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http://www.ipodphysics.com/math-right-hand-rule.php	Cross Product / The Right Hand Rule Important Equation a x b = c Cross Product Right Hand Rule In mathematics and physics, the right-hand rule is a common mnemonic for understanding notation conventions for vectors in 3 dimensions. It was invented for use in electromagnetism by British physicist John Ambrose Fleming in the late 19th century. When choosing three vectors that must be at right angles to each other, there are two distinct solutions, so when expressing this idea in mathematics, one must remove the ambiguity of which solution is meant. There are variations on the mnemonic depending on context, but all variations are related to the one idea of choosing a convention. Definition by Wikipedia One form of the right-hand rule is used in situations in which an ordered operation must be performed on two vectors a and b that has a result which is a vector c perpendicular to both a and b. The most common example is the vector cross product. The right-hand rule imposes the following procedure for choosing one of the two directions. $\vec{a} \times \vec{b} = \vec{c}$ • With the thumb, index, and middle fingers at right angles to each other (with the index finger pointed straight), the middle finger points in the direction of c when the thumb representsa and the index finger represents b. Other (equivalent) finger assignments are possible. For example, the first (index) finger can represent a, the first vector in the product; the second (middle) finger, b, the second vector; and the thumb, c, the product. Definition by Wikipedia	{"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": 0.8739866018295288, "perplexity": 620.7653626252533}, "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/1518891807660.32/warc/CC-MAIN-20180217185905-20180217205905-00263.warc.gz"}
https://homework.cpm.org/category/CON_FOUND/textbook/mc2/chapter/5/lesson/5.1.3/problem/5-36	Home > MC2 > Chapter 5 > Lesson 5.1.3 > Problem5-36 5-36. Alden found a partially completed 5-D chart: Define Do Decide Target $74$ Trial 1: $15$ $2(15) = 30$ $15 + 2 = 17$ $15 + 30 + 17 =$ $62$ too small Trial 2: $18$ $2(18) = 36$ $18 + 2 = 20$ $18 + 36 + 20 =$ $74$ just right 1. Create a word problem that could have been solved using this chart. Examine the chart in the Define and Do sections to figure out what the expression from the 5-D chart is, and the relationship between each of the units. If the first unit is the variable, $x$, and the three units are added together for the target value, $74$, think of any possible word problems that could illustrate this 5-D chart. The farthest house is twice as far as the closest house. The middle house is two more miles farther than the closest house. If the total distance is $74$ miles, how far is the closest house? 2. What words would you put above the numbers in the three empty sections in the "Trial" and "Define" parts of the chart? Based on the expression and the word problem you created, which sections are the shortest, medium, or longest distance? 3. What word(s) would you put above the "Do" column? Based on the word problem you created, what would this total represent? From the example answer, the ''Do'' column could be the total distance for all three houses.	{"extraction_info": {"found_math": true, "script_math_tex": 14, "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": 0.8042080402374268, "perplexity": 1424.866409001209}, "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-2022-33/segments/1659882570692.22/warc/CC-MAIN-20220807181008-20220807211008-00704.warc.gz"}
http://math.stackexchange.com/questions/144267/wkb-approximation-question/144277	# WKB approximation question I was reading some stuff on asymptotic analysis, but how do you get from the 1st line to the 2nd line? $y \sim \frac{1+x}{2\lambda}\exp\left(\frac{\lambda x}{1+x}\right) - \frac{1+x}{2\lambda}\exp\left(\frac{-\lambda x}{1+x}\right)\\y(1) \sim\frac{1}{\lambda}\exp\left(\frac{\lambda}{2}\right) \text{as } \lambda \rightarrow \infty$ I see they substituted $x=1$, but where does the $- \frac{1+x}{2\lambda}\exp\left(\frac{-\lambda x}{1+x}\right)$ term go? What does $y(1) \sim\frac{1}{\lambda}\exp\left(\frac{\lambda}{2}\right) \text{as } \lambda \rightarrow \infty$ actually mean? The main part im not sure about is the $\sim$. - $$f\sim g \quad (\text{as }x\rightarrow\infty)$$ reads "$f$ is asymptotic to $g$ as $x$ goes to infinity" . This means basically that $\lim_{x\rightarrow \infty} \frac{f}{g} = 1$ . Because $e^x$ dominates $e^{-x}$ as $x$ becomes large (i.e. $e^{-x} \in \mathcal{o}(e^{x})$) they neglected the term with the negative exponent in the second line.	{"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": 0.9529242515563965, "perplexity": 118.4114564855042}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042987135.9/warc/CC-MAIN-20150728002307-00157-ip-10-236-191-2.ec2.internal.warc.gz"}
https://open.library.ubc.ca/cIRcle/collections/48630/items/1.0366264	# Open Collections ## BIRS Workshop Lecture Videos ### Stabilizing Weighted Graphs Koh, Cedric #### Description An edge-weighted graph G is called stable if the value of a maximum-weight matching equals the value of a maximum-weight fractional matching. Stable graphs play an important role in some interesting game theory problems, such as network bargaining games and cooperative matching games, because they characterize instances which admit stable outcomes. Motivated by this, in the last few years many researchers have investigated the algorithmic problem of turning a given graph into a stable one, via edge- and vertex-removal operations. However, all the algorithmic results developed in the literature so far only hold for unweighted instances, i.e., assuming unit weights on the edges of G. We give the first polynomial-time algorithm to find a minimum cardinality subset of vertices whose removal from G yields a stable graph, for any weighted graph G. The algorithm is combinatorial and exploits new structural properties of basic fractional matchings, which may be of independent interest. In contrast, we show that the problem of finding a minimum cardinality subset of edges whose removal from a weighted graph G yields a stable graph, does not admit any constant-factor approximation algorithm, unless P=NP. In this setting, we develop an O(Delta)-approximation algorithm for the problem, where Delta is the maximum degree of a node in G.	{"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": 0.8932477235794067, "perplexity": 389.13780678454816}, "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/1656103943339.53/warc/CC-MAIN-20220701155803-20220701185803-00442.warc.gz"}
https://fr.maplesoft.com/support/help/maple/view.aspx?path=DEtools%2Ffirtest	DEtools - Maple Programming Help DEtools firtest test a given first integral Calling Sequence firtest(first_int, ODE, y(x)) Parameters first_int - first integral ODE - ordinary differential equation y(x) - (optional) indeterminate function of the ODE Description • The firtest command checks whether a given expression is a first integral of a given ODE. Similar to odetest, firtest returns $0$ when the result is valid, or an algebraic expression obtained after simplifying the PDE for the first integral associated with the given ODE (see odepde). Among other things, firtest can be used to test the results obtained using the command firint. • If the result returned by firtest is not zero, the expression might nevertheless be a first integral. Sometimes, with further simplification, you can obtain the desired $0$ using commands such as expand, combine, and so on. • This function is part of the DEtools package, and so it can be used in the form firtest(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[firtest](..). Examples A first order ODE > $\mathrm{with}\left(\mathrm{DEtools}\right):$ > $\mathrm{ODE}≔\mathrm{diff}\left(y\left(x\right),x\right)=y\left(x\right)a\left(x\right)$ ${\mathrm{ODE}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{y}{}\left({x}\right){}{a}{}\left({x}\right)$ (1) An integrating factor for ODE above > $\mathrm{Μ}≔\mathrm{intfactor}\left(\mathrm{ODE}\right)$ ${\mathrm{Μ}}{≔}{{ⅇ}}^{{\int }{-}{a}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}$ (2) A related (to Mu) first integral for ODE above > $\mathrm{ans}≔\mathrm{firint}\left(\mathrm{Μ}\mathrm{ODE}\right)$ ${\mathrm{ans}}{≔}{{ⅇ}}^{{\int }{-}{a}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}{}{y}{}\left({x}\right){+}{\mathrm{_C1}}{=}{0}$ (3) Testing the first integral > $\mathrm{firtest}\left(\mathrm{ans},\mathrm{ODE}\right)$ ${0}$ (4) A second order ODE example > $\mathrm{ODE}≔\mathrm{diff}\left(y\left(x\right),x,x\right)=-\frac{2\left(2\mathrm{diff}\left(y\left(x\right),x\right)+5x{y\left(x\right)}^{2}+2{x}^{2}y\left(x\right)\mathrm{diff}\left(y\left(x\right),x\right)\right)}{x}$ ${\mathrm{ODE}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{-}\frac{{2}{}\left({2}{}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{5}{}{x}{}{{y}{}\left({x}\right)}^{{2}}{+}{2}{}{{x}}^{{2}}{}{y}{}\left({x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)\right)}{{x}}$ (5) > $\mathrm{first_int}≔2{x}^{5}{y\left(x\right)}^{2}+{x}^{4}\mathrm{diff}\left(y\left(x\right),x\right)+\mathrm{_C1}=0$ ${\mathrm{first_int}}{≔}{2}{}{{x}}^{{5}}{}{{y}{}\left({x}\right)}^{{2}}{+}{{x}}^{{4}}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{\mathrm{_C1}}{=}{0}$ (6) > $\mathrm{firtest}\left(\mathrm{first_int},\mathrm{ODE}\right)$ ${0}$ (7)	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 17, "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": 0.9986535906791687, "perplexity": 1361.2215027652294}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370521876.48/warc/CC-MAIN-20200404103932-20200404133932-00559.warc.gz"}
https://www.spp2026.de/members-guests/15-member-pages/prof-dr-anand-dessai/	# Members & Guests ## Prof. Dr. Anand Dessai Université de Fribourg E-mail: anand.dessai(at)unifr.ch Telephone: +41 26 300-9184 Homepage: http://homeweb1.unifr.ch/dessaia/pub/... ## Publications within SPP2026 We show that the moduli space of metrics of nonnegative sectional curvature on every homotopy RP^5 has infinitely many path components. We also show that in each dimension 4k+1 there are at least 2^{2k} homotopy RP^{4k+1}s of pairwise distinct oriented diffeomorphism type for which the moduli space of metrics of positive Ricci curvature has infinitely many path components. Examples of closed manifolds with finite fundamental group with these properties were known before only in dimensions 4k+3≥7. Journal to appear in Transactions of the AMS Link to preprint version We show that in each dimension 4n+3, n>1, there exist infinite sequences of closed smooth simply connected manifolds M of pairwise distinct homotopy type for which the moduli space of Riemannian metrics with nonnegative sectional curvature has infinitely many path components. Closed manifolds with these properties were known before only in dimension 7, and our result also holds for moduli spaces of Riemannian metrics with positive Ricci curvature. Moreover, inconjunction with work of Belegradek, Kwasik and Schultz, we obtain that for each such M the moduli space of complete nonnegative sectional curvature metrics on the open simply connected manifold M × R also has infinitely many path components. Journal Bulletin of the London Math. Society Volume 50 Pages 96-107 Link to preprint version Link to published version Let M be a Milnor sphere or, more generally, the total space of a linear S^3-bundle over S^4 with H^4(M;Q) = 0. We show that the moduli space of metrics of nonnegative sectional curvature on M has infinitely many path components. The same holds true for the moduli space of m etrics of positive Ricci curvature on M. Journal preprint arXiv Pages 11 pages Link to preprint version • 1	{"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": 0.9002240896224976, "perplexity": 327.7527992568781}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875146342.41/warc/CC-MAIN-20200226115522-20200226145522-00113.warc.gz"}
http://mathhelpforum.com/advanced-statistics/193391-probability-distribution.html	## Probability distribution 1. You go to the supermarket with your friend. After picking up something, each of you arrives at a different queue at the same time. Assume that the time you need to wait(measured in minutes) has a geometric distribution with mean 2 and the waiting time of your friend is also geometrically distributed but with mean 4. a) Find the pmf of the difference between the waiting times of you and your friend. b) Find the probability that you wait longer than your friend. 2. Suppose that 40% of voters are in favor of certain legislation. A large number of voters are polled and a relative frequency estimate for the above proportion is obtained. Use the Chebyshev inequality to determine how many voters should be polled in order to make sure that the probability that the relative frequency estimate differs from the actual probability by less than 0.01 is at least 0.95. 3. Let the number of widgets tested in an assembly line in 1 hour be a binomial random variable with parameter n=600 and p . Suppose that the probability that a widget is faulty is q. Denote S as the number of widgets that are found faulty in a 1-hour period. Find the mean and variance of S.	{"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": 0.9260799288749695, "perplexity": 326.1033046354893}, "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/1410657139669.58/warc/CC-MAIN-20140914011219-00177-ip-10-234-18-248.ec2.internal.warc.gz"}
http://nrich.maths.org/6949/solution	### Number Detective Follow the clues to find the mystery number. ### Eight Dominoes Using the 8 dominoes make a square where each of the columns and rows adds up to 8 ### Online A game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter. # Which Numbers? (2) ##### Stage: 2 Challenge Level: In a similar way to Which Numbers? (1) the solutions we had tended to identify correctly two of the sets but struggled with the third. Joshua of Crookhill Primary School said: For the red group it is all the multiples of $6$. For the blue group it is $+ 13$ every time. and for the black we have no idea what so ever! Sophie and Jo of Huish Primary continued: The blue set's give away numbers are $26, 39, 65$ and $91$. We first looked at the end digits and saw they were going up by $3$ each time. We then knew it was a multiple of somthing with a $3$ on the end. We then knew they were going up by $10$ each time. We added the $10$ and the $3$ together to get $13$. So the blue set is going up by $13$ each time: $\{13,26,39,52,65,78,91\}$. There are $7$ numbers in the blue which is the same as on the sheet. The red set's give away numbers are $12, 18, 30, 42, 66, 78, 84$. We knew they were even, so it would be in either the $2$s, $4$s, $6$s or $8$s. We narrowed it down to the $6$s and the $2$s. The $2$s has $50$ numbers less than $101$, so we knew it was the $6$s. There were $16$ numbers in the red set like it said on the sheet: $\{6,12,18,24,30,36,42,48,54,60,66,72,78,84,90,96\}$. The black set's give away numbers are $14, 17, 33, 38, 51, 57, 74, 79, 94, 99$. We thought a long time about what it could be. As we looked closer we realised that the $10$s digit was always odd. We also realised that there are $50$ numbers with an odd $10$s digit before $101$. So the black set is all the $10$s, $30$s, $50$s, $70$s and $90$s. Do you agree with Sophie and Jo?	{"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": 0.8343479633331299, "perplexity": 572.2404594345279}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320915.38/warc/CC-MAIN-20170627032130-20170627052130-00077.warc.gz"}
