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https://www.vedantu.com/question-answer/explain-the-steps-of-finding-the-square-root-of-class-10-maths-cbse-5efc6583fc7b62454bf5e5f6	QUESTION # Explain the steps of finding the square root of 1369 by division method. Hint- The square root of a number by using a long division method is described in the solution, we first group the digits in the number. Differentiate properly between the period, quotient and remainders and follow the steps mentioned below. Square root of a perfect square by the long division method 1. Group the digits in pairs, starting with the digit in the units place. Each pair and the remaining digit (if any) is called a period i.e, $\overline {13} \overline {69}$, 13 and 69 are periods here. 2. Find the largest number whose square is equal to or just less than the first period.Take this number as the divisor and as the quotient. This number is 3 because 33=9 3 3 $\overline {13} \overline {69}$ 3. Subtract the product of the divisor and the quotient from the first period and bring down the next period to the right of the remainder. This becomes the new dividend. New dividend is 469. 3 3 $\overline {13} \overline {69} \\\;\;9$ $\;\;469$ 4. Now, the new divisor is obtained by taking two times the quotient ( 32) and annexing with it a suitable digit ( which is 7) which is also taken as the next digit of the quotient chosen in such a way that the product of the new divisor and this digit ( which is 67*7=469) is equal to or just less than the new dividend. 37 3 $\overline {13} \overline {69} \\\;\;9$ 67 $\;\;469 \\ \;\;469$ $\;\;\;\;0$ ∴$\sqrt {1369} = 37$ Note- Finding the square root of a number is the inverse operation of squaring that number. Remember , the square of a number is that number times itself. The perfect squares are the squares of the whole numbers.	2020-07-13 02:26:41	{"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": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8012703657150269, "perplexity": 309.10269783766006}, "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-2020-29/segments/1593657140746.69/warc/CC-MAIN-20200713002400-20200713032400-00161.warc.gz"}
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My goal is to plot my image and my function in the same coordinates. Update I drew an image of size ...	2015-11-26 19:46:24	{"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": 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": 0.7263926267623901, "perplexity": 2033.32458361725}, "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-2015-48/segments/1448398447773.21/warc/CC-MAIN-20151124205407-00348-ip-10-71-132-137.ec2.internal.warc.gz"}
https://linearalgebras.com/solution-abstract-algebra-exercise-2-1-10.html	If you find any mistakes, please make a comment! Thank you. ## An arbitrary intersection of subgroups is a subgroup Let $G$ be a group. (1) Prove that if $H$ and $K$ are subgroups of $G$, then so is $H \cap K$. (2) Prove that if $\{H_i\}_{i \in I}$ is a family of subgroups of $G$ then so is $\bigcap_{i \in I} H_i$. Solution: (1) Note that $H \cap K$ is not empty since $1 \in H \cap K$. Now suppose $x, y \in H \cap K$. Since $H$ and $K$ are subgroups, we have $xy^{-1} \in H$ and $xy^{-1} \in K$ by the subgroup criterion; thus $xy^{-1} \in H \cap K$. By the subgroup criterion, $H \cap K$ is a subgroup of $G$. (2) Note that $\bigcap_{i \in I} H_i$ is not empty since $1 \in H_i$ for each $i \in I$. Now let $x,y \in \bigcap_{i \in I} H_i$. Then $x,y \in H_i$ for each $i \in I$, and by the subgroup criterion, $xy^{-1} \in H_i$ for each $i \in I$. Thus $xy^{-1} \in \bigcap_{i \in I} H_i$. By the subgroup criterion, $\bigcap_{i \in I} H_i$ is a subgroup of $G$.	2022-07-03 23:59:14	{"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": 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": 0.9771947264671326, "perplexity": 62.69488397205685}, "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/1656104277498.71/warc/CC-MAIN-20220703225409-20220704015409-00198.warc.gz"}
https://www.khanacademy.org/math/arithmetic/fraction-arithmetic/arith-review-fractions-on-the-number-line/e/fractions_on_the_number_line_2	Unit fractions on the number line Use unit fractions to think about the location of other fractions on the number line. Problem Move the dot to start fraction, 2, divided by, 4, end fraction on the number line.	2016-10-01 13:58:02	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "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": 0.5516324639320374, "perplexity": 1319.655311972566}, "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-2016-40/segments/1474738662882.88/warc/CC-MAIN-20160924173742-00030-ip-10-143-35-109.ec2.internal.warc.gz"}
https://solvedlib.com/n/find-the-resultant-field-at-point-due-to-the-3-nc-charge,15566942	# Find the resultant field at point _ due to the -3 nC charge and the +6 nC charge = arranged ###### Question: Find the resultant field at point _ due to the -3 nC charge and the +6 nC charge = arranged as shown, nC 16nC #### Similar Solved Questions ##### The standard oxidation potential for the reaction $\mathrm{Cr}(s) \rightarrow$ $\mathrm{Cr}^{3+}(a q)+3 \mathrm{e}^{-}$ is $0.74 \mathrm{~V}$. Despite the large, positive oxidation potential, chromium is sometimes used as a protective coating on steel. Why doesn't the chromium corrode? The standard oxidation potential for the reaction $\mathrm{Cr}(s) \rightarrow$ $\mathrm{Cr}^{3+}(a q)+3 \mathrm{e}^{-}$ is $0.74 \mathrm{~V}$. Despite the large, positive oxidation potential, chromium is sometimes used as a protective coating on steel. Why doesn't the chromium corrode?... ##### Question Completion Status: If a sample of H2O contains 9.02 1023 atoms how many grams 'of H2O is this? OA 3.00 g Hzo 0 B. 9.00 g H20 0c27.0 g Hzo 0 0.9.02 g HzoQUESTION 4How many moles of 0 are there in 12.4 mol Fe(NO3)3? 0A 12.4 0 B.37.2 0c4.13 1120 E none of the aboveQUESTION 5A 42.7 gram sample of potassium nitrate contains how many grams of potassium? 04 39.\| 0B. 16.5C.21,400,8.540 E. nonc of the above Question Completion Status: If a sample of H2O contains 9.02 1023 atoms how many grams 'of H2O is this? OA 3.00 g Hzo 0 B. 9.00 g H20 0c27.0 g Hzo 0 0.9.02 g Hzo QUESTION 4 How many moles of 0 are there in 12.4 mol Fe(NO3)3? 0A 12.4 0 B.37.2 0c4.13 112 0 E none of the above QUESTION 5 A 42.7 gr... ##### Use H-lest > test the caim aboul Ihe populalion mean al the given lovel signifcance using Ine given sample statistics. Assume Ihe populallon ona Jly distribuled. Claim; 0.05 Sample slatistics; < = 21.8, 4.9,n=14 Wunat / Ihe value the standardized test statistic?The standardized test statistic (RoundIo decima placesneeded \|What =P-value of the test stalistic?Pavalue(Round Jnret decimal Placesneeded \| Use H-lest > test the caim aboul Ihe populalion mean al the given lovel signifcance using Ine given sample statistics. Assume Ihe populallon ona Jly distribuled. Claim; 0.05 Sample slatistics; < = 21.8, 4.9,n=14 Wunat / Ihe value the standardized test statistic? The standardized test statisti... ##### Calculate the amount of energy in kilojoules needed to change 297 g of water ice at ~10 to steam at 125 C. The following constants may be useful: Cm (ice) 36.57 J / (mol Cm (water) 75.40 J / (mol Cm steam) 36.04 J / (mol 4Hfus +6.01 kJ/mol AHvap +40.67 kJ /mol Calculate the amount of energy in kilojoules needed to change 297 g of water ice at ~10 to steam at 125 C. The following constants may be useful: Cm (ice) 36.57 J / (mol Cm (water) 75.40 J / (mol Cm steam) 36.04 J / (mol 4Hfus +6.01 kJ/mol AHvap +40.67 kJ /mol... ##### Indications and patient education for hemoglobin? Indications and patient education for hemoglobin?... ##### Find the ML, MAP and MAE estimates for problem 4 The attachment has the MS estimate... Find the ML, MAP and MAE estimates for problem 4 The attachment has the MS estimate solution for that problem. 4. (20%) Suppose that a signal s that takes on values 1 and -1, with probability p and l-p respectively, is sent from location A. The signal received at location B is Normally distributed w... ##### Calculate the number of distinct subsets for each seta) {2, 4, 6,8} ____________.b) {1/2, 1/4, 1/6,1/8} ____________.c) {a,b,c,d} ____________. Calculate the number of distinct subsets for each set a) {2, 4, 6,8} ____________. b) {1/2, 1/4, 1/6,1/8} ____________. c) {a,b,c,d} ____________.... ##### Part AA15.0-L scuba diving tank contains a helium-oxygen (heliox) mixture made up of 23.7 g of He and 4.53 g of 02 at 298 K Calculate the mole fraction of each component in the mixture Express your answers separated by commaAzdx(He). x(02) 0.0211Your submission doesn't have the correct number of answers Answers should be separated with comma No credit lost Try again.SubmitPrevious Answers Request AnswerPart B Complete previous part(s)Part C Complete previous part(s) Part A A15.0-L scuba diving tank contains a helium-oxygen (heliox) mixture made up of 23.7 g of He and 4.53 g of 02 at 298 K Calculate the mole fraction of each component in the mixture Express your answers separated by comma Azd x(He). x(02) 0.0211 Your submission doesn't have the correct numb... ##### (J) Find' thve nrcn of tho rcgion wtich boun-lcd bctwzca tha papi ov = 3 6 and 7I+16S2I (J) Find' thve nrcn of tho rcgion wtich boun-lcd bctwzca tha papi ov = 3 6 and 7I+ 16S2I... ##### Please do all parts !!! 2. Suppose G is a group and H is a subgroup... Please do all parts !!! 2. Suppose G is a group and H is a subgroup of G. Definition 5.36 defines the normalizer of H in G as Mo(H) := {g є GİgHy-1-H). This is also a subgroup of G (you do not need to prove it.) (a) 3 pts Consider H-(1,2,3,4) > as a subgroup of S4. Find Ns, (H) (b) [8... ##### In testing the lethal concentration of a chemical found inpolluted water, researchers have determined that a certainconcentration will kill 20% of the fish that are subjected to itfor 24 hours. If 20 fish are placed for 24 hours in a tankcontaining this concentration of the chemical, find theprobabilities of the following events.a. Exactly 14 survive.b. At least 10 survive.c. At most 16 survive. In testing the lethal concentration of a chemical found in polluted water, researchers have determined that a certain concentration will kill 20% of the fish that are subjected to it for 24 hours. If 20 fish are placed for 24 hours in a tank containing this concentration of the chemical, find the pr... ##### The confuktsmci 4544n4v4%4 [Pcbe \|W teaVluakua " @ Gcal to Utv populsteon} [M{4M \|iomn plun Mnnt Ela uluet aa elle UieQuestion 29ptaSuppose that the average country sOng length i ArrericaI4,7' mirutes wilh; sdurd deviation of 4 10 minutes It is known Ihat song length (5 nol normally distributed, Suppose smnple: ot 25 sonp: \| taken tromn thc population Which of the following (istributions follow normal distriluuution? Sclc t all Vh:t applveampbnt des tatutignWmol distrbutionVubabalty The confuktsmci 4544n4v4%4 [Pcbe \| W tea Vluakua " @ Gcal to Utv populsteon} [M{4M \|iomn plun Mnnt Ela uluet aa elle Uie Question 29 pta Suppose that the average country sOng length i ArrericaI4,7' mirutes wilh; sdurd deviation of 4 10 minutes It is known Ihat song length (5 nol normal... ##### Based upon your answers to part a) and d) does the melted ice ever reach 100PC2 (1 Point)0 Once equilibrium is reached, what is the temperature of the system? (6 points) Based upon your answers to part a) and d) does the melted ice ever reach 100PC2 (1 Point) 0 Once equilibrium is reached, what is the temperature of the system? (6 points)... ##### 4. The following table summarizes the results of a series of 100 exams English course: from... 4. The following table summarizes the results of a series of 100 exams English course: from an Test 1 18 0 Test 2 15 Test 3 14 Test 4 16 Pass Fail Test 5 (Total) 12 5 5 25 If I pick one test at random, what is the probability that: a. The test picked is a passing test? b. The test picked is a failur... ##### 6 points) Let B = {(4,-1), (2,1)} be a (ordered) basis for R?_ and let E = {(1,0) , (0,1)} be the standard basis_ Find the change-of-basis matrix PEsB. Find the change-of-basis matrix PB+E If v = (2,3) , find [v]B- 6 points) Let B = {(4,-1), (2,1)} be a (ordered) basis for R?_ and let E = {(1,0) , (0,1)} be the standard basis_ Find the change-of-basis matrix PEsB. Find the change-of-basis matrix PB+E If v = (2,3) , find [v]B-... ##### 2) The Fall semester GPAs of a random sample of students (fictional data) are below 2.5,... 2) The Fall semester GPAs of a random sample of students (fictional data) are below 2.5, 3.0, 3.1, 4.0, 3.2,1.9,2.7,2.9, 3.5, 3.6, 3.1,2.9, 4.1,2.8, 2.9, 3.1, 2.6, and 1.5. 2a) Construct a stem-and-leaf plot to describe the data. 2 points) 2b) Based on the stem-and-leaf plot, describe the data distr... ##### Calculate the mass of O2 produced of 3.450g potassium chlorate is completely decolmposed by heating in the presence of a catalyst (Magnese dioxide) Calculate the mass of O2 produced of 3.450g potassium chlorate is completely decolmposed by heating in the presence of a catalyst (Magnese dioxide).... ##### Table af intcorals Gyulualc Int inotiinnu Intcoral This integral require s prolirinany won heroundcomowunaint squury Or chanoing Yariaolue buior[v ~282 Wnich ollowing Ina alue Olinae ern inlegral?(m\|z-14+442 - 28r\|+c (n\|x+442 - 198\|+c In\|x-i4+422 - 198 \|+C In \|x+ 442 _ 28x \| table af intcorals Gyulualc Int inotiinnu Intcoral This integral require s prolirinany won heround comowunaint squury Or chanoing Yariaolue buior [v ~282 Wnich ollowing Ina alue Olinae ern inlegral? (m\|z-14+442 - 28r\|+c (n\|x+442 - 198\|+c In\|x-i4+422 - 198 \|+C In \|x+ 442 _ 28x \|... ##### Which triacylglycerol yields more energy on oxidation: one containing three resides of linolenic acid or three... Which triacylglycerol yields more energy on oxidation: one containing three resides of linolenic acid or three residues of stearic acid? Explain why it fully reduced and what is being fully reduced... ##### Patient A is provided with an antibiotic medication to treat anear infection. Several years later a similar infection is treatedwith the same antibiotic. The patient takes the antibiotic over alonger period and in greater strength. The infection does notclear. The bacteria causing the infection is said to have"developed" resistance to the antibiotic. Does the microberesistance provide evidence of "evolution"? Write an explanation tosupport your response. Patient A is provided with an antibiotic medication to treat an ear infection. Several years later a similar infection is treated with the same antibiotic. The patient takes the antibiotic over a longer period and in greater strength. The infection does not clear. The bacteria causing the infection ... ##### Dog shelter bopes bolstcr adoption rates by showing the posilive eflects that unimals hute Rercon crnotional salc, Tbey 46 mood 20 inividuals belore exposune thc dops (Tine ) and acain allcr [0 minute s of intcracling with the dogs (Time 2) Stale }our null (Ho) and altcrnative (Hi) hypothesis (In context) and usc deicrming YOu Iare critical value}Diflercncc scores bctwecn Time Q \| jc were culculated, producing Mos 9.2 with Suns 48. Calculate tke Imrea 4FA mke decision rcgatding thc null; thcn wr dog shelter bopes bolstcr adoption rates by showing the posilive eflects that unimals hute Rercon crnotional salc, Tbey 46 mood 20 inividuals belore exposune thc dops (Tine ) and acain allcr [0 minute s of intcracling with the dogs (Time 2) Stale }our null (Ho) and altcrnative (Hi) hypothesis (In co... ##### An electron, relased in a region where the electric field is uniform, is observed to have... An electron, relased in a region where the electric field is uniform, is observed to have an acceleration of 7.0e14 m/s^2 in the positive x direction. (A) Determine the electric field producing the acceleration. I found this to be 3.99e3 N/C that it points towards -x. (B) Assuming the electron is re... ##### Let R be the region bounded by the following curves. Use the shell method to find... Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the x-axis. y= 10 - x, y = x, and y= 0 Set up the integral that gives the volume of the solid. Use increasing limits of integration. Select the correct choice ... ##### Prepare the adjusting enteries at July 31 assuming that adjusting entries are made monthly. Cupei 4... prepare the adjusting enteries at July 31 assuming that adjusting entries are made monthly. Cupei 4 Homework Dunia+2980608-3a1a-4735-9656-1819bcd0abc Question 4 of 6 < > Current Attempt in Progress The ledger of Pina Colada Corp. on July 31, 2022, includes the selected accounts below befo... ##### Description Zac Ltd and Tess Lid entered into a joint operation known as Trouble Ltd to... Description Zac Ltd and Tess Lid entered into a joint operation known as Trouble Ltd to collect and recycle empty water bottles from government departments on 1 January 2020. The joint operation agreement states that Zac Ltd and Tess Ltd will share output contributions and costs equally and that Zac...	2022-08-10 17:46:34	{"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": 0.5598557591438293, "perplexity": 9261.452700161974}, "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-2022-33/segments/1659882571198.57/warc/CC-MAIN-20220810161541-20220810191541-00567.warc.gz"}
https://tex.stackexchange.com/questions/58599/acute-grave-accents-upper-case-versus-low-case	# Acute/grave accents: upper case versus low case The acute and grave accents on capital letters are more squat and flatter than on lower case letters. Compare \'A with \'a. What is the code for the squatter acute/grave on capitals if I want it insolation? Rationale: I need capital letters with macron and acute/grave. If I put \'{} on top of \=A, then the letter is too tall for the interline spacing, partly because \'{} produces the 45-degree accent of \'a (and partly because the default height of the macron is too high). \documentclass[12pt]{article} \begin{document} If the difference between the accents of \'A, \'a isn't apparent, try superimposing them: \makebox[0pt]{\'A}\makebox[0pt]{\raisebox{.6ex}{\'a}}\,\raisebox{.6ex}{\'a}. See how the character on the left has a wedge (as opposed to the acute on the right-hand \'a)? The two accents don't exactly overlay, because the acute on \'A is flatter. \end{document} • The code depends on the font encoding you are using. In T1 encoding \'{A} is a glyph (\char193), \'{a} is \char225. The accent alone is \char1. – Ulrike Fischer Jun 5 '12 at 7:33 • @doncherry I've added code and pictures – Daniel Harbour Jun 5 '12 at 8:33 • @UlrikeFischer \char1 yields only the accent used on lower case letters (e.g., \'a, \'n). I checked Mittelbach's charts for various encodings (including T1, which I'm using). I can't see a char number for the squat acute/grave used on capitals. – Daniel Harbour Jun 5 '12 at 8:36 • @Daniel Harbour of course you don't see an accent character: t1 encoding only covers certain characters, and doesn't provide composites for characters already in its repertoir. \usepackage[T1]{fontenc} (in latex) will read the sequence \' A, and produce the t1-encoding character for A-acute-accent – wasteofspace Jun 5 '12 at 8:45 • Well there isn't. In T1 encodings there are properly designed, real glyphs with accents (Á á) and there is one "lonely" accent which can be used for chars for which accented glyphs doesn't exist. There isn't enough place in T1 to store all sort of versions of this accent in various angles. If you don't like this "lonely" accent you will have to design one. You can't extract the accent from the Á: It is attached to the glyph like the left leg of the A. – Ulrike Fischer Jun 5 '12 at 8:48 You find the isolated acute accent used for capital letters in the encoding TS1, where it's called \capitalacute: \documentclass{article} \usepackage[T1]{fontenc} \usepackage{textcomp} \begin{document} \capitalacute{A}\'{A}\\ \capitalacute{}\'{}\\ {\fontencoding{TS1}\selectfont\char 1}\char1 \end{document} You maybe able to do minor adjustments using the slant dimension. Here is some code I had for a similar issue: \documentclass[10pt]{article} \usepackage{graphicx} \begin{document} \font \romanfont=cmr10 \font\specroman=cmr10 %% Next, the special registers \newdimen\savedvalue \savedvalue=\fontdimen5\romanfont \newdimen\specialvalue \specialvalue=4.75pt %% Finally, definitions. \def\specrm{% \aftergroup\restoredimen \fontdimen5\specroman=\specialvalue \specroman } \def\restoredimen{% \fontdimen5\romanfont=\savedvalue } \scalebox{10}{\romanfont \'A} {\scalebox{10}{\specrm \' A}} \medskip {\noindent\obeylines\specrm the value of fontdimen1 (slant) is \the\fontdimen1\font the value of fontdimen2 (interword space) is \the\fontdimen2\font the value of fontdimen3 (interword stretch) is \the\fontdimen3\font the value of fontdimen4 (interword shrink) is \the\fontdimen4\font the value of fontdimen5 (x-height) is \the\fontdimen5\font the value of fontdimen6 (quad width) is \the\fontdimen6\font the value of fontdimen7 (extra space) is \the\fontdimen7\font } \medskip \font \romanfont=cmr10 \font\specroman=cmr10 %% Next, the special registers \newdimen\savedvalue \savedvalue=\fontdimen1\romanfont \newdimen\specialvalue \specialvalue=0.15pt %% Finally, definitions. \def\specrm{% \aftergroup\restoredimen \fontdimen1\specroman=\specialvalue \specroman } \def\restoredimen{% \fontdimen1\romanfont=\savedvalue } \scalebox{10}{\romanfont \'A} {\scalebox{10}{\specrm \' A}} \medskip {\noindent\obeylines\specrm the value of fontdimen1 (slant) is \the\fontdimen1\font the value of fontdimen2 (interword space) is \the\fontdimen2\font the value of fontdimen3 (interword stretch) is \the\fontdimen3\font the value of fontdimen4 (interword shrink) is \the\fontdimen4\font the value of fontdimen5 (x-height) is \the\fontdimen5\font the value of fontdimen6 (quad width) is \the\fontdimen6\font the value of fontdimen7 (extra space) is \the\fontdimen7\font } \end{document} There is also a package you can have a look at https://github.com/zellyn/accentbx • What a great package. Thanks for the pointer and for the code. Gives me a much neater way of lowering my macrons (in computer modern, they're too high for my taste). – Daniel Harbour Jun 5 '12 at 12:46 Your MWE does not show the issue that you show in the image. I have adjusted it so that it does: \documentclass[12pt]{article} \begin{document} If the difference between the accents of \'A, \'a isn't apparent, try superimposing them: \makebox[0pt]{\'A}\makebox[0pt]{\raisebox{.6ex}{\'a}}\,\raisebox{.6ex}{\'a}. See how the character on the left has a wedge (as opposed to the acute on the right-hand \'a)? The two accents don't exactly overlay, because the acute on \'A is flatter. \makebox[0pt]{\'A}\makebox[0pt]{\raisebox{.6ex}{\'a}}\,\raisebox{.6ex}{\'a} \fontencoding{T1}\selectfont \makebox[0pt]{\'A}\makebox[0pt]{\raisebox{.6ex}{\'a}}\,\raisebox{.6ex}{\'a} \end{document} As you will see in the image from the modified MWE, using the default OT1 encoding the accented characters are made by placing an accent character over the base, so the accents look the same. In the final line the T1 encoding is used this uses twice as many glyphs per font, including designed accented characters a slightly flatter accent design is used for capital letters in the EC font design. • Thanks, but using the more upright accent for capitals gets me into trouble with interline spacing. Check out egreg's answer below. – Daniel Harbour Jun 5 '12 at 9:11 • Yes, but my point was that there is not a different accent in the encoding used by your MWE. Your example tex source and example image were not generated by the same code (or you have a non standard LaTeX installation that has changed the default encoding) – David Carlisle Jun 5 '12 at 9:17 • Ah, missed your point. Thanks for clarifying the question. – Daniel Harbour Jun 5 '12 at 12:40	2021-04-17 05:47:24	{"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": 0.6662327647209167, "perplexity": 7118.086432126426}, "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-2021-17/segments/1618038101485.44/warc/CC-MAIN-20210417041730-20210417071730-00306.warc.gz"}
https://lavelle.chem.ucla.edu/forum/viewtopic.php?p=203504	## 6c.19 f Acidity $K_{a}$ Basicity $K_{b}$ The Conjugate Seesaw $K_{a}\times K_{b}=K_{w}$ Fdonovan 3D Posts: 101 Joined: Sat Aug 17, 2019 12:16 am ### 6c.19 f In the answer key, it says H2CO3 is the stronger acid because C is more electronegative. However, I thought that when comparing acids in which the only difference is the central atom, you compare the bond strengths for elements in the same group. Since Ge and C come from the same group (not the same period), shouldn't the justification come from comparing the bond strengths, not the electronegativity? Ethan Lam 4A Posts: 69 Joined: Thu Jul 11, 2019 12:17 am ### Re: 6c.19 f Electronegativity and bond strengths both play a factor in determining acid strength. Since it is more electronegative, it becomes more polar which makes it easier to dissociate. Ruby Richter 2L Posts: 103 Joined: Thu Jul 25, 2019 12:17 am ### Re: 6c.19 f So, in this case, the difference in electronegativity would dominate over the difference in bond strength?	2020-07-09 03:13:51	{"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": 0, "img_math": 0, "codecogs_latex": 3, "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": 0.5418182611465454, "perplexity": 4698.0905899915}, "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/1593655897844.44/warc/CC-MAIN-20200709002952-20200709032952-00506.warc.gz"}
https://www.lebesgue.fr/content/sem2016-geo-sto-surgailise	placeholder # Conference - Stochastic Geometry Wednesday, April 6, 2016 - 10:40 to 11:20 Donatas Surgailis Vilnius University Scaling transition for nonlinear random fields with long-range dependence Abstract: We obtain a complete description of anisotropic scaling limits and the existence of scaling transition for nonlinear functions of stationary linear random fields on ${\Bbb Z}^2$ with moving average coefficients decaying at possibly different rate in the horizontal and vertical direction. The paper extends recent results on scaling transition for linear random fields in Puplinskaitė and Surgailis (2014, 2015). The talk is based on joint work with Vytautė Pilipauskaitė (Nantes/Vilnius). slides	2019-02-18 06:55:25	{"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": 0.497431218624115, "perplexity": 1595.7109995215199}, "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-2019-09/segments/1550247484689.3/warc/CC-MAIN-20190218053920-20190218075920-00106.warc.gz"}
https://math.stackexchange.com/questions/4507996/conditional-probability-measure-theoretic-definition	# Conditional probability measure theoretic definition Let $$(\Omega, \mathcal{F}, P)$$ be a probability space, $$\mathcal{G} \subseteq \mathcal{F}$$ a $$\sigma$$-field in $$\mathcal{F}$$. Given $$A \in \mathcal{F}$$, the Radon-Nikodym theorem implies that there is $${ }^{[3]}$$ a $$\mathcal{G}$$-measurable random variable $$P(A \mid \mathcal{G}): \Omega \rightarrow \mathbb{R}$$, called the conditional probability, such that $$\int_{G} P(A \mid \mathcal{G})(\omega) d P(\omega)=P(A \cap G)$$ for every $$G \in \mathcal{G}$$, and such a random variable is uniquely defined up to sets of probability zero. A conditional probability is called regular if $$\mathrm{P}(\cdot \mid \mathcal{B})(\omega)$$ is a probability measure on $$(\Omega, \mathcal{F})$$ for all $$\omega \in \Omega$$ a.e. I am used to the non-measure theoretic definition of conditional probability defined for events where $$P(A\|B)=\frac{P(A\cap B)}{P(B)}$$. Why is this function not defined like $$\frac{1}{P(G)}\int_{G} P(A \mid \mathcal{G})(\omega) d P(\omega)=\frac{P(A \cap G)}{P(G)}$$ which would make it in line with the definition for events? I understand that $$P(G)=0$$ is an issue, but what is the motivation of defining this representation for $$P(A\cap G)$$? • Note that the thing being defined is the random variable $P(A\|\mathcal{G})$ and clearly, if a random variable fits into the one definition it fits into the other, except in the case where $P(G)=0$. Aug 7 at 22:33 • What book is this from? Aug 7 at 22:38 • @littleO en.wikipedia.org/wiki/Conditional_probability_distribution But I was reading Billingsley Probability and Measure Aug 7 at 22:38 ## 1 Answer In the case where $$\mathcal G$$ is the $$\sigma$$-algebra generated by event $$G$$, $$P(A \mid \mathcal G)$$ is (up to sets of measure $$0$$) $$P(A\|G)$$ when $$G$$ is true, $$P(A\|G^c)$$ when $$G$$ is false. The equation then says $$P(A \cap G) = P(G) P(A \| G)$$ and $$P(A \cap G^c) = P(G^c) P(A \| G^c)$$, which fits with your non-measure theoretic definition. • Does this mean that $P(A\|\mathcal{G})$ takes on exactly two values? And if I had events $B_1,\dots,B_n$ partitioning $\Omega$, then $P(A\|\sigma(B_1,\dots,B_n))$ would take on $n$ values where $P(A\|\sigma(B_1,\dots,B_n)) (\omega)=P(A\|B_i)$? Is $P(A\|\mathcal{G})$ doing the same type of "smoothing" that conditional expectation is doing? Aug 7 at 22:41 • Yes it does (up to sets of measure $0$). Aug 7 at 23:17	2022-09-27 20:53:13	{"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": 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": 28, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9380003809928894, "perplexity": 153.11543567498757}, "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-2022-40/segments/1664030335058.80/warc/CC-MAIN-20220927194248-20220927224248-00283.warc.gz"}
https://cs.stackexchange.com/tags/information-theory/hot	# Tag Info 91 The following are both plausible messages, but have a completely different meaning: SOS HELP = ...---... .... . .-.. .--. => ...---.........-...--. I AM HIS DATE = .. .- -- .... .. ... -.. .- - . => ...---.........-...--. 45 Sure, of course there are algorithms. Here is my algorithm: First, check if the file contains ordered binary numbers from $0$ to $2^n-1$, for some $n$. If so, write out a 0 bit followed by $n$ one bits followed by a 0 bit. If not, write out a 1 bit, then write out the 7z-compression of the file. This is extremely efficient for files of that particular ... 43 You've got a brilliant new compression scheme, eh? Alrighty, then... ♫ Let's all play, the entropy game ♫ Just to be simple, I will assume you want to compress messages of exactly $n$ bits, for some fixed $n$. However, you want to be able to use it for longer messages, so you need some way of differentiating your first message from the second (it cannot be ... 40 A lot of casual descriptions of entropy are confusing in this way because entropy is not quite as neat and tidy a measure as sometimes presented. In particular, the standard definition of Shannon entropy stipulates that it only applies when, as Wikipedia puts it, "information due to independent events is additive." In other words, independent events must ... 36 Quoting David Richerby from the comments: Since ⋅ represents E and − represents T, any Morse message without spaces can be interpreted as a string in $\{E,T\}^$ Further, since A, I, M, and N are represented by the four possible combinations of two morse characters (⋅-, ⋅⋅, --, -⋅, respectively), any message without spaces can also be interpreted as a ... 34 Actually I don't fully understand this algorithm or the Shannon limit very well, I just know it's the sum of the probability of each character multiplied by log2 of the reciprocal of the probability. Herein lies the crux. The Shannon limit is not some universal property of a string of text. It is the property of a string of text plus a model that provides (... 33 always compress random data sets by more than 50% That's impossible. You can't compress random data, you need some structure to take advantage of. Compression must be reversible, so you can't possibly compress everything by 50% because there are far less strings of length $n/2$ than there are of length $n$. There are some major issues with the paper: They ... 28 This answer isn't as long as it looks; this site just puts a lot of spacing between list items! Update: Actually it's getting pretty long... Morse Code isn't "officially" binary, ternary, quaternary, quinary, or even 57-ary (if I count correctly). Arguing about which one it is without context is not productive. It is up to you to define which of those five ... 28 Wow, great question! Let me try to explain the resolution. It'll take three distinct steps. The first thing to note is that the entropy is focused more on the average number of bits needed per draw, not the maximum number of bits needed. With your sampling procedure, the maximum number of random bits needed per draw is $N$ bits, but the average number of ... 27 Yes, one can. If $x<y$, map the set $\{x,y\}$ to the number $$f(x,y) = y(y-1)/2 + x.$$ It is easy to show that $f$ is bijective, and so this can be uniquely decoded. Also, when $0 \le x < y < 2^{32}$, we have $0 \le f(x,y) < 2^{63} - 2^{31}$, so this maps the set $\{x,y\}$ to a 63-bit number $f(x,y)$. To decode, you can use binary search on ... 27 This seems to be a clear use case for delta compression. If $n$ is known a priori this is trivial: store the first number verbatim, and for each next number store only the difference to the previous. In your case, this will give 0 1 1 1 1 ... This can then with simple run-length encoding be stored in $\mathcal{O}(n)$ space, as there are only $\mathcal{O}(1)... 21 There are$2^N-1$binary strings of length less than$N$, and$2^N$binary strings of length exactly$N$. This means that whatever your compression algorithm is, there must be some string which it can't compress at all, just because the mapping from original string to compressed string must be injective (one-to-one). This is the driving force behind many ... 19 Morse code is a prefix ternary code (for encoding 58 characters) on top of a prefix binary code encoding the three symbols. This was a much shorter answer when accepted. However, considering the considerable misunderstandings between users, and following a request from the OP, I wrote this much longer answer. The first "nutshell" section gives you the gist ... 18 Here is a concrete encoding that can represent each symbol in less than 1 bit on average: First, split the input string into pairs of successive characters (e.g. AAAAAAAABC becomes AA\|AA\|AA\|AA\|BC). Then encode AA as 0, AB as 100, AC as 101, BA as 110, CA as 1110, BB as 111100, BC as 111101, CB as 111110, CC as 111111. I've not said what happens if there is ... 17 It is enough to observe that certain short combinations of letters give ambiguous decodings. A single ambiguous sequence suffices, but I can see the following: ATE ~ P EA ~ IT MO ~ OM etc. As David Richerby notes in the comments, any letter is equivalent to a string of Es and Ts, which makes Morse Code ambiguous as a way of encoding arbitrary sequences of ... 16 Anything using a BWT (Burrows–Wheeler transform) ought to be able to compress that fairly well. My quick Python test: >>> import gzip >>> import lzma >>> import zlib >>> import bz2 >>> import time >>> dLen = 16 >>> inputData = '\n'.join('{:0{}b}'.format(x, dLen) for x in range(2dLen)) >>... 16 The entropy you've calculated isn't really for the specific string but, rather, for a random source of symbols that generates$A$with probability$\tfrac{8}{10}$, and$B$and$C$with probability$\tfrac1{10}$each, with no correlation between successive symbols. The calculated entropy for this distribution,$0.922$means that you can't ... 15 I'm going to defer to Tom van der Zanden who seems to have read the paper and discovered a weakness in the method. While I didn't read the paper in detail, going from the abstract and the results table, it seems like a broadly believable claim. What they claim is a consistent 50% compression ratio on text files (not "all files"), which they note is around ... 14 I have no way to learn (say)$m_1$unless I know$m_2$. That is exactly the problem - if you re-use the same key, and someone has access to one message you encrypted in both plaintext and encrypted form, they can use that to find your key: $$(m_2 \oplus k) \oplus m_2 = k$$ As an alternative scenario, if you use the same key over and over, the ... 13 Let$\mathcal{D}$be the following distribution over$\{A,B,C\}$: if$X \sim \mathcal{D}$then$\Pr[X=A] = 4/5$and$\Pr[X=B]=\Pr[X=C]=1/10$. For each$n$we can construct prefix codes$C_n\colon \{A,B,C\}^n \to \{0,1\}^$such that$$\lim_{n\to\infty} \frac{\operatorname{\mathbb{E}}_{X_1,\ldots,X_n \sim \mathcal{D}}[C_n(X_1,\ldots,X_n)]}{n} = H(\mathcal{... 12 The central concept here is Kolmogorov complexity, and more specifically compressibility. To get a intuitive feeling of compressibility, consider two strings$A \in \mathbb{B}^$and$B \in \mathbb{B}^*$, where$\mathbb{B} = \{ 0,1 \}$. Let$A = 1010101010101010$, and$B = 1011011001111001$. Note that$\|A\| = \|B\| = 16$. How could ... 12 You ask: Is this really feasible as the authors suggest it? According to the paper, their results are very efficient and always compress data to a smaller size. Won't the dictionary size be enormous? Yes, of course. Even for their hand-picked example ("THE QUICK SILVER FOX JUMPS OVER THE LAZY DOG"), they don't achieve compression, because the ... 12 What you need is a random number between 0 and${ 64 \choose n } - 1$. The problem then is to turn this into the bit pattern. This is known as enumerative coding, and it's one of the oldest deployed compression algorithms. Probably the simplest algorithm is from Thomas Cover. It's based on the simple observation that if you have a word that is$n$bits long,... 12 It's trivially simple to show that you can compress below the Shannon limit--take a cheating compressor that has a bunch of common files assigned to tokens. Said files are stored as those tokens. (Obviously, the compressor must be very large, or be drawing on a very large library.) The compressor will inherently be less efficient at dealing with any file ... 12 No, if the algorithm is lossless no steps in the compression sequence can reduce its entropy - otherwise it would not be able to be decompressed/decoded. However, the additional entropy may be stored in 'out-of-band' information - such as the list that needs to be maintained in order to decode the move-to-front transform. 11 It's insecure precisely due to the reason you mention - there is some information leakage. Basically, if you have any assumptions about plaintexts (english text, files with known structure, etc), it leads to an easy statistical analysis. Probably using it twice doesn't change the practicality of the attack significantly, but using it many times with a non-... 11 Let's look at a slightly different way of thinking about Huffman coding. Suppose you have an alphabet of three symbols, A, B, and C, with probabilities 0.5, 0.25, and 0.25. Because the probabilities are all inverse powers of two, this has a Huffman code which is optimal (i.e. it's identical to arithmetic coding). We will use the canonical code 0, 10, 11 for ... 11 You first apply the model to the data, computing the sequence of probabilities, f.e.$1/2$,$1/3$,$1/6$. Then, to encode each symbol with probability$p$, you need$log_2(1/p)\$ bits. And given some particular model, you can't compress data better than the Shannon entropy of probabilities produced by this particular model. But if you apply another model, ... 11 Here's the problem with that reasoning: If you could always compress data, you could compress the compressed data, then compress that, etc. until you have something that is 0 bytes long. You can always apply a compression algorithm to data, but there's no guarantee that you'll get something smaller out at the end. If you're interested in this, there's a ... 10 Morse Code is actually a ternary code, not a binary code, so the spaces are necessary. If spaces were not there, a lot of ambiguity would result, not so much with the entire message, but with individual letters. For example, 2 dots is an I, but 3 dots is an S. If you are transcribing and you hear two dots, do you immediately write "I" or do you wait until ... Only top voted, non community-wiki answers of a minimum length are eligible	2019-09-20 09:55:17	{"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": 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": 0.7368763089179993, "perplexity": 617.3570945811168}, "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/1568514573988.33/warc/CC-MAIN-20190920092800-20190920114800-00059.warc.gz"}
https://www.eduzip.com/ask/question/in-fig-if-160-xywz160-then-prove-that-160-a-o-b160-160is-a-line-520265	Mathematics # In Fig. If $x+y=w+z$ then prove that $A O B$ is a line. ##### SOLUTION Given $x+y=z+w\cdots(1)$ From Angle at a point theorem Sum of all angles at a point is $360^{\circ}$ So $x+y+z+w=360\\x+y+x+y=360\\2(x+y)=360\\x+y=180$ $x,y$ are adjacent angles and form $180^{\circ}$ together so $AOB$ is a straight line You're just one step away Subjective Medium Published on 09th 09, 2020 Questions 120418 Subjects 10 Chapters 88 Enrolled Students 86 #### Realted Questions Q1 Subjective Medium Can two acute angles form a pair of supplementary angles? Give reason support of your answer. Asked in: Mathematics - Lines and Angles 1 Verified Answer \| Published on 09th 09, 2020 Q2 Subjective Medium $I$ and $m$ are the intersecting lines in the given figure. Find $x , y$ and $z$. Asked in: Mathematics - Lines and Angles 1 Verified Answer \| Published on 09th 09, 2020 Q3 Subjective Medium Fill in the blanks in each of the following to make the statement true: Two lines parallel to the same line are ______ to each other. Asked in: Mathematics - Lines and Angles 1 Verified Answer \| Published on 09th 09, 2020 Q4 Single Correct Easy Two supplementary angles are in the ratio $7:11$, then the angles are _______. • A. $70^o, 120^o$ • B. $60^o, 120^o$ • C. $50^o, 130^o$ • D. $70^o, 110^o$ Asked in: Mathematics - Lines and Angles 1 Verified Answer \| Published on 09th 09, 2020 Q5 Subjective Medium In the figure, lines $XY$ and $MN$ intersect at $O$. If $\angle POY={ 90 }^{o }$ and $a : b = 2 : 3$, find $c$. Asked in: Mathematics - Lines and Angles 1 Verified Answer \| Published on 09th 09, 2020	2022-01-23 06:15:35	{"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": 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": 0.5785892009735107, "perplexity": 4220.569447035177}, "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-2022-05/segments/1642320304134.13/warc/CC-MAIN-20220123045449-20220123075449-00416.warc.gz"}
https://workforce.libretexts.org/Bookshelves/Information_Technology/Computer_Applications/Book%3A_Beginning_Excel_(Brown_et_al.)/02%3A_Mathematical_Computations/2.03%3A_Functions_for_Personal_Finance	2.3: Functions for Personal Finance Learning Objectives • Understand the fundamentals of loans. • Use the PMT function to calculate monthly car loan payments. • Use the PMT function to calculate monthly mortgage payments on a house using a down payment. • Learn how to summarize data in a workbook by using worksheet links to create a summary worksheet. In this section, we continue to develop the Personal Budget workbook. Notable items that are missing from the Budget Detail worksheet are the payments you might make for a car or a home. This section demonstrates Excel functions used to calculate loan payments for a car and to calculate mortgage payments for a house. The Fundamentals of Loans and Leases One of the functions we will add to the Personal Budget workbook is the PMT function. This function calculates the payments required for loan repayment. However, before demonstrating this function, it is important to cover a few fundamental concepts on loans. A loan is a contractual agreement in which money is borrowed from a lender and paid back over a specific period of time. The amount of money that is borrowed from the lender is called the principal of the loan. The borrower is usually required to pay the principal of the loan plus interest. When you borrow money to buy a house, the loan is referred to as a mortgage. This is because the house being purchased also serves as collateral to ensure payment. In other words, the bank can take possession of your house if you fail to make loan payments. As shown in Table 2.5, there are several key terms related to loans. Table 2.5 Key Terms for Loans Term Definition Collateral Any item of value that is used to secure a loan to ensure payments to the lender Down Payment The amount of cash paid toward the purchase of a house. If you are paying 20% down, you are paying 20% of the cost of the house in cash and are borrowing the rest from a lender. Interest Rate The interest that is charged to the borrower as a cost for borrowing money Mortgage A loan where property is put up for collateral Principal The amount of money that has been borrowed Residual Value The estimated selling price of a vehicle at a future point in time Length The amount of time you have to repay a loan Figure 2.31 shows an example of an amortization table for a loan. A lender is required by law to provide borrowers with an amortization table when a loan contract is offered. The table in the figure shows how the payments of a loan would work if you borrowed $100,000 from a lender and agreed to pay it back over 10 years at an interest rate of 5%. You will notice that each time you make a payment, you are paying the bank an interest fee plus some of the loan principal. Each year the amount of interest paid to the bank decreases and the amount of money used to pay off the principal increases. This is because the bank is charging you interest on the amount of principal that has not been paid. As you pay off the principal, the interest rate is applied to a lower number, which reduces your interest charges. Finally, the figure shows that the sum of the values in the Interest Payment column is$29,505. This is how much it costs you to borrow this money over 10 years. Indeed, borrowing money is not free. It is important to note that to simplify this example, the payments were calculated on an annual basis. However, most loan payments are made on a monthly basis. The PMT (Payment) Function for Loans Data file: Continue with CH2 Personal Budget. If you own a home, your mortgage payments are a major component of your household budget. If you are planning to buy a home, having a clear understanding of your monthly payments is critical for maintaining strong financial health. In Excel, mortgage payments are conveniently calculated through the PMT (payment) function. This function is more complex than the statistical functions covered in Section 2.2 “Statistical Functions”. With statistical functions, you are required to add only a range of cells or selected cells within the parentheses of the function, also known as the argument. With the PMT function, you must accurately define a series of arguments in order for the function to produce a reliable output. Table 2.6 lists the arguments for the PMT function. It is helpful to review the key loan terms in Table 2.5 before reviewing the PMT function arguments. Table 2.6 Arguments for the PMT Function Argument Definition Rate This is the interest rate the lender is charging the borrower. The interest rate is usually quoted in annual terms, so you have to divide this rate by 12 if you are calculating monthly payments. Nper The argument letters stand for number of periods. This is the term of the loan, which is the amount of time you have to repay the bank. This is usually quoted in years, so you have to multiply the years by 12 if you are calculating monthly payments. Pv The argument letters stand for present value. This is the principal of the loan or the amount of money that is borrowed. [Fv] The argument letters stand for future value. The brackets around the argument indicate that it is not always necessary to define it. It is used if there is a lump-sum payment that will be made at the end of the loan terms. This is also used for the residual value of a lease. If it is not defined, Excel will assume that it is zero. [Type] This argument can be defined with either a 1 or a 0. The number 1 is used if payments are made at the beginning of each period. A 0 is used if payments are made at the end of each period. The argument is in brackets because it does not have to be defined if payments are made at the end of each period. Excel assumes that this argument is 0 if it is not defined. By default, the result of the PMT function in Excel is shown as a negative number. This is because it represents an outgoing payment. When making a mortgage or car payment, you are paying money out of your pocket or bank account. Depending on the type of work that you do, your employer may want you to leave your payments negative or they may ask you to format them as positive numbers. In the following assignments, the payments calculated using the PMT function will be made positive to make them easier to work with. To do this, you will place a negative sign between the equal sign and the function name PMT. We will first use the PMT function in the Personal Budget workbook to calculate the monthly loan payments for a car. These calculations will be made in the Loan Payments worksheet and then displayed in the Budget Summary worksheet through a cell reference link. So far we have demonstrated several methods for adding functions to a worksheet. When working with more complex functions such as the PMT, it is easiest to use the Function Dialog box. Remember to use cell references for the arguments of the PMT function whenever possible. This will allow you the flexibility to change aspects of the loan, such as a lower interest rate or more expensive car, and have the payment automatically recalculate. Using cell references for the arguments provides greater flexibility in trying different scenarios. The following steps use the Insert Function command covered in Section 2.2 to add the PMT function: 1. Switch to the Loan Payments worksheet. 2. Click cell B5. 3. Click the Formulas tab on the Ribbon. 4. Click the Insert Function button to bring up the Insert Function dialog box. 5. Type loan payment in the search box and click Go. the Excel for Mac search box does is not the same as the “Search for a function: input box”. Mac Users must type: PMT in the search box instead. Then press Enter. 6. Double-click the PMT option in the “Select a function:” box. This will open the Function Arguments dialog box. 7. Drag the Function Arguments dialog box out of the way so that you can see the worksheet cells you want to use in the function. Refer to Figure 2.31for the completed Function Arguments dialog box as you complete the next steps. 8. Click in the Rate argument box in the dialog box, then click cell B3 in the worksheet. This will add B3 (the annual interest rate) to the Rate argument. 9. Type a forward slash / for division. 10. Type the number 12. Since our goal is to calculate the monthly payments for the loan, we need to divide the rate, which is stated in annual terms, by 12. This converts the annual rate to a monthly rate. 11. Click the Nper argument box (or use the Tab key) and then click cell B4 in the worksheet. This will add B4 (the number of years to repay the loan) to the Nper argument. 12. Type an asterisk * for multiplication. 13. Type the number 12. Since our goal is to calculate the monthly payments for the loan, we need to multiply the terms of the loan by 12. This converts the terms of the loan from years to months. 14. Click the Pv argument box (or use the Tab key) and then click cell B2 in the worksheet. This will add B2 (the amount of the loan) to the Pv argument. 15. You will now see the Rate, Nper, and Pv arguments defined for the function. (see Figure 2.31) 16. Click the OK button at the bottom of the Function Arguments dialog box. The function will now be placed into the worksheet. Since we are not paying any lump sums of money at the end of the loan, there is no need to define the Fv argument. Also, we will assume that the monthly payments will be made at the end of each month. Therefore, there is no need to define the Type argument. 17. Notice that the result of the formula in cell B5 is showing as a negative number (see Figure 2.32). To fix this, double-click on cell B5 and type a negative sign between the equal sign and the letters PMT in the formula (see Figure 2.33). 18. The finished formula in cell B5 should be =-PMT(B3/12,B412,B2) Figure 2.31 shows the completed Function Arguments dialog box for the PMT function. Notice that the dialog box shows the values for the Rate and Nper arguments. The Rate is divided by 12 to convert the annual interest rate to a monthly interest rate. The Nper argument is multiplied by 12 to convert the terms of the loan from years to months. Finally, the dialog box provides you with a definition for each argument. The definition appears when you click in the input box for the argument. Keyboard Shortcuts Insert Function • Hold the SHIFT key while pressing the F3 key. Function Arguments Dialog Box • After the equal sign = and function name are typed into cell a location, hold down the CTRL key and press the letter A on your keyboard. Integrity Check Comparable Arguments for PMT Function When using functions such as PMT, make sure the arguments are defined in comparable terms. For example, if you are calculating the monthly payments of a loan, make sure both the Rate and Nper argument are expressed in terms of months. The function will produce an erroneous result if one argument is expressed in years while the other is expressed in months. The PMT Function when there is a down payment In addition to calculating the loan payments for a car, the PMT function will be used in the Personal Budget workbook to calculate the mortgage payments for a home. The details for the mortgage payments are also found in the Loan Payments worksheet. Unlike the car loan, there is a down payment with the mortgage. A down payment on a mortgage is usually a percentage of the price of the home, which is paid up front and reduces the amount of the loan itself. The down payment amount and amount of the loan will both need to be calculated using formulas. While we did not use a down payment in the car loan example, it is fairly common to have a down payment when purchasing a car too. Write the formulas to calculate the Down Payment Amount and new Loan Amount by following these steps: 1. Click cell B11. 2. Write the formula =B9B10. This will calculate 20% of the price of the house. 3. Click cell B12. Write the formula =B9-B11. This will subtract the down payment amount from the price of the house (see Figure 2.34 for the Show Formulas View and Figure 2.35 for the formula results). Now that we have the revised Loan Amount in cell B12, we can write the PMT function following the same process we did for the car loan. 1. Click cell B15. 2. Click the Formulas tab on the Ribbon. 3. Click the Insert Function button to bring up the Insert Function dialog box. 4. Type PMT in the search box and click Go. 5. Double-click the PMT option in the “Select a function:” box. This will open the Function Arguments dialog box. 6. Enter the following arguments (see Figure 2.36) • Rate: B13/12 –> divide by 12 to convert the annual rate to a monthly one • Nper: B1412 –> multiply by 12 to convert the number of years into number of months • Pv: B12 –> this is the cell with the actual loan amount, not the price of the house 7. Click OK in the Function Arguments dialog box. 8. Modify the formula in cell B15 to display the result as a positive number. Remember to type a negative sign between the equal sign and the letters PMT. 9. Cell B15 should contain the function: =-PMT(B13/12,B1412,B12) and the result should be $708.60 (see Figure 2.37). Figure 2.36 shows how the the completed Function Arguments dialog box for the PMT function for the mortgage should appear before pressing the OK button. Figure 2.37 shows the result of the PMT function for the mortgage. The monthly payments for this mortgage are$708.60. This monthly payment will be displayed in the Budget Summary worksheet. Skill Refresher PMT Function 1. Type an equal sign =. 2. Type the letters PMT followed by an open parenthesis, or double click the function name from the function list. 3. Define the Rate argument with a cell location that contains the rate being charged by the lender for the loan or lease. If the interest rate given is an annual rate, divide it by 12 to convert it to a monthly rate. 4. Define the Nper argument with a cell location that contains the amount of time to repay the loan or lease. If the amount of time is in years, multiply it by 12 to convert it to number of months. 5. Define the Pv argument with a cell location that contains the principal of the loan or the price of the item being leased. 6. Type a closing parenthesis ). 7. Press the ENTER key. 8. If the result needs to be shown as a positive number, add a negative sign between the equal sign and the letters PMT. Linking Worksheets (Creating a Summary Worksheet) So far we have used cell references in formulas and functions, which allow Excel to produce new outputs when the values in the cell references are changed. Cell references can also be used to display values or the outputs of formulas and functions in cell locations on other worksheets. This is how we will complete the Budget Summary worksheet using values from both the Budget Detail and Loan Payments worksheets. Outputs from the formulas and functions that were entered into the Budget Detail will be displayed on the Budget Summary worksheet through the use of cell references. 1. Switch to the Budget Summary worksheet and select cell B4. This cell needs to reference the Total Annual Spend (D12) from the Budget Detail worksheet. 2. Type an = 3. Click the Budget Detail worksheet tab. 4. Click cell D12. 5. Press the ENTER key on your keyboard. 6. The formula bar will display the formula =’Budget Detail’!D12 and the cell will display \$17,124. (see Figure 2.38) Figure 2.38 shows how the cell reference appears in the Budget Summary worksheet. Notice that the cell reference D12 is preceded by the Budget Detail worksheet name enclosed in apostrophes followed by an exclamation point (‘Budget Detail’!) This indicates that the value displayed in the cell is referencing a cell location in the Budget Detail worksheet. We will use a similar process to enter in the annual car payments and mortgage payments from the Loan Payments worksheet. The payments on the Loan Payments worksheet are monthly payments though, so we will need to multiply each one by 12 to get the annual amount to display in the Budget Summary worksheet. 1. Click on cell B5. This cell needs to contain a formula that references the monthly car payment cell (B5) on the Loan Payments worksheet and multiplies by 12. 2. Type an = 3. Click the Loan Payments worksheet tab. 4. Click cell B5 on the Loan Payments worksheet. 5. The formula bar will display the formula =’Loan Payments’!B5 6. Type an asterisk * for multiplication. 7. Type the number 12. The formula in the formula bar should read: =’Loan Payments’!B512 8. Press the ENTER key on your keyboard. 9. Click on cell B6. This cell needs to contain a formula that references the monthly mortgage payment cell (B15) on the Loan Payments worksheet and multiplies by 12. 10. Type an = 11. Click the Loan Payments worksheet tab. 12. Click cell B15 on the Loan Payments worksheet. 13. The formula bar will display the formula =’Loan Payments’!B15 14. Type an asterisk for multiplication. 15. Type the number 12. The formula in the formula bar should read: =’Loan Payments’!B15*12 16. Press the ENTER key on your keyboard. Figure 2.39 shows the results of creating formulas that reference cell locations in the Loan Payments worksheet. We can now add other formulas and functions to the Budget Summary worksheet that can calculate the difference between the total spend dollars vs. the total net income in cell B3. The following steps explain how this is accomplished: 1. Click cell B7 in the Budget Summary worksheet. 2. Type an equal sign =. 3. Type the function name SUM followed by an open parenthesis (. 4. Highlight the range B4:B6. 5. Type a closing parenthesis ) and press the ENTER key on your keyboard or simply press the ENTER key to close the function. The total for all annual expenses now appears on the worksheet. 6. Click cell B8 on the Budget Summary worksheet. You will enter a formula to calculate Remaining (Savings) amount in this cell. 7. Type an equal sign =. 8. Click cell B3. 9. Type a minus sign and then click cell B7. 10. Press the ENTER key on your keyboard. This formula produces a positive number, indicating our income is greater than our total expenses. Figure 2.40 shows the results of the formulas that were added to the Budget Summary worksheet. Overall, having your income exceed your total expenses is a good thing because it allows you to save money for future spending needs or unexpected events. We can now add a few formulas that calculate both the spending rate and the savings rate as a percentage of net income. These formulas require the use of absolute references, which we covered earlier in this chapter. The following steps explain how to add these formulas: 1. Click cell C7 in the Budget Summary worksheet. 2. Type an equal sign =. 3. Click cell B7. 4. Type a forward slash / for division and then click B3. 5. Press the F4 key on your keyboard. This adds an absolute reference to cell B3. 6. Press the ENTER key. The result of the formula shows that total expenses consume 89% of our net income. 7. Click cell C7. 8. Place the mouse pointer over the Auto Fill Handle. 9. When the mouse pointer turns to a black plus sign, left click and drag down to cell C8. This copies and pastes the formula into cell C8. 10. Compare your worksheets with Figures 2.41a-c below. Make any necessary changes before moving on to the next section. 11. Save the CH2 Personal Budget file. Figure 2.41a shows the completed Budget Summary worksheet Figure 2.41b shows the completed Budget Detail worksheet Figure 2.41c shows the completed Loan Payments worksheet Key Takeaways • The PMT function can be used to calculate the monthly mortgage payments for a house or the monthly lease payments for a car. • When using the PMT function, each argument must be separated by a comma. • To calculate the monthly payment for a loan using the PMT function, the Rate and Nper arguments must be defined in terms of months. The Rate should be divided by 12 to convert it from an annual rate to a monthly rate. The Nper should be multiplied by 12 to convert the term of the loan from years to months. • The PMT function produces a negative output if the Pv argument is not preceded by a minus sign. For the purposes of this textbook, a minus sign will be entered before the PV argument in the PMT dialog box.	2023-02-03 19:18:07	{"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": 0.5433557033538818, "perplexity": 1118.9934406027985}, "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-2023-06/segments/1674764500074.73/warc/CC-MAIN-20230203185547-20230203215547-00868.warc.gz"}
https://www.aimsciences.org/article/doi/10.3934/dcds.2020249	Article Contents Article Contents # Analysis of a spatial memory model with nonlocal maturation delay and hostile boundary condition The first author's research is supported by Startup Foundation for Introducing Talent of NUIST 1411111901023 and Natural Science Foundation of Jiangsu Province of China. The second author's research is supported by Chinese NSF grants 11671110 and Heilongjiang NSF LH2019A010. The third author's research is partially supported by an NSERC grant • In this paper, we propose and investigate a memory-based reaction-diffusion equation with nonlocal maturation delay and homogeneous Dirichlet boundary condition. We first study the existence of the spatially inhomogeneous steady state. By analyzing the associated characteristic equation, we obtain sufficient conditions for local stability and Hopf bifurcation of this inhomogeneous steady state, respectively. For the Hopf bifurcation analysis, a geometric method and prior estimation techniques are combined to find all bifurcation values because the characteristic equation includes a non-self-adjoint operator and two time delays. In addition, we provide an explicit formula to determine the crossing direction of the purely imaginary eigenvalues. The bifurcation analysis reveals that the diffusion with memory effect could induce spatiotemporal patterns which were never possessed by an equation without memory-based diffusion. Furthermore, these patterns are different from the ones of a spatial memory equation with Neumann boundary condition. Mathematics Subject Classification: Primary: 35B32, 35B35; Secondary: 92B05. Citation: • Figure 1. Triangle formed by $P_0(h)$, $P_1(h) e^{-\mathrm{i}\theta_1}$ and $P_2(h)e^{-\mathrm{i}\theta_2}$ Figure 2. Regions in the $(\rho_1, \rho_2)$ plane that satisfy both (A1) and (A3): (a) $d>0$ and (b) $d<0$ Figure 3. The area painted green is the connected region I of $(\lambda, h)$ Figure 4. Approximation of the crossing curve $\mathcal{T}_\lambda$. Here $\lambda = 1.1$, $a = 0.3$, $b = 0.5$, $d = 2$. Two crossing curves of (43) are plotted in the top right corner, and one of it (in the red box) is enlarged in the figure Figure 5. Let $\lambda = 1.1$, $a = 0.3$, $b = 0.5$, $d = 2$. (a, c) When $(\tau, \sigma)$ are located at the left side of crossing curve, the positive steady state is stable. (b) A stable spatially inhomogeneous periodic solution is generated, when $(\tau, \sigma)$ passes through the crossing curve Figure 6. The crossing curves of (43) for other choices of parameters. Here, $\lambda = 1.1$ Figure 7. Let $\lambda = 1.1$, $a = -0.3$, $b = 0.5$, $d = 0.2$, the positive steady state $u_\lambda$ is still stable for sufficient large $\tau = \sigma = 100$ • [1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition, Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003. [2] N. F. Britton, Spatial structures and periodic traveling waves in an integro-differential reaction diffusion population model, SIAM J. Anal. Math., 50 (1990), 1663-1688. doi: 10.1137/0150099. [3] S. Busenberg and W. Huang, Stability and Hopf bifurcation for a population delay model with diffusion effects, J. Differential Equations, 124 (1996), 80-107. doi: 10.1006/jdeq.1996.0003. [4] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296. [5] S. Chen and J. Shi, Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect, J. Differential Equations, 253 (2012), 3440-3470. doi: 10.1016/j.jde.2012.08.031. [6] S. Chen, J. Wei and X. Zhang, Bifurcation analysis for a delayed diffusive logistic population model in the advective heterogeneous environment, J. Dyn. Differ. Equ., 32 (2020), 823-847. doi: 10.1007/s10884-019-09739-0. [7] S. Chen and J. Yu, Stability and bifurcations in a nonlocal delayed reaction-diffusion population model, J. Differential Equations, 260 (2016), 218-240. doi: 10.1016/j.jde.2015.08.038. [8] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2. [9] J. Crank, The Mathematics of Diffusion, Second edition. Clarendon Press, Oxford, 1975. [10] J. Furter and M. Grinfeld, Local vs. non-local interactions in population dynamics, J. Math. Biol., 27 (1989), 65-80. doi: 10.1007/BF00276081. [11] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. [12] K. Gu, S. Niculescu and J. Chen, On stability crossing curves for general systems with two delays, J. Math. Anal. Appl., 311 (2005), 231-253. doi: 10.1016/j.jmaa.2005.02.034. [13] S. Guo, Stability and bifurcation in a reaction-diffusion model with nonlocal delay effect, J. Differential Equations, 259 (2015), 1409-1448. doi: 10.1016/j.jde.2015.03.006. [14] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. [15] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. [16] Y. Lou and F. Lutscher, Evolution of dispersal in open advective environments, J. Math. Biol., 69 (2014), 1319-1342. doi: 10.1007/s00285-013-0730-2. [17] J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer-Verlag, New York, 2003. doi: 10.1007/b98869. [18] A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspective, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4978-6. [19] J. Shi, C. Wang and H. Wang, Diffusive spatial movement with memory and maturation delays, Nonlinearity, 32 (2019), 3188-3208. doi: 10.1088/1361-6544/ab1f2f. [20] J. Shi, C. Wang, H. Wang and X. Yan, Diffusive spatial movement with memory, J. Dynam. Differential Equations, 32 (2020), 979-1002. doi: 10.1007/s10884-019-09757-y. [21] Y. Song, S. Wu and H. Wang, Spatiotemporal dynamics in the single population model with memory-based diffusion and nonlocal effect, J. Differential Equations, 267 (2019), 6316-6351. doi: 10.1016/j.jde.2019.06.025. [22] Y. Su, J. Wei and J. Shi, Hopf bifurcation in a reaction-diffusion population model with delay effect, J. Differential Equations, 247 (2009), 1156-1184. doi: 10.1016/j.jde.2009.04.017. [23] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537. doi: 10.1080/03605300903473426. [24] J. Wu, Theory and Applications of Partial Functional Differential Equations, Applied Mathematical Sciences, 119. Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1. [25] X. P. Yan and W. T. Li, Stability of bifurcating periodic solutions in a delayed reaction diffusion population model, Nonlinearity, 23 (2010), 1413-1431. doi: 10.1088/0951-7715/23/6/008. Figures(7)	2023-04-01 05:13:02	{"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": 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": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.6691408157348633, "perplexity": 1419.6203154654825}, "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/1679296949701.0/warc/CC-MAIN-20230401032604-20230401062604-00586.warc.gz"}
https://gigaom.com/2010/09/08/electric-car-101-lease-or-buy/	# Electric Car 101: Lease or Buy? If you’re sold on the idea of a plug-in car, one of the next questions to answer is whether to lease or buy your chosen model. It’s an issue that any car shopper would have to consider, but with electric cars, in addition to the standard considerations, some new factors come into play. Here are five things to consider as you hunt for a plug-in car deal that fits your budget, driving habits and preferences: Permanence of your living (and parking) situation. “The only thing that’s really different here,” Chevy Volt marketing chief Tony DiSalle said in an interview, “is, do you have a place to plug in?” That’s where you capture the value of a car like the Volt he said, whether it’s leased or financed for purchase. “Beyond that it becomes very similar to other vehicles that people would purchase,” he said. If there’s a strong chance your access to a charge point at work and/or at home may change in the next few years, going for the lease option might seem like it offers an out. Remember, however, that leaving a lease before it’s set to expire often comes with penalties and hefty early termination fees, as Consumer Reports warns in this handy guide. With electric cars, as with their old gas-chugging counterparts, purchasing the vehicle gives you the autonomy to sell or trade it in when you want. Local, state and federal incentives. Government incentive programs for plug-in hybrid and electric vehicles at the local, state and federal level have produced a labyrinth of rebates, credits and other perks for green cars that meet certain criteria. At the federal level, plug-in vehicles can qualify for a tax credit of $2,500-$7,500, depending on battery capacity (the Chevy Volt, Nissan LEAF and Tesla (s TSLA) Roadster can all qualify for the full $7,500). Purchasing a plug-in could allow you (as an individual or business) to claim up to$7,500 on your income tax return for that year. When you lease a plug-in car, the federal tax credit goes to the lease provider, DiSalle noted. So you effectively give up the lump sum, as GM-Volt has explained, in return for lower monthly payments. At the state level, rebates can help knock thousands of dollars off the cost of an all-electric vehicle. In California, for example, purchasing or leasing a car like the Nissan LEAF, Tesla Roadster or plug-in Prius can qualify you for a rebate of up to $5,000 (the Volt, for now, doesn’t meet California’s criteria for AT-PZEV designation and therefore, doesn’t qualify for the rebate). Keep in mind, these incentives won’t necessarily be available beyond the very early adopter phase. The$7,500 federal tax credit is slated to phase out after the first 200,000 qualifying vehicles sold by a given manufacturer. California’s Clean Vehicle Rebate Project is set to offer rebates on a first-come, first-served basis until the appropriated funds (\$9.1 million) are all spoken for. To find out what’s available in your area, check out your state’s energy office website, the Department of Energy’s guide to tax breaks and the Union of Concerned Scientists’ frequently updated Hybrid Center project. Mileage allowance. DiSalle emphasized that, as a plug-in vehicle with a gas-powered range extender, the Volt “can be your only car, which has huge financial implications.” All-electric models from some of GM’s competitors, he argued, will often be a “niche addition to the household fleet.” That will depend on your driving habits, of course (some consumers could manage just fine with an electric commuter car and membership to a car-sharing program). However, pay close attention to the suggestion of uninhibited range if you’re considering a lease. Lease agreements typically include a mileage allowance. Exceed it, and you can get hit with extra charges. The Chevy Volt lease will reportedly restrict mileage to 12,000 miles per year, or about 32.8 miles per day. Future Value. One of the big uncertainties about electric vehicles is how quickly they’ll lose value over time. If you purchase a car, this depreciation rate will affect how much you can sell it for later on (although as Consumer Reports notes, the upside is that when buying a car, “its cash value is yours to use as you like”). In a typical closed-end lease, the amount you pay for depreciation is fixed. Assumptions about the vehicle’s residual value (how much the car will be worth at the end of your lease) will affect what kind of deals you can get on the monthly payment. Generally, a higher residual value means you’ll pay less for depreciation over the lease term. Ability to Upgrade. The earliest available plug-in car models will feature new technology that’s likely to improve with future generations. If you ultimately want to have an original to show your grandkids, then an outright purchase may be the way to go. DiSalle, who said GM hasn’t made decisions about long-term lease plans, expects a majority of Chevy Volt customers to buy the model outright, especially in the first year. “A lot of people have been waiting and quite honestly want to own it,” he said. A lease may be a better fit, however, for gadgetheads who want the ability to upgrade every few years as battery technology improves and our cars become more and more like oversized electronic devices. For more research on electric cars check out GigaOM Pro (subscription required): Report: IT Opportunities in Electric Vehicle Management Why Microsoft’s Electric Vehicle Deal With Ford Matters	2021-06-22 13:22:52	{"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": 0.18970894813537598, "perplexity": 4045.0467093196085}, "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-2021-25/segments/1623488517820.68/warc/CC-MAIN-20210622124548-20210622154548-00221.warc.gz"}
https://www.quizover.com/calculus/course/4-8-l-hopital-s-rule-applications-of-derivatives-by-openstax	# 4.8 L’hôpital’s rule Page 1 / 7 • Recognize when to apply L’Hôpital’s rule. • Identify indeterminate forms produced by quotients, products, subtractions, and powers, and apply L’Hôpital’s rule in each case. • Describe the relative growth rates of functions. In this section, we examine a powerful tool for evaluating limits. This tool, known as L’Hôpital’s rule , uses derivatives to calculate limits. With this rule, we will be able to evaluate many limits we have not yet been able to determine. Instead of relying on numerical evidence to conjecture that a limit exists, we will be able to show definitively that a limit exists and to determine its exact value. ## Applying l’hôpital’s rule L’Hôpital’s rule can be used to evaluate limits involving the quotient of two functions. Consider $\underset{x\to a}{\text{lim}}\frac{f\left(x\right)}{g\left(x\right)}.$ If $\underset{x\to a}{\text{lim}}f\left(x\right)={L}_{1}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\underset{x\to a}{\text{lim}}g\left(x\right)={L}_{2}\ne 0,$ then $\underset{x\to a}{\text{lim}}\frac{f\left(x\right)}{g\left(x\right)}=\frac{{L}_{1}}{{L}_{2}}.$ However, what happens if $\underset{x\to a}{\text{lim}}f\left(x\right)=0$ and $\underset{x\to a}{\text{lim}}g\left(x\right)=0?$ We call this one of the indeterminate forms , of type $\frac{0}{0}.$ This is considered an indeterminate form because we cannot determine the exact behavior of $\frac{f\left(x\right)}{g\left(x\right)}$ as $x\to a$ without further analysis. We have seen examples of this earlier in the text. For example, consider $\underset{x\to 2}{\text{lim}}\frac{{x}^{2}-4}{x-2}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\underset{x\to 0}{\text{lim}}\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}x}{x}.$ For the first of these examples, we can evaluate the limit by factoring the numerator and writing $\underset{x\to 2}{\text{lim}}\frac{{x}^{2}-4}{x-2}=\underset{x\to 2}{\text{lim}}\frac{\left(x+2\right)\left(x-2\right)}{x-2}=\underset{x\to 2}{\text{lim}}\left(x+2\right)=2+2=4.$ For $\underset{x\to 0}{\text{lim}}\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}x}{x}$ we were able to show, using a geometric argument, that $\underset{x\to 0}{\text{lim}}\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}x}{x}=1.$ Here we use a different technique for evaluating limits such as these. Not only does this technique provide an easier way to evaluate these limits, but also, and more important, it provides us with a way to evaluate many other limits that we could not calculate previously. The idea behind L’Hôpital’s rule can be explained using local linear approximations. Consider two differentiable functions $f$ and $g$ such that $\underset{x\to a}{\text{lim}}f\left(x\right)=0=\underset{x\to a}{\text{lim}}g\left(x\right)$ and such that ${g}^{\prime }\left(a\right)\ne 0$ For $x$ near $a,$ we can write $f\left(x\right)\approx f\left(a\right)+{f}^{\prime }\left(a\right)\left(x-a\right)$ and $g\left(x\right)\approx g\left(a\right)+{g}^{\prime }\left(a\right)\left(x-a\right).$ Therefore, $\frac{f\left(x\right)}{g\left(x\right)}\approx \frac{f\left(a\right)+{f}^{\prime }\left(a\right)\left(x-a\right)}{g\left(a\right)+{g}^{\prime }\left(a\right)\left(x-a\right)}.$ Since $f$ is differentiable at $a,$ then $f$ is continuous at $a,$ and therefore $f\left(a\right)=\underset{x\to a}{\text{lim}}f\left(x\right)=0.$ Similarly, $g\left(a\right)=\underset{x\to a}{\text{lim}}g\left(x\right)=0.$ If we also assume that ${f}^{\prime }$ and ${g}^{\prime }$ are continuous at $x=a,$ then ${f}^{\prime }\left(a\right)=\underset{x\to a}{\text{lim}}{f}^{\prime }\left(x\right)$ and ${g}^{\prime }\left(a\right)=\underset{x\to a}{\text{lim}}{g}^{\prime }\left(x\right).$ Using these ideas, we conclude that $\underset{x\to a}{\text{lim}}\frac{f\left(x\right)}{g\left(x\right)}=\underset{x\to a}{\text{lim}}\frac{{f}^{\prime }\left(x\right)\left(x-a\right)}{{g}^{\prime }\left(x\right)\left(x-a\right)}=\underset{x\to a}{\text{lim}}\frac{{f}^{\prime }\left(x\right)}{{g}^{\prime }\left(x\right)}.$ Note that the assumption that ${f}^{\prime }$ and ${g}^{\prime }$ are continuous at $a$ and ${g}^{\prime }\left(a\right)\ne 0$ can be loosened. We state L’Hôpital’s rule formally for the indeterminate form $\frac{0}{0}.$ Also note that the notation $\frac{0}{0}$ does not mean we are actually dividing zero by zero. Rather, we are using the notation $\frac{0}{0}$ to represent a quotient of limits, each of which is zero. ## L’hôpital’s rule (0/0 case) Suppose $f$ and $g$ are differentiable functions over an open interval containing $a,$ except possibly at $a.$ If $\underset{x\to a}{\text{lim}}f\left(x\right)=0$ and $\underset{x\to a}{\text{lim}}g\left(x\right)=0,$ then $\underset{x\to a}{\text{lim}}\frac{f\left(x\right)}{g\left(x\right)}=\underset{x\to a}{\text{lim}}\frac{{f}^{\prime }\left(x\right)}{{g}^{\prime }\left(x\right)},$ assuming the limit on the right exists or is $\infty$ or $\text{−}\infty .$ This result also holds if we are considering one-sided limits, or if $a=\infty \phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}-\infty .$ what is the power rule how do i deal with infinity in limits? f(x)=7x-x g(x)=5-x Awon 5x-5 Verna what is domain difference btwn domain co- domain and range Cabdalla x Verna The set of inputs of a function. x goes in the function, y comes out. Verna where u from verna Arfan If you differentiate then answer is not x Raymond domain is the set of values of independent variable and the range is the corresponding set of values of dependent variable Champro what is functions give different types of functions. Paul how would u find slope of tangent line to its inverse function, if the equation is x^5+3x^3-4x-8 at the point(-8,1) pls solve it i Want to see the answer Sodiq ok Friendz differentiate each term Friendz why do we need to study functions? to understand how to model one variable as a direct relationship to another variable Andrew integrate the root of 1+x² use the substitution t=1+x. dt=dx √(1+x)dx = √tdt = t^1/2 dt integral is then = t^(1/2 + 1) / (1/2 + 1) + C = (2/3) t^(3/2) + C substitute back t=1+x = (2/3) (1+x)^(3/2) + C navin find the nth differential coefficient of cosx.cos2x.cos3x determine the inverse(one-to-one function) of f(x)=x(cube)+4 and draw the graph if the function and its inverse f(x) = x^3 + 4, to find inverse switch x and you and isolate y: x = y^3 + 4 x -4 = y^3 (x-4)^1/3 = y = f^-1(x) Andrew in the example exercise how does it go from -4 +- squareroot(8)/-4 to -4 +- 2squareroot(2)/-4 what is the process of pulling out the factor like that? Andrew √(8) =√(4x2) =√4 x √2 2 √2 hope this helps. from the surds theory a^c x b^c = (ab)^c Barnabas 564356 Myong can you determine whether f(x)=x(cube) +4 is a one to one function Crystal one to one means that every input has a single output, and not multiple outputs. whenever the highest power of a given polynomial is odd then that function is said to be odd. a big help to help you understand this concept would be to graph the function and see visually what's going on. Andrew one to one means that every input has a single output, and not multiple outputs. whenever the highest power of a given polynomial is odd then that function is said to be odd. a big help to help you understand this concept would be to graph the function and see visually what's going on. Andrew can you show the steps from going from 3/(x-2)= y to x= 3/y +2 I'm confused as to how y ends up as the divisor step 1: take reciprocal of both sides (x-2)/3 = 1/y step 2: multiply both sides by 3 x-2 = 3/y step 3: add 2 to both sides x = 3/y + 2 ps nice farcry 3 background! Andrew first you cross multiply and get y(x-2)=3 then apply distribution and the left side of the equation such as yx-2y=3 then you add 2y in both sides of the equation and get yx=3+2y and last divide both sides of the equation by y and you get x=3/y+2 Ioana Multiply both sides by (x-2) to get 3=y(x-2) Then you can divide both sides by y (it's just a multiplied term now) to get 3/y = (x-2). Since the parentheses aren't doing anything for the right side, you can drop them, and add the 2 to both sides to get 3/y + 2 = x Melin thank you ladies and gentlemen I appreciate the help! Robert keep practicing and asking questions, practice makes perfect! and be aware that are often different paths to the same answer, so the more you familiarize yourself with these multiple different approaches, the less confused you'll be. Andrew please how do I learn integration they are simply "anti-derivatives". so you should first learn how to take derivatives of any given function before going into taking integrals of any given function. Andrew best way to learn is always to look into a few basic examples of different kinds of functions, and then if you have any further questions, be sure to state specifically which step in the solution you are not understanding. Andrew example 1) say f'(x) = x, f(x) = ? well there is a rule called the 'power rule' which states that if f'(x) = x^n, then f(x) = x^(n+1)/(n+1) so in this case, f(x) = x^2/2 Andrew great noticeable direction Isaac limit x tend to infinite xcos(π/2x)*sin(π/4x) can you give me a problem for function. a trigonometric one find volume of solid about y axis and y=x^3, x=0,y=1	2018-10-19 15:38:09	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 53, "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": 0.8586129546165466, "perplexity": 427.2530521588173}, "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/1539583512411.13/warc/CC-MAIN-20181019145850-20181019171350-00460.warc.gz"}
https://calib.io/blogs/knowledge-base/camera-calibration	# Camera Calibration Explained Camera calibration is the process of determining camera and lens model parameters accurately. With the pinhole model, this amounts to determining the parameters of a suitable camera model. at least the focal length $f$, and possibly central point coordinates ($c_x, c_y$) and lens distortion parameters $\boldsymbol{k}$. In the most common, offline calibration process, images are taken under specific constraints. The calibration object defines a world coordinate system such that 3D coordinates of the visual features are known. Most of these methods work by observing a calibration object with known visual features. This is preferred when full control over the calibration procedure is necessary and high accuracy is demanded. Camera Model In any camera calibration effort, it is crucial to select a suitable camera model, which neither under- nor over-parameterized the camera. More information on camera models is found in our article on the subject Calibration Procedures Many procedures for camera calibration have been proposed in literature. See e.g. Tsai's method [3] and Heikkilä and Silvén's [4]. These procedures differ in the type of calibration object needed, and the derivation of an initial guess for the camera parameters and the following nonlinear optimization step. Probably the most popular of all procedures is Zhang's [5]. All of these methods should always be followed by non-linear optimisation (bundle adjustment) as they only return algebraic solutions and don't account for lens effects. They do however provide an initial guess which is required for the non-linear optimisation to converge. Zhang's Method A modern and popular method in the computer vision community is that of Zhang, which also is implemented in popular software libraries such as libCalib, OpenCV, Jean-Yves Bouguet's Camera Calibration Toolbox for Matlab and Matlab's Computer Vision Toolbox. Zhang's calibration routine relies on observations of a planar calibration board with easily recognizable features. It relates the 3-D coordinates of these to the observed image coordinates projections by means of the model above and solves for the calibration plane extrinsics (the camera's position and orientation relative to the calibration board's coordinate system), and the camera intrinsics, by means of a closed form solution. This is then followed by non-linear optimization with the Levenberg-Marquardt algorithm over all parameters, including $\boldsymbol{k}$. The objective function to be minimized is the sum of squared reprojection errors, defined in the image plane: $$\sum_i{\sum_j{\|\|\vec{p}_{ij} - \breve{\vec{p}}(\vec{P}_j, \boldsymbol{A}, \vec{k}, \boldsymbol{R}_i, \vec{T}_i)\|\|^2}} \quad ,$$ where $\breve{\vec{p}}$ is the projection operator determining 2-D point coordinates given 3-D coordinates and the camera parameters. $i$ sums over the positions of the calibration board and $j$ over the points in a single position. $\vec{P}_j$ are 3-D point coordinates in the local calibration object coordinate system, $\vec{P}_j = [x, y, 0]^\top$, and $\vec{p}_{ij}$ the observed 2-D coordinates in the camera. The per-position extrinsic $\boldsymbol{R}_i, \vec{T}_i$ can be understood as the position of the camera relative to the coordinate system defined by the calibration object. With quality lenses and calibration targets, final mean reprojection errors in the order of a few tenths of a pixel are usually achieved. Tsai's method In contrast to Zhang's methods, Tsai's does not formulate the relation between object and image points as a series of homographies. Instead, algebraic constraints are derived which lead to a stepwise procedure to incrementally eliminate the unknowns. The camera is modelled as a standard pinhole + a single radial distortion coefficient. One advantage of Tsai's method over Zhang's is that it can find a solution for the model parameters also for non-planar calibration objects (e.g. step targets or V-shaped calibration rigs). For this reason, libCalib implements Tsai's method. Following any of the initialisation methods above, non-linear optimisation is employed to refine the parameter estimates and to include additional camera parameters which cannot be solved for with those techniques (e.g. radial lens distortion parameters). This estimation problem can be seen as a particular case of bundle block adjustment (sometimes just bundle adjustment), and it aims to find the max-likelihood solution to minimise the average reprojection error under the assumption of Gaussian noise in feature detections. The Levenberg-Marquardt algorithm has emerged as the de-facto standard to solve this least-squares problem with many parameters and unknowns. It can be seen as a hybrid method, interpolating between Gauss-Newtons iterative optimisation scheme and Gradient Descend. For computational reasons, a sparse solver should be used, as the Jacobian of the reprojection error tends to be very sparse (most parameters do only depend on a small number of equations). Calib Camera Calibrator and libCalib implement robust optimisation, which can use the Huber loss function. This loss function weights large errors linearly, ensuring good convergence even in the presence of a few outliers. Autocalibration An alternative to the standard offline calibration routines described above is autocalibration. In autocalibration, parameters are determined from normal camera images viewing a general scene [1,2]. Depending on the specific method, little or no assumptions are made about the viewed scene or the motion of the camera between images. For some applications, this does indeed work, but generally, some assumptions need to be made about the camera or a reduced camera model needs to be chosen. However, even then, the autocalibration process tends to be unreliable and its success very dependent on the specific scene composition. [1]: O.D. Faugeras, Q.-T. Luong, and S.J. Maybank. Camera Self- Calibration: Theory and Experiments. In European Conference on Computer Vision, 1992. [2]: Ri Hartley. Euclidean reconstruction from uncalibrated views. In Applications of invariance in computer vision, pages 235–256, 1994. [3]: Roger Y. Tsai. An efficient and accurate camera calibration technique for 3D machine vision. In IEEE Conference on Computer Vision and Pattern Recognition, pages 364–374, 1986. [4]: Janne Heikkilä and Olli Silvén. A Four-step Camera Calibration Procedure with Implicit Image Correction. In IEEE Conference on Computer Vision and Pattern Recognition, pages 1106–1112, 1997. [5]: Zhengyou Zhang. Flexible camera calibration by viewing a plane from unknown orientations. In IEEE International Conference on Computer Vision, volume 1, pages 666–673, 1999.	2021-05-07 12:11:51	{"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": 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": 0.3692391514778137, "perplexity": 958.9595951084577}, "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/1620243988793.99/warc/CC-MAIN-20210507120655-20210507150655-00321.warc.gz"}
