosition = inverseProjectionView * clipSpacePosition;
worldPosition = vec4((worldPosition.xyz / worldPosition.w ), 1.0f);
return worldPosition;
}
Which I am sure is how many other sources do it...
But the positions seem distorted and get worse as i move the camera away from origin it seems, which of course then breaks all of my lighting...
Please see attached image to see the difference between depth reconstructed world space position and the actual world space position
Any help would be much appreciated!
K
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Which is not what you want since ur in shader you xould precompute z coord or use glfragcoord z value
Amyway( model * view) * projection = mat1
Vertex * mat1 = a
A.xyz = a.xyz / a.w
A = a * 0.5 + 0.5
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Your code looks slow but sane. How are screenUVs computed?
Are you passing the exact same view matrix that the scene was rendered with? Maybe try computing the inverse-view-proj matrix on the CPU and passing it in.
Which is not what you want since ur in shader you xould precompute z coord or use glfragcoord z Value
Reconstructing position from the depth buffer is common in post-processing / deferred shading systems, as glfragcoord is not available and there's no other way to determine the per pixel world-space coordinates.
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screenuvs are just 0-1 uvs for a screen space quad like the usual deferred setup
When i use an explicit world space position buffer everything renders perfectly. So I am sure it is reading from the correct uvs
I tried cpu inverse view proj and it didn't make any difference.
It is so weird, because it looks almost okay until the camera is moved far away!
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I can't believe this - it was something so simple:
my glDepthRangef was set to 0.1f and 1000.0f
setting it to 0.0f and 1000.0f
fixed the position reconstruction
argh!
Thanks for the help anyway! :)
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glDepthRange specifies the NDC->depth-texture transform. Generally you want to leave it on 0,1 and never touch it.
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oh so that explains it then.
Thank you!
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"The setting of (0,1) maps the near plane to 0 and the far plane to 1. With this mapping, the depth buffer range is fully utilized."
This says you have no need to touch it. | 2018-04-21 19:18:13 | {"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": 0.20270244777202606, "perplexity": 3078.156099653082}, "config": {"markdown_headings": true, "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-2018-17/segments/1524125945317.36/warc/CC-MAIN-20180421184116-20180421204116-00563.warc.gz"} |
https://www.physicsforums.com/threads/bells-derivation-socks-and-jaynes.581193/ | # Bell's derivation; socks and Jaynes
1. ### harrylin
Hello,
For this little discussion I base myself on Bell's paper on Bertlmann's socks:
http://cdsweb.cern.ch/record/142461
Although I have participated in a number of discussions about Bell's theorem, I always had the uneasy feeling not to fully understand the definitions of symbols and the notation - in particular how to account for lambda in probability calculations.
So, although I intend to discuss here the validity (or not) of Jayne's criticism of Bell's equation no.11, I'll start very much more basic. Using Bell's example of socks, I think that we could write for example:
P1(pink) = 0.5
Here P1(pink) stands for the probability to observe a pink sock on the left foot on an arbitrary day. An experimental estimation of it is found by taking the total from many observations, divided by the number of observations.
As the colour depends on Bertlmann's mood, we can then account for that mood as an unknown variable "lambda" (here I will just put X, for unknown). However, any local realistic theory that proposes such an unknown variable as explanation, still must predict the same observed result. Therefore, I suppose that if we include X as causal factor, we must still write:
P1(pink|X) = 0.5
Thus far correct?
Last edited: Feb 25, 2012
2. ### IsometricPion
177
I don't think this is quite correct. Rather, if X correlates with Bertlmann wearing a pink sock then P1(pink|X)=(P1(pink)/P(X))>0.5. Instead, $\int{}P_1(pink|X)P(X)dX=P_1(pink)=0.5$ (obviously if X is a causal factor it must correlate with P1(pink)). I think what Bell is saying in equation 11 is that if one knew λ (in addition to the local conditions) there would be no residual correlations between the distributions of the measurements (after accounting for its effects).
3. ### harrylin
Thanks for that clarification! I had not looked at it that way. However, X is like EPR's hidden function: Bertlmann's unknown and unpredictable mood determines what socks he will wear. X stands for the physical model, which is here an invisible random function (indeed, it happens in his head) that delivers one of {pink, not pink}. Obviously the chance to observe a Bertlmann pair of socks on Bertlmann's feet is simply 1. Then we must have, for the case that half of the time a pink sock is observed on the left foot:
P1(pink|X)=P1(pink)/P(X) = P1(pink)/1 =0.5
It's exactly the same as for a fair coin: P(head | fair coin) = 0.5.
I can imagine that someone would like to split the probability estimation up into unknown "knowns": then we can separate it into the cases that Bertlmann decides to put a pink sock on his left leg, and the cases that he decides to put another colour on his left leg. However, what we are interested in the result over many times, and then we are necessarily back at where we were here above. Thus, I don't see any use for that.
Last edited: Feb 25, 2012
4. ### IsometricPion
177
I misinterpreted what you meant by X. I took it to mean a variable taking on values from the set of moods, some subset of which would correlate with Bertlmann wearing a pink sock instead of the mood model itself. In the latter case, I certainly agree with your results.
Edit: A couple of papers that may be relevent to this discussion: Jaynes' view of EPR, a critque of Jaynes' view
Last edited: Feb 25, 2012
5. ### harrylin
@IsometricPion: thanks for the links! I suspect that our own discussion here, which is based on http://cdsweb.cern.ch/record/142461, will show that the Arxiv paper misses the point; we'll see!
instead of running to eq.11, I will first work out the example that Bell gave in his introduction, as he did not do so himself.
Note that in Bell's paper the pictures come after the text. I'll start with a partial re-take.
Elaborating on Bell's example of Bertlmann's socks, we could write for example:
P1(pink) = 0.5
Here P1(pink) stands for the probability to observe a pink sock on the left foot on an arbitrary day. An experimental estimation of it is found by taking the total from many observations, divided by the number of observations.
As the colour depends on Bertlmann's mood, we can account for that mood as an unknown function "lambda" (here I will just put X, for unknown). However, any "classical" theory that proposes such a physical model, still must predict the same observed result. Therefore, if we include X as invisible cause for the outcome, we must still write:
P1(pink|X) = 0.5
(Compare: P(head | fair coin) = 0.5)
Similarly we can write for the right leg:
P2(pink|X) = 0.5
Bell remarks:
The fact that "pink" on the left foot implies "not pink" on the right foot implies a strong correlation between results. We can acknowledge that correlation as follows, with for convenience a slight change of notation:
P(L,R|X) =/= P1(L|X) P2(R|X)
Here L stands for "pink on left leg", and R stands for "pink on right leg".
Ok so far?
Last edited: Feb 26, 2012
6. ### alsor
5
pink = 1, not pink = -1
then:
P(LR) = 0, because L and R are always different;
formally:
P(LR) = P(L|R)*P(R); P(R) = P(L) = 1/2;
but: P(L|R) = 0 <> P(L); (both socks have never the same colour)
corr = 0 - 1 = -1, full anti-correlation.
And using Bell reasoning:
P(LR) = P(L)*P(R) = 1/2 * 1/2 = 1/4;
corr = 2*1/4 - 2*1/4 = 0.
Two random socks, and completely independent, of course.
7. ### harrylin
@ alsor: it appears that in this matter we both agree with Jaynes.:tongue2:
However, again you ran far ahead of me and I'm not sure if everyone who, so far, didn't "see" this point of Jaynes etc., could follow you. So, I'll continue my slow pace to make sure that everyone who watches this topic can follow me and that we all agree on the basic facts as well as notation. I'll catch up with you later.
Last edited: Feb 26, 2012
8. ### IsometricPion
177
This misrepresents Bell's model of local hidden variables. Equation 11 of his paper assumes one knows the values of the hidden variables, in this case Bartlmann's mood. So, P(L|bartlmann feels like wearing a pink sock on his right foot)=0 (since beyond his mood one also knows that he does not wear the same color socks) and P(R|bartlmann feels like wearing a pink sock on his right foot)=1. Thus P(L|mood=right, pink)*P(R|mood=right, pink)=1*0=0.
If instead one does not know his mood (or anything about it other than it can take on one of two sets of values), P(L)=P(L|RP)P(RP)+P(L|R¬P)P(R¬P)=0*0.5+1*0.5=0.5, by exchangeability. The problem Jaynes sees with Bell's reasoning is not his statistical or mathematical procedure/ability, rather he thinks Bell is to restrictive in what he (Bell) consideres to be valid variables for the probability distributions for a theory upholding local realism.
9. ### harrylin
Thanks for the correction; however, although indeed Bell doesn't make a blunder of that proportion, Jaynes certainly points out a subtle error in Bell's equation; according to Jaynes it is not correct.
Anyway, we're not there yet: the problem with the illustration of Bertlmann's socks is that it by far doesn't catch the complexity of the problem at hand. If the observations would always be perfectly anti-correlated, there wouldn't be a riddle.
Now, I'm afraid that his next illustration of Lille and Lyon matches it even less well; thus, for this discussion I have been trying to come up with a variant of Bertlmann's socks that addresses the fact that the local conditions affect the observed correlation, but I didn't come up with a good looking one (I thought of observation of white or yellow socks in daylight/artificial light, as well as mud on his socks, but I'm not satisfied). Any better suggestion? If not, we should perhaps move on to the introduction of eq.11.
Last edited: Feb 27, 2012
10. ### ThomasT
Metaphors are unnecessary and sometimes confusing, imho. Why not just refer to Bell's original formulation of a local realistic QM expectation value. Where does lambda appear and what does it refer to?
11. ### harrylin
While it may appear that he defines it very precisely, different people interpret it slightly differently in the literature. Moreover, I wasn't in the clear about notation. However, this discussion is already making it quite clear (I just needed a memory refresh!); we're now moving on to Bell vs Jaynes.
Last edited: Feb 27, 2012
### Staff: Mentor
I'm pretty much a spectator in these discussions, but I'd like to point out that there was a long thread here about three years ago, about Jaynes's objections to Bell:
This was before you joined PF, so you may not have seen this. It may or may not fit in with the direction you were planning to go.
It was split off from another thread, by the way, which is why it appears to start in the middle of a discussion.
13. ### harrylin
Thank you! Indeed I had not seen that one... BTW I was also very much a spectator of another current thread in which I saw the suggestion to start this topic. Now I'll first check out the old thread.
14. ### harrylin
Ok, I'm afraid that I will need some time to work through that old thread; and I'm very busy this week.
Still, I started reading it and I notice some disagreement about what Bell claimed to prove. There is no use getting into arguments about the meaning of "local realism" and philosophy. What the "local realist" Einstein insisted on, and what Bell claimed to be incompatible with QM, was "no spooky action at a distance". Or, as Bell put it in his first paper:
Those who deviate from that issue are shooting at straw men.
Bell puts it this way in his Bertlmann's socks paper:
The focus of this discussion is Bell's attempt to prove that Einstein's "no action at a distance" principle is incompatible with QM, in the light of Jayne's first criticism.
Last edited: Feb 28, 2012
15. ### harrylin
I now had a better look at it, and I think that in particular posts #26 and #31 are important. Anyway I'll give a short summary of how I now see it.
If Jaynes' criticism focuses on Bell's equation no.11 in his "socks" paper, it was perhaps due to a misunderstanding about what Bell meant (his comments were based on an earlier paper).
P(AB|a,b,x) = P(A|a,x) P(B|b,x) (Bel 11)
Here x stands for Bell's lambda, which corresponds to the circumstances that lead to a single pair correlation (in contrast to my earlier X, which causes the overall correlation for many pairs).
According to Jaynes it should be instead, for example:
P(AB|a,b,x) = P(A|B,a,b,x) P(B|a,b,x)
Perhaps Jaynes thought that Bell meant:
P(AB|a,b,X) = P(A|a,X) P(B|b,X)
in which case Jaynes claimed that:
P(AB|a,b,X) = P(A|B,a,b,X) P(B|a,b,X)
This is really tricky. :uhh:
However, he really was disagreeing with the integral equation.
According to him, it should not be:
P(AB|a,b) = ∫ P(A|a,x) P(B|b,x) p(x) dx
but:
P(AB|a,b) = ∫ P(AB|a,b,x) P(x|a,b) dx
and thus:
P(AB|a,b) = ∫ P(AB|a,b,x) p(x) dx = ∫ P(A|B,a,b,x) P(B|a,b,x) p(x) dx
Is my summary of the disagreement correct?
What is the significance of little p(x) instead of P(x)?
16. ### DrChinese
5,681
http://bayes.wustl.edu/etj/articles/cmystery.pdf
As I read it, this is one of Jaynes's arguments. However, I think it is attacking a straw man. The essence of Bell's argument does not require the factorization so much as a definition of what realism is.
For a SINGLE photon, not a pair: does it have a well-defined polarization at 0, 120, and 240 degrees independent of the act of observation? Once you answer this in the affirmative, as any local realist must, the Bell conclusion (a contradiction between the assumption and QM's predictions) follows quickly. If you answer as no, then you already deny local realism so it is moot.
So I really don't see the significance here of Jaynes' argument. The only people that take it seriously are local realists looking for support for their position. The vast majority of scientists see it for what it is, something of a technicality with no serious implications for the Theorem whatsoever.
In other words, it would be helpful to see an example that somehow related specifically to photon polarization rather than urns (which does not seem to be much of an analogy).
Last edited: Mar 1, 2012
17. ### lugita15
Speaking of this paper, does anyone know what Jaynes is talking about in the end of page 14 and going on to page 15, concerning "time-alternation theories"? He seems to be endorsing a local realist model which makes predictions contrary to QM, and he claims that experiments peformed by "H. Walther and coworkers on single atom masers are already showing some resemblance to the technology that would be required" to test such a theory. Does anyone one know whether such a test has been peformed in the decades since he wrote his paper?
18. ### IsometricPion
177
I think it is clear that λ in Bell's paper corresponds to x here (rather than X). While I am less certain of Jaynes' meaning I think it is probably x as well (since it appears on the left side of the | indicating that it is a variable in some of his equations). Jaynes is pretty consistent, so I would expect everything he denotes by λ to refer to the same thing (i.e., all his λ's should correspond with x's rather than X's).
Jaynes refers to probabilities essentially as logical statements of uncertain truth value. His P(y|Y) correspond to logical statements where Y is the predicate and y is the antecedent the truth value of which one is uncertain (the amount of (rational) belief one has that y has a value between u and v is P(u≤y≤v|Y)=∫uv P(y|Y)dy). He refers to any probability not of this form as p(y), since one cannot ascribe a logical statement to such a probability without more information regarding its context. Since the context here is clear and consistently applied, I think it is just a matter of formalism (i.e., there is no substantial difference). (Jaynes defines what he means by these symbols in Appendix B of Probability Theory: The Logic of Science.)
Jaynes states what he thinks are Bell's hidden assumptions:
He goes on to mention a type of local hidden variable theory he does not think Bell's theorm covers, though I do not yet understand his arguement as to why it isn't covered.
I think a key point to this discussion is how to define local realism in terms of the functional dependence of probability distributions of outcomes of the EPR (thought) experiment. Once this is agreed upon (i.e., all the variables and symbols we are using are well-defined) the rest should just be a matter of mathematics (about which I think we all should be able to agree).
Last edited: Mar 1, 2012
19. ### lugita15
Yes, that's what I was asking about in my previous post. He claims that ordinary Bell tests won't be able to test this "time-alternating" model, but some other experiments could test it.
20. ### Delta Kilo
273
1. Urn example is a red herring. The cases are not equivalent. There is no correspondence established between A,B,a,b on one hand and R1,R2 on the other. Specifically, λ was lost along the way. Let's try to put it back.
λ is going to be the complete state of the urn before the first draw - that's our hidden random variable. A would be the location of the ball to be drawn first - a freely chosen parameter, mutually independent from λ. a=a(A,λ) is the outcome - deterministic function of A and λ. Now the state of the urn after the first draw is γ=γ(A,λ) - another deterministic function of A and λ. And finally b=b(B,γ) - is yet another deterministic function.
Now b=b(B,γ)=b(B,γ(A,λ))=b(A,B,λ), and b clearly depends on A, that is ∃A,B1,B2,λ: b(A,B1,λ) ≠ b(A,B2,λ). Here b(...) is a deterministic function and A,B1,B2,λ are merely placeholders, arguments of ∃, loop variables is you wish. There is a clear causal link: given the same initial state of the urn, the choice of ball in the first draw causally affects the results of the second draw.
In contrast, in Bell's case we have explicitly denied this causal link as violating locality: ∀A,B1,B2,λ: b(A,B1,λ) = b(A,B2,λ). Note we are not talking here about randomness, conditional probabilities, observer's state of knowledge etc., we are simply trying to establish a link between the value of a deterministic function and its parameter. See the difference?
So the two cases have different physical models behind them and it is not a surprise the results of one are not applicable to another. In fact, the urn would be a great example provided first and second draw can be spacelike separated
2. Regarding time-dependence. In Bell's case λ includes everything that might possibly affect the experiment, except for settings A and B. I think it is safe to assume that absolute value of t does not matter (otherwise we're in for a rough ride). Any relative time delay in the experiment will appear as yet another random factor collectively included in λ and the integral in (12) will include integration over the whole range of it. As long as these delays are independent from choices A and B, it's within Bell's framework.
Last edited: Mar 2, 2012 | 2015-07-29 02:50:15 | {"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": 0.7633182406425476, "perplexity": 1226.9728112347213}, "config": {"markdown_headings": true, "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-2015-32/segments/1438042985647.51/warc/CC-MAIN-20150728002305-00205-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://sac.founderit.com/ | # ICDAR 2019 Competition on Table Detection and Recognition
The competition will contain two tasks:
1. Task 1: Table Region Detection
The ICDAR 2019 cTDaR evaluates two aspects of table analysis: table detection and recognition. The objective is to detect the table region within a document and to correctly detect the table structure. The participating methods will be evaluated on a modern dataset and additionally, on archival documents with printed and handwritten tables present.
The dataset consists of modern documents and archival ones with various formats, including document images and born-digital formats such as PDF. The annotated contents contain the table entities and cell entities in a document, while we do not deal with nested tables. We gathered 1000 modern ones and 1000 archival ones as table region detection task's test dataset and 80 documents as table recognition task's test dataset (see Figure 1).
Figure 1: Example of table dataset
Detailed description will be announced shortly together with the input format.
Regarding the evaluation of table detection, most of the current work uses the precision & recall rates to measure the experiment results. There are two well-known metrics for evaluating the performance of algorithms, (i) the metric based on the table regions [1], and (ii) the metric based on the text regions [2],[3]. We choose the metric (i) to evaluate the performance of table region detection, and apply the metric (ii) to evaluate that of table recognition. Based on these measures, an overall performance of various algorithms can be compared with each other.
1. Metric for table region detection task
The task is evaluated by a traditional method. Intersection over Union (IOU)[1] is calculated to estimate whether a table region detected by the participant is correctly located. Let A denote the region detected by a participant and B denote the corresponding region described in the groundtruth file. The IOU is calculated as follows:
$$IOU=\frac{A\bigcap B}{A+B-A\bigcap B}$$
Average Precision (AP) is the metric to evaluate the task. For the task, the precision/recall curve is computed from a method’s ranked output. Recall is defined as the proportion of all positive examples ranked above a given rank. Precision is the proportion of all examples above that rank which are from the positive class. The AP summarizes the shape of the precision/recall curve, and is defined as the mean precision at a set of eleven equally spaced recall levels [0,0.1,0.2,...,1].
2. Metric for table recognition task
Firstly, the structure of a table is defined as a matrix of cells. The groundtruth provides row bounding box list、column bounding box list、cell’s bounding box、 textual content and its start and end column and row positions. We propose the following metric:
1. Cell’s adjacency relation-based table structure evaluation[2]
Blank cells are not represented in the grid. A benefit of such a representation is that each cell is independent from what has previously occurred in the table definition. For comparing two cell structures, we use the method: for each table region we generate a list of adjacency relations between each content cell and its nearest neighbor in horizontal and vertical directions. No adjacency relations are generated between blank cells or a blank cell and a content cell. This 1-D list of adjacency relations can be compared to the groundtruth by using precision and recall measures, as shown in Figure 2. If both cells are identical and the direction matches, then it is marked as correctly retrieved; otherwise it is marked as incorrect.
Figure 2: Comparison of an incorrectly detected cell structure with the groundtruth [2]
3. We will also release a number of tools to enable the participants to automatically compare their result to the groundtruth.
Release of website and samples: March 1, 2019 Release of the training dataset: March 10, 2019 Release of the test dataset and evaluation tools: March 25, 2019 Deadline of result submission: April 20, 2019 Release of the annotations of the test dataset: April 30, 2019
[1] L. Gao, X. Yi, Z. Jiang, L. Hao and Z. Tang, “ICDAR 2017 POD Competition,” in ICDAR, 2017, pp. 1417-1422.
[2] M. C. Gobel, T. Hassan, E. Oro, G. Orsi, ”ICDAR2013 Table Competition,” in Proc. of the 12th ICDAR (IEEE, 2013), pp. 1449-1453.
[3] A. C. e Silva, “Metrics for evaluating performance in document analysis: application to tables,” IJDAR, vol. 14, no. 1, pp. 101–109, 2011.
Hervé Déjean, Naver Labs Europe, France
herve.dejean@naverlabs.com
Jean-Luc Meunier, Naver Labs Europe, France
jean-luc.meunier@naverlabs.com
Florian Kleber, Computer Vision Lab, TU Wien, Austria
kleber@cvl.tuwien.ac.at
Eva Lang, Archiv des Bistums Passau, Germany
eva.lang@ieee.org
Liangcai Gao, Institute of Computer Science & Technology, Peking University, China
Document Image Analysis and Recognition Technical Committee，China Society of Image and Graphics(DIAR-CSIG)
glc@pku.edu.cn
Yilun Huang, Institute of Computer Science & Technology, Peking University，China
huangyilun@pku.edu.cn
Yu Fang, State Key Laboratory of Digital Publishing Technology, Founder Group Co. LTD., China
fangyu@founder.com | 2019-02-20 22:09:48 | {"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": 1, "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": 0.4215531647205353, "perplexity": 2614.0877472032566}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2019-09/segments/1550247496694.82/warc/CC-MAIN-20190220210649-20190220232649-00268.warc.gz"} |
https://eprints.soton.ac.uk/256010/ | The University of Southampton
University of Southampton Institutional Repository
# Long-range forces and data compression in Vortex State simulations
Fangohr, Hans, Cox, Simon, Price, Andrew, Daniell, Geoffrey J., Robinson, Alexa M. and de Groot, P.A.J. (2000) Long-range forces and data compression in Vortex State simulations.
Record type: Other
## Abstract
We report on two aspects of simulations of the vortex state in high-temperature superconductors. Firstly, we cover the treatment of the involved long-range forces. Secondly, we suggest improvements on how to compress vortex position data effectively to enable visualisation of complex systems. A number of problems arise when long-range forces such as $K_1(x)$ or $1/x$ are used in particle-particle simulations. If a simple cut-off for the interaction is used, the system may find an equilibrium configuration at zero temperature that is not a regular lattice yet has an energy lower than the theoretically predicted minimum for the physical system: this is an artefact of the cut-off [1]. We have developed two methods to overcome these problems in Monte Carlo and molecular dynamics simulations. The first uses a smoothed potential to truncate the interaction in a single unit cell: this is appropriate for phenomenological characterisations, but may be applied to any potential. The second is a new method for the $K_0(x)$ potential and sums the energy in an infinitely tiled periodic system, which is in excess of 20,000 times faster than previous naive methods which add periodic images in shells of increasing radius: this is suitable for quantitative studies. Finally, we demonstrate how tree methods and space filling curves can be used to store particle data more efficiently. [1] H. Fangohr et. al., Peter A.J. de Groot and Geoffrey J. Daniell and Ken S. Thomas (2000) Efficient Methods for Handling Long-Range Forces in Particle-Particle Simulations. Journal of Computational Physics, 162 p.372-384.
Full text not available from this repository.
Published date: December 2000
Additional Information: Poster Presentation, Condensed Matter and Materials Physics Conference, Bristol, United Kingdom.
Organisations: Electronics & Computer Science
## Identifiers
Local EPrints ID: 256010
URI: http://eprints.soton.ac.uk/id/eprint/256010
PURE UUID: 2a8c235b-0156-47f6-901c-cfd98ffd6523
ORCID for Hans Fangohr: orcid.org/0000-0001-5494-7193
## Catalogue record
Date deposited: 26 Aug 2001
## Contributors
Author: Hans Fangohr
Author: Simon Cox
Author: Andrew Price
Author: Geoffrey J. Daniell
Author: Alexa M. Robinson
Author: P.A.J. de Groot | 2021-09-28 17:06:32 | {"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": 0.37414315342903137, "perplexity": 3585.6954975205654}, "config": {"markdown_headings": true, "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-2021-39/segments/1631780060877.21/warc/CC-MAIN-20210928153533-20210928183533-00684.warc.gz"} |
https://cstheory.stackexchange.com/questions/4409/which-integer-linear-programs-are-easy/4410 | # Which Integer Linear Programs are easy?
While trying to solve a problem, I ended up expressing part of it as the following integer linear program. Here $\ell,m,n_{1},n_{2},\ldots,n_{\ell},c_{1},c_{2},\ldots,c_{m},w$ are all positive integers given as part of the input. A specified subset of the variables $x_{ij}$ is set to zero, and the rest can take positive integral values:
Minimize
$\sum_{j=1}^{m}c_{j}\sum_{i=1}^{\ell}x_{ij}$
Subject to:
$\sum_{j=1}^{m}x_{ij}=n_{i}\,\,\forall i$
$\sum_{i=1}^{\ell}x_{ij}\ge w\,\,\forall j$
I would like to know if this integer program is solvable in polynomial time; my original problem is solved if it is, and I have to try some other way if it isn't. So my question is:
How do I figure out if a certain integer linear program can be solved in polynomial time? Which integer linear programs are known to be easy? In particular, can the above program be solved in polynomial time? Could you point me to some references on this?
It is a special case of the transportation problem (or the minimum-cost flow problem), and so can be solved in polynomial time. The coefficient matrix is totally unimodular since it is the incidence matrix of a bipartite graph.
The following Wikipedia articles could be useful.
• @Yoshio : Thank you, that answers my particular problem instance (once I have verified it for myself). Do you know of conditions other than total unimodularity which guarantee a polynomial-time solution? – gphilip Jan 20 '11 at 3:58
• @gphilip : I would summarize theses questions by the term "integrality of polyhedra" and the literature on this subject is huge. The book "Combinatorial Optimization: Packing and Covering" by Gerard Cornuejols (published in 2001) describes several results along this line. – Yoshio Okamoto Jan 20 '11 at 4:37
• @Yoshio : Could you tell me why you think that the coefficient matrix is the incidence matrix of a bipartite graph? Pardon my ignorance, but to speak of a coefficient matrix, do we not have to first convert all constraints to the standard form ($Ax\le b$)? Once we do that, the matrix will have -1 entries, and then it does not match the definition of an incidence matrix (AFAIK). Or is it the case that we can speak of the coefficient matrix without first converting the constraints to the standard form? – gphilip Jan 20 '11 at 13:15
• @gphilip : Excuse me. I made an implicit short-cut, and I was speaking of the coefficient matrix without converting to the standard form. I used the following short-cuts. (1) If $A$ is totally unimodular (TU, for short), then $-A$ is also TU, which means that we don't have to care about the direction of inequalities. (2) If $A$ is TU, then $[A \mid -A]$ is also TU, which means that we don't have to care about equality constraints. (3) Every submatrix of a TU matrix is TU. Applying these rules to the incidence matrix of a bipartite graph should prove the property for the standard form. – Yoshio Okamoto Jan 20 '11 at 14:32
• Let me change my short-cut rules as follows. (1) Duplication of a row maintains total unimodularity. (2) Reversal of the sign of a row maintains total unimodularity. They should do the job. – Yoshio Okamoto Jan 20 '11 at 15:07
In general, it's hard to say. But a sufficient condition is your constraint matrix is totally unimodular and right-hand side is always integer (in this case the right hand side is integer, but you still have to check about unimodularity)
You should take a look at this: http://en.wikipedia.org/wiki/Linear_program#Integer_unknowns
• I was thinking about your matrix and it looks totally unimodular. – Vinicius dos Santos Jan 20 '11 at 0:19
• @Vinicius : Could you tell me why the matrix looks totally unimodular to you? I could not figure this out, in spite of Yoshio's comment (please see my response there). – gphilip Jan 20 '11 at 13:45
• @gphilip: At en.wikipedia.org/wiki/Unimodular_matrix in the "Common totally unimodular matrices" section, the first item list 4 sufficient conditions for a matrix to be unimodular. I think that these conditions, together with the shortcuts Yoshio commented, are enough to show that the problem can be solved in polynomial time. – Vinicius dos Santos Jan 20 '11 at 15:16
• @gphilip: What is the motivation for this Linear Program? – Vinicius dos Santos Jan 20 '11 at 15:17
• @Vinicius : We are trying to solve a problem phrased in terms modifying an input matrix in a certain way to obtain another matrix with some good properties. This LP came out of one sub-problem during the process. – gphilip Jan 21 '11 at 6:52
An integer program with only equalities can be solved by linear program.
• this seemed important for its own sake. – T.... Aug 19 '17 at 20:32
• I wouldn't call that an integer program. It's a system of linear equations over the integers, solvable efficiently by computing the Hermite normal form. – Sasho Nikolov Aug 19 '17 at 22:34
• @SashoNikolov a degenerate case but definitely a valid one though. – T.... Aug 19 '17 at 22:39
• why negative vote? – T.... Aug 20 '17 at 0:44 | 2020-05-25 02:13:03 | {"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": 0.5978191494941711, "perplexity": 329.55484698626975}, "config": {"markdown_headings": true, "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-2020-24/segments/1590347387155.10/warc/CC-MAIN-20200525001747-20200525031747-00052.warc.gz"} |
https://physics.codidact.com/users/53922/activity | # Activity for Istiak
Type On... Excerpt Status Date
Edit Post #281581 Post edited:
Comment Post #285133 @#8056 I will welcome any answer of my question. Usually, I like derivation rather than he had done it. A derivation can clear up any confusion no matter who done it what way. Everyone doesn’t think same way. The way I was learnt derivation of E= mc^2 from high school's book was not the way how Einst...
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Comment Post #285133 When putting -1 in Minkowskian metric I just get the spacetime invariant which is $ds^2=c^2t^2-dx^2$. Even I get the same spacetime invariant if I put ict into time coordinate. My first question was, how Minkowski found that? Or he just wrote it curiously. Second question was, how imaginary nu...
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Comment Post #285133 Yep! It is about putting -1 in the Minkowski metric and putting factor i into the time coordinate.
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Question Clear up confusion on Minkowski signature
All given metrics are for orthonormal-basis. 2 dimensional spacetime : I saw that Minkowski Metric looks like this : $$\pmatrix{-1 & 0 \\\\ 0 & 1}$$ or $$\pmatrix{1 & 0 \\\\ 0 & -1}$$ I was wondering why it's not written like identity metric. To define a vector in spacetime it's written li...
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To understand virtual work you should understand virtual displacement first, The wiki I have linked in first paragraph describes virtual displacement deeply. Usually we write $\delta$ or $d$ to say infinitesimal, now the question is infinitesimal what? As the OP wrote in question that $\sumi Fi\cd... (more) 4 days ago Comment Post #284890 He was asking why it was happening which is for friction or something similar to it. But if he wants us to say how to repair it than I will remark it as off-topic. (more) 4 days ago Edit Post #285090 Initial revision 4 days ago Answer A: What does Laplace operator represent? Laplacian acts like Divergence but not completely. If you take a function (called$\vec{A}$) and write that laplacian of that function is$0$than it will be flat space. $$\nabla^2\vec{A}=0$$ But if you write Laplacian like this : $$\Delta \vec{A}=\text{c}\frac{\partial^2 \vec A}{\partial ... (more) 4 days ago Edit Post #285089 Initial revision 4 days ago Answer A: What is \mathcal{O} in integration of potential? Actually \mathcal O is some standard point and r is where potential depends on.$$V(b)-V(a)=-\int{\mathcal{O}}^B \vec E \cdot d\vec l +\int{\mathcal{O}}^A \vec E \cdot d\vec l =-\int{\mathcal{O}}^B \vec E \cdot d\vec l-\int^{\mathcal{O}}B \vec E \cdot d\vec l=-\intA^B \vec E \cdot ... (more) 4 days ago Edit Post #285088 Initial revision 4 days ago Question What is$\mathcal{O}$in integration of potential? I saw following equation in Griffiths EM $$V(r)=-\int\mathcal{O}^r \vec E \cdot d\vec l$$ While the surface was closed but not symmetrical circular. At first I was thinking$\mathcal O$was representing closed surface and$r$was representing radius of that circle. But when I noticed the circle ... (more) 4 days ago Comment Post #285044 I have shared in a WhatsApp group and 3 facebook groups, I had in lots of groups but I left them cause they were unhelpful for me. If I can see new activities here than I will market the site.. :) Interested to do anything for the community (more) 6 days ago Edit Post #285044 Initial revision 6 days ago Answer A: How can we grow this community? I am not a marketer so my idea may not be good. As a physics enthusiast I can give some thoughts. Recently, I was just reading questions and answers in PSE just to learn new things. And what I have seen that is most of questions were asked by undergraduate students and high students or colleg... (more) 6 days ago Edit Post #284761 Initial revision about 1 month ago Question What does Laplace operator represent? I was wondering what's the physical meaning of Laplace operator.$\vec\nabla$actually represent a field. I had seen that Laplace operator is written as$\vec \nabla \cdot \vec\nabla=\nabla^2=\Delta$So Laplace operator is scalar quantity. I had found usage of Laplace operator in a book and Wiki-pag... (more) about 1 month ago Comment Post #284739 I agree with the idea. I believe professional might stop visiting the site when they see that there's no more question. Seems like, the site might be dead. If we don’t get questioners than the site will be dead for less question. Later, when there will be more questioner then the site will be dead fo... (more) about 1 month ago Edit Post #284728 Initial revision about 1 month ago Answer A: Who should the temporary moderators be? Is it the correct time to have a moderator? It's already a year we have the Physics community. But it's really inactive. There's only 42 question in Q&A which proves that we still don't have lots of member to build the community yet. So it's not the better time to have a mod either. Mithrandir w... (more) about 1 month ago Comment Post #284645 Instead of deleting my earlier comment, I am just adding new comment. Since **it's not you who misread**. It's me who misread. Cause the youtuber had wrote Entropy but I read Energy. Sorry again! (more) about 2 months ago Comment Post #284645 Sorry! Did you misread? I said energy. But you said Entropy. There is difference. And I had read it in First law of thermodynamics. (more) about 2 months ago Edit Post #284645 Post edited: about 2 months ago Comment Post #284645 At the current time of the video : https://youtu.be/sPveM04XCxI?t=143 he wrote that "Energy can't be destroyed but it can created". I don't believe it but the person wrote in that video's description that "this video is specially made for the S.Y.Bsc Physics students". My mind can't agree with th... (more) about 2 months ago Edit Post #284645 Initial revision about 2 months ago Question What is "order" and "disorder" in entropy? What is "disorder" in entropy? >Entropy is measurement of "disorder". (Some says it's not "disorder") I had read "disorder" and "order" of entropy in my book. But I was wondering what "order" and "disorder" actually represent. As Boltzmann said, $$S=k\ln \Omega$$ Where$\Omega$is number of... (more) about 2 months ago Comment Post #284434 It is about general.. (more) 2 months ago Edit Post #284434 Initial revision 2 months ago Question Should I always write units in equation no matter if it looks like variable? I am not sure if it is suitable in the site. I am asking the question here cause Olin and Celtschk said it here. They told me to use unit in every "single line" (Nope! They didn't say like it. Olin said to always separate number and units. And Celtschk said to write units in middle lines). I wa... (more) 2 months ago Edit Post #283894 Post edited: 3 months ago Edit Post #283894 Post edited: 3 months ago Edit Post #284073 Post edited: 3 months ago Edit Post #284075 Initial revision 3 months ago Answer A: Find the values of A, B, and C such that the action is a minimum >The Euler-lagrangian equation gives the equations of motion that once solved give you a family of solutions that minimize the action. A unique solution is given by specifying boundary conditions. It is just a case of inputing those boundary conditions. > >Wlog let$ x(0)=0 $and$x(t0)=a $. Integ... (more) 3 months ago Edit Post #284073 Initial revision 3 months ago Question What's the importance of Poisson brackets? $$F=F(q,p,t)$$ $$\frac{dF}{dt}=\frac{\partial F}{\partial q}\frac{\partial q}{\partial t}+\frac{\partial F}{\partial p}\frac{\partial p}{\partial t}+\frac{\partial F}{\partial t}$$ $$=\frac{\partial F}{\partial q}\dot{q}+\frac{\partial F}{\partial p}\dot{p}+\frac{\partial F}{\partial t}$$ $$=\frac... (more) 3 months ago Edit Post #284050 Initial revision 3 months ago Question Find the values of A, B, and C such that the action is a minimum > A particle is subjected to the potential V (x) = −F x, where F is a constant. The particle travels from x = 0 to x = a in a time interval t0 . Assume the motion of the particle can be expressed in the form x(t) = A + B t + C t^2 . Find the values of A, B, and C such that the action is a ... (more) 3 months ago Edit Post #283954 Post edited: 3 months ago Edit Post #283954 Post edited: 3 months ago Edit Post #283954 Initial revision 3 months ago Answer A: What does Lagrangian actually represent? Lagrangian is no energy. It’s just the Lagrangian. It's perhaps more fundamental than energy in a certain sense. In general, you can think of it as a function that minimizes the action. That's the definition of Lagrangian. said by Golam Ishtiak (You can find him right here also) For reference ... (more) 3 months ago Edit Post #283919 Initial revision 3 months ago Answer A: Find initial velocity when a stuntman jump from 1.25 \ m height The method is correct. >when you write s=vt, s is the horizontal distance and v is the horizontal component of the initial velocity (and it happens to be that the initial velocity has only horizontal component but it could have been different), which does not change through the motion, that's ... (more) 3 months ago Edit Post #283918 Initial revision 3 months ago Question Find equation of motion using. Lagrangian given equation is L'=\frac{m}{2}(a\dot{x}^2+2b\dot{x}{y}+c\dot{y}^2)-\frac{k}{2}(ax^2+2bxy+cy^2) >$$L'=\frac{m}{2}(a\dot{x}^2+2b\dot{x}{y}+c\dot{y}^2)-\frac{k}{2}(ax^2+2bxy+cy^2)$$where a, b, and c are arbitrary constants but subject to the condition that b^2 − ac \ne 0. What are the equations of motion? Examine particularly the two cases a = 0 = c and b = 0, c = −a. What is the physic... (more) 3 months ago Edit Post #283894 Post edited: 3 months ago Edit Post #283894 Initial revision 3 months ago Answer A: Book suggestion for Classical Mechanics For an absolute beginner : If you don't have any idea of any theories than, I would suggest to study theories at first. Just practice beginner problems at first which contains no-calculus(It's OK if that book contains few beginner calculus). I suggest to read High Schools Physics Book for studying... (more) 3 months ago Comment Post #283783 added... currently, it's second question in list of Q&A (more) 3 months ago Edit Post #283783 Post edited: 3 months ago Edit Post #283783 Initial revision 3 months ago Question Question is showing closed instead of last_activity or reopened Magic.. :D list The question was reopened but, showing closed instead of reopened (It should show lastactivity according to code). question link (more) 3 months ago Comment Post #283703 I was looking at your profile then, I saw that moderator ability isn't showing in Physics but, showing in CD Meta, Software and others (not visited all communities). (more) 4 months ago Comment Post #283293 Is there something else which I should add? (more) 4 months ago Comment Post #283695 In your last second equation you had \delta q beside dt. I know that dt is used for representing changes respect to something. While integral goes away, than dt goes away either. Hence, dt is no more. But, I wonder where \delta t had gone? It should be right there. If not than, it is because th... (more) 4 months ago Comment Post #283680 Somehow I can see it now... Could you write an answer? So, it might be helpful for future visitor if they see something related to this... (more) 4 months ago Edit Post #283684 Initial revision 4 months ago Question What does Lagrangian actually represent? L=T-U Here, L is Lagrangian. T is kinetic energy. U is potential energy. But, what Lagrangian actually is? I know what Holonomic and non-holonomic is. But, I was thinking what the Lagrangian represent. Like, F represent force applied on some body. To me, Lagrangian is just representing some kind ... (more) 4 months ago Edit Post #283683 Post edited: 4 months ago Edit Post #283683 Post edited: typo 4 months ago Edit Post #283683 Initial revision 4 months ago Question Prove differential form of Lagrangian How to derive the Lagrangian differential force?$$\frac{d}{dt}(\frac{\partial L}{\partial \dot{x}})+\frac{\partial L}{\partial x}=0$$I was trying to do something.$$L=T-U=\frac{1}{2} m\dot{x}^2-\frac{1}{2}kx^2\frac{\partial L}{\partial \dot{x}}=m\dot{x}\frac{\partial L}{\partia... (more) 4 months ago Edit Post #283293 Post edited: 4 months ago Edit Post #283680 Post edited: 4 months ago Edit Post #283680 Post edited: 4 months ago Edit Post #283680 Initial revision 4 months ago Question Why ability doesn't show in every users profile? Why abilities don't show in every user page? Take a look at @Mithrandir's profile according to his profile he doesn't have any ability. But, I don't think that he doesn't have ability. I was checking other users if they also don't have ability. But, I saw they had. What's the reason of it? I had earl... (more) 4 months ago Edit Post #283293 Post edited: 4 months ago Edit Post #283656 Post edited: 4 months ago Edit Post #283656 Post edited: 4 months ago Edit Post #283656 Initial revision 4 months ago Question What is virtual work? $$\sumi Fi \cdot \delta ri$$ is virtual work when internal force is$0. For that reason, $$\sumi Fi \cdot \delta ri = 0$$ Here internal force stands for what? When a object's displacement is imaginary then, that object's work is virtual work. But, how a object's displacement can be imaginary? ... (more) 4 months ago Comment Post #283511 [Done](https://physics.codidact.com/posts/283618) (more) 4 months ago Edit Post #283618 Initial revision 4 months ago Question What happens if an electron collides with another electron? We know that electrons always repel electrons. But, if somehow they collide, then what will happen? I found a related question of above one in Quora. >An incoming electron can instead make them both bounce away. > >Two free electrons will bounce off each other unless perfectly aligned. In whi... (more) 4 months ago Edit Post #283511 Post edited: 4 months ago Comment Post #283573 Gotcha! Thanks. (more) 4 months ago Comment Post #283573 Newton said,g=\frac{Gm}{r^2}$. OK! I am taking$x=r$and,$a=g$.$a=\frac{Gm}{x^2}$. Although, acceleration is positive. I also know that when we push objects upward then, acceleration becomes negative. But, OP doesn't say we are pushing up or down. (more) 4 months ago Edit Post #283293 Post edited: 4 months ago Edit Post #281633 Post edited: 4 months ago Edit Post #283511 Post edited: 4 months ago Comment Post #283511 @#8049 Isn't electron, proton and positron noun? First later of noun is forever capital. But, in your suggestion it was small letter why? (more) 4 months ago Comment Post #283517 That should be$mg\cos \theta$instead of$mg$.. I made the mistake while drawing (more) 4 months ago Edit Post #283517 Initial revision 4 months ago Answer A: Why used$\cos\theta$for$\text{y}$axis or, gravitational force? picture Force is perpendicular to the surface. And, green color "rod" is parallel to the surface. So, $$F = N - mg\cos \theta=0$$ Reference (more) 4 months ago Edit Post #283397 Post edited: 4 months ago Edit Post #283511 Initial revision 4 months ago Question What happens if an electron collides with a proton? What will happen if an electron and a proton collide? They attract each other inside an atom. But, why they don't collide inside atoms? Does the "boundary" of the nucleus push them away? >depends on the energy of the electron. For low energies, a bound state will be formed due to electromagnetic i... (more) 4 months ago Edit Post #283448 Post edited: 4 months ago Comment Post #283410 @#8056 It is well-fitted in Q&A but, I found the problem in "Problems" that's why I asked it here. If you think that Q&A is better than Problems for the question than, you can flag for mod attention. They will move to Q&A (more) 4 months ago Edit Post #283448 Initial revision 4 months ago Answer A: Why answer matched but, unit? The error was on mass. In question : $$m=20 \ g$$ And, I wrote following one in answer $$m=20\times 10^3 \ kg$$ which is completely wrong. Cause,$1 \ kg=1000 \ g$(more) 4 months ago Edit Post #283420 Post edited: 4 months ago Comment Post #283420 Ohh! Actually, I didn't know that writing units in intermediate steps is required. I always write in last steps. So, Thanks for the information. And, I have completed the question also... :) (more) 4 months ago Edit Post #283420 Post edited: 4 months ago Comment Post #283420 I have edited... It was actually$kgm^{-3}$(more) 4 months ago Edit Post #283420 Post edited: 4 months ago Comment Post #283293 @#52997 So, I have to choose a random book which is looking good to me. If I can understand that properly then, I should move to another book. So, I have to read lots of books? (more) 4 months ago Edit Post #283420 Post edited: 4 months ago Edit Post #283420 Initial revision 4 months ago Question Why answer matched but, unit? >Mass of a timber is$20 \ g$. And, density of that timber is$0.27 \ g/cc$. That timber was bind to a metallic materials and, it was released to$0.970 \ g/cc$water. How much the wood was submerged in water? I was trying to solve the problem following way. $$F=Ah\rho g$$ $$=V\rho g$$ $$=V \... (more) 4 months ago Edit Post #283410 Initial revision 4 months ago Question Why used \cos\theta for \text{y} axis or, gravitational force? >figure 3.2figure 3.1 >Mass M1 is held on a plane with inclination angle θ, and mass M2 hangs over the side. The two masses are connected by a massless string which runs over a massless pulley (see Fig. 3.1). The coefficient of kinetic friction between M1 and the plane is µ. M1 is released from r... (more) 4 months ago Comment Post #283293 @#52997 Usually, I am not a school, college or University student. I am interested in learning Physics. That's why I am looking for a classical mechanics book which contains most of Classical topics and, little bit of Lagrangian, GR, SR (But, It's OK if that classical Mechanics book doesn't contain t... (more) 4 months ago Comment Post #283395 Since, photons always travel at constant speed. Hence, photons acceleration is forever 0. speed of photon changes depending on medium. (more) 4 months ago Edit Post #283397 Initial revision 4 months ago Question Find initial velocity when a stuntman jump from 1.25 \ m height >figure >A stuntman jumped from 1.25 \ \text{m} height and, landed at distance 10 \ \text{m}. Find velocity when he jumped. (Take \text{g}=10 \ ms^{-2}) I had solved it following way.$$h=\frac{1}{2}gt^2=>1.25=5\cdot t^2=>t=\frac{1}{2} \ s$$And,$$s=vtv=\frac{s}{t}$$... (more) 4 months ago Edit Post #283386 Post edited: \cdot or, direct cross product is better than asterisk 4 months ago Edit Post #283392 Post edited: The phrase (your moving at a constant speed) isn’t looking good to me 4 months ago Suggested Edit Post #283386 Suggested edit: \cdot or, direct cross product is better than asterisk (more) helpful 4 months ago Suggested Edit Post #283392 Suggested edit: The phrase (your moving at a constant speed) isn’t looking good to me (more) helpful 4 months ago Edit Post #283391 Post edited: 4 months ago Edit Post #283391 Post edited: 4 months ago Edit Post #283391 Initial revision 4 months ago Question Why we can't find a particle accelerating unless there's some other particle accelerating somewhere else? I was reading "Introduction to classical Mechanics" by David Morin. In that book they wrote that >The third law says we will never find a particle accelerating unless there’s some other particle accelerating somewhere else. The other particle might be far away, as with the earth–sun system, but it... (more) 4 months ago Comment Post #283304 Anyone can answer on that post thousand years later also. "That person" just want to find the perfect book available right now... (more) 4 months ago Comment Post #283258 @#8176 Yes! But, there's no way to migrate to another community :( So, we can flag it now (more) 4 months ago Edit Post #283293 Initial revision 4 months ago Question Book suggestion for Classical Mechanics Since, there's no book category currently so, I am asking it here. And, It can be moved to that category later. I want to start learning now so, I don't want to wait. Which book is best for Classical Mechanics? I have studied Classical Mechanics little bit. But, I want a book which explains th... (more) 4 months ago Edit Post #283272 Initial revision 4 months ago Question Book suggestion category proposal I was thinking for a new category called Book suggestion (not only book suggestion either). Usually, most of Science students like to study Science books (their favorite subject) by themselves. Some of them self-study rather than getting to University. And, some of them looks for book suggestion. If ... (more) 4 months ago Comment Post #283258 Seems, like computer science 🤔 (more) 4 months ago Edit Post #283072 Post edited: 4 months ago Edit Post #283072 Post edited: 4 months ago Comment Post #283046 History related question is off-topic (more) 4 months ago Edit Post #283072 Initial revision 4 months ago Question When selecting needs author attention it doesn't reach to moderator I had flagged the post as off-topic. There was a comment automatically created (Post Feedback). When I visited the flag raised page. I noticed it's not available anymore. Seems like, if I choose needs author attention than, it doesn't reach to moderator. But, some kind of questions can't be made on... (more) 4 months ago Comment Post #283046 I don’t think history related question is on-topic (more) 4 months ago Comment Post #283046 The question is off-topic (more) 4 months ago Suggested Edit Post #282727 Suggested edit: improve formatting (more) pending 5 months ago Edit Post #282668 Initial revision 5 months ago Answer A: What is inflation in cosmology? I haven't studied to much. I don't have any idea of Physics. Last few moments, I was just passing my time by reading theories. I have little bit idea of inflationary theory. History of the universe If you think of a balloon than, if it suddenly burst then, it will spread. Something happened... (more) 5 months ago Comment Post #281840 @Canina suppose, if i code a software and, i take the mobile(which have that software) in a "silent" room(where i can't hear anything) and, if i put the dB level 0 or less than 0 somehow than is that correct measurement? (more) 7 months ago Comment Post #281840 @Canina what is actually reference level? (more) 7 months ago Comment Post #281840 http://www.cochlea.org/en/hear/human-auditory-range (more) 7 months ago Edit Post #281840 Initial revision 7 months ago Question decibel level of human hearing(human-auditory-range) Decibel level listed of solfege. | Do | Re | Mi| Fa | So | La | Ti | Do |- | - | - |- |- |- |- |- | | 24 | 27 | 30 | 32 | 36 | 40 | 45 | 80 noise intensity I saw that whispering decibel level is around 40. And, normal voice level is around 60. And, Human can hear 0 - (120-130). So, my que... (more) 7 months ago Comment Post #281647 To reviewer, this post is off-topic in Q&A site. So, i would to request migrate it to problems if possible otherwise close it as off-topic (more) 7 months ago Suggested Edit Post #281649 Suggested edit: (more) declined 7 months ago Comment Post #281649 In my other post I had added main question. I forgot to add it here. EE site will be better for the problem. But, EE is a part of Physics. That's why I ask any Physics related question here. What current flows thru the 4 Ω resistor? I don't have the value. If I think of the circuit without 4 Ω than... (more) 7 months ago Edit Post #281647 Post edited: 7 months ago Edit Post #281647 Post edited: 7 months ago Edit Post #281647 Initial revision 7 months ago Question Delta to Wye conversion ![enter image description here][1] I have a circuit like this. And, I want to calculate resistance from here. I was following these steps. I was calculating resistance for left side circuit.$$R1=\frac{2 × 2}{2+2+4}=0.5\OmegaR2=\frac{2 × 4}{2+2+4}=1\OmegaR3=\frac{2 × 4}{2+2+4}=1... (more) 7 months ago Edit Post #281633 Post edited: mistook tag 7 months ago Edit Post #281633 Initial revision 7 months ago Question showing a downvote in statistics while I haven't got any downvote statistics In my Physics Codidact profile I am getting that I got 1 downvotes. I am not sure if I asked a question earlier in which I got 1 downvotes. In my current posts I don't have any downvotes or upvotes. Even, my reputation is OK. So, I think it is a bug in Statistics. For better understa... (more) 7 months ago Comment Post #281631 Yes! I had solved those problem. I just added them as sample. I have invited once my friend. That's why I had to ask this type question. (more) 7 months ago Edit Post #281629 Post edited: quoting 7 months ago Edit Post #281629 Initial revision 7 months ago Question Is "homework-and-exercises" off-topic in Codidact? In SE sites, homework-and-exercises question isn't allowed in Physics SE. So, I want to know does Codidact also mark home-and-exercises question as off-topic? I have earlier asked twice question on homework-and-exercises but, I didn't get any response cause, there's just few active people in Ph... (more) 7 months ago Edit Post #281581 Post edited: 7 months ago Edit Post #281581 Post edited: 7 months ago Edit Post #281581 Post edited: 7 months ago Edit Post #281581 Initial revision 7 months ago Question Slipping and rotation >A disk of mass$M$and radius$R$is initially rotating at angular velocity of$\omega$. While rotating, it is placed on a horizontal surface whose coefficient of friction is$\mu=0.5$. How long will it take for the disk to stop rotating? Someone had solved the problem following way \begin{al... (more) 7 months ago Edit Post #281568 Initial revision 7 months ago Answer A: Calculating frequency of oscillation While force is vector. So, we should use vector addition not "direct addition". From vector addition formula $$C=\sqrt{A^2+B^2+2ABcos\alpha}$$ For force, $$F=Eq$$ $$F=mg$$ $$F=\sqrt{(Eq)^2+(mg)^2+2Eqmgcos\alpha}$$$$f=\frac{1}{2\pi} \sqrt{\frac{\sqrt{(Eq)^2+(mg)^2+2Eqmgcos\alpha}}{m... (more) 7 months ago Comment Post #281541 @Olin Lathrop Yes! I understood the question(maybe, 1 or 2 hours ago) after asking the question. (more) 7 months ago Edit Post #281541 Initial revision 7 months ago Question Calculating frequency of oscillation > A 4C point charge of mass 2kg is suspended by a string of length 6m. The charge is placed in the Earth's Gravitational field and a uniform horizontal electric field of strength$\sqrt{11}NC^{-2}\$. If the charge is displaced slightly from equilibrium, what will be the frequency of oscillation? (g=10... | 2021-12-08 14:52:16 | {"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": 1, "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": 0.6764591336250305, "perplexity": 2162.2969370766923}, "config": {"markdown_headings": true, "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-2021-49/segments/1637964363515.28/warc/CC-MAIN-20211208144647-20211208174647-00432.warc.gz"} |
https://florida-mortgage-lenders.com/rcmp-civilian-fujp/article.php?id=state-diagram-calculator-b466e1 | Show Instructions. It also shows plots of the function and illustrates the domain and range on a number line to enhance your mathematical intuition. In general, you can skip parentheses, but be very careful: e^3x is e^3x, and e^(3x) is e^(3x). Free online beam calculator for generating the reactions, calculating the deflection of a steel or wood beam, drawing the shear and moment diagrams for the beam. By using this website, you agree to our Cookie Policy. VP Online features a powerful UML diagram tool that lets you create state machine diagram and other UML diagrams easily and quickly. b press. At the start of a design the total number of states required are determined. Built with Noam, Bootstrap, Viz.js, and jQuery. State Diagram is made with the help of State Table. IP SI altitude. For example, in the left figure above (due to R. Abbott), a car must traverse a town while obeying all traffic laws, and making no U-turns. Created by Ivan Zuzak and Vedrana Jankovic. State Diagrams and State Tables. Fundamental to the synthesis of sequential circuits is the concept of internal states. Use our online Venn diagram calculator to create Venn diagram for two sets or two circles. The Mohr's Circle calculator provides an intuitive way of visualizing the state of stress at a point in a loaded material. Thus, the output of the circuit at any time depends upon its current state and the input. T dry bulb rH rel.humidity TDP dew point W abs. This is achieved by drawing a state diagram, which shows the internal states and the transitions between them. Free functions calculator - explore function domain, range, intercepts, extreme points and asymptotes step-by-step This website uses cookies to ensure you get the best experience. Enter the value of set A and B, and also both of their union and intersection values to create Venn diagram. A state diagram is a labeled directed graph together with state information that can be used to indicate that certain paths on in a system may be traversed only in a certain way. Venn Diagram: Venn Diagram can also be referred as primary diagram, set diagram or logic diagram. Use this beam span calculator to determine the reactions at the supports, draw the shear and moment diagram for the beam and calculate the deflection of a steel or wood beam. See the reference section for details on the methodology and the equations used. End Behavior Calculator. You can construct your diagrams with drag and drop, save your work in cloud workspace, output and share your design via numerous formats such as PNG, JPG, SVG, PDF, etc. This calculator will determine the end behavior of the given polynomial function, with steps shown. More than just an online function properties finder. We have examined a general model for sequential circuits. Print. humidity h enthalpy Tw wet bulb sp. FSM simulator is a demo of using noam, a JavaScript library for working with finite-state machines, grammars and regular expressions. These also determine the next state of the circuit. State diagrams are also known as problem space models (Atallah 1998, p. 36-2). Free online Psychrometric Calculator . In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. Make accept state: double-click on an existing state; Type numeric subscript: put an underscore before the number (like "S_0") Type greek letter: put a backslash before it (like "\beta") This was made in HTML5 and JavaScript using the canvas element. The arrows represent the Next State of Flip-Flops. State Tables and State Diagrams. Wolfram|Alpha is a great tool for finding the domain and range of a function. The Binary number inside the circle represent the Present State of Flip-Flop . Code available on GitHub and licensed under Apache License v2.0. In this model the effect of all previous inputs on the outputs is represented by a state of the circuit. For working with finite-state machines, grammars and regular expressions line to enhance your mathematical intuition and regular expressions and! Calculator will determine the next state of the function and illustrates the domain and range of function! Dry bulb rH rel.humidity TDP dew point W abs states required are determined states and the between. Atallah 1998, p. 36-2 ) of state Table the Binary number inside Circle. And jQuery on the methodology and the transitions between them at the start of a function model. The transitions between them at any time depends upon its current state and the equations used are also as. 5 * x Viz.js, and also both of their union and intersection to! The Circle represent the Present state of Flip-Flop you can skip the multiplication,! 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X ` JavaScript library for working with finite-state machines, grammars and regular expressions fsm simulator is a demo using... Point in a loaded material regular expressions polynomial function, with steps shown sequential circuits is the concept internal. Union and intersection values to create Venn diagram can also be referred primary! And also both of their union and intersection state diagram calculator to create Venn diagram for sets... Both of their union and intersection values to create Venn diagram for two sets two. Output of the function and illustrates the domain and range on a number line to enhance your mathematical intuition to. A point in a loaded material machines, grammars and regular expressions visualizing state... Reference section for details on the methodology and the transitions between them material!