https://export.arxiv.org/list/math-ph/new	# Mathematical Physics ## New submissions [ total of 23 entries: 1-23 ] [ showing up to 2000 entries per page: fewer \| more ] ### New submissions for Fri, 4 Dec 20 [1] Title: On the Near-Critical Behavior of Continuous Polymers Comments: arXiv admin note: text overlap with arXiv:2008.04493 Subjects: Mathematical Physics (math-ph) The aim of this paper is to investigate the distribution of a continuous polymer in the presence of an attractive finitely supported potential. The most intricate behavior can be observed if we simultaneously and independently vary two parameters: the temperature, which approaches the critical value, and the length of the polymer chain, which tends to infinity. We describe how the typical size of the polymer depends on the two parameters. [2] Title: On the Lyapunov-Perron reducible Markovian Master Equation Comments: 22 pages, no figures Subjects: Mathematical Physics (math-ph); Quantum Physics (quant-ph) We consider an open quantum system in $M_{d}(\mathbb{C})$ governed by quasiperiodic Hamiltonian with rationally independent frequencies and under assumption of Lyapunov-Perron reducibility of associated Schroedinger equation. We construct the Markovian Master Equation and resulting CP-divisible evolution in weak coupling limit regime, generalizing our previous results from periodic case. The analysis is conducted with application of projection operator techniques and concluded with some results regarding stability of solutions and existence of quasiperiodic global steady state. [3] Title: A geometric approach to Wigner-type theorems Subjects: Mathematical Physics (math-ph) Let $H$ be a complex Hilbert space and let ${\mathcal P}(H)$ be the associated projective space (the set of rank-one projections). Suppose that $\dim H\ge 3$. We prove the following Wigner-type theorem: if $H$ is finite-dimensional, then every orthogonality preserving transformation of ${\mathcal P}(H)$ is induced by a unitary or anti-unitary operator. This statement will be obtained as a consequence of the following result: every orthogonality preserving lineation of ${\mathcal P}(H)$ to itself is induced by a linear or conjugate-linear isometry ($H$ is not assumed to be finite-dimensional). As an application, we describe (not necessarily injective) transformations of Grassmannians preserving some types of principal angles. ### Cross-lists for Fri, 4 Dec 20 [4] arXiv:2012.01453 (cross-list from quant-ph) [pdf, other] Title: Constructing quantum codes from any classical code and their embedding in ground space of local Hamiltonians Comments: 29 pages + references ; 7 figures Subjects: Quantum Physics (quant-ph); Strongly Correlated Electrons (cond-mat.str-el); Mathematical Physics (math-ph) We introduce a framework for constructing a quantum error correcting code from {\it any} classical error correcting code. This includes CSS codes and goes beyond the stabilizer formalism to allow quantum codes to be constructed from classical codes that are not necessarily linear or self-orthogonal (Fig. 1). We give an algorithm that explicitly constructs quantum codes with linear distance and constant rate from classical codes with a linear distance and rate. As illustrations for small size codes, we obtain Steane's $7-$qubit code uniquely from Hamming's [7,4,3] code, and obtain other error detecting quantum codes from other explicit classical codes of length 4 and 6. Motivated by quantum LDPC codes and the use of physics to protect quantum information, we introduce a new 2-local frustration free quantum spin chain Hamiltonian whose ground space we analytically characterize completely. By mapping classical codewords to basis states of the ground space, we utilize our framework to demonstrate that the ground space contains explicit quantum codes with linear distance. This side-steps the Bravyi-Terhal no-go theorem because our work allows for more general quantum codes beyond the stabilizer and/or linear codes. We hesitate to call this an example of {\it subspace} quantum LDPC code with linear distance. [5] arXiv:2012.01550 (cross-list from math.DG) [pdf, ps, other] Title: Bochner-Kodaira Formulas and the Type IIA Flow Comments: 36 pages, comments welcome! Subjects: Differential Geometry (math.DG); Mathematical Physics (math-ph); Analysis of PDEs (math.AP); Symplectic Geometry (math.SG) A new derivation of the flow of metrics in the Type IIA flow is given. It is adapted to the formulation of the flow as a variant of a Laplacian flow, and it uses the projected Levi-Civita connection of the metrics themselves instead of their conformal rescalings. [6] arXiv:2012.01593 (cross-list from math.DS) [pdf, ps, other] Title: Logarithmic capacity of random $G_δ$-sets Subjects: Dynamical Systems (math.DS); Mathematical Physics (math-ph); Probability (math.PR) We study the logarithmic capacity of $G_\delta$ subsets of the interval $[0,1].$ Let $S$ be of the form \begin{align} S=\bigcap_m \bigcup_{k\ge m} I_k, \end{align} where each $I_k$ is an interval in $[0,1]$ with length $l_k$ that decrease to $0$. We provide sufficient conditions for $S$ to have full capacity, i.e. $\mathop{\mathrm{Cap}}(S)=\mathop{\mathrm{Cap}}([0,1])$. We consider the case when the intervals decay exponentially and are placed in $[0,1]$ randomly with respect to some given distribution. The random $G_\delta$ sets generated by such distribution satisfy our sufficient conditions almost surely and hence, have full capacity almost surely. This study is motivated by the $G_\delta$ set of exceptional energies in the parametric version of the Furstenberg theorem on random matrix products. We also study the family of $G_\delta$ sets $\{S(\alpha)\}_{\alpha>0}$ that are generated by setting the decreasing speed of the intervals to $l_k=e^{-k^\alpha}.$ We observe a sharp transition from full capacity to zero capacity by varying $\alpha>0$. [7] arXiv:2012.01818 (cross-list from math.DG) [pdf, ps, other] Title: Port-Hamiltonian Modeling of Ideal Fluid Flow: Part I. Foundations and Kinetic Energy Comments: This is a preprint submitted to the journal of Geometry and Physics. Please do not CITE this version, but only the published manuscript Subjects: Differential Geometry (math.DG); Mathematical Physics (math-ph); Fluid Dynamics (physics.flu-dyn) In this two-parts paper, we present a systematic procedure to extend the known Hamiltonian model of ideal inviscid fluid flow on Riemannian manifolds in terms of Lie-Poisson structures to a port-Hamiltonian model in terms of Stokes-Dirac structures. The first novelty of the presented model is the inclusion of non-zero energy exchange through, and within, the spatial boundaries of the domain containing the fluid. The second novelty is that the port-Hamiltonian model is constructed as the interconnection of a small set of building blocks of open energetic subsystems. Depending only on the choice of subsystems one composes and their energy-aware interconnection, the geometric description of a wide range of fluid dynamical systems can be achieved. The constructed port-Hamiltonian models include a number of inviscid fluid dynamical systems with variable boundary conditions. Namely, compressible isentropic flow, compressible adiabatic flow, and incompressible flow. Furthermore, all the derived fluid flow models are valid covariantly and globally on n-dimensional Riemannian manifolds using differential geometric tools of exterior calculus. [8] arXiv:2012.01827 (cross-list from physics.flu-dyn) [pdf, ps, other] Title: Port-Hamiltonian Modeling of Ideal Fluid Flow: Part II. Compressible and Incompressible Flow Comments: This is a prevprint submitted to the journal of Geometry and Physics. Please DO NOT CITE this version, but only the published manuscript Subjects: Fluid Dynamics (physics.flu-dyn); Mathematical Physics (math-ph); Differential Geometry (math.DG) Part I of this paper presented a systematic derivation of the Stokes Dirac structure underlying the port-Hamiltonian model of ideal fluid flow on Riemannian manifolds. Starting from the group of diffeomorphisms as a configuration space for the fluid, the Stokes Dirac structure is derived by Poisson reduction and then augmented by boundary ports and distributed ports. The additional boundary ports have been shown to appear naturally as surface terms in the pairings of dual maps, always neglected in standard Hamiltonian theory. The port-Hamiltonian model presented in Part I corresponded only to the kinetic energy of the fluid and how its energy variables evolve such that the energy is conserved. In Part II, we utilize the distributed port of the kinetic energy port-Hamiltonian system for representing a number of fluid-dynamical systems. By adding internal energy we model compressible flow, both adiabatic and isentropic, and by adding constraint forces we model incompressible flow. The key tools used are the interconnection maps relating the dynamics of fluid motion to the dynamics of advected quantities. [9] arXiv:2012.01858 (cross-list from math.RT) [pdf, ps, other] Title: Local opers with two singularities: the case of $\mathfrak{sl}(2)$ Comments: 56 pages. Comments are very welcome! Subjects: Representation Theory (math.RT); Mathematical Physics (math-ph); Algebraic Geometry (math.AG) We study local opers with two singularities for the case of the Lie algebra sl(2), and discuss their connection with a two-variables extension of the affine Lie algebra. We prove an analogue of the Feigin-Frenkel theorem describing the centre at the critical level, and an analogue of a result by Frenkel and Gaitsgory that characterises the endomorphism rings of Weyl modules in terms of functions on the space of opers. [10] arXiv:2012.01889 (cross-list from gr-qc) [pdf, ps, other] Title: Null infinity as an open Hamiltonian system Authors: Wolfgang Wieland Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph) When a system emits gravitational radiation, the Bondi mass decreases. If the Bondi energy is Hamiltonian, it can thus only be a time dependent Hamiltonian. In this paper, we show that the Bondi energy can be understood as a time-dependent Hamiltonian on the covariant phase space. Our derivation starts from the Hamiltonian formulation in domains with boundaries that are null. We introduce the most general boundary conditions on a generic such null boundary, and compute quasi-local charges for boosts, energy and angular momentum. Initially, these domains are at finite distance, such that there is a natural IR regulator. To remove the IR regulator, we introduce a double null foliation together with an adapted Newman--Penrose null tetrad. Both null directions are surface orthogonal. We study the falloff conditions for such specific null foliations and take the limit to null infinity. At null infinity, we recover the Bondi mass and the usual covariant phase space for the two radiative modes at the full non-perturbative level. Apart from technical results, the framework gives two important physical insights. First of all, it explains the physical significance of the corner term that is added in the Wald--Zoupas framework to render the quasi-conserved charges integrable. The term to be added is simply the derivative of the Hamiltonian with respect to the background fields that drive the time-dependence of the Hamiltonian. Secondly, we propose a new interpretation of the Bondi mass as the thermodynamical free energy of gravitational edge modes at future null infinity. The Bondi mass law is then simply the statement that the free energy always decreases on its way towards thermal equilibrium. [11] arXiv:2012.01943 (cross-list from math.CA) [pdf, other] Title: Finite-Part Integration in the Presence of Competing Singularities: Transformation Equations for the hypergeometric functions arising from Finite-Part Integration Comments: 44 pages, 3 figures Subjects: Classical Analysis and ODEs (math.CA); Mathematical Physics (math-ph); Complex Variables (math.CV) Finite-part integration is a recently introduced method of evaluating convergent integrals by means of the finite part of divergent integrals [E.A. Galapon, {\it Proc. R. Soc. A 473, 20160567} (2017)]. Current application of the method involves exact and asymptotic evaluation of the generalized Stieltjes transform $\int_0^a f(x)/(\omega + x)^{\rho} \, \mathrm{d}x$ under the assumption that the extension of $f(x)$ in the complex plane is entire. In this paper, the method is elaborated further and extended to accommodate the presence of competing singularities of the complex extension of $f(x)$. Finite part integration is then applied to derive consequences of known Stieltjes integral representations of the Gauss function and the generalized hypergeometric function which involve Stieltjes transforms of functions with complex extensions having singularities in the complex plane. Transformation equations for the Gauss function are obtained from which known transformation equations are shown to follow. Also, building on the results for the Gauss function, transformation equations involving the generalized hypergeometric function $\,_3F_2$ are derived. [12] arXiv:2012.01995 (cross-list from math.CO) [pdf, other] Title: Multicritical random partitions Comments: 12 pages, 3 figures Subjects: Combinatorics (math.CO); Mathematical Physics (math-ph); Probability (math.PR) We study two families of probability measures on integer partitions, which are Schur measures with parameters tuned in such a way that the edge fluctuations are characterized by a critical exponent different from the generic $1/3$. We find that the first part asymptotically follows a "higher-order analogue" of the Tracy-Widom GUE distribution, previously encountered by Le Doussal, Majumdar and Schehr in quantum statistical physics. We also compute limit shapes, and discuss an exact mapping between one of our families and the multicritical unitary matrix models introduced by Periwal and Shevitz. [13] arXiv:2012.02079 (cross-list from cond-mat.stat-mech) [pdf, other] Title: Effective free-fermionic form factors and the XY spin chain Subjects: Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph) We introduce effective form factors for one-dimensional lattice fermions with arbitrary phase shifts. We study tau functions defined as series of these form factors. On the one hand we perform the exact summation and present tau functions as Fredholm determinants in the thermodynamic limit. On the other hand simple expressions of form factors allow us to present the corresponding series as integrals of elementary functions. Using this approach we re-derive the asymptotics of static correlation functions of the XY quantum chain at finite temperature. ### Replacements for Fri, 4 Dec 20 [14] arXiv:1801.05183 (replaced) [pdf, ps, other] Title: Riemannian exponential and quantization Comments: Important changes have been made with respect to the previous version, including 1) An improved demonstration of the equivalence between the two proposed quantizations and 2) A major extension of one of them to a much broader set of functions. The title, the abstract and the introduction have been modified, making them more in line with the new content Subjects: Mathematical Physics (math-ph) [15] arXiv:1811.02551 (replaced) [pdf, ps, other] Title: Topological defects in lattice models and affine Temperley-Lieb algebra Comments: 44 pages, v2: much improved version with few sections rewritten, new result in Theorem 2.1, many typos fixed Subjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Quantum Algebra (math.QA); Representation Theory (math.RT) [16] arXiv:2003.10526 (replaced) [pdf, ps, other] Title: Hessian metric via transport information geometry Authors: Wuchen Li Subjects: Differential Geometry (math.DG); Information Theory (cs.IT); Mathematical Physics (math-ph) [17] arXiv:2004.08934 (replaced) [pdf, ps, other] Title: Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications Comments: 70 pages Subjects: Metric Geometry (math.MG); Mathematical Physics (math-ph); Differential Geometry (math.DG); Optimization and Control (math.OC) [18] arXiv:2008.11884 (replaced) [pdf, ps, other] Title: Orthogonal rational functions with real poles, root asymptotics, and GMP matrices Subjects: Spectral Theory (math.SP); Mathematical Physics (math-ph); Classical Analysis and ODEs (math.CA) [19] arXiv:2009.04783 (replaced) [pdf, ps, other] Title: Bounds on amplitude damping channel discrimination Comments: 15 pages. 7 figures Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph) [20] arXiv:2011.08830 (replaced) [pdf, other] Title: Stable maps to Looijenga pairs Comments: 114 pages (80pp+appendices), 40 figures. v2: minor changes, references added Subjects: Algebraic Geometry (math.AG); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph) [21] arXiv:2011.10554 (replaced) [pdf, other] Title: Multi-orbital Flat Band Ferromagnetism with a Provable Percolation Representation Subjects: Strongly Correlated Electrons (cond-mat.str-el); Mathematical Physics (math-ph) [22] arXiv:2011.11402 (replaced) [pdf, other] Title: The linear and nonlinear instability of the Akhmediev breather Authors: P. G. Grinevich (1 and 2), P. M. Santini (3 and 4) ((1) Steklov Mathematical Institute of Russian Academy of Sciences, (2) L.D. Landau Institute for Theoretical Physics of Russian Academy of Sciences, (3) Dipartimento di Fisica, Università di Roma "La Sapienza", (4) Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Roma) Comments: 31 pages, 4 figures, Simplification of final formulas was made in this version Subjects: Pattern Formation and Solitons (nlin.PS); Mathematical Physics (math-ph); Fluid Dynamics (physics.flu-dyn); Optics (physics.optics) [23] arXiv:2011.13499 (replaced) [pdf, ps, other] Title: Contact Geometry in Superconductors and New Massive Gravity Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph) [ total of 23 entries: 1-23 ] [ showing up to 2000 entries per page: fewer \| more ] Disable MathJax (What is MathJax?) Links to: arXiv, form interface, find, math-ph, recent, 2012, contact, help (Access key information)	{"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": 0.8644097447395325, "perplexity": 1253.8547626797408}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "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-2020-50/segments/1606141743438.76/warc/CC-MAIN-20201204193220-20201204223220-00220.warc.gz"}