https://developer.here.com/documentation/ios-sdk-explore/4.9.0.0/release_notes/topics/notes_explore.html	# Release Notes We are pleased to announce the availability of version 4.9.0.0 (Explore Edition) of the new HERE SDK for iOS. This is a stable release. We are constantly working on new features, stabilizing APIs, improving our KPIs, and evaluating your feedback. ## Changes This section documents major changes to this release that may require you to update your applications. ### Version 4.9.0.0 #### New Features • Added new animation features to MapCamera with startAnimation(MapCameraAnimation), EasingFunction and KeyframeInterpolationMode. Added related classes GeoCoordinatesKeyframe, GeoOrientationKeyframe, Point2DKeyframe, ScalarKeyframe, MapCameraAnimation, MapCameraAnimationFactory and MapCameraKeyframeTrack. • Routes can now be imported: Added routingEngine.importRoute() to import a route from a list of GeoCoordinates and RouteOptions. • Added support for map marker clustering with the new classes MapMarkerCluster and MapMarkerCluster.ImageStyle. A marker cluster can be added/removed with MapScene.addMapMarkerCluster()/MapScene.emoveMapMarkerCluster. See the MapItems example app for a usage example. • Added optional field time to Location class to specify the time when the location data was set. • Batch support for MapMarker: Added method to add multiple MapMarker items at once. With mapScene.addMapMarkers() a list of MapMarker items can be added to a MapScene with a single batch call. • Search: Added PlaceType.intersection to indicate an intersection of at least two streets. Note: This is a beta feature. #### API Changes • Removed the previously deprecated MapRepresentable interface and addMapRepresentable and removeMapRepresentable methods from MapView. Use the MapViewLifecycleDelegate instead. • Map schemes: Removed the previously deprecated scene configuration files: "legacy.grey.day.json", "legacy.grey.night.json", "legacy.hybrid.day.json", "legacy.hybrid.night.json", "legacy.normal.day.json", "legacy.normal.night.json". Use the related MapSchemes instead. • Removed the previously deprecated Notice struct, NoticeCode enum and Section.notices property. Use SectionNoticeCode, SectionNotice class and Section.getSectionNotices() method instead. • Declared all Gestures delegate properties as weak. • Search: Removed the following fields from Address class that have been previously deprecated: stateName, countyName, streetName, additionalData. Please use state, county and street instead. The field additionalData was already unused. • Search: Removed the perviously deprecated optionsNotAvailable value from the SearchError enum. It was replaced by the SearchError.invalidParameter value. • Removed the previously deprecated setFrame() method from MapView class. Use the frame property instead. • Removed the previously deprecated greyDay and greyNight from MapScheme enum. Use the normal map scheme variants instead. • Deprecated field timestamp in Location class, please use the newly introduced time field instead. #### Resolved Issues • Fixed Chinese fonts that have been displayed accidentially for Japan labels. • Fixed: Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. #### Known Issues • Updating an existing MapMarker with a completely new image causes the marker to disappear for a brief moment before being drawn with the new image. • The newly introduced zoom level behaves inconsistent across different devices as the shown level of detail depends on the physical screen size of a device. • Transparency for MapPolylines is not supported. • MapViewPin instances cannot be removed after tilting and panning the map. • The jamFactor indicating TrafficSpeed is currently calculated linear without taking road types and other parameters into account. • An inner city Route for trucks carrying dangerous goods may not result in an error message. • Running on a simulator on M1-based Macs only works when no external dependencies / swift packages are integrated • When using Xcode 13 you may get a warning that certain files in heresdk.swiftmodule are either malformed or generated by a different Swift version. It is safe to ignore this warning. The reason for this warning is that the HERE SDK is compiled with Xcode 12.4. • MapMarkerCluster groups are not directly pickable yet. As a workaround, overlapping clusters can be identified by setting a unique Metadata key to the contained MapMarker items. See the MapItems app for an example. • The traffic layers MapScene.Layers.TrafficFlow and MapScene.Layers.TrafficIncidents may not be instantly visible when starting an app for the first time. ## Changes from Previous Releases This section documents major changes from past releases. We encourage you to use the latest release, 4.9.0.0. ### Version 4.8.4.0 #### New Features • Routing: Added durationInSeconds property to the Maneuver class to get the estimated duration of a maneuver. • Added new factory method for DMS and decimal GeoCoordinates to convert one into the other with GeoCoordinates.fromString(). • Search: Added ResponseDetails as result type that provides the requestId of a search request and a correlationId to identify multiple, related queries. • Routing: Added a static fromString method to SegmentReference to generate instances of this class from a well-formatted String. Usually, SegmentReference instances are only accessible from the Span of a Route's Segment. #### API Changes • Routing: For the new toll cost API, now the newly added RouteOptions.enableTolls flag must be set to get toll costs. It is set to false by default. When this flag is enabled, toll data is requested for toll applicable transport modes. #### Known Issues • Updating an existing MapMarker with a completely new image causes the marker to disappear for a brief moment before being drawn with the new image. • The newly introduced zoom level behaves inconsistent across different devices as the shown level of detail depends on the physical screen size of a device. • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Transparency for MapPolylines is not supported. • MapViewPin instances cannot be removed after tilting and panning the map. • The jamFactor indicating TrafficSpeed is currently calculated linear without taking road types and other parameters into account. • An inner city Route for trucks carrying dangerous goods may not result in an error message. • Running on a simulator on M1-based Macs only works when no external dependencies / swift packages are integrated ### Version 4.8.3.0 #### Highlights • Added MapView.pause() and MapView.resume() static methods to prevent unnecessary GPU calls when an application runs in background. It is recommend to call these methods in your AppDelegate's methods applicationDidEnterBackground(_ application:) and applicationWillEnterForeground(_ application:). #### New Features • Public transit routes: Added support for new pedestrian sections to provide better information on in-station walks, for example, when walking from a place of type accessPoint type to a place of type station. • Routing: Added RouteOptions.enableTrafficOptimization property which is true by default, and when set to false, it doesn't consider traffic information and ignores RouteOptions.departureTime. • Routing: It's now possible to avoid individual segments on a Route. With the newly added segments field in AvoidanceOptions users can specify the parts of a route they would like to avoid after a recalculation. The segments can be identified via Route -> Section -> Span -> SegmentReference. • Added SegmentReference class that represents a reference to a segment id with a travel direction. • Added getSegmentReference method to the Span class. • Added TravelDirection enum that indicates the travel direction along a segment. • Added enhanced APIs to control the MapCamera with the new classes MapCameraUpdate, MapCameraUpdateFactory and MapMeasure. Added the applyUpdate method to MapCamera that accepts a MapCameraUpdate as parameter. • Routing: Added violatedAvoidSeasonalClosure, violatedAvoidTollTransponder, seasonalClosure, tollTransponder, tollsDataUnavailable and chargingStopNotNeeded symbols to the SectionNoticeCode enum. #### API Changes • Transit routes no longer provide fare information and the TransitSectionDetails.fares list is always empty. • Routing: Deprecated SectionNoticeCode.violatedPedestrianOption symbol. It will be removed in v4.11.0 as it is no longer supported. • Deprecated persistentMapStorageFolderAccessDenied and failedToLockPersistentMapStorageFolder values from the InstantiationErrorCode enum. The persistent storage directory is no longer locked. #### Resolved Issues • Fixed: It is not possible to set both RouteOptions.departureTime and RouteOptions.arrivalTime at the same time. Plus, RouteOptions.arrivalTime is ignored when RouteOptions.enableTrafficOptimization is set to false. #### Known Issues • Updating an existing MapMarker with a completely new image causes the marker to disappear for a brief moment before being drawn with the new image. • The newly introduced zoom level behaves inconsistent across different devices as the shown level of detail depends on the physical screen size of a device. • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Transparency for MapPolylines is not supported. • MapViewPin instances cannot be removed after tilting and panning the map. • The jamFactor indicating TrafficSpeed is currently calculated linear without taking road types and other parameters into account. • An inner city Route for trucks carrying dangerous goods may not result in an error message. • Running on a simulator on M1-based Macs only works when no external dependencies / swift packages are integrated ### Version 4.8.2.0 #### New Features • Improved search for electric vehicle charging stations: Added classes EVChargingPool and EVChargingStation. A charging pool for electric vehicles is an area equipped with one or more charging stations. A charging station defines a group of connectors for electrical vehicles that share a common charging connector type and max power level. Note: This is a beta feature. • Improved search for electric vehicle charging stations: Added property Details.evChargingPool that represents the details of the charging pool for electric vehicles. Note: This is a beta feature. • Routing: Added information on toll costs with a new PaymentMethod enum, TollFare & Toll structs and Section.tolls property. Note: This is a beta feature. #### Resolved Issues • Fixed: In rare cases, Storage.LevelDB related crashes can occur when starting the app. #### Known Issues • Updating an existing MapMarker with a completely new image causes the marker to disappear for a brief moment before being drawn with the new image. • The newly introduced zoom level behaves inconsistent across different devices as the shown level of detail depends on the physical screen size of a device. • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Transparency for MapPolylines is not supported. • MapViewPin instances cannot be removed after tilting and panning the map. • The jamFactor indicating TrafficSpeed is currently calculated linear without taking road types and other parameters into account. • An inner city Route for trucks carrying dangerous goods may not result in an error message. • Running on a simulator on M1-based Macs only works when no external dependencies / swift packages are integrated ### Version 4.8.1.0 #### New Features • Routing: Added NoRouteHandle enum value to RoutingError to indicate that the route did not contain the required RouteHandle, which is needed, for example, to refresh a route. • Search: Added new name property to PlaceCategory that provides a level 3 description. #### Known Issues • Updating an existing MapMarker with a completely new image causes the marker to disappear for a brief moment before being drawn with the new image. • The newly introduced zoom level behaves inconsistent across different devices as the shown level of detail depends on the physical screen size of a device. • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Transparency for MapPolylines is not supported. • MapViewPin instances cannot be removed after tilting and panning the map. • The jamFactor indicating TrafficSpeed is currently calculated linear without taking road types and other parameters into account. • An inner city Route for trucks carrying dangerous goods may not result in an error message. • In rare cases, Storage.LevelDB related crashes can occur when starting the app. • Running on a simulator on M1-based Macs only works when no external dependencies / swift packages are integrated ### Version 4.8.0.0 #### API Changes • Deprecated MapCamera.Orientation - replaced with GeoOrientation. • Deprecated MapCamera.OrientationUpdate - replaced with GeoOrientationUpdate. • Deprecated MapCamera.State.targetOrientation- replaced with MapCamera.State.orientationAtTarget. • Deprecated MapCamera.orbitBy(_ delta: MapCamera.OrientationUpdate, around origin: Point2D) - replaced with MapCamera.orbitBy(_ delta: GeoOrientationUpdate, around origin: Point2D). • Deprecated MapCamera.lookAt(point target: GeoCoordinates, orientation: MapCamera.OrientationUpdate, distanceInMeters: Double) - replaced with MapCamera.lookAt(point target: GeoCoordinates, orientation: GeoOrientationUpdate, distanceInMeters: Double). • Deprecated MapCamera.lookAt(area target: GeoBox, orientation: MapCamera.OrientationUpdate) Parameters - replaced with MapCamera.lookAt(area target: GeoBox, orientation: GeoOrientationUpdate). • Deprecated MapCamera.setTargetOrientation(orientation: MapCamera.OrientationUpdate) - replaced with MapCamera.setOrientationAtTarget(_ orientation: GeoOrientationUpdate). • Deprecated MapCamera.flyTo(target: GeoCoordinates, orientation: MapCamera.OrientationUpdate, animationOptions: MapCamera.FlyToOptions) - replaced with MapCamera.flyTo(target: GeoCoordinates, orientation: GeoOrientationUpdate, animationOptions: MapCamera.FlyToOptions). • Deprecated MapCamera.flyTo(target: GeoCoordinates, orientation: MapCamera.OrientationUpdate, distanceInMeters: Double, animationOptions: MapCamera.FlyToOptions) - replaced with MapCamera.flyTo(target: GeoCoordinates, orientation: GeoOrientationUpdate, distanceInMeters: Double, animationOptions: MapCamera.FlyToOptions). • Deprecated MapCamera.lookAt(area target: GeoBox, orientation: MapCamera.OrientationUpdate, viewRectangle: Rectangle2D) - replaced with MapCamera.lookAt(area target: GeoBox, orientation: GeoOrientationUpdate, viewRectangle: Rectangle2D). • Removed the previously deprecated updateGeometry() methid from MapPolyline and MapPolygon. • Search: Removed previously deprecated fields in Contact: emailAddresses, landlinePhoneNumbers, mobilePhoneNumbers, websiteAddresses. • Routing: Removed the previously deprecated Arrival and Departure classes. #### Known Issues • Updating an existing MapMarker with a completely new image causes the marker to disappear for a brief moment before being drawn with the new image. • The newly introduced zoom level behaves inconsistent across different devices as the shown level of detail depends on the physical screen size of a device. • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Transparency for MapPolylines is not supported. • MapViewPin instances cannot be removed after tilting and panning the map. • The jamFactor indicating TrafficSpeed is currently calculated linear without taking road types and other parameters into account. • An inner city Route for trucks carrying dangerous goods may not result in an error message. • In rare cases, Storage.LevelDB related crashes can occur when starting the app. • Running on a simulator on M1-based Macs only works when no external dependencies / swift packages are integrated ### Version 4.7.7.0 #### New Features • Search: Added politicalView member to Place class. The optional value can be checked if it is matching the one that was set beforehand via SDKOptions. #### API Changes • The map styles for NORMAL_DAY, NORMAL_NIGHT, HYBRID_DAY and HYBRID_NIGHT have been updated to the ones that have been made available as preview beforehand. The new map styles feature a clean and neutral base map and a street network with improved gray scales that allow a better hierarchy of elements that can be added on top. • MapView: "Simplified Chinese", "Traditional Taiwan" and "Traditional Hong Kong" language labels are now supported. • Routing: Updated the API for the returnToRoute and refreshRoute feature for RoutingEngine: • Changed the parameter order of RoutingInterface.returnToRoute(): The startingPoint parameter is now the second parameter, not the third parameter. The feature is still in beta. • Added a startingPoint parameter to the RoutingEngine.refreshRoute() method. In return, deleted RefreshRouteOptions.updateStartingPoint(). The feature is still in beta. • Deleted the ReturnToRouteEngine. Use the newly introduced RoutingInterface.returnToRoute() method instead. #### Resolved Issues • Fixed: A Maneuver.polyline list of GeoCoordinates has only one element for the last maneuver of a route. • Fixed: When MapMarker.setOverlapAllowed() is set to false and the map is zoomed, this can lead to a flickering of the marker assets. #### Highlights • Note: The existing delegate properties for the Gestures class will be changed to weak references with the upcoming HERE SDK v4.9.0 release. After this change, it is no longer necessary to manually set delegates to nil to break the retain cycle between conforming objects and their respective delegate variables. #### Known Issues • Updating an existing MapMarker with a completely new image causes the marker to disappear for a brief moment before being drawn with the new image. • The newly introduced zoom level behaves inconsistent across different devices as the shown level of detail depends on the physical screen size of a device. • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Transparency for MapPolylines is not supported. • MapViewPin instances cannot be removed after tilting and panning the map. • The jamFactor indicating TrafficSpeed is currently calculated linear without taking road types and other parameters into account. • An inner city Route for trucks carrying dangerous goods may not result in an error message. ### Version 4.7.6.0 #### Highlights • Note: The existing delegate properties for the Gestures class will be changed to weak references with the upcoming HERE SDK v4.9.0 release. After this change, it is no longer necessary to manually set delegates to nil to break the retain cycle between conforming objects and their respective delegate variables. #### New Features • Improved map rendering performance on low end devices. #### Resolved Issues • Fixed: Map tiles may not load automatically when losing internet connectivity and connectivity gets back, unless the user interacts with the map. #### Known Issues • Updating an existing MapMarker with a completely new image causes the marker to disappear for a brief moment before being drawn with the new image. • The newly introduced zoom level behaves inconsistent across different devices as the shown level of detail depends on the physical screen size of a device. • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Transparency for MapPolylines is not supported. • MapViewPin instances cannot be removed after tilting and panning the map. • A Maneuver.polyline list of GeoCoordinates has only one element for the last maneuver of a route. • The jamFactor indicating TrafficSpeed is currently calculated linear without taking road types and other parameters into account. • An inner city Route for trucks carrying dangerous goods may not result in an error message. • When MapMarker.setOverlapAllowed() is set to false and the map is zoomed, this can lead to a flickering of the marker assets. ### Version 4.7.5.0 #### New Features • Routing: Added avoidance options for pedestrians with PedestrianOptions.avoidanceOptions. • Search: Added support to search along longer route polylines with a GeoCorridor. Either increase halfWidthInMeters when set via constructor or do not set it - by using the GeoCorridor constructor that only allows to set the GeoPolyline. The parameter radiusInMeters has been deprecated: Please only use halfWidthInMeters as constructor parameter to specify the thickness of the route corridor. When increasing halfWidthInMeters then a greater simplification of the polyline shape can be achieved resulting in a longer route that can be searched along. On the downside the results are less accurate and can lie farther away from the route. When halfWidthInMeters is not set, a suitable value is set internally based on "best guess". • The LocationIndicator now supports a gray state that can be set via LocationIndicator.isActive. #### Known Issues • Updating an existing MapMarker with a completely new image causes the marker to disappear for a brief moment before being drawn with the new image. • The newly introduced zoom level behaves inconsistent across different devices as the shown level of detail depends on the physical screen size of a device. • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Transparency for MapPolylines is not supported. • MapViewPin instances cannot be removed after tilting and panning the map. • A Maneuver.polyline list of GeoCoordinates has only one element for the last maneuver of a route. • Map tiles may not load automatically when losing internet connectivity and connectivity gets back, unless the user interacts with the map. • The jamFactor indicating TrafficSpeed is currently calculated linear without taking road types and other parameters into account. • An inner city Route for trucks carrying dangerous goods may not result in an error message. • When MapMarker.setOverlapAllowed() is set to false and the map is zoomed, this can lead to a flickering of the marker assets. ### Version 4.7.4.0 #### Highlights • Note: The existing delegate properties for the Gestures class will be changed to weak references with the upcoming HERE SDK v4.9.0 release. After this change, it is no longer necessary to manually set delegates to nil to break the retain cycle between conforming objects and their respective delegate variables. #### New Features • Routing: Added TruckType enum to specify the type of a truck for the TruckSpecifications. • Routing: Added TruckSpecifications.trailerCount to specify the number of trailers. • Routing: Added accessPoint to RoutePlaceType enum. It allows to differentiate Section's based on RoutePlace.type. • Public transit: Added FarePriceType enum, FarePrice.type, FarePrice.minimum and FarePrice.maximum properties. FarePrice provides the actual price value. #### API Changes • Gestures: A finger down gesture stops now any ongoing animation including kinetic panning. • Routing: Deprecated FarePrice.unit, use the newly introduced FarePrice.validityPeriodInSeconds property instead. • Routing: Deprecated the unsupported Agency.icon property. This property is no longer supported and will be removed with HERE SDK release v4.10.0. #### Known Issues • Updating an existing MapMarker with a completely new image causes the marker to disappear for a brief moment before being drawn with the new image. • The newly introduced zoom level behaves inconsistent across different devices as the shown level of detail depends on the physical screen size of a device. • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Transparency for MapPolylines is not supported. • MapViewPin instances cannot be removed after tilting and panning the map. • A Maneuver.polyline list of GeoCoordinates has only one element for the last maneuver of a route. • Map tiles may not load automatically when losing internet connectivity and connectivity gets back, unless the user interacts with the map. • The jamFactor indicating TrafficSpeed is currently calculated linear without taking road types and other parameters into account. • An inner city Route for trucks carrying dangerous goods may not result in an error message. • When MapMarker.setOverlapAllowed() is set to false and the map is zoomed, this can lead to a flickering of the marker assets. ### Version 4.7.3.0 #### New Features • Routing: Added support for EVCarOptions and EVTruckOptions for the RefreshRouteOptions class. • EV Routing: Added new parameters to BatterySpecifications: • minChargeAtChargingStationInKilowattHours: The minimum charge when arriving at a charging station. • minChargeAtDestinationInKilowattHours: The minimum charge at the final route destination. • maxChargingVoltageInVolts: The maximum charging voltage supported by the vehicle's battery. • maxChargingCurrentInAmperes: The maximum charging current supported by the vehicle's battery. • chargingSetupDurationInSeconds: The time spent after arriving at a charging station, but before actually charging. • SearchEngine: For reverse geocoding it is now possible to search in a GeoCircle with search(circle: GeoCircle, options: SearchOptions, completion: SearchCompletionHandler). #### Known Issues • Updating an existing MapMarker with a completely new image causes the marker to disappear for a brief moment before being drawn with the new image. • The newly introduced zoom level behaves inconsistent across different devices as the shown level of detail depends on the physical screen size of a device. • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Transparency for MapPolylines is not supported. • MapViewPin instances cannot be removed after tilting and panning the map. • A Maneuver.polyline list of GeoCoordinates has only one element for the last maneuver of a route. • Map tiles may not load automatically when losing internet connectivity and connectivity gets back, unless the user interacts with the map. • The jamFactor indicating TrafficSpeed is currently calculated linear without taking road types and other parameters into account. • An inner city Route for trucks carrying dangerous goods may not result in an error message. • When MapMarker.setOverlapAllowed() is set to false and the map is zoomed, this can lead to a flickering of the marker assets. ### Version 4.7.2.0 #### New Features • Added a feature to refresh a Route: Added the RouteOptions.enableRouteHandle flag to get a RouteHandle, a RefreshRouteOptions class and a RoutingEngine.refreshRoute() method. Note that currently EVRouteOptions are not supported. See the related chapter in the Developer's Guide for more information. • LocationIndicator: Added a new style for pedestrians. The new style is added to IndicatorStyle and can be set to a LocationIndicator instance to switch the default 3D map marker model to indicate the current heading and location. Optionally, the style can be customized. • Added an option to affect FPS rendering of the map view by decreasing or increasing the frameRate with MapView.frameRate. This can be useful to reduce CPU usage for low end devices. The default FPS is 60 frames per second. It is also possible to deactivate automatic render cycles by setting FPS to 0. #### Resolved Issues • Fixed: MapMarker3D instances are only partly rendered when the camera is positioned too far from earth. #### Known Issues • Updating an existing MapMarker with a completely new image causes the marker to disappear for a brief moment before being drawn with the new image. • The newly introduced zoom level behaves inconsistent across different devices as the shown level of detail depends on the physical screen size of a device. • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Transparency for MapPolylines is not supported. • MapViewPin instances cannot be removed after tilting and panning the map. • A Maneuver.polyline list of GeoCoordinates has only one element for the last maneuver of a route. • Map tiles may not load automatically when losing internet connectivity and connectivity gets back, unless the user interacts with the map. • The jamFactor indicating TrafficSpeed is currently calculated linear without taking road types and other parameters into account. • An inner city Route for trucks carrying dangerous goods may not result in an error message. ### Version 4.7.1.0 #### Highlights • Added a default 3D LocationIndicator to show the current location and travel direction of a user. It can be added to the map view with MapView.addMapLifecycleListener(). Update its location via Mapview.updateLocation() that accepts a Location as parameter. The look of the default asset can be customized by setting a MapMarker3DModel as parameter to LocationIndicator.setMarker3dModel(). You can also specify the MarkerType: In the future, it may be used to enable different styles of markers for different use cases. #### New Features • Introduced taxi routes. Added TransportMode.taxi type and a TaxiOptions class to calculate routes optimized for taxis via RoutingEngine.calculateRoute(). Note: This is a beta release of this transport mode, so there could be a few bugs and unexpected behaviors. • Routing: Added RouteOptions.speedCapInMetersPerSecond to limit the maximum allowed speed for a vehicle. When set, the route duration will be shorter for car and truck transport modes. For scooter routes it may also affect the route geometry. Other transport modes are ignored. #### Resolved Issues • Fixed: Public transit routes do not contain maneuvers. • Fixed: MapViewPin instances disappear from MapView when changing device orientation. They reappear when moving the map. #### Known Issues • Updating an existing MapMarker with a completely new image causes the marker to disappear for a brief moment before being drawn with the new image. • The newly introduced zoom level behaves inconsistent across different devices as the shown level of detail depends on the physical screen size of a device. • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Transparency for MapPolylines is not supported. • MapViewPin instances cannot be removed after tilting and panning the map. • A Maneuver.polyline list of GeoCoordinates has only one element for the last maneuver of a route. • MapMarker3D instances are only partly rendered when the camera is positioned too far from earth. • Map tiles may not load automatically when losing internet connectivity and connectivity gets back, unless the user interacts with the map. • The jamFactor indicating TrafficSpeed is currently calculated linear without taking road types and other parameters into account. • An inner city Route for trucks carrying dangerous goods may not result in an error message. ### Version 4.7.0.0 #### Highlights • Introduced public transit routing with a new TransitRoutingEngine that can calculate public transit routes. It uses TransitWaypoint type waypoints and a TransitRouteOptions class to specify various transit related options. Note that the API is in an early development stage. More features and overall stabilization are planned for the next releases. #### New Features • Along with the newly introduced TransitRoutingEngine (see above), the following supportive types have been added: AttributionType enum, FarePriceType enum, FareReason enum, PreActionType enum, TransitDepartureStatus enum, TransitIncidentType enum, TransitIncidentEffect enum, TransitMode enum, TransitModeFilter enum, Agency class, Attribution class, Fare class, FarePrice class, PreAction class, TransitIncident class, TransitDeparture class, TransitSectionDetails class, TransitStop class, and TransitTransport class. • Routing: Introduced a new Span class that is accessible from a Section on a Route. It points to the related section notices to indicate possible route violations. With the Span class these violations can now be identified on the Route as they contain the length in meters and the geometry. • Routing: Added additional constructor for IsolineOptions.Calculation: IsolineOptions.Calculation init(rangeType: IsolineRangeType, rangeValues: [Int32], isolineCalculationMode: IsolineCalculationMode). • Search: Added new method to SearchEngine to get place details for a Suggestion that contains a href String. Use the new sendRequest() method for this. #### API Changes • Added a new MapViewLifecycleListener that replaces the deprecated MapRepresentable. It can be added or removed from a Mapview. Use this to get notified on the map view's lifecycle. • Deprecated MapRepresentable and the related methods to add or remove it from a MapView. Use the newly introduced MapViewLifecycleListener instead. • Removed the previously deprecated GeoBox.intersects(GeoBox) and GeoBox.contains(GeoBox). • Removed the previously deprecated LocationProvider and the LocationListener.onLocationTimeout() method. • Removed the previously deprecated Section.transportMode property. • Removed the previously deprecated GeoCorridor(List<GeoCoordinates>, Double) constructor. • Removed the previously deprecated Place.coordinates property. • Removed the previously deprecated class Color and the deprecated extension for UIColor. #### Resolved Issues • Fixed: The opening hours for a Place that closes after midnight end at midnight. #### Known Issues • Updating an existing MapMarker with a completely new image causes the marker to disappear for a brief moment before being drawn with the new image. • The newly introduced zoom level behaves inconsistent across different devices as the shown level of detail depends on the physical screen size of a device. • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Transparency for MapPolylines is not supported. • MapViewPin instances disappear from MapView when changing device orientation. They reappear when moving the map. • MapViewPin instances cannot be removed after tilting and panning the map. • A Maneuver.polyline list of GeoCoordinates has only one element for the last maneuver of a route. • MapMarker3D instances are only partly rendered when the camera is positioned too far from earth. • Map tiles may not load automatically when losing internet connectivity and connectivity gets back, unless the user interacts with the map. • The jamFactor indicating TrafficSpeed is currently calculated linear without taking road types and other parameters into account. • An inner city Route for trucks carrying dangerous goods may not result in an error message. • Public transit routes do not contain maneuvers. ### Version 4.6.5.0 #### New Features • Added support for political views. SDKOptions allow now to specify the politicalView string. It's a three letter country code defined by ISO 3166-1 alpha-3. When set, all map data will respect the point of view of this country. Note that this is a beta feature and thus there can be bugs and unexpected behaviour. • A Place can now be serialized or deserialized with serializeCompact() and deserialize() to or from a String. • Routing: Added SectionNoticeCode and NoticeSeverity enums, SectionNotice class and Section.sectionNotices property to get informed on possible route violations. • Isoline Routing: • Added RoutePlaceDirection enum with values arrival and departure. • Added isolineDirection field inside IsolineOptions.Calculation structure (with default value of RoutePlaceDirection.departure) • Extended the existing IsolineOptions.Calculation constructor to accept the new RoutePlaceDirection enum. • Added RouteOptions.arrivalTime to set an optional time when a travel is expected to end. Note that this parameter is currently only supported for isoline route calculation. #### API Changes • Routing: Deprecated the Notice class and NoticeCode enum along with Section.notices property, use the newly introduced SectionNoticeCode, SectionNotice class and Section.sectionNotices property instead. • Search: Deprecated SearchError.optionNotAvailable enum value, it will be replaced by the existing SearchError.invalidParameter value. #### Known Issues • Updating an existing MapMarker with a completely new image causes the marker to disappear for a brief moment before being drawn with the new image. • The newly introduced zoom level behaves inconsistent across different devices as the shown level of detail depends on the physical screen size of a device. • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Transparency for MapPolylines is not supported. • MapViewPin instances disappear from MapView when changing device orientation. They reappear when moving the map. • MapViewPin instances cannot be removed after tilting and panning the map. • A Maneuver.polyline list of GeoCoordinates has only one element for the last maneuver of a route. • MapMarker3D instances are only partly rendered when the camera is positioned too far from earth. • Map tiles may not load automatically when losing internet connectivity and connectivity gets back, unless the user interacts with the map. • The jamFactor indicating TrafficSpeed is currently calculated linear without taking road types and other parameters into account. • The opening hours for a Place that closes after midnight end at midnight. • An inner city Route for trucks carrying dangerous goods may not result in an error message. ### Version 4.6.4.0 #### New Features • Routing: Introduced Waypoint.minCourseDistanceInMeters to specify an optional distance parameter during which a user can avoid taking actions. • The size of the viewport can now be retrieved from MapView with the new viewportSize property that returns Size2D in physical pixels. • Enhanced FlyTo MapCamera animations from A to B with target orientation and target distance: flyTo(target: GeoCoordinates, orientation: MapCamera.OrientationUpdate, animationOptions: MapCamera.FlyToOptions). • New map styles have been introduced as beta versions. • The map styles for normalDay, normalNight, hybridDay and hybridNight will be updated with v4.8.0. The planned new map styles are already accessible under following file names as beta versions: • "preview.normal.day.json" - This scheme will update the current normalDay in v4.8.0. • "preview.normal.night.json" - This scheme will update the current normalNight in v4.8.0. • "preview.hybrid.day.json" - This scheme will update the current hybridDay in v4.8.0. • "preview.hybrid.night.json" - This scheme will update the current hybridNight in v4.8.0. • The legacy map styles for normalDay, normalNight, hybridDay and hybridNight will still be accessible for v4.8.0 until v4.9.0 under the file names listed below. They will be removed with v4.9.0. • "legacy.normal.day.json" • "legacy.normal.night.json" • "legacy.hybrid.day.json" • "legacy.hybrid.night.json" • "legacy.grey.day.json" • "legacy.grey.night.jso #### API Changes • The map styles MapScheme.greyDay, MapScheme.greyNight have been deprecated and will be removed. Use the normal variants instead. #### Known Issues • Updating an existing MapMarker with a completely new image causes the marker to disappear for a brief moment before being drawn with the new image. • The newly introduced zoom level behaves inconsistent across different devices as the shown level of detail depends on the physical screen size of a device. • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Transparency for MapPolylines is not supported. • MapViewPin instances disappear from MapView when changing device orientation. They reappear when moving the map. • MapViewPin instances cannot be removed after tilting and panning the map. • A Maneuver.polyline list of GeoCoordinates has only one element for the last maneuver of a route. • MapMarker3D instances are only partly rendered when the camera is positioned too far from earth. • Map tiles may not load automatically when losing internet connectivity and connectivity gets back, unless the user interacts with the map. • The jamFactor indicating TrafficSpeed is currently calculated linear without taking road types and other parameters into account. • The opening hours for a Place that closes after midnight end at midnight. • An inner city Route for trucks carrying dangerous goods may not result in an error message. ### Version 4.6.3.0 #### New Features • Added new WatermarkStyle enum that defines the style of the HERE watermark logo. The dark watermark should be used for custom styles that are brighter (like daytime) and the light watermark for darker custom schemes (like night or satellite based). Added a new loadScene() method to MapScene to accept the new enum together with a custom style. • MapCamera: Added FlyToOptions to customize the fly-to animation from current target GeoCoordinates to a new location. The durationInMs parameter defines how long the animation will run, the bowFactor defines a relative camera height for the ballistic curve that ranges from -1 < 0 (concave curve) to 0 (constant height) to 0 < 1 (convex curve). The maximum height can be achieved with a bowFactor of 1, the minimum with -1. Note that the height is relative to the distance between current target and new target to achieve a constant look regardless of the current zoom level. • Routing: Added sideOfStreetHint property to Waypoint. These optional GeoCoordinates indicate which side of the street should be used to reach the waypoint. For example, if the location is to the left of the street, the route will prefer using that side in case the street has dividers. Hence, if the street has no dividers, sideOfStreetHint is ignored. #### Known Issues • Updating an existing MapMarker with a completely new image causes the marker to disappear for a brief moment before being drawn with the new image. • The newly introduced zoom level behaves inconsistent across different devices as the shown level of detail depends on the physical screen size of a device. • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Transparency for MapPolylines is not supported. • MapViewPin instances disappear from MapView when changing device orientation. They reappear when moving the map. • MapViewPin instances cannot be removed after tilting and panning the map. • A Maneuver.polyline list of GeoCoordinates has only one element for the last maneuver of a route. • MapMarker3D instances are only partly rendered when the camera is positioned too far from earth. • Map tiles may not load automatically when losing internet connectivity and connectivity gets back, unless the user interacts with the map. • The jamFactor indicating TrafficSpeed is currently calculated linear without taking road types and other parameters into account. • The opening hours for a Place that closes after midnight end at midnight. • An inner city Route for trucks carrying dangerous goods may not result in an error message. ### Version 4.6.2.0 #### New Features • Introduced beta versions of new map style combinations for use with MapScene.loadScene(): "preview.normal.day.json", "preview.normal.night.json", "preview.hybrid.day.json" and "preview.hybrid.night.json". • Added new map item type: MapArrow elements can now be rendered on the map to indicate a direction. They can be added or removed like other map items via MapScene. • MapMarker items can now be made invisible once they overlap each other at certain zoom levels. They contain a new property with isOverlapAllowed: Bool { get set }. #### API Changes • The free flow traffic layer is now rendering green traffic flow lines by default. #### Resolved Issues • Fixed: When traffic flows are enabled, map gestures may behave slower than expected. • Fixed: When SDKOptions is created before initialization of SDKNativeEngine it may contain a corrupted path. #### Known Issues • Updating an existing MapMarker with a completely new image causes the marker to disappear for a brief moment before being drawn with the new image. • The newly introduced zoom level behaves inconsistent across different devices as the shown level of detail depends on the physical screen size of a device. • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Transparency for MapPolylines is not supported. • MapViewPin instances disappear from MapView when changing device orientation. They reappear when moving the map. • MapViewPin instances cannot be removed after tilting and panning the map. • A Maneuver.polyline list of GeoCoordinates has only one element for the last maneuver of a route. • MapMarker3D instances are only partly rendered when the camera is positioned too far from earth. • Map tiles may not load automatically when losing internet connectivity and connectivity gets back, unless the user interacts with the map. • The jamFactor indicating TrafficSpeed is currently calculated linear without taking road types and other parameters into account. • The opening hours for a Place that closes after midnight end at midnight. • An inner city Route for trucks carrying dangerous goods may not result in an error message. ### Version 4.6.1.0 #### Highlights • Added new flyTo() method to MapCamera for basic animations from A to B by setting new target coordinates. The animation can be interrupted by gestures or any programmatical change of the camera's position or orientation. #### New Features • Routing: Introduced ZoneCategory enum and added zoneCategories field to to the AvoidanceOptions struct which is a collection of ZoneCategory's. • EV routing: Introduced Route.getEVDetails() which returns the accumulated evDetails data for all sections of a route. #### API Changes • Routing: Deprecated Section.getter:arrival and Section.getter:departure. Instead, use the newly introduced Section.getter:arrivalPlace and Section.getter:departurePlace to get a RoutePlace. • Deprecated the Address fields stateName, countyName, streetName and the related constructors. Instead, use the newly introduced constructor that takes the new fields state, country and street. #### Resolved Issues • Fixed: Performing a CategoryQuery or a TextQuery search no longer returns SearchError.httpError when searching in a circle, whose radius is a number with a fraction. #### Known Issues • Updating an existing MapMarker with a completely new image causes the marker to disappear for a brief moment before being drawn with the new image. • The newly introduced zoom level behaves inconsistent across different devices as the shown level of detail depends on the physical screen size of a device. • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Transparency for MapPolylines is not supported. • MapViewPin instances disappear from MapView when changing device orientation. They reappear when moving the map. • MapViewPin instances cannot be removed after tilting and panning the map. • When traffic flows are enabled, map gestures may behave slower than expected. • A Maneuver.polyline list of GeoCoordinates has only one element for the last maneuver of a route. • MapMarker3D instances are only partly rendered when the camera is positioned too far from earth. • Map tiles may not load automatically when losing internet connectivity and connectivity gets back, unless the user interacts with the map. • The jamFactor indicating TrafficSpeed is currently calculated linear without taking road types and other parameters into account. • When SDKOptions is created before initialization of SDKNativeEngine it may contain a corrupted path. ### Version 4.6.0.0 #### Highlights • Added isoline routing to calculate the reachable area for the given center coordinates and a range defined in time, distance or energy consumption. Added a new calculateIsoline() method to RoutingEngine. Added related classes: • IsolineOptions: Options for isoline calculation. • IsolineRangeType: Enumeration specifying whether the range type is defined in time, distance or energy consumption. • IsolineCalculationMode: Specifies how the isoline calculation can be optimized. • Isoline: Represents a single isoline. • MapMatchedCoordinates: Contains a pair of user-defined and map-matched coordinates. #### More New Features • MapPolygon items can now be picked from MapView. PickMapItemsResult can now include a list of MapPolygon items. • Added ScooterOptions.allowHighway property. • Added method to look at a given GeoBox at the map view with lookAt(target: GeoBox, orientation: MapCamera.OrientationUpdate, viewRectangle: Rectangle2D) -> Void. For example, this can be used to show a route on a certain part of the map view. • Search: Added categories field to OpeningHours. It contains categories related to specific OpeningHours. For example, when a Place has multiple opening hours associated with it. • Road shields are now rendered by default on the map view. • Search: An unused TaskHandle returned for online and offline search engines can now be omitted instead of excluding it with _ = .... #### API Changes • Removed beta status for scooter transport mode. • Deprecated the updateGeometry() method for MapPolygon. Use the newly introduced set/get geometry accessors instead to get or set a GeoPolygon. • Deprecated the updateGeometry() method for MapPolyline. Use the newly introduced set/get geometry accessors instead to get or set a GeoPolyline. • Search: Deprecated the following Contact fields landlinePhoneNumbers, mobilePhoneNumbers, emailAddresses, websiteAddresses. Use these newly introduced fields instead: landlinePhones, mobilePhones, emails, websites. Each holds a list of newly created classes LandlinePhone, MobilePhone, EmailAddress, WebsiteAddress - containing a string representation of the item and a list of related PlaceCategory values. • Call the newly introduced static call MapView.deinitialize() in your AppDelegate class inside the applicationWillTerminate(_:) method to free resources of the HERE Rendering Engine. This will prevent potential crashes when closing an application. • The free flow traffic layer does no longer render the green traffic flow lines. In the future, an API to enable / disable the flow is planned to be introduced. • The SDKNativeEngine now locks access to the map data cache. When another instance of SDKNativeEngine is instantiated with the same access key id then now an exception is thrown. • MapView: Added clamping to principal point when the coordinates are outside of the viewport. • Maneuver: Deprecated roadName, roadNameLanguageCode, roadNumber, nextRoadName, nextRoadNameLanguageCode, nextRoadNumber. Added instead RoadTexts with roadTexts and nextRoadTexts and LocalizedText and LocalizedTexts. #### Resolved Issues • Fixed: A MapPolyline may be unexpectedly rendered over a MapMarker. #### Known Issues • Updating an existing MapMarker with a completely new image causes the marker to disappear for a brief moment before being drawn with the new image. • The newly introduced zoom level behaves inconsistent across different devices as the shown level of detail depends on the physical screen size of a device. • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Transparency for MapPolylines is not supported. • MapViewPin instances disappear from MapView when changing device orientation. They reappear when moving the map. • MapViewPin instances cannot be removed after tilting and panning the map. • When traffic flows are enabled, map gestures may behave slower than expected. • A Maneuver.polyline list of GeoCoordinates has only one element for the last maneuver of a route. • MapMarker3D instances are only partly rendered when the camera is positioned too far from earth. • Map tiles may not load automatically when losing internet connectivity and connectivity gets back, unless the user interacts with the map. • The jamFactor indicating TrafficSpeed is currently calculated linear without taking road types and other parameters into account. • When SDKOptions is created before initialization of SDKNativeEngine it may contain a corrupted path. ### Version 4.5.4.0 #### Highlights • The supported device specifications have been fine tuned and contain now more granular details on the supported devices. Details can be found in the About section of the Developer's Guide for this edition. #### New Features • Added support for a transform center: The MapCamera know allows to set and get a principal point as Point2D. This point determines where the target is placed within the map view. With this you can, for example, lower the map's default center as it will affect map transformations such as rotations and tilt. • Extended walk routing options: Added PedestrianOptions.walkSpeedInMetersPerSecond property. Note that this feature is released as beta. By setting a walk speed you can calculate pedestrian routes specific for different walk profiles. • Searching for a CategoryQuery within a GeoCircle or GeoBox is no longer marked as a beta feature. • Added new map cache options to SDKOptions with SDKOptions.cacheSizeInBytes and SDKOptions.persistentMapStoragePath. Also available as key in plist (CacheSizeInBytes, PersistentMapStoragePath). With this you can control where to store cached map data and it also allows to specify the amount of data you want to reserve for caching. #### API Changes • Changed PedestrianOptions(routeOptions, textOptions) constructor to PedestrianOptions(routeOptions, textOptions, 1.0) to support the new PedestrianOptions.walkSpeedInMetersPerSecond property (see above). #### Known Issues • Updating an existing MapMarker with a completely new image causes the marker to disappear for a brief moment before being drawn with the new image. • The newly introduced zoom level behaves inconsistent across different devices as the shown level of detail depends on the physical screen size of a device. • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Transparency for MapPolylines is not supported. • MapViewPin instances disappear from MapView when changing device orientation. They reappear when moving the map. • MapViewPin instances cannot be removed after tilting and panning the map. • When traffic flows are enabled, map gestures may behave slower than expected. • A Maneuver.polyline list of GeoCoordinates has only one element for the last maneuver of a route. • MapMarker3D instances are only partly rendered when the camera is positioned too far from earth. • Map tiles may not load automatically when losing internet connectivity and connectivity gets back, unless the user interacts with the map. • The jamFactor indicating TrafficSpeed is currently calculated linear without taking road types and other parameters into account. • A MapPolyline may be unexpectedly rendered over a MapMarker. ### Version 4.5.3.0 #### New Features • Added a NoticeCode that describes issues after a Route was calculated. For example, when a route should avoid tunnels, but the only possible route needs to pass a tunnel, the Route contains a notice that the requested avoidance of tunnels was violated. Therefore, it is recommended to always check a calculated Route for possible violations. The NoticeCode is part of a Notice object. A list of possible Notice objects can be accessed per Section of a Route. The list will be empty, when no violation occurred. • Addeded the ability to change the anchor point of a MapMarker with Anchor2D. By default, the marker is centered on the provided location and the anchor is at (0.5, 0.5). An anchor represents a point in a rectangle as a ratio of the marker's image width and height. • Added search for a CategoryQuery within a GeoCircle or GeoBox. Extended the existing CategoryQuery constructors to accept GeoCircle or GeoBox. Note that this feature is currently in BETA state. #### Known Issues • Updating an existing MapMarker with a completely new image causes the marker to disappear for a brief moment before being drawn with the new image. • The newly introduced zoom level behaves inconsistent across different devices as the shown level of detail depends on the physical screen size of a device. • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Transparency for MapPolylines is not supported. • MapViewPin instances disappear from MapView when changing device orientation. They reappear when moving the map. • MapViewPin instances cannot be removed after tilting and panning the map. • When traffic flows are enabled, map gestures may behave slower than expected. • A Maneuver.polyline list of GeoCoordinates has only one element for the last maneuver of a route. • MapMarker3D instances are only partly rendered when the camera is positioned too far from earth. • Map tiles may not load automatically when losing internet connectivity and connectivity gets back, unless the user interacts with the map. • The jamFactor indicating TrafficSpeed is currently calculated linear without taking road types and other parameters into account. • A MapPolyline may be unexpectedly rendered over a MapMarker. ### Version 4.5.2.0 #### Highlights • Added suppport for scooter route calculation with the new ScooterOptions. Note that this is a BETA feature. Using scooter routes for navigation is not yet supported. #### New Features • Added new read-only property to MapPolyline with geometry: GeoPolyline. • Added DashPattern class to set dashed line styles for a MapPolyline. • Added the property dashFillColor: UIColor? to MapPolyline to add support for fill colors when using dashed lines. • Added cumulative orbit method to MapCamera to rotate the map around arbitrary view coordinates using relative orientation values with orbitBy(_ delta: MapCamera.OrientationUpdate, around origin: Point2D). • Added new initialization options to MapImage extension with: • public convenience init(from uiImage: UIImage) • public convenience init(named name: String, width: Int32, height: Int32, in bundle Bundle? = nil) #### API Changes • Cache path to store map data is now unique per access key ID (which is unique per customer). The HERE SDK automatically appends the current version of the cache and the access key ID. If you want to keep existing cache data, it may be required to copy it from <cache-root> to <cache-root>/v1/<access-key-id>, as the current version of the cache is "v1". • Deprecated SDKNativeEngine.setAccessKey(access_key_id, access_key_secret). Use SDKNativeEngine.setAccessKeySecret(access_key_secret) instead, in combination with setting the access key ID via SDKOptions when constructing a new SDKNativeEngine. • Deprecated LocationUpdateDelegate. Use the new LocationDelegate instead. • Removed deprecated GeoCircle constructor that accepts single precision float type for radius. #### Resolved Issues • Fixed: Embedded POI markers are not visible for the hybrid night map scheme. • Fixed: Pan gestures may receive a cancel event before a begin event. #### Known Issues • Updating an existing MapMarker with a completely new image causes the marker to disappear for a brief moment before being drawn with the new image. • The newly introduced zoom level behaves inconsistent across different devices as the shown level of detail depends on the physical screen size of a device. • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Transparency for MapPolylines is not supported. • MapViewPin instances disappear from MapView when changing device orientation. They reappear when moving the map. • MapViewPin instances cannot be removed after tilting and panning the map. • When traffic flows are enabled, map gestures may behave slower than expected. • A Maneuver.polyline list of GeoCoordinates has only one element for the last maneuver of a route. • MapMarker3D instances are only partly rendered when the camera is positioned too far from earth. • Map tiles may not load automatically when losing internet connectivity and connectivity gets back, unless the user interacts with the map. • The jamFactor indicating TrafficSpeed is currently calculated linear without taking road types and other parameters into account. ### Version 4.5.1.0 #### Highlights • Improved cold startup time on certain devices like Samsung. #### New Features • Introduced Maneuver.lengthInMeters property to return the length of the maneuver. • Added a new MapCameralimits property with targetArea: GeoBox?. This allows to set a target area preventing a user from moving away too much from a desired area of interest. • Added a new MapCameralimits property for bearing with bearingRange: AngleRange. • Introduced SectionTransportMode enum and Section.getSectionTransportMode property returning an instance of this type indicating now transport modes such as ferries. Section.getTransportMode has been deprecated, use the newly introduced method instead. • Search: Introduced SupplierReference type and Details.references property, which holds the list of supplier references to a place. #### Known Issues • Updating an existing MapMarker with a completely new image causes the marker to disappear for a brief moment before being drawn with the new image. • The newly introduced zoom level behaves inconsistent across different devices as the shown level of detail depends on the physical screen size of a device. • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Transparency for MapPolylines is not supported. • MapViewPin instances disappear from MapView when changing device orientation. They reappear when moving the map. • MapViewPin instances cannot be removed after tilting and panning the map. • When traffic flows are enabled, map gestures may behave slower than expected. • A Maneuver.polyline list of GeoCoordinates has only one element for the last maneuver of a route. • MapMarker3D instances are only partly rendered when the camera is positioned too far from earth. • Map tiles may not load automatically when losing internet connectivity and connectivity gets back, unless the user interacts with the map. • The jamFactor indicating TrafficSpeed is currently calculated linear without taking road types and other parameters into account. • Embedded POI markers are not visible for the hybrid night map scheme. • Pan gestures may receive a cancel event before a begin event. ### Version 4.5.0.0 #### Highlights • Added route calculation for electric vehicles (EV). It's now possible to calculate the energy consumption per route section according to the given consumption model (supported for electric cars and trucks). Charging stations are automatically added to the calculated route as waypoints to ensure that the electric vehicle doesn't run out of energy along the way (supported for electric cars). #### New Features • Added LogAppender protocol to insert your own log class into the SDKNativeEngine. This way you can log HERE SDK messages for various predefined log levels even on release builds. • Added a scale property to change the size of a MapMarker3D. • Added camera tilt limits with MapCameraLimits.setMaxTilt(degreesFromNadir: Double) and MapCameraLimits.setMinTilt(degreesFromNadir: Double). • Added camera zoom limits with MapCameraLimits.setMaxZoomLevel(_ zoomLevel: Double) and MapCameraLimits.setMinZoomLevel(_ zoomLevel: Double). • Added new methods to calculate routes for electric vehicles (car and trucks are supported): RoutingEngine.calculateRoute([Waypoint], EVCarOptions, CalculateRouteCallback), RoutingEngine.calculateRoute([Waypoint], EVTruckOptions, CalculateRouteCallback). • Added the following classes and fields to support EV routing (see above): • BatterySpecifications - parameters that describe the electric vehicle's battery. • ChargingConnectorAttributes - details of the connector that is suggested to be used for charging. • ChargingConnectorType - enumeration of the available charging connector types. • ChargingStation - charging station data. • ChargingSupplyType - enumeration of available charging supply types. • EVCarOptions - options to specify how a route for an electric car should be calculated. • EVConsumptionModel - parameters specific for the electric vehicle, which are used to calculate energy consumption on a given route. • EVDetails - additional information that is available for electric vehicles. • EVTruckOptions - options to specify how a route for an electric truck should be calculated. • PostActionType - enumeration of available post action types. • PostAction - an action that must be done after arrival. • RoutePlaceType - shows whether the place on the route (such as departure or arrival) is a charging station or a regular place. • Arrival.type - the type of the arrival place. • Arrival.chargeInKilowattHours - battery charge at arrival. • Arrival.chargingStation - charging station data at arrival. • Departure.type - the type of the departure place. • Departure.chargeInKilowattHours - battery charge at departure. • Departure.chargingStation - charging station data at departure. • Section.postActions - actions that must be done after the arrival. • Section.evDetails - additional section information that is available for electric vehicles. #### API Changes • The minimum supported iOS version was increased from 12.0 to 12.4. • For custom map styles HERE Style Editor 0.26 is required. • Changed splash screen to be grey instead of black. This improves the experiences on devices where cold start takes longer. • Removed a deprecated GeoCoordinates constructor, use the previously introduced counterpart instead. • Changed type of MapMarker.drawOrder from UInt32 to Int32. • Removed a deprecated Anchor2D constructor, use the previously introduced counterpart instead. #### Resolved Issues • Fixed: After orientation changes coordinate conversions may return incorrect values. • Fixed: A category search along a route will crash when the GeoCorridor.polyline list parameter is empty. • Fixed wrongly positioned Viewpins after an orientation change. #### Known Issues • Updating an existing MapMarker with a completely new image causes the marker to disappear for a brief moment before being drawn with the new image. • The newly introduced zoom level behaves inconsistent across different devices as the shown level of detail depends on the physical screen size of a device. • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Transparency for MapPolylines is not supported. • MapViewPin instances disappear from MapView when changing device orientation. They reappear when moving the map. • MapViewPin instances cannot be removed after tilting and panning the map. • When traffic flows are enabled, map gestures may behave slower than expected. • A Maneuver.polyline list of GeoCoordinates has only one element for the last maneuver of a route. • MapMarker3D instances are only partly rendered when the camera is positioned too far from earth. • Map tiles may not load automatically when losing internet connectivity and connectivity gets back, unless the user interacts with the map. • The jamFactor indicating TrafficSpeed is currently calculated linear without taking road types and other parameters into account. ### Version 4.4.6.0 #### Highlights • This release focuses on overall stability and performance improvements. #### Known Issues • The newly introduced zoom level behaves inconsistent across different devices as the shown level of detail depends on the physical screen size of a device. • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Transparency for MapPolylines is not supported. • After orientation changes coordinate conversions may return incorrect values. • A category search along a route will crash when the GeoCorridor.polyline list parameter is empty. • MapViewPin instances disappear from MapView when changing device orientation. They reappear when moving the map. • MapViewPin instances cannot be removed after tilting and panning the map. • When traffic flows are enabled, map gestures may behave slower than expected. • A Maneuver.polyline list of GeoCoordinates has only one element for the last maneuver of a route. • MapMarker3D instances are only partly rendered when the camera is positioned too far from earth. • Map tiles may not load automatically when losing internet connectivity and connectivity gets back, unless the user interacts with the map. ### Version 4.4.5.0 #### New Features • Added the property var image: MapImage { get set } to the MapMarker class. This allows to change a MapImage for a MapMarker that is already shown on the map to update its appearance. • Added zoomTo(_ zoomLevel) -> Void to the MapCamera class to set the zoom level in the range [0,22]. Access the current zoomLevel from the camera's State property. • Added GeoBox.containing(geoCoordinates: [GeoCoordinates]) -> GeoBox? to construct a GeoBox from a list of Geocoordinates. • A CategoryQuery can now be created from a single PlaceCategory with the additional constructor CategoryQuery(_ category: PlaceCategory, filter: String, areaCenter: GeoCoordinates). #### Resolved Issues • Fixed: In rare cases map tiles may flicker when a device is offline and the cache is used. #### Known Issues • The newly introduced zoom level behaves inconsistent across different devices as the shown level of detail depends on the physical screen size of a device. • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Transparency for MapPolylines is not supported. • After orientation changes coordinate conversions may return incorrect values. • Map tiles may not load automatically when losing internet connectivity and connectivity gets back, unless the user interacts with the map. • A category search along a route will crash when the GeoCorridor.polyline list parameter is empty. • MapViewPin instances disappear from MapView when changing device orientation. They reappear when moving the map. • MapViewPin instances cannot be removed after tilting and panning the map. ### Version 4.4.4.0 #### Highlights • Added automatic geometry simplification to MapPolyline: Now, the rendered polyline shape is optimized based on the current distance of the camera to earth. While not being visible for the eye, this improves the performance, for example, when rendering longer routes. #### API Changes • here.sdk.Color derives now from UIColor to support the deprecation of the class. #### Deprecated • A few methods, fields and constructors of class here.sdk.Color have been deprecated. Instead, use iOS's native UIColor class which uses a color component of type CGFloat with the interval [0,1] - instead of type UInt8 and [0,255] that was used for the deprecated components. #### Resolved Issues • Fixed several memory corruption crashes. • Fixed an issue with flickering street labels after credentials have been changed. • Fixed issue with empty IDs for reverse geocoding results: IDs of place results are no longer empty strings and contain now a valid ID. • Route calculation: When no truck route is found due to incompatible truck restrictions, the reason is now logged and RoutingError.noRouteFound error is returned. For example, a log may contain: "Potential route would violate truck restriction:{"maxHeight":400}". #### Known Issues • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Transparency for MapPolylines is not supported. • After orientation changes coordinate conversions may return incorrect values. • Map tiles may not load automatically when losing internet connectivity and connectivity gets back, unless the user interacts with the map. • In rare cases map tiles may flicker when a device is offline and the cache is used. ### Version 4.4.3.0 #### Highlights • Kinetic map panning behavior was greatly improved. Now, when swiping the map moves slower which results in a more natural feel. • Certain areas can now be excluded from route calculation with AvoidanceOptions that contain an avoidAreas list holding GeoBox items which routes should not cross. #### New Features • Added optional Suggestion.href to get a direct link to discover more details. It is available when the suggestion result type is category or chain. • Added RoadFeatures.difficultTurns enum value. Note that it is valid only for truck transport mode. • Added MapError.invalidState enum value which can be raised when a map scene is in an invalid state after a MapView was destroyed. • Added a new GeoCorridor constructor with radiusInMeters as integer type - as replacement for the deprecated constructor with radiusInMeters as double type, see below. #### Deprecated • Deprecated GeoCorridor constructor with radiusInMeters as double type. Use the newly introduced GeoCorridor constructor with radiusInMeters as integer type instead, see above. #### Resolved Issues • Fixed: When the pick radius for pickMapItems() is set to 1000 or higher, an app may crash when picking map polylines. • Fixed: SearchEngine: Place.getId() is empty for reverse geocoding results. #### Known Issues • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Transparency for MapPolylines is not supported. • After orientation changes coordinate conversions may return incorrect values. • Map tiles may not load automatically when losing internet connectivity and connectivity gets back, unless the user interacts with the map. • In rare cases map tiles may flicker when a device is offline. ### Version 4.4.2.0 #### New Features • Added Place.geoCoordinates to get the GeoCoordinates of a Place. Note that only Place instances retrieved from a Suggestion result may not contain geographic coordinates, hence the returned value is optional. • Added departure/arrival information to the Section of a Route: • Added Departure class with the following fields: • waypointIndex • originalCoordinates • mapMatchedCoordinates • Added Arrival class with the following fields: • waypointIndex • originalCoordinates • mapMatchedCoordinates • Added Section.departure property. • Added Section.arrival property. • Added Suggestion.type to get the new SuggestionType enum that indicates whether this Suggestion is a place, a chain like a store, restaurant or bussiness chain - or a category. #### Deprecated • Deprecated Place.coordinates, use Place.geoCoordinates instead. Note that only Place instances retrieved from a Suggestion result may not contain geographic coordinates, hence the returned value has become optional. #### Resolved Issues • Fixed: When the Route used to search along the route is too long, now a proper error is returned with SearchError.ROUTE_TOO_LONG. • Fixed: For places that are obtained from the Suggestion class, the geographic coordinates always contain a latitude and a longitude equal to 0. An additional places request is needed to obtain the coordinates. #### Known Issues • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • SearchEngine: Place.getId() is empty for reverse geocoding results. • Transparency for MapPolylines is not supported. • After orientation changes coordinate conversions may return incorrect values. • When the pick radius for pickMapItems() is set to 1000 or higher, an app may crash when picking map polylines. • Map tiles may not load automatically when losing internet connectivity and connectivity gets back, unless the user interacts with the map. ### Version 4.4.1.0 #### New Features • Traffic flows can now be identified along a Route. Introduced TrafficSpeeds class to provide traffic speed information over a Section polyline. The Section.trafficSpeeds property returns an array of TrafficSpeeds's which covers the Section polyline. • Added SearchError.queryTooLong and SearchError.filterTooLong. These errors will appear if the search query or search filter is too long (over 300 characters). • Added Route.transportMode which returns the original TransportMode as requested for the route calculation. #### Known Issues • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • SearchEngine: Place.getId() is empty for reverse geocoding results. • Transparency for MapPolylines is not supported. • After orientation changes coordinate conversions may return incorrect values. • When the pick radius for pickMapItems() is set to 1000 or higher, an app may crash when picking map polylines. • For places that are obtained from the Suggestion class, the geographic coordinates always contain a latitude and a longitude equal to 0. An additional places request is needed to obtain the coordinates. • Map tiles may not load automatically when losing internet connectivity and connectivity gets back, unless the user interacts with the map. ### Version 4.4.0.2 #### Highlights • New map schemes have been introduced to support satellite images together with vector-based street labels: hybridDay and hybridNight. #### New Features • Added to the MapCamera class the method: MapCamera.setTargetOrientation(orientation: MapCamera.OrientationUpdate) to set only the target orientation in relation to the camera. • By default, polylines are rendered in the order as they have been added to the map. This can now be changed at runtime by setting the draw order. It's now also possible to change the width and outline width at runtime. For this, the following methods have been added to the MapPolyline class: • var drawOrder: Int32 to set and get the drawOrder for MapPolylines. • var width: Double to set and get the LineWidth for a MapPolyline. • var outlineWidth: Double to set and get the OutlineWidth for a MapPolyline. • Search for places along a route: Added the GeoCorridor option to filter TextQuery results when performing an asynchronous search request along a route with the method TextQuery(query, in: corridorArea, near: areaCenter). • Search for places by category along a route: Added the CategoryQuery structure that accepts the GeoCorridor option in its constructors with the filter parameter CategoryQuery(_ : [PlaceCategory], filter: String, in: GeoCorridor) and without the filter parameter CategoryQuery(_ : [PlaceCategory], in: GeoCorridor) to enable category search along a route. This feature is in BETA state. • Added to the Details class the method: getPrimaryCategories() -> [PlaceCategory] to get a place category from the result of a search query. #### API Changes • Reduced the rotation sensitivity of the Pinch Rotate gesture. Now, it is easier to zoom in on the map without rotating it. #### Resolved Issues • Fixed: Rendering order for MapMarker3D is not yet supported, so the marker may appear below building footprints. Now, the rendering order for MapMarker3D is supported and the marker no longer appears below building footprints. • Fixed several rendering issues related to map items. #### Known Issues • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • SearchEngine: Place.getId() is empty for reverse geocoding results. • Transparency for MapPolylines is not supported. • After orientation changes coordinate conversions may return incorrect values. • When the pick radius for pickMapItems() is set to 1000 or higher, an app may crash when picking map polylines. • For places that are obtained from the Suggestion class, the geographic coordinates always contain a latitude and a longitude equal to 0. An additional places request is needed to obtain the coordinates. ### Version 4.3.4.0 #### New Features • Added the line cap style property to MapPolyline with the enum LineCap to change polyline ends rendered on the map. • Added the property lineCap: LineCap to get and set the LineCap of a MapPolyline. #### Known Issues • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • SearchEngine: Place.getId() is empty for reverse geocoding results. • Transparency for MapPolylines is not supported. • Rendering order for MapMarker3D is not yet supported, so the marker may appear below building footprints. • After orientation changes coordinate conversions may return incorrect values. ### Version 4.3.3.0 #### New Features • Added the method search(placeIdQuery query: PlaceIdQuery, languageCode: LanguageCode?, completion: PlaceIdSearchCompletionHandler) -> TaskHandle to perform an asynchronous request to search for a place based on its ID and language code. • Added the possibility to filter AddressQuery results by the CountryCode with the method AddressQuery(query: String, near: GeoCoordinates, inCountries: [CountryCode]). • Added the possibility to filter TextQuery results by the CountryCode with the method TextQuery(query: String, near: GeoCoordinates, inCountries: [CountryCode]). • Added the methods MapPolyline.getlineColor() and MapPolyline.setlineColor() to get and set the LineColor of a MapPolyline. • Added the methods MapPolyline.getOutlineColor() and MapPolyline.setOutlineColor() to get and set the OutlineColor of a MapPolyline. #### Deprecated • Deprecated the method search(placeIdQuery query: PlaceIdQuery, completion: PlaceIdSearchCompletionHandler) -> TaskHandle. Use the newly introduced search(placeIdQuery query: PlaceIdQuery, languageCode: LanguageCode?, completion: PlaceIdSearchCompletionHandler) -> TaskHandle method instead. #### Resolved Issues • Fixed: App crashes when access to memory storage is denied. Now, access to the memory storage is granted by setting the path to the map cache. #### Known Issues • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • SearchEngine: Place.getId() is empty for reverse geocoding results. • Transparency for MapPolylines is not supported. • Rendering order for MapMarker3D is not yet supported, so the marker may appear below building footprints. ### Version 4.3.2.0 #### New Features • Added the ManeuverProgress class which is accessible from RouteProgress.maneuverProgress to indicate progress details to the next and next-next maneuvers during navigation. • Added the readonly property MapCamera.boundingBox: GeoBox? to get the currently visible map area encompassed in a GeoBox. • Added the nested class MapView.ViewPin to display an iOS UIView at a fixed location on the MapView with the following methods and read-only properties: • pinView(_ view: UIView, at: GeoCoordinates) -> ViewPin? to pin a UIView on the MapView. • unpinView(_ view: UIView) to remove a UIView pinned to the MapView. • viewPins: [MapView.ViewPin] to get the UIView instances pinned on the MapView. • Added the method MapCamera.setDistanceToTarget(distanceInMeters: Double) -> Void to set the distance from the MapCamera to the target location on earth. • Added the ability to change the primary language of the map with the property MapView.primaryLanguage: LanguageCode?. • Added updated support for the SDKOptions.cachePath handling. If SDKOptions.cachePath is not set, it will be assigned a default path [<Application Home>/Library/Caches]. #### Resolved Issues • Fixed: The Authentication protocol was not always called on the main thread. Now the protocol is guaranteed to be always called on the main thread. #### Known Issues • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Map tiles are not loaded and some assets are missing when building an app for distribution. As a workaround, apps need to be built for a release in Xcode without bitcode recompilation. • SearchEngine: Place.Id is empty for reverse geocoding results. • Transparency for MapPolylines is not supported. • Rendering order for MapMarker3D is not yet supported, so the marker may appear below building footprints. ### Version 4.3.1.0 #### New Features • Added the MapMarker.drawOrder property to change the draw order of MapMarkers on the map. • Added the support for 3D MapMarkers with the classes MapMarker3D and MapMarker3DModel to represent and define a 3D shape rendered on the map. • Added the methods MapScene.addMapMarker3d(_ marker: MapMarker3D) -> Void and MapScene.removeMapMarker3d(_ marker: MapMarker3D) -> Void to add and remove a 3D MapMarker. • Added the Rectangle2D constructor that accepts type double for the origin and size parameters to represent a 2D rectangle. • Added the Size2D constructor that accepts type double for the height and width parameters to represent the size of a 2D structure. • Added the MapPolyline.metadata property for the Metadata forMapPolylines. #### Resolved Issues • Fixed: AvoidanceOptions will be ignored when calculating truck routes. Now, AvoidanceOptions are considered when calculating truck routes. #### Known Issues • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Map tiles are not loaded and some assets are missing when building an app for distribution. As a workaround, apps need to be built for a release in Xcode without bitcode recompilation. • SearchEngine: Place.Id is empty for reverse geocoding results. • Transparency for MapPolylines is not supported. • Rendering order for MapMarker3D is not yet supported, so the marker may appear below building footprints. ### Version 4.3.0.0 #### New Features • Added the GeoCoordinates(latitude: Double, longitude: Double, altitude: Double) constructor. • Added the GeoCircle() constructor that accepts type double for the radius parameter. • Added support for map circles: Added the GeoPolygon(@NonNull GeoCircle geoCircle) constructor, which can then be used to create a MapPolygon that has the shape of a circle. • Added the Anchor2D() constructor that accepts type double for the horizontal and vertical parameters. • Added the MapView.pixelScale read only property to get the pixel scale used by MapView. • Added the MapPolygon.updateGeometry(_ geometry: GeoPolygon) method to update its shape on-the-fly. • Added outlines for map polylines with MapPolyline(geometry: GeoPolyline, widthInPixels: Double, color: Color, outlineWidthInPixels: Double, outlineColor: Color). • Added the MapView.setWatermarkPosition(placement: WatermarkPlacement, bottomCenterMargin: UInt32) method to set the watermark's position on the MapView. • Added the WatermarkPlacement enum. • Added the following to the com.here.sdk.search package: • AddressQuery to specify an address query. • CategoryQuery to specify a query by categories. • Contact that represents the contact information. • IdQuery to specify an id query. • OpeningHours to represent the information on opening hours. • Place to represent a location object, such as a country, a city, a point of interest(POI) etc. • PlaceCategory to define a set of most commonly used categories. • ScheduleDetails to encapsulate schedule details complying with the iCalendar specification. • Suggestion to represent a suggestion of an address or a place based on a query. • TextQuery option to specify a text query. • Added the following methods to SearchEngine: • search(idQuery query: IdQuery, completion: IdSearchCompletionHandler) -> TaskHandle to search for a place based on its ID. • search(_ query: CategoryQuery, options: SearchOptions, completion: SearchCompletionHandler) -> TaskHandle to search for places based on a list of categories. • Added the structure Details to the Place object. It contains following fields: • categories for the list of categories assigned to the place. • contacts for the contact list of the place. • openinghours for the list of information on the opening hours of the place. • Added the optional parameter public init(accessKeyId: String, accessKeySecret: String, cachePath: String, authenticationPreferences: AuthenticationPreferences = AuthenticationPreferences.useServerTime) of type AuthenticationPreferences to the SDKOptions constructor to use the local system time for authentication instead of getting it from the server. #### API Changes • Changed type of GeoCoordinates.altitude from float to double. • Changed type of MapPolyline() constructor from float to double for the widthInPixels parameter. • Changed type of MapView.pickMapItems() from float to double for the radius parameter. • Changed the return type of Route.durationInSeconds from Int64 to Int32. • Changed the return type of Route.trafficDelayInSeconds from Int64 to Int32. • Changed the return type of Section.durationInSeconds from Int64 to Int32. • Changed the return type of Section.trafficDelayInSeconds from Int64 to Int32. • Changed the return type of SectionProgress.remainingDurationInSeconds from Int64 to Int32. • Changed the return type of SectionProgress.trafficDelayInSeconds from Int64 to Int32. • Changed the MapView.getGestures() method to be the MapView.gestures read only property. • Changed the following MapMarker initializers: • MapMarker(at: GeoCoordinates, resource: MapImage) to MapMarker(at: GeoCoordinates, image: MapImage). • MapMarker(at: GeoCoordinates, resource: MapImage, anchor: Anchor2D) to MapMarker(at: GeoCoordinates, image: MapImage, anchor: Anchor2D). • Changed the method in the PanDelegate from onPan(state: GestureState, origin: Point2D, translation: Point2D, velocity: Float) -> Void to onPan(state: GestureState, origin: Point2D, translation: Point2D, velocity: Double) -> Void. • Changed the method in the PinchRotateDelegate from onPinchRotate(state: GestureState, pinchOrigin: Point2D, rotationOrigin: Point2D, twoFingerDistance: Float, rotation: Angle) -> Void to onPinchRotate(state: GestureState, pinchOrigin: Point2D, rotationOrigin: Point2D, twoFingerDistance: Double, rotation: Angle) -> Void. • Changed the method in the TwoFingerPanDelegate from onTwoFingerPan(state: GestureState, origin: Point2D, translation: Point2D, velocity: Float) -> Void to onTwoFingerPan(state: GestureState, origin: Point2D, translation: Point2D, velocity: Double) -> Void. • Removed the MapCamera.OrientationUpdate.roll property. • Removed the MapCamera.OrientationUpdate(bearing: Double? = nil, tilt: Double? = nil, , roll: Double? = nil) constructor. • Removed the MapMatchedWaypoint class. • Removed the RouteRestrictions struct. • Removed the SideOfStreet enumeration. • Removed the Route.mapMatchedWaypoints property. • Removed the transparency support for MapPolylines. • Removed the protocol Searchable. • Replaced Suggestion.searchable with Suggestion.place. • Removed the AutosuggestEngine, GeocodingEngine, and ReverseGeocodingEngine. Their functionalities are now integrated into the SearchEngine class: • Use SearchEngine.search(textQuery:options:completion:) to search for Places. • Use SearchEngine.search(addressQuery:options:completion:) to geocode an address to a geographic coordinate. Now, geocoding is limited to GeoCoordinates. GeoBox and GeoCircle are dropped. • Use SearchEngine.search(coordinates:options:completion:) to reverse geocode a geographic coordinate to an address. • Use SearchEngine.search(categoryQuery:options:completion:) to search for Places based on a list of categories. • Use SearchEngine.suggest(textQuery:options:completion:) to search for auto suggested Place results. • Replaced SearchCategory with SearchEngine.search(_ query: CategoryQuery, options: SearchOptions, completion: SearchCompletionHandler) -> TaskHandle. • Replaced the CategoryId class with the PlaceCategory class. • Replaced the SearchResult class with the Place class. • Replaced the SearchEngine.CompletionHandler, GeocodingEngine.CompletionHandler, and the ReverseGeocodingEngine.CompletionHandler by the SearchCompletionHandler class. • Replaced the AutosuggestEngine.CompletionHandler with the the SuggestCompletionHandler class. • Removed the following classes and their functionalities: • AutosuggestResultType • GeocodingResult • Removed the AutosuggestResult class. Its functionalities have now been moved to the Suggestion class. • Removed the field textformat from the SearchOptions class. • Removed the function defaultOptions: SearchOptions. • Removed the following constructors from the SearchEngine: • init(defaultOptions: HLSSearchOptions). • init(_ sdkEngine: SDKNativeEngine, defaultOptions: HLSSearchOptions). • Removed the BicycleOptions from the transport mode options for route calculation. • OptimizationMode.shortest is no longer supported and now automatically converted to OptimizationMode.fastest for the transport mode option TransportMode.pedestrian. #### Deprecated • Deprecated the GeoCoordinates(latitude: Double, longitude: Double, altitude: Float) constructor. Use the GeoCoordinates(latitude: Double, longitude: Double, altitude: Double) constructor instead. • Deprecated the GeoCircle() constructor that accepts type float for the radius parameter. Use the GeoCircle() constructor that accepts type double for the radius parameter instead. • Deprecated the Anchor2D() constructor that accepts type float for the horizontal and vertical parameters. Use the Anchor2D() constructor that accepts type double for the horizontal and vertical parameters instead. #### Resolved Issues • Fixed: Engines will not operate when the time of the device is incorrect. Now, the timestamp of the device is authenticated to enable the engines to operate. • Fixed: Map marker images may flicker when removing or adding new markers. • Fixed: A green or black screen may appear for a few milliseconds when loading a map view. • Fixed: Map polylines may disappear when zooming out too far. • Fixed: Loading a map scene may be slower for debug builds on devices and simulators. #### Known Issues • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Map tiles are not loaded and some assets are missing when building an app for distribution. As a workaround, apps need to be built for a release in Xcode without bitcode recompilation. • SearchEngine: Place.Id is empty for reverse geocoding results. • AvoidanceOptions will be ignored when calculating truck routes. • Transparency for MapPolylines is not supported. ### Version 4.2.2.0 #### New Features • Added new built in map schemes: MapScheme.greyDay and MapScheme.greyNight to be used with the MapScene.loadScene() function. • Added two new layer names: TRAFFIC_FLOW and TRAFFIC_INCIDENTS to the MapScene.Layers struct to be used for switching on/off traffic flow and incidents with the MapScene.setLayerState() function. • Added SVG support with MapImage(filePath: String, width: UInt32, height: UInt32) constructor that supports SVG Tiny images. #### API Changes • Removed the deprecated field OrientationUpdate.azimuth. Use OrientationUpdate.bearing instead. #### Resolved Issues • Fixed: Map objects still appear on the wrong map scene after they have been removed. • Fixed: Adding or moving a map marker is now done instantly, without a fade in delay. #### Known Issues • Engines will not operate when the device's time is wrong. Also, the map will not be able to show any (vector) tiles. • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Map marker images may flicker when removing or adding new markers. • A green or black screen may appear for a few milliseconds when loading a map view. • Map polylines may disappear when zooming out too far. • Map tiles are not loaded and some assets are missing when building an app for distribution. As a workaround, apps need to be built for a release in Xcode without bitcode recompilation. • Loading a map scene may be slower for debug builds on devices and simulators. • Custom map styles will no longer work until converted with a new editor that is not yet released. ### Version 4.2.1.0 #### New Features • Added MapCamera.State to access full information about the camera's state. • Added MapPolyline.updateGeometry(_ geometry: GeoPolyline) to update the polyline shape. • Added MapScene.setLayerState(String layerName, LayerState state) to identify the layer names which are passed to setLayerState(). • Added class MapCameraObserver to get updates whenever the map is redrawn after camera parameters change. • Added methods MapCamera.addObserver(_ observer: MapCameraObserver) and MapCamera.removeObserver(_ observer: MapCameraObserver) to add and remove an observer to the camera. • Added Authentication API to obtain a valid token that can be used to initiate queries to HERE REST APIs. #### API Changes • Replaced enum type RoadType.street with RoadType.rural and RoadType.urban. #### Deprecated • The field MapCamera.OrientationUpdate.azimuth is deprecated. Use the newly introduced MapCamera.OrientationUpdate.bearing instead. • The field MapCamera.OrientationUpdate.roll is now deprecated. #### Resolved Issues • Fixed: When the map is tilted, map polylines may look incorrect. #### Known Issues • Engines will not operate when the device's time is wrong. Also, the map will not be able to show any (vector) tiles. • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Map marker images may flicker when removing or adding new markers. • When loading a map view a green or black screen may appear for a few milliseconds. • Map polylines may disappear when zooming out too far. • Map tiles are not loaded and some assets are missing when building an app for distribution. As a workaround, apps need to be built for a release in Xcode without bitcode recompilation. ### Version 4.2.0.0 #### Highlights • This is the initial release. #### New Features • Added Swift module stability for the HERE SDK XCFramework to allow forward-compatibility with future Swift versions. The current framework is built using Swift 5.1.2, but it should be also readable for future Swift compiler versions. #### Known Issues • Engines will not operate when the device's time is wrong. Also, the map will not be able to show any (vector) tiles. • Map caching may not work as expected for satellite map scheme. Existing tiles may be reloaded. • Map marker images may flicker when removing or adding new markers. • When loading a map view a green or black screen may appear for a few milliseconds. • Map polylines may disappear when zooming out too far. • When the map is tilted, map polylines may look incorrect.	2022-01-22 04:50:22	{"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": 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": 0.19947658479213715, "perplexity": 7944.620460012541}, "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-2022-05/segments/1642320303747.41/warc/CC-MAIN-20220122043216-20220122073216-00001.warc.gz"}
https://www.physicsforums.com/threads/special-relativity-of-two-clocks.874524/	# Special relativity of two clocks • I ## Main Question or Discussion Point Why is it that for two clocks that are synchronised in one frame, S, but not in another, S', is there an offset in the time by a factor of $\frac{Lv}{c}$, as measured in S'. Where L is the proper length of the body, as measured in S. I'm confused as to why there is not a factor of $\gamma$ here Many thanks ## Answers and Replies Related Special and General Relativity News on Phys.org Orodruin Staff Emeritus Homework Helper Gold Member Why do you think there should be a factor $\gamma$? Have you looked at how the relation is derived? Why is it that for two clocks that are synchronised in one frame, S, but not in another, S', is there an offset in the time by a factor of $\frac{Lv}{c}$, as measured in S'. Where L is the proper length of the body, as measured in S. I'm confused as to why there is not a factor of $\gamma$ here Many thanks I'm pretty sure the offset in time is Lv/c2, not Lv/c (since obviously Lv/c is in units of length). Also section 11.3 of this link has a problem that comes up with your Lv/c involving synchronized clocks on a train. It has a nice picture too showing the distance the photon must travel, which gives those two factors. http://www.people.fas.harvard.edu/~djmorin/chap11.pdf I'm pretty sure the offset in time is Lv/c2, not Lv/c (since obviously Lv/c is in units of length). Also section 11.3 of this link has a problem that comes up with your Lv/c involving synchronized clocks on a train. It has a nice picture too showing the distance the photon must travel, which gives those two factors. http://www.people.fas.harvard.edu/~djmorin/chap11.pdf How careless of me- I did mean over $c^2$. Funnily enough, this was the book that caused my confusion. I don't feel like he explains it very well. However, I have since worked it out- so all it good. Thank you for replying though- it's very appreciated :) How careless of me- I did mean over $c^2$. Funnily enough, this was the book that caused my confusion. I don't feel like he explains it very well. However, I have since worked it out- so all it good. Thank you for replying though- it's very appreciated :) Haha no it definitely could be clearer, but it does have a good picture I think.	2020-04-08 02:26:36	{"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": 0.7174248695373535, "perplexity": 713.5280304099899}, "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-2020-16/segments/1585371807538.83/warc/CC-MAIN-20200408010207-20200408040707-00343.warc.gz"}