Active Directory Security Group Permissions, Powershell Suppress Error, Dow Lab Johar Branch Contact Number, How To Block Youtube On Toshiba Smart Tv, Affidavit Of Consideration Definition, Largest Russian Battleship Ww2, Mn Dnr Deer Hunting, Rickey Smiley Daughter Update, | 2021-06-19 22:01:47 | {"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": 0.5049153566360474, "perplexity": 1522.590872489279}, "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-2021-25/segments/1623487649731.59/warc/CC-MAIN-20210619203250-20210619233250-00611.warc.gz"} |
https://answerriddle.com/answer-where-is-the-narrowest-house-in-america/ | # Answer: Where is the narrowest house in America?
The Question: Where is the narrowest house in America?
Idaho
Virginia
California
New York | 2021-06-18 15:16:08 | {"extraction_info": {"found_math": false, "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": 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": 0.9169520735740662, "perplexity": 10100.656513646574}, "config": {"markdown_headings": true, "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-2021-25/segments/1623487637721.34/warc/CC-MAIN-20210618134943-20210618164943-00626.warc.gz"} |
https://collegephysicsanswers.com/openstax-solutions/indigenous-people-sometimes-cook-watertight-baskets-placing-hot-rocks-water | Change the chapter
Question
Indigenous people sometimes cook in watertight baskets by placing hot rocks into water to bring it to a boil. What mass of $500^\circ\textrm{C}$ rock must be placed in 4.00 kg of $15.0^\circ\textrm{C}$ water to bring its temperature to $100^\circ\textrm{C}$, if 0.0250 kg of water escapes as vapor from the initial sizzle? You may neglect the effects of the surroundings and take the average specific heat of the rocks to be that of granite.
$4.38\textrm{ kg}$
Solution Video | 2021-10-25 23:21:46 | {"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": 0.3017360270023346, "perplexity": 890.8339853370069}, "config": {"markdown_headings": true, "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-2021-43/segments/1634323587770.37/warc/CC-MAIN-20211025220214-20211026010214-00370.warc.gz"} |
https://tex.stackexchange.com/questions/145413/gb4e-example-with-two-paragraphed-footnote-throws-error | # gb4e example with two-paragraphed footnote throws error
gb4e is a useful package for writing linguistics papers, as it allows one to append judgments to examples, via the optional argument of \ex{}.
As such, I would like to retain the use of this package, but I have just run into a problem. I'm not sure what is going on internally, but it seems that gb4e prevents paragraph breaks from occurring anywhere within the scope of the required argument of \ex{}.
In particular, I have a footnote attached to an example to clarify an aspect of the example. If I try to put a paragraph break inside of this footnote, using either a blank line or the TeX primitive \par, an error is thrown.
Here's an MWE:
\documentclass{article}
\usepackage{gb4e}
\begin{document}
\begin{exe}
\ex[]{An example requiring a footnote\footnote{A footnote with two paragraphs.\par Here's the second paragraph.}}
\end{exe}
\end{document}
And the resulting error:
Runaway argument? {An example requiring a footnote.\footnote {A footnote with two parag\ETC. ./test.tex:8: Paragraph ended before \@ex was complete. \par l.8 ...ootnote{A footnote with two paragraphs.\par Here's the second paragra...
?
Is it possible to achieve a multiple-paragraphed footnote inside of a gb4e example, or is there a workaround that would allow me to retain the use of gb4e?
You can change \par onto \endgraf. (The height of the page is intentionally reduced on a picture).
\documentclass{article}
\usepackage{gb4e}
\begin{document}
\begin{exe}
% \ex[]{An example requiring a footnote\footnote{A footnote with two paragraphs.\par Here's the second paragraph.}}
\ex[]{An example requiring a footnote\footnote{A footnote with two paragraphs.\endgraf Here's the second paragraph.}}
\end{exe}
\end{document} | 2019-10-17 05:33:23 | {"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": 0.9128407835960388, "perplexity": 3156.878935501191}, "config": {"markdown_headings": true, "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-2019-43/segments/1570986672723.50/warc/CC-MAIN-20191017045957-20191017073457-00080.warc.gz"} |
https://eprint.iacr.org/2017/1260/20190908:065559 | ## Cryptology ePrint Archive: Report 2017/1260
Collision Resistant Hashing from Sub-exponential Learning Parity with Noise
Yu Yu and Jiang Zhang and Jian Weng and Chun Guo and Xiangxue Li
Abstract: The Learning Parity with Noise (LPN) problem has recently found many cryptographic applications such as authentication protocols, pseudorandom generators/functions and even asymmetric tasks including public-key encryption (PKE) schemes and oblivious transfer (OT) protocols. It however remains a long-standing open problem whether LPN implies collision resistant hash (CRH) functions. Based on the recent work of Applebaum et al. (ITCS 2017), we introduce a general framework for constructing CRH from LPN for various parameter choices. We show that, just to mention a few notable ones, under any of the following hardness assumptions (for the two most common variants of LPN)
1) constant-noise LPN is $2^{n^{0.5+\epsilon}}$-hard for any constant $\epsilon>0$;
2) constant-noise LPN is $2^{\Omega(n/\log n)}$-hard given $q=poly(n)$ samples;
3) low-noise LPN (of noise rate $1/\sqrt{n}$) is $2^{\Omega(\sqrt{n}/\log n)}$-hard given $q=poly(n)$ samples.
there exists CRH functions with constant (or even poly-logarithmic) shrinkage, which can be implemented using polynomial-size depth-3 circuits with NOT, (unbounded fan-in) AND and XOR gates. Our technical route LPN$\rightarrow$bSVP$\rightarrow$CRH is reminiscent of the known reductions for the large-modulus analogue, i.e., LWE$\rightarrow$SIS$\rightarrow$CRH, where the binary Shortest Vector Problem (bSVP) was recently introduced by Applebaum et al. (ITCS 2017) that enables CRH in a similar manner to Ajtai's CRH functions based on the Short Integer Solution (SIS) problem.
Furthermore, under additional (arguably minimal) idealized assumptions such as small-domain random functions or random permutations (that trivially imply collision resistance), we still salvage a simple and elegant collision-resistance-preserving domain extender that is (asymptotically) more parallel and efficient than previously known. In particular, assume $2^{n^{0.5+\epsilon}}$-hard constant-noise LPN or $2^{n^{0.25+\epsilon}}$-hard low-noise LPN, we obtain a polynomially shrinking collision resistant hash function that evaluates in parallel only a single layer of small-domain random functions (or random permutations) and produces their XOR sum as output.
Category / Keywords: foundations / Learning Parity with Noise, Collision Resistant Hashing, binary Shortest Vector Problem
Original Publication (with minor differences): IACR-ASIACRYPT-2019
Date: received 27 Dec 2017, last revised 8 Sep 2019
Contact author: yuyuathk at gmail com
Available format(s): PDF | BibTeX Citation
Short URL: ia.cr/2017/1260
[ Cryptology ePrint archive ] | 2019-11-16 21:30:55 | {"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": 0.6710257530212402, "perplexity": 9401.757263933156}, "config": {"markdown_headings": true, "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-2019-47/segments/1573496668765.35/warc/CC-MAIN-20191116204950-20191116232950-00536.warc.gz"} |
https://leanpub.com/arc42inpractice/read | ## Acknowledgments
Many people helped in the last few years with constructive comments and critique, advice and questions around arc42.
Ralf D. Müller is the good soul and tireless committer in the arc42 open-source universe: He answers support questions, maintains the tool-chain that generates arc42 from its AsciiDoc sources and keeps our Github-issues in check. Thank you so much!
We like to thank Martin Dungs, Uwe Friedrichsen, Phillip Ghadir, Mahbouba Gharbi, Franz Hofer, Prof. Arne Koschel, Jürgen Krey, Anton Kronseder, Prof. Bernd Müller, Alex Nachtigall, Axel Noellchen, Robert Reiner, Markus Schärtel, Roland Schimmack, Michael Simons, Boris Stumm, Daniel Takai, Eberhard Wolff, Oliver Wronka and Stefan Zörner for their support during preparation of this book.
Special thanx to Pedro Lafuente Blanco, Stefan Paal, Christopher Schmidt, Silvia Schreier, Per Starke and Oliver Tigges for their constructive and insightful reviews and suggestions.
innoQ supports arc42 by hosting the arc42 Confluence wiki and the arc42.org website. Christian Sarazin clears away all the technical debris. Thanx to my skillful, knowledgeable, competent and critical colleagues.
Cover design by Andreas Steinbrecher.
Parts of translation from the German original were supported by Sven Johann and Per Starke.
Gernot: thanks Uli, Lynn und Per: You are superb, the best family in the universe. My time with you is always too short.
Peter: Special thanks to my wife Monika. You not only survived yet another book project but enriched it by your helpful insights and comments from the non-IT world.
#### Status
the current version of this book is a pre-release: Only a few parts have been finished, many others (especially chapter IV) are still under heavy work.
We do our best to fill the missing gaps asap - and we will continue to publish early, publish often. Leanpub will notify you of updates.
#### Revisions
• V 0.2.0: March 26th 2016: first published version, containing I-III and V, skeletons of VI and VII.
• March 2016: new cover (thanx, Andreas), chapter 1, 2, 3
• January 2016: initial German version, start of translation on Leanpub.
• March 2016: Sven Johann joined the translation effort (thx!)
• April 2016: moved to Github with webhooks
## I. Introduction
May the force of the proper word and diagram be with you.
Often software systems are in use for many years - and are constantly maintained and improved. Sometimes the life of software comes to a tragic end: missing maintainability and insufficient documentation are a risky game with uncertain outcome even for small changes. Marginal extension can only be done with massive efforts. The business value of the system fades away.
As a software architect, design your systems in a way that they are maintainable, flexible and understandable, thus thoroughly avoiding their „decay“. To achieve that you should not only fulfill the functional requirements but also concentrate on inner qualities of your system. One of your jobs is to adequately communicate its architecture in both written form and by word of mouth.
In this book we introduce you to arc42, a proven, practical standard for documenting and communicating software architecture – available free of charge! arc42 is based on years of experience and used successfully since 2005 in many companies and organizations in different application domains.
First of all arc42 is a template for architecture documentation.
It answers the following two questions in a pragmatic way, but can be tailored to your specific needs:
• What should we document/communicate about our architecture?
• How should we document/communicate?
Figure I.1 gives you the big picture: It shows a (slightly simplified) overview of the structure of arc42.
In case you are impatient and immediately want to see a worked-out example of arc42-documentation, you can jump to chapter II. It will show you how arc42-documentation feels like.
For the rest of you (or readers returning from chapter II) we will give you background information that will make your work with arc42 easier.
### I.1 Basic Principles of arc42
##### Clear Structure
Compare arc42 to a cabinet1 with drawers: the drawers are clearly marked with labels indicating the content of each drawer. arc42 contains 12 such drawers (a few more than you see in the picture above). The meaning of these arc42 drawers is easy to understand.
Therefore, arc42 offers you a simple and clear structure to document and communicate your (complex!) system. Starting with the goals and requirements for your system and its embedding
into its environment you can provide the important stakeholder of your system with adequate information about the architecture.
arc42 is optimized for understandability and adequacy. It naturally guides you to explain any kind of architecture information or decision in an understandable and reproducible context.
Individuals and organizations using arc42 especially like two things about it:
1. the understandability of the documentation resulting from its standardized structure (the drawers) and
2. the manageable effort to create such documentation. We call it “painless documentation”.
##### Independent of process models
You can work on the drawers of the arc42 cabinet in any order – whatever seems useful and adequate under your constraints. When you start a development from scratch you will probably start with requirements and goals; when you maintain or extend existing systems you may dive into the details of the building blocks immediately.
With arc42 you can start and stop working on your architecture documentation any time. The fixed structure of your cabinet enables you to continue any time. The only prerequisite is an agreed understanding among all team members about the meaning of the arc42-drawers.
Therefore arc42 is completely process agnostic.
##### Architecture documentation with little effort
The arc42 template is a framework for creativity and architecture work, a basis for discussion and feedback among your stakeholders, support for familiarization of rookies and much more.
arc42 is available under a liberal open source license. Therefore you can use it free of charge, even in commercial environment. Working with the arc42 template does not require additional effort for you and your team:
• You only describe things that your stakeholders really have to know.
• You explain facts and issues that are necessary to understand the system or individual design decisions.
• You only keep track of important architecture decision that you had to make anyhow.
arc42 helps you to find the most adequate drawer where facts, features and decisions are kept, so that all stakeholders can easily find them again.
##### Risk: Template Zombies2
We want to warn you about one risk when using arc42 template: users could interpret the template as a form and could be tempted to fill every field . We seriously dislike terms like “fill in” and “form”.
arc42 is intended to be a lightweight tool that can easily be adapted to your specific need. It is not nearly a “fill-in all fields” form.
Our deep aversion against forms and our fear of misuse of arc42 has motivated us to describe a series of basic tips (in chapter III). You should especially obey tip III-2 (economy) and tip III-3 (adequacy).
### I.2 Why This Book
Organizations apply arc42 since 2005 to document software architectures. Since that time we (the authors) helped to introduce arc42, restructure existing documentations according to the clear structure of arc42 or develop and document new systems with arc42.
The downloadable template contains short hints and tips for each parts (the drawers). The overall structure is easy to understand; there are no hurdles to use the template.
Despite this simple and clear structure we have been confronted with a lot of practical every-day questions. We have answered those questions in this book – in the style of the missing manual3.
### I.3 What This Book is NOT
In this book we focus on effective and efficient use of arc42 to document and communicate software architectures. We explicitly excluded a lot of other topics or disciplines.
This book is no introduction to:
• Software architecture and design. We assume that you have basic methodology know-how about software design and development. You are able to distinguish problem (requirements) from solutions (architecture, implementation). You know principles like separation of concern, information hiding, loose coupling, strong cohesion, simplicity, high consistency, and you separate domain and technical aspects in your implementation. You have learned about different views of your software architecture and cross-cutting concepts. We collected a few resources in the appendix.
• Your favorite technology: Sometimes people confuse specific technologies or frameworks with software architecture. We won’t intr#oduce any implementation technologies here, although you need to know about these when you design and implement systems.
• Architecture and Design Patterns: Productive software developers and architects immensely benefit by reusing well-proven solution approaches also known as patterns. You can find patterns (== proven solutions) for various problem domains, described in excellent printed and online sources. Fire up your favorite search engine…
• Modeling: You should be able to use models to abstract static and dynamic aspects of your system. We assume that you know how static models (building block, components, modules and their relationships) relate to the source code. And you also know basics about dynamic models (runtime models, process models)
• UML (Unified Modeling Language): We will explain in many of the chapters of this book how to use UML pragmatically and effectively. But we assume that you have basic knowledge about class diagrams, component diagrams, sequence and activity diagrams as well as deployment diagrams. More information can be found in [Booch+05] or [Pilone-05].
• Requirements Engineering and Business Analysis: software architects have to understand and potentially clarify requirements for a system. They may have to elicit, document and manage requirements – i.e. do work that ideally would have been done by requirements analysts. We assume that you know about functional requirements and quality requirements and are able to discuss them with your stakeholders. A good source for more information is [Robertson-12].
Although our tips mainly refer to effective usage of arc42, many of them will help you and your team to design and implement better systems.
### I.4 Our Assumptions About You
When writing this book, we (the authors) had several (potentially silly) assumptions about you (the readers) in mind:
• First of all, you have loads of work to do. Therefore, you only want to read those parts of the book that you consider to be relevant for your work. For this purpose we have written the navigation guide in section I.5.
• You are an experienced software architect or developer. Therefore, you are aware of adequate documentation. You have practical experience with the development of architectures and software systems and therefore you are aware what adequate documentation should be.
• You are developing or maintaining mid-size to large, sometimes complex software systems.
• You work under timing constraints and only want to read the relevant parts of the book. Therefore, we offer you a navigation guide in section I.5.
• We assume that sometimes in your professional life you have suffered from missing or excessive documentation.
• You want to communicate or document information about the architecture, the structure and the implementation of your system.
• You do not want to waste time; you want to keep the effort on an adequate level: for some systems you will need detailed information; for other systems it is sufficient to concentrate on core topics.
• Maybe you are already aware of arc42 and want to know how to apply the template more easily, more effective or with more pragmatism.
### I.5 Quick Navigation
Since you are very busy in your everyday life you might be interested in identifying the parts of this book that are most relevant for your concrete problems.
The following diagram gives an overview of this book. We have numbered the main chapters with roman numerals (I, II, III) so that you can easily distinguish book chapters from the section numbers in the arc42 template - for which we use the normal Arabic numerals 1 to 12.
Chapter II demonstrates the use of the arc42 template via a small open-source system. In this example you will see how a concrete architecture documentation could look like. For each of the 12 sections of the template you will also find a short motivation explaining why this section is in the template and what should be captured. You can read chapter II independently from the rest of the book.
Chapter III explains some basis rules of adequate architecture documentation, especially our pledge for “systematic thriftiness”.
Chapter IV contains tons of practical tips for each section of arc42 (i.e. for each drawer – to pick up the metaphor from the introduction). We briefly motivate each arc42 section. Therefore, there is deliberate (small) redundancy with Chapter II. Here we give answers to the questions that we received from our users and seminar participants. This is the most extensive part of the book.
Chapter V explains how to use arc42 in everyday life. You will find hints for creating new systems or amending existing systems, for agile projects and for very large projects with many subsystems.
Chapter VI introduces tools and tool categories you can use to bring arc42 to life.
Chapter VII answers frequently asked questions in various categories (sometimes referring to the tips of Chapter III to VI). We maintain and update this FAQ chapter online.
### I.6 Conventions
##### Tip I-1: Our numbering scheme
In this book we offer more than 200 different tips around arc42. The numbering scheme aligns them with the corresponding book chapters: the roman prefixes (III, IV, V etc.) refers to the book chapter, the arabic number sequentially numbers tips within chapters.
## II. arc42 by Example
Chapter IV contains practical tips for each part of arc42.
The system documented here is a small open source system hosted on Github.
##### Tip II-1: Improve your arc42 skills by reading additional examples
Luckily there are additional examples for arc42 architecture documentation available from several different authors.
• There’s a Leanpub book, arc42 by Example, that contains several examples.
• Managing bike tours, by Michael Simons: Java based app, productive since several years.
• Chess engine, by Stefan Zörner (in German).
• Customer relationship management (CRM).
• VENOM, a large scale (artificial) e-commerce system. Contains fragments from various real systems, combined into a single legacy tragedy.
If you like YOUR example to be listed here, just drop us an email.
‘nough preamble. Let’s get started…
### II.1. Introduction and Goals
HtmlSC supports authors creating digital formats by checking hyperlinks, images and similar resources.
#### 1.1 Requirements Overview
The overall goal of HtmlSC is to create neat and clear reports, showing errors within HTML files. Below you find a sample report.
HtmlSanityCheck (HtmlSC) checks HTML for semantic errors, like broken links and missing images.
It has been created to support authors who create HTML as output format.
1. Authors write in formats like AsciiDoc, Markdown or other formats, which are transformed to HTML by the corresponding generators.
2. HtmlSC checks the generated HTML for broken links, missing images and other semantic issues.
3. HtmlSC creates a test report, similar to the well-known unit test report.
##### Basic Usage
1. A user configures the location (directory and filename) of an HTML file, and the corresponding images directory.
2. HtmlSC performs various checks on the HTML and
3. reports its results either on the console or as HTML report.
HtmlSC can run from the command line or as Gradle plugin.
##### Basic Requirements
ID Requirement Explanation
G-1 Read HTML files HtmlSC shall read one or several (configurable) HTML files as input.
G-2 Gradle Plugin usage HtmlSC can be run/used as Gradle plugin
G-3 Command line usage HtmlSC can be run from the command line (shell)
G-4 Open source license All required dependencies/libraries shall be compatible with a CreativeCommons license.
G-5 Public repositories HtmlSC shall be available via public repos, like the Gradle plugin portal.
G-6 Multiple input files Configurable for a set of files, processed in a single run, HtmlSC produces a joint report.
G-7 Suggestions When HtmlSC detects errors, it shall identify suggestions or alternatives that would repair the error
##### Required Functions
HtmlSC shall provide the following checks in HTML files:
ID Requirement Explanation
C-1 Missing images Check all image tags if the referenced image files exist.
C-2 Broken internal links Check all internal links from anchor-tags (href=”#XYZ”) if the link targets “XYZ” are defined.
C-3 Missing local resources Check if referenced files (e.g. css, js, pdf) are missing.
C-4 Duplicate link targets Check all link targets (… id=”XYZ”) if the id’s (“XYZ”)are unique.
C-5 Malformed links Check all links for syntactical correctness.
C-6 Unused images Check for files in image-directories that are not referenced in any HTML files in this run.
C-7 Illegal link targets Check for malformed or illegal anchors (link targets).
C-8 Broken external links Check external links for both syntax and availability.
C-9 Broken ImageMaps Though ImageMaps are a rarely used HTML construct, HtmlSC shall shall identify the most common errors in their usage.
##### Reporting and Output Requirements
ID Requirement Explanation
R-1 Various output formats Checking output in plain text and HTML
R-2 Output to stdout HtmlSC can output results on stdout (the console)
R-3 Configurable output directories HtmlSC can store results in file in configurable output directories.
#### 1.2 Quality Goals
Priority Quality-Goal Scenario
1 Correctness Every broken internal link (cross reference) is found.
1 Correctness Every missing local image is found.
2 Flexibility Multiple checking algorithms, report formats and clients. At least Gradle, command-line and a graphical client have to be supported.
2 Safety Content of the files to be checked is never altered.
2 Correctness Correctness of every checker is automatically tested for positive AND negative cases.
2 Correctness Every reporting format is tested: Reports must exactly reflect checking results.
3 Performance Check of 100kB html file performed under 10 secs (excluding Gradle startup)
#### 1.3 Stakeholders
Remark: For our simple HtmlSC example we have an extremely limited number of stakeholders, in real-life you will most likely have many more stakeholders!
Role Description Goal, Intention
Documentation author writes documentation with HTML output wants to check that the resulting document contains good links, image references.
arc42 user uses the arc42 template for architecture documentation wants a small but practical example of how to apply arc42.
software developer wants an example of pragmatic architecture documentation
• Section IV-1 contains additional tips regarding requirements in general.
• In reality, the quality requirements or pretty often neglected and/or remain implicit.
• A more complete overview of quality goals can be given in section-iv-10 on quality scenarios
### II.2 Constraints
HtmlSC shall be:
• platform-independent and should run on the major operating systems (Windows(TM), Linux, and Mac-OS(TM))
• implemented in Java or Groovy
• integrated with the Gradle build tool
• runnable from the command line
• developed under a liberal open-source license
• Section IV-2 contains additional tips regarding constraints.
### II.3 Context
#### 3.1 Business Context
Neighbor Description
user documents software with toolchain that generates html. Wants to ensure that links within this HTML are valid.
build system mostly Gradle
local HTML files HtmlSC reads and parses local HTML files and performs sanity checks within those.
local image files HtmlSC checks if linked images exist as (local) files.
external web resources HtmlSC can be configured to optionally check for the existence of external web resources. Due to the nature of web systems, this check might need significant time and might yield invalid results due to network and latency issues.
#### 3.2 Deployment Context
The following diagram shows the participating computers (nodes) with their technical connections plus the major artifacts of HtmlSC, the hsc-plugin-binary.
Node / Artifact Description
hsc-development where development of HtmlSC takes place
hsc-plugin-binary compiled and packaged version of HtmlSC including required dependencies.
artifact repository global public cloud repository for binary artifacts, similar to MavenCentral. HtmlSC binaries are uploaded to this server.
hsc user computer where arbitrary documentation takes place with html as output formats.
build.gradle Gradle build script configuring (among other things) the HtmlSC plugin to perform the HTML checking.
For details see the deployment-view.
• Section IV-3 contains additional tips regarding the business and/or technical context.
• Details of the external interfaces shown in the context might be elaborated in the building block view. See section IV-5
### II.4 Solution Strategy
1. Implement HtmlSC mostly in the Groovy programming language and partially in Java with minimal external dependencies.
2. We wrap this implementation into a Gradle plugin, so it can be used within automated builds. Details are given in the Gradle userguide.
3. Apply the template-method-pattern to enable:
• Section IV-4 contains additional tips regarding the solution strategy.
• Keep this part of your documentation very short, move extensive explanations, examples etc. to arc42-section 8 (concepts).
### II.5 Building Block View
#### 5.1 Whitebox HtmlSanityChecker
Rationale: We used functional decomposition to separate responsibilities:
• HSC Core shall encapsulate checking logic and HTML parsing/processing.
• HSC Gradle Plugin encapsulates all Gradle specific stuff
• Various kinds of UI (console, graphical) are handled by separate components.
Contained Blackboxes:
Building block Description
HSC_Core HTML parsing and sanity checking, file handling
HSC Gradle Plugin integrates the Gradle build tool with HtmlSC, enabling arbitrary gradle builds to use HtmlSC.
HSC Command Line Interface (not documented)
HSC Graphical Interface (planned, not implemented)
(Gradle based) Build System builds the output artifacts (plugin, jar)
##### 5.1.1 HSC Core (Blackbox)
Intent/Responsibility: HSC_Core contains the core functions to perform the various sanity checks. It parses the html file into a DOM-like in-memory representation, which is then used to perform the actual checks.
Interfaces:
Interface (From-To) Description
Command Line Interface -> Checker Uses the AllChecksRunner class.
Gradle Plugin -> Checker Exposes HtmlSC via a standard Gradle plugin, as described in the Gradle user guide.
Details are described in the HSC-Core Whitebox.
#### 5.2 Building Blocks - Level 2
##### 5.2.1 HSC-Core (Whitebox)
Rationale: This structures follows a strictly functional decomposition:
• parsing and handling HTML input,
• checking,
• creating suggestions and
• collecting checking results
Contained Blackboxes:
Building block Description
Checker Abstract class, used in form of the template-pattern. Shall be subclassed for all checking algorithms.
AllChecksRunner Facade to the different Checker instances. Provides a (parameter-driven) command-line interface.
ResultsCollector Collects all checking results.
Reporter Reports checking results to either console or file.
HtmlParser Encapsulates HTML parsing, provides methods to search within the (parsed) DOM tree. We use the open source JSoup, see the corresponding design decision.
Suggester In case of checking issues, suggests alternatives (did you mean xyz?). Suggestions are included in results.
Source Files:
• org.aim42.htmlsanitycheck.AllChecksRunner
• org.aim42.htmlsanitycheck.HtmlSanityCheckGradlePlugin
###### 5.2.1.1 Checker and <xyz>Checker Subclasses
The abstract Checker provides a uniform interface (public void check()) to different checking algorithms. It is based upon the concept of extensible checking algorithms.
#### 5.3 Building Blocks - Level 3
##### 5.3.1 ResultsCollector (Whitebox)
Rationale: This structures follows the hierarchy of checks, managing results for:
1. a number of pages/documents
2. a single page, each containing many
3. single checks within a page
Contained Blackboxes:
Building block Description
Per-Run Results Results for potentially many HTML pages/documents.
SinglePageResults Results for a single HTML page
SingleCheckResults Results for a single type of check (e.g. missing-images check or broken-internal-link check)
Finding A single finding, (e.g. “image ‘logo.png’ missing”). Can contain suggestions.
##### 5.3.2 Interface Results
The Result interface is used by all clients (especially Reporter subclasses, graphical and command-line clients) to access checking results. It consists of three distinct methods for:
1. overall RunResults,
2. single-page results (PageResults) and
3. single-check results (SingleCheckResults).
See the interface definitions below - taken from the Groovy source code:
##### 5.3.3 Suggester (Whitebox)
For a give input (target), Suggester searches within a set of possible values (options) to find the n most similar values. For example:
• Target = “McDown”
• Options = {“McUp”, “McDon”, “Mickey”}
• The resulting suggestion would be “McDon”, because it has the greatest similarity to the target “McDown”.
Suggester is used in the following cases:
• Broken image links: Compares the name of the missing image with all available image file names to find the closest match.
• Missing cross references (broken internal links): Compares the broken link with all available link targets (anchors).
• Section IV-4 contains additional tips regarding the building block view.
### II.6 Runtime View
#### II.6.1 Execute all checks
A typical scenario within HtmlSC is the execution of all available checking algorithms on a set of HTML pages.
Precondition: HtmlSC has been called from a Gradle build and is properly configured.
Scenario:
1. User or build-server calls gradle htmlSanityCheck
2. Gradle executes build-target htmlSanityCheck via the HtmlSC plugin
3. The plugin executes method AllChecksRunner.performAllChecks with parameter Collection<File> filesToCheck andFile resultsDir
4. performAllChecks:
1. creates PerRunResults instance
2. creates a list of Checkerinstances, one for every available checking algorithm
5. iterate over all these Checkerinstances. Every instance:
1. executes its own checks
2. adds one instance of SingleCheckResult containing its own results.
6. reportCheckingResultsAsHTML creates the final report.
#### II.6.2 Report checking results
Reporting is done in the natural hierarchy of results (see the corresponding concept in section 8.2.1 for an example report).
1. per “run” (PerRunResults): date/time of this run, files checked, some configuration info, summary of results
2. per “page” (SinglePageResults):
3. create page result header with summary of page name and results
4. for each check performed on this page create a section with SingleCheckResults
5. per “single check on this page” report the results for this particular check
• Section IV-6 contains additional tips regarding the runtime view.
• You often use runtime scenarios only to find or verify building blocks, not that much for documentation.
### II.7 Deployment view
Node / Artifact Description
hsc plugin binary Compiled version of HtmlSC, including required dependencies.
hsc-development Development environment
artifact repository Global public cloud repository for binary artifacts, similar to mavenCentral HtmlSC binaries are uploaded to this server.
hsc user computer Where documentation is created and compiled to HTML.
build.gradle Gradle build script configuring (among other things) the HtmlSC plugin.
The three nodes (computers) shown in the diagram above are connected via Internet.
Prerequisites:
• HtmlSC developers need a Java development kit, Groovy, Gradle plus the JSoup HTML parser.
• HtmlSC users need a Java runtime (> 1.6) plus a build file named build.gradle. See below for a complete example.
• Section IV-7 contains additional tips regarding the deployment view.
• Sometimes it’s useful to describe high-level deployment or infrastructure in the technical context in arc42 section 3.2, corresponding tips are given in this book
### II.8 Technical and Crosscutting Concepts
#### 8.1 Domain Model
Term Description
Anchor Html element to create ->Links. Contains link-target in the form <a href="link-target">
Cross Reference Link from one part of the document to another part within the same document. Special form of ->Internal Link, with a ->Link Target in the same document.
External Link Link to another page or resource at another domain.
Finding Description of a problem found by one ->Checker within the ->Html Page.
Html Element HTML pages (documents) are made up by HTML elements .e.g., <a href=”link target”>, <img src=”image.png”> and others. See the definition from the W3-Consortium
Html Page A single chunk of HTML, mostly regarded as a single file. Shall comply to standard HTML syntax. Minimal requirement: Our HTML parser can successfully parse this page. Contains ->Html Elements. Synonym: Html Document.
id Identifier for a specific part of a document, e.g. <h2 id="#someHeader">.Often used to describe ->Link Targets.
Internal Link Link to another section of the same page or to another page of the same domain. Also called ->Cross Reference or Local Link.
Link Any a reference in the ->Html Page that lets you display or activate another part of this document (->Internal Link) or another document, image or resource (can be either ->Internal (local) or ->External Link). Every link leads from the Link Source to the Link Target.
Link Target Target of any ->Link, e.g. heading or any other a part of ->Html Documents, any internal or external resource (identified by URI). Expressed by ->id.
Local Resource local file, either other Html files or other types (e.g. pdf, docx)
Run Result The overall results of checking a number of pages (at least one page).
Single Page Result A collection of all checks of a single ->Html Page.
URI Universal Resource Identifier. Defined in RFC-2396, the ultimate source of truth concerning link syntax and semantic.
#### 8.2 Structure of HTML Links
Remark: For many web developers or HTML experts the following information on URI syntax might be completely evident. As we wrote this book also for different kind of people, we included this information anyhow.
HtmlSC performs various checks on HTML links (hyperlinks), which usually follow the URI syntax specified by RFC-2396. URIs are generally used to link to arbitrary resources (documents, files or parts within documents).
Their general structure is depicted in the following figure - you also find a unit test below.
#### 8.3 Multiple Checking algorithms
HtmlSC uses the template-method-pattern to enable flexible checking algorithms:
“The Template Method defines a skeleton of an algorithm in an operation, and defers some steps to subclasses”.
We achieve that by defining the skeleton of the checking algorithm in one operation (performCheck), deferring the specific checking algorithm steps to subclasses. The invariant steps are implemented in the abstract base class, while the variant checking algorithms have to be provided by the subclasses.
Component Description
Checker abstract base class, containing the template method check() plus the public method performCheck()
Image File ExistC Checker checks if referenced local image files exist
Internal Links Checker checks if cross references (links referenced within the page) exist
DuplicateIdChecker checks if any id has multiple definitions
#### 8.4 Reporting
HtmlSC supports the following output (== reporting) formats and destinations:
• formats (HTML and text) and
• destinations (file and console)
The reporting subsystem uses the template method pattern to allow different output formats (e.g. Console and HTML). The overall structure of reports is always the same.
The (generic and abstract) reporting is implemented in the abstract Reporter class as follows:
1. initialize the report, e.g. create and open the file, copy css-, javascript and image files.
2. create the overall summary, with the overall success percentage and a list of all checked pages with their success rate.
3. iterate over all pages
4. write report footer - in HTML report also create back-to-top-link
5. for a single page, report the number of checks and problems plus the success rate
6. for every singleCheck on that page, report a summary and
7. all detailed findings for a singleCheck.
8. for every checked page, create a footer, page break or similar to graphically distinguish pages between each other.
The sample report below illustrates this.
• Section IV-8 contains additional tips regarding crosscutting concepts.
### II.9 Design Decisions
In the current version of HtmlSC we won’t check external links. These checks have been postponed to later versions.
#### 9.2 HTML Parsing with jsoup
To check HTML we parse it into an internal (DOM-like) representation. For this task we use Jsoup, an open-source parser without external dependencies.
To quote from the their website:
[quote] jsoup is a Java library for working with real-world HTML. It provides a very convenient API for extracting and manipulating data, using the best of DOM, CSS, and jQuery-like methods.
Goals of this decision: Check HTML programmatically by using an existing API that provides access and finder methods to the DOM-tree of the file(s) to be checked.
Decision Criteria:
• few dependencies, so the HtmlSC binary stays as small as possible.
• accessor and finder methods to find images, links and link-targets within the DOM tree.
Alternatives:
• HTTPUnit: a testing framework for web applications and -sites. Its main focus is web testing and it suffers from a large number of dependencies.
• jsoup: a plain HTML parser without any dependencies (!) and a rich API to access all HTML elements in DOM-like syntax.
Find details about usage of this parser in the HTML encapsulation.
#### 9.3 String Similarity Checking using Jaro-Winkler-Distance
The small java string similarity library (by Ralph Allen Rice) contains implementations of several similarity-calculation algorithms. As it is not available as public binary, we use the sources instead, primarily: net.ricecode.similarity.JaroWinklerStrategy.
• Section IV-9 contains additional tips regarding decisions.
### II.10 Quality Scenarios
Remark: For our small example, such a quality tree is overly extensive… whereas in real-live systems we’ve seen quality trees with more than 100 scenarios.
#### 10.2 Quality Scenarios
ID Description
10.2.1 Every broken internal link will be found.
10.2.2 Every missing (local) image will be found.
10.2.3 Correctness of all checks is ensured by automated positive and negative tests.
10.2.4 The results-report must contain all results (aka findings)
10.2.5 HtmlSC shall be extensible with new checking algorithms and new usage scenarios (i.e. from different build systems)
10.2.6 HtmlSC leaves its source files completely intact: Content of files to be checked will never be modified.
10.2.7 HtmlSC performs all checks on a 100kByte HTML file in less than 10 seconds.
• The most important quality goals (expressed as scenarios) have been summarized in the introductory section IV-1.2.
• Section IV-10 contains additional tips regarding quality scenarios.
### II.11 Risks and technical debt
Remark: In our small example we don’t see any real risks for architecture and implementation. Therefore the risks shown below are a bit artificial…
#### 11.1 Technical risks
Risk Description
Bottleneck with access rights on public repositories Currently only one single developer has access rights to deploy new versions of HtmlSC on public servers like Bintray or Gradle plugin portal.
High effort required for new versions of AsciiDoc Upgrading AsciiDoc from v-0.x to v-1.x required significant effort for HtmlSC due to several breaking changes in Asciidoc. Such high effort might be needed again for future upgrades of the AsciiDoc API
#### 11.2 Business or domain risks
Risk Description
System might become obsolete In case AsciiDoc or Markdown processors implement HTML checking natively, HtmlSC might become obsolete.
• Section IV-11 contains additional tips regarding risks.
• Help your management by informing them about technical risks - and options for their mitigation.
### II.12 Glossary
In the case of our small example, the terms given here should be good friends to most developers. You find a more interesting version of the glossary in section II-8.1.
Term Definition Link A reference within an →HTMLPage. Points to →LinkTarget Cross Reference Link from one part of a document to another part within the same document. External Hyperlink Link to another HTML-page or to a resource within another domain or site. Run Result Combined checking results for multiple pages (→HTMLPages) SinglePageResults Combined results of all Checker instances for a single HTML page.
• Section IV-12 contains additional tips regarding the glossary.
• Many terms in the glossary correspond to parts of the “Ubiquitous Language” from Eric Evans’ Domain Driven Design
## III. Fundamentals of Effective Documentation
When designing and implementing complex systems, it is very helpful to distinguish between requirements (the problem) and the solution. When you improve your understanding of the problem, you can align the solution way better to these requirements.
Let’s begin by discussing the requirements to technical documentation and communication, independent of arc42. Afterwards we’ll present numerous pragmatic tips how you can create effective documentation simple, fast and with less effort than you ever imagined. Painless, nonviolent, without sophisticated tools or elaborate processes.
### III.1 Documentation Requirements
Requirements for technical documentation will come from various stakeholders:
• Consumers (readers), who want to get their specific tasks done with help of the documentation. This group includes software developers, who want to read the documentation to implement new features in their system easily and quickly.
• Producers (authors), who produce and maintain documentation. They want to make changes in the system with minimal effort to update the doucemtation.
The following paragraphs summarize the seven commandments4 or rules for architecture documentation. The rest of the book focusses on how you can achieve these.
##### Rule 1: Helpful
Documentation has to be help all readers to perform their concrete tasks. It should ease or facilitate their work. Therefore the following relation must be true:
Even during system development architecture documentation should be helpful and not be regarded as an unnecessarily burdening extra effort.
##### Rule 2: Correct
Incorrect information in (technical) documentation can be as bad as software bugs. Even worse: Readers who find mistakes in one part of the documentation quickly loose their trust in the rest: The perceived value of the documentation vastly decreases.
Therefore, one important requirement for documentation is its correctness: Never (ever!) allow incorrect information in documentation. Correctness is the highest goal that you should never jeopardize!
##### Rule 3: Current
Correctness of documentation changes over time. What was correct yesterday could already be wrong today. You want your documentation to be current.
##### Rule 4: Easy to find
A specialization of “easy to use”: Documentation consumers shall be able to find information easily and quickly. Fixed structures (like arc42) and conventions help with that.
##### Rule 5: Easy to understand
Consumers (readers) have to understand documentation easily. Documentation has to fulfil their expectations in many dimensions: language, notation, form and tooling. Easier said than done - because sometimes the producers of documentation do not (yet) know all consumers that might need that documentation in the future.
##### Rule 6: Easy to change
Every change (enhancements, reconstructions, maintenance, and even bugfixes) can lead to necessary changes in the documentations. The easier it is for developers to adapt documentation, the higher the chances that the documentation really gets updated.
On the other hand, if changing the documentation is difficult and costly it is simple not done. Documentation then becomes outdated (loosing its correctness).
Many tips in this book address this requirement and show how to improve changeability of documentation.
##### Rule 7: Adequat
We don’t know how much documentation you really need for your system, how detailed you need it, and which notation you prefer. The stakeholders of your system may have special requirements and wishes concerning documentation based upon their specific tasks and experiences. arc42 suggests a pragmatic structure, but you have to determine the level of detail, adequate notations, and sufficient formalism for your system.
See also section VII.2 on the minimal amount of documentation.
### III.2 Fundamental Tips for Documentation
##### Tip III-1: Appoint a Responsible Person (The Docu-Gardener)
In real life, gardeners have at least two different tasks: to plant (create new documentation)and to weed (remove outdated or unnecessary documentation). For (software architecture) documentation your gardener needs to:
1. care for the adequate form and content and
2. proactively search for unnecessary or outdated parts and remove them.
Please note: care does not mean your gardener shall create all content by her- or himself, but identify appropriate producers within the team or among associated stakeholders.
##### Tip III-2: Document economically (“Less is often more”)
When you create documentation, you implicitly create a mortgage for future generation: Whenever you modify your system, someone might need to maintain or adjust the corresponding documentation.
We really believe that documentation can be helpful and ease development work - but only in an extend and degree appropriate for the system and its stakeholders.
Our tips for appropriate (economical or thrifty) documentation:
• Less (shorter) documentation can be often read and digested in shorter time (but beware of overly cryptic brevity, so no Perl, APG or regular expressions).
• Less documentation implies fewer future changes or modifications.
• Explicitly decide what kind and amount of documentation is appropriate, and with what level of detail.
• Differentiate between short-lived, volatile documentation (i.e. flipcharts for your project work) and long-lived system documentation. See [tip V-6]{#tip-v-6}
• Dare to leave gaps: Deliberately leave certain parts of your (arc42) documentation empty. Especially within arc42-section 8 (crosscutting concepts) you can often remove numerous subsections that might not be relevant for your specific system.
##### Tip III-3: Clarify appropriateness and needs through early feedback
Type, amount and level of details of your documentation should be appropiate,
in relation to the system, the affected people, domain, risks, criticality, effort/cost and possibly even other factors. arc42 supports you with suggesting a documentation/communication structure, but does not suggests a certain form, notation, level of detail or target audience.
What appropriateness means in your specific case and for your system you have to find out for yourself, with help of your stakeholders:
At first, describe only a little part or aspect of your system - and get specific feedback for this documentation from involved stakeholders.
• What parts of this documentation is helpful?
• Which aspects need improvement? What kind of improvement?
• Which parts of the documentation should we enhance?
• Where can we shorten the documentation?
• Is the way of presenting, level of details and notation appropriate?
• Is the type of output (e.g.: pdf, html, docx, Wiki) acceptable?
• Do we have to document the parameters at the interface X in the architecture, or are the unit-tests enough?
For example, create diagrams in several iterations or refinements: Your first draft can be handwritten and really schematic. Through feedback you should quickly find out, if either your figure actually goes in the right direction (good) - or if your documentation does in no way fulfill the information demand of the stakeholders (bad). This way, you can sort out expectations through feedback. The more concrete you ask affected people for specific feedback and/or points of view, the more concrete their answers will turn out.
Avoid closed questions like “Is the documentation okay this way?” - because neither after a “yes” nor after a “no” you concretely know what to do!
Better ask open questions like “What do we have to change or enhance so that this documentation can support your work better?”
##### Tip III-4: Communicate top-down, work differently if necessary
Top-down means presenting circumstances starting from a high level of abstraction (few details), stepwise refining up to more details and concretisation. Such top-down communication generally facilitates understanding for all kinds of stakeholders.
On the other hand, a top-down communication structure should never enforce a specific developement method or a specific order of development activities.
Especially during developement it is often more effective to solve problems bottom-up. Communication and documentation of results should nevertheless be organized top-down.
arc42 supports this advice with it’s strict top-down scheme, see figure arc42 top-down.
1. arc42 sections 1 & 2 contain the introduction and goals, a brief summary of the systems’ requirements. This is the birds’ eye perspective on the system.
2. arc42 section 3 describes the context with the external interfaces. It’s an high-level overview and adds only little detail.
3. arc42 section 4, the solution strategy, gives a glimpse into the systems’ most influential decisions, concepts or approaches.
4. The building block view in section 5 is inherently organized top-down. It adds an arbitrary amount of detail to describe the how the source code of the system is structured.
5. The runtime view in arc42 section 6 shows typical or important scenarios that show how the building blocks interact at runtime.
6. Often the crosscutting concepts in arc42 section 8 contain detailed descriptions or specifications how certain recurring problems within the system are solved. Quite often this will be the most detailed part of architecture documentation.
7. The decisions in arc42 section 9 might be detailed, although several real-world documentations we encountered remained quite brief here, and instead focussed on the concepts in the preceeding section.
##### Tip III-5: Focus on Explanation and Rationale, Not Only Facts
Most facts about a software system can be found in its source code - but not their explanation, reasoning and rationale:
• Why is a specific white box structured as it is? Why does it consist of five blackboxes, not seven or eleven?
• Why was this (and not another) library/framework chosen to generate pdf documents?
• Why is one building block deployed on different hardware than all the others?
For arc42 documentation we propose to (briefly) explain at least the whitebox structures of the building block view. Furthermore you should elaborate surprising, special or risky decisions or concepts.
##### Tip III-6: Rate Requirements Higher than Principles
Specific requirements or goals shall always be more important than generic or global principles - as most of use are valued by reaching our goals, and only very few of for adhering to principles.
You should therefore treat arc42 or similar templates as pragmatic means, and definitely not as strict regulation.
Never treat arc42 as a form (like the yearly income-tax-form), but as a well-formed and prestructured cabinet with drawers (see figure “arc42-metaphor”).
Use these drawers as appropriate in your current and specific situation.
Stefan Zörner calls this approach (in his German book [Zörner-15]): „Use templates pragmatically“.
##### Tip III-7: Separate Volatile and Stable Documentation
We assume that you work in a project (rather short-lived) on a system (rather long-lived):
During development or maintenance of a system the development team needs efficient means for ad-hoc and often short-lived, temporary communication. For such project documentation, refrain from any formal or process requirements, if possible.
• Keep the entrance barrier for (written) communication as low as possible,
• Prefer low-tech tools over sophisticated electronic or online gadgets.
We call such documentation volatile, as teams need it only to support their development work - and many of these topics will not be relevant for the system on the long run.
Eine verantwortliche Person (der Doku-Gärtner aus Tipp 1‑1) sollte diese Struktur (gemäß arc42) festlegen und gemeinsam mit dem Team angemessen mit Inhalt füllen.
##### Tip III-8: Don’t Repeat Yourself, If Possible
Vermeiden Sie unnötige Wiederholungen. Mehrfache Darstellung identischer Sachverhalte erzeugt überhöhten Pflegeaufwand bei Änderungen, und macht obendrein die Nutzung von Dokumentation schwieriger. Bei Redundanz ist die Chance recht groß, dass Sachverhalte an einer Stelle leicht anders beschrieben werden als an anderen Stellen. Das verwirrt Leser: Sind diese Unterschiede gewollt oder ein Versehen? Auf welche dieser Stellen kann ich mich verlassen, und auf welche eher nicht? Warum schränken wir dann bereits im Namen dieses Tipps auf „falls möglich“ ein, und fordern keine absolute Redundanzfreiheit? Der Grund liegt im Lesekomfort: An manchen Stellen möchten Sie Ihren Lesern Sachverhalte „als Ganzes“ vermitteln. Beispiel: Der arc42-Abschnitt 1 (Einführung und Ziele) erläutert die Anforderungen an das System. Hoffentlich haben Sie die Anforderungen in einem Requirements-Dokument, einem Lastenheft, einer Sammlung von User-Stories oder ähnlicher Dokumentation beschrieben. In der Architekturdokumentation möchten wir die wesentlichen Inhalte dieser Anforderungen komprimiert wiedergeben (Gewollte Redundanz!) – weil wir Lesern komfortabel diesen Extrakt der Anforderungen vorstellen möchten. Würden wir in diesem Abschnitt lediglich Hyperlinks auf andere Dokumente einfügen, wären wir einerseits redundanzfrei, andererseits zwängen wir die Leser dazu, die Informationen an völlig anderer Stelle nachzulesen… Also: Erlauben Sie Redundanz nur dort, wo sie den Lesekomfort oder das Verständnis erhöht.
##### Tip III-9: Document Unmistakably
Dokumentation sollte von allen Konsumenten gleichermaßen (eindeutig, non-ambiguous) verstanden werden. Nennen Sie gleiche Dinge überall gleich. Unterschiedliche Dinge brauchen unterschiedliche Namen. Eine Spezialisierung dieses Tipps lautet: „Erklären Sie Ihre Notation“: Sorgen Sie dafür, dass alle Beteiligten die verwendeten Notationen kennen. Bei Bedarf verwenden Sie Legenden oder verweisen (im Falle von Standardnotationen wie UML oder ER-Diagrammen) auf die gängige Literatur. Das hört sich vielleicht überflüssig an („ist doch jedem klar, was hier gemeint ist…“) – aber andere Personen könnten Ihre Symbole auf ganz unterschiedliche Arten interpretieren:
Ohne Erklärung der Symbole könnten Foo, Bar und bazz beispielsweise folgende (unterschiedliche) Bedeutung haben: ¥ Vom Hardware-Server Foo fließen bazz-Daten zum DB-Server Bar. ¥ Die fachliche Aktivität Foo sendet einen bazz-Event an den Prozess Bar. ¥ Das Skript Foo startet die Funktion bazz im Modul Bar… ¥ Die Funktion bazz übersetzt Foo in Bar
Sie sehen – ohne weitere Erklärung droht Missverständnis. Agieren Sie in dieser Hinsicht präventiv – und stellen Eindeutigkeit a priori sicher!
##### Tip III-10: Establish a Positive Documentation Culture
Ensure that documentation is a positive and friendly term within your team. Documentation shall support your stakeholders, never hinder them.
Persisting important architectural decisions, structures or concepts shall be regarded as something good, it’s positive for you and other stakeholders of your system.
For example, incorporate documentation in your Definition-of-Done (DoD), at least in case you’re developing in agile ways (what we really hope for you).
##### Tip III-11: Create Documentation From The Reader’s Point of View
Erstellen Sie Dokumentation für und aus der Sicht von deren Konsumenten ([Clements-11] nennt das „write documentation from the reader’s point of view“). Nehmen Sie die Stakeholder-Tabelle von arc42 (siehe arc42-Abschnitt 1.3) ernst: Darin sammeln Sie die konkreten Wünsche und Anforderungen Ihrer Stakeholder an das System und dessen Dokumentation!
## IV. Effective use of arc42
In chapter II you got to know arc42 by means of a simple example. For all arc42 sections you learned why it exists and what you can communicate with it.
In this section you might encounter conflicting advice: We do that with intention:
Let’s look at one concrete example: In section IV-1 we recommend the use of activity diagrams, but also the use of numbered lists. What sounds like a contradiction is intended to demonstrate various alternatives. In some places we will offer explicit criteria for making a specific selection, in other places, the choice is just a matter of taste. In case of doubt, please use the incremental approach: get feedback on a (coarse or preliminary) version of a arc42-section and adjust detail or notation based on the feedback (see Tip III-2).
### IV.1 Tips for introduction and goals
#### IV.1.1 Requirements Overview
##### Tip IV-1: Compact summary of the functional requirements and driving forces.
This section is for many people the first thing they learn about the system. Here you should express the business or project goals clearly and concisely. Give a brief overview of what problem the systems solves. Note:
Note: Sometimes a requirements document relates to more than the one system that we focus on in our architecture documentation. If project scope and system scope are different, you focus in the particular document only the parts of the requirements that apply to this system.
• For systems with complex or extensive business requirements
• For systems without an existing (and reasonable) requirements documentation
##### Tip IV-2: Limit yourself to the essential tasks and use cases
List here the fundamental use cases, procedures, processes or user stories (however you call it in your organization). Limit yourself to a level of abstraction so that also outsiders are able to get an overview of the major tasks in a short period of time.
##### Tip IV-3: Highlight the business objectives of the system
You should ensure that the business objectives are explicitly known. Actually, this should already be in the project contract, or in the requirements specifications … Business objectives are often more global and on a higher level compared to the more detailed system requirements.
##### Tip IV-4: Create an overview by grouping or clustering requirements
In order to provide an overview of the functions of your system, describe in your architecture documentation only the importance of these groups without going into detailed individual requirements.
The figure below shows an example: some ellipses group (cluster) multiple requirements or use cases. Some of them can be found in the table
Building block Description
Import Handling Import-from-Mandator, Import-von-PrintShop, Import-from-Scanner, Import-from-CallCenter, Import-from-CAMS, …
Configuration Configure-Person, Configure-PrintJob, Configure-ScanOCR, Configure-Reports, …
##### Tip IV-5: Make sure you can reference the requirements
In case you are referencing requirements in the architecture documentation, e.g. to justify a design decision, you need to make sure that these requirements can be identified by a short key or something similar
• Sometimes you can take those IDs from the requirements documentation.
• If your requirements are managed by a tool (e.g. an issue tracker), you can use those ID’s - with some tools you even have stable URLs.
##### Tip IV-7: Use BPMN diagrams to describe functional requirements
In case your stakeholders perceive activity diagrams as too technical, you may use BPMN diagrams. The Business Process Model Notation explicitly addresses business stakeholders and can therefore be seen as an alternative to activity diagrams for describing business or technical processes.
##### Tip IV-8: Use a numbered list to describe functional requirements
In this context you could also use a numbered lists as simple and pragmatic way to describe activities, procedures or processes. The activity
mentioned earlier could then be exemplified as follows:
1. Authentication
2. Select a product
3. Check customer type
1. Private customer, add the VAT (value-added tax)
2. Business customer, ask for the VAT-ID
4. Create Invoice
In case you have to describe concurrent processes, activity diagrams see tip IV-6 are the better choice
##### Tip IV-9: Use (semi) formal text to describe functional requirements
We used PlantUml (http://plantuml.com/) to create the activity
mentioned earlier: This open source tool generated the diagram from the following textual description:
Activities are between: and;, branches can be read as pseudo code and that way you combine the benefits of plain text with graphical representation.
#### IV.1.2 Quality Goals
You must know the relevant quality requirements of your stakeholders - preferably concrete and measurable. As an architect you need to know your success metrics …
##### Tip IV-11: Always work with explicit quality requirements
Requirement documents often focus on functional requirements, quality goals remain implicit (and therefore unclear, uncertain, interpretable …). However, you can captures the desired quality attributes relatively easy,
##### Tip IV-12: Explain quality requirements through scenarios
Scenarios explain in short sentences how the system should react in certain situations at certain events. There are several categories of these scenarios:
• Usage scenarios: how does the system react in certain types of use? In the example below: The execution time of an HTML-validation must not exceed 5 seconds.
• Change scenarios: how does the system behave when you change it or extend it? This allows you to identify how fast certain kinds of changes or extensions can be done or how much effort is probably needed.
• Failure or downtime scenarios: how does the system behave when a serious problem occurs, such as the failure of central hardware or software components.
Scenarios may relate to a variety of possible quality attributes, which are structured hierarchically by current quality models (e.g. ISO-25010). Some examples:
Change scenarios:
• A new algorithm for the routing of robots in a high-bay warehouse needs to be integrated. One developer can make this change within 4 hours including the modifications of the build and the unit and integration tests
• At the end of the year, the output format of the (annual) reports must be adjusted to fit new legal requirements. All required data are already in the database and the changes affect layout, formatting and aggregations. These changes can be fully implemented within at most 60 person-hours.
Usage scenarios
• The system selects the necessary data for the XY-process within 1 second (up to 100 concurrent users) or within 3 seconds (up to 1,000 concurrent users).
• After switching on, it takes at most 4 seconds until the navigation system accepts input from the GUI.
##### Tip IV-13: If you do not get quality requirements, make your assumptions explicit
We experience again and again that the development team receives no quality requirements from customers or key stakeholders. That leaves the quality goals implicit leading to a high risk of misunderstanding and dissatisfaction of all the people involved.
Our advice: based on your knowledge and experience, you can make assumptions about appropriate quality goals, the so-called educated guess.