https://www.physicsforums.com/threads/orthogonality-of-functions.47923/	# Orthogonality of Functions 1. Oct 15, 2004 ### theFuture We were doing examples in class today and showed that sin and cos were orthogonal functions. In general, is true that even and odd functions are orthogonal? I was unsure where a proof of this might begin, mostly how to generalize the notion of an even or odd function. 2. Oct 15, 2004 ### matt grime This depends on what your "inner product" is. Let's assume it is $$<f,g> = \int_{-a}^a f(x)g(x)dx$$ an odd function is one that satisfies f(x) = -f(-x) an even one satisfies f(x)=f(-x) 1. show that the product of an even and an odd function is odd 2. show that the integral of an odd function over any interval [-a,a] is zero. 3. Oct 15, 2004 ### theFuture Thanks. Now that I see it like that I can't believe I couldn't come up with that.	{"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": 0.9329403042793274, "perplexity": 497.4610468375372}, "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-2016-44/segments/1476988719273.37/warc/CC-MAIN-20161020183839-00490-ip-10-171-6-4.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/165671/convergence-of-the-sequence-sqrt12-sqrt1-sqrt12-sqrt13-sqrt1-sq	# Convergence of the sequence $\sqrt{1+2\sqrt{1}},\sqrt{1+2\sqrt{1+3\sqrt{1}}},\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1}}}},\cdots$ I recently came across this problem Q1 Show that $\lim\limits_{n \rightarrow \infty} \underbrace{{\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots+n\sqrt{1}}}}}}}_{n \textrm{ times }} = 3$ After trying it I looked at the solution from that book which was very ingenious but it was incomplete because it assumed that the limit already exists. So my question is Q2 Prove that the sequence$$\sqrt{1+2\sqrt{1}},\sqrt{1+2\sqrt{1+3\sqrt{1}}},\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1}}}},\cdots,\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots+n\sqrt{1}}}}}$$ converges. Though I only need solution for Q2, if you happen to know any complete solution for Q1 it would be a great help . If the solution from that book is required I can post it but it is not complete as I mentioned. Edit: I see that a similar question was asked before on this site but it was not proved that limit should exist. - possible duplicate of Limit of Nested Radical – Gerry Myerson Jul 2 '12 at 12:15 Did you look at all the links in all the answers to that earlier question? – Gerry Myerson Jul 2 '12 at 12:27 @GerryMyerson I think it's hard to search for the earlier posts. – Frank Science Jul 2 '12 at 13:18 @Frank While it is true that the SE search function leaves much to be desired (instead try googling with site:MSE), it does work well for reasonable unique terms like "nested radical". Indeed, it yields the cited duplicate question as first match. But, of course, one does need to know the English buzzwords for these objects. – Bill Dubuque Jul 2 '12 at 14:46 Let $f_n(0)=\sqrt{1+n}$ and $f_n(k)=\sqrt{1+(n-k)f_n(k-1)}$. Then $0<f_n(0)<n+1$ when $n>0$. Assume that $f_n(k)<n+1-k$ and we can show by induction that $$f_n(k+1) < \sqrt{1+(n-k-1)(n-k+1)} = \sqrt{1+(n-k)^2-1} = n+1-(k+1)$$ for all k. Your expression is $f_n(n-2)$ which is increasing in $n$ and bounded above by $3$, so converges. - Note that $$x+1=\sqrt{1+x(x+2)}\tag{1}$$ Iterating $(1)$, we get $$x+1=\sqrt{1+x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{1+(x+3)\color{#C00000}{(x+5)}}}}}\tag{2}$$ Note that $$s_3=\sqrt{1+x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{1+(x+3)\color{#C00000}{\sqrt{1}}}}}}\tag{3}$$ Instead of $\color{#C00000}{\sqrt{1}}$ as in the last term of $(3)$, $(2)$ has $\color{#C00000}{(x+n+2)}$. Thus, the increasing sequence in $(3)$ is bounded above by $x+1$. Thus, the sequence in $(3)$ has a limit. - I try to compliment this every time - but I love the custom-dulled colors. – mixedmath Dec 15 '12 at 19:40 $x + 1 = \sqrt {1 + x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{}.....}}}$. Put $x=2$ gives you the solution. For proof see http://zariski.files.wordpress.com/2010/05/sr_nroots.pdf - It is the same proof that I know but discussion of convergence is not complete still +1 for dicussion of numerical convergence. – sabertooth Jul 2 '12 at 12:52 Note that $\frac{x+y}{x+z} < \frac{y}{z}$ for positive $x, y, z$; thus $\frac{\sqrt{x+y}}{\sqrt{x+z}} < \frac{\sqrt{y}}{\sqrt{z}}$ for $y > z$. Thus we get $\frac{a_{n+1}}{a_n} = \frac{\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots+n\sqrt{1+(n+1)\sqrt{1}}}}}}}{\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots+n\sqrt{1}}}}}} < \frac{\sqrt{\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots+n\sqrt{1+(n+1)\sqrt{1}}}}}}}{\sqrt{\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots+n\sqrt{1}}}}}} < \frac{\sqrt{\sqrt{\cdots\sqrt{n+2}}}}{\sqrt{\sqrt{\cdots\sqrt{1}}}} = O(n^{(2^{-n})})$. Next, $\ln{a_{n+1}}-\ln{a_n} = O(\frac{\ln_n}{2^n})$. Summing the equations we get $\ln{a_n} - \ln{a_1} = O(\frac{\ln_1}{2^1} + \cdots + \frac{\ln_n}{2^n})$; letting $n$ to approach $\infty$, we get $\ln{a} - \ln{a_1} = O(1)$, thus there is a finite limit of $a_i$. -	{"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": 0.950838565826416, "perplexity": 360.693942906651}, "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/1448398459333.27/warc/CC-MAIN-20151124205419-00172-ip-10-71-132-137.ec2.internal.warc.gz"}
https://yutsumura.com/exponential-functions-form-a-basis-of-a-vector-space/	# Exponential Functions Form a Basis of a Vector Space ## Problem 590 Let $C[-1, 1]$ be the vector space over $\R$ of all continuous functions defined on the interval $[-1, 1]$. Let $V:=\{f(x)\in C[-1,1] \mid f(x)=a e^x+b e^{2x}+c e^{3x}, a, b, c\in \R\}$ be a subset in $C[-1, 1]$. (a) Prove that $V$ is a subspace of $C[-1, 1]$. (b) Prove that the set $B=\{e^x, e^{2x}, e^{3x}\}$ is a basis of $V$. (c) Prove that $B’=\{e^x-2e^{3x}, e^x+e^{2x}+2e^{3x}, 3e^{2x}+e^{3x}\}$ is a basis for $V$. ## Proof. ### (a) Prove that $V$ is a subspace of $C[-1, 1]$. Note that each function in the subset $V$ is a linear combination of the functions $e^x, e^{2x}, e^{3x}$. Namely, we have $V=\Span\{e^x, e^{2x}, e^{3x}\}$ and we know that the span is always a subspace. Hence $V$ is a subspace of $C[-1,1]$. ### (b) Prove that the set $B=\{e^x, e^{2x}, e^{3x}\}$ is a basis of $V$. We noted in part (a) that $V=\Span(B)$. So it suffices to show that $B$ is linearly independent. Consider the linear combination $c_1e^x+c_2 e^{2x}+c_3 e^{3x}=\theta(x),$ where $\theta(x)$ is the zero function (the zero vector in $V$). Taking the derivative, we get $c_1e^x+2c_2 e^{2x}+3c_3 e^{3x}=\theta(x).$ Taking the derivative again, we obtain $c_1e^x+4c_2 e^{2x}+9c_3 e^{3x}=\theta(x).$ Evaluating at $x=0$, we obtain the system of linear equations \begin{align} c_1+c_2+c_3&=0\\ c_1+2c_2+3c_3&=0\\ c_1+4c_2+9c_3&=0. \end{align} We reduce the augmented matrix for this system as follows: \begin{align} \left[\begin{array}{rrr\|r} 1 & 1 & 1 & 0 \\ 1 &2 & 3 & 0 \\ 1 & 4 & 9 & 0 \end{array} \right] \xrightarrow[R_3-R_1]{R_2-R_1} \left[\begin{array}{rrr\|r} 1 & 1 & 1 & 0 \\ 0 &1 & 2 & 0 \\ 0 & 3 & 8 & 0 \end{array} \right] \xrightarrow[R_3-3R_2]{R_1-R_2}\6pt] \left[\begin{array}{rrr\|r} 1 & 0 & -1 & 0 \\ 0 &1 & 2 & 0 \\ 0 & 0 & 2 & 0 \end{array} \right] \xrightarrow{\frac{1}{2}R_3} \left[\begin{array}{rrr\|r} 1 & 0 & -1 & 0 \\ 0 &1 & 2 & 0 \\ 0 & 0 & 1 & 0 \end{array} \right] \xrightarrow[R_2-2R_2]{R_1+R_3} \left[\begin{array}{rrr\|r} 1 & 0 & 0 & 0 \\ 0 &1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array} \right]. \end{align} It follows that the solution of the system is c_1=c_2=c_3=0. Hence the set B is linearly independent, and thus B is a basis for V. #### Anotehr approach. Alternatively, we can show that the coefficient matrix is nonsingular by using the Vandermonde determinant formula as follows. Observe that the coefficient matrix of the system is a Vandermonde matrix: \[A:=\begin{bmatrix} 1 & 1 & 1 \\ 1 &2 &3 \\ 1^2 & 2^2 & 3^2 \end{bmatrix}. The Vandermonde determinant formula yields that $\det(A)=(3-1)(3-2)(2-1)=2\neq 0.$ Hence the coefficient matrix $A$ is nonsingular. Thus we obtain the solution $c_1=c_2=c_3=0$. ### (c) Prove that $B’=\{e^x-2e^{3x}, e^x+e^{2x}+2e^{3x}, 3e^{2x}+e^{3x}\}$ is a basis for $V$. We consider the coordinate vectors of vectors in $B’$ with respect to the basis $B$. The coordinate vectors with respect to basis $B$ are $[e^x-2e^{3x}]_B=\begin{bmatrix} 1 \\ 0 \\ -2 \end{bmatrix}, [e^x+e^{2x}+2e^{3x}]_B=\begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix}, [3e^{2x}+e^{3x}]_B=\begin{bmatrix} 0 \\ 3 \\ 1 \end{bmatrix}.$ Let $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ be these vectors and let $T=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$. Then we know that $B’$ is a basis for $V$ if and only if $T$ is a basis for $\R^3$. We claim that $T$ is linearly independent. Consider the matrix whose column vectors are $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$: \begin{align} \begin{bmatrix} 1 & 1 & 0 \\ 0 &1 &3 \\ -2 & 2 & 1 \end{bmatrix} \xrightarrow{R_3+2R_1} \begin{bmatrix} 1 & 1 & 0 \\ 0 &1 &3 \\ 0 & 4 & 1 \end{bmatrix} \xrightarrow[R_3-4R_1]{R_1-R_2}\6pt] \begin{bmatrix} 1 & 0 & -3 \\ 0 &1 &3 \\ 0 & 0 & -11 \end{bmatrix} \xrightarrow{-\frac{1}{11}R_3} \begin{bmatrix} 1 & 0 & -3 \\ 0 &1 &3 \\ 0 & 0 & 1 \end{bmatrix} \xrightarrow[R_2-3R_3]{R_1+3R_3} \begin{bmatrix} 1 & 0 & 0 \\ 0 &1 &0 \\ 0 & 0 & 1 \end{bmatrix}. \end{align} Thus, the matrix is nonsingular and hence the column vectors \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 are linearly independent. As T consists of three linearly independent vectors in the three-dimensional vector space \R^3, we conclude that T is a basis for \R^3. Therefore, by the correspondence of the coordinates, we see that B’ is a basis for V. ## Related Question. If you know the Wronskian, then you may use the Wronskian to prove that the exponential functions e^x, e^{2x}, e^{3x} are linearly independent. See the post Using the Wronskian for Exponential Functions, Determine Whether the Set is Linearly Independent for the details. Try the next more general question. Problem. Let c_1, c_2,\dots, c_n be mutually distinct real numbers. Show that exponential functions \[e^{c_1x}, e^{c_2x}, \dots, e^{c_nx} are linearly independent over $\R$. The solution is given in the post ↴ Exponential Functions are Linearly Independent ### More from my site • Exponential Functions are Linearly Independent Let $c_1, c_2,\dots, c_n$ be mutually distinct real numbers. Show that exponential functions $e^{c_1x}, e^{c_2x}, \dots, e^{c_nx}$ are linearly independent over $\R$. Hint. 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Let $W=C^{\infty}(\R)$ be the vector space of all $C^{\infty}$ real-valued functions (smooth function, differentiable for all degrees of differentiation). Let $V$ be the vector space of all linear transformations from $W$ to $W$. The addition and the scalar multiplication of $V$ […] • The Subset Consisting of the Zero Vector is a Subspace and its Dimension is Zero Let $V$ be a subset of the vector space $\R^n$ consisting only of the zero vector of $\R^n$. Namely $V=\{\mathbf{0}\}$. Then prove that $V$ is a subspace of $\R^n$. Proof. To prove that $V=\{\mathbf{0}\}$ is a subspace of $\R^n$, we check the following subspace […] ### 1 Response 1. 10/20/2017 […] the post ↴ Exponential Functions Form a Basis of a Vector Space for the […] This site uses Akismet to reduce spam. Learn how your comment data is processed. ##### Use Coordinate Vectors to Show a Set is a Basis for the Vector Space of Polynomials of Degree 2 or Less Let $P_2$ be the vector space over $\R$ of all polynomials of degree $2$ or less. Let $S=\{p_1(x), p_2(x), p_3(x)\}$,... 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https://www.groundai.com/project/akari-infrared-observations-of-the-supernova-remnant-g292018-unveiling-circumstellar-medium-and-supernova-ejecta/	AKARI Infrared Observations of G292.0+1.8 AKARI Infrared Observations of the Supernova Remnant G292.0+1.8: Unveiling Circumstellar Medium and Supernova Ejecta Ho-Gyu Lee1 2 , Bon-Chul Koo3 , Dae-Sik Moon4 , Itsuki Sakon1 , Takashi Onaka1 , Woong-Seob Jeong5 , Hidehiro Kaneda6 , Takaya Nozawa7 , and Takashi Kozasa8 1affiliation: Department of Astronomy, Graduate School of Science, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan; [email protected] [email protected], [email protected] 2affiliationmark: 2affiliation: Department of Physics and Astronomy, Seoul National University, Seoul, 151-742, Korea; [email protected] 3affiliation: Department of Astronomy and Astrophysics, University of Toronto, Toronto, ON M5S 3H4, Canada; [email protected] 4affiliation: Korea Astronomy and Space Science Institute, 61-1, Whaam-dong, Yuseong-gu, Deajeon 305-348, Korea; [email protected] 5affiliation: Graduate School of Science, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan; [email protected] 6affiliation: Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Chiba 277-8568, Japan; [email protected] 7affiliation: Department of Cosmosciences, Graduate School of Science, Hokkaido University, Sapporo 060-0810, Japan; [email protected] Abstract We present the results of observations of the O-rich supernova remnant G292.01.8 using six IRC and four FIS bands covering 2.7–26.5 m and 50–180 m, respectively. The images show two prominent structures; a bright equatorial ring structure along the east-west direction and an outer elliptical shell structure surrounding the remnant. The equatorial ring structure is clumpy and incomplete with its western end opened. The outer shell is almost complete and slightly squeezed along the north-south direction. The central position of the outer shell is 1 northwest from the embedded pulsar and coincides with the center of the equatorial ring structure. In the northen and southwestern regions, there is also faint emission with a sharp boundary beyond the bright shell structure. The equatorial ring and the elliptical shell structures were partly visible in optical and/or X-rays, but they are much more clearly revealed in our images. There is no evident difference in infrared colors of the two prominent structures, which is consistent with the previous proposition that both structures are of circumstellar origin. However, we have detected faint infrared emission of a considerably high 15 to 24 m ratio associated with the supernova ejecta in the southeastern and northwestern areas. Our IRC spectra show that the high ratio is at least partly due to the emission lines from Ne ions in the supernova ejecta material. In addition we detect a narrow, elongated feature outside the SNR shell. We derive the physical parameters of the infrared-emitting dust grains