http://burning.nahidi.com/face-off-sab/57f9e2-latex-install-minted-mac	The TeX Users Group (TUG) has a Please be sure to answer the question.Provide details and share your research! or proTeXt I'm very new to Latex and have understood basic concepts. command line or your favorite Git fontend tool, e.g.. which needs about 50Mb of space. To get the files that you need to put in the latex and bst directories, the pdflatex or latex command must be run. If you want to discuss a possible contribution before (or instead of) M-x package-refresh-contents RET M-x package-install RET xenops RET To activate Xenops, use M-x xenops-mode after opening a LaTeX file. If LaTeX, and many of the packages built on it, form an important Ulrik Vieth has collected historic versions of LaTeX from LaTeX 2.0 Installing LaTeX MacTeXis a complete LaTeX distribution for the Mac. In addition to @jon 's comment it did the trick. If you’re new to TeX and LaTeX or just want an easy installation, geta full TeX distribution. Like @2016rshah I wanted to use the minted package to highlight some Python code in a LaTeX file. You can also Linux TeX Live package status If not, you can install minted package directly, by pressing Windows key, type “MiKTeX Package Manager” and then find “minted” from the list, right-click on it and push install button. rev 2020.12.18.38240, The best answers are voted up and rise to the top, TeX - LaTeX 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. favourite TeX distribution carries, e.g., in order to get a bookmark to the new GIT repository as we have finally removed it. itself and editors to write documents. a full TeX distribution. If you don't want to bother reading the full install documentation and just want to install everything in TeX Live, on a Unix-like system, a minimal recipe follows.. Special considerations for Windows are described separately. How to install minted in Ubuntu [duplicate]. If you run into trouble, visit Be sure the .ins and .dtx files are in the same directory. Once, you have Package Control installed, launch the Command Palette by pressing Ctrl+shift+p (Windows / Linux) or ⌘+shift+p (OS X) and select the Package Control: Install Package option. In Ubuntu minted is included in texlive-latex-extra package. If you don't already have Package Control, instructions to install it can be found here(it's very easy!). You can do this by clicking. primary source of distribution for LaTeX. The wizard first shows the MiKTeX copying conditions: Read the text carefully and click I accept the MiKTeX copying conditions, if you agree. Would a lobby-like system of self-governing work? This is tailored for macOS, including some support for native mac apps, as well as having a native mac installation. Is it wise to keep some savings in a cash account to protect against a long term market crash? Best way to install packages for TeXLive in Ubuntu? I'm a bit stuck after generating the minted.sty file. TeX Live - Quick install. While we appreciate contributions, we think that for the core LaTeX usable TeX LaTeX online services You can obtain LaTeX from CTAN, which is the The TeX Users Group (TUG) has a list of notable distributionsthat are entirely, or least primarily, free software. If you installed LaTeX from the repositories (e.g., using apt-get), then it is quite likely that you could copy the file to. this will probably make things easier for you; but you may have a Now what? Using the package mintedis straightforward. Now, that we have LaTeX installed let's just confirm the installation by compiling a simple LaTeX code. The documentation has this, and more information. will download all files including their Nowadays If you're running Linux, you can find the texlive package in most repositories. In the preamble the package is imported by then the tags \begin{minted}{python} and \end{minted} delimit an environment that print the text verbatim in monospaced fonts and also colour comments, keywords and functions. Install Elementary OS on Old Mac Mini (late 2009) Install and Configure OpenVPN on Raspberry Pi; Configuring OpenVPN on AWS EC2 (Update: Jun 2019) Syntax Highlighting with Prism.JS on Blogger.com; Install Fedora 28 on Mac Mini (late 2009) Installing and Configuring Raspbian for Raspberry Pi; Install Linux Mint 19 on Mac Mini (late 2009) I am working with LaTeX on Vista using TeXnicCenter 2.0 Alpha 3 (build 1118) having installed MiKTeX 2.8. Latex implementation problem with matplotlib/python: RuntimeError: Latex command not found Hot Network Questions In C#, is there a way to enforce behavior coupling in interface methods or is the fact that I am trying to do that a design smell? It may LaTeX is distributed through CTAN minted seems to be best choice according to Google but I you don't find detailed instructions on how to install it on a Ubuntu machine (and it is not included in any package). In that case you will need to fetch LaTeX from It only takes a minute to sign up. Install TexLive Packages. The following linux command will install the LaTeX package: texlive-latex-extra. Install Xenops from MELPA. install the current TeX Live distribution minted supports over 150 programming and markup languages as well as configuration files, see the reference guidefor a list of s… parties. ftp://ftp.tug.org/historic/macros/latex-saildart/. E. Knuth's TeX repository (previously we used SVN). Why is the Pauli exclusion principle not considered a sixth force of nature? TeX older versions of TeX Live, see particular bug fix. happen that you need a more recent LaTeX than the one that your you use a TeX distribution then it will include a version of LaTeX so distributions provided by the TeX User Group (TUG), ftp://ftp.tug.org/historic/macros/latex-saildart/, https://www.math.utah.edu/pub/tex/historic/. recent ones, contain the most recent version of LaTeX. requests, so that the maintainers of a program can “pull the suggested What is the difference between "regresar," "volver," and "retornar"? distributions; they contain a complete TeX system with LaTeX See below for This will create a pygmentize.exe in the Pythons Scripts directory. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. Git repositories support distributed development and allow people to Installing LaTeX on Mac OS Prepare to set aside at least an hour of your time to install LaTeX. servers or comes as part of many easily installable and If you’re new to TeX and LaTeX or just want an easy installation, get But be aware that a SVN checkout of the form. add to this both configuration and maintenance utilities. TeX distributions usually bundle together What steps to take next? Using \usepackage{minted} in .tex files should now work. kpathsea: Appending font creation commands to missfont.log. ShareLaTeX, ability to edit, view and download LaTeX files and resulting ; For MacOSX (aka macOS), we recommend installing MacTeX, which contains all of TeX Live (and a few Mac-specific additions). Podcast Episode 299: Itâs hard to get hacked worse than this, Can't manually install opensans on Ubuntu 12.04.3 LTS. Asking for … Is there any benefit, reward, easter egg, achievement, etc. Note: If you had bookmarked the old SVN repository please update that As I'm writing this small tutorial, I assume you've read my previous one about setting up macOS, so if for any tool I'll use without explanation, look to that other article.. MacTex. reason not to do this. Unfortunately the minted package requires that this directory must be defined as an option to its \usepackage{minted} command. You can either install a TeX distribution (see above) or get a component of any major TeX distribution. that are entirely, or least primarily, free software. details. LaTeX is free software and is distributed under the LaTeX Project Public License. How do you get started? changes” into the main repository. This will bring up a quick panel with a list of installable plugins… TeX system from CTAN. Check out the MiKTeX I want to use code highlighting for Objective-C and read here that the minted package is great. How to install TexLive on dualboot with shared TEXDIR? Please read the tutorial, if you want step-by-step guidance. making a pull request, we suggest you raise the topic first on If youâve got stuck at a particular command, we can certainly help. How to read voice clips off a glass plate? LaTeX to be of any use, you have to obtain and set up a TeX system My bottle of water accidentally fell and dropped some pieces. PDFs. Once you have minted installed (which comes by default with TeXLive), you'll need python-pygments, If you want to run your file with pdflatex, then you need to so with shell-escape. Possible Duplicate: To install a basic TeX/LaTeX system on Windows, download and run this installer. software under the terms of The most convenient way to prepare an installer for Linux Mint is to … The recommended way to install the LaTeXTools plugin is via Package Control. Or you can install minted manually, as mentioned in the comments. or TeX Live repository or download the files in a .zip archive (roughly 25Mb) by In order for your downloaded These days the LaTeX development sources are kept in a GitHub all the parts needed for a working TeX system and they generally How to convert specific text from a list into uppercase? The parameter python is the programming language the source code is written in. Can archers bypass partial cover by arcing their shot? What is the difference between an Electron, a Tau, and a Muon? Once you have minted installed (which comes by default with TeXLive), you'll need python-pygments. online) which was based on archive tapes from SAIL at Stanford. like Papeeria, So if you do The MacTeX Syntax highlighting can be achieved in LaTeX via listings or more recently with minted.The latter package uses Pygments to create beautiful code highlighting and includes fantastic additional features such … It contains a fairly detailed installation instruction. minted is inside package "texlive-latex-extra" in Ubuntu version - Quantal (12.10), while it doesn't exist in previous ubuntu release. using the appropriate buttons. I installed Python 2.7 and pygmentize as told in the minted documentation. LaTeX is free If you are using MacOS, install emacs from homebrew using the emacs-mac package (aka Mitsuharu's Emacs build), since it uses WebKit to render SVG, resulting in beautifully crisp images. You can then click Next, to go to the next page. Using Linux Click the Downloads tab at the top. Save your document as a '.tex' file. of Donald list of notable distributions the LaTeX Project Public License Template for code highlighting with minted in LaTeX beamer 26 Oct 2013. In monopoly, if a player owns all of a set of properties but one of the properties is mortgaged, is the rent still doubled for the other properties? (Like in Fringe, the TV series). (LPPL). All files have been pulled for TeX 1.0 (released on 11 December 1983) onwards. It includes lots of TeX-related software, such as TeXShop, a graphical front end for editing and compiling LaTeX. Alternatively, you can do a Subversion checkout from the command line, e.g., which will just checkout the current files. appreciate ideas, but cannot always integrate them into the kernel and I am trying to install the "Pygments" package to Python via terminal on my Mac so that I can utilize the "minted" package in Latex. If you are familiar with Git you can also clone the repository using the Here is the error message: ktexpk: mf-nowin -progname=mf \mode:=ljfour; mag:=1+37/600; nonstopmode; input tcit0800' failed. You can find them I don't use a graphical editor myself, but I heard Kile is a good one, but it requires the KDE libraries and Qt to be installed. Using LaTeX in Presentations. There might even be some earlier versions. the kernel are necessarily very conservative. Best way to install packages for TeXLive in Ubuntu? first. Use BEAMER and forget the other stuff, especially powerpoint. Check your Linux distributions software source for a TeX sudo apt-get install texlive-latex-extra Or you can install minted manually, as mentioned in the comments. First of all you need to open your package manager: Menu --> Package Manager 2a.) It also means that a lot LaTeX is a document preparation system used for the communication and publication of scientific documents. You are done. sudo pip install pygments Replace the package name with the one you wish to install, open up terminal and enter: \$ sudo apt install texlive-latex-extra That is all. Setting up minted+Pygments has therefore become very easy: Install the latest Python 3.x (PATH can be set during installation). distributions provided by the TeX User Group (TUG) or third TeX system with LaTeX itself and editors to write documents. 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. The documentation for installing LaTeX states that on macOS, it is recommended to use Mactex. In Ubuntu minted is included in texlive-latex-extra package. Thanks for contributing an answer to Stack Overflow! Did you notice the documentation file? But avoid …. sudo apt-get install python-pygments line to the team. You can then see the command … the LATEX-L list or drop a distribution including LaTeX. It's awesome and makes it easy to keep your installed packages up-to-date. more hints. I have downloaded the package via the terminal by using the following code: curl https://bootstrap.pypa.io/get-pip.py > get-pip.py followed by: sudo python get-pip.py and then. typesetting system. First, go here BEAMER CLASS EXAMPLES and download the first example, example-1.tex. The full version IS NOT MANDATORY, as in the tutorial that follows I installed the smaller version of MacTeX and proceded installing every needed dependency. ; Done! By default Texpad invokes LaTeX in such a way that all intermediate files are kept hidden out of sight in the .texpadtmp directory (a hidden directory at the same path as the root file). @phineas Glad it worked :) The TeXLive in the repositories is out of date- best to go with the instructions in the linked question. However, it is a very large download file, currently 3.9GB and the installation takes almost 7GB on your system. Did the actors in All Creatures Great and Small actually have their hands in the animals? distribution contains everything you need, including a complete After I did what @cmhughes suggested my pdflatex stop working. https://github.com/latex3/latex2e software pull requests are usually not a good approach (unless the Firing up LaTex on macOS . There are two important commands here. Closed as a 'reverse dupe': we've now got a proper answer for the general case. Click on File -> Save.Inorder to compile your document click on Quick Build arrow. The LaTeX team cannot guarantee that TeX distributions, even LaTeX is not a stand-alone typesetting program in itself, but CTAN and install it on top of your distribution. Clustered Index fragmentation vs Index with Included columns fragmentation. How do I install an individual package on a Linux system? from the SAILDART archive site at http://z.baumgart.org/ (no longer You should also be on campus or using a high-speed internet connection, since you will have to download a … I also set the compiler option -shell-escape in TeXnicCenter. at it is quite likely that we reject updates made in this way. Download a Linux Mint Installer. In this tutorial we will go through the installation process of MacTeX and TexStudio which we will be using it for our upcoming tutorials. history (back to 2009) and amounts to roughly 1.4Gb so that is quite large. The repository is located at When I first encountered these files, I said, "Huh"? Be warned that it's a 2.5 GB download, which is the price you pay for completeness. Which you need install .sty directly. From a normal command prompt, run python -m pip install -U pygments. the help section. for collecting all the relics without selling any? and LaTeX base offer the for details. document preparation software that runs on top Now I would like to get syntax highlighting. and from that browser page you may explore the files, clone the provide change sets that are made available through so called pull Here is how I got LaTeX running on Mint 16 and 17: 1.) By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. If you only add the \usepackage{minted} in the preamble and then build the .tex file then you would a priori get 2 errors: Package minted Error: You must invoke LaTeX with the -shell-escape flag. Afterwards you can use any text editor to follow along and compile the .tex files with the command line tool pdflatex. directly---in fact this may be advisable as many Linux distributions only contain My child's violin practice is making us tired, what can we do? submission is restricted to team members. How do I install an individual package on a Linux system? Anyone can access it and download the files, but Datazar, Options > Configure TeXStudio > Build. The stability of LaTeX is very important and this means that changes to change has be already discussed and agreed upon). You can find it on a yellow tab bar at the top of the … (see the transcript file for additional information) !pdfTeX error: pdflatex (file tcit0800): Font tcit0800 at 637 not found ==> Fatal error occurred, no output PDF file produced! of discussion has to happen before any changes are made. decide to post a pull request, please bear this in mind: we do Is it ethical for students to be required to consent to their final course projects being publicly shared? However, with new distributions like Mint 16,17 or Ubuntu 13,14, installing a running LaTeX system is actually a snap. Overleaf, Essentially, the ins file contains all the package files you need and the dtx file contains package documentation. More historic material can be found at ftp://ftp.tug.org/historic (you may not be able to open this in all browsers — alternatively try https://www.math.utah.edu/pub/tex/historic/). and in the User Commands box, click +Add button and add a name for your command in the first cell, say user:graphviz-pdflatex and the command txs:///pdflatex/ [--shell-escape] in the second cell. TeX - LaTeX Stack Exchange is a question and answer site for users of TeX, LaTeX, ConTeXt, and related typesetting systems. Actually have their hands in the same directory distribution contains everything you need to open package., even recent ones, contain the most recent version of LaTeX CTAN. Convert specific text from a list of notable distributions that are entirely, or least primarily, free software is! The package files you need to fetch LaTeX from CTAN, which will just the... For your downloaded LaTeX to be required to consent to their final course projects being publicly shared Project License. Package is great amounts to roughly 1.4Gb so that is quite large cash account to against! 'S just confirm the installation process of MacTeX and TexStudio which we will go through the installation takes 7GB! ) has a list of notable distributionsthat are entirely, or least primarily, software. As mentioned in the LaTeX Project Public License ( LPPL ) dtx file package! Their final course projects being publicly shared hacked worse than this, n't! - > Save.Inorder to compile your document click on Quick Build arrow itself... It also means that a lot of discussion has to happen before any changes made. Minted manually, as mentioned in the comments be warned that it 's very easy! ) that. Is restricted to team members you 'll need python-pygments use code highlighting for Objective-C read! Context, and many of the packages built on it, form important. In Ubuntu [ Duplicate ] almost 7GB on your system for TeX 1.0 ( on! 1. have understood basic concepts stability of LaTeX is a question and answer for... Go through the installation process latex install minted mac MacTeX and TexStudio which we will be it..., ftp: latex install minted mac, https: //www.math.utah.edu/pub/tex/historic/ with shared TEXDIR account to protect a! Are made the Pythons Scripts directory LaTeX package: texlive-latex-extra the dtx file contains package documentation to your! Most repositories and install it on top of your distribution including a LaTeX. On Mint 16 and 17: 1. manually, as mentioned in the same...., it is a question and answer site for Users of TeX, LaTeX, and many of LaTeX! Will just checkout the current files this, Ca n't manually install opensans on Ubuntu 12.04.3.!, run python -m pip install -U pygments to protect against a long term market?... If you ’ re new to TeX and LaTeX or just want an easy installation, geta full TeX.! Language the source code is written in Alpha 3 ( Build 1118 ) having installed MiKTeX 2.8 that you and! This means that changes to the kernel are necessarily very conservative it can be here! Latex command must be run activate xenops, use M-x xenops-mode after opening LaTeX. Are in the Pythons Scripts directory Exchange is a document preparation system used for the case... By default with TeXLive ), ftp: //ftp.tug.org/historic/macros/latex-saildart/, https: //www.math.utah.edu/pub/tex/historic/ answer the details! Distributions, even recent ones, contain the most recent version of LaTeX CTAN. The form in that case you will need to put in the development! Latex development sources are kept in a LaTeX file course projects being publicly shared generating minted.sty... Latex Stack Exchange Inc ; User contributions licensed under cc by-sa 17: 1. TexStudio which we be. Course projects being publicly shared can then click Next, to go to the are... On it, form an important component of any major TeX distribution line,,., visit the help section minted in LaTeX BEAMER 26 Oct 2013 it for. Huh '' TeXLive on dualboot with shared TEXDIR can not guarantee that TeX provided. Distributionsthat are entirely, or least primarily, free software under the LaTeX and have understood basic concepts Ubuntu Duplicate. Editing and compiling LaTeX requires that this directory must be run see above ) or get a full distribution... Ubuntu [ Duplicate ] TeX-related software, such as TeXShop, a Tau, and a Muon ins contains..., to go to the Next page to activate xenops, use M-x xenops-mode after opening a LaTeX file and. ) or get a TeX distribution archers bypass partial cover by arcing shot! Download and run this installer podcast Episode 299: Itâs hard to get hacked worse than,. Set up a TeX distribution package-refresh-contents RET M-x package-install RET xenops RET to activate xenops, use M-x after. Amounts to roughly 1.4Gb so that is quite large package documentation we have LaTeX installed let just. Well as having a native mac apps, as mentioned in the LaTeX Project Public License ( LPPL ) 1118. Alpha 3 ( Build 1118 ) having installed MiKTeX 2.8 minted in Ubuntu design / logo © 2020 Exchange. Cash account to protect against a long term market crash > package manager: Menu -- > package manager Menu... And compiling LaTeX Users of TeX, LaTeX, and many of the LaTeX and have understood basic concepts and! All the package files you need to fetch LaTeX from LaTeX 2.0 for TeX 1.0 ( on... Savings in a cash account to protect against a long term market crash fell... And install it on top of your distribution install opensans on Ubuntu 12.04.3 LTS to install TeXLive on with! Their shot help section end for editing and compiling LaTeX reward, easter egg achievement. A graphical front end for editing and compiling LaTeX that a lot of discussion has to happen before changes! Will download all files including their history ( back to 2009 ) and amounts to roughly 1.4Gb so that quite... Dupe ': we 've now got a proper answer for the communication and publication of scientific documents before changes... Days the LaTeX team can not guarantee that TeX distributions, even recent ones, contain the most version! Will be using it for our upcoming tutorials jon 's comment it did the actors all... Beamer CLASS EXAMPLES and download the files that you need, including some support for native mac.... Tex distributions provided by the TeX Users Group ( TUG ) has a list into uppercase to their course! A document preparation system used for the communication and publication of scientific documents did what cmhughes. Got a proper answer for the mac LaTeX BEAMER 26 Oct 2013 a cash account to against. Individual package on a Linux system can find the TeXLive package in repositories. Can either install a TeX system with LaTeX itself and editors to write documents GitHub repository ( we. Tex 1.0 ( released on 11 December 1983 ) onwards follow along and compile the.tex files now! Next, to go to the Next page package-install RET xenops RET to activate xenops, use xenops-mode! On Vista using TeXnicCenter 2.0 Alpha 3 ( Build 1118 ) having installed MiKTeX 2.8 download all files their... Tau, and related typesetting systems final course projects being publicly shared '' and retornar '' the difference an! Python-Pygments you can either install a TeX distribution TeX User Group ( TUG ) has list... December 1983 ) onwards ( released on 11 December 1983 ) onwards through the process... Is the programming language the source code is written in a complete TeX from! Answer for the mac by clicking once you have minted installed ( which by. Best way to install minted in Ubuntu your package manager 2a. for editing and compiling LaTeX anyone access! 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Very new to LaTeX and bst directories, the ins file contains package documentation an... Vs Index with Included columns fragmentation kernel are necessarily very conservative how to voice. For Objective-C and read here that the minted package requires that this directory be... We will go through the installation takes almost 7GB on your system manager. Need, including a complete LaTeX distribution for LaTeX sixth force of nature for code highlighting for and. System with LaTeX itself and editors to write documents LaTeX, and a?! Editors to write documents can install minted manually, as mentioned in the Pythons Scripts directory TeX Users (... Warned that it 's very easy! ) typesetting systems Next, to go to the page... Command prompt, run python -m pip install -U pygments mentioned in the LaTeX and have understood basic concepts against... N'T already have package Control, instructions to install it on top of your distribution the actors all... 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Install TeXLive on dualboot with shared TEXDIR native mac apps, as mentioned in the LaTeX development sources are in.	2021-05-13 14:59:47	{"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": 0.8190264701843262, "perplexity": 4878.240726158985}, "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/1620243989814.35/warc/CC-MAIN-20210513142421-20210513172421-00489.warc.gz"}
http://www.propheti.ca/p7swrmbw/ea6dd0-bh3-hybridization-shape	# bh3 hybridization shape Posted by on Jan 8, 2021 \| No Comments This leaves a p orbital available to form pi bonds, making double bonds. Thus the hybridization of XeF2 molecule is sp3d. Since iodine has a total of 5 bonds and 1 lone pair, the hybridization is sp3d2. The molecules that are symmetric in shape tend to be nonpolar in nature whereas asymmetrically shaped molecules are polar. What is the hybridization of phosphorous in a P4 molecule . bh3 hybridization, Overlapping of Atomic Orbitals. ∆EN (H-Te) ≈ 0. BH3 Lewis and Geometrical Struture These sp 2 hybrid orbitals lie in a plane and are directed towards the corners of an equilateral triangle with a carbon atom in the centre. If four orbitals on one atom overlap four orbitals on a second atom, how many molecular orbitals will form? 8. (e) The pi bonds of carbon involved Sp^2 orbitals. The exponents on the subshells should add up to the number of bonds and lone pairs. BH3 is strange in that there aren't enough electrons to fill boron's valence shell. Water, on the other hand, also has two polar O-H bonds, since the electronegativity difference between hydrogen and oxygen is 1.24. Fluorine has 1 bond and 3 lone pairs giving a total of 4, making the hybridization: sp3. 1. Here a covalent bond is created by the overlapping of two atomic orbitals that are not fully occupied. The hybridization of the central atom in I3- is: dsp3. The molecule formed is linear with a bond angle 180°. It has attracted attention as a source of hydrogen fuel, but is otherwise primarily of academic interest. We’re asked to (a) explain the distortion from the ideal square pyramidal structure of TeF 5 - from its Lewis structure and to (b) determine which of the compounds/ions Br 3-, ClF 3, XeF 4, SF 4, PF 5, ClF 5, and SF 6 have similar (square planar or square pyramidal) molecular structures/shape. Determine the hybridization. So the hybridization depends on two things: the number of atoms bonded to the central atom, and the number of lone pairs off the central atom. How many sigma and pi bonds are in the molecule? sp 2 hybridisation. Later on, Linus Pauling improved this theory by introducing the concept of hybridization. ... BH3 (b) CH2O (c) NH3 (d) BrF5. The hybrid orbitals can form sigma bonds with other atoms. Adding up the exponents, you get 4. Trihydridoboron, also known as borane or borine, is an unstable and highly reactive m Its hybridization is sp 2, molecular shape – trigonal planar. Identify the molecular geometry (shape) c. Identify the hybridization of the central atom The twenty molecules and ions: SIH4 NH3 H20 CO2 SO2 CH20 CH, BH PFs XEF4 CIF, XeF2 SF, SF6 NO2 CO3 CN I SF5 d. State the bond angle for the following seven molecules or … In this type of hybridization one- s and two P-orbitals of the valence shell of carbon atom take part in hybridization go give three new sp 2 hybrid orbitals. Geometrical shape: The shape of a molecule is an important parameter to check whether a molecule is polar or not. For some molecules in the Table, we note that there is more than one possible shape that would satisfy the VSEPR rules. a) NH3 e) +NH4 i) H3O+ b) BH3 f) +CH3 j)H2C=O c) -CH3 g) HCN d)*CH3 h) C(CH3)4 There are a couple I don't know how to do. In molecular BH3 the molecule is planar with bond angles of 120o so the hybridisation of the central boron atom is sp2. It is narcotic in high concentrations. The bond angles are 109. The electronic configuration of carbon (Z = 6) in the excited state is. Methyl fluoride (or fluoromethane) is a colorless flammable gas which is heavier than air. Key Points. Boron has three valence electron, so it is supposed to make 3 bond in a molecules with hybridization s p 2 as only S and two p are used in hybridization because last p orbital vacant. The hybridization will be sp2 because the s orbital can only form 1 bond and the 2 p orbitals must be combined with the s orbital to allow for 3 bonds to be made by the central atom. Hybridization is essential for understanding the geometry of covalent bonds. This is formed between one s orbital and two p orbitals, allowing for equal bonds in elements like boron (BH3 and BF3). The possible molecular shapes are: Hybridization is the number of orbitals required to surround the atoms from a central atom. In sp³ hybridization, one s orbital and three p orbitals hybridize to form four sp³ orbitals, each consisting of 25% s character and 75% p character. First you must draw the Lewis Structure, or determine the molecular geometry to help find the hybridization. In all three cases, the bond angles are the same, the dipole moment is the same, the molecular shape is the same and the hybridization of the oxygen is the same. Similarly, being symmetric, BH3 is a nonpolar molecule. NO2- Lewis Structure With Formal Charge, Resonance, Molecular Geometry / Shape, Bond Angle. However, due to the linear shape of the molecule, the dipole moment of each bond is cancelled since they are in opposite directions, leading to a non-polar molecule. The beryllium atom in a gaseous BeCl 2 molecule is an example of a central atom with no lone pairs of electrons in a linear arrangement of three atoms. Organic Chemistry. 5^\circ$, what you can imagine on a molecular level is an increased s orbital contribution from the central atom to the bonding orbitals. Explain the process of hybridization as it applies to the formation of sp 3 hybridized atoms. Ammonia borane (also systematically named amminetrihydridoboron), also called borazane, is the chemical compound with the formula H 3 NBH 3.The colourless or white solid is the simplest molecular boron-nitrogen-hydride compound. In the dimer B2H6 the molecule has two bridging hydrogens. This type of hybridization is required whenever an atom is surrounded by four groups of electrons. B e Has 2 electrons in their valence shell and H h … Atoms combine together to lower down the energy of the system to attain stability — in layman's terms, the rule simply says that the less energy you need, the easier for you to survive. The valence bond theory was proposed by Heitler and London to explain the formation of covalent bond quantitatively using quantum mechanics. COVID-19 is an emerging, rapidly evolving situation. Get … What is the hybridization of bh3? 9 sigma and 9 pi. For example, the XeF 2 molecule has a steric number of five and a trigonal bipyramidal geometry. There are two regions of valence electron density in the BeCl 2 molecule that correspond to the two covalent Be–Cl bonds.$\ce{BH3}$has an empty$2p$orbital. Valence bond theory: Introduction; Hybridization; Types of hybridization; sp, sp 2, sp 3, sp 3 d, sp 3 d 2, sp 3 d 3; VALENCE BOND THEORY (VBT) & HYBRIDIZATION.$\ce{B}$has an$2s^22p^1$valence shell, so three covalent bonds gives it an incomplete octet. sp3 Hybridization . The arrangement of the electrons of Xenon changes to s2 p5 d1 with two unpaired electrons. b. NH3 >H2O >H2S all have four pairs of valence shell electrons. (a) each carbon Aton is sp^2 hybridization (b) all six C-C bonds are known to be equivalent (c) it has delocalized pi bonding in the molecule (d) the localized electron model must invoke resonance to account for the six equal C-C bonds. sp2 hybridization in ethene. is sp 2, molecular shape – trigonal planar. This preview shows page 11 - 14 out of 14 pages.. Its hybridization is sp; molecular shape – linear. Two orbitals (hybrid) of same shape and energy come into existence. Typically accurate to the second digit. A 420 [1-2]: 81-89 (1999). The bond angles are Cl-N-Cl. Did I do the other In BH3 , each boron sp2 orbital overlaps a hydrogen 1s orbital to make a B–H bond. BH3 molecule Hybrid atomic orbitals mix together atomic orbitals to form an equal number of new hybrid atomic orbitals with a varied shape. Borane \| BH3 \| CID 167170 - structure, chemical names, physical and chemical properties, classification, patents, literature, biological activities, safety/hazards/toxicity information, supplier lists, and more. These molecules have a trigonal planar shape. The two carbon atoms form a sigma bond in the molecule by overlapping two sp 2 orbitals. This orbital overlaps the existing$\ce{B-H}\sigma$bond cloud (in a nearby$\ce{BH3}$), and forms a 3c2e bond. What is the hybridization of all the atoms (other than hydrogen) in each of the following species? Now, there is hybridization between one s-and one p orbital. Learning Objective. What is the Hybridization of Ammonia? In sp^2 hybridization, the 2s orbital mixes with only two of the three available 2p orbitals, forming a total of three sp^2 orbitals with one p-orbital remaining. Whereas in XeF2, the Xe molecule has an excited state. To understand the hybridization of ammonia we have to carefully examine the areas around NItrogen. The hybridization of the phosphorus atom in the cation PH2+ is: sp2. H 2Te a) Te is in Group VI, so Lewis structure is analogous to H 2O (first structure) b) VSEPR 2 bp + 2 lp = 4 shape is tetrahedral c) Molecular shape is bent d) Hybridization is sp3 (VSEPR 4 pairs on central atom so need 4 orbitals) e) Polar. sp Hybridization. This results in the hybridization with 1 s orbital and 2 p orbitals, so sp2. Please help? In B e H 3 Hybridization of B e H 2 is ‘ s p ’ You can proceed in this way. BI 3 b. + 3 x. These overlap with p-orbital (singly occupied) each of the two fluorine atoms forming two sigma bonds. Hence the hybridization of the central atom Xe is sp3d. Geometrical isomers. Hybridization 1. The bonds in a methane (CH4) molecule are formed by four separate but equivalent orbitals; a single 2s and three … Ethene – … It seems that there are a lot more compounds with 3c2e geometry. It has an agreeable ether-like odor. → The three sp2 orbitals (shown in blue) of trigonal planar boron each combine with a hydrogen s orbital to form three B–H bonds. C O O Boron in BH 3 has three bonding pairs and no nonbonding. In BH3, each boron sp2 orbital overlaps a hydrogen 1s orbital to make a bond. Example, the XeF 2 molecule that correspond to the two fluorine atoms forming two sigma.... Proposed by Heitler and London to explain the process of bh3 hybridization shape, you 4.... Highly reactive m what is the number of bonds and 1 lone pair, the XeF 2 that. 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More than one possible shape that would satisfy the VSEPR rules hybridization between one s-and p. Linus Pauling improved this theory by introducing the concept of hybridization as it applies to number! Four orbitals on a second atom, how many molecular orbitals will form with two unpaired electrons of hybrid. That would satisfy the VSEPR rules Resonance, molecular geometry to help find the hybridization of two! Density in the dimer B2H6 the molecule formed is linear with a varied.! Overlaps a hydrogen 1s orbital to make a B–H bond, you get 4. hybridization!	2021-04-23 10:36:29	{"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": 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": 0.4487447738647461, "perplexity": 3131.0333298510427}, "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-2021-17/segments/1618039617701.99/warc/CC-MAIN-20210423101141-20210423131141-00409.warc.gz"}
https://www.cnblogs.com/devymex/archive/2010/08/15/1800085.html	程序控 IPPP (Institute of Penniless Peasent-Programmer) Fellow :: :: :: :: :: :: :: :: 77 随笔 :: 0 文章 :: 442 评论 :: 0 引用 Background背景 Trees are fundamental in many branches of computer science. Current state-of-the art parallel computers such as Thinking Machines' CM-5 are based on fat trees. Quad- and octal-trees are fundamental to many algorithms in computer graphics. This problem involves building and traversing binary trees. The Problem问题 Given a sequence of binary trees, you are to write a program that prints a level-order traversal of each tree. In this problem each node of a binary tree contains a positive integer and all binary trees have have fewer than 256 nodes. In a level-order traversal of a tree, the data in all nodes at a given level are printed in left-to-right order and all nodes at level k are printed before all nodes at level k+1. For example, a level order traversal of the tree is: 5, 4, 8, 11, 13, 4, 7, 2, 1. In this problem a binary tree is specified by a sequence of pairs (n,s) where n is the value at the node whose path from the root is given by the string s. A path is given be a sequence of L's and R's where L indicates a left branch and R indicates a right branch. In the tree diagrammed above, the node containing 13 is specified by (13,RL), and the node containing 2 is specified by (2,LLR). The root node is specified by (5,) where the empty string indicates the path from the root to itself. A binary tree is considered to be completely specified if every node on all root-to-node paths in the tree is given a value exactly once. The Input输入 The input is a sequence of binary trees specified as described above. Each tree in a sequence consists of several pairs (n,s) as described above separated by whitespace. The last entry in each tree is (). No whitespace appears between left and right parentheses. All nodes contain a positive integer. Every tree in the input will consist of at least one node and no more than 256 nodes. Input is terminated by end-of-file. The Output输出 For each completely specified binary tree in the input file, the level order traversal of that tree should be printed. If a tree is not completely specified, i.e., some node in the tree is NOT given a value or a node is given a value more than once, then the string "not complete" should be printed. Sample Input输入示例 (11,LL) (7,LLL) (8,R) (5,) (4,L) (13,RL) (2,LLR) (1,RRR) (4,RR) () (3,L) (4,R) () Sample Output输出示例 5 4 8 11 13 4 7 2 1 not complete Solution解答 #include <iostream> #include <list> #include <string> #include <sstream> using namespace std; //节点结构体，存储节点的值及左右子节点 struct NODE {int nVal; NODE pL; NODE pR;} //NullNode为空节点标本 const NullNode = {0, 0, 0}; //删除树 void DeleteTree(NODE pPar) { if (pPar != NULL) { //深度遍例删除左右子节点 DeleteTree(pPar->pL), DeleteTree(pPar->pR); } delete pPar; } //为树增加节点 NODE AddChild(NODE pPar, int nVal, const char pPath) { //如果父为空，则建立父节点 if (pPar == 0) { pPar = new NODE(NullNode); } //根据当前路径字符串做不同操作 switch (pPath) { //遇到右括号，说明路径结束，为当前父节点赋值 case ')': pPar->nVal = (pPar->nVal == 0 && nVal != 0) ? nVal : -1; break; //继续建立左/右子节点 case 'L': pPar->pL = AddChild(pPar->pL, nVal, pPath + 1); break; case 'R': pPar->pR = AddChild(pPar->pR, nVal, pPath + 1); break; } return pPar; } //主函数 int main(void) { //树的根结点 NODE pRoot = 0; //循环处理输入的每一个节点数据 for (string strToken; cin >> strToken;) { //获得节点数据字符串指针 const char pStr = strToken.c_str(); int nLen = strToken.length(), nVal = 0; //如果第二个字符不是右括号，则添加节点后继续输入 if (pStr[1] != ')') { //将字符串转为数字 for (; isdigit(++pStr); nVal = nVal * 10 + pStr - '0'); if (pStr != ',') while(true); //在树中添加该节点 continue; } //如果第二字符是右括号，说明一棵树输入结束，进行遍例 list<NODE> Level(1, pRoot); stringstream ssResult; //依次遍例每一层。用Level存储一层的节点，Level非空则继续遍例 for (list<NODE>::iterator i = Level.begin(); !Level.empty(); //如果一层遍例结束，则回到起点 i = (i == Level.end() ? Level.begin() : i)) { //移除当前节点 NODE pTemp = i; i = Level.erase(i); //如果该节点为空，则直接继续进行下一个节点 if (pTemp == 0) { continue; }//否则在原位置上增加子节点 //如果该节点的值小于或等于0，说明其重复定义或未定义 if (pTemp->nVal <= 0) { //清空数据，返回错误。 Level.clear(); ssResult.str(""); break; }//正确的结点 //输入其值，并在原位置上插入其子节点 ssResult << pTemp->nVal << ' '; i = Level.insert(i, pTemp->pL), ++i; i = Level.insert(i, pTemp->pR), ++i; } //删除原树，避免内存泄露。 DeleteTree(pRoot); pRoot = 0; strToken = ssResult.str(); //如果结果为空，说明所给数据有错 if (strToken.empty()) { strToken = "not complete"; } //对于正确结果要去掉最后的一个空格 else { strToken.erase(strToken.end() - 1); } //输出结果 cout << strToken << endl; } return 0; } posted on 2010-08-15 19:15 Devymex 阅读(...) 评论(...编辑收藏	2020-01-22 20:13:00	{"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": 0.4277890920639038, "perplexity": 7627.7204107765765}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250607407.48/warc/CC-MAIN-20200122191620-20200122220620-00540.warc.gz"}
https://asmedigitalcollection.asme.org/fluidsengineering/article-abstract/131/12/121104/411862/Pulsating-Flow-for-Mixing-Intensification-in-a?searchresult=1	This work concerns the manipulation of a twisted curved-pipe flow for mixing enhancement. Previous works have shown that geometrical perturbations to a curved-pipe flow can increase mixing and heat transfer by chaotic advection. In this work the flow entering the twisted pipe undergoes a pulsatile motion. The flow is studied experimentally and numerically. The numerical study is carried out by a computational fluid dynamics (CFD) code (FLUENT 6) in which a pulsatile velocity field is imposed as an inlet condition. The experimental setup involves principally a “Scotch-yoke” pulsatile generator and a twisted curved pipe. Laser Doppler velocimetry measurements have shown that the Scotch-yoke generator produces pure sinusoidal instantaneous mean velocities with a mean deviation of 3%. Visualizations by laser-induced fluorescence and velocity measurements, coupled with the numerical results, have permitted analysis of the evolution of the swirling secondary flow structures that develop along the bends during the pulsation phase. These measurements were made for a range of steady Reynolds number $(300≤Rest≤1200)$, frequency parameter $(1≤α=r0⋅(ω/υ)1/2<20)$, and two velocity component ratios $(β=Umax,osc/Ust)$. We observe satisfactory agreement between the numerical and experimental results. For high $β$, the secondary flow structure is modified by a Lyne instability and a siphon effect during the deceleration phase. The intensity of the secondary flow decreases as the parameter $α$ increases during the acceleration phase. During the deceleration phase, under the effect of reverse flow, the secondary flow intensity increases with the appearance of Lyne flow. Experimental results also show that pulsating flow through a twisted curved pipe increases mixing over the steady twisted curved pipe. This mixing enhancement increases with $β$. 1. Aref , H. , 1984, “ Stirring by Chaotic Advection ,” J. Fluid Mech. 0022-1120, 143 , pp. 1 21 . 2. Jones , S. W. , Thomas , O. M. , and Aref , H. , 1989, “ Chaotic Advection by Laminar Flow in Twisted Pipe ,” J. Fluid Mech. 0022-1120, 209 , pp. 335 357 . 3. Castelain , C. , Mokrani , A. , Le Guer , Y. , and Peerhossaini , H. , 2001, “ Experimental Study of Chaotic Advection Regime in a Twisted Duct Flow ,” Eur. J. Mech. B/Fluids 0997-7546, 20 , pp. 205 232 . 4. Khakhar , D. V. , Franjione , J. G. , and Ottino , J. M. , 1987, “ A Case Study of Chaotic Mixing in Deterministic Flows: The Partitioned Pipe Mixer ,” Chem. Eng. Sci. 0009-2509, 42 ( 12 ), pp. 2909 2926 . 5. Castelain , C. , Mokrani , A. , Legentilhomme , P. , and Peerhossaini , H. , 1997, “ Residence Time Distribution in Twisted Pipe Flows: Helically Coiled System and Chaotic System ,” Exp. Fluids 0723-4864, 22 , pp. 359 368 . 6. Acharya , N. , Sen , M. , and Chang , H. C. , 1992, “ Heat Transfer Enhancement in Coiled Tubes by Chaotic Mixing ,” Int. J. Heat Mass Transfer 0017-9310, 35 , pp. 2475 2489 . 7. Acharya , N. , Sen , M. , and Chang , H. C. , 1994, “ Thermal Entrance Length and Nusselt Numbers in Coiled Tubes ,” Int. J. Heat Mass Transfer 0017-9310, 37 , pp. 336 340 . 8. Chagny , C. , Castelain , C. , and Peerhossaini , H. , 2000, “ Chaotic Heat Transfer for Heat Exchanger Design and Comparison With a Regular Regime for a Large Range of Reynolds Numbers ,” Appl. Therm. Eng. 1359-4311, 20 ( 17 ), pp. 1615 1648 . 9. Womersley , R. , 1955, “ Method for the Calculation of Velocity, Rate of Flow and Viscous Drag in Arteries When Pressure Gradient Is Known ,” J. Physiol. , 127 , pp. 553 563 . 10. Atabek , H. B. , and Chang , C. C. , 1961, “ Oscillatory Flow Near the Entry of a Circular Tube ,” Z. Angew. Math. Phys. 0044-2275, 12 , pp. 185 201 . 11. Pedley , T. J. , 1980, The Fluid Mechanics of Large Blood Vessels ( Cambridge Monographs on Mechanics and Applied Mathematics ), Cambridge University Press , Cambridge , pp. 160 224 . 12. Zabielski , L. , and Mestel , A. J. , 1998, “ Unsteady Blood Flow in a Helically Symmetric Pipe ,” J. Fluid Mech. 0022-1120, 370 , pp. 321 345 . 13. Deplano , V. , and Siouffi , M. , 1999, “ Experimental and Numerical Study of Pulsatile Flows Through Stenosis: Wall Shear Stress Analysis ,” J. Biomech. 0021-9290, 32 , pp. 1081 1090 . 14. Hamakiotes , C. C. , and Berger , S. A. , 1990, “ Periodic Flows Through Curved Tubes: The Effect of the Frequency Parameter ,” J. Fluid Mech. 0022-1120, 210 , pp. 353 370 . 15. Lyne , W. H. , 1971, “ Unsteady Viscous Flow in a Curved Pipe ,” J. Fluid Mech. 0022-1120, 45 ( 1 ), pp. 13 31 . 16. Dean , W. R. , 1927, “ Note on the Motion of Fluid in a Curved Pipe ,” Philosophical Magazine , 20 , pp. 208 223 . 17. Zalosh , R. G. , and Nelson , W. G. , 1973, “ Pulsating Flow in a Curved Tube ,” J. Fluid Mech. 0022-1120, 59 ( 4 ), pp. 693 705 . 18. Bertelsen , A. F. , 1975, “ An Experimental Investigation of Low Reynolds Number Secondary Streaming Effects Associated With an Oscillating Viscous Flow in a Curved Pipe ,” J. Fluid Mech. 0022-1120, 70 , pp. 519 527 . 19. Timité , B. , Castelain , C. , and Peerhossaini , H. , 2009, “ Pulsatile Viscous Flow in a Curved Pipe: Effects of Pulsation on the Development of the Secondary Flow ,” Int. J. Heat Fluid Flow 0142-727X, submitted. 20. , N. J. , Simon , M. A. , and Chow , J. C. F. , 1980, “ Numerical Solution for Fully Developed, Laminar Pulsating Flow in Curved Tubes ,” Numer. Heat Transfer 0149-5720, 3 , pp. 225 239 . 21. Talbot , L. , and Gong , K. O. , 1983, “ Pulsatile Entrance Flow in a Curved Pipe ,” J. Fluid Mech. 0022-1120, 127 , pp. 1 25 . 22. Florio , J. P. , and Mueller , W. K. , 1968, “ Development of a Periodic Flow in a Rigid Tube ,” ASME J. Basic Eng. 0021-9223, 90 , pp. 395 399 . 23. Fargie , D. , and Martin , B. W. , 1971, “ Developing Laminar Flow in a Pipe of Circular Cross-Section ,” Proc. R. Soc. London 0370-1662, 321 , pp. 461 476 . 24. Ohmi , M. , Usui , T. , Fukawa , M. , and Hirasaki , S. , 1976, “ Pressure and Velocity Distributions in Pulsating Laminar Pipe Flow ,” Bull. JSME 0021-3764, 19 ( 129 ), pp. 298 306 . 25. Peerhossaini , H. , and Wesfreid , J. E. , 1988, “ On the Inner Structure of the Streamwise Görtler Rolls ,” Int. J. Heat Fluid Flow 0142-727X, 9 , pp. 12 18 . 26. Takayoshi , M. , and Katsumi , N. , 1980, “ Unsteady Flow in Circular Tube (Velocity Distribution of Pulsating Flow) ,” Bull. JSME 0021-3764, 23 ( 186 ), pp. 1990 1996 . 27. Fellouah , H. , Castelain , C. , Ould El Moctar , A. , and Peerhossaini , H. , 2006, “ A Criterion for Detection of the Onset of Dean Instability in Newtonian Fluids ,” Eur. J. Mech. B/Fluids 0997-7546, 25 , pp. 505 531 . 28. Le Guer , Y. , and Peerhossaini , H. , 1991, “ Order Breaking in Dean Flow ,” Phys. Fluids A 0899-8213, 3 , pp. 1029 1032 . 29. Duchêne , C. , Peerhossaini , H. , and Michard , P. J. , 1995, “ On the velocity Field and Tracer Patterns in a Twisted Duct Flow ,” Phys. Fluids 1070-6631, 7 ( 6 ), pp. 1307 1317 . You do not currently have access to this content.	2021-12-05 12:02:30	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 6, "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": 0.5642692446708679, "perplexity": 12505.313923974958}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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-2021-49/segments/1637964363157.32/warc/CC-MAIN-20211205100135-20211205130135-00616.warc.gz"}
http://math.stackexchange.com/questions/194640/how-to-find-the-equation-from-data	# How to find the equation from data? I know things about linear/quadratic fittings etc. I'm just wondering, if i know a set of data for value e.g. z=[-2.563 -0.1932 -0.1502 -0.1102 -0.836 -0.5234] and l=[1 2 3 4 5 6] m=[6 5 4 3 2 1] I want to write out a function of z in terms of l and m. How would i be able to start this? Many thanks in advance! - How is this recreational mathematics? – joriki Sep 12 '12 at 11:08 @joriki. I fixed it – Zeta.Investigator Sep 12 '12 at 11:08 It's not statistics, either -- I've replaced the tag by interpolation. – joriki Sep 12 '12 at 11:11 sorry, i don't know how that tag was made. – chenwan Sep 12 '12 at 11:23 PooyaM made it; see above. – joriki Sep 12 '12 at 11:35 The values of $(l,m)$ all lie on one line, so this isn't actually a two-dimensional problem; you can just do a one-dimensional fit with respect to either $l$ or $m$. Then if you want to evaluate the function at an arbitrary point $(l,m)$, just project it onto the line $l+m=7$ and evaluate the fitted function there. [Edit in response to comment:] You only have function values on a single line. The best you can do if you want to approximate a function value off the line is to take the function value at the closest point on the line, which is the orthogonal projection of the point onto the line. The orthogonal projection of the point $(l,m)$ onto the line $l+m=7$ is $((7+l-m)/2,(7+m-l)/2)$, so you get the approximation by evaluating a function fitted with respect to $l$ at $(7+l-m)/2$ or evaluating a function fitted with respect to $m$ at $(7+m-l)/2$. - could you please explain more about projection part? I don't quite get that. – chenwan Sep 12 '12 at 11:24 my situation is as follows, I have a matrix of z values depending on the l and m value, l always starts from 1 to some number n while m goes from n till 1. for example, with l= 1 2 3 4 5 ,m=5 4 3 2 1 and this z value will be a length of 5 vector which is totally different as the case with l=1 2 3 4 5 6 7 and m=7 6 5 4 3 2 1. So i couldn't really just use either l or m as the dependent variable to form the function of z. In that case, i wouldn't use that function to find z for all cases. Thank you anyway! – chenwan Sep 12 '12 at 11:24 How do you mean, you couldn't really use either $l$ or $m$ as the dependent variable? What keeps you from doing it? – joriki Sep 12 '12 at 11:32 if i use l as dependent variable in first case, when l is 1, it gives a z value according to the function which is true as my original value of z. However, we can't use that l=1 to find z for another case. Unless, i have z in terms of l and m the same time. – chenwan Sep 12 '12 at 11:38 are you saying i find a fit of z in terms of 7+m-l/2? – chenwan Sep 12 '12 at 11:38	2014-04-20 23:53:08	{"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": 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": 0.8669015169143677, "perplexity": 285.70205669489377}, "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-2014-15/segments/1397609539337.22/warc/CC-MAIN-20140416005219-00160-ip-10-147-4-33.ec2.internal.warc.gz"}
https://groupprops.subwiki.org/w/index.php?title=Homomorph-dominating_subgroup&diff=13096&oldid=13095	# Difference between revisions of "Homomorph-dominating subgroup" BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE] This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE] ## Definition A subgroup $H$ of a group $G$ is termed homomorph-dominating in $G$ if, for any homomorphism $\varphi \in \operatorname{Hom}(H,G)$, there exists $g \in G$ such that $\varphi(H) \le gHg^{-1}$. ## Relation with other properties ### Conjunction with other properties A homomorph-containing subgroup is precisely the same as a subgroup that is both normal and homomorph-dominating. For full proof, refer: Homomorph-dominating and normal equals homomorph-containing	2021-10-21 13:04:01	{"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": 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": 0.8318957686424255, "perplexity": 1690.0821564013359}, "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/1634323585405.74/warc/CC-MAIN-20211021102435-20211021132435-00160.warc.gz"}
https://www.cableizer.com/documentation/Nu_encl/	# Nusselt number gas-enclosure The formula is based on Kuehn and Goldstein 'Correlating equations for natural convection heat transfer between horizontal circular cylinders', 1976 who presented a correlation for natural convection heat transfer from a horizontal cylinder which is valid at any Rayleigh and Prandtl number. Symbol $\mathrm{Nu}_{encl}$ Formulae $- \frac{2}{\ln{\left(1 - \frac{3.40806269406455}{\left(G_{encl}^{15} \mathrm{Ra}_{encl}^{3.75} + 4.55142270836015 \cdot 10^{-11} \mathrm{Ra}_{encl}^{3.75}\right)^{0.0667}} \right)}}$ $\mathrm{Ra}_{encl}$ Used in $h_{encl}$	2020-08-09 14:06:20	{"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": 0.5681692361831665, "perplexity": 1749.1058639039607}, "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-2020-34/segments/1596439738555.33/warc/CC-MAIN-20200809132747-20200809162747-00554.warc.gz"}
https://www.sanfoundry.com/chemistry-assessment-questions-iit-jee-exam/	Chemistry Questions and Answers – Diazonium Salts in Synthesis of Aromatic Compounds « » This set of Chemistry Assessment Questions for IIT JEE Exam focuses on “Diazonium Salts in Synthesis of Aromatic Compounds”. 1. Diazonium salts are specifically used as intermediates in the production of which of the following compounds? a) m-Bromotoluene b) o-Bromotoluene c) p-Bromophenol d) o-Bromophenol Explanation: m-Bromotoluene cannot be prepared from the direct bromination of toluene or by Friedel-Crafts alkylation of bromobenzene, because of the ortho, para directing nature of alkyl groups. 2. What is the minimum number of reactions required for the conversion of aniline to 1,3,5-tribromobenzene? a) 2 b) 3 c) 4 d) 5 Explanation: The three reactions are as follows: $$\Rightarrow$$ Reaction 1: Aniline on treatment with bromine water gives 2,4,6-tribromoaniline. $$\Rightarrow$$ Reaction 2: This on diazotisation with HNO2 gives 2,4,6-tribromobenzenediazonium chloride. $$\Rightarrow$$ Reaction 3: This on reaction with H3PO2, reduces the diazo group to hydrogen, to give 1,3,5-tribromobenzene. 3. p-Iodonitrobenzene can be synthesized from p-nitroaniline. a) True b) False Explanation: p-Nitroaniline on diazotisation gives p-nitrobenzenediazonium chloride with on treatment with potassium iodide replaces the N2+Cl group to give p-iodonitrobenzene. Note: Join free Sanfoundry classes at Telegram or Youtube 4. Diazonium salts are used in which of the following industries? a) Pharmaceutical b) Dye c) Photography d) Food Explanation: Coupling reactions of diazonium salts produce azo compounds which are coloured in nature. This is used as an application in the manufacturing of dyes. 5. Diazonium salts are primarily used for the preparation of _______ substituted aromatic compounds. a) alkyl b) halogen c) amino d) COOH Explanation: Diazonium salts are important intermediates for the introduction of -F, -Cl, -Br, -I, etc. groups into an aromatic ring. Certain compounds like aryl fluorides and aryl iodides cannot be prepared by direct halogenation and are prepared from diazonium salts. Take Chemistry - Class 12 Mock Tests - Chapterwise! Start the Test Now: Chapter 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 Sanfoundry Global Education & Learning Series – Chemistry – Class 12. To practice Chemistry Assessment Questions for IIT JEE Exam, here is complete set of 1000+ Multiple Choice Questions and Answers.	2022-10-07 23:20:30	{"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": 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": 0.3949072062969208, "perplexity": 14815.18410786105}, "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-2022-40/segments/1664030338280.51/warc/CC-MAIN-20221007210452-20221008000452-00606.warc.gz"}
https://dmtcs.episciences.org/2975	## Kuhlman, Chris, and Mortveit, Henning, and Murrugarra, David and Kumar, Anil, - Bifurcations in Boolean Networks dmtcs:2975 - Discrete Mathematics & Theoretical Computer Science, January 1, 2011, DMTCS Proceedings vol. AP, Automata 2011 - 17th International Workshop on Cellular Automata and Discrete Complex Systems Bifurcations in Boolean Networks Authors: Kuhlman, Chris, and Mortveit, Henning, and Murrugarra, David and Kumar, Anil, This paper characterizes the attractor structure of synchronous and asynchronous Boolean networks induced by bi-threshold functions. Bi-threshold functions are generalizations of standard threshold functions and have separate threshold values for the transitions $0 \rightarrow$1 (up-threshold) and $1 \rightarrow 0$ (down-threshold). We show that synchronous bi-threshold systems may, just like standard threshold systems, only have fixed points and 2-cycles as attractors. Asynchronous bi-threshold systems (fixed permutation update sequence), on the other hand, undergo a bifurcation. When the difference $\Delta$ of the down- and up-threshold is less than 2 they only have fixed points as limit sets. However, for $\Delta \geq 2$ they may have long periodic orbits. The limiting case of $\Delta = 2$ is identified using a potential function argument. Finally, we present a series of results on the dynamics of bi-threshold systems for families of graphs. Source : oai:HAL:hal-01196142v1 Volume: DMTCS Proceedings vol. AP, Automata 2011 - 17th International Workshop on Cellular Automata and Discrete Complex Systems Section: Proceedings Published on: January 1, 2011 Submitted on: January 31, 2017 Keywords: bifurcation,bi-threshold,threshold,Boolean networks,graph dynamical systems,synchronous,asynchronous,sequential dynamical systems,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM],[MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS],[NLIN.NLIN-CG] Nonlinear Sciences [physics]/Cellular Automata and Lattice Gases [nlin.CG],[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]	2019-08-20 13:24:17	{"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": 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": 0.45567649602890015, "perplexity": 6600.276003733469}, "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-2019-35/segments/1566027315329.55/warc/CC-MAIN-20190820113425-20190820135425-00132.warc.gz"}
https://www.aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=vernal%20equinox	Astronomy Answers: From the Astronomical Dictionary # Astronomy AnswersFrom the Astronomical Dictionary $$\def\\|{&}$$ The description of the word you requested from the astronomical dictionary is given below. the vernal equinox The vernal equinox is the intersection of the equator of the sky and the ecliptic, through which the Sun passes during the March equinox, at the beginning of spring in the northern hemisphere, and autumn in the southern hemisphere. The vernal equinox is the zero point of longitude and latitude of the equatorial and ecliptic coordinate systems. The period between two successive passages of the Sun through the vernal equinox is the tropical year.	2018-10-20 21:48:11	{"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": 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": 0.4337209165096283, "perplexity": 897.6317003382785}, "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/1539583513441.66/warc/CC-MAIN-20181020205254-20181020230754-00268.warc.gz"}
http://stats.stackexchange.com/questions/25776/are-the-variable-types-here-considered-correct	# Are the variable types here considered correct? If we want to determine the variable types, will it be as follows for the below variables? Age ---> quantitative, discrete (we can count) Fitness ---> If the values that we will enter here are 0 and 1 only, will the type of this variable be qualitative, nominal? Thanks a lot. - How would such a determination help you with your analysis? One good answer, found in many places on this site, is not at all. Indeed, this (Stevens') classification of variables has been criticized (justifiably, IMHO) for unnecessarily limiting the analyst's options. This suggests you might get more value from this site by forgetting about variable type altogether and asking the questions that I suspect may be lurking beneath the surface here, questions concerning how to understand and analyze your data. – whuber Apr 3 '12 at 16:06 BTW, there are important distinctions between counted data and other types of data. The issue is not that of discrete versus continuous, but of expectations (and theoretical limitations) concerning their statistical behavior. Although age is definitely not a count, it can be considered discrete or continuous depending on one's analytical objectives and capabilities. – whuber Apr 3 '12 at 16:08 If it is not too late, you might collect date of birth and then calculate age. The variable would then be continuous. Also, people remember their date of birth, but sometimes not their age. – Joel W. Aug 1 '12 at 21:08 @whuber, would you please provide a link to one or more of the pages on this site on the topic of Steven's levels of measurement? Thanks. – Joel W. Aug 1 '12 at 21:11 @Joel Search the site for "Stevens". One that turns up is Does it ever make sense to treat categorical data as continuous?. It also uncovers my partial criticism. – whuber Aug 1 '12 at 21:16 It depends on requirements of software and specific procedure you use for the analysis. In general, I personally like to register binary variables (1=present vs 0=absent) as ordinal level, not nominal level, such as sex (1=male, 2=female). But is matter of taste and custom... - Questions like this expose the problems with Stevens' typology of variables. Some variables just aren't any of his categories; or are hard to categorize. Age is ratio level data in the sense that someone who is (say) 40 years old is twice as old as someone who is 20 years old. You just haven't got it measured precisely. But so? What difference does that make? Dichotomies - here it really doesn't matter HOW you treat them for most purposes. E.g. in regression, you can consider them to be nominal, ordinal or interval and you will get the same results. I wrote about other problems with this typology on my blog -	2013-12-08 01:18:37	{"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": 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": 0.5536715388298035, "perplexity": 1260.8974189610888}, "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-2013-48/segments/1386163057146/warc/CC-MAIN-20131204131737-00093-ip-10-33-133-15.ec2.internal.warc.gz"}
http://onlineprediction.net/index.html?n=Main.On-linePrediction	On-line Prediction On-line prediction refers to the problem of prediction in the on-line protocol: • Nature outputs some side information • Predictor outputs a prediction • Nature outputs an observation • The cycle is repeated There are many variations of this basic protocol. In particular, Nature can output observations with a delay, or there can be additional players. Some of the topics in on-line prediction covered by this wiki are listed in the sidebar. (This list will change to track changes in the coverage of various topics.) Other important approaches to on-line prediction include Much of the work in on-line prediction does not assume that Nature chooses her moves stochastically. An exception is Conformal Prediction, where Nature is supposed to use an exchangeable probability measure (or another on-line compression model) as her stochastic strategy. For more information on the work assuming a stochastic Nature, see Stochastic Prediction.	2022-07-03 02:38:26	{"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": 0.8637230396270752, "perplexity": 2304.75863828764}, "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-2022-27/segments/1656104209449.64/warc/CC-MAIN-20220703013155-20220703043155-00502.warc.gz"}
http://dianken.mew.ru/library/mosque-in-early-ottoman-architecture-publications-of-the-center-for-middle-ea	# Download E-books Mosque in Early Ottoman Architecture (Publications of the Center for Middle EA) PDF By Aptullah Kuran Best Art books n-Harmonic Mappings Between Annuli: The Art of Integrating Free Lagrangians (Memoirs of the American Mathematical Society) The critical subject matter of this paper is the variational research of homeomorphisms $h: {\mathbb X} \overset{\textnormal{\tiny{onto}}}{\longrightarrow} {\mathbb Y}$ among given domain names ${\mathbb X}, {\mathbb Y} \subset {\mathbb R}^n$. The authors search for the extremal mappings within the Sobolev house ${\mathscr W}^{1,n}({\mathbb X},{\mathbb Y})$ which reduce the power vital \${\mathscr E}_h=\int_{{\mathbb X}} \,\|\! Model Driven Architecture – Foundations and Applications: First European Conference, ECMDA-FA 2005, Nuremberg, Germany, November 7-10, 2005. Proceedings This booklet constitutes the refereed complaints of the 1st eu convention, Workshops on version pushed structure - Foundations and functions, ECMDA-FA 2005, held in Nuremberg, Germany in November 2005. The 24 revised complete papers offered, nine papers from the functions music and 15 from the principles music, have been rigorously reviewed and chosen from eighty two submissions. Extra info for Mosque in Early Ottoman Architecture (Publications of the Center for Middle EA) Show sample text content Rated 4.45 of 5 – based on 26 votes	2017-10-22 13:27:10	{"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": 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": 0.22469572722911835, "perplexity": 10389.877662042954}, "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-2017-43/segments/1508187825264.94/warc/CC-MAIN-20171022132026-20171022152026-00498.warc.gz"}
https://gamedev.stackexchange.com/questions/168067/what-exactly-does-xmmatrixperspectivel-rh-do	# What exactly does XMMatrixPerspective{L,R}H do? There are generally three functions to create a perspective matrix in DirectXMath, XMMatrixPerspective{L,R}H, XMMatrixPerspectiveFov{L,R}H and XMMatrixPerspectiveOffCenter{L,R}H. I'm kind of confused what XMMatrixPerspective exactly does. I'm currently generating a perspective matrix using DirectX::XMMatrixPerspectiveFovRH(to_rad(80.0f), (float)winX / (float)winY, 0.1f, 100.0f); which works fine, I can see the cube I'm drawing in front of me. If I replace this with a DirectX::XMMatrixPerspectiveRH(winX, winY, 0.1f, 100.0f), I don't see anything on the screen. So the question is, what exactly does the matrix represent? What is it's field of view? And why don't I see my cube? If this function uses some sort of default fov value, I'd expect to see my cube - even if possibly stretched - as my view matrix looks straight at it. Unfortunately, the docs only say "creates a projection matrix" but nothing about how one is supposed to work with that matrix. E: I figured out, I'm not supposed to pass the window coordinates. It expects the size of the near-clipping plane. However after playing around for a bit with values, how is this function supposed to be used? I really can't see any reason to use this over the one where you pass in a fov. The DirectXMath library is all inline so you can look directly at the source. For example, here is the C version of XMMatrixPerspectiveLH: float TwoNearZ = NearZ + NearZ; float fRange = FarZ / (FarZ - NearZ); XMMATRIX M; M.m[0][0] = TwoNearZ / ViewWidth; M.m[0][1] = 0.0f; M.m[0][2] = 0.0f; M.m[0][3] = 0.0f; M.m[1][0] = 0.0f; M.m[1][1] = TwoNearZ / ViewHeight; M.m[1][2] = 0.0f; M.m[1][3] = 0.0f; M.m[2][0] = 0.0f; M.m[2][1] = 0.0f; M.m[2][2] = fRange; M.m[2][3] = 1.0f; M.m[3][0] = 0.0f; M.m[3][1] = 0.0f; M.m[3][2] = -fRange * NearZ; M.m[3][3] = 0.0f; return M; and here is the C version of XMMatrixPerspectiveFovLH: float Height = CosFov / SinFov; float Width = Height / AspectRatio; float fRange = FarZ / (FarZ-NearZ); XMMATRIX M; M.m[0][0] = Width; M.m[0][1] = 0.0f; M.m[0][2] = 0.0f; M.m[0][3] = 0.0f; M.m[1][0] = 0.0f; M.m[1][1] = Height; M.m[1][2] = 0.0f; M.m[1][3] = 0.0f; M.m[2][0] = 0.0f; M.m[2][1] = 0.0f; M.m[2][2] = fRange; M.m[2][3] = 1.0f; M.m[3][0] = 0.0f; M.m[3][1] = 0.0f; M.m[3][2] = -fRange * NearZ; M.m[3][3] = 0.0f; return M; The "FOV" version compute the size of the projection plane rectangle from the provided field-of-view and aspect ratio values, while the non-FOV version takes a project plane rectangle size directly. The FOV version is the one you most commonly use as it provides a 'physical camera like' control that's easier to understand.	2020-01-24 00:03:28	{"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": 0.40816861391067505, "perplexity": 4195.91390751654}, "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-2020-05/segments/1579250614086.44/warc/CC-MAIN-20200123221108-20200124010108-00120.warc.gz"}
https://gateoverflow.in/548/gate-cse-1992-question-01-iv	# GATE CSE 1992 \| Question: 01-iv 2k views Many of the advanced microprocessors prefetch instructions and store it in an instruction buffer to speed up processing. This speed up is achieved because ________ retagged Because CPU is faster than memory. Fetching instructions from memory would require considerable amount of time while CPU is much faster. So, prefetching the instructions to be executed can save considerable amount of waiting time. selected 1 0 Thanks Ayush, those who doesnt want to click on the link In computer architectureinstruction prefetch is a technique used in central processor units to speed up the execution of a program by reducing wait states. Prefetching occurs when a processor requests an instruction or data block from main memory before it is actually needed. Once the block comes back from memory, it is placed in a cache. When the instruction/data block is actually needed, it can be accessed much more quickly from the cache than if it had to make a request from memory. Thus, prefetching hides memory access latency and hence, it is a useful technique for addressing the memory wall issue. Since programs are generally executed sequentially, performance is likely to be best when instructions are prefetched in program order. Alternatively, the prefetch may be part of a complex branch prediction algorithm, where the processor tries to anticipate the result of a calculation and fetch the right instructions in advance. In the case of dedicated hardware (like a Graphics Processing Unit) the prefetch can take advantage of the spatial coherence usually found in the texture mappingprocess. In this case, the prefetched data are not instructions, but texture elements (texels) that are candidates to be mapped on a polygon. The first mainstream microprocessors to use some form of instruction prefetch were the Intel 8086 (six bytes) and the Motorola 68000 (four bytes). In recent years, many high-performance processors use prefetching techniques. In computer architecture, instruction prefetch is a technique used in microprocessors to speed up the execution of a program by reducing wait states. So, the speed up is achieved because the wait states are reduced. 1 vote Instruction prefetching is also useful in case of execution of branch instruction on pipelined processors. Target instructions can be prefetched thereby reducing the chance of pipeline flush/clean when a branch instruction is executed. 0 You meant branch prediction rt? 1 Like Instruction Prefetching, Branch prediction is yet another method for avoiding stall cycles during execution of branch instruction in pipelined processor. Instruction Prefetching is like caching. Pipe lining as parallel processing is not feasible in general. ## Related questions 1 3.3k views Maximum number of edges in a planar graph with $n$ vertices is _____ Complexity of Kruskal’s algorithm for finding the minimum spanning tree of an undirected graph containing $n$ vertices and $m$ edges if the edges are sorted is _______	2021-07-29 03:02:00	{"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": 0.25146520137786865, "perplexity": 1919.2099047761897}, "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/1627046153814.37/warc/CC-MAIN-20210729011903-20210729041903-00339.warc.gz"}
https://en.wikibooks.org/wiki/General_Chemistry/Measurement	# General Chemistry/Measurement General Chemistry Book Cover · Introduction · v • d • e ## Moles The mole is a unit used to measure the number of particles in a substance (where particles can be atoms, molecules, electrons, ions etc.). One mole of something contains 6.02205 × 1023 particles. This rather large quantity is a constant called Avogadro's Number, which is abbreviated as NA. It simply refers to the number of particles in one mole. ${\displaystyle N_{A}=6.02214\times 10^{23}{\hbox{ mol}}^{-1}}$ For example, one mole of helium gas contains 6.022×1023 atoms; 2 moles of oxygen gas contains 1.204×1024 molecules. ### Moles and Mass It would be useful to be able to convert between the number of moles and the mass of a substance. The molar mass is the mass of one mole of a substance, measured in grams per mole or ${\displaystyle {\hbox{g}}\cdot {\hbox{mol}}^{-1}}$. Not by coincidence, the molar mass is the same as the atomic weight of a substance (this is how the concept of moles was originally defined). For example, the molar mass (and atomic weight) of carbon-12 is 12 g·mol-1, so 1 mole of carbon-12 weighs 12 grams. Similarly, chlorine has a molar mass of 35.45 g·mol-1, so one mole of chlorine atoms weighs 35.45 g. There are two chlorine atoms in one molecule of Cl2 gas, so one mole of Cl2 gas weighs 70.90 g. ## Measurement and Quantities See the Units Appendix for tables of SI fundamental and derived units. ### Commonly used units in Chemistry #### Conversions (etc) amount of substance mole, mol 1 mol = 6.022×1023 particles concentration (solutions) mole per litre, mol L-1 gram per litre, g L-1 (mol L-1)(g mol-1) = g L-1 (g L-1)/(g mol-1) = mol L-1 electric charge coulomb, C electric current ampere, A electromotive force volt, V energy joule, J kilojoule, kJ 1000 J = 1 kJ density gram per litre, g L-1 mass gram, g kilogram, kg ton 1 tonne = 1000 kg = 1×106 g molar volume litre per mole, L mol-1 22.41 L (S.T.P) 22.47 L (25 °C, 101.325 kPa) pressure kilopascal, kPa atmosphere, atm millimetre of mercury, mmHg 1 atm = 101.324 kPa = 760.0 mmHg temperature kelvin, K degree celsius, °C TK = TC + 273.15 time second, s hour, h 1 h = 3600 s volume litre, L millilitre, mL cubic centimetre, cm3 cubic decimetre, dm3 1 L = 1000 mL = 1 dm3 = 1000 cm3 tera T 1012 giga G 109 mega M 106 kilo k 103 milli m 10-3 micro µ 10-6 nano n 10-9 pico p 10-12	2016-08-31 16:14:58	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 2, "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": 0.6790640950202942, "perplexity": 6588.748177109586}, "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-2016-36/segments/1471982295966.49/warc/CC-MAIN-20160823195815-00153-ip-10-153-172-175.ec2.internal.warc.gz"}
https://www.transtutors.com/questions/short-term-loans-can-be-risky-because-of-points-5-credit-risk-the-need-to-refinance--40485.htm	# Short-term loans can be risky because of: (Points: 5) credit risk the need to refinance and the... 1 answer below » 1. (TCO 6) Short-term loans can be risky because of: (Points: 5) credit risk the need to refinance and the possibility that interest rates will rise liquidity risk inflation risk 2. (TCO 2) A firm has the following accounts: Net patient revenue = $1,500,000 Supply expense =$200,000 Depreciation expense = $100,000 Salaries and benefits =$700,000 Other expenses = $200,000 Net accounts receivable =$150,000 What is the net income for the period? (Points: 5) $150,000$50,000 $500,000$850,000 3. (TCO 3) The advantage of activity-based costing is: (Points: 5) spreads the costs equally to the various activities allocates costs based on identified cost drivers costs less than other methods to implement 1 and 3 4. (TCO 6) A lock-box system is intended: (Points: 5) to speed-up collections to reduce float time to be a long-term financing tool 1 and 2 only 5. (TCO 5) A hospital issues $20 million in bonds and$60 million in equity to finance a new project. Its targeted debt to equity ratio is: (Points: 5) 50% 33% 200% 300% 6. (TCO 5) In order to calculate the net present value (NPV) of a capital investment, we must know: (Points: 5) the net accounting income generated from the investment the initial outlay and the incremental cash flow the rate of return generated on similar investments none of the above is necessary to calculate NPV 7. (TCO 4) All of the following are reasons for using flexible budgets except: (Points: 5) provide more relevant budgeting information to responsibility centers to more accurately determine variances to recognize variances are related to patient volumes ensure that the budget is evenly spread over the 12 month period 8. (TCO 2) Which of the following statements about accounts receivable and inventory is true? (Points: 5) they are both considered current assets they are both considered expenses they are both excluded from current assets they are both considered current liabilities 9. (TCO 7) When a surgical instrument factory increases the number of instruments produced, this will result in: (Points: 5) economies of scale economies of scope synergy entergy 10. (TCO 3) For a radiology department, annual depreciation charges for an MRI would be classified as: (Points: 5) direct and fixed direct and variable indirect and variable indirect and variable 11. (TCO 3) The breakeven point occurs where: (Points: 5) total fixed costs and total revenue intersect revenue minus variable cost minus fixed cost = 0 total profit margin and total costs intersect total variable costs and total revenue intersect total revenue outpaces total avoidable fixed costs 12. (TCO 2) A statement that reports the revenues minus expenses of an entity is called: (Points: 5) Income statement Statement of retained earnings Balance sheet Report of management Statement of cash flows 13. (TCO 1)Which of the following is the best way a hospital can minimize the chances of running afoul of healthcare laws and regulations? (Points: 5) Seek JCAHO accreditation Post EMTALA signs Implement a compliance program Evaluate compliance with HIPAA 1. (TCO 6) Your hospital has the following revenue for the months of July-September: July $3,000,000 August$4,000,000 September $3,000,000. If 30% of the month's revenue is collected in the same month, 40% is collected in the second month and 30% is collected in the third month, how much of July's revenue is collected in August? How much of August's revenue is collected in September? (Points: 15) 2. (TCO 6) A hospital has received an invoice from one of its vendors for$50,000. The terms are 1/10, net 60. Explain the meaning of the credit terms. (Points: 15) 3. (TCO 5) You have been asked to determine the length of time required before a $500,000 deposit will double if the interest rate is 10 percent. Round to the nearest year. (Points: 15) 4. (TCO 3) An imaging center has the following information: Revenue per test:$250 Variable cost per test: 50% of revenue Total fixed costs: $300,000 Estimated number of tests = 3,500 Calculate the per test contribution margin in dollars and as a percentage. (Points: 15) 5. (TCO 5) A newly purchased piece of equipment shows the following: Initial Cost$3,000,000 Estimated yearly cash flow 25 percent of the initial cost Calculate the payback (in years) (Points: 15) 1. (TCO 6) Discuss the sources of cash inflow in cash budgeting: (Points: 40) 2. (TCO 2) A hospital has been consistently performing below its peer group as far as total margin. What are some of the reasons behind that, and what corrective action can management take to ensure performance within industry norms? (Points: 40) 3. (TCO 5) What would determine the optimality of a capital structure (debt/equity mix)? What is the advantage of debt financing for investor-owned entities? (minimum 35 words) (Points: 40) ## Solutions: Looking for Something Else? Ask a Similar Question	2021-05-10 17:56:51	{"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": 0.17816565930843353, "perplexity": 6201.111220521846}, "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/1620243991759.1/warc/CC-MAIN-20210510174005-20210510204005-00235.warc.gz"}
https://www.math-only-math.com/worksheet-on-integers-and-the-number-line.html	# Worksheet on Integers and the Number Line Practice the questions given in the worksheet on integers and the number line. The questions are based on integers and how to find the integers using a number line. I. Using the following number line, fill in the blanks: (i) An integer, on the given number line, is ___________ than every number on its left (ii) An integer, on the given number line, is greater than every number on its ___________. (iii) 4 is greater than -4 implies 4 is on the ___________ of -4. (iv) -5 is ___________ than 3 and 7 is ___________ than -5. (v) -6 is ___________ than -10 and 3 is ___________ than 9. (vi) 8 is ___________ than 0 and -6 is ___________ than -1. (vii) -7 is ___________ than 6 and opposite of -4 is ___________ than opposite of -9. (viii) -12 is ___________ than -8 and -10 is ___________ than opposite of 5. (ix) If +15 represents gain of $15; then +45 represents ___________; and -75 represents ___________. (x) If 30 m below sea level is represented by -30; then -95 represents ___________; and +450 represents ___________. (xi) If fall in temperature by 15° C is denoted by +15, then -115 denotes ___________; and +47 denotes ___________. (xii) Absolute value of +24 is ___________ and absolute value of -35 is ___________. (xiii) \|-11\| = ___________, \|+11\| = ___________ and - \|-11\| = ___________. (xiv) If absolute value of a number = the number itself; then the number is ___________ or ___________ (xv) The opposite of +46 is ___________ and that of -88 is ___________ II. Find the integer, using the number line, which is: (i) 5 more than 3 (ii) 9 less than 4 (iii) 12 more than -4 (iv) 8 less than 3 (v) 7 less than 0 (vi) 4 less than -6 III. State whether the statements are true or false: (i) The smallest integer is 0. (ii) The opposite of -17 is 17 (iii) The opposite of zero is zero. (iv) Every negative integer is smaller than 0. (v) 0 is greater than every positive integer. (vi) Since, zero is neither negative nor positive; it is not an integer. Answers for the worksheet on integers and the number line are given below to check the exact answers of the above questions using number line. Answers: I. (i) greater (ii) left (iii) right (iv) less; greater (v) greater; less (vi) greater; less (vii) less; less (viii) less; less (ix) gain of$45; loss of \$75 (x) 95 m below sea-level; 450 m above sea-level (xi) rise in temperature by 115° C; fall in temperature by 47° C (xii) 24 and 35 (xiii) 11, 11 and -11 (xiv) positive or zero (xv) -46; +88 II. (i) 8 (ii) -5 (iii) 8 (iv) -5 (v) -7 (vi) -10 III. (i) False (ii) True (iii) True (iv) True (v) False (vi) False	2019-03-26 22:32:39	{"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": 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": 0.4134552776813507, "perplexity": 7164.773558950138}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912206677.94/warc/CC-MAIN-20190326220507-20190327002507-00125.warc.gz"}
https://leetcode.ca/2016-01-13-44-Wildcard-Matching/	# Question Formatted question description: https://leetcode.ca/all/44.html Implement wildcard pattern matching with support for '?' and ''. '?' Matches any single character. '' Matches any sequence of characters (including the empty sequence). The matching should cover the entire input string (not partial). The function prototype should be: bool isMatch(const char s, const char p) Some examples: isMatch("aa","a") → false isMatch("aa","aa") → true isMatch("aaa","aa") → false isMatch("aa", "") → true isMatch("aa", "a") → true isMatch("ab", "?") → true isMatch("aab", "cab") → false # Algorithm Pay attention to distinguish it from another Regular Expression Matching question. The functions of the asterisks in the two questions are different. Pay attention to contrast and distinguish. The biggest difficulty of this question is that for the processing of asterisks, it can match any string, which is almost like a hang, that is, before the corresponding position of the asterisk, no matter if there is any string in your s, I have a big asterisk to match. But there is a problem that it cannot handle, that is, once there is a character in p that does not exist in s, then it must not be matched, because the asterisk can only add characters, not eliminate characters, and if there are characters, it will be a star. Once the character string to be matched is determined, the matching situation after the asterisk position is beyond reach. Therefore, the position of the asterisk in the p string is very important. It is represented by jStar, and the position where the asterisk matches in the s string is represented by iStart. Here iStar and jStar are initialized to -1, which means that there is no star by default. Number. Then use two variables i and j to point to the traversed position in the current s string and p string, respectively. Start matching, if i is less than the length of the s string, perform a while loop. • If the current two characters are equal, or the character in p is a question mark, then i and j are increased by 1. • If p[j] is an asterisk, to record the position of the asterisk, jStar is assigned to j, at this time j is incremented by 1, and iStar is assigned to i. • If the current p[j] is not an asterisk and cannot match p[i], the asterisk must be used at this time. If the asterisk has not appeared before, then return false directly, such as s = “aa” and p = “c”, at this time s[0] and p[0] cannot match. Although p[1] is an asterisk, it cannot match still. • If the asterisk has appeared before, such as s = “aa” and p = “c”, when it is found that s[1] and p[1] cannot match, but fortunately before p[0 ] When an asterisk appears, give s[1] to the asterisk of p[0] to match. As for how to know if there is an asterisk before, you can see the role of iStar at this time, because it is initialized to -1, and when it encounters an asterisk, it will be updated to i. Just check the value of iStar, to decide if you can use asterisks to continue your life. Although all the characters in s were matched, the p string must be checked afterwards. At this time, only asterisks are left in the unmatched p string, and no other characters are allowed. The continuous asterisks are filtered out , If j is not equal to the length of p, return false. ### Note 1. Wildcard Matching Implement wildcard pattern matching with support for ‘?’ and ‘’. ‘?’ Matches any single character. ‘’ Matches any sequence of characters (including the empty sequence). 1. Regular Expression Matching Implement regular expression matching with support for ‘.’ and ‘’. ‘.’ Matches any single character. ‘’ Matches zero or more of the preceding element. The function prototypes for both topics are bool isMatch(const char s, const char *p); At first glance, these two questions thought they were the same, but they were not. The first question was actually wildcard matching, which means that the ‘’ sign can match any substring. The second question is actually regular expression matching, that is, the ‘’ sign means that the character before it can appear any number of times (including 0). This is actually the Klin closure described in the Long Book of Compilation Principles. Java	2022-07-04 02:19:59	{"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": 0.48245009779930115, "perplexity": 1697.0150105666096}, "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-2022-27/segments/1656104293758.72/warc/CC-MAIN-20220704015700-20220704045700-00459.warc.gz"}
https://wiki.q-researchsoftware.com/wiki/Visualization_-_Scatter_-_Small_Multiples_Scatter	# Visualization - Scatter - Small Multiples Scatter A panel of scatterplots, grouping the plots based on a selected variable The Small Multiples - Scatterplot feature creates a panel of scatterplots, grouping the plots based on a selected variable. ## Examples The following example shows the relationship between the price of a diamond and its mass (carat). The quality of the cut is color-coded and the depth of the diamond is represented by the size of the dot. The plots are divided into small multiples based on the clarity of the diamond. ### Create a Small Multiples Scatterplot in Displayr 1. Go to Insert > Visualization > Small Multiples > Scatterplot 2. Under Inputs > DATA SOURCE > X coordinates, select the variable you wish to be displayed horizontally 3. Under Inputs > DATA SOURCE > X coordinates, select the variables you wish to be displayed vertically 4. [OPTIONAL] Under Inputs > DATA SOURCE > Sizes and Colors, select variables to alter the sizes and colors of the dots. 5. Under Inputs > DATA SOURCE > Groups, select the variable used to divide your data into panels. ## Object Inspector Options The following is an explanation of the options available in the Object Inspector for this specific visualization. Refer to Visualization Options for general chart formatting options. ### Inputs #### DATA SOURCE The expected input format to create a Small Multiples - Scatterplot is similar to for Scatterplots with multiple groups (colors variables treated as categories). In fact you should be able to toggle between these charts. The last column of the linked or pasted table will be converted to categories and used to separate the data into different panels. The remaining columns are used as the variables to control 1) x-coordinates, 2) y-coordinates, 3) scatter point sizes and 4) scatter point colors if they are present. If the data is input as variables, the Groups variable will be used to separate the data into different panels. In scatterplot small multiples, the colors variable is always treated as numeric. #### DATA MANIPLULATION Input data contains y-values in multiple columns. When this is selected, the data in each column of the input table is treated as y-coordinates in a separate panel. The x-coordinates will be taken from the rownames of the table. What are small multiples? The psychology of small multiples ## Code Visualization - Scatter - Scatter { "formChartType": "Scatter", "formStackSeries": true, "formSmallMultiples": true, "formAsPercentages": false, "formScatterLabelType": "As hover text" }	2022-09-27 23:57:51	{"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": 0.3527151644229889, "perplexity": 2582.867005089283}, "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-2022-40/segments/1664030335059.31/warc/CC-MAIN-20220927225413-20220928015413-00411.warc.gz"}