Together with 2-3 team members, you could write down those assumptions as scenarios and discuss these educated guesses with your stakeholders. These assumptions are always better than having no explicit quality requirements!
##### Tip IV-14: Use checklists for quality requirements
With its hierarchical representation (see ISO 25010 figure below, the ISO standard 25010 provides a good checklist. Alternatively shows arc42-QA common quality characteristics.
• availability
• modifiability or maintainability
• performance
• security
• safety
• usability
• testability
##### Tip IV-15: Use examples to work out quality goals together with your stakeholders
The arc42-QA subproject contains more than 50 exemplary quality scenarios that you can use as a template for defining quality goals and requirements of your system.
##### Tip IV-16: Keep the introduction short! Show only the “hit list” of the quality requirements
Although you must meet all quality requirements that are demanded in the requirements and by the stakeholders, you should keep this section short. Show only a handful (“hit list”) of these requirements, possibly with only brief explanations and not only limited to a list of keywords. (all the other qualities can either be found in the specification or in the quality tree in arc42 section 10.)
##### Tip IV-17: Combine quality goals with the action points of the “solutions strategy” section
Sometimes you make decisions based on concrete and specific quality requirements. In such cases, it helps to document these quality goals and the resulting decisions in a consolidated table. We propose that you put this into the arc42 section 4 (solution strategy). In arc42-section 1.2 (quality goals), you only add a reference.
##### Tip IV-18: Show the detailed quality requirements in arc42 section 10
You should list a detailed overview of all quality requirements (quality tree and scenarios) in arc42 section 10 (quality tree). You can also represent there the relationship between the quality goals.
#### IV.1.3 Stakeholder
Content Explicit overview of all the stakeholders of the system, i.e., all people, roles or organizations who
• should know the architecture or
• must be convinced of the architecture,
• work with architecture or code (e.g., use interfaces),
• need the documentation of the architecture for their own work
• make decisions about the system and its development
Motivation
You should know the people who are participating the project or are concerned by it, otherwise you will experience surprises later in the development process. These stakeholders determine among other things the scope and level of detail of your work and results.
##### Tip IV-19: Search broadly for stakeholders
In order to give you some ideas, we came up with a frightening long list of possible stakeholders studying [Clements-11] as well as from looking at the template arc42. All these people or roles may have an interest in the architecture or its documentation …
Analyst, Business Analyst, (other) architects, auditor, supervisory board, project sponsor, public authority, council, build manager, business manager, database administrator, end users, enterprise architect, developer, external service providers, external partners, department, technical administrator, hardware engineer, hotline / support, infrastructure planners, integrator, IT strategy, lawyer, configuration manager, control panel, customer, steering committee, management, neighboring systems, network administrator, operator, product managers, product owners, project managers, QA department, release manager, rollout manager, Scrum Master, security officers, system integrator, tester, UX designers, related projects, maintenance team, designers, suppliers
##### Tip IV-20: Describe the expectations of the architecture and documentation stakeholders
Clarify the expectations of the stakeholders regarding the architecture and its documentation. Ask for expected shape, content and necessary detailing. Asking the stakeholders for their expectations can result in a lot of beenfits, even if it does not necessarily sound like architectural work:
• You respond to the specific needs of the stakeholders and thus achieve greater satisfaction among your target audience
• You avoid unnessessary work, because you focus on the content / topics that really matter to your stakeholders. You avoid to document things which might become important
##### Tip IV-21: Maintain a stakeholder table
You should explicitly represent the expectations of these stakeholders (see above) with respect to the architecture and its documentation in the form of a table.
You can find a minimal version in table below, which only outlines the expectations or required artifacts.
Role Expectation
Administrator Deployment-overview, installation and operations details, firewalls
QA department Description of the interfaces for load testing, possible measuring points for performance testing, technical concept for security and reliability
The table below shows a more detailed version including the relevance for approval and contact information. Please pay attention to any changes in the project teams and add next to the concrete persons, if necessary, also their replacement and work areas / departments / organization respectively.
Role Contact Relevance for approval Expectation
Project leader Ms. Foobar, Ph.D. High Overview technical risk, external interfaces
Project sponsor Mrs. Lovelace, Ph.D. High Proof that the top-3 quality goals can be achieved
Backend Developer Bruno Batch None Persistenz and reporting concept, Details DWH interface
##### Tip IV-22: Do not use the stakeholder table if your management already maintains a consistent stakeholder overview
In case your management (e.g. project management or product owner) takes the overview of stakeholders seriously and therefore maintains a stakeholder table, you should only cross-reference it in the architecture documentation.
But be careful: in our experience, project managers focus rather on organizational information about stakeholders (e.g., contact details and contact persons). In arc42 we need information about the specific expectations of the stakeholders with regards to the architecture and its documentation. We therefore propose that an architect should maintain the stakeholder table himself …
##### Tip IV-23: Classify your stakeholders by interest and influence
Especially while working under time pressure, you can only take care of a few stakeholders. In such cases, you could use a visual classification by interest and influence instead of a stakeholder table. Such a classification can be done very informally by gathering the team around a flipchart and put the outcome into the documentation as needed.
• High impact, great interest: you should involve these people intensively and/or take care of them. Do everything to satisfy this category of stakeholders in all respects. Communication pro-actively with them.
• High impact, no strong interest: invest just as much effort as needed to satisfy these stakeholders
• Little impact, strong interest: these stakeholders can be very helpful for the work on the system: involve them actively and keep them appropriately informed. They can particularly provide feedback on all types of technical or domain details.
• Little effect, little interest: here you could minimize your effort and only provide the information these stakeholders would consume when necessary
Ideally, you have a stakeholder table plus such a classification.
Note: some stakeholders might assess their own influence / power higher than it reflects your classification. Publishing this matrix could therefore lead to some of your stakeholders become quite frustrated.
### IV.2 Tips for constraints
Content Restrictions, handycaps and company guidelines, which limit your freedom regarding design, implementation or development process. These contraints apply sometimes organization- or company-wide across the boundaries of individual systems.
Motivation
##### Tip IV-24: Look at the contraints of other systems within the organization
If you don’t know any contraints of your system, start looking on other systems within the organization.
##### Tip IV-25: Clarify the consequences of constraints
You should clarify the consequences of boundary conditions, e.g. resulting (additional) costs or effort. If constraints bring unreasonable consequences (e.g., can only be satisfied with excessively high costs), you should negotiate about them with the relevant stakeholders.
##### Tip IV-26: Describe organizational constraints
Organizational constraints like time and budget are for good reason unpopular with development teams, because they limit the freedom of design or implementation decisions very strongly. Therefore disclose these types of constraints.
Sometimes such constraints relate to development processes, third party contracting or legal concerns. Discuss them with your management.
##### Tip IV-27: Describe design and development constraints
In addition to the organisational constraints, technical contraints will probably apply as well to the design and development of your system. That can be guidelines from the managements or the organisation itself regarding hardware, operations, technology selection, use of products, frameworks or reference architectures. Disclose such constraints so that the development team can adjust to them in time.
##### Tip IV-28: Differentiate different categories of constraints
If necessary, differentiate between technical, organizational and political constraints or overlapping conventions (e.g., programming guidelines, documentation-, naming- or organisational conventions).
### IV.3 Tips for the context view
The next few tips hold for both kinds of context. You find specific tips in section IV.3.1 (business context) and section IV.3.2 (technical context).
##### Tip IV-29: Explicitly demarcate your system from its environment
You should demarcate your system from the other IT systems in order to show how it fits into the existing IT environment, which external interfaces are offered or consumed and what users or roles are using the system.
That way you show the scope of your system: what are the responsibilities your system takes and what are the responsibilities of the other systems. The following diagram shows the context of a webshop which delegates the payment handling to an external provider (and this way removed from the scope of the webshop).
##### Tip IV-30: Show the context graphically
In the graphic context boundary, you should show the whole system as a black box, e.g., as a single component. Below are some examples of context diagrams. You have a variety of graphical options:
• Our favorite: UML component or package diagrams
• UML use case diagrams
• “Boxes and lines”
In any case, you should place your system in the middle and the other systems, roles, users, etc., around it.
##### Tip IV-31: Combine the context diagram with a table
You should always supplement the context diagram with a table. This way you can reduce the amount of labels in the diagram and easily add explanations, rationale or cross-references. The comprehensive context diagram above requires such a tabular explanation.
We only show excerpts of the corresponding table, but you should consider some characteristics of this typical “graphics/table” pair:
• You should use short identifiers within the diagram in terms of abstractions or generic terms. We use “Services” or “Market Data” in the example, which are then described in more detail in the table
• Reference within the table to more detailed explanations, e.g. if you are using terms from your domain language, you don’t need to explain them in the context, but refer to the relevant chapter (probably 8.1, domain modell).
Element Meaning
User Summarizes all types of users, internal (backoffice), external (customers, partners)
Product data Product data consists of the catalog data, figures, as well as availability, configuration rules, order- and delivery information, and in some cases prices and sources.
Services Contains possible transport- and installation services, offers, dates and binding orders
##### Tip IV-32: Explicitly indicate risks in the context
You could use flashy colors or symbols to indicate risks. The diagram shows that the webshop uses a payment provider to process payments. Obviously, this is a very important service: the availability of the webshop may depend on the availability of the payment service.
You should highlight those risks in the diagram and explain them in the accompanying text. Ideally, you refer to the risk list of your project, or rather arc42 section 11.
Further examples of risks that you should point out clearly within the context:
• Availability risk: if external systems are down: an external system heavily influences the availability of your system.
• Cost risk: the usage of an external system is expensive, individual calls or other types of use cost money. Examples are credit card checks or payment/booking services.
• Security risks: you receive/send sensible data from/to external systems. That could make these interfaces particularly interesting for a potential attacker.
• Volatility (high probability of change) of external systems: Interfaces of external systems are changed often (they are “work in progress”). The syntax and semantics of the transmitted data could be changed on short notice, which means that you either have effort adapting to these changes or you need to develop a flexible consumer for these interfaces.
• Complexity risks: using this interface is exceptionally complex or difficult, because it might have complex data structures, uses esoteric frameworks, complicated handshakes or an arbitrary mixture of those.
##### Tip IV-33: The context should give an overview and should not contain any details
Show Details, e.g., from external interfaces or external systems, in the block view or external interface documents.
The diagram above contains too many details of the interfaces - you could replace the seven individual interfaces to the billing component by a single “Billing” interface and show the details later in the block view.
Too many details make communication and feedback about the context boundary complicated. However, you should here at least mention domain-specific inputs/outputs and protocols.
##### Tip IV-34: Simplify the context boundary through categorization
Keep the context boundary simple: summarize external interfaces, systems or user roles that have strong similarities. Show explicitly that they are categories or summaries.
In diagram above, the system “Foo” sends reports to users. In the refinement of level 1 on the right you can see two different types of reports for two different groups of users. This type of refinement is desirable and consistent. The context boundary marked reports accordingly as a category. You can find such categories for many criteria (see tip below).
##### Tip IV-35: If many external systems are involved, merge them by explicit criteria
If your system interacts with many external systems, you can combine several of these external systems by explicit criteria. You should explicitly state these criteria. Such criteria may include:
• Systems with whom we share common or similar data
• Systems with whom we communicate within the same use case or at the same time
• Systems with whom we communicate using identical or similar technologies (e.g. categorized by ftp- and WebService)
• Systems that belong to the same or similar organization, such as: all systems of garages, insurances, glass manufacturing and tax advisors
• Systems that have similar domain or technical tasks to solve (all systems printing documents, dealing with chip cards, etc.)
The example below marks the “Logistics” external system on the left as << category >>. On the right side, the detailed version shows three specific instances of the category << logistics >>.
##### Tip IV-36: Summarize “many external systems” with ports
If your system interacts with many external systems, you could use ports at the external interface of your system instead of displaying all external systems as separate icons. See below.
Das sieht auf den ersten Blick ungewöhnlich aus, kann aber Aufwand sparen. Vergleichen Sie die Diagramme aus Bild 7 und Bild 12 – es handelt sich um dasselbe System. This looks at first glance unusual, but can save effort. Compare the graphs of Figure 7 and Figure 12 - is the same system.
##### Tip IV-37: Show all (all!) external interfaces
We believe that aiming for completeness is usually a bad goal - with (at least) one exception: you should put all, really all, external systems surrounding your system explicitly into the context boundary. In order to save expenses, you could create categories, groups or clusters of external systems (see tip 4 34 and 4 Tip 35)
##### Tip IV-38: If you offer external interfaces to other systems, create discrete interface documents
The development teams building these external interfaces like compact and appropriate documentation about them. Therefore explain the usage (!) of the interfaces and not so much the fundamental concepts or justifications.
You can document REST API’s rudimentarily with tools like swagger (http://swagger.io/) or Apiary (https://apiary.io/)[^no_hypermedia_support]. Unit tests are a good choice to describe other types of external interfaces.
##### Tip IV-39: Differentiate domain and technical context
Especially with information systems, you can often ignore technical details of the infrastructure within the context boundary and focus on the domain related parts.
However, if hardware or technology plays an important role for your system, you need both, a domain as well as a technical context. See section 4.3.2 (Technical context).
In the figure after next below you can find a simplified example from the embedded world: the left side shows the domain context, the right side the technical context.
The figure below shows a small example of a (web-based) information system with domain and technical context.
You recognize protocols such as HTTPS or SSH within the technical context, which are intentionally not mentioned in the domain context.
Further, the (from a domain point of view external) component “Reporting” is deployed in the same technical infrastructure as the main system.
Wo ist der Hinweis zu (4 - In den Abschnitten IV.5 (Bausteinsicht) und IV.6 (Laufzeitsicht) gehen wir auf diesen Rat näher ein. Siehe Seite 30.)?
#### IV.3.1 Domain Context
##### Tip IV-40: Show rather data flows within the domain context
Apply this tip for information systems and not so much for real-time or hardware-oriented and strongly process-oriented systems. Since you are going to discuss the (domain) context boundary often with a variety of stakeholders, you cannot assume that detailed UML or modeling skills are known.
Intuitively, many people understand an arrow between software systems as data flow. (Directed) UML relationships like dependency, association, etc. are rather confusing for these individuals. Therefore: within the context, show data flows by turning around the direction of the arrow for conventional UML method or service calls.
Note: this advice is controversial in some teams, because we so explicitely do not comply with the standard UML notation, and also doing that only in the context. However, we found that data flows are often easier to communicate with (non-technical) stakeholders.
For the formalists among you: use the normal, dotted UML dependency arrow and annotate it with the stereotype << flow >> to indicate it as data flow and not as a control flow.
For less puristic UML users: invent a new arrow (e.g., solid line with arrowhead) and declare it - as suggested above - in the legend as data flow. IT veterans recognize here the the good old context diagrams from the structured analysis.
##### Tip IV-41: Show external influences in the context
Your system can handle different types of dependencies to external systems, e.g.:
• data or information dependencies
• temporal dependencies
• local or spatial dependencies
• hardware dependencies
• dependencies on persons, organisations or roles
• transitive (indirect) dependencies
Often you will only show user roles and data/information dependencies in the domain context. However, sometimes other types of dependencies can also be important for your system. Such dependencies can be described in the diagram itself or in an additional explanation.
An example is shown in the next figure: users must register before using the system (step 1), the system orders the text messaging (step 2), which an external text message provider sends via the mobile network (step 3). The user enters the code (step 4), and in step 5, the system verifies the identity of a person against an external branch identity server, which, in turn, triggers a verification at a registry office (step 6).
Steps 3 and 6 are transitive (also called indirect) dependencies. The system depends on the registry office, even if it doesn’t use it’s interface directly. The system is maintained by administrators, who have only access via a hardware device (Hardware Access Control, HAC). The system and the HAC must be physically in the same room - a spatial dependency. Some of these dependencies have impact on the achievement of the quality requirements for your system - and therefore can be risks (see tip IV 32).
We are currently at number 42: if you like science fiction, and at the same time have a quirky sense of humor … then Douglas Adams’ book “The Hitchhiker’s Guide to the Galaxy” would be a reading recommendation. After all, the number from arc42 comes from there…
##### Tip IV-43: Show transitive dependencies in the context
Systems have sometimes indirect (transitive) dependencies that may be relevant for communication with some stakeholders. Include those dependencies in your context. An example is shown in the figure above: The dependencies 3 and 6 as well as the registry office in the upper left corner are indirect.
Caution: transitive dependencies in the context contradict the rule of economicalness (as you get additionel elements by showing transitive dependencies). Therefore you should only describe them if it is necessary for understanding the actual situation.
##### Tip IV-44: Pay attention to quality requirements for interfaces
Special quality requirements may apply to interfaces of your own system and maybe as well to your external systems, e.g. safety, throughput or availability requirements. These so-called “service levels” need to be treated with care, because they may have high implementation and operating expenses and -risks.
Consider also (tip IV 32) with respect to risks of external interfaces.
#### IV.3.2 Technical Context
##### Tip IV-45: Show the technical context
You should show hardware and infrastructure of your system, such as computers, processors, networks or buses. Especially for integrated hardware/software systems (embedded systems) you almost always need a technical context. The embedded systems context diagram above (right side) is such an example. However, it may also be important for information systems to show some aspects of hardware and infrastructure already in the context, e.g. for security issues (see the webshop example above).
##### Tip IV-46: Use the technical context to describe protocols or channels
As you have already seen in the webshop and embedded software examples above, you can explicitly label protocols (SSH, HTTPS, VPN) or channels (CAN bus) within the context.
##### Tip IV-47: Explain the relationship between a functional interface and its technical realization
(wie übersetzen wir fachlich und technisch am besten?))
In case your functional interfaces are realized in the technical infrastructure through different technical channels, protocols or interfaces, you should explicitely describe the functional-technical mapping.
A simple example of such a situation can be found graphically here (Link-TODO). The relationships (mapping) can be described easily with a table:
Function Technical
Brake interrupt CAN bus
Driver oversteer
Revolutions per minute
Throttle position
Control value
User commands Keys in cockpit
Status information Audio output
Display in cockpit
##### Tip IV-48: Show the technical context in the deployment view
You can move technology and infrastructure into the deployment view (arc42 chapter VII) if your focus is on functional topics and abandon the technical context (arc42 chapter III-2)
But if technology and infrastructure is important to you, then you should give this overview already in the technical context.
##### Tip IV-49: Show variants of the technical context
If your system is used in different working or application environments, you can represent this in the technical context boundary. In the following figure you can see a system in a test environemt (left side) and in a significantly more complex production environment (right side). Alternatively, you can show such variants in the deployment view in arc42 section VII.
[^no_hypermedia_support] Silvia Schreier noted correctly that these tools unfortunately ignore hypermedia.
### IV.4 Tips for solution strategy
##### Tip IV-50: Explain the solution strategy as compact as possible (e.g. as list of keywords)
Explain your solution strategy in keywords, e.g. as a short list of relevant decisions or approaches. You explain the details as a concept in arc42 chapter VIII (cross-functional concepts).
You should emphasize on creating an overview, not so much on completeness or a long explanations.
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##### Tip IV-53: Refer to concepts, views or code
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##### Tip IV-78: Frugally document internal interfaces
As external interfaces are often more important or critical than
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##### Tip IV-79: Document / specify interfaces with unit-tests
We hope that you would create such unit tests anyway… so re-use them as part of your documentation.
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##### Tip IV-81: Use building block view level-1 for “further” information
e.g. management information, completion status, technologies used, release/deployment information
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##### Tip IV-83: If you refine several building blocks at once, ensure uniqe mapping
Every blackbox in the refined whitebox shall belong to exactly ONE blackbox on the previous level.
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##### Tip IV-85: Document concepts instead of building blocks
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### IV.6 Tips for the runtime view
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##### Tip IV-89: Explain concrete (specific)
scenarios in the runtime view
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##### Tip IV-115: Create at least a domain data model
Yes - we know that DDD is way more than just a data model - but if you don’t get at least your data right, your system is likely to fail…
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##### Tip IV-119: Use stereotypes to document which concepts are applied within which building blocks.
Show in building blocks which concepts are or have to be applied.
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##### Tip IV-128: Keep the introduction (arc42 section 1) brief
Move details, especially of quality requirement, to this section 10.
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### IV.12 Tips for the glossary
##### Tip IV-140: Take the glossary seriously
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## V. arc42 in Everyday Work
The main goal of arc42 is collecting useful and valuable documentation about an IT system, relevant for development and architecture. arc42 helps to achieve these goals, independent of your development process and the kind of your systems.
However, it’s not the goal to fill out arc42 completely. Remember the fundamental tips of chapter III: Document economically, oriented at concrete stakeholder requirements.
Secondly, work and document in the order appropriate for your specific system, usually not in the sequence of the arc42. In other words: Don’t let the arc42 template dictate a task order.
### V.1 Good Start
Remember tip III-1, that proposes one responsible person (documentation gardener). This person can add documentation tasks to the backlog or delegate them to the team. The gardener shall not create all content by herself! In tip III-7 we formulated:
The responsible person (the documentation gardener from tip III-1) shall create … arc42 content together with the team.
##### Tip V-1: Explicitely clarify the target audience of your documentation
Develop an overview of all relevant stakeholders5 - at best in cooperation with your management. Put the result in arc42 section 1.2 (Stakeholder).
With this target audience clarify their concrete information requirements with respect to architecture and its documentation. See tip III-3 (Fast feedback).
##### Tip V-2: Use existing examples as jumpstart
You can find numerous public examples for arc42 documentation of different systems. Use an example to help your team get up to speed…
• Biking: Manage your favorite biking tracks, plus photos. Author: Michael Simons.
• Chess engine: Author: Stefan Zoerner.
• eCommerce (VENOM): Large scale eCommerce system for configuring and selling of complex systems. Author: Gernot Starke
• Customer-Relationship-Management (CRM): Managing customer relationsships in mass-market situations, like insurance cards or mobile phone tariffs. Mostly batch, complex interactions with external partners.
##### Tip V-3: Remove unneeded arc42-sections
Tailor the template to your specific needs: Adjust your documentation to stakeholder needs or requirements by removing parts or sections that nobody needs.
Tailoring absolutely requires a thorough understanding of documentation requirements - which you gain by applying tip V-1.
Please note: By tailoring you might alter the overall structure and/or section numbering of your arc42 documentation. This can make recognition of the structure a little more difficult for certain stakeholders, therefore apply tailoring with care.
##### Tip V-4: Augment arc42 by required sections or topics
In contrast to the preceeding tip V-3 you might need to augment the arc42 documentation of your system by some topics.
We have seen the following examples of adaptions in practice:
• Business or technical data models. In case you work primarily data oriented, or your domain contains complex data structures, a top-level section on those structures can signilize their importance.
• Screen or user interface layout: Create a top-level UI section if you base important architectural or technical decisions on your UI topics (like layout, typography or other visual topics). Might also be helpful if you need to intensively discuss layout or other UI topics with stakeholders.
• Required information for developers (developer guide): In case you have no separate documentation that describes important developer stuff (like environment, conventions, programming style and the like), an additional section in your arc42 documentation might help. We recommend to keep this info separate from the architecture documentation, though.
##### Tip V-5: Indicate currently unneeded arc42 sections
For systems demanding more formal documentation you should indicate those arc42 sections you don’t currently need or which are currently not required (as opposed to tip v-3). Add statements of the following kind to the corresponding arc42 sections:
• currently not needed
• will be documented for version/release x.y
• deliberately left out / omitted
##### Tip V-6: Prepare an arc42 repository for the team
Your development team shall focus on creating software, not on setting up documentation tools. You should prepare the required tools in advance: Prepare everything the team needs for lean, efficient and pragmatic documentation.
The concrete embodiment of this tip depends on your toolchain, see chapter VI.
The term repository means all media or tools your team needs to create or maintain documentation, the more centralized and homogeneous, the better.
For example, attend to the following:
• Create a wiki space for the project related (short-term) information. Part of this shall be structured analog to arc42. XXXX (link missing)
• Create a (prestructured) arc42 model within your favorite UML or modeling tool (in case your team sports such legendaries).
• Create a document for arc42, hopefully with something textbased like AsciiDoc or Markdown, or in the text processor of your choice.
Prefill such a repository with existing content, or content that stakeholders need to know.
In tip-iii-7 we proposed to separate system- and project documentation, like shown in the following diagram:
##### Tip V-7: Prepare a sandbox for documentation
Prepare a demo system, which the team can use to try out all documentation tools without risk.
Such a sandbox is especially important for (feature-rich) modeling tools: Their usage is often non-intuitive and non-trivial. Typical developers seldomly use such tools, therefore it is important to have small systems at hand to experiment with the tools.
##### Tip V-8: Document (and work) from different perspectives
Deliberately change perspectives from time to time to improve your understanding of relationships between different aspects. Reflect (some of) these perspectives in your documentation.
These perspectives can complement, augment and support each other, for example can a runtime scenario support understanding of a complex building block (or code) structure.
arc42 proposes a number of useful perspectives:
• Structural views (building blocks, runtime scenarios, infrastructure and deployment), arc42 sections 5, 6 and 7.
• Crosscutting or technical concepts, arc42 section 8.
• Black- and whitebox perspectives within the building block view, both part of arc42 section 5.
• Domain and technical perspective. The former helps to build the right system (do the right things), the latter to build it right (do things right).
##### Tip V-9: Documents belong into version control
When setting up tools for documentation, always keep version control in mind (exception: Wikis, which usually have their own version and/or change control). Documentation without both change history and reliable backup is not acceptable. We don’t care wether you use git, subversion, mercurial, TFS or other such tools - main point is you actively use one of those.
This tip might sound trivial - but we still (in the year 2016, deep in the 21st century!) find organizations that maintain Word(tm) documents on shared network drives and do versioning by appending consecutive numbers to filenames.
You should, of course, version binary documents (diagrams, images, office documents) too.
Please note: Maintaining models from UML tools within your VCS XXXX
##### Tip V-10: Conduct brief and thematically focussed documentation reviews
To ensure your documentation is adequate for your stakeholders you should ask for feedback as early as possible. Prerequisite is an overview of these stakeholders (see tip v-1).
Give selected, small parts of the documentation to selected persons for review. Ask specific questions instead of broad ones, similar to the following examples:
• Which additional information do you require in the description of interface X?
• Which parts of diagram Y are ok/correct/good, and which aren’t?
• Which changes/additions to the documentation do you propose?
Incorporate their feedback in a timely manner and repeat the review.
##### Tip V-11: Treat documentation tasks equal to developent tasks
Manage missing, insufficient or inappropriate parts of your documentation with the same processes and tools like any other development tasks, like features, tests or bugs.
For example, include reviewer feedback (see tip V-10) in your backlog or issue tracker. Even if the team cannot currently complete such tasks you keep an overview of open documentation issues.
##### Tip V-12: Document continously - parallel to development
In case you ever tried to move documentation to the end of a project, you know what happens in such cases: Documentation will never be created.
Therefore you should document continously - a little bit every day. In our experience even 15-20 minutes per day can lead to good results.
Both the fixed structure and the proactively prepared repository (see tip V-6) facilitate effective documentation even in small timeboxes.
##### Tip V-13: Always document in timeboxes
Timeboxes are a gread methodical tool to really focus on (and reach!) a specific goal. They help you concentrate on what’s currently important.
For every documentation task allocate a fixed time period (timebox) to that specific task: The more value that task, the more important the topic, the longer the timebox.
Such timeboxes should be at least 15 minutes long, but rarely longer than 4 hours. Only for huge documentation tasks your timeboxes should be even longer (see also tip v-15 on follow-up documentation).
At the beginning of every timebox (be it long or short!), invest a little time for planning and priorization. Answer the following questions for yourself:
• In the current situation, what’s the most important topic I need to document?
• Which approaches (textual description, bullet-list, table, diagrams etc) seem fit for this topic? What does arc42 propose? Do I see lean alternatives?
• For which stakeholders do I document? Who needs/requires/wants this documentation? (If the answer to this question is nobody, you better use this timebox to improve your source code or write some awesome integration tests.)
##### Tip V-14: Formulate like “The system {has|does|is|…}”
Sometimes you will write specifications or even requirements for a system or component to be built in the future.
An obvious option to formulate such topics is “the system { shall | should | will|…} “, to verbally indicate the futuristic character of your statements.
But: When the system will be build according to your specs or reqs, you will need to change such sentences, or rewrite them for your documentation. You can easily save this additional effort:
Therefore: Principally stick to indicative formulations (like has, does or is) instead of using conditional or future forms (like shall, should or will).
### V.2 arc42 for Existing Systems
Software teams spend most of their time maintaining (modifying, correcting, improving, enhancing, tuning, patching, fixing…) existing systems - and not performing greenfield development (see section V-3) from scratch.
arc42 is well suited to support these maintenance tasks. You can use arc42 to effectively document your existing system a posteriori or in hindsight.
Your approach will depend on your current situation, wether you have:
• no documentation,
• too much documentation,
• outdated documentation,
• inhomogeneous documentation of varying quality,
• demand for specific topics/aspects
or any mix of these.
##### Tip V-15: Create hindsight documentation iteratively in short timeboxes
In case you currently have not usable documentation at all, you can create an acceptable overview with 3-5 persons within 2-4 hours (obviously depending on the size of your system and its criticality…)
Yes - you read correctly: We’re talking about a very small timebox here: Invest just a few hours instead of weeks:
Prepare a flip chart with the arc42 structure and provide sticky notes and pens. Choose a few (3-5) knowledgeable persons and proceed as follows:
• within 3-5 minutes (!) brainstorm answers to the following questions:
• Which stakeholders are interested in documentation?
• What is the minimal information those stakeholders need/require?
• Which of these information do we already have?
• within 10-15 minutes work along the arc42 sections:
• What are the most important information that we (developers, architects) need to keep?
• Which of these information do we already have?
• Consolidate the results of these short brainstorming sessions: For every arc42 section, note:
• the stakeholders interested in this section
• where this specific information is already existing or
• which stakeholders might contribute/create this information.
In the diagram below you find an example: Concerning the context, information already exists in the wiki, whereas building-block and deployment views are completely missing. Some concepts are documented in wiki and subversion…
From this comparison (“what you need” versus “what you have”) you can infer concrete documentation tasks, that participants can resolve in the remaining time of your documentation timebox. Focus on content, not on form. Refrain from using high-end tools but write on paper and draw sketches on paper.
The documentation gardener (see tip III-1) can selectively improve certain parts of this documentation: create beautiful UML diagrams, move content to your wiki or other documentation tool.
In the documentation timebox you should focus on the deltas between “need” and “have” and start addressing those!
##### Tip V-16: Create documentation before you continue development
As an alternative to the iterative documentation of tip V-15, you might work a priori instead: Document the following topics for your existing system before you continue development:
• Building block view level 1, source code structure on highest abstraction level (arc42 section 5.1)
• Solution strategy, essential technical, conceptual or structural decisions (arc42 section 4)
• Concept, crosscutting concerns, fundamental principles or implementation rules (arc42 section 8)
• Quality requirements (arc42 section 1.2) and decisions/solutions derived from those (solution strategy, concepts)
• Context with external interfaces (arc42 section 3)
Again - refine and detail iteratively, based upon stakeholder feedback.
##### Tip V-17: Document those parts you’re anyhow working on
To change small parts of a system in the course of bug fixing or enhancing, you often need to understand additional parts around the actual culprit. Developing this understanding usually requires more effort than performing the actual change.
Use this opportunity to enhance the documentation of the affected parts.
For example, if you modify a level-2 building block, document responsibility and interface of this component. Explain the context of this building block by means of embedding it into a level-1 whitebox - without striving for completeness. You might go even further, and show the external interfaces (or interface changes) required for this modification within the system context (arc42 section 3).
Dare to leave gaps in your documentation: Leave other parts on level-1 or even the context undocumented. You’ll be positively surprised how good your documentation will become ofter a few of such iterations.
##### Tip V-18: Make other documents unnecessary as quickly as possible
A real danger when creating hindsight (a posteriori) documentation is the (potentially incorrect) existing documentation.
Ensure existing documentation becomes completely superfluous:
• move valid information from existing documents into your arc42 repository
• tag this existing information with “needs care” or “to be overhauled” if this old information needs to be reworked
• if you cannot delete old documents, at least tag/mark them as “invalid” or “outdated”
##### Tip V-19: Remove superfluous or outdated documents
In some organizations we experienced documentation overflow: An abundance of documents and information, spread over dozens or hundrets of files on different network shares, content-management-systems or even modeling tools.
In such organizations it’s common that stakeholders create their own documentation and ignore other peoples’ documentation.
If you apply tip V-18 in the large, you should move the content of many of these existing documents or sources into your arc42 repository.
Initially, retrievability or findability is more important than consistency: Remove as many of the existing documents as possible and move documentation into the central arc42 repository.
You can improve the consistency of documentation iteratively over time.
##### Tip V-20: Explicitly reveal problems and risks
Collect technical and structural problems and risks in arc42 section 11 (risks and technical debt). Use those as basis for future improvements within the system and its documentation.
See the aim42 open source projects, which describes a systematic approach to software improvement, evolution and modernization.
### V.3 arc42 for New Systems
When developing new systems from scratch you should create documentation to support the ongoing development.
Documentation shall enable and support communication between stakeholders, especially to avoid implicit assumptions. It shall facilitate stakeholders’ tasks.
Within the following tips we propose a number of alternative perspectives to successfully start software architecture documentation.
Depending on your situation you should invest in each of these perspectives. We don’t propose any specific order - use feedback and iteration to adjust form, detail and depth.
Follow the domain driven design (DDD) and model domain entities and services. That’s an elegant bottom-up approach to create building blocks. In many domains, these entities and services are long-lasting and fundamental parts of any architecture. Therefore a model (for example as UML or entity-relationship model) of the domain should be part of your architecture documentation.
• Usually the domain models are part of the crosscutting concepts, located in arc42-section 8.1 (business models). That’s a good home for business service models too.
• Alternatively you include the (entity or data-oriented) part of your domain into a detailed layer of the building block view (e.g. arc42-section 5.2 or 5.3)
##### Tip V-22: Start with quality requirements
From the beginning of development you should care for the required quality of your system. This especially holds if these kinds of requirements are missing from your requirements documentation.
As soon as you have worked out some quality scenarios based upon major quality goals (see arc42 section 1.2 and section 10), you should collect appropriate solution approaches together with your team. Like explained in tip iv-51, arc42 section 4 (solution strategy) is an ideal place for quality goals plus corresponding architectural decisions or approaches.
XXX: example table of scenarios plus approaches
##### Tip V-23: Start with the context and external interfaces
We cannot repeat it often enough:
It’s always the goddamned interfaces.
Start early in development by demarcating your system against its neighbours (the context, in arc42-section 3). Iteratively improve this early version of the context over time.
See also tips iv-29 up to tip iv-49.
##### Tip V-24: Start with the solution strategy
A brief brainstorming with essential stakeholders (especially the development team) will usually result in ideas and proposals how to tackle the problem at hand: What technologies or patterns to use and how to apply them.
Briefly sketch these ideas in arc42 section 4 (solution strategy). Keep this section short and compact. In case you need more extensive documentation, include it in the appropriate part of arc42 section 8 (crosscutting concepts).
You can document the solution strategy within hours (for small to medium systems) or a couple of days (for larger systems) - but you definitely need to have the appropriate stakeholders on board - who can really explain the nitty-gritty details of the solution approaches.
### V.4 arc42 for Agile Projects
We already argumented in favor of lean, pragmatic, agile and economical (that is, appropriate) documentation.
If it is possible within your organization, try to strive for economicalness, frugality, brevity and a clear stakeholder focus for your documentation.
Refrain from producing waste that nobody will use in the future.
##### Tip V-25: Take documentation serious even in agile projects
Even in agile projects a certain amount of documentation will be useful or required. Even well functioning agile teams will encounter some change, even agile team players sometimes forget some details.
Good agile teams that are successful on the long run work very disciplined and properly. They value inner quality of their systems, both code and documentation are maintained appropriately.
You read about appropriateness in several occasions within this book, especially tip III-3 (appropriateness) and tip III-6 (requirements over principles).
##### Tip V-26: Make documentation part of the Definition-of-Done
The Definition-of-Done (DoD) is a team specific understanding of all expectations that software and accompanying artefacts must live up to in order to be done (meaning: releasable into production). DoD is a well established practice from agile methods, especially Scrum:
DoD consists of a list of tasks and artifacts that deliver value to the system, create appreciation from corresponding stakeholders.
You should convince your team to add appropriate parts of technical documentation to your DoD. Important stakeholders might support your argumentation by stating the value they gain from proper documentation.
##### Tip V-27: Apply an iterative and incremental approach to documentation
Document iteratively and incrementally: Start with very little content, focus on important topics, concepts, decisions, interfaces or building blocks. Add content (== increment) based upon feedback from stakeholders.
Rinse and repeat.
Such iterations assure that you document lean and economically, according to tip III-2 and tip III-3.
See also tip V-10 (review documentation), tip V-12 (document continously) and tip V-13 (document in timeboxes).
##### Tip V-28: Use walls instead of electronic tools
During the time a team works cooperatively on a system, it’s highly useful to continously keep imporant information (diagrams, structures, models, concepts or similar) in sight, so everybody can see and comment it.
Print out (or sketch by hand) the context diagram plus the topmost level(s) of the building block view. Pin them to a wall in your project room. If that’s forbidden, grab a cork- or notice board for these diagrams.
Group discussions are way easier with pens in front of a wall as within a complex UML tool. Periodically include changes from the wall into your (electronic) repository. Replace overly scribbled paper by new and adapted printouts. That requires only minimal effort from your side.
For parts or aspects that are currently under heavy diskussion, boards with moveable notes are ideally suited. We love stattys6 to symbolize building blocks and (erasable) pens for dependencies.
##### Tip V-29: Enable cooperative work, even for distributed teams
Paper and whiteboards are the simplest tools for cooperative creation of ideas - not that much for documentation.
These low ceremony tools help you with planning tasks and projects, but you should provide additional tools for your documentation.
Tools like Google-docs, Dropbox-paper or similar web based apps allow concurrent access and coordination of distributed work.
### V.5 arc42 for Large Scale Systems
Very large systems are often maintained by many persons in parallel. Consistent and appropriately-economical documentation of such systems poses a number of organizational and technical challenges.
We summarized our most important suggestion (modularization) in the following tip V-30. Nevertheless there are additional tool requirements for large systems (see chapter VI)
##### Tip V-30: Modularize extensive documentation
Factor out (extract) documentation commonalities of the system and create separate documentation for relevant subsystems.
Link up these different parts of the documentation by hyperlinks, so that consumers can navigate freely between special and common parts.
A toolchain supporting modularization is an important prerequisite: it’s a nuisance with typical office products, but works quite smoothly with wikis (see section VI-4) or markup-languages (see section VI-3).
For a schematic example of modularized documentation see figure arc42-modularized: The overall systems is made up from three subsystems. The common architecture documentation contains the introduction and goals, strategic decisions, building blocks level 1 plus crosscutting concepts. All these informations are valid and useful for the other three (smaller) documents.
Each subsystems’ documentation contains only parts of the arc42 template, and each subsystem focusses on different aspects.
The common part of your architecture documentation will most often contain the global system goals and the overall business context. Eventually it will also contain the solution strategy plus some crosscutting concepts.
##### Tip V-31: Extract goals, context and concepts
As proposed in the preceeding tip V-30 you should factor out documentation commonalities of the system and create separate documentation for relevant subsystems.
A central document appoint (or documentation hub) for the overall system should at least contain the systems’ goals, its context with the external neighbours and some crosscutting concepts.
Extracting these commonalities saves effort for the subsystems’ documentation.
##### Tip V-32: Appoint a responsible person responsible for the overall documentation
In case you follow our advice to modularize documentation, you should appoint a responsible person for the extracted (factored-out) part of the documentation - similar to the documentation gardener we proposed in tip III-1.
<t.b.d.>
<t.b.d.>
<t.b.d.>
## VII. Frequently Asked Questions
Category Section Keywords
General A Cost, license, contributing
Methodology B minimal amount of documentation,
where-does-what-info-belong,
notations, UML
arc42 sections C quality requirements, context,
building blocks, runtime scenarios,
deployment, concepts etc.
Modelling D UML, diagrams, interfaces, ports
understandability, consistency,
clarity
arc42 and Agility E Scrum, Kanban, definition-of-done
minimal, lean, economical
Tools F Tools and their application,
source code and documentation
Versioning and G versioning documents,
Variants variants of systems
Traceability H tracing requirements to solution
decisions and vice-versa
Management J very large systems, standardization,
governance, checklists, access-rights
Customizing K tailoring and customizing,
known adaptions of arc42
Just in case your question(s) regarding arc42 and its usage in practical situations are still missing - please let us know: Either as an email to info@arc42.de or by opening an issue on our github project.
### VII.A General questions
##### Question A-1: What does the 42 mean?
“42” is a quotation from the science-fiction novel Hitchhikers Guide to the Galaxy by the famous Douglas Adams.
In his book the number 42 is the “answer to the meaning of life, the universe, and everything” - calculated by a very big computer over a very very long time:
Software developers like to use 42 as magic number, that’s a number without any real significance, which you could replace by any other number. Of course the name arc42 wouldn’t sound half as cool if we had choosen another number.
##### Question A-2: What’s the license?
Creative-Commons Sharealike 4.0
Under this license, you’re free to:
• Share: copy and redistribute the material in any medium or format
• Adapt: remix, transform, and build upon the material for any purpose, even commercially.
We, the licensors, (Gernot Starke and Peter Hruschka, the creators of arc42) cannot (and surely will not) revoke these freedoms as long as you follow the license terms.
You must give appropriate credit, provide a link to the license, and indicate if changes to arc42 were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
If you remix, transform, or build upon material from arc42, you must distribute your contributions under the same license as the original.
##### Question A-3: What’s the pricing model of arc42?
arc42 is completely free to use (and will remain so!), for arbitrary kinds of system, even in commercial context.
##### Question A-4: How widely is arc42 used?
In the D.A.CH. region (Germany, Austria, Switzerland), arc42 is used by many organizations and corporations in various industries. We don’t know any statistically relevant numbers, though…
In these countries, alternatives like Simon Browns’ SA4D/C4 model are not well-established.
In the international scene, we guess that arc42 is way behind SA4D/C4.
The template is available in English, German and Spanish. Further translations are most welcome (but will not be provided by the current maintainers of arc42).
##### Question A-5: What’s the target audience of architecture documentation?
(Potentially) all stakeholders of a system, that require information about the architecture, e.g. the internal structure, crosscutting concepts or fundamental decisions.
Which persons or roles are interested in what parts of the architecture depends on the specific context. Some typical targets for such documentation:
• Software developers who actually implement within the system.
• Software developers of neighbour systems, who need to know about external interfaces and/or their technical details.
• Software architects who need to prepare, shepard or implement architectural decisions.
• Operators or administrators of software systems, who either need to configure the system for specific infrastructure needs, optimize it or perform administrative tasks requiring architectural understanding. This often includes security relevant tasks, like firewall or database settings.
• Technical managers who organize maintenance or evolution of the system.
• Auditors or reviewers, who valid, current and accurate architectural information to assess or evaluate the system.
• Testers and QA-engineers who need to plan, impelement or perform whitebox-, performance- or security testing.
• Consumers of services offered by the system, in case they need to understand additional details apart from the interface (blackbox) specification.
In our opinion, software architects should maintain an overview of current stakeholders together with their specific needs for architectural information.
##### Question A-6: What are possible alternatives to arc42?
• Simon Brown’s SA4D/C4 model. You find a brief introduction in one of his blog posts
• The View-Model of Paul Clements et. al from the Software Engineering Institute. From our point of view, their approach is very heavyweight and can become quite formal when applied to real-world projects.
• In case your organization has created its own “Software Architecture Documentation Template”: It should contain quality requirements, a scope/context section and information about building blocks and crosscutting concepts. If it does, you’re pretty close to arc42 anyway…
##### Question A-7: For what kinds of systems is arc42 applicable?
Due to its flexibility concerning tools, processes and extend, arc42 is well-suited for all kinds of IT-systems (small to large, simple to complex, all kinds of implementation technologies).
For highly-critical systems, (e.g. safety-critical systems, where health or life are at stake), you need to seriously consider documentation requirements. In such cases, arc42 might not be the adequate approach (see question A-8).
We recommend to a-priori evaluate arc42, and enhance it according to your specific needs in such cases. Let domain experts or appropriate auditors review the resulting documentation structure.
For all other normal systems you can apply arc42: Depending on risk, complexity, scope or size you should assess both the arc42 structure and your toolchain for fitness regarding your specific purpose.
We successfully worked with dozens of clients in small, medium and large organizations on many different systems with arc42 - and it never let us down :-)
##### Question A-8: In which cases shall we NOT use arc42?
If you’re creating safety-critical systems, e.g. medical or avionics systems, where health or life are at stake, you will most likely encounter comprehensive and formal specification and documentation requirements, that arc42 will most likely not fulfill.
Many of the hints and tips we give for economical documentation might not be valid in such cases.
##### Question A-9: How can I contribute or report bugs in arc42?
• We maintain the template itself at Github. You can fork the repository and send us pull requests…
• You can open issues in this repo
• You can email us, or contact us over Twitter.
### VII.B Questions on methodology
##### Question B-1: Which parts of arc42 are “essential”, which are “optional”?
Please don’t fill in everything. Document only what your stakeholders need (you should ask them, one-by-one).
Everything you document will potentially require maintenance effort in the future. With every piece of documentation, you’re effectively piling up maintenance debt - and in parallel creating some information value…
Please only document things, facts and reasons that you or your stakeholders consider important enough to justify written documentation.
##### Question B-2: Does arc42 prescribe or enforce specific notations?
No, arc42 works completely (!!) independent of notation or syntax.
##### Question B-3: Shall I use UML for arc42 documentation?
You don’t need to - but:
UML notation is well-supported by various (free and commercial) tools, has a clear semantic and is pretty much standardized - therefore we (Peter + Gernot) like it.
On the other hand, UML does not guarantee any positive effects for your architecture or your team. Other notations might work as good (or bad) as UML.
You need to ensure that all stakeholders have a mutual understanding of your notation. That can become really difficult, if you draw only boxes and lines - as those have no semantic at all (at least without further explanation)
People tend to interpret such symbols in a way they like - most likely different than the original author intented…
See question D-4.
##### Question B-4: What is the minimal amount of an arc42 documentation?
There is no general rule - as the minimal amount depends on the system, its stakeholders, criticality, complexity, size and/or risks.
We suggest to always explain, document or specify the following aspects:
• 3-5 central quality requirements, expressed in scenarios.
• Context view and external interfaces
• Brief explanation of your solution strategy, the most important decisions influencing the implementation or operation of the system.
• Building block view, level 1.
• Most important crosscutting concepts.
##### Question B-5: Where to document external interfaces?
Both scope/context (arc42 section 3) and building block view (arc42 section 5) provide adequate room for external interfaces.
How and where you describe external interfaces depends on several factors:
• If your system provides external interfaces to external consumers and you and your team can decide about any details of these interfaces - then you have to care for their documentation.
• Do you need to provide external systems with information on how to use your interfaces? You could provide example code and other interface documentation, but need to make sure this information is easily accessible to the external stakeholders.
• Do you just provide data at your external interfaces, and no services? Then you might stick to just a documentation of data formats (and maybe semantics).
• In case you use interfaces provided by external systems, make sure their documentation is easily accessible by your development and maintenance teams. Include references in the arc42 context or building block view.
##### Question B-6: How to document external interfaces?
Document / specify interfaces by providing unit-tests. We hope that you would create such unit tests anyway, so you can re-use them as part of your interface documentation.
Describe syntax, semantic, protocoll/interaction of your interfaces, show sample interactions, provide sample data. Add some unit tests and/or other example code.
##### Question B-7: Where to put links to external systems (“neighbors”) documentation?
Use the context view to reference external systems’ documentation or to provide references to the parts of your arc42 documentation containing appropriate information.
If an external system/interface is especially important or risky for your architecture, it’s sometimes adequate to copy parts of this interface document into your own arc42…
##### Question B-8: What is a blackbox / the blackbox template?
A blackbox is a building block or component of the system which can be explained or defined in terms of its inputs and outputs, without knowledge of its internal workings.
You can easily document or specify a blackbox with a simple template. We propose a table for it, in the following form (required information in bold, the rest is optional):
part explanation
Name name or id of this blackbox. The name should be the title of the blackbox template.
Responsibility What does this blackbox do, what task or function does it fulfill? What use-cases, use-case-clusters does it handle?
Interface What is the interface (input, output) of this blackbox, what is its API, what does it require as input and what does it provide as output.
Source code Where to find the source code of this blackbox. That might be the most important entry point, a number of files, directories or packages. Anything that might help a developer to find details.
Qualities What quality requirements does this blackbox satisfy? How fast/secure/robust is it? Does it offer any guarantees or service-level-agreements?
Open issues Are there known issues, problems or risks associated with this blackbox?
Requirements What are the requirements satisfied by this blackbox? Such information is important if you need traceability. See section H (Traceability)
##### Question B-9: What is a whitebox?
A whitebox contains multiple building blocks (blackboxes) with their relationships. It is a more detailed view of a blackbox of a higher abstraction level. You find an example in the diagram below:
This whitebox contains three blackboxes (A,B,C) and some relationships - a few of which are labelled.
##### Question B-10: What is the building block hierarchy?
The building block view explains your source code in a hierarchical way:
The topmost level (level 1) represents the white box description of the overall system, made up of black box descriptions of the system’s top level building blocks.
Level 2 zooms into the building blocks of Level 1 and is thus made up of the white box descriptions of all building blocks of Level 1 together with the black box descriptions of the building blocks of Level 2.
The following diagram shows a (rather generic) example - starting the hierarchical decomposition from the context view (called level-0 in arc42 terms)
##### Question B-11: How to document a whitebox with the whitebox template
The whitebox template is a structured means of specifying or describing whiteboxes. It contains the following elements (required information in bold, the rest is optional):
part explanation
Diagram Shows the contained blackboxes plus their relationships
Rationale The reason why this whitebox is structured like it is.
Contained blackboxes Names and responsibilities of contained blackboxes, plus references to their detailed description.
Internal relationships (interfaces) Brief description of the relationships between internal blackboxes, plus references to their detailed description.
Risks and issues Know issues or risks with this whitebox, information about eventual technical debt or ideas for improvement
##### Question B-12: Where shall I describe important (blackbox) components?
The building block view should be a hierarchical explanation of your source code, of the static (code) structure of your system. It contains blackboxes of various level-of-detail.
It can be very understandable to make your important blackboxes part of your topmost refinement level, which we call level-1 in arc42.
If your important blackbox is something quite small which you don’t want to explicitly contain in level-1, then you could contain it in any refinement level - and mark it in its containing whitebox in color or by other visual means: Draw it larger than others, attach a graphical label/icon to it, give it a colored border - anything to let it stand out. Be pragmatic!
See also the questions in section C-5 (Building block view).
##### Question B-14: Can I update documentation incrementally? Or shall I document after the implementation is finished?
Create and update documentation continuously - in sync with your sprints or iterations.
If you defer documentation to “later”, I am reasonably sure it might never be done. Don’t procrastinate - but make (brief, short, compact, economical) documentation a habit, like checking-in source code in version control or writing automated tests (you test, don’t you?).
In my opinion it’s way better to have short up-to-date and current documentation of only some aspects of the system, than to just have a plan to sometimes document the whole system…
Reserve a brief period of time every day (or at least every week) to catch up with your documentation tasks. Close your door, your email reader and your messenger apps, flip your phone to silent mode… and jot down some important, interesting, risky or otherwise valuable information about your current topics.
##### Question B-15: What kind of information shall I provide for which stakeholder?
You need to inquire with your specific stakeholders, we can only give you rough approximations from our experience… that will most likely miss the goals of people involved in or with your system.
Show them examples what you could deliver with low effort. Incorporate their feedback. See #tip-iii-3 (appropriateness).
Read the short answer above! Ask your specific stakeholders, what artifacts or information they expect or require from the architecture documentation (or more general, the technical documentation).
Don’t ask what they require from the system (hopefully you already know that…)
Typical answers we received are summarized in the following table:
Stakeholder Required information/artifacts
Project management Quality goals, (business) context with external interfaces, solution strategy (most important architectural decisions), building block view level-1, overview of important concepts. Sometimes an overview of infrastructure and deployment.
Top- or upper management Top-3 quality goals, major business requirements, business context with neighbor systems, compliance to IT strategy, compliance to security (and maybe other) standards, operation/deployment concepts, cost-relevant architectural decisions.
Product owner Quality goals, building block view with interfaces, overview of important architectural decisions, overview of crosscutting concepts, known issues and risks
Software developers Quality goals with detailed scenarios, solution strategy with links to detailed concept descriptions and/or example solutions, building block view with interface descriptions, important runtime scenarios, domain model, crosscutting concepts, deployment view with possible variants, glossary.
Operators, administrators Infrastructure and deployment, external interfaces with technical and organizational details, logging and monitoring concepts.
Tester, QS-team Detailed business and quality requirements, solution strategy, building blocks with interfaces, crosscutting concepts influencing testing or testability.
Developers of neighbor systems Business and/or technical context, details description of external interfaces, end-to-end interaction/runtime scenarios at these interfaces.
##### Question B-16: What does economical documentation mean?
Some tips for appropriate (economical or thrifty) documentation:
• Less (shorter) documentation can be often read and digested in shorter time (but beware of overly cryptic brevity, so no Perl, APG or regular expressions).
• Less documentation implies fewer future changes or modifications.
• Explicitly decide what kind and amount of documentation is appropriate, and with what level of detail.
• Differentiate between short-lived, volatile documentation (i.e. flipcharts for your project work) and long-lived system documentation. See [tip V-6]{#tip-v-6}
• Dare to leave gaps: Deliberately leave certain parts of your (arc42) documentation empty. Especially within arc42-section 8 (crosscutting concepts) you can often remove numerous subsections that might not be relevant for your specific system.
### VII.C Questions on arc42 sections
#### 1. Requirements, quality-goals, stakeholder
##### Question C-1-1: How extensive shall I describe the requirements of the system?
So that readers of your (architecture) documentation can understand why the system exists and what its major requirements are.
• Briefly explain the major (max 3-5) use-cases, features or functions.
• Document the top 3-5 quality requirements by showing scenarios (see question C-10-2 (quality scenarios)).
• Make sure the most important domain terms are commonly understood among the development team (arc42 provides a glossary section for that purpose, but you better let your product owner or other stakeholders create and maintain those definitions.)
• Link to additional requirements documentation, if it exists.
##### Question C-1-2: What are quality goals (aka quality attributes)?
We use the term quality goals synonym to architecture goal:
A quality attribute that the system needs to achieve. These goals often have long term character in contrast to (short term) project goals.
Software quality in general is the degree to which a system possesses the desired combination of attributes (see: software quality).
Important quality attributes are performance, robustness, security, safety, flexibility, dependability, maintainability etc..
It’s helpful to distinguish between
• runtime quality attributes (which can be observed at execution time of the system),
• non-runtime quality attributes_ (which cannot be observed as the system executes) and
• business quality attributes (cost, schedule, marketability, appropriateness for organization)
Examples of runtime quality attributes are functionality, performance, security, availability, usability and interoperability.
Examples of non-runtime quality attributes are modifiability, portability, reusability, testability.
It sometimes helps to take a look at the (very generic) ISO-25010 software product quality tree, which summarizes most practically relevant potential quality goals on the abstraction level of “quality attributes (see below)
##### Question C-1-3: Why does arc42 propose a stakeholder table?
1. System stakeholders are potential consumers of information about the software or system architecture.
2. Several of the stakeholders might impose quality requirements on the system, that the development team needs to know about.
3. Architecture stakeholders are sometimes forgotten during conventional requirements analysis, e.g. dev-teams of external interfaces, auditors or developers themselves - these people or organizations will not have requirements concerning the system itself, but its architecture or architecture documentation.
4. Stakeholders are the most important source for any kind of feedback that can help you and the development team to evolve and improve the architecture…
##### Question C-1-4: What shall be included in a stakeholder table?
Bold information should be present, the other parts are optional.
Information Meaning
Name/Role who or which part of an organization has an interest in the system or its architecture? Sometimes you name specific people, quite often you’ll stick to roles
Knowledge What do these stakeholders know about the system or its associated processes?
Expected deliverables What do these stakeholders expect from the architecture or its documentation? Please don’t confuse this with the system requirements. See question C-1-5 (deliverables) below.
Relevance (priority) This is critical information: Some stakeholders will relevant or required for production acceptance or sign-off - but: Explicitly stating relevance or priority might frustrate, irritate or even instigate those with lower priorities… You need to consider the political consequences of putting this information in your documentation - but the team should definitely know about stakeholders’ relevance.
Contact As trivial as a phone number or email address, so you or the team can contact this stakeholder.
Comment Any other information people might need concerning this stakeholder.
##### Question C-1-5: What are the expected deliverables in the stakeholder table?
What documents, artifacts, deliverables, reports or other means of communication do stakeholders expect or require from the architecture or development team?
Please don’t confuse this with the system requirements.
Some examples might clarify the term:
• Developers of the fooBarexternal interface might expect a detailed description with sample data and code snippets, so they know how to implement this interface (explained in the building block view, arc42 section 5).
• All developers need to know how the system stores cost-relevant furpsdata into our auto-replicated NoSQL datastore (most likely explained in on of the crosscutting concepts in arc42 section 8)
• The product owner calls for a decision log of architecturally relevant decisions (arc42 section 9).