in the shocked circumstellar medium and compare the result with model calculations of dust destruction by a SN shock. The results suggest that the progenitor was at the center of the infrared circumstellar shell in red supergiant stage and that the observed asymmetry in the SN ejecta could be a result of either a dense circumstellar medium in the equatorial plane and/or an asymmetric explosion. ISM: individual(G292.0+1.8 (catalog )) — infrared: ISM — ISM: dust — shock waves — supernova remnants slugcomment: Submitted: August 18, 2019 1 Introduction Young core-collapse supernova remnants (SNRs) in the Galaxy provide an unique opportunity to study fine details of ejecta from supernova (SN) explosions as well as those of the circumstellar medium (CSM) produced over the final evolutionary stages of massive stars. An obvious group of such young core-collapse SNRs is O-rich SNRs that show optical spectra featuring strong O and Ne lines, with lines from lighter elements (e.g., He) either weak or absent. Since the ejecta from a progenitor and the swept-up CSM in the O-rich SNRs are not completely mixed together yet, observation of the O-rich SNRs is promising to study the ejecta and CSM related to the explosion of a progenitor and its late-stage evolution. G292.01.8 (MSH11–54), together with Cassiopeia A and Puppis A, forms a rare group of O-rich SNRs in the Galaxy. The O-rich nature of G292.01.8 was discovered by the optical detection of fast-moving O-rich and Ne-rich ejecta (Goss at al., 1979; Murdin & Clark, 1979). Recent wide-field observations of the optical [O iii] line in G292.01.8 have revealed that the ejecta velocities range from 1400 km s to 1700 km s and also that distant ejecta knots are located primarily along the north-south direction (Ghavamian et al., 2005; Winkler & Long, 2006). The dynamical center of G292.01.8, based on the kinematics of the [O iii] line emission from the ejecta, is roughly coincident with the geometrical center of the radio emission, but shows an apparent offset from its pulsar position discovered at the southeast (Winkler et al., 2009, see below). Ghavamian et al. (2009) reported that the mid-infrared (MIR) spectrum of the ejecta is dominated by ionic lines and a broad bump around 17 m using the Space Telescope (). They proposed that the latter is produced either by Polycyclic Aromatic Hydrocarbons (PAHs) along the line of sight or newly-formed dust within the ejecta. Besides there is also an equatorial MIR continuum component from the dust associated with the shocked CSM partly overlapping with the ejecta emission. A previous infrared (IR) study based on the data obtained with Infrared Astronomical Satellite (), on the other hand, identified enhanced far-IR (FIR) emission in the southwestern part of the SNR, suggesting that the source has encountered adjacent clouds (Braun et al., 1986; Park et al., 2007). In radio, G292.01.8 mainly consists of bright emission from a central pulsar wind nebula (PWN) of 4′ (in diameter) and relatively fainter outer plateau emission of 9′ (Lockhart et al., 1977; Braun et al., 1986; Gaensler & Wallace, 2003). The plateau emission declines sharply outward, and there is no apparent surrounding radio shell structure. The pulsar PSR J1124–5916, which is located 46″ from the dynamical center of the SNR in the southeast direction, was discovered by radio timing (Camilo et al., 2002), confirming the PWN nature of the central emission of the SNR. The pulsar and PWN were also identified in X-ray emission (Hughes et al., 2003), where the pulsar has a nearby jet and a torus of 5″ scale (Park et al., 2007). The jet axis is slightly tilted to the northeast direction (Park et al., 2007). There are two clear differences between the soft and hard X-ray emission of G292.01.8. First, while the former is produced by shocked gas of normal composition distributed as equatorial filaments, indicating it is likely from CSM, the latter is dominated by bright metallic lines from the ejecta (Park et al., 2002, 2004). Next, the X-ray emission is harder in the northwestern part than the southeastern part, implying density variation or asymmetric SN explosion (Park et al., 2007). The distance to G292.01.8, on the other hand, was estimated to be 6.2 kpc from H I absorption observations (Gaensler & Wallace, 2003). In this paper we present an extensive IR study of G292.01.8 using multi-band imaging and spectroscopic data obtained with the space telescope (Murakami et al, 2007). We describe the details of our observations and present the results of the observations in §2 and 3, respectively. In §4 we discuss IR emission in G292.01.8 from circumstellar dust with a comparison to model calculations of dust destruction by a SN shock. We also discuss the IR emission from the ejecta and we suggest that there is a negligible amount of dust associated with the ejecta. Then, we make a comparison to the results of obserations. We finally discuss the SN explosion in G292.01.8 based on our observations followed by our conclusions in §5. 2 Observations The multi-band imaging observations of G292.01.8 were carried out using its six near-IR (NIR) and MIR bands in the 2.7–26.5 m range as well as four FIR bands in the 50–180 m range on 2007 January 17 and 19. The NIR and MIR images were obtained with Infrared Camera (IRC) that has three simultaneously operating NIR, MIR-S, and MIR-L channels. The three channels have comparable field of views of . The NIR and MIR-S channels share the same pointing direction, while that of the MIR-L channel is 25′ away. The IRC observations were conducted in the two-filter mode that produced images of two bands for each channel (Onaka et al., 2007). The total on-source integration times were 178 s for the NIR (N3, N4) observations and 196 s for both the MIR-S (S7, S11) and MIR-L (L15, L24) observations. The basic calibration and data reduction, including dark subtraction, linearity correction, distortion correction, flat fielding, image stacking, and absolute position determination were performed by the standard IRC Imaging Data Reduction Pipeline (version 20071017). Tables 1 and 2 present a journal of our observations, including the spectroscopic observations, and the basic parameters of the imaging bands, respectively. The FIR images were obtained with Far-Infrared Surveyor (FIS) in two round-trip scans using the cross-scan shift mode (Kawada et al., 2007). The scan speed and length were 15″ s and 240″, respectively, creating images of 40′ 12′ size elongated along the scanning direction. All the four FIS band (N60, Wide-S, Wide-L, and N160) images were obtained simultaneously by a single observing run (Table 1). The initial data calibration and reduction such as glitch detection, dark subtraction, flat fielding and flux calibration were processed with FIS Slow-Scan Toolkit (version 20070914), and the final image construction was performed with a refined sampling mechanism. These IRC and FIS multi-band imaging observations were followed by NIR and MIR spectroscopic observations and L18W-band (13.9–25.6 m, combining both L15 and L24 bands) imaging observations carried out in 2007 July 20–22. The spectroscopic observations were conducted using four (NG, SG1, SG2, and LG2) grisms. Similar to the aforementioned imaging observations, the NG, SG1, and SG2 mode observations were conducted simultaneously, while LG2 mode observations were done separately (Ohyama et al., 2007). Table 3 lists the characteristics of the spectroscopic observations. Spectra from the peaks of the equatorial emission and ejecta identified in the ratio between the L15- and L24-band images (L15/L24 hereafter) were obtained together with that from a reference background position at the southeastern part of the source (Table 4). The data calibration and reduction were processed with IRC Spectroscopy Toolkit (version 20070913). The obtained flux was converted to the surface brightness based on the measured slit size. IRC spectroscopy is made at the slits located at the edge of the large imaging field-of-view (FoV). There is internal scattered light in the array of the MIR-S and the light diffusing from the imaging FoV affects SG1 and SG2 spectra to some extent. This effect has been corrected for according to the method given in Sakon et al. (2008). A similar scattered light is also recognized in LG2 spectra and has been corrected for in a similar way. In addition there is a contribution from the second order light from the edges of the imaging FoV in LG2 spectra, but we expect that it is not significant in the background-subtracted spectra. These spectroscopic observations were simultaneously accompanied by supplemental short (49 s 3) L18W-band imaging observations of a field 5′ away from the central slit position in the southeast direction in order to check the pointing of the satellite. This supplemental, short-exposure L18W-band image of the southeast, which fortuitously revealed interesting structures beyond the SNR boundary (see § 3.4), was combined with deep (442 s) L18W-band image of the source to generate a final larger-area map. The L18W-band imaging data were processed with the same procedures that we used for other imaging data sets as described above. 3 Results 3.1 Multi-band Infrared Images of G292.0+1.8 Figure 1 presents our IRC multi-band images of G292.01.8 together with the ATCA 20 cm radio continuum image (Gaensler & Wallace, 2003), X-ray (0.3–8 keV) image (Park et al., 2002), and the background-subtracted S11-band image (S11–S7; see below) for comparison. The IR emission associated with the SNR is most apparent in the L15- and L24-band images as two prominent features: first, there is a ring-like structure composed of two clumpy, narrow, and long filaments crossing the central part of the SNR along the east-west direction – we name this “Equatorial Ring” (ER); secondly, the outer boundary of the SNR appears as an almost-complete shell structure with its eastern part opened – we name this “Outer Elliptical Shell” (OES). The radius of the ER is 3 (or 5 pc). Its southern filament is brighter than the northern one by a factor of on average and has the brightest clump at (RA, decl.) = (, ) close to the filament center. The southern filament also shows prominent X-ray and [O iii] emission and has been called “equatorial bar” or “equatorial belt” (Park et al., 2002; Gonzalez & Safi-Harb, 2003; Ghavamian et al., 2005). The northern filament is 1′ away from the southern filament and both filaments are elongated roughly parallel to each other. The OES shows an almost-complete ellipse of 7′ 6′ ( 12 pc 10 pc) with its major axis aligning roughly with the plane of the ER. The center of the ellipse determined by elliptical fits weighted by the 24 m surface brightness of the OES is (, ) located at the middle of two filaments of the ER. In the west the OES appears to be connected with the two filaments of the ER and its emission is enhanced there, especially at the southwest near the southern filament of the ER. In contrast the eastern part of the OES is open, and also the emission from the southern filament of the ER is truncated in the east before it reaches the boundary of the SNR. In addition to the ER and OES, there appear to be at least two more clearly identifiable features in the images. First, the L15-band image shows a separate structure that extends 5 southward from the ER on the southeastern side of the SNR. It is elongated vertically with the bright portion located 0.5 below the southern equatorial filament. Next, there is faint emission extending beyond the northern and southeastern edges of the OES. This faint emission is contained within the outermost boundary of radio emission observed in G292.01.8 (Gaensler & Wallace, 2003). As in Figure 1, G292.01.8 is not detected in the S7- and S11-band images: while some of the S11-band emission appears to arise from locations of strong L15- and L24-band emission, there is almost no corresponding emission in the S7-band image. This suggests that the S7-band emission is dominated by background emission, probably emission from PAHs in the line of sight interstellar medium (ISM). In the S11–S7 difference image (Figure 1) we subtract out scaled S7-band emission from the S11-band image and confirmed that the two filaments of the ER and southwestern part of the OES have appreciable S11-band emission. (Note that the three bright point-like sources close to the eastern end of the southern filament of the ER in the S11–S7 difference image are stellar sources.) The NIR N3- and N4-band images, on the other hand, show only stellar emission without any apparent feature brighter than 16 Jy at 3 m that might be associated with the SNR. Table 5 presents IR measurements of G292.01.8. In the Table, we list the fluxes and ratios of whole remnant area and IR-Ejecta region (see befow for IR-Ejecta). In addition, we also list the peak intensities and ratios of the ER, the southweastern OES, and the IR-Ejecta. 3.2 Infrared Colors and Ejecta Identification Figure 2 (left), which is a three-color (7, 15 and 24 m) MIR image of G292.01.8, shows that most of the prominent features have roughly similar MIR colors. One exception is the feature in the southeastern part of the SNR with significant excess in the shorter wavebands – we name this feature “IR-Ejecta” because it is believed to originate in shocked SN ejecta as we describe below. It is bright near the southern equatorial filament and stretching directly southward beyond the SNR boundary. This feature is clearly seen in Figure 2 (right) which shows the L15/L24 emission ratio. The ratio image was produced by dividing the L15-band image by the L24-band image after background subtraction. The background was estimated by fitting a slanted plane to the areas surrounding the SNR. The resulting background planes were almost flat, i.e., the brightness differences over the entire images were only 1.7 % and 2.4 % in L15 and L24, respectively. We applied a Gaussian convolution to the L15-band image in order to match the final spatial resolution to that of the L24-band image and masked out the pixels with small ( 30.2 MJy sr) L24-band intensities. The resulting L15/L24 color image is significantly different from the original band images. In Figure 2, the L15/L24 ratio is almost uniform ( 0.25), and there is no feature corresponding to the prominent ER and OES. The most notable feature is the IR-Ejecta, where L15/L24 ratio raises to 0.8, which is in fact consistent with what we identified in the three-color image (left panel of Figure 2). It appears to be of a triangular shape with one of its vertices towards the direction to the SNR center. The southern vertex passes though the southern boundary of the OES and extends over the SNR boundary, while the northeastern vertex is located just above the ER. The region of high color ratio of the IR-Ejecta covers a large portion of the southeastern area of the SNR and positionally coincides with the O- and Ne-dominant ejecta identified by previous optical and X-ray observations (Ghavamian et al., 2005; Winkler & Long, 2006; Park et al., 2002). There is also a wispy patch of emission around (, ) far ( 1′) beyond the SNR boundary, which has the color ratio similar to that of the IR-Ejecta. It, however, is not shown in the [O iii] images (Winkler & Long, 2006). Besides the IR-Ejecta in the southeastern region, the northwestern region shows extended emission of the high color ratio, although it is not as significant as the southeastern region. Overall the two regions of the high color ratio are roughly symmetric with respect to the center of the OES. Figure 3 compares the pixel values of the L24- and L15-band emission, where most of the points are distributed along the thick, solid line of a slope of 0.25. (Note that pixels of stars and pixels with small L24-band intensities are masked out.) If L24- and L15-band fluxes are from the same region and if they are both thermal dust emission, then the color ratio and the dust temperatures are related as Iν(15)Iν(24)=κν(15)Bν(15,T)κν(24)Bν(24,T) , (1) where is the surface brightness at (m) in Jy sr, is the Planck function, and is the dust opacity in cm g. The slope of 0.25 corresponds to the dust temperature of 126 K for a mixture of carbonaceous and silicate interstellar grain of = 3.1 (Draine, 2003). In Figure 3 there are two regions, where the data points show a high L15/L24 ratio compared to the thick, solid line: first, the thin line of the slope of 0.88 represents one group whose L24-band surface brightness is less than 33 MJy sr; secondly, there is another group of data points of higher L15-band emission whose L24-band surface brightness lies in the range of 33–40 MJy sr. The inset in the lower-right corner of Figure 3 shows that the data points with high L15/L24 ratios are all from the IR-Ejecta as expected. The ones in the second group, i.e., ones with high 24 m surface brightness, are superposed on the ER and therefore their emission is partly from the swept-up CSM while the data points in the first group should represent the emission only from the ejecta. The slope of 0.88 corresponds to a dust color temperature of 240 K; however the L15-band flux is largely from line emission at this position so that the physical dust temperature should be lower than this (see §3.4). 