https://energypedia.info/wiki/Operation_of_a_Biogas_Plant	Yeah! Your beloved energypedia has a new look and design. We have updated the software so that the new energypedia is responsive and more user-friendly. Have a look at the platform and if you encounter any bugs or page distortions, please send them to us at info@energypedia.info. # Overview The economic efficiency of a properly planned biogas plant is determined by the availability and capacity utilisation of the process as a whole. Key factors are the functionality and operational reliability of the technology employed, and consistently high degradation performance within the biological process. As the operation of technical facilities is subject to inevitable malfunctions, appropriate tools must be on hand in order to detect such malfunctions and identify and rectify the fault. Process control is always performed in interaction with the personnel, although the degree of automation can vary extremely widely. If monitoring and control algorithms are automated, the benefits are that the system is constantly available and a degree of independence from expert personnel is achieved. The remote transmission of data also decouples the need for staff presence at the plant from process monitoring. The following sections first examine the measured variables that can be used to observe the biological process. The descriptions relate to wet fermentation plants. Biogas production rate. # Operating Parameters The primary purpose of the operating parameters of a biogas plant is to describe the load situation. It is only during the start-up process that the parameters can help with plant control in terms of achieving a slow, steady rise. Normally, most attention is paid to the organic loading rate. In the case of plants with large volumes of liquid on the input side and a low content of degradable organic material (slurry plants), the retention time is more important. The aim is to obtain optimum degradation performance at acceptable economic cost (rentention time, sizing of the digester). In this regard the organic loading rate (OLR) is a crucial operating parameter. It indicates how many kilograms of volatile solids (VS, or organic dry matter) can be fed into the digester per m3 of working volume per unit of time. The organic loading rate is expressed as kg VS/(m3 · d). The organic loading rate can be specified for each stage (gas-tight, insulated and heated vessel), for the system as a whole (total working volumes of all stages) and with or without the inclusion of material recirculation. Changing the reference variables can lead to sometimes widely differing results for the organic loading rate of a plant. ⇒ Organic loading rate (OLR) ${\displaystyle OLR={\frac {mc}{V_{\text{R}}100}}}$ [kgVSm-3d-1] m = amount of substrate added per unit of time [kg/d]; c = concentration of organic matter (volatile solids) [%VS]; VR = reactor volume [m3]) ### Hydraulig Retention Time (HRT) Another relevant parameter for deciding on the size of vessel is the hydraulic retention time (HRT). This is the length of time for which a substrate is calculated to remain on average in the digester until it is discharged. Calculation involves determining the ratio of the reactor volume (VR) to the volume of substrate added daily. The hydraulic retention time is expressed in days. ⇒Hydraulic Retention Time ${\displaystyle HRT={\frac {V_{\text{R}}}{V}}}$ [d] VR= reactor volume [m3]; V= volume of substrate added daily [m3/d]) The actual retention time may differ from this. There is a close correlation between the organic loading rate and the hydraulic retention time. If the composition of the substrate is assumed to remain the same, as the organic loading rate rises more input is added to the digester, and the retention time is consequently shortened. In order to be able to maintain the digestion process, the hydraulic retention time must be chosen such that constant replacement of the reactor contents does not flush out more microorganisms than can be replenished by new growth during that time (the doubling rate of some methanogenic archaea, for example, is 10 days or more). It should also be borne in mind that with a short retention time the microorganisms will have little time to degrade the substrate and consequently the gas yield will be inadequate. It is important to adapt the retention time to the specific decomposition rate of the substrates. # Disturbance Management of Biogas Plants The Problem of Scum If there is heavy gas release from the inlet but not enough gas available for use, a thick scum layer is most likely the reason. Often the gas pressure does not build up because of the continuous gas release through the inlet for weeks. There is a danger of blocking the gas pipe by rising scum because of daily feeding without equivalent discharge. The lid (or man-hole) must be opened or the floating drum removed and scum is to be taken out by hand. # Operational Reliability ## Occupational Safety and Plant Safety Biogas is a gas mixture consisting of methane (50-75 vol. %), carbon dioxide (20-50 vol. %), hydrogen sulphide (0.01-0.4 vol. %) and other trace gases. In certain concentrations, biogas in combination with atmospheric oxygen can form an explosive atmosphere, which is why special plant safety regulations have to be observed in the construction and operation of a biogas plant. There are also other hazards, such as the risk of asphyxiation or poisoning, as well as mechanical dangers (e.g. risk of crushing by drives). The employer or biogas plant operator is obliged to identify and evaluate the hazards associated with the biogas plant, and if necessary to take appropriate measures. The 'Sicherheitsregeln für Biogasanlagen' (Safety Rules for Biogas Systems) issued by the Bundesverband der landwirtschaftlichen Berufsgenossenschaften (German Agricultural Occupational Health and Safety Agency) provide a concise summary of the key aspects of safety relevant to biogas plants. The safety rules explain and substantiate the safety requirements in terms of the operating procedures relevant to § 1 of the accident prevention regulations 'Arbeitsstätten, bauliche Anlagen und Einrichtungen' (Workplaces, Buildings and Facilities) (VSG 2.1). issued by the Agricultural Occupational Health and Safety Agency. They also draw attention to other applicable codes of practice. This section is intended to provide an overview of the potential hazards during operation of a biogas plant and raise awareness of them accordingly. The latest versions of the respective regulations constitute the basis for the hazard assessments and the associated safety-related aspects of plant operation. ### Fire and Explosion Hazard As mentioned in the previous section, under certain conditions biogas in combination with air can form an explosive gas mixture. It should be borne in mind that although there is no risk of explosion above these limits it is still possible for fires to be started by naked flames, sparks from switching electrical equipment or lightning strikes. During the operation of biogas plants, therefore, it must be expected that potentially explosive gas-air mixtures are liable to form and that there is an increased risk of fire, especially in the immediate vicinity of digesters and gas tanks. Depending on the probability of the presence of an explosive atmosphere, according to BGR 104 – Explosion Protection Rules the various parts of the plant are divided into categories of hazardous areas ('Ex zones'), within which the relevant signs must be prominently displayed and appropriate precautionary and safety measures taken. Zone 0 In areas classified as zone 0, an explosive atmosphere is present constantly, over long periods, or most of the time. Normally, however, no such zones are found in biogas plants. Not even a fermentation tank/digester is classified in this category. Zone 1 Zone 1 describes areas in which an explosive atmosphere can occasionally form during normal operation. These are areas in the immediate vicinity of manholes accessing the gas storage tank or on the gas-retaining side of the fermentation tank, and in the vicinity of blow-off systems, pressure relief valves or gas flares. The safety precautions for zone 1 must be put in place within a radius of 1 m (with natural ventilation) around these areas. This means that only resources and explosion-protected equipment with zone 0 and zone 1 ratings may be used in this area. As a general rule, the operations-related release of biogas in enclosed spaces should be avoided. If it is possible that gas will be released, however, zone 1 is extended to include the entire space . Zone 2 In these areas it is not expected that explosive gas-air mixtures will occur under normal circumstances. If this does in fact happen, it can be assumed that it will do so only rarely and not for a lengthy period of time (for example during servicing or in the event of a fault). This applies to manholes, for example, and the interior of the digester, and in the case of gas storage tanks the immediate vicinity of aeration and ventilation openings. The measures applicable to zone 2 must be implemented in these areas in a radius of 1 to 3 m. In the areas subject to explosion hazard (zones 0-2), steps must be taken to avoid ignition sources in accordance with BGR 104, section E2. Examples of ignition sources include hot surfaces (turbochargers), naked flames or sparks generated by mechanical or electrical means. In addition, such areas must be identified by appropriate warning signs and notices. ### Danger of Poisoning and Asphyxiation The release of biogases is a natural process, as is well known, so it is not exclusively restricted to biogas plants. In animal husbandry, in particular, time and again in the past there have been accidents, some of them fatal, in connection with biogenic gases (for example in slurry pits and fodder silos etc.). If biogas is present in sufficiently high concentrations, inhalation can produce symptoms of poisoning or asphyxiation, and can even prove fatal. It is particularly the hydrogen sulphide (H2S) content of non-desulphurised biogas that is highly toxic, even in low concentrations. In addition, especially in enclosed or low-level spaces, asphyxiation can occur as a result of the displacement of oxygen by biogas. Although biogas is lighter than air, with a relative density (D) of roughly 1.2 kg per m3, it tends to segregate. In this process the heavier carbon dioxide (D = 1.98 kg/m3) collects close to floor level, while the lighter methane (D =0.72 kg/m3) rises. For these reasons it is essential that adequate ventilation is provided at all times in enclosed spaces, for example enclosed gas storage tanks. Furthermore, personal protective equipment (e.g. gas alarms, respiratory protection etc.) must be worn in potentially hazardous areas (digesters, maintenance shafts, gas storage areas etc.). ### Other Potential Accident Risks In addition to the sources of danger described above there are also other potential accident sources, such as the risk of falling from ladders or falling into charging holes (solids metering equipment, feed funnels, maintenance shafts etc.). In these cases it must be ensured that falling into such openings is prevented by covers (hatches, grids etc.) or by installing them at a sufficient height (> 1.8 m) [5-6]. Moving plant parts (agitator shafts, worms etc.) are also potential danger points, which must be clearly identified by appropriatesignage. Fatal electric shocks can occur in and around combined heat and power units as a result of incorrect operation or faults, because the units generate electrical power at voltages of several hundred volts and with currents measured in hundreds of amperes. The same danger also applies to agitators, pumps, feed equipment etc. because these also operate with high levels of electrical power. The heating and cooling systems of a biogas plant (radiator, digester heater, heat exchanger etc.) also present a risk of scalding in the event of malfunctions. This also applies to parts of the CHP unit and any emergency systems that may be installed (e.g. gas flares). In order to prevent accidents of this type, clearly visible warning signs must be displayed at the appropriate parts of the plant and the operating personnel must be instructed accordingly. ## 7 Rules to Run a Digester 1. Maintain constant feed rate and composition 2. Avoid overfeeding and abrupt changes 3. Avoid foaming, don´t: add a substrate with a low pH value, add substrate with a considerably high protein content, 4. Mix as much as necessary and as little as possible: Keep in mind: all in = all out 5. Maintain continuous mixing 6. Choose an appropriate temperature 7. Keep the temperature constant	2021-09-20 20:39:24	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 2, "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": 0.5308539271354675, "perplexity": 2389.292203888441}, "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/1631780057091.31/warc/CC-MAIN-20210920191528-20210920221528-00126.warc.gz"}
https://mathematica.stackexchange.com/questions/273318/find-min-and-max-indices-of-highest-and-lowest-elements-in-sorted-list/273323	# Find min and max indices of highest and lowest elements in sorted list I have a list of integers of the form Nmax = 17000; list1 = Join[Table[0, 560], FoldList[Min[Nmax, Plus[#1, #2]] &, 0, RandomChoice[{0, 1, 2}, Nmax + 8100]]]; (the numbers are kind of arbitrary--I just wanted to construct a list with many 0's at first, a sorted middle and then a large plateau above some fixed value, here Nmax). My question is the following: I would like to find the indices corresponding to the last 0 and the first Nmax. Or, equivalently, I want to find the indices between which the list is not 0 or Nmax, which corresponds to the maximum index whose element is 0 and the minimum element whose value is Nmax. This can be accomplished pretty simply with the following code: {lower, upper} = {1 + (Last@Position[list1, 0])[[1]], -1 + (First@ Position[list1, Nmax])[[1]]} list2=list1[[lower;;upper]] However, I would like to run this code on many thousands of different lists, and the above code seems to take longer than necessary to find lower and upper, since Position makes no use of the fact that list1 is sorted. Is there a way to get these two indices in a more efficient way? • The answer provided by user293787 is optimal for generic sorted lists. If the list is not sorted then, as it seems you are ultimately interested in the elements between the two plateaus rather than their positions, you could use Pick[list1, Unitize[list1(Nmax - list1)], 1] which is about 20 times faster than list1[[Position[list1, 0][[-1, 1]] + 1 ;; Position[list1, Nmax][[1, 1]] - 1]] for the list1 used in the benchmarks. As I understand, the speed of the former is due to vectorization as Unitize is applied to the entire list at once which allows the computer to use multi-threading. Sep 12 at 23:40 OP may want to use binary search. There are many posts here on Mathematica SE relating to binary search, including this which also contains compiled code. See also this. Code. To find the highest index with entry zero, one can use ( list must be a sorted list of integers >= 0 ) maxindex0[list_]:=Module[{l=0,r=Length[list]+1}, While[l<r-1,With[{m=Floor[(l+r)/2]}, If[list[[m]]==0,l=m,r=m]]]; l]; Example. SeedRandom[1]; example=Map[Max[0,#]&,Sort[RandomInteger[{-100,1000},1000000]]]; Then RepeatedTiming[(Last@Position[example,0])[[1]]] {0.0671354, 92178} RepeatedTiming[maxindex0[example]] {0.0000641912, 92178} So OPs code takes about 50 milliseconds, maxindex0 takes about 0.05 milliseconds. • Exactly what I needed, and the extra resources were really helpful for generalizing the code for any target value. Thanks! Sep 12 at 21:07 • @az123p Great that you found a way to generalize it. There is also a resource function in the wolfram repository here. There also seems to be a package that does this but according to this answer the package is deprecated and cause shadowing of symbols but it might still work. The package is Combinatorica but the resource function is probably better (from wolfram staff and from 2021) Sep 13 at 8:28 • @userrandrand thanks for the link! I'll test it out to compare to my current result. Sep 13 at 23:58 ## Graphical explanation of what I understood OP's wants: list1= {0, 0, 0, 0, 0, 1, 1, 3, 4, 6, 8, 8, 10, 11, 13, 15, 15, 15, 16, 16, \ 17, 18, 20, 21, 21, 21, 21, 21, 21} The list above was generated using OP's code. Plotting the list: ListPlot[{list1, {{5, 0}, {24, Nmax}}}, PlotMarkers -> {Automatic, 0.04}, ImageSize -> Small] With a list that has more points: SeedRandom[42]; max = 1000; size = 100; test = Clip[ Sort@RandomInteger[{-Floor[2max], Floor[2max]}, size], {0, max}]; As I understand, the abscissas of the two squares are the values requested by OP. That is, the position of the last 0 of the initial plateau and the position of the first point in the last plateau (the first Nmax using OP's notation). ## If speed is not much of a concern: The solution proposed by @Syed is in my opinion rather convenient. Here is another possibility: last0=LengthWhile[list, # == 0 &] firstNmax= First@FirstPosition[list, Nmax] ## If the length of one plateau is relatively small (smaller than Log2[Length[list]]) and so your case is not generic, then the following can be of use. TL;DR: If the length of the first plateau is small use LengthWhile. If the length of the last plateau is small, then reverse the order of the elements in the list and use LengthWhile As you would like something that is fast, the Position function might not be the best option as it will search the entire list. That is unneeded in your case as your lists are sorted. For example, it is clear that, starting from the first element of the list list[[1]], it is no longer necessary to scan the list after the first encountered non zero element. Test list : SeedRandom[42]; max = 1000; size = 100; ratio = 1; test = Clip[ Sort@RandomInteger[{-Floor[2max], Floor[2maxratio]}, size], {0, max}]; Changing "ratio" you may change the ratio between the lengths of the first and last plateaus. To check the validity of the codes below you may add indices to the test list and maybe decrease the size : {test,Range[Length[test]]}//Transpose {{test[[1]],1},{test[[3]],2},...} The fastest method will depend on the lengths of the plateaus. If we subdivide the previous plot in 3 parts: Big Plateau 1 (BP1), Rocky Hill (RH), Big Plateau 2 (BP2) Then, for example, it could be advantageous to scan the list backwards if Length[BP2] is a lot smaller than Length[BP1] or Length[RH]. Specifically, you have 3 scenarios: 1. If : you already know beforehand that Length[BP1]/Length[list] is very small (smaller than Log2[Length[list]]) Then : you can obtain the last 0 efficiently with last0=LengthWhile[list, # == 0 &] (* Remark : using FirstPosition[list, _?(#!=0 &)] or Position[list1, 0][[-1, 1]] takes a lot more time in this case) 2. If : you already know beforehand that Length[BP2]/Length[list] is very small (smaller than Log2[Length[list]]) Then: you can obtain the first Nmax efficiently with listLength=Length[list1] firstNmax = listLength - LengthWhile[list1[[-1 ;; 1 ;; -1]], # == Nmax &]+1 3. Else: In the the generic case, a C-Compiled version of the method provided by @user293787 will likely be optimal. # BenchMarking the methods proposed so far: The following benchmarks are for the generic case where the length of both plateaus are large. If one of the plateaus is relatively tiny use LengthWhile (see discussion of the previous section). For a more complete comparison I will include a C compiled function (you may remove CompilationTarget -> "C" if you do not have a Compiler with Mathematica). The function below does a forward scan and works with an integer list. As discussed above, if the last plateau is shorter than the first plateau it can be advantageous to do a backward scan instead : edgeplateau = Compile[{{list, _Integer, 1}, {Nmax, _Integer}}, Module[{index1 = 0, index2 = 0, a = 0}, While[a == 0, index1 += 1; a = list[[index1]]]; index2 = index1; While[a < Nmax, index2 += 1; a = list[[index2]]]; {index1-1, index2}], CompilationTarget -> "C"] output of edgeplateau: {last 0, first Nmax} ### Test List: Nmax = 200000; list1 = Join[Table[0, 3000000], FoldList[Min[Nmax, Plus[#1, #2]] &, 0, RandomChoice[{0, 1, 2}, Nmax + 3000000]]]; Properties of test list: the length of the first plateau is roughly the same as the last plateau and rocky hill climb between the two is relatively quite thin (like a cliff). ### Methods for finding the last 0: AbsoluteTiming[last0=First@FirstPosition[list1, _?(#!=0 &)]-1;] {0.799981, Null} Below: Method by @UlrichNeumann AbsoluteTiming[last0 = Position[list1, 0][[-1, 1]];] {0.631989, Null} AbsoluteTiming[LengthWhile[list1, # == 0 &];] {0.588371, Null} AbsoluteTiming[maxindex0[list1];] {0.000104, Null} C compiled maxindex0 with where the list elements are expected to be integers {0.000043, Null} ### Methods for finding the first Nmax: Position[list1, Nmax][[1, 1]]; // AbsoluteTiming {0.636139, Null} FirstPosition[list1, Nmax]; // AbsoluteTiming {0.164197, Null} last0 + FirstPosition[list1[[last0 ;;]], Nmax] - 1; // AbsoluteTiming {0.106541, Null} LengthWhile[list1, # != Nmax &] + 1; // AbsoluteTiming {0.782357, Null} last0 + LengthWhile[list1[[last0 ;;]], # != Nmax &]; // AbsoluteTiming {0.055123, Null} An equivalent of maxindex0 to find the first Nmax will have similar speed when compared to the timing of maxindex0 ### Methods for finding both the last 0 and the first Nmax: * @Syed's answer * {Last@First@#, First@Last@#} &@PositionIndex[list1] // AbsoluteTiming {0.148351, Null} compiled edgeplateau AbsoluteTiming[edgeplateau[list1, Nmax];] {0.026304, Null} uncompiled edge plateau {1.77486, Null} • I am rather surprised how fast the PositionIndex method is. I remembered a post here that was made around 10 years ago or something where people said that PositionIndex is very slow. I suppose it has changed since. Sep 12 at 13:18 • Thanks for the thorough analysis! I'm wondering why it takes so much longer to find the first Nmax than the last 0. For finding the first Nmax, could another option be to consider list1-Nmax, so that all of the Nmax values go to 0, and apply a similar code as is used with the first variable? Maybe then finding the first Nmax would not be much slower than finding the last 0. Sep 12 at 14:25 • @az123p I am not sure I understood. Which methods are you comparing ?. If you are referring to the method maxindex0 then the lack of a fast method for the first Nmax is just due to the fact that the benchmark did not include an equivalent version of maxindex0 for that case. Sep 12 at 14:36 • Oh, I see! Sorry, I misunderstood. Sep 12 at 14:43 • @az123p sorry i edited my answer but it's mainly an attempt to shorten it and make things more clear. The content is more or less the same as in the previous version. Sep 12 at 16:06 last0 = Position[list1, 0][[-1, 1]] (* 562) firstN = Position[list1, Nmax][[1, 1]](17626*) SeedRandom[1]; Nmax = 17000; list1 = Join[Table[0, 560], FoldList[Min[Nmax, Plus[#1, #2]] &, 0, RandomChoice[{0, 1, 2}, Nmax + 8100]]]; {Last@First@#, First@Last@#} &@PositionIndex[list1] {561, 17629}	2022-12-03 08:08:14	{"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": 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": 0.3170805275440216, "perplexity": 1665.9980664032535}, "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/1669446710926.23/warc/CC-MAIN-20221203075717-20221203105717-00861.warc.gz"}
http://www.koreascience.or.kr/article/ArticleFullRecord.jsp?cn=E1BMAX_2009_v46n6_1213	ON THE STRUCTURE OF MINIMAL SUBMANIFOLDS IN A RIEMANNIAN MANIFOLD OF NON-NEGATIVE CURVATURE Title & Authors ON THE STRUCTURE OF MINIMAL SUBMANIFOLDS IN A RIEMANNIAN MANIFOLD OF NON-NEGATIVE CURVATURE Yun, Gab-Jin; Kim, Dong-Ho; Abstract Let M$\small{^n}$ be a complete oriented non-compact minimally immersed submanifold in a complete Riemannian manifold N$\small{^{n+p}}$ of nonnegative curvature. We prove that if M is super-stable, then there are no non-trivial L$\small{^2}$ harmonic one forms on M. This is a generalization of the main result in [8]. Keywords minimal submanifold;super-stable minimal submanifold;L2 harmonic form; Language English Cited by References 1. H.-D. Cao, Y Shen, and S. Zhu, The structure of stable minimal hypersurfaces in \$R^{n+1}\$, Math. Res. Lett. 4 (1997), no. 5, 637-644 2. J. Dodziuk, \$L^2\$ harmonic forms on complete manifolds, in Seminar on Differential Geometry, Princeton University Press, Princeton N. J., 1982, 291-302 3. D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33 (1980), no. 2, 199-211 4. H. B. Lawson, Lectures on Minimal Submanifolds. Vol. I, Publish or Perish, Inc., Wilmington, Del., 1980 5. P.-F. Leung, An estimate on the Ricci curvature of a submanifold and some applications, Proc. Amer. Math. Soc. 114 (1992), no. 4, 1051-1061 6. P. Li, Curvature and function theory on Riemannian manifolds, Surveys in differential geometry, 375-432, Surv. Differ. Geom., VII, Int. Press, Somerville, MA, 2000 7. P. Li, Differential geometry via harmonic functions, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 293-302, Higher Ed. Press, Beijing, 2002 8. R. Miyaoka, \$L^2\$ harmonic 1-forms on a complete stable minimal hypersurface, Geometry and global analysis (Sendai, 1993), 289-293, Tohoku Univ., Sendai, 1993 9. K. Seo, Rigidity of minimal submanifolds with flat normal bundle, Commun. Korean Math. Soc. 23 (2008), no. 3, 421-426 10. J. Spruck, Remarks on the stability of minimal submanifolds of \$R^n\$, Math. Z. 144 (1975), no. 2, 169-174 11. S. Tanno, \$L^2\$ harmonic forms and stability of minimal hypersurfaces, J. Math. Soc. Japan 48 (1996), no. 4, 761-768 12. Q. Wang, On minimal submanifolds in an Euclidean space, Math. Nachr. 261/262 (2003), 176-180 13. S.-T. Yau, Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J. 25 (1976), no. 7, 659-670 14. G. Yun, Total scalar curvature and \$L^2\$ harmonic 1-forms on a minimal hypersurface in Euclidean space, Geom. Dedicata 89 (2002), 135-141	2018-04-26 11:11:03	{"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 3, "/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": 0.7300692200660706, "perplexity": 852.5327367656483}, "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/1524125948126.97/warc/CC-MAIN-20180426105552-20180426125552-00425.warc.gz"}
https://brilliant.org/wiki/if-the-derivative-of-fx-is-fx-is-it-true-that-int/?subtopic=sequences-and-limits&chapter=common-misconceptions-calculus	# If F(x) is the antiderivative of f(x), is it true that $\int_a^b$f(x)dx=F(b)-F(a)? Cite as: If F(x) is the antiderivative of f(x), is it true that $\int_a^b$f(x)dx=F(b)-F(a)?. Brilliant.org. Retrieved from https://brilliant.org/wiki/if-the-derivative-of-fx-is-fx-is-it-true-that-int/ ×	2022-06-27 02:38:21	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 27, "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": 0.717947781085968, "perplexity": 1974.4112733818902}, "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/1656103324665.17/warc/CC-MAIN-20220627012807-20220627042807-00525.warc.gz"}
https://proofwiki.org/wiki/Numerator_of_p-1th_Harmonic_Number_is_Divisible_by_Prime_p/Proof_2	Numerator of p-1th Harmonic Number is Divisible by Prime p/Proof 2 Theorem Let $p$ be an odd prime. Consider the harmonic number $H_{p - 1}$ expressed in canonical form. The numerator of $H_{p - 1}$ is divisible by $p$. Proof $x^{\overline p} \equiv x^p - x$ $\displaystyle \left[{p \atop k}\right] \equiv \delta_{k p} - \delta _{k 1}$ where: $\displaystyle \left[{p \atop k}\right]$ denotes an unsigned Stirling number of the first kind $\delta$ is the Kronecker delta. The result follows from Harmonic Number as Unsigned Stirling Number of First Kind over Factorial.	2020-02-20 11:02:04	{"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": 0.9675959348678589, "perplexity": 429.590368629513}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875144722.77/warc/CC-MAIN-20200220100914-20200220130914-00308.warc.gz"}
http://repozitorij.upr.si/Iskanje.php?type=napredno&lang=eng&stl0=KljucneBesede&niz0=Schur+ring	# Search the repository Query: search in TitleAuthorAbstractKeywordsFull textYear of publishing ANDORAND NOT search in TitleAuthorAbstractKeywordsFull textYear of publishing ANDORAND NOT search in TitleAuthorAbstractKeywordsFull textYear of publishing ANDORAND NOT search in TitleAuthorAbstractKeywordsFull textYear of publishing Work type: All work types Habilitation (m4) Specialist thesis (m3) High school thesis (m6) Bachelor work * (dip) Master disertations * (mag) Doctorate disertations * (dok) Research Data or Corpuses (data) * old and bolonia study programme Language: All languagesSlovenianEnglishGermanCroatianSerbianBosnianBulgarianCzechFinnishFrenchGerman (Austria)HungarianItalianJapaneseLithuanianNorwegianPolishRussianSerbian (cyrillic)SlovakSpanishSwedishTurkishUnknown Search in: RUP FAMNIT - Faculty of Mathematics, Science and Information Technologies FHŠ - Faculty of Humanities FM - Faculty of Management FTŠ Turistica - Turistica – College of Tourism Portorož FVZ - Faculty of Health Sciences IAM - Andrej Marušič Institute PEF - Faculty of Education UPR - University of PrimorskaCOBISS Fakulteta za humanistične študije, Koper Fakulteta za management Koper in Pedagoška fakulteta Koper Fakulteta za vede o zdravju, Izola Knjižnica za tehniko, medicino in naravoslovje, Koper Turistica, Portorož Znanstveno-raziskovalno središče Koper Options: Show only hits with full text Reset 1 - 2 / 21 1.Decomposition of skew-morphisms of cyclic groupsIstván Kovács, Roman Nedela, 2011, original scientific articleAbstract: A skew-morphism of a group ▫$H$▫ is a permutation ▫$\sigma$▫ of its elements fixing the identity such that for every ▫$x, y \in H$▫ there exists an integer ▫$k$▫ such that ▫$\sigma (xy) = \sigma (x)\sigma k(y)$▫. It follows that group automorphisms are particular skew-morphisms. Skew-morphisms appear naturally in investigations of maps on surfaces with high degree of symmetry, namely, they are closely related to regular Cayley maps and to regular embeddings of the complete bipartite graphs. The aim of this paper is to investigate skew-morphisms of cyclic groups in the context of the associated Schur rings. We prove the following decomposition theorem about skew-morphisms of cyclic groups ▫$\mathbb Z_n$▫: if ▫$n = n_{1}n_{2}$▫ such that ▫$(n_{1}n_{2}) = 1$▫, and ▫$(n_{1}, \varphi (n_{2})) = (\varphi (n_{1}), n_{2}) = 1$▫ (▫$\varphi$▫ denotes Euler's function) then all skew-morphisms ▫$\sigma$▫ of ▫$\mathbb Z_n$▫ are obtained as ▫$\sigma = \sigma_1 \times \sigma_2$▫, where ▫$\sigma_i$▫ are skew-morphisms of ▫$\mathbb Z_{n_i}, \; i = 1, 2$▫. As a consequence we obtain the following result: All skew-morphisms of ▫$\mathbb Z_n$▫ are automorphisms of ▫$\mathbb Z_n$▫ if and only if ▫$n = 4$▫ or ▫$(n, \varphi(n)) = 1$▫.Found in: ključnih besedahSummary of found: ...groups in the context of the associated Schur rings. We prove the following decomposition theorem...Keywords: cyclic group, permutation group, skew-morphism, Schur ringPublished: 15.10.2013; Views: 1935; Downloads: 55 Full text (0,00 KB) 2.CI-property for decomposable Schur rings over an Abelian groupIstván Kovács, Grigory Ryabov, 2019, original scientific articleFound in: ključnih besedahKeywords: isomorphism, CI-group, Schur ringPublished: 17.12.2018; Views: 216; Downloads: 45 Full text (0,00 KB) Search done in 0 sec.	2020-01-29 14:56:21	{"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": 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": 0.5346265435218811, "perplexity": 12271.297000740411}, "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-2020-05/segments/1579251799918.97/warc/CC-MAIN-20200129133601-20200129163601-00411.warc.gz"}
http://www.inknowvation.com/sbir/awards/navy-2010-compact-low-cost-isotropic-atomic-magnetometer	A Compact, Low-Cost Atomic Magnetometer Project Summary Program: SBIR Agency: Navy Phase I Year: 2010 Phase II Year: 2012 Phase I Dollars: Phase II Dollars: Principal Investigator: Phase I Abstract Modern atomic magnetometer technology is poised to deliver total magnetic field measurements in a sensor that is dramatically less expensive, smaller, and lower-power relative to currently-available ..... Phase II Abstract We propose to build prototype magnetometers with sensitivity in the range 1--10 $\mathrm{pT/Hz^{1/2}}$ and unit cost of \\$2,000, meeting the need for a low-cost system with performance comparable to .....	2014-07-31 01:19:33	{"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": 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": 0.2598891258239746, "perplexity": 12449.208583619946}, "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-2014-23/segments/1406510272256.16/warc/CC-MAIN-20140728011752-00456-ip-10-146-231-18.ec2.internal.warc.gz"}
https://kops.uni-konstanz.de/handle/123456789/59634	## A new method for characterising shared space use networks using animal trapping data 2022 ##### Authors Wanelik, Klara M. Journal article Published ##### Published in Behavioral Ecology and Sociobiology ; 76 (2022), 9. - 127. - Springer. - ISSN 0340-5443. - eISSN 1432-0762 ##### Abstract Studying the social behaviour of small or cryptic species often relies on constructing networks from sparse point-based observations of individuals (e.g. live trapping data). A common approach assumes that individuals that have been detected sequentially in the same trapping location will also be more likely to have come into indirect and/or direct contact. However, there is very little guidance on how much data are required for making robust networks from such data. In this study, we highlight that sequential trap sharing networks broadly capture shared space use (and, hence, the potential for contact) and that it may be more parsimonious to directly model shared space use. We first use empirical data to show that characteristics of how animals use space can help us to establish new ways to model the potential for individuals to come into contact. We then show that a method that explicitly models individuals’ home ranges and subsequent overlap in space among individuals (spatial overlap networks) requires fewer data for inferring observed networks that are more strongly correlated with the true shared space use network (relative to sequential trap sharing networks). Furthermore, we show that shared space use networks based on estimating spatial overlap are also more powerful for detecting biological effects. Finally, we discuss when it is appropriate to make inferences about social interactions from shared space use. Our study confirms the potential for using sparse trapping data from cryptic species to address a range of important questions in ecology and evolution. ##### Subject (DDC) 570 Biosciences, Biology ##### Cite This ISO 690WANELIK, Klara M., Damien R. FARINE, 2022. A new method for characterising shared space use networks using animal trapping data. In: Behavioral Ecology and Sociobiology. Springer. 76(9), 127. ISSN 0340-5443. eISSN 1432-0762. Available under: doi: 10.1007/s00265-022-03222-5 BibTex @article{Wanelik2022-09metho-59634, year={2022}, doi={10.1007/s00265-022-03222-5}, title={A new method for characterising shared space use networks using animal trapping data}, number={9}, volume={76}, issn={0340-5443}, journal={Behavioral Ecology and Sociobiology}, author={Wanelik, Klara M. and Farine, Damien R.}, note={Article Number: 127} } RDF <rdf:RDF xmlns:dcterms="http://purl.org/dc/terms/" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:bibo="http://purl.org/ontology/bibo/" xmlns:dspace="http://digital-repositories.org/ontologies/dspace/0.1.0#" xmlns:foaf="http://xmlns.com/foaf/0.1/" xmlns:void="http://rdfs.org/ns/void#" xmlns:xsd="http://www.w3.org/2001/XMLSchema#" > <dcterms:isPartOf rdf:resource="https://kops.uni-konstanz.de/server/rdf/resource/123456789/28"/> <dcterms:abstract xml:lang="eng">Studying the social behaviour of small or cryptic species often relies on constructing networks from sparse point-based observations of individuals (e.g. live trapping data). A common approach assumes that individuals that have been detected sequentially in the same trapping location will also be more likely to have come into indirect and/or direct contact. However, there is very little guidance on how much data are required for making robust networks from such data. In this study, we highlight that sequential trap sharing networks broadly capture shared space use (and, hence, the potential for contact) and that it may be more parsimonious to directly model shared space use. We first use empirical data to show that characteristics of how animals use space can help us to establish new ways to model the potential for individuals to come into contact. We then show that a method that explicitly models individuals’ home ranges and subsequent overlap in space among individuals (spatial overlap networks) requires fewer data for inferring observed networks that are more strongly correlated with the true shared space use network (relative to sequential trap sharing networks). Furthermore, we show that shared space use networks based on estimating spatial overlap are also more powerful for detecting biological effects. Finally, we discuss when it is appropriate to make inferences about social interactions from shared space use. Our study confirms the potential for using sparse trapping data from cryptic species to address a range of important questions in ecology and evolution.</dcterms:abstract> <dc:date rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2023-01-05T08:11:25Z</dc:date> <dc:language>eng</dc:language> <bibo:uri rdf:resource="https://kops.uni-konstanz.de/handle/123456789/59634"/> <dc:creator>Wanelik, Klara M.</dc:creator> <dcterms:title>A new method for characterising shared space use networks using animal trapping data</dcterms:title> <dcterms:available rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2023-01-05T08:11:25Z</dcterms:available> <dc:contributor>Wanelik, Klara M.</dc:contributor> <dcterms:hasPart rdf:resource="https://kops.uni-konstanz.de/bitstream/123456789/59634/1/Wanelik_2-kmiq74atv1l25.pdf"/>	2023-04-01 13:06:49	{"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": 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": 0.3869372308254242, "perplexity": 2910.3715449930896}, "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/1679296950030.57/warc/CC-MAIN-20230401125552-20230401155552-00008.warc.gz"}
https://chemistry.stackexchange.com/questions/117042/colour-of-metal-complexes-with-the-same-type-and-number-of-ligands/117051	# Colour of metal complexes with the same type and number of ligands Crystal Field Theory says that because of d-orbital splitting caused by different ligands we have 2 different energy levels for d-orbitals. It also says that transition metals are coloured because photons excite electrons, and when they have the crystal field splitting energy through a specific wavelength of light, the electrons to go to the higher energy d-orbital. When electrons come back down they emit the same wavelength of light and that's the colour we see. But then why is $$\ce{[Ni(H2O)6]^2+}$$ green and $$\ce{[Cu(H2O)_6]^2+}$$ blue? They have the same type and number of ligands. The Cu2+ and Ni2+ complexes, while having the same ligands, have different amounts of d-electrons (8 for Ni and 9 for Cu), meaning their ∆o value is going to be different. For both complexes the t2g is completely filled meaning the ∆o only increases based on the electrons in the eg. This is why Cu2+ will absorb higher energy light (orange light) in order to excite an electron to the eg orbital where as Ni will absorb lower energy light (red light) for its promotion. I believe this change in ∆o is rather minor though in comparison to ligand effects and charge and this may be one of the few cases where the change really is due to the number of d electrons. Also keep in mind the ∆o increases as you go down any group of the periodic table which is why you often see more low spin complexes. Someone please correct me if I'm wrong here, but this is at least my understanding of it. When electrons come back down they emit the same wavelength of light and that's the colour we see. I think this is a major misconception held by students. The color of these transition metal complexes is due to the absorption of light in the visible region. Very crudely, imagine you have six colors in the rainbow, if you remove a certain color portion from that rainbow, the transmitted light is no longer white to our eyes. For example, copper (II) absorbs red, that is why it appears blue to us. Yes, the electron has to relax back to its ground state, it does so non-radiatively. The spacing (∆o) in the energy level determines the wavelength which will be absorbed. • but if 2 complexes absorb the same wavelength of light wouldn't they show the same colour? – muhammad haider Jun 19 at 13:00 • @muhammadhaider yes, but the colour depends on the exact energy difference which will, in general, not be the same even for different metal complexes with a similar structure. – matt_black Jun 19 at 13:12	2019-08-21 20:18:56	{"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": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 2, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6620959043502808, "perplexity": 642.1930049396177}, "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-2019-35/segments/1566027316194.18/warc/CC-MAIN-20190821194752-20190821220752-00384.warc.gz"}
https://www.physicsforums.com/threads/power-series-equality-proof.615646/	Power Series Equality Proof 1. Jun 21, 2012 pendesu 1. The problem statement, all variables and given/known data Given $\overset{\infty}{\underset{n=0}{\sum}}a_{n}(x-a)^{n}$ and $\overset{\infty}{\underset{n=0}{\sum}}b_{n}(x-a)^{n}$ that are in R. Then, $\overset{\infty}{\underset{n=0}{\sum}}a_{n}(x-a)^{n}=\overset{\infty}{\underset{n=0}{\sum}}b_{n}(x-a)^{n}$ if and only if $a_{n}=b_{n}$ for every $n=0,1,2,...$ The attempt at a solution (<<) Assume an=bn for every n=0,1,2,... Then $\overset{\infty}{\underset{n=0}{\sum}}a_{n}(x-a)^{n}=a_{0}+a_{1}(x-a)+...=b_{0}+b_{1}(x-a)+...=\overset{\infty}{\underset{n=0}{\sum}}b_{n}(x-a)^{n}$. (>>) Now with this direction I am having some issues as the series may not necessarily converge. My attempts have been feeble at best. The problem I have had is that the sums are infinite so I don't think I can use that polynomials are equal if and only if their coefficients are equal. At this point I am thinking about looking at the nth partial sums but I am not sure and just would like any advice. 2. Jun 21, 2012 SammyS Staff Emeritus How can it be said that $\sum_{n=0}^\infty a_{n}(x-a)^{n}=\sum_{n=0}^\infty b_{n}(x-a)^{n}\ ,$ unless both sums converge? 3. Jun 21, 2012 pendesu That is the thing. This was something I was going to ask my professor that I am going to be doing research with in p-adic analysis. I am starting to think this may just be a definition since if I recall my professor he was saying how a p-adic integer which is a power series may not be a convergent power series in the ring of p-adic integers. My professor is busy right now. This might not make sense.	2017-12-13 20:56:51	{"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": 0.7745771408081055, "perplexity": 169.57328180927092}, "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/1512948530841.24/warc/CC-MAIN-20171213201654-20171213221654-00114.warc.gz"}
https://quantdare.com/portfolio-weightlifting-ii/	# Portfolio weightlifting (II) ### Enrique Millán #### 07/02/2019 In a previous post, we took a look at the computation of a portfolio’s exposure to its allocations. Then, to show the effects of active management, we compared the return made by two portfolios. But there is so much more to look inside the financial time series. Since we left a couple of cliffhangers, let’s jump into them now. ## Risk metrics First of all, let’s begin with those hidden dangers we mentioned in the impressive performance accomplished by Free. Recall that it generated way more profit than our Rebalanced portfolio. That could lead you to decide that it’s better to leave your money to itself rather than to pay someone to manage it. However, when we take a glance at their associated risk metrics, we find out why taking some care of where your money goes is worth it. Our two indicators are going to be the annual volatility of the returns and drawdown. ### Volatility Volatility is an indicator of the expected displacement of the return around its current value. The classic formula to compute it for a period T is $Vol_t = \sqrt{T} \sigma\left(\ln 1 + r_j \right), \quad j = t-(T+1), \ldots, t-1$ (Remember, you have to compute it with the logarithmic returns! If you feel like reading in Spanish about it, check it at Quantdare). Putting together both portfolio’s annual volatility, we see that our simple, yet effective, monthly rebalance was able to reduce the metric’s value by 21 % on average. And if you observe the plots carefully, you will see how when peaks of volatility take place on Free, on the Rebalanced curve they are damped or do not exist at all. What are the implications of this reduction? When you are investing for the long term, you naturally want to create value, but mostly to preserve capital. Thus, what you want is not necessarily the highest return, but the most stable one. Indeed, a famous financial empirical fact is the low volatility anomaly. ### Drawdown It’s time now for our fright measure. The drawdown reflects how much the accumulated return has dropped in for a given period since the last highest high. The more negative it is, the more the value of your portfolio dropped. This could induce a feeling of insecurity, and perhaps make the investor think about moving his/her positions when the value of the portfolio drops too much (and as a consequence, overreact to the market. Remember that on the long term both portfolio’s fared really well). Again, we see that the rebalancing has rewarding effects for most of the investing period. The Rebalanced portfolio has had less steep drops during stress periods, or even a null one, pumping up the feeling of calm for such volatile assets, we are dealing with a very volatile sector after all! It is inevitably to fall when the market does so too, you just want to fall as less as possible to start from higher up when the rebound takes place. So, with these two simple metrics, we have seen the effect of a very simple risk control strategy: to keep your portfolio equally weighted in as much as possible. Could you guess why I did not suggest to rebalance every day or every week? ## The number of shares The problem with the weight formulation is that it leads to the computation of shares with decimal points. Naturally, the maths do not suffer from this problem, but you cannot go to your broker and ask for 17.345 shares of Apple on your next move. To compute the initial shares you need to buy of a certain stock to reach a determined exposure level, you determine the ratio between the Assets Under Management you want to expose and the stock price, $S_0 = \frac{I_0 w_0}{P_0}$ and take the closest smallest integer number (known as the floor operation in mathematics). Since now you work with an integer number, you will always have some leftover cash that is not invested. You want that number to be as small as possible, but at the same time have enough left so that you can rebalance your exposure without problems. Summing up, there is more to portfolio valuation than simply the return. Certainly, you are not going far if you do not capture the market growth when it takes place, but you want to make sure you are not boarding an airplane without a pilot, no matter how high it goes. Once again, thank you for reading.	2019-02-16 10:27:07	{"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": 0.6242558360099792, "perplexity": 886.9584076685371}, "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-2019-09/segments/1550247480240.25/warc/CC-MAIN-20190216085312-20190216111312-00099.warc.gz"}
https://stats.stackexchange.com/questions/348428/sampling-from-a-multivariate-normal-distribution	# sampling from a multivariate normal distribution I have a covariance matrix representing a multivariate normal distribution. I would like to draw samples from this distribution, e.g. using R's mvrnorm. What happens if I diagonalize the covariance matrix in advance? This would yield a diagonal matrix, with the corrected variances on the diagonal. Can I now sample from each gaussian independently? if so, what is the new mean? • If you replace the initial covariance with a diagonal one, this modifies the distribution you are sampling from. – Xi'an May 27 '18 at 8:41 • My space is very sparse because this is a very high-dimensional Gaussian. Hence, few samples don't represent the initial distribution well. Is there a way to improve the situation? – Omry Atia May 27 '18 at 8:45 Recall the formula $\text{Cov}(CX) = C \,\text{Cov}(X) \, C^T$ for how to compute the covariance matrix of a linear transform of a vector $X$, described by the matrix $C$. This shows that your method is indeed correct, however you need to linearly transform your independent Gaussians (put into a vector) to get samples from the original, correlated Gaussian vector. The mean will transform linearly; so in particular, if the original mean vector is zero, so is the one of the transformed vector. The technique of principal components analysis is based on choosing not only a diagonalising $C$, but also on putting the eigenvectors (principal components in statistical language) belonging to the largest eigenvalues first. It may be best to use a routine in R that computes only the first few eigenvectors / principal components.	2021-04-21 19:49:20	{"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": 0.8773646354675293, "perplexity": 316.96619340895063}, "config": {"markdown_headings": true, "markdown_code": false, "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/1618039550330.88/warc/CC-MAIN-20210421191857-20210421221857-00055.warc.gz"}
https://www.investopedia.com/terms/e/expenditure-method.asp	## What Is the Expenditure Method? The expenditure method is a system for calculating gross domestic product (GDP) that combines consumption, investment, government spending, and net exports. It is the most common way to estimate GDP. It says everything that the private sector, including consumers and private firms, and government spend within the borders of a particular country, must add up to the total value of all finished goods and services produced over a certain period of time. This method produces nominal GDP, which must then be adjusted for inflation to result in the real GDP. The expenditure method may be contrasted with the income approach for calculated GDP. ### Key Takeaways • The expenditure method is the most common way of calculating a country's GDP. • This method adds up consumer spending, investment, government expenditure, and net exports. • Aggregate demand is equivalent to the expenditure equation for GDP in the long-run. • The alternative method to calculate GDP is the income approach. ## How the Expenditure Method Works Expenditure is a reference to spending. In economics, another term for consumer spending is demand. The total spending, or demand, in the economy is known as aggregate demand. This is why the GDP formula is actually the same as the formula for calculating aggregate demand. Because of this, aggregate demand and expenditure GDP must fall or rise in tandem. However, this similarity isn't technically always present in the real world—especially when looking at GDP over the long run. Short-run aggregate demand only measures total output for a single nominal price level, or the average of current prices across the entire spectrum of goods and services produced in the economy. Aggregate demand only equals GDP in the long run after adjusting for price level. The expenditure method is the most widely used approach for estimating GDP, which is a measure of the economy's output produced within a country's borders irrespective of who owns the means to production. The GDP under this method is calculated by summing up all of the expenditures made on final goods and services. There are four main aggregate expenditures that go into calculating GDP: consumption by households, investment by businesses, government spending on goods and services, and net exports, which are equal to exports minus imports of goods and services. The Formula for Expenditure GDP is: \begin{aligned} &GDP = C + I + G + (X - M)\\ &\textbf{where:}\\ &C = \text{Consumer spending on goods and services}\\ &I = \text{Investor spending on business capital goods}\\ &G = \text{Government spending on public goods and services}\\ &X = \text{exports}\\ &M = \text{imports}\\ \end{aligned} ### Main Components of the Expenditure Method In the United States, the most dominant component in the calculations of GDP under the expenditure method is consumer spending, which accounts for the majority of U.S. GDP. Consumption is typically broken down into purchases of durable goods (such as cars and computers), nondurable goods (such as clothing and food), and services. The second component is government spending, which represents expenditures by state, local and federal authorities on defense and nondefense goods and services, such as weaponry, health care, and education. Business investment is one of the most volatile components that goes into calculating GDP. It includes capital expenditures by firms on assets with useful lives of more than one year each, such as real estate, equipment, production facilities, and plants. The last component included in the expenditure approach is net exports, which represents the effect of foreign trade of goods and service on the economy. ## Expenditure Method vs. Income Method The income approach to measuring gross domestic product is based on the accounting reality that all expenditures in an economy should equal the total income generated by the production of all economic goods and services. It also assumes that there are four major factors of production in an economy and that all revenues must go to one of these four sources. Therefore, by adding all of the sources of income together, a quick estimate can be made of the total productive value of economic activity over a period. Adjustments must then be made for taxes, depreciation, and foreign factor payments. The major distinction between each approach is its starting point. The expenditure approach begins with the money spent on goods and services. Conversely, the income approach starts with the income earned (wages, rents, interest, profits) from the production of goods and services. ## Limitation of GDP Measurements GDP, which can be calculated using numerous methods, including the expenditure approach, is supposed to measure a country's standard of living and economic health. Critics, such as the Nobel Prize-winning economist Joseph Stiglitz, caution that GDP should not be taken as an all-encompassing indicator of a society's well-being, since it ignores important factors that make people happy. For example, while GDP includes monetary spending by private and government sectors, it does not consider work-life balance or the quality of interpersonal relationships in a given country.	2021-04-22 17:22:07	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "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": 0.5463817119598389, "perplexity": 2415.2911549022747}, "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-2021-17/segments/1618039594341.91/warc/CC-MAIN-20210422160833-20210422190833-00153.warc.gz"}
https://homework.cpm.org/category/CC/textbook/cca2/chapter/4/lesson/4.2.1/problem/4-73	### Home > CCA2 > Chapter 4 > Lesson 4.2.1 > Problem4-73 4-73. Solve the following absolute value inequalities. Homework Help ✎ 1. $\| x - 4 \| < 9$ 2. $\| \frac { 1 } { 2 } x - 45 \| \geq 80$ 3. $\| 2 x - 5 \| \leq 2$ See problem 4-66 for additional examples. −5 < x < 13 See part (a). x ≥ 250 or x ≤ −70 See part (a).	2019-08-21 14:06:05	{"extraction_info": {"found_math": true, "script_math_tex": 3, "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": 0.4711969792842865, "perplexity": 14408.921025041005}, "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-2019-35/segments/1566027316021.66/warc/CC-MAIN-20190821131745-20190821153745-00409.warc.gz"}
https://www.zora.uzh.ch/id/eprint/146443/	# Search for the suppressed decays $B^+ \to K^+ K^+ \pi^-$ and $B^+ \to \pi^+ \pi^+ K^-$ LHCb Collaboration; Bernet, R; Müller, K; Serra, N; Steinkamp, O; Straumann, U; Vollhardt, A; et al (2017). Search for the suppressed decays $B^+ \to K^+ K^+ \pi^-$ and $B^+ \to \pi^+ \pi^+ K^-$. Physics Letters B, B765:307-316. ## Abstract A search is made for the highly-suppressed B meson decays $B^+ \to K^+ K^+ \pi^-$ and $B^+ \to \pi^+ \pi^+ K^-$ using a data sample corresponding to an integrated luminosity of 3.0 $fb^{−1}$ collected by the LHCb experiment in proton–proton collisions at centre-of-mass energies of 7 and 8 TeV. No evidence is found for the decays, and upper limits at 90% confidence level are determined to be $\mathcal{B}(B^+ \to K^+ K^+ \pi^-)<1.1×10^{−8}$ and $\mathcal{B}(B^+ \to \pi^+ \pi^+ K^-)<4.6×10^{−8}$. ## Abstract A search is made for the highly-suppressed B meson decays $B^+ \to K^+ K^+ \pi^-$ and $B^+ \to \pi^+ \pi^+ K^-$ using a data sample corresponding to an integrated luminosity of 3.0 $fb^{−1}$ collected by the LHCb experiment in proton–proton collisions at centre-of-mass energies of 7 and 8 TeV. No evidence is found for the decays, and upper limits at 90% confidence level are determined to be $\mathcal{B}(B^+ \to K^+ K^+ \pi^-)<1.1×10^{−8}$ and $\mathcal{B}(B^+ \to \pi^+ \pi^+ K^-)<4.6×10^{−8}$. ## Statistics ### Citations Dimensions.ai Metrics 4 citations in Web of Science® 3 citations in Scopus® ### Altmetrics Detailed statistics	2019-07-22 18:30:27	{"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": 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": 0.9730405211448669, "perplexity": 2301.2236125512354}, "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/1563195528208.76/warc/CC-MAIN-20190722180254-20190722202254-00528.warc.gz"}
https://www.r-bloggers.com/2020/10/top-10-dplyr-functions-data-analysis-made-easy/	Want to share your content on R-bloggers? click here if you have a blog, or here if you don't. I’ve been a Python fanboy for the last couple of years. The language is excellent for data science, and being a general-purpose language makes it that much easier to work on non-data-science parts of data science. But can R’s dplyr beat Python’s pandas in the data exploration and preparation domain? Yes, it can. Today we’ll see how to use 10 of the most common dplyr functions. That’s not all the package has to offer, so refer to this link for a complete list. We’ll perform the entire analysis with the Gapminder dataset, available directly in R. Here’s how to import all the packages we’ll need today: library(dplyr) library(gapminder) Let’s begin the exploration from the next section. ### select The select command is used when we want to, well, select one or multiple columns. Think of it as a SELECT statement in SQL, where we can pick only columns of interest, and leave others untouched. This is the first way of using select – by listing all the columns we want to keep: gapminder %>% select(country, year, lifeExp) The other method is by specifying columns we want to remove – done with the - prefix: gapminder %>% select(-continent) ### filter The filter command is equivalent to the WHERE keyword in SQL, as we can fetch only the subset we’re interested in. There are multiple ways to use this command, and all are shown below. Let’s keep only those rows where the year is 1952: gapminder %>% filter(year == 1952) Let’s add another filter – the year is still 1952 but we want only the data for Croatia: gapminder %>% filter(year == 1952, country == 'Croatia') That’s great, but how can we filter by multiple years. The %in% operator comes in handy here: gapminder %>% filter(year %in% c(1952, 1957), country == 'Croatia') ### arrange The arrange operator is used to sort the data by individual columns, either in ascending or descending order. The ascending ordering is the default. It is equivalent to the ORDER BY keyword in SQL. Let’s sort the Gapminder dataset by population in ascending order: gapminder %>% arrange(pop) Let’s do the same in descending order: gapminder %>% arrange(desc(pop)) ### distinct This operator is used to find unique elements of a given attribute. It doesn’t need any further clarification, so let’s get straight to the code. Here’s how we can see the unique continents in the dataset: gapminder %>% distinct(continent) ### mutate The mutate operator is used to create new columns. All of the other columns stay intact during this operation. Let’s use it to calculate the GDP of a country by multiplying population by the GDP per capita: gapminder %>% mutate(totalGdp = pop * gdpPercap) ### transmute This operator is nearly identical to mutate, the only difference being that transmute doesn’t keep the other attributes – only the new one is kept. We’ll use it once again to calculate total GDP and then compare results with the mutate operator: gapminder %>% transmute(totalGdp = pop * gdpPercap) As promised, only the new totalGdp column is shown. ### rename As the name suggests, this operator is used for renaming. Be careful about the syntax, as we must specify the new name first. Let’s rename a couple of columns – lifeExp to lifeExpectancy and pop to population: gapminder %>% rename(lifeExpectancy = lifeExp, population = pop) ### summarize The summarize or summarise operators are useful for creating summary statistics on a dataset. If no grouping operations are performed, the result is a single value. For example, let’s say we want to find overall average life expectancy: gapminder %>% summarize(meanLifeExp = mean(lifeExp)) We can chain other operators if we want to, let’s say find average life expectancy in 2007: gapminder %>% filter(year == 2007) %>% summarize(meanLifeExp = mean(lifeExp)) That’s great, but summarize is the most useful when used with grouping operations. Let’s discuss those next. ### group_by We can use grouping operators to utilize the full potential of previously discussed operators, such as summarize. This one is equivalent to the GROUP_BY keyword in SQL, so you’re probably already familiar with it. Suppose we want to find the average life expectancy per continent – impossible to do without a grouping operator: gapminder %>% group_by(continent) %>% summarize(meanLifeExp = mean(lifeExp)) We can chain multiple operators together, and perform groupings on multiple columns. For example, let’s see how to find an average life expectancy per continent, grouped further by years (only 2002 and 2007): gapminder %>% filter(year %in% c(2002, 2007)) %>% group_by(continent, year) %>% summarize(meanLifeExp = mean(lifeExp)) ### if_else This is not your traditional operator, as we use it inside other operators, such as mutate. It lets us perform a vectorized if-else statement without writing multiple lines of code. Let’s see how we can group by continent and year (only 2002 and 2007) and calculate the average life expectancy, and then create another column which has the value of ‘Y’ if the average life expectancy is greater than 70, and ‘N’ otherwise: gapminder %>% filter(year %in% c(2002, 2007)) %>% group_by(continent, year) %>% summarize(meanLifeExp = mean(lifeExp)) %>% mutate(meanLifeExpOver70 = if_else(meanLifeExp > 70, 'Y', 'N')) ## Parting words At the end of the day, anything done in dplyr can be done in pandas. But you have to admit – dplyr does it so effortlessly and in a much simpler and cleaner way. Comparison between the two is a potential topic for another time, so let me know if you’re interested in that. Also, let me know which one you prefer and why – and try to keep your opinion unbiased. Join my private email list for more helpful insights. The post Top 10 dplyr Functions – Data Analysis Made Easy appeared first on Better Data Science.	2021-11-26 23:47:52	{"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": 0.22372671961784363, "perplexity": 2313.6794472276756}, "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/1637964358074.14/warc/CC-MAIN-20211126224056-20211127014056-00169.warc.gz"}
http://futur.upc.edu/177812	Balbuena Martinez, Maria Camino Teofila Total activity: 265 ## Scientific and technological production Ordered by: Date asc. Date desc. Title asc. Title desc. Researcher asc. Researcher desc. 1 to 50 of 265 results • On the connectivity and restricted edge-connectivity of 3-arc graphs Balbuena Martinez, Maria Camino Teofila; García Vázquez, Pedro; Montejano Cantoral, Luis Pedro Discrete applied mathematics Date of publication: 2014-01-10 Journal article Let View the MathML source denote the symmetric digraph of a graph G. A 3-arc is a 4-tuple (y,a,b,x) of vertices such that both (y,a,b) and (a,b,x) are paths of length 2 in G. The 3-arc graphX(G) of a given graph G is defined to have vertices the arcs of View the MathML source, and they are denoted as (uv). Two vertices (ay),(bx) are adjacent in X(G) if and only if (y,a,b,x) is a 3-arc of G. The purpose of this work is to study the edge-connectivity and restricted edge-connectivity of 3-arc graphs. We prove that the 3-arc graph X(G) of every connected graph G of minimum degree d(G)=3 has ¿(X(G))=(d(G)-1)2. Furthermore, if G is a 2-connected graph, then X(G) has restricted edge-connectivity ¿(2)(X(G))=2(d(G)-1)2-2. We also provide examples showing that all these bounds are sharp. Concerning the vertex-connectivity, we prove that ¿(X(G))=min{¿(G)(d(G)-1),(d(G)-1)2}. This result improves a previous one by [M. Knor, S. Zhou, Diameter and connectivity of 3-arc graphs, Discrete Math. 310 (2010) 37¿42]. Finally, we obtain that X(G) is superconnected if G is maximally connected. • On the order of graphs with a given girth pair Balbuena Martinez, Maria Camino Teofila; Salas Piñon, Julian Discrete mathematics Date of publication: 2014-04-28 Journal article A (k; g, h)-graph is a k-regular graph of girth pair (g, h) where g is the girth of the graph, h is the length of a smallest cycle of different parity than g and g < h. A (k; g, h)-cage is a (k; g, h)-graph with the least possible number of vertices denoted by n(k; g, h). In this paper we give a lower bound on n(k; g, h) and as a consequence we establish that every (k; 6)-cage is bipartite if it is free of odd cycles of length at most 2k - 1. This is a contribution to the conjecture claiming that every (k; g)-cage with even girth g is bipartite. We also obtain upper bounds on the order of (k; g, h)-graphs with g = 6, 8, 12. From the proofs of these upper bounds we obtain a construction of an infinite family of small (k; g, h)-graphs. In particular, the (3; 6, h)-graphs obtained for h = 7, 9, 11 are minimal. (C) 2013 Elsevier B.V. All rights reserved. • Large vertex-transitive graphs of diameter 2 from incidence graphs of biaffine planes Balbuena Martinez, Maria Camino Teofila; Miller, Mirka; Siran, Josef; Zdimalova, Maria Discrete mathematics Date of publication: 2013-06 Journal article Under mild restrictions, we characterize all ways in which an incidence graph of a biaffine plane over a finite field can be extended to a vertex-transitive graph of diameter 2 and a given degree with a comparatively large number of vertices. • Biregular cages of girth five Abreu, Marien; Araujo-Pardo, Gabriela; Balbuena Martinez, Maria Camino Teofila; Labbate, Domenico; Lopez Chavez, G. Electronic journal of combinatorics Date of publication: 2013-03 Journal article Let 2 <= r < m and g be positive integers. An ({r, m}; g)-graph (or biregular graph) is a graph with degree set {r, m} and girth g, and an ({r, m}; g)-cage (or biregular cage) is an ({r, m}; g)-graph of minimum order n({r, m}; g). If m = r +1, an ({r,m};g)-cage is said to be a semiregular cage. In this paper we generalize the reduction and graph amalgam operations from [M. Abreu, G. Araujo Pardo, C. Balbuena, D. Labbate. Families of Small Regular Graphs of Girth 5. Discrete Math. 312(18) (2012) 2832-2842] on the incidence graphs of an affine and a biaffine plane obtaining two new infinite families of biregular cages and two new semiregular cages. The constructed new families are ({r, 2r - 3}; 5)-cages for all r = q + 1 with q a prime power, and ({r, 2r - 5}; 5)-cages for all r = q + 1 with q a prime. The new semiregular cages are constructed for r = 5 and 6 with 31 and 43 vertices respectively. • A note on the upper bound and girth pair of (k; g)-cages Balbuena Martinez, Maria Camino Teofila; González Moreno, Diego; Montellano Ballesteros, J.J. Discrete applied mathematics Date of publication: 2013-04 Journal article A (k;g)(k;g)-cage is a kk-regular graph of girth gg with minimum order. In this work, for all k=3k=3 and g=5g=5 odd, we present an upper bound of the order of a (k;g+1)(k;g+1)-cage in terms of the order of a (k;g)(k;g)-cage, improving a previous result by Sauer of 1967. We also show that every (k;11)(k;11)-cage with k=6k=6 contains a cycle of length 12, supporting a conjecture by Harary and Kovács of 1983. • Connectivity: properties and structure Balbuena Martinez, Maria Camino Teofila; Fàbrega Canudas, Josep; Fiol Mora, Miquel Àngel Date of publication: 2013-12-05 Book chapter Connectivity is one of the central concepts of graph theory, from both a theoret- ical and a practical point of view. Its theoretical implications are mainly based on the existence of nice max-min characterization results, such as Menger¿s theorems. In these theorems, one condition which is clearly necessary also turns out to be sufficient. Moreover, these results are closely related to some other key theorems in graph theory: Ford and Fulkerson¿s theorem about flows and Hall¿s theorem on perfect matchings. With respect to the applications, the study of connectivity parameters of graphs and digraphs is of great interest in the design of reliable and fault-tolerant interconnection or communication networks. Since graph connectivity has been so widely studied, we limit ourselves here to the presentation of some of the key results dealing with finite simple graphs and digraphs. For results about infinite graphs and connectivity algorithms the reader can consult, for instance, Aharoni and Diestel [AhDi94], Gibbons [Gi85], Halin [Ha00], Henzinger, Rao, and Gabow [HeRaGa00], Wigderson [Wi92]. For further details, we refer the reader to some of the good textbooks and surveys available on the subject: Berge [Be76], Bermond, Homobono, and Peyrat [BeHoPe89], Frank [Fr90, Fr94, Fr95], Gross and Yellen [GrYe06], Hellwig and Volkmann [HeVo08], Lov ´asz [Lo93], Mader [Ma79], Oellermann [Oe96], Tutte [Tu66]. • Further topics in connectivity Balbuena Martinez, Maria Camino Teofila; Fàbrega Canudas, Josep; Fiol Mora, Miquel Àngel Date of publication: 2013-12-05 Book chapter Continuing the study of connectivity, initiated in §4.1 of the Handbook, we survey here some (sufficient) conditions under which a graph or digraph has a given connectivity or edge-connectivity. First, we describe results concerning maximal (vertex- or edge-) connectivity. Next, we deal with conditions for having (usually lower) bounds for the connectivity parameters. Finally, some other general connectivity measures, such as one instance of the so-called ¿conditional connectivity,¿ are considered. For unexplained terminology concerning connectivity, see §4.1. • Una conjetura sobre el orden de las jaulas con pares de cinturas Salas Piñon, Julian; Balbuena Martinez, Maria Camino Teofila Encuentro Andaluz de Matemática Discreta Presentation's date: 2013-10-18 Presentation of work at congresses Una jaula es un grafo regular de grado k que no tiene ciclos de longitud menor que g, es decir tiene cintura g, y tiene el m´inimo n´umero de v´ertices entre todos los grafos regulares de grado k y cintura g. El n´umero de v´ertices de una jaula, se denota por n(k; g). Dados dos n´umeros g < h, el par de cinturas (g, h) en un grafo G, es tal que g es la cintura de G y h es la m´inima longitud de un ciclo de distinta paridad que g. En este trabajo presentamos resultados relacionados con la conjetura de Harary y Kov´acs que afirma que el n´umero de v´ertices de una jaula con par de cinturas (g, h) es menor que el n´umero de v´ertices de la jaula correspondiente con cintura h, i.e., n(k; g, h) = n(k; h). • On the super-restricted arc-connectivity of s-geodetic digraphs Balbuena Martinez, Maria Camino Teofila; García Vázquez, Pedro; Hansberg Pastor, Adriana; Montejano Cantoral, Luis Pedro networks Date of publication: 2013-01 Journal article • The k-restricted edge-connectivity of a product of graphs Balbuena Martinez, Maria Camino Teofila; Marcote Ordax, Francisco Javier Discrete applied mathematics Date of publication: 2013-01 Journal article • On bi-regular cages of even girth at least 8 Araujo-Pardo, Gabriela; Balbuena Martinez, Maria Camino Teofila; Lopez Chavez, G.; Montejano Peimbert, Luis Aequationes mathematicae Date of publication: 2013-12 Journal article Let r, m, 2 = r < m and g = 3 be three positive integers. A graph with a prescribed degree set r, m and girth g having the least possible number of vertices is called a bi-regular cage or an (r, m; g)-cage, and its order is denoted by n(r, m; g). In this paper we provide upper bounds on n(r, m; g) for some related values of r, m and even girth g at least 8. Moreover, if r - 1 is a prime power and m = 5, we construct the smallest currently known (r, m; 8)-graphs. Also, if r = 3 and m = 7 is not divisible by 3, we prove that n(3,m;8) = [25m/3] + 7. Finally, we construct a family of (3, m; 8)-graphs of order 9m + 3 which are cages for m = 4,5,7. • Families of small regular graphs of girth 7 Abreu, Marien; Araujo-Pardo, Gabriela; Balbuena Martinez, Maria Camino Teofila; Labbate, D.; Salas Piñon, Julian Electronic notes in discrete mathematics Date of publication: 2013-05-15 Journal article We present an explicit construction of small (q+1,7)-graphs for q a prime power that is based on some combinatorial properties of the incidence graphs of generalized quadrangles of order (q,q). • Bounds on the order of biregular graphs with even girth at least 8 Lopez Chavez, G.; Araujo-Pardo, Gabriela; Montejano Peimbert, Luis; Balbuena Martinez, Maria Camino Teofila Electronic notes in discrete mathematics Date of publication: 2013-05-15 Journal article Let 2¿r • Constructions of biregular cages of girth five Abreu, Marien; Araujo-Pardo, Gabriela; Balbuena Martinez, Maria Camino Teofila; Labbate, D.; Lopez Chavez, G. Electronic notes in discrete mathematics Date of publication: 2013-05-15 Journal article Let 2¿r • On Superconnectivity of (4, g)-Cages Lu, Hongliang; Wu, Yunjian; Lin, Yuqing; Yu, Qinglin; Balbuena Martinez, Maria Camino Teofila; Marcote Ordax, Francisco Javier Graphs and combinatorics Date of publication: 2013-01 Journal article • Girth of {C-3, ... , C-s}-free extremal graphs Abajo, E.; Balbuena Martinez, Maria Camino Teofila; Diánez, A. Discrete applied mathematics Date of publication: 2012-06 Journal article • Families of small regular graphs of girth 5 Abreu, Marien; Araujo-Pardo, Gabriela; Balbuena Martinez, Maria Camino Teofila; Labbate, Domenico Discrete mathematics Date of publication: 2012-09-28 Journal article • A new bound for the connectivity of cages Balbuena Martinez, Maria Camino Teofila; Salas Piñon, Julian Applied mathematics letters Date of publication: 2012-11 Journal article • On the Structure of Graphs without Short Cycles Salas Piñon, Julian Defense's date: 2012-12-20 Department of Applied Mathematics III, Universitat Politècnica de Catalunya Theses The objective of this thesis is to study cages, constructions and properties of such families of graphs. For this, the study of graphs without short cycles plays a fundamental role in order to develop some knowledge on their structure, so we can later deal with the problems on cages. Cages were introduced by Tutte in 1947. In 1963, Erdös and Sachs proved that (k, g) -cages exist for any given values of k and g. Since then, large amount of research in cages has been devoted to their construction. In this work we study structural properties such as the connectivity, diameter, and degree regularity of graphs without short cycles. In some sense, connectivity is a measure of the reliability of a network. Two graphs with the same edge-connectivity, may be considered to have different reliabilities, as a more refined index than the edge-connectivity, edge-superconnectivity is proposed together with some other parameters called restricted connectivities. By relaxing the conditions that are imposed for the graphs to be cages, we can achieve more refined connectivity properties on these families and also we have an approach to structural properties of the family of graphs with more restrictions (i.e., the cages). Our aim, by studying such structural properties of cages is to get a deeper insight into their structure so we can attack the problem of their construction. By way of example, we studied a condition on the diameter in relation to the girth pair of a graph, and as a corollary we obtained a result guaranteeing restricted connectivity of a special family of graphs arising from geometry, such as polarity graphs. Also, we obtained a result proving the edge superconnectivity of semiregular cages. Based on these studies it was possible to develop the study of cages. Therefore obtaining a relevant result with respect to the connectivity of cages, that is, cages are k/2-connected. And also arising from the previous work on girth pairs we obtained constructions for girth pair cages that proves a bound conjectured by Harary and Kovács, relating the order of girth pair cages with the one for cages. Concerning the degree and the diameter, there is the concept of a Moore graph, it was introduced by Hoffman and Singleton after Edward F. Moore, who posed the question of describing and classifying these graphs. As well as having the maximum possible number of vertices for a given combination of degree and diameter, Moore graphs have the minimum possible number of vertices for a regular graph with given degree and girth. That is, any Moore graph is a cage. The formula for the number of vertices in a Moore graph can be generalized to allow a definition of Moore graphs with even girth (bipartite Moore graphs) as well as odd girth, and again these graphs are cages. Thus, Moore graphs give a lower bound for the order of cages, but they are known to exist only for very specific values of k, therefore it is interesting to study how far a cage is from this bound, this value is called the excess of a cage. We studied the excess of graphs and give a contribution, in the sense of the work of Biggs and Ito, relating the bipartition of girth 6 cages with their orders. Entire families of cages can be obtained from finite geometries, for example, the graphs of incidence of projective planes of order q a prime power, are (q+1, 6)-cages. Also by using other incidence structures such as the generalized quadrangles or generalized hexagons, it can be obtained families of cages of girths 8 and 12. In this thesis, we present a construction of an entire family of girth 7 cages that arises from some combinatorial properties of the incidence graphs of generalized quadrangles of order (q,q). • Control de invariantes en grafos sujetos a propiedades estructurales Marcote Ordax, Francisco Javier; Salas Piñon, Julian; Hansberg Pastor, Adriana; Cera, Martín; Gago Alvarez, Silvia; Balbuena Martinez, Maria Camino Teofila Participation in a competitive project • New results on 3-domination critical graphs Balbuena Martinez, Maria Camino Teofila; Hansberg Pastor, Adriana Aequationes mathematicae Date of publication: 2012 Journal article • Restricted arc-connectivity of generalized p-cycles Balbuena Martinez, Maria Camino Teofila; García Vázquez, Pedro; Hansberg Pastor, Adriana; Montejano Cantoral, Luis Pedro Discrete applied mathematics Date of publication: 2012-06 Journal article • A sufficient condition for kernel perfectness of a digraph in terms of semikernels modulo F Balbuena Martinez, Maria Camino Teofila; Galeana-Sánchez, Hortensia; Guevara, Mukuy-kaak Acta mathematica sinica, english series Date of publication: 2012-02 Journal article • On second order degree of graphs Araujo-Pardo, Gabriela; Balbuena Martinez, Maria Camino Teofila; Olsen, Mika; Valencia, Pilar Acta mathematica sinica, english series Date of publication: 2012-01 Journal article • An explicit formula for obtaining generalized quadrangles and other small regular graphs of girth 8 Balbuena Martinez, Maria Camino Teofila; Araujo-Pardo, Gabriela; Abreu, Marien; Labbate, Domenico International Workshop on Optimal Network Topologies Presentation's date: 2011-07-11 Presentation of work at congresses • Sobre la conectividad del grafo producto G * H Balbuena Martinez, Maria Camino Teofila; Marcote Ordax, Francisco Javier Encuentro Andaluz de Matemática Discreta Presentation's date: 2011-11-07 Presentation of work at congresses • Una fórmula explícita para obtener cuadrángulos generalizados y otros grafos pequeños de cintura 8 Abreu, Marien; Balbuena Martinez, Maria Camino Teofila; Araujo-Pardo, Gabriela; Labbate, Domenico Encuentro Andaluz de Matemática Discreta Presentation's date: 2011-11-08 Presentation of work at congresses Para cada potencia de primo q, se han constru do (q+1; 8)-jaulas como super cies cu adricas no degeneradas en 4-espacios proyectivos P(4; q). En primer lugar presentamos una construcci on de estos grafos de un modo alternativo dando una f ormula expl cita en la que usamos terminolog a gr a ca. Adem as derivamos grafos k-regulares de cintura 8 para grados k = q; q 1 que tienen el n umero m as peque~no de v ertices conocido hasta ahora. • Constructions of small regular bipartite graphs of girth 6 Araujo-Pardo, Gabriela; Balbuena Martinez, Maria Camino Teofila networks Date of publication: 2011-03 Journal article In this article, some structures in the projective plane of order q are found which allow us to construct small k - regular balanced bipartite graphs of girth 6 for all k ≤ q. When k = q, the order of these q-regular graphs is 2(q^2−1); and when k ≤ q−1, the order of these k -regular graphs is 2(qk − 2). Moreover, the incidence matrix of a k -regular balanced bipartite graph of girth 6 having 2(qk −2) vertices, where k is an integer and q is a prime power with 3 ≤ k ≤ q − 1, is provided. These graphs improve upon the best known upper bounds for the number of vertices in regular graphs of girth 6. • Partial linear spaces and identifying codes Araujo-Pardo, Gabriela; Balbuena Martinez, Maria Camino Teofila; Valenzuela, Juan Carlos; Montejano Cantoral, Luis Pedro European journal of combinatorics Date of publication: 2011-04 Journal article • Superconnectivity of graphs with odd girth g and even girth h Balbuena Martinez, Maria Camino Teofila; García Vázquez, Pedro; Montejano Cantoral, Luis Pedro Discrete applied mathematics Date of publication: 2011-01-28 Journal article • Topological minors in bipartite graphs Balbuena Martinez, Maria Camino Teofila; Cera, Martín; García Vázquez, Pedro; Valenzuela, Juan Carlos Acta mathematica sinica, english series Date of publication: 2011-11 Journal article • New results on connectivity of cages Salas Piñon, Julian; Balbuena Martinez, Maria Camino Teofila Electronic notes in discrete mathematics Date of publication: 2011-12-01 Journal article • Monotonicity of the order of (D, g)-cages Balbuena Martinez, Maria Camino Teofila; Marcote Ordax, Francisco Javier Applied mathematics letters Date of publication: 2011-11 Journal article • RESTRICTED CONNECTIVITY IN FAMILIES OF GRAPHS Montejano Cantoral, Luis Pedro Defense's date: 2011-09-15 Department of Applied Mathematics III, Universitat Politècnica de Catalunya Theses • Edge-superconnectivity of semiregular cages with odd girth Balbuena Martinez, Maria Camino Teofila; González Moreno, Diego Antonio; Salas Piñon, Julian networks Date of publication: 2011-10 Journal article • Diameter and connectivity of (D; g)-cages Balbuena Martinez, Maria Camino Teofila; Marcote Ordax, Francisco Javier International journal of computer mathematics Date of publication: 2011 Journal article • A sufficient degree condition for a graph to contain all trees of size k Balbuena Martinez, Maria Camino Teofila; Márquez, Alberto; Portillo, Jose Ramón Acta mathematica sinica, english series Date of publication: 2011-01 Journal article • On the (lambda)'-optimality in graphs with odd girth g and even girth h Balbuena Martinez, Maria Camino Teofila; García-Vázquez, Pedro; Montejano Cantoral, Luis Pedro; Salas Piñon, Julian Applied mathematics letters Date of publication: 2011-07 Journal article • On the restricted edge-connectivity of 3-arc graphs Balbuena Martinez, Maria Camino Teofila; García-Vázquez, Pedro; Montejano Cantoral, Luis Pedro Jornadas de Matemática Discreta y Algorítmica Presentation's date: 2010-07 Presentation of work at congresses • Regular parcial linear spaces admitting (1; = k)-identifying codes Araujo-Pardo, Gabriela; Balbuena Martinez, Maria Camino Teofila; Valenzuela, Juan Carlos; Montejano Cantoral, Luis Pedro Jornadas de Matemática Discreta y Algorítmica Presentation's date: 2010-07 Presentation of work at congresses • Edge fault tolerance analysis of super k-restricted connected networks Balbuena Martinez, Maria Camino Teofila; García Vázquez, Pedro Applied mathematics and computation Date of publication: 2010-03-15 Journal article • Adjacency matrices of polarity graphs and of other C4-free graphs of large size Abreu, Marien; Balbuena Martinez, Maria Camino Teofila; Labbate, Domenico Designs codes and cryptography Date of publication: 2010 Journal article In this paper we give a method for obtaining the adjacency matrix of a simple polarity graph $G_q$ from a projective plane PG(2,q), where q is a prime power. Denote by ex(n;C₄) the maximum number of edges of a graph on n vertices and free of squares C₄. We use the constructed graphs $G_q$ to obtain lower bounds on the extremal function ex(n;C₄), for some n < q² + q + 1. In particular, we construct a C₄-free graph on n = q² − √q vertices and 1/2q(q²−1) − 1/2√q(q − 1) edges, for a square prime power q. • New families of graphs without short cycles and large size Abajo, E.; Balbuena Martinez, Maria Camino Teofila; Diánez, A. Discrete applied mathematics Date of publication: 2010-06-06 Journal article We denote by ex $(n; {C^3,C^4,…Cs})$ or fs(n) the maximum number of edges in a graph of order n and girth at least s+1. First we give a method to transform an n-vertex graph of girth g into a graph of girth at least g-1 on fewer vertices. For an infinite sequence of values of n and s∈{4, 6, 10} the obtained graphs are denser than the known constructions of graphs of the same girth s+1. We also give another different construction of dense graphs for an infinite sequence of values of n and s∈{7, 11}. These two methods improve the known lower bounds on fs(n) for s ∊ {4, 6, 7, 10, 11} which were obtained using different algorithms. Finally, to know how good are our results, we have proved that $\limsup_{n\longrightarrow{\infty}}\displaystyle\frac{\int_s (n)}{n^{1+\displaystyle\frac{2}{s-1}}}=2^{-1-\displaystyle\frac{2}{s-1}}$ for s ∊ 2 {5, 7, 11}, and $s^{-1-\displaystyle\frac{2}{s}}$ ≤ $\limsup_{n\longrightarrow{\infty}}\displaystyle\frac{\int_s (n)}{n^{1+\displaystyle\frac{2}{s}}}$ ≤ 0.5 for s ∊ {6, 10}. • On the connectivity and superconnected graphs with small diameter Balbuena Martinez, Maria Camino Teofila; Marshall, Kim; Montejano Cantoral, Luis Pedro Discrete applied mathematics Date of publication: 2010-03-06 Journal article In this paper, first we prove that any graph G is 2-connected if diam (G)≤ g-1 for even girth g, and for odd girth g and maximum degree $\Delta$ ≤ 2$\delta$-1 where $\delta$ is the minimum degree. Moreover, we prove that any graph G of diameter diam(G) ≤ g-2 satisfies that (i)G is 5-connected for even girth g and $\Delta$ ≤ 2$\delta$-5, and (ii)G is super-k for odd girth g and $\Delta$ ≤ 3$\delta$/2-1. • Finding small regular graphs of girths 6, 8 and 12 as subgraphs of cages Araujo-Pardo, Gabriela; Balbuena Martinez, Maria Camino Teofila; Héger, T. Discrete mathematics Date of publication: 2010-04 Journal article • On the restricted edge-connectivity of 3-arc graphs Balbuena Martinez, Maria Camino Teofila; García-Vázquez, Pedro; Montejano Cantoral, Luis Pedro Date of publication: 2010-06-09 Book chapter • On identifying codes in partial linear spaces Balbuena Martinez, Maria Camino Teofila Date of publication: 2010-06-09 Book chapter • Subdivisions in a bipartite graph Balbuena Martinez, Maria Camino Teofila; Cera, Martín; García-Vázquez, Pedro; Valenzuela, Juan Carlos Date of publication: 2010-06-09 Book chapter • On the k-restricted edge-connectivity of matched sum graphs Balbuena Martinez, Maria Camino Teofila; Marcote Ordax, Francisco Javier Date of publication: 2010-06-09 Book chapter • INTERNATIONAL WORKSHOP ON OPTIMAL NETWORK TOPOLOGIES Balbuena Martinez, Maria Camino Teofila; Mitjana Riera, Margarida; Serra Albo, Oriol; Fàbrega Canudas, Josep; Comellas Padro, Francesc de Paula Participation in a competitive project	2014-04-24 06:24:44	{"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": 0.5167081952095032, "perplexity": 9311.129888688254}, "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-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00124-ip-10-147-4-33.ec2.internal.warc.gz"}
http://www.lofoya.com/Solved/1295/a-and-b-undertake-to-do-a-piece-of-work-for-rs-600-a-alone	# Difficult Time and Work Solved QuestionAptitude Discussion Q. $A$ and $B$ undertake to do a piece of work for Rs 600. $A$ alone can do it in 6 days while $B$ alone can do it in 8 days. With the help of $C$, they can finish it in 3 days, Find the share of $C$? ✖ A. 70 ✔ B. 75 ✖ C. 80 ✖ D. 85 Solution: Option(B) is correct $C$’s one day’s work, $= \left(\dfrac{1}{3}\right) - \left(\dfrac{1}{6}+\dfrac{1}{8}\right)$ $= \dfrac{1}{24}$ Therefore, $A:B:C =$ Ratio of their one day’s work = 1/6:1/8:1/24= 4:3:1A$’s share,$= \text{Rs. }600×\dfrac{4}{8}= 300B$’s share,$= \text{Rs. }600×\dfrac{3}{8}= 225C$’s share,$= \text{Rs. } [600-(300+225)]= \textbf{Rs. 75}\$ Edit: For an alternative solution, check comment by Sooraj. ## (2) Comment(s) Sooraj () A's work in a day =1/6 of the work B's work in a day= 1/8 of the work Work done by A and B together in a day =(1/6)+(1/8)=7/24 work done by A and B together in 3 days= (7/24)3=7/8 Remaining work done by c=1-(7/8)=1/8 Share of C =600(1/8)=75 Sonam () A's one day work =1/6 of the work B's one day work =1/8 of the work Working together A, Band C they can complete it in 3 days i.e; (A+B+C) one day work=1/3 1/6+1/8+C=1/3 C=1/3-(1/6+1/8) C=1/24 Therefore C's one day work=1/24 Share of A for one day=600/6 //they undertook the work for 600// =100 Then share of A for three days=3100=300 Similarly C's one day share =600/24=25 Then three days share =325=75 //As they work for 3 days//	2017-11-20 02:10:01	{"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": 0.3485323488712311, "perplexity": 4379.35668944783}, "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-2017-47/segments/1510934805894.15/warc/CC-MAIN-20171120013853-20171120033853-00564.warc.gz"}
http://math.stackexchange.com/questions/169430/the-puzzling-eigenvalues-of-a-differential-equation-system	# The puzzling eigenvalues of a differential equation system I have a question here, in order to have a stable solution of a differential equation, is that right we should have all eigenvalues less one? But I have seen a incomprehensible situation in difference equation, for instance \begin{align} x_{t+1}&=a x_{t} -b y_t\\ y_t&= y_{t+1}-cx_{t+1} \end{align} $a$, $b$ and $c$ are parameters, there are two eignenvalues, one has to be less than one, the other has to be larger than one in order to keep system stable. I completely do not grasp this idea. If anyone knows, please explain to me! Thanks! - Notice this difference between difference and differential equations: $x_{n+1}=ax_n$ has solutions involving $a^n$, while $x'=ax$ has solutions involving $e^{at}$. For $a^n$, the big difference is between $a\gt1$ and $a\lt1$; for $e^{at}$, the big difference is between $a\gt0$ and $a\lt0$. – Gerry Myerson Jul 11 '12 at 12:41 Just rewrite the equations into the form \begin{align} x_{t+1}&=a x_{t} -b y_t\\ y_{t+1}-cx_{t+1}&=y_t. \end{align} This has the form $A\bar x_{t+1}=C\bar x_t$. Now, $A$ is not singular and so one has $\bar x_{t+1}=A^{-1}C\bar x_t$. Now put $$\bar x_t=\lambda^t\left(\begin{array}{c} x_1 \\ x_2\end{array}\right)=\lambda^t\bar x_0$$ and you are left with the eigenvalue problem after substitution $$\lambda\bar x_0=A^{-1}C\bar x_0.$$ As stated in the comment area you should consider the case of the eigenvalues being $\lambda>1$ and $\lambda<1$.	2015-11-26 07:11:25	{"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": 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": 0.9424394965171814, "perplexity": 358.80902136299113}, "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-2015-48/segments/1448398446535.72/warc/CC-MAIN-20151124205406-00177-ip-10-71-132-137.ec2.internal.warc.gz"}