• Operation or administration need to know technical detail of the required database and middleware configuration.
#### 2. Constraints
##### Question C-2-1: What are constraints?
Constraints restrict your freedom in decisions, concerning the system or any associated processes.
Such constraints are often imposed by organizations across several IT systems.
Examples:
• System has to be operated by data-center XYZ, as our company has a long-term contract with them.
• Part of software/hardware development has to be off-shored to our Asian subdivision.
• Our operational database system needs to be IBM DB2.
Please consider also questions C-2-2, C-2-3 and C-2-4.
##### Question C-2-2: What types of constraints exist?
Three major types of constraints exist:
• Organizational constraints: E.g. compliance to (standard) processes, budget, time, required information flows, required (management) reporting structures, adherence to certain documentation templates or conventions…
• Technical constraints: Usage of specific hardware, middleware, software-components, adherence to specific technical or architectural decisions (e.g. use of specific frameworks or libraries)
• Conventions, e.g. programming style, naming conventions or similar stuff.
##### Question C-2-3: What kind of constraints shall I document?
At first - try to avoid documentation of constraints, as somebody else might already have documented them. Refer or link to existing documentation.
Document the constraints that:
• shape or shaped important architectural or technical decisions,
• help people to better understand your architecture
##### Question C-2-4: Can/shall we negotiate constraints?
Should some constraints prove to be especially unfavorable, risky or even expensive for your system, you should definitely try to negotiate them.
An example: Let’s say the company-wide constraint is: “Our software has to be developed in Java”. One of your tasks is the development of a specific smart-card reader, including the appropriate hardware device driver - you should convince your management that such software is better written in either C, Forth or Lua.
#### 3. Context
##### Question C-3-1: What is the context (scope)?
“The context defines the relationships, dependencies, and interactions between the system and its environment: People, systems, and external entities with which it interacts.”
Quoted from Rozanski-Woods
The context shows the complete system as one within its environment, either from a business perspective (business context) or from a technical or deployment perspective (technical context). The context view (or context diagram) shows the boundary between a system and its environment, showing the entities in its environment (its neighbors) with which it interacts.
It thereby identifies the system’s relevant external interfaces. Make sure that the interfaces are specified with all their relevant aspects (what is communicated, in which format is it communicated, what is the transport medium, …).
##### Question C-3-2: How shall I document the context (scope)?
Combine a component diagram with a table. In that diagram, your system shall be a black box. All external interfaces (neighbor systems, user roles, participating actors and sensors) shall be visible.
In case you have many neighbor systems (or external interfaces), you may categorize them by finding proper abstractions or generalizations. For example, for an e-commerce system your system will likely have external interfaces to numerous payment providers (credit card companies, banks etc). In the context diagram, you can combine all those into a single payment providerexternal interface, and defer the details to either a detailed building block level or even a payment privider concept.
Accompany your context diagram by a brief table, explaining the names/identifiers from the diagram. Refrain from too many details here, but refer or hyperlink to corresponding parts of your arc42 documentation.
See question C-3-7 (simplify documentation of business context) for some options of working more efficiently with the business context.
##### Question C-3-3: What is the “business context”?
All neighboring systems and business data that is exchanged with the system.
The business context should avoid overly technical information (like detailed protocol or data type information, communication channels or hardware information).
You find an example on the left side of the diagram below. Contrast it with the technical context of the right side of this image (described in question C-3-4 (technical context)).
##### Question C-3-4: What is the “technical context”?
Specification of the technical communication channels or protocols linking your system to neighboring systems and the environment.
You will find technology and physical infrastructure information in a technical context.
##### Question C-3-5: In which cases shall I document the “business context”?
Always (as in “in every single system you ever work on”).
There are, though, several options for you to save effort in documenting the business context - see question C-3-7 (simplify documentation of business context).
##### Question C-3-6: In which cases shall I document the “technical context”?
When hardware, protocols or deployment is of utmost importance for many stakeholders, then you can use the technical context to convey such information.
In most business systems you can ignore the technical context and document or specify that information in the deployment view in arc42-section 7.
##### Question C-3-7: How can I simplify documentation of the “business context”?
Some ideas:
• Reduce precision or the amount of detail you show in the business context. Show abstractions, aggregate neighbors or even interfaces.
• Show categories of neighbors, not every single neighbor system.
• Use UML port symbols on a component diagram to avoid drawing too many boxes and lines
• Combine multiple types of users or user-roles into appropriate abstractions or more generic roles (e.g. combine private and corporate users simply into “users”).
##### Question C-3-8: Shall I document risks or problems in the context?
External interfaces are especially sensitive to problems or risks, often functionality or availability of your system depends on some of these external interfaces.
Problems or risks associated with external interfaces therefore need special attention or countermeasures - so it’s a great idea to explicitly show those risks or problems in the context view.
Below you find some examples of risks or problems that might occur at external interfaces:
• Availability risk: if external systems are down: an external system heavily influences the availability of your system.
• Cost risk: the usage of an external system is expensive, individual calls or other types of use cost money. Examples are credit card checks or payment/booking services.
• Security risks: you receive/send sensible data from/to external systems. That could make these interfaces particularly interesting for a potential attacker.
• Volatility (high probability of change) of external systems: Interfaces of external systems are changed often (they are “work in progress”). The syntax and semantics of the transmitted data could be changed on short notice, which means that you either have effort adapting to these changes or you need to develop a flexible consumer for these interfaces.
• Complexity risks: using this interface is exceptionally complex or difficult, because it might have complex data structures, uses esoteric frameworks, complicated handshakes or an arbitrary mixture of those.
#### 4. Solution strategy
##### Question C-4-1: What is the “solution strategy”?
A short summary and explanation of the fundamental solution ideas and strategies.
An architecture is often based upon some key solution ideas or strategies. These ideas should be familiar to everyone involved in architecture work.
##### Question C-4-2: How to document the “solution strategy”?
As brief as possible. As many different stakeholders will read the “solution strategy”, its explanation shouldn’t be overly technical.
State these decisions which gravely influence(d) your architecture. Refrain from overly detailed explanation, don’t describe possible alternatives or even implementation guidelines.
You can (or should) dive into such technical detail in arc42 section 8 (crosscutting concepts), where you can elaborate on the how and why of your approaches. If only single building blocks are concerned, even the building block view (arc42 section 5) might be the appropriate location for this information.
Provide hyperlinks or at least references to sections or documents giving additional information or detail.
##### Question C-4-3: Can you provide examples for the solution strategy?
Yes - of course :-)
The following examples are taken from the arc42 by Example Leanpub book.
###### From the Html-Sanity-Checker example architecture
1. Implement HtmlSC mostly in the Groovy programming language and partially in Java with minimal external dependencies.
2. Wrap this implementation into a Gradle plugin, so it can be used within automated builds. Details are given in the Gradle userguide.
3. Apply the template-method-pattern to enable:
(Remark: Some hyperlinks in the paragraph above might not work, as they were only meant to be examples.)
###### From a Mass-Market CRM system
Goal/Requirement Architectural Approach Details
Flexible Data Structure Database structure + persistence code is completely (100%) generated from UML-model Persistence concept, section 8.1
Flexibility in Transmission Formats (CSV and fix-record-formats Create domain-specific languages for CSV and fix-format import/export configurations. Build an ANTLR based parser for these languages plus the corresponding interpreters. Custom Eclipse Editor, Section 8.2
Flexibility (Configurable CSV/fix formats) Implement customized editor for CSV/fix DSL as Eclipse plugin Custom Eclipse Editor, Section 8.2
(Remark: Again - no links to details are given in last column of the table - in any real architecture documentation you should prefer hyperlinks to just naming the sections. )
#### 5. Building block view
##### Question C-5-1: What is a “building block”?
Any piece of source code of your system.
Any programming or source code construct (e.g. module, component, subsystem, class, interface, package, library, framework, layer, partition, tier, function, macro, operation, data structure, …) that you implemented to make your system work.
Further examples of building blocks which might be relevant for your system:
• configuration files or items
• UI specific files, styles or definitions, like css-files in web development.
• Any kind of templates used to generate other artifacts at compile-, build- or runtime
• Build- or makefiles
• Deployment- or installation-related artifacts (e.g. deployment- or container descriptors)
##### Question C-5-2: Do third-party libraries, frameworks or tools count among building blocks?
Limit building blocks to things you implement or maintain yourself.
Some external software (like middleware, database, UI-toolkit or similar) might be essential to understanding the structure of your system. You can include those in the building block view.
You definitely should show those elements in the deployment view (arc42-section 7)!.
##### Question C-5-3: How does source code relate to building blocks?
Building blocks represent your source code. Your building blocks should be an abstraction of your source code.
The problem: There can be several possibilities how you can aggregate source code constructs to building blocks. I’m afraid you have to read the longer answer below…
Mapping code to building blocks is a challenge, which we like to demonstrate by a small example, see the following figure. In its center you see a directory listing of (Python) source code files making up the system. Both on the left and the right side of the images you find different, but perfectly valid building block structures for these files.
Both versions were created with different abstraction criteria, both are possible.
In reality, you should organize your source code along your abstraction criteria. Some technologies or frameworks impose certain directory structures that can suggest completely different building block structures than those intended by the architects.
Especially layer structures (view, application, infrastructure) found in some technologies can obfuscate any business- or domain structures that could be useful in understanding the overall organization of the system.
Every single piece of source code should be contained in one of your level-1 building blocks.
##### Question C-5-4: How detailed shall we document the building block view?
Show at least level-1, the top-level structure of the implementation.
As detailed as your stakeholders need it. In many cases, refining just a few of the top-level building blocks will be sufficient - with safety-critical system (as always) being an exception!
##### Question C-5-5: Can I refine a group of building blocks together?
Yes, sometimes it can be helpful (although it somewhat destroys the symmetrical structure of the building block hierarchy).
You find an example below:
##### Question C-5-6: How can I document or specify building blocks?
It depends on wether you need an overview (black box) or details (white box):
For the overview (black box), you should document or specify building blocks by giving the following information:
• Responsibility: What does this blackbox do, what task or function does it fulfill? What use-cases, use-case-clusters does it handle?
• Interface: What is the interface (input, output) of this blackbox, what is its API, what does it require as input and what does it provide as output.
• Source code: Where to find the source code of this blackbox. That might be the most important entry point, a number of files, directories or packages. Anything that might help a developer to find details.
We call this the black box template, see also question B-8 (black box template).
For details (white box), you should use a diagram with some explanation (aka white box template, see question B-9 (white box template).
##### Question C-5-7: How shall I document building blocks for non-object-oriented systems?
Wether object-oriented, procedural, functional or any other programming paradigm: Building blocks are an abstraction of your source code. You shall primarily describe or specify their respective responsibilities (and a few other details).
arc42 does not depend on a particular development approach or programming paradigm…
For other hints to describe or specify building block structures, see question C-5-6 (document building blocks).
##### Question C-5-8: How do I describe internal interfaces?
By internal interface we mean the interface between building blocks within a system, in contrast to external interfaces that connects our system to the external world.
You have a number of options, shown below with increasing degree of detail and effort:
1. Show the interface by any connection (line) within a whitebox diagram.
2. Give the interface a (hopefully self-describing) name (aka aptonym).
3. Describe the name informally (e.g. within a table below the respective whitebox diagram) with one or two sentences.
4. Document the interface and its usage by one or more unit-tests. The idea behind those tests should be “test-as-documentation”. On one hand these tests are precise, on the other hand they shouldn’t add much overhead to your work, as you will write some unit tests anyway (at least we hope so…)
5. Add the programming interface (API): list methods or functions with the required/optional parameters.
6. Add further UML details to your whitebox diagram, e.g. ball-socket, ports, and describe those within your modelling tool.
7. Add quality attributes of this interface to the API description, e.g. performance or throughput guarantees. Some people call those the “service level agreements” of the interface.
I like to emphasize the usefulness of test-as-documentation: It’s a developer-friendly and pragmatic way of documenting potentially critical elements of your architecture. Those tests will (hopefully) be automatically included into your documentation - so the documentation is always correct.
##### Question C-5-9: How do I describe external interfaces?
Basically similar to internal interfaces (see question C-5-8 (internal interfaces)).
The major distinction might be the fact that external people (beyond the scope and influence of your systems’ stakeholders) can be involved, both as consumers and as providers of interfaces (or data or events at these interfaces).
As these external people will likely not have access to your internal arc42 documentation, you might be required to provide distinct documents for your external interfaces.
##### Question C-5-10: How can I avoid redundancy with interfaces in the building block view?
Especially external interfaces might occur several times within the building block hierarchiy - how can I avoid to document or specify them at several places redundantly?
See the diagram below - the interface X (marked with red circles) occures three times in the hypothetical system shown there.
Handle such situations in the following manner:
1. In the context diagram, give the interface an appropriate name and briefly explain its business relevance or significance in just a few words.
2. Describe the interface in detail at the level where it is actually handled (e.g. the service is implemented or data is consumed). In the diagram above, this would be whitebox “B”.
3. At all other occurances (especially in the context view), add references to the detailed description. In the scenario above, such references should occur in the context and the level-1 whitebox.
##### Question C-5-11: How does the hierarchy of building blocks relate to the arc42 sections (5.1, 5.2 etc.)?
Level-n of the building block view shall be documented in section 5-n of the building block view, where level-0 (zero) is the context view and level-1 your topmost system whitebox.
See the following diagram.
##### Question C-5-12: What kind of building blocks don’t I need to document or specify?
In case you want or need to save documentation effort, you have some good opportunities within the building block view. You can possibly omit the following kinds of building blocks from your documentation:
• Really small blocks that will easily be understood by reading the source code.
• Blocks that serve general purpose and are not specific to your system.
• Blocks that only apply (thoroughly defined) crosscutting concepts and don’t do much else.
• Blocks that implement purely technical functions, like persistence, logging, communication, data transformation, data validation or data dispatching.
Tendencially avoid documenting lower-level building blocks.
##### Question C-5-13: What kind of building blocks shall I document or specify?
This is the opposite of question C-5-12 (blocks you don’t need to document).
You should document or specify the following kinds of building blocks:
• The level-1 whitebox (aka the top-level decomposition of your system).
• Blocks adressing specific or important functionality.
• Blocks that adress important quality attributes of the system.
• Blocks that handle complicated or complex functionality.
• Blocks that mitigate or handle risks.
• Blocks that contain surprises or unusual ideas.
• Blocks that somehow deviate from the rest of the system.
• Blocks that deviate from typical developers’ expectation.
• Blocks that are required or important for creating business value.
Focus on higher-level building blocks. Level-1 should always be documented.
Remember to be lazy and document economically: Smaller documentation for fewer building blocks are easier and cheaper to maintain than large documentation for many blocks…
#### 6. Runtime view
##### Question C-6-1: What is a runtime scenario?
The runtime view describes the behavior and interaction of the system’s building blocks with one or more scenarios.
A runtime scenario shows specific interactions or behaviors of building blocks or building block instances, either with each other or external elements.
It shows how the system fulfills certain tasks.
##### Question C-6-2: What do I document in the runtime view?
How the system executes, the dynamics of the system, its behavior in terms of its building blocks.
The following types of scenarios might be relevant for you:
• The most important use cases, features or functions
• System startup
• The systems´ behavior on its most important external interfaces
• The systems´ behavior in important error or failure situations
##### Question C-6-3: How can I describe scenarios or execution flows?
You have several options (many more than for the static building block view..). The following list is ordered by increasing documentation and maintenance effort. Options further down that list are more fine-grained and usually contain more details.
1. Document scenarios in plain text by enumerations or numbered lists. Include precise hints which building block executes which step(s) of use cases, processes or functions.
2. Use activity diagrams or flowcharts with swimlanes.
3. Use UML sequence diagrams. They are time-consuming to create and maintain with most interactive tools, but are an excellent means to show the mapping between building blocks and their actions. See question F-10 (tools for sequence diagrams) for some tips on tools.
UML has some additional options (e.g. state transition or object diagrams) to describe behavioral aspects of systems or building blocks. Those can be sometimes be useful, but are less often used that activity- or sequence diagrams.
##### Question C-6-4: What are partial scenarios?
Partial scenarios describe parts or extracts of complete scenarios or processes. They show only interesting, difficult, risky or important aspects of some greater process.
Concentrating on these essentials brings several benefits:
• You avoid trivial parts of scenarios
• You work more economically by leaving out unneccessary, non-important or low-priority aspects. That saves time/effort in creating and maintaining documentation!
• Smaller scenarios might be easier to understand (if (!) you make very clear which parts of the overall scenario you left out!)
A risk of partial scenarios might be consumers that don’t understand the prerequisites or preconditions of a partial scenario. Use annotations within your diagrams to explicitly clarify such required knowledge or facts.
In the figure shown below, you find a complete scenario first, and a nice partial scenario as an extract afterwards.
As you see, the first interactions seem quite trivial. Therefore, we can simple leave them out in a partial scenario:
|
##### Question C-6-5: Which scenarios shall I describe or document?
The following types of scenarios are candidates for runtime scenarios:
• The general case (aka the rule) of the most important use cases, features or functions.
• Important interactions with external systems, neighbours or user categories.
• Dynamic behavior of important external interfaces.
• Interactions that affect important quality goals or requirements.
• Error or failure conditions that might influence overall system behavior.
• Bootstrap-, startup or shutdown procedures, especially in distributed systems.
• Interactions that somehow deviate from normal stakeholder expectation, especially deviate from developer expectation.
• Interactions that work in non-standard ways.
• Scenarios or processes that are subject to timing constraints
#### 7. Deployment view
##### Question C-7-1: What does the deployment view show?
1. Your hardware structure(s), also called technical infrastructure.
2. The mapping(s), also called deployment, of your software to this hardware.
For part 1, hardware, there might be several variants. Therefore part 2, deployment, might be different, depending on these hardware structures.
##### Question C-7-2: Why do I need a deployment view?
The deployment (mapping of software onto hardware elements) can be rather complicated, see the example in question C-7-5. See also the second part of question C-7-1 (deployment).
##### Question C-7-3: Who shall describe/document the deployment view?
The first part of the deployment view, (see question C-7-1 (deployment).) will sometimes be created and maintained by stakeholders responsible for technical infrastructure and/or hardware.
The second part, the mapping of software to hardware, will often be documented and maintained by architects and/or the development team.
##### Question C-7-4: Shall I use UML deployment diagrams for the deployment view?
The UML provides only this type of diagram for deployment and infrastructure, therefore you have no practical alternative.
On the other hand, you can of course use any free form of diagram to depict your technical infrastructure.
Make sure that stakeholders understand such notations - and that architecture- and infrastructure decisions can be understood based upon these diagrams.
##### Question C-7-5: Can there be different deployment scenarios or variants?
Yes, for example when you have different stages from development to your production environment.
See the following example, where the (static) building blocks A, B, C can be deployed in three different alternatives:
##### Question C-7-6: What shall I do when my building blocks get dynamically assigned an execution environment (node) - so I cannot statically assign them to infrastructure nodes?
If some part of your system or your infrastructure decides at runtime where a particular instance of a building block gets executed, then the deploment view should at least explain this behavior.
It might be useful to create a “dynamic deployment concept” in arc42 section 8 and refer to this concept from the deployment view.
#### 8. Crosscutting concepts
##### Question C-8-1: What is a crosscutting concept?
We use the term “concept” for rules, principles or other decisions, guidelines, processes that influence one or more elements of the architecture.
• Decisions, or concepts that cannot adequatly be assigned to a single building block
• Decisions or rules that influence several
• building blocks
• parts of the implementation
• runtime scenarios
• interfaces
• several developers
##### Question C-8-2: Our arc42 section 8 (on concepts) is a melting pot of information we couldn’t put elsewhere? How can we keep an overview?
The variety of topics in section 8 might seem chaotic, but all topics contained have something in common: They are all crosscutting (for an explanation, see question C-8-1 (what is crosscutting)).
You shall remove (!) all concepts not relevant for your system (see question C-8-3 (deal with many concepts)).
Then you can group/order the remaining topics, either to system-specific criteria or after the following proposal (based upon ideas by Stefan Zörner):
• Business or domain aspects: domain, entity or data models, the ubiquitous language from domain-driven design
• Architectural patterns: What (recurring) patterns have been applied, and how?
• Development: Build and build management, code generation, configuration, (automated) testing, migration
• Under the hood: persistence, distribution, transactions, session-handling, caching, threading, exception and error handling, security
• Interactions with users or external systems: user interface aspects, i18n, validation, accessibility, communication, integration
• Operations: deployment, installation, monitoring
##### Question C-8-3: How shall I deal with the multitude of potentially crosscutting topics?
At first, treat arc42 section 8 as a checklist and work iteratively:
1. Remove every topic that is not relevant for your system.
2. Prioritize the remaining subsections, criteria should be importance and risk.
3. Work on the highest priorities and briefly (!) document the corresponding decisions.
Many crosscutting concepts will be highly technical, therefore you document or specify those for developers. Source code with brief explanations can sometimes be sufficient - and can save you from writing awkward documents!
##### Question C-8-4: How shall I describe my business-/domain model?
You have several options, shown in order of increasing effort and degree-of-detail
1. Create an (tabular) glossary of relevant business terms.
2. Create an informal outline, where you show data types and their relationships. Define the terms in a glossary.
3. Create a data or entity model (either as UML class diagram or Entity-Relationship diagram), where you show business entities with their attributes and relationsships. Define the terms in a glossary.
4. Define a rich object model, where you add methods, functions or services to the entity model.
5. Work according to the Domain Driven Design (see [Evans-2004]): Create a ubiquitous domain language with several bounded contexts, model business entities and aggregates, use value objects, services, repositories and factories to enable communication within and about the domain. (Obviously this FAQ does not strive to provide detailed introduction to the fascinating topic of DDD, you have to look elsewhere7.)
##### Question C-8-5: Are there any general rules how to describe a concept?
Often developers (or other technical people) have to adhere or comply to concepts, or obey them. Therefore understandability, clarity, and applicability are primary goals, together with a shot of consistency, of course.
Some rules you might apply:
• Be practical and use source code examples to explain and demonstrate. Automated test can accompany the plain code, giving developers a feel how to do things.
• Write your concepts in the form of developer use cases”: “A developer wants to achieve XYZ” - and explain step by step what people have to do.
• Explain reasons why the concept is like it is. If you deviate from established standards or procedures, explain why and how you came to your solution.
• You can combine text with static and dynamic diagrams to describe your more complicated concepts.
• Describe the applicability: In which or for what cases shall the concept be applied?
• Describe the limits: In what cases, under which circumstances will the concept fail or cease to work?
#### 9. Architectural decisions
In software engineering literature you find both “architecture decision” and “architectural decision”.
##### Question C-9-1: What kind of decisions shall I describe or document?
Describe or document the following kind of decisions:
• risky
• with expensive consequences
• with long-lasting effects
• affecting either
• a large number of stakeholders
• very special or important stakeholders
• that took a long time or much effort to decide
• astonishing
(document) architecturally significant” decisions: those that affect the structure, non-functional characteristics, dependencies, interfaces, or construction techniques.
Quoted from Michael Nygard
##### Question C-9-2: How can I document an architectural decision?
• Write a blog for your decisions, every decisions becomes a blog post.
• Use a text format similar to an (Alexandrian) pattern, like explained in question C-9-3 (Architecture Decision Record).
• For important decisions, the following topics might be interesting:
• subject of the decision
• affected stakeholders
• decision criteria (with priorities)
• alternatives
• evaluation of alternatives for the various criteria
• who took the decision?
• reason for chosing this alternative in favor of others
##### Question C-9-3: What’s an Architecture Decision Record (ADR)?
In 2011 Michael Nygard proposed to document important architecture decisions in the following pattern-like format:
• Title: A short phrase with an ID, e.g. “ADR 9: LDAP for Multitenant Integration”
• Context: Forces at play, including technological, political, social, and project organizational. Forces might be conflicting.
• Decision: How do we deal with these forces, what do we do.
• Status: A decision may be “proposed” (if stakeholders haven’t yet agreed), or “accepted” (once it is agreed). Later it might be marked “deprecated” or “superseded” (you might include a reference to its replacement).
• Consequences: What happens after the decision has been applied. All consequences should be listed here, not just the “positive” ones. A particular decision may have positive, negative, and neutral consequences.
The ADR format lacks decision criteria, which I (Gernot) sometimes regard as really important… but maybe I’m prejudiced.
##### Question C-9-4: How can we handle a large number of architecture decisions?
Create a blog (RSS-feed) and write a brief entry for your important decisions. Tag those with labels (e.g.: frontend, backend, SAP-interface or similar), so stakeholders can filter them.
Such a blog shows the history of your system quite well. You can combine the blog with the idea of architecture decision records (see question C-9-3).
#### 10. Quality scenarios
##### Question C-10-1: What is Software Quality?
(from IEEE Standard 1061): Software quality is the degree to which software possesses a desired combination of attributes. This desired combination of attributes need to be clearly defined; otherwise, assessment of quality is left to intuition.
##### Question C-10-2: What is a quality scenario?
Quality scenarios document required quality attributes. They help to describe required or desired qualities of a system, in pragmatic and informal manner, yet making the abstract notion of “quality” concrete and tangible.
• Event/stimulus: Any condition or event arriving at the system
• System (or part of the system) is stimulated by the event.
• Response: The activity undertaken after the arrival of the stimulus.
• Metric (response measure): The response should be measurable in some fashion.
There are different kinds of scenarios:
1. Usage scenarios: The system is used (any use case or system function is executed). Such scenarios describe how the system behaves in these cases, e.g. in terms of runtime performance, memory consumption, throughput or similar.
2. Change (or modification) scenarios: Any component within the system, its environment or its operational infrastructure changes or is being changed.
3. Failure scenarios: Some part of the system, its infrastructure or neighbors fail.
##### Question C-10-3: What is a quality tree?
(syn: quality attribute utility tree). A hierarchical model to describe product quality: The root “quality” is hierarchically refined in areas or topics, which itself are refined again. Quality scenarios form the leaves of these tree.
• Standards for product quality, like ->ISO 25010, propose generic quality trees. You find this generic quality tree in question C-1-2 (quality goals).
• The quality of a specific system can be described by a specific quality tree (see the example below).
##### Question C-10-4: Are there examples for quality scenarios?
• A new price calculation rule can be added to the pricing engine within 4 hours.
• The daily sales report for a single product category is generated within 10 seconds.
• When storage devices fail, the system gracefully shuts down (instead of crashing uncontrollably).
• The web client requires <5MB per browser session.
• Should the system run out of memory while processing the xyz algorithm, it will not crash, but will report the situation to an administrative user, stop the xyz process and return control to the interactive user within 30 seconds.
#### 11. Risks and technical debt
##### Question C-11-1: What are risks and technical debt?
Short answer: The currently known problems.
The known risks (things that might become problems) and problems in the system, its related organizational, operational and development processes. They can refer to source code, structures, concepts, decisions and all other aspects of the system.
#### 12. Glossary
You should ensure that all participating people have a common understanding of the important business (and technical) terminology which they use in context with the system.
The glossary is one manifestation of the general rule of “better explicit than implicit”.
### VII.D Questions on modeling
Here we collect questions regarding UML and other modeling notations, plus general questions regarding understandability and consistency of models.
##### Question D-0: Why do I need a model? I have source code.
For cute, neat, small and mignion systems, you won’t need any model.
For larger, distributed, risky or otherwise complicated systems, architectural information in addition to source code can support both development and evolution of systems.
Models in our sense are arbitrary abstractions of architecture or architecture decisions, relating to either structure or (crosscutting) concepts.
Examples of such models we found useful:
• Context of the system, showing external neighbours and interfaces.
• Building blocks (e.g. subsystems, modules or such), representing potentially large chunks of source code.
• Other views, e.g. runtime or deployment
• A domain model, representing the core of your domain.
Models can be expressed using diagrams plus explanations, but might also be textual or tabular…
##### Question D-1: What alternatives to UML exist for architecture models?
Several alternatives compete with UML, the list below gives an overview (just an arbitrary selection, not exhaustive):
• informal box- and line diagrams. They can be helpful and are often ok, but please explain your notation, otherwise your diagrams are open for (mis-)interpretation.
• Fundamental-Modeling-Notation (FMC), an (academic) approach to communicate dynamic systems with block-diagrams, petri-nets and entity-relationship diagrams. See the FMC notation reference for details. FMC is rarely used in business or industrial systems, although it’s quite promising.
• SysML is an UML dialect, created to support systems engineering. Supported by numerous modeling tools, but in my opinion not practically relevant.
• Simon Browns pragmatic C4 model of software architecture description.
##### Question D-2: How to arc42 and UML relate to each other?
They don’t really need each other. You can very well use arc42 with and without UML.
Graphical modelling with UML does by no means make you a better architect, nor does it (automatically) improve your systems.
You might use UML to describe or specify the following aspects of your architecture:
• static structure (building block view)
• runtime behavior or runtime scenarios (runtime view)
• deployment or infrastructure
##### Question D-3: How can I use UML to communicate a hierarchy of building blocks?
Component- and package diagrams can communicate hierarchies, as both UML components and packages can be nested.
For examples, see
##### Question D-4: How can I describe interfaces with UML?
The trivial option (usually not recommended): Just draw a line between two boxes to indicate that in interface between those two exist.
If you need more serious options, you have at least the following options (orderd by required effort for creating and maintaining such descriptions):
1. Give the line a label (to make it referencable) and explain its meaning in a table below the diagram. This should be sufficient for many non-safety-critical systems.
2. Use the provided/required notation (aka “ball/socket”), explicitely denoting which services/data/events the providing building block offers. There’s a nice explanation by Martin Fowler on this option.
3. Use distinct interface building blocks.
4. Use ball-and-socket notation in combination with port symbols.
The following figure shows options 1-4.
##### Question D-5: What can I use UML ports for?
Ports, these little rectancular boxes attached to components, packages or even nodes, represent an (optionally named) collection of interfaces. They come in handy for several reasons:
• Ports can support the detailed mapping of internal resp. external view of white- and blackboxes: Use ports to describe which internal building block of any whitebox communicates with an interface of the corresponding blackbox. In the following diagram, Foo communicates with Bar over a port. In the refining whitebox, the component BarA handles that interaction.
• I (Gernot) often used ports to denote the transmission protocoll for a particular interface: For an interface (e.g. inFoo) I can show that its available over http, https and ftp by attaching the same interface ball/socket to several ports, each port representing a distinct “access option” (ftp, http, https)…
Note: This is not what the original inventors of UML intented…
• In hardware- and deployment diagrams, ports can represent input/output channels, network adapters, virtual networks, IP-addresses or similar .
##### Question D-6: How can I improve the understandability of my diagrams/models?
Understandable diagrams contain 5-15 elements with their relationships - normal people simply cannot grasp more than that number.
If you have overloaded or large diagrams, reduce the number of elements by abstraction:
• Aggregate several elements into a named blackbox. These elements should be cohesive, they should somehow belong together. The criteria should be an explicit decision.
• Refrain from showing loads of details, e.g. attributes or methods of single classes. Those details can often be left to source code.
• Especially in runtime scenarios, don’t always start with the beginning of a scenario, but dive-right-into the interesting parts.
##### Question D-8: How can I improve the consistency of models?
Consistency (or homogeneity) implies recognizability and often understandability. We like to explain it with the term “(inner) order”
Consistent systems or models treat identical situations in the same manner, coherent or uniform.
Consistency in models includes topics like layout, color, naming or uniform usage of symbols.
### VII.E Questions on arc42 and agile methods
The Agile Manifesto says: “Working software over comprehensive documentation”. It does not say “no documentation”.
##### Question E-1: Does arc42 work with Scrum, Kanban or other lean/agile approaches?
Yes, it surely does. arc42 is supposed to be used “on demand” - you provide only the type or amount of information required for your specific documentation purpose.
This interpretation fits nicely with lean and agile approaches - where you shall avoid waste and focus on fulfilling requirements… arc42 helps you to fulfill your documentation requirements easily in a pre-structured and well-thought-out way.
##### Question E-2: Shall our team make arc42 part of the definition of done (DoD)?
Yes, of course: If documentation is not part of the DoD, then it surely won’t be done.
Remember: arc42 proposes lean, economical and compact documentation that can be incrementally or iteratively enhanced.
##### Question E-3: How can arc42 help us in reducing documentation effort?
• Document crosscutting concepts instead of too many building blocks.
• Use abstractions in the building block view and runtime view.
##### Question E-4: We use Scrum. How can we integrate arc42 into our sprints?
Documentation shall be part of your Definition-of-Done. Therefore arc42 is automatically included in Scrum processes.
##### Question E-5: In agile projects, what kind of information shall we write down and what shall we communicate orally?
Imagine you’re a future stakeholder who then needs to work on the system (implement, architect, deploy, test, operate, monitor or other). Then ask yourself what information you need to have…
##### Question E-6: In agile projects, can we skip (defer) documentation for a while and focus only on implementing?
In theory, yes. In practice such a deferral means that this documentation will never ever be created - and stakeholders might need to dig deep into source code to gather the appropriate information.
We urge you - document economically (short, brief, compact), but do this continously.
##### Question E-7: What are the most important tips for documentation in agile projects?
• Even in a Scrum-team, appoint a person responsible for documentation. We like to call her/him the “The Docu-Gardener”, whos’ tasks shall be:
1. care for the adequate form and content and
2. proactively search for unnecessary or outdated parts and remove them. Please note: care does not mean your gardener shall create all content by her- or himself, but identify appropriate producers within the team or among associated stakeholders.
• Document economically (“Less is often more”): We really believe that documentation can be helpful and ease development work - but only in an extend and degree appropriate for the system and its stakeholders.
• Clarify appropriateness and needs through early feedback: The type of your documentation should be appropiate with respect to the system, the affected people, domain, risks, criticality, effort/cost and possibly even other factors.
• Focus on explanation and rationale, not only facts: Most facts about a software system can be found in its source code - but not their explanation, reasoning and rationale.
• Rate requirements higher than principles.
### VII.F Questions on tools for arc42
As authors, we reserve the right to explain our personal experience and opinion. You should not base your tool selection solely on the information given here! There might be awesome tools available which we failed to mention…
##### Question F-1: What tools are well-suited for arc42?
This is a classical it depends type of question. Ask three different people, get at least five different (and most likely conflicting) answers…
• As arc42 documentation should always be a combination of text, tables and diagrams, a combination of different tools will often be a better choice than trying to get everything from a single tool (although tool vendors will tell you a different story)
• Often, (UML) modeling tools come with an abundance of functions and very limited usability. Their learning curve is high and might frustrate many. Overall, team acceptance tends to be low. Despite these usabilty shortcomigs we usually prefer real modeling tools over a pure graphics editor, especially for their better model consistency.
• Text processors (like Microsoft Word(R) or Open/Libre-Office) are omnipresent (practically all stakeholders can use them), but developers and other techies often don’t love them. Text processing tools provide limited team- and collaboration features, no real diff/merge, difficult to automate and are difficult to integrate with other tools. Nevertheless arc42 works with this category of tools without hassle.
• Plaintext-based approaches (like Markdown or AsciiDoc) in combination with graphics-tools (for smaller systems) or modelling tools (for medium to large systems) allow for documentation close-to-source-code and a high degree of automatability - therefore they tend to enjoy high acceptance with development teams.
• We love the combination of wiki plus graphics- or modelling tool. You can generate stakeholder-specific artifacts or pdf from modern wiki systems (like Confluence(R)).
##### Question F-2: What are useful criteria for selecting a toolset for arc42?
Some other criteria to watch out for:
• Teamwork: several people should be able to work in parallel
• Versioning: documentation artifacts need to be version-controlled. That’s quite easy for single files, but some modeling tools make versioning harder than it should be.
• Artifact generation: Sometimes you need beautiful or stakeholder-specific output (e.g. pdf, html). Some tools excel in this, others fail miserably.
• Robustness: You don’t want to re-create diagrams or documentation because your boss made you use that (supposedly) cheap free-and-open-source documentation tool… I had especially bad experiences with immature graphic/UML editors.)
• Availability of know-how: For several of the more common tools it’s quite easy to find people with corresponding experience. Some tool vendors provide excellent support, also in methodical questions.
##### Question F-3: Can I automatically include important code artifacts in arc42 documentation?
Short answer: Never (!) copy/paste code into your documentation - as it will be outdated before you hit the save key.
Longer answer (for AsciiDoc):
You can include code fragments be the following scheme:
Instead of explicitely giving the line numbers in the include statements, you can annotate the corresponding source files with comment tags, as shown in the following example:
and then reference this snippet in your documentation as follows:
See also the awesome blogpost from mrhaki and another on partial includes
##### Question F-4: How do I use arc42 with modeling tools?
Most modeling tools lack out-of-the-box arc42 support, you have to create your desired arc42 (sub-)structures by yourself.
arc42 does provide an empty arc42-structured model for Sparx Enterprise Architect(R), although not very elaborated.
We advice to restrict the use of modeling tools to the pure graphical part of architecture documentation, leave text and tables to other tools that are more suited for these kinds of information. In this case, create the following arc42-sections in your modeling tool:
• Context view
• Building block view
• Runtime view
• Deployment view
• Crosscutting concepts
<t.b.d.>
##### Question F-6: Are there (free or open-source) modeling tools available?
In principal, yes. For example: * UMLet * Modelio * Violet
We have, at least until November 2016, not gotten to know any free modeling tool that can really compete with commercial tools with respect to robustness, useability, feature completeness and availability-of-know-how. Before you go for any free modeling tool, please make reasonably sure that your candidate fulfills the basic requirements we explained in Question F-2, concerning teamwork, robustness, versioning and artifact generation.
Actually, compliance to any specific UML standard is very often overrated.
##### Question F-7: Can we use arc42 with the Atlassian Confluence wiki?
Yes - it’s a (near-perfect) combination: Confluence8 is a powerful, yet easy-to-use collaboration platform, a “wiki-on-steroids”.
You can easyli map the arc42 structure on Confluence pages, and even use predefined templates to setup the complete arc42 structure in a breath (see question F-8 (Confuence tools) for details on available tooling).
In Confluence, you can tag (label, mark) pages, so it becomes very simple to add arbitrary meta-information to certain parts of your documentation. Use this e.g. to add version/release-specific information, or to distinguish between “already-implemented” and “planned-for-the-future” information.
Confluence does not (!) provide out-of-the-box diagramming features, but numerous tools are available to the rescue, see question F-9 (Confluence Diagramming tools) for details).
##### Question F-8: What tools can I use to setup arc42 in Confluence?
There a a few options to simplify the setup of the arc42 structure within your Confluence:
• The Networked-Assets ATC macro to insert the arc42 template into any Confluence space.
• The smartics Blueprints for arc42
• (Within the near future…) You can get a Confluence space generated out of the arc42 Golden Master from the arc42-template github repository. We’re currently working hard on making this available for download. Please contact us for a pre-release version if you’re interested.
See question F-9 (Diagrams in Confluence) for an overview of diagramming tools available for Confluence.
##### Question F-9: How can I create diagrams in Confluence?
• You can create and maintain your diagrams with any modeling tool and export your diagrams in jpg or png. Concluence can import these files, and will even keep a history of updates, if you ever upload newer versions. Beware: This requires maximal manual effort…
• You can create and maintain diagrams with a Confluence graphics plugin. I (Gernot) have positive experience with the following:
• Gliffy, a well-known plugin for creating arbitrary diagrams within Confluence pages. The editor is complety integrated into the web browser. Supports different versions of diagrams, stable and robust, requires a commercial license.
• Draw.io, a powerful browser-based graphics editor, also available as Confluence plugin. I (Gernot) have used the plain-browser version of draw.io - which can be used completely offline and export diagrams to various formats.
##### Question F-10: What tools shall I use for sequence diagrams?
Sequence diagrams (SDs) can be really useful to document or specify runtime scenarios, see question C-6-3 (how to document scenarios), but they are time-consuming to create and maintain with graphical modeling tools.
We will briefly discuss the following options:
###### UML tool for sequence diagrams
With your favorite UML modeling tool, you can create and maintain sequence diagrams by dragging static building blocks onto the drawing canvas. When using these kind of tools, it’s relatively easy to maintain integrity between static and dynamic views, but the effort for creating diagrams is really high.
###### Sequence diagrams generated from textual description
PlantUML is a free tool that can render sequence diagrams from a textual description.
Consider an example: In the following listing you find the description of a simple sequence, shown in the figure below:
Charming: Such textual descriptions can be merged and versioned like any other source code!
PlantUML supports most UML SD constructs, like interaction references, loops, alternatives and so on. You can apply some styles to diagrams and export in several graphic formats (png, jpg, svg). There are numerous plugins available for wikis, development environments, build tools or the command line.
###### SDs build with browser apps
You may have a look at:
• DrawIO, an interactive (browser-based) graphic editor that can draw arbitrary (vector) graphics. Quite cool is the offline feature.
• Web-Sequence-Diagrams, similar to PlantUML, but only browser-based. For high-quality output, you need a (paid) subscription. The free online version can export png files. An example is given below:
##### Question F-10: Can I use PlantUML for building block / component diagrams?
No, you should not use PlantUML for static diagrams - although you could. Read the longer answer for an explanation.
We really encourage the use of PlantUML for sequence diagrams (see question F-9 (Tools for sequence diagrams)), but don’t like it for static diagrams. That’s because in static diagrams we propose to carefully layout your building blocks, so that important elements are focussed, or elements somehow belonging together are drawn close to each other.
In PlantUML you can give formating hints, but the autolayout algorithms decide for themselves wether they can apply your suggestions or ignore them. I (Gernot) tried numerous times for real-world situations - and was always really unhappy with the results.
From my experience, static diagrams, like class- or component diagrams, look well only in hello-world like situations…
### VII.G Questions on arc42 and versions / variants
##### Question G-1: Shall I version control our architecture documentation?
Yes, of course!. At best with the same strategy, processes and tools you use for your source code.
(Gernot’s opinion): If it’s not under version control, it does not exist (with the sole exception of well-managed wiki systems, that easily allow you to read older versions of documents.)
If you cannot use your version control system for your (architecture) documentation, see Question G-2.
##### Question G-2: We cannot use version control for documents. What to do?
Really (really!) make sure you find a (mostly automated) way to get your documentation under version control.
One way which helped me several times is the following:
• When you release your software (in continous deployment scenarios, you might stick to major releases…), create a pdf file from your documentation.
• Put this pdf under version control, in the same branch, tag/label or release info as your code-to-be-released.
##### Question G-3: How does versioning work with graphical models?
Although some tool vendors argue differently, recognizing the meaning of differences in diagrams imho requires human intelligence: I (Gernot) have not yet found any automatism supporting meaningful diff of diagrams or images. In my opinion somebody needs to care for new versions of diagrams.
It’s often a good idea to (automatically) include the modification date in a diagram (some modeling tools support this feature).
<t.b.d.>
### VII.H Questions on arc42 and traceability
##### Question H-1: What does traceability mean (with arc42)?
“Traceability (or Requirements Traceability) refers to the ability to link product requirements back to stakeholders’ rationales and forward to corresponding design artifacts, code, and test cases.” (quoted from Wikipedia)
Full-scale traceability needs every single requirement to be referenced in any part of the architecture/solution documentation.
From a more pragmatical architectural (arc42) perspective, traceability can also refer to the understandability or explanation of important or fundamental architecture or solution decisions: “We took the decision Dec-X because of the requirements Req-Y.”
##### Question H-2: Shall we strive for traceability in our documentation?
If you can, you should definitely avoid it: It’s horribly expensive, takes an awful lot of time and is really difficult.
In case you’re working in safety critical systems, like medical, pharmaceutical or avionics, you are most likely required to provide traceability.
One nice hook within arc42 are blackboxes within the building block view: Use the blackbox template to document:
• the requirements fulfilled or satisfied by this particular building block,
• the source code needed to implement this building block.
##### Question H-3: How can I keep architecture documentation in sync with source code?
We’ve several suggestions, but the most important one first:
• Document economically, especially be frugal with building block details: Many developers will understand many details of building blocks by digesting source code - so don’t create a large number of whitebox diagrams.
• Level-1 of the building block view will be one of your best friends: That top-level structure of your system will most often remain quite stable over time, and won’t need to be updated often. Many source code changes don’t affect level-1!
• Defer documentation of volatile parts: In case you can anticipate structural changes or volatility in certain parts of your system, leave the documentation of these parts as abstract and high-level as possible until these parts have reached a stable state.
• Prefer documenting “crosscutting concepts” (arc42 section 8) over detailed building blocks (section 5) or runtime scenarios (section 6).
### VII.J Questions on managing documentation
##### Question J-1: How to document (very) large systems with arc42
Modularize extensive documentation:
Factor out (extract) documentation commonalities from different parts of the system and create separate documentation for relevant subsystems.
Link up these different parts of the documentation by hyperlinks, so that consumers can navigate freely between special and common parts.
A toolchain supporting modularization is an important prerequisite: it’s a nuisance with typical office products, but works quite smoothly with wikis or markup-languages.
For a schematic example of modularized documentation see figure arc42-modularized: The overall systems is made up from three subsystems. The common architecture documentation contains the introduction and goals, strategic decisions, building blocks level 1 plus crosscutting concepts. All these informations are valid and useful for the other three (smaller) documents.
Each subsystems’ documentation contains only parts of the arc42 template, and each subsystem focusses on different aspects.
The common part of your architecture documentation will most often contain the global system goals and the overall business context. Eventually it will also contain the solution strategy plus some crosscutting concepts.
##### Question J-2: Does a documentation governance make sense for architecture documentation?
It depends: On one hand, homogeneity, consistency and standardization can be really useful and efficient for documentation, on the other hand can cost and effort for such standardized documentation become very high.
Answer the following questions for your situation. The more “yes” answers the more useful a regulated documentation might be:
• Do you have many (i.e. more than 15) IT-systems you develop or maintain?
• Do many of these systems have technical similarities (use common frameworks, implementation approaches, technical concepts)?
• Do you work with different implementation or maintenance teams?
• Do you have a lot of change within your development team?
• Do different organizations work on your systems (i.e. external service providers)?
• Do you work in regulated domains (i.e. medicine, aerospace, pharmaceutics etc.)?
• Will the documentation of your systems be audited, validated or otherwise (formally) checked?
• Are some of your systems highly critical for success or survival of your organization?
• Are your systems maintained by a large development team (>100 people)?
##### Question J-3: Is there a checklist for arc42 documentation?
See question J-4.
##### Question J-4: Is there a general checklist for architecture documentation?
Counterquestion: What do you want to achieve with such a checklist?
• Improve the content of the documentation: You won’t get that by a checklist, but only by good examples, coaching and feedback.
• Checking formal criteria: Can you improve by checklists - but there’s no general consensus on formal criteria. We suggest you refrain from overly formal criteria - as documentation won’t get much more useful when it’s formal.
• You want to remind your team of some important parts of the documentation: Determine the subset of arc42 that your stakeholders really need - and then include those in your definition-of-done (see also question E-2 )
##### Question J-5: How can I create delta-documentation with arc42?
In other words: How can I document the changes (deltas) between two consecutive versions of a system?
<t.b.d.>
##### Question J-6: What level of access permissions shall stakeholders have on the architecture documentation?
Keep the administrative effort as low as possible: If you accord details access permissions, you create administrative overhead.
Usually, all stakeholders can read and write the documentation. Some cases justify exceptions from this rule:
• For large teams, you should modularize your documentation and restrict write-access to certain parts, to avoid accidental changes.
• You might differentiate (short-term) project and (long-term) system documentation. Everybody can write in project documentation, only few people might modify system documentation.
• In case of regulated documentation, you might need detailed permissions (and access logs) to avoid manipulation.
### VII.K Questions on customizing arc42
##### Question K-1: Are we allowed to customize arc42 for our (company/organization/system…) purpose?
Yes, of course. See Question A-2 (arc42 License).
You may modify the structure of the template, re-create it in any tool of your choice or even rename it for your own purpose (one of our clients called it SAT42, what we really enjoyed.)
To be fair, you should reference Gernot Starke and Peter Hruschka as creators of arc42 in case you use or modify the template…
##### Question K-2: What kinds of customizations are common for arc42?
Customize or change arc42 only if you absolutely need to. The standardized structure of arc42 has a high recall value - many people are accustomed to it.
Keep arc42 modifications to subsections and leave the high-level structure unchanged.
I (Gernot) encountered the following modifications during real-world projects:
• User-interface view (UI-design, UI-forms, common layout etc). Sometimes, UI details can be highly relevant for other architectural aspects, e.g. handling application state, integrating with workflow technology or similar.
• Data view, explaining technical data models or concrete table structures. Often relevant in very data-centric organisations, where portfolios of applications are built around these data structures.
• Developer guide, containing detailed development information, sometimes called development use cases.
## Appendix A: Further References and Resources
Agile Modeling
Scott Ambler: Agile Modeling. Tips and practices to pragmatic, agile modelling and documentation.
arc42
The template for software architectures, http://arc42.org (German website http://arc42.de)
arc42@Github
The arc42 Github repository: https://github.com/arc42/arc42-template. The (AsciiDoc) master version of the template is maintained here.
arc42 Examples of Quality Requirements
Examples for quality requirements with explanations and definitions: (currently only in German) https://github.com/arc42
ATAM,
Architecture Tradeoff Analysis Method. Systematic approach to evaluate the quality of software architectures by identifying risks and tradeoffs with respect to system-specific quality attributes.
Bass+2012
Bass, L/Clements, P/Kazman, R. Software Architecture in Practice, 3rd edition, Addison-Wesley, 2012. Although the title suggests otherwise, a quite fundamental (and sometimes abstract) book, written by members of the Software Engineering Institute. The authors have a strong background in ultra-large scale (often military) systems - so their advice sometimes conflicts with smaller kinds of projects. In our opinion it’s fundamental, but not very practical.
Brown2015
Brown, Simon. Software Architecture For Developers, https://leanpub.com/software-architecture-for-developers Technical leadership by coding, coaching, collaboration, architecture sketching and just enough up front design. Available on Leanpub. Readable, pragmatic and developer-oriented overview.
Buschmann+1996
Buschmann, Frank/Meunier, Regine/Rohnert, Hans/Sommerlad, Peter. A System of Patterns: Pattern-Oriented Software Architecture 1, 1st edition, 1996, John Wiley & Sons.
DeMarco+07
De Marco, Tom/Hruschka, Peter/Lister, Tim: Adrenaline Junkies and Template Zombies, http://www.amazon.com/Adrenaline-Junkies-Template-Zombies-Understanding, Dorset-House, 2007
Evans2004
Evans, Eric. Domain-Driven Design: Tackling Complexity in the Heart of Software, 1st edition, 20 August 2003, Addison-Wesley, 2003. The seminal and exhaustive work on constructing software with focus on the domain. One of my all-time favorite IT books, although you need some patience to read through it.
HtmlSanityChecker
A very small open source project to semantically check HTML files, e.g. for missing links, missing images, duplicate link targets and other kinds of errors.
Clements+2002
Clements, Paul/Kazman, Rick/Klein, Mark. Evaluating Software Architectures: Methods and Case Studies, Addison-Wesley, 2001.
Hargis+2004
Hargis, Gretchen/Carey, Michelle/Hernandez, Ann. Developing Quality Technical Information: A Handbook for Writers and Editors, IBM Press, 2nd edition, Prentice Hall, 2004. If you need to write lots of documentation, you should have a look at this book.
Hofmeister+2000
Hofmeister, Christine/Nord, Robert/Soni, Dilip. Applied Software Architecture, 1st edition, Addison-Wesley, 1999
Quian+2010
Qian, K/Fu, X/Tao, L/Xu, C/Herrera, J. Software Architecture and Design Illuminated, 1st edition, Jones and Bartlett, 2010. Well-structure and readable collection of architecture styles and patterns.
## Appendix B: Also by This Author
If you liked the way we explained, communicated and documented software architectures, you might want to see more arc42 examples.
#### arc42 by Example
Also published on Leanpub, this collection of real-world systems, documented and explained with the proven and pragmatic arc42 template.
It contains the software architecture documentation of several concrete systems, how to improve your architecture communication and documentation: You find it on https://leanpub.com/arc42byexample.
#### iSAQB CPSA-Foundation Study Guide
Also published on Leanpub, this book helps you prepare for the iSAQB examination “Certified Professional for Software Architecture - Foundation Level”.
## Appendix C: About Gernot Starke
Dr. Gernot Starke (innoQ Fellow) is co-founder and longstanding user of the (open source) arc42 documentation template. For more than 20 years he works as software architect, coach and consultant, conquering the challenges of creating effective software architectures for clients from various industries.
In 2008 Gernot co-founded the International Software Architecture Qualification Board (iSAQB e.V.) and since then supports it as an active member.
2014 he founded the (open source) Architecture Improvement Method aim42.
Gernot has authored several (German) books on software architecture and related topics.
Gernot studied computer science at the Institute of Technology in Aachen (RWTH Aachen) and finished with a Diploma. He then worked as a developer and consultant for smaller software companies, before coming back to university for international research on methodical software engineering. 1995 he received his PhD from Johannes Kepler University of Linz, Austria (Prof. Gerhard Chroust for his thesis on “Software Process Modeling”.
He then joined Schumann AG in Cologne and did consulting and development work for several years. He became technical director of the “Object Reality Center”, a joint-venture of Sun Microsystems and Schumann Consulting AG and lead the first European Java Project (the Janatol project for Hypobank in Munich).
Since then he has consulted and coached numerous clients from various domains, mainly finance, insurance, telecommunication, logistics, automotive and industry on topics around software engineering, software development and development process organization.
Gernot was an early adopter of the agile movement and has successfully worked as Scrum master in agile projects.
He lives in Cologne with his wife (Cheffe Uli) and his two (nearly grown-up) kids, two cats and a few Macs.
Email Gernot Starke or contact him via Twitter @gernotstarke.
## Appendix D: Table of Tips
Here you find an overview of all tips contained in this book. The following table is generated9 from the complete book by parsing the markdown source.
Tip III-1 Appoint a Responsible Person (The Docu-Gardener)
Tip III-2 Document economically (“Less is often more”)
Tip III-3 Clarify appropriateness and needs through early feedback
Tip III-4 Communicate top-down, work differently if necessary
Tip III-5 Focus on Explanation and Rationale, Not Only Facts
Tip III-6 Rate Requirements Higher than Principles
Tip III-7 Separate Volatile and Stable Documentation
Tip III-8 Don’t Repeat Yourself, If Possible
Tip III-9 Document Unmistakably
Tip III-10 Establish a Positive Documentation Culture
Tip III-11 Create Documentation From The Reader’s Point of View
Tip IV-128 Keep the introduction (arc42 section 1) brief
Tip IV-129 Document and explain the specific quality tree
Tip IV-130 Document the quality tree as mindmap
Tip IV-131 Use the quality tree as checklist
Tip IV-132 Consider usage-scenarios in the quality tree
Tip IV-133 Consider change-scenarios in the quality tree
Tip IV-134 Consider stress- and error scenarios in the quality tree
Tip IV-135 Use the quality tree for systematic architecture assessment/evaluation
Tip IV-136 Search for risks with different stakeholders
Tip IV-137 Examine external interfaces for risks
Tip IV-138 Identify risks by qualitative architecture evaluation
Tip IV-139 Examine (development/deployment/etc) processes for risks
Tip IV-140 Take the glossary seriously
Tip IV-141 Document the glossary as a table
Tip IV-142 Keep the glossary brief and compact
Tip IV-143 Restrict the glossary to relevant terms, avoid trivia
Tip IV-144 Make the product owner be responsible for the glossary
Tip IV-1 Compact summary of the functional requirements and driving forces.