3.3 Mid-Infrared Spectroscopy of Ejecta and Equatorial Peak Figure 4 shows the background-subtracted MIR grism spectra of the peak positions of the ER and the IR-Ejecta.444We only present the MIR spectra because NIR spectra were heavily contaminated by emission from nearby field stars. (Note that the LG2 spectra at 17 m were obtained from positions slightly shifted from those of the SG spectra of 5–14 m as we described in § 2.) The spectrum from the IR-Ejecta peak clearly shows the [Ne ii] line at 12.8 m, which confirms the nature of the radiatively shocked ejecta. No Ar lines (i.e., [Ar ii] at 7.0 m and [Ar iii] at 9.0 m) were detected, while neither [Ne iii] line at 15.6 m nor [O iv] line at 25.9 m was covered. Table 6 lists the flux of the [Ne ii] line and upper limits of the [Ar ii] and [Ar iii] lines. The flux of the [Ne ii] line at 12.8 m is 2.82.510 erg cm s sr, corresponding to the surface brightness of 0.240.21 MJy sr in the L15-band image. Given the L15-band surface brightness of 1.60.1 MJy sr, 15 % of the L15-band emission is due to the [Ne ii] line emission. For the S11 band, the observed [Ne ii] line emission flux corresponds to the S11-band surface brightness of 0.120.11 MJy sr. This is equivalent to 55 % contribution to the total S11-band emission, which is much larger than the case of the L15-band emission. The difference is because there is no strong emission line other than the [Ne ii] line in the S11 band while there is additional [Ne iii] 15.5 m line in the L15 band (see the spectral responses in Figure 4). (There could be some dust emission too. See § 4.3.) On the other hand, the spectrum at the ER peak is dominated by the continuum emission, although the LG2 spectra have a lower signal to noise ratio. 3.4 Large-scale Infrared Emission around G292.0+1.8 Figure 5 presents a combined L18W-band image (§ 2) covering both the SNR and an extended area in the southeast. Besides the features of the SNR that we already described in previous sections, there is a notable feature of the L18-band emission in the south. The elongated “Narrow tail” is located around (, ), 7′ apart from the center of the SNR. It is close to the IR-Ejecta in the southeastern part of the SNR and its elongation direction is roughly towards the center of the SNR. Its extent is or 2.6 pc. In addition, there is faint, diffuse emission toward the south and southeast too. The emission toward the southeast appers distinct - it appears to protrude from the open portion of the OES having similar outer boundary with radio plateau at the east and south, but extends southeast beyond the radio boundary of the SNR. We consider that this “Wide tail” could be associated with the SNR. There is large, diffuse H ii region superposed on G292.01.8 on the sky (RCW 54; Rodgers et al. 1960), but its H emission is elongated along the northeast-southwest direction without any emission corresponding to the Wide tail in the Southern H-Alpha Sky Survey Atlas (Gaustad et al., 2001). Therefore, the Wide tail is not associated with H ii regions and its association with the SNR is likely. Figure 6 presents the FIS FIR-band images of G292.01.8. The images cover the entire SNR with a scan direction of northeast to southwest. The FIR emission of the SNR associated with the ER and OES is clearly detected in the N60- and Wide-S-band images with strong concentration in the southwestern part, although fine details are difficult to see because of their low spatial resolutions. The N160- and Wide-L-band images are, however, clearly different from images of the other bands. They show an elongated feature extended in the northwest-southeast direction which shows no correlation with the emission associated with the SNR. Also its peak position is located outside the boundary of the SNR. These indicate that the N160- and Wide-L-band emission is not directly associated with G292.01.8, although it is possible that the enhanced brightness in the SW part of OES is due to the interaction of the remnant with this extended structure (cf. Braun et al., 1986). The bottom panels of Figure 6 present the background-subtracted N60- and Wide-S-band images. The background emission was estimated by calculating scaling factors between the N60- and N160-band images and also between the Wide-S- and N160-band images from the areas outside the SNR. Compared to the N60- and Wide-S-band images of the top panels, the background-subtracted images show more clearly the emission associated with the SNR, including the northern part of the OES that is not clear in the orignial images. 4 Discussions 4.1 Destruction of Circumstellar Dust 4.1.1 Infrared Emission from Shocked Circumstellar Dust The ER and OES are the most prominent features in our multi-band IR images (Figures 1 and 6). The ER is composed of two filaments, where the southern one is brighter than the northern counterpart. The southern filament appears to be of the normal composition in soft X-rays (Park et al., 2002) without any apparent radial motion in the optical (Ghavamian et al., 2005), which led the authors to conclude that it is CSM from the progenitor of the SN in G292.01.8. We suggest that the northern filament, which has been clearly found by observations, could be part of the same structure based on its similar distribution to the southern filament. We note that the bright clump at (, ) near the eastern end of the northern filament has a counterpart in the [O iii] image of Ghavamian et al. (2005). Its velocity is near zero, which supports that the northern and southern filaments form a single structure. Furthermore, it is most clearly visible at the 120 km s frame of the Rutgers Fabry-Perot velocity scan images and absent at the 0 km s frame where the southern filament is brightest (Figure 2 of Ghavamina et al. 2005). We interpret this velocity difference as an indication suggesting that the northern and southern filaments are parts of a tilted, expanding ring structure produced by the progenitor of the SN. The location of the center point of the OES at the middle of the two filaments also reconciles with the interpretation, given the CSM-nature of the OES (see below). We note that such a, but smaller, ring structure of the CSM was found in SN 1987A and also possibly in the Crab nebula (Bouchet et al., 2004; Green et al., 2004). The OES, on the other hand, is relatively fainter in X-rays than the ER (Park et al., 2002). The MIR brightness and color of the OES, however, are similar to those of the ER (Figure 2). Toward the southern filament of the ER, Ghavamian et al. (2005) reported the detection of the optical radiative lines produced by the partially radiative shocks starting to develop cooling zones. It implies that the X-ray emitting gas in the ER is cooler than that of the OES, while they have somewhat similar dust properties. And there is a possibility that the ER in S11-S7 difference image also contains the [Ne ii] line emission produced in the CSM region where the shock has started to cool down to 100,000 K. However, its contribution might be small, because we have not detected the [Ne ii] 12.8 m line at the ER. In addition to the ER and OES, there are faint MIR emissions beyond the northern and southeastern edges of the OES (Figure 2). They have sharp outer boundaries, representing the current locations of the SN blast wave. Faint X-ray emission was detected in those areas too (Park et al., 2002). It is possible that the SN blast wave has overtaken the OES and produced those emission features while propagating into a more diffuse medium. On the other hand, it is also possible that the remnant has a front-back asymmetry and they are just projected boundaries of the more-extended shell. In any case, their asymmetric spatial distribution suggests that either the ambient density distribution or the mass ejection from progenitor was asymmetric. As in Figure 4, the MIR emission of G292.01.8 is dominated by continuum emission, not by line emission. This implies that the IR emission is from shock-heated dust grains in the CSM. Figure 7 presents the spectral energy distribution (SED) of the SNR in Table 5. We applied modified blackbody fits composed of two dust components to the observed SED in order to obtain the best-fit dust temperatures. The dust model based on a mixture of carbonaceous and silicate interstellar grain of = 3.1 (Draine, 2003) gave dust temperatures of 103 K (warm dust) and 47 K (cold dust), corresponding to the mass of 4.5 0.9 10 M and 4.8 10 M, respectively. (Note that the lower limit of the cold dust temperature comes from the upper limit of the flux at 140 m.) For the case of the dust model based on the graphite and silicate grain of 0.001 to 0.1 m size (Draine & Lee, 1984; Laor & Draine, 1993), the derived total mass is in the range of (1.0–3.4) 10 M, comparable to the total mass obtained for the former model. The derived dust mass corresponds to the dust-to-gas ratio of 1.6 10, if we use the 30.5 M swept-up mass (at distance of 6.2 kpc) of the gas in the CSM obtained in X-ray observations (Gonzalez & Safi-Harb, 2003). This is lower than the ratio 6.2 10 found in the local ISM (Zubko et al., 2004). The low dust-to-gas ratios in the swept-up materials by the shock destruction were also obtained with Spitzer observations on the core-collapse SNRs in the Large Magellanic Cloud (Williams et al., 2006) and the Kepler (Blair et al., 2007). According to our result, 75 % of the dust in G292.01.8 might have been destroyed by the SN shock, or the initial dust-to-gas ratio surrounding G292.01.8 might be lower than the local value. It is comparable to the fraction derived in other SNRs, such as 64 % in Cas A, (Dwek et al., 1987) and 78 % in Kepler (Blair et al., 2007). 4.1.2 Model Calculations of Shock-heated Dust Emission We perform model simulations for the destruction of dust by SN shock waves and the thermal emission from shock-processed dust. The physical processes of dust in shocks have been so far discussed in many works (e.g., Tielens et al., 1994; Vancura et al., 1994; Dwek et al., 1996; Jones, 2004). Once the circumstellar dust grains are swept up by the blast wave, they acquire high velocities relative to the gas and are eroded by kinetic and/or thermal sputtering in the shock-heated gas. Dust grains are also heated by collisions with energetic electrons in the hot gas and radiate thermal emission at IR wavelengths. Dynamics, erosion, and temperature of dust depend on the temperature and density of the gas as well as the chemical composition and size of dust grains. We adopt the model of dust destruction calculation by Nozawa et al. (2006), in which the motion, destruction, and heating of dust in the shocked gas are treated in a self-consistent manner by following the time evolution of the temperature and density of the gas for the spherically symmetric shock wave. As the initial condition of the SN ejecta we consider the freely expanding ejecta with the velocity profile of and the density profile of at and at , where and are the velocity and radius of the outermost ejecta, respectively. Taking the kinetic energy of ergs and the ejecta mass of 8 (Gaensler & Wallace, 2003) for the density profile of core-collapse SNR of and (e.g., Chevalier, 1982; Matzner & McKee, 1999; Pittard et al., 2001), we obtain g cm, cm s, and cm at 10 yrs after explosion, when the simulations are started. Since most of the dust grains swept up by the forward shock during the later epoch of the evolution, the calculation results are not sensitive to the ejecta structure. For the ambient medium we consider the constant hydrogen number density of 0.1, 0.5, 1, and 10 cm. The circumstellar dust is assumed to be amorphous carbon or silicate (forestrite) with the power-law size distributions () ranging from m to m (Mathis et al., 1977). The optical constants are taken from Edo (1983) and Semenov et al. (2003). Based on the time evolution of the size distribution and temperature of dust given by the simulation and the assumption of the dust grains being in thermal equilibrium, we calculate the IR SED by thermal emission from the shock-heated dust. The detailed description for calculating the IR SED from shocked dust will be given elsewhere (T. Nozawa et al. 2009 in preparation). Figure 8 compares the observed fluxes of G292.01.8 with the calculated IR SEDs at 3,000 yrs for 0.1, 0.5, 1, and 10 cm. We present the results with the initial dust-to-gas mass ratio to best reproduce parts of the observed SED, for amorphous carbon (Figure 8a) and silicate (Figure 8b). The typical temperatures of dust are 35–55, 45–65, 50–70, and 60–80 K for 0.1, 0.5, 1, and 10 cm, respectively, and the resulting dust masses are in the range of (0.3–5) for carbon and (0.4–8) for silicate with the higher values for lower ambient density (thus lower temperature of dust). It can be seen that the results of the silicate grains with 0.5 cm, which coincides with the density estimated from X-ray observations (Gonzalez & Safi-Harb, 2003), can reasonably reproduce the overall shape of the IR SED for G292.01.8. In this case, the mass of grains radiating IR emission is . Note that the initial dust-to-gas mass ratio corresponding this result is , which is smaller than that in the local Galaxy. If the IR emission originates in the swept-up materials containing the mass-loss wind of RSG with the solar metallicity, the condensation of silicate is expected, and, according to our results, its condensation efficiency could be low. It should be noted here that the simulation results for 10 cm significantly underestimates the flux at short (11 ) wavelength. However, the SED at shorter wavelengths could be resolved by including the effect of a stochastic heating of small grains; stochastically heated dust produces more emission at shorter wavelengths than the dust with equilibrium temperature, and it may also allow acceptable fits for lower initial density and/or different size distribution of dust than the current best fit. Alternatively, this disagreement may be caused by the difference in the assumed composition of dust. To gain deeper insights into the properties of dust in G292.01.8, we need further investigations by taking account of the stochastic heating and changing the composition and size distribution of dust as well as the density profile in the ambient medium. 