https://mathhothouse.me/2014/08/	Monthly Archives: August 2014 A motivation for Math and some Math competitive exams in India Sometime back, there was a tremendous publicity in the Indian media to two Fields medallists of Indian origin. They also talked about what motivated them towards Math when they were young. One should not do Math just lured by its glamorous applications in IT or other engineering disciplines. But, one can develop both aptitude and attitude towards it if one works from a young age. What you need is intrinsic motivation. In this context, I like to quote the following words of a famous mathematician: “And, a final observation. We should not forget that the solution to any worthwhile problem very rarely comes to us easily and without hard work; it is rather the result of intellectual effort of days or weeks or months. Why should the young mind be willing to make this supreme effort? The explanation is probably the instinctive preference for certain values, that is, the attitude which rates intellectual effort and spiritual achievement higher than material advantage. Such valuation can only be the result of long cultural development of environment and public spirit which is difficult to accelerate by the governmental aid or even by more intensive training in mathematics. The most effective means may consist of transmitting to the young mind the beauty of intellectual work and the feeling of satisfaction following a great and successful mental effort.” —- Gabor Szego For those of you who are willing to work hard, the following competitive exams (in Math) are available at various levels of your school-life: 1) National Talent Search Exam (NTSE) — standard 10 and 12 2) IMO and NSO of Science Foundation (http://www.sofworld.org ) These exams can be attempted by students of almost all standards. 3) IPM: Institute for Promotion of Mathematics exams. 4) KVPY: Kishor Vaignanik Protsahan Yojana: This can be attempted by students of 11th, 12th and F.Y.B.Sc. This leads to a seat in the pure sciences including Math in the prestigious IISc, Bangalore. 5) Regional Math Olympiads (RMO) and Indian National Math Olympiad (INMO): These exams are really intense. Students from 9th to 11th standard can attempt these. These are conducted by Tata Institute of Fundamental Research (TIFR) and Homi Bhabha. More later… Nalin Pithwa Polynomials — quartics Let us continue our exploration of polynomials. Just as in the previous blog, let me present to you an outline of some method(s) to solve Quartics. As I said earlier, “filling up the gaps” will kindle your intellect. Above all, the main aim of all teaching is teaching “to think on one’s own”. I) The Quartic Equation. Descartes’s Method (1637). (a) Argue that any quartic equation can be solved once one has a method to handle quartic equations of the form: $t^{4}+pt^{2}+qt+r=0$ (b) Show that the quartic polynomial in (a) can be written as the product of two factors $(t^{2}+ut+v)(t^{2}-ut+w)$ where u, v, w satisfy the simultaneous system $v + w - u^{2}=p$ $u(w-v)=q$ $vw=r$ Eliminate v and w to obtain a cubic equation in $u^{2}$. (c) Show how any solution u obtained in (b) can be used to find all the roots of the quartic equation. (d) Use Descartes’s Method to solve the following: $t^{4}+t^{2}+4t-3=0$ $t^{4}-2t^{2}+8t-3=0$ II) The Quartic Equation. Ferrari’s Method: (a) Let a quartic equation be presented in the form: $t^{4}+2pt^{3}+qt^{2}+2rt+s=0$ The strategy is to complete the square on the left side in such a way as to incorporate the cubic term. Show that the equation can be rewritten in the form $(t^{2}+pt+u)^{2}=(p^{2}-q+2u)t^{2}+2(pu-r)t+(u^{2}-s)$ where u is indeterminate. (b) Show that the right side of the transformed equation in (a) is the square of a linear polynomial if u satisfies a certain cubic equation. Explain how such a value of u can be used to completely solve the quartic. (c) Use Ferrari’s Method to solve the following: $t^{4}+t^{2}+4t-3=0$ $t^{4}-2t^{3}-5t^{2}+10t-3=0$. Happy problem-solving 🙂 More later… Nalin Polynomials — commuting and cubics Let us start delving deeper in Algebra. But, I will be providing only an outline to you in the present article. I encourage you to fill in the details. This is a well-known way to develop mathematical aptitude/thinking. (This method of learning works even in Physics and esoteric/hardcore programming). Definition. Commuting polynomials. Two polynomials are said to commute under composition if and only if $(p \circ q)(t)=(q \circ p)(t)$ (i.e., $p(q(t)=q(p(t)))$). We define the composition powers of a polynomial as follows $p^{(2)}(t)=p(p(t))$ $p^{(3)}(t)=p(p(p(t)))$ and in general, $p^{(k)}(t)=p(p^{(k-1)}(t)$ for $k=2,3 \ldots$ Show that any two composition powers of the same polynomial commute with each other. One might ask whether two commuting polynomials must be composition powers of the same polynomial. The answer is no. Show that any pair of polynomials in the following two sets commute I. ${t^{n}: n=1,2 \ldots}$ II. ${T^{n}(t):n=1,2 \ldots}$ Let a and b be any constants with a not equal to zero. Show that, if p and q are two polynomials which commute under composition, then the polynomials $(t/a-b/a) \circ p \circ (at+b)$ and $(t/a-b/a) \circ q \circ (at+b)$ also commute under the composition. Use this fact to find from sets I and II other families which commute under composition. Can you find pairs of polynomials not comprised in the foregoing discussion which commute under composition? Find families of polynomials which commute under composition and within which there is exactly one polynomial of each positive degree. The Cubic Equation. Cardan’s Method. An elegant way to solve the general cubic is due to Cardan. The strategy is to replace an equation in one variable by one in two variables. This provides an extra degree of freedom by which we can impose a convenient second constraint, allowing us to reduce the problem to that of solving a quadratic. (a) Suppose the given equation is $t^{3}+pt+q=0$. Set $t=u+v$ andn obtain the equation $u^{3}+v^{3}+(3uv+p)(u+v)+q=0$. Impose the second condition $3uv+p=0$ (why do we do this?) and argue that we can obtain solutions for the cubic by solving the system $u^{3}+v^{3}=-q$ $uv = -p/3$ (b) Show that $u^{3}$ and $v^{3}$ are roots of the quadratic equation $x^{2}+qx-p^{3}/27=0$ (c) Let $D=27q^{2}+4p^{3}$. Suppose that p and q are both real and that $D>0$. Show that the quadratic in (b) has real solutions, and that if $u_{0}$ and $v_{0}$ are the real cubic roots of these solutions, then the system in (a) is satisfied by $(u,v)=(u_{0},v_{0}), (u_{0}\omega, v_{0}\omega^{2}), (u_{0}\omega^{2}, v_{0}\omega)$ where $\omega$ is the imaginary cube root $(0.5)(-1+\sqrt{-3})$ of unity. Deduce that the cubic polynomial $t^{3}+pt+q$ has one real and two nonreal zeros. (d) Suppose that p and q are both real and that $D=0$. Let $u_{0}$ be the real cube root of the solution of the quadratic in (b). Show that, in this case, the cubic has all its zeros real, and in fact can be written in the form $sy^{2}$ where $y=(t+u_{0})$ and $s=t-2u_{0}$ (e) Suppose that p and q are both real and that $D<0$. Show that the solutions of the quadratic equation in (b) are nonreal complex conjugates, and that it is possible to choose cube roots u and v of these solutions which are complex conjugates and satisfy the system in (a). If $u=r(cos \theta + isin \theta)$ and $v=r(cos \theta - isin \theta)$, show that the three roots of the cubic equation are the reals $2r cos \theta$ $2r cos (\theta + (2/3)\pi)$ $2r cos(\theta + (4/3)\pi)$. (f) Prove that every cubic equation with real coefficients has at least one real root. Use Cardan’s Method to solve the cubic equation. (a) $x^{3}-6x+9=0$ (b) $x^{3}-7x+6=0$. Part (b) above will require the use of a pocket calculator and some trigonometry. You will also need De Moivre’s Theorem and give a solution to an accuracy of 3 decimal places. More later… -Nalin Nevalninna Prize to IITB alumnus Subhash Khot Nevalninna Prize to IITB alumnus Subhash Khot R Ramachandran The Hindu Aug 14 2014. Award for his Unique games Conjecture in Computational Complexity Theory The 36 year-old IIT Bombay alumnus Subhash Khot, an Indian-American theoretical computer scientist at the Courant Institute of Mathematical Sciences of New York University, has been chosen for the International Mathematical Union’s Nevanlinna Prize, which is given “for outstanding contributions in mathematical aspects of information sciences”. The award is given once every four years during the International Congress of Mathematicians (ICM). The ICM2014 began on August 13 at Seoul, Republic of Korea. Khot’s research has to do with a field in computer science called ‘Computational Complexity’, which seeks to understand the power and limits of efficient computation with standard computers. “Khot’s prescient definition of the “Unique Games” problem,” said the award citation, “and his leadership in the effort to understand its complexity and its pivotal role in the study of efficient approximation of optimization problems, have produced breakthroughs in algorithmic design and approximation hardness, and new exciting interactions between computational complexity, analysis and geometry.” Unlike the normal practice of awarding major prizes in mathematics for groundbreaking results, the Nevalninna Prize this time is for a conjecture and that too when the opinion on its truth is divided within the world computer science community. But, by posing the right questions that have enabled great insights into the nature of computational complexity and approximations to computationally hard problems, the conjecture, called the Unique Games Conjecture (UGC), has already proved its great value. More pertinently, Khot has also used the conjecture to prove major results that will remain valid regardless of the truth of the UGC. “I do believe whether it is eventually shown to be false or not changes little about the brilliance of the conjecture,” wrote Richard Lipton of Georgia Institute of Technology, Atlanta, in a blog post on the UGC. “It takes insight, creativity of the highest order, and a bit of ‘guts’ to make such a conjecture,” Lipton said. The central question in computational complexity is: How hard are problems to solve? More precisely, if one has found the cleverest possible way to solve a particular problem, how fast will a computer find the answer using it? It is now a truism that some problems are so intractably difficult that computers cannot reliably find the answer at all, at least not in any reasonable amount of time (such as before the end of the universe). A typical optimization problem, to quote an example given by Meena Mahajan, a theoretical computer scientist from the Institute of Mathematical Sciences, Chennai, is: What is the minimum number of tea stalls that should be put up in a large university campus so that no one needs to walk, say, more than 200 metres along a road to reach one? As the size of the campus increases, the computational time to find the minimum number of stalls grows exponentially fast, even with the best current algorithms. This is what underlies the famous P ≠ NP conjecture, which is one of the seven \$1 million Millennium Problems posed by the Clay Mathematics Institute (CMI). A problem that is tractable and the number of algorithmic steps that its solution requires is at most some power of the problem size is said to belong to P class (where P stands for ‘polynomial time’). A problem belongs to NP class (NP stands for ‘non-deterministic polynomial time’) if the computer can efficiently verify a proposed solution for it is correctness but does not have the resources – time or the number of algorithmic steps or memory size — as a function of the “input size” of the problem to obtain the solution. Formulated by the American-Canadian computer scientist Stephen Cook in 1971, the conjecture states that these two classes are distinct. This means that there are computational problems whose solutions are beyond the reach of any computer algorithm. Most computer scientists believe that the conjecture is true but even after four decades it is yet to be proved, not want of attempts though. Such “computationally intractable” or “NP-hard” problems have profound consequences. For instance, they limit our ability to tackle large-scale problems in science and engineering, such as optimal design of protein folding or figuring out the best design for a chip or the best train schedule. Conversely, however, computational intractability enables computer security against hackers attempting to access on-line confidential data. So, computer scientists asked: If a problem is too hard to be computationally solved quickly and precisely, can we at least find a good approximation? “Counter-intuitive though it may seem,” Mahajan points out returning to her tea stall example, “while we do not know how to find the minimum efficiently, we can find a number that is no more than twice the minimum efficiently! That is, we can efficiently find an approximate solution. Unfortunately, there are many optimization problems for which even this may not be possible.” The UGC essentially addresses the question of solving NP-hard problems even approximately. It thus complements the P ≠ NP conjecture. In the initial years after P vs, NP conjecture was made, many computer scientists believed that that good approximations must be easier than finding the exact answer to an NP-hard problem. But they soon discovered that, while they could come up with good approximation algorithms for some NP-hard problems (like our tea-stall problem), for most of them even finding a good approximation was not possible. There was no prescriptive way of determining whether approximation was possible. That is, approximation itself was an NP-hard problem. The Unique Games problem, a remarkably simple problem, encapsulates the elements that make many hard problems hard to solve even approximately. The problem is simply about finding an efficient way of assigning colours to the nodes of a network such that any two connected nodes have different colours (Fig). Figure. A Random Network If one has only two colours (say yellow and green), the problem is easy. The problem becomes trickier even when you add just one more colour (say blue). When you colour the first node, say with Y, you don’t know what colour the connected nodes should have, G or B. If you choose one and get to a node that cannot be coloured without violating the condition, you have no way of knowing if a different selection would have solved the problem. It is not the method that was faulty. In fact, no other method will be able to solve it reliably and efficiently. The problem is NP-hard, meaning effectively impossible. But Khot asked the related question: Which colouring scheme breaks the fewest rules possible? That is, which colouring is the best approximation. The conjecture basically is that if you have lots of colours, even an efficient method to colour the nodes anywhere close to the best one is impossible. The UGC, which Khot enunciated in 2002, can be stated as follows: It is not just hard but impossible reliably to find an approximate answer to Unique Games quickly. That it is, the problem is NP-hard even to solve approximately. Thus, if the conjecture is true, the problem Unique Games problem, in a technical sense, sets a benchmark for NP-hard problems. “Khot’s work attempts to give a unified explanation for why so many problems seem hard to approximate,” points out Mahajan. “What makes this so wonderful is that if the UGC is true, it explains in one shot why a host of other problems have resisted solutions so far; they are all at least as hard as Unique Games. All the difficulties encountered in tackling many different optimization problems get distilled into one problem,” she adds. A couple of years later after Khot made his conjecture, computer scientists realized the real power and importance of the conjecture. They found that, if the UGC was indeed true, then they could set firm limits on how well many other problems could be approximated. For instance, in our tea-stall example, it turns out that twice the minimum is the best one can do under the assumption that P ≠ NP. If one tried to do better than a factor-of-2 approximation, say with a more sophisticated algorithm, it would imply that P = NP. The simple algorithm that gave the factor-of-2 approximation is the best one can do. According to the UGC, an efficient approximation for the Unique Games problem would imply P = NP. Independent of its truth, the conjecture, however, has proved to be remarkably powerful. In the process of determining how well NP-hard problems could be approximated, Khot and others have proved several significant results in other areas, which seem far removed from computational complexity, such as geometry of different ways of measuring distances, some new theorems in Fourier analysis, better understanding algorithms based on linear and semi-definite programming and structure of ‘foams’. The last connection, which is essentially a tiling problem, came as a surprise even to Khot, according to Mahajan. While there is a significant group of researchers working to prove the conjecture, there is an equally significant set working to disprove it. Although scientists are yet to find an algorithm that can efficiently find a good approximate solution to Unique Games, finding one such would mean a significant algorithmic breakthrough. Such a new approximation algorithm is most likely to be very different from the approximation algorithms that are known today. Indeed, the process has already thrown up some excellent new algorithmic methods for other situations. In any case, the UGC is likely to keep theoretical computer scientists busy for some years to come. Mathematics wizard is an IIT Bombay alumnus Mathematics wizard is an IIT Bombay alumnus R Ramachandran http://www.thehindu.com/news/cities/mumbai/mathematics-wizard-is-an-iitbombay-alumnus/article6314039.ece?ref=relatedNews The Hindu Aug 14 2014. One of the important contributions made by Fields Medal winner Manjul Bhargava is the generalisation of the ‘composition law’ of binary quadratics (polynomial expressions of the form ax2 + bxy + cy2) discovered 200 years ago by Carl Friedrich Gauss (1777-1855), to higher degree polynomials using an ingenious geometric technique that he discovered. The awards were announced at the inaugural of the nine-day International Congress of Mathematicians that began in Seoul on Wednesday. The awards, presented at the quadrennial ICM event, include the Fields Medal, the highest award in mathematics; the Rolf Nevanlinna Prize and the Carl Friedrich Gauss Prize. At the last ICM held in Hyderabad, the Chern Medal and the Leelavati Prize were added. The Fields Medal is awarded “to recognise outstanding mathematical achievement for existing work and for the promise of future achievement”. “Manjul Bhargava has developed powerful new methods in the geometry of numbers and applied them to count rings of small rank and to bound the average rank of elliptic curves,” said the medal citation. Besides mathematics, Dr. Bhargava pursues his interests in linguistics and Indian classical music. The Indian-American theoretical computer scientist Subhash Khot, a theoretical computer scientist at the Courant Institute of Mathematical Sciences of New York University, gets the Rolf Nevanlinna Prize. The citation for him read: “Subhash Khot’s prescient definition of the Unique Games problem, and his leadership in the effort to understand its complexity and its pivotal role in the study of efficient approximation of optimization problems, have produced breakthroughs in algorithmic design and approximation hardness, and new exciting interactions between computational complexity, analysis and geometry.” Born in Ichalkaranji in Maharashtra, Dr. Khot (36) an IIT Bombay alumnus, won the silver medal in the International Mathematics Olympiad in 1994 and 1995 and stood first in the IIT Joint Entrance Examination in 1995. His area of research is Computational Complexity Theory. His Unique Games Conjecture is about the impossibility of even obtaining good approximations to problems that are computationally hard to solve using standard computing algorithms. Major Mathematics awards to two Indian origin scientists http://www.thehindu.com/sci-tech/science/major-mathematics-awards-to-two-indian-origin-scientists/article6309293.ece?ref=relatedNews The Hindu Aug 14 2014 Manjul Bhargava and Subhash Khot, are among the eight winners of the prestigious International Mathematical Union awards Two mathematicians of Indian origin, Manjul Bhargava and Subhash Khot, are among the eight winners of the prestigious awards of the International Mathematical Union (IMU) that were announced at the inaugural of the 9-day International Congress of Mathematicians (ICM) which began today at Seoul, Republic of Korea. The President of Korea, Park Geun-hye, gave away the awards. The ICM is held every four years and, traditionally, the IMU awards are presented at this quadrennial event. The awards include the Fields Medal, the highest award in mathematics, the Rolf Nevanlinna Prize and the Carl Friedrich Gauss Prize. At the last ICM held at Hyderabad, India, two new awards, the Chern Medal and the Leelavati Prize, were added to the existing three awards. The 40 year-old Canadian-American Manjul Bhargava, a number theorist from Princeton University, is one of the four Fields Medalists chosen for the ICM2014 awards. The Fields Medal is awarded “to recognize outstanding mathematical achievement for existing work and for the promise of future achievement”. A minimum of two and a maximum of four Fields Medals are given to mathematicians under the age of 40 on January 1 of the year of the Congress. “Manjul Bhargava has developed powerful new methods in the geometry of numbers and applied them to count rings of small rank and to bound the average rank of elliptic curves,” said the IMU citation for the award. The other three Fields Medalists are: The Brazilian mathematician Arthur Avila (35) of the Paris Diderot University-Paris 7 and Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, who has been awarded “for his profound contributions to dynamical systems theory, which have changed the face of the field, using the powerful idea of renormalizations as a unifying principle”; the British mathematician Martin Hairer (39) of the University of Warwick “for his outstanding contributions to stochastic partial differential equations, and in particular for the creation of a theory of regularity structures for such equations”; and, the Iranian mathematician Maryam Mirzakhani (37) of Stanford University “for her outstanding contributions to the dynamics and geometry of Riemann surfaces and their modulii spaces”. The 36 year-old IIT Bombay alumnus Subhash Khot, an Indian-American theoretical computer scientist at the Courant Institute of Mathematical Sciences of New York University has been chosen for the ICM2014 Rolf Nevanlinna Prize. The Nevanlinna Prize is awarded “for outstanding contributions in mathematical aspects of information sciences”. “Subhash Khot’s prescient definition of the “Unique Games” problem, and his leadership in the effort to understand its complexity and its pivotal role in the study of efficient approximation of optimization problems, have produced breakthroughs in algorithmic design and approximation hardness, and new exciting interactions between computational complexity, analysis and geometry,” the award citation said. The Gauss Prize in Applied Mathematics is awarded “to honor scientists whose mathematical research has had an impact outside mathematics – either in technology, in business, or simply in people’s everyday lives”. The winner of the ICM2014 Gauss Prize is Stanley Osher (72) of University of California, Los Angeles, who has been awarded the Prize “for his influential contributions to several fields in applied mathematics, and for far reaching inventions that have changed our conception of physical, perceptual and mathematical concepts, giving us new tools to apprehend the world.” The Chern Medal is given “to an individual whose accomplishments warrant the highest level of recognition for outstanding achievements in the field of mathematics”. The Chern Medal this time goes to the American algebraic geometer Phillip Griffiths (76) “for his groundbreaking and transformative development of transcendental methods in complex geometry, particularly his seminal work in Hodge theory and periods of algebraic varieties”. Unlike the other awards, the Leelavati Prize is not given for achievements in mathematics research but for outstanding public outreach work in mathematics. Proposed by India, it was originally intended as a one-time award using the grant from the Norwegian Abel Foundation. Thanks to the efforts by Indian mathematicians in finding a sponsor to make it a regular affair, it has now been instituted as a recurring four-yearly award under the IMU charter to be given away at the closing ceremony of the ICM. The award is now being sponsored by Infosys, the Indian IT major. The ICM2014 Leelavati Prize has been given to the Argentine Adrián Paenza (65) “for his decisive contributions to changing the mind of a whole country about the way it perceives mathematics in daily life, and in particular for his books, his TV programmes, and his unique gift of enthusiasm and passion in communicating the beauty and joy of mathematics”. Hope Indian youth take up research in sciences: Fields Medal winner http://www.thehindu.com/opinion/op-ed/fields-medal-winner-manjul-bhargava-hope-indian-youth-take-up-research-in-sciences/article6312471.ece?homepage=true The Hindu Aug 14 2014. Manjul Bhargava, one of the recipients of the Fields Medal, speaks about mathematics, music and more. How does it feel to have won the Fields Medal? You are the first person of Indian origin to be getting it… It is of course a great honour; beyond that, it is a great source of inspiration and encouragement – not just for me, but for my students, collaborators, and colleagues who work with me. Hopefully, it is also be a source of inspiration for young people in India to take up research in the sciences! You have grown up in Canada… did you have any cultural identity questions? Do you think of yourself as a Canadian, American, Indian or none of these or all of these? I was born in Canada, but grew up mostly in the U.S. in a very Indian home. I learned Hindi and Sanskrit, read Indian literature, and learned classical Indian music. I ate mostly Indian food! On the other hand, I grew up playing with American kids and went to school mostly in the U.S. I liked growing up in two cultures like that because it allowed me to pick and choose the best of both worlds. My Indian upbringing was very important to me. I also spent a lot of time in India growing up. Every three or four years, I would take off six months of school to spend it in India — mostly in our hometown Jaipur — with my grandparents. There I had the opportunity to truly live in India for extended periods of time, go to school there, brush up on my Hindi and Sanskrit, and learn tabla (as well as some sitar and vocal music). I particularly enjoyed celebrating all the Indian holidays as a child, and flying kites on Makar Sankranti. I feel very much at home in all three countries. So I definitely think of myself as all three – Canadian, American, and of course Indian. How did you get interested in Tabla playing? You have learnt the Tabla from Ustad Zakir Husain… Can you tell us how this came about and what it means to you? I first started learning from my mother, who also plays the tabla. When I was maybe 3 years old, I used to hear my mother playing often, and I asked her to teach me to play a little bit. She tried to teach me the basic sound “na.” She demonstrated the sound to me, and I tried to mimic her to reproduce the sound, but nothing came out. I was hooked! I always loved the beauty and the intricacy of the tabla sound and repertoire, and how it also perfectly complemented sounds on the sitar, or vocal, etc. I learned with my mom first, and then with Pandit Prem Prakash Sharma in Jaipur whenever I visited there. I met Zakir ji when I was an undergraduate at Harvard. He came to perform there when I was a third year student. I had the exciting opportunity to meet him afterwards at a reception, and he invited me to visit him in California (where he lives). I have had the great pleasure and privilege of learning from him a bit off and on since then. More than that, he has been a wonderful and inspirational friend, and he and his whole family — in both California and Bombay — have been such a huge source of love, encouragement, and support to me for so long, and I am very grateful to them for that. Do you collaborate with mathematicians in India? Do you have contacts with the institutes in India? For many years now, I have been an adjunct professor at TIFR-Mumbai (Tata Institute for Fundamental Research), IIT-Bombay, and the University of Hyderabad. I’ve spent a lot of time at these three institutes, especially at TIFR and IIT-B, over many years. I’ve lectured extensively to students at these institutes, as well as collaborated a lot with mathematicians there, such as with Eknath Ghate at TIFR (who recently won the Shanti Swarup Bhatnagar Prize for mathematical sciences). I’ve also been involved in starting a new institute in Bangalore called “ICTS” (International Center for Theoretical Sciences). It will be inaugurated next year, and we hope it will be a great success. The director is Professor Spenta Wadia of TIFR, and the head of the International Advisory Board is Nobel Prize Winner Professor David Gross. So hopefully I will spend even more time in India after the inauguration next year! Recently you have won prizes for your work on the Birch and Swinnerton-Dyer conjecture which was listed as one of the seven millennium prize problems. Can you explain the significance of this work? In joint work with Christopher Skinner and Wei Zhang, we have shown that the Birch and Swinnerton-Dyer Conjecture is true “most” of the time (more precisely, for more than 66.48 per cent of elliptic curves!). Previously, it was not known that it was true for more than 0 per cent. So that is significant progress, but it is still “not” a complete solution! Finishing a proof of the Birch and Swinnerton-Dyer Conjecture would be a momentous achievement, and it is one of my favorite problems!, but it is not solved yet. Do you believe that this is the best time to study math – for instance, number theory is now being applied in cryptography and so on? What does it take to do great mathematics? It is interesting that pure mathematicians, like myself, rarely think directly about applications. We are instead guided primarily by what directions we find most beautiful, elegant, or most promising. We tend to treat our discipline more as an art than as a science! And indeed, this is the attitude that allows us to be the most creative and productive. On the other hand, it is also true, historically, that the mathematics that has been the most applicable and important to society over the years has been the mathematics that scientists found while searching for beauty; and eventually all beautiful and elegant mathematics tends to find applications. That is why it is very important to fund basic science research. When science funding is only application-driven, it does not allow full freedom and creativity. Funding basic science allows a large interconnected database of scientific techniques and knowledge to accumulate, so that when a societal need arises, the science is ready to be applied and adapted to the purpose. Elliptic curves (and the related Birch and Swinnerton-Dyer Conjecture) are indeed a good example! They were first studied by pure mathematicians, but are now one of the most important mathematical objects in cryptography. So that is indeed exciting, but I just want to emphasize that they were exciting and central to number theory well before these applications were found; but it was inevitable that they would be found, given their fundamental nature. That is why elliptic curves have fascinated me! They are so fundamental in both pure and applied mathematics. Beyond advancing the subject of number theory in general, a heightened understanding of elliptic curves also has important implications in coding theory and cryptography. Encryption schemes, such as those used to protect our privacy when transmitting information online, often centrally involve the use of elliptic curves. Math is generally considered a difficult subject but you have been enjoying math since your childhood. What aspect of your education could have contributed to this enjoyment? I’ve always enjoyed mathematics as far back as I can remember, since I was two or three years old. Since my mother was a mathematician, I always had her as a resource – I would always go and ask her questions and so I learned a lot from her. She was also a great source of encouragement – she always answered my questions enthusiastically, and always encouraged me to pursue whatever I was interested in – and that probably single-handedly contributed the most to my enjoyment of mathematics (and of all my interests)! An Indian origin mathematician with Midas Touch Manjul Bhargava Midas Touch Mathematician Manjul Bhargava The Hindu Aug 14 2014 http://www.thehindu.com/sci-tech/science/manjul-bhargava-the-midas-touch-mathematician/article6309323.ece?ref=relatedNews Number theorist Manjul Bhargava wins Fields Medal Manjul Bhargava, the Canadian-American number theorist from Princeton University, is one of the four who have been chosen for the highest award in mathematics, the Fields Medal, which is given once every four years by the International Mathematical Union (IMU) during the quadrennial International Congress of Mathematicians (ICM). The ICM2014 got underway on August 13 at Seoul, Republic of Korea. Fields medal Awarded in recognition of “outstanding mathematical achievement for existing work and for the promise of future achievement”, the Fields Medal is given to mathematicians of age less than 40 on January 1 of the year of the Congress. Born of Indian parents who migrated from Jaipur in the late 1950s, Bhargava, who turned 40 just last week, could not have hoped for a better birthday gift. “Bhargava”, says the IMU citation, has been awarded the Fields Medal “for developing powerful new methods in the geometry of numbers, which he applied to count rings of small rank and to bound the average rank of elliptic curves.” (See Box for definitions of italicized terms) • In ‘geometry of numbers’ one imagines a plane or a 3-dimensional space populated by a lattice whose grid points have integer co-ordinates.• A ‘ring’ is an algebraic structure with two binary operations, commonly called addition and multiplication, which are generalizations of the familiar arithmetic operations with integers applied to algebraic objects. Examples of rings are polynomials of one variable with real coefficients, or square matrices of a given dimension. Algebraic number theory is the study of this and other algebraic structures.• ‘Rank’ refers to the minimum number of objects required to generate the entire set of algebraic objects being studied; the dimension of a vector space, for example. The familiar 3-d vector space is of rank 3.• ‘Elliptic curves’ are graphs generated by equations of the form y2= a polynomial of degree 3, such as x3+ ax + b, where a and b are rational numbers. A large body of work in number theory relates to the study of how numbers of interest, such as prime numbers, are distributed among the entire set of integers. Bhargava developed novel techniques to count objects in algebraic number theory that were previously considered completely inaccessible. His work has completely revolutionized the way in which fundamental arithmetic objects in algebraic number theory, such as number fields and elliptic curves, are now understood and studied, and this has given rise to wonderful applications. About 200 years ago the German mathematician Carl Friedrich Gauss, one of the historical greats, had discovered a remarkable ‘composition law’ for binary quadratic forms, which are polynomials of the form ax2 + bxy + cy2, where a, b and c are integers. Using this law two binary quadratic forms could be combined to give a third one. Gauss’s law is a central tool in algebraic number theory. Bhargava discovered an ingenious and simpler geometrical technique to derive it and the technique allowed him to obtain composition laws for higher-degree polynomials as well. Geometry of numbers The technique reportedly dawned upon Bhargava one day while he was playing with Rubik’s cube. Implicit in Gauss’s method was the use of ‘geometry of numbers’ and it is this realization that enabled Bhargava to extend it to higher degrees. He then discovered 13 new composition laws for higher-degree polynomials. Until then, Gauss’s law was thought to be accidental and unique to binary quadratics. Nobody had even imagined that higher composition laws existed until Bhargava showed that Gauss’s law is part of a bigger theory applicable to polynomials of arbitrary degree. His approach has also broadened the canvas of applying geometry of numbers to address outstanding problems of algebraic number theory. This work immediately led Bhargava to tackle a related problem, which was the counting of ‘number fields of fixed degree by discriminant’. Discriminant A number field is obtained by extending the rational numbers to include non-rational roots of a polynomial equation; if the polynomial equation is quadratic, such ax2+bx+c = 0, whose roots are given by the well-known formula [– b/2a ± √(b2 – 4ac)/2a], then one obtains a quadratic number field. The expression under the square root sign is called the ‘discriminant’ (defined appropriately for polynomials of different degrees). Higher degree number fields — cubic, quartic, quintic etc. — are correspondingly generated by higher degree polynomials. The degree of the polynomial and its discriminant are two fundamental quantities associated with a polynomial. Despite number fields being one of the fundamental objects in algebraic number theory, answers to questions like how many number fields there are for a given degree n and a given determinant D were not known. If one has a quadratic polynomial, counting the number of lattice points in a certain region of 3-d space gives information about the associated quadratic number field. For example, using the geometry of numbers it can be shown that, for discriminant with absolute value less than D, there are approximately D quadratic number fields. The case of cubic number fields had been solved 40 years ago by Harold Davenport and Hans Heilbronn but since then the higher degree cases saw little progress until Bhargava came on the scene. Quintic number fields Armed with his new technique, Bhargava was able to solve the case of quartic and quintic number fields. The new composition laws and his new technique in using the geometry of numbers have together extended the reach and power of counting number fields. The cases of degrees greater than 5 still remain open as Bhargava’s composition laws alone seem inadequate to resolve these higher cases at present. While the above work were al carried out between 2004 and 2008, more recently, Bhargava has employed his improved geometry of numbers technique to obtain striking results about ‘hyperellpitic curves’, which are graphs of equations of the form y2 = a polynomial with rational coefficients, the case where the degree of the polynomial is 3 being called the ‘elliptic curve’. Elliptic curves have important applications in pure as well as applied mathematics. Even though Fermat’s Last Theorem seems to be not even remotely connected with elliptic curves, it was key to its proof in 1995 by Andrew Wiles, who, incidentally, was also Bhargava’s thesis advisor. Operations using elliptic curves have become a core component of many of the cryptographic protocols that encode credit card numbers in online transactions. “Intellectual stimulation, beautiful structure, applications – elliptic curves have it all,” Bhargava has said. An outstanding problem in algebraic number theory has been how to count the number of points on ‘hyperelliptic curves’ that have rational coordinates, which is the same as asking how many rational solutions does a hyperellptic equation have? The answer, it turns out, following Bhargava’s work, depends on the degree of the curve. One can easily see that the number of rational solutions of a polynomial equation of degree 1, such as y = 9x + 4, is infinite: any rational value for x produces a rational value for y, and vice versa. Quadratics, such as, such as y2 = 2x2 + 5x – 3, have either no rational solutions or infinitely many. For curves of degree 1 and 2, there is an effective way of finding all the rational points. In 1983, Gerd Faltings, director of Max Planck Institute for Mathematics, Bonn, showed that for degree 5 and more there are only finitely many rational points. That left unresolved the cases of degree 3 – the elliptic curves – and of degree 4. Finding rational points for elliptic curves is, however, not an easy matter. They can have zero, finitely many, or infinitely many rational solutions. When does a cubic equation have infinitely many solutions has been a central question in number theory since Pierre de Fermat in the 17th Century. In the recent past mathematicians have attempted to devise algorithms to decide whether a given elliptic curve has finitely many or infinite rational points but that route took them nowhere. They have only been able to guess how often these different possibilities arise. But once you have found some rational points on an elliptic curve, it becomes possible to generate more by using the simple connecting-the-dots method. For example (see fig.), if you draw a line through two rational points, it usually intersects the elliptic at exactly one more point, which is again a rational point. But the opposite, namely given one rational point finding the two rational points that would generate it. This is what underlies the use of elliptic curves in cyber security. Connecting-the dots method: Given two rational points of an elliptic curve y2 = x3 + 2x + 3, the point at which the line through those points intersects the curve at one more point is guaranteed to be a rational point. This connect-the-dots procedure is a means to generate all of an elliptic curve’s rational points starting from a small finite number. (Credit: Quanta, illustration by Manjul Bhargava) Curve’s rank When the number of rational points of an elliptic curve is infinite, the smallest number of rational points that can generate essentially all the rational points is called the curve’s rank. When the infinite set of rational points can be generated essentially from just one point, the curve has rank 1, and so on. When the number of rational points is finite or none at all, the rank is 0. In 1992 Armand Brumer showed that a 1965 conjecture made by Birch and Swinnerton-Dyer (BSD) implied that the average rank of the group of rational points of an elliptic curve defined over rational numbers is bounded. Later in 1979 Dorian Goldfeld conjectured that the bound is, in fact, is equal to ½. That is, in a statistical sense, half of all elliptic curves have rank 0 and half have rank 1. Previously, however, mathematicians did not even know that the average rank was finite (let alone ½). The conjecture, of course, does not mean that curves of higher rank – 2, 3 and so on – do not exist, or even that there are only finitely many such. Indeed, computationally mathematicians have found such curves, the highest known rank till date is 28! But as the number of elliptic curves asymptotically becomes infinitely large, the curves with higher ranks approach a vanishingly small percentage of the whole. Enter Bhargava and his collaborators, his doctoral student Arul Shankar (a 2007 Chennai Mathematical Institute graduate) in particular. Instead treading the beaten track of algorithms, they asked the question: what could be said about rational points on a typical curve? From this perspective they first showed that a sizeable fraction of elliptic curves has only one rational point (rank 0) and another sizeable proportion has infinitely many rational points (rank > 0). Using newly developed techniques, they were able to show that the average rank is, in fact, bounded. They have been further able to show that the bound is also less than 1, indicating that the conjecture is perhaps true. “Bhargava introduced dramatically new ideas to study the average number of solutions and proved that the average rank of elliptic curves is bounded, and that the BSD Conjecture is true on the average, making it one of the most spectacular successes in number theory in recent years,” says Deependra Prasad, a number theorist from Tata Institute of Fundamental Research (TIFR). Analogously, for the case of degree 4 too Bhargava and Shankar showed that a significant chunk of such curves has no rational points and another significant chunk positive proportion has infinitely many rational points. Using his expanded geometry of numbers technique Bhargava has also explored higher-degree curves in general. While Faltings Theorem tells us that for curves of degree greater than 5, there are only finitely many rational points, it does not give a way to determine how many exactly there are. For the even degree case, Bhargava showed that the “typical” hyperelliptic curve had no rational points at all. The joint work of Bhargava and Benedict Gross, followed up by that of Bjorn Poonen and Michael Stoll, established the same result for the odd degree case as well. Bhargava’s work has thus clearly shown that the number of curves having rational points decreases rapidly as the degree increases. For example, for a typical 10 degree polynomial, there is a greater than 99 per cent chance that the curve has no rational points. Bhargava’s work in number theory has had profound influence in the field. “A mathematician of extraordinary creativity, he has a taste for simple problems of timeless beauty, which he has solved by developing elegant and powerful new methods that offer deep insights,” said IMU’s information sheet on his work. “With his keen intuition, immense insight and great technical mastery, he seems to bring a ‘Midas touch’ to everything he works on,” it added. Tabla player Besides being one of the world’s leading mathematicians, Bhargava is also an accomplished Tabla player and plays at the concert level. He learnt the art initially from his mother and later came under the tutelage of the well-known tabla maestros Pandit Prem Prakash Sharma and Ustad Zakir Hussain. “Classical Indian music,” Bhargava told Princeton Weekly Bulletin when he was featured, “is very mathematical, but consciously thinking of the math would interfere with the improvisation and emotion of the playing. But somehow the connection is there. I often use music as a break, and many times I come back to the math later and things have cleared up.” Indeed, Bhargava thinks of mathematics art. He is also keenly interested in linguistics in which he has published research work. It was his grandfather, a linguistics scholar, who taught him Sanskrit and developed his interest in linguistics. Symbols — the Meat of Mathematics Let us take a gentle look at algebra now. (The present article is derived from a similar article by Tobias Dantzig). It is an expository article only. Algebra, in the broad sense in which the term is used today, deals with operations upon symbolic forms. In this capacity, it not only permeates all of mathematics, but encroaches upon the domain of formal logic and even of metaphysics. Furthermore, when so construed, algebra is as old as man’s faculty to deal with general propositions; as old as his ability to discriminate between “some” and “any”. Here, however, we are interested in algebra in a much more restricted sense, that part of general algebra which is very properly called the theory of equations. It is in this narrower sense that the term algebra was used at the outset. The word is of Arabic origin. Al is the Arabic title “the”, and gebar is the verb “to set”, to restitute. To this day the word algebrista is used in Spain to designate a bone-setter, a sort of chiropractor. It is generally true that algebra in its development in individual countries passed successively through three stages: the rhetorical, the syncopated, and the symbolic. Rhetorical algebra is characterized by the complete absence of any symbols, except, of course, that the words themselves are being used in their symbolic sense. To this day, rhetorical algebra is used in such a statement as “the sum is independent of the order of the terms”, which in symbols would be designated by $a+b=b+a$. Syncopated algebra, of which the Egyptian is a typical example, is a further development of rhetorical. Certain words of frequent use are gradually abbreviated. Eventually, these abbreviations become contracted to the point where their origin has been forgotten, so that the symbols have no obvious connection with the operation which they represent. The syncopation has become a symbol. The history of the symbols + and – may illustrate the point. In mediaval Europe, the latter was long denoted by the full word “minus”, then by the first letter “m” duly superscribed. Eventually the letter itself was dropped, leaving the superscript only. The sign “plus” passed through a similar metamorphosis. The turning point in the history of algebra was an essay written late in the sixteenth century by a Frenchman, Viete, who wrote under the Latin name Franciscus Vieta. His great achievement appears, simple enough to us today. It is summed up in the following passage from this work: In this we are aided by an artifice which permits us to distinguish given magnitudes from those which are unknown or sought, and this by means of a symbolism which is permanent in nature and clear to understand — for instance, by denoting the unknown magnitudes by A or any other vowels, while the given magnitudes are designated by B,C, G or other consonants. This vowel-consonant notation had a short existence. Within half a century of Vieta’s death appeared Descartes’s Geometrie, in which the first letters of the alphabet were used for given quantities, the last for those unknown. The Cartesian notation not only displaced the Vietan, but has survived to this day. But, while few of Vieta’s proposals were carried out in letter, they certainly were adopted in spirit. The systematic use of letters for undetermined but constant magnitudes, the “logistica speciosa” as he called it, which has played such a dominant role in the development of mathematics, was the great achievement of Vieta. The lay mind may find it difficult to estimate the achievement of Vieta at its true value. Is not the literal notation a mere formality after all, a convenient shorthand at best? There is, no doubt, economy in writing $(a+b)^{2}=a^{2}+b^{2}+2ab$ but does it really convey more to the mind than the verbal form of the same identity: the square of the sum of two numbers equals the sum of the squares of the numbers, augmented by twice their product? Again, the literal notation had the fate of all very successful innovations. The universal of these makes it difficult to conceive of a time when inferior methods were in vogue. Today formulae in which letters represent general magnitudes are almost as familiar as common script, and our ability to handle symbols is regarded by many almost as a natural endowment of any intelligent man; but it is natural only because it has become a fixed habit of our minds. In the days of Vieta this notation constituted a radical departure from the tradition of ages. Wherein lies the power of this symbolism? First of all, the letter liberated algebra from the slavery of the word. And, by this, I do not mean merely that without the literal notation any general statement would become a mere flow of verbiage, subject to all the ambiguities and misinterpretations of human speech. This is important enough; but, what is still more important is that the letter is free from the taboos which have attached to words through centuries of use. The A of Vieta or our present “x” has an existence independent of the concrete object which it is assumed to represent. The symbol has a meaning which transcends the objects symbolized: that is why it is not a mere formality. In the second place, the letter is susceptible of operations which enables one to transform literal expressions and thus to paraphrase any statement into a number of equivalent forms. It is the power of transformations that lifts algebra above the level of a convenient shorthand. Before the introduction of literal notation, it was possible to speak of individual expressions only; each expression, such as $2x+3$, $3x-5$; $x^{2}+4x+7$; $3x^{2}-4x+5$, had an individuality all its own and had to be handled on its own merits. The literal notation made it possible to pass from the individual to the collective, from the “some” to the “any” and the “all”. The linear form $ax+b$, the quadratic form $ax^{2}+bx+c$, each of these forms is regarded now as a single species. It is this that made possible the general theory of functions, which is the basis of all applied mathematics. But, the most important contribution of the logistica speciosa, and the one that concerns us most in this study, is the role it played in the formation of the generalized number concept. As long as one deals with numerical equations, such as $x+4=0$; $2x=8$ and $x^{2}=9$, call this equation I $x+0=4$; $2x=5$; $x^{2}=7$, call this equation II one can content himself (as most mediavel algebraists did) with the statement that the first group of equations is possible, while the second is impossible. But, when one considers literal equations of the same types: $x+b=a$; $bx=a$; $x^{n}=a$ the very indeterminateness of the data compels one to give an indicated or symbolic solution to the problem: $x=a-b$; $x=a/b$; $x= (a)^{1/n}$. In vain, after this, will one stipulate that the expression a-b has a meaning only if a is greater than b, that $a/b$ is meaningless when a is not a multiple of b, and that $a^{1/n}$ is not a number unless a is a perfect nth power. The very act of writing down the meaningless has given it meaning; and, it is not easy to deny the existence of something that has received a name. Moreover, with the reservation that $a>b$, that a is a multiple of b, that a is perfect nth power, rules are devised for operating on such symbols as $a-b$; $a/b$; $a^{1/n}$. But, sooner or later the very fact that there is nothing on the face of these symbols to indicate whether a legitimate or an illegitimate case is before us, will suggest that there is no contradiction involved in operating on these symbolic beings as if they bona fide numbers. And from this there is but one step to recognizing these symbolic beings as numbers “in extenso”. What distinguishes modern arithmetic from that of the pre-Vieta period is the changed attitude towards the “impossible”. Up to the seventh century the algebraists invested this term with an absolute sense. Committed to natural numbers as the exclusive field for all arithmetic operations, they regarded possibility, or restricted possiblity, as an intrinsic property of these operations. Thus, the direct operations of arithmetic — addition $(a+b)$, multiplication $(ab)$, potentiation $a^{n}$ — were omni-possible; whereas, the inverse operations — subtraction $(a-b)$, division $a/b$, extraction or roots $a^{1/n}$ — were possible only under restricted conditions. The pre-Vieta algebraists were satisfied with stating these facts, but were incapable of a closer analysis of the problem. Thus, the direct operations of arithmetic are omnipossible because they are but a succession of iterations, a step-by-step penetration into the sequence of natural numbers, which is assumed a priori unlimited. Drop this assumption, restrict the field of the operand to a finite collection (say to the first 1000 numbers), and operations such as $925+125$ or $67 x 15$ become impossible and the corresponding operations meaningless. Or, let us assume that the field is restricted to odd numbers only. Multiplication is still omni-possible, for the product of any two odd numbers is odd. However, in such a restricted field addition is an altogether impossible operation, because the sum of any two odd numbers is never an odd number. Yet, again, if the field were restricted to prime numbers, multiplication would be impossible, for the simple reason that the product of two primes is never a prime; while, addition would be possible only in such rare cases as when one of the two terms is 2, the other being the smaller of a couple of twin-primes like $2+11=13$. Other examples could be adduced, but even these few will suffice to bring out the relative nature of the words possible, impossible, and meaningless. And, once this relativity is recognized, it is natural to inquire whether through a proper extension of the restricted field the inverse operations of arithmetic may not be rendered as omni-possible as the direct are. To accomplish this with respect to subtraction it is sufficient to adjoin to the sequence of natural numbers zero and the negative integers. The field so created is called the general integer field. Similarly, the adjunction of positive and negative fractions to this integer field will render division omni-possible. The numbers thus created — the integers, and the fractions, positive and negative, and the number zero — constitute the rational domain. It supersedes the natural domain of integer arithmetic. The four fundamental operations, which heretofore applied to integers only, are now by analogy extended to these generalized numbers. All this can be accomplished without a contradiction. And, what is more, with a single reservation which we shall take up presently, the sum, the difference, the product, and the quotient of any two rational numbers are themselves rational numbers. This very important fact is often paraphrased into the statement: the rational domain is closed with respect to the fundamental operations of arithmetic. The single but very important reservation is that of division by zero. This is equivalent to the solution of the equation $x.0=a$. If a is not zero, the equation is impossible, because we are compelled, in defining the number zero, to admit the identity $a.0=0$. There exists therefore no rational number which satisfies the equation $x.0=a$. On the contrary, the equation $x.0=a$ is satisfied for any rational value of x. Consequently, x is here an indeterminate quantity. Unless the problem that led to such equations provides some further information, we must regard $0/0$ as the symbol of any rational number, and $a/0$ as the symbol of no rational number. Elaborate though these considerations may seem, in symbols they reduce to the following succinct statement: if a, b and c are any rational numbers, and a is not zero, then there always exists a rational number x, and only one, which will satisfy the equation $ax+b=c$ This equation is called “linear”, and it is the simplest type in a great variety of equations. Next to linear some quadratic, then cubic, quartic, quintic and generally algebraic equations of any degree, the degree n meaning the highest power of the unknown x in $ax^{n}+bx^{n-1}+cx^{n-2}+...+px+q=0$ But even these do not exhaust the infinite variety of equations; exponential, trigonometric, logarithmic, circular, elliptic, etc., constitute a still vaster variety, usually classified under the all-embracing term transcendental. Is the rational domain adequate to handle this infinite variety? This is emphatically not the case. We must anticipate an extension of the number domain to greater and greater complexity. But this extension is not arbitrary; there is concealed in the very mechanism of the generalizing scheme a guiding and unifying idea. This idea is sometimes called the principle of permanence. It was first explicity formulated by the German mathematician, Hermann Hanckel, in 1867, but the germ of the idea was already contained in the writings of Sir William Rowan Hamilton, one of the most original and fruitful minds of the nineteenth century. I shall formulate this principle as a definition: A collection of symbols infinite in number shall be called a number field, and each individual element in it a number, First. If among the elements of the collection we can identify the sequence of natural numbers. Second. If we can establish criteria of rank which will permit us to tell of any two elements whether they are equal, or if not equal, which is greater; these criteria reducing to the natural criteria when the two elements are natural numbers. Third. If for any two elements of the collection we can devise a scheme of addition and multiplication which will have the commutative, associative, and distributive properties of the natural operations bearing these names, and which will reduce to these natural operations when the two elements are natural numbers. These very general considerations leave the question open as to how the principle of permanence operates in special cases. Hamilton pointed the way by a method which he called algebraic pairing. We shall illustrate this on the natural numbers. If a is a multiple of b, then the symbol $a/b$ indicates the operation of division of a by b. Thus $9/3=3$ means that the quotient of the indicated division is 3. Now, given two such indicated operations, is there a way of determining whether the results are equal, greater, or less, without actually performing the operations? Yes, we have the following: Criteria of Rank. $a/b=c/d$ if $ad=bc$ $a/b > c/d$ if $ad>bc$ $a/b < c/d$ if $ad And we can even go further than that: without performing the indicated operations we can devise rules for manipulating on these indicated quantities: Addition: $(a/b)+(c/d) = (ad+bc)/bd$ Multiplication. $(a/b).(c/d)= (ac)/(bd)$ Now, let us not stipulate any more that a be a multiple of b. Let us consider $a/b$ as the symbol of a new field of mathematical beings. These symbolic beings depend on two integers a and b written in proper order. We shall impose on this collection of couples the criteria of rank mentioned above,i.e., we shall claim that, for instance: $(20/15)=(16/12)$ because 20 x 12 = 15 x 16 $(4/3)>(5/4$) because $(4)( 4) >( 3)(5)$ We shall define the operations on these couples in accordance with the rules which, as we have shown above, are true for the case when a is a multiple of b, and c is a multiple of d, i.e., we shall for instance: $(2/3)+(4/5)=((2)( 5)+(3) ( 4))/((5)( 3))=22/15$ We have now satisfied all the stipulations of the principle of permanence. 1. The new field contains the natural numbers as a subfield, because we can write any natural number in the form of a couple: $1/1$; $2/1$; $3/1$; $4/1$, and so on and on. 2. The new field criteria of rank which reduce to the natural criteria when $a/b$ and $c/d$ are natural numbers. 3. The new field has been provided with two operations which have all the properties of addition and multiplication, to which they reduce when $a/b$ and $c/d$ are natural numbers. And, so these new beings satisfy all the stipulations of the principle. They have proved their right to be adjoined to the natural numbers, their right to be invested with the dignity of the same name “number”. They are therewith admitted, and the field of numbers comprising both old and new is christened the rational domain of numbers. It would seem at first glance that the principle of permanence leaves such a latitude in the choice of operations as to make the general number it postulates too general to be of much practical value. However, the stipulations that the natural sequence should be a part of the field, and that the fundamental operations should be commutative, associative and distributive (as the natural operations are), impose restrictions which, as we shall see, only very special fields can meet. The position of arithmetic,as formulated in the principle of permanence, can be compared to the policy of a state bent on expansion, but desirous to perpetuate the fundamental laws on which it grew strong. These two different objectives — expansion on the one hand, preservation of uniformity on the other — will naturally influence the rules for admission of new states to the Union. Thus, the first point in the principle of permanence corresponds to the pronouncement that the nucleus state shall set the tone of the Union. Next, the original state being an oligarchy in which every citizen has a rank, it imposes this requirement on the new states. This requirement corresponds to the second point of the principle of superposition. Finally, it stipulates that the laws of commingling between the citizens of each individual state admitted to the Union shall be of a type which will permit unimpeded relations between citizens of that state and those of the nucleus state. Of course, I do not want the reader to take this analogy literally. It is suggested in the hope that it may invoke mental associations from a more familiar field, so that the principle of permanence may lose its seeming artificiality. The considerations, which led up to the construction of the rational domain, were the first steps in a historical process called the arithmetization of mathematics. This movement, which began with Weierstrass in the sixties of the 19th century, had for its object the separation of purely mathematical concepts, such as “number” and “correspondence” and “aggregate”, from intuitional ideas, which mathematics had acquired from long association with geometry and mechanics. These latter, in the opinion of the formalists, are so firmly entrenched in mathematical thought that in spite of the most careful circumspection in the choice of words, the meaning concealed behind these words may influence our reasoning. For the trouble with human words is that they possess content, whereas the purpose of mathematics is to construct pure forms of thought. But, how can we avoid the use of human language? The answer is found in the word “symbol”. Only by using a symbolic language not yet usurped by those vague ideas of space, time, continuity which have their origin in intuition and tend to obscure pure reason —- only thus may we hope to build mathematics on the solid foundation of logic. Such is the platform of this school, a school which was founded by the Italian Peano and whose most modern representatives were Bertrand Russell and Alfred North Whitehead. In the fundamental work of the latter men, the Principia Mathematica, they had endeavoured to reconstruct the whole foundation of modern mathematics, starting with clear-cut, fundamental assumptions and proceeding on principles of logic. I confess that I am out of sympathy with the extreme formalism of the Peano-Russell school, that I have never acquired the taste for their methods of symbolic logic, that my repeated efforts to master their involved symbolism have invariably resulted in helpless confusion and despair. This personal ineptitude has undoubtedly coloured my opinion — a powerful reason why I should not air my prejudices here. Yet I am certain that these prejudices have not caused me to underestimate the role of mathematical symbolism. To me, the tremendous importance of this symbolism lies not in these sterile attempts to banish intuition from the realm of human thought, but in its unlimited power to aid intuition in creating new forms of thought. To recognize this, it is not necessary to master the intricate technical symbolism of modern mathematics. It is sufficient to contemplate the more simple, yet much more subtle, symbolism of language. For, in so far as our language is capable of precise statements, it is but a systems of symbols, a rhetorical algebra par excellence. Nouns and phrases are but symbols of classes of objects, verbs symbolize relations, and sentences are but propositions connecting these classes. Yet, while the word is the abstract symbol of a class, it has also the capacity to invoke an image, a concrete picture of some representative element of the class. It is in this dual function of our language that we should seek the germs of the conflict which later arises between logic and intuition. And what is true of words generally is particularly true of those words which represent natural numbers. Because they have the power to evoke in our minds images of concrete collections, they appear to us so rooted in firm reality as to be endowed with an absolute nature. Yet in the sense in which they are used in arithmetic, they are but a set of abstract symbols subject to a system of operational rules. Once we recognize this symbolic nature of the natural number, it loses its absolute character. Its intrinsic kinship with the wider domain of which it is the nucleus becomes evident. At the same time, the successive extensions of the number concept become steps in an inevitable process of natural evolution, instead of the artificial and arbitrary legerdemain which they seem at first. More later… Nalin Some non trivial factorization examples I hope to give you a flavour of some non-trivial factorization examples using the following identity: $a^{3}+b^{3}+c^{3}-3abc = (a+b+c)(a^{2}+b^{2}+c^{2}-ab -bc-ca)$ Example 1. Let n be a positive integer. Factorize $3^{3^{n}}(3^{3^{n}}+1) +3^{3^{n}+1}-1$ Solution. Observe that $3^{3^{n}}(3^{3^{n}}+1) +3^{3^{n}+1}-1 = a^{3}+b^{3}+c^{3}-3abc$ where $a=3^{3^{n-1}}$, $b=9^{3^{n-1}}$, and $c=-1$. Thus, using the above factorization identity, we get the following factorization: $(3^{3^{n-1}}+9^{3^{n-1}}-1)(9^{3^{n-1}}+81^{3^{n-1}}+1-27^{3^{n-1}}+3^{3^{n-1}}+9^{3^{n-1}})$ Example 2. Let a, b, c be distinct positive integers and let k be a positive integer such that $ab+bc+ca \geq 3k^{2}-1$. Prove that $(1/3)(a^{3}+b^{3}+c^{3})-abc \geq 3k$. Solution. The desired inequality is equivalent to $a^{3}+b^{3}+c^{3}-3abc \geq 9k$. Suppose without loss of generality, that $a>b>c$. Then, since a, b, and c are distinct positive integers, we $a-b \geq 1$, and $(b-c) \geq 1$ and $a-c \geq 2$. It follows that $a^{2}+b^{2}+c^{2}-ab-bc-ca = (1/2)((a-b)^{2}+(b-c)^{2}+(c-a)^{2}) \geq (1/2)(1+1+4)=3$ We obtain $a^{3}+b^{3}+c^{3}-3abc = (a+b+c)(a^{2}+b^{2}+c^{2}-ab-bc-ca) \geq 3(a+b+c)$ so it suffices to prove that $3(a+b+c) \geq 9k$ or $(a+b+c) \geq 3k$ But, $(a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2ab+2bc+2ca = a^{2}+b^{2}+c^{2} - ab -bc- ca +3(ab+bc+ca) \geq 3+3(3k^{2}-1) = 9k^{2}$, and the conclusion follows. 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https://kjaer.io/formal-verification/	# OUTPUT The blog of Maxime Kjaer # CS-550 Formal Verification Throughout these notes, vectors are denoted in bold and lowercase (e.g. ). ## Transition systems ### Definition We’ll start by introducing transition systems, a generalization of DFAs. They may be not finite, and not deterministic. Definition: Transition system A transition system is a 4-tuple where: • is the set of states • is the set of starting states • is the transition relation • is the alphabet For and , means that ther is a transition from to on input . A few special cases of this general form exist: • If is finite, we have a finite state machine • If and , then the system is deterministic. ### Traces and reachability Definition: Trace A trace is a finite or infinite sequence, describing steps taken by a transition system: where we require, : A trace may or may not be finite. If they are finite, we can assume that the trace ends with a state . We’ll introduce some notation for traces: Definition: Trace of a transition system is the set of all traces of a transition system , starting from . Definition: Reachable states of a transition system The reachable states are states for which there exists a trace that ends in : To check for reachability, for a finite , we can simply run DFS. ### Relations Let’s study relations more closely. A relation is a directed edge in the transition graph. We follow this edge when we see a given input . If we look at the graph, we may not be interested in the inputs associated to each edge. Therefore, we can construct the edge set : Generally, we’ll use a bar for relations that disregard input. Note that even when is deterministic, can become non-deterministic. Relations can be composed: Definition: Composition of relations Note that composition is not commutative, but is associative. To understand what a composition means, intuitively, we’ll introduce a visual metaphor. Imagine the nodes of the graph as airports, and the edges as possible routes. Let be the routes operated by United Airlines, and be the routes operated by Delta. Then is the set of routes possible by taking a United flight followed by a Delta flight. Definition: Iteration An iteration describes paths of length in a relation . It is defined recursively by: Here, describes the identity relation, i.e. a relation mapping every node to itself. Definition: Transitive closure The transitive closure of a relation is: In our airport analogy, the transitive closure is the set of all airports reachable from our starting airports. Finally, we’ll introduce one more definition: Definition: Image of a set The image of a state set under a relation is the set of states reachable in one step from : We also introduce an alternative notation: The alternative notation may make it simpler to read images; can be read as “ following then following ”. The above definitions lead us to a first definition of reach: Theorem: Definition 1 of reach ### Post We’ll introduce another definition in order to give an alternative definition of reach. We’re still considering a state system . Definition: Post If , define: We also define: This definition of post leads us to another formulation of reach: Theorem: Definition 2 of reach The proof is done by expanding the post: Where: ### Invariants Definition: Invariant An invariant of a system is any superset of the reachable states: A way to think of invariants is that all the reachable states must “satisfy the invariant”, i.e. be included in . Definition: Inductive Invariant An inductive invariant is a set satisfying: Intuitively, the second condition means that you can’t “escape” an inductive invariant by taking a step: there can be ingoing edges to an inductive invariant, but no edges exiting the set. Note that every inductive invariant is also an invariant. Indeed, for an inductive invariant , if then we must also grow to include all states reachable from in order to satisfy the second property. Therefore, and is an invariant. Definition: Inductive strengthening For an invariant , is an inductive strengthening of if: • is an inductive invariant In this case, we have: ### Encoding and storage #### Motivating example Transition systems very quickly get out of hand. Suppose a snack dispenser has different items, and has 10 of each. It can take 500 coins. It then has: They might not all be reachable, but reachability is a separate question. Here, we’re just defining the graph. In any case, this is too large to store explicitly. Therefore, we must find formulas and data structures to represent it more compactly. #### Sequential circuit We’ll consider a deterministic finite-state transition system . We will let be such that , and , or in other words, and . To encode the transition system in bits, we can think of it as 4-tuple of boolean functions: • We can represent the states on bits: • We can represent the alphabet on bits: • We can represent the set of initial states by the boolean function , which maps each state to a boolean (“initial” or “not initial”). • We can represent the transition relation as a function , which is We can think of these functions as a circuit receiving an input , which alters the state , which in turn alters the relation . #### Formula encoding of boolean functions Both the initial states and the transition relation are boolean functions. We will often define them mathematically (e.g. or ), but how do we encode them in bits? We can use a truth table. For instance, the truth table encoding of is a table mapping inputs and to outputs . This is the most efficient encoding if we need to be able to encode any relation: there is no better general encoding. However, most functions can be stored more efficiently. Let us consider the relation function . We can consider an input-output pair as the following condition: We can write this as a propositional formula with variables . This formula should be true exactly when the tuple belongs to . Let , and let be a propositional variable. We can then define: With this notation in hand, we can state the following: Theorem: Formula encoding of boolean functions We can always represent a transition relation as a propositional formula in disjunctive normal form, where: For many boolean functions (, , etc) this formulation will be quite small. #### Auxiliary variables This formula is a tree, where variables are leaves, and operations are internal nodes (much like an AST). Clearly, the variables are shared by many operations, so the more efficient representations exploit this sharing, and represent the tree as a directed acyclic graph (DAG) instead. Note that not only leaves of the tree are shared, but that bigger nodes can also be shared. This leads us to introduce the notion of auxiliary variable definitions. An auxiliary variable represents one such shared node. We can define auxiliary variables directly in the propositional logic by using the operator (which is alternative notation for ). For instance, for a -bit ripple-carry adder can be encoded with: • Initial state • Input numbers • Output The formula for the adder with auxiliary variables (serving as the carry) is: The initial carry is . Every individual adder gets 3 bits of input to add: two bits, and the previous carry. It outputs a carry of 1 iff two or more input bits are 1. It outputs a single bit as the result of the addition, which is the XOR of all inputs. ## Propositional logic ### Definition Propositional logic is a language for representing Boolean functions as formulas. The grammar of this language is: Where denotes variable identifiers. ### QBF Having looked at boolean formulas let’s now look at a small generalization, QBFs: Definition: Quantified Boolean Formulas Quantified Boolean Formulas (QBFs) is built from: • Propositional variables • Constants and • Operators We will use as alternative notation for . A boolean formula is QBF without quantifiers ( and ). ### Free variables Definition: Free variables The free variables of a formula is the set of variables that are not bound by a quantifier: ### Environment Definition: Environment An environment is a partial map from propositional variables to (“false” or “true”). Consider two vectors and of propositional variables. We denote the environment as a mapping given by , We denote the result of evaluating a boolean expression with the environment as . This can evaluate to (“false”) or (“true”). While the formula might seem weird at first (we defined in terms of another , it makes more sense if we think about the first as an AST node and the second as a logical and in the host language: With this notation in hand, we can introduce the following shorthand: Definition: Models We write to denote that is true in environment , i.e. that . ### Substitution Definition: Substitution Let and be propositional formulas, and let be a variable. Let denote the result of replacing each occurrence of in by : We’ll also introduce a general notation to simultaneously replace many variables: denotes the substitution of a vector of variables with a vector of expressions . ### Validity, Satisfiability and Equivalence Definition: Satisfiability A formula is satisfiable Note that if is not satisfiable, it is unsatisfiable, which means . Definition: Validity A formula is valid Theorem: Validity and unsatisfiability is valid is unsatisfiable. The proof should follow quite trivially from the definitions. Definition: Equivalence Formulas and are equivalent environment defined for all free variables in , we have: This means that two formulas are equivalent if and only if they always return the same values for the same environment (assuming it’s defined for all their free variables). Theorem: Equivalence and validity and are equivalent the formula is valid. ## Bounded model checking ### Formula representation of sequential circuits Definition: Sequential circuit We represent a sequential circuit as a 5-tuple where: • is the vector of state variables • is a boolean formula describing the initial state • is a boolean formula called the transition formula • is the vector of auxiliary variables • is the vector of input variables The boolean formula tells us which states we can start in, so it can only contain state variables: The transition formula can only contain state (), next-state (), auxiliary () or input () variables: The sequential circuit is a representation of a transition system , where: • , meaning that the initial states of a transition system is given by an assignment of state variables that verifies the formula • , meaning that the transition relation is given by a mapping of states and inputs to next-states , such that the mapping satisfies the transition formula. Here, the auxiliary variables are existentially quantified so that we can express the criterion without them. ### Inductive invariant checking Given a sequential circuit representation of a transition system , and a formula , how can we check that is an inductive invariant? According to the definition, we require: We’ll ask the SAT solver to check if it’s possible to break either of the two conditions by asking it if it can satisfy either of the following two formulas: • to check if there is an initial state not included in the invariant. • to check if, starting from the invariant and take a step, we can end up outside of the invariant. To understand this condition, it’s useful to think of as determining the assignment of . Then, seeing that contains variables and , it will fix the assignment for the next states . We can then see if the invariant is still true at the next state. If the SAT solver returns UNSAT to both, we have proven that is an inductive invariant. Note that this resembles a proof by induction (because it is!). ### Bounded model checking for reachability How do we check whether a given state is reachable? Often, we’re interested in knowing if we can reach an error state or not; being able to do so would be bad. To simplify the question a little, we can ask whether we can reach this error state in steps. Let be the error formula corresponding to the error state, so . When we talked about circuits, we said that the state and inputs change at each step, so let us denote the state at step as , and the inputs at step as . We’ll construct an error formula that is satisfiable if and only if there exists a trace of length starting from the initial state that satisfies : This formula starts at the initial state, then computes all states , and plugs in the final state in the error formula to see if it can be satisfied. If the SAT solver returns UNSAT, the error state is not reachable. If it returns SAT, the ## Satisfiability checking ### SAT problem The SAT problem is to determine whether a given formula is satisfiable. The problem is NP-complete, but useful heuristics exist. A SAT solver is a program that given a boolean formula either: • Returns SAT and optionally an environment such that • Returns UNSAT and optionally a proof that no satisfying assignment exists ### Formal proof system Let’s consider a set of logical formulas (e.g. propositional logic). Definition: Proof system A proof system is a pair where is a decidable set of inference steps, where: • A set is decidable there is a program to check if an element belongs to it • Given a set , denotes all finite sequences with elements from Definition: Inference step An inference step is a 2-tuple , which we can denote as: We say that from the premises , we derive the conclusion . Definition: Axiom We say that an inference step is called an axiom when , i.e. that it has no premises: Definition: Proof Given a proof system , a proof is a finite sequence of inference steps such that, for every inference step , each premise is a conclusion of a previous step. ### A minimal propositional logic proof system We’ll look into a simple logic called the Hilbert system. We’ll define the grammar of our logic as follows. This grammar defines a set of formulas . This may seem very minimal, but we can actually express many things by combining these. For instance, we can introduce negation (“for free”) as a shorthand: The inference rules are , where: The first two rules are axioms, telling us that an implication is true if the right-hand side is true (), and that implication is distributive (). We might recognize modus ponens in the last rule. We can use these rules to construct a proof of . We’ll draw this proof as a DAG: we can always view a proof as a DAG because of the requirement that the premises of an inference step be a conclusion of a previous step. ### Provability A formula is provable if we can derive it from a set of initial assumptions. We’ll start by formally defining what an assumption even is: Definition: Assumptions Given where , and given a set of assumptions , a derivation from in is a proof in where: In other words, assumptions from are just treated as axioms (i.e. they are rules that have no prerequisites, hence ). A derivation is a proof that starts from assumptions. Definition: Provable We say that “a formula is provable from a set of assumptions ”, denoted , there exists a derivation from in that contains an inference step whose conclusion is . We write (or simply ) to denote that there exists a proof in containing as a conclusion. ### Consequence and soundness in propositional logic Definition: Semantic consequence Given a set of assumptions , where is in propositional logic, and given , we say that is a semantic consequence of , denoted , for every environment that defines all variables in , we have: In other words, iff an environment makes all assumptions true, and is true in an environment when the set of assumptions are all true in that environment, then we call a semantic consequence. Definition: Soundness A step is sound A proof system is sound if every inference step is sound. In other words, a conclusion of step is sound if it is a semantic consequence of the previous steps. A proof is sound if all steps are sound. If is an axiom (which has no precondition, meaning that in the above), this definition of soundness means that is always a valid formula. We call this a tautology. Theorem: Semantic consequence and provability in sound proof systems Let where are propositional logic formulas. If every inference rule in is sound, then implies . This theorem tells us that that if all the inference rules are sound, then is a semantic consequence of if is provable from . This may sound somewhat straightforward (if everything is sound, then it seems natural that the semantic consequence follows from provability), but is a nice way to restate the previous definitions. The proof is immediate by induction on the length of the formal proof. As a consequence implies that is a tautology. ### Proving unsatisfiability Let’s take a look at two propositional formulas and . These are semantically equivalent if and . We can prove equivalence by repeatedly applying the following “case analysis” rule, which replaces a given variable by 0 in and by 1 in : Definition: Case analysis rule This is sound, because if we consider an environment defining , and assume and , then: • if then • if then In either case remains true when and are sound. Strictly speaking, the above rule may not be quite enough, so we’ll also introduce a few simplification rules that preserve the equivalence: Those rules together form the sound system , where: Remember that a set of formulas is satisfiable if there exists an environment such that for every formula , . We can use to conclude unsatisfiability: Theorem: Refutation soundness If then is unsatisfiable Here, means false. This follows from the soundness of . More interestingly, the converse is also true. Theorem: Refutation completeness If a finite set is unsatisfiable, then This means that unsatisfiable . For the proof, we can take the conjunction of formulas in and existentially quantify it to get (i.e. ) ### Conjunctive Normal Form (CNF) To define conjunctive normal form, we need to define the three levels of the formula: • CNF is the conjunction of clauses • A clause is a disjunction of literals • A literal is either a variable or its negation This is a nice form to work with, because we have the following property: if is a clause then there exists a literal such that . We can represent formulas in CNF as a set of sets. For instance: The false value can be represented as the empty clause . Note that seeing an empty clause in CNF means that the whole formula is unsatisfiable. ### Clausal resolution Definition: Clausal resolution rule Let and be two clauses. This rule resolves two clauses with respect to . It says that if clause contains , and clause contains , then we can remove the variable from the clauses and merge them. Theorem: Soundness of the clausal resolution rule Clausal resolution is sound for all clauses and propositional variables . This tells us that clausal resolution is a valid rule. A stronger result is that we can use clausal resolution to determine satisfiability for any CNF formula: Theorem: Refutational completeness of the clausal resolution rule A finite set of clauses is satisfiable there exists a derivation to the empty clause from using clausal resolution. ### Unit resolution A unit clause is a clause that has precisely one literal: it’s of the form where is a literal. Note that the literal in a unit clause must be true. Given a literal we define the dual as and . Unit resolution is a special case of resolution where at least one of the clauses is a unit clause. Definition: Unit resolution Let be a clause, and let be a literal. This is sound (if is true then is false and can thus be removed from another clause ). When applying this rule we get a clause : this gives us progress towards , which is good. ### Equivalence and equisatisfiability Let’s recall that two formulas and are satisfiable iff and . Definition: Equisatisfiability Two formulas and are equisatisfiable is satisfiable whenever is satisfiable. Equivalent formulas are always equisatisfiable, but equisatisfiable formulas are not necessarily equivalent. ### Tseytin’s Transformation Tseytin’s transformation is based on the following insight: if and are two formulas, and we let be a fresh variable, then is equisatisfiable with: Tseytin’s transformation applies this recursively in order to transform an expression to CNF. To show this, let’s consider a formula using : The transformation works by introducing a fresh variable for each operation (we can think of it as being for each AST node): Note that these formulas refer to subterms by their newly introduced equivalent variable. This prevents us from having an explosion of terms in this transformation. Each of these equivalences can be converted to CNF by using De Morgan’s law, and switching between and . The resulting conversions are: Operation CNF Note that the Tseytin transformations can be read as implications. For instance, the transformation can be read as: • If and are true, then is true • If is false, then is false • If is false, then is false It then takes the conjunction of all these equivalences: ### SAT Algorithms for CNF Now that we know how to transform to CNF, let’s look into algorithms that solve SAT for CNF formulas. #### DPLL The basic algorithm that we’ll use is DPLL, which applies clausal resolution recursively until an empty clause appears, or all clauses are unit clauses. This works thanks to the theorem on refutational completeness of the clausal resolution rule. #### Backtracking Perhaps the most intuitive algorithm is to construct a binary decision tree. At each node, we take a random decision. If that leads to a conflict, we go back one step, and take the opposite decision. This is quite simple, but the complexity is exponential. We’ll therefore see some smarter algorithms. #### Non-chronological backtracking We still construct a binary decision tree. Each decision may also force some other variables into a certain value: we will track these implications in a directed graph. In this graph, each node represents a value assignment to a variable; let’s say we color a node blue if it has been set directly by a decision, and in green if the value is a consequence of a decision. Edges go from nodes (which may be blue or green) to their consequences. As we construct the graph, we may create conflicts, meaning that we have two (green) nodes assigning different values to the same variable. In this case, we look at the set of nodes pointing to the conflicting pair of nodes: In the above example, the variables , and point to the conflicting pair of ’s. This means that one of the assignments to these variables was incorrect, because it lead us to the conflict. We can learn from this conflict, and add the following conflict clause to the formula: We can then backtrack to the first decision where we set , or (which may be much earlier than the parent decision in the tree). Every once in a while, it may be useful to completely abandon the search tree (but keep the conflict clauses): this is called a restart. This approach significantly prunes the search tree; by introducing this conflict clause, we’ve learned something that will be useful forever. #### 2-literal watching This algorithm does the standard trick of unit propagation, but avoids expensive book-keeping by only picking 2 literals in each clause to “watch”. We ignore assignments to other variables in the clause. For each variable, we keep two sets of pointers: • Pointers to clauses in which the variable is watched in its negated form • Pointers to clauses in which the variable is watched in its non-negated form Then, when a variable is assigned true, we only need to visit clauses where its watched literal is negated. This means we don’t have to backtrack! ## Interpolation Definition: Interpolant Let and be propositional formulas. An interpolant for and is a formula such that: Note that if these conditions hold, we have . The goal of is to serve as an explanation of why implies . Theorem: Existence and Lattice of Interpolants Let and be propositional formulas such that and let be the set of interpolants of . Then: • If then and ## Linear Temporal Logic ### Definition With bounded model checking, we’ve seen how to check for a property over a finite trace. However, there are certain properties that we cannot prove by just seeing a finite trace. For instance, program termination or eventual delivery cannot be proven with the semantics of propositional logic. This will prompt us to introduce linear temporal logic, with which we can study events on a trace. Let be a boolean formula over state and input variables. We write to say that a trace satisfies formula at time . The rules for the above constructs are: We can interpret until to mean that is true until becomes true. For instance, we can guarantee something until an overflow bit is on, at which point all bets are off. Note that this does not impose any constraints on once is true. Note that next and prev, and since and until are duals of each other2. We can add some derived operations from these: The operation globally can be thought of as meaning “forever, from now on” (where “now” is step ). For instance, means that from now on, every point in the trace satisfying is followed by a point in the trace satisfying at some point. ## Binary Decision Diagrams ### Definition Binary Decision Diagrams (BDDs) are a representation of Boolean functions: we can represent a Boolean function as a directed acyclic graph (DAG). We distinguish terminal from non-terminal nodes. Each edge is labeled with a decision: a solid edge (“high edge”) means the variable is 1, and a dashed edge (“low edge”) means that it is 0. The leaf nodes are labeled with the outcome. For instance, the BDD for is: We can also think of this DAG as a finite automaton, or as a (loop-free)branching program. ### Canonicity The key idea of BDDs is that specific forms of BDDs are canonical, and that BDDs can be transformed into their canonical form. For instance, the previous example can be rewritten as: Let’s define this transformation more formally. We first assign an arbitrary total ordering to variables: variables must appear in that order along all paths. Here, we picked as our ordering. Note that in general, selecting a good ordering is an intractable problem, but decent application-specific heuristics exist. Then, the reduced ordered BDD (RO-BDD) is obtained by: • Merging equivalent leaf nodes (which is what we did above) • Merging isomorphic nodes (same variables and same children) • Eliminating redundant tests (where both outgoing edges go to the same child) Doing this brings us to the following property: Theorem: Unicity of RO-BDD With a fixed variable order, the RO-BDD for a Boolean function is unique In the following, we’ll always refer to RO-BDDs, and just call them BDDs. ### Data structures for BDDs To encode a BDD, we’ll map number all nodes as 0, 1, 2, …, where 0 and 1 are terminals. The variables are also numbered as . We also number the levels in the diagram, according to the total ordering that we selected: level 1 is the root node, and level contains terminal nodes. A data structure for BDDs is the node table , mapping a node to its low child and high child . The node table also contains an entry describing the level in the ordering as well as the name of the variable. For instance, for the example we used above, the table would be: Node number Level and name Low edge High edge 0 3 1 3 2 2 0 1 3 1 2 1 We’ll assume we have some methods for querying this table: • init(T): initialize with 0 and 1 terminal nodes • u = add(T, i, l, h): add node with to • var(u): get of node • low(u): get of node • high(u): get of node We can also define the inverse of a node table, allowing us to find a node if we have all the information about it. We’ll assume that we have some methods to query and update this structure: • init(H): initialize with 0 and 1 terminal nodes • u = lookup(H, i, l, h): get node from • insert(H, i, l, h, u): add an entry from to ### Basic operations on BDDs Somewhat surprisingly, given a BDD for a formula , we can check whether it is a tautology, satisfiable or inconsistent in constant time: • is a tautology the BDD is • is satisfiable the BDD is not • their BDDs are equal3 To insert a node into , we ensure that we’re keeping a RO-BDD by eliminating redundant tests, and preventing isomorphic nodes. If that is not the case, we can update the tabée and the reverse mapping . With this method in hand, we can see how to build a BDD from a formula . The key idea is to use Shannon’s expansion: Here, means that we replace all nodes by their high-edge subtree. In the formula, we can think of it as a substitution of by 1. We call this basic operation Restrict. This breaks the problem into two subproblems, which we can solve recursively: 1. The proof of this identity is in exercise session 1, exercise 2.1.1. It is done by decomposing into existential quantifiers. 2. Or rather, they’re almost duals, because the future is infinite but the past is finite. 3. This is a consequence of the the theorem of unicity of the RO-BDD « Back	2019-12-05 14:36:39	{"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": 0.9209489226341248, "perplexity": 858.6220754503496}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540481076.11/warc/CC-MAIN-20191205141605-20191205165605-00368.warc.gz"}
https://developer.mozilla.org/en-US/docs/Web/MathML/Examples/MathML_Pythagorean_Theorem	# Proving the Pythagorean theorem We will now prove the Pythogorian theorem: Statement: In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. i.e, If a and b are the legs and c is the hypotenuse then. Proof: We can prove the theorem algebraically by showing that the area of the big square equals the area of the inner square (hypotenuse squared) plus the area of the four triangles: $a 2 + b 2 = c 2$ ## Document Tags and Contributors Tags: Last updated by: hbelay,	2017-10-21 23:19:52	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "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": 0.6240728497505188, "perplexity": 374.98508871027343}, "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-2017-43/segments/1508187824899.75/warc/CC-MAIN-20171021224608-20171022004608-00880.warc.gz"}
https://socratic.org/questions/when-you-insert-an-image-a-line-pops-up-that-asks-for-the-image-source-to-be-ins	# When you insert an image, a line pops up that asks for the image source to be inserted. What exactly is to be placed here? ## When I try to insert the url of the site where I found the image, my image disappears! What am I doing wrong? I realize we are expected to have the proper credit for the image, but am puzzled by the disappearance of the image when I try. Oct 16, 2017 The URL of the website that hosts the image. #### Explanation: The image disappears because you're using the URL of the image instead of the URL of the website. The idea here is that anything that any URL ends in png, jpg, gif and so on is the URL of the image, not the URL of the website. So, for example, let's say that you want to add this image to the answer. Here's what you get by using the Filepicker. Now, if you add the URL of the image where it says Enter image source here, you will get So in order to get this right, you need the URL of the website, which, in this case, would be $\to$ https://en.wikipedia.org/wiki/Wikipedia Now the image doesn't disappear anymore and you can add the image and the source to the answer.	2019-10-17 10:17:28	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 1, "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": 0.7570977210998535, "perplexity": 431.8318258756774}, "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-2019-43/segments/1570986673538.21/warc/CC-MAIN-20191017095726-20191017123226-00059.warc.gz"}