Tip IV-2 Limit yourself to the essential tasks and use cases
Tip IV-3 Highlight the business objectives of the system
Tip IV-4 Create an overview by grouping or clustering requirements
Tip IV-5 Make sure you can reference the requirements
Tip IV-6 Use activity diagrams to describe functional requirements
Tip IV-7 Use BPMN diagrams to describe functional requirements
Tip IV-8 Use a numbered list to describe functional requirements
Tip IV-9 Use (semi) formal text to describe functional requirements
Tip IV-11 Always work with explicit quality requirements
Tip IV-12 Explain quality requirements through scenarios
Tip IV-13 If you do not get quality requirements, make your assumptions explicit
Tip IV-14 Use checklists for quality requirements
Tip IV-15 Use examples to work out quality goals together with your stakeholders
Tip IV-16 Keep the introduction short! Show only the “hit list” of the quality requirements
Tip IV-17 Combine quality goals with the action points of the “solutions strategy” section
Tip IV-18 Show the detailed quality requirements in arc42 section 10
Tip IV-19 Search broadly for stakeholders
Tip IV-20 Describe the expectations of the architecture and documentation stakeholders
Tip IV-21 Maintain a stakeholder table
Tip IV-22 Do not use the stakeholder table if your management already maintains a consistent stakeholder overview
Tip IV-23 Classify your stakeholders by interest and influence
Tip IV-24 Look at the contraints of other systems within the organization
Tip IV-25 Clarify the consequences of constraints
Tip IV-26 Describe organizational constraints
Tip IV-27 Describe design and development constraints
Tip IV-28 Differentiate different categories of constraints
Tip IV-29 Explicitly demarcate your system from its environment
Tip IV-30 Show the context graphically
Tip IV-31 Combine the context diagram with a table
Tip IV-32 Explicitly indicate risks in the context
Tip IV-33 The context should give an overview and should not contain any details
Tip IV-34 Simplify the context boundary through categorization
Tip IV-35 If many external systems are involved, merge them by explicit criteria
Tip IV-36 Summarize “many external systems” with ports
Tip IV-37 Show all (all!) external interfaces
Tip IV-38 If you offer external interfaces to other systems, create discrete interface documents
Tip IV-39 Differentiate domain and technical context
Tip IV-40 Show rather data flows within the domain context
Tip IV-41 Show external influences in the context
Tip IV-43 Show transitive dependencies in the context
Tip IV-44 Pay attention to quality requirements for interfaces
Tip IV-45 Show the technical context
Tip IV-46 Use the technical context to describe protocols or channels
Tip IV-47 Explain the relationship between a functional interface and its technical realization
Tip IV-48 Show the technical context in the deployment view
Tip IV-49 Show variants of the technical context
Tip IV-50 Explain the solution strategy as compact as possible (e.g. as list of keywords)
Tip IV-51 Describe important solution approaches as a table
Tip IV-52 Describe solution approaches in context of quality requirements
Tip IV-53 Refer to concepts, views or code
Tip IV-54 Let the solution strategy grow iteratively / incrementally
Tip IV-55 Justify the solution strategy
Tip IV-56 Use a common format for sections of the building block view
Tip IV-57 Organize the building block view hierarchically
Tip IV-58 Always describe level-1 of the building block view (“Level-1 is your friend”)
Tip IV-59 Describe the responsibility or purpose of every (important) building block
Tip IV-60 Hide the inner workings of blackboxes
Tip IV-61 Document blackboxes as table using the minimal blackbox template
Tip IV-62 Document blackboxes with the extensive blackbox template
Tip IV-63 Document blackboxes as structured text
Tip IV-64 Justify every whitebox (blackbox decomposition)
Tip IV-65 Document whiteboxes with the minimal whitebox template
Tip IV-66 Document whiteboxes with the extensive blackbox template
Tip IV-67 Use runtime views to explain or specify whiteboxes
Tip IV-68 Use inheritance to describe similar behavior of multiple building blocks
Tip IV-69 Use crosscutting concepts to describe or specify of multiple building blocks
Tip IV-70 Document multiple levels of the building block view
Tip IV-71 Keep your external interfaces in context and building block view consistent
Tip IV-72 Explain the mapping of source code to building blocks
Tip IV-73 Describe where to find the source code of building blocks
Tip IV-74 Organize the mapping of source code to building blocks according to directory structure
Tip IV-75 Organize the mapping of source code to building blocks according to the modularization of the programming language(s)
Tip IV-76 Ensure that every piece of source code has its proper place within the building block view
Tip IV-77 In exceptional cases, include thirt-party software (libraries, frameworks, middleware etc) in the building block view
Tip IV-78 Frugally document internal interfaces
Tip IV-79 Document / specify interfaces with unit-tests
Tip IV-80 Document / specify interfaces with runtime scenarios
Tip IV-81 Use building block view level-1 for “further” information
Tip IV-82 Refine several building blocks at once, if that’s adequate
Tip IV-83 If you refine several building blocks at once, ensure uniqe mapping
Tip IV-84 Refine only a few building blocks
Tip IV-85 Document concepts instead of building blocks
Tip IV-86 Always map existing building blocks to the activities within runtime scenarios
Tip IV-87 Limit the runtime view to very few scenarios
Tip IV-88 Explain schematic (generic) processes in the runtime view
Tip IV-89 Explain concrete (specific)
Tip IV-90 Use runtime scenarios primarily to detect or identify building blocks - rarely for documentation
Tip IV-91 Use sequence diagrams to describe or specify runtime scenarios
Tip IV-92 Document or specify parial scenarios
Tip IV-93 Use plain activity diagrams to describe or specify runtime scenarios
Tip IV-94 Use activity diagrams with partitions to describe or specify runtime scenarios
Tip IV-95 Use text to describe runtime scenarios
Tip IV-96 Mix large and small building blocks within single runtime scenarios
Tip IV-97 Document or specify the technical (hardware) infrastructure
Tip IV-98 Document and explain the infrastructure decisions
Tip IV-99 Document or specify the various (technical) runtime or staging environments
Tip IV-100 Show the deployment view as hierarchy
Tip IV-101 Document or specify the mapping of building blocks to hardware
Tip IV-102 Use UML deployment diagrams for software-hardware-mapping
Tip IV-103 Use building blocks instead of artifacts in deployment diagrams
Tip IV-104 Use artifacts instead of building blocks in deployment diagrams
Tip IV-105 Use tables to document or specify software-hardware mapping
Tip IV-106 Explain intention or relevance of infrastructure nodes
Tip IV-107 Document or specify everything which might be neccessary for operating the system
Tip IV-108 Leave hardware decisions to infrastructure experts
Tip IV-109 Restrict documentation of concepts to the most important topics
Tip IV-110 In concepts, explain “how it should be done”
Tip IV-111 Keep foundations of background information to a bare minmum
Tip IV-112 Document or specify domain or business models
Tip IV-113 Combine domain models with the glossary
Tip IV-114 Document or specify the domain model
Tip IV-115 Create at least a domain data model
Tip IV-116 Document concepts by showing source code
Tip IV-117 Document crosscutting concepts with the 4-quadrant-model
Tip IV-118 Document decisions instead of concepts
Tip IV-119 Use stereotypes to document which concepts are applied within which building blocks.
Tip IV-120 Use the arc42 topic proposals as checklist for crosscutting concepts
Tip IV-121 Document the criteria for important decisions
Tip IV-122 Explain reasons for important decisions
Tip IV-123 Document or explain decisions as mindmap
Tip IV-124 Document or explain decisions as table
Tip IV-125 Document or explain decisions as “Architecture Decitions Record (ADR)”
Tip IV-126 Document or explain discarded alternatives
Tip IV-127 Document or explain decisions informally as blog
Tip V-1 Explicitely clarify the target audience of your documentation
Tip V-2 Use existing examples as jumpstart
Tip V-3 Remove unneeded arc42-sections
Tip V-4 Augment arc42 by required sections or topics
Tip V-5 Indicate currently unneeded arc42 sections
Tip V-6 Prepare an arc42 repository for the team
Tip V-7 Prepare a sandbox for documentation
Tip V-8 Document (and work) from different perspectives
Tip V-9 Documents belong into version control
Tip V-10 Conduct brief and thematically focussed documentation reviews
Tip V-11 Treat documentation tasks equal to developent tasks
Tip V-12 Document continously - parallel to development
Tip V-13 Always document in timeboxes
Tip V-14 Formulate like “The system {has|does|is|…}”
Tip V-15 Create hindsight documentation iteratively in short timeboxes
Tip V-16 Create documentation before you continue development
Tip V-17 Document those parts you’re anyhow working on
Tip V-18 Make other documents unnecessary as quickly as possible
Tip V-19 Remove superfluous or outdated documents
Tip V-20 Explicitly reveal problems and risks
Tip V-23 Start with the context and external interfaces
Tip V-24 Start with the solution strategy
Tip V-25 Take documentation serious even in agile projects
Tip V-26 Make documentation part of the Definition-of-Done
Tip V-27 Apply an iterative and incremental approach to documentation
Tip V-28 Use walls instead of electronic tools
Tip V-29 Enable cooperative work, even for distributed teams
Tip V-30 Modularize extensive documentation
Tip V-31 Extract goals, context and concepts
Tip V-32 Appoint a responsible person responsible for the overall documentation
Tip V-33 Unify the documentation toolchain
Tip V-34 Leave documentation of IT-landscape to enterprise architects
Tip V-35 Structure your overall documentation along organizational boundaries
Tip II-1 Improve your arc42 skills by reading additional examples
## Appendix E: Table of Questions
Here you find an overview of frequently asked questions. The following table is generated10 from the complete book by parsing the markdown source.
Question A-1 What does the 42 mean?
Question A-2 What’s the license?
Question A-3 What’s the pricing model of arc42?
Question A-4 How widely is arc42 used?
Question A-5 What’s the target audience of architecture documentation?
Question A-6 What are possible alternatives to arc42?
Question A-7 For what kinds of systems is arc42 applicable?
Question A-8 In which cases shall we NOT use arc42?
Question A-9 How can I contribute or report bugs in arc42?
Question B-1 Which parts of arc42 are “essential”, which are “optional”?
Question B-2 Does arc42 prescribe or enforce specific notations?
Question B-3 Shall I use UML for arc42 documentation?
Question B-4 What is the minimal amount of an arc42 documentation?
Question B-5 Where to document external interfaces?
Question B-6 How to document external interfaces?
Question B-7 Where to put links to external systems (“neighbors”) documentation?
Question B-8 What is a blackbox / the blackbox template?
Question B-9 What is a whitebox?
Question B-10 What is the building block hierarchy?
Question B-11 How to document a whitebox with the whitebox template
Question B-12 Where shall I describe important (blackbox) components?
Question B-13 Can I use arc42 in agile projects, e.g. with Scrum?
Question B-14 Can I update documentation incrementally? Or shall I document after the implementation is finished?
Question B-15 What kind of information shall I provide for which stakeholder?
Question B-16 What does economical documentation mean?
Question C-1-1 How extensive shall I describe the requirements of the system?
Question C-1-2 What are quality goals (aka quality attributes)?
Question C-1-3 Why does arc42 propose a stakeholder table?
Question C-1-4 What shall be included in a stakeholder table?
Question C-1-5 What are the expected deliverables in the stakeholder table?
Question C-2-1 What are constraints?
Question C-2-2 What types of constraints exist?
Question C-2-3 What kind of constraints shall I document?
Question C-2-4 Can/shall we negotiate constraints?
Question C-3-1 What is the context (scope)?
Question C-3-2 How shall I document the context (scope)?
Question C-3-3 What is the “business context”?
Question C-3-4 What is the “technical context”?
Question C-3-5 In which cases shall I document the “business context”?
Question C-3-6 In which cases shall I document the “technical context”?
Question C-3-7 How can I simplify documentation of the “business context”?
Question C-3-8 Shall I document risks or problems in the context?
Question C-4-1 What is the “solution strategy”?
Question C-4-2 How to document the “solution strategy”?
Question C-4-3 Can you provide examples for the solution strategy?
Question C-5-1 What is a “building block”?
Question C-5-2 Do third-party libraries, frameworks or tools count among building blocks?
Question C-5-3 How does source code relate to building blocks?
Question C-5-4 How detailed shall we document the building block view?
Question C-5-5 Can I refine a group of building blocks together?
Question C-5-6 How can I document or specify building blocks?
Question C-5-7 How shall I document building blocks for non-object-oriented systems?
Question C-5-8 How do I describe internal interfaces?
Question C-5-9 How do I describe external interfaces?
Question C-5-10 How can I avoid redundancy with interfaces in the building block view?
Question C-5-11 How does the hierarchy of building blocks relate to the arc42 sections (5.1, 5.2 etc.)?
Question C-5-12 What kind of building blocks don’t I need to document or specify?
Question C-5-13 What kind of building blocks shall I document or specify?
Question C-6-1 What is a runtime scenario?
Question C-6-2 What do I document in the runtime view?
Question C-6-3 How can I describe scenarios or execution flows?
Question C-6-4 What are partial scenarios?
Question C-6-5 Which scenarios shall I describe or document?
Question C-7-1 What does the deployment view show?
Question C-7-2 Why do I need a deployment view?
Question C-7-3 Who shall describe/document the deployment view?
Question C-7-4 Shall I use UML deployment diagrams for the deployment view?
Question C-7-5 Can there be different deployment scenarios or variants?
Question C-7-6 What shall I do when my building blocks get dynamically assigned an execution environment (node) - so I cannot statically assign them to infrastructure nodes?
Question C-8-1 What is a crosscutting concept?
Question C-8-2 Our arc42 section 8 (on concepts) is a melting pot of information we couldn’t put elsewhere? How can we keep an overview?
Question C-8-3 How shall I deal with the multitude of potentially crosscutting topics?
Question C-8-4 How shall I describe my business-/domain model?
Question C-8-5 Are there any general rules how to describe a concept?
Question C-9-1 What kind of decisions shall I describe or document?
Question C-9-2 How can I document an architectural decision?
Question C-9-3 What’s an Architecture Decision Record (ADR)?
Question C-9-4 How can we handle a large number of architecture decisions?
Question C-10-1 What is Software Quality?
Question C-10-2 What is a quality scenario?
Question C-10-3 What is a quality tree?
Question C-10-4 Are there examples for quality scenarios?
Question C-11-1 What are risks and technical debt?
Question D-0 Why do I need a model? I have source code.
Question D-1 What alternatives to UML exist for architecture models?
Question D-2 How to arc42 and UML relate to each other?
Question D-3 How can I use UML to communicate a hierarchy of building blocks?
Question D-4 How can I describe interfaces with UML?
Question D-5 What can I use UML ports for?
Question D-6 How can I improve the understandability of my diagrams/models?
Question D-7 How can I reduce the number of elements in a component /building block diagram?
Question D-8 How can I improve the consistency of models?
Question E-1 Does arc42 work with Scrum, Kanban or other lean/agile approaches?
Question E-2 Shall our team make arc42 part of the definition of done (DoD)?
Question E-3 How can arc42 help us in reducing documentation effort?
Question E-4 We use Scrum. How can we integrate arc42 into our sprints?
Question E-5 In agile projects, what kind of information shall we write down and what shall we communicate orally?
Question E-6 In agile projects, can we skip (defer) documentation for a while and focus only on implementing?
Question E-7 What are the most important tips for documentation in agile projects?
Question F-1 What tools are well-suited for arc42?
Question F-2 What are useful criteria for selecting a toolset for arc42?
Question F-3 Can I automatically include important code artifacts in arc42 documentation?
Question F-4 How do I use arc42 with modeling tools?
Question F-5 How can I use arc42 together with Sparx Enterprise Architect(r)
Question F-6 Are there (free or open-source) modeling tools available?
Question F-7 Can we use arc42 with the Atlassian Confluence wiki?
Question F-8 What tools can I use to setup arc42 in Confluence?
Question F-9 How can I create diagrams in Confluence?
Question F-10 What tools shall I use for sequence diagrams?
Question F-10 Can I use PlantUML for building block / component diagrams?
Question G-1 Shall I version control our architecture documentation?
Question G-2 We cannot use version control for documents. What to do?
Question G-3 How does versioning work with graphical models?
Question G-4 How can I describe several variants of a system?
Question H-1 What does traceability mean (with arc42)?
Question H-2 Shall we strive for traceability in our documentation?
Question H-3 How can I keep architecture documentation in sync with source code?
Question J-1 How to document (very) large systems with arc42
Question J-2 Does a documentation governance make sense for architecture documentation?
Question J-3 Is there a checklist for arc42 documentation?
Question J-4 Is there a general checklist for architecture documentation?
Question J-5 How can I create delta-documentation with arc42?
Question J-6 What level of access permissions shall stakeholders have on the architecture documentation?
Question K-1 Are we allowed to customize arc42 for our (company/organization/system…) purpose?
Question K-2 What kinds of customizations are common for arc42?
## Notes
1Cabinet image from Openclipart
2The term „template zombies“ has been coined by Tom DeMarco, Peter Hruschka et al. in the award-winning book „Adrenalin Junkies and Templates Zombies“ (Dorset House 2007)
3The missing manuals® is a series of book from O’Reilly publishing with the nice subtitle „The book that should have been in the box“.
4These are more than just requirements, therefore we choose this somehow stronger title for them…
5Stakeholders might be project management, product owner, developers, admins or other persons/roles/organizations responsible for the product, system, budget and/or schedule.]
6Adhesive notes made from recyclable plastic material that attaches to almost any surface without glue. They can be removed and reused multiple times. See http://www.stattys.com/.
7Eric Evans: Domain-Driven Design (Addision Wesley, 2004): The original source, 700+ pages of dense content. Vaugn Vernon, another veteran of DDD, has written “Domain-Driven Design Distilled” (Addision Wesley 2016), with only 170 pages - brief intro.
8The name and logo of Confluence is copyrighted by Atlassian Software. arc42 is neither affiliated with or in any way sponsored by Atlassian Corporation.
9Table generated: November/24/2016
10Table generated: November/24/2016 | 2023-03-22 08:34:42 | {"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": 0.25223594903945923, "perplexity": 6582.493511560461}, "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-2023-14/segments/1679296943809.22/warc/CC-MAIN-20230322082826-20230322112826-00746.warc.gz"} |
http://www.optimization-online.org/DB_HTML/2017/12/6372.html | - Convergence Rates for Deterministic and Stochastic Subgradient Methods Without Lipschitz Continuity Benjamin Grimmer(bdg79cornell.edu) Abstract: We generalize the classic convergence rate theory for subgradient methods to apply to non-Lipschitz functions via a new measure of steepness. For the deterministic projected subgradient method, we derive a global $O(1/\sqrt{T})$ convergence rate for any function with at most exponential growth. Our approach implies generalizations of the standard convergence rates for gradient descent on functions with Lipschitz or H\"older continuous gradients. Further, we show a $O(1/\sqrt{T})$ convergence rate for the stochastic projected subgradient method on functions with at most quadratic growth, which improves to $O(1/T)$ under strong convexity. Keywords: Subgradient Method, Convex Optimization, Convergence Rates Category 1: Convex and Nonsmooth Optimization (Convex Optimization ) Citation: Download: [PDF]Entry Submitted: 12/11/2017Entry Accepted: 12/11/2017Entry Last Modified: 12/11/2017Modify/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 Optmization Society. | 2019-03-24 02:50:49 | {"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": 0.5018380284309387, "perplexity": 2301.361224806942}, "config": {"markdown_headings": true, "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-2019-13/segments/1552912203168.70/warc/CC-MAIN-20190324022143-20190324044143-00349.warc.gz"} |
https://mrtrix.readthedocs.io/en/dev/reference/commands/population_template.html | # population_template¶
## Synopsis¶
Generates an unbiased group-average template from a series of images
## Usage¶
population_template input_dir template [ options ]
• input_dir: The input directory containing all images used to build the template
• template: The output template image
## Description¶
First a template is optimised with linear registration (rigid and/or affine, both by default), then non-linear registration is used to optimise the template further.
## Options¶
• -type registration_stage(s) Specify the types of registration stages to perform. Options are “rigid” (perform rigid registration only which might be useful for intra-subject registration in longitudinal analysis), “affine” (perform affine registration), and “nonlinear”, as well as combinations of registration types: “rigid_affine”, “rigid_nonlinear”, “affine_nonlinear”, “rigid_affine_nonlinear”. Default: rigid_affine_nonlinear
• -voxel_size Define the template voxel size in mm. Use either a single value for isotropic voxels or 3 comma separated values.
• -initial_alignment mode Method of alignment to form the initial template. Options are “mass” (default), “geometric” and “none”.
• -mask_dir Optionally input a set of masks inside a single directory, one per input image (with the same file name prefix). Using masks will speed up registration significantly
• -warp_dir Output a directory containing warps from each input to the template. If the folder does not exist it will be created
• -transformed_dir Output a directory containing the input images transformed to the template. If the folder does not exist it will be created
• -linear_transformations_dir Output a directory containing the linear transformations used to generate the template. If the folder does not exist it will be created
• -template_mask Output a template mask. Only works if -mask_dir has been provided. The template mask is computed as the intersection of all subject masks in template space.
• -noreorientation Turn off FOD reorientation in mrregister. Reorientation is on by default if the number of volumes in the 4th dimension corresponds to the number of coefficients in an antipodally symmetric spherical harmonic series (i.e. 6, 15, 28, 45, 66 etc)
### Options for the non-linear registration¶
• -nl_scale Specify the multi-resolution pyramid used to build the non-linear template, in the form of a list of scale factors (default: 0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0). This implicitly defines the number of template levels
• -nl_lmax Specify the lmax used for non-linear registration for each scale factor, in the form of a list of integers (default: 2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4). The list must be the same length as the nl_scale factor list
• -nl_niter Specify the number of registration iterations used within each level before updating the template, in the form of a list of integers (default: 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5). The list must be the same length as the nl_scale factor list
• -nl_update_smooth Regularise the gradient update field with Gaussian smoothing (standard deviation in voxel units, Default 2.0 x voxel_size)
• -nl_disp_smooth Regularise the displacement field with Gaussian smoothing (standard deviation in voxel units, Default 1.0 x voxel_size)
### Options for the linear registration¶
• -linear_no_pause Do not pause the script if a linear registration seems implausible
• -linear_estimator Choose estimator for intensity difference metric. Valid choices are: l1 (least absolute: |x|), l2 (ordinary least squares), lp (least powers: |x|^1.2), Default: l2
• -rigid_scale Specify the multi-resolution pyramid used to build the rigid template, in the form of a list of scale factors (default: 0.3,0.4,0.6,0.8,1.0,1.0). This and affine_scale implicitly define the number of template levels
• -rigid_lmax Specify the lmax used for rigid registration for each scale factor, in the form of a list of integers (default: 2,2,2,4,4,4). The list must be the same length as the linear_scale factor list
• -rigid_niter Specify the number of registration iterations used within each level before updating the template, in the form of a list of integers (default:50 for each scale). This must be a single number or a list of same length as the linear_scale factor list
• -affine_scale Specify the multi-resolution pyramid used to build the affine template, in the form of a list of scale factors (default: 0.3,0.4,0.6,0.8,1.0,1.0). This and rigid_scale implicitly define the number of template levels
• -affine_lmax Specify the lmax used for affine registration for each scale factor, in the form of a list of integers (default: 2,2,2,4,4,4). The list must be the same length as the linear_scale factor list
• -affine_niter Specify the number of registration iterations used within each level before updating the template, in the form of a list of integers (default:500 for each scale). This must be a single number or a list of same length as the linear_scale factor list
### Additional standard options for Python scripts¶
• -nocleanup do not delete intermediate files during script execution, and do not delete scratch directory at script completion.
• -scratch /path/to/scratch/ manually specify the path in which to generate the scratch directory.
• -continue <ScratchDir> <LastFile> continue the script from a previous execution; must provide the scratch directory path, and the name of the last successfully-generated file.
### Standard options¶
• -info display information messages.
• -quiet do not display information messages or progress status. Alternatively, this can be achieved by setting the MRTRIX_QUIET environment variable to a non-empty string.
• -debug display debugging messages.
• -force force overwrite of output files. | 2019-11-13 05:59:17 | {"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": 0.189596489071846, "perplexity": 4716.735829140153}, "config": {"markdown_headings": true, "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-2019-47/segments/1573496665985.40/warc/CC-MAIN-20191113035916-20191113063916-00101.warc.gz"} |
https://www.physicsforums.com/threads/factor-group-question.765104/ | # Factor Group Question
1. Aug 7, 2014
### PsychonautQQ
1. The problem statement, all variables and given/known data
if K is normal in G and has index m, show g^m is an element of K for all g in G
Work (I haven't done much with proofs so bear with me):
|G/K| = |G| / |K| = m
|G| = x
|K| = y
g^m must be an element of G since m|x
if g^m is an element of G and K is normal to G then
(g^m)K = K(g^m) --> (g^m)K(g^m)^-1 = K for all g^m in G
is this work legit?
Last edited: Aug 7, 2014
2. Aug 7, 2014
### HallsofIvy
Is this what you meant to say? g to any integer power is an element of G because G is closed under multiplication.
This is pretty much the definition of "normal subgroup". But you haven't even said, much less proved, that g is in K.
3. Aug 7, 2014
### PsychonautQQ
Ahh okay yeah I don't know what I thought I was doing there lol. Help a noob out? Hint hint possibly?
4. Aug 7, 2014
### jbunniii
Hint: $g^m \in K$ if and only if $g^mK = K$.
5. Aug 9, 2014
### PsychonautQQ
I just saw your clue and will think about it and try to use it to help me figure the problem out, but meanwhile here is what I've done so far on my own.
|G| / |K| = m
|G| = x
|K| = y
x = my
G = {1,g,g^2.....,g^(x-1)} are all elements of G so
{1,g,g^2.....,g^(x-1)}^x are all elements of G so
{1,g,g^2.....,g^(x-1)}^my are all elements of G
is this going in the right direction? I'll start working on this problem again now with the hint you've given
Last edited: Aug 9, 2014
6. Aug 9, 2014
### jbunniii
I don't see where this is taking you. Try using my hint to reword the problem in terms of the group $G/K$, i.e. work with cosets of $K$ instead of individual elements of $G$.
7. Aug 9, 2014
### PsychonautQQ
Solution: There are m cosets of K in G. This means that the group G/K has m elements in it, thus any element of that group raised to the m power would equal the identity element, K. (Kg)^m = Kg^m = K
Is this correct? If it is then wow,, I overthinking this
8. Aug 9, 2014
### jbunniii
Yes, that's all there is to it. | 2018-02-22 07:34:55 | {"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": 0.745622456073761, "perplexity": 1458.7734975655462}, "config": {"markdown_headings": true, "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-2018-09/segments/1518891814036.49/warc/CC-MAIN-20180222061730-20180222081730-00196.warc.gz"} |
https://tex.stackexchange.com/questions/348846/conditional-printing-of-solutions-in-exams-package | # Conditional printing of solutions in exams package
I am using exam package and want to print solutions conditionally. I can print solutions by using answers option in exam. I can get two lines for answers using \fillwithlines{1.0in} in each question. But this is a very naive approach. I wonder how can I get two lines automatically for answers if I don't want to print solutions. Any help will be highly appreciated.
\documentclass[12pt, addpoints, answers]{exam}
\usepackage{amsmath, amsfonts, amssymb, xhfill, float}
\usepackage[margin=0.5in]{geometry}
\setlength{\topmargin}{-0.2in}
\setlength\linefillheight{0.5in}
\begin{document}
\begin{questions}
\question
Where does XYZ live?
\fillwithlines{1.0in}
\begin{solution}
XYZ lives in home.
\end{solution}
\end{questions}
\end{document}
What about using the environment solutionorlines instead of the environment solution?
%\documentclass[12pt, addpoints, answers]{exam}
\usepackage{amsmath, amsfonts, amssymb, xhfill, float}
\usepackage[margin=0.5in]{geometry}
\setlength{\topmargin}{-0.2in}
\setlength\linefillheight{0.5in}
\begin{document}
\begin{questions}
\question
Where does XYZ live?
\begin{solutionorlines}[1.0in]
XYZ lives in home.
\end{solutionorlines}
\end{questions}
\end{document}
there was a similar question with beamer in normal and handout mode.
I think of the same solution with the {subfiles} package:
You basically create 3 files:
exam-with-answers.tex
\documentclass[12pt, addpoints, answers]{exam}
\usepackage{subfiles}
\begin{document}
\subfile{your-former-main.tex}
\end{document}
exam-without-answers.tex
\documentclass[12pt, addpoints]{exam}
\usepackage{subfiles}
\begin{document}
\subfile{your-former-main.tex}
\end{document}
your-former-main.tex
\documentclass[exam-with-answers.tex]{subfile}
%% ...
you can successfully compile either one. your-former-main.tex and exam-with-answers.tex will result in containing the answeres, exam-without-answers.tex will have no answeres. Plus you get both documents with different names without any additional effort. | 2019-08-17 18:07:56 | {"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": 0.475321501493454, "perplexity": 6029.210012639535}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "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-2019-35/segments/1566027313436.2/warc/CC-MAIN-20190817164742-20190817190742-00351.warc.gz"} |
http://curiousily.com/data-science/2017/04/24/tensorflow-for-hackers-part-3.html | Have you ever stood still, contemplating about how cool would it be to build a model that can distinguish cats from dogs? Don’t be shy now! Of course you did! Let’s get going!
# Cat or a dog?
We have 25,000 labeled pictures of dogs and cats. The data comes from Kaggle’s Dogs vs Cats challenge. That’s how a bunch of them look like:
source: subsubroutine.com
Let’s focus on a specific image. Each picture can be represented as a 3-dimensional array. We will resize all training image to 50 x 50 pixels. Here’s a crazy example:
source: subsubroutine.com
Additionally, we will remove all color and turn them pictures into grayscale ones. First things first, let’s prepare our environment:
# Setting up
Download the train and test zip files from Kaggle and extract them into your current working directory.
import cv2
import numpy as np
import os
from random import shuffle
from tqdm import tqdm
import tensorflow as tf
import matplotlib.pyplot as plt
import tflearn
from tflearn.layers.conv import conv_2d, max_pool_2d
from tflearn.layers.core import input_data, dropout, fully_connected
from tflearn.layers.estimator import regression
%matplotlib inline
TRAIN_DIR = 'train'
TEST_DIR = 'test'
IMG_SIZE = 50
LR = 1e-3
MODEL_NAME = 'dogs-vs-cats-convnet'
tf.__version__
'1.1.0'
# Image preprocessing
We have 25,000 images for training and 12,500 for testing. Let’s create a function that encodes the labels of the training images:
def create_label(image_name):
""" Create an one-hot encoded vector from image name """
word_label = image_name.split('.')[-3]
if word_label == 'cat':
return np.array([1,0])
elif word_label == 'dog':
return np.array([0,1])
Now, for the actual reading of training and test data. Every image will be resized to 50 x 50 pixels and read as grayscale:
def create_train_data():
training_data = []
for img in tqdm(os.listdir(TRAIN_DIR)):
path = os.path.join(TRAIN_DIR, img)
img_data = cv2.resize(img_data, (IMG_SIZE, IMG_SIZE))
training_data.append([np.array(img_data), create_label(img)])
shuffle(training_data)
np.save('train_data.npy', training_data)
return training_data
def create_test_data():
testing_data = []
for img in tqdm(os.listdir(TEST_DIR)):
path = os.path.join(TEST_DIR,img)
img_num = img.split('.')[0]
img_data = cv2.resize(img_data, (IMG_SIZE, IMG_SIZE))
testing_data.append([np.array(img_data), img_num])
shuffle(testing_data)
np.save('test_data.npy', testing_data)
return testing_data
Now, let’s split the data. 24,500 images for training and 500 for testing. We also need to reshape the data appropriately for TensorFlow:
# If dataset is not created:
train_data = create_train_data()
test_data = create_test_data()
# If you have already created the dataset:
train = train_data[:-500]
test = train_data[-500:]
X_train = np.array([i[0] for i in train]).reshape(-1, IMG_SIZE, IMG_SIZE, 1)
y_train = [i[1] for i in train]
X_test = np.array([i[0] for i in test]).reshape(-1, IMG_SIZE, IMG_SIZE, 1)
y_test = [i[1] for i in test]
100%|██████████| 25000/25000 [01:04<00:00, 386.67it/s]
100%|██████████| 12500/12500 [00:32<00:00, 387.32it/s]
# Convolutional Neural Nets
How will we do it? Isn’t that just too hard of a task? Convolutional Neural Networks to the rescue!
In the past, people had to think of and code different kinds of features that might be relevant to the task at hand. Examples of that would be whiskers, ears, tails, legs, fur type detectors. These days, we can just use Convolutional NNs. They can learn features from raw data. How do they work?
Ok, got it? It was a great explanation. You can think of convolutions as small sliding lenses (let’s say a 5 x 5) that are “activated” when are placed above some feature that is familiar to them. That way, convolutions can make sense of larger portions of the image, not just single pixels.
# Building our model
Finally, the fun part begins! We will use tflearn to build our Convolutional Neural Network. One additional bonus will be the use of a Dropout layer. Here’s the model:
tf.reset_default_graph()
convnet = input_data(shape=[None, IMG_SIZE, IMG_SIZE, 1], name='input')
convnet = conv_2d(convnet, 32, 5, activation='relu')
convnet = max_pool_2d(convnet, 5)
convnet = conv_2d(convnet, 64, 5, activation='relu')
convnet = max_pool_2d(convnet, 5)
convnet = fully_connected(convnet, 1024, activation='relu')
convnet = dropout(convnet, 0.8)
convnet = fully_connected(convnet, 2, activation='softmax')
convnet = regression(convnet, optimizer='adam', learning_rate=LR, loss='categorical_crossentropy', name='targets')
model = tflearn.DNN(convnet, tensorboard_dir='log', tensorboard_verbose=0)
model.fit({'input': X_train}, {'targets': y_train}, n_epoch=10,
validation_set=({'input': X_test}, {'targets': y_test}),
snapshot_step=500, show_metric=True, run_id=MODEL_NAME)
Training Step: 3829 | total loss: [1m[32m11.45499[0m[0m | time: 35.818s
| Adam | epoch: 010 | loss: 11.45499 - acc: 0.5025 -- iter: 24448/24500
Training Step: 3830 | total loss: [1m[32m11.49676[0m[0m | time: 36.938s
| Adam | epoch: 010 | loss: 11.49676 - acc: 0.5007 | val_loss: 11.60503 - val_acc: 0.4960 -- iter: 24500/24500
--
We resized our images to 50 x 50 x 1 matrices and that is the size of the input we are using.
Next, a convolutional layer with 32 filters and stride = 5 is created. The activation function is ReLU. Right after that, a max pool layer is added. That same trickery is repeated again with 64 filters.
Next, a fully-connected layer with 1024 neurons is added. Finally, a dropout layer with keep probability of 0.8 is used to finish our model.
We use Adam as optimizer with learning rate set to 0.001. Our loss function is categorical cross entropy. Finally, we train our Deep Neural Net for 10 epochs.
All that is great, but our validation accuracy doesn’t seem that good. Flipping a coin might be a better model than the one we created. Let’s go bigger and better (hopefully):
# Building our (bigger) model
tf.reset_default_graph()
convnet = input_data(shape=[None, IMG_SIZE, IMG_SIZE, 1], name='input')
convnet = conv_2d(convnet, 32, 5, activation='relu')
convnet = max_pool_2d(convnet, 5)
convnet = conv_2d(convnet, 64, 5, activation='relu')
convnet = max_pool_2d(convnet, 5)
convnet = conv_2d(convnet, 128, 5, activation='relu')
convnet = max_pool_2d(convnet, 5)
convnet = conv_2d(convnet, 64, 5, activation='relu')
convnet = max_pool_2d(convnet, 5)
convnet = conv_2d(convnet, 32, 5, activation='relu')
convnet = max_pool_2d(convnet, 5)
convnet = fully_connected(convnet, 1024, activation='relu')
convnet = dropout(convnet, 0.8)
convnet = fully_connected(convnet, 2, activation='softmax')
convnet = regression(convnet, optimizer='adam', learning_rate=LR, loss='categorical_crossentropy', name='targets')
model = tflearn.DNN(convnet, tensorboard_dir='log', tensorboard_verbose=0)
model.fit({'input': X_train}, {'targets': y_train}, n_epoch=10,
validation_set=({'input': X_test}, {'targets': y_test}),
snapshot_step=500, show_metric=True, run_id=MODEL_NAME)
Training Step: 3829 | total loss: [1m[32m0.34434[0m[0m | time: 44.501s
| Adam | epoch: 010 | loss: 0.34434 - acc: 0.8466 -- iter: 24448/24500
Training Step: 3830 | total loss: [1m[32m0.35046[0m[0m | time: 45.619s
| Adam | epoch: 010 | loss: 0.35046 - acc: 0.8432 | val_loss: 0.50006 - val_acc: 0.7860 -- iter: 24500/24500
--
That is pretty much the same model. One difference is the number of convolutional and max pool layers we added. So, our model has much more parameters and can learn more complex functions. One proof of that is the validation accuracy that is around 0.8. Let’s take our model for a spin!
# Did we do well?
Let’s have a look at a single prediction:
d = test_data[0]
img_data, img_num = d
data = img_data.reshape(IMG_SIZE, IMG_SIZE, 1)
prediction = model.predict([data])[0]
fig = plt.figure(figsize=(6, 6))
ax.imshow(img_data, cmap="gray")
print(f"cat: {prediction[0]}, dog: {prediction[1]}")
cat: 0.8773844838142395, dog: 0.12261549383401871
That doesn’t look right. How about some more predictions:
fig=plt.figure(figsize=(16, 12))
for num, data in enumerate(test_data[:16]):
img_num = data[1]
img_data = data[0]
orig = img_data
data = img_data.reshape(IMG_SIZE, IMG_SIZE, 1)
model_out = model.predict([data])[0]
if np.argmax(model_out) == 1:
str_label='Dog'
else:
str_label='Cat'
y.imshow(orig, cmap='gray')
plt.title(str_label)
y.axes.get_xaxis().set_visible(False)
y.axes.get_yaxis().set_visible(False)
plt.show()
# Conclusion
There you have it! Santa is a dog! More importantly, you built a model that can distinguish cats from dogs using only raw pixels (albeit, with a tiny bit of preprocessing). Additionally, it trains pretty fast on relatively old machines! Can you improve the model? Maybe change the architecture, keep probability parameter of the Dropout layer or the optimizer? What results did you get?
The only thing left for you to do is snap a photo of your cat or dog and run it through your model. Was the net correct? | 2017-12-15 09:55:39 | {"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": 0.31090301275253296, "perplexity": 10297.981075413336}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2017-51/segments/1512948568283.66/warc/CC-MAIN-20171215095015-20171215115015-00070.warc.gz"} |
https://archive.lib.msu.edu/crcmath/math/math/p/p018.htm | If is a Power series which is regular for except for Poles within this Circle and except for , at which points the function is assumed continuous when only points are considered, then at least a subsequence of the Padé Approximants are uniformly bounded in the domain formed by removing the interiors of small circles with centers at these Poles and uniformly continuous at for .
Baker, G. A. Jr. The Padé Conjecture and Some Consequences.'' §II.D in Advances in Theoretical Physics, Vol. 1 (Ed. K. A. Brueckner). New York: Academic Press, pp. 23-27, 1965. | 2021-11-27 17:52:29 | {"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": 0.9310974478721619, "perplexity": 498.0564987071683}, "config": {"markdown_headings": true, "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-2021-49/segments/1637964358208.31/warc/CC-MAIN-20211127163427-20211127193427-00457.warc.gz"} |
https://www.epfl.ch/labs/transp-or/semester-projects/ | # Semester projects
Total: 226
Schedule repair in liner shipping
Large liner shipping companies operate several hundred ships worldwide. These ships carry multi-modal containers on pre-established routes with a regular schedule (typically weekly). A schedule is thus an ordered list of port calls (i.e. stops in a harbour) with associated time windows. Since there are several thousands possible ports in the world, designing routes is a complex problem. Furthermore, since a rotation takes weeks or even months, roughly four journeys out of five end up requiring adjustments to accommodate delays, breakdowns or other unforeseeable but frequent events. Due to the extent of the operations, improving from an adequate schedule to an optimal one may yield enormous benefits, not only economically but also in terms of environmental impact and of quality of life for workers. Given the nominal schedule and an additional constraint stemming from an unforeseen event, the aim of the project is to use geographical information and demand statistics in the various ports of a liner shipping network, the student applies Operational Research methodologies to generate a new optimal schedule, completing as much of the original mission as possible while minimising the ecological and economic impacts.
Student: Benoit Pahud (GC), January 29, 2021
Supervision: Stefano Bortolomiol, Nour Dougui, Michel Bierlaire
Generating choice sets of transport modes for activity based applications
Understanding human mobility behavior is essential to estimate and forecast transport demand. Activity-based models (ABM) are an increasingly popular approach to integrate complex behav- ioral dimensions influencing mobility choices, such as habits or social interactions. A signicant challenge in ABM research is the choice set for each individual (consisting of the type of activities, the schedules, the locations and the modes to reach them), which is highlycombinatorial and thus impossible to fully enumerate. This pre-study will specically investigate the choice set of transport modes for individuals' daily activities, guided by (but not limited to) the following research questions: (1) How can we generate a plausible set of modes for an individual to consider, for each of their activities? (2) What are the main determinants for including/excluding a mode in the choice set? (3) What are the data requirements for these tasks? What methods or tools can be used to augment datasets which lack this essential information? The objective is to develop a methodology to generate a choice set of modes, by means of theoretical and data-driven approaches. The mode choice itself is not included in the scope of this project.
Student: Benoit Pahud (GC), January 29, 2021
Supervision: Janody Pougala, Tim Hillel, Michel Bierlaire
Generating choice sets of destinations for activity based applications
Understanding human mobility behavior is essential to estimate and forecast transport demand. Activity-based models (ABM) are an increasingly popular approach to integrate complex behavioral dimensions influencing mobility choices, such as habits or social interactions. A signicant challenge in ABM research is the choice set for each individual (consisting of the type of activities, the schedules, the locations and the modes to reach them), which is highly combinatorial and thus impossible to fully enumerate. This pre-study will specically investigate the choice set of locations for individuals' daily activities, guided by (but not limited to) the following research questions: (1) How can we generate a plausible set of locations for an individual to consider, for each of their activities? (2) Which attributes can be used to characterize a location or a space (e.g. opening hours, public transport level of service, job density...) ? (3) What is the required granularity or scale to yield consistent results in terms of activity-based estimation, while maintaining a computational tractability ? (4) What are the data requirements for these tasks? What methods or tools can be used to augment datasets which lack this essential information ? The objective is to develop a methodology to generate a choice set of locations, by means of theoretical and data-driven approaches. The choice of location itself is not included in the scope of this project.
Student: Nicolas Salvadé (GC), January 29, 2021
Supervision: Janody Pougala, Tim Hillel, Michel Bierlaire
Optimizing Organizational Chart using local search method
The aim of this project is to produce a near optimal Organisational Chart for a real case study. This Organisational Chart should optimize the span of control of Management in the company by maximizing a coverage metric using clearly-de ned manipulations of the organizational design. To achieve this goal a local search method will be used and the results should be compared to those given by an exact method (Simplex). We will in a rst phase apply the optimization method to a small sample of Organisational Chart without considering the role of the employees. In a second phase, the role of the employees should be considered and a larger Organisational Chart should be considered.
Student: Hugo Bocquet, June 30, 2020
Supervision: Nour Dougui, Selin Atac, Michel Bierlaire
Optimal regulation of oligopolistic markets with discrete choice models of demand
This project is inspired by the ongoing research project titled 'New generation of demand-supply interaction models', funded by the Swiss National Science Foundation. In particular, the project aims at studying models to find optimal policies to regulate markets characterized by oligopolistic competition and in which consumers make a discrete choice among a finite set of alternatives. In this framework, consumers are modelled as utility maximizers, according to random utility theory. Suppliers are modelled as profit maximizers, according to the traditional microeconomic treatment. Market competition is modelled as a non-cooperative game, for which an epsilon-equilibrium solution is sought. Finally, the regulator can affect the behavior of all other agents, for instance by giving subsidies or imposing taxes. In transport markets, these interventions might target specific alternatives, to reduce externalities such as congestion or emissions, or specific segments of the population, to achieve social welfare objectives. The objective of the project is to analyze, implement and evaluate algorithms to solve realistic applications of this problem, with particular focus on the transportation sector.
Student: Elodie Duliscouet, Paulin Raison, Yahya Basiouny (GM), June 12, 2020
Supervision: Stefano Bortolomiol, Selin Atac, Michel Bierlaire
Robust routing and scheduling in liner shipping
Large liner shipping companies operate several hundred ships worldwide. These ships carry multi-modal containers on pre-established routes with a regular schedule (typically weekly). A schedule is thus an ordered list of port calls (i.e. stops in a harbour) with associated time windows. Since there are several thousands possible ports in the world, designing routes is a complex problem. Furthermore, since a rotation takes weeks or even months, roughly four journeys out of five end up requiring adjustments to accommodate delays, breakdowns or other unforeseeable but frequent events. Due to the extent of the operations, improving from an adequate schedule to an optimal one may yield enormous benefits, not only economically but also in terms of environmental impact and of quality of life for workers. Given geographical information and demand statistics in the various ports of a network, the project aims at applying Operational Research methodologies to automatically generating shipping schedules that make sense technically, economically and environmentally, as well as resisting perturbations. This project is proposed by Mediterranean Shipping Company (MSC).
Student: Yannis Voet (GC), June 12, 2020
Supervision: Stefano Bortolomiol, Nour Dougui, Michel Bierlaire
Bayesian analysis of multinomial discrete choice model with t-distributed kernel errors
Student: Thomas Gasos, June 09, 2020
Supervision: Rico Krueger, Michel Bierlaire
Welfare-maximizing design of a transportation system
The simulation-based linear representation of a discrete choice introduced allows to linearly approximate the expected maximum utility associated with each individual and simulation draw, which provides a measure of consumer surplus. This is of special interest in the case of advanced choice models (e.g., mixture of logit models), as the expected maximum utility presents a highly nonlinear expression. Hence, a linear formulation for the consumer surplus facilitates the derivation of a generalized objective function of some measure of social welfare as the policy objective, which can be employed to assess the performance of different urban transportation policies. The objective of the project is to provide linear (or piecewise linear) formulations to the different aspects that are included in a published non-linear optimization problem, and compare their performance with respect to the original approach, both in terms of computational time and quality of the obtained solution. Additionally, the impact of the assumptions that lead to nonlinear expressions will be evaluated.
Student: Benjamin Nicolas--Noir, May 29, 2020
Supervision: Meritxell Pacheco, Michel Bierlaire
Analysis of the value of demand forecasting within vehicle sharing systems
Vehicle sharing systems (VSSs) are becoming increasingly popular, primarily due to their financial and environmental advantages. However, VSSs face many operational challenges, including inventory management of vehicles and parking spots, vehicle load balancing and redistribution, pricing strategies, and demand forecasting. If these challenges are not addressed properly, the VSS risks experiencing a significant loss of customers and revenue. Recently, new VSSs have been introduced which use light electric vehicles (LEVs). These systems face a number of unique challenges. For instance, demand forecasting for LEV sharing systems is more complex, as locations are not fixed and journeys can start and end at any allowable location. LEV sharing systems also serve a higher portion of the population since these type of vehicles do not require a driving license. As such, the existing techniques for analysing VSSs are not sufficient for these new systems. To address this need, one needs to forecast the future demand of this novel transportation mode. Since it is exhausting to collect the data to develop a demand model, this project will aim to identify the value of a demand model by using mathematical models tailored for rebalancing operations from the literature and simulations. As LEV systems are still in their infancy, data describing them is not yet available. Therefore, the student will use alternative data, including PubliBike bike-sharing system data, accounting for any potential diff erences in the analysis. Based on the findings from the literature, the student will analyse diff erent mathematical models.
Student: Jasso Espadaler (CE), January 31, 2020
Supervision: Selin Atac, Stefano Bortolomiol, Michel Bierlaire
Investigating daily activity patterns
This project will involve working with real world trip-diary data for Switzerland (2015 Mobility and Transport microcensus) to investigate how people schedule activities (e.g. work, lunch, sport, shopping etc) in their day. The project will involve: -Extracting and standardising daily activity schedules from the data, including representing the activity types, locations, start times, and durations with appropriate precision and/or scale, -Establishing a metric to define the similarity (or distance) between different activity schedules, -Using unsupervised machine learning (clustering) algorithms to identify prototypical activity schedules.
Student: Sergey Gasparovich (GC), January 31, 2020
Supervision: Janody Pougala, Tim Hillel, Michel Bierlaire
Tour-based mode choice modelling
This project will involve working with real world trip-diary data for London and/or Switzerland to develop sequential tour-based mode-choice models (i.e. where the choice of mode for a trip-leg is dependent on the full tour). The project will involve: • Extracting tour itineraries from the trip-diary data, • Investigating different tour patterns, e.g. primary return-tours, subtours, complex multilegged tours, • Defining different tour-based mode-choice situations which are able to fully represent the tour patterns existing in the data, • Developing appropriate sequential choice models for each mode-choice situation using Discrete Choice Models and/or machine learning.This project will involve working with real world trip-diary data for London and/or Switzerland to develop sequential tour-based mode-choice models (i.e. where the choice of mode for a trip-leg is dependent on the full tour). The project will involve: -Extracting tour itineraries from the trip-diary data, -Investigating different tour patterns, e.g. primary return-tours, subtours, complex multilegged tours, -Defining different tour-based mode-choice situations which are able to fully represent the tour patterns existing in the data, -Developing appropriate sequential choice models for each mode-choice situation using Discrete Choice Models and/or machine learning.
Student: Adrien Nicolet (SGC), January 31, 2020
Supervision: Tim Hillel, Janody Pougala, Michel Bierlaire
Risk assessment of pedestrian crossings
Student: Adrien Nicolet (IC), January 31, 2020
Supervision: Nicholas Molyneaux, Gael Lederrey, Michel Bierlaire
Estimation of Discrete Choice Models using HAMABS 2.0
Improvement of HAMABS for choice modeling estimation using automatic switch.
Student: Linah Charif (GC), December 20, 2019
Supervision: Gael Lederrey, Nicola Ortelli, Michel Bierlaire
Calibrating pedestrian control strategies.
The objective of this semester project is to calibrate two pedestrian control strategies using heuristic methods and/or optimization methods like simulated annealing or genetic algorithms. Two strategies are under investigation: gating to prevent a gridlock situation and flow separators which prevent counter flow. The student will have to use the pedestrian simulator I have implemented and integrate it with an optimization approach like simulated annealing.
Student: Léopold Bouraux (IC), June 30, 2019
Supervision: Nicholas Molyneaux, Gael Lederrey, Michel Bierlaire
Demand forecasting for a novel transportation mode
The vehicle sharing systems (VSSs) are becoming more and more popular due to both economic and environmental effects. However, these systems mainly focus on two specific types of vehicles: bicycles and cars. Recently, some other types of vehicles (e.g. light electric vehicles) with some specific properties that distinguish them from cars and bicycles are introduced for use in such kind of systems. Therefore, they lack research in each component of the framework. The aim of this project is to forecast the demand for a novel transportation mode. The student should survey different types of approaches, such as ARIMA, to forecast demand in similar systems and compare them by identifying performance measures. Since the data for this specific transportation mode does not exist yet the student will be directed to use the taxi data, which shares many properties with the system of interest.
Student: Denis Steffen (MATH), June 30, 2019
Supervision: Selin Atac, Tim Hillel, Michel Bierlaire
Disruption-caused railway timetable rescheduling problem and its solution
In railway networks, unexpected disruptions may occur for different reasons and cause delays, service denial, and, consequently, passenger inconvenience. This planned master project will tackle the railway timetable rescheduling problem from a macroscopic and mesoscopic point of view in case of unexpected disruptions. The master project will consider several remediation strategies such as canceling, delaying or rerouting the trains, or introduction of other transportation modes. The other important aspect of the planned master project is its applicability in practice. Hence, the outcome of the project will be the algorithm for timetable rescheduling which will be implemented in the commercial software suite Viriato. The main functionality of Viriato is the train timetabling and it is produced by the company SMA und Partner AG. In order to prepare the candidate for the master project, this pre-study project will include several introductory tasks related to the problem comprehension, solutions approach and algorithm implementation. Therefore, the tasks of this project will be: 1) review of the relevant literature, 2) demand data preprocessing with the purpose of matching infrastructure and demand data, and 3) obtaining the knowledge of the Viriato software and its Algorithm Platform which enables extension of Viriato with third party algorithms.
Student: Oliver Mathias Buschor, June 21, 2019
Supervision: Nikola Obrenovic, Meritxell Pacheco, Michel Bierlaire
In collaboration with SMA und Partner
Location choice equilibrium - pedestrian demand analysis at EPFL campus
Modeling location choice is fundamental to understand travel behavior and to predict travel demand in urban spaces. Location choice models are often developed based on the discrete choice framework and allow the analyst to predict to which place (and when) an individual travels. While the location choice behavior has been studied a lot in the city scale, the literature in pedestrian facilities is relatively limited. In this project, we develop a pedestrian location choice model in the EPFL campus, using the data collected from Wi-Fi traces. We start with developing a simple location choice model and test its speci cations using the sample. Then the student collects additional data of population for aggregate forecasting, i.e. actual demand prediction at each location. Using this technique, we then extend the model to consider the congestion e ect. This involves in a xed point problem to achieve a Nash equilibrium, solved by the nested xed point (NFXP) algorithm or the nested pseudo likelihood (NPL) algorithm.
Student: Tianyang Dong (MA), June 03, 2019
Supervision: Yuki Oyama, Zhengchao Wang, Michel Bierlaire
Passenger satisfaction maximization under budget constraints
This project aims at characterizing a passenger satisfaction application relying on a demand-based optimization framework, which integrates discrete choice models (state-of-the-art for the mathematical modeling of the demand) n Mixed Integer Linear Programming (MILP) models, which are usually considered to address supply decisions such as the price of a service or the number of units to produce of a certain item. This formulation allows us to express passenger satisfaction directly in terms of the expected maximum utility of the future scenario, which simplifi es notably the common representation relying on the consumer surplus. The objective is the maximization of the passenger satisfaction in a short-distance commuting context while accounting for different settings with respect to road tolling and investment in public transportation.
Student: Tatiana Moavensadeh-Ghasnavi, June 03, 2019
Supervision: Meritxell Pacheco, Yuki Oyama, Michel Bierlaire
Variantes d'horaires sur la tangentielle Nantes-Lyon
Environ 69% de la longueur de la tangentielle ferroviaire Nantes-Lyon sont aujourd'hui électrifiés en 25kV/50Hz et 9% en 1,5kV=. Toutefois, les 143 km séparant St.Germain-des-Fossés à St.Germain-au-Mt-d'Or restent non électrifiés. Le projet consiste à développer des variantes d'horaires entre Nantes et Lyon en tenant compte notamment des correspondances dans les noeuds de Nantes, Angers, Tours et Bourges. La problématique des correspondances avec Clermont-Ferrand sera prise en compte et l'insertion des circulations dans le noeud lyonnais sera sommairement analysée. Les éventuelles difficultés pour cette tangentielle à fournir à la fois une succession de services relativement locaux et une desserte de bout en bout performante seront mises en évidence.
Student: Axel Valentin Gabriel Curis (SGC), January 31, 2019
Supervision: Daniel Emery
Railway infrastructure maintenance
Following this summer tragedy in Genoa, Italy, European states and governments became more aware of the importance of keeping their countries’ infrastructure modern and maintained. This interdisciplinary project will particularly focus on the topic of railway infrastructure maintenance and will cover some legal, economic, social, political aspects related to this socio-economic problem. The goal of this project is to study the topic of railway infrastructure maintenance from two perspectives. Initially, a qualitative analysis will be performed to precisely define the scope of the research and the main factors that affect the planning of investments to maintain railways infrastructures. Based on the findings of the qualitative analysis, the students will then critically analyze existing mathematical models that generate maintenance schedules and optimal railway investments, and will propose a framework which will be applied on a real-life case study.
Student: Ludovica Sessa and Robert Abboud (CDM), January 31, 2019
Supervision: Stefano Bortolomiol, Selin Atac, Michel Bierlaire
Dynamic optimization of self-service vehicles
This project is a pré-étude project that will prepare the student to working on the Master thesis. The proposed topic is related to the dynamic optimization of self-service vehicle fleets, with a possible focus on dynamic pricing.
Student: Mohamed Detsouli (GC), January 31, 2019
Supervision: Stefano Bortolomiol, Nikola Obrenovic, Michel Bierlaire
Formulating and solving a dial-a-ride problem
The Dial-a-Ride Problem (DARP) is a problem to design a vehicular route and schedule, given the passenger requests that are characterized by origins (pickup points) and destinations (delivery points) often with the time windows. In recent years, solving the DARP is increasingly demanded, re ecting new technologies for mobility. On-demand transportation for elderly or disabled people is a typical example of application. The project is mainly dedicated to the following speci c tasks. First, the literature review will be done so that the student gets familiar with the model concepts and formulations of the DARP. The student is expected to understand the di erence among several types of models, such as between static and dynamic models, or between single-vehicle and multi-vehicle models. At the same time, the solution methodologies that are relevant to each type of DARP will be investigated. Second, the student should acquire the basic skills for optimization problem. Though several exercises, she will get used to coding and using appropriate softwares for solving the problem. Given the knowledge and skills, nally, she should de ne a DARP for a speci c example. The problem will be solved by at least two methodologies and be analyzed.
Student: Rym Karime (GC), January 31, 2019
Supervision: Yuki Oyama, Nour Dougui, Michel Bierlaire
Passage au 1/4h de l'exploitation du FMA
Les Transports Public Fribourgois (TPF) exploite la ligne à voie normale Fribourg-Morat-Anet (FMA). Les trains assurant ce service utilise les voies des CFF à Fribourg, entre Fribourg et Givisiez, à Morat, entre Morat et Muntelier et les voies du Bern-Neuchâtel (BN) expoité par les BLS à Anet. L'horaire de cette ligne est donc difficile à mettre en place. Le projet consiste à proposer une offre au 1/4h sur toute la ligne et/ou une offre combinée en pseudo-1/4h en alternant des trains REG et RE. Les mesures d'infrastructures seront mises en évidence.
Student: Félix Boesch (SGC), January 31, 2019
Supervision: Daniel Emery
Exploitation optimale de la ligne à voie métrique des TPF Palézieux-Bulle- Montbovon
La ligne des TPF à voie métrique Palézieux-Bulle-Montbovon doit offrir de bonnes correpondances dans ces trois noeuds ferroviaire. Le projet consiste à proposer des horaires optimaux en fonction des variantes d'horaire CFF entre Palézieux et Fribourg d'une part et de l'exploitation MOB (Montreux-Oberland Bernois) d'autre part. Une desserte plus dense au Sud de Bulle et certains prolongements jusqu'à Chateau d'Oex peuvent être envisagés, en fonction des besoins et des temps de rebroussement disponibles.