4.2 Infrared Emission from Supernova Ejecta 4.2.1 Ejecta Neon Line Emission We have detected MIR emission associated with the SN ejecta: IR-Ejecta. It is prominent in the 15/24 m ratio image by its high L15/L24 ratio (Figure 2), but is marginally seen in the total intensity maps of the MIR (11–24 m) and FIR (65–90 m) too. The emitting area coincides with the fast-moving ( km s), O-rich SN ejecta: the triangle-shaped bright portion near the equatorial filament was called “spur” and the extension to the south was called “streamers” by Ghavamian et al. (2005). In the high-resolution [O iii] 5007 image, the spur is crescent-shaped with a sharp boundary toward the SNR center, while the streamer appears to be composed of clumps embedded in diffuse emission. Recent measurement of their proper motions showed that they are expanding systematically from a point near the geometrical center of the OES (Winkler et al., 2009, see § 4.4 too). According to our spectroscopic result, the IR-Ejecta shows [Ne ii] 12.8 m emission but no [Ar ii] 7.0 or [Ar iii] 9.0 m emission. Note that the latter lines from Ar ions were also detected at the metal-rich ejecta in young SNRs (Douvion et al., 2001; Smith et al., 2009; Williams et al., 2008). The lack of IR line emission from Ar ions in this area is consistent with the results of optical or X-ray studies which showed that there is no line emission from elements heavier than S in this area (Ghavamian et al. 2005 and references therein; Park et al. 2002). The absence of Ar lines supports the claim by Ghavamian et al. (2005) that we are not seeing the ejecta in the inner-most region accelerated by pulsar wind nebula but seeing the He-burning-synthesized, O-rich ejecta swept-up by reverse shock. The IR-Ejecta shows high L15/L24 ratio, i.e., L15/L24=0.88 compared to 0.25 of the shocked CSM. The high L15/L24 ratio is at least partly due to Ne lines. (There is [O iv] 25.9 m line in the L24 band, but it is weak, i.e., % of the [Ne ii]+[Ne iii] lines. See § 4.3.) According to our estimation, [Ne ii] 12.8 m line contributes 15 % to the L15 flux. In the L15 band, there is another strong Ne line, [Ne iii] 15.6 m line. We may estimate the possible contribution of [Ne iii] 15.6 m line in G292.0+1.8 as follows. The strength of a forbidden Ne line is given by where is the frequency of the line, is the column density of Ne or Ne ions in the upper state along the line of sight, and is the Einstein coefficient, so that the [Ne iii] 15.6 m/[Ne ii] 12.8 m ratio is given by I15.6 μmI12.8 μm=0.57N(Ne++)N(Ne+)f(Ne++,3P1)f(Ne+,2P1/2) , (2) where and are column densities of Ne and Ne ions, and are the fractions of Ne and Ne ions in the upper states of the corresponding emission lines (see Glassgold et al. 2007 for atomic parameters of these lines). Assuming statistical equilibrium, the last factor, , can be easily calculated (e.g., Glassgold at al., 2007), and varies from 1 to 2 as the electron density increases from a low-density limit to densities much higher than the critical density ( cm at 10,000 K). Therefore, the intensity ratio is unless , which happens at temperatures K in collisional equilibrium (e.g., Allen & Dupree, 1969). Ghavamian et al. (2005) suggested that ejecta undergoes radiative shocks with a velocity of 50–200 km s from the [O iii] 5007 line widths. The temperature of shocked ejecta gas immediately behind the shock of velocity should be high, i.e., K, assuming pure neon preshock gas, but since the shocked gas element cools fast due to enhanced heavy elements and the column density of the gas at might vary roughly with , we expect and therefore . Toward the SN ejecta in young core-collapse SNRs such as Crab, Cas A, the SNR B054069.3 in the LMC, or the SNR 1E010272.3 in the SMC, the [Ne iii]15.6 m/[Ne ii]12.8 m ratio varies from 0.3 to too (Douvion et al., 2001; Temin et al., 2006; Smith et al., 2009; Williams et al., 2008; Rho et al., 2009). We may conclude that the contribution of the [Ne iii]15.6 m line to the L15 band should be at most comparable to that of the [Ne ii]12.8 m line unless the shock is truncated at relatively high temperatures. 4.2.2 Dust Emission Associated with Ejecta We have derived IR fluxes associated with the ejecta and the results are summarized in Table 5. It is difficult to extract the ejecta emission in IR observations, although our results unveiled the ejecta region unambiguously. The IR-Ejecta is clearly divided into two regions (Figures 2 and 3); one that coincides with the ejecta-only region and has a tight correlation in the L15 vs L24 plot and the other superposed on the ER and has a loose correlation in the L15 vs L24 plot. These IR-Ejecta regions are part of the structure defined as the [O iii] “Spur” by Ghavamian et al. (2005), with the ejecta-only region being the bottom of the Spur, and the ejecta superimposed on the ER being the top portion of the Spur. We measure the flux at the ejecta-only region masking the emission from stars. The area of the ejecta-only region is more than four times larger than the superposed ejecta region and it contains most of the flux. In order to remove the background emission, we subtract the average of nearby ( 2) low intensity region. The ejecta spectrum in Figure 4 shows that the 11 and 15 m fluxes are not from dust continuum. We believe that the S11-band is mostly dominated by line emission, while the L15-band needs an additional component (see § 3.3). In the next paragraph, we show that some of the L15 flux is due to a bump at 15–25 m. On the other hand, FIR emission is certainly dominated by dust continuum. The L24-band contains the [O iv] 25.9 m line emission, whose wavelength is not covered with our spectroscopic observations. Its contribution in L24-band is expected to be less than 20 % in our estimation using the [O iv] 25.9 m line intensity in Ghavamian et al. (2009). At first, we fit 24–140 m band fluxes using a modified-blackbody for silicate and graphite grain models of 0.001–0.1 m size (Draine & Lee, 1984; Laor & Draine, 1993). The best fits give temperature of 64–88 K and dust mass of 2.0–8.2 10 M. If we exclude the L24-band data point, the temperature drops to 44–52 K and the dust mass increases by a factor of 5. This IR-emitting dust mass is much smaller ( 4 10) than the theoretically predicted dust mass (0.1–1 M) to be formed in the core-collapse SN explosion (e.g., Nozawa et al., 2003). 4.3 Comparison with Spitzer Spectroscopic Results Recently the results from MIR observations of G292.01.8 with the satellite have been published (Ghavamian et al., 2009) based on low-resolution spectroscopy of the ejecta and the southern filament of the ER where the ejecta emission is superimposed on the emission from the CSM. The former is located 0′.5 north from our IR-Ejecta slit position, while the latter, which has a high L15/L24 ratio, is 1′ apart in the northeast direction. Their ejecta spectrum shows strong emission from the [Ne ii] 12.8 m, [Ne iii] 15.6 m, [O iv] 25.9 m lines and relatively weak [Ne v] 24.4 m and [Ne iii] 36.0 m lines, but no lines of Mg, Si, S, Ar or Fe are identified. (Note that the Si and S lines in their spectrum are not from the ejecta but from the background.) The observed ratio of [Ne ii] 12.8 m to [Ne iii] 15.6 m is 2.1–2.7. The non-detection of Ar lines and the observed ratio of Ne lines are consistent with our results and the prediction in § 4.2. The observed flux of [O iv] 25.9 m line is % of the [Ne ii] plus [Ne iii] line fluxes (Ghavamian et al., 2009). If the L15 and L24 fluxes are entirely due to lines, we estimate that the L15/L24 ratio should be 2.3–2.6 using the line fluxes in Ghavamian et al. (2009). (It becomes higher if we consider the 15-25 m bump in the next paragraph.) The observd ratio, however, is 0.25 for the ER and OES and 0.88 for the ejecta. Therefore, the contribution of [O iv] 25.9 m line flux to the observed L24-band flux should be small, particularly toward the ER where (§ 3.2). We, however, note that the IRS spectrum toward the ejecta in Ghavamian et al. (2009) does not show any obvious dust continuum emission between 24 and 36 m. We consider that it could be because there was dust continuum emission in the IRS background. In the 24 m image, there is faint filament coincident with the IRS background region (LL1 Sky in Ghavamian et al. 2009). This filament is associated with the OES and its 24 m emission might be dominated by dust continuum, so that, by subtracting its spectrum from the ejecta spectrum, the continuum feature could have been removed. An interesting feature in the ejecta spectrum is a weak 15–25 m bump, which was suggested to be produced by newly-formed dust or swept-up PAHs along the line of sight. First of all, this bump explains the high L15/L24 ratio at the ejecta region described in § 4.2.1. The peak of the bump feature appears in the L15 band. Therefore, it contributes mainly to the 15 m band. Secondly, our observations show that the areas with a high L15/L24 ratio are coincident with those of the optical O-rich ejecta region. This supports the interpretation that the bump feature is related to the newly-formed dust in association with the SN ejecta, not to the swept-up PAHs. According to Ghavamian et al. (2009), the spectrum from the southern filament of the ER is consistent with the emission from two dust components: a warm (or hot) component of 114 K and a cold component of 35 K. The temperature of the warm dust is higher than what we estimated with the data by 10 K, while that of the cold dust is lower than the temperature by a comparable amount. It is possible that the presence of the bump feature in the spectrum resulted in a higher temperature for the warm dust. For the cold dust temperature, as explained in Ghavamian et al. (2009), the absence of the longer wavelength ( 30 m) data in the spectrum is likely the primary reason of the lower temperature compared to the results. IR spectroscopic observations covering a broad band are necessary to clearly resolve the issues. 4.4 Supernova Explosion in G292.0+1.8 4.4.1 Circumstellar Shell and the Explosion Location One interesting result that we obtained in this study is the difference ( 42″ 1 pc) between the center of the CSM and the dynamical center of the ejecta (Figure 2). The former corresponds to the center of the ER and OES that we determined with the results; the latter was determined by the distribution of the O-rich ejecta in the optical, which is close to the center of the SNR in the radio emission. In addition, the position of the pulsar is different from the both positions – it is shifted in the southeast direction from the dynamical center by 46″. One possibility is that the progenitor star was at the center of OES during its RSG phase but exploded at the position of the dynamical center of optical knots. This is possible because the RSG wind could be confined by external pressure while the central star is moving. According to Chevalier (2005), the RSG wind from a 25–35 M star, which explodes as SN IIL/b, would be pressure confined while its outer radius becomes pc. The radius of OES (6 pc) is comparable to what the theory predicts. If the progenitor star was moving at 10 km s and exploded after 10 yrs of the pressure confinement of the shell, then the explosion center would be close to the dynamical center of optical knots. Another possibility is that the SN exploded at the center of ER and OES, not at the dynamical center of the ejecta or the center of the radio emission. This is motivated by the fact that the OES shows a very well-defined shell structure surrounding the SNR, thereby the center of the OES may pinpoint to the real location of the progenitor, which later exploded as a SN. On the other hand, the centers of the radio emission and the O-rich ejecta motion could have been weighted toward the southeast: Firstly, the geometrical center of the radio nebula could be significantly weighted toward the southeast because of the existence of the bright PWN. Secondly, the dynamical center of optical knots is also weighted to the southeast because the bright optical knots in the southeastern area are moving relatively slowly and the center is derived using an assumption of an unhindered constant velocity since the explosion (Winkler et al., 2009). If the SN explosion in G292.01.8 indeed occurred at the center of the OES, then the tangential velocity of the pulsar needs to be 1,000 km s. Although 1,000 km s is somewhat large as a pulsar kick velocity, it is still within the acceptable range (e.g., Ng & Romani, 2007), and the pulsar in another O-rich SNR Puppis A may also have such a high velocity (Hui & Becker, 2006). 4.4.2 Ejecta Distribution and Explosion Asymmetry The optical and X-ray studies of the SN ejecta in G292.01.8 have shown that their spatial distribution and physical properties are not symmetric, including the northwest-southeast concentration of the O and Ne ejecta, and the distinctive difference of the X-ray plasma temperature between the northwestern and southeastern areas (Park et al., 2002; Ghavamian et al., 2005; Park et al., 2007). Also the ejecta in the northern and southern boundaries move faster than those in the eastern and western areas (Winkler et al., 2009). It is worth to note that the pulsar jet axis is also along the northeast-southwest direction (Park et al., 2007). Our result on the Ne-line emitting ejecta material identified by their high L15/L24 ratio is consistent with the spatial distribution seen in [O iii] 5007: most of the / ions are distributed in the southeastern area called spur and streamers, several isolated ones coincide with the optical knots, and the other group of Ne-rich knots is distributed in the northwestern area and coincides with [O iii] optical emission knots (fast-moving knots, or FMKs, as reported by Ghavamian et al. (2005) and Winkler & Long (2006)). The northwest ejecta is less obvious in optical but it is easily recognized in the X-ray image of ionized O and Ne elements (Park et al., 2002). This type of bipolar ejecta distribution is also found in other young core-collapse SNRs, e.g., Cas A and G11.20.3 (Smith et al., 2009; Koo et al., 2007; Moon et al., 2009). Cas A shows a similar Ne ejecta distribution, which is almost perpendicular to the well-known narrow northeast-southwest jet, but is roughly aligned to the bipolar ionic ejecta, which sometimes is suggested be the major direction of explosion (Hwang et al., 2004; Wheeler et al., 2008; Smith et al., 2009). In G11.20.3, which is the remnant of the historical SN AD 386, the iron ejecta is found to be distributed mainly along northwest-southeast direction (Koo et al., 2007; Moon et al., 2009). The symmetry axis in the spatial and kinematical distribution of ejecta in G292.01.8, therefore, is either along northwest-southeast or north-south, which is perpendicular to the plane of the ER. We consider that either the explosion was asymmetric and/or the CS wind was denser in the equatorial plane so that the ejecta expanding in this plane was slowed down more compared to those expanding to the other directions. An interesting feature is the Narrow tail in the wide L18W-band image that extends from the end of the streamers to outside the remnant. A possible explanation is that the Narrow tail is a part of the O-rich streamer. Note that the Narrow tail is connected to the streamers by a southern patch of emission near the boundary of L15-band image. The distance from the center to the end of the Narrow tail is 7 (13 pc), which is 2 (3 pc) farther than the outermost O-rich clump in this area (Winkler et al., 2009). It, however, does not has an optical counterpart in the [O iii] image (Winkler et al., 2009). If we assume a constant velocity and adopt an age of 3,000 yrs (Ghavamian et al., 2005; Winkler et al., 2009), the transverse velocity of the Narrow tail is 4,000 km s. This gives a possibility that its radial velocity is also very large, beyond the velocity range of previous optical narrow-band imaging observations (e.g., 2,000 km s in Winkler & Long, 2006). An alternative explanation is that the Narrow tail is a reradiated IR light echo similar to the echoes identified in Cas A (Krause et al., 2005; Dwek & Arendt, 2008). The lack of counterpart in the optical [O iii] and X-ray images, together with the faint feature in the FIR images (Figure 6), could be consistent with the continuum origin of the IR emission. In case of a light echo, the location of the echo can be derived from the geometrical equation of ellipse whose two focuses are the SN and the observer (e.g., Couderc, 1939; Dwek & Arendt, 2008). Applying a projected radius of 13 pc and a time delay of 3,000 yrs, we obtain a location of the IR echo at 450 pc behind G292.01.8 and its angular offset of 2 from the line of sight. It appears that this alignment is too tight at a first glance, but it can be a selection effect caused by our limited imaging area. For example, Krause et al. (2005) discovered IR echoes along the scan direction in Cas A for the first time, but Dwek & Arendt (2008) reported that the echoes were distributed in many positions in large ( 2) area. More observations are definitely necessary to inspect the nature of the Narrow tail. On the other hand, the sharp boundary of the SN blast wave was detected only toward the north and southwest (Figure 2). The absence of SN blast wave in the other directions implies that either the shock is trapped within the shell because that part of the shell has a larger column density or the SN blast wave has propagated far beyond because the ambient density is lower toward that direction. There is no indication in the images that the OES is denser, where the SN blast wave is missing. Instead those parts are fainter in the FIR images which indicates a lower column density. In this regard, the faint MIR emission that extends far beyond the bright shell to the southeast (Figure 5) is interesting because if it is part of the SNR, it indicates that the SN blast wave has propagated much further out toward this direction probably due to the lower ambient density. A deep radio observation could reveal faint features associated with this IR structure. 