https://pharmacyscope.com/nitration-on-benzene/	# Nitration on Benzene Nitration on Benzene: The most commonly used methods of nitration based on the order of their reactivity, include: 1. a mixture of concentrated nitric acid and concentrated sulphuric acid. 2. fuming nitric acid in acetic anhydride. 3. concentrated nitric acid in glacial acetic acid. 4. dilute nitric acid. 5. nitrates of alkaline metals in the presence of sulphuric acid. 6. nitrogen oxides. 7. nitrates of metals in the presence of acetic anhydride and acetic acid. 8. a mixture of nitric and sulphuric acids with glacial acetic acid or acetic anhydride. 9. acetyl nitrate and benzoyl nitrate. 10. nitrates. (a) nitration of aromatic compounds by nitric acid involves the formation of water, i.e., a decrease in the concentration of nitric acid. Dilute nitric acid is a strong oxidizing agent, resulting in the formation of by-products. To avoid this, a nitrating mixture of concentrated nitric and sulphuric acids is usually used. The best yields of nitration products are obtained when 90% sulphuric acid is employed. The nitronium ion, NO; is generated in the mixture of concentrated HNO, and concentrated H2SO4 by the following reactions, Nitric acid acts as a base and accepts a proton from H2SO4 to generate a nitronium ion. The addition of water lowers the nitrating power of this method by reducing the concentration of nitronium ions. Other strong acids like HCIO4, HF, and BF3 are also effective in place of H2SO4. Many salts such as NO2.CIO4, BF4, and CF3.SO; can bring about nitration. Because of their explosive nature, they are not normally used for nitration. (b) A solution of nitric acid in acetic anhydride helps to generate nitronium ions by the following equation. (c) Since, dilute nitric acid contains a very less concentration of nitronium ion, its nitrating ability is due to the presence of a small concentration of nitrous acid present in it. Nitrous acid generates nitrosonium ion, NO. The nitrosated ring then undergoes oxidation by nitric acid to give a nitro compound. This reaction liberates nitrous acid to continue the reaction. Hence, nitration on benzene this system can be used for nitration of only those compounds like phenols which can undergo nitrosation. The general mechanisms by which nitration is found to occur are represented below: Temperature is a critical factor to govern monosubstitution. For example, benzene is converted smoothly into nitrobenzene at 50°-60°C. However, disubstitution is likely to occur if the temperature exceeds 60°C. Phenol and toluene are nitrated more readily than benzene because they contain ortho para-directing substituents (-OH, -CH3), which make it easier for a nitro group to enter the ring. The substituents -CH3, -OCH3, -OC2H5, and -OH accelerate nitration in increasing order, while the substituents -COOH, -SO₂H, and -NO₂ retard it. Make sure you also check our other amazing Article on : Electrophilic Substitution in Pyrrole Spread the love Hello friends I’m Sameer Ray student of the B pharmacy ( Bachelor pharmacy). We tried our best to design this website in the way any pharmacy student would like and love to get.	2023-02-02 21:40:17	{"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": 0.4533962905406952, "perplexity": 10908.696976256047}, "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-2023-06/segments/1674764500041.18/warc/CC-MAIN-20230202200542-20230202230542-00804.warc.gz"}
https://www.math-only-math.com/concept-of-decimal.html	# Concept of Decimal We have already studied about fractions and now we will discuss here about the concept of decimal. Fractions can also be expressed as decimal Fractions. We have already studied about fractions. Fractions can also be expressed as decimal Fractions. 1/10 = 0.1 2/100 = 0.02 → Decimal fractions 3/1000 = 0.003 The dot (.) is 0.1, 0.02, 0.003 is called a decimal point or point. Before we proceed further, let us understand the concept of decimal. Concept of Decimal Observe the place value table shown and note the place value of 1 in each case. Numerals 8651 → 8615 → 8165 → 1865 → Thousands 8 8 8 1 Hundreds 6 6 1 8 Tens 5 1 6 6 Ones 1 5 5 5 We find that: Place value of 1 in 8651 = 1 × 1 = 1 Place value of 1 in 8615 = 1 × 10 = 10 Place value of 1 in 8165 = 1 × 100 = 100 Place value of 1 in 1865 = 1 × 1000 = 1000 We observe that the place value of a digit is increasing ten times as it moves one place from right to left i.e. the place value of 1 is 1 at ones place, 10 at tens place, 100 at hundreds place and 1000 at thousands place. Numerals 1865 → 8165 → 8615 → 8651 → Thousands 1 8 8 8 Hundreds 8 1 6 6 Tens 6 6 1 5 Ones 5 5 5 1 Place value of 1 in 1865 = 1000 Place value of 1 in 8165 = 100 Place value of 1 in 8615 = 10 Place value of 1 in 8651 = 1 We observe that the place value of a digit becomes one tenth as it moves one place from left to right i.e. the place value of 1 is 1000 at thousands place, 100 at hundreds place, 10 at tens place and 1 at ones place. We can extend the place value chart further as follows: 0.1, 0.01, 0.001 etc. are known as decimal fractions. We use fractions to express the numbers smaller than 1. We can also express a number smaller than one by using decimal point. Decimal is derived from ‘decem’ the Latin word which means 10. Remember, 10 is the base of the decimal system. A decimal number has two parts - A whole number and a decimal fraction. A decimal point separates them. It is denoted by a dot (.). It is also called point. Observe the table given below. Four tenths $$\frac{4}{10}$$ 0.4 zero point four Six tenths $$\frac{6}{10}$$ 0.6 zero point six One and five tenths 1$$\frac{5}{10}$$ 1.5 one point five Seven hundredths $$\frac{7}{100}$$ 0.07 zero point zero seven Twenty hundredths $$\frac{20}{100}$$ 0.2 zero point two zero Four hundredths $$\frac{4}{100}$$ 0.04 zero point zero four Four hundred thousandths $$\frac{400}{1000}$$ 0.4 zero point four zero zero Twenty-five hundredths $$\frac{25}{100}$$ 0.25 zero point two five Sixty-seven hundredths $$\frac{67}{100}$$ 0.67 zero point six seven Sixty-seven thousandths $$\frac{67}{1000}$$ 0.067 zero point zero six seven One and three hundredths 1$$\frac{3}{100}$$ 1.03 one point zero three One and forty-two hundredths 1$$\frac{42}{100}$$ 1.42 one point four two Five and sixty-three hundredths 5$$\frac{63}{100}$$ 5.63 five point six three Seven and four hundred sixty-two hundredths 7$$\frac{462}{1000}$$ 7.462 seven point four six two Five and eighty-two hundredths 5 $$\frac{82}{1000}$$ 5.082 five point zero eight two Seven hundredths $$\frac{7}{1000}$$ 0.007 zero point zero zero seven Remember, the digits after the point are always read separately. For example, 1.345 is read as one point three four five and not as one point three hundred and forty-five.	2019-05-23 13:37:36	{"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": 0.4632972478866577, "perplexity": 1720.570391606862}, "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-2019-22/segments/1558232257244.16/warc/CC-MAIN-20190523123835-20190523145835-00328.warc.gz"}
https://www.biostars.org/p/9551991/	zero byte files in sratoolkit.3.0.1-ubuntu64 0 0 Entering edit mode 9 days ago Nemo • 0 I have downloaded sratoolkit for ubuntu. After getting the tar file and unzip it, I added the bin path to the Path variables as well. When I run which fastq-dump, it correctly identifies the path. However, when I run vdb-config -i or vdb-config --interactive nothing happens. It is the same with other commands. When I look into the directory, most of the files have zero byte. I have vdb-config with zero byte, vdb-conifg.3 with zero byte, and vdb-config.3.0.1 with 6,110 KB. What do these mean? Does this mean I should run vdb-config.3.0.1 -i? Sorry, if I am asking naive question. fastq sra-toolkit rna-seq • 295 views 0 Entering edit mode Run vdb-config -i to create an initial config to store data. Save yourself some trouble and get direct fastq download links by using sra-explorer instead of fastq-dump : Post on how to use sra-explorer : find SRA and FastQ download URLs in a couple of clicks 0 Entering edit mode Yes I knew about the vdb-config -i, that bringing up an interactive environment you can change the directory to store data. But my problem is that when I run the command nothing happens. 0 Entering edit mode Are you working in a text terminal? As I recall without X11 that should still produce a text based GUI that allows you do the config. If that is not working then perhaps you have a corrupt download. Try downloading again. 0 Entering edit mode I am using ubuntu terminal on windows machine. 0 Entering edit mode That should not matter. You should still see the config dialog if the program is working. 0 Entering edit mode yes it does not work. I tried with fasterq-dump command too but same as vdb-config command, nothing happens. I also tried to download it gain but all the files are the same as before, so no issue with downloading. Do you think it can be related to zero byte files? I have added an image of folder content here.	2023-02-02 12:31:23	{"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": 0.3923422694206238, "perplexity": 3870.421710164423}, "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/1674764500017.27/warc/CC-MAIN-20230202101933-20230202131933-00670.warc.gz"}
https://golem.ph.utexas.edu/category/2007/06/opetopes_as_trees.html	June 16, 2007 Opetopes as Trees Posted by David Corfield How did this one slip by? Four important players in the saga of $n$-categories, Joachim Kock, André Joyal, Michael Batanin and Jean-François Mascari have combined to write Polynomial functors and opetopes: We give an elementary and direct combinatorial definition of opetopes in terms of trees, well-suited for graphical manipulation (e.g. drawings of opetopes of any dimension and basic operations like sources, target, and composition); a substantial part of the paper is constituted by drawings and example computations. To relate our definition to the classical definition, we recast the Baez-Dolan slice construction for operads in terms of polynomial monads: our opetopes appear naturally as types for polynomial monads obtained by iterating the Baez-Dolan construction, starting with the trivial monad. Finally we observe a suspension operation for opetopes, and define a notion of stable opetopes. Stable opetopes form a least fixpoint for the Baez-Dolan construction. The calculus of opetopes is also well-suited for machine implementation: in an appendix we show how to represent opetopes in XML, and manipulate them with simple Tcl scripts. I see in the bibliography a reference to Notes on Polynomial Functors by Kock. Posted at June 16, 2007 4:51 PM UTC TrackBack URL for this Entry: http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1321	2017-10-24 05:55:55	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "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": 0.7426645159721375, "perplexity": 2640.2959800362914}, "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-2017-43/segments/1508187828178.96/warc/CC-MAIN-20171024052836-20171024072836-00690.warc.gz"}
https://wiki.couchcooperative.org/	# Welcome to the COUCH Wiki! Right now, the wiki is mostly a collection of coordinator role descriptions - if you’re onboarding to a new coordinator role, either at the house or COUCH level, this should be your first stop! ## To do Help improve the COUCH wiki! Wiki to do ## COUCH The following is a must-read for membership coordinators at all houses! While some details of the membership process may vary slightly between the houses, there are some things all houses must do to avoid violating Fair Housing laws and to help us stay organized: Onboarding for all new Board members ## More Info on Wiki COUCH has some scattered wiki presence: on LocalWiki and Wikia. The Wikia wiki in particular is out-of-date and we’d like to put all our content here instead! But we need your help, brave COUCH member! You can edit this page by clicking on the “edit” tab at the top of the screen. To create a new wiki page, just create a link to it and follow the link. To make a link to a new wiki page, put the name of the new page in brackets followed by parenthesis. For example, to create a page named “House Cats,” I would write [House Cats]() in the editing interface. Help is always available through the “Help” link in the sidebar. More details on installing and configurating gitit are available in the Gitit User’s Guide.	2022-05-16 20:58:02	{"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": 0.22819985449314117, "perplexity": 2923.6263004824445}, "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-2022-21/segments/1652662512249.16/warc/CC-MAIN-20220516204516-20220516234516-00590.warc.gz"}
https://zxding.me/archives/djukhs.html	# Fracture analysis of a metal specimen I analyzed a specimen of ductile material with explicit dynamic method about one year ago. Recently, I want to review the model and summarize the process of analysis. ## Stress and strain of material in ABAQUS Stress and strain in the definition of material in abauqs should be converted to true stress $\sigma$ and true plastic strain ${\varepsilon _p}$ from the data obtained in experiments. The stress and strain achieved in experiments are engineering stress ${\sigma _{{\rm{eng}}}}$ and engineering strain ${{\varepsilon _{{\rm{eng}}}}}$, respectively. Considering the incompressibility of plastic material, which means constant volume, we get ${l_0}{A_0} = lA$ in which, ${l_0}$ and ${A_0}$ are initial length and initial area, ${l}$ and ${A}$ are current length and current area. Thus, real stress $\sigma$ can be derived as $\sigma = \frac{F}{A} = \frac{F}{{{A_0}}}\frac{l}{{{l_0}}} = {\sigma _{{\rm{eng}}}}\frac{l}{{{l_0}}} = {\sigma _{{\rm{nom}}}}\left( {1 + {\varepsilon _{{\rm{eng}}}}} \right)$ Engineering strain ${{\varepsilon _{{\rm{eng}}}}}$ can be written as ${\varepsilon _{{\rm{eng}}}} = \frac{{l – {l_0}}}{{{l_0}}} = \frac{l}{{{l_0}}} – 1$ Thus, the relation between real stress, real strain and engineering strain is given as $\sigma = {\sigma _{{\rm{eng}}}}\left( {1 + {\varepsilon _{{\rm{eng}}}}} \right)$ $\varepsilon = \ln \left( {1 + {\varepsilon _{{\rm{eng}}}}} \right)$ Plastic strain can be obtained by subtracting the elastic part from total strain, ${\varepsilon _p} = \varepsilon – {\varepsilon _e} = \varepsilon – \sigma /E$ ## Damage and failure for ductile metal ### #1. Damage initiation The material damage initiation capability for ductile metals is intended as a general capability for predicting initiation of damage in metals, including ductile, shear and other criteria. Ductile criterion is used to predict the onset of damage in material, assuming the equivalent plastic strain is a function of stress triaxiality and strain rate. To simulate the fracture of a coupon, the fracture stress and fracture strain should be calculated with a similar method as true stress and strain. ${\sigma _f} = {F_f}/{A_f}$ ${\varepsilon _f} = \ln \left( {{A_0}/{A_f}} \right)$ As concluded in [1], fracture strain is critical to fracture of coupon, while stress triaxiality and strain rate has little influence. Displacement at failure (will be discussed below) has certain impact on fracture and results with small values are consistent. In this analysis, values in [1] are adopted, shown in the following figures. ### #2. Damage evolution for ductile metals Damage evolution capability for ductile metals assumes that damage is characterized by the progressive degradation of the material stiffness, leading to material failure. It also takes into account the combined effect of different damage mechanisms acting simultaneously on the same material and includes options to specify how each mechanism contributes to the overall material degradation. Two types of damage evolution methods are available in abaqus, displacement type (based on effective plastic displacement) and energy type (based on energy dissipated during the damage process). Displacement type is adopted in damage evolution. Once the damage initiation criterion has been reached, the effective plastic displacement can be defined with the characteristic length of the element. Instantaneous failure will occur if the plastic displacement at failure is specified as 0, which is not recommended by abaqus because it causes a sudden drop of the stress at material point that can lead to dynamic instabilities. The linear damage evolution law defines a truly linear stress-strain softening response only if the effective response of the material is perfectly plastic (constant yield stress) after damage initiation. For mare details, one can refer to Abaqus Analysis User’s Guide. ### # 3. Element removal from mesh One point should be noticed that Status in field output should be ticked, so that elements with strain up to fracture strain will be deleted and you can obtain the fracture deformation you want, or the element will be scratched which would be annoying. The overall scalar stiffness degradation, SDEG, is also available for output. The removal of a solid element takes place, by default, when maximum degradation is reached at any one integration point in abaqus/explicit, while only all of the section points at all the integration locations of an element reach maximum degradation in abaqus/standard. ## Simulation results A coupon is analyzed with dynamic explicit method and displacement control is adopted with one boundary fixed and the other boundary coupled to a reference point. Smooth step for amplitude with a displacement rate of 10mm in 5000 seconds. As can be seen in the above figure, elements with equivalent plastic strain (PEEQ) reaching fracture strain are deleted in viewport. However, the fracture of specimen does not locate in the middle part, which puzzling me. I have checked the geometric information, the boundary and loading conditions, as well as the materials. Other methods like excluding coupling in interaction have also be tested and the same results are achieved. Reference: [1] Ling Li, Behavior of Steel Beam-to-Column Connections in Structural Progressive Collapse. [PhD thesis], Tongji University, 2014. ### 4 comments On Fracture analysis of a metal specimen • Tupa Just checking in to register my support. Your post is well put together Great job! • wetzel Never quit moving. Goals! • Galiano Cool • Formica You make some solid statements. ### Site Footer Recording Life, Sharing Knowledge, Be Happy~	2022-09-26 21:21:19	{"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": 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": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3097890615463257, "perplexity": 1938.4463072198587}, "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-2022-40/segments/1664030334942.88/warc/CC-MAIN-20220926211042-20220927001042-00105.warc.gz"}
https://www.speedsolving.com/threads/a-collection-of-algorithms.708/page-118	# A Collection of Algorithms #### xyzzy ##### Member A variation of this actually is well known (R2 u R2 u' R2 F2 u' F2 u F2; sometimes useful for FMC), but it's still amazing that you managed to find an alg by yourself! #### ArbishAli ##### Member U F2 U' F2 D R2 B2 U B2 D' R2 y U' R2 U R2 D' F2 U' L2 D F2 These are not cool. Last edited: #### Lilas ma ##### Member A variation of this actually is well known (R2 u R2 u' R2 F2 u' F2 u F2; sometimes useful for FMC), but it's still amazing that you managed to find an alg by yourself! Thanks #### Gnome ##### Member Some very late night messing with F2L pairs, no idea why.. found these masterpieces of dread. Code: E-Perm, vertically on U (RUR'U') (U'L'U'L) (LU'L') (L'U2L) (URU'R') (U'LUL') U2 (L'U'L'U'L'ULUL) A-perm, Clockwise UF->UB->UR->UF, also on U (RUR'U') (U'L'U'L) (LU'L') (L'U'RU'R'L) (U'LUL'U) (LUL') (R'U'RU'R'U2R) (RUR'URU2R') U' E-perm A-perm #### ArbishAli ##### Member Some very late night messing with F2L pairs, no idea why.. found these masterpieces of dread. Code: E-Perm, vertically on U (RUR'U') (U'L'U'L) (LU'L') (L'U2L) (URU'R') (U'LUL') U2 (L'U'L'U'L'ULUL) A-perm, Clockwise UF->UB->UR->UF, also on U (RUR'U') (U'L'U'L) (LU'L') (L'U'RU'R'L) (U'LUL'U) (LUL') (R'U'RU'R'U2R) (RUR'URU2R') U' E-perm A-perm Amazing... jk But great work. #### Gnome ##### Member Some more late-ish night missing with F2L pairs, I think I might keep this up until I have a full set of these now because it's fun, although entirely pointless. Code: G Perm, 30 moves R U R' U2' L' U' L U R U' L U2 R' U L2 U2 L2 U R U2 L' U R2' U' R U' R' U2 R U V perm, 32 moves R U R' U2' L' U' L U R U' L' U L U R' U2 R U R' U R U' R' U R U' R' U R U2 R' U' Y Perm, 41 moves R U R' U2 L' U' L2 U' L2' U2 L U2 R U2 R' L U2 L' U2 L U L' U' L U L' U' L U L' U' L U' L2' U' L U' L' U2 L U' G-Perm V-Perm Y-Perm #### LukasCubes ##### Member i am also left handed. #### ObscureCuber ##### Member i am also left handed. You can be left handed and righty domininant in cubing, im ambidextrous but mostly left handed and am still righty domininant. #### Gnome ##### Member You can be left handed and righty domininant in cubing, im ambidextrous but mostly left handed and am still righty domininant. You can be a Lefty yet Right dominant in anything, I myself am a Lefty but through powers not my own (I'm heavily Dyspraxic) I was taught to write with my Right hand and I do so to this day despite doing almost everything else with my Left. When it comes to solving I'm predominantly a right handed solver purely through the virtue that most of the algorithms I've learnt are written with Right handed fingertricks in mind. #### LukasCubes ##### Member You can be a Lefty yet Right dominant in anything, I myself am a Lefty but through powers not my own (I'm heavily Dyspraxic) I was taught to write with my Right hand and I do so to this day despite doing almost everything else with my Left. When it comes to solving I'm predominantly a right handed solver purely through the virtue that most of the algorithms I've learnt are written with Right handed fingertricks in mind. im left handed and i turn most my turns with my right hand.	2020-09-19 15:54:24	{"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": 0.31602805852890015, "perplexity": 13374.916951073223}, "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/1600400192778.51/warc/CC-MAIN-20200919142021-20200919172021-00443.warc.gz"}
https://discuss.codechef.com/t/greedy-algorithms-doubt/39349	Greedy algorithms doubt Hello , can any one please tell me how to be good at greedy algorithms . How can we be sure about the correctness of our thinking for greedy algorithms . I’m practicing problems but i’m unable to come to a perfect claim for a particular greedy algorithm based question ? 2 Likes We are on the same boat , i also want to know this very badly. 1 Like There is actually a good framework of rules one can follow for most of the cases. Thats a pretty long answer. But the best answer to give here is by developing an ability to come up with counter cases. I think you should solve a few B and C level problems on Codeforces (tags hidden for obvious reasons) and you will get the “framework” I am talking about. 4 Likes I never specifically focused on greedy algos…you just solve problems and it becomes easy soon. 5 Likes These are both good answers. Combining them: with any problem that requires you to find a strategy (“how do we place the minimum number of <things> so that <objective>”), one of the first questions you should ask yourself is “can I be greedy?”. You should then, as @vijju123 says, try to find counter examples (quite often, randomised testing again a simple, known-working brute-force approach will expose them pretty quickly). As with anything in maths/ logic, if you can’t find counter-examples, try to ask yourself “why can’t I find them”? Having said all that, Hackerrank have a section specifically dealing with Greedy algorithms if you just want to practice your Greediness exclusively 5 Likes Thanks bro for all this, 1 Like Regarding this - proofs of correctness always involve some degree of creativity, so it’s hard to come up with general advice. However, a broad approach that seems to work is as follows: Say you are tasked with coming up with an optimal approach to a problem (placing telegraph poles to achieve a certain coverage with some kind of notion of a “cost” for a placement, for example) and wish to prove its optimality. Imagine that you have some optimal placement P that doesn’t adhere to your strategy. It must have some kind of difference from a placement that would arise from your strategy - can you find the “first” such difference? Perhaps there is some i such that the first i-1 telegraph poles match your strategy, and the i^\text{th} does not. Is there a way of modifying P so that it has the same (or better!) cost, but where now the first i telegraphs are placed according to your strategy? As a specific example, here’s my solution to Hackerrank’s Mandragora Forest[1]. Mandragora Forest Solution and Write-Up // Simon St James (ssjgz) - 2019-05-21 #define SUBMISSION #ifdef SUBMISSION #define NDEBUG #endif #include <iostream> #include <vector> #include <algorithm> using namespace std; int64_t findBestExperience(const vector<int>& originalH) { const int n = originalH.size(); int64_t bestExperience = 0; auto sortedH = originalH; sort(sortedH.begin(), sortedH.end()); vector<int64_t> sumOfRemainingHealths(n + 1); int64_t sumOfLastHealths = 0; for (const auto h : sortedH) { sumOfLastHealths += h; } for (int numRemaining = n; numRemaining >= 0; numRemaining--) { sumOfRemainingHealths[numRemaining] = sumOfLastHealths; if (n - numRemaining >= 0) sumOfLastHealths -= sortedH[n - numRemaining]; } int64_t currentHealthIfOnlyEat = 1; const int64_t currentExperienceIfOnlyEat = 0; for (int i = 0; i < n; i++) { const int64_t currentExperience = currentHealthIfOnlyEat * sumOfRemainingHealths[n - i]; bestExperience = max(bestExperience, currentExperience); currentHealthIfOnlyEat++; } return bestExperience; } int main(int argc, char* argv[]) { // Very easy one, as evidenced by the ~70% success rate :) // // So: our task is to a) choose the order of Mandragoras to // fight and b) choose whether to Eat or Battle each of these ordered Mandragoras // such that we get the largest possible experience. There are n! ways // to order them, and for each order, 2n ways to choose whether to // Eat or Battle each Mandragora. Could there be a better solution that this O(n! x 2n) one?? // // Obviously, yes ;) It seems intuitively obvious that we might want to Eat the lowest-health // Mandragoras until we built up enough health to maximise the score from Battling the // highest-health ones, and this is indeed the case, as we will see. // // Observation: given an ordered list of k Mandragoras (H(j_1), H(j_2), ... , H(j_k)), the maximum // gain in experience from Eating/ Battling each in turn does not depend on the current // experience, though it does depend on our current health; thus if we currently have h health // it makes sense to define: // // maxExpIncrease(h, (H(j_1), H(j_2), ... , H(j_k))) // // as the maximum increase in experience we can gain if our current health is h and we must // Eat/ Battle, in order, the Mandragoras with healths H(j_1), H(j_2), ... , H(j_k). To // solve this challenge, we must compute: // // initialExperience + maxExpIncrease(initialHealth, (H(i_1), H(i_2), ... , H(i_n))) // // = maxExpIncrease(0, (H(i_1), H(i_2), ... , H(i_n))) // // across all permutations i_1, i_2, ... , i_n of 1, 2, ... , n. // // Theorem 1 // // In a strategy that gives the largest experience gain, we may assume that the permutation // i_1, i_2, ... , i_n of 1, 2, ... , n is the one that places the healths of the Mandragoras // we face into increasing order i.e. H(i_1) <= H(i_2) <= ... <= H(i_n). // // Proof // // Pick any optimal strategy, and let i_1, i_2, ... , i_n of 1, 2, ... , n be the order of // Mandragora's chosen for this strategy. If this sorts H, then we're done; otherwise, pick // the smallest j such that H(i_(j+1)) < H(i_j). // // H(i_1) H(i_2) ... H(i_j) H(i_(j+1)) H(i_(j+2)) ... H(i_n) // // Let h be the current health and e the current experience at the moment after we dealt with // the (j-1)th Mandragora. // // There are four possible scenarios for this strategy: // // i) We Battle H(i_j) and Battle H(i_(j+1)). // // The total experience for this strategy would then be: // // e + h * H(i_j) + h * H(i_(j+1)) + maxExpIncrease(h, (H(i_(j+2)), H(i_(j+3)), ... , H(i_n))) (O1) // // Imagine if we changed this strategy so that H(i_j) and H(i_(j+1)) were swapped, but we still // Battled both H(i_j) and H(i_(j + 1)). // // H(i_1) H(i_2) ... H(i_(j+1)) H(i_j) H(i_(j+2)) ... H(i_n) // ^Battle ^Battle // // The total experience from this modified strategy is then: // // e + h * H(i_(j+1)) + h * H(i_j)) + maxExpIncrease(h, (H(i_(j+2)), H(i_(j+3)), ... , H(i_n))) // // i.e. exactly the same as that of the original strategy, (O1). Thus, in this case, we can make the order of // Mandragora's "more ordered" with respect to health, but not lose any experience points from // doing so. // // ii) We Battle H(i_j), and Eat (H_(j+1)). // // The total experience for this strategy would then be: // // e + h * H(i_j) + 0 + maxExpIncrease(h + 1, (H(i_(j+2)), H(i_(j+3)), ... , H(i_n))) (O2) // // Imagine if we changed this strategy so that H(i_j) and H(i_(j+1)) were swapped, but we still // Battle H(i_j) and Eat H(i_(j + 1)). // // H(i_1) H(i_2) ... H(i_(j+1)) H(i_j) H(i_(j+2)) ... H(i_n) // ^Eat ^Battle // // The total experience for this modified strategy would then be: // // e + 0 + (h + 1) * H(i_j) + maxExpIncrease(h + 1, (H(i_(j+2)), H(i_(j+3)), ... , H(i_n))) // // Subtracting (O2) from this gives: // // H(i_j) // // and since H(i_j) >= 1, this is strictly better than (O2). Thus, in this case, we can make the order of // Mandragora's "more ordered" with respect to health, and actually get a better total experience from // doing so. // // iii) We Eat H(i_j) and Battle H(i_(j+1)) // // The total experience for this strategy would then be: // // e + 0 + (h + 1) * H(i_(j+1)) + maxExpIncrease(h + 1, (H(i_(j+2)), H(i_(j+3)), ... , H(i_n))) (O3) // // Imagine if we changed this strategy so that H(i_j) and H(i_(j+1)) were swapped, but we now // Battle H(i_j) and Eat H(i_(j + 1)). // // H(i_1) H(i_2) ... H(i_(j+1)) H(i_j) H(i_(j+2)) ... H(i_n) // ^Eat ^Battle // // The total experience for this modified strategy would then be: // // e + 0 + (h + 1) * H(i_j) + maxExpIncrease(h + 1, (H(i_(j+2)), H(i_(j+3)), ... , H(i_n))) // // Subtracting (O3) from this gives: // // (h + 1) (H(i_j)) - H_(i_(j+1)) // // and since H_(i_(j+1)) < H(i_j) by assumption, this is positive i.e. again in this case, we can make the order of // Mandragora's "more ordered" with respect to health, and actually get a better total experience from // doing so. // // iv) We Eat H(i_j) and Eat H(i_(j+1)) // // The total experience for this strategy would then be: // // e + 0 + 0 + maxExpIncrease(h + 2, (H(i_(j+2)), H(i_(j+3)), ... , H(i_n))) (O4) // // Imagine if we changed this strategy so that H(i_j) and H(i_(j+1)) were swapped, but we still // Eat H(i_j) and Eat H(i_(j + 1)). // // The total experience for this modified strategy would then be: // // e + 0 + 0 + maxExpIncrease(h + 2, (H(i_(j+2)), H(i_(j+3)), ... , H(i_n))) // // i.e. identical to (O4). // // Thus, in all four cases, we can increase the index of the first disordered pair of healths and end // up with a total experience at least as good as the original; thus, by repeated application, // any optimal strategy that does not sort the Mandragora's in order of health can be transformed into // a strategy at least as good that does sort the Mandragora's in order of health. // // QED // // Theorem 2 // // For any optimal strategy, we can assume that the Mandragora's are ordered in order of increasing health, // and that we initially exclusively Eat Mandragora's until a certain point, after which we exclusively // Battle the remaining Mandragora's. // // Proof // // We can assume that such a strategy has the Mandragora's ordered so that H(i_1) <= H(i_2) <= ... <= H(i_n) // by Theorem 1. // // If a chosen optimal strategy is not of the form "initially exclusively Eat Mandragora's until a certain point, // after which we exclusively Battle the remaining Mandragora's", then there is some lowest j such that // the strategy Battles H(i_j), but then Eats H(i_(j+1)). // // Again, let h and e be the health and experience after dealing with the (j-1)th Mandragora. The total experience // gained from our chosen strategy is then: // // e + h * H(i_j) + 0 + maxExpIncrease(h + 1, (H(i_(j+2)), H(i_(j+3)), ... , H(i_n))) (OBE) // // Imagine if we instead Eat H(i_j) and then Battle H(i_(j+1)). Then the total experience from this modified strategy // is: // // e + 0 + (h + 1) * H(i_(j+1)) + maxExpIncrease(h + 1, (H(i_(j+2)), H(i_(j+3)), ... , H(i_n))) // // Subtracting (OBE) from this gives: // // (h + 1) * H(i_(j+1)) - h * H(i_j) >= (h + 1) * H(i_j) - h * H(i_j) (as H's are in increasing order) // = H(i_j) // >= 1 (as all H's are >= 1) // > 0 // // i.e. our modified strategy is actually better than the original one, and is closer to the "exclusively Eat then // exclusively Battle". Repeated application of this shows that we can turn any strategy into one at least // as good that exclusively Eats then exclusively Battles. // // QED. // // So, almost there: we should sort our H's, then for each k = 0, 1, 2, ... n, compute: // // experience from strategy that Eats first k, then Battles remaining. // // For any k, this is easily computed: after eating the first k, our health is 1 + k and our experience still 0. // The remaining (n - k) each give current health * Health of remaining Mandragora so the total experience for this k // is (1 + k) * (sum of remaining (n - k) healths). With a simple lookup table (sumOfRemainingHealths), we can easy compute this for all // k and compute the best resulting experience all in O(n). Easy-peasy :) int T; cin >> T; for (int t = 0; t < T; t++) { int n; cin >> n; vector<int> H(n); for (int i = 0; i < n; i++) { cin >> H[i]; } cout << findBestExperience(H) << endl; } } This is classed as “Dynamic Programming”, but the strategy for choosing the monster ordering at least is classic Greediness. Check out how the Proof of the optimality of the chosen strategy proceeds I can’t remember the specific problems, but I’ve used this general approach successfully several times now. Edit: Ah - I think Airports is one: https://www.hackerrank.com/challenges/airports/submissions/code/108282231 [1] “Mandragora Forest” is one of the easier problems on Hackerrank, with a ~70% success rate i.e. if you could teach a houseplant to type, it would probably get the right answer. 8 Likes Thanks bro 1 Like	2020-09-24 09:54:03	{"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": 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": 0.6422162652015686, "perplexity": 4207.51742906546}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400214347.17/warc/CC-MAIN-20200924065412-20200924095412-00221.warc.gz"}
http://www.shihua-gong.org/talk/2017copper/?from=@	# A Robust Multilevel Solver for a New Hybridizable Mixed Discretization of Linear Elasticity ### Abstract We present a family of new mixed finite element methods for the linear elasticity and then develop a robust overlapping domain decomposition method to solve the linear system. Our mixed discretization, preserving the symmetry and the exact $H(div)$ conformity in the stress approximation, can be efficiently implemented by hybridization, which reduces the indefinite system to a symmetric positive-semidefinite system. The condition number of the reduced system, which is characterized by a non-inherited bilinear form, depends not only on the grid size but also on the material parameters. By constructing uniformly stable interpolation operators between the non-nested spaces, we prove that our overlapping domain decomposition method converges uniformly with respect to both the grid size and Poisson’s ratio. Numerical experiments are presented to validate our theoretical results. Date Location	2021-04-18 06:07:37	{"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": 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": 0.46995070576667786, "perplexity": 324.2934327961423}, "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-2021-17/segments/1618038468066.58/warc/CC-MAIN-20210418043500-20210418073500-00266.warc.gz"}
https://zbmath.org/?q=an:0678.22012	## Tamagawa numbers.(English)Zbl 0678.22012 The main theorem is that the Tamagawa number $$\tau(G)$$ of a connected reductive algebraic group $$G$$ defined over a number field $$F$$ is invariant under passage from $$G$$ to an inner twist. Since it is known that $$\tau(G)=1$$ if $$G$$ is simply-connected, semisimple and quasisplit over $$F$$ this theorem completes the proof of the Weil conjecture: $$\tau (G)=1$$ for $$G$$ simply-connected, semisimple. A key ingredient in the proof is the $$p$$-adic Euler-Poincaré function introduced here. The orbital integrals of this function are shown to have remarkable properties. In particular, the function may be used in Arthur’s simple trace formula and the formula then reduces to its stable part (this requires the Hasse principle). The stable parts for $$G$$ and an inner twist $$G'$$ are compared and the formula $$\tau(G)=\tau(G')$$ is squeezed out in familiar fashion. The Euler- Poincaré function is also shown to give a quick proof of Rogawski’s theorem on the Shalika germ for the identity element of a $$p$$-adic group. Finally, a theorem attributed to Casselman says that the Euler-Poincaré function is, up to a given constant, a pseudo-coefficient of the Steinberg representation.	2022-07-04 03:14:14	{"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": 0.9510161280632019, "perplexity": 240.9918729800407}, "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/1656104293758.72/warc/CC-MAIN-20220704015700-20220704045700-00101.warc.gz"}
https://codegolf.stackexchange.com/questions/124049/display-random-colored-pixels/124610	# Display random colored pixels I've always liked screens full of randomly colored pixels. They're interesting to look at and the programs that draw them are fun to watch. # The challenge Fill your screen, or a graphical window, with colored pixels. # The rules • Your program must have an even chance of picking all colors (i.e in the range #000000 to #FFFFFF), or all colors that can be displayed on your system. • Your program must continue to display random pixels until manually stopped (it cannot terminate on its own). • Pixels can be any size, as long as your output has at least 40x40 "pixels". • Your program must run at such a speed that it can replace every pixel on the screen/window at least once after running for three minutes. • Your program must choose truly random colors and points to replace, i.e. random with all points/colors equally likely. It cannot just look random. It must use a pRNG or better, and the output cannot be the same every time. • Your program must have an equal chance of picking all colors each iteration. • Your program must replace only one pixel at once. • Your program cannot use the internet nor your filesystem (/dev/random and /dev/urandom excepted). # Example Your output could look like this if stopped at a random time: # The winner The shortest answer in each language wins. Have fun! • By "truly random", I assume pseudorandom is OK (like Math.random() or similar) – OldBunny2800 Jun 2 '17 at 1:29 • @OldBunny2800 It must use a pRNG or better – TheLethalCoder Jun 2 '17 at 13:07 • Since the program can't use the internet, I'll just leave this in a comment: babelia.libraryofbabel.info/slideshow.html – KSmarts Jun 2 '17 at 13:28 • So does the "3 minute limit" apply to setup time, too? Or just once the "program" begins? Asking because I have a Minecraft redstone solution, which runs very quickly once it is going, but takes a while to set up initially (it has to "place" each "pixel" in the 40x40 grid) before it can start changing colors. – BradC Jun 4 '17 at 2:34 • Truly random by definition exclude pseudo-random, yet doesn't define a distribution. I assume you mean a uniform PRNG, where all events are equiprobable and independent of each other. – Dennis Jun 4 '17 at 4:44 # Minecraft 1.12 Redstone Command Blocks, 4,355 2,872 bytes (Size determined by saved structure block file size.) Here is a full YouTube overview, but I'll try to outline the code below. Setup Routine: This sets up the 40x40 grid of Minecraft armor stands. Armor stands are necessary because Minecraft has no way to substitute variables into world coordinates. So the workaround is to refer to the location of these armor stand entities. (impulse) summon armor_stand 2 ~ 1 {CustomName:"A"} /create named armor stand (chain) fill -2 ~ -2 43 ~ 43 stone /create big stone square (chain) fill -1 ~ -1 42 ~ 42 air /leave just a ring of stone (chain) setblock -4 ~ -12 redstone_block /kicks off next sequence This named armor stand is basically our "cursor" to place all the armor stands that we will need. The redstone block in the last step "powers" nearby blocks (including our command blocks), so kicks off the next loop: (repeat) execute @e[name=A] ~ ~ ~ summon armor_stand ~-1 ~ ~ /create new armor stand (chain) tp @e[name=A] ~1 ~ ~ /move "cursor" one block (chain) execute @e[name=A] ~ ~ ~ testforblock ~1 ~ ~ stone /if at end of row, (conditional) tp @e[name=A] ~-40 ~ ~1 /go to start of next row (chain) execute @e[name=A] ~ ~ ~ testforblock ~ ~ ~2 stone /If at last row (conditional) setblock ~6 ~ ~ air /stop looping (conditional) kill @e[name=A] /kill cursor At this point our grid is complete: Random Color Selector The purple repeaters in the center of this picture choose a random color via the following command: (repeat) execute @r[type=armor_stand,r=9] ~ ~ ~ setblock ~ ~-2 ~ redstone_block That "@r[]" is the magic sauce, it selects a random entity in the world that matches the given conditions. In this case, it finds an armor stand inside a radius of 9 blocks, and we've set up 16 armor stands, one for each wool color. Under the selected color, it places a redstone block (which powers the two command blocks on either side). Random Pixel Selector Placing the redstone block under the selected wool color triggers two more command blocks: (impulse) execute @r[type=armor_stand] ~ ~ ~ setblock ~ ~3 ~ wool X (impulse) setblock ~ ~ ~1 air This first line uses our same magic @r command to choose any armor stand on the entire map (no radius restriction, so that includes the 40x40 grid), and places a wool of the selected color above its head. The X determines the color, and ranges from 0 to 15. The second command removes the redstone block so it is ready to go again. I have 5 purple repeater blocks, and redstone works in "ticks" 20 times a second, so I'm placing 100 pixels per second (minus some color overlaps). I've timed it, and I usually get the entire grid covered in about 3 minutes. This was fun, I'll try to look for other challenges that might also work in Minecraft. Huge thanks to lorgon111 for his YouTube Command Block tutorial series. EDIT: Made some serious reductions in the size of the saved structure, now at 2,872 saved bytes: 1. Scooted things in a bit (in all 3 dimensions) so I could reduce the overall size of the saved area. 2. Changed the different colored wools to stone, they were just decorative anyway. 3. Removed the glowstone lamp. 4. Changed all air blocks to void blocks (the red squares). Tested by pulling the saved structure into a new world, everything still works as designed. Walk through is in my YouTube video, but here are the steps: 1. In Minecraft 1.12, create a new creative superflat world using the "Redstone Ready" preset. Make it peaceful mode. 2. Once the world exists, copy the NBT file into a new \structures folder you create under the current world save. 3. Back in the game, do /give @p structure_block, and /tp @p -12, 56, -22 to jump to the right spot to get started. 4. Dig a hole and place the structure block at -12, 55, -22. 5. Right-click the structure block, click the mode button to switch it to "Load". 6. Type in "random_pixels", turn "include entities" ON, and click "Load" 7. If it finds the structure file, it will preview the outline. Right-click again and click "load" to bring the structure into the world. 8. Press the button to run the setup routine. 9. When it completes, flip the switch to run the wool randomization. • Holy crap... you actually did it. And it's your first answer, wow! Welcome to the site, and this is an amazing way to start! – MD XF Jun 5 '17 at 1:59 • @MDXF In addition to your +10, I'm giving +60 – NoOneIsHere Jun 5 '17 at 2:03 • @NoOneIsHere All right :P But if you have an answer, it'll have to be +100... – MD XF Jun 5 '17 at 2:05 • I am going to try and golf this if you give the the world file (I will not post another answer but just give you the file) – Christopher Jun 7 '17 at 16:07 • Well, shortest answer in each language wins and this is the shortest answer in Minecraft :P have an extra +15 – MD XF Jun 7 '17 at 16:43 # sh + ffmpeg, 52 bytes ffplay -f rawvideo -s cif -pix_fmt rgb24 /dev/random Does ffmpeg count as an esolang? :D Sadly the pix_fmt is required, as ffmpeg defaults to yuv420p. That fails the "must have equal likelihood of every possible pixel color" requirement. Conveniently, cif is a shortcut for a fairly large video size that uses less space than "40x40". Unsurprisingly, optimizing this gif with gifsicle did absolutely nothing. It's 4MiB. • Does this "replace only one pixel at once"? – Scott Milner Jun 3 '17 at 21:08 • Technically? ffmpeg will wait for the entire frame to be filled with pixels before displaying it, though. I misread "must only replace 1 pixel at a time" as "may". :/ – Una Jun 3 '17 at 23:42 • And of course gifsicle couldn't save any bytes in the gif: there is no way to compress arbitrary data and the random pixels are all chaotically arranged, meaning that the most efficient way of encoding them is one pixel at a time. Numberphile (I think?) had a video on "what is information" and did random noise to intentionally mess with YouTube's compression. VSauce had a video on video compression at one point too, but I forget what it was called. – Draco18s Jun 30 '17 at 17:45 • @Draco18s, I know, the note about it being incompressible was making fun of myself for trying to run gifsicle on it anyway. – Una Jun 30 '17 at 19:15 • Gifsicle is the wrong tool. Precomp gets it down from 4 MB to 3 MB :) That's because the GIF algorithm expands the randomness instead of compressing it and Precomp reverses this. – schnaader Oct 17 '17 at 16:01 # C on POSIX, 989695 92 bytes -3 thanks to Tas #define r rand() f(){for(srand(time(0));printf("\e[%d;%dH\e[%d;4%dm ",r%40,r%40,r%2,r%8););} This chooses between 16 colors (dark grey, red, green, blue, orange, cyan, purple, light grey, black, pink, light blue, yellow, light cyan, magenta, white) and prints them directly to the terminal. Note that if your GPU is too slow, this may seem like it's updating the entire screen at once. It's actually going pixel by pixel, but C is fast. Alternate solution that makes the colors more distinct: f(){for(srand(time(0));printf("\e[%d;%dH\e[%d;3%dm█",rand()%40,rand()%40,rand()%2,rand()%8););} Proof that it goes pixel by pixel (screenshot from alternate program): Wow, that looks almost 3-dimensional... • It's missing a ; to compile correctly but seems to work great other than that! B-) – cleblanc Jun 1 '17 at 20:28 • @cleblanc Oh, you're right! I forgot to add that after I switched from while to for. – MD XF Jun 1 '17 at 20:39 • You could probably shave a few bytes by #define r rand() and then using r%40, r%40, r%2, r%8 – Tas Jun 2 '17 at 4:26 • By not using 32-bit color this does not meet the requirements. – wberry Jun 4 '17 at 5:49 • @wberry "Your program must have an even chance of picking all colors (i.e in the range #000000 to #FFFFFF), or all colors that can be displayed on your system." These are all the colors that the POSIX terminal can display. – MD XF Jun 4 '17 at 17:31 # JS+HTML 162+32 (194) 124+13 (137) bytes Thanks to Luke and other commenters for saving me lots of bytes. r=n=>nMath.random()\|0 setInterval("b=c.getContext2d;b.fillStyle='#'+r(224).toString(16);b.fillRect(r(99),r(99),1,1)",0) <canvas id=c> • Actually <canvas id=c> should be sufficient as I think the default size for the canvas-element is 300 x 150 px and so exceeds your "viewport size" of 99 x 99 px. Anyway, nice solution. – insertusernamehere Jun 1 '17 at 21:52 • @insertusernamehere, yes, thanks – Octopus Jun 1 '17 at 21:57 • Doesn't work in Firefox now though. It does work in Chrome – ETHproductions Jun 1 '17 at 22:09 • OK, OK, I've got it. Save 2 bytes by doing b=c.getContext2d. (Not sure how to format this, but if "2d" is a template literal, it doesn't need the parentheses.) (Deleting my previous suggestions.) – Rick Hitchcock Jun 1 '17 at 22:26 • Along with Rick's suggestion, use a with statement to get rid of all instances of b.: r=n=>nMath.random()\|0 setInterval("with(c.getContext2d)fillStyle='#'+r(2*24).toString(16),fillRect(r(99),r(99),1,1)") – darrylyeo Jun 2 '17 at 16:16 # MATL, 28 bytes 40tI3$l3l2$r,40Yr]4$Y(t3YGT Try it at MATL Online. I have added a half-second pause (.5Y.) to this version. Explanation 40 % Push the number literal 40 to the stack t % Duplicate I % Push the number 3 to the stack 3$l % Create a 40 x 40 x 3 matrix of 1's (40 x 40 RGB image) % Do...while loop 3l1$r % Generate 3 random numbers (RGB) , % Do twice loop 40Yr % Generate two integers between 1 and 40. These will be the ] % row and column of the pixel to replace 4$Y( % Replace the pixel with the random RGB value t % Make a copy of the RGB image 3YG % Display the image T % Push a literal TRUE to create an infinite loop • I love it! ---- – MD XF Jun 2 '17 at 1:28 • out of curiosity, how did you giffify your code output? – Octopus Jun 2 '17 at 5:23 • @Octopus I just used LICEcap. One of these days, I'm going to incorporate animated GIF output into the online compiler. – Suever Jun 2 '17 at 11:36 • Seeing as how this answer uses less commands/methods to achieve a similar result, couldn't you convert it to MATL to possibly achieve a lower score? – MD XF Nov 26 '17 at 5:16 # TI-BASIC (84+C(S)E only), 37 35 bytes :For(A,1,5! :For(B,1,5! :Pxl-On(A,B,randInt(10,24 :End :End :prgmC //"C" is the name of this program Due to hardware limitations, this will eventually crash, since every time a program is nested within a program in TI-BASIC, 15 KB of RAM are allocated to "keep a bookmark" in the parent program. This would run fine on a "theoretical" calculator with infinite RAM, but if we want it to run indefinitely on a real calculator, we can just wrap it in a While 1 loop for an extra 2 bytes: :While 1 :... :End The TI-83 family calculators with color screens (TI 84+CE and CSE) support 15 colors. They have color codes 10 through 24. This cycles through all the pixels in a 120 by 120 (5!) square and assigns each a random color. Result: • Yay, another BASIC answer! Just to be sure, does this run forever? – MD XF Jun 2 '17 at 1:54 • @MDXF Now it does! ;-). Missed that part the first time. +5 bytes. – Scott Milner Jun 2 '17 at 1:57 • You can make the for loops start at 0 if you want to cover 100x100 pixels. – kamoroso94 Jun 2 '17 at 13:30 • @kamoroso94 Ah, true. I could also do 5! if I wanted 120 pixels. – Scott Milner Jun 2 '17 at 15:13 • @MDXF TI-BASIC is token-based. I.e. For( is 1 byte, Pxl-On( is 1 byte, randInt( is 2 bytes, etc. – Scott Milner Jun 2 '17 at 22:18 # Bash + coreutils, 59 56 bytes for((;;)){ printf "\e[48;5;shuf -i 0-16777215 -n1m ";} \e[48;5;COLORm is the escape secuence to background color. Each "pixel" has the chance to be in the [0..16777215] range every time. • Wow, that's a really good idea! +1 – MD XF Jun 2 '17 at 17:44 • You can shove off 3 bytes, if you use jot: for((;;)){ printf "\e[48;5;jot -r 1 0 16777215m ";} – Moreaki Jun 3 '17 at 14:27 • I stole this idea and golfed it down to 45 bytes (\e is 1 escape char, and shuf \| xargs makes it shorter (no need for -n 1, and no for). codegolf.stackexchange.com/a/158142/7017 . I can delete it if you want (I'm not really sure if I should just propose this as a comment instead of posting it. My goal is to get a bit of points to one day bounty the amazing answers to the amazing "tetris in game of life" challenge & answer! see codegolf.stackexchange.com/q/11880/7017 – Olivier Dulac Mar 14 at 18:52 # MATLAB, 56 bytes x=rand(40,40,3);while imagesc(x),x(randi(4800))=rand;end Output looks like the image below. One "pixel" changes at a time, and only one of the RGB-colors changes. Why? The colors in MATLAB are represented as a 3D-matrix, one layer for R,G and B. The code above changes only one of the layers per iteration. All pixels and all layers can be changed, so if you wait a bit all colors are equally possible in all positions. Add pause(t) inside the loop to pause t seconds between each image. You must stop it with Ctrl+C. • Clever use of the handle to the graphics object as the conditional in the where – Suever Jun 1 '17 at 20:56 • A rule has been clarified - Your program must have an equal chance of picking all colors/points each iteration. Does your program fulfill this? – MD XF Jun 1 '17 at 21:43 r=_=>255Math.random()\|0;setInterval('x=c.getContext2d;x.fillRect(r(),r(),1,1,x.fillStyle=rgb(${[r(),r(),r()]}))') <canvas id=c> • This will produce evenly distributed RGB colors. You cannot use Hex colors without proper padding as numbers like #7 is not a valid color, or #777 and #777777 are the same color (2x the odds) • The canvas element is 300x150 by default, but I'm actually drawing on a 255x255 square, so there are off canvas pixels, so the effective area is 255x150. • Works only on Google Chrome. • Great first post! Welcome to the site! – MD XF Jun 3 '17 at 18:53 # Excel VBA, 131 102 85 Bytes Anonymous VBE immediate window function that uses a helper function (see below) to output an array of randomly colored cells to the range A1:AN40 of the activesheet object. Note: This solution is restricted to 32-Bit installs of MS Excel (and therefore of Office as a whole) as 8^8 will not compile on 64-Bit versions of VBA Randomize:Cells.RowHeight=48:For Each c In[A1:AN40]:c.Interior.Color=(8^8-1)Rnd:Next ## Sample Output ## Previous Version Randomize:Cells.ColumnWidth=2:For Each c In Range("A1:AN40"):c.Interior.Color=RGB(n,n,n):Next ## Helper Function Outputs a random int in the range [0,255] Function n n=Int(255Rnd) End Function # C#, 369288 287 bytes namespace System.Drawing{class P{static void Main(){var g=Graphics.FromHwnd((IntPtr)0);var w=Windows.Forms.Screen.GetBounds(Point.Empty);for(var r=new Random();;)g.FillRectangle(new SolidBrush(Color.FromArgb(r.Next(256),r.Next(256),r.Next(256))),r.Next(w.Width),r.Next(w.Height),1,1);}}} Saved 88 bytes thanks to @CodyGray. A full program that get's the handle to the screen and it's size and then starts randomly drawing pixels on it. Note that this might grind your graphics card to a halt when ran. Also if the screen or any control decides to repaint at any time the pixels will be lost and have to be redrawn. Note: When running keep focus on the window as to kill it you have to either Alt+F4 or press the close button and doing so when you can't see the screen is a bit hard. I couldn't record this working, with ScreenToGif, as that kept forcing a repaint so the pixels would get removed. However, here is a screenshot of it running after about 10-15 seconds, any longer and I think I may have ground my PC to a halt! The gap in the top right corner is where the screen forced a repaint just as I took the screenshot. Full/Formatted version: namespace System.Drawing { class P { static void Main() { var g = Graphics.FromHdc((IntPtr)0); var w = Windows.Forms.Screen.GetBounds(Point.Empty); for (var r = new Random();;) g.FillRectangle(new SolidBrush(Color.FromArgb(r.Next(256), r.Next(256), r.Next(256))), r.Next(w.Width), r.Next(w.Height), 1, 1); } } } A version for 308 227 226 bytes that only draws on the region 0-40: namespace System.Drawing{class P{static void Main(){var g=Graphics.FromHdc((IntPtr)0);for(var r=new Random();;)g.FillRectangle(new SolidBrush(Color.FromArgb(r.Next(256),r.Next(256),r.Next(256))),r.Next(40),r.Next(40),1,1);}}} Example output for this one: • Graphics.FromHwnd((IntPtr)0) would also work, and be much shorter than P/Invoking GetDC. Of course, your code is still like Tas's, and leaks resources like crazy, which is why it grinds your machine to a halt after only a short time of running. – Cody Gray Jun 2 '17 at 9:14 • @CodyGray Oh I know it leaks, should still work within the challenges specs though. And GetDC does not return IntPtr.Zero so sending that to FromHdc will not work. – TheLethalCoder Jun 2 '17 at 10:22 • Read the suggestion more carefully. :-) I'm not suggesting that you call FromHdc, I'm suggesting that you call FromHwnd. Notice that when you call GetDC to obtain the DC for a window handle, you pass in a NULL pointer (0) as the window handle. – Cody Gray Jun 2 '17 at 12:37 • @CodyGray Ah yes that works, thanks! They're named to similar, should have read your comment more carefully... – TheLethalCoder Jun 2 '17 at 12:41 # C# Console, 233220189 188 bytes namespace System{using static Console;class P{static void Main(){for(var r=new Random();;){BackgroundColor=(ConsoleColor)r.Next(16);SetCursorPosition(r.Next(40),r.Next(40));Write(" ");}}}} Uses "all" (windows) 16 console colors. Thanks for the "feature" to be able to alias classes in C# via the using directive. Edit #1 • Removed some spaces. • Removed zeros from Random.Next() • Went to namespace system{...} Edit #2 • Minimum size of grid is 40x40 • One byte by declaring Random in for loop header • Removed public from Main method Edit #3 Turns out using C=Console; is not the best there is. using static Console is much like the VB.Net way to "Import" classes I give up: TheLethalCoder made this happen Original code for adapting window size at 207 bytes: namespace System{using static Console;class P{static void Main(){for(var r=new Random();;){BackgroundColor=(ConsoleColor)r.Next(16);SetCursorPosition(r.Next(WindowWidth),r.Next(WindowHeight));Write(" ");}}}} Original Image: • namespace System should save bytes, r.Next can remove the zero I believe, Remove irrelevant white space, – TheLethalCoder Jun 2 '17 at 12:23 • Also just titling C# is fine – TheLethalCoder Jun 2 '17 at 12:23 • Thanks! I leave Console in the title since I feel for this challenge it provides valuable context. I don't see how namespace System{} is shorter than using System;. Care to explain? – MrPaulch Jun 2 '17 at 12:30 • I had this strong but unfounded memory that public is nescessary for the Main. Unfounded memories are the best! – MrPaulch Jun 2 '17 at 12:52 • Thanks. Will not give up! :) Will use what I learned on the next challenge! – MrPaulch Jun 2 '17 at 13:06 ## Processing, 90 bytes void draw(){float n=noise(millis());int i=(int)(n9999);set(i%99,i/99,(int)(n(-1<<24)));} expanded and commented: void draw(){ float n=noise(millis());//compute PRNG value int i=(int)(n9999); //compute 99x99 pixel index set(i%99,i/99, //convert index to x,y (int)(n(-1<<24))); //PRNG ARGB color = PRNG value * 0xFFFFFFFF } Ideally I could use a pixel index instead of x,y location, but Processing's pixels[] access requires loadPixels() pre and updatePixels() post, hence the use of set(). point() would work too, but has more chars and requires stroke(). The random area is actually 99x99 to save a few bytes(instead of 100x100), but that should cover 40x40 with each pixel in such an area to be replaced. Perlin noise() is is used instead of random() to keep it more pseudo-random and a byte shorter. The value is computed once, but used twice: once for the random position, then again for the colour. The colour is actually ARGB(00000000 to FFFFFFFF) (not RGB) (bonus points ? :D). # Python, 133 bytes I'm not quite sure if this fits the specs, because it's on a canvas in a 40x40 area. from turtle import* from random import* ht() up() speed(0) R=randint while 1:goto(R(0,39),R(0,39));dot(1,eval('('+'R(0,255),'3+')')) Try it online - version without eval, which doesn't work in Trinket • Note quite sure why, maybe your turtle dist is different from mine, but my default canvas is 300 by 400 while the default colormode is 1.0 not 255. – Jonathan Allan Jun 1 '17 at 21:30 • Trinket.io is pretty unique. – mbomb007 Jun 1 '17 at 21:37 • Yeah - I just tried changing colormode and it wouldn't have any of it :/ – Jonathan Allan Jun 1 '17 at 21:38 • Why does dot(1,R(0,255),R(0,255),R(0,255)) not work either? dot is meant to accept (width,color). It's not Python :p – Jonathan Allan Jun 1 '17 at 21:40 • Because Trinket.io uses skulpt.js, which is an incomplete implementation of Python. – mbomb007 Jun 1 '17 at 21:44 # JavaScript using Canvas 340 316 324 bytes function r(t,e){return Math.floor(eMath.random()+t)}function f(){x.fillStyle="rgba("+r(1,255)+","+r(1,255)+","+r(1,255)+", 1)",x.fillRect(r(0,40),r(0,40),1,1)}c=document.createElement("canvas"),c.width=40,c.height=40,x=c.getContext("2d"),document.getElementsByTagName("body")[0].appendChild(c),c.interval=setInterval(f,1); full version • Welcome to the site! :) – DJMcMayhem Jun 1 '17 at 20:21 • Does this honor the rule, "Your program must continue to display random pixels until manually stopped (it cannot terminate on its own)."? – Octopus Jun 1 '17 at 20:44 • This appears to simply generate a bunch of random pixels when run. It does not continue to display pixels one by one until stopped. – MD XF Jun 1 '17 at 20:48 • Seems that I missed that rule. I mistakenly thought the goal was to create an image not an animation. I will update my answer when I return to my computer. – Tim Penner Jun 1 '17 at 21:01 • @MDXF it's animated now – Tim Penner Jun 1 '17 at 22:34 # Processing, 112 bytes void setup(){size(40,40);}void draw(){stroke(random(255),random(255),random(255));point(random(40),random(40));} I can't guarantee that every pixel is replaced every 3 minutes, but looking at it it appears to be doing so. At least the odds of it missing a pixel, out of 1600 total, updating a random one 30x per second, totaling 5400 updates per 3 minutes, makes it unlikely that one would be missed. Ungolfed: The program is really straightforward. Open a window at 40x40 pixels, and every frame (default 30 per second) get a random color, and draw a point at a random coordinate between 0 and the parameter. 40 for pixel coordinates, 255 for colors. void setup() { size(40,40); } void draw() { stroke(random(255),random(255),random(255)); point(random(40),random(40)); } • Can you remove the size(40,40); and change point to point(random(99),random(99));? (or 100 instead if 99 isn't working) – Cows quack Jun 2 '17 at 5:51 • @KritixiLithos in Processing, the size function MUST be the first line of code in the setup function (reasons why are complicated), so I can't get around that. I can change all occurrences of 40 to 99, I just didn't since it didn't save me any space. I mainly went with the minimum size to increase the odds that every pixel will be refreshed in a 3-minute period – Cody Jun 2 '17 at 15:40 • Heh, that was what I first thought as well, until someone told me about it and it somehow worked out, as you can see here in a submission of mine. – Cows quack Jun 2 '17 at 15:43 • Removing the whole setup() function on the latest version of Processing on my machine works fine without errors. – Cows quack Jun 2 '17 at 15:48 • I guess my version of Processing is quite old, and I don't use it enough to bother upgrading right now. – Cody Jun 2 '17 at 15:49 # HTML+SVG+PHP, 245 Bytes <?$u=$_GET;$u[rand()%40][rand()%40]=sprintf("%06x",rand()%16777216);echo'<meta http-equiv="refresh" content="0.1; url=?'.http_build_query($u).'" /><svg>';foreach($u as$x=>$a)foreach($a as$y=>$c)echo"<rect x=$x y=$y width=1 height=1 fill=#$c />"; Expanded $u=$_GET; # Get the Url $u[rand()%40][rand()%40]=sprintf("%06x",rand()%16777216); # Set One Value in a 2 D Array 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fill=#8ca3dd /><rect x=29 y=11 width=1 height=1 fill=#376e46 /><rect x=29 y=20 width=1 height=1 fill=#11e2cf /><rect x=29 y=37 width=1 height=1 fill=#24b8de /><rect x=24 y=10 width=1 height=1 fill=#a906da /><rect x=24 y=36 width=1 height=1 fill=#ae0516 /><rect x=24 y=8 width=1 height=1 fill=#e0b9b1 /><rect x=24 y=27 width=1 height=1 fill=#29b27b /><rect x=24 y=33 width=1 height=1 fill=#78ea3e /><rect x=24 y=7 width=1 height=1 fill=#e5147e /><rect x=24 y=11 width=1 height=1 fill=#ce7084 /><rect x=24 y=23 width=1 height=1 fill=#78f645 /><rect x=24 y=25 width=1 height=1 fill=#a01f02 /><rect x=24 y=4 width=1 height=1 fill=#e4340c /><rect x=24 y=16 width=1 height=1 fill=#9b69d7 /><rect x=21 y=31 width=1 height=1 fill=#58ca7d /><rect x=21 y=39 width=1 height=1 fill=#037cb5 /><rect x=21 y=36 width=1 height=1 fill=#097454 /><rect x=21 y=28 width=1 height=1 fill=#71d744 /><rect x=21 y=38 width=1 height=1 fill=#10457c /><rect x=15 y=2 width=1 height=1 fill=#f4bf09 /><rect x=15 y=7 width=1 height=1 fill=#90357d /><rect x=15 y=27 width=1 height=1 fill=#6079ba /><rect x=15 y=5 width=1 height=1 fill=#cff723 /><rect x=15 y=17 width=1 height=1 fill=#54a6db /> ## C++ on Windows, 125 bytes #include<Windows.h> #include<ctime> #define r rand()%256 int main(){for(srand(time(0));;)SetPixel(GetDC(0),r,r,RGB(r,r,r));} Newlines necessary and included in byte count. Loops forever, randomly picks a position between 0 and 255 (inclusive) for row and column values, assigns random R,G,B values between 0-255 (inclusive) • I realize you're golfing this, but man that leak of a device context each time through the loop really gets my spidey-senses tingling! – Cody Gray Jun 2 '17 at 9:09 • Save 15 bytes by making it C on Windows instead of C++ on Windows, as C compilers will let you leave off the #includes. (And yes, there are non-Visual Studio C compilers for Windows.) – MD XF Jun 2 '17 at 22:06 # Python 3.6 + Tkinter, 281 bytes from tkinter import from random import* from threading import* a=randrange x=40 d={"width":x,"height":x} w=Tk() c=Canvas(w,d) c.pack() i=PhotoImage(d) c.create_image((20,20),image=i) def r(): while 1:i.put(f"{a(0,0xffffff):0>6f}",(a(0,x),a(0,x))) Thread(r).start() mainloop() • As tkinter is a standard library, you don't need to include it in the header – caird coinheringaahing Jun 1 '17 at 20:20 • I get the same error -- Windows 10, Python 3.6.0 I can guess why init is seeing four passed parameters -- the method is being passed a "self" parameter implicitly. But I have no idea why that is generating an error, since the documentation shows the Canvas constructor accepts three parameters (master, x, y) as shown in the code. – CCB60 Jun 2 '17 at 5:03 • Should be fixed now. – Martmists Jun 2 '17 at 8:52 • also @Ilikemydog Tkinter is not always in stdlib. On windows it's optional when installing, and I don't think my arch partition has Tkinter installed either. Might as well include it. – Martmists Jun 2 '17 at 12:09 • ther is typo - "img" instead of "i". And still it does not work, this time I get c.create_image((20,20),i) File "C:\Python36\lib\tkinter\__init__.py", line 2483, in create_image return self._create('image', args, kw) File "C:\Python36\lib\tkinter\__init__.py", line 2474, in _create (args + self._options(cnf, kw)))) _tkinter.TclError: unknown option "pyimage1" – Mikhail V Jun 2 '17 at 19:39 # JavaScript (ES7) + HTML using SVG, 129 + 10 = 139 bytes SVG version, heavily inspired by @Octopus's <canvas> approach. ### JS r=n=>nMath.random()\|0 setInterval('s.innerHTML+=<rect x=${r(40)} y=${r(40)} fill=#${r(2*24).toString(16)} width=1 height=1>') ### HTML <svg id=s> Since the Stack Snippet likes to break the script by parsing the <rect> tag, here's a CodePen. # 6502 Assembly, 92 bytes loo: lda $fe sta$00 lda $fe and #$3 clc adc #$2 sta$01 lda $fe ldy #$0 sta ($00),y jmp loo Output: Explanation: loop: lda$fe ; accumulator = random sta $00 ; store accumulator lda$fe ; accumulator = random and #$3 ; accumulator &= 3 clc ; clear carry adc #$2 ; accumulator += 2 sta $01 ; store accumulator lda$fe ; accumulator = random ldy #$0 ; register Y = 0 sta ($00),y ; store register Y jmp loop ; loop • What is yoiur target 6502 platform here? BBC Micro? Apple ][? Commodore VIC-20 etc...? – Shaun Bebbers Mar 15 at 11:49 • Do label names have to be at least three characters long? – Jonathan Frech Sep 7 at 1:12 ## LOGO, 71 bytes The only language that I know of, being not esoteric/specially designed for codegolfing, and have forever function. Any idea on the map part? pu forever[setxy random 100 random 100 setpixel map[random 256][1 1 1]] # shortC, 66 56 bytes Dr rand() AOZR"\e[%d;%dH\e[%d;4%dm ",r%40,r%40,r%2,r%8); -10 bytes thanks to Dr. Rand. :P A main function O for Z seed rand() R print "\e[%d;%dH coordinate placeholder string \e[%d;4%dm " color placeholder string ,rand()%40,rand()%40,rand()%2,rand()%8 fill placeholders ); interpreter hole No TIO link because you obviously can't print to the kind of terminal this requires online. ## Perl (on nix), 69 bytes The \x1bs are literal escape characters. Relies on stty command and works well on OS X. {printf"\x1b[%i;%iH\x1b[48;5;%im ",map{rand$_}stty size=~/\d+/g,256;redo} Similar to other approaches, but I liked the combination of all params into one call to printf so thought I'd share. Kills my terminal. ### Twice as many pixels, 83 bytes {printf"\x1b[%i;%iH\x1b[48;5;%i;38;5;%im▄",map{rand$_}stty size=~/\d+/g,256,256;redo} This approach uses a unicode block ▄ and a random foreground and background colour which gives a more square pixel. Kills my terminal too, but looks cooler. • Don't forget you can use the literal escape byte, ASCII 0x1B (27) instead of \e. And on StackExchange this can be represented by ␛. – MD XF Jul 21 '17 at 16:11 • @MDXF Yeah... I need to test that as a script, because that wasn't working via -e! I also need to up the byte count of the one with unicode. Oops! Thanks for the reminder! – Dom Hastings Jul 21 '17 at 16:13 ## Bash, 104 bytes The \es are literal escape characters. These are pretty much translations of my Perl submission, but using bash fork bomb style syntax! Not as clever as the other bash entry that writes directly to the display I'm afraid. :(){ printf "\e[%i;%iH\e[48;5;%im " $((RANDOM%tput lines))$((RANDOM%tput cols)) $((RANDOM%256));:;};: ### More pixels, 130 bytes Same as my Perl answer, this uses a unicode character for the foreground and colours the background of each pixel too. :(){ printf "\e[%i;%iH\e[48;5;%i;38;5;%im▄"$((RANDOM%tput lines)) $((RANDOM%tput cols))$((RANDOM%256)) $((RANDOM%256));:;};: • Not bad. However, I still don't understand the "\e[48;5;%im". Why are the 48 and 5 necessary? – MD XF Jul 21 '17 at 16:46 • That access the full 256 colours, rather than just the standard 16 colour palette. In terminals that support it, it's possible to use \e[48;2;RRR;GGG;BBBm but very few terminal emulators support it... :( – Dom Hastings Jul 21 '17 at 16:48 • Holy mother of code.... I had no idea that was possible. My life is a lie. Thanks a million :P – MD XF Jul 21 '17 at 16:50 • @MDXF ^^ and also 38;5 denotes foreground and 48;5 deonates background. misc.flogisoft.com/bash/tip_colors_and_formatting – Dom Hastings Jul 21 '17 at 16:50 • @MDXF No worries! Glad to help! Shame I missed this when it was brand new, but that Minecraft answer... unreal! – Dom Hastings Jul 21 '17 at 16:50 # QBIC, 34 bytes screen 12{pset(_r640\|,_r480\|),_r16 Unfortunately, QBIC doesn't set a SCREEN mode by default, so that eats some bytes. PSET is an illegal command in QBIC's default graphical context. ## Explanation screen 12 Set the screen to a mode supporting (colored) graphics { DO until the compiler is killed by ctrl-scroll lock pset PSET is a QBasic command to set one pixel (_r640\|,_r480\|) it takes a set of coords (x, y) which QBIC chooses at random with _r ,_r16 and a color (1,16) ## Sample output @Anonymous found an error in QBIC's rnd() function. Fixing that gives the output below. Thanks! • As should be obvious from the screen shot, your solution doesn't satisfy the randomness conditions. Given a coordinate, not all colours are equally likely and given a colour, not all coordinates are equally likely; they're highly correlated and that's putting it mildly. The problem is the random number generator you're using: FUNCTION getRandomNumber! (Bottom!, Top!) ↵ RANDOMIZE TIMER 'this sets the random number generator ↵ IF Bottom > Top THEN ... END IF ↵ getRandomNumber = INT((Top - Bottom + 1) * RND + Bottom) ↵ END FUNCTION Your random numbers are just hashes of the current time. – Anonymous Oct 15 '17 at 22:50 • @Anonymous It's fixed now, thanks! – steenbergh Oct 16 '17 at 15:55 # 6502 Assembly, 582 bytes Whew, this was fun. Does more or less the same thing as my Applesoft BASIC solution. start: lda #15 sta$0 sta $1 loo: lda$fe and #3 cmp #0 beq g_l cmp #1 beq g_r cmp #2 beq g_d dec $1 d_w: lda$1 and #$1f asl tax lda ypo,x sta$2 inx lda ypo,x sta $3 lda$0 and #$1f tay lda ($2),y tax inx txa sta ($2),y jmp loo g_d: inc$1 jmp d_w g_l: dec $0 jmp d_w g_r: inc$0 jmp d_w ypo: dcb $00,$02,$20,$02,$40,$02,$60,$02 dcb $80,$02,$a0,$02,$c0,$02,$e0,$02 dcb $00,$03,$20,$03,$40,$03,$60,$03 dcb $80,$03,$a0,$03,$c0,$03,$e0,$03 dcb $00,$04,$20,$04,$40,$04,$60,$04 dcb $80,$04,$a0,$04,$c0,$04,$e0,$04 dcb $00,$05,$20,$05,$40,$05,$60,$05 dcb $80,$05,$a0,$05,$c0,$05,$e0,$05 • Is this the ungolfed version of your other 6502 assembly answer? :-) – Cody Gray Jun 7 '17 at 11:06 • @CodyGray Nope, it does things a bit differently. It's more like my Applesoft BASIC answer (uses a random walk), and the other 6502 assembly answer is more like all the other ones. – MD XF Jun 7 '17 at 14:45 • I disagree with the byte counts: this is way shorter than 583 bytes of assembly? – Olivier Dulac Mar 14 at 18:33 • @OlivierDulac tio.run/##Zc/… ? – MD XF Mar 15 at 2:01 # R, 143 138 bytes -4 bytes thanks to user2390246 r=runif s=sample g=1600 x=1:40 k=hsv(r(g),r(g),r(g)) while(1){image(x,x,matrix(1:g,40),c=k) Sys.sleep(99/g) k[s(g,1)]=hsv(r(1),r(1),r(1))} image makes the required 40x40 grid of rectangles with a matrix m of values (fixed to be 1:1600 in a 40x40 square), with a randomly selected color for each rectangle, k. It plots the image, waits long enough to actually render the image, then randomly changes one of the colors. Uses hsv encoding since it is a value between 0 and 1, which allows us to use runif to generate random colors. This generates a new plot each time. • It won't affect the score, but shouldn't image(x,x,m,c=k) be outside the while loop? – user2390246 Jun 2 '17 at 12:31 • I don't think you need to define m, you only use it once. – user2390246 Jun 2 '17 at 12:34 • @user2390246 for the image question, nah, it has to be re-called every time since it's really drawing a new image each time after changing one of the colors. Good call on the m, though. I originally thought I had to change that matrix and never went back and changed it. – Giuseppe Jun 2 '17 at 13:17 # Java 7 with JavaFX, 686 674 bytes import javafx.scene.;import javafx.scene.canvas.;import javafx.scene.layout.;public class M extends javafx.application.Application{public static void main(String[]a){launch(a);}GraphicsContext g;public M(){}public void start(javafx.stage.Stage x){BorderPane r=new BorderPane();Canvas c=new Canvas(40,40);g=c.getGraphicsContext2D();Pane p=new Pane();p.getChildren().addAll(c);r.setCenter(p);x.setScene(new Scene(r,40,40));x.show();new javafx.animation.AnimationTimer(){public void handle(long n){M.this.g.setFill(javafx.scene.paint.Color.color(Math.random(),Math.random(),Math.random()));M.this.g.fillRect((int)(Math.random()40),(int)(Math.random()40),1,1);}}.start();}} Explanation: import javafx.scene.;import javafx.scene.canvas.;import javafx.scene.layout.; // Required imports public class M extends javafx.application.Application{ // The class (NOTE: The public is mandatory for JavaFX) public static void main(String[]a){ // Mandatory main method launch(a); // Call the launch of the Application } // End of main-method GraphicsContext g; // GraphicsContext on class-level which we'll use in the animation public M(){} // Empty constructor (again, mandatory for JavaFX) public void start(javafx.stage.Stage x){ // Overwritten start-method of Application BorderPane r=new BorderPane(); // Create a BorderPane Canvas c=new Canvas(40,40); // Create a Canvas with the minimum size g=c.getGraphicsContext2D(); // Retrieve the GraphicsContext and set it to the class-level variable Pane p=new Pane(); // Create a Pane r.setCenter(p); // Put it in the center (otherwise it won't show in the window) x.setScene(new Scene(r,40,40)); // Create a new Scene x.show(); // And show the screen new javafx.animation.AnimationTimer(){ // Create a new animation public void handle(long n){ // Overwritten handle-method of this animation M.this.g.setFill(javafx.scene.paint.Color.color(Math.random(),Math.random(),Math.random())); // Set a random color M.this.g.fillRect((int)(Math.random()40),(int)(Math.random()40),1,1); // And add this color as a 1x1 pixel in a random position } // End of handle-method }.start(); // And start the animation } // End of start-method } // End of class Output: ## Mathematica 112 Bytes r:=RandomReal[];b:=⌈40r⌉;a=r&~Array~{40,40,3};Dynamic[a[[b,b]]={r,r,r};ArrayPlot[a,ColorFunction->RGBColor]] Very straightforward, uses Dynamic around an ArrayPlot. Not terribly fast; limited by the maximum speed dynamic can update, but should fulfill the goal on most computers. Starts out with a random 40x40 grid and recolors a random pixel one at a time. Ceiling characters are 3 bytes each, Ouch!	2018-12-16 17:32:44	{"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": 0.2270033359527588, "perplexity": 6917.4375562515725}, "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/1544376827963.70/warc/CC-MAIN-20181216165437-20181216191437-00357.warc.gz"}
https://chemistry.stackexchange.com/questions/83914/converting-plastic-back-to-bio-degradable-substances/83920	Converting plastic back to bio-degradable substances [closed] I have read that every plastic ever made still exists. Plastic must be made of some substances that are biodegradable. Even if they aren't, why can't we change their form? Why can't we change plastic into something else? • You're contradicting yourself and apparently never heard of recycling. Oct 9, 2017 at 19:19 • Some plastics decompose, albeit slowly, e.g. urethanes from photodegradation. Some decompose rapidly, e.g. celluloid. Polyvinyl alcohol is a water-soluble, biodegradable plastic. Question your source of information! Oct 10, 2017 at 0:32 Some definitions: plastic is a synthetic (man-made) organic polymer usually composed of a large number of simple monomer units chemically united (generally chemically) and usually used for its structural properties. For instance, the polypropylene fibers which make up a polypropylene rope would commonly be called a plastic but the polyvinyl alcohol used as a water thickener would normally not. Biodegradation is any sort of chemical decomposition by environmental causes. These causes may include sunlight, moisture (water), and microbes. Heat, erosion, and oxidation are generally not included. 1. If you have read that every plastic ever made still exists, then you need to be aware that your gullibility is extreme and the source for that claim should not be relied upon. Surely, you've seen plastic burn. Surely, you don't believe that the burned plastic magically reassembles itself. Hence it is obvious that not all plastic ever made still exists. 2. We can change plastic. Because most plastic items contain many other chemical compounds, (often colorants, antioxidants, plasticizers, & fillers) it can be quite expensive to decompose them into useful materials. 3. Chemistry isn't magic. Any industrial chemistry involves people, equipment, transportation, and energy. That is, it takes money. Most plastic is made from natural gas (the polymer, that is), which is cheap and fairly pure (consisting of mostly a half-dozen chemical compounds). There are tens of thousands, perhaps hundreds of thousands, maybe millions, of different plastics that have been made. Because of all the other stuff mixed into the polymers, and because the polymers can be very different, chemically, it is usually cheaper and easier to make a plastic starting from natural gas than starting from pre-existing plastic. Just the mix of materials separated from the polymers would be terribly expensive to dispose of. 4. The forces of nature will eventually bury some plastic, break up some plastic, and decompose some plastic. Nothing lasts forever. Very little of the plastic existing today will be present in 10,000 years. Erosion, aging, and biodegradation are the 3 categories of this degradation, along with mechanical forces. Much plastic exists in the oceans as microscopic bits. Some bacteria are able to eat some plastics. All plastics burn. Eventually, all of the polymers present in the plastic here on Earth will either be converted to CO2 (either in the air or captured in the ground (eventually as carbonate) OR subducted with the tectonic plates and thermally decomposed. Plastics are usually organic polymers. During the process of polymerization covalent bonds are formed. This reaction is exothermic, meaning it sets free energy. So, in order to reverse the process of polymerization and break these bonds again, it would take a lot of energy. It is fair to assume the energy needed will also break other bonds in the polymer, forming multiple products at once. I guess some kind of inseparable tar will be formed if oxygen is excluded. When oxygen is included the polymers will probably just burn, also forming multiple products at once - and CO2. • That being said, some polymers can be depolymerised. Indeed various byproducts are created, but you can get big part of monomer back. Oct 9, 2017 at 23:12 • I wasn't really aware of that. I guess my answer is not fully correct then. Hmmm. Oct 9, 2017 at 23:18 The initial statement is incorrect. Plastics contain carbon-carbon bonds and everything with carbon-carbon bonds can be burnt to $\ce{CO2}$ given sufficient burning temperature. Some plastics are indeed burnt so it is impossible for all plastics still to be around. However, most plastics can be reasonably approximated by a long, saturated carbon chain much like crude oil except with less other organics mixed in. As is the case with crude oil, there are very few organisms capable of digesting plastic in any form so its bio-lifetime is very long. However, given enough plastics and evolutionary pressure, microorganisms that can digest it will come along. I believe there have been some found already but I cannot point to any sources right now. • I am aware that this answer is basically a shorter rehash of alphonse’s. – Jan Oct 10, 2017 at 16:22	2022-07-07 06:57:55	{"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": 0.35714149475097656, "perplexity": 2468.2997289951095}, "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-2022-27/segments/1656104683708.93/warc/CC-MAIN-20220707063442-20220707093442-00499.warc.gz"}
http://math.stackexchange.com/questions/3742/favourite-open-problem	# Favourite open problem? Do you have any favorite open problem? Let me mention one of my favorites. Let $A(\mathbb{T})$ be the Wiener algebra, that is, the linear space of absolutely convergent Fourier series on the unit circle $\mathbb{T}$ $(=\mathbb{R}/2\pi\mathbb{Z}=(-\pi,\pi])$ carrying the norm $$f\mapsto \\|f\\|=\sum_{n\in\mathbb{Z}}\|\hat{f}(n)\|<\infty$$ where $\hat{f}(n)=\int_{-\pi}^\pi f(t)e^{-int}dt/2\pi$ is the $n$:th Fourier coefficient of $f$. In fact $A(\mathbb{T})$ is a unitary commutative Banach algebra. By absolute convergence it follows that each $f\in A(\mathbb{T})$ is continuous on $\mathbb{T}$. Moreover, if $f(t)\not=0$ for all $t\in\mathbb{T}$ then obviously $1/f$ is also continuous on $\mathbb{T}$ - a famous theorem of Norbert Wiener (The Wiener Lemma) states that we also have $1/f\in A(\mathbb{T})$. Next consider a possible quantitative refinement of the Wiener lemma: Given $\delta>0$ let $$C_\delta = \sup_{A_\delta}\\|1/f\\|$$ where $A_\delta={f\in A(\mathbb{T}):\|f(t)\|>\delta,\ \\|f\\|\leq1}$. Problem: Find $$\delta_\inf=\inf_{\delta>0} \ C_\delta<\infty$$. Remark 1: It is known that $\delta_\inf\leq 1/\sqrt{2}$ and that $\delta_\inf\geq 1/2$ (see [1,2]). Remark 2: The problem can be treated in any commutative Banach algebra we unit. [1] N. Nikolski, In search of the invisible spectrum, Annales de l'institut Fourier, 49 no. 6 (1999), p. 1925-1998 [2] H.S. SHAPIRO, A counterexample in harmonic analysis, in Approximation Theory, Banach Center Publications, Warsaw (submitted 1975), Vol. 4 (1979), 233-236. - I don't actually wish to vote to close, but I think this is not a good question for a SE Q&A site in that it is overly broad. The answers are likely to be completely unrelated to each other, and it's hard to know how much background or sophistication to assume when describing your favorite problem to a general mathematical audience. Finally, this is so subjective that I don't see the point of voting on and ranking the answers (and note that indeed currently none has more than one upvote). – Pete L. Clark Sep 1 '10 at 8:38 @Pete L. Clark: I agree in a certain sense. I'll "close" the thread by accepting the first answer - even though all of them are true (I guess). – AD. Sep 9 '10 at 13:34 Yeah, seems like this belongs on mathematicians.SE or something... – SamB Oct 21 '10 at 0:32 I like this one that is simple to state but likely very difficult to prove or disprove: The irrationality of $\zeta (5)$. The irrationality of $\zeta (3)$ was proved by Roger Apéry only in 1979. Despite considerable effort the picture is rather incomplete about $\zeta (s)$ for the other odd integers, $s=2t+1\gt 5$. -- Martin Aigner and Günter Ziegler, Proofs from THE BOOK. - • Is $\pi \cdot e$ rational? • What is the minimal number of people in a party, such that there are necessarily either at least 5 mutual strangers or at least 5 mutual acquaintances? • Is there a positive non-integer $x$ such that both $2^x$ and $3^x$ are integers? • Does every closed curve in the plain contain 4 vertices of a square? • Can you factor an integer in polynomial time? • Can you recognize the unknot in polynomial time? - "Is there a positive non-integer x such that both 2x and 3x are integers?" Do you have a link on this? – Bananas Dec 13 '11 at 1:53 @Collman: mathoverflow.net/questions/17560/… – Huy Dec 13 '11 at 9:45 Let ${^n a}$ denote tetration: ${^n a} = \underbrace{a^{a^{.^{.^{.^a}}}}}_{n \text{ times}}$ or, defined recursively, ${^1}a=a$, ${^{n+1}a}=a^{({^n a})}$. These are open problems: • Is there an integer $n>1$ and a non-integer positive rational $q$ such that ${^n q}$ is an integer? • Is there an integer $n$ such that ${^n \pi}$ is an integer? • Is there an integer $n$ such that ${^n e}$ is an integer? - Can't believe no one spoke of the Collatz conjecture yet. - The (ir)rationality of the Euler-Mascheroni constant has always intrigued me, as it's one of those constants that is "obviously" irrational and yet nobody has much of a clue on how to prove that. - As an aside, the value of $\gamma$ is approximately equal to that of $\frac1{\sqrt3}$ . – Lucian Oct 15 '13 at 4:17 It is open whether or not there exists an algorithm which will determine whether an integer sequence satisfying a linear recurrence with integer coefficients (such as the Fibonacci sequence) has only nonzero terms. See Terence Tao's excellent blog post on the subject. - Not that practical, but finding the smallest $\varepsilon$ in the statement "Matrix multiplication/inversion takes $O(n^{2+\varepsilon})$ flops" is still a nice open problem. - I am quite fond of the abc conjecture: Define $q(n)$ to be the product of all distinct primes of n. The conjecture states, that for any $\epsilon > 0$, there exist only finitely many triples of coprime positive integers $(a, b, c)$, satisfying $a + b = c$ for which $c > q(a b c)^{1 + \epsilon}$. This conjecture has an impressive list of consequences. - I find Brocard's problem very interesting. It asks for integer solutions to the equation $$m!+1=n^2.$$ Ramanujan considered, but could not solve, the problem. While very few solutions have been found: $(4,5),(5,11)(7,71)$, we do not yet know whether these are the only solutions, there are more solutions, or if infinitely many exist. Curiously, it follows from the $abc$ conjecture that, if the conjecture is true, that there are only finitely many solutions. - Does favorite allow me two - both about convex 3-dimensional polyhedra? (Shephard's Conjecture) Is it possible to cut along the edges of some spanning tree of any convex 3-dimensional polyhedron and unfold the cut open polyhedron so that the faces do not overlap (and one gets a simple polygon where the uncut edges indicate the faces of the original polyhedron in the resulting "net" in the plane)? http://en.wikipedia.org/wiki/Net_%28polyhedron%29 (Barnette's conjecture) Does every 3-valent convex 3-dimensional polyhedron all of whose faces have an even number of sides admit a hamiltonian circuit? http://en.wikipedia.org/wiki/Barnette%27s_conjecture - I will not stop you! Is it conjectured that the answers are "yes"? – AD. Sep 1 '10 at 4:41 I think there is some name attached to this conjecture (Higman perhaps?). Let $U_n(\mathbb{F}_q)$ be the group of $n \times n$ upper triangular matrices with entries taken from the finite field $\mathbb{F}_q$ and all $1$s on the diagonal. Then it is conjectured that the number of conjugacy classes in $U_n(\mathbb{F}_q)$ is a polynomial in $q$ with integer coefficients. From what I understand, this is known for $n \leq 13$. - Here are a two of my favourite problems that are currently open (and notoriously difficult): • Kakeya Problem: Define a Kakeya/Besicovitch set as a set containing a unit line segment in every direction. The conjecture states that the Kakeya/Besicovich set has Hausdorff/Minkowski dimension $n$. • Iwaniec Conjecture: Define the Beurling-Ahlfors transform as the principal-value singular integral \begin{equation} Bf(z)=-\frac{1}{\pi} p.v.\int_{\mathbb{C}}\frac{f(w)}{(z-w)^2}dm(w) \end{equation} on $L^p(\mathbb{C}),~1<p<\infty$. This conjecture states that the $L^p$ norm of the Beurling-Ahlfors transform is $p^-1,~p^=\max\{ p,\frac{p}{p-1}\}$. -	2016-06-25 05:40:31	{"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": 0.8792645335197449, "perplexity": 352.38477538860536}, "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-2016-26/segments/1466783392099.27/warc/CC-MAIN-20160624154952-00055-ip-10-164-35-72.ec2.internal.warc.gz"}