Student: Fabien Jacot-Descombes Jonas Gschwend (SGC), January 31, 2019
Supervision: Daniel Emery
Optimal taxi charging decision given the real-time charging station and taxi states and future uncertainties
This project aims at optimizing the charging decision of taxi drivers. Di erent from conventional taxi drivers, electric taxi drivers are more influenced by their refuelling strategy. This is because the refuelling process of electric taxis is longer than conventional taxis, especially in the case that electric taxis need to wait before charging. This longer charging duration will decrease the serving time of electric taxis which will, in return, influence the profi ts of electric taxis. Therefore, an intelligent charging decision strategy which can help electric taxis to charge optimally (maximizing the profi t of taxi) is highly desirable. To build the optimal charging decision program, we assume that a taxi will make a decision at each time step. When the taxi drivers make a decision, we assume that the taxi driver knows that, in the future hours, what are the expectation and variance of revenue during each time period (let's say 30 minutes). In addition to that, the taxi driver also knows the exact position of charging station and the chargers that are installed in the station. Besides, we also assume that the taxi driver also receives information about the availability of chargers from all stations by the time he makes a decision. Moreover, the taxi driver also has a prediction of the congestion condition of the charging station in the following hours. Based on the above assumption, the taxi drivers make his charging decision at each time step, i.e. charge or not, if charge, where, and when to charge for how much. The students who work on this project should build an optimization program which can make the charging decision for the taxi driver. This project is composed of two stages. First, the students will look into the taxi tracking data to understand the taxi request characteristics. This includes the mean and variance of a taxi running distances and revenue from served taxi demands across the taxis. After, the students will learn to come up with an optimization program which can help the taxi drivers to make the charging decision given the current states of taxis and its knowledge about the future world.
Student: Julien Johan Haan (Section of GC), Loic Senser (Section of MTE) (GC), January 25, 2019
Supervision: Zhengchao Wang, Yuki Oyama, Michel Bierlaire
Towards a techno-economic evaluation framework for regional train propulsion architectures
Railway can be the most environmental mode for land transport. However the sector faces cost challenges. Apart from main routes, tracks are often not electrified, requiring pollutive and operational expensive diesel propulsion systems. In contrary to road vehicles, there are no commercial large-scale applications of hybrid drivetrains. Furthermore, the dependencies of hybrid drivetrains on energy supplying infrastructure, like overhead wires or recharge points, has not been researched yet. In order to assess the potential of hybrid drivetrains holistically, the project Toolbox for Optimal Railway Propulsion Architectures (TORPA) has set up a framework to define drivetrain solutions, optimize them, and compare them among each other. Prior to this semester project, 58 drivetrain architectures have been defined. One of them was optimized toward the objectives of driving CO2 emissions and investment costs, on a specific use case. However, these bare results were inconclusive and expensive to compute for more than one use case. In this semester project, we downselect the number of possible architectures, outlining the currently most relevant ones. Furthermore, we include infrastructure requirements in the definition of architectures. The metric of investment cost is extended to display the full life cycle costs of architectures. The model of driving CO2 emissions is attempted to display all vehicle life cycle emissions, but left at the current state after qualitatively stating significant CO2 contributors not included in the calculation models. In order to assess the potential of the extended framework, it was required to state test cases, define tangible experiments, and prioritize them. Previously, it is researched which parameters should be chosen to be relevant for such experiments. Thereby, representative generic test tracks are conceived. We choose the question of viability of track electrification as first test case for the software framework. Thus, the currently operating vehicle architectures for diesel or full electric operation are applied. Especially the impacts of currently existing electrification infrastructure and track parameters, like distance and stop frequency, are investigated. Firstly, we imply a track with average constitution and operation. It is found, that the break even of track electrification is reached when a third of it is already electrified due to e. g. intersection with other tracks. Furthermore, we find that stop frequency is decisive for electrification break even: Tracks with shortened stop distance may be cheaper to be operated electrically, even if there is no catenary infrastructure built up yet.
Student: Florian Mueller, January 22, 2019
Supervision: Nour Dougui, Nikola Obrenovic, Michel Bierlaire
Locating charging station for electric taxis
This project aims at locating the position of charging station for electric taxis. Currently, transportation is transforming to be more sustainable. The electric vehicle is a promising way to achieve such a goal. However, charger unavailability is among one of the main issues that hindered the adoption of the electric vehicle. To contribute to mitigating such issue, this project is de ned accordingly to optimize the charging station location. The location problem is a classic problem in the operational research domain. It is usually solved using a p-median, p-mean, set covering, or flow covering method. The goal of this research is to apply these methods to decide the location of charging stations in San Francisco using taxi tracking data and compare the di erence in the end. This project is composed of three stages. First, the student should work on mapping the taxi tracking data on the map using map-matching techniques. Then, the student should apply point-based location methods (p-median, p- mean, set-covering) to find the optimal location. The student is not required to apply all the relevant methods but chooses one or two in which he is interested. After, the students will apply and characterize a flow-based method. Finally, an analysis of the results is performed to evaluate the results obtained.
Student: Oliver Mathias Buschor, Younes Bensaid, December 21, 2018
Supervision: Zhengchao Wang, Meritxell Pacheco, Michel Bierlaire, not applicable
Optimization of school accessibility in developing countries
The Global Program for Safer Schools (GPSS) focuses on risks linked to education infrastructure. The program aims to save lives and reduce the physical impact of disasters on school infrastructure. We are working with the Kyrgyz Republic Government to improve the capacity to respond to disasters, providing safer and quality learning environment for children, and managing the cost of disasters and climate shocks. This semester project/master thesis proposes a methodology to optimize educational infrastructure networks based on accessibility; and proposes solutions to improve the performance of educational infrastructure networks. In other words, it optimizes the location and the dimension of school buildings. Several factors are included such as transportation network, home location of the students and mobility patterns. The results is a methodology to suggest investments to reduce the risks in case of disaster. The cities of Bishkek in the Kyrgyz Republic is used as a case study. Methodologies from optimization, programming (Java) and geographical information system (GIS) will be used for completing this project.
Student: Yassine El Ouazzani, Oliver Mathias Buschor (GCMA3), December 21, 2018
Supervision: Riccardo Scarinci, Meritxell Pacheco, Michel Bierlaire
In collaboration with World Bank
Goodness of fit in DCMs
In these last years, with the arrival and wide-spreading of Big data, the discrete choice modeling community has gained access to larger datasets, more computing power and the possibility to drastically increase model complexity. It has therefore become crucial to establish precise measures of the goodness of fit as well as techniques to detect and reduce overfitting. We propose to apply known techniques from machine learning – such as cross-validation – to discrete choice models. We investigate the existence of overfitting in discrete choice models using a linear and a polynomial multinomial logit model on the Optima dataset. We adopted a train-test approach to evaluate model performance. Model estimates for both models were computed on N = 100, 200 and 500 runs of random train-test splits, using an 80-20 ratio. The results show that the polynomial model has better fit on the training samples, but performs worse than the linear model on the testing set, indicating the presence of overfitting. The use of K-fold cross-validation for simple multinomial logit models has also been explored. However, model estimates obtained using K-fold cross-validation did not differ from the model estimates estimates obtained from a single model fitting, due to underfitting. The distributions of the results obtained on the Optima dataset motivated the exploration of an empirical hypothesis test to determine the presence of overfitting. The test is based on the assumption that training and testing samples of a model which is not overfitting will have the same mean. This test was implemented using the t-test for two samples with unequal variances, also known as Welch test. This empirical test presented in this report did not differentiate the models in terms of overfitting. However, it could be a starting point of our future research, which will be aimed at establishing a statistical definition of overfitting.
Student: Jessica Hopkins (MA), December 21, 2018
Supervision: Gael Lederrey, Michel Bierlaire, Nicholas Molyneaux
Sustainable & intelligent transportation evaluation and plan
This project is composed of two stages. First, the student will search for and make a summary of emerging transportation technologies that are proposed in both literature and industries. After, the student will learn to come up with feasible and promising transportation systems composed of the emerging technologies, and then quantitatively evaluate the proposed systems based on travel demand simulations.
Student: David Gunter, July 20, 2018
Supervision: Zhengchao Wang, Yuki Oyama
Cost Reduction Uisng Passenger Centric Timetabling
"To design their timetables, train-operating companies mostly focus on operational aspects and cost. In Switzerland, a paradigm of transportation planning is to create regular-interval timetables (a.k.a. cyclic timetables) which aim for maximal transfer connections, simplicity and hence user friendlyness. SBB uses travel simulation models to predict the impact of timetable changes on travel demand and revenue. Mathematical timetable optimization methods are not yet used by SBB. But a recent EPFL thesis (2016) shows that the timetable itself has a significant impact on the performance of the operator in terms of the number of transported passengers: a timetable design that considers the behavior of passengers leads to higher revenue(s), market share(s), higher value of passenger-km etc. In this project, the aim is to use the optimization methods developed by the TRANSP-OR Lab consisting in combining cyclic and non-cyclic timetables and apply it to the Swiss Federal Railways’ timetable design. The goal is to evaluate the performance of the current Swiss interval timetable and to compare it to the optimal one. Suitability of hybrid timetables for the Swiss railway network will be investigated".
Student: Robert Abboud, June 29, 2018
Supervision: Virginie Lurkin, Michel Bierlaire
Design of a stated-preferences survey for a high-speed vacuum transportation mode
Since the high-speed vacuum transportation technology represents an innovative transportation mode, it is necessary to obtain data from surveys of hypothetical market/situations, the so-called stated-preferences (SP) surveys. The goal of this project is to design an SP questionnaire to evaluate the impact of this innovative transportation mode and to measure some indicators (such as the willingness-to-pay). A pilot test of the preliminary survey in a small sample will be carried out in order to evaluate the quality of the questionnaire. The main findings will be used to define the final survey that will be distributed to a representative sample of the population by a specialized company.
Student: Thibaut Richard et Martí Montesinos Ferrer, June 20, 2018
Supervision: Meritxell Pacheco, Yuki Oyama, Michel Bierlaire
Development of a heuristic algorithm for a hub location problem
In railway networks, the location of marshalling and shunting yards, i.e. facilities used for sorting and consolidation of transported goods, determines the costs of cargo transport to a great extent. Hence, each railway company is highly motivated to determine the optimal location of these yards. Such task can be represented as a multi-level facility location problem or a hub location problem. The aim of this project is the development of an heuristic algorithm for solving a facility location problem with facilities split in multiple hierarchical layers. The project will be a part of a larger research project done in the cooperation with SBB. The student will start from understanding the exact mathematical model of the mentioned problem and encode it into a heuristic algorithm, e.g. a local search algorithm or VNS. Also, the student will have to solve the problem on a smaller data set using an exact method and compare the solution with the results of the heuristic algorithm. The student needs to have good programming skills (Java or some other OO language) and knowledge of mixed integer linear programming and heuristics.
Student: Thibaut Guillaume Marie Richard, June 19, 2018
Supervision: Nikola Obrenovic, Nicholas Molyneaux, Michel Bierlaire
In collaboration with SBB Cargo
A solution approach for the Multicommodity Flow Problem within rail freight transportation
This project aims at studying the Multicommodity Flow Problem (MFP), which has several real life applications, particularly within the transportation sector. The project has been inspired by an ongoing research project conducted by TRANSP-OR in collaboration with SBB Cargo. In the context of railway freight operations, companies have limited availability of both railway lines and trains. Therefore, they have the goal to optimize as much as possible the usage of their resources in order to minimize the cost of transporting the goods. Such optimization problem can be modelled as a MFP. This project is composed of two stages. First, the student will learn the mathematical model of the MFP, in its two variants: node-arc formulation and path-based formulation. The two variants will be coded and tested on small instances by using a Mixed-Integer Linear Programming (MILP) solver such as CPLEX. Further, the student will implement a solution method to solve larger instances of the problem, by using either an exact method (such as column generation) or a heuristic method.
Student: Nicolas Pradignac (IC), June 19, 2018
Supervision: Stefano Bortolomiol, Nikola Obrenovic, Michel Bierlaire
In collaboration with SBB Cargo
Modelling competition in demand-based optimization models
The project aims at studying and understanding the interactions between supply and demand in an oligopolistic market, in which multiple operators compete for the same pool of customers. This is a common situation in the transportation sector as well as in other markets. Operators take the supply-side decisions that optimize their own performance function (e.g. maximization of revenues or profits). Such decisions are influenced both by the decisions of their competitors and by the preferences of the customers who consider purchasing one of the services offered on the market. The latter ones are modelled at a disaggregate level according to the random utility theory. The starting point of the project is a recent modelling framework that allows to include any random utility model in a mixed integer optimization formulation. In this framework, a single operator exploits its knowledge of the demand to maximize its objective function, while assuming that the decisions of its competitors are held fixed. The goal of this project is to extend the existing framework by including the case in which two operators simultaneously optimize their decisions. Such problem falls into the category of competitive games called two-player non-cooperative games. The student will refer to the concept of Nash equilibrium to answer questions such as "how does competition affect prices?", "what products should each competitor offer and in which quantity?", "should two competitors fight to attract the same group of customers or should they each target a different market segment?", among others.
Student: Charlotte Darné (BS), June 19, 2018
Supervision: Stefano Bortolomiol, Virginie Lurkin, Michel Bierlaire
Safer schools thanks to an improved transport access in developing countries
The Global Program for Safer Schools (GPSS) focuses on risks linked to education infrastructure. The program aims to save lives and reduce the physical impact of disasters on school infrastructure. We are working with the Kyrgyz Republic Government to improve the capacity to respond to disasters, providing safer and quality learning environment for children, and managing the cost of disasters and climate shocks. This semester project/master thesis proposes a methodology to assess and optimize educational infrastructure networks based on accessibility; and proposes solutions to improve the performance of educational infrastructure networks. In other words, it evaluates the risks based on the school network, including parameters like transportation network, home location of the students and mobility patterns. The results is a methodology to suggest investments to reduce the risks in case of disaster. Two case studies will be carry on in the cities of Bishkek and Osh in the Kyrgyz Republic. Methodologies from transportation studies, geographical information system (GIS) and programming will be used for completing this project.
Student: Zora Oswald, June 16, 2018
Supervision: Riccardo Scarinci, Meritxell Pacheco, Michel Bierlaire
In collaboration with The World Bank
Usage d'un sous-cantonnement pour densifier la circulation aux abords d'un noeud ferroviaire important
Le projet consiste, sur un découpage des cantons en sous-cantons virtuels, à faire circuler plusieurs combinaisons de trains à signalisation latérale et de trains à signalisation en cabine (ETCS_L2/3) pour déterminer les gains en débit pouvant être attendus.
Student: Cloé Lafaye, June 08, 2018
Supervision: Daniel Emery
Building offline and online optimization algorithms for dispatchment of teams at Nez Rouge.
Nez Rouge (http://nezrouge.ch) is a swiss charitable association with the purpose of driving people back home safely during the cold nights of December. They have to find many volunteers across the whole country and the demand never stops growing. Instead of helping them find new volunteers, I propose to solve one difficult task: dispatching efficiently the volunteers, the idea being to reduce the waiting-time of customers. Using data provided by Nez Rouge, we will first build an offline optimization problem to find a lower bound on the estimated waiting-time and compare it to the actual waiting-time. The second step will be to build an online optimization problem using the API of OpenStreetMap to solve this problem in real-time. The third and final step will be to build a webpage for the volunteers at Nez Rouge. Note: This project can be done as a master project or split between multiple semester projects, depending on the number of credits you have to do.
Student: Colin Ducommun, June 08, 2018
Supervision: Gael Lederrey, Nicholas Molyneaux, Michel Bierlaire
Qualité de l'offre nationale des projets d'horaire EA2030 et EA2035
Des deux projets d'horaire nationaux (EA2030 et EA2035) sont à disposition des décideurs pour choisir entre deux variantes d'aménagement du réseau ferroviaires suisse. Le projet consiste à notamment à évaluer la qualité de l'offre nationale offerte par ces projets en terme de temps de parcours et de fréquence, l'horaire 2018 servant de référence. Les valeurs absolues et relatives des critères retenus (temps de parcours, fréquence, nombre de correspondance, ...) seront notamment disposées sous la forme de matrice OD, chaque O(resp. D) étant une gare significative sur la plan démographique et politique (p.ex. chef-lieu de canton)
Student: Zora Oswald (SGC), June 08, 2018
Supervision: Daniel Emery
Offre nationale ferroviaire 2035 avec tronçons SwissMetro
Un projet d'horaire ferroviaire national 2035 a été publié avec les aménagements des infrastructures nécessaires exploiter un tel horaire. Le projet consiste à développer des variantes de réalisation de tronçons SwissMetro. Pour chacune de ces variantes il sera recensé les aménagement d'infrastructure prévus devenant superflus d'une part, et le projet d'horaire 2035 sera alors adapté à la présence de ces tronçons d'autre part. Les avantages/inconvénients de chaque variante seront comparés, le projet d'aménagement 2035 et l'horaire 2035 sans SwissMetro servant de référence.
Student: Lauriane Masson (SGC), June 08, 2018
Supervision: Daniel Emery
Automatic utility specification using machine learning techniques
The objective of this project is to help building a utility function using different Data Analysis functions. It is well known that modeler spend a lot of time creating their utility function for DCMs. The goal of this project is to build an automated procedure to generate Utility functions based on the data. The tools used are Data Analysis and Machine Learning (Decision Tree, Random Forest, Linear Regression, Clustering, etc.). The utility functions found using these tools will then be compared to standard and more advanced utility functions. Another direction that we can take is using Machine Learning to extract the possible nests for a Nested Logit Model. This direction, however, requires a deeper knowledge in Discrete Choice Modelling. We are open to the discussion concerning the available directions.
Student: Nicola Ortelli, June 08, 2018
Supervision: Gael Lederrey, Tim Hillel, Virginie Lurkin
Conduite Automatique des Trains entre Neuchâtel et La Chaux-de-Fonds
L'European Train Control System Level 2 (ETCS_L2) autorise la conduite automatique des trains via le "paquet 44". Le projet consiste à introduire dans ce "paquet" l'ensemble des instructions nécessaires pour aller, en conduite automatique, de A à B tout en conservant un conducteur à bord (GoA2). Le parcours entre Neuchâtel et La Chaux-de-Fonds via un nouveau tracé servira de "démonstrateur".
Student: Guillaume Sauvin (SGC), June 08, 2018
Supervision: Daniel Emery
Augmentation de la réserve de capacité m1 des TL par actions limitées au matériel roulant
Le métro m1 des Transports Publics Lausannois (TL) relie le Centre-Ville de Lausanne à la Gare de Renens située dans l'Ouest Lausannois en passant notamment par l'Université (UNIL) et l'École Polytechnique Fédérale de Lausanne (EPFL). En heure de pointe en période estudiantine il circule avec des intervalles de 5 minutes, soit les intervalles les plus faibles possibles compte tenu de l'infrastructure actuelle à simple voie. Depuis de nombreuses année, de nombreuses mesures touchant la demande ont été prises en vue de réduire les pointes afin de dé-saturer cette ligne (décalage des heures de début de cours entre UNIL et EPFL, nouvelle ligne de bus passant par Renens-Gare et les Hautes Écoles, …). Toutefois, avec la croissance continue de la population se rendant sur les sites universitaires, la saturation se profile à moyen terme. Cette pré-étude de master quantifiera l'apport des moyens d'actions pour accroître la capacité du matériel roulant sans avoir a modifier l'infrastructure.
Student: David Moy de Vitry (SGC), June 08, 2018
Supervision: Daniel Emery
Offre nationale ferroviaire 2035 avec un tronçon SwissMetro
Le projet consiste, notamment, à développer des scénarios de réseau pour le mode de transport SwissMetro, à en définir les caractéristiques principales ainsi qu'un ordre de priorité de réalisation des tronçons. Dans une seconde étape, l'horaire GK12 sera adapté pour tenir compte de la présence du tronçon-pilote du(des) scénario(s) le(s) plus prometteur(s). Une comparaison exhaustive des avantages/inconvénients des horaires GK12 original et GK12 modifié(s) clôturera ce travail.
Student: Lauriane Masson (SGC), June 08, 2018
Supervision: Daniel Emery
On the optimization of CAPEX and OPEX for the design of a full electric large capacity urban bus system
During the last few decades, environmental impact of the fossil fuel-based transportation infrastructure has led to renewed interest in electric transportation infrastructure, especially in urban public mass-transportation sector. The deployment of battery-powered electric bus systems within the public transportation sector plays an important role to increase energy efficiency and to abate emissions. RAn efficient feeding stations installation and an appropriate dimensioning of battery capacity are crucial to minimize the total cost of ownership for the citywide bus transportation net- work. The objective of this project is to extend an existing optimization model to the multiple lines case and to come up with an objective function that better reflect the real costs incurred by the operator. The objective function should include Capital Expenditures (CAPEX) and Operational Expenditures (OPEX). For the multiple line case, the central issue is to deal with the feeding stations that are shared among different lines.
Student: Guillaume Mollard, January 31, 2018
Supervision: Virginie Lurkin, Stefan Binder, Michel Bierlaire
Integrating demand and supply in the context of airlines
Student: Thibaut Richard et Gabriel Curis (GC), January 24, 2018
Supervision: Meritxell Pacheco, Anna Fernandez Antolin, Michel Bierlaire
Offre 2030 entre Berne et Lausanne répondant aux attentes des cantons (BE, FR et VD)
Student: Benoît Corday (SGC), January 12, 2018
Supervision: Daniel Emery, Jean-Daniel BURI
From public transport vehicles to pedestrian flows.
The objective of this semester project is to extend the notion of "train induced flows" to other modes of public transport. When public transport vehicles arrive in the station, many passengers disembark. From the infrastructure's point-of-view, there is a flow of pedestrians arriving. The characteristics of such flows depend on the vehicle's characteristics (door width for example) and the characteristics of the infrastructure (width of the corridors for example). The student will explore the different interactions and characterize the different flows by using models similar to the "train induced flows" model, already published. The specifications will be different based on the various modes and platform setups. The second objective is to explore the feasibility of using a Poisson distribution to model the pedestrians arriving onto platforms. Similarly to alighting flows, different configurations can lead to different model specifications. The student will start by exploring the literature then, based on the pre-existing models, some new formulations can be proposed. Data is available for calibration of pedestrian flows boarding trains. Synthetic data can be used for the calibration of other models. This project can be done during the spring 2018 semester.
Student: Rodolphe Farrando, January 12, 2018
Supervision: Nicholas Molyneaux, Gael Lederrey, Michel Bierlaire
Mobilité lors des JO d’hiver et perspectives pour Sion 2026
La pré-étude se propose premièrement de recenser les démarches d’organisation de la mobilité lors de JO présentées par les villes candidates récentes, notamment Sion 2002 et Torino 2006, et de déterminer et de récolter les valeurs dimensionnantes de mobilité en lien avec les différentes disciplines sportives. Dans une seconde partie, la pré-étude se focalisera sur le cas de la ville candidate Sion 2026. Elle présentera en particulier un panorama de l’offre TP actuelle pour atteindre actuellement les sites retenus, ainsi que l’offre envisagée par le programme PRODES 2025. Le Projet De Master (PDM) proprement dit cherchera premièrement à atteindre les objectifs de la pré-étude non totalement atteints. Son objectif principal sera de générer des concepts d’offre TC, voire même des variantes d’horaires. Les lignes à étudier plus particulièrement seront choisies en février 2018. La liste de ces lignes, non exhaustive, comporte notamment les lignes « extérieures » («Olympic Ring», raccords aux aéroports internationaux GVA et ZRH) et les lignes « intérieures » au Valais (Brig-Ulrichen, Martigny-Le Châble,).
Student: Clément Sintes (SGC), January 12, 2018
Supervision: Daniel Emery, Stefano MANELLI Pierre FAVRE
Aménagements futurs du complexe ferroviaire de Clermont-Ferrand
Le travail consiste premièrement à établir un état des lieux (plan et caractéristiques des voies, postes d’aiguillages, IFTE, GOV, types de mouvements et volumétrie, roulements de matériel, équipements de maintenance et de remisage, etc.) Une analyse de l’adéquation des équipements aux besoins engendrés par le remplacement des rames tractées par des rames automotrices sera alors menée; et d’éventuelles mesures seront proposées. Si le temps le permet, les conséquences sur les GOV de Clermont seront estimées si un scénario ambitieux d’électrification devait voir le jour autour de Clermont (Volvic/Le Cendre/Aulnat). de référence.
Student: Axel Valentin (SGC), January 12, 2018
Supervision: Daniel Emery
Infrastructure ferroviaire entre Lausanne et Genève à l'horizon 2030
Le projet consiste à déterminer les infrastructures nécessaires et les horaires possibles sur la ligne Lausanne-Genève à l'horizon 2030. Un projet d'offre 2030 prévoit pas moins de dix paires de trains voyageurs (4 IC, 2 IR, 4 RE). L'OFT pour sa part souhaite qu'une paire de trains marchandises au minimum puisse circuler chaque heure.
Student: Cloé Lafaye et Julien Thiriot (SGC), January 12, 2018
Supervision: Daniel Emery
Analysis of pedestrian group behavior based on tracking data and pattern recognition methods
The objective of this project is to analyze the group behavior among pedestrians based on individual trajectory data. The data is collected in Lausanne train station, where a large-scale network of smart sensors has been used to track pedestrians. The project aims to improve the understanding of group behavior among pedestrians and its impact on pedestrian dynamics. It involves the following steps: (i) Development/selection of suitable methods (e.g. data mining/pattern recognition) for identification of individuals walking together, based on pedestrian trajectory data, and their implementation. (ii) Analysis focused on the behavior of identified groups; (iii) Comparison of the findings with the existing empirical basis, as well as with the proposed theories and models that take group dynamics into account. The student needs to have good programming skills (Scala/Matlab), and knowledge of statistical analysis.
Student: Montesinos Ferrer Martí, December 22, 2017
Supervision: Marija Nikolic, Evanthia Kazagli, Zhengchao Wang
Identifying the objectives of car drivers route choice behavior
The goal of this project is to shed light on the underlying objectives of car drivers' behavior --- pertaining to their route choices --- when traveling from one location of a transportation network to another. The student will use clustering methods, defined on the basis of different measures of similarity, that are associated with potential objectives such as the minimization of the travel time, the length or the complexity of a route. The results of the clustering will be analyzed to draw insights into the use of specific objectives depending on the features of the trip, such as the origin and destination points, and the departure time. The project involves an initial stage of data processing. The student needs to have good programming skills (Matlab or equivalent) and good knowledge of statistics. Familiarity with SQL is a plus.
Student: Nicola Marco Ortelli, December 22, 2017
Supervision: Evanthia Kazagli, Marija Nikolic, Michel Bierlaire
Alternative activity pattern generation for stated preference surveys
Student: Nicola Marco Ortelli (GC), December 22, 2017
Supervision: Anna Fernandez Antolin, Gael Lederrey, Michel Bierlaire
Modeling purchases of new cars for 2015: a comparison between countries
We are interested in analyzing the behavior of people when they face the decision to buy a new car. We have a large dataset containing information of new car purchases, where respondents answered to several questions related to their socioeconomic characteristics and to the attributes of their recently purchased car. We have developed a framework that has been applied to the data corresponding to France in 2014. The objective of this semester project is to apply it to 2015 for several countries (France, Germany, Italy and Spain) and to obtain and predict market shares in each of the countries as well as to compute willingness to pay towards different attributes. Moreover, the student will analyze the difference between the different countries and between the different years.
Student: Martí Montesinos (GC), June 19, 2017
Supervision: Anna Fernandez Antolin, Meritxell Pacheco, Michel Bierlaire
Planning of feeding station installment for a full electric large capacity urban bus system
The integration of customer behavioural models in optimization provides a better understanding of the preferences of clients (the demand) to policy makers while planning for their systems (the supply). These preferences are formalized with discrete choice models, and the corresponding optimization models where supply and demand closely interact are associated with (mixed) integer optimization problems. One concrete application of this integration consists of an operator selling services to a market, each service at a given price to a finite number of customers, called the capacity of the service. We are interested in finding the best strategy in terms of pricing and capacity allocation in order to maximize the revenues of the operator. The project objectives can be tailored for both Master or Semester project students.
Student: Adrien Ruault (IN), June 16, 2017
Supervision: Virginie Lurkin, Marija Nikolic, Michel Bierlaire
Speed profile of an innovative catenary-free electric bus
The TOSA bus system is a revolutionary catenary-free electric bus concept that includes small short-range on-board batteries and a series of fully automated fast charging stations installed at some bus stops. The automatic fast-charging stations partially replenish the bus batteries in a few seconds whenever a bus arrives at the bus stops, while avoiding any interference with the bus schedules and operations. This system has been defined during the project myTOSA 1.0 and implemented in a pilot test in the city of Geneva in 2013. The project myTOSA 1.0 is currently extended by the project myTOSA 2.0. myTOSA 2.0 is composed of several modules, the most relevant for this semester project is the traffic simulation. The traffic simulation represents the movement of the buses in a network, and requires a representation of the infrastructure as an input. This project aims to define the typical speed profile of an electric bus moving in an urban network. The speed profile includes acceleration and deceleration patters for different driver behaviour, network configurations and traffic levels. The influence of traffic lights, bus priority, roundabouts on the speed profile should be investigated. The first step to achieve this aim is to define the probability distribution of the number of intersections crossed by the bus. This step is necessary to complete this project successfully. Further steps in the directions of the definition of the speed profile are considerate optional. Methodologies from transportation modelling, geographical information system (GIS) and programming (MATLAB) will be used for completing this project.
Student: Valentin Axel Olivier & Nicolet Adrien (GC), June 16, 2017
Supervision: Riccardo Scarinci, Virginie Lurkin, Michel Bierlaire
Organisation de l’amélioration de la performance de l’offre ferroviaire dans les métropoles d’Auvergne Rhône Alpes: Retour d’expérience du nœud ferroviaire lyonnais et enseignements à tirer pour les autres métropoles
L’État français et SNCF Réseau ont signé un contrat de performance permettant de donner une trajectoire financière pour la décennie à venir et qui demande d’organiser l’amélioration de la performance, notamment de la régularité. D’autre part, dans le cadre de Schéma Régionaux d’Aménagement, de Développement Durable et d’Egalité des Territoires (SRADDET), SNCF Réseau entend positionner l’infrastructure ferroviaire comme véritable pivot de la mobilité au niveau régional. L’étude (pré-étude et travail pratique de Master) doit permettre de s’approprier les méthodes de travail utilisées pour le NFL ( audit, diagnostic, bilan des circulations, mobilisation des acteurs internes ou externes …) et les outils correspondants permettant à tous les acteurs de participer à l’amélioration de la performance en appliquant des plans d’action à court et moyen terme ( organisation du projets, identification des phases décisives, formalisation ders plans d’action, ...) Une analyse dans la cohérence des plans d’action court terme mais également long terme sera développé en prenant en compte la stabilité des évolutions des hypothèses de l’offre ferroviaire. La méthode de travail ainsi conceptualisée par un travail personnel et autonome et une reformulation des facteurs clés de succès et les outils correspondants seront appliqués, à titre de « démonstration » pour la métropole grenobloise ou clermontoise. Des plans d’actions à court, moyen et long terme seront déduits et proposés
Student: Jean-Baptiste Landes (SGC), June 09, 2017
Supervision: Daniel Emery
Emotions and risky discrete choices
This CSE project aims at understanding, analyzing and quantitative modeling of the roles of emotions in risky choice making
Student: Le Sueur Cécile Christianne Anny Déborah Suzanne (SMA), June 02, 2017
Supervision: Matthieu de Lapparent, Michel Bierlaire
Modeling public transport transfers in the “new” Lausanne train station
Exploration of the tracking data collected in the main station of Basel. The objective is to calculate passenger-centric variability indicators and perform some simple modelling tasks.
Student: José Ramón Rodriguez, June 02, 2017
Supervision: Nicholas Molyneaux, Riccardo Scarinci, Michel Bierlaire
Building an integrated model for modelling pedestrian movements inside hubs
The objective of this project is to build a simulator for measuring the impact of management strategies inside transportation hubs.
Student: Charles Jeanbart, June 02, 2017
Supervision: Nicholas Molyneaux, Riccardo Scarinci, Michel Bierlaire
Models for pedestrian movements based on integrated and sequential clustering
The focus of this study is the modeling of speed-density relationship for pedestrian movements using the potential of available data. The data set considered in this project contains pedestrian trajectories collected in the train station in Lausanne. To track pedestrians, a large-scale network of tracking sensors is installed in the station (Alahi et al., 2014). The analysis we have performed reveals a high scatter in the data. To characterize the observed scatter we have developed a multi-class model of the speed-density relationship based on the latent class modeling approach (Nikolic et al., 2017). The latent class approach allows for modeling the segmentation in the population and the movement behavior simultaneously. The aim of this project is the derivation of the model representing speed-density relationship based on a two-stage (sequential) approach. The first stage involves segmentation of pedestrian trajectory data, by using machine learning techniques (clustering). In the second stage, a separate speed-density model is to be estimated for each cluster discovered in the first stage. The performance of the approach will be tested using real data, and compared to the more integrated, latent class approach.
Student: Konde Romain Olivier Bondo, June 02, 2017
Supervision: Marija Nikolic, Iliya Markov, Michel Bierlaire
Investigating the role of attitudes in the purchase of new cars
Attitudes and perceptions play an important role in decision-making processes. We are interested in new car purchases, and how attitudes and perceptions play a role in this context. We have a large dataset containing information of new car purchases, where respondents answered to several attitudinal questions. The objective of this semester project is to analyze these attitudinal questions, starting with principal component analysis. The study will be expanded to see how these attitudes affect purchases of new cars, and of electric vehicles in particular.
Student: Nicola Ortelli, January 23, 2017
Supervision: Anna Fernandez Antolin, Meritxell Pacheco, Michel Bierlaire
Mobilité Rail+Bus 2020 dans les Alpes vaudoises
Une communauté d’études, comprenant notamment la CITAV, a étudié de manière globale un territoire géographique appelé « Alpes vaudoises 2020 ». Le volet mobilité n’a été traité que de manière globale, tel que l’on peut notamment le constater dans le rapport final de juillet 2013. Après lecture attentive des documents « Alpes vaudoises 2020 » et des horaires actuels, les étudiants développeront et évalueront des variantes d’offre Rail+Bus sur la base du projet d’horaire trains 2025 le plus actuel.
Student: David Moy de Vitry Jean-Baptiste Landes, January 13, 2017
Supervision: Daniel Emery
Régulation de vitesse par la signalisation pour garantir un croisement actif
Dans le cadre de la densification d’horaires cadencés sur lignes à simple voie, il y a souvent nécessité que les trains se croisent hors gare. Il est donc intéressant de prévoir un îlot de croisement actif, c’est-à-dire sans qu’aucun des deux trains n’ait besoin de s’arrêter. L'étude consiste premièrement à établir une démarche pour établir les différentes topologies devant être étudiées. Dans la partie principale, le projet consistera à faire varier de manière systématique et logique, sur un tronçon de ligne suisse à voie étroite fictif, la vitesse de ligne, la longueur de l'ilot à deux voies, les endroits de détection des trains, les endroits de réduction de la vitesse pour le train en avance sur l'autre, et la valeur de cette(ces) réduction(s) de vitesse. La signalisation sera de type latéral à trois aspects avec, si besoin, chiffre complémentaire d'annonce ou d'exécution de vitesse. L'outil de simulation OpenTrack sera utilisé.
Student: Benoît Corday Marc Zimmermann, January 13, 2017
Supervision: Daniel Emery
Energy consumption of an innovative catenary-free electric bus
Electric busses help to decrease pollution in city centers. However, they need to be constantly attached to a power source. This limits their mobility and brings visual pollution due to continuous catenaries throughout the city. Focus of this project is a revolutionary catenaryfree electrical bus that includes short-range on-board batteries and a series of fast charging stations installed at some bus stops capable to charge the bus batteries in a few seconds without interference with the bus schedules. This project aims to develop a traffic simulation of this electric bus moving on a simple road network. A discrete event simulation representing the fundamental components of the system (e.g. bus, passengers load, on-board battery, charging station, storage level) should be developed, calibrated and validated using traffic data (such as number passengers on board, traffic and congestion information). Then, key indexes representing the system performance should be defined (e.g. travel time, dwell time, battery charge level). Methodologies from transportation modelling, simulation and programming (Java) will be used for completing this project.
Student: Xiaoran Yu, December 23, 2016
Supervision: Riccardo Scarinci, Yousef Maknoon, Michel Bierlaire
Accounting for dynamics in pedestrian multi-class speed-density relationship
The relationship between speed and density plays an important role in modeling of pedestrian traffic. It is useful for planning and design of pedestrian facilities, and it is also a required input or calibration criterion for models of pedestrian dynamics. The relationship is specified under the assumption that the traffic system is at equilibrium (stationary and homogenous). The analysis we have performed, based on the data collected in the Lausanne train station, rules out the use of a unique equilibrium relationship due to a high scatter in the data. This scatter may be explained by the violation of the equilibrium assumptions, as documented in the literature. To characterize the observed scatter we have developed a multi-class model of the speed-density relationship based on the latent class modeling approach. The model is derived by relaxing the homogeneity assumption of equilibrium relationships. It is assumed that pedestrian population is heterogeneous (e.g. different trip purpose, different time to departure, etc.) and that this heterogeneity leads to the existence of multiple pedestrian classes that are characterized by different behavior. There are two specification issues related to the panel data set (data collected over multiple time periods for the same sample of individuals) that we use in our analysis. The first is serial correlation across the observations of the same individual due to unobserved individual factors that persist over time. The second is related to dynamics, meaning that the speed in one period may depend on the speed values in the past. We have addressed the first issue by introducing an agent effect in the model that captures individual related unobserved factors. We term this model the static model with agent effect. The aim of this project will be to deal with the second issue, or dynamics. We will start with the simplified assumption that the speed at time t is influenced by the speed at time t - 1 only. We term such a model a dynamic model with agent effect. The performance of the approach will be tested using real data.
Student: Marc-Edouard Schultheiss (GC), December 23, 2016
Supervision: Marija Nikolic, Matthieu de Lapparent, Michel Bierlaire
Étude détaillée des contours de référence ferroviaires et dérogations
Dans le cadre du projet Clip-Air (http://clipair.epfl.ch/) il est envisagé d'acheminer des capsules par le rail. La pré-étude au projet de Master consiste à étudier très précisément les gabarits de référence ferroviaires pour voie à écartement UIC tant au niveau suisse, qu'européen et américain non seulement en alignement mais aussi en courbe. Les relations avec les gabarits limites des obstacles seront mises en évidence. Les procédures suisses pour les transports exceptionnels par rail sur voie à écartement UIC seront décrites.
Student: Martin Ellwanger (SGC), December 23, 2016
Supervision: Daniel Emery
4. Modeling Route Choice in Quebec City Using Mental Representations
In this project, we build upon previous work proposed by Kazagli et al., 2016 to model route choice for a big network and dataset concerning the city of Quebec. We aim first to define the mental representation items (MRIs) for the case study and then to derive operational route choice models based on them. A GPS dataset from the city of Quebec is available and will be used for this purpose. The focus of the project is methodological.
Student: Mathieu Plourde, December 23, 2016
Supervision: Evanthia Kazagli, Matthieu de Lapparent, Michel Bierlaire
Generalization and policy analysis for a rich inventory routing problem
We solve a rich logistical problem inspired from practice, in which a set of trucks is used for collecting recyclable waste from large containers over a finite planning horizon. Each container is equipped with a sensor, which communicates its level at the start of the day. Given a history of observations, a forecasting model is used to estimate the point demand forecasts as well as a forecasting error representing the level of uncertainty. The problem falls under the framework of the stochastic inventory routing problem. We introduce dynamic probabilistic information in the solution process, which impacts the cost through the probability of container overflows on future days and the probability of route failures. To solve the problem, we implement an adaptive large neighborhood search algorithm, which integrates a specialized forecasting model, tested and validated on real data. The student will analyze the performance of the algorithm and its results for various scenarios and collection and routing policies. In particular, we would like to see the impact on the best solution of the full probabilistic model with various cost parameters against simpler policies, such as one using buffer truck and container capacities to handle stochastic demand. We would like to apply the model on a rolling horizon basis and analyze the expected value of perfect information, the properties of the stochastic solution, etc. Other practical features such as open tours, multi-product problems, etc. may also be explored. We can also work on improving the algorithmic performance per se through new operators and more intelligent search strategies in the current implementation. This project is appropriate for a student with excellent coding skills in Java and knowledge of a good software for plotting and data analysis such as Matlab or R. The focus of the project will be suited to the student's interests, coding abilities, and the number of credits he or she needs.
Student: Prisca Aeby (IC), December 23, 2016
Supervision: Iliya Markov, Yousef Maknoon, Michel Bierlaire
Pricing and capacity allocation strategies for a demand-based revenues maximization problem
The integration of customer behavioural models in optimization provides a better understanding of the preferences of clients (the demand) to policy makers while planning for their systems (the supply). These preferences are formalized with discrete choice models, and the corresponding optimization models where supply and demand closely interact are associated with (mixed) integer optimization problems. One concrete application of this integration consists of an operator selling services to a market, each service at a given price to a finite number of customers, called the capacity of the service. We are interested in finding the best strategy in terms of pricing and capacity allocation in order to maximize the revenues of the operator. The project objectives can be tailored for both Master or Semester project students.
Student: Jonathan Lachkar (GC), December 23, 2016
Supervision: Meritxell Pacheco, Virginie Lurkin, Michel Bierlaire
Demand based rolling stock allocation
Providing a high level of service for the passengers is one of the most important requirements of passenger railway company. In practice during rush hours passengers cannot be transported according to usual service standards because of a shortage of the rolling stock capacity. The purpose of this pre-project is to determine an efficient method to allocate the rolling stock taking into account passengers demand.
Student: Salma Derouiche (IC), December 23, 2016
Supervision: Yousef Maknoon, Tomás Robenek, Michel Bierlaire, Simon Landureau
Price of anarchy in public transit networks
Student: Marc-Edouard Schultheiss (GC), December 23, 2016
Supervision: Stefan Binder, Tomás Robenek, Michel Bierlaire
Measure of user-oriented service variability in hubs
This project's goal is to explore the service variability (reliability) in public transport hubs. Three operator oriented service characteristics are common in literature (robustness, reliability and punctuality). As these measures are operator oriented, they might not reflect actual level-of-service (LOS) experienced by users (pedestrians). To account for this difference, user-centered measures will be defined then observed using an already secured data set. Some examples of variability measures are OD travel time, passenger transfer times, passenger waiting times, train punctuality, etc. The main data sets used for this project is the pedestrian tracking data from the Lausanne and Basel train stations and possibly some effective train time tables.
Student: Anouk Allenspach and Dieynaba Dia (GC), December 23, 2016
Supervision: Nicholas Molyneaux, Riccardo Scarinci, Michel Bierlaire
Passengers' connection time preferences in airline itinerary choice
As noted by Theis et al. (2006), "Network airlines traditionally attempt to minimize passenger connecting times at hub airports, assuming that passengers prefer minimum scheduled elapsed time for their trips. However, minimizing connecting times creates schedule peaks at hub airports. These peaks are extremely cost-intensive in terms of additional personnel, resources, runway capacity, and schedule recovery. Consequently, passenger connecting times should be minimized only if the anticipated revenue gain of minimizing passenger connection times is larger than the increase." Prior work has used (small) stated preference surveys to estimate customers' connection time preferences, and explicitly whether this function is nonlinear with connection time (i.e., customers avoid very short connections and very long connections). However, little is known as to how these connection time preferences vary as a function of other characteristics, including flight frequency (or schedule delay), length of haul, whether the flight is the last flight of the day, prior on-time performance of flight legs. As part of this project, the student will estimate airline itinerary choice models using a large ticketing database provided by the Airlines Reporting Corporation (ARC). The student will build off of prior MNL models that have been estimated which have corrected for price endogeneity and focus explicitly on refining the utility function related to connection time preferences.
Student: Elisabeth Zbinden (GC), December 23, 2016
Supervision: Virginie Lurkin, Matthieu de Lapparent, Michel Bierlaire, GARROW
Grandes mobilités réversibles. Approche par la modélisation dynamique de choix discrets.
Student: Alexis Gumy, December 23, 2016
Supervision: Matthieu de Lapparent, Michel Bierlaire, E. Ravalet, V. Kaufman (LASUR)
Une modélisation de la grande mobilité
Student: Pauline Hosotte, December 23, 2016
Supervision: Matthieu de Lapparent, Michel Bierlaire, E. Ravalet, V. Kaufman (LASUR)
Traffic simulation model of an innovative catenary-free electric bus
Electric busses help to decrease pollution in city centres. However, they need to be constantly attached to a power source. This limits their mobility and brings visual pollution due to continuous catenaries throughout the city. Focus of this project is a revolutionary catenary-free electrical bus that includes short-range on-board batteries and a series of fast charging stations installed at some bus stops capable to charge the bus batteries in a few seconds without interference with the bus schedules. This project aims to develop a traffic simulation of this electric bus moving on a simple road network. A discrete event simulation representing the fundamental components of the system (e.g. bus, passengers load, on-board battery, charging station, storage level) should be developed, calibrated and validated using traffic data (such as number passengers on board, traffic and congestion information). Then, key indexes representing the system performance should be defined (e.g. travel time, dwell time, battery charge level). Methodologies from transportation modelling, simulation and programming (MATLAB) will be used for completing this project.
Student: Romain Meyer (SGC), June 17, 2016
Supervision: Riccardo Scarinci, Yousef Maknoon, Michel Bierlaire
Modeling evolution of sales of alternative fuel vehicles in Europe
Student: Christophe Paillard (MA), June 17, 2016
Supervision: Matthieu de Lapparent, Michel Bierlaire
Solar Decathlon: strategies for a sustainable mobility to achieve the goal of a 2000-Watt society
Solar Decathlon is an international competition that challenges twenty university teams from around the world to build and operate solar-powered houses. This project is part of the EPFL multidisciplinary team of students participating at this competition. The global objective is to transform a regular district in Fribourg into a sustainable eco-district using the solar-powered building as an activator. This building would promote sustainable actions, reduction in energy consumption and soft mobility. This project aims to evaluate the necessary strategies to incentive sustainable mobility inside the district. The goal is to suggest concrete actions that would incentive the modal shift toward more sustainable modes of transport. The effects of these strategies should be quantified and analyzed. The total energy consumed for mobility should respect the limit suggested by the 2000-Watt society. This concept proposes a way of living where nobody consumes more than a continuous 2000 W in comparison with the 6000 W currently consumed in Europe. Knowledge of transportation systems, environmental impacts and strong quantitative analysis skills are required to complete this project. The student will also interact with the other members of the Solar Decathlon competition.
Student: Charles Albert Jeanbart (SGC), June 17, 2016
Supervision: Riccardo Scarinci, Michel Bierlaire
Analysis of External Effects on Cyclicity in Passenger Railway Service
In the passenger railway service, cyclicity is in general perceived as a beneficial attribute. The main allegation being memorability of such timetables by the passengers. However no quantitative proof has been presented to support such statement. In this project, the student will become familiar with a passenger satisfaction (that is based on utility theory) and further extend this concept by the attribute expressing the perception of the cyclicity. Student will then solve an optimization problem maximizing the passenger satisfaction and/or train operating company's profit. The benchmark will be done on a small part of Swiss network, followed by a test on a full scale network of Israeli Railways. Basic knowledge of programming is required.
Student: Lucia Montero (SCS), June 17, 2016
Modeling route choice in Québec using mental representations
In this project, we aim at simplifying the route choice problem by modeling the strategic decisions of people --represented by the mental representations of their itineraries-- instead of the operational ones --represented by paths. We define an abstracted graph based on what we denote mental representation item (MRI) and we derive operational route choice models based on it. A GPS dataset from the city of Quebec is available and will be used for this purpose. Hence, the project is divided in two main parts; the first and technical part which involves data processing, and the second methodological part which involves the development of a model for the MRI network.
Student: Mathieu Plourde, June 03, 2016
Supervision: Evanthia Kazagli, Matthieu de Lapparent, Michel Bierlaire
Strategic Energy Planning under Uncertainty
The main goal of the project is to classify parameter uncertainty (i.e. defining ranges of variations or probability distribution functions) for the Swiss energy system in the year 2035 and evaluate its impact on energy planning decisions.
Student: Cyprien Say (SGC), June 01, 2016
Hedonic pricing of car attributes: a comparison across European countries
Hedonic pricing is a revealed preference approach that is used for valuation of constituent characteristics of a good or a service. This project aims at pricing car attributes using disaggregate data on car purchases in 5 European countries from 2010 to 2014. Attention will be paid on estimation of market price gradients and willingness-to-pay for engine types, fuel consumption, weight to power ratio, etc., while accounting for heterogenous preferences of consumers and controlling for market segments, changes in quality, and brand effects. The candidate will take a strong interest in nonlinear regression methods, issues in econometric modeling, analysis and preparation of large datasets, and programming.
Student: Anna-Katharina Clodong (GC), January 30, 2016
Supervision: Matthieu de Lapparent, Anna Fernandez Antolin, Michel Bierlaire
Specification testing of fundamental diagrams for an anisotropic pedestrian network loading model
Student: Joel Mateus Fonseca (SC), January 30, 2016
Supervision: Flurin Hänseler, Marija Nikolic, Michel Bierlaire
Online estimation of pedestrian origin-destination demand in train stations using Kalman Filtering
Student: Marc Solsona Bernet (GC), January 30, 2016
Supervision: Flurin Hänseler, Stefan Binder, Michel Bierlaire
Accelerating moving walkways as a transport mode of the future: system optimization and management
In a hypothetical future where the use of private cars will be limited in cities, the need for movement will be satisfied by a mix of transport modes such as public transport, cycling, walking and other innovative systems. One of these possible futuristic systems, focus of this project, is an urban network of Accelerating Moving Walkways (AMW), i.e. a moving conveyor system for pedestrian similar to the one used in airports capable to reach speeds up to 15km/h. This project aims to optimize the network design of this innovative transport system of AMWs on the city of Geneva. Given the origin-destination demand, pedestrians are assigned to the city road network in order to obtain route choice and trip distributions. This information is used to identify the optimal configuration of link equipped with AMWs and their capacity. For this, a specifically developed optimization framework is used. Empirical mobility data and the road network (both already available) should be used as input of the optimization framework. Methodologies from mathematical optimisation, traffic assignment, programming (MATLAB) and Geographic Information Systems (GIS) will be used for completing this project.
Student: Alexandre Petit, January 15, 2016
Supervision: Riccardo Scarinci, Michel Bierlaire
Development of a novel pedestrian walking model applicable to congested flows
We are in the process of developing a novel pedestrian walking model that can describe multi-directional and congested pedestrian flow. Given a certain demand, the model is supposed to predict travel times and density levels as accurately as possible. The basic idea thereby consists in discretizing walkable space into cells and links, and to compute for each link a speed based on an empirical density-speed relation. The goal of this project, which can be carried out as a Bachelor's, semester, or Master's thesis, is (i) to understand and improve the existing mathematical model, (ii) to accordingly update the computational model (written in Java), and (iii) to consider a real-world case study involving either a Swiss railway station, a Dutch bottleneck experiment or a pedestrian crossing in Hong Kong, China. In the long term, this model will be useful for real-time crowd monitoring and control, as well as for infrastructure dimensioning of e.g. a train station. Skills in object-oriented programming and a basic knowledge of statistical mathematics are required; knowledge of parallel computing is a plus. Semester: Spring 2015. If interested, please contact us at {flurin.haenseler,marija.nikolic}@epfl.ch. We're looking forward to hearing from you.
Student: Gael LEDERREY, June 19, 2015
Supervision: Flurin Hänseler, Marija Nikolic, Michel Bierlaire
In collaboration with SBB-CFF-FFS
Pedestrian movement in train stations: modeling speed-density relationship for different classes of passengers
The increased number of passengers in train stations is causing congestion not only on trains, but also on platforms and underpasses. To better design and manage these infrastructures, pedestrian movement should be understood and modeled accurately. Fundamental Diagram (FD) plays an important role in the representation of pedestrian dynamics. FD models the relationship between density of pedestrians and the speed at which they are able to move. This macroscopic model represents the average behavior and does not take into account differences among pedestrians. They can differ in terms of their purpose of the trip (e.g. business/leisure), personal characteristics (e.g. age, gender), presence of luggage, etc. This heterogeneity can lead to different walking behavior of pedestrians and should be adequately modeled by the FD. This project aims to develop a Multi-Class Fundamental Diagram (MC-FD) assuming the existence of different classes of passengers in a train station. As a case study we will use the Lausanne train station where detailed pedestrian trajectories have been collected. The key elements of the project are: (i) literature review of MC-FD for both pedestrians and vehicles; (ii) development of the modeling assumptions employing empirical, qualitative and sociological considerations; (iii) model specification; (iv) sensitivity analysis, calibration and validation of the proposed model based on empirical data. Knowledge of statistical mathematics and familiarity with programming (MATLAB) are required.
Student: Laure Emma Rosine, June 19, 2015
Supervision: Marija Nikolic, Riccardo Scarinci, Michel Bierlaire
Development of an aggregate route choice model for a big network
The use of random utility models for route choice analysis involves challenges stemming from the high requirements in data and data processing, the physical overlap of paths, and the large size of the choice set. These factors increase the complexity of the models significantly. In order to simplify the problem, a novel approach based on an aggregate representation of route choices has been proposed. The conventional representation and modeling approach is based on path alternatives constructed as link-by-link sequences on the network. This approach entails a very large number of possible paths connecting a given origin and destination (OD), and high correlation among the alternative paths. In this work, we claim that a path is solely the manifestation of the route choice, i.e. the way the traveler implements her decision to take a specific route, and we replace the paths with aggregate elements that we denote as Mental Representation Items (MRIs). This key feature allows us to reduce the complexity of the model and at the same time is more behaviorally realistic. The aim of this project is to extend this approach and apply it in a big network for which information about congestion is available. A GPS dataset from the city of Quebec is available and will be used for this purpose. Knowledge of statistical mathematics and familiarity with MATLAB and SQL are required. Geographic Information Systems (GIS) software will be used for completing this project.
Student: Arriagada Diego Alexandre, June 19, 2015
Supervision: Evanthia Kazagli, Matthieu de Lapparent, Michel Bierlaire
In collaboration with McGill University
Routing of a mixed fleet of electric and diesel trucks: Analysis of solution approaches
A transportation company wants to study the potential benefit of replacing some of the diesel trucks in its fleet with electric ones. Electric and diesel trucks have comparable characteristics in terms of capacity and electric trucks are capable of performing an average daily itinerary on a single charge. The price of an electric truck is higher, while its operating cost is much lower compared to a diesel truck. Therefore, we want to find an intelligent deployment of the mixed fleet so as to minimize cost. The goal is the development of a vehicle routing model and/or heuristic, taking into account the characteristics of the various trucks, including fuel/energy consumption, speed, capacity, compatibilities, working time, service times, etc. The integration of the truck load's impact on fuel/energy consumption and the analysis of battery replacement/recharging strategies could also be considered if time permits. Solution quality can be compared against benchmarks results and data from the state of practice. As a previous semester project, a student has already worked on defining the problem, developing a heuristic algorithm and obtaining some results. The current project is intended to do a deeper analysis and implementation of solution methodologies. This project is appropriate for a student with an interest in operations research and experience in programming (Java preferred). The workload can be adjusted to the number of credits.
Student: Noortje Verstegen (IN-S), May 29, 2015
Supervision: Iliya Markov, Stefan Binder, Michel Bierlaire
In collaboration with HES-Fribourg, CREM
Implementation of a futuristic transport system based on accelerated moving walkways: optimization on a real case study
In a hypothetical future where the use of private cars will be limited in cities, the need for movement will be satisfied by a mix of transport modes such as public transport, cycling, walking and other innovative systems. One of these possible futuristic systems, focus of this project, is an urban network of Accelerated Moving Walkways (AMWs), i.e. a moving conveyor system for pedestrian similar to the one used in airports. This project aims to apply an optimization framework to identify the optimal network design of this innovative transport system of AMWs on a real case study. Given the origin-destination demand, pedestrians are assigned to a road network in order to have route choice and trip distributions. On this scenario, the optimal configuration of link equipped with AMWs and their capacity is define. For this purpose, a specifically developed optimization framework is used, and it should be applied to a real city. Lausanne presents appropriate transportation and geographical characteristic to be used as a case study. Empirical mobility data and the city network should be used as input of the optimization framework. Methodologies from mathematical optimization, traffic assignment, programming (MATLAB) and Geographic Information Systems (GIS) will be used for completing this project.
Student: Raphaël Luthi (SIE-S), May 29, 2015
Supervision: Riccardo Scarinci, Iliya Markov, Michel Bierlaire
Planning tool for the admission of medical graduates into the Otolaryngology training program in the CHUV
Student: Guillaume Lopez, January 15, 2015
A destination choice model for EPFL Campus
Based on WiFi traces from access points, the goal of this semester project consists in developing binary discrete choice models for the choice of attending classes.
Student: Loïc Tinguely (SGC), December 19, 2014
Supervision: Antonin Danalet, Matthieu de Lapparent
3D routing approach for air navigation planning for small sized planes
This projects aims to study and visualize the aviation data using factors such as, geographical information of the plane as well as weather condition for a flight most specifically during cruise phase. MATLAB is a powerful software with high quality of visualization of data, however, when the size of data gets bigger, the performance of MATLAB degrades and it becomes slower. Recently, some new software such as, Tableau has been introduced that provide powerful tools to visualize the big data.