5 Conclusions We have presented the NIR to FIR imaging and MIR spectroscopic observations of the O-rich SNR G292.01.8 using the IRC and FIS instruments aboard satellite. The almost continuous multiband imaging capability of covering wide IR wavelengths together its wide field of view enabled us to clearly see the distinct IR emission from the entire SNR and to derive its IR charactersitics. We derive the physical parameters of IR-emitting dust grains in the swept-up circumstellar medium and compare the result with the model calculations of dust destruction by a SN shock. The overall shape of the observed SED can be explained by a simple model using characteristic SNR parameters with a bit lower initial dust-to-gas ratio. At 11 m, the model flux is significantly smaller, which may indicate the importance of stochastic heating. We have not detected any signficant amount of freshly-formed dust associated with the SN ejecta. The AKARI results in this paper give new insights into the explosion dynamics of G292.01.8. We have discovered an almost symmetric IR shell probably produced by the circumstellar wind from the progenitor star in the RSG phase. Its center is significantly offset from the previously suggested explosion centers. We consider that either OES represents the circumstellar shell pressure-confined by external medium or the SN exploded close to the center of the OES. In the latter case, the pulsar in G292.01.8 may be traveling at a speed of km s. At the same time, the ejecta distribution is unveiled by their high 15 to 24 m ratio. The ejecta are mainly distributed along the northwest-southeast direction. This symmetric pre-supernova structure and asymmetric ejecta distribution appear to be rather common in the remnants of SN IIL/b, which suffer strong mass-loss like Cas A (Hines et al., 2004; Smith et al., 2009). There is also a Narrow tail outside the SNR shell, which might be similar to the feature observed in Cas A. A detailed study is necessary to understand the nature of this feature. We suggest that multi-band IR imaging observations are powerful tools to explore both the ejecta and CSM emission in young core-collapse SNRs. Especially the 15 and 24 m images are useful to reveal the detailed structure of IR features, which leads to better understanding of environments of the progenitor and the SN explosion. This work is based on observations with , a JAXA project with the participation of ESA. We wish to thank all the members of the project. We also thank B. Gaensler for providing the ATCA 20 cm image and S. Park for providing the X-ray image. This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-357-C00052) and the Korea Science and Engineering Foundation (R01-2007-000-20336-0). This work was also supported in part by a Grant-in-Aid for Scientific Research for the Japan Society of Promotion of Science (18204014). T.N. has been supported in part by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, and by the Grant-in-Aid for Scientific Research of the Japan Society for the Promotion of Science (19740094). Facility: Akari References • Allen & Dupree (1969) Allen, J. W., & Dupree, A. K. 1969, ApJ, 155, 27 • Blair et al. (2007) Blair, W. P., Ghavamian, P., Long, K. S., Williams, B. J., Borkowski, K. J., Reynolds, S. P., & Sankrit, R. 2007, ApJ, 662, 998 • Bouchet et al. (2004) Bouchet, P., De Buizer, J. M., Suntzeff, N. B., Danziger, I. J., Hayward, T. 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http://www.newworldencyclopedia.org/entry/Thermal_conductivity	# Thermal conductivity Fire test used to test the heat transfer through firestops and penetrants used in construction listing and approval use and compliance. In physics, thermal conductivity, $k$, is the property of a material that indicates its ability to conduct heat. It appears primarily in Fourier's Law for heat conduction. Conduction is the most significant means of heat transfer in a solid. By knowing the values of thermal conductivities of various materials, one can compare how well they are able to conduct heat. The higher the value of thermal conductivity, the better the material is at conducting heat. On a microscopic scale, conduction occurs as hot, rapidly moving or vibrating atoms and molecules interact with neighboring atoms and molecules, transferring some of their energy (heat) to these neighboring atoms. In insulators the heat flux is carried almost entirely by phonon vibrations. ## Mathematical background First, heat conduction can be defined by the formula: $H=\frac{\Delta Q}{\Delta t}=k\times A\times\frac{\Delta T}{x}$ where $\frac{\Delta Q}{\Delta t}$ is the rate of heat flow, k is the thermal conductivity, A is the total surface area of conducting surface, ΔT is temperature difference and x is the thickness of conducting surface separating the two temperatures. Thus, rearranging the equation gives thermal conductivity, $k=\frac{\Delta Q}{\Delta t}\times\frac{1}{A}\times\frac{x}{\Delta T}$ (Note: $\frac{\Delta T}{x}$ is the temperature gradient) In other words, it is defined as the quantity of heat, ΔQ, transmitted during time Δt through a thickness x, in a direction normal to a surface of area A, due to a temperature difference ΔT, under steady state conditions and when the heat transfer is dependent only on the temperature gradient. Alternately, it can be thought of as a flux of heat (energy per unit area per unit time) divided by a temperature gradient (temperature difference per unit length) $k=\frac{\Delta Q}{A\times{} \Delta t}\times\frac{x}{\Delta T}$ Typical units are SI: W/(m·K) and English units: Btu·ft/(h·ft²·°F). To convert between the two, use the relation 1 Btu·ft/(h·ft²·°F) = 1.730735 W/(m·K).[1] ## Examples In metals, thermal conductivity approximately tracks electrical conductivity according to the Wiedemann-Franz law, as freely moving valence electrons transfer not only electric current but also heat energy. However, the general correlation between electrical and thermal conductance does not hold for other materials, due to the increased importance of phonon carriers for heat in non-metals. As shown in the table below, highly electrically conductive silver is less thermally conductive than diamond, which is an electrical insulator. Thermal conductivity depends on many properties of a material, notably its structure and temperature. For instance, pure crystalline substances exhibit very different thermal conductivities along different crystal axes, due to differences in phonon coupling along a given crystal axis. Sapphire is a notable example of variable thermal conductivity based on orientation and temperature, for which the CRC Handbook reports a thermal conductivity of 2.6 W/(m·K) perpendicular to the c-axis at 373 K, but 6000 W/(m·K) at 36 degrees from the c-axis and 35 K (possible typo?). Air and other gases are generally good insulators, in the absence of convection. Therefore, many insulating materials function simply by having a large number of gas-filled pockets which prevent large-scale convection. Examples of these include expanded and extruded polystyrene (popularly referred to as "styrofoam") and silica aerogel. Natural, biological insulators such as fur and feathers achieve similar effects by dramatically inhibiting convection of air or water near an animal's skin. Thermal conductivity is important in building insulation and related fields. However, materials used in such trades are rarely subjected to chemical purity standards. Several construction materials' k values are listed below. These should be considered approximate due to the uncertainties related to material definitions. The following table is meant as a small sample of data to illustrate the thermal conductivity of various types of substances. For more complete listings of measured k-values, see the references. ## List of thermal conductivities This is a list of approximate values of thermal conductivity, k, for some common materials. Please consult the list of thermal conductivities for more accurate values, references and detailed information. Material Thermal conductivity W/(m·K) Cement, portland [2] 0.29 Concrete, stone [2] 1.7 Air 0.025 Wood 0.04 - 0.4 Alcohols and oils 0.1 - 0.21 Silica Aerogel 0.004-0.03 Soil 1.5 Rubber 0.16 Epoxy (unfilled) 0.19 Epoxy (silica-filled) 0.30 Water (liquid) 0.6 Thermal grease 0.7 - 3 Thermal epoxy 1 - 4 Glass 1.1 Ice 2 Sandstone 2.4 Stainless steel[3] 12.11 ~ 45.0 Aluminum 237 Gold 318 Copper 401 Silver 429 Diamond 900 - 2320 LPG 0.23 - 0.26 ## Measurement Generally speaking, there are a number of possibilities to measure thermal conductivity, each of them suitable for a limited range of materials, depending on the thermal properties and the medium temperature. There can be made a distinction between steady-state and transient techniques. In general the steady-state techniques perform a measurement when the temperature of the material that is measured does not change with time. This makes the signal analysis straight forward (steady state implies constant signals). The disadvantage generally is that it takes a well-engineered experimental setup. The Divided Bar (various types) is the most common device used for consolidated rock samples. The transient techniques perform a measurement during the process of heating up. The advantage is that measurements can be made relatively quickly. Transient methods are usually carried out by needle probes (inserted into samples or plunged into the ocean floor). For good conductors of heat, Searle's bar method can be used. For poor conductors of heat, Lees' disc method can be used. An alternative traditional method using real thermometers can be used as well. A thermal conductance tester, one of the instruments of gemology, determines if gems are genuine diamonds using diamond's uniquely high thermal conductivity. ### Standard Measurement Techniques • IEEE Standard 442-1981, "IEEE guide for soil thermal resistivity measurements" see als soil_thermal_properties.[4] • IEEE Standard 98-2002, "Standard for the Preparation of Test Procedures for the Thermal Evaluation of Solid Electrical Insulating Materials"[5] • ASTM Standard D5470-06, "Standard Test Method for Thermal Transmission Properties of Thermally Conductive Electrical Insulation Materials"[6] • ASTM Standard E1225-04, "Standard Test Method for Thermal Conductivity of Solids by Means of the Guarded-Comparative-Longitudinal Heat Flow Technique"[7] • ASTM Standard D5930-01, "Standard Test Method for Thermal Conductivity of Plastics by Means of a Transient Line-Source Technique"[8] • ASTM Standard D2717-95, "Standard Test Method for Thermal Conductivity of Liquids"[9] ## Difference between US and European notation In Europe, the k-value of construction materials (e.g. window glass) is called λ-value. U-value used to be called k-value in Europe, but is now also called U-value. K-value (with capital k) refers in Europe to the total isolation value of a building. K-value is obtained by multiplying the form factor of the building (= the total inward surface of the outward walls of the building divided by the total volume of the building) with the average U-value of the outward walls of the building. K-value is therefore expressed as (m2.m-3).(W.K-1.m-2) = W.K-1.m-3. A house with a volume of 400 m³ and a K-value of 0.45 (the new European norm. It is commonly referred to as K45) will therefore theoretically require 180 W to maintain its interior temperature 1 degree K above exterior temperature. So, to maintain the house at 20°C when it is freezing outside (0°C), 3600 W of continuous heating is required. ## Related terms The reciprocal of thermal conductivity is thermal resistivity, measured in kelvin-metres per watt (K·m·W−1). When dealing with a known amount of material, its thermal conductance and the reciprocal property, thermal resistance, can be described. Unfortunately there are differing definitions for these terms. ### First definition (general) For general scientific use, thermal conductance is the quantity of heat that passes in unit time through a plate of particular area and thickness when its opposite faces differ in temperature by one degree. For a plate of thermal conductivity k, area A and thickness L this is kA/L, measured in W·K−1 (equivalent to: W/°C). Thermal conductivity and conductance are analogous to electrical conductivity (A·m−1·V−1) and electrical conductance (A·V−1). There is also a measure known as heat transfer coefficient: the quantity of heat that passes in unit time through unit area of a plate of particular thickness when its opposite faces differ in temperature by one degree. The reciprocal is thermal insulance. In summary: • thermal conductance = kA/L, measured in W·K−1 • thermal resistance = L/kA, measured in K·W−1 (equivalent to: °C/W) • heat transfer coefficient = k/L, measured in W·K−1·m−2 • thermal insulance = L/k, measured in K·m²·W−1. The heat transfer coefficient is also known as thermal admittance ### Thermal Resistance When thermal resistances occur in series, they are additive. So when heat flows through two components each with a resistance of 1 °C/W, the total resistance is 2 °C/W. A common engineering design problem involves the selection of an appropriate sized heat sink for a given heat source. Working in units of thermal resistance greatly simplifies the design calculation. The following formula can be used to estimate the performance: $R_{hs} = \frac {\Delta T}{P_{th}} - R_s$ where: • Rhs is the maximum thermal resistance of the heat sink to ambient, in °C/W • $\Delta T$ is the temperature difference (temperature drop), in °C • Pth is the thermal power (heat flow), in Watts • Rs is the thermal resistance of the heat source, in °C/W For example, if a component produces 100 W of heat, and has a thermal resistance of 0.5 °C/W, what is the maximum thermal resistance of the heat sink? Suppose the maximum temperature is 125 °C, and the ambient temperature is 25 °C; then the $\Delta T$ is 100 °C. The heat sink's thermal resistance to ambient must then be 0.5 °C/W or less. ### Second definition (buildings) When dealing with buildings, thermal resistance or R-value means what is described above as thermal insulance, and thermal conductance means the reciprocal. For materials in series, these thermal resistances (unlike conductances) can simply be added to give a thermal resistance for the whole. A third term, thermal transmittance, incorporates the thermal conductance of a structure along with heat transfer due to convection and radiation. It is measured in the same units as thermal conductance and is sometimes known as the composite thermal conductance. The term U-value is another synonym. In summary, for a plate of thermal conductivity k (the k value[10]), area A and thickness L: • thermal conductance = k/L, measured in W·K−1·m−2; • thermal resistance (R value) = L/k, measured in K·m²·W−1; • thermal transmittance (U value) = 1/(Σ(L/k)) + convection + radiation, measured in W·K−1·m−2. ## Textile industry In textiles, a tog value may be quoted as a measure of thermal resistance in place of a measure in SI units. ## Origins The thermal conductivity of a system is determined by how atoms comprising the system interact. There are no simple, correct expressions for thermal conductivity. There are two different approaches for calculating the thermal conductivity of a system. The first approach employs the Green-Kubo relations. Although this employs analytic expressions which in principle can be solved, in order to calculate the thermal conductivity of a dense fluid or solid using this relation requires the use of molecular dynamics computer simulation. The second approach is based upon the relaxation time approach. Due to the anharmonicity within the crystal potential, the phonons in the system are known to scatter. There are three main mechanisms for scattering (Srivastava, 1990): • Boundary scattering, a phonon hitting the boundary of a system; • Mass defect scattering, a phonon hitting an impurity within the system and scattering; • Phonon-phonon scattering, a phonon breaking into two lower energy phonons or a phonon colliding with another phonon and merging into one higher energy phonon. ## Notes 1. Perry's Chemical Engineers' Handbook, 7th ed., Table 1-4. 