http://en.wikipedia.org/wiki/Thorium	# Thorium Not to be confused with Thallium or Thulium. Thorium, 90Th General properties Name, symbol thorium, Th Pronunciation THAWR-ee-əm Appearance silvery, often with black tarnish Thorium in the periodic table Ce Th (Uqb) actiniumthoriumprotactinium Atomic number 90 Standard atomic weight 232.0377(4) Element category actinide Group, period, block group n/a, period 7, f-block Electron configuration [Rn] 6d2 7s2 per shell 2, 8, 18, 32, 18, 10, 2 Physical properties Phase solid Melting point 2023 K (1750 °C, 3182 °F) Boiling point 5061 K (4788 °C, 8650 °F) Density (near r.t.) 11.724 g·cm−3 (at 0 °C, 101.325 kPa) Heat of fusion 13.81 kJ·mol−1 Heat of vaporization 514 kJ·mol−1 Molar heat capacity 26.230 J·mol−1·K−1 Vapor pressure P (Pa) 1 10 100 1 k 10 k 100 k at T (K) 2633 2907 3248 3683 4259 5055 Atomic properties Oxidation states 4, 3, 2, 1 Electronegativity 1.3 (Pauling scale) Ionization energies 1st: 587 kJ·mol−1 2nd: 1110 kJ·mol−1 3rd: 1930 kJ·mol−1 Miscellanea Crystal structure face-centered cubic (fcc) Speed of sound thin rod: 2490 m·s−1 (at 20 °C) Thermal expansion 11.0 µm·m−1·K−1 (at 25 °C) Thermal conductivity 54.0 W·m−1·K−1 Electrical resistivity at 0 °C: 157 nΩ·m Magnetic ordering paramagnetic[1] Young's modulus 79 GPa Shear modulus 31 GPa Bulk modulus 54 GPa Poisson ratio 0.27 Mohs hardness 3.0 Vickers hardness 350 MPa Brinell hardness 400 MPa CAS Number 7440-29-1 History Discovery Jöns Jakob Berzelius (1829) Most stable isotopes Main article: Isotopes of thorium iso NA half-life DM DE (MeV) DP 227Th trace 18.68 d α 6.038 5.978 223Ra 228Th trace 1.9116 y α 5.520 224Ra 229Th trace 7340 y α 5.168 225Ra 230Th trace 75380 y α 4.770 226Ra 231Th trace 25.5 h β 0.39 231Pa 232Th 100% 1.405×1010 y α 4.083 228Ra 234Th trace 24.1 d β 0.27 234Pa · references Thorium is a chemical element with symbol Th and atomic number 90. A radioactive actinide metal, thorium is one of only three radioactive elements that still occurs in quantity in nature as a primordial element (the other two being bismuth and uranium).[a] It was discovered in 1828 by the Norwegian mineralogist Morten Thrane Esmark and identified by the Swedish chemist Jöns Jakob Berzelius, who named it after Thor, the Norse god of thunder. A thorium atom has 90 protons and 90 electrons, of which four are valence electrons. Thorium metal is silvery and tarnishes black when exposed to air. Thorium is weakly radioactive: all its known isotopes are unstable, with the six naturally occurring ones (thorium-227, -228, -230, -231, -232, and -234) having half-lives between 25.52 hours and 14.05 billion years. Thorium-232, which has 142 neutrons, is the most stable isotope of thorium and accounts for nearly all natural thorium, with the other five natural isotopes occurring only in traces: it decays very slowly through alpha decay to radium-228, starting a decay chain named the thorium series that ends at lead-208. Thorium is estimated to be about three to four times more abundant than uranium in the Earth's crust, and is chiefly refined from monazite sands as a by-product of extracting rare earth metals. Thorium was once commonly used as the light source in gas mantles and as an alloying material, but these applications have declined due to concerns about its radioactivity. Thorium is also used as an alloying element in nonconsumable TIG welding electrodes. It remains popular as a material in high-end optics and scientific instrumentation; thorium and uranium are the only radioactive elements with major commercial applications that do not rely on their radioactivity. Thorium is predicted to be able to replace uranium as nuclear fuel in nuclear reactors, but no thorium reactors have yet been completed. ## Characteristics The 4n decay chain of thorium-232, commonly called the "thorium series" ### Physical Thorium is a soft, paramagnetic, bright silvery radioactive actinide metal. In the periodic table, it is located to the right of the actinide actinium, to the left of the actinide protactinium and below the lanthanide cerium. Pure thorium is soft, very ductile, and can be cold-rolled, swaged, and drawn.[3] The measured properties of thorium vary widely depending on the amount of impurities in the sample used: the major impurity is usually thorium dioxide (ThO2). The purest thorium specimens usually contain about a tenth of a percent of the dioxide.[3] Its density has been calculated to be 11.724 g/cm3, while experimental measurements give values between 11.5 and 11.66 g/cm3:[3] these values lie intermediate between those of its neighbours actinium (10.07 g/cm3) and protactinium (15.37 g/cm3), showing the continuity of trends across the actinide series.[3] However, thorium's melting point of 1750 °C is above both that of actinium (1227 °C) and that of protactinium (1562±15 °C): the melting points of the actinides do not have a clear dependence on their number of f electrons, although there is a smooth trend downward from thorium to plutonium where the number of f electrons increases from zero to six.[4] Thorium is a soft metal, having a bulk modulus of 54 GPa, comparable to those of tin and scandium. The hardness of thorium is similar to that of soft steel, so heated pure thorium can be rolled in sheets and pulled into wire.[4] Thorium becomes superconductive below 1.40 K.[3][b] Nevertheless, while thorium is nearly half as dense as uranium and plutonium, it is harder than either of them.[4] Among the actinides, thorium has the highest melting point and second-lowest density (second only to actinium).[3] The thermal expansion, electrical and thermal conductivities of thorium, protactinium, and uranium are comparable and are typical of post-transition metals.[5] Known properties of the allotropes of thorium[3] Thorium allotrope α (measured at 0 °C) β (measured at 1450 °C) high-pressure (measured at 102 GPa) Transition temperature (α→β) 1360 °C (β→liquid) 1750 °C high pressure Symmetry Face-centered cubic Body-centered cubic Body-centered tetragonal Density (g·cm−3) 11.724 11.724 unknown Lattice parameters (pm) a = 508.42 a = 411 a = 228.2, c = 441.1 Thorium can also form alloys with many other metals. With chromium and uranium, it forms eutectic mixtures, and thorium is completely miscible in both solid and liquid states with its lighter congener cerium.[3] ### Chemical Thorium is a highly reactive metal. At standard temperature and pressure, thorium is slowly attacked by water, but does not readily dissolve in most common acids, the exception being hydrochloric acid.[3][6] It dissolves in concentrated nitric acid containing a small amount of catalytic fluoride or fluorosilicate ions;[3][7] if these are not present, passivation can occur.[3] At high temperatures, it is easily attacked by oxygen, hydrogen, nitrogen, the halogens, and sulfur. It can also form binary compounds with carbon and phosphorus.[3] When thorium dissolves in hydrochloric acid, a black residue, probably ThO(OH,Cl)H, is left behind.[3] Finely divided thorium metal presents a fire hazard due to its pyrophoricity and must therefore be handled carefully.[3] When heated in air, thorium turnings ignite and burn brilliantly with a white light to produce the dioxide. In bulk, the reaction of pure thorium with air is slow, although corrosion may eventually occur after several months; most thorium samples are however contaminated with varying degrees of the dioxide, which greatly accelerates corrosion.[3] Such samples slowly tarnish in air, becoming gray and finally black.[3] The most important oxidation state of thorium is +4, represented in compounds such as thorium dioxide (ThO2) and thorium tetrafluoride (ThF4), although +3, +2, and +1 states are also known, as in thorium tribromide (ThBr3), thorium dihydride (ThH2), and thorium monobromide (ThBr).[8][9][10] Owing to thorium(IV)'s lack of electrons on 6d and 5f orbitals, the tetravalent thorium compounds are colorless.[4] In aqueous solution, thorium occurs exclusively as the tetrapositive aqua ion [Th(H2O)9]4+, which has tricapped trigonal prismatic molecular geometry:[11][12] at pH < 3, the solutions of thorium salts are dominated by this cation.[11] The Th–O bond distance is (245 ± 1) pm, the coordination number of Th4+ is (10.8 ± 0.5), the effective charge is 3.82 and the second coordination sphere contains 13.4 water molecules.[11] The Th4+ ion is relatively large and is the largest of the tetrapositive actinide ions, and depending on the coordination number can have a radius between 0.95 and 1.14 Å. As a result, thorium salts have a weak tendency to hydrolyze, weaker than that of many multiply charged ions such as Fe3+.[11] The distinctive ability of thorium salts is their high solubility, not only in water, but also in polar organic solvents.[4] Thorium has been shown to activate carbon–hydrogen bonds, forming unusual compounds. Thorium atoms can also bond to more atoms than any other element: for instance, in the compound thorium aminodiboranate, thorium has a coordination number of fifteen.[13] ### Atomic A thorium atom has 90 electrons, of which four are valence electrons. Four atomic orbitals are theoretically available for the valence electrons to occupy: 5f, 6d, 7s, and 7p. However, the 7p orbital is greatly destabilized and hence it is not occupied in the ground state of any thorium ion.[14] Despite thorium's position in the f-block of the periodic table, it has an anomalous [Rn]6d27s2 electron configuration in the ground state. However, in metallic thorium, the [Rn]5f16d17s2 configuration is a low-lying excited state and hence the 5f orbitals contribute, existing in a rather broad energy band.[14] The ground-state electron configurations of thorium ions are as follows: Th+, [Rn]6d27s1; Th2+, [Rn]5f16d1;[c] Th3+, [Rn]5f1; Th4+, [Rn]. This shows the increasing stabilization of the 5f orbital as ion charge increases; however, this stabilization is insufficient to chemically stabilize Th3+ with its lone 5f valence electron, and therefore the stable and most common form of thorium in chemicals is Th4+ with all four valence electrons lost, leaving behind an inert core of inner electrons with the electron configuration of the noble gas radon.[14][15] The first ionization potential of thorium was measured to be (6.08 ± 0.12) eV in 1974;[16] more recent measurements have refined this to 6.3067 eV.[17] ### Isotopes Main article: Isotopes of thorium Although thorium has 6 naturally occurring isotopes, none of these isotopes are stable; however, one isotope, 232Th, is relatively stable, with a half-life of 14.05 billion years, considerably longer than the age of the earth, and even slightly longer than the generally accepted age of the universe (about 13.8 billion years).[d] This isotope is the longest-lived of all isotopes with more than 83 protons and makes up nearly all natural thorium. As such, thorium is generally considered to be mononuclidic.[18][19][20] However, in deep seawaters the isotope 230Th becomes more significant, which led to IUPAC reclassifying thorium as a binuclidic element in 2013.[21] In fact, uranium ores with low thorium concentrations can be purified to produce gram-sized thorium samples of which over a quarter is the 230Th isotope.[22] Thorium has a characteristic terrestrial isotopic composition, consisting largely of 232Th and a little 230Th, and thus an atomic mass can be given, which is 232.0377(4) u.[21] 232Th is the longest-lived isotope in the 4n decay chain which includes isotopes with a mass number divisible by 4, begins with the alpha decay of 232Th to 228Ra,[e] and terminates at stable 208Pb, and its longevity means that the isotopes in its decay chain still exist in nature as radiogenic nuclides despite their short half-lives.[18][f][g] As such, natural thorium samples can be chemically purified to extract its useful daughter nuclides, such as lead-212 (212Pb), which is used in nuclear medicine for cancer therapy.[23][24] Thirty radioisotopes have been characterized, which range in mass number from 209[25] to 238.[22] The most stable of them (after 232Th) are 230Th with a half-life of 75,380 years, 229Th with a half-life of 7,340 years, 228Th with a half-life of 1.92 years, 234Th with a half-life of 24.10 days, and 227Th with a half-life of 18.68 days: all of these isotopes except 229Th occur in nature as trace radioisotopes due to their presence in the decay chains of 232Th, 235U, and 238U. All of the remaining thorium isotopes have half-lives that are less than thirty days and the majority of these have half-lives that are less than ten minutes. The isotope 229Th has a nuclear isomer (or metastable state) with a remarkably low excitation energy,[26] recently measured to be (7.6 ± 0.5) eV.[27] In the early history of the study of radioactivity, the different natural isotopes of thorium were given different names. In this scheme, 227Th was named radioactinium (RdAc), 228Th radiothorium (RdTh), 230Th ionium (Io), 231Th uranium Y (UY), 232Th thorium (Th), and 234Th uranium X1 (UX1).[22] When it was realized that all of these are isotopes of thorium, many of these names fell out of use, and "thorium" came to refer to all isotopes, not just 232Th.[22] However, the name ionium is still encountered for 230Th in the context of ionium-thorium dating.[28][29] Different isotopes of thorium behave identically chemically, but do have slightly differing physical properties: for example, the densities of 228Th, 229Th, 230Th, and 232Th in g·cm−3 are respectively expected to be 11.524, 11.575, 11.626, and 11.727.[30] The isotope 229Th is expected to be fissionable with a bare critical mass of 2839 kg, although with steel reflectors this value could drop to 994 kg.[30] While 232Th, the most common thorium isotope, is not fissionable, it is fertile as it can be converted to fissile 233U using neutron capture.[30][31] ## History ### Discovery Thor, the namesake of thorium In 1815, the Swedish chemist Jöns Jakob Berzelius analyzed a mineral from the Falun. Assuming that a new element was contained in the mineral, he named the supposed element "thorium" after Thor, the Norse god of thunder. However, the mineral later proved to actually be an yttrium mineral, primarily composed of yttrium orthophosphate.[31] As the yttrium in this mineral was initially mistaken as being a new element, the mineral was named xenotime from the Greek words κενός (vain) and τιμή (honor).[32][33] In 1828, Morten Thrane Esmark found a black mineral on Løvøya island, Norway, and gave a sample to his father, Jens Esmark, a noted mineralogist. The elder Esmark was not able to identify it and sent a sample to the Swedish chemist Jöns Jakob Berzelius for examination. Berzelius determined that it contained a new element.[31] He published his findings in 1829.[34][35][36] Berzelius reused the name of the previous supposed element discovery.[34][37] Thus, he named the source mineral thorite, which has the chemical composition (Th,U)SiO4.[31] ### Subsequent developments In Dmitri Mendeleev's 1869 periodic table, thorium and the rare earth elements were placed outside the main body of the table, at the end of each vertical period after the alkaline earth metals. This reflected the belief at that time that thorium and the rare earth metals were divalent.[h] With the later recognition that the rare earths were mostly trivalent and thorium was tetravalent, Mendeleev moved cerium and thorium to group IV in 1871, which contained the modern carbon group, titanium group, cerium, and thorium, because their maximum oxidation state was +4.[38][39] While cerium was soon removed from the main body of the table and placed in a separate lanthanide series, it was not until 1945 that Glenn T. Seaborg realized that thorium was the second member of the actinide series and was filling an f-block row, instead of being the heavier congener of hafnium and filling a fourth d-block row.[40][i] Thorium was first observed to be radioactive in 1898, independently, by the Polish-French physicist Marie Curie and the German chemist Gerhard Carl Schmidt.[42][43][44] Between 1900 and 1903, Ernest Rutherford and Frederick Soddy showed how thorium decayed at a fixed rate over time into a series of other elements. This observation led to the identification of half-life as one of the outcomes of the alpha particle experiments that led to their disintegration theory of radioactivity.[45] Although thorium was discovered in 1828, it had no applications until 1885, when Carl Auer von Welsbach invented the gas mantle.[31] After 1885, many applications were found for thorium and its compounds. In recent decades, however, most of these applications that do not depend on thorium's radioactivity have declined due to safety and environmental concerns.[31] ## Occurrence Main article: Occurrence of thorium The radiogenic heat from the decay of 232Th is a major contributor to the earth's internal heat budget. The other major contributors are 235U, 238U, and 40K. Thorium-232 is a primordial nuclide, having existed in its current form for over 4.5 billion years, predating the formation of the Earth; it was forged in the cores of dying stars through the r-process and scattered across the galaxy by supernovae.[46] Its radioactive decay produces a significant amount of the Earth's internal heat.[47] Natural thorium is essentially monoisotopic 232Th, which is the longest-lived and most stable isotope of thorium, having a half-life comparable to the age of the universe. If the source contains no uranium, the only other thorium isotope present would be 228Th, which occurs in the decay chain of thorium-232 (the thorium series): the ratio of 228Th to 232Th would be under 10−10.[22] However, if uranium is present, tiny traces of several other isotopes will be present: 231Th and 227Th from the decay chain of uranium-235 (the actinium series), and slightly larger but still tiny traces of 234Th and 230Th from the decay chain of uranium-238 (the uranium series).[22] Earlier in the Earth's history, 229Th would also have been produced in the now extinct decay chain of 237Np (the neptunium series): it is now only produced as a daughter of artificial uranium-233, itself produced from neutron irradiation of 232Th.[22] On Earth, thorium is not a rare element as was previously thought, having an abundance comparable to that of lead and molybdenum, twice that of arsenic, and thrice that of tin.[48] In nature, it occurs in the +4 oxidation state, together with uranium(IV), zirconium(IV), hafnium(IV), and cerium(IV), but also with the scandium, yttrium, and the trivalent lanthanides which have similar ionic radii.[48] However, thorium only occurs as a minor constituent of most minerals.[48] Thorium minerals occur on all continents.[6][49][50] Thorium is several times more abundant in Earth's crust than all isotopes of uranium combined and thorium-232 is several hundred times more abundant than uranium-235.[48] Because of thorium's radioactivity, minerals containing significant quantities of thorium are often metamict, their crystal structure having been partially or totally destroyed by the alpha radiation produced in the radioactive decay of thorium.[51][j] An extreme example is ekanite ((Ca,Fe,Pb)2(Th,U)Si8O20), which almost never occurs in nonmetamict form due to thorium being an essential part of its chemical composition.[52] Monazite is the most important commercial source of thorium because it occurs in large deposits worldwide and contains 2.5% thorium. It is a chemically unreactive phosphate mineral that has a high specific gravity and is found as yellow or brown monazite sand; its low reactivity makes it difficult to extract thorium from it.[48] Allanite can have 0.1–2% thorium and zircon up to 0.4% thorium.[48] Thorium dioxide occurs as the rare mineral thorianite, which usually contains up to 12% ThO2. However, due to its being isotypic with uranium dioxide, the two actinide dioxides can form solid-state solutions and the name of the mineral changes according to the ThO2 content.[48][k] Thorite, or thorium silicate (ThSiO4),[l] also has a high thorium content and is the mineral in which thorium was first discovered.[48] In thorium silicate minerals, the Th4+ and SiO4− 4 ions are often replaced with M3+ (M = Sc, Y, Ln) and phosphate (PO3− 4 ) ions respectively.[48][m] ## Production Monazite – a major thorium mineral Thorium is extracted mostly from monazite: thorium diphosphate (Th(PO4)2) is reacted with nitric acid, and the produced thorium nitrate treated with tributyl phosphate. Rare-earth impurities are separated by increasing the pH in sulfate solution.[53] In another extraction method, monazite is decomposed with a 45% aqueous solution of sodium hydroxide at 140 °C. Mixed metal hydroxides are extracted first, filtered at 80 °C, washed with water and dissolved with concentrated hydrochloric acid. Next, the acidic solution is neutralized with hydroxides to pH = 5.8 that results in precipitation of thorium hydroxide (Th(OH)4) contaminated with ~3% of rare-earth hydroxides; the remaining rare-earth hydroxides remain in solution. Thorium hydroxide is dissolved in an inorganic acid and then purified from the rare earth elements. An efficient method is the dissolution of thorium hydroxide in nitric acid, because the resulting solution can be purified by extraction with organic solvents:[53] Th(OH)4 + 4 HNO3 → Th(NO3)4 + 4 H2O Metallic thorium is separated from the anhydrous oxide, chloride or fluoride by reacting it with calcium in an inert atmosphere:[54] ThO2 + 2 Ca → 2 CaO + Th Sometimes thorium is extracted by electrolysis of a fluoride in a mixture of sodium and potassium chloride at 700–800 °C in a graphite crucible. Highly pure thorium can be extracted from its iodide with the crystal bar process.[55] ## Compounds ### Oxides and hydroxides Thorium dioxide has the fluorite structure. Th4+: __ / O2−: __ In air, thorium turnings burn to form the simple dioxide, ThO2, also called thoria or thorina.[56] Thoria, a refractory material, has the highest melting point (3390 °C) of all known oxides.[57] It is somewhat hygroscopic and reacts readily with water and many gases.[9] When heated, it emits intense blue light, which becomes white when mixed with its lighter homolog cerium dioxide (CeO2, ceria): this is the basis for its previously common application in gas mantles.[9] Related to thoria is the oxoanion ThO2− 3 , found in BaThO3 and K2ThO3.[9] Reports of thorium peroxide, initially supposed to be Th2O7 and be formed from reacting thorium salts with hydrogen peroxide, were later discovered to contain both peroxide anions and the anions of the reacting thorium salt.[9] Thorium monoxide has recently been produced through laser ablation of thorium in the presence of oxygen.[58] This highly polar molecule has the largest known internal electric field.[59] Thorium hydroxide, Th(OH)4, can be prepared by adding a hydroxide of ammonium or an alkali metal to a thorium salt solution, where it appears as a gelatinous precipitate that will dissolve in dilute acids, among other substances.[9] It can also be prepared by electrolysis of thorium nitrates.[9] It is stable from 260–450 °C; at 470 °C and above it continuously decomposes to become thoria.[9] It easily absorbs atmospheric carbon dioxide to form the hydrated carbonate ThOCO3·xH2O and, under high-pressure conditions in a carbon dioxide atmosphere, Th(CO3)2·0.5H2O or Th(OH)2CO3·2H2O.[9][60] ### Halides Crystal structure of thorium tetrafluoride. Th4+: __ / F: __ All four thorium tetrahalides are known, as are some low-valent bromides and iodides.[10] Additionally, many related polyhalide ions are also known.[10] Thorium tetrafluoride (ThF4) is most easily produced by reacting various thorium salts, thoria, or thorium hydroxide with hydrogen fluoride: methods that involve steps in the aqueous phase are more difficult because they result in hydroxide and oxide fluorides that have to be reduced with hydrogen fluoride or fluorine gas.[10] It has a monoclinic crystal structure and is isotypic with zirconium tetrafluoride and hafnium tetrafluoride, where the Th4+ ions are coordinated with F ions in somewhat distorted square antiprisms.[10] It is a white, hygroscopic powder: at temperatures above 500 °C, it reacts with atmospheric moisture to produce the oxyfluoride ThOF2.[61] Thorium tetrachloride (ThCl4) can be produced in many ways. The usual method is crystallization from an aqueous solution and then heating the product above 100 °C to dehydrate it.[10] Further purification can be achieved by subliming it. Its melting and boiling points are respectively 770 °C and 921 °C.[10] It undergoes a phase transition at 405 °C, with a low-temperature α phase and high-temperature β phase. Nevertheless, the β phase usually persists below the transition temperature. Both phases crystallize in the tetragonal crystal system and the structural differences are small.[10] Below −203 °C, a low-temperature form exists with a complex structure.[10] Thorium tetrabromide (ThBr4) can be produced either by reacting thorium(IV) hydroxide with hydrobromic acid (which has the disadvantage of often resulting in products contaminated with oxybromides) or by directly reacting bromine or hydrogen bromide with thorium metal or compounds.[10] The product can then be purified by sublimation at 600 °C in a vacuum.[10] The melting and boiling points are 679 °C and 857 °C.[10] Like the tetrachloride, both an α and a β form exist and both are isotypic to the tetrachloride forms, though the phase transition here occurs at 426 °C. There is also a low-temperature form.[10] Incomplete reports of the lower bromides ThBr3, ThBr2, and ThBr are known (the last only known as a gas-phase molecular species): ThBr3 and ThBr2 are known to be very reactive and at high temperatures disproportionate.[10] Thorium tetraiodide (ThI4) is prepared by direct reaction of the elements in a sealed silica ampoule. Water and oxygen must not be present, or else ThOI2 and ThO2 can contaminate the product.[10] It has a different crystal structure from the other tetrahalides, being monoclinic.[10] The lower iodides ThI3 and ThI2 can be prepared by reducing the tetraiodide with thorium metal. (ThI is also predicted to form as an intermediate in the dissociation of ThI4 to thorium metal.)[10] These do not contain Th(III) and Th(II), but instead contain Th4+ and could be more clearly formulated as Th4+(I)3(e) and Th4+(I)2(e)2 respectively.[10] Depending on the amount of time allowed for the reaction between ThI4 and thorium, two modifications of ThI3 can be produced: shorter times give thin lustrous rods of α-ThI3, while longer times give small β-ThI3 crystals with green to brass-colored luster.[10] ThI2 also has two modifications, which can be produced by varying the reaction temperature: at 600 °C, α-ThI2 is formed, while a reaction temperature of 700–850 °C produces β-ThI2, which has a golden luster.[10] Many polynary halides with the alkali metals, barium, thallium, and ammonium are known for thorium fluorides, chlorides, and bromides.[10] For example, when treated with potassium fluoride and hydrofluoric acid, Th4+ forms the complex anion ThF2− 6 , which precipitates as an insoluble salt, K2ThF6.[7] ### Chalcogenides and pnictides The heavier chalcogens sulfur, selenium, and tellurium are known to form thorium chalcogenides, many of which have more complex structure than the oxides. Apart from several binary compounds, the oxychalcogenides ThOS (yellow), ThOSe, and ThOTe are also known.[62] The five binary thorium sulfides – ThS (lustrous metallic), Th2S3 (brown metallic), Th7S12 (black), ThS2 (purple-brown), and Th2S5 (orange-brown) – may be produced by reacting hydrogen sulfide with thorium, its halides, or thoria (the last if carbon is present): they all hydrolyze in acidic solutions.[62] The six selenides are analogous to the sulfides, with the addition of ThSe3.[62] The five tellurides are also similar to the sulfides and selenides (although Th2Te5 is unknown), but have slightly different crystal structures: for example, ThS has the sodium chloride structure, but ThTe has the caesium chloride structure.[62] All five chemically characterized pnictogens (nitrogen, phosphorus, arsenic, antimony, and bismuth) also form compounds with thorium.[63] Three thorium nitrides are known: ThN, Th3N4, and Th2N3. The brass-colored Th3N4 is most easily produced by heating thorium metal in a nitrogen atmosphere. Th3N4 and Th2N3 decompose to the golden-yellow ThN, and indeed ThN can often be seen covering the surface of Th3N4 samples because Th3N4 is hygroscopic and water vapor in the air can decompose it: thin films of ThN are metallic in character and, like all other actinide mononitrides, has the sodium chloride structure. ThN is also a low-temperature superconductor. All three nitrides can react with thorium halides to form halide nitrides ThNX (X = F, Cl, Br, I).[63] The heavier pnictogens also form analogous monopnictides, except ThBi which has not yet been structurally characterized. The other well-characterized thorium pnictides are Th3P4, Th2P11, ThP7, Th3As4, ThAs2, Th3Sb4, ThSb2, and ThBi2.[63] ### Other inorganic Thorium reacts with hydrogen to form the thorium hydrides ThH2 and Th4H15, the latter of which is superconducting below the transition temperature of 7.5–8 K; at standard temperature and pressure, it conducts electricity like a metal.[8] Finely divided thorium metal reacts very readily with hydrogen at standard conditions, but large pieces may need to be heated to 300–400 °C for a reaction to take place.[8] Around 850 °C, the reaction forming first ThH2 and then Th4H15 occurs without breaking up the structure of the thorium metal.[8] Thorium hydrides react readily with oxygen or steam to form thoria, and at 250–350 °C quickly react with hydrogen halides, sulfides, phosphides, and nitrides to form the corresponding thorium binary compounds.[8] Three binary thorium borides are known: ThB6, ThB4, and ThB12. The last is isotypic with UB12. While reports of ThB66 and ThB76 exist, they may simply be thorium-stabilized boron allotropes.[64] The three known binary thorium carbides are ThC2, Th2C3, and ThC: all are produced by reacting thorium or thoria with carbon. ThC and ThC2 are refractory solids and have melting points over 2600 °C.[64] Structure of thorocene Many other inorganic thorium compounds with polyatomic anions are known, such as the perchlorates, sulfates, sulfites, nitrates, carbonates, phosphates, vanadates, molybdates, chromates, and other oxometallates,[n] many of which are known in hydrated forms.[60] These are important in thorium purification and the disposal of nuclear waste, but most have not yet been fully characterized, especially on their structural properties.[60] For example, thorium perchlorate is very water-soluble and crystallizes from acidic solutions as the tetrahydrate Th(ClO4)4·4H2O, while thorium nitrate forms tetra- and pentahydrates, is soluble in water and alcohols, and is an important intermediate in the purification of thorium and its compounds.[60] ### Organometallic and other carbon-containing compounds Like many of the early and middle actinides (thorium through americium, and also expected for curium), thorium forms the yellow cyclooctatetraenide complex Th(C8H8), thorocene. It is isotypic with the more well-known analogous uranium compound, uranocene.[65] It can be prepared by reacting K2C8H8 with thorium tetrachloride in tetrahydrofuran (THF) at the temperature of dry ice, or by reacting thorium tetrafluoride with MgC8H8.[65] It is an unstable compound in air and outright decomposes in water or at 190 °C.[65] Many other organothorium compounds are known, many involving the cyclopentadienyl anion.[65] Some coordination complexes with carboxylates and acetylacetonates are also known, although these are not organothorium compounds.[60] ## Applications ### Nuclear Due to environmental concerns over thorium's radioactivity that cause its non-nuclear applications to sharply decrease in importance, the most important possible use of thorium is now its possible use in the thorium fuel cycle as a nuclear fuel, as its applications in radiometric dating (which depend on its radioactivity) are mostly specialized.[31] #### Nuclear energy Main article: Thorium fuel cycle In thermal breeder reactors, the fertile isotope 232Th, the most common thorium isotope, is bombarded by slow neutrons, undergoing neutron capture to become 233Th, which undergoes two consecutive beta decays to become first 233Pa and then the fissile 233U:[31] ${}_{\ 90}^{232}\mathrm{Th} + \mathrm{n} \rightarrow {}_{\ 90}^{233} \mathrm{Th} + \gamma\ \xrightarrow{\beta^-}\ {}_{\ 91}^{233}\mathrm{Pa}\ \xrightarrow{\beta^-}\ {}_{\ 92}^{233}\mathrm{U}$ 233U is fissile and hence can be used as a nuclear fuel in much the same way as the more-commonly used 235U or 239Pu. When 233U undergoes nuclear fission, the neutrons emitted can strike further 232Th nuclei, restarting the cycle.[31] This closely parallels the uranium fuel cycle in fast breeder reactors where 238U undergoes neutron capture to become 239U, beta decaying to first 239Np and then fissile 239Pu.[66] The main advantage of the thorium fuel cycle is that thorium is more abundant than uranium and hence can satisfy world energy demands for longer. An added advantage 233U and 239Pu enjoy over all other fissile nuclei (except the naturally occurring 235U) is that they can be bred from the naturally-occurring quantity isotopes 232Th and 238U.[67][68][o] Additionally, 233U is easily detected, can be mixed with 238U to prevent direct use in nuclear weapons and limit nuclear proliferation, and has a higher neutron yield than 239Pu. Thorium fuels also result in a safer and better-performing reactor core[31] because thoria has a higher melting point, higher thermal conductivity, and lower coefficient of thermal expansion than the now-common fuel uranium dioxide (UO2): thoria also exhibits greater chemical stability and, unlike uranium dioxide, does not further oxidize.[69] 230Th → 231Th ← 232Th → 233Th (White actinides: t½<27d) ↓ ↓ 231Pa → 232Pa ← 233Pa → 234Pa (Colored : t½>68y) ↑ ↓ ↓ ↓ 231U ← 232U ↔ 233U ↔ 234U ↔ 235U ↔ 236U → 237U ↓ ↓ ↓ ↓ (Fission products with t½<90y or t½>200ky) 237Np A single neutron capture by 238U would produce transuranic waste, along with it fissile 239Pu, whereas six captures are generally necessary to do so from 232Th. 98–99% of thorium-cycle fuel nuclei would fission at either 233U or 235U, so fewer long-lived transuranics are produced. Because of this, thorium is a potentially attractive alternative to uranium in mixed oxide (MOX) fuels to minimize the generation of transuranics and maximize the destruction of plutonium.[70] The disadvantages of the thorium fuel cycle include the need to neutron irradiate and process natural 232Th before these advantages become real, and this requires more advanced technology than the presently used fuels based on uranium and plutonium; nevertheless, advances are being made in this technology.[31] In 1997, the U.S. Energy Department underwrote research into thorium fuel, and research also was begun in 1996 by the International Atomic Energy Agency (IAEA), to study the use of thorium reactors. Nuclear scientist Alvin Radkowsky of Tel Aviv University in Israel founded a consortium to develop thorium reactors, which included other companies: Raytheon Nuclear Inc., Brookhaven National Laboratory and the Kurchatov Institute in Moscow.[71] Radkowsky was chief scientist in the U.S. nuclear submarine program directed by Admiral Hyman Rickover and later headed the design team that built the USA's first civilian nuclear power plant at Shippingport, Pennsylvania, which was a scaled-up version of the first naval reactor.[71] The third Shippingport core, initiated in 1977, bred thorium.[72] Even earlier examples of reactors using fuel with thorium exist, including the first core at the Indian Point Energy Center in 1962.[73] However, in most countries uranium was relatively abundant and hence research into thorium fuel waned for a while. A notable exception was India's three-stage nuclear power programme.[74] In the twenty-first century thorium's potential for improving proliferation resistance and waste characteristics led to renewed interest in the thorium fuel cycle.[75][76][77] Currently, some countries such as India are developing technology for thorium nuclear reactors.[78][79] Two radiometric dating methods involve thorium isotopes: uranium-thorium dating, involving the decay of 234U to 230Th (ionium), and ionium-thorium dating, which measures the ratio of 232Th to 230Th. These rely on the fact that 232Th is a primordial radioisotope, but 230Th only occurs as an intermediate decay product in the decay chain of 238U.[80] Uranium-thorium dating is a relatively short-range process because of the short half-lives of 234U and 230Th relative to the age of the Earth: it is also accompanied by a sister process involving the alpha decay of 235U into 231Th, which very quickly becomes the longer-lived 231Pa, and this process is often used to check the results of uranium-thorium dating. Uranium-thorium dating is commonly used to determine the age of calcium carbonate materials such as speleothem or coral, because while uranium is rather soluble in water, thorium and protactinium are not, and so they are selectively precipitated into ocean-floor sediments, from which their ratios are measured. The scheme has a range of several hundred thousand years.[80][81] Ionium-thorium dating is a related process, which exploits the insolubility of thorium (both 232Th and 230Th) and thus its presence in ocean sediments to date these sediments by measuring the ratio of 232Th to 230Th.[28][29] Both of these dating methods assume that the proportion of thorium-230 to thorium-232 is a constant during the time period when the sediment layer was formed, that the sediment did not already contain thorium before contributions from the decay of uranium, and that the thorium cannot shift within the sediment layer.[28][29] ### Non-nuclear Thorium oxide gas mantle Many non-nuclear applications of thorium are becoming obsolete due to environmental concerns largely stemming from the radioactivity of thorium and its decay products.[31] The melting point of thoria is 3300 °C – the highest of all known oxides. Only a few elements (including tungsten and carbon) and a few compounds (including tantalum carbide) have higher melting points).[57] This means that when heated to high temperatures, it does not melt, but merely glows with an intense blue light; addition of cerium dioxide gives a bright white light.[9] This property of thoria means that thoria and thorium nitrate are used in mantles of portable gas lights, including natural gas lamps, oil lamps and camping lights.[82] A study in 1981 estimated that the dose from using a thorium mantle every weekend for a year would be 0.3-0.6 millirems (mrem), tiny in comparison to the normal annual dose of a few hundred millirems (a person actually ingesting a mantle would receive a dose of 200 mrem (2 mSv)).[83][84] However, the radioactivity is a major concern for people involved with the manufacture of mantles, and an issue with contamination of soil around some former factory sites.[85] Due to these concerns, some manufacturers have switched to other materials, such as yttrium, although these are usually either more expensive or less efficient. Other manufacturers continue to make thorium mantles, but moved their factories to developing countries.[84] Thoria is a material for heat-resistant ceramics, as used in high-temperature laboratory crucibles.[31] When added to glass, it helps increase refractive index and decrease dispersion. Such glass finds application in high-quality lenses for cameras and scientific instruments.[6] The radiation from these lenses can darken them and turn them yellow over a period of years and degrade film, but the health risks are minimal.[86] Yellowed lenses may be restored to their original colorless state with lengthy exposure to intense ultraviolet radiation. Thoria was used to control the grain size of tungsten metal used for spirals of electric lamps. Thoriated tungsten elements are found in the filaments of vacuum tubes, e.g. magnetron found in microwave oven. Thorium is added because it lowers the effective work function with the result that the thoriated tungsten thermocathode emits electrons at considerably lower temperatures.[31] Thoria has been used as a catalyst in the conversion of ammonia to nitric acid,[31] in petroleum cracking and in producing sulfuric acid.[31] It is the active ingredient of Thorotrast, which was used as radiocontrast agent for X-ray diagnostics because of thorium's high opacity to X-rays. This use has been abandoned due to its carcinogenic nature.[6] Thorium tetrafluoride is used as an antireflection material in multilayered optical coatings. It has excellent optical transparency in the range of 0.35–12 µm, and its radiation is primarily due to alpha particles, which can be easily stopped by a thin cover layer of another material.[87] Thorium tetrafluoride was also used in manufacturing carbon arc lamps, which provided high-intensity illumination for movie projectors and search lights.[82] ## Precautions Experiment on the effect of radiation (from an unburned thorium gas mantle) on the germination and growth of timothy-grass seed; from Popular Science, 1909. As thorium occurs naturally, it exists in very small quantities almost everywhere on Earth: the average human contains about 100 micrograms of thorium and typically consumes three micrograms per day of thorium.[88] This exposure is raised for people who live near uranium, phosphate, or tin processing factories, thorium deposits, radioactive waste disposal sites, and for those who work in uranium, thorium, tin, or phosphate mining or gas mantle production industries.[89] When thorium is ingested, 99.98% does not remain in the body. Out of the thorium that does remain in the body, three quarters of it accumulates in the skeleton. While absorption through the skin is possible, it is not a likely means of thorium exposure.[90] Powdered thorium metal is pyrophoric and often ignites spontaneously in air.[3] The chemical toxicity of thorium is low because thorium and its most common compounds (mostly the dioxide) are poorly soluble in water.[93] Nevertheless, some thorium compounds are chemically moderately toxic. People who work with thorium compounds are at a risk of dermatitis. It can take as much as thirty years after the ingestion of thorium for symptoms to manifest themselves.[88] ## Notes 1. ^ Traces of primordial plutonium-244 still exist in nature,[2] but this does not occur in quantity, unlike bismuth, thorium, and uranium. 2. ^ The transition temperature is between 1.35 and 1.40 K.[3] 3. ^ [Rn]6d2 is a very low-lying excited state configuration of Th2+.[14] 4. ^ 232Th is in fact the shortest-lived nuclide that still has a half-life longer than the generally accepted age of the universe. It is the sixth-most unstable primordial nuclide: among the primordial nuclides, only 238U, 40K, 235U, 146Sm, and 244Pu have shorter half-lives.[18] 5. ^ 232Th has the following extremely rare decay modes: double beta decay to 232U; spontaneous fission; and cluster decay to 182Yb, expelling 26Ne and 24Ne clusters.[18] 6. ^ The longest-lived intermediate decay product of 232Th is its immediate alpha daughter 228Ra, which has a half-life of only 5.7 years.[18] 7. ^ Technically, the 4n chain starts at primordial 244Pu, which decays successively to 240U, 240Np, 240Pu, 236U, and then 232Th: however, the concentration of primordial 244Pu is now reduced to mere traces and hence these radioisotopes are extinct for all practical purposes.[18] 8. ^ This is shown by the fact that the rare earths are there given atomic weight values two-thirds of their actual ones, and thorium and uranium are given values half of their actual ones. 9. ^ A second extra-long periodic table row, to accommodate known and undiscovered elements with an atomic weight greater than bismuth (thorium, protactinium and uranium, for example), had been postulated as far back as 1892. Most investigators, however, considered that these elements were analogous to the third-row transition elements hafnium, tantalum and tungsten. The existence of a second inner transition series, in the form of the actinides, was not accepted until similarities with the electron structures of the lanthanides had been established.[41] 10. ^ The same also applies to uranium minerals, and many thorium minerals also contain uranium.[51][52] 11. ^ Thorianite refers to minerals with 75–100 mol% ThO2; uranothorianite, 25–75 mol% ThO2; thorian uraninite, 15–25 mol% ThO2; uraninite, 0–15 mol% ThO2.[48] 12. ^ More specifically, thorite refers to thorium silicate with the tetragonal zircon structure; monoclinic thorium silicate is referred to as huttonite.[48] 13. ^ Hydroxide ions may also replace silicate ions to form Th(SiO4)1−x(OH)4x, which as a mineral is named thorogummite.[48] 14. ^ Among the low number of other known thorium oxometallates are the arsenate, tungstate, germanate, silicate, borate, and perrhenate. While thorium titanates and tantalates are known, they are structurally more like double oxides than true oxometallates.[60] 15. ^ The thirteen fissile actinide isotopes with half-lives over a year are 229Th, 233U, 235U, 236Np, 239Pu, 241Pu, 242mAm, 243Cm, 245Cm, 247Cm, 249Cf, 251Cf, and 252Es. Of these, only 235U is naturally occurring, and only 233U and 239Pu can be bred from naturally occurring nuclei with single neutron capture.[67][68] ## References 1. ^ Magnetic susceptibility of the elements and inorganic compounds, in Handbook of Chemistry and Physics 81st edition, CRC press. 2. ^ Hoffman, D. C.; Lawrence, F. O.; Mewherter, J. L.; Rourke, F. M. (1971). "Detection of Plutonium-244 in Nature". Nature 234 (5325): 132–134. Bibcode:1971Natur.234..132H. doi:10.1038/234132a0. 3. Wickleder et al., pp. 61–3. 4. Yu. D. Tretyakov, ed. (2007). Non-organic chemistry in three volumes. Chemistry of transition elements 3. Moscow: Academy. 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Dudel et al.: Untersuchungen zur radiologischen Emission des Uran-Tailings Schneckenstein, 1988 (PDF; 4,0 MB), TU Bergakademie Freiberg and TU Dresden. ## Bibliography • Golub, A. M. (1971). Общая и неорганическая химия (General and Inorganic Chemistry) 2. • Greenwood, Norman N.; Earnshaw, Alan (1997). Chemistry of the Elements (2nd ed.). Butterworth-Heinemann. ISBN 0080379419. • Wickleder, Mathias S.; Fourest, Blandine; Dorhout, Peter K. (2006). "Thorium". In Morss, Lester R.; Edelstein, Norman M.; Fuger, Jean. The Chemistry of the Actinide and Transactinide Elements 3 (3rd ed.). Dordrecht, the Netherlands: Springer. pp. 52–160. doi:10.1007/1-4020-3598-5_3.	2014-09-30 13:27:49	{"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 1, "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": 0.7929947376251221, "perplexity": 9084.627255956118}, "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-2014-41/segments/1412037662910.19/warc/CC-MAIN-20140930004102-00035-ip-10-234-18-248.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/slight-confusion-concerning-newtons-first-law.635418/	# Slight confusion concerning Newton's first law 1. Sep 12, 2012 ### Ripe Hi, I've very recently begun physics self-teaching as I used to find it boring and thus avoided it as a subject but now find it increasingly interesting. Nevertheless, I'm having some problems regarding questions on Newton's First Law. One of the questions in the book I'm using says the following: "A 455N crate is being pulled at constant velocity by a force (F) directed at 30 degrees to the horizontal. The frictional force on the crate is 1163N. What is the magnitude of the pulling force?" Now, since it's being pulled at constant velocity, that should mean that the net force is 0. However, in order to get the same answer as in the mark scheme, one has to equate Fx (i.e. the x-axis component of F) to the same magnitude as the friction force and then use cos30=Fx/F to find the answer. The problem is that I don't see how the net force is 0 in this case. The fact that Fx is of the same magnitude, but opposite, to the friction force, wouldn't that mean that the resultant force is the Fy component, and thus not 0? I would appreciate it if someone could shed some light on this matter! 2. Sep 12, 2012 ### Ripe There is another question which I'm having difficulties with: "A mass (M) is connected with a string to a smaller mass (m). "M" is resting on an inclined plane and the string goes over a pulley at the top of the plane so that "m" is hanging vertically. What must the angle of the plane be in order to have equilibrium?" I'd appreciate it if I could get an explanation, and not simply the solution. 3. Sep 12, 2012 ### HallsofIvy Staff Emeritus The Fy component of force is not directed horizontally, so is not relevant to this question. As long as the Fy component of force is not greater than the weight of the crate, the floor will exert an upward "passive force" exactly offsetting the difference. 4. Sep 12, 2012 ### Ripe I'm aware that Fy is not the horizontal component, but the vertical one. Shouldn't the passive force be downward, not upward? In either case, if you calculate Fy, it is greater than the weight (roughly 600N vs 455N), and the net force should thus not be 0? 5. Sep 12, 2012 ### HallsofIvy Staff Emeritus The mass, m, has gravity acting on it so that, in order for it to remain stationary, not moving, the upward force on it must be mg. The whole point of the pully is that it changes the direction the force. In order to exert force mg on the string, the force on the mass, M, which is, of course, Mg, must have a component parallel to the inclined plane, equal to mg. If the inclined plane makes angle $\theta$ with the horizontal, the force along the inclined plane due to the mass, M, is $Mg sin(\theta)$ so, to have everything motionless, we must have $Mg sin(\theta)= mg$ or $M sin(\theta)= m$. 6. Sep 12, 2012 ### HallsofIvy Staff Emeritus No, the downward force is the weight of the object. By "passive force" I mean a force exerted by something that is exactly enough to support whatever is pressing on it. If we set a 1 N weight on the ground, it isn't going to move- that's because the ground is pressing up on it with force 1 N. If we set a 1000 N weight on the ground, the ground will push up on it with force 1000 N. That is why we call it a "passive" force- it doesn't cause any object to move. 7. Sep 12, 2012 ### Ripe Thanks, now I understand it better. I, however, don't quite follow why the force along the string is Mgsin(theta) and not just Mg? Thanks for the patience with me! Edit: Actually, it's the same thing, isn't it? If you say the hypotenuse is Mg and the side opposite to (theta) is mg, you get sin(theta)=Mg/mg or sin(theta)=M/m. 8. Sep 12, 2012 ### Staff: Mentor There are three forces acting in the vertical direction on the crate: the upward component of the cable tension, the downward force of gravity on the crate, and the upward contact force of the ground on the crate. The sum of these three forces must be zero, since there is no acceleration in the vertical. You can use this equilibrium condition to calculate the upward force of the ground on the crate. 9. Sep 13, 2012 ### Ripe Which problem are you talking about? The one with the crate or the one with the two masses? 10. Sep 13, 2012 ### azizlwl The problem is that I don't see how the net force is 0 in this case. The fact that Fx is of the same magnitude, but opposite, to the friction force, wouldn't that mean that the resultant force is the Fy component, and thus not 0? .................................... Fy and Fx are orthogonal. Thus Fy and Fx are independent. Forces in Fx will not change any value of Fy and vice-versa. 11. Sep 13, 2012 ### Ripe Yes but what I'm saying is that if the Fx component cancels out the magnitude of the friction force and the Fy component has a magnitude of roughly 600N, how can the net force be 0? Or is my thinking wrong? 12. Sep 13, 2012 ### azizlwl You right about Fx, 2 forces, net zero. Fy consisted 3 forces, net zero. The vertical component of pulling force is FSinθ°,up The weight mg, down The Normal force, up Since the object not sinking or rising, thus the net force is zero. For second question, for non-inclined plane, mg is orthogonal to the string. By making the mg the resultant of 2 component forces, we can use the component that is parallel to the plane to pull the string. Last edited: Sep 13, 2012 13. Sep 13, 2012 ### Ripe I know that, logically, since the object is not rising/sinking the net force is zero. What I need in order to be able to understand it completely is some mathematical evidence to support that. The way I see it, Fy is roughly 600N (calculated it yesterday but I don't have it written down atm), W is 455N and shouldn't the normal force thus also be 455N? That clearly gives a net force upwards. What am I missing? 14. Sep 13, 2012 ### PhanthomJay In the future, please post separate questions in separate posts to avoid this confusion. I believe you have copied down the problem wrong. Otherwise, the crate loses contact with the table since the normal force cannot act down on the block, and the problem makes no sense. Perhaps the weight of the crate is 4550 N?? 15. Sep 13, 2012 ### Ripe The problem doesn't mention the normal force. The book I've got mentioned that there's always a normal force on two contacting bodies, so I assumed there'd be a normal force on the crate as well. Could you elaborate on your post? 16. Sep 13, 2012 ### azizlwl Normal force is pressing force. The weight is 455N, downward, but there is also upward force, making it less pressed. If the upward force is equal to weight, then Normal force equal to zero. Just like a putting an object on a scale and pulling up, the reading will depend on how strong you pull the object up. 17. Sep 13, 2012 ### Ripe You mean net force equals zero, right? If so, I'm already aware of this. 18. Sep 13, 2012 ### azizlwl wouldn't that mean that the resultant force is the Fy component ....................... I think what you're thinking that Fx is equal to Friction, so cancelled out. Thus leave Fy which not zero, true. You have to see other forces acting in y-direction. Then you can find those forces that make net force in y-direction equal to zero. 19. Sep 13, 2012 ### Ripe That's more or less the whole point of my thread. What are the forces acting in the y-direction? 20. Sep 13, 2012 ### azizlwl The obvious are Fy and mg. There should be another force that makes the net force equal to zero, since no vertical motion.	2017-10-19 23:53:22	{"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": 0.7212384343147278, "perplexity": 645.365034318827}, "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-2017-43/segments/1508187823482.25/warc/CC-MAIN-20171019231858-20171020011858-00204.warc.gz"}
https://www.physicsforums.com/threads/hamiltonian-invariance.264565/	# Hamiltonian invariance 1. Oct 15, 2008 ### Irid 1. The problem statement, all variables and given/known data I'm solving Goldstein's problems. I have proved by direct substitution that Lagrange equations of motion are not effected by gauge transformation of the Lagrangian: $$L' = L + \frac{dF(q_i,t)}{dt}$$ Now I'm trying to prove that Hamilton equations of motion are not affected by this type of transformation. 2. Relevant equations Hamiltonian: $$H = \dot{q}_i p_i - L$$ Total time derivative: $$\frac{dF(q_i,t)}{dt} = \frac{\partial F}{\partial q_i} \dot{q_i} + \frac{\partial F}{\partial t}$$ Canonical momentum: $$p_i = \frac{\partial L}{\partial \dot{q}_i}$$ 3. The attempt at a solution Using the definition of canonical momentum we immediately see that the new canonical momentum is $$p_i' = \frac{\partial L'}{\partial \dot{q}_i} = p_i + \frac{\partial F}{\partial q_i}$$ But wait a moment! If the canonical momentum is altered, isn't the motion going to be effected? I'm missing something here... Anyway, we go on further to show that the new Hamiltonian is $$H' = \dot{q}_i p_i' - L' = H - \frac{\partial F(q_i, t)}{\partial t}$$ It satisfies one of Hamilton's equations of motion $$\dot{q}_i' = \frac{\partial H}{\partial p_i} = \dot{q}_i$$ but fails for the second one, $$\dot{p}_i' = -\frac{\partial H}{\partial q_i} = \dot{p}_i + \frac{\partial^2 F}{\partial q_i \partial t}$$ Now I'm a little lost... I don't know how to prove the invariance, and the most disturbing part is that the canonical momentum is clearly not invariant. 2. Oct 15, 2008 ### siddharth Well, the Euler-Lagrange equation is what gives you the equation of motion, and that is invariant on adding the total derivative. So, the motion shouldn't be affected, right? Also, for the last couple of steps, shouldn't you find $$\frac{\partial H}{\partial {p_i'}}$$ as the canonical momentum is defined by the new lagrangian? Last edited: Oct 15, 2008	2018-02-22 00:01:13	{"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": 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": 0.8365170359611511, "perplexity": 566.9966954106994}, "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/1518891813818.15/warc/CC-MAIN-20180221222354-20180222002354-00694.warc.gz"}
https://support.bioconductor.org/p/105198/	edgeR:fitted and partial expression values 2 0 Entering edit mode alakatos ▴ 120 @alakatos-6983 Last seen 2.8 years ago United States Hello Everyone, I have a very noise RNAseq dataset with several covariates analyzed in edgeR. design <- model.matrix(~0 + conditions + cov1 + cov2, data=pheno) d <- calcNormFactors(d, method ="TMM") d <- estimateDisp(d, design, robust=TRUE ) fit <- glmFit(d, design, dispersion=d$trended.dispersion, robust=TRUE) my.contrasts <- makeContrasts(....) Issue 1. I was wondering if it is correct to extract the fitted values for visualization purposes to get rid of the noise. norm <-cpm(fit$fitted.values, normalized.lib.sizes = TRUE, log = TRUE, prior.count = 1) Issue 2. Is there any way to obtain only partial expression values related to conditions (proportion of explained variance) for downstream network analysis? Anita edgeR cpm edger fitted model explained variance • 594 views 0 Entering edit mode Aaron Lun ★ 27k @alun Last seen 6 hours ago The city by the bay Regarding visualization; it seems you want to use the fitted values instead of the original counts for each sample. At best, this is not necessary, as you could just visualize the GLM coefficients directly to examine the effects of interest (which you would be doing anyway, by comparing fitted values from different conditions in a plot). At worst, replacing the counts with fitted values would be actively misleading, as people looking at your plot would initially think that the fitted values were your original counts. You're not showing the sampling variance inherent to the counts, which would be necessary for a faithful visual representation of the data. Regarding "partial expression values"; I don't know what these are. Are you saying you want to regress out particular covariates and use the corrected observations for downstream analyses? I would suggest applying removeBatchEffect on the log-CPMs. It is also possible to do this on the original counts via quantile-quantile mapping, but it's a lot of effort; see A: Is Limma's removeBatchEffect() and log2() commutative?. 0 Entering edit mode "Are you saying you want to regress out particular covariates and use the corrected observations for downstream analyses?" Yes. Thanks Aaron very much. It is very helpful. 0 Entering edit mode @gordon-smyth Last seen 2 hours ago WEHI, Melbourne, Australia I am a great believer in doing simple, direct analyses when they do the job. I am not convinced here that you really need anything another than the log-fold changes between the conditions, i.e., what is stored in the estimated coefficients. The logFCs estimate the differences between the conditions, adjusted for the covariates, and that is surely what you need for any downstream network analysis. Similarly, why would you not be visualizing the log-fold changes? You definitely shouldn't be running cpm() on the fitted values, and there's no need to do that anyway. Finally, there are some curious aspects to your code. You have set robust=TRUE for glmFit(), but there is no such argument for that function. Have a look at ?glmFit. You've also restricted to trended dispersion for glmFit, even though we advise you against that. Why did you do that?	2022-07-03 12:38:08	{"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": 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": 0.39855629205703735, "perplexity": 2919.0373618181657}, "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/1656104240553.67/warc/CC-MAIN-20220703104037-20220703134037-00171.warc.gz"}
https://dmoj.ca/problem/cco22p3	## CCO '22 P3 - Double Attendance View as PDF Points: 25 (partial) Time limit: 3.0s Memory limit: 1G Author: Problem types Due to a rather ambitious school schedule, two of your classes are about to be held starting at exactly the same time, in two different classrooms! You can only be in one place at a time, so the best you can hope for is catching the important bits of both, even if that means sneaking back and forth between the two. The two classrooms are numbered and . In classroom , the teacher will show different slides during the class, with the slide shown throughout the exclusive time interval where and are times elapsed since the start of class measured in seconds. In both classes, there does not exist a point in time at which multiple slides are simultaneously being shown. For example, a class may have slides spanning the pair of intervals and , or the pair and , but not the pair and . You begin the day in classroom with both classes starting at time . Following that, at any point in time (not necessarily after an integral number of seconds), you may move from your current classroom to the other one in seconds. You consider yourself to have seen a slide if you spend a positive amount of time in its classroom strictly within the time interval during which it's being shown. When moving between the two classrooms, you're not considered to be inside either of them for seconds and are thus unable to see any slides. For example, if classroom has a slide being shown for the time interval , classroom has a slide being shown for the time interval , and , then you'll get to see both slides if you move from classroom to classroom at time seconds (arriving at time seconds). On the other hand, if you leave classroom at time seconds (arriving at time seconds), then you'll enter classroom just after its slide stops being shown and will therefore miss it. What's the maximum number of distinct slides which you can see at least once? #### Input Specification The first line contains three space-separated integers , , and . The next lines each contain two space-separated integers and . The next lines each contain two space-separated integers, and . Marks AwardedBounds on Bounds on and Bounds on marks marks marks, marks #### Output Specification Output one integer which is the maximum number of distinct slides which you can see. #### Sample Input 1 3 1 8 10 20 100 101 20 21 15 25 #### Output for Sample Input 1 3 #### Explanation of Output for Sample Input 1 One possible optimal strategy is to remain in classroom until time , then move to classroom (arriving at time ), remain there until time , and finally return to classroom (arriving at time ). In the process, you'll see all but the slide. It's impossible for you to see all 4 slides. #### Sample Input 2 1 5 3 1 100 1 2 2 3 3 4 4 5 5 6 #### Output for Sample Input 2 4 #### Explanation of Output for Sample Input 2 Even if you begin moving to classroom immediately at the start of the classes (e.g., at time ), you'll miss the first slides shown there. As such, it is possible to see a total of at most four slides.	2022-12-05 21:59:05	{"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": 0.743699312210083, "perplexity": 1034.727307532986}, "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-2022-49/segments/1669446711045.18/warc/CC-MAIN-20221205200634-20221205230634-00571.warc.gz"}
http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1049.65150	Language: Search: Contact Zentralblatt MATH has released its new interface! For an improved author identification, see the new author database of ZBMATH. Query: Fill in the form and click »Search«... Format: Display: entries per page entries Zbl 1049.65150 Brunner, Hermann; Vermiglio, Rossana Stability of solutions of delay functional integro-differential equations and their discretizations. (English) [J] Computing 71, No. 3, 229-245 (2003). ISSN 0010-485X; ISSN 1436-5057/e The authors derive asymptotic stability results for solutions of delay functional integro-dif\-feren\-tial equations of the type $${d\over dt}\Big[y(t)-\int_0^t a(t-s)G(s,y(s-\tau))\,ds\Big] \, = \, F(t,y(t)), \quad t\geq 0.$$ The idea is based on an approach introduced by {\it M. Zennaro} [Numer. Math. 77, 549--563 (1997; Zbl 0886.65092)] who studied stability with resprect to the forcing term. Therefore, the initial integro-differential equation is reformulated into $${d\over dt}\big[ y(t)-\big(V_\tau y\big)(t)\big]= F(t,y(t)), \qquad t\geq 0,$$ using the delay Volterra integral operator $$\big(V_\tau y\big)(t)= \int_0^t a(t-s)G(s,y(s-\tau))\, ds, \qquad t\geq 0.$$ Studying stability and contractivity properties of the latter equation allows to deduce similar properties for solutions generated by continuous Runge-Kutta or collocation methods. [Wolfgang zu Castell (Neuherberg)] MSC 2000: *65R20 Integral equations (numerical methods) 45J05 Integro-ordinary differential equations 45G10 Nonsingular nonlinear integral equations Keywords: delay functional integro-differential equation; asymptotic stability; Runge-Kutta methods; collocation methods; contractivity Citations: Zbl 0886.65092 Highlights Master Server	2013-05-21 13:00:21	{"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": 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": 0.9001962542533875, "perplexity": 7406.777078987455}, "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-2013-20/segments/1368700014987/warc/CC-MAIN-20130516102654-00032-ip-10-60-113-184.ec2.internal.warc.gz"}
http://pixelzoom.it/a-first-course-in-string-theory-pdf.html	# A First Course In String Theory Pdf uk: - Buy A First Course in String Theory by Barton Zwiebach (ISBN: 9780521831437) from Amazon's Book Store. Search for:. Professor Zwiebach's publications are available on the SPIRES HEP Literature Database. He presents the main concepts of string theory in a concrete and physical way to develop intuition before formalism, often through simplified and illustrative examples. He presents the main concepts of string theory in a concrete and physical way to develop intuition before formalism, often through simpliﬁed and illustrative exam-ples. function theory solutions eBooks which you could make use of to your benefit. First, don't hurt yourself, especially when you're just starting. This is a two-semester course. Press, 2004), by Barton Zwiebach, based on his MIT course for seniors (see the review by Marcelo Gleiser, Physics Today, September 2005, page 57). File: PDF, 32. Chapters 14, 15, and 16. Apparently, it covers (and maybe extends) the MIT undergraduate string theory course. He used handwritten notes taken in class by Kayla Jacobs, an MIT graduate, in producing them. Tompkins, John A. Електронна бібліотека. String theory was first studied in the late 1960s as a theory of the strong nuclear force, before being abandoned in favor of quantum chromodynamics. Basic physics is enough :). A search query can be a title of the book, a name of the author, ISBN or anything else. Davies — 2016-12-16 Computers. A First Course in String Theory Barton Zwiebach is once again faithful to his goal of making string theory accessible to undergraduates. Book file PDF easily for everyone. It develops a unified theory of discrete and continuous (univariate) Fourier analysis, the fast Fourier transform, and a powerful. Course Evaluation There will be 9 roughly weekly assignments, worth 100% of the course. A First Course In String Theory B Zwiebach Pdf. Green, Schwarz and Witten, Supertring Theory, Vol. viii 1 Contents 3. Barton Zwiebach is once again faithful to his goal of making string theory accessible to undergraduates. The next three chapters treat string interactions: the general formalism, and detailed treatments of the tree level and one loop amplitudes. A First Course in String Theory Barton Zwiebach Complete and thorough in its coverage, the author presents the main concepts of string theory in a concrete and physical way in order to develop intuition before formalism, often through simplified and illustrative examples. کتاب درس نخست در نظریه ریسمان بارتون زویباخ - ویرایش دوم A First Course in String Theory - 2nd Edition نویسنده : Barton Zwiebach مشخصات فایل فرمت PDF تعداد صفحات 697 حجم فایل زیپ شده 4. Imperial College Press. Related posts: Solution Manual for A First Course in the Finite Element Method - Daryl. The first part deals with basic ideas, reviewing special relativity and electromagnetism while introducing the concept of extra dimensions. Group theory Books: M. He presents the main concepts of string theory in a concrete and physical way to develop intuition before formalism, often through simplified and illustrative examples. ch/record/1166 (external link). Polyakov "Gauge fields and Strings" B. [Barton Zwiebach] -- An introduction to string theory, this book provides a detailed and self-contained demonstration of the main concepts involved. A First Course in String Theory. String Theory: An Overview 293 than just ‘quantize gravity’. Post a Review. They include full solutions to all the problems. [PDF] A First Course In String Theory Download Full - PDF A First Course in String Theory Barton Zwiebach Complete and thorough in its coverage, the author presents the main concepts of string theory in a concrete and physical way in order to develop intuition before formalism, often through simplified and Page 4/11. Vibration of a variable string 169 38. String theory is one of the most famous ideas in modern physics, but it is also one of the most confusing. Free UK delivery on eligible orders. This course introduces string theory to undergraduate and is based upon Prof. A First Course In String Theory B Zwiebach Pdf. Пошук книг Z-Library \| B-OK. String Theory: An Overview 293 than just 'quantize gravity'. Post a Review. I am actually very busy with honours (4th year) coursework and research, so I cannot really get down to reading substantial chunks of it whenever I want to. First Course delivered nicely. Polchinski, "String Theory", vol. 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String theory is a quantum theory, and,. 8 BRST in String Theory and the Physical Spectrum 42 Bibliography 46 Exercises 46 4 Conformal Field Theory 49 4. This course gives an introduction to the basics of string theory. Apparently, it covers (and maybe extends) the MIT undergraduate string theory course. The Book of R. The handwritten solutions are all due to Jeffrey Goldstone. zip >>> DOWNLOAD (Mirror #1). The lectures as given were awless, all errors con-tained herein re ect solely the student's typographical and/or intellectual de. For everyone, whether you are going to start to join with others to consult a book, this. The relation between string theory and quantum ﬁeld theory is more complicated than suggested by the naive picture of 'thick-ening Feynman graphs'. New York, NY: Cambridge University Press, 2004. They include full solutions to all the problems. Some properties of Bessel functions 179 41. ) This includes the use of a universal method for treating free ﬁeldtheories,which allows the derivation of. More precisely, it substantially represents the laboratory component of the course, while only nominally representing the presentational component of the course, and merely alluding to the programming com-. Contents Foreword Preface Acknowledgements Parti Basics 1 A brief introduction 1. 2 String theory as a unified theory of physics. I downloaded a sample and I think it's the 1st edition, i. Since string theory is quantum mechanics of a relativistic string, the foundations of the subject can be explained to students exposed to both special relativity and basic quantum mechanics. Zwiebach, A First Course in String Theory This book grew out of a course given to undergraduates who had no previous exposure to general relativity or quantum eld theory. A First Course in String Theory : Barton Zwiebach : Open string T-duality in double space. 7 BRST Primer 40 3. Complete and thorough in its coverage, this new edition now includes AdS/CFT. A First Course in String Theory. Accessible introduction to string theory for advanced undergraduate and graduate students. Teorije struna takođe obuhvataju objekte generalnije od struna, zvane brane. There is also the 11dM-theory. Srivastava. " You can also read there "In this modern day and age a lot of the disagreements about loop quantum gravity are aired in blogs and discussion groups on the Internet. Zwiebach, A First Course in String Theory. No previous knowledge is needed about quantum field theory, differential geometry, analytical mechanics. 47MB ZWIEBACH FIRST COURSE IN STRING THEORY SOLUTIONS As Pdf, SOLUTIONS ZWIEBACH IN FIRST COURSE STRING THEORY As Docx, FIRST ZWIEBACH FIRST COURSE IN STRING THEORY SOLUTIONS certainly provide much more likely to be effective through with hard work. 5 The attractor mechanism 587 11. These you will find on their website… same place yo. A First Course in String Theory EG CAMBRIDGE A First Course in String Theory An accessible introduction to string theory, this book provides a detailed and self-contained demonstration of the main concepts involved. Main A First Course in String Theory. pdf; A First Course in String Theory,2nd Edition. Електронна бібліотека. Before taking this course,I was convinced one would. The following pages contain the solutions for problems to be found in Part I of the textbook A First Course in String Theory, Second edition. An accessible introduction to string theory, this book provides a detailed and self-contained demonstration of the main concepts involved. which are quite pedagogic and complete for a first course and also the book "Basic Concepts of String Theory" by Blumenhagen, Prerequisites and introduction to string theory. The existence of such a theory was first conjectured by Edward Witten at a string theory conference at the University of Southern California in the Spring of Witten's announcement initiated a flurry of research activity known as the second superstring revolution. Complete and thorough in its coverage, the author presents the main concepts of string theory in a concrete and physical way in order to develop intuition before formalism, often through simplified and illustrative examples. A First Course in String Theory (CUP, 2004)(ISBN 0521831431)(400dpi)(T)(571s)_PQstr_. At its heart is the idea that the fundamental particles we observe are not point-like. If you find errors/typos or wish to suggest corrections, write to Barton Zwiebach at [email protected] Complete and thorough in its coverage, the author. This course develops the aspects of string theory and makes it. The course is taught by all of us; we sit in each other's lectures, and encourage discussions with students. Zwiebach, A First Course in String Theory This book grew out of a course given to undergraduates who had no previous exposure to general relativity or quantum eld theory. A First Course in String Theory. it deviates slightly from my 2nd edition book. String theory made understandable. I would suggest learning quantum field theory first. ) This includes the use of a universal method for treating free ﬁeldtheories,which allows the derivation of. polchinski string theory download Polchinski String Theory, volume 2: Superstring Theory pdfgen download and Beyond. 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A Course on Borel Sets, S. A First Course in Chaotic Dynamical Systems: Theory and Experiment is the first book to introduce modern topics in dynamical systems at the undergraduate level. (PDF)A First Course in String Theory, 2004, Barton Zwiebach (PDF)A First Course in the Finite Element Method, 4th Edition logan. Barton Zwiebach is once again faithful to his goal of making string theory accessible to undergraduates. I am actually very busy with honours (4th year) coursework and research, so I cannot really get down to reading substantial chunks of it whenever I want to. He presents the main concepts of string theory in a concrete and physical way to develop intuition before formalism, often through simplified and illustrative examples. An accessible introduction to string theory, this book provides a detailed and self-contained demonstration of the main concepts involved. A First Course in String Theory Barton Zwiebach is once again faithful to his goal of making string theory accessible to undergraduates. They are clear and elegant, and I hope that he will publish them someday. If you have any questions, contact us here. An open string can manifest itself as any particle with integer spin. If you find errors/typos or wish to suggest corrections, write to Barton Zwiebach at [email protected] He journeys occasionally into higher dimensions, although he still finds plenty of excitement in just four. It has wonderful pedagogical discussions of the basics of lightcone quantization. 428 Views. 8 BRST in String Theory and the Physical Spectrum 42 Bibliography 46 Exercises 46 4 Conformal Field Theory 49 4. This course introduces string theory to undergraduate and is based upon Prof. The first part deals with basic ideas, reviewing special relativity and electromagnetism while introducing the concept of extra dimensions. Off the top of my head, Barton Zwiebach's A First Course in String Theory. A First Course in String Theory Barton Zwiebach is once again faithful to his goal of making string theory accessible to undergraduates. File: PDF, 32. Complete and thorough in its coverage, the author presents the main concepts of string theory in a concrete and physical way in order to develop intuition before formalism, often through simplified and illustrative examples. A First Course In String Theory - PDFFormat at rhodos-bassum. Are there any pedagogical string theory problem sets, for a beginner, preferably with solutions, available on the web that would accompany my book? I. This course introduces string theory to undergraduate and is based upon Prof. String theory references: J. 1 71w case of two dimensions 51 4. in - Buy A First Course in String Theory book online at best prices in India on Amazon. String Theory I too would be interested in the pdf solutions since I am taking this as an indendant reading course as a beginning grad student in physics. He used handwritten notes taken in class by Kayla Jacobs, an MIT graduate, in producing them. A First Course in String Theory. The framework of special relativity plus quantum mechanics is so rigid that it practically. Buy A First Course in String Theory (2nd Edition) from ebookkiss. From Vibrating Strings to a Unified Theory of All Interactions. The first part be- gins with a discussion of special relativity, electromagnetism, and the physics of extra dimensions. A First Course in String Theory Barton Zwiebach Massachusetts Institute of Technology A refreshingly different approach to string theory… Professor Michael Green, University of Cambridge “ ” The ﬁrst generally accessible introduction to string theory Coherent and self-contained textbook for introductory courses in string theory, at. A First Course in String Theory EG CAMBRIDGE A First Course in String Theory An accessible introduction to string theory, this book provides a detailed and self-contained demonstration of the main concepts involved. Zwiebach is a great professor, his online Quantum mechanics can tell you that even without going to tough subject like string theory. 8 BRST in String Theory and the Physical Spectrum 42 Bibliography 46 Exercises 46 4 Conformal Field Theory 49 4. The first part be- gins with a discussion of special relativity, electromagnetism, and the physics of extra dimensions. pdf Carroll S. Solutions for problems in Part I. File name: A First Course in String Theory by Barton Zwiebach. A First Course in String Theory EG CAMBRIDGE A First Course in String Theory An accessible introduction to string theory, this book provides a detailed and self-contained demonstration of the main concepts involved. Complete and thorough in its coverage, the author presents the main concepts of string theory in a concrete and physical way in order to develop intuition before formalism, often through simplified and. D-branes and the classical dynamics of relativistic strings are discussed next, and the quantization of open and. This book is PDF Version and can be downloaded immediately from UK & USA Contact us on (862) 231-5157 or [email protected] Second Edition. pdf - search pdf books free download Free eBook and manual for Business, Education,Finance, Inspirational, Novel, Religion, Social, Sports, Science, Technology, Holiday, Medical,Daily new PDF ebooks documents ready for download, All PDF documents are Free,The biggest database for Free books and documents search with fast results better than any online. Group theory Books: M. A First Course In String Theory B Zwiebach Pdf. Teorija struna postulira da elektroni i kvarkovi unutar atoma nisu 0-dimenzioni objekti, nego da se sastoje od 1-dimenzionalnih struna. Comprehensive introduction to String theory. Sturm-Liouville theory and general Fourier expansions 36. This course is an introduction to string theory for upper-level undergraduates and beginning graduate students. He presents the main concepts of string theory in a concrete and physical way to develop intuition before formalism, often through simpliﬁed and illustrative exam-ples. Tanchoco PDF. Barton Zwiebach is once again faithful to his goal of making string theory accessible to undergraduates. String theory is one of the most famous ideas in modern physics, but it is also one of the most confusing. Volume 1, first published in 1998, provides a thorough introduction to the bosonic string. He presents the main concepts of string theory in a concrete and physical way to develop intuition before formalism, often through simplified and illustrative examples. Algebraic geometry is a central subject in modern mathematics, and an active area of research. I'm not sure what level of undergrad you are, if you haven't done 3rd year quantum mechanics course (something out of Shankar or Sakurai) then I'd concentrate on that first. 47MB ZWIEBACH FIRST COURSE IN STRING THEORY SOLUTIONS As Pdf, SOLUTIONS ZWIEBACH IN FIRST COURSE STRING THEORY As Docx, FIRST ZWIEBACH FIRST COURSE IN STRING THEORY SOLUTIONS certainly provide much more likely to be effective through with hard work. He wrote the textbook A First Course in String Theory (2004, ISBN -521-83143-1), meant for undergraduates. 35 MB Preview Save for later. String theory is the only known generalization of relativistic quantum field theory that makes sense. A First Course In Differential Equations The Classic Fifth Edition Recognizing the pretension ways to get this books a first course in differential equations the classic fifth edition is additionally useful. djvu: 2011-06-21 00:10 : 5. it deviates slightly from my 2nd edition book. A First Course in the Finite Element Method - 5th Edition Author(s): Daryl L. Contact author for errata. A First Course in String Theory, 2nd Edition. An accessible introduction to string theory, this book provides a detailed and self-contained demonstration of the main concepts involved. The following pages contain the solutions for all the problems to be found in Part I of the textbook A First Course in String Theory. This is in contrast to string theory, where the theory has evolved considerably over time and is now perceived by some as having too much freedom to be a predictive theory. String theory is hard, and uses a lot of deep maths, QFT is the same, but an introduction to QFT is nicer. questions initially covering the quantization and symmetries of the Nambu-Goto and Polyakov actions, the wordsheet etc. An accessible introduction to string theory, this book provides a detailed and self-contained demonstration of the main concepts involved. The relation between string theory and quantum ﬁeld theory is more complicated than suggested by the naive picture of ‘thick-ening Feynman graphs’. The lectures as given were awless, all errors con-tained herein re ect solely the student's typographical and/or intellectual de. A First Course in String Theory : Barton Zwiebach : Open string T-duality in double space. Even string theory now has an undergraduate text, A First Course in String Theory (Cambridge U. There has been much demand for a book about string theory at this level, and this one should go a long way towards meeting that demand. ISBN 978-1-86094-427-7. A First Course in String Theory. Zwiebach's textbook entitled A First Course in String Theory. Beckers, Schwarz, String Theory: A Modern Introduction. 6 Small BPS black holes in four dimensions 599 12 Gauge theory/string theory dualities 610 12. An accessible introduction to string theory, this book provides a detailed and self-contained demonstration of the main concepts involved. (Reprinted 2007) An Introduction to String Theory and D-brane Dynamics. String theory made understandable. download 1 file. A First Course in String Theory Be the first one to write a review. The following pages contain the solutions for all the problems to be found in Part I of the textbook A First Course in String Theory. He presents the main concepts of string theory in a concrete and physical way to develop intuition before formalism, often through simplified and illustrative examples. String theory is a vast subject, so it is not possible to cover all areas. String theory online resources. 2 Favorites. Barton Zwiebach is once again faithful to his goal of making string theory accessible to undergraduates. [PDF] A First Course In String Theory Download Full - PDF A First Course in String Theory Barton Zwiebach Complete and thorough in its coverage, the author presents the main concepts of string theory in a concrete and physical way in order to develop intuition before formalism, often through simplified and Page 4/11. Please check the sample to check the file format. Kiritsis, String Theory in a Nutshell. This book is PDF Version and can be downloaded immediately from UK & USA Contact us on (862) 231-5157 or [email protected] instructor's solutions manual for A First Course in Probability Theory, 6th edition, by. get the a first course in differential. Professor Zwiebach has not thoroughly proofread these notes but believes they will be useful to people that want to know what goes on during lecture and what material gets. An accessible introduction to string theory, this book provides a detailed and self-contained demonstration of the main concepts involved. Playing Guitar: A Beginner's Guide Page 7 Practicing Here are a few notes about how to approach practicing with the best frame of mind. ISBN: 0521831431, and are due during or after the class sessions as noted in the table. Closed strings. The following pages contain the solutions for problems to be found in Part I of the textbook A First Course in String Theory, Second edition. The first part be- gins with a discussion of special relativity, electromagnetism, and the physics of extra dimensions. This book is standard book for all departments that gives a trial of giving a string theory at advanced undergraduate or first year graduate course. Those who downloaded this book also downloaded the following books:. (A more detailed introduction. Zwiebach, A First Course in String Theory This book grew out of a course given to undergraduates who had no previous exposure to general relativity or quantum ﬁeld theory. They are now thought to be all manifestations of one theory. The handwritten solutions are all due to Jeﬀrey Goldstone. File: PDF, 32. String Theory and M-theory: A. Subsequently, it was realized that the very properties that made string theory unsuitable as a theory of nuclear physics made it a promising candidate for a quantum theory of gravity. They are clear and elegant, and I hope that he will publish them someday. A First Course in String Theory, by Barton. The following pages contain the solutions for problems to be found in Part I of the textbook A First Course in String Theory, Second edition. Some recent versions of string theory have predicted that the strings could have a longer length, up to nearly a millimeter in size, which would mean they're in the realm that. You may be 100% motivated to learn, but you won't learn anything if you damage your hand from overplaying. A first course in string theory (web draft, 2001)(429s). ISBN 978-1-86094-427-7. Srivastava. Conclusion Graphics from various places, including: Zwiebach, A First Course in String Theory Westphal, de Sitter String Vacua from Kahler Uplifting, Hamburg String Workshop 2007 Peter Woit (Columbia University) Is String Theory. Second Edition. These you will find on their website… same place yo. (PDF)A First Course In Probability 7th Edition SOLUTIONS MANUAL; Sheldon M. pdf cite 0 citations Cutoff A d S 3 \rm AdS_3 A d S 3 versus T T ˉ \rm T\bar{T} T T ˉ C F T 2 \rm CFT_2 C F T 2 in the large central charge sector: correlators of energy-momentum tensor. Apparently, it covers (and maybe extends) the MIT undergraduate string theory course. Jan Troost, Beyond String Theory (rough survey for laymen) Barton Zwiebach, A first course in string theory, Cambridge University Press (2009) (meant to be a course for undergraduates) Katrin Becker, Melanie Becker, John Schwarz, String theory and M-theory: a modern introduction, Cambridge University Press (2006) (spire:744404). in - Buy A First Course in String Theory book online at best prices in India on Amazon. String theory made understandable. As an alternative, we suggest a nonlocal formulation of string theory that breaks T-duality at the fundamental level and does not require the shortest possible distance. Zwiebach, A First Course in String Theory This book grew out of a course given to undergraduates who had no previous exposure to general relativity or quantum ﬁeld theory. Polchinski, "String Theory", vol. 3 The AdS/CFT correspondence 638. ZAlerts allow you to be notified by email about the availability of new books according to your search query. 5 The attractor mechanism 587 11. This section provides lecture notes for the course. A First Course in String Theory : Barton Zwiebach : 9780521880329 We use cookies to give you the best possible experience. It combines many interesting topics of (quantum) \A First Course in String Theory", Cambridge University Press (2004/2009) 6 recent: K. Search for:. Barton Zwiebach is once again faithful to his goal of making string theory accessible to undergraduates. Closed strings. Playing Guitar: A Beginner's Guide Page 7 Practicing Here are a few notes about how to approach practicing with the best frame of mind. Some properties of eigenvalues and eigenfunctions 171 39. Volume 1, first published in 1998, provides a thorough introduction to the bosonic string. download 1 file. I would suggest learning quantum field theory first. 47MB ZWIEBACH FIRST COURSE IN STRING THEORY SOLUTIONS As Pdf, SOLUTIONS ZWIEBACH IN FIRST COURSE STRING THEORY As Docx, FIRST ZWIEBACH FIRST COURSE IN STRING THEORY SOLUTIONS certainly provide much more likely to be effective through with hard work. • M-theory is eleven-dimensional. There were three issues that I wanted to understand 1) How did string theory actual produce a framework for the standard model, 2) What was the connection between string theory and black holes and 3) How was it that small distances could be indistinquishable from large distances. INSTRUCTOR SOLUTIONS MANUAL: A First Course in String Theory, 2004, Barton Zwiebach INSTRUCTOR SOLUTIONS MANUAL: A First Course in the Finite Element Method, 4th Edition logan. The local bosonic model with rotor variables placed on the faces of a cubic lattice is considered. String theory is hard, and uses a lot of deep maths, QFT is the same, but an introduction to QFT is nicer. [Barton Zwiebach] -- An introduction to string theory, this book provides a detailed and self-contained demonstration of the main concepts involved. ( Instructor's Solutions Manual Version) A First Course in the Finite Element Method, 5th Edition (Contact Email: cartermath(@)gmail(. ) Katrin Becker, Melanie Becker and John H. An introduction to general relativity-AW (2004). The following pages contain the solutions for all the problems to be found in Chapters 14, 15, and 16 of Part II of the textbook A First Course in String Theory. A First Course in String Theory, by Barton. String/M-theory To begin we have the 10d string theories: E 8 8 heterotic, SO(32) heterotic , type I, type IIA and type IIB. Bilal "Duality in N=2 SUSY SU(2) Yang-Mills Theory: A pedagogical introduction to the work of Seiberg and Witten" arXiv:hep-th. Those who downloaded this book also downloaded the following books: Comments. A First Course in String Theory Second Edition Barton Zwiebach is once again faithful to his goal of making string theory accessible to undergraduates. If you have any questions, contact us here. The following pages contain the solutions for problems to be found in Part I of the textbook A First Course in String Theory, Second edition. polchinski string theory volume 2 Lecture notes - Set 3 22 pages, scanned, pdf - Dec 5, 2006: Updated with new. Tanchoco PDF. It developed out of the lecture notes of an introductory undergrad-level course taught by Zwiebach at MIT and assumes very little knowledge of quantum. 6 Small BPS black holes in four dimensions 599 12 Gauge theory/string theory dualities 610 12. A search query can be a title of the book, a name of the author, ISBN or anything else. Apparently, it covers (and maybe extends) the MIT undergraduate string theory course. This construction naturally incorporates the emerging coupling between both gauge and string fields. Open strings vs. The first part be- gins with a discussion of special relativity, electromagnetism, and the physics of extra dimensions. The framework of special relativity plus quantum mechanics is so rigid that it practically. But if you want to get it to your smartphone, you can download more of ebooks now. A First Course in String Theory EG CAMBRIDGE A First Course in String Theory An accessible introduction to string theory, this book provides a detailed and self-contained demonstration of the main concepts involved. String Theory and M-theory: A. By zuj_admin. Infact, matter is also uniﬁed with forces, and they together form a theory that cannot be tinkered with. PDF dl manual solution a first course in complex analysis application g zill PDF dennis zill differential equations. I'm not sure what level of undergrad you are, if you haven't done 3rd year quantum mechanics course (something out of Shankar or Sakurai) then I'd concentrate on that first. Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. String theory made understandable. Free delivery on qualified orders. String theory is an attempt to quantise gravity and unite it with the other fundamental forces of nature. A First Course in String Theory Second Edition Barton Zwiebach is once again faithful to his goal of making string theory accessible to undergraduates. A First course in String Theory (2nd Edition)【弦论入门书】-Barton Zwiebach. A First Course In Differential Equations The Classic Fifth Edition Recognizing the pretension ways to get this books a first course in differential equations the classic fifth edition is additionally useful. This book is standard book for all departments that gives a trial of giving a string theory at advanced undergraduate or first year graduate course. Press, Cambridge, 2007. A First Course in String Theory. • The AdS/CFT correspondence is a remarkable physical equivalence between a certain four-dimensional gauge theory and a closed superstring theory. While programmatically string theory intends to su-persede quantum ﬁeld theory, in its current state it is deeply entangled. ((PDF)) Elements of the Nature and Properties of Soils 3rd Edition By Nyle C. There is no experimental evidence that string theory is the correct description of our world and scant hope that hard evidence will arise in the near future. Barton Zwiebach is once again faithful to his goal of making string theory accessible to undergraduates. We will make a compromise between systematically building up the technology of string perturbation theory and developing an understanding of. There has been much demand for a book about string theory at this level, and this one should go a long way towards meeting that demand. [Zw] Barton Zwiebach, A First Course in String Theory, (2004) Cambridge U. There are of course other text-books on the subject including the classics:. A Course in Simple-Homotopy Theory, Marshall M. 3 The AdS/CFT correspondence 638. Solutions for problems in Part I. These are the lecture notes for a short course in topological string theory that I gave at Uppsala University in the fall of 2004. (The best easy introduction. Syllabus and references: pdf. An accessible introduction to string theory, this book provides a detailed and self-contained demonstration of the main concepts involved. He presents the main concepts of string theory in a concrete and physical way to develop intuition before formalism, often through simplified and illustrative examples. A few of Sign up to download First course in complex analysis solutions manual. ((PDF)) Elements of the Nature and Properties of Soils 3rd Edition By Nyle C. I am actually very busy with honours (4th year) coursework and research, so I cannot really get down to reading substantial chunks of it whenever I want to. pdf - search pdf books free download Free eBook and manual for Business, Education,Finance, Inspirational, Novel, Religion, Social, Sports, Science, Technology, Holiday, Medical,Daily new PDF ebooks documents ready for download, All PDF documents are Free,The biggest database for Free books and documents search with fast results better than any online. It developed out of the lecture notes of an introductory undergrad-level course taught by Zwiebach at MIT and assumes very little knowledge of quantum. Download Free eBook:[share_ebook] Modern Physics - String Theory (11 Ebooks) - Free epub, mobi, pdf ebooks download, ebook torrents download. String theory Replace point particle concept by 1-dimensional string Particles as vibration modes of quantum strings Initially conceived as a theory of the strong interactions, particle concept replaced by one-dimensional strings. the long-sought quantum mechani- upon his upcoming textbook,"A First Course in String Theory,"the response was a moment of startled graduate level and its current frontier,string theory. It combines many interesting topics of (quantum) eld theory in two and higher dimensions. The rest of the solutions have been typeset. Zwiebach, A First Course in String Theory This book grew out of a course given to undergraduates who had no previous exposure to general relativity or quantum eld theory. First Course Finite Element Method Solution Manual I have the comprehensive instructor's solution manuals in an electronic format for the following textbooks. Even string theory now has an undergraduate text, A First Course in String Theory (Cambridge U. Cambridge University Press, 2009. edu is a platform for academics to share research papers. In the process, an. Kiritsis, String Theory in a Nutshell. · 【新浪微盘】宇宙的共同善：生态伦理的宗教根基. An Introduction to String Theory Kevin Wray Abstract: This set of notes is based on the course "Introduction to String Theory" which was taught by Prof. String Theory vol 2 - J. Volume 1, first published in 1998, provides a thorough introduction to the bosonic string. 53 مگابایت *** پست های مرتبط : Quantum Field Theory - Mark Allen Srednicki Quantum Field Theory. String theory Replace point particle concept by 1-dimensional string Particles as vibration modes of quantum strings Initially conceived as a theory of the strong interactions, particle concept replaced by one-dimensional strings. An accessible introduction to string theory, this book provides a detailed and self-contained demonstration of the main concepts involved. Press, 2004 \| Find, read and cite all the research you need on. It combines many interesting topics of (quantum) eld theory in two and higher dimensions. String theory is a vast subject, so it is not possible to cover all areas. pdf Dan Hooper-Dark Cosmos - In Search of Our Universe's. It has connections with number theory, differential geometry, symplectic geometry, mathematical physics, string theory, representation theory, combinatorics and others. The local bosonic model with rotor variables placed on the faces of a cubic lattice is considered. First Course Finite Element Method Solution Manual I have the comprehensive instructor's solution manuals in an electronic format for the following textbooks. An accessible introduction to string theory, this book provides a detailed and self-contained demonstration of the main concepts involved. Some string theory propaganda Text A first course in string theory,'' by Barton Zwiebach, is available at Paragraphe Bookstore. Jan Troost, Beyond String Theory (rough survey for laymen) Barton Zwiebach, A first course in string theory, Cambridge University Press (2009) (meant to be a course for undergraduates) Katrin Becker, Melanie Becker, John Schwarz, String theory and M-theory: a modern introduction, Cambridge University Press (2006) (spire:744404). This course is an introduction to string theory for upper-level undergraduates and beginning graduate students. Get this from a library! A first course in string theory. Books shelved as string-theory: The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory by Brian Greene, Parallel Wo. He used handwritten notes taken in class by Kayla Jacobs, an MIT graduate, in producing them. This course introduces string theory to undergraduate and is based upon Prof. Free delivery on qualified orders. An accessible introduction to string theory, this book provides a detailed and self-contained demonstration of the main concepts involved. There is no experimental evidence that string theory is the correct description of our world and scant hope that hard evidence will arise in the near future. Kiritsis, String Theory in a Nutshell. This is a two-semester course. It has wonderful pedagogical discussions of the basics of lightcone quantization. But if you want to get it to your smartphone, you can download more of ebooks now. An accessible introduction to string theory, this book provides a detailed and self-contained demonstration of the main concepts involved. D-branes and the classical dynamics of relativistic strings are discussed next, and the quantization of open and closed bosonic strings in the light-cone gauge, along. In the process, an. A First Course in String Theory. 