Student: Jamal El Rhazi (SGC), December 19, 2014
Routing of a mixed fleet of electric and internal combustion trucks
A transportation company has introduced a number of electric trucks into their current fleet of internal combustion trucks. Electric and internal combustion trucks have similar characteristics in terms of capacity and electric trucks are capable of performing an average daily itinerary on a single charge. The goal is the development of a vehicle routing model and heuristic, taking into account the characteristics of the various trucks, including fuel/energy consumption, speed, capacity, compatibilities, working time, service times, etc. Various objectives could be considered and analyzed, such as the minimization of time, energy consumption and cost. The integration of the truck load's impact on fuel/energy consumption and the analysis of battery replacement/recharging strategies could also be considered if time permits. Data is available for benchmarking with the state of practice and assessing the heuristic's performance against the mathematical model. This project is appropriate for a student with an interest in operations research (mixed integer linear programming), some experience with mathematical programming languages (e.g. AMPL or CPLEX OPL) and good programming skills (e.g. C++, Java or Matlab).
Student: Thomas Cibils (MTE), December 19, 2014
Supervision: Iliya Markov, Stefan Binder, Michel Bierlaire
In collaboration with HES-Fribourg, CREM
Accounting for attitudes in modeling demand for electric vehicles
The purpose is to characterize and to model determinants of electric vehicle acceptance and adoption by individuals. Special attention will be paid to additional integration of attitudinal drivers and barriers to adherence to such technology. It aims at understanding how consumers accept the financial and lifestyle investments associated with the leap from traditional to electric powertrains, focusing on battery electric (BEV) and plug-in hybrid electric (PHEV) vehicles. To this extent, the 2012 Renault-Nissan Alliance survey on electric vehicles will be used. Data are collected over 5 European countries: France, Italy, Germany, Spain, UK. It focuses on current owners of a car bought new between 6 months and 5 years ago and who intend to buy a new car in the next 5 years. State-of-the-art and extended discrete choice models will be developed to account for formation of latent processes and existence of endogeneity issues.
Student: Maurin Baillif (GC), December 19, 2014
Supervision: Matthieu de Lapparent, Anna Fernandez Antolin, Michel Bierlaire
In collaboration with Nissan
Development and Implementation of a Decision Making Tool in Quality Control Networks using Quantitative System Modeling Techniques
In an increasingly complex business environment, managers have to grapple with problems and issues, which range from relatively trivial to the strategic. This project proposes to develop and implement a decision making tool to optimize quality control networks. To do so, several metaheuristics will be used as an efficient approach to improve global value chains towards lean manufacturing applied to a production process of Nestle. The control mechanism of each control point determines the conformity (or non-conformity) of the product in the supply chain defining a sample randomly chosen units and indicating the maximum number of bad units for each batch to pass control points. As control points become decision variables, the detailed examination of control mechanisms at each control point requires the use of nonlinear algorithms. The following decision variables have to be optimized in the considered networks: (1) determine the set of control points; (2) determine the control plan parameters of each control point. That point requires understanding basics of statistical quality control. The objective function contains a performance and a risk component, and a budget constraint has to be satisfied. The following metaheuristics will at least be investigated: tabu search, variable neighborhood search and adaptive memory algorithms. In addition, an exact method relying on CPLEX should be compared to the proposed metaheuristics.
Student: Pierre Jullien de Pommerol (SGC), July 18, 2014
Supervision: Michel Bierlaire, Prof. Nicolas Zufferey, EPFL STI LGPP
A two-step approach for estimating pedestrian demand in a congested network
Student: Eduard Rojas (IC), May 30, 2014
Supervision: Flurin Hänseler, Antonin Danalet, Michel Bierlaire
In collaboration with SBB-CFF-FFS
Optimisation of the network design of a futuristic transport system based on moving walkways
In a hypothetical future where the use of private cars will be limited in cities, the need of movement will be satisfied by a mix of transport modes like public transport, cycling, walking and other innovative systems. One of these possible futuristic systems, focus of this project, is an urban network of moving walkways, i.e. a moving conveyor system for pedestrian like the one used in airports. This project aims to study and optimize the network design of this innovative transport system of moving walkways, exploring various network configuration and system characteristics to understand which specification could satisfy at best the increased demand. Given the origin destination demand and trip distribution, the variables of the optimization problem should be defined such that indexes of performance, e.g. overall traveling time and cost, are minimized. Example of design variables for this innovative transport system are: number of links to be equipped with moving walkways, moving pathway speed and system capacity that can be related to energy consumption and construction cost. Methodologies from mathematical optimization, transport network design, simulation and scripting will be used for complete this project.
Student: Guillaume Lopez, May 30, 2014
Supervision: Riccardo Scarinci, Jianghang Chen, Michel Bierlaire
Train Management in SNCF Application
Student: Cao Huu-Ân (CS), May 30, 2014
Supervision: Tomás Robenek, Stefan Binder, Michel Bierlaire
Schedule-based estimation of pedestrian travel demand within a quasi-uncongested railway station
In many railway stations, capacity limits are reached at peak hours and congestion in pedestrian facilities occurs. There is thus a general need to analyze and model pedestrian flows in train stations. In this process, estimating pedestrian origin-destination demand and, more precisely, route flows is a major challenge. The ultimate goal of this research project is to dynamically predict pedestrian travel demand within a train station based on train time table and train track assignment. To this end, a preliminary methodology has been developed. This framework needs to be refined and extended in many ways, which can be an interesting and suitable task for a semester or Master's thesis. Your thesis would be part of a comprehensive project employing pedestrian tracking, flow modeling and infrastructure optimization. All ideas and model concepts will be tested, parametrized and validated on a real case study.
Student: Quentin Mazars-Simon (SSC), January 31, 2014
Supervision: Flurin Hänseler, Amanda Stathopoulos
In collaboration with SBB-CFF-FFS
Demand/supply coupling in pedestrian traffic estimation
The main purpose of this paper is to show a mathematical framework for solving the fixed-point arising in joint demand estimation / traffic assignment problems. As a case study. a congested corridor with known outflow is considered, for which the inflows is predicted using the developed framework. The model is based on a coupling between the travel demand and the network supply (the infrastructure available) in an analogous way as what is done in economics.
Student: Jérémy Rabasco (CS), January 31, 2014
Supervision: Flurin Hänseler, Michel Bierlaire
Optimisation des tournées de ramassage des employés de l'aéroport de Genève (Pre-Project)
Student: Isabel Tovar (SGC), January 20, 2014
Supervision: Tomás Robenek, Stefan Binder, Michel Bierlaire, Philippe Quaglia
In collaboration with Geneva Airport
Implementation of Tabu Search in Quality Control Networks
This project proposes to optimize quality control networks using the tabu search algorithm as a novel solution to improve global supply chains towards lean manufacturing applied to the Nestle chocolate supply chain. The control mechanism of each control point determines the conformity (or non-conformity)) of the product in the supply chain defining a sample of randomly chosen units and indicating the maximum number of bad units for each batch to pass a control point. As control points become decision variables, the detailed examination of control mechanisms at each control point require the use of non-linear algorithms. In a first phase, control points as decision variables necessitate the examination and identification of (sub)optimal control mechanisms. This is followed by the proposition of a first model using the tabu search algorithm. This model will demonstrate links among suboptimal solutions, but also synergies among risk, performance and budget distribution in quality control networks. In order to efficiently allocate quality control network resources, this project proposes to globally optimize the control mechanism for equi-distribution of risk, performance and budget using the tabu search algorithm.
Student: Pierre Jullien de Pomerol (SGC), January 20, 2014
Supervision: Michel Bierlaire, Prof. Nicolas Zufferey, EPFL STI LGPP
Vehicle dispatching problem in physical internet hub and spoke model
The call for sustainability in logistics section promotes the idea of Physical Internet (PI). The approach of PI aims to universally interconnect logistics networks as the digital internet did with computer networks. In the world of physical internet, consumer goods are encapsulated in smart and secured containers (i.e., PI containers) and are routed in logistics networks with the similar pattern that digital packets are handled in the internet.
Student: Alexis Dubil (SGC), January 17, 2014
Supervision: Jianghang Chen
Identify User’s Locations of Interest from Smartphone WiFi Data
Smartphone is a powerful and convenient tool for collecting a variety of data (location, social interaction, and more) that can be useful for individual mobility analysis. Most of the ongoing research relies on GPS to acquire accurate location. GPS though is expensive in terms of battery consumption. In addition, the GPS sensor embedded in mobile phones fail in practice more often than dedicated GPS devices. Yet, smartphones are endowed with multiple sensors from which location can be inferred. Wifi sensors provide stable indoor location data and have been identified as a potential alter- native to indicate location that can well serve in addition and complementary to GPS data to advance mobility learning. The location extraction from Wi-fi records permits the detection of users’ regular places of visit, the frequency and intensity (number of times) of visiting these places, as well as patterns such as the time of day and duration of visit. This kind of information is relevant for the identification of activity locations making use of clustering techniques, and after further analysis they can support measuring the activity space and intensity of activity participation of the user. By fusing the information about activity locations with land use and points of interest (POI) data, as well as with more data acquired from the smartphone sensors (e.g. phone status, Bluetooth, charging, phone interaction etc.), and taking into account temporal dimensions, it is then possible to infer the type of activity at these location (including home and work locations) and subsequently trip/ activity purposes can be disclosed. In addition to the above, and building on the identification of meaningful clusters, WiFi data have the potential to assist the identification of origins and destinations of trips (often missing in case of GPS). This information can then be used for improving and validating trip detection algorithms, which currently appear to be problematic and hence introduce biases in the subsequent stage of map-matching (e.g. one a known trip is segmented in several due to the limitations of the algorithm and GPS data). This project aims at exploiting the advantages of the WiFi data discussed above for individual mobility analysis.
Student: Amélie Buisson (GC) (SGC), January 17, 2014
Supervision: Evanthia Kazagli, Antonin Danalet, Michel Bierlaire, Francisco Pereira
Vehicle routing problem coupling with bin packing
The call for sustainability in logistics section promotes the idea of Physical Internet (PI). The approach of PI aims to universally interconnect logistics networks as the digital internet did with computer networks. In the world of physical internet, consumer goods are encapsulated in smart and secured containers (i.e., PI boxes) and are routed in logistics networks with the similar pattern that digital packets are handled in the internet. In this student project, we focus on the last mile problem in the PI. After PI boxes arrive their final designated PI hub, given clients’ addresses and the information of the trucks (e.g., truck weight and volume capacities) available at this hub, the decision makers aim to assign the minimal number of trucks to deliver all the arrived PI containers to their destinations in a cost-effective way. This is a typical Vehicle Routing Problem (VRP) but with the side constraint that we need to take the dimensions of the PI boxes (in 3D) into account when loading and/or unloading occur. This problem is termed vehicle routing problem coupling with bin packing. Its essential constraints that the student need to consider are: 1. The available number of vehicles is bounded; 2. Vehicle weight and volume capacities cannot be exceeded; 3. The loading and unloading need to follow the policy of “Last-in-first-out”.
Student: Buytaert Gabrielle (SSC), January 17, 2014
Supervision: Jianghang Chen
Exploring pedestrian mobility using video tracking data in Lausanne train station
Pedestrians, contrary to the other modes, do not have defined network and they do not follow strict constraints. A characteristic feature of pedestrian route choice is that routes are continuous trajectories in time and space - pedestrians choose a route from an infinite set of alternatives. As a consequence of these facts we have observed the distribution of walked distances and travel times for specific origin-destination pairs at the train station in Laussane, during the morning rush hour. The goal of this project is to analyse the distribution of walked distances and corresponding travel times and speed values based on pedestrian trajectories collected at the train station in Lausanne. Here we would like to answer the questions such as: 1) Is the motion of pedestrians random or does it follow a specific pattern? How can we characterize it? 2) What is the impact of congestion on the mentioned observables? 3) Do familiarity with the place, the attractiveness of shops and signs found along the corridors affect pedestrian motion behaviour and how? What are the other factors that cause deviations from a straight line motion? Programming skills and knowledge of statistical mathematics are required.
Student: Babel Hugo Louis, January 17, 2014
Supervision: Marija Nikolic, Evanthia Kazagli
Mode choice analysis from a large smartphone dataset
Smartphone is a powerful and convenient tool for collecting a variety of data (location, social interaction, and more) that can be useful for individual mobility analysis. One of the advantages of smartphone data is the ability to collect data over longer periods of time (panel data) without burdening the respondent. Having such resolution of information enables us to gain better insight into the general mobility style of people than it would be possible by means of traditional one- or two-day travel surveys. It becomes clear that smartphone data have great potential with respect to analysing travel- ers’ profiles and disclose systematic mobility (e.g habit, morning or afternoon routines etc.) and switching behavior patterns (departure time, chosen route, mode). This project focuses on such higher-order mobility styles in order to shed light on travel behavior.
Student: Mikael Nicolas Xavier Friederich (SGC), January 17, 2014
Supervision: Evanthia Kazagli, Marija Nikolic
Exploration of smartphone users trip data to investigate travel behavior
Smartphone is a powerful and convenient tool for collecting a variety of data (location, social interaction, and more) that can be useful for individual mobility analysis. One of the advantages of smartphone data is the ability to collect data over longer periods of time (panel data) without burdening the respondent. Having such resolution of information enables us to gain better insight into the general mobility style of people than it would be possible by means of traditional one- or two-day travel surveys. This project aims at unveiling frequent behaviours in terms of space, i.e., the regions of space traversed during movements and understand the processes generating them. The work elaborated in the project is expected to support our effort in incorporating travelers’ mental/ spatial represen- tation items (MRI) in route choice modeling. More specifically, the goal of this bigger project is to develop a modeling framework where the route choice decisions will take place in a higher/ concep- tual level. Path alternatives will be constructed as –replaced by– sequences of mental geo-marked items –akin to the concept of anchor points as elements of travelers’ mental maps in cognitive science. We are interested in car route choice, hence the scope of this semester project will also focus on car trips. In this context, processing and analysing the trips of the smartphone users can reveal the most frequently traversed points or segments of the network (both in an aggregate but also in an individual level) that can signify such items in the network.
Student: Mikael Nicolas Xavier Friederich (SGC), January 17, 2014
Supervision: Evanthia Kazagli, Marija Nikolic
Modeling the demand for financial products
This MA pre-project focuses on developing a demand model for financial products. The work is carried out in collaboration with a trading service partner an based on real data on commodity portfolio investments. The project will cover the following: i) a literature review on the demand for financial products, behavioural finance on trading choices, ii) development of a demand model on the client data that is able to qualify the impact of different determinant (price, spread, client features, external events) on decisions to buy/sell the commodity, iii) testing the model on different scenarios (sensitivity to different changes in conditions) and validation. The work comprises: conceptual modeling, data elaboration/exploration and empirical model development, hence this pre-project and the related MA project requires skills in mathematics, management of large data, statistics/econometrics related to discrete choice models and some knowledge of behavioural theories in finance.
Student: Billal Mahoubi (SGC), January 10, 2014
Supervision: Amanda Stathopoulos
Activities in Paléo Music Festival from Bluetooth
Can we guess the activities of spectators in a music festival from Bluetooth traces collected from 10 people with smartphones used as antennas? This project will study an existing dataset and existing map data from Paléo music festival in Nyon, Switzerland. Depending on the results, the report will either draw conclusions about spectators' behavior, or recommend other data collection specifications for this particular case study. For more information, don't hesitate to contact the assistant of the project.
Student: Elisaveta Kondratieva (SSC), June 07, 2013
Supervision: Antonin Danalet
Visualization of pedestrian demand in a 3D graph
This project aims at visualizing flows in a pedestrian graph in 3D. The graph is coded as a postGIS database (postgresql). The goal consists in representing the density at each node and eventually the flows between the nodes (data coming from our own algorithm). The visualization needs to be both in 3D and in 2D for printing. An IC student would ideally fit the goal of this project. The first step would be to compare the pros and cons of Processing and Google Map/Earth. Then it would consists in visualizing the pedestrian graph in the chosen tool. As a final step, the student will include density data of signals (without treatment) and the results of our algorithm in the map (treated data) and generate final maps and animations from data. As an inspiration of how it could possibly look like this (but with pedestrian as dots instead of trains)
Student: Javier Lopez-Montenegro Ramil (IN), June 07, 2013
Supervision: Antonin Danalet, Bilal Farooq
Modelling car ownership duration
Household vehicle ownership influences many aspects of travel demand, with pronounced impact for energy consumption, travel mode distribution, residential location patterns and city attractiveness. Despite a large number of vehicle choice models focusing on the type of car chosen, formal modelling of the timing and duration of ownership has received relatively little attention. Statistical models known as duration (hazard) models can be used to estimate the distribution of vehicle ownership lengths in a population. A central interest is to identify the determinants of the time that elapses between two automobile transactions, including characteristics of the car and household and macroeconomic/policy variables. In addition, the influence of less tangible factors such as expectations regarding the market, future vehicle launches, propensity for planning and personal aspirations may have a large impact on the timing of renewal and vehicle ownership duration. We have a large dataset available containing rich information on vehicle acquisitions and are looking for a student to collaborate on data elaboration, exploration and development of a duration model for vehicle ownership. Several assumptions for the hazard function and innovative determinants of ownership spells will be considered. The student should be familiar with management of large data, statistics/econometrics and have some notion of behavioral theories of decision-making.
Student: Natalie Sauerwald (SMA), June 07, 2013
Supervision: Amanda Stathopoulos, Aurélie Glerum
In collaboration with PSA
Optimisation des transports scolaires à Grandson II
Analyser les problématiques, actuelles et futures, et proposer des solutions d'amélioration découlant de l'organisation et de la gestion de l'ensemble des transports nécessaires à la bonne marche du groupement et de l'arrondissement scolaire de Grandson (18 communes, 1'350 élèves, 70 classes, 12 sites scolaires). Ces problématiques résultent principalement d'une augmentation significative des coûts liés à l'introduction de la communauté tarifaire vaudoise (Mobilis), de contraintes liées à l'entrée en vigueur d'HarmoS, des projets de constructions de l'association intercommunal, de l'évolution du réseau régional des transports voire également des synergies à développer avec le milieu parascolaire.
Student: Franka Tholen (SMA), June 07, 2013
Supervision: Michel Bierlaire, Tomás Robenek, Bilge Atasoy, Yves Guilloud, Association Intercommunale du Groupement et de l'Arrondissement Scolaires de Grandson
Associations generation in synthetic population
In recent years significant advancements have been made in the research related to agent based urban systems modelling, especially in the area of activity based large scale travel demand models. Synthesis of different types of agents (person, family, and household) is an integral part of the input preparation for such models and micosimulations. In addition to agents generation, these recent developments also require synthesizing associations between di fferent types of agents. For instance, which individual person is married to whom, who is the father of a 7 years old male going to a particular school, etc. Such information is very useful in realistic modelling and microsimulation of short term decisions like who gets to take the car for work and long term decisions for instance, what type of dwelling a particular household will buy. A Markov chain Monte Carlo simulation based agents synthesis approach has been proposed by Farooq et al. (2012). At core, the approach used Gibbs sampler to draw from the joint distribution of agent attributes using the available data. Among other advantages, they also demonstrated that the approach can reproduce the joint distribution better than the conventional fitting based approaches. Here we extend the simulation based synthesis approach so that it can also generate the association among drawn agents of di fferent types. Required skill set: Experience in data analysis and pre-processing; probability and statistic; coding in Java or C#
Student: Paul Anderson, June 07, 2013
Supervision: Bilal Farooq, Dimitrios Efthymiou
Optimization of Paris' Metro System
The Metrolab (a joint venture between RATP (Régie Autonome des Transports Parisiens) and Alstom) would like to conduct analysis of their current metro system in terms of metro schedules and pedestrian congestion in the stations. The aim of this project is to use optimization techniques to design more flexible dynamic timetables and partially to handle the pedestrian congestion in the stations. This project will be offered only for spring semester 2013.
Student: Thomas Cibils (SMA), June 07, 2013
Supervision: Tomás Robenek, Jianghang Chen, Michel Bierlaire
In collaboration with MetroLab
Enhancement of Naville's Press Delivery Regulation Tool: An Exploratory Analysis
The aim of this project is to explore data driven complementary strategies to the existing allocation strategy of Naville, in order to provide guidance for potential improvements of the existing allocation tool of the company. In this project the student will investigate the potential benefits of including a wider set of data in order to identify press demand patterns over time. Accounting for consumer profiles, detailed sale points data, geographical area, external events, etc., will enable a more accurate understanding and anticipation of demand fluctuations. Drawing from the results of this analysis we will provide recommendations for improvements of the existing tool used by Naville. The proposed exploratory project can form the basis for a further work on the challenging theme of modeling and predicting demand and optimizing distribution strategies in the context of declining and uncertain evolution of the press market.
Student: Lea Kissling (SMA), June 07, 2013
Supervision: Evanthia Kazagli, Amanda Stathopoulos
In collaboration with Naville
Mobility learning from smartphone WIFI data
Smartphone is a powerful and convenient platform for collecting location data for mobility behavior research. Most of the research relies on GPS to get accurate location, or GSM to get massively available but imprecise location. This project investigates the feasibility of using WIFI data for mobility behavior research. The dataset is collected from 200 smartphones over 2 years. The data collection application (EPFLScope) records all nearby WIFI access points (AP) every 3-5 minutes. Some of the WIFI access points are associated with location information. Therefore, it is possible to identify the location of the smartphone according to the nearby WIFI AP. With this location information, we can further understand spatial and temporal mobility patterns.
Student: Amélie Buisson (SGC), June 07, 2013
Supervision: Jingmin Chen, Evanthia Kazagli, Marija Nikolic
Movement patterns of pedestrians on platforms
To understand how people access a train station by means of trains, the arrival and departure pattern of pedestrians on platforms is of interest. In particular, the embarkation/disembarkation behavior of train passengers is an important factor for congestion in pedestrian facilities. The departure pattern of pedestrians awaiting to board a train has been shown to follow a beta distribution. In contrast to that, arriving passengers usually enter a train station as a very dense crowd, causing a theoretical inflow that exceeds the capacity of pedestrian facilities. The goal of this work is to mathematically describe the arrival and departure patterns caused by single trains as well as for platforms on which multiple trains follow each other in close succession. Very detailed pedestrian tracking data of platform 3/4 in Gare de Lausanne will form the basis of this analysis. Basic programming skills or a strong willingness to acquire such knowledge is desired.
Student: Nicholas Alan Molyneaux, May 31, 2013
Supervision: Flurin Hänseler, Amanda Stathopoulos
In collaboration with SBB-CFF-FFS
Development of a novel pedestrian flow simulator
Due to a general increase in travel demand, pedestrian flows in railway stations are gaining importance. Space is getting scarce, pedestrian density is reaching critical levels, and walking times are getting longer. To better understand these phenomena, we are currently developing a cell-based pedestrian flow simulator. Key features of this framework include its ability to i) realistically reproduce pedestrian density waves caused by arriving trains, ii) adequately describe multi-directional flows often present in public spaces, iii) consider differences in walking speeds among characteristic groups of pedestrians (such as passengers with luggage, handicapped people, travelers in a hurry, etc). To extend and implement this framework, basic programming skills or a strong willingness to acquire such knowledge is needed. Familiarity with fundamental fluid dynamics would be a plus. Extensive support and insight into ongoing research are provided.
Student: Thomas Mühlematter, May 31, 2013
Supervision: Flurin Hänseler, Bilal Farooq
Dynamic estimation of pedestrian origin-destination within train stations: Exploitation of pedestrian tracking data and comparison to travel surveys
OD demand will be estimated dynamically (i.e., as a function of time) based on observed pedestrian trajectories. Key in this process is a diligent choice of origins/destinations and their cordons, as well as a smart handling of ‘lost’ pedestrians, i.e., people of which the algorithm looses track due to overcrowding or bad light conditions. Another challenge is finding a way to deal with multi-destination trips. For instance, a person entering a train station which goes to a ticket vending machine before boarding a train falls into this category. To help us tackle these issues, VisioSafe grants us access to some of their codes such that we can see how they have solved similar problems. Besides developing an OD estimation algorithm based on tracking data, visualization of OD demand is an important part of this thesis. Furthermore, it might be interesting to investigate how the estimated OD demand compares to previous estimates based on travel surveys (Anken et al., 2012) and pedestrian count data (ASE).
Student: Maëlle Zimmermann (MA), January 12, 2013
Supervision: Flurin Hänseler, Antonin Danalet
Movement patterns of pedestrians on platforms prior to/after train departures/arrivals in Gare de Lausanne: Exploitation of pedestrian tracking data
In order to better understand how people access and leave a train station by means of trains, the arrival and departure pattern of pedestrians on platform is of interest. In particular, the embarkation/disembarkation behavior of train passengers is an important factor for congestion in pedestrian facilities. The departure pattern of pedestrians awaiting to board a train is expected to follow a beta distribution. In contrast to that, arriving passengers usually enter a train station as a very dense crowd, causing a theoretical inflow that exceeds the capacity of pedestrian facilities. This saturation effect needs to be taken into account when modeling arrival patterns. The goal of this work is to mathematically describe the arrival and departure patterns caused by single trains as well as for platforms on which multiple trains follow each other in close succession. Case studies for RER, RE and ICE/EC trains will be considered in order to better understand the influence of train size and type on these patterns.
Student: Isabel Tovar, January 12, 2013
Supervision: Flurin Hänseler, Marija Nikolic
Optimisation des transports scolaires à Grandson I
Analyser les problématiques, actuelles et futures, et proposer des solutions d'amélioration découlant de l'organisation et de la gestion de l'ensemble des transports nécessaires à la bonne marche du groupement et de l'arrondissement scolaire de Grandson (18 communes, 1'350 élèves, 70 classes, 12 sites scolaires). Ces problématiques résultent principalement d'une augmentation significative des coûts liés à l'introduction de la communauté tarifaire vaudoise (Mobilis), de contraintes liées à l'entrée en vigueur d'HarmoS, des projets de constructions de l'association intercommunal, de l'évolution du réseau régional des transports voire également des synergies à développer avec le milieu parascolaire.
Student: Clement Massart (SMA), January 11, 2013
Supervision: Tomás Robenek, Bilge Atasoy, Michel Bierlaire, Yves Guilloud, Association Intercommunale du Groupement et de l'Arrondissement Scolaires de Grandson
Robustness and recovery in airline scheduling
Student: Jonathan Blaiberg (SGC), January 11, 2013
Supervision: Michel Bierlaire, Bilge Atasoy
Job Shop Scheduling in a Medical Parts Production Factory
Student: Nathan Scheinmann (SMA), January 11, 2013
Supervision: Michel Bierlaire, Nitish Umang
Process Optimization and School Schedules: New Legistlation, New Constraints
Student: Ailin Zhang (SMA), January 11, 2013
Supervision: Michel Bierlaire, Nitish Umang
Population synthesis for large-scale agent based microsimulation
Dynamic microsimulation of cities, including transportation, land use, and energy, require an initial dissaggregate population of agents (households and persons) as a key input. Due to privacy reasons, most of the times, governments do not provide access to full information on the population in census. This requires synthesizing the population from available datasets. In principle two datasets are necessary for such synthesis: a) disaggregate sample of households/persons; b) demographic summaries for all the zones in the study area. Techniques like Iterative Proportional Fitting (IPF) method are developed to use these datasets and generate a synthetic population of households and persons. A new technique for population synthesis has been developed at Transportation and Mobility lab. The scope of this project is to a) prepare datasets from the census and various other micro data samples b) run the simulations code based of the developed technique, for various scenarios (code is written in c++ and will be provided to the student. Only a very basic programming knowledge is required from the student) c) analysis and comparison of the results with other methods. This project will require developing understanding of the general methodology of population synthesis, preprocessing the datasets, familiarity with MySQL, GIS software, and some programming experience is recommended.
Student: Lovisa Arnesson, January 10, 2013
Supervision: Bilal Farooq
In collaboration with European Union funded, SustainCity project
Pedestrian flow simulation in Lausanne train station
This project is about modelling, simulating, and analyzing the future expansion scenarios, in the context of pedestrian flows within Lausanne Gare (train station). We have already developed the modelling and simulation test-bed in a pedestrian flow simulator, called VisWalk. Here we would like to use more detailed data for existing and future demand in our models and analysis. A considerable amount of time will be spent to perform the calibration of various parameters using aggregate level speed and density related data. If time permits, we would also be interested in doing small infrastructure changes, specifically tapering of the intersections between PIs and ramps in order to evaluate their effects of the pedestrian flows. Ideal student should have some experience in using CAD software (AutoCAD, Microstation); should have some knowledge of databases; and be able to do basic programming. The student working on the project will have a great and unique opportunity to work directly with SBB CFF FFS, a major transportation software company called PTV AG, and a rapidly growing startup from EPFL called VisioSafe.
Student: Nicolas de Lamberterie, January 10, 2013
Supervision: Bilal Farooq, Marija Nikolic
In collaboration with SBB CFF FFS, PTV AG, VisoSafe
Modelling the choice of vehicle in an extended framework
It is becoming increasingly clear that models of vehicle choice need to find a way to measure complex consumer substitution patterns across products and over time, while accounting for a wealth of features that modulate the buying experience. This means that planning, acquisition and use of the vehicle is linked through features such as attitudes, life-event changes, status aspirations and perceptions of making a good deal. We have a large dataset available containing rich information on vehicle acquisitions and are looking for a student to collaborate on data elaboration, exploration and modelling that factors in richer behavioral variables than is common in traditional vehicle choice models. Methodologically the choice of vehicle will be modeled using random utility discrete choice demand systems. The student should be familiar with management of large data, statistics/econometrics and have some notion of behavioral theories of decision-making. The work will start with exploring the structure in the data, formulating simple models to identify some constructs and segments in the data then move towards a more formal choice model with perceptual/attitudinal components.
Student: Areg Gevorgyan (MTE), January 07, 2013
Supervision: Amanda Stathopoulos
Generation and Simulation of MATSim plans for Brussels
The objective of this project is to generate travel plans for a synthetic population for the city of Brussels and simulate their travel behavior with the transport microsimulation software MATSim. The plans will be generated based on available census and travel survey data. The network for Brussels will be provided. The project considers the implementation of a prototype (simplified) model and a sensitivity analysis of the simulation results
Student: Sona Hunanyan (MA), July 30, 2012
Supervision: Ricardo Hurtubia
Optimizing Security Staff Operations at Geneva Airport
Geneva airport is the second largest airport in Switzerland. Airport is capable of easily handling 80-100 landings and take-offs per hour. However one of the major bottlenecks today is the passenger handling capacity. The airport terminal area can handle up to 3000 passengers per hour. In recent times, passenger congestion is frequently building up at the airport. Because of the geographical reasons, it is not easy to expand the terminal building. As a result, it may be fruitful to attempt reducing congestion by efficient handling of the passengers and not letting the passengers wait too long to be serviced. Security screening is the most important area for departing passengers. Optimizing the operating costs for security personnel at airports is a complex problem due to a number of reasons. Each passenger and their corresponding hand baggage need to undergo security screening before entering the boarding gates. Even though the passengers and the baggage need to move together, the rate of service for the two can be vastly different. Security personnel work over specific shift times and need to be provided with suitable meal breaks. Staff members can work full-time or part-time, with a minimum of four hours per day. Passenger arrival pattern at an airport can be extremely uneven. Flight activities also tend to be non-uniform while passenger service expectations and staffing inflexibilities due to shift durations can make the problem even more complex. In this work, we would use the flight schedule and service criteria to develop a method to find optimal shift timings and the mix of full-time and part-time workers such that the overall costs are minimized and the service criteria conditions are fulfilled. Term Project (Master Project Pre-Study) Deliverables • Determine global parameters for passenger arrival rates, service times and shift times at different times of the day and days of the week for both passengers and baggages • Develop and implement a cost optimization MIP model with discrete time intervals Masters Project Deliverables • Adapt the deterministic linear model to account for non-linear passenger arrival and service rates and implement the same • Build a robust optimization model to capture vastly varying arrival and service rates. Determine the various ways to capture model stability with special emphasis on recoverability • Perform sensitivity analysis and compare the results of the stochastic model with simulation • Use the results to make recommendations: For example, create a separate queue or sub-queue for passengers without baggage, or passengers whose flights depart in 20 mins should be brought ahead of the queue, etc. • Develop and implement the model for staff rostering at the security screening Apart from expertise in modeling, the student is expected to be able to implement his model and write his own codes in a standard programming language, such as C or C++ Desired Masters Project Deliverables • Evaluate the tendency for staff to take leaves (including sick leaves) and vacation (subject to data availability) • Plan for contingencies in your model Project Organization The student would be expected to update us on the progress and revert to us for queries on a pre-decided day of the week. The student would be expected to update us on the progress and revert to us for queries on a pre-decided day of the week. A midterm presentation would be scheduled on Thursday, 10.11.2011. At the completion of the term project, a report will be delivered (both in paper as well as electronic form) before the 13.01.2012. The final oral presentation for the term project will be scheduled on 11.01.2012. Similarly, the Masters project would be due before the 01.06.2012 while the oral presentation for Masters project will be scheduled between 18.06.2012 and 29.06.2012 or a mutually convenient and agreed date. All the files, programs, codes, data and report associated with the project must be delivered before these date. All reports will be submitted in two copies to TRANSP-OR Lab and GVA Airport. References • Dowling, D., Krishnamoorthy, M., Mackenzie, H., and Sier, D. (1997). Staff rostering at a large international airport. Annals of Operations Research, 72:125-147. • Ernst, A., Jiang, H., Krishnamoorthy, M., and Sier, D. (2004). Staff scheduling and rostering: A review of applications, methods and models. European journal of operational research, 153(1):3-27. • Gilliam, R. (1979). An application of queueing theory to airport passenger security screening. Interfaces, 9(4):117-123.
Student: MAHMOUD KHAROUF, June 20, 2012
Supervision: Michel Bierlaire, Prem Kumar, Nitish Umang, VINCENT-RUBEN JIMENEZ, GVA AIRPORT
Positioning Clip-Air among other transportation systems as a multi-modal flexible aircraft
Clip-Air is an innovative air transportation system based on modularity. By design, loading units (capsules) can be detached from the carrying unit (wing) which has several advantages in terms of fleeting operations. To quantify these advantages an integrated schedule design and fleet assignment model is developed for both standard planes and Clip-Air. The comparative analysis has shown that there is a potential increase in the transportation capacity thanks to modularity of Clip-Air. To see the impacts of Clip-Air with a systematic perspective it is important to build analogies with existing transportation systems. The novel feature of Clip-Air is its flexible transportation capacity and it is similar to the case of railways where the train cars are assigned to the locomotives. Moreover, the multi-modal aspect of Clip-Air has strong similarities with maritime transportation where standard load units are transported in different transportation modes. Starting from these analogies we can analyze the similarities and differences between the existing transportation systems and new air transportation system Clip-Air. Similarities will strengthen the basis of this new idea and the differences will justify the need for new models and methodologies to analyze the added value of Clip-Air.
Student: Jonathan Blaiberg (SGC), June 08, 2012
Supervision: Bilge Atasoy
Tracking Pedestrians with WiFi Traces
Gathering data about pedestrian localization and tracking indoor is a hot topic today. WiFi traces to track pedestrian paths have been collected on EPFL Campus. The poor quality and the scarcity of WiFi localization precludes the use of traditional map matching methods. This project aims at adapting methods recently developed for Smartphone GPS data.
Student: Yusen Bian (MA), June 08, 2012
Supervision: Antonin Danalet
Transportation mode choice models including word data
Recently, there has been an emphasis in the discrete choice literature on the introduction of attitudes and perceptions into discrete choice models. We are interested to see how these factors impact on choice.
A joint work with social scientists has led to the development of new surveys which include questions on the perception of transportation modes. In the framework of a survey conducted with CarPostal, the following question was asked:
"For each of the following transportation modes, indicate 3 adjectives that characterize them the best:
The car is: 1)________ 2)________ 3)________
The train is: 1)________ 2)________ 3)________
etc."
After this survey, a second online survey was conducted to collect information on how individuals would situate the adjectives reported in the first survey (e.g. stressful, relaxing, full, etc.) on a scale of comfort. For example, respondents were ask to rate adjective 'relaxing' on a scale from -2 to 2, with -2 indicating a total discomfort and 2 indicating a total comfort. In addition to this, they had to report some socio-economic information (languages spoken at home or work, mother tongue, level of education, etc.).
This project has two goals:
1. Model of the effect of the socio-economic information of the respondents of the second survey on their ratings of the adjectives.
2. Integrate the model developed in 1. into a discrete choice model to explain the transportation mode preferences.
Student: Peng Cui (SMA), June 08, 2012
Supervision: Aurélie Glerum
Sensitivity analysis for a new generation of aircraft: Clip-Air
Clip-Air is an innovative air transportation system based on modularity. By design, loading units (capsules) can be detached from the carrying unit (wing) which has several advantages in terms of fleeting operations. To quantify these advantages an integrated schedule design and fleet assignment model is developed for both standard planes and Clip-Air. The comparative analysis has shown that there is a potential increase in the transportation capacity thanks to modularity of Clip-Air. Since Clip-Air is only exists in a simulation environment, the parameters regarding the design of the Clip-Air are based on estimation. A sensitivity analysis should be carried out to see the impact of the estimated parameters on the results. The objective of this project is to identify the parameters to which the model is more sensitive.
Student: Joseph Abisaleh (SGC), June 08, 2012
Supervision: Bilge Atasoy
Analyse des fréquentations de bus en Suisse
Ce projet de semestre porte sur une analyse des données de fréquentation de plusieurs lignes de bus à travers la Suisse. Il s'agit d'analyser le lien entre la fréquentation et différents autres facteurs tels que la fréquence de la ligne, des variables temporelles (jour de la semaine, heure de la journée), le type de ligne, le bassin (densité de population, emplois) ou encore le lien avec une gare CFF (présence, mais aussi éventuellement fréquence des trains dans cette gare, ou du moins type de gare). Ce projet permet de travailler sur des données d'une grande richesse, au sein d'un projet de recherche multidisciplinaire en cours en lien avec l'industrie et d'autres laboratoires de l'EPFL, en particulier la CEAT.
Student: Suzy Polka (SSC), February 17, 2012
Supervision: Antonin Danalet, Ythier Jeanne
Mode choice model for the city of Nice
The goal of this project is to implement a mode choice model for the city of Nice. This model will be used by a public transport operator in order to identify the potentials for increasing market shares for public transport in this area, based on a better understanding of the demand structure between public transport and car. The model will be estimated on data collected from a recent Household Travel Survey which sampled around 6500 households. Information on travel times and costs is available for trips performed by public transport but similar information has to be collected for private transport modes. The model will be implemented and estimated using BIOGEME, an estimation software developed by the Transport and Mobility Laboratory. The project will be conducted in collaboration with Veolia Transport.
Student: My Hang Nguyen (GC), January 17, 2012
Supervision: Ricardo Hurtubia
In collaboration with Veolia Transport
Multi-modal transportation modeling for a new generation of aircraft: Clip-Air
Clip-Air is an innovative air transportation system based on modularity. By design, loading units (capsules) can be detached from the carrying unit (wing) which has several advantages in terms of fleeting operations. To quantify these advantages an integrated schedule design and fleet assignment model is developed for both standard planes and Clip-Air. The comparative analysis has shown that there is a potential increase in the transportation capacity thanks to modularity of Clip-Air. As a further investigation, the performance of Clip-Air is needed to be analyzed from a network perspective. Repositioning of Clip-Air capsules by other means of transport, specifically railways, will give this multi-modal network viewpoint. The repositioning is believed to increase the profit especially in case of unbalanced demand between airport pairs. Furthermore, since Clip-Air capsules are completely detached from the wing, the storage and transfer costs are expected to be reduced. The aim of this project is to develop an appropriate model for the repositioning of Clip-Air capsules and obtain an integrated fleeting model. The tasks of this semester project are the following: 1. Get familiar with the concept of fleet assignment and already developed models. 2. Make a literature search on empty container management models. 3. Develop an appropriate model for the repositioning of capsules to be integrated into the existing fleet assignment model. Repositioning model will determine the flow of empty capsules throughout the network considering the carrying costs of capsules by rail and storage cost of capsules at airports. 4. Obtain preliminary results for a few data instances. 5. Evaluate the effect of repositioning.
Student: Jonathan Blaiberg (SGC), January 13, 2012
Supervision: Bilge Atasoy, Matteo Salani
Robustness and Recovery in Berth Allocation Problem
Student: Wei Li (CDM), January 13, 2012
Supervision: Michel Bierlaire, Nitish Umang
The problems of Train Design, Railroad Blocking and Train Assignment are fairly complex and often observed in the railroad industry. Efficient solution methods to these problems bring a huge potential to achieve enhanced operational performance and dramatically reduced costs. Blocking is defined as an activity where a set of shipments arriving at or commencing from a certain node station and departing to another particular node station, or further, are grouped together and sent across as the same train to minimize costs and exploit economies of scale. This problem has marked similarities with the airline scheduling which operates flights across a predetermined hub and spoke network. The problem considered here not only necessitates determining the “right” hubs and “right” trains to be scheduled on the network, but also scheduling the shipments on appropriate trains between the hub station yards and spoke station yards so that the overall costs are minimized. In the problem considered by us, we are given a network comprising a set of nodes and arcs. We are also given a set of shipments with their origin and destination nodes. We are given a range of costs such as the cost of car travel per mile, cost of train travel per mile, cost of starting a train, cost of grouping (also referred to as classifying or blocking) shipments at an intermediate station of a train, cost of train imbalance, cost of crew imbalance and the cost of a missed car that is not transported. It is also required that each train path overlaps one or more crew segments completely. Crew segment between two nodes will also always follow the shortest path between those two nodes. Thus a train cannot run on a section which is not on the path of a crew. Train imbalance is defined as the imbalance generated due to the fact that the number of outgoing and incoming trains at a node differ. Total train imbalance for the network is computed as the sum of imbalances at every node. Crew imbalance is generated due to the fact that a crew operates on a segment, but cannot find an operational train to return to their base. In addition to these considerations, there are specific requirements relating to the maximum number of trains that a shipment can travel on, the maximum number of blocking (or classifications or work events – as referred in the problem statement) allowed for each train, the maximum number of trains allowed on any arc and the limitation on the maximum train length and tonnage. The main objective of our efforts would be to find a cost minimizing set of feasible trains that operate on one or multiple crew segments completely. We would also need to determine the least cost assignment of shipments to these trains. Term Project Deliverables • Develop and implement a cost optimization single or multi-stage mixed integer program model • Solve the model using a mix of relaxations and heuristics and report results on two data sets • Prepare a comprehensive report on solution methods and future research tracks on this problem Apart from expertise in modeling, the student is expected to be able to implement his model and write his own codes in a standard programming language, such as Java or C# Project Organization The student would be expected to update us on the progress and revert to us for queries on a pre-decided day of the week. A midterm presentation would be scheduled on Thursday, 10.11.2011. At the completion of the term project, a report will be delivered (both in paper as well as electronic form) before the 13.01.2012. The final oral presentation for the term project will be scheduled on 11.01.2012. All the files, programs, codes, data and report associated with the project must be delivered before these date. All reports will be submitted in five copies to TRANSP-OR Lab, LUTS Lab, Dr. Prem Kumar and Mr. Burak Boyaci. One copy is for the student.
Student: Stefan Binder, January 12, 2012
Supervision: Michel Bierlaire, Prem Kumar, BURAK BOYACI, LUTS
Mobility identification from smartphone GPS data
We record data from smarpthones over 2 years from about 180 N95 smartphone user. The data includes, GPS, nearby Bluetooth devices, nearby WIFI spots, calendar, etc. This project aims at identifying mobility history (trips and destinations) from raw data. The project will be based on some existing solutions (with existing Java and Python code). And the student will improve or develop a new software to process the data, and generate and visualize the data. You can choose among Python, Java or C++ for the project. The objectives of the project are following (You can choose among them): 1. Identify trips (origin, destination and time) from the GPS data. 2. Detect transportation mode (car, or public transport, etc) from the trips.
Student: Denis Garcia (SMA), January 11, 2012
Supervision: Jingmin Chen, Ricardo Hurtubia
Battery life modelling
In a context where electric vehicles are going to be released soon on the market, a high importance is given to the analysis of the degradation of their batteries. A key aspect of this research is to identify the factors that are responsible for a fast battery degradation, in order to give drivers advices on a battery-friendly driving.
The aim of the project is two-fold:
1) Refine an existing model of the state of charge of the battery, in order to identify which are the factors explaining the battery discharge and how they affect it.
2) Analyze the impact of different driving patterns and other factors on battery degradation. The student will have to get familiar with an existing battery life model owned by ZEM.
Student: François Anken (SMA), June 30, 2011
Supervision: Aurélie Glerum, Responsable externe ZEM
In collaboration with ZEM
Measuring Passenger Satisfaction using [email protected] Survey
M1nd-set is an international market research agency specialized in air travel surveys and research. The company conducts regular passenger satisfaction surveys among the major fullservice airlines on different international market areas. M1nd-set manages reasonable passenger response rates and over a third of their survey is administered on business-class travelers who value passenger service.
At present, [email protected] survey captures the passenger satisfaction levels for different parameters, such as waiting times for check-in, boarding and in-flight services, staff friendliness, seat comfort, in-flight entertainment, meals, etc., that are perceived to drive the overall satisfaction of flying with a particular airline company. M1nd-set uses a specific method of calculating weights of these parameters, in the order of importance, and arriving at overall airline satisfaction levels. However we see some scope in applying more advanced scientific models to this data to gain a deeper perspective on the relative significance of these parameters. In addition, it is critical to understand if higher satisfaction levels with a particular carrier are actually translating into increased demands and thus increased revenues. This could also mean that we can test the sufficiency of the currently captured parameters and if there are some additional parameters that could be silently driving the demand.
M1nd-set is seeking fresh, innovative and out-of-the-box ideas to analyze the data mine at hand. They are willing to give access to the data base (under the umbrella of the enclosed NDA) to the student who brings an interesting research idea. Ideally the outcome of this research / analysis should be presented to airlines on conference(s) and/or published.
This project will be composed of the following stages: 1. Understanding basic airline industry domain, various players and [email protected] survey in that perspective 2. Literature review and discussion on new ideas 3. Preliminary data analysis 4. Application of fresh methodologies for data analysis, such as using Discrete Choice Models for analysis 5. Inclusion of additional questions on the survey that could provide new insights 6. Final presentation and report
Project will commence around mid February and would continue for about 4 1/2 months
Student: Lidija Stankovikj (SMA), June 30, 2011
Supervision: Bilge Atasoy, Aurélie Glerum, M1nd-set
Travel behavior models: an Abu Dhabi case study
In this project, the student aims at studying the state-of-the-art demand modeling method- ologies in the transportation field. She will learn discrete choice models, and relevant tech- niques, such as simulation, and apply them in the study of mode choice and/or driving behavior. An Abu Dhabi case study will be carried out.
Student: Laurène Aigrain (SGC), January 30, 2011
Supervision: Michel Bierlaire, Jingmin Chen
Estimation and simulation of bid-auction and choice location models
Student: Thibaut Dubernet (SGC), January 20, 2011
Supervision: Ricardo Hurtubia, Thomas Robin
The urban development effects of the construction of a metro line
Student: Aurélien Odobert (SGC), January 20, 2011
Supervision: Ricardo Hurtubia
Prototype MATSim model for Brussels
Student: Sohrab Sahaleh (SGC), January 20, 2011
Supervision: Gunnar Flötteröd, Ricardo Hurtubia
Quantitative analysis of urban sustainability indicators
Student: Timothée Vincent (SGC), January 20, 2011
Supervision: Gunnar Flötteröd, Ricardo Hurtubia
An analysis of a potential implementation and investigation Singapore's road pricing system within the MATSim transportation microsimulation
Student: Youssef Mezdani, January 20, 2011
Supervision: Gunnar Flötteröd, Alexander Erath
Optimizing Staffing Plans at Airports
Minimizing operating costs for maintaining ground personnel at airports is a complex problem due to uneven flight activities, passenger service expectations and staffing inflexibilities due to shift durations. In this work, we would use the flight schedule and service criteria to develop a method to find optimal shift timings that considers non-productive time due to activity changeovers, the mix of full-time and part-time workers and passenger waiting time criteria.
Student: François Anken (SMA), January 05, 2011
Supervision: Prem Kumar
Integrating the latent attitudes into mode choice
In the classical mode choice modeling we have the modal attributes like time and cost, and the socio-economic information as explanatory variables. However there are unobservable variables like attitudes, perceptions, lifestyle etc. which are effective in travel behavior. In this project aim is to come up with integrated choice models including these latent variables.
Student: Lidija Stankovikj (SMA), January 05, 2011
Supervision: Bilge Atasoy, Aurélie Glerum
Analysis of Electric Vehicle Data
The project aims to apply statistical analytics and basic mathematical techniques to analyze a unique data set of electric vehicle usage, and a cutting edge battery life model. The student will analyze the impact of different driving patterns and other factors on battery degradation – a key question facing the transportation industry today. The student will be exposed to the latest developments in mobility and electric vehicle development, and will have constant access to supervisors that will guide him / her to better understand the technical and industrial context of the project. The student will have great autonomy in choosing the mathematical and analytic techniques he/she is familiar with to run simulations, stress the data sets, and incorporate new analytical models as necessary.
Student: Parmeet Singh Bhatia, January 05, 2011
Supervision: Aurélie Glerum, Responsable externe ZEM
In collaboration with ZEM
Integrating demand functions inside optimization model
For the new air-transportation system CLIP-AIR we want to have a fleeting model which includes a demand model. The project requires 1. understanding of the concept, 2. implementing the model in a general purpose math language, 3. using a solver for the nonlinear mixed integer problem, 4. performing preliminary tests with given instances.
Student: Nancy Moret (SMA), January 05, 2011
Supervision: Bilge Atasoy, Matteo Salani
Insertion d'un nouveau moyen de transport aérien sur la base d'avions de transport existants
Insertion d'un nouveau moyen de transport aérien sur la base d'avions de transport existants Il s'agit de créer une base de données comprenant les avions de ligne les plus représentatifs utilisés par les différentes compagnies dans les transports de passagers et de fret. Il faudra remplir un tableau par types et constructeurs d'avions: en tonnages, surfaces alaires, masses max et min, charges, nombre de passagers, altitudes, rayon maximal de chacun, ainsi que les vitesses de décollage, de croisière et d'atterrissage. La recherche, la mise en place et le classement de l'ensemble de ces données permettront de faire des correspondances par type de missions, comme les courts courriers, moyen-courriers et long-courriers. Ceci constitue l'étape N°1 Cette classification permettra de définir l'étape N°2, qui sera exprimée par une série de graphiques faisant correspondre les caractéristiques aux missions. Par exemple rechercher des correspondances de surfaces alaires et de poids ou de surfaces et de masses en corrélation à un rayon d’action maximum, etc. Les étudiants chercheront et exprimeront ces concordances entre ces types d'avions par une série de graphiques explicites sachant que les données collectées sont optimisées aux maximums par les lois de la mécanique des fluides, véritable base commune à tout constructeur d'avions. L'étape N°3 consistera, avec les données qui seront remises aux étudiants, à faire un tableau réunissant un maximum d’éléments concernant le projet Clip-Air. Ceci au travers des deux publications existantes qui seront remises aux étudiants. Ces données seront complétées avec l'aide des personnes travaillant dans le projet. L'étape N°4 sera d'insérer le projet Clip-Air dans les graphiques de l'étape N°2. Ceci permettra aux étudiants de tirer des conclusions qui situeront le projet Clip-Air par comparaison, dans: - la classe opérationnelle la plus proche -son potentiel de transports -d'évaluer "les manques" ou "les trop", qui permettrait de correspondre à une classe opérationnelle donnée (en tenant compte de facteurs spécifiques au projet Clip-Air qui seront discutés à cette étape).
Student: Laurene Aigrain & Dethier Daphné (SGC), January 05, 2011
Supervision: Claudio Leonardi, Bilge Atasoy
Destination choice models for a free bicycle system
Several cities around the world are implementing “free bicycle systems” that consist of a network of stations were bicycles are available for users to “pay and ride”. Usually these systems allow users to return the bicycle in a different station from the one where it was taken. The project consists in estimating destination choice models that should allow to predict the station where a bicycle will be returned, given the origin, the time of the day, the length (in time) of the trip and the characteristics of the surroundings of each station. The project considers 3 main stages: Data analysis, model estimation (with the software BIOGEME) and model validation.
Student: Zehra Onen (SMA), June 30, 2010
Supervision: Ricardo Hurtubia, Thomas Robin
Simulateur de mouvements de piétons
Les modèles de mouvement de piétons permettent de simuler le comportement de foules. Ils sont utiles dans le domaine de la sécurité, de l'évacuation, mais également pour la plannification urbaine. En effet de nos jours pour des raisons environementales, de nombreux centres-ville sont interdits aux voitures et transformés en zones piétonnes. Ces modèles servent à simuler le comportement des piétons dans des contextes urbains prédéfinis afin de choisir l'aménagement qui sera le mieux adapté. Ce projet concerne l'implémentation d'un modèle de mouvements de piétons, modèle développé à l'EPFL au sein du laboratoire Transp-or
Student: Viljami Laurmaa (SMA), June 20, 2010
Supervision: Thomas Robin, Javier Cruz, Mamy Fetiarison
Algorithme de génération de vols de repositionnement pour améliorer la réparation d'un horaire perturbé
Dans le domaine du trafic aérien, un problème récurrent est de réparer un horaire perturbé suite à des évènements tels que de mauvaises conditions météorologiques, une défaillance technique d'un avion, etc. La problème de réparation d'horaire est largement étudié dans la littérature. Malheureusement, si la majorité des approches existantes considère la possibilité d'intégrer des vols dits "de repositionnement" (des vols additionnels qui ne sont pas prévus initialement), aucune d'entre elles n'étudie la génération des vols à considérer. Il s'agit donc d'élaborer un algorithme permettant de générer de bons candidats à améliorer la solution de réparation, sans pour autant considérer tous les vols de repositionnement possibles.
Student: Sabine Luisier (SMA), June 04, 2010
Supervision: Niklaus Eggenberg
Optimisation d'horaires aériens
Dans le cadre du transport aérien, qui est sujet à de nombreux retards, il est essentiel de planifier les horaires de manière à mieux contrôler ces retards. Le but de ce projet est d'étudier, implémenter et tester un modèle d'optimisation qui maximise le temps de connexion des passagers, afin de réduire le nombre de connections ratées dans le cas où des retards sont observés.
Student: Sezin Afsar (SMA), June 04, 2010
Supervision: Niklaus Eggenberg
The Tactical Berth Allocation Problem: hierarchical vs integrated models in the context of container terminal operations
Container terminal operations have received increasing interest in the scientific literature over the last years and operations research techniques are more and more used to improve terminal's efficiency and productivity. In particular, the simultaneous optimization of decision problems that are usually solved hierarchically by terminal's planners represents nowadays a promising research trend. The Tactical Berth Allocation Problem (TBAP) deals with the integration of the berth allocation problem (BAP) and the quay crane assignment problem (QCAP). It aims to schedule incoming ships over a time horizon, assigning them a berthing position and a certain quay crane profile (i.e. number of quay cranes per working shift). These decisions are strictly correlated, since the number of quay cranes assigned to a ship affects its expected handling time, and thus has impact on the scheduling in the berth allocation plan. The problem has been modeled as a mixed integer program and housekeeping costs generated by the berth assignment are taken into account by a quadratic term in the objective function. The aim of this project is twofold: (i) to compare the integrated approach (modeled in TBAP) to the standard hierarchical approach (first solve BAP, then QCAP); (ii) to analyze the impact of different BAP objective functions on yard operations, by taking into account housekeeping costs.
Student: Luca Furrer (SMA), June 04, 2010
Supervision: Ilaria Vacca
Un modèle préliminaire d'UrbanSim pour Lausanne
Depuis longtemps l'importance de l'interaction entre la forme urbaine, les infrastructures de transports et la demande de transport a été reconnu. Par contre ce n'est que récemment que des méthodes analytiques tel que des systèmes de modélisation de transport et d'occupation du sol ont été développées pour pouvoir mieux comprendre ces interactions. UrbanSim est un système de modélisation d'occupation du sol qui devient de plus en plus utilisé. L'objectif de ce projet est de monter un modèle préliminaire d'UrbanSim pour la ville de Lausanne.
Student: Sarah Droz (SGC), January 15, 2010
Supervision: Ricardo Hurtubia
Visualization of Cell Phone Data on Google Earth
This project aims at providing to our data collection campaign participants a convenient and lively way of visualizing cell phone data which is collected from their cell phones and stored on web servers. Various kinds of data are display on google earth to show chains of users’ activities, with time-space information provided by GPS data.
Student: Raoul Neu (SIN), January 15, 2010
Supervision: Jingmin Chen
Testing the algorithm for generating path observation from GPS data
This project aims at using and testing an innovative probabilistic path observation generation algorithm on location data and comparing against state-of-the-art map matching algorithms.
Student: Jensen Anders Fjendbo (SGC), January 15, 2010
Supervision: Michel Bierlaire, Jingmin Chen
Modeling the link between transport and land-use with UrbanSim
Location of activities in the city has an important effect on travel demand and on the transport system’s performance. Land-use models are used to forecast location in the city, in order to help decision-making for urban and transport planning. Because of its flexibility, UrbanSim is an increasingly popular alternative for integrated land-use and transport modeling. However, UrbanSim is not exactly an integrated model but a land-use model that works together with a transportation model. This makes relevant to understand if the interaction between transport and land use is properly modeled by UrbanSim. The objective of this project is to implement the latest stable version of UrbanSim for the city of Brussels, using data that was already collected to implement an older version of the model. The project also considers a deeper analysis on how UrbanSim accounts for the relation between transport and land use.
Student: Peter Goodings (SGC), January 15, 2010
Supervision: Ricardo Hurtubia
Modeling the effects of spatio-temporal flexibility in activity scheduling
In travel demand modeling there is an increasing interest in understanding and modeling the planning or scheduling of activities over space and time. This is due to the fact that a better understanding of activity scheduling processes will contribute to the development of activity scheduling models and an understanding of the short (week) dynamics. In the literature, very little is known about the activity scheduling processes and the spatio-temporal flexibility of activities. For an individual, each (work, leisure, shopping) activity in a day can be classified according to its level of flexibility in routine, pre-arrange and spontaneous in the two dimensions, space and time. Based on his spatio-temporal constraints, an individual is making a decision to assign its degree of flexibility. The objective of this research is to investigate how socio-demographics of individual/households, life-stage, ICT access (cell phone/internet), accessibility to services, and activity attributes (time, duration, week day) affect the spatio-temporal flexibility of out-of home activities. The analysis will be done using data from a survey conducted in Québec City from 2002-2006 to estimate activity-choice models.
Student: Laetitia Bettex (SGC), January 15, 2010
Supervision: Ricardo Hurtubia
Game theory applied to ambush avoidance
This project continues along the stream of the previous project "Minimizing risk of ambush for vehicle routes". The goal of the project is to apply game theory to determine the optimal mixed strategy to select as set of vehicle routes. Each route comes along with the expected maximal payoff when the route is implemented. A strategy to determine the game matrix is to be conceived.
Student: Peter Goodings Swartz (SGC), January 15, 2010
Supervision: Matteo Salani
Analysis of Transport Mode Choice in Trieste
In the transport planning context, survey data have been collected between 2002 and 2003 in Trieste. Three modes of transport have been considered : car, motorcycle and bus. Transportation is a major application field of Discrete Choice Models (DCM) since they can capture lots of situations where a choice is performed. In order to study the mode choice of transport users, DCMs have been developed and calibrated with these collected data. Analysis of the results pointed out some prediction inaccuracy. This means that improvement of the model still has to be made. The aim of this project is to study the travelers choice behavior with the existing models and investigate possible improvements of the models specification.
Student: Alexandre Khelifa (SGC), January 15, 2010
Supervision: Mamy Fetiarison
Mode choice modelling with qualitative aspects inclusions
Discrete Choice Models (DCM) are widely used in transportation to explain mode choice. Objective attributes have been commonly used to describe the different alternatives, such as travel time or travel cost. Kaufmann et al. (2001) have conducted some transportation mode choice surveys. In addition to the classical data collection, they have included questions about experiences and perceptions of transportation modes. The obtained database is large and contains lots of respondents sociological aspects, difficult to apprehend in a classical DCM, due to their qualitative nature. In order to go through such issues, J.L.Walker (2001) has proposed a new modelling framework based on DCM and latent concepts. She used schemes to describe links between variables. The aim of the project is to identify complex causal links between sociological aspects and transportation mode choice, based on the data cited above, and then derive a new modelling framework.
Student: Zehra Onen (SMA), January 10, 2010
Supervision: Thomas Robin
Finance and Discrete Choice Models
Default risk evaluation for credit cards and mortgages, credit ratings determination, probabilities estimation of acquisitions and mergers of firms (used to guide investment), are very complex financial tasks. They require a very good expertise of the field. This is due to the high number of information that decision makers need to account for. In addition, information could be very heterogeneous in term of source and impact on the decision itself. Models can help financial decision maker in their tasks. For instance, classification methods are mainly used to model financial decisions. Discrete Choice Models (DCM) have been successively applied to many fields in the past 40 years, such as transportation or marketing. Compared to machine learning methods they have the advantages to explicitly model causal relations between dependant variables and exogenous attributes, and to avoid data over-fitting. DCMs start to be used in the financial field. The aim of this project is to investigate DCM use in a financial context, then find relevant problem to which it can be applied and finally illustrate those concepts on a real case study.
Student: Inès Azaiez (SMA), January 10, 2010
Supervision: Thomas Robin
Expérience des moyens de transport et choix modal
Lorsque l’on choisit d’utiliser un moyen de transport plutôt qu’un autre, indépendamment des temps de déplacements et des coûts comparés des alternatives en présence, l’usager a des préférences. Celles-ci sont liées à son expérience des moyens de transport et à la perception qu’il en a. Les modélisations du choix modal ne rendent qu’imparfaitement compte de cette dimension, intrinsèquement qualitative. Pour la traiter, le LaSUR a développé depuis une dizaine d’années un dispositif de citation d’adjectifs, appliqué à de gros échantillons de population lors d’enquêtes de mobilité. Nous proposons comme sujet de Master de modéliser sur la base des données disponibles, l’impact de l’expérience sensorielle des moyens de transports sur le choix modal.
Student: Seyed Tavakoli (SGC), June 30, 2009
Supervision: Michel Bierlaire, Thomas Robin
Minimizing risk of ambush for vehicle routes
Vehicle routing problems are known problems in which a fleet of vehicles is routed through a network to collect (or deliver) items from customers. The objective of routing problems in an hostile environment is to prevent ambushes (for valuables deliveries). The goal of routing under potential attacks is to design safe routes in order to minimize the risk of being exposed to dangers. This project is designed to investigate and implement innovative routing strategies.
Student: Gaëtan Duyckaerts, Peter Goodings Swartz (SGC), June 19, 2009
Supervision: Matteo Salani
Analysis of consumer behaviour in terms of product preference/choice.
This project will be conducted in collaboration with the Nestlé Research Center. Quantitative studies are frequently used in the food industry in order to determine whether a product is superior to its competitor(s). Many types of studies exist, but this project focuses on a specific type of study involving only two products and in which consumers are asked to evaluate both products with regard to different criteria (such as flavour, appearance, texture, etc.) and to choose the product they prefer. Socio-demographic characteristics as well as information on consumption habits are also collected during the test. The aim of this project is to explore a database containing about 100 studies that have been conducted over the past years on a same product category in different countries. Such studies are usually analyzed by the means of descriptive statistics in order to answer the two following questions: -Which product do consumers prefer? -Why do consumers prefer this product? In this project, we propose to analyze the studies by making use of discrete choice modelling. This technique allows understanding and modelling the behaviour of consumers in a quantitative way when they are exposed to such choice situations, i.e. the choice between two products. Discrete choice models should allow identifying and quantifying the effect of both product attributes and consumer characteristics on the preference.
Student: Aurélie Glerum (SMA), June 12, 2009
Supervision: Thomas Robin, Michaël Thémans
In collaboration with Nestlé
Effets sur le trafic d'une nouvelle jonction autoroutière à Chavannes
Actuellement, il n’y a qu’une seule jonction autoroutière entre celles de la Blécherette et de Malley pour desservir l’Ouest Lausannois : la jonction de Crissier. Or deux nouvelles jonctions sont en projet à Ecublens et Chavannes. Selon les choix effectués pour les voies d’accès à l’autoroute, ces jonctions peuvent diminuer les nuisances en réduisant les kilomètres parcourus, surtout hors autoroute ou les augmenter en amenant du trafic de transit indésirable dans des zones à préserver. Le projet consiste à examiner des variantes d’aménagement du réseau dans le sud de Renens, éventuellement à en suggérer de nouvelles, et à évaluer ces diverses variantes en s’aidant du logiciel Emme de simulation du trafic.
Student: Annie Faniry Andriamanorohasinjafiniarivo Ravalitera (SGC), June 05, 2009
Supervision: Jean-Pierre Leyvraz
Extending the framework for MEV discrete choice models
Student: Lorenza Santini (SMA), February 03, 2009
Supervision: Michel Bierlaire, Mogens Fosgerau, DTU, Denmark
Implementation of attribute processing strategies in advanced discrete choice models
Student: Carol Kirchhofer (SMA), February 03, 2009
Supervision: Michel Bierlaire, John Rose, ITLS, University of Sydney
Estimation of household location choice models
Location of activities in the city has an important effect on travel demand and on the transport system’s performance. Land-use models are used to forecast location in the city, in order to help decision-making for urban and transport planning. To predict the location of agents it’s first necessary to estimate “location choice models” (which are usually a part of the land-use model). These models are estimated through maximum-likelihood methods from observed location data. The aim of the project is to estimate several different household location choice models for the same city, using different specifications for the household’s utility function and testing different methods to deal with “attribute’s thresholds” in the utility. The estimations will be made using Biogeme, databases will be provided.
Student: Antonin Danalet (SMA), January 12, 2009
Supervision: Ricardo Hurtubia
Minimizing risk of vehicle routes in valuables collection from banks
Vehicle routing problems are well known combinatorial problems in which a fleet of vehicles is routed through a network to collect (or deliver) items from customers. The objective of routing problems is to minimize operational costs in terms of distance traveled such that all customers are visited and all operational constraints are respected. In the context of logistic services delivered to credit institutes, e.g. the collection of valuables or distribution of coins, the total distance traveled is no more the primary objective to optimize. For logistic companies is more convenient to design {\em safe} routes in order to minimize the risk of being robbed. Beside risk minimization, logistic companies commit to provide a certain level of service to credit institutes in terms of pick-up frequencies.
Student: Fabrice Piat (SGC), January 11, 2009
Supervision: Matteo Salani
Est-il possible de réduire le nombre de panélistes dans une étude DTS (Dominance Temporelle des Sensations) sans altérer la qualité des résultats?
La méthode de la Dominance Temporelle des Sensations (DTS) permet de suivre l’évolution des perceptions sensorielles en bouche au cours de la mastication d’un produit alimentaire. Elle consiste simplement à demander au panéliste de sélectionner la caractéristique dominante du produit au cours de la mastication. Par exemple, si le produit est une barre de céréales, le panéliste doit indiquer au cours de la mastication d’une bouchée si le produit est plutôt croquant, friable, collant ou pâteux. Cette caractéristique dominante peut changer au cours du temps. Sur l’ensemble du panel, cette méthode permet de dessiner les courbes d’évolution de chaque descripteur sensoriel au cours du temps.
Cette méthode est relativement récente et les premières études ont été réalisées avec un nombre de panélistes très important (environ 50 évaluations de chaque produit). L’objectif de ce projet est de tester si cette méthode donne des résultats aussi satisfaisants avec un nombre de panélistes réduit.
Le travail consistera donc à :
• se familiariser avec la méthode DTS et la construction des courbes
• développer un programme permettant de construire ces courbes pour un sous-ensemble de panélistes
• générer les résultats pour plusieurs tailles de panel en utilisant les données de plusieurs études DTS existantes au centre de recherche Nestlé
• proposer une méthode pour comparer les résultats des panels de taille réduite avec le panel complet
• appliquer cette méthode et analyser les résultats
• rédiger un rapport présentant la démarche et mettant en avant les principales conclusions
Student: Glerum Aurélie (SMA), January 11, 2009
Supervision: Thomas Robin, Michaël Thémans, Nicolas Pineau (Nestlé Research Center)
In collaboration with Nestlé
Is it possible to reduce the number of panelists in a TDS (Temporal Dominance of Sensations) study without altering the quality of the results?
Student: Aurélie Glérum (SMA), January 06, 2009
Supervision: Thomas Robin, Michel Bierlaire, Michaël Thémans
In collaboration with Nestlé Research Center
La modélisation de transport de la région Lausannoise avec PTV Vision
La modélisation de transport est un aspect fondamental de la planification de transport. Plusieurs différents logiciels existent pour modéliser la performance des réseaux de transports urbains. Les logiciels EMME, TransCAD et PTV Vision sont parmi les plus utilisés au monde pour modéliser le trafic. Au sein du laboratoire TRANSP-OR existent déjà des modèles EMME et TransCAD de Lausanne. Pour faire connaître aux étudiants de l'EPFL les différentes options disponibles pour la modélisation de transport, le développement d'un modèle PTV Vision de Lausanne est prévu. Le but de ce projet est d'utiliser les données existantes pour monter un modèle PTV Vision de la ville de Lausanne.
Student: Chen Lu (SSC), July 15, 2008
Supervision: Zachary Patterson
Déconvolution de signaux géochimiques
Le but général de la méthode est de connaître, à partir du relevé de la concentration d'un élément chimique dans l'air, la part naturelle, et la part anthropique(celle dûe à l'homme, par exemple la pollution). Concrètement à partir de données existantes, nous voulons implémenter et résoudre une maximisation de vraisemblance, afin de déterminer les deux composantes du signal.
Student: Gfeller Nicolas (SMA), June 30, 2008
Supervision: Thomas Robin
Dynamic Traffic Assignment in Lausanne
Dynameq is a new breed of equilibrium dynamic traffic assignment (DTA) for use on large, congested networks, distributed by INRO (www.inro.ca). Dynameq gives planners a view into dynamic traffic conditions, and provides rational scenario comparisons that are only possible with an equilibrium-based solution. It has been successfully used in cities like Montréal. The purpose of the project is to develop a Dynameq model of Lausanne. The challenge consist in gathering relevant available data, and identify additionnal data which would need to be collected. The objective is to obtain a prototype model for which the missing pieces are well identified. The project will be conducted at the Centre de Recherche sur les Transports (University of Montréal), in close collaboration with the staff of INRO.
Student: Vidaud Marine (SGC), June 20, 2008
Supervision: Michel Bierlaire, Mahut Michael
Development of mode choice models in Trieste
The discrete choice models are very relevant concerning the transport mode choice. A survey on the daily trips has been conducted in Trieste, Italy, in 2000-2001, led by the municipality. The study of the resultant data base will permit to understand the Triestan’s habits, along with developing transport mode choice models. The aim of this project is to analyse real data (the dataset described above), perform a complete modelling process, together with statistical tests, and prepare appropriate documentation.
Student: Violin Alessia (SMA), June 17, 2008
Supervision: Michel Bierlaire
Optimization of Lausanne's traffic signal timings
The aim of this project is to optimize the traffic signal settings of the network of the city of Lausanne.
The phases of the project are as follows. The student will:
- provide an overview of the existing formulations for this optimization problem;
- propose a formulation, this formulation may be taken from exisiting studies;
- implement the problem on a subnetwork of the city of Lausanne, the student will be given access to a Matlab code which may need to be adapted according to the formulation;
- analyse and discuss the results.
Basic knowledge of Matlab is required for this project.
Student: Yanjun Zhang (SGC), June 06, 2008
Supervision: Carolina Osorio Pizano
Analyse géographique pour l'implémentation d'un prototype de modèle intégré de transport et d'occupation du sol pour la région lausannoise
UrbanSim est un système de modélisation transport et occupation du sol qui devient de plus en plus utilisé. Se projet se fera dans le cadre d'un projet plus large pour développer un prototype d'un modèle intégré de transport et d'occupation du sol pour la région lausannoise. Ces modèles demandes beaucoup de données socio-démographiques ainsi que géographiques pour la région d'étude. L'objectif de ce projet est de contribuer au développement des données pour le tableau Gridcells' du modèle en développement. En particulier le projet aura comme but : l'identification des données manquantes pour ce tableau, l'identification des sources pour les données manquantes, le traitement de ces données et l'incorporation dans le tableau Gridcells.'
Student: Bettex Laetitia (SGC), May 30, 2008
Supervision: Zachary Patterson
Algorithmes On Line pour le CVRP avec Demandes Aléatoires
EN - The project copes with the capacitated vehicle routing problem (CVRP) in the case of unknown demand volumes of the customers. The aim is to abord the problem with an on line strategy. The student will first familiarize with CVRP and on line algorithms and then develop and test an on line strategy.
FR -Le projet propose d'aborder le problème de trounées de véhicules avec contraintes de capacité (CVRP) dans le cas où les volumes des demandes des clients ne sont pas connus. Il s'agit ici d'aborder le problème avec une approche on line. L'étudiant devra d'abord se familiariser avec le CVRP et les problèmes on line puis développer et tester une stratégie.
Student: Boukriba Sami (SSC), May 23, 2008
Supervision: Niklaus Eggenberg
Déterminer les vols de repositionnements pour un horaire aérien perturbé
Lors de l'exécution d'un horaire, il arrive souvent que ledit horaire devienne irréalisable. Dans le cas du transport aérien, pour réagir à un tel évènement, il s'agit de retarder ou d'annuler certains vols, voir même faire des trajets "à vide" pour repositionner les avions. Dans ce projet, il s'agit d'élaborer une technique permettant d'identifier, étant donné l'horaire devenu irréalisable et l'état des avions, quel(s) vol(s) de repositionnement sont à considérer pour minimiser autant les coûts que les annulations de vols et les retards.
Student: Karker Amin, Tournier Sebastien (SSC), May 23, 2008
Supervision: Niklaus Eggenberg
Calibration of on Integrated Transportation Land-use Model - UrbanSim for Brussels
Transportation and infrastructure planning requires a good understanding of future transportation demand. Traditionally, transportation demand analysis has focused primarily on the transportation system independent of its relationship with land-use and urban form. Integrated Transportation and Land-use Models (ILUMs) explicitly model these interactions. The TRANSP-OR laboratory currently has an operational land-use model (UrbanSim) for the city of Brussels in Belgium. The purpose of this project is to is to fine-tune UrbanSim for Brussels by improving the location, developer and land-price models. The analysis will take advantage of GIS land-use data for Brussels. The project will include an application of the modeling system to a practical transportation and land-use planning problem. The project will be undertaken in cooperation with Stratec of Brussels.
Student: Zemzemi Fatima and Stoitzev Iordanka (SMA), January 18, 2008
Supervision: Zachary Patterson
Analyse de données requises pour un modèle intégré de Lausanne
Depuis longtemps l'importance de l'interaction entre la forme urbaine, les infrastructures de transports et la demande de transport a été reconnu. Par contre ce n'est que récemment que des méthodes analytiques tel que des systèmes de modélisation de transport et d'occupation du sol ont été développées pour pouvoir mieux comprendre ces interactions. UrbanSim est un système de modélisation qui devient de plus en plus utilisé. Ce projet représente une phase préparatoire pour le développement d'un modèle UrbanSim pour la région lausannoise.
Student: Maret Jonathan (SGC), January 18, 2008
Supervision: Zachary Patterson
Dynamic Traffic Assignment in Lausanne (préétude)
Le logiciel Emme permet de comparer des variantes de réseaux routiers urbains en se basant sur une affectation des déplacements des automobilistes de leur origine à leur destination. Cette affectation est macroscopique (on considère des flux globaux de déplacements) et statique (les conditions d’écoulement sont censées être constantes durant toute la période considérée). D’autres modèles comme Dynameq et AIMSUN se basent sur une affectation microscopique (chaque véhicule est modélisé) et dynamique (les conditions peuvent changer au cours du temps). Ces modèles demandent plus de données, mais permettent en échange d’évaluer des mesures comme des changements dans la gestion des carrefours et de représenter l’évolution au cours du temps. L’agglomération lausannoise dispose d’une banque de données Emme et, pour sa partie la plus centrale, d’une banque AIMSUN. Le projet de master, qui se déroulera chez INRO à Montréal, consiste à créer à partir de ces données et éventuellement d’autres données disponibles, une banque de données Dynameq, de documenter le processus utilisé et d’identifier les données supplémentaires nécessaires pour améliorer la qualité de cette banque.
Student: Vidaud Marine (SGC), January 14, 2008
Supervision: Michel Bierlaire, Jean-Pierre Leyvraz
Modèle de choix discret
Les modèles de choix de discret sont très pertinents concernant le choix de mode de transport. Une enquête sur les trajets a été menée d’août à décembre 2003 par l’ETHZ, à Frauenfeld et ses abords dans le canton de Thurgovie, en Suisse. Celle-ci concerne 230 personnes, suivies pendant 6 semaines et provenant de 99 ménages. L’étude de la base de données résultante permettra de mieux connaître les habitudes des voyageurs, ainsi que de développer un modèle de choix de mode de transport. Le but de ce projet est de s’initier à l’analyse des modèles de choix discret leurs applica- tions sur des données réelles, en l’occurence les données décrites ci-dessus.
Student: Weber Caroline (SMA), January 09, 2008
Supervision: Thomas Robin
Modèle de classes latentes en analyse de choix discret
Le but de ce projet est de s'initier à l'analyse des modèles de choix de classes latentes et leurs application sur des données réelles en vue de préparation d'un projet de master concernant le choix de billets d'avion.
Student: Kirchhofer Carol (SMA), January 07, 2008
Supervision: Michel Bierlaire
Modèles GEV et MEV
Les modèles de choix discrets ont été prédominants dans l'analyse de transport durant ces dernières années. Le modèle MNL(Multinomial Logit) est particulièrement apprécié grâce à ses capacités calculatoires et sa forme analytique. Il est obtenuà partir de la loi de Gumbel, appelée aussi loi des valeurs extrêmes de type I. D'une part, une version multivariée (MEV) a été proposé par McFadden (1978). D'autre part, différentes lois monovariées aux valeurs extrêmes peuvent être rassemblées en utilisant la spécification dite généalisée (GEV). Le but de ce projet est d'une part de maîtriser les concepts liés aux modèes MEV (Mul- tivariate Extreme Value) et GEV, afin d'investiguer la possibilité de combiner les deux. D'autre part, les modµeles d'utilité multiplicatifs (Fosgerau & Bierlaire 2007) permettent de faire un lien avec les modèles GEV ( Fosgerau, short note). Ce lien sera aussi investigué, en vue de la réalisation d'un projet de master lié à ce sujet avec Mogens Fosgerau à Copenhague.
Student: Santini Lorenza (SMA), January 07, 2008
Supervision: Michel Bierlaire
Graphical interface for a JAVA simulator
Project aim: The aim of this project is to implement a graphical interface for an existing discrete event simulator.
Description:
Numerical simulators are often used to evaluate the impact of changing the characteristics of an already existing system (e.g infrastructure improvements in highways). Such computer models can also be used to evaluate modifications under hypothetical scenarios that would be difficult to observe in the real world (e.g. predict future congestion levels, based on demographic forecasts.) These numerical simulators mimic the behavior of complex systems, yielding performance measures that are rich in information. In order to take full advantage of this detailed information, and to be able to summarize it in a user friendly manner, a graphical interface is crucial.
The aim of this project is to implement a graphical interface for a numerical simulator.
The student will be given a JAVA implemented discrete event simulator that studys the flow of units (e.g. vehicles, pedestrians) through a network (e.g. highways, corridors). The aim will be to implement an attractive graphical interface that will summarize and illustrate the main performance measures that the simulator yields.
Student: Fetiarison Mamy Nirina, August 31, 2007
Supervision: Carolina Osorio Pizano
A Survey of Active Integrated Land-use Models around the World
Traditional travel demand modeling has tended to ignore the important interactions between urban form and transportation demand and the performance of the transportation system. Integrated transportation land-use models (ILUMs) explicitly model these interactions. These models are becoming increasingly popular. Better understanding these models requires a better knowledge of how they are applied in different contexts. The purpose of this project is to undertake research to provide a summary of active ILUM applications around the world.
Student: Vignon Olayitan (SGC), June 29, 2007
Supervision: Zachary Patterson
The spillback phenomenon frequently arises in urban traffic networks under congestion. If the network contains loops then spillback is a potential source of dealocks, also known as gridlocks. In this project the student will be given acces to a network simulation tool. Based on empirical studies of simple networks under deadlock the student will:
1) identify preformance measures that indicate the presence of deadlocks
2) propose analytical performance measures that may help detect deadlocks
3) validate the proposed performance measures
Student: Anken Nicolas (SMA), June 29, 2007
Supervision: Carolina Osorio Pizano
Study of Optical Flow techniques for motion estimation in video sequences
In image processing, when dealing with video sequences, it is usually very useful to have an estimation of the motion in order to obtain spatio-temporal information. To achieve this, the most common approach is to compute the optical flow, which is the velocity field of the apparent motion between frames.
The aim of this project is to study the most relevant approaches to compute the optical flow and implement them using C, C++ or JAVA.
Student: Epely-Chauvin Gaël (SGC), June 29, 2007
Supervision: Javier Cruz
Etude du comportement d'achat des consommateurs
Le comportement des consommateurs durant la démarche d'achat est un phénomène compliqué et propre à chaque personne, qui regroupe des paramètres observables, comme le choix final, et des composants latents, comme p. ex. le raisonnement avant ce choix. La modélisation de l'impact de marketing ciblé (via design des panneaux publicitaires) est du plus grand intérêt pour la compréhension de ce comportement. Le but de ce projet est d'unifier les données venant des sources différentes et d'améliorer le modèle de la démarche complète d'achat basé sur les données d'une étude réelle de traçage des yeux.
Student: Alexandre Xavier (SMA), June 29, 2007
Supervision: Michel Bierlaire
Simulation of finite capacity queueing networks
The aim of this project is to reproduce via numerical simulation the behaviour of a network of finite capacity queues. The simulation tool is to be both implemented and validated. The tool shall be used to analyse the HUG (Geneva University Hospital) hospital room network.
Student: Meier Pirmin (SMA), February 17, 2007
Supervision: Carolina Osorio Pizano
An Integrated Land Use Model Application to Brussels using UrbanSim
The interrelationship between urban form, transportation infrastructure and transportation demand has long been recognized. More recently, analysts have been developing Integrated Transportation and Land-use Models (ILUMs) to better understand and quantify these interrelationships. As a result, several modeling options are now available to develop ILUMs. One such model has already been applied to the city of Brussels in Belgium. UrbanSim is an increasingly popular alternative for integrated land-use modeling. The current study is intended to test the feasibility of applying UrbanSim to a city for which a fair bit of transportation and land-use data already exist, within a medium-term project timeline. The project is being undertaken in collaboration with Stratec in Brussels.
Student: Samartzis Lefteris (SMA), February 17, 2007
Supervision: Zachary Patterson
Analysis of a Recovery Network for an Airline Recovery Method
Airline schedules are often disrupted because of unforeseen events like lateness or bad weather. If the schedule becomes unfeasible, the scheduler has to recover from the actual state in order to get a new feasible schedule, either by canceling or postponing flights or even rerouting planes. The model often used is a time-space recovery network which encodes several possible recovery decisions for each plane in the fleet. It's main disadvantage is its size. The student will have to familiarize with airline scheduling and network modeling in order to analyze and simplify as much as possible the recovery network. The aim would be be able to end up with a reduction algorithm that throws network of smaller size by means of number of nodes and arcs amd eventually test the reduction algorithm on existing instances.
Student: Messina Daniele (SMA), February 16, 2007
Supervision: Niklaus Eggenberg
Etude des modèles de mouvements de piétons
Etude des modèles de mouvements de piétons, revue de la littérature, recensement des modèles existants dans divers domaines tels que l'architecture, planification urbaine, gestion de grands évènements, évacuation...etc. Le but est d'identifier ces divers modèles, déterminer leurs hypothèses, contextes d'utilisation, et faire une étude comparative.
Student: Li Xiangchun (SMA), February 16, 2007
Supervision: Thomas Robin
Modèles de choix discret pour la reconnaissance des expressions faciales statiques
Student: Danalet Antonin (SMA), February 16, 2007
Supervision: Michel Bierlaire
Validation de modèles de choix de route
Student: Anken Nicolas (SMA), February 15, 2006
Supervision: Emma Frejinger
Detection of behavioral inconsistency in Revealed Preferences surveys
Student: Laurence de Torrenté and Ariane Wenger (SMA), February 01, 2006
Supervision: Michaël Thémans, Michel Bierlaire
Detection of behavioral inconsistency in Revealed Preferences surveys
Student: Julie Marc (SMA), June 23, 2005
Supervision: Michaël Thémans, Michel Bierlaire
Discrete choice models: development of case studies
Student: Frédéric Anken (SMA), June 22, 2005
Supervision: Michaël Thémans, Michel Bierlaire
Analyse de l'approche sous-réseau pour la modélisation de choix de route
Student: Gilliéron Fanny (SMA), June 20, 2005
Supervision: Emma Frejinger
Analysis of choice sets for route choice models
Student: Fortón Garcia Verònica, June 20, 2005
Supervision: Emma Frejinger
Deterministic Correction of the Multinomial Logit Model for Route Choice Analysis
Student: Regis Céline (SSC), June 20, 2005
Supervision: Emma Frejinger
Different strategies for trust-region size management
Student: Emmanuel Leclercq (SMA), June 20, 2005
Supervision: Michaël Thémans, Michel Bierlaire
Stated Preferences Survey for the choice of exchange university
Student: Eric Von Aarburg (SMA), June 20, 2005
Supervision: Michaël Thémans, Michel Bierlaire
Optimisation of the Operating Suit
Student: Pivin Edward (SMA), February 15, 2005
Supervision: Emma Frejinger
Utilisation d'un système géographique pour un problème de logistique
Student: Emery Sarah and Roth Isabelle (SMA), February 15, 2005
Supervision: Emma Frejinger
Route Choice Analysis using GPS Data
Student: Lindbäck Rolf (SSC), February 15, 2005
Supervision: Emma Frejinger, Michaël Thémans
Calibration of models for transportation demand in the context of real-time applications
Student: Mohamed Rhmari Tlemçani (SSC), June 25, 2004
Supervision: Michaël Thémans, Michel Bierlaire
On the convergence of multi-dimensional filter methods
Student: Lionel Dumartheray (SMA), June 23, 2004
Supervision: Michaël Thémans, Michel Bierlaire
Adaptation of GSM for unconstrained nonlinear optimization
Student: Anouck Brossard and Sarah Degallier (SMA), June 23, 2004
Supervision: Michaël Thémans, Frank Crittin, Michel Bierlaire
Identification of Coherent Behavior using Linear Programming
Student: Ittig Oliver (SMA), June 20, 2004
Supervision: Emma Frejinger, Michaël Thémans
Dynamic O-D matrices estimation with iGSM
Student: Olivier Grandjean (SMA), February 01, 2004
Supervision: Frank Crittin, Michaël Thémans, Michel Bierlaire
Discrete choice models to capture drivers behavior in response to real-time traffic information
Student: Thomas Quentin Maillard (SMA), June 20, 2003
Supervision: Michaël Thémans, Michel Bierlaire
Testing GSM on unconstrained nonlinear optimization problems
Student: Mina Adel Latif (SSC), June 20, 2003
Supervision: Michaël Thémans, Frank Crittin, Michel Bierlaire
Empirical analysis of the correlation in discrete choice models
Student: Steve Salom (SSC), February 01, 2003
Supervision: Michaël Thémans, Michel Bierlaire
Optimisation de tournées de collecte et de distribution de linges industriels
Une entreprise, leader romand, spécialisée dans le lavage de linges industriels (hôpitaux, homes, hôtellerie, restaurants, industries, ...) désire optimiser ces tournées de collecte de linge sale et de distribution de linge propre. Il s'agira ainsi d'étudier la logique et la complexité des processus, et proposer un algorithme qui doit permettre d'optimiser ces tournées et le plan de chargement des camions en fonction de contraintes multiples. Selon les résultats de ce travail, la direction de l'entreprise est ouverte à toute proposition de modifications de sa stratégie ou/et de ses processus industriels et logistiques.
Student: , October ,
Supervision: Michel Bierlaire, Philippe Wieser, EPFL, CDM
Activity organization and ICT
Selon indications ultérieures.
Student: Laetitia Bettex (SGC), October ,
Supervision: Michel Bierlaire | 2020-10-01 22:51:46 | {"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": 0.3477434515953064, "perplexity": 5960.912990040793}, "config": {"markdown_headings": true, "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-2020-40/segments/1600402132335.99/warc/CC-MAIN-20201001210429-20201002000429-00508.warc.gz"} |
http://tex.stackexchange.com/questions/65089/im-trying-to-redefine-the-plain-pagestyle-as-empty | # I'm trying to redefine the plain pagestyle as empty
I tried to use this command, but doesn't work.
\makeatletter
\let\ps@plain\ps@empty
\makeatother
I put it in the beginning, in the middle and just before the \begin{document}.
-
Welcome to TeX.SE. It is always best to compose a fully compilable MWE that illustrates the problem including the \documentclass and the appropriate packages so that those trying to help don't have to recreate it. – Peter Grill Jul 29 '12 at 1:17
If you set the pagestyle to plain \pagestyle{plain} you should then get the desired result. – Peter Grill Jul 29 '12 at 1:18
You could use
\usepackage{nopageno}
-
Page styles are merely a collection of macros which change the style of the page. Using
\makeatletter
\let\ps@plain\ps@empty
\makeatother
merely equates the two page styles in terms of what they do, but technically does not issue any page style. That is, you have to specify the page styles to be plain before they're actually changed. Therefore, also adding
\AtBeginDocument{\pagestyle{plain}}
(anywhere in your preamble) should make all your pages be styled as plain (which is empty).
- | 2016-05-30 22:24:24 | {"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": 0.8740450143814087, "perplexity": 1762.0382390837094}, "config": {"markdown_headings": true, "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-2016-22/segments/1464051114647.95/warc/CC-MAIN-20160524005154-00025-ip-10-185-217-139.ec2.internal.warc.gz"} |
https://unapologetic.wordpress.com/2007/07/12/ | # The Unapologetic Mathematician
## The Temperley-Lieb Category
Okay, after last week’s shake-ups I’m ready to get back into the swing of things. I mentioned yesterday something called the “Temperley-Lieb Category”, and it just so happens we’re right on schedule to explain it properly.
We’ve seen the category of braids and how the braided coherence theorem makes it the “free braided monoidal category on one object”. That is, it has exactly the structure needed for a braided monoidal category — no more, no less — and if we pick any object in another such category $\mathcal{C}$ we get a unique functor from $\mathcal{B}raid$ to $\mathcal{C}$.
So of course we want the same sort of thing for monoidal categories with duals. We’ll even draw the same sorts of pictures. A point on a horizontal line will be our generating object, but we also need a dual object. So specifically we’ll think of the object as being “going up through the point” and the dual as “going down through the point”. Then we can draw cups and caps to connect an upward line and a downward line and interpret it as a duality map. Notice, though, that we can’t make any curves cross each other because we have no braiding! Here’s an example of such a Temperley-Lieb diagram:
Again, we read this from bottom to top, and from left to right. On the bottom line we have a downward line followed by an upward line, which means we start at the object $X^*\otimes X$. Then we pass through a cap, which corresponds to the transformation $\epsilon_{X^*}$. Then we go through a cup ($\eta_X^*$) to get to $X\otimes X^*$, and another cup to get to $X\otimes X^*\otimes X\otimes X^*$. A cap in the middle ($\epsilon_{X^*}$) is followed by a cup ($\eta_X$), and then another pair of caps ($\epsilon_X\otimes\epsilon_X$). Then we have a cup $\eta_{X^*}$ and another $\eta_X$ to end up at $X\otimes X^*\otimes X\otimes X^*$.
We could simplify this a bit by cancelling two cup/cap pairs using the equations we imposed on the natural transformations $\eta$ and $\epsilon$. In fact, this is probably a much easier way to remember what those equations mean. The equations tell us in algebraic terms that we can cancel off a neighboring cup and cap, while the topology of diagram says that we can straighten out a zig-zag.
Incidentally, one feature that’s missing from this diagram is that it’s entirely possible to have an arc (pointing either way) start at the bottom of the diagram and leave at the top.
Now if we have any category $\mathcal{C}$ with duals and an object $C$ we can build a unique functor from the category $\mathcal{OTL}$ of oriented Temperley-Lieb diagrams to $\mathcal{C}$ sending the upwards-oriented line to the object $C$. It sends the above diagram (for example) to the morphism
$\left[(1_C\otimes\eta_C\otimes1_{C^*})\circ\eta_{C^*}\circ(\epsilon_C\otimes\epsilon_C)\circ(1_C\otimes\eta_C\otimes1_{C^*})\circ(1_C\otimes\epsilon_{C^*}\otimes1_{C^*})\circ(1_C\otimes1_C^*\otimes\eta_{C^*})\circ\eta_{C^*}\circ\epsilon_{C^*}\right]:$
$C^*\otimes C\rightarrow C\otimes C^*\otimes C\otimes C^*$.
Another useful category is the free monoidal category with duals on a single self-dual object. This is the Temperley-Lieb category $\mathcal{TL}$ which looks just the same as $\mathcal{OTL}$ with one crucial difference: since the object $X$ is its own dual, we can’t tell the difference between the two different directions a line could go. Up and down are the same thing. In the algebra this might seem a little odd, but in the diagram all it means is we get to drop the little arrows that tell us which way to go.
And now if we have any category $\mathcal{C}$ with duals and any self-dual object $C=C^*$ we have a unique functor from $\mathcal{TL}$ to $\mathcal{C}$ sending the strand to $\mathcal{C}$. This is how Temperley-Lieb diagrams are turned into (categorified) $\mathfrak{sl}_2$ representations in Khovanov homology.
July 12, 2007 | 2015-08-01 18:13:38 | {"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": 0, "img_math": 32, "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": 0.7998284697532654, "perplexity": 289.1256077288858}, "config": {"markdown_headings": true, "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-2015-32/segments/1438042988860.4/warc/CC-MAIN-20150728002308-00077-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://ask.sagemath.org/question/24892/all-ideals-of-ring/?sort=votes | # All Ideals of Ring
For given group G, by G.subgroups(), we can list all subgroups. now my question is :
Is there something for subgroups() for finding all left ideals of a given ring. For instance, k = GF(5); M = MatrixSpace(k,2,2) How can I have all left ideals?
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There is not a single command to achieve that, but in this particular case you can do the following:
sage: k = GF(5)
sage: M = MatrixSpace(k,2,2)
sage: units = [m for m in M if m.is_invertible()]
sage: nonunits = [m for m in M if not m.is_invertible()]
sage: len(units)
480
sage: len(nonunits)
145
Since every ideal must be generated by some nonunits, but also changing a generator by a unit times it does not change the ideal, let's check how many essentially distinct generators can we have:
sage: associated = []
sage: for m in nonunits:
....: if not True in [a*m in associated for a in units]:
....: associated.append(m)
....:
sage: associated
[
[0 0] [1 0] [0 1] [1 1] [2 1] [1 2] [4 1]
[0 0], [0 0], [0 0], [0 0], [0 0], [0 0], [0 0]
]
That is, you only want to check for subsets of that set as possible generators of your ideals. But it is clear that two of those elements actually generate the other four, so you just need to check for ideals generated by up to two generators.
You have the trivial ideal, the other 6 principal ideals generated by the other elements in associated, and the only thing you need to check is if the ideal generated by two of them is the total one. It is easy to see that it is, but in case you don't notice at first sight you can just check it directly:
sage: for (a,b) in Tuples(units,2):
....: if (a*associated[1] + b*associated[2]).is_invertible():
....: print a
....: print b
....: break
....:
[0 1]
[1 0]
[1 0]
[0 1]
There you are: only six nontrivial ideals, that happen to be principal.
more
Nice. Note that instead of True in [a*m in associated for a in units], you can use the more pythonic any([a*m in associated for a in units]).
( 2014-11-28 20:34:23 -0600 )edit | 2019-11-14 07:31:40 | {"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": 0.4092777371406555, "perplexity": 830.3224360370684}, "config": {"markdown_headings": true, "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-2019-47/segments/1573496668004.65/warc/CC-MAIN-20191114053752-20191114081752-00006.warc.gz"} |
https://www.physicsforums.com/threads/minimum-velocity-of-an-object-thrown-directly-up-to-never-fall-down.752224/ | Minimum Velocity of an object thrown directly up to never fall down?
1. May 5, 2014
Nathanael
EDIT: This is in the wrong section isn't it? How do I move it to the General Physics section? (My bad.)
This isn't a homework problem (I'm not in a physics class) so hopefully this isn't the wrong section.
My question is about finding the minimum velocity needed for an object (directly upwards) to never return to Earth.
(**Ignoring air resistance and other subtleties such as other sources of gravity and such**)
I have an idea for finding it, but I've gotten stuck. My idea was to make it "choppy" (turn acceleration/velocity into being piece-wise-linear) and then take the limits as the piece size gets infinitely small.
I've turned it into an equation that I think would solve it, but the problem is that it's an infinite sum and I can't make progress towards evaluating it because I can't figure out how to generalize the nth term in terms of n. Perhaps that makes this a bit more of a math problem, (but most physics is, to an extent,) so sorry if this is in the wrong section (this is my first post).
I've posted a picture of my "solution" (I don't know where it is but I'll assume it will appear).
The equation in the red circle (at the top) is my "solution"
G is the universal constant of gravity, M is the mass of Earth, r is the radius of Earth and V is the great thing I'm trying to solve for.
My idea behind this equation:
b(subscript)1 is the change in velocity over the first height interval (Δh) because Δh divided by v is the time spent moving through that interval, and a(subscript)1 is the acceleration at that height. Then for the next height interval you replace v with v+change-in-v (in other words, replace v with v+b(subscript)1) and apply the same logic (and so on for all terms). Taking the infinite sum of this would give you the change in velocity as you got infinitely far. Taking the limit as Δh→0 would make it "more true" (because acceleration is continuous, not "choppy") and setting it equal to -v would mean you only reach zero velocity at an "infinite distance" (and that would therefore make v the minimum velocity, right?)
My questions are:
1.) Is the logic/equation correct? Would it give you the appropriate minimum velocity?
(I think it would, but then again, I ALWAYS think my logic is right, and it very often isn't.)
2.) Is the equation solvable? Is this an adequate approach to the problem? Can you solve it for v?
(I think you can, since there's only constants and v, but the infinite sum might cause problems, I'm not very mathematically advanced so I don't know)
3.) Is it possible to generalize b(subscript)n in terms of n? Is it possible to evaluate the sum without generalizing it in terms of n? (If so how?) Are there other methods of evaluating the sum?
Question 3.) is the "wall" that i've run into. I can't solve the problem because I can't figure out how to generalize b(subscript)n in terms of n. (I've generalized it in the only concise way I know how, but I don't think it's solvable like as it is. It would take some genius maths I think.)
Can anyone help with these questions? Thank you and sorry for the lengthy post.
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Last edited: May 5, 2014
2. May 5, 2014
Nathanael
Also, I'm up for discussing alternative methods.
I have one other method in mind, but I think I need to know more calculus or differential equations. The other method goes like this:
Create a function for the impact velocity depending on the initial distance away (assuming initial velocity of zero) then take the limit of that function as the initial distance (the independent variable) approaches infinity.
I THINK this would give you the same answer "v" (even though it seems like a different process) because I'm assuming it would be "symmetrical" (meaning, if you start at zero velocity at an infinite distance, then your final ("impact") velocity would be the same as the initial velocity (at earth) that ended up being zero at an infinite distance). But I don't quite know that it's true that that would give you the same answer, it just seems to be so.
But, to generalize the impact velocity as a function of the initial distance, (and therefore to solve it using this method,) I would need to integrate (integration is just the "continuous sum," correct?) the acceleration against time, but I don't know the acceleration as a function of time, I only know the acceleration as a function of the distance (newtons law of gravity). So I would need to know the distance as a function of time, but that would depend on the acceleration, and thus starts the endless circle.
I don't know how to integrate functions yet (except by anti-derivative) let alone integrate a "circular function" like that one.
(I call it a "circular function" because you need to plug your distance into the acceleration to see how your distance changes then continuously replug it in over and over at every instant of time to solve it. (I can picture 3d system (time/accl./dist.), but I don't know how to go through the circle with numbers and especially not continuously, that takes some hardcore math skill) In theory, it would be solvable, because all the information is there (I think?) but I have no idea how to tackle that sort of problem. I think that these so-called "circular functions" are the nature of what are called "differential equations," right?)
So, this second method is not really a feasible method to solving the problem (at least not for me and my limited mathematical abilities).
Last edited: May 5, 2014
3. May 5, 2014
ehild
Your idea, that the initial speed is the same as the impact speed if the stone returns, is correct. That comes from conservation of energy. It holds for gravity.
Also, you think correctly that the minimum initial speed corresponds to infinite distance of travel and infinity is never reached.
So you can find the minimum upward speed to determine the energy of the stone on the surface of the Earth: E= KE +PE= const, KE=0.5mv2, and the potential energy is PE=-GMm/r where M is the mass of Earth, m is the mass of the stone and r is the distance from the Earth centre. At the surface of the Earth, r=R, the radius of Earth. At infinity, r→ ∞ PE=0, and KE should be also zero, so KE+PE=0. It should be zero also on the surface of Earth. From that, you get the speed.
Your next question is if the stone never returns if it goes to infinity. That looks plausible but you want to prove it mathematically.
The motion of planets and satellites are described by the Universal Law of Gravitation. From that, the planets move along closed orbit (ellipse) if their energy is negative. And it was observed by Kepler and also can be derived mathematically that the relation between the time period T of the orbits and the semimajor axis "a" of the ellipse is T2/a3=constant for the same central mass, Earth in this case. if a is infinite, T is also infinite.
Throwing up a stone, it goes along a straight line and back. You can consider it as a very thin ellipse, with zero minor axis.
You can also derive the time dependence directly, and there were such problems solved here in PF, but it involves quite hard integration.
ehild
4. May 5, 2014
Nathanael
So you're saying that an up-down motion can be viewed as a special case of an elliptical orbit, and because of this, (and because of the T2/a3=c relationship,) that the only way to never return (a.k.a. to have an infinite period) is to have an infinite distance? Is that a legitimate proof?
I'm not a proof kind of person (frankly because I'm not clever enough to find them) but this is quite a brilliantly simple way of proving that.
(I am completely lost when it comes to proving something that I only know from intuition.)
KE+PE=Constant... that's a law of physics (I feel dumb )... (I'm very new to physics, I didn't think about this)
Is that where the formula KE=0.5mv2 comes from?
So you're saying that you can literally just find the v that satisfies the equation -GMm/r + mv2/2 = 0 and that will give you the same answer as the limit of my infinite sum (assuming no mistakes in my equation) and it will give you the same answer as solving that differential equation????
-GMm/r + mv2/2 = 0 ............ Wow. That's amazing, what a genius use of the KE+PE=Constant law, that is amazingly simple.... I always try to find the simplest way to solve something, but that, that's just remarkably simple.
Will that seriously give you the answer?? Or am I misunderstanding what you meant?
I did that and got 11,180 meters per second, that's quite a bit higher than I expected. (Although I suppose expectations are pretty useless in physics, and it is a some-what reasonable answer)
The idea that initially began me thinking about this problem was that I thought, "(In a hypothetical universe with just Earth and an object,) No matter how unimaginably far away an object is it will always eventually fall to the earth (Or should I say the Earth and the object will fall to each other)." Then I thought, "I wonder if, if the object is moving away from the Earth at any speed, will it still always fall to the Earth?"
One part of me said Yes, because, even though the gravity will be very weak, it will be acting against the objects motion for infinitely long. If it is acting for infinitely long, it shouldn't matter how weak it is.
Then another part of me said No, it wouldn't fall to Earth, because the rate at which the acceleration is slowing down will be far faster than the rate at which the speed is slowing down.
But then another part of me said "But it will still be a finite deceleration for an infinite time, so it will still EVENTUALLY slow to zero velocity then turn around and come back."
But then part of me said "but the velocity could be decreasing for infinitely long yet still never get below a certain value, (like an asymptote) because the acceleration is decreasing so much faster that there must be some limit involved (like an asymptote)"
This train of thought eventually led to the question I asked in my original post. I concluded that if the minimum velocity to never fall back down were infinite, then an object moving away from Earth at ANY speed and ANY distance would *EVENTUALLY* fall back to Earth, and if the minimum velocity were finite, then it didn't necessarily have to fall back to Earth (I'm still talking about this hypothetical Universe with just Earth and the object).
I also concluded that the minimum velocity would in fact be finite (and you could potentially never return to earth if you're moving away quick enough (which at a reasonable distance, probably wouldn't need to be very quick))
However, all my "conclusions" are actually just intuitions....
So I want to ask you (and anyone else) how would you approach these conflicting ideas? (since you're an actual physicist and I'm just a naive kid)
Last edited: May 5, 2014
5. May 5, 2014
ehild
Nathanael,
It is very good that you are trying to understand Physical phenomena, and your method is very similar to those of ancient Greeks.... But a lot of things have been discovered since then. Physics became 90 percent Mathematics. There are some basic laws, as Newton's Laws in Mechanics, which, combined with Calculus (founded also by Newton) describe quite accurately how a thrown stone, a spaceship, a satellite, the planets move. Applying the laws of Physics, we were able to send satellites orbiting around the Earth and seeing even your car with the GPS; keeping the International Space Station (ISS) on orbit and sending supply to the people working on board there. You can see the ISS with naked eye moving among the stars on the heaven if you look at right place in right time, as the motion of the ISS can be accurately predicted. It was also possible sending rockets to the Mars or even father, to the Saturn, landing on its moon Titan, and sending back data and pictures from its surface. All our present life depends on tools which were invented and created by applying the laws of Physics and Maths.
You need to learn the basics of Physics and Maths. Do it systematically, starting from the beginning.
ehild
6. May 5, 2014
ehild
Well, it can be proved formally, by setting up and solving equations. That was an explanation without Maths.
You asked about an object thrown up vertically. It would return as it falls back on the same way as it started, if it is launched from one of the poles. Otherwise, it would have some side-way component of velocity, equal to the velocity of the surface of Earth.
If the stone could penetrate and fall through the Earth without loss of energy, it would oscillate away and back forever.
Allowing enough horizontal velocity, the stone never falls down, but orbits above the surface of the Earth. For such circular orbit, the force of gravity equals the centripetal force for the circular motion. The minimum horizontal speed is mv2/R=GmM/R2. But the gravitational force at the surface of Earth is equal to mg=9.81m. So the minimum orbital velocity is just v=√(gR)
A Physicist can not exist without intuition, but he needs also a solid knowledge.
KE is abbreviation of kinetic energy and PE stands for potential energy. The kinetec energy of a small particle is 0.5 mv2. The kinetic energy of an extended body is the sum of the KE-s of all its particles. The potential energy is related to force. If the force is conservative, the sum of the potential energy and the kinetic energy of the body is constant.
You have the equation mass times acceleration = force. That means a differential equation for the position vector. If you solve it, you get the same relation between distance and velocity as you get from conservation of energy. Actually, conservation of energy is derived from Newton's second equation F=ma by integration.
I suggest to get a Calculus-based Mechanics book and read... You will find how the motion in gravitational field can be solved.
ehild | 2017-12-12 09:04:00 | {"extraction_info": {"found_math": false, "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": 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": 0.8146528601646423, "perplexity": 395.5562797694777}, "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-2017-51/segments/1512948515311.25/warc/CC-MAIN-20171212075935-20171212095935-00191.warc.gz"} |
http://codeforces.com/blog/entry/53054 | ### altruist's blog
By altruist, history, 4 years ago, translation,
Hello everybody. Recently, I was thinking about a one problem and now I want to share it with you:
You have two arrays of the same length n, and you have to calculate minimum number of swaps of two arbitrary indexes which transform the first array A into the second B. ( All elements in arrays are not neccessery distinct ) I know how to solve this problem when all elements are distinct in O(n). Do you know how to solve common problem in effective asymptotics?
• +64
» 4 years ago, # | ← Rev. 2 → +9 Check out this problem from AtCoder. It's quite possible that you can use a very similar approach (and the answer will be slightly smaller as you can swap arbitrary elements).
» 4 years ago, # | 0 If all elements are distinct then solution is with dfs (cycle counting), am I right?
• » » 4 years ago, # ^ | +3 Yes
• » » 4 years ago, # ^ | 0 i know o(nlogn) solution by sorting and using fenwick tree, no matter if elements distinct or not ?may you explain your o(n) solution for the problem if all elements is different ?
• » » » 4 years ago, # ^ | ← Rev. 2 → +1 Here check my code for finding minimum swaps to sort (I think it is understandable from code, if you didn't get, write): intt n, arr[maxx]; intt used[maxx]; intt cycle = 0; void dfs (intt v) { used[v] = 1; if (used[arr[v]]) cycle ++; else dfs (arr[v]); } int main() { cin >> n; for (intt i = 1; i <= n; i++) cin >> arr[i]; /// ONLY PERMUTATION OF NUMBERS 1 to n for (intt i = 1; i <= n; i++) if (!used[i]) dfs (i); cout << n - cycle << endl; return 0; }
• » » » » 4 years ago, # ^ | -6 he didn't ask for converting the first array to a sorted array.he asks that he will give you two different arrays, and you want to convert the first one to the second in minimum number of swaps.like 1 6 3 4 2 to 6 3 1 2 4ans will be 3is there o(n) solution for that !
• » » » » » 4 years ago, # ^ | +4 These problems are almost same, just fix array B to {1, 2, 3, .., n}
• » » » » » » 4 years ago, # ^ | 0 i think there are not the same, and by the way, i tested your solution for 3 2 5 1 4 and it output 2, what that means, is it required 2 swaps to sort the element ? how ?
• » » » » » » » 4 years ago, # ^ | ← Rev. 2 → +1 almost same ≠ fully sameCheck code belowBtw, my code prints 3 for your array...
• » » » » » 4 years ago, # ^ | ← Rev. 2 → +14 Here I just edited code for 2 arrays :) Codeintt n, arr[maxx], brr[maxx]; intt pos[maxx], used[maxx]; intt cycle = 0; void dfs (intt v) { used[v] = 1; if (used[pos[arr[v]]]) cycle ++; else dfs (pos[arr[v]]); } int main() { cin >> n; for (intt i = 1; i <= n; i++) cin >> arr[i]; for (intt i = 1; i <= n; i++) cin >> brr[i], pos[brr[i]] = i; for (intt i = 1; i <= n; i++) if (!used[i]) dfs (i); cout << n - cycle << endl; return 0; }
• » » » » 4 years ago, # ^ | 0 Can we count the inversions of a permutation on N numbers in O(N) complexity?
• » » » » » 4 years ago, # ^ | 0 No
• » » » » 4 years ago, # ^ | 0 Would you please elaborate that technique to me or share some links? ( I am TooNewbie :) )
» 4 years ago, # | +57 Let's build directed graph G = (V, E). V ={a[i] | i = 1 .. n}. E = {(b_i, a_i) | i = 1 .. n}. Now we have to cover this graph with maximal number of edge-disjoint cycles.Also we can generate arrays for each graph with income degrees equals to outcome degrees.I think that this problem is NP-hard. But I would happy to know solution if I am not correct.I couldn't found appropriate article. But for example here http://www.math.ucsd.edu/~jverstra/cycle2.pdf in Introduction is information that problem of packing maximal count of edge-disjoint cycles in graph (both directed and undirected) is NP-hard. I think that our problem is similar.
• » » 4 years ago, # ^ | ← Rev. 6 → +15 Yea, I think you can do a reduction from the problem you linked. Let's make a distinction:Maximal cycle packing: Find a maximal number of simple edge-disjoint cycles in (V, A), not necessarily covering all arcs. Link claims this is NP-Complete*.Maximal cycle cover: Find a maximal number of simple edge-disjoint cycles in (V, A), such that each edge is covered by a cycle. We want to prove this is NP-Complete.Lemma 1: a graph has a cycle cover if and only if indeg(v) = outdeg(v) for all v (trivial)Suppose we can solve the maximal cycle cover problem efficiently. Given a digraph (V, A) to solve the maximal cycle packing problem on, we add one extra vertex u', and for every , if indeg(v) > outdeg(v) we add indeg(v) - outdeg(v) arcs** from v to u', and vice versa if the inequality reverses. This gives us a new digraph (V', A'). Note: clearly indeg(v) = outdeg(v) for all v (including u') in the end, so (V', A') has a cycle cover.Now suppose that you have an optimal cycle packing on (V, A) with k simple cycles, then you can extend this to a solution for the cycle cover problem on (V', A') with k + indeg(u'): first we remove all these cycles from (V', A') (recall that (V, A) is contained in (V', A')). This doesn't change that fact that indeg(v) = outdeg(v) for all v, so the resultant graph still has a cycle cover. In addition, there are no cycles that don't pass through u' (otherwise, we could add this cycle to our packing for (V, A)). Thus, after removing the cycle packing on (V, A) from (V', A'), we can decompose the remains of the graph into exactly indeg(u') simple cycles (quickly, using e.g. Hierholzers algorithm). So we proved:Thm 1: A optimal packing of k cycles in (V, A) gives rise to a cover of k + indeg(u') cycles in (V', A').Conversely, suppose we have an optimal cover of k' simple cycles over (V', A'), then there must be exactly indeg(u') simple cycles going through u' (_exactly_ indeg(u') since this is a cover). Remove these cycles, and you end up with a cycle packing with k' - indeg(u') simple cycles in (V, A). Thus:Thm 2: An optimal cover of k' cycles in (V', A') gives rise to a packing of k' - indeg(u') cycles in (V, A).This clearly implies that the optimal solution to the cycle packing problem on (V, A) differs from the cycle cover problem on (V', A') by indeg(u') (a constant that depends only on (V, A)), and that we can efficiently reconstruct such a packing, given an optimal cover. Thus, the cycle cover problem is also NP-Complete.*: The linked paper claims it is NP-Complete but doesn't provide a proof. I can't find any either (other than two more papers making the claim without a proof or citation).**: Not sure if we're allowing multi-edges, but clearly you can replace k copies of (u, v) with (u, wi) and (wi, v) for k new vertices wi, without really affecting the problem. | 2021-01-23 13:54:43 | {"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": 0.4917525351047516, "perplexity": 1133.1005993989115}, "config": {"markdown_headings": true, "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-2021-04/segments/1610703538082.57/warc/CC-MAIN-20210123125715-20210123155715-00317.warc.gz"} |