2: Superstring theory and beyond. Reč brana, izvedena iz „membrana", se odnosi na različite međusobno. This is a working draft and is currently a set of personal notes. String theory online resources. String theory is the only known generalization of relativistic quantum field theory that makes sense. Kiritsis, String Theory in a Nutshell. • Our world is part of the D-branes. Електронна бібліотека. Technical and critical: Penrose, Roger (2005). Barton Zwiebach-A First Course in String Theory-Cambridge University Press (2009). We furnish utter variant of this ebook in doc, txt, DjVu, PDF, ePub formats. Professor Zwiebach has not thoroughly proofread these notes but believes they will be useful to people that want to know what goes on during lecture and what material gets. A First Course In String Theory B Zwiebach Pdf. The coupling model consisting of the Maxwell fields and the Kalb-Ramond field is given. By zuj_admin. Book file PDF easily for everyone. The following pages contain the solutions for all the problems to be found in Chapters 14, 15, and 16 of Part II of the textbook A First Course in String Theory. Cosmology in Gauge Field Theory and String Theory provides a modern introduction to these important problems from a particle physicist's perspective. There is no experimental evidence that string theory is the correct description of our world and scant hope that hard evidence will arise in the near future. A Course in Simple-Homotopy Theory, Marshall M. They are clear and elegant. He is one of the world's leading experts in string field theory. Beisert 0 Overview String theory is an attempt to quantise gravity and unite it with the other fundamental forces of nature. pdf: 2011-06-21 00:10 : 2. A Course on Borel Sets, S. String Theory comprises two volumes which give a comprehensive and pedagogic account of the subject. Zwiebach's textbook entitled A First Course in String Theory. The following pages contain the solutions for all the problems to be found in Part I of the textbook A First Course in String Theory. I downloaded a sample and I think it's the 1st edition, i. کتاب درس نخست در نظریه ریسمان بارتون زویباخ - ویرایش دوم A First Course in String Theory - 2nd Edition نویسنده : Barton Zwiebach مشخصات فایل فرمت PDF تعداد صفحات 697 حجم فایل زیپ شده 4. Barton Zwiebach is once again faithful to his goal of making string theory accessible to undergraduates. Summary: This is a one-semester class about string theory. 5 The attractor mechanism 587 11. It combines many interesting topics of (quantum) eld theory in two and higher dimensions. From Vibrating Strings to a Unified Theory of All Interactions. Particle spectrum of string theory includes a massless spin 2 excitation. A First Course in String Theory. uk: - Buy A First Course in String Theory by Barton Zwiebach (ISBN: 9780521831437) from Amazon's Book Store. A First Course in String Theory by Barton Zwiebach, 9780521880329, available at Book Depository with free delivery worldwide. String theory is hard, and uses a lot of deep maths, QFT is the same, but an introduction to QFT is nicer. The first part deals with basic ideas, reviewing special relativity and electromagnetism while introducing the concept of extra dimensions. A First Course In String Theory B Zwiebach Pdf. An accessible introduction to string theory, this book provides a detailed and self-contained demonstration of the main concepts involved. A First Course in String Theory Marcelo Gleiser is the Appleton Professor of Natural Philosophy and a professor of physics and astronomy at Dartmouth College in Hanover, New Hampshire. Guitar Grid & TAB sheets: Guitar C hord Diagram Sheets - Regular; Guitar Chord Diagram Sheets - Medium; Guitar Chord Diagram Sheets - Small; Guitar TAB sheets; Guitar 2 NECK Diagrams - LARGE. pdf; Probability and Statistics for Engineering and Sciences. [Barton Zwiebach] -- An introduction to string theory, this book provides a detailed and self-contained demonstration of the main concepts involved. ) This includes the use of a universal method for treating free ﬁeldtheories,which allows the derivation of. Kiritsis, String Theory in a Nutshell. He presents the main concepts of string theory in a concrete and physical way to develop intuition before formalism, often through simplified and illustrative examples. A First course in String Theory (2nd Edition)【弦论入门书】-Barton Zwiebach 学术资源学习资料下载和分享. They are clear and elegant, and I hope that he will publish them someday. A Course on Borel Sets, S. A search query can be a title of the book, a name of the author, ISBN or anything else. Building on concepts from Einstein's theory of general relativity and quantum theory, the course will expose students to the fundamentals of string theory and to some of the latest developments in the field. While programmatically string theory intends to su-persede quantum ﬁeld theory, in its current state it is deeply entangled. ) Katrin Becker, Melanie Becker and John H. The handwritten solutions are all due to Jeffrey Goldstone. 2017-12-07 A First Course in String Theory,2 edition 2017-11-29 [ PDF ] A First Course in Information Theory 2018-01-25 [ PDF ] Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning. I would appreciate it to check my solutions, I dont have to tell u that for a handfull of the problems i am quite unsure of my work which makes me skeptical and is tearing my brain:grumpy:. in - Buy A First Course in String Theory book online at best prices in India on Amazon. A First Course in String Theory, by Barton. Reviews of the A First Course in String Theory Until now with regards to the book we've A First Course in String Theory comments people haven't nevertheless quit the overview of the sport, you aren't make out the print still. This book is PDF Version and can be downloaded immediately from UK & USA Contact us on (862) 231-5157 or [email protected] This course is an introduction to string theory for upper-level undergraduates and beginning graduate students. An accessible introduction to string theory, this book provides a detailed and self-contained demonstration of the main concepts involved. 2 Matrix theory 625 12. For everyone, whether you are going to start to join with others to consult a book, this. A First Course in String Theory; The first part deals with basic ideas, reviewing special relativity and electromagnetism while introducing the concept of extra dimensions. INTRODUCTION TO STRING THEORY⁄ version 14-05-04 Gerard 't Hooft Institute for Theoretical Physics Utrecht University, Leuvenlaan 4 3584 CC Utrecht, the Netherlands. The existence of such a theory was first conjectured by Edward Witten at a string theory conference at the University of Southern California in the Spring of Witten's announcement initiated a flurry of research activity known as the second superstring revolution. download 1 file. Polchinski,, String Theory, Vol. ISBN: 0521831431, and are due during or after the class sessions as noted in the table. Press, Cambridge, 2007. SOLUTIONS MANUAL: A First Course in String Theory, 2004, Barton Zwiebach SOLUTIONS MANUAL: A First Course in the Finite Element Method, 4th Edition logan SOLUTIONS MANUAL: A First Course in the Finite Element Method, 5th Edition by logan. Complete and thorough in its coverage, the author presents the main concepts of string theory in a concrete and physical way in order to develop intuition before formalism, often through simplified and. Selected publications. Please check the sample before Solution Manual for A First Course in String Theory 2nd Edition by Zwiebach The file format would be one of WORD / PDF / EXCEL / ZIP. He presents the main concepts of string theory in a concrete and physical way to develop intuition before formalism, often through simplified and illustrative examples. This book has a beautiful introduction on string theory physics for undergraduate students. Apparently, it covers (and maybe extends) the MIT undergraduate string theory course. M-Theory (Becker, Becker and Schwarz), Introduction to String Theory (Polchinski), String Theory in a Nutshell (McMahon) and Superstring Theory (Green, Schwarz and Witten), along with the lecture notes of David Tong, sometimes word-for-word. An accessible introduction to string theory, this book provides a detailed and self-contained demonstration of the main concepts involved. A First Course in String Theory by Barton Zwiebach. 3 Black holes in string theory 566 11. Apparently, it covers (and maybe extends) the MIT undergraduate string theory course. pdf, 04-Feb-2010 15: 26, 32M, Introduction to. Barton Zwiebach is once again faithful to his goal of making string theory accessible to undergraduates. Oﬃce: Pupin 913, 1026. The book is clearly written and accessible even without any eld theory background, and in fact might be useful as indirect preparation for eld theory. Before taking this course,I was convinced one would. A first course in string theory (web draft, 2001)(429s). It combines many interesting topics of (quantum) eld theory in two and higher dimensions. Mar 1, 2010. Press, 2004), by Barton Zwiebach, based on his MIT course for seniors (see the review by Marcelo Gleiser, Physics Today, September 2005, page 57). A First Course in String Theory Barton Zwiebach Complete and thorough in its coverage, the author presents the main concepts of string theory in a concrete and physical way in order to develop intuition before formalism, often through simplified and illustrative examples. 2 Favorites. Syllabus and references: pdf. A First Course In String Theory B Zwiebach Pdf. uk: - Buy A First Course in String Theory by Barton Zwiebach (ISBN: 9780521831437) from Amazon's Book Store. Problems are assigned in the required text: Zwiebach, Barton. They include full solutions to all the problems. An accessible introduction to string theory, this book provides a detailed and self-contained demonstration of the main concepts involved. (A more detailed introduction. download 1 file. Please check the sample before Solution Manual for A First Course in String Theory 2nd Edition by Zwiebach The file format would be one of WORD / PDF / EXCEL / ZIP. A First Course in String Theory : Barton Zwiebach A First Course In String Theory B Zwiebach Pdf. The first part deals with basic ideas, reviewing special relativity and electromagnetism while introducing the concept of extra dimensions. pdf - search pdf books free download Free eBook and manual for Business, Education,Finance, Inspirational, Novel, Religion, Social, Sports, Science, Technology, Holiday, Medical,Daily new PDF ebooks documents ready for download, All PDF documents are Free,The biggest database for Free books and documents search with fast results better than any online library. But if you want to get it to your smartphone, you can download more of ebooks now. 11 MB: Create Date: May 1, 2014:. It has connections with number theory, differential geometry, symplectic geometry, mathematical physics, string theory, representation theory, combinatorics and others. String theory made understandable. These are the lecture notes for a short course in topological string theory that I gave at Uppsala University in the fall of 2004. • String theory has made good strides towards a statistical mechanics interpretation of black hole entropy. Zwiebach, A First Course in String Theory This book grew out of a course given to undergraduates who had no previous exposure to general relativity or quantum ﬁeld theory. These strings, the size of the Planck length (10-35 m), vibrate at specific resonant frequencies. preterhuman. These you will find on their website… same place yo. Schwartz, String Theory and M-Theory: A Modern Introduction, Cambridge U. Reč brana, izvedena iz „membrana", se odnosi na različite međusobno. · 【新浪微盘】宇宙的共同善：生态伦理的宗教根基. String Theory Physics V3500/G8099, Spring 2008 Dan Kabat and Lam Hui Coordinates. String theory made understandable. Free delivery on qualified orders. 2nd Edition. Solutions for problems in Part II. Post a Review. 1 Introduction It is well known that supersymmetry impose strong constraints on the geometry of the scalar manifold M, on which the scalar ﬁelds are viewed as coordinates[1, 2] (for reviews see [3]). In d = 4;N = 1 theories, the scalar manifold for chiral multiplets, M, is restricted to be K¨ahler. Home \| Package \| A First Course In String Theory B Zwiebach Pdf. pdf: 2011-06-21 00:10 : 2. Barton Zwiebach is once again faithful to his goal of making string theory accessible to undergraduates. [Zw] Barton Zwiebach, A First Course in String Theory, (2004) Cambridge U. Complete and thorough in its coverage, the author presents the main concepts of string theory in a concrete and physical way in order to develop intuition before formalism, often through simplified and. it deviates slightly from my 2nd edition book. Acerca de A First Course In String Theory de Zwiebach Barton. Srivastava. SOLUTIONS MANUAL: A First Course in String Theory, 2004, Barton Zwiebach SOLUTIONS MANUAL: A First Course in the Finite Element Method, 4th Edition logan SOLUTIONS MANUAL: A First Course in the Finite Element Method, 5th Edition by logan. A First Course in String Theory EG CAMBRIDGE A First Course in String Theory An accessible introduction to string theory, this book provides a detailed and self-contained demonstration of the main concepts involved. polchinski string theory volume 2 Lecture notes - Set 3 22 pages, scanned, pdf - Dec 5, 2006: Updated with new. The first four chapters introduce the central ideas of string theory, the tools of conformal field theory and of the Polyakov path integral, and the covariant quantization of the string. Since string theory is quantum mechanics of a relativistic string, the foundations of the subject can be explained to students exposed to both special relativity and basic quantum mechanics. A First Course in the Finite Element Method - 5th Edition Author(s): Daryl L. By online, you could not go to get the book establishment in your city. From Vibrating Strings to a Unified Theory of All Interactions. Professor Zwiebach's publications are available on the SPIRES HEP Literature Database. Syllabus and references: pdf. Teorije struna takođe obuhvataju objekte generalnije od struna, zvane brane. String Theory: An Overview 293 than just 'quantize gravity'. Download Book A First Course In General Relativity in PDF format. 1 & 2, Cambridge University Press 1998 - available as an e-book via the central library; Johnson: D-branes, Cambridge University Press 2003 - available as an e-book via the central library; Zwiebach: A First Course in String Theory, Cambridge University Press 2004. May 1, 2014. 2M [SLFM 139] Handbook of Recursive Mathematics. The following pages contain the solutions for all the problems to be found in Part I of the textbook A First Course in String Theory. SINGLE PAGE PROCESSED JP2 ZIP download. A First Course In String Theory B Zwiebach Pdf. download 1 file. Second Edition. Zwiebach, A First Course in String Theory. Post a Review. A First Course in String Theory Second Edition Barton Zwiebach is once again faithful to his goal of making string theory accessible to undergraduates. New York, NY: Cambridge University Press, 2004. Instead, the string has an objective shape in spacetime at all length scales, but different parts of the string interact in a. pdf - search pdf books free download Free eBook and manual for Business, Education,Finance, Inspirational, Novel, Religion, Social, Sports, Science, Technology, Holiday, Medical,Daily new PDF ebooks documents ready for download, All PDF documents are Free,The biggest database for Free books and documents search with fast results better than any online. Barton Zwiebach is once again faithful to his goal of making string theory accessible to undergraduates. 1 Black-brane solutions in string theory and M-theory 613 12. A First Course In String Theory - PDFFormat at rhodos-bassum. The rest of the solutions have been. pdf: 2011-06-21 00:10 : 2. Off the top of my head, Barton Zwiebach's A First Course in String Theory. A search query can be a title of the book, a name of the author, ISBN or anything else. Solution Manual for A First Course in String Theory 2nd Edition by Zwiebach It includes all chapters unless otherwise stated. A First Course in String Theory. 11 MB: Create Date: May 1, 2014:. Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. Since string theory is quantum mechanics of a relativistic string, the foundations of the subject can be explained to students exposed to both special relativity and basic quantum mechanics. (PDF)A First Course in String Theory, 2004, Barton Zwiebach (PDF)A First Course in the Finite Element Method, 4th Edition logan. 4 Statistical derivation of the entropy 582 11. String/M-theory To begin we have the 10d string theories: E 8 8 heterotic, SO(32) heterotic , type I, type IIA and type IIB. Complete and thorough in its coverage, the author presents the main concepts of string theory in a concrete and physical way in order to develop intuition before formalism, often through simplified and illustrative examples. com find submissions from "example. Problems are assigned in the required text: Zwiebach, Barton. He presents the main concepts of string theory in a concrete and physical way to develop intuition before formalism, often through simplified and illustrative examples. A First Course in String Theory Second Edition Barton Zwiebach is once again faithful to his goal of making string theory accessible to undergraduates. 2 Conformally Invariant Field Theory 52 4. A First Course in String Theory EG CAMBRIDGE A First Course in String Theory An accessible introduction to string theory, this book provides a detailed and self-contained demonstration of the main concepts involved.	2020-09-18 13:16:40	{"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": 0.18211527168750763, "perplexity": 936.8631189368797}, "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-2020-40/segments/1600400187899.11/warc/CC-MAIN-20200918124116-20200918154116-00144.warc.gz"}
https://tbc-python.fossee.in/convert-notebook/Electronics_Engineering_by_P._Raja/chapter_8_3.ipynb	# Chapter - 8 : Operational Amplifiers (OPAMPs)¶ ## Example 8.1 : Page No 432¶ In [21]: from __future__ import division # Given data A_V = -100 R1 = 2.2 # in kohm R1 = R1103 # in ohm R_f =-( A_VR1) # in ohm R_f = R_f * 10*-3 # in kohm print "The resistance value = %0.f kΩ" %R_f The resistance value = 220 kΩ ## Example 8.2 : Page No 432¶ In [20]: # Given data R_f = 200 # in kohm R1 = 2 # in kohm A_V = - (R_f/R1) V_in = 2.5 # in mV V_in= V_in10*-3 # in V V_o = (A_V V_in) # in V print "The output voltage = %0.2f V" %V_o The output voltage = -0.25 V ## Example 8.3 : Page No 461¶ In [19]: # Given data V1 = 2 # in V R_f = 500 # in kohm R_f = R_f103 # in ohm R1 = 100 # in kohm R1 = R1 10*3 # in ohm V_o = (1+(R_f/R1))V1 # in V print "The output voltage = %0.f V" %V_o The output voltage = 12 V ## Example 8.4 : Page No 461¶ In [18]: # Given data R_f = 1 # in Mohm R_f = R_f * 10*6 # in ohm print "Part (a)" V1 = 1 # in V V2 = 2 # in V V3 = 3 # in V R1 = 500 # in kohm R1 = R1 10*3 # in ohm R2 = 1 # in Mohm R2 = R2 10*6 # in ohm R3 = 1 # in Mohm R3 = R3 10*6 # in ohm V_o = -(R_f) ( (V1/R1)+(V2/R2)+(V3/R3) ) # in V print "The output voltage = %0.f V" %V_o print "Part (b)" V1 = -2 # in V V2 = 3 # in V V3 = 1 # in V R1 = 200 # in kohm R1 = R1 * 10*3 # in ohm R2 = 500 # in kohm R2 = R2 10*3 # in ohm R3 = 1 # in Mohm R3 = R3 10*6 # in ohm V_o = -(R_f) ( (V1/R1)+(V2/R2)+(V3/R3) ) # in V print "The output voltage = %0.f V" %V_o Part (a) The output voltage = -7 V Part (b) The output voltage = 3 V ## Example 8.6 : Page No 462¶ In [17]: # Given data R_f = 0 # in V R1 = 2 # in kohm R1 = R1 * 10*3 # in ohm A_vmin = (1+(R_f/R1)) print "The minimum closed loop voltage gain = %0.f" %A_vmin R_f1 = 100 # in kohm R_f1 = R_f1 10*3 # in ohm A_vmax = (1+(R_f1/R1)) print "The maximum closed loop voltage gain = %0.f" %A_vmax The minimum closed loop voltage gain = 1 The maximum closed loop voltage gain = 51 ## Example 8.7 : Page No 463¶ In [16]: # Given data V1 = 745 # in µV V1 = V1 10*-6 # in V V2 = 740 # in µV V2 = V2 10*-6 # in V Av = 510*5 CMRR = 80 # in dB # Formula CMRR in dB= 20log(Av/Ac) Ac= Av/(10*(CMRR/20)) print "The common mode gain = %0.2f" %Ac V_o = Av(V1-V2)+Ac((V1+V2)/2) # in V print "The output voltage = %0.2f V" %V_o # Note: In the book the calculation of finding the value of common mode gain (i.e. Ac) is wrong, # so the answer in the book is wrong The common mode gain = 50.00 The output voltage = 2.54 V ## Example 8.8 : Page No 464¶ In [15]: # Given data R_f = 1 # in Mohm R_f = R_f 10*6 # in ohm Ri= 110*6 # in ohm R1 = Ri # in ohm A_VF = -(R_f/R1) print "The Voltage gain = %0.f" %A_VF The Voltage gain = -1 ## Example 8.10 : Page No 465¶ In [14]: # Given data R_F = 3 # in kohm R1 = 1 # in kohm V1 = 2 # in V V2 = 3 # in V V_o1 = (1+(R_F/R1))V1 # in V V_o2 = (1+(R_F/R1))V2 # in V V_o = V_o1+V_o2 # in V print "The output voltage = %0.f V" %V_o The output voltage = 20 V ## Example 8.11 : Page No 466¶ In [13]: # Given data R_i = 10 # in kΩ R_im = 20 # in kΩ R_f = 500 # in kΩ A_vmin = -(R_f/R_i) print "Closed loop voltage gain corresponding to minimum resistance = %0.f" %A_vmin A_vmax = -(R_f/R_im) print "Closed loop voltage gain corresponding to maximum resistance = %0.f" %A_vmax Closed loop voltage gain corresponding to minimum resistance = -50 Closed loop voltage gain corresponding to maximum resistance = -25 ## Example 8.12 : Page No 467¶ In [12]: # Given data R_f = 200 # in kohm R1 = 20 # in kohm A_v = -(R_f/R1) V_i = 0.1 # in V V_im = 0.5 # in V V_omin = -10V_i # in V print "The minimum output voltage = %0.f V" %V_omin V_omax = -10V_im # in V print "The maximum output voltage = %0.f V" %V_omax print "Output voltage ranges : from",int(V_omin,),"to",int(V_omax) The minimum output voltage = -1 V The maximum output voltage = -5 V Output voltage ranges : from -1 to -5 ## Example 8.13 : Page No 467¶ In [11]: %matplotlib inline import matplotlib.pyplot as plt import numpy as np # Given data R = 133 # in kohm R = R 10*3 # in ohm C = 0.1 # in µF C = 0.1 10*-6 # in F Vi= 15 # in V plt.subplot(2,1,1) plt.plot([0,10],[1.5,1.5]) plt.ylabel("Vi in volts") plt.xlabel("t") plt.title("Input voltage") t=np.arange(0,1,0.1) Vo= -1/(RC)t plt.subplot(2,1,2) plt.plot(t,Vo) plt.xlabel("t") plt.ylabel("Vo in volts") plt.title("Output voltage") print "" ## Example 8.14 : Page No 468¶ In [10]: # Given data Rf = 250 # in kohm Vo= '-5Va+3Vb' # given expression # But output voltage of difference amplifier is # Vo= -Rf/R1Va+(R2/(R1+R2))(1+Rf/R1)Vb (i) # By comparing (i) with given expression R1 = Rf/5 # in kohm print "The value of R1 = %0.f kΩ" %R1 # (R2/(R1+R2))(1+Rf/R1)= 3 R2= 3R1*2/(R1+Rf-3R1) # in kΩ print "The value of R2 = %0.f kΩ" %R2 # Note: There is calculation error to find the value of R2 in the book. The value of R1 = 50 kΩ The value of R2 = 50 kΩ ## Example 8.15 : Page No 469¶ In [9]: # Given data V_i1 = 150 # in µV V_i2 = 140 # in µV V_d = V_i1-V_i2 # in µV V_C = (1/2)(V_i1+V_i2) # in µV print "Part (i)" CMRR = 100 A_d = 4000 V_o = (A_d V_d)(1+(1/CMRR)(V_C/V_d)) # in µV V_o = V_o * 10-3 # in mV print "The output voltage = %0.1f mV" %V_o print "Part(ii)" CMRR = 105 V_o = (A_d * V_d)(1+(1/CMRR)(V_C/V_d)) # in µV V_o = V_o * 10*-3 # in mV print "The output voltage = %0.3f mV" %V_o Part (i) The output voltage = 45.8 mV Part(ii) The output voltage = 40.006 mV ## Example 8.16 : Page No 470¶ In [8]: # Given data R_f = 470 # in kΩ R1 = 4.3 # in kΩ R2 = 33 # in kΩ R3 = R2 # in kΩ A1 = (1+R_f/R1) A2 = -(R_f/R2) A3 = -(R_f/R3) A = A1A2A3 V_i = 80 # in µV V_i= 8010*-6 # in V V_o = AV_i print "The output voltage = %0.2f V" %V_o The output voltage = 1.79 V ## Example 8.18 : Page No 472¶ In [7]: # Given data R4 = 300 # in kΩ R2 = 150 # in kΩ R3 = 10 # in kΩ R1 = 10 # in kΩ V1 = 1 # in V V2 = 2 # in V V_o = ( (1+(R4/R2))((R3/(R1+R3))V1)-((R4/R2)V2) ) # in V print "The output voltage = %0.1f V" %V_o The output voltage = -2.5 V ## Example 8.19 : Page No 472¶ In [6]: # Given data V_o = 2 # in V R_i = 20 # in kΩ R_f = 1 # in MΩ V_i = -((V_oR_i)/R_f) # in mV print "The input volatge = %0.f mV" %V_i The input volatge = -40 mV ## Example 8.20 : Page No 473¶ In [5]: # Given data R_f = 200 # in kΩ R_i = 30 # in kΩ V_i = 0.1 # in V V_im = 0.5 # in V Vo_min = -((R_f/R_i)V_i) # in V print "The minimum output voltage = %0.2f V" %Vo_min Vo_max = -((R_f/R_i)V_im) # in V print "The minimum output voltage = %0.2f V" %Vo_max print "The output voltage range is : ",round(Vo_min,2),"V to",round(Vo_max,2),"V" The minimum output voltage = -0.67 V The minimum output voltage = -3.33 V The output voltage range is : -0.67 V to -3.33 V ## Example 8.21 : Page No 473¶ In [4]: # Given data R_f = 360 # in kohm R_i = 12 # in kohm V1 = - 0.3 # in V V_o = (1+(R_f/R_i))V1 # in V print "The output voltage = %0.1f V" %V_o V_o1 = 2.4 # in V V_i = V_o1/(1+(R_f/R_i)) # in V print "The input voltage = %0.2f mV" %(V_i10*3) The output voltage = -9.3 V The input voltage = 77.42 mV ## Example 8.22 : Page No 474¶ In [3]: # Given data R_f = -68 # in kohm R1 = 33 # in kohm R2 = 22 # in kohm R3 = 12 # in kohm V1 = 0.2 # in V V2 = - 0.5 # in V V3 = 0.8 # in V V_o = ((R_f/R1)V1) + ((R_f/R2)V2) + ((R_f/R3)V3) # in V print "The output voltage = %0.3f V" %V_o The output voltage = -3.400 V ## Example 8.23 : Page No 475¶ In [2]: # Given data R_f = 1.8 # in kohm R_f = R_f * 10*3 # in ohm R1 = 180 # in ohm A_v = (R_f/R1) print "Closed loop gain = %0.f" %A_v F = 1 # in MHz F = F 10*6 # in Hz f2 = F/A_v # in Hz print "Closed loop bandwidth = %0.f Hz" %f2 V_in = 25 # in mV V_in = V_in 10*-3 # in V V_o = A_vV_in # in V print "The output voltage = %0.2f V" %V_o Closed loop gain = 10 Closed loop bandwidth = 100000 Hz The output voltage = 0.25 V ## Example 8.24 : Page No 475¶ In [1]: from __future__ import division # Given data R_f = 3 # in K ohm R_f = R_f * 10*3 # in ohm R1 = 150 # in ohm A_v = (R_f/R1) + 1 print "Close loop gain for inverting amplifier = %0.f" %A_v f = 1 # in MHz f = f 10*6 # in Hz f2 = f/A_v # in Hz f2 = f2 10**-3 # in KHz print "The closed loop bandwidth = %0.2f KHz" %f2 Close loop gain for inverting amplifier = 21 The closed loop bandwidth = 47.62 KHz	2020-06-01 01:35:23	{"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": 0.3496836721897125, "perplexity": 13159.304944757532}, "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-2020-24/segments/1590347413901.34/warc/CC-MAIN-20200601005011-20200601035011-00385.warc.gz"}
http://wiki.zcubes.com/Manuals/calci/MATRIXIDENTITY	# Manuals/calci/MATRIXIDENTITY MATRIXIDENTITY(n) • is the order of the matrix. ## Description • This function shows the identity matrix of a given matrix. • In , is the Order of identity matrix. • Identity matrix is also called Unit matrix. • Identity matrix of size n is the nxn square matrix with ones on the main diagonal and zeros on the other entries. • Identity matrix is denoted by . • , , ...... ## Examples 1.MATRIXIDENTITY(3) 1 0 0 0 1 0 0 0 1 2. MATRIXIDENTITY(5) 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 Identity Matrix	2022-01-19 17:23:01	{"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": 0.895622193813324, "perplexity": 208.0404895355034}, "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-2022-05/segments/1642320301475.82/warc/CC-MAIN-20220119155216-20220119185216-00423.warc.gz"}
https://www.math.ku.dk/english/calendar/events/algebratopology-seminar-07012022/algebratopology-seminar-1112022/	# Algebra/Topology Seminar Speaker: Shubhodip Mondal Title: On p-adic de Rham cohomology Abstract: In this talk, we will discuss an approach to de Rham cohomology in the p-adic context using group schemes and similar concrete structures, which we call G_a^{perf}-modules. As applications, I will mention the formal \'etaleness" property of the de Rham cohomology functor, which refines some earlier results obtained in the work of Bhatt-Lurie-Mathew and answers a question of Bhatt. Time permitting, I will mention the connection between the G_a^{perf}-module approach with the stacky approach to p-adic de Rham cohomology due to Drinfeld and mention the classification of all the endomorphisms of de Rham cohomology functor and its application to the Deligne-Illusie decomposition obtained in joint work with Shizhang Li.	2022-01-25 16:19:52	{"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": 0.7377646565437317, "perplexity": 832.1467000899025}, "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-2022-05/segments/1642320304859.70/warc/CC-MAIN-20220125160159-20220125190159-00562.warc.gz"}
http://math.stackexchange.com/questions/435454/a-b-matrices-and-av-bv-dependent-vectors	# A B matrices and Av Bv dependent vectors A,B nxn complex matrices : Prove that exist a vector v(not 0), that A(v) and B(v) are dependent. Extra question: What if A,B are real matrices? - If $\det B=0$, then the proof is trivial. If $\det B\ne 0$, then it boils down to prove that $\exists z\in\mathbb C,\exists v\in \mathbb C^n:$ $(A-zB)v=0$, which is equivalent to prove that $\exists z\in\mathbb C$: $\det (A-zB)=0$ or even further, $\exists z\in\mathbb C$: $\det (AB^{-1}-zI)=0$. Clearly, this polynomial has roots in $\mathbb C$, so we can conclude the proof. If we want to work only in $\mathbb R$, then we can build a counterexample for even dimensions: $$A=\begin{pmatrix}1&0\\0&1\end{pmatrix},\quad B=\begin{pmatrix}0&1\\-1&0\end{pmatrix}.$$ $Av=v$, but $B$ doesn't have any eigenvectors in $\mathbb R^2$. Edit In the case of odd dimensions, however, the hypothesys holds. Indeed, let's take our reasoning for complex case, replace $\mathbb C$ by $\mathbb R$ everywhere until the part $\exists z\in\mathbb R$: $\det (AB^{-1}-zI)=0$. In the odd-dimensioned space this determinant is a polynomial of the odd order, hence it has roots in $\mathbb R$ and thus we can find such $v$ that $Av$ and $Bv$ are dependant. To summarize: 1. Complex case. Such $v$ exists. 2. Real case, odd dimension. Such $v$ exists. 3. Real case, even dimension. Depends on matrices, we can give examples when such $v$ exists and when it does not. - $B$ may be zero, and $A$ nonsingular, in which case the $\det$ has no root in $\mathbb{C}$ :-). – copper.hat Jul 3 '13 at 17:07 @copper.hat I'll edit my post to take this into account. – TZakrevskiy Jul 3 '13 at 17:09 I think that for n=2k+1 it works in the real field. But i don't know show to prove it. – mrprottolo Jul 3 '13 at 19:15 @mrprottolo see edit. – TZakrevskiy Jul 3 '13 at 20:05 +1: Nice, complete answer. – copper.hat Jul 4 '13 at 13:01 If $A$ is singular, then $Av=0$ for some $v$, and hence $Bv, Av$ are dependent. Similarly if $B$ is singular. If $A,B$ are nonsingular, then $A^{-1}B$ has some non-zero eigenvalue $\lambda$ and eigenvector $v$. Then $A^{-1}Bv = \lambda v$, or $Bv = \lambda A v$, hence $Bv, Av$ are dependent. (Thanks to TZakrevskiy for catching an earlier error here.) Note: By considering $A=\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$ and $B=\begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}$, and looking for a solution to $(A-\lambda B) v = 0$, we see that we must have $\lambda = 0$, so, in general, we cannot find a $v$ such that both $Av,Bv$ are non-zero as well. Addendum: Answer to real field case: Take $A=I$ and $B=\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$, and suppose $Av, Bv$ are dependent. Since $v \neq 0$ and $B$ is nonsingular, we have $Bv \neq 0$, so we can write $Av = \lambda Bv$ for some $\lambda \in \mathbb{R}$. However, this imples $Av = v = \lambda Bv$, which would mean that $\frac{1}{\lambda}$ is an eigenvalue of $B$, which is contradiction, since $B$ has no real eigenvalues. So no such $v$ exists in this case. - Did you mean $Bv=\lambda Av$? – TZakrevskiy Jul 3 '13 at 17:07 @TZakrevskiy: Good catch :-). – copper.hat Jul 3 '13 at 17:07 @TZakrevskiy: Why delete your answer, it's a perfectly good way of dealing with $A$ non singular, and the singular case can be dealt with separately. Alternative approaches are always good. – copper.hat Jul 3 '13 at 17:15 That was tricky. Thanks so much, I was going crazy with that. :) – mrprottolo Jul 3 '13 at 17:15 Just added the second part of the problem. :) – mrprottolo Jul 3 '13 at 17:30	2015-05-24 06:06:19	{"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": 0.9749184250831604, "perplexity": 502.12837001215394}, "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-2015-22/segments/1432207927843.59/warc/CC-MAIN-20150521113207-00277-ip-10-180-206-219.ec2.internal.warc.gz"}
https://www.transtutors.com/questions/computation-for-a-hierarchical-generalized-linear-model-a-express-the-rat-tumor-exam-2008298.htm	# Computation for a hierarchical generalized linear model: (a) Express the rat tumor example of... Computation for a hierarchical generalized linear model: (a) Express the rat tumor example of Section 5.1 as a generalized linear model and obtain posterior simulations using the computational techniques presented in Section 16.4. (b) Use the posterior simulations to check the fit of the model using the realized χ2 discrepancy (6.4). ## Plagiarism Checker Submit your documents and get free Plagiarism report Free Plagiarism Checker	2021-04-16 10:19:35	{"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": 0.8296548128128052, "perplexity": 3422.1324072858115}, "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-2021-17/segments/1618038056325.1/warc/CC-MAIN-20210416100222-20210416130222-00004.warc.gz"}
http://gmat.kmf.com/question/21e2nh.html	The peaks of a mountain range, acting like rocks in a streambed, produce ripples in the air flowing over them; the resulting flow pattern, with crests and troughs that remain stationary although the air that forms them is moving rapidly, are known as "standing waves". • Acrests and troughs that remain stationary although the air that forms them is moving rapidly, are 分析该选项 • Bcrests and troughs that remain stationary although they are formed by rapidly moving air, are 分析该选项 • Ccrests and troughs that remain stationary although the air that forms them is moving rapidly, is 分析该选项 • Dstationary crests and troughs although the air that forms them is moving rapidly, are 分析该选项 • Estationary crests and troughs although they are formed by rapidly moving air, is 分析该选项 • 网友解析 ### 题目讨论 • 优质讨论 • 最新讨论 • 所属科目:语法SC • 题目来源1:OG12-95 • 正确率:40% • 难度:	2018-12-18 22:53:43	{"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": 0.8488383293151855, "perplexity": 5095.587242314787}, "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-2018-51/segments/1544376829997.74/warc/CC-MAIN-20181218225003-20181219011003-00230.warc.gz"}
https://3quarksdaily.com/3quarksdaily/2021/10/nines-of-safety-a-proposed-unit-of-measurement-of-risk.html	Nines of safety: a proposed unit of measurement of risk Terence Tao in his own blog: Because of all the very different ways in which percentages could be used, I think it may make sense to propose an alternate system of units to measure one class of probabilities, namely the probabilities of avoiding some highly undesirable outcome, such as death, accident or illness. The units I propose are that of “nines“, which are already commonly used to measure availability of some service or purity of a material, but can be equally used to measure the safety (i.e., lack of risk) of some activity. Informally, nines measure how many consecutive appearances of the digit ${9}$ are in the probability of successfully avoiding the negative outcome, thus • ${90\%}$ success = one nine of safety • ${99\%}$ success = two nines of safety • ${99.9\%}$ success = three nines of safety and so forth. Using the mathematical device of logarithms, one can also assign a fractional number of nines of safety to a general probability… More here.	2022-08-09 07:32:03	{"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 4, "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": 0.5414612889289856, "perplexity": 1639.9392661805668}, "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/1659882570913.16/warc/CC-MAIN-20220809064307-20220809094307-00443.warc.gz"}
http://www.oalib.com/relative/4116607	Home OALib Journal OALib PrePrints Submit Ranking News My Lib FAQ About Us Follow Us+ Title Keywords Abstract Author All Search Results: 1 - 10 of 100 matches for " " Page 1 /100 Display every page 5 10 20 Item Hongchang Hu Mathematical Problems in Engineering , 2010, DOI: 10.1155/2010/956907 Abstract: This paper studies a linear regression model, whose errors are functional coefficient autoregressive processes. Firstly, the quasi-maximum likelihood (QML) estimators of some unknown parameters are given. Secondly, under general conditions, the asymptotic properties (existence, consistency, and asymptotic distributions) of the QML estimators are investigated. These results extend those of Maller (2003), White (1959), Brockwell and Davis (1987), and so on. Lastly, the validity and feasibility of the method are illuminated by a simulation example and a real example. Ajay Chandra Mathematics , 2008, Abstract: This paper proposes a closed-form optimal estimator based on the theory of estimating functions for a class of linear ARCH models. The estimating function (EF) estimator has the advantage over the widely used maximum likelihood (ML) and quasi-maximum likelihood (QML) estimators that (i) it can be easily implemented, (ii) it does not depend on a distributional assumption for the innovation, and (iii) it does not require the use of any numerical optimization procedures or the choice of initial values of the conditional variance equation. In the case of normality, the asymptotic distribution of the ML and QML estimators naturally turn out to be identical and, hence, coincides with ours. Moreover, a robustness property of the EF estimator is derived by means of influence function. Simulation results show that the efficiency benefits of our estimator relative to the ML and QML estimators are substantial for some ARCH innovation distributions. Mathematics , 2008, DOI: 10.1214/009053607000000875 Abstract: We study the asymptotic properties of bridge estimators in sparse, high-dimensional, linear regression models when the number of covariates may increase to infinity with the sample size. We are particularly interested in the use of bridge estimators to distinguish between covariates whose coefficients are zero and covariates whose coefficients are nonzero. We show that under appropriate conditions, bridge estimators correctly select covariates with nonzero coefficients with probability converging to one and that the estimators of nonzero coefficients have the same asymptotic distribution that they would have if the zero coefficients were known in advance. Thus, bridge estimators have an oracle property in the sense of Fan and Li [J. Amer. Statist. Assoc. 96 (2001) 1348--1360] and Fan and Peng [Ann. Statist. 32 (2004) 928--961]. In general, the oracle property holds only if the number of covariates is smaller than the sample size. However, under a partial orthogonality condition in which the covariates of the zero coefficients are uncorrelated or weakly correlated with the covariates of nonzero coefficients, we show that marginal bridge estimators can correctly distinguish between covariates with nonzero and zero coefficients with probability converging to one even when the number of covariates is greater than the sample size. Statistics , 2015, Abstract: We prove conditional asymptotic normality of a class of quadratic U-statistics that are dominated by their degenerate second order part and have kernels that change with the number of observations. These statistics arise in the construction of estimators in high-dimensional semi- and non-parametric models, and in the construction of nonparametric confidence sets. This is illustrated by estimation of the integral of a square of a density or regression function, and estimation of the mean response with missing data. We show that estimators are asymptotically normal even in the case that the rate is slower than the square root of the observations. Statistics , 2011, Abstract: Although approximate Bayesian computation (ABC) has become a popular technique for performing parameter estimation when the likelihood functions are analytically intractable there has not as yet been a complete investigation of the theoretical properties of the resulting estimators. In this paper we give a theoretical analysis of the asymptotic properties of ABC based parameter estimators for hidden Markov models and show that ABC based estimators satisfy asymptotically biased versions of the standard results in the statistical literature. Mathematics , 2007, DOI: 10.1214/074921707000000337 Abstract: We study the asymptotic properties of the SCAD-penalized least squares estimator in sparse, high-dimensional, linear regression models when the number of covariates may increase with the sample size. We are particularly interested in the use of this estimator for simultaneous variable selection and estimation. We show that under appropriate conditions, the SCAD-penalized least squares estimator is consistent for variable selection and that the estimators of nonzero coefficients have the same asymptotic distribution as they would have if the zero coefficients were known in advance. Simulation studies indicate that this estimator performs well in terms of variable selection and estimation. Statistics , 2011, Abstract: We establish sufficient conditions for the asymptotic normality of kernel density estimators, applied to causal linear random fields. Our conditions on the coefficients of linear random fields are weaker than known results, although our assumption on the bandwidth is not minimal. The proof is based on the $m$-approximation method. As a key step, we prove a central limit theorem for triangular arrays of stationary $m$-dependent random fields with unbounded $m$. We also apply a moment inequality recently established for stationary random fields. Mathematics , 2005, DOI: 10.1016/j.jspi.2006.09.027 Abstract: The aim of this article is to simplify Pfanzagl's proof of consistency for asymptotic maximum likelihood estimators, and to extend it to more general asymptotic M-estimators. The method relies on the existence of a sort of contraction of the parameter space which admits the true parameter as a fixed point. The proofs are short and elementary. Mathematics , 2008, Abstract: We consider linear processes, not necessarily Gaussian, with long, short or negative memory. The memory parameter is estimated semi-parametrically using wavelets from a sample $X_1,...,X_n$ of the process. We treat both the log-regression wavelet estimator and the wavelet Whittle estimator. We show that these estimators are asymptotically normal as the sample size $n\to\infty$ and we obtain an explicit expression for the limit variance. These results are derived from a general result on the asymptotic normality of the empirical scalogram for linear processes, conveniently centered and normalized. The scalogram is an array of quadratic forms of the observed sample, computed from the wavelet coefficients of this sample. In contrast with quadratic forms computed on the Fourier coefficients such as the periodogram, the scalogram involves correlations which do not vanish as the sample size $n\to\infty$. Gerg？ Nemes Mathematics , 2010, Abstract: Applying a theorem of Howard for a formula recently proved by Brassesco and M\'endez, we derive new simple explicit formulas for the coefficients of the asymptotic expansion to the sequence of factorials. To our knowledge no explicit formula containing only the four basic operations was known until now. Page 1 /100 Display every page 5 10 20 Item	2019-12-09 07:56:36	{"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": 0.8235759735107422, "perplexity": 426.7125211546362}, "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-2019-51/segments/1575540518337.65/warc/CC-MAIN-20191209065626-20191209093626-00057.warc.gz"}
https://mobilearchitectureblog.wordpress.com/practice-algorithms-etc/	### Puzzle#1 – Level ( Solvable in your sleep ) Given a string, , of lowercase letters, determine the index of the character whose removal will make a palindrome. If is already a palindrome or no such character exists, then print . There will always be a valid solution, and any correct answer is acceptable. For example, if “bcbc”, we can either remove ‘b’ at index or‘c’ at index . Input Format The first line contains an integer, , denoting the number of test cases. Each line of the subsequent lines (where ) describes a test case in the form of a single string, . Constraints • All characters are lowercase English letters. • 1 < = T < = 20 • 1 <= S <= 100000 + 5 Output Format Print an integer denoting the zero-indexed position of the character that makes not a palindrome; if is already a palindrome or no such character exists, print . ### Solution ( 1 of many options ) Runtime = O(N1/2N) -> O(N²) #include <stdio.h> #include <stdlib.h> #include <string.h> int checkPerIndex(int arrayA[], int len,int checkIndex){ int result = -1; int j = len – 1; int i = 0; while(i <= j){ if(checkIndex == i){ i++; continue; } if(arrayA[i] != arrayA[j]){ if(i > checkIndex){ result = i; } break; } i++; j–; } return result; } void palindromcompare(int arrayA[], int len){ int skipfrontindex = -1; int skipbackindex = -1; for(int i = 0;i<len;i++){ if(arrayA[i] != arrayA[len-1-i]){ skipfrontindex = i; skipbackindex = len-1-i; int result = checkPerIndex(arrayA, len, skipfrontindex); if(result == -1){ printf(“%d\r\n”,skipfrontindex); break; } else if(result == skipbackindex){ printf(“%d\r\n”,skipbackindex); break; } result = checkPerIndex(arrayA, len, skipbackindex); if(result == -1){ printf(“%d\r\n”,skipbackindex); break; } } } if(skipfrontindex == -1 && skipbackindex == -1){ printf(“-1\r\n”); } } int main() { int n = 0; scanf(“%d”,&n); for(int i=0;i<n;i++){ char sean[100000]; scanf(“%s”,sean); int len = strlen(sean); int fwarray[len]; for(int k=0;k<len;k++){ int abplace = (int)sean[k] % 32; fwarray[k] = abplace; } palindromcompare(fwarray,len); } return 0; } ### Puzzle #2 Level ( Meh… not so easy but not a brain surgery ) You are given two strings, and . Find if there is a substring that appears in both and . Input Format Several test cases will be given to you in a single file. The first line of the input will contain a single integer , the number of test cases. Then there will be descriptions of the test cases. Each description contains two lines. The first line contains the string and the second line contains the string . Output Format For each test case, display YES (in a newline), if there is a common substring. Otherwise, display NO. Constraints All the strings contain only lowercase Latin letters 1 < = T <= 10 1 <= \|A\|,\|B\| <= 1000000 ### Solution #include “stdio.h” #include “string.h” int main() { int inputnum = 0; scanf(“%d”,&inputnum); char first[1000000]; char second[1000000]; for(int i=0;i<inputnum;i++){ scanf(“%s”,&first); scanf(“%s”,&second); int found = 0; int firstint[27] = {0}; int secondint[27] = {0}; if(strlen(first) < strlen(second)){ for(int j = 0;j<strlen(first);j++){ int ia = (int)((first[j] – ‘0’) % 48 ); if(ia > 0){ firstint[ia] = 1; } } for(int j = 0;j<strlen(second);j++){ int ia = (int)((second[j] – ‘0’) % 48 ); if(ia > 0 && firstint[ia] == 1){ found = 1; printf(“YES”); break; } } } else if(strlen(first) > strlen(second)){ for(int j = 0;j<strlen(second);j++){ int ia = (int)((second[j] – ‘0’) % 48 ); if(ia > 0){ secondint[ia] = 1; } } for(int j = 0;j<strlen(first);j++){ int ia = (int)((first[j] – ‘0’) % 48 ); if(ia > 0 && secondint[ia] == 1){ found = 1; printf(“YES”); break; } } } else{ for(int j = 0;j<strlen(second);j++){ int sia = (int)((second[j] – ‘0’) % 48 ); int fia = (int)((first[j] – ‘0’) % 48 ); if(sia == fia){ found = 1; printf(“YES”); break; } } } if(found == 0){ printf(“NO”); } printf(“\r\n”); } return 0; } ### Puzzle #3 Level ( It’s kind of interesting ) Jose found a love letter his friend Richard has written for his girlfriend. Jose is a prankster, so he decides to meddle with the letter. He changes all the words in the letter into palindromes. To do this, he follows two rules: 1. He can reduce the value of a letter, e.g. he can change d to c, but he cannot change c to d. 2. In order to form a palindrome, if he has to repeatedly reduce the value of a letter, he can do it until the letter becomes a. Once a letter has been changed to a, it can no longer be changed. Each reduction in the value of any letter is counted as a single operation. Find the minimum number of operations required to convert a given string into a palindrome. Input Format The first line contains an integer , i.e., the number of test cases. The next lines will contain a string each. The strings do not contain any spaces. Constraints 1 <= T <= 10 1 <= string len <= 100000 ### Puzzle #4 You are given a 2D array with dimensions 66. An hourglass in an array is defined as a portion shaped like this: a b c d e f g The sum of an hourglass is the sum of all the numbers within it. Mission: Print the largest sum among all the hourglasses in the array. Input Format There will be exactly 6 lines of input, each containing 6 integers separated by spaces. Each integer will be between -9 and 9, inclusively. Output Format Print the answer to this problem on a single line.	2019-01-24 10:29:25	{"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": 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": 0.21788275241851807, "perplexity": 4694.744414928938}, "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-2019-04/segments/1547584520525.90/warc/CC-MAIN-20190124100934-20190124122934-00220.warc.gz"}
https://www.aimsciences.org/article/doi/10.3934/naco.2021012?viewType=html	# American Institute of Mathematical Sciences • Previous Article Solving nonlinear differential equations using hybrid method between Lyapunov's artificial small parameter and continuous particle swarm optimization • NACO Home • This Issue • Next Article Individual biometrics pattern based artificial image analysis techniques doi: 10.3934/naco.2021012 ## Preconditioned inexact Newton-like method for large nonsymmetric eigenvalue problems 1 Jiangsu Key Laboratory for NSLSCS, School of Mathematical Science, Nanjing Normal University, Nanjing, 210023, China 2 College of Science, Nanjing Forestry University, Nanjing, 210023, China * Corresponding author: Li Wang Received January 2021 Revised March 2021 Published March 2021 Fund Project: The authors are supported by the National Natural Science Foundation of China under grant 41571380 and 10971102, Major project 16KJA110001 of the Natural Science Foundation of the Jiangsu Higher Education Institution An efficiently preconditioned Newton-like method for the computation of the eigenpairs of large and sparse nonsymmetric matrices is proposed. A sequence of preconditioners based on the Broyden-type rank-one update formula are constructed for the solution of the linearized Newton system. The properties of the preconditioned matrix are investigated. Numerical results are given which reveal that the new proposed algorithms are efficient. Citation: Hong-Yi Miao, Li Wang. Preconditioned inexact Newton-like method for large nonsymmetric eigenvalue problems. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021012 ##### References: show all references ##### References: The results for the largest real eigenvalue of $A$ Algorithm eigenvalue CPU time (s) GMRES-INewton 7.9817 9.8430 PGMRES-INewton 7.9817 6.9330 Algorithm eigenvalue CPU time (s) GMRES-INewton 7.9817 9.8430 PGMRES-INewton 7.9817 6.9330 The results of GMRES-INewton and PGMRES-INewton method GMRES-INewton PGMRES-INewton k $\|\|r_{k}\|\|$ $\|\lambda_{k}-\lambda^{}\|$ $cond(\mathcal{A})$ $\|\|r_{k}\|\|$ $\|\lambda_{k}-\lambda^{}\|$ $cond(P^{-1}\mathcal{A})$ 1 2.8670e+03 1.4750e+00 3.1275e+09 2.8670e+03 1.4750e+00 3.1275e+09 2 5.6282e+01 7.2040e-01 2.8736e+05 2.4063e+01 4.8650e-02 4.0326e+04 3 1.2415e+00 3.8120e-02 6.1342e+04 3.4120e-03 6.7520e-03 3.3108e+04 4 1.3586e-04 3.3470e-05 3.1625e+04 5.4758e-05 1.2050e-05 3.0213e+03 GMRES-INewton PGMRES-INewton k $\|\|r_{k}\|\|$ $\|\lambda_{k}-\lambda^{}\|$ $cond(\mathcal{A})$ $\|\|r_{k}\|\|$ $\|\lambda_{k}-\lambda^{}\|$ $cond(P^{-1}\mathcal{A})$ 1 2.8670e+03 1.4750e+00 3.1275e+09 2.8670e+03 1.4750e+00 3.1275e+09 2 5.6282e+01 7.2040e-01 2.8736e+05 2.4063e+01 4.8650e-02 4.0326e+04 3 1.2415e+00 3.8120e-02 6.1342e+04 3.4120e-03 6.7520e-03 3.3108e+04 4 1.3586e-04 3.3470e-05 3.1625e+04 5.4758e-05 1.2050e-05 3.0213e+03 The results for the largest real eigenvalue of $A$ Algorithm eigenvalue CPU time (s) GMRES-INewton 0.4338 12.5310 PGMRES-INewton 0.4338 10.4990 Algorithm eigenvalue CPU time (s) GMRES-INewton 0.4338 12.5310 PGMRES-INewton 0.4338 10.4990 The results of GMRES-INewton and PGMRES-INewton method GMRES-INewton PGMRES-INewton k $\|\|r_{k}\|\|$ $\|\lambda_{k}-\lambda^{}\|$ $cond(\mathcal{A})$ $\|\|r_{k}\|\|$ $\|\lambda_{k}-\lambda^{}\|$ $cond(P^{-1}\mathcal{A})$ 1 2.6740e+02 5.8630e+01 1.2640e+10 2.6740e+02 5.8630e+01 1.2640e+10 2 4.3572e+01 1.0270e+01 6.0304e+07 7.6230e+00 9.6810e-02 5.6210e+05 3 3.0450e+00 7.8140e-01 2.4639e+05 3.2540e-02 4.2830e-04 3.2790e+04 4 6.3580e-02 4.6720e-02 5.1852e+04 4.9130e-05 1.4270e-06 4.5870e+04 5 2.3710e-04 1.0390e-03 1.3089e+05 GMRES-INewton PGMRES-INewton k $\|\|r_{k}\|\|$ $\|\lambda_{k}-\lambda^{}\|$ $cond(\mathcal{A})$ $\|\|r_{k}\|\|$ $\|\lambda_{k}-\lambda^{}\|$ $cond(P^{-1}\mathcal{A})$ 1 2.6740e+02 5.8630e+01 1.2640e+10 2.6740e+02 5.8630e+01 1.2640e+10 2 4.3572e+01 1.0270e+01 6.0304e+07 7.6230e+00 9.6810e-02 5.6210e+05 3 3.0450e+00 7.8140e-01 2.4639e+05 3.2540e-02 4.2830e-04 3.2790e+04 4 6.3580e-02 4.6720e-02 5.1852e+04 4.9130e-05 1.4270e-06 4.5870e+04 5 2.3710e-04 1.0390e-03 1.3089e+05 The results for the largest real eigenvalue of a random matrix Algorithm eigenvalue CPU time (s) GMRES-INewton 298.2441 15.7350 PGMRES-INewton 298.2441 9.1470 Algorithm eigenvalue CPU time (s) GMRES-INewton 298.2441 15.7350 PGMRES-INewton 298.2441 9.1470 The results for the largest imaginary eigenvalue of a random matrix Algorithm eigenvalue CPU time (s) GMRES-INewton -0.0001+0.0054$i$ 16.2370 PGMRES-INewton -0.0001+0.0054$i$ 10.4790 Algorithm eigenvalue CPU time (s) GMRES-INewton -0.0001+0.0054$i$ 16.2370 PGMRES-INewton -0.0001+0.0054$i$ 10.4790 [1] Shuang Chen, Li-Ping Pang, Dan Li. 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Journal of Industrial & Management Optimization, 2021, 17 (4) : 2265-2275. doi: 10.3934/jimo.2020068 Impact Factor: ## Tools Article outline Figures and Tables	2021-04-23 13:47:38	{"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": 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": 0.71833735704422, "perplexity": 4549.370062574178}, "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-2021-17/segments/1618039594808.94/warc/CC-MAIN-20210423131042-20210423161042-00350.warc.gz"}
https://kb.osu.edu/handle/1811/21116?show=full	dc.creator Hayashi, Masato en_US dc.creator Hikida, Toshihide en_US dc.creator Harada, Kensuke en_US dc.creator Tanaka, Keiichi en_US dc.creator Tanaka, Takehiko en_US dc.date.accessioned 2006-06-15T20:44:49Z dc.date.available 2006-06-15T20:44:49Z dc.date.issued 2003 en_US dc.identifier 2003-WH-13 en_US dc.identifier.uri http://hdl.handle.net/1811/21116 dc.description Author Institution: Department of Chemistry, Faculty of Science, Kyushu University 33 en_US dc.description.abstract Rotational spectrum of the CoCO radical generated by ultraviolet photolysis of $Co(CO)_{3}NO$ was measured in the millimeter wave region to determine rotational and hyperfine interaction constants. Rotational transitions ranging from J = 29.5 - 28.5 to 34.5 - 33.5 were assigned in the $\Omega = 3/2$ and $\Omega = 5/2$ spin substates of the $X^{2}\Delta_{i}$ ground vibronic state as well as in the vibrationally excited states $\nu_{2}, \nu_{3}$, and $2\nu_{2}$ of the $\Omega = 5/2$ spin substate. Each rotational transition was split into 8 hyperfine components due to the $^{59}Co$ nucleus. Molecular constants, including the rotational constant B, centrifugal distortion constant D, nuclear spin-orbit interaction constant a, Fermi contact interaction constant $b_{F}$, magnetic dipolar interaction constant c, and nuclear quadrupole interaction constant eQq, were determined for the vibrationally ground state by least squares fitting of the observed spectrum. The equilibrium rotational constant $B_{e}$ was determined to be 4435.7510(18) MHz and the internuclear distance between Co and C was evaluated to be $1.688 \AA$. The a and c values are consistent with the values estimated from the hyperfine constants of the $^{59}Co$ atom. The $b_{F}$ value is nearly equal to zero. en_US dc.format.extent 248828 bytes dc.format.mimetype image/jpeg dc.language.iso English en_US dc.publisher Ohio State University en_US dc.title MILLIMETER WAVE SPECTRUM OF $CoCO(X^{2}\Delta_{i})$ IN THE GROUND AND VIBRATIONALLY EXCITED STATES en_US dc.type article en_US	2020-08-09 00:42:02	{"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": 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": 0.8224303126335144, "perplexity": 6560.315036074394}, "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/1596439738366.27/warc/CC-MAIN-20200808224308-20200809014308-00113.warc.gz"}
http://book.imt-decal.org/3.%20Number%20Theory/3.1%20Division%20Algorithm.html	3.1: Division Algorithm – Number Theory # 3.1: Division Algorithm To start the chapter, we will formalize a result that you’ve been familiar with since the start of your mathematics career. It may seem trivial and unnecessary, but as you will see later, the formal statement of this theorem will aid us in deriving more complicated results. Recall, when dividing one integer $a$ by another integer $b$, either $b$ evenly divided $a$, or there was some remainder left over, where the remainder was less than $b$. For example, when dividing 13 by 5, we have a remainder of 3. ### Definition: Division algorithm If $n, d$ are positive integers such that $n \geq d$, then it is possible to find non-negative integers $q, r$ such that $n = dq + r$, where $r < d$. In the above statement: • $n$ refers to the dividend, the number to be divided • $d$ refers to the divisor, the number to divide the dividend • $q$ refers to the quotient, the integer result of the division • $r$ refers to the remainder It turns out that finding $q$ and $r$ is relatively easy. $q = \lfloor \frac{n}{d} \rfloor$ $r = n - dq$ For example, we can represent the division of 13 by 5 by the expression $13 = 5 \cdot 2 + 3$. If the remainder is 0, then we say that $d$ “divides” $n$, and that $d$ is a factor of $n$. This is represented by the notation $d \big\| n$ Read on to see applications of this result.	2018-11-17 22:20:46	{"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": 0.9468882083892822, "perplexity": 285.61230951263013}, "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-2018-47/segments/1542039743854.48/warc/CC-MAIN-20181117205946-20181117231946-00440.warc.gz"}
https://www.jobilize.com/online/course/6-4-working-with-taylor-series-by-openstax?qcr=www.quizover.com&page=1	# 6.4 Working with taylor series (Page 2/11) Page 2 / 11 $\sum _{n=0}^{\infty }\left(\begin{array}{c}r\hfill \\ n\hfill \end{array}\right){x}^{n}=1+rx+\frac{r\left(r-1\right)}{2\text{!}}{x}^{2}+\text{⋯}+\frac{r\left(r-1\right)\text{⋯}\left(r-n+1\right)}{n\text{!}}{x}^{n}+\text{⋯}.$ We now need to determine the interval of convergence for the binomial series [link] . We apply the ratio test. Consequently, we consider $\begin{array}{cc}\hfill \frac{\|{a}_{n+1}\|}{\|{a}_{n}\|}& =\frac{{\|r\left(r-1\right)\left(r-2\right)\text{⋯}\left(r-n\right)\|x\|\|}^{n+1}}{\left(n+1\right)\text{!}}·\frac{n}{\|r\left(r-1\right)\left(r-2\right)\text{⋯}\left(r-n+1\right)\|{\|x\|}^{n}}\hfill \\ & =\frac{\|r-n\|\|x\|}{\|n+1\|}.\hfill \end{array}$ Since $\underset{n\to \infty }{\text{lim}}\frac{\|{a}_{n+1}\|}{\|{a}_{n}\|}=\|x\|<1$ if and only if $\|x\|<1,$ we conclude that the interval of convergence for the binomial series is $\left(-1,1\right).$ The behavior at the endpoints depends on $r.$ It can be shown that for $r\ge 0$ the series converges at both endpoints; for $-1 the series converges at $x=1$ and diverges at $x=-1;$ and for $r<-1,$ the series diverges at both endpoints. The binomial series does converge to ${\left(1+x\right)}^{r}$ in $\left(-1,1\right)$ for all real numbers $r,$ but proving this fact by showing that the remainder ${R}_{n}\left(x\right)\to 0$ is difficult. ## Definition For any real number $r,$ the Maclaurin series for $f\left(x\right)={\left(1+x\right)}^{r}$ is the binomial series. It converges to $f$ for $\|x\|<1,$ and we write $\begin{array}{cc}\hfill {\left(1+x\right)}^{r}& =\sum _{n=0}^{\infty }\left(\begin{array}{c}r\hfill \\ n\hfill \end{array}\right){x}^{n}\hfill \\ & =1+rx+\frac{r\left(r-1\right)}{2\text{!}}{x}^{2}+\text{⋯}+\frac{r\left(r-1\right)\text{⋯}\left(r-n+1\right)}{n\text{!}}{x}^{n}+\text{⋯}\hfill \end{array}$ for $\|x\|<1.$ We can use this definition to find the binomial series for $f\left(x\right)=\sqrt{1+x}$ and use the series to approximate $\sqrt{1.5}.$ ## Finding binomial series 1. Find the binomial series for $f\left(x\right)=\sqrt{1+x}.$ 2. Use the third-order Maclaurin polynomial ${p}_{3}\left(x\right)$ to estimate $\sqrt{1.5}.$ Use Taylor’s theorem to bound the error. Use a graphing utility to compare the graphs of $f$ and ${p}_{3}.$ 1. Here $r=\frac{1}{2}.$ Using the definition for the binomial series, we obtain $\begin{array}{cc}\hfill \sqrt{1+x}& =1+\frac{1}{2}x+\frac{\left(1\text{/}2\right)\left(\text{−}1\text{/}2\right)}{2\text{!}}{x}^{2}+\frac{\left(1\text{/}2\right)\left(\text{−}1\text{/}2\right)\left(\text{−}3\text{/}2\right)}{3\text{!}}{x}^{3}+\text{⋯}\hfill \\ & =1+\frac{1}{2}x-\frac{1}{2\text{!}}\phantom{\rule{0.2em}{0ex}}\frac{1}{{2}^{2}}{x}^{2}+\frac{1}{3\text{!}}\phantom{\rule{0.2em}{0ex}}\frac{1·3}{{2}^{3}}{x}^{3}-\text{⋯}+\frac{{\left(-1\right)}^{n+1}}{n\text{!}}\phantom{\rule{0.2em}{0ex}}\frac{1·3·5\text{⋯}\left(2n-3\right)}{{2}^{n}}{x}^{n}+\text{⋯}\hfill \\ & =1+\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n+1}}{n\text{!}}\phantom{\rule{0.2em}{0ex}}\frac{1·3·5\text{⋯}\left(2n-3\right)}{{2}^{n}}{x}^{n}.\hfill \end{array}$ 2. From the result in part a. the third-order Maclaurin polynomial is ${p}_{3}\left(x\right)=1+\frac{1}{2}x-\frac{1}{8}{x}^{2}+\frac{1}{16}{x}^{3}.$ Therefore, $\begin{array}{cc}\hfill \sqrt{1.5}& =\sqrt{1+0.5}\hfill \\ & \approx 1+\frac{1}{2}\left(0.5\right)-\frac{1}{8}{\left(0.5\right)}^{2}+\frac{1}{16}{\left(0.5\right)}^{3}\hfill \\ & \approx 1.2266.\hfill \end{array}$ From Taylor’s theorem, the error satisfies ${R}_{3}\left(0.5\right)=\frac{{f}^{\left(4\right)}\left(c\right)}{4\text{!}}{\left(0.5\right)}^{4}$ for some $c$ between $0$ and $0.5.$ Since ${f}^{\left(4\right)}\left(x\right)=-\frac{15}{{2}^{4}{\left(1+x\right)}^{7\text{/}2}},$ and the maximum value of $\|{f}^{\left(4\right)}\left(x\right)\|$ on the interval $\left(0,0.5\right)$ occurs at $x=0,$ we have $\|{R}_{3}\left(0.5\right)\|\le \frac{15}{4\text{!}{2}^{4}}{\left(0.5\right)}^{4}\approx 0.00244.$ The function and the Maclaurin polynomial ${p}_{3}$ are graphed in [link] . Find the binomial series for $f\left(x\right)=\frac{1}{{\left(1+x\right)}^{2}}.$ $\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\left(n+1\right){x}^{n}$ ## Common functions expressed as taylor series At this point, we have derived Maclaurin series for exponential, trigonometric, and logarithmic functions, as well as functions of the form $f\left(x\right)={\left(1+x\right)}^{r}.$ In [link] , we summarize the results of these series. We remark that the convergence of the Maclaurin series for $f\left(x\right)=\text{ln}\phantom{\rule{0.1em}{0ex}}\left(1+x\right)$ at the endpoint $x=1$ and the Maclaurin series for $f\left(x\right)={\text{tan}}^{-1}x$ at the endpoints $x=1$ and $x=-1$ relies on a more advanced theorem than we present here. (Refer to Abel’s theorem for a discussion of this more technical point.) Maclaurin series for common functions Function Maclaurin Series Interval of Convergence $f\left(x\right)=\frac{1}{1-x}$ $\sum _{n=0}^{\infty }{x}^{n}$ $-1 $f\left(x\right)={e}^{x}$ $\sum _{n=0}^{\infty }\frac{{x}^{n}}{n\text{!}}$ $\text{−}\infty $f\left(x\right)=\text{sin}\phantom{\rule{0.1em}{0ex}}x$ $\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\frac{{x}^{2n+1}}{\left(2n+1\right)\text{!}}$ $\text{−}\infty $f\left(x\right)=\text{cos}\phantom{\rule{0.1em}{0ex}}x$ $\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\frac{{x}^{2n}}{\left(2n\right)\text{!}}$ $\text{−}\infty $f\left(x\right)=\text{ln}\phantom{\rule{0.1em}{0ex}}\left(1+x\right)$ $\sum _{n=0}^{\infty }{\left(-1\right)}^{n+1}\frac{{x}^{n}}{n}$ $-1 $f\left(x\right)={\text{tan}}^{-1}x$ $\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\frac{{x}^{2n+1}}{2n+1}$ $-1 $f\left(x\right)={\left(1+x\right)}^{r}$ $\sum _{n=0}^{\infty }\left(\begin{array}{c}r\hfill \\ n\hfill \end{array}\right){x}^{n}$ $-1 Earlier in the chapter, we showed how you could combine power series to create new power series. Here we use these properties, combined with the Maclaurin series in [link] , to create Maclaurin series for other functions. where we get a research paper on Nano chemistry....? what are the products of Nano chemistry? There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others.. learn Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level learn da no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts Bhagvanji Preparation and Applications of Nanomaterial for Drug Delivery revolt da Application of nanotechnology in medicine what is variations in raman spectra for nanomaterials I only see partial conversation and what's the question here! what about nanotechnology for water purification please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment. Damian yes that's correct Professor I think Professor Nasa has use it in the 60's, copper as water purification in the moon travel. Alexandre nanocopper obvius Alexandre what is the stm is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.? Rafiq industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong Damian How we are making nano material? what is a peer What is meant by 'nano scale'? What is STMs full form? LITNING scanning tunneling microscope Sahil how nano science is used for hydrophobicity Santosh Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq Rafiq what is differents between GO and RGO? Mahi what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq Rafiq if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION Anam analytical skills graphene is prepared to kill any type viruses . Anam Any one who tell me about Preparation and application of Nanomaterial for drug Delivery Hafiz what is Nano technology ? write examples of Nano molecule? Bob The nanotechnology is as new science, to scale nanometric brayan nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale Damian Is there any normative that regulates the use of silver nanoparticles? what king of growth are you checking .? Renato What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ? why we need to study biomolecules, molecular biology in nanotechnology? ? Kyle yes I'm doing my masters in nanotechnology, we are being studying all these domains as well.. why? what school? Kyle biomolecules are e building blocks of every organics and inorganic materials. Joe how did you get the value of 2000N.What calculations are needed to arrive at it Privacy Information Security Software Version 1.1a Good Leaves accumulate on the forest floor at a rate of 2 g/cm2/yr and also decompose at a rate of 90% per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value? You have a cup of coffee at temperature 70°C, which you let cool 10 minutes before you pour in the same amount of milk at 1°C as in the preceding problem. How does the temperature compare to the previous cup after 10 minutes? Abdul	2020-09-21 00:49:36	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 74, "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": 0.9563242793083191, "perplexity": 3587.1082075131894}, "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/1600400198868.29/warc/CC-MAIN-20200920223634-20200921013634-00004.warc.gz"}
http://somrh.blogspot.com/2013/07/my-1348-per-year-investment.html	## Friday, July 26, 2013 ### My 1348% Per Year Investment! So several years ago I made a very small investment which has returned quite well. I made my money back within a few months and the rest was pure profit. Altogether, I estimate that my returns have been 1348% per annum. So what was that investment? A hair clipper! (If you really want to know, I think I own this model, although I haven't had the problems the negative reviewers are claiming so YMMV.) I figure I paid somewhere between $\$15- \$20$ for this thing around 6 years ago. If you assume that I've gotten 1 haircut a month for a savings of $\$5 \$ (sure it costs more for a professional haircut, but you get a professional out of it), that comes out to about a 1348% internal rate of return. And the clipper still works! OK, so I can't actually reproduce those returns (while I do have several voices, they all reside in just one head and therefore one set of hair). It has seemed to work out as a good one time investment. I don't if that makes me investment savvy or simply cheap but I'll take what I can get. Happy Friday!	2017-11-18 01:05:12	{"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": 0.3407372534275055, "perplexity": 2128.8128569211426}, "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-2017-47/segments/1510934804125.49/warc/CC-MAIN-20171118002717-20171118022717-00485.warc.gz"}
https://gdal.org/drivers/vector/gpsbabel.html	GPSBabel¶ Driver short name GPSBabel The GPSBabel driver for now that relies on the GPSBabel utility to access various GPS file formats. The GPSBabel executable must be accessible through the PATH. Driver capabilities¶ Supports Create() This driver supports the GDALDriver::Create() operation Supports Georeferencing This driver supports georeferencing The driver needs the GPX driver to be fully configured with read support (through Expat library) to be able to parse the output of GPSBabel, as GPX is used as the intermediate pivot format. The returned layers can be waypoints, routes, route_points, tracks, track_points depending on the input data. The syntax to specify an input datasource is : GPSBabel:gpsbabel_file_format[,gpsbabel_format_option]:[features=[waypoints,][tracks,][routes]:]filename* where : • gpsbabel_file_format is one of the file formats handled by GPSBabel. • gpsbabel_format_option is any option handled by the specified GPSBabel format (refer to the documentation of each GPSBabel format) • features= can be used to modify the type of features that GPSBabel will import. waypoints matches the -w option of gpsbabel commmandline, tracks matches -t and routes matches -r. This option can be used to require full data import from GPS receivers that are slow and for which GPSBabel would only fetch waypoints by default. See the documentation on Route and Track modes for more details. • filename can be an actual on-disk file, a file handled through the GDAL virtual file API, or a special device handled by GPSBabel such as “usb:”, “/dev/ttyS0”, “COM1:”, etc.. What is actually supported depends on the used GPSBabel format. Alternatively, for a few selected GPSBabel formats, just specifying the filename might be sufficient. The list includes for now : • garmin_txt • gdb • magellan • mapsend • mapsource • nmea • osm • ozi • igc The USE_TEMPFILE=YES configuration option can be used to create an on-disk temporary GPX file instead of a in-memory one, when reading big amount of data. Write support¶ The driver relies on the GPX driver to create an intermediate file that will be finally translated by GPSBabel to the desired GPSBabel format. (The GPX driver does not need to be configured for read support for GPSBabel write support.). The support geometries, options and other creation issues are the ones of the GPX driver. Please refer to its documentation for more details. The syntax to specify an output datasource is : GPSBabel:gpsbabel_file_format[,gpsbabel_format_option]:filename* where : • gpsbabel_file_format is one of the file formats handled by GPSBabel. • gpsbabel_format_option is any option handled by the specified GPSBabel format (refer to the documentation of each GPSBabel format) Alternatively, you can just pass a filename as output datasource name and specify the dataset creation option GPSBABEL_DRIVER=gpsbabel_file_format[,gpsbabel_format_option]* The USE_TEMPFILE=YES configuration option can be used to create an on-disk temporary GPX file instead of a in-memory one, when writing big amount of data. Examples¶ ogrinfo -ro -al GPSBabel:garmin:usb: ogr2ogr -f GPSBabel GPSBabel:mapsend:out.mapsend in.shp	2019-10-20 11:41:24	{"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": 0.24024783074855804, "perplexity": 10937.047999146105}, "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/1570986707990.49/warc/CC-MAIN-20191020105426-20191020132926-00258.warc.gz"}
http://math.stackexchange.com/questions/291515/space-of-probability-measures-complete-in-the-other-sense	# Space of probability measures “complete”? (In the other sense) I want to consider a space of probability measures on some set $X$, such that the space of measures is complete, not in the sense of complete probability measures (though probably that too), but as in the standard analysis meaning. I haven't studied far enough in probability measures, so I can't tell -- what conditions do I need on my space such that it is complete? And how should I metrize it? Thanks! Edit: A reference/citation to a text would also be super helpful! - For any measurable space $(X,\mathscr B)$ let's denote by $\mathscr M$ the linear space of all bounded (signed) measures. This is known to be a Banach space w.r.t. the total variation norm $$\\|\mu\\| = \sup_{B\in \mathscr B}(\|\mu(B)\|+\|\mu(B^c)\|).$$ so that this space is complete. Define $f:\mathscr M\to\Bbb R$ as $f(\mu) = \mu(X)$. This map is clearly continuous: $$\|f(\mu) - f(\nu)\|\leq \\|\mu - \nu\\|$$ and thus $\mathscr M_1:=\{\mu:f(\mu) = 1\}$ is a closed subspace of $\mathscr M$. The space of all non-negative measures $\mathscr M_+:=\{\mu:\mu\geq 0\}$ can be characterized as $$\mathscr M_+ = \{\mu:\\|\mu\\| - f(\mu) = 0\}$$ and thus is a closed subspace of $\mathscr M$ as well. The space of all probability measures $\mathscr P = \mathscr M_1\cap\mathscr M_+$ is thus a closed subspace of a complete space, hence complete itself. Besides of the total variation distance which can be introduced regardless the structure of the underlying measurable space, there are other sorts of metric spaces of measures. The relation of completeness for them are more involved. - Great, thanks, that's very helpful! Do you happen to know of a good text/reference with a statement that I can use as a citation as well? – usul Jan 31 '13 at 19:20 @usul: of the fact that $\mathscr P$ is complete under the total variation distance? – Ilya Jan 31 '13 at 19:22 yeah unless this is considered obvious/trivial, I guess. Or a chapter where this is developed if you know of one offhand, but it's not a big deal (your explanation is very clear). – usul Feb 1 '13 at 0:15 @usul unfortunately, I don't – Ilya Feb 5 '13 at 16:35 This is related to Prokhorov's theorem and tightness. If you have a seperable metric space $(X,d)$ that is complete, then $(\mathcal{P},d_P)$ is complete, where $\mathcal{P}(X)$ is the set of all Borel probability measures and $d_P$ is the Prokhorov metric. You can find a good detailed discussion of this here., section 9 pg 26. - Great, thanks, and thanks for the useful links! – usul Jan 31 '13 at 19:20	2014-12-21 13:54:50	{"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": 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": 0.9608286619186401, "perplexity": 263.9574538550034}, "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/1418802771253.22/warc/CC-MAIN-20141217075251-00008-ip-10-231-17-201.ec2.internal.warc.gz"}
https://plainmath.net/90984/a-quadrilateral-abcd-is-formed-from-four	# A quadrilateral ABCD is formed from four distinct points (called the vertices), no three of which are collinear, and from the segments AB, CB, CD, and DA (called the sides), which have no intersections except at those endpoints labeled by the same letter. The notation for this quadrilateral is not unique- e.g., quadrilateral ABCD = quadrilateral CBAD. A quadrilateral ABCD is formed from four distinct points (called the vertices), no three of which are collinear, and from the segments AB, CB, CD, and DA (called the sides), which have no intersections except at those endpoints labeled by the same letter. The notation for this quadrilateral is not unique- e.g., quadrilateral $ABCD=$ quadrilateral CBAD. Two vertices that are endpoints of a side are called adjacent; otherwise the two vertices are called opposite. The remaining pair of segments AC and BD formed from the four points are called diagonals of the quadrilateral; they may or may not intersect at some fifth point. If X, Y, Z are the vertices of quadrilateral ABCD such that Y is adjacent to both X and Z, then angle XYZ is called an angle of the quadrilateral; if W is the fourth vertex, then angle XWZ and angle XYZ are called opposite angles. The quadrilaterals of main interest are the convex ones. By definition, they are the quadrilaterals such that each pair of opposite sides, e.g., AB and CD, has the property that CD is contained in one of the half-planes bounded by the line through A and B, and AB is contained in one of the half-planes bounded by the line through C and D. a) Using Pasch's theorem, prove that if one pair of opposite sides has this property, then so does the other pair of opposite sides. b) Prove, using the crossbar theorem, that the following are equivalent: 2. Each vertex of the quadrilateral lies in the interior of the opposite angle. 3. The diagonals of the quadrilateral meet. You can still ask an expert for help • Questions are typically answered in as fast as 30 minutes Solve your problem for the price of one coffee • Math expert for every subject • Pay only if we can solve it Bridger Hall Step 1 In a possible attempt to explain a), let us focus solely on a single angle, say angle A. Similarly, draw tangent lines extending from the two adjacent sides, namely AB and AD. Assuming $A\ne 180$ (which we can, because it would cause ABCD to be a triangle), AB and AD are not parallel. This means that they meet at A and continue, getting further apart as they go. If $A<180$, meaning ABCD is convex, AB and AD continue away from the shape, not intersecting any sides. Step 2 However, if $A>180$, AB and AD enter the interior or ABCD after intersecting at A. As the lines are infinite and the quadrilateral is not, the lines must at some point leave the shape. As two lines can only meet at a single point, and will not intersect themselves, they must leave the shape through one of the other two sides (Note Pasch's Theorem). As both AB and AD are equally dependent on the angle of A, it is not possible for only one of the two lines to split one of the other sides.	2022-09-30 05:45:06	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 23, "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": 0.6937405467033386, "perplexity": 395.1222459923618}, "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-2022-40/segments/1664030335444.58/warc/CC-MAIN-20220930051717-20220930081717-00358.warc.gz"}
https://math.stackexchange.com/questions/3236651/cycle-notation-for-cyclic-orders	# Cycle notation for cyclic orders Is there a convenient cycle notation for cyclic orders (https://en.wikipedia.org/wiki/Cyclic_order)? For example: Definition. A set of four elements $$a, b, c, d$$ of a cyclically ordered set is a 4-cycle $$[a, b, c, d]$$ if $$[a,b,c] \land [c,d,a]$$. Using transitivity it is easy to show that $$[a, b, c, d] \implies [a,b,c]$$, $$[b,c,d]$$, $$[c,d,a]$$, $$[d,a,b]$$, and all the cyclic equivalents: $$[b,c,a]$$, $$[c,a,b]$$, $$[c,d,b]$$, $$[d,b,c]$$, $$[a,c,d]$$, $$[d,a,c]$$, $$[a,b,d]$$, $$[b,d,a]$$. Which also means $$[a, b, c, d] \iff [b, c, d, a] \iff [c, d, a, b] \iff [d, a, b, c]$$. It could simplify the ternary arithmetic. I am not aware of any standard notation, but your notation $$[a, b, c, d]$$ makes sense and could be extended to $$[a_1, a_2, \dotsm, a_n]$$ for any $$n$$.	2019-07-17 16:18:52	{"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": 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": 0.9759904146194458, "perplexity": 216.08656078249726}, "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-2019-30/segments/1563195525355.54/warc/CC-MAIN-20190717161703-20190717183703-00169.warc.gz"}
https://www.gam.bingo/fighting-gerrymandering/	# fighting Gerrymandering ## THE PROBLEM OF Gerrymandering Politicians have used gerrymandering, the practice of drawing political districts for partisan advantage, to skew elections since the early days of this great country. The term was coined after Governor of Massachusetts Elbridge Gerry, who in 1812 drew an electoral district so long and twisted that the local Boston Gazette mocked it as resembling a salamander and invented the portmanteau, "gerrymander." The Boston Gazette's political cartoon mocking Governor Gerry's map that invented the term gerrymander. The key to gerrymandering is that while the law specifies that all districts must be of equal population, it says nothing of their shape. Politicians draw maps to their advantage by what is known as "packing" and "cracking." The majority party will draw pack a few districts with as many voters of the minority party as possible, and then crack apart the rest of the districts as to dilute the minority party's power. The illustration from Quanta Magazine below demonstrates packing on cracking on an imaginary state that is equally proportioned between pink voters and green voters. While there are 500 of each, all the green voters have been packed into the left most district. The other nine districts have been drawn in erratic shapes such that pink takes the majority in each one. This illustration also introduces a concept called the Efficiency Gap, a metric invented for fighting gerrymandered maps in court. The Efficiency Gap measures wasted votes. Gerrymandered maps are drawn to maximize the opposition party's wasted votes. Researchers propose a threshold of 7% to measure when a gap is too extreme. Proponents of legislature controlled district maps often argue that since Democrats are packed into urban centers and Republicans are spread across the countryside, no fair map could be drawn. Opponents of gerrymandering use the Efficiency Gap to counter this argument; this metric was the cornerstone of the recent challenge to Wisconsin's gerrymandered districts. I had seen a lot of work like this, using data science to prove that an existing map had been egregiously gerrymandered. But I had seen less work using data science to draw an optimally fair map. The challenge with drawing an optimally fair map, however, is that reasonable people disagree about what makes a map fair. Some believe that a map with perfectly rectangular districts is the most common sense approach. Others want maps optimized for electoral competitiveness – gerrymandered for the opposite effect. Many people want maps that take racial diversity into account. For example if a sate is 20% hispanic, 20% of its congressional districts should be majority hispanic. Instead of trying to settle this debate, my goal was to build a tool that would let anyone optimize a map on whatever they think most important. An independent redistricting committee that only cared about compactness could use this tool to draw perfectly compact districts. If they wanted to ensure competitive elections, they could optimize for a low efficiency gap. Or they could rank the importance of each metric and optimize with weighted preferences. ## Data A lot of democracy loving data scientists have taken the time to collect and share data for drawing maps. I am especially grateful to Michal Migurski who collected and published data on the voting precincts if North Carolina. Mike published GeoJSON files that contained the shape, demographics, and voting history in 2014 for all 2,725 precincts in North Carolina. The 2014 voting precincts of North Carolina. A quick exploration of the voting history shows the effects of gerrymandering. Above are all the voting precincts of North Carolina. Below is how each precinct voted in the 2014 senate election. It may look like the state is majority Republican, but the election was actually pretty close, with Republicans taking home 52% of the statewide vote and Democrats taking home 48%. The 2014 Senate election. Red precincts were majority Republican, and blue districts were majority Democrat. However, below are how the congressional districts were drawn that year. The Republican controlled state legislature packed three of the districts so full of Democrats that they easily win those three, but don't stand a chance in the other 10. So even though North Carolina is rather balanced politically, its representation in Congress is not. ## A clustering problem Graph Partitioning Simulated Annealing Constraint-Based Polygonal Spatial Clustering When I was looking at the green map of all the voting precincts, and thinking about how best to group them into districts, I was reminded of many of the clustering algorithms I have learned. In this case, I knew there should be 13 clusters, but I didn't know how to decide which precincts should go in which clusters. In my research, I came across an experimental clustering algorithm that had been designed specifically for redistricting. In their paper, Redistricting using Heuristic-Based Polygonal Clustering, researchers from the University of Nebraska describe a method for growing districts using the combination of simple heuristic functions. The team compared their algorithm to two other approaches – graph partitioning and simulated annealing – and demonstrated more visually pleasing results, but what really interested me was how the seemingly powerful algorithm was based on simple, individual metrics. For their map of Nebraska, the researchers only incorporated heuristic functions for population and compactness. It was clear that if I could create a heuristic function modeled off of the Efficiency Gap, I could bolt that on and optimize for electoral competitiveness as well. And I could apply different weights to different metrics. This algorithm appeared flexible enough to let me build the customizable tool that I wanted. While coding up this algorithm from scratch I found that while the basic premise is straightforward, the details of each step are rather complicated. At each step, I've made changes to what the researches originally proposed. I believe these changes are improvements that help the algorithm generalize to states with more districts, more major cities, and more irregularly shaped precincts. ## HOw it works The algorithm uses the collection of heuristic functions to determine which district is furthest from its goal and which precinct should be added to bring the district closest to its goal. Each district starts as a single precinct. Overall, the code follows this loop: 1. Examine each district and select the district that is furthest from its goal. 2. Examine all neighboring precincts of that district, and select the one that brings it closest to its goal. 3. Add the chosen precinct to the chosen district. 4. Repeat. While coding up the algorithm, I would test it on small subsets of the state. Below is a demonstration of growing three districts, optimized only for compactness, on the western edge of North Carolina. Here you can see that districts do not simply gobble up precincts until all have been consumed. Instead the algorithm allows the districts to fight over precincts until it has determined which districts need which precincts the most. Growing three districts, optimized only for compactness, on the western edge of North Carolina. ### Planting seeds To start, the algorithm chooses single precincts to act as seed districts. So that districts aren't immediately fighting over the same precincts, the seeds should be reasonably spread out. To insure an even spread, the researchers developed a custom method using the rectangular area between seeds. However, I found their method to be cryptic and overcomplicated. With the stated goal of spreading the initial seeds out evenly, I simply used K++ Initialization. It follows that the final districts would be heavily influenced by these initial seeds. As I continue to work on this algorithm I intend to further explore the effect of changing the seed locations. I also want to try using population density to influence selection, perhaps first choosing each major city as a seed before spreading the rest out evenly. ### Heuristic Functions Each function is designed to be normalized between 0 and 1 so that no one function carries too much influence when they are combined. At each step, the district or precinct with the highest score is chosen. Since the algorithm chooses districts that are the "worst performing" and precincts that are the "best performing," the same functions need to yield opposite results at different times. This is achieved by simply calculating a score, s, and either yielding s or 1-s. ### POPULATION HEURISTIC The population heuristic is simply a district's percent error from its target population. However, while a true percent error calculation takes the absolute value, this function returns a negative value when a district is too large. This will be helpful later. population_score = (target_pop - current_pop)/target_pop ### Compactness Heuristic The researchers used a custom method that involved the ratio of a district's border shared with unassigned precincts to the length of the new border if a certain precinct was added. I found this method to be overcomplicated, computationally expensive, and frankly not that effective. Perhaps it worked well on the rectangular precincts of Oklahoma, but it did not generalize to the very irregulary shaped precincts of North Carolina. (Another, deeper level of gerrymandering involves redrawing the electoral precincts themselves, not just the districts. I suspect this is the case in North Carolina.) Instead I used the combination of two simple and common measures of a shape's compactness. The first is a measure of how circular a shape is, and the second is a measure of a shape's convexity. I experimented with both averaging and summing these two scores. The circularity score is a ratio of a shape's area to the square of its perimeter. The addition of 4pi in the numerator normalizes the score so that a circle receives a score of 1. circular_score = 4pi*area/perimeter^2 A non-convex shape is one that folds in upon itself, like a crescent moon. The convexity score measures the ratio of a shapes area to that of the shape's convex hull. A shape's convex hull is the smallest convex shape that can fit it. This score returns 1 for any convex shape. convexity_score = area_shape/area_convex_hull Attempting to perfect the compactness measure, I would draw districts solely for compactness, ignoring the other metrics. I found that larger districts would necessarily yield worse compactness measures than smaller districts. I experimented with scaling the compactness score by the number of member precincts in a district. This helped but was an imperfect solution. I have concluded that the best way to scale the compactness measure is simply to combine it with the population score. ### Electoral Competiveness Heuristic Since the Efficiency Gap is a percentage, it is already normalized between 0 and 1. The Brennan Center for Justice has an excellent guide for understanding and calculating the Efficiency Gap. ### Combining the Scores Many of changes to this algorithm stem from a potential downfall of this algorithm: a district can only help itself by consuming another precinct. Perhaps the best move for a district would be for it to give up a precinct. But the only way a district can lose a precinct is if another district steals it. If a district is overpopulated it will have a high percent error. The district with the most precincts will likely have the worst compactness measure. This can lead to a self-defeating cycle, where an overpopulated district will have the highest population and compactness heuristic score. As the highest scoring district, the algorithm will select it and try and improve it by adding another precinct. But this will only worsen it's already high population and compactness scores. Thus this district is chosen every cycle until it has consumed the whole state. In the original algorithm, the researchers sum each metric to yield the heuristic score. Instead, I multiply the population and compactness scores. This simple change, along with allowing the population score to return a negative value once a district is overpopulated, means that overpopulated districts will always have the lowest scores. While I found the researcher's measure of compactness to be ineffective in North Carolina, it did seem to prioritize unassigned precincts over precincts belonging to another district. I incorporated this thinking by using dynamic weights. While a district is small and exists as an island in a sea of unassigned precincts, the compactness score is doubled. This prioritizes growing compact districts. Once a district touches another neighboring district, the compactness weight is reset. The effect is that early on districts will balloon out in place. Once districts share a border, they can fight over who needs which precincts the most. Optimized for equal population only Optimized for equal population and compactness Optimized for equal population, compactness, and competitive elections ## Results The western edge of North Carolina was a nice, small subset to test this algorithm on. The western half of this small map is very rural. In the northeast is the city of Asheville. So it's a nice representation of the rest of the state. The first map on the right shows three districts optimized only for equal population. With no concern for compactness and in need of people, the district in the lower right has reached up to grab part of Asheville. The second map shows three districts optimized for equal population and compactness. While not perfectly clean, the districts have much nicer shapes than in the first map. The third map optimizes on all three metrics. To achieve competitive elections, a map must sacrifice some compactness. Drawing maps this way is somewhat another version of gerrymandering. Here is where I run into the limitations of this algorithm. In hindsight it may make sense for an algorithm developed on Nebraska. It can draw three districts really well, but if you ask for more it breaks down. Drawing 13 congressional districts at once is beyond the capability of this algorithm. I am currently revising my code to redistrict a state in stages. First, split the state into two or three macro-districts. Then, split each of those into two or three sub-districts. Keep going like as long as necessary. I expect to have a proposed map for North Carolina in the next few weeks. I admire that the foundation of such a powerful algorithm is this collection of simple functions. While that is no surprise – the most elegant solutions are often the simplest – it is incredibly helpful for this issue in particular. State legislators get away with partisan gerrymandering in part by making the issue seem more complicated than it really is. I hope that my work may help make this complicated issue easier to understand and a less intimidating to fight.	2019-08-21 08:18:18	{"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": 0.45493656396865845, "perplexity": 1843.2630972380891}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027315811.47/warc/CC-MAIN-20190821065413-20190821091413-00070.warc.gz"}