2. 2.0 2.1 Thermal Conductivity of some common Materials Retrieved May 26, 2008. 3. Thermal Conductivity of Metals Retrieved May 26, 2008. 4. IEEE guide for soil thermal resistivity measurements Retrieved May 26, 2008. 5. Standard for the Preparation of Test Procedures for the Thermal Evaluation of Solid Electrical Insulating Materials Retrieved May 26, 2008. 6. Standard Test Method for Thermal Transmission Properties of Thermally Conductive Electrical Insulation Materials Retrieved May 26, 2008. 7. Standard Test Method for Thermal Conductivity of Solids by Means of the Guarded-Comparative-Longitudinal Heat Flow Technique Retrieved May 26, 2008. 8. Standard Test Method for Thermal Conductivity of Plastics by Means of a Transient Line-Source Technique Retrieved May 26, 2008. 9. Standard Test Method for Thermal Conductivity of Liquids Retrieved May 26, 2008. 10. Definition of k value from Plastics New Zealand Retrieved May 26, 2008. ## References • Baierlein, Ralph. 2003. Thermal Physics. Cambridge: Cambridge University Press. ISBN 0521658381 • Halliday, David, Robert Resnick, and Jearl Walker. 1997. Fundamentals of Physics, 5th ed. New York: Wiley. ISBN 0471105589 • Serway, Raymond A. and John W. Jewett. 2004. Physics for Scientists and Engineers. Belmont, CA: Thomson-Brooks/Cole. ISBN 0534408427 • Srivastava, G. P. 1990. The Physics of Phonons. Bristol: A. Hilger. ISBN 0852741537 • Young, Hugh D. and Roger A. Freedman. 2003. Physics for Scientists and Engineers. San Fransisco, CA: Pearson. 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http://math.stackexchange.com/questions/61920/if-a-graph-has-no-cycles-of-odd-length-then-it-is-bipartite-is-my-proof-correc	If a graph has no cycles of odd length, then it is bipartite: is my proof correct? I came up with a proof of Graph $G$ has no cycles of odd length $\implies$ $G$ is bipartite. like this: Without loss of generality, let's only consider a connected component, because if every connected component of a graph is bipartite, then the whole graph is bipartite. Pick up a random vertex $v$ in $G$, calculate the length of the shortest simple path from $v$ to any other node, call this value distance from $v$, and divide nodes into 2 groups according to the parity of their distance to $v$. If we can prove that nodes belong to the same group can not be adjacent, then we know that we actually get a partition of the $G$ that fulfill the definition of bipartite graph. Now, to introduce contradiction, assume two nodes $x, y$ with both even or odd distance from $v$ are adjacent, then the shortest simple path $\langle v, x \rangle$, $\langle v, y \rangle$ and edge $\{x, y\}$ contains a cycle with odd length, which is contradictory to that $G$ has no cycles of odd length. In other words, nodes both with even or odd distance from $v$ can not be adjacent, which is exactly what we need. So my question is, is my proof correct? And is there simpler method to prove the proposition? Edit: (to address comment from Srivatsan Narayanan) To prove that $\langle v, x \rangle$ and $\langle v, y \rangle$, together with $\langle x, y \rangle$ contains a cycle with odd length is obvious when $\langle v, x \rangle$ and $\langle v, y\rangle$ are disjoint. When that's not the case, let's give the last node shared by $\langle v, x\rangle$ and $\langle v, y\rangle$ the name $v'$. So the three nodes $v', x, y$ forms a cycle with length $\newcommand{\len}{\operatorname{len}}$ $$L = \len(\langle v', x \rangle) + \len(\langle v', y \rangle) + 1 = \len(\langle v, x \rangle) + \len( \langle v, y \rangle) - 2 \cdot \len(\langle v, v'\rangle) + 1 .$$ where $\len()$ means the length of the shortest path. As $\len(\langle v, x \rangle)$ and $\len(\langle v , y \rangle)$ are both even or odd, then $L$ must be odd. Therefore, in both cases, disjoint or not, $\langle v, x \rangle$, $\langle v, y \rangle$ and $\langle x, y\rangle$ contains a cycle with odd length. Edit2 To see $\len(\langle v, x \rangle) = \len(\langle v, v'\rangle) + \len(\langle v', x \rangle)$, we can simply prove that both $\langle v, v' \rangle$ and $\langle v', x \rangle$ are both shortest path. And that's obvious, because if it's not the case, there exist a path shorter than $\langle v, v' \rangle$ from $v$ to $v'$, or there exist a path shorter than $\langle v', x\rangle$ from $v'$ to $x$, then $\langle v, x \rangle$ can not be a shortest path. - You will need to argue that the shortest paths from $v$ to $x$ and $y$, together with the edge $xy$, forms an odd cycle more carefully. This is clear when the shortest paths are disjoint; what would happen otherwise? (Also, a typo: your very first line should read connected graph, not a path.) – Srivatsan Sep 4 '11 at 22:38 Also: it should be "if every connected component", not "if any connected component" (I would understand the latter as saying that if at least one connected component is bipartite, then the graph is bipartite, that is clearly not what you mean to say). – Arturo Magidin Sep 4 '11 at 22:47 @ablmf: I don't think you are addressing Srivatsan's comment: just saying that the paths from $v$ to $x$, from $v$ to $y$, and the edge $[x,y]$ "contains a cycle of odd length" is not very informative. It's reasonably clear that they contain a cycle, and that if the paths $v\to x$ and $v\to y$ are disjoint, then the cycle will be of odd length; but you have to prove that it contains a cycle of odd length even if the paths are not disjoint, and that is not quite so obvious that you can get away with not saying anything about it. – Arturo Magidin Sep 5 '11 at 1:15 @ablmf I think that should do. One more nitpick: You still must justify that the distances from $v$ to $v'$ along the paths $\langle v, x \rangle$ and $\langle v,y \rangle$ are both equal to $len(v,v')$, the shortest distance from $v$ and $v'$. – Srivatsan Sep 5 '11 at 1:36 Thanks! This is my first proof on math.stackexchange.com Although it's answering my own question. – ablmf Sep 5 '11 at 1:40 I believe the question is resolved to the satisfaction of the OP. See the comments and the revisions to the question for the relevant discussions. Here I present a different, and--in my mind--conceptually cleaner proof of the same fact. Assume $G$ is a connected graph such that all of whose cycles are of even length. We generalize this slightly to the following Proposition. Any closed walk in $G$ has even length. Proof. Towards a contradiction, suppose not. Let $W$ be a closed walk of odd length such that the length of $W$ is as small as possible. By hypothesis, $W$ cannot be a cycle; i.e., $W$ visits some intermediate vertex at least twice. Hence we can write $W$ as the "concatenation" of two non-trivial closed walks $W_1$ and $W_2$, each of which is shorter than $W$. Further, $\len W_1 + \len W_2 = \len W$, which is odd. Thus at least one of $W_1$ and $W_2$ is of odd length, contradicting the minimality of $W$. Thus there cannot be any closed walk in $G$ of odd length. $\quad\quad \Box$ Partitioning the graph into even and odd vertices. Now, fix a vertex $v$, and define $E$ (resp. $O$) be the set of vertices $x$ in $G$ such that there is an even-length (resp. odd-length) walk from $v$ to $x$. The sets $E$ and $O$ partition $V$: • Assuming $G$ is connected, then clearly $E \cup O = V$. • We now show that $E \cap O = \emptyset$. To the contrary, suppose $x$ is in both $E$ and $O$. Then there is a $v$-$x$ walk $W_1$ of even length and another one $W_2$ of odd length. Then the walk $W_1 \circ \operatorname{reverse} (W_2)$ is a closed walk in $G$ of odd length, a contradiction. Finally, we show that every edge crosses the cut $(E, O)$: • Assume $x \in E$ and $xy$ is an edge. Then there exists a $v$-$x$ walk $W$ of even length. Therefore, $W \circ xy$ is a $v$-$y$ walk and it has odd length. Therefore, $y \in O$. • Similarly, if $x \in O$ and $xy$ is an edge, we can show that $y$ is in $E$. This proof is similar to the above case. This establishes that $G$ is bipartite, as desired. -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9053668975830078, "perplexity": 126.95290922486468}, "config": {"markdown_headings": false, "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-2016-30/segments/1469257823670.44/warc/CC-MAIN-20160723071023-00315-ip-10-185-27-174.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/hydrogen-atom.102374/	# Hydrogen atom 1. Dec 2, 2005 ### asdf1 hydrogen atoms in states of high quantum number have been created in the labortatory and observed in space. Find the quantum number of the Bohr orbit in a hydrogen atom whose radius is 0.01mm. my problem: n=(0.00001/(5.29*10^-11))^(1/2)=434.7 i think that n should be 434, because the electron doesn't have enough energy to move up to 435 but the correct answer is 435... 2. Dec 2, 2005 ### alfredblase I'm not sure but I think this may explain it. Orbits in the Bohr model are quantized. If they were not then the answer could be 434.7. Which means that the orbit was measured (inevitably) with some uncertainty as if the Bohr model is correct then it couldn't possibly be of 0.01mm radius. Now if we know that our value of 434.7 must be either 434 or 435, we look to see which "true" value our answer is closest to. And we find the value of n = 435. The point is we are not sure of the exact value of the atoms energy, but we know it is much more likely that our atom is in the 435 orbit than the 434 orbit. Last edited: Dec 2, 2005 3. Dec 3, 2005 ### asdf1 that's logical~ thank you very much!!! :) Similar Discussions: Hydrogen atom	{"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": 0.9601314663887024, "perplexity": 639.7928318838523}, "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-2017-22/segments/1495463612399.20/warc/CC-MAIN-20170529145935-20170529165935-00178.warc.gz"}
http://mathoverflow.net/questions/69815/how-to-motivate-and-interpret-the-geometric-solutions-of-hamilton-jacobi-equatio/69821	# How to motivate and interpret the geometric solutions of Hamilton-Jacobi equation? Studying the Hamilton-Jacobi equation, I meet a generalization of the notion of its solutions, which is found already in the work of Sophus Lie. For an H-J eqn, I mean a first order pde $H\circ dS=0$ in an unknown scalar function $S$ defined on a smooth manifold $M$, where $H\in C^\infty (T^\ast M,\mathbb{R})$. If $S$ is a solution then the image $\Lambda$ of its differential $dS$ is included in $H^{-1}(0)$ and has the following properties: 1. $\Lambda$ is a lagrangian submanifold of $(T^\ast M,d\theta_M)$, 2. $\Lambda$ is transversal to the fibers of $\tau_M^{\ast}:T^\ast M\to M$, 3. the restriction of $\tau_M^{\ast}$ to $\Lambda$ is injective. Conversely, if a submanifold $\Lambda$ of $T^\ast M$, included in $H^{-1}(0)$, satisfies the properties 1, 2, and 3, then it is equal to the image of the differential of a solution, unique up to a constant. But if a submanifold $\Lambda$ of $T^\ast M$, included in $H^{-1}(0)$, satisfies only the conditions 1 and 2, then, around each of its points, it is again equal to the image of the differential of a solution, but this can fail to holds globally. The idea of Sophus Lie was to give up both conditions 2 and 3. Adopting this point of view, we define a generalized (or geometric) solution of $H\cic dS=0$ to be any lagrangian submanifold $\Lambda$ of $(T^\ast M,d\theta_M)$ which is included in $H^{-1}(0)$. I don't think that this generalization is only due to the sake of abstractness. Infact, considering generalized solutions, it is possible, arguing with tecniques from symplectic geometry, to prove the local existence and uniqueness theorem, at the same time, for generalized and usual solutions. But I am hoping to find "more" practical applications which illustrate the meaningfulness of geometric solutions. I would like to learn if ther is some physical or geometrical problem involving an H.-J. eqn, whose comprehension is sensibly augmented by the consideration of generalized solutions. So my question is: What are the possible arguments and applications that motivate and help to interpret the notion of geometric solutions for an Hamilton-Jacobi equation? - @Mathphysicist: I have merged two of your tags in one, hoping to be more descriptive of the content. – Giuseppe Tortorella Jul 9 '11 at 8:23 @Giuseppe: there's really no point for an geometric-theory-of-pdes tag. If you must, you should use the already existing geometric-analysis tag. – Willie Wong Jul 16 '11 at 2:04 A very interesting practical application is the problem of state estimation - for linear systems the answer is called the Kalman filter. Given a vector field $\dot{x} = a(x,v)$ and a measurement equation $y=c(x,w)$, compute the initial condition $x(t_0)$, the perturbation $v(t)$, and the measurement error $w(t)$ that minimize a cost function $J$. The cost is usually expressed as an integral over time of some function of $v$ and $w$. Using Pontryagin's maximum principle or Bellman's dynamic programming, one arrives at a HJ equation which is used to find $v$. The additional step needed is to determine $x(t_0)$. It is a static minimization problem, which however needs to be repeated at each instant $t$ in the interval of interest. This is not a very practical answer. For linear systems with quadratic costs, the Kalman filter provides a recursive solution to the complete problem. In more general cases, the problem is much less studied either by engineers or by mathematicians. This is unlike the optimal control problem which has been studied extensively. I think the geometry of the solutions is crucial. My understanding is that the filter equation is a particular symmetry of the Hamilton-Jacobi-Bellman partial differential equation - at least when everything is smooth. Meanwhile, the Hamiltonian vector field is a characteristic of the partial differential equation - also a particular symmetry, but not the one that gives a recursive solution to the estimation problem. - Dear Pait, I appreciate very much your thoughtful responce. I have given a look at the corresponding sections of the book of Agrachev and Sachkov, but I have not found the lagrangian submanifolds not transversal to the fibers considered as solution generalized for the H.-J. eqn for the optimal cost. Where could I look for such objects in the context of control theory? I would like to learn if considering generalized solutions is possible obtain more information rather than using only usual solution. Thank you. – Giuseppe Tortorella Jul 9 '11 at 7:50 Would you mind helping me with (or pointing to) explanations for the terms "lagrangian submanifolds not transversal to the fibers" and "generalized solutions"? That would help me translate between the two sides of the literature, the mathematical and the engineering. Thanks! – Pait Jul 10 '11 at 21:38 I described this notion already in the text of my question. Please I woulde like to know the points of my question that are not enough clear, or are not written in proper english, so that I could correct them. Thank you in advance. – Giuseppe Tortorella Jul 11 '11 at 13:37 Given the HJ eqn $H\circ dS=0$ in the unknown function $S$ on the smooth manifold $M$, where $H\in C^\infty(T^\ast M)$. A generalized solution is defined to be a submanifold of $T^\ast M$, the cotangent space of $M$, which is included in $H^{−1}(0)$ and lagrangian w.r.t. the canonical symplectic form $dθ_M$. Here $θ_M$ is the tautological, or Liouville, 1-form on $T^\ast M$. – Giuseppe Tortorella Jul 11 '11 at 17:46 I think I wanted a reference, maybe to a book, with a more leisurely explanation. It's not that your text is in any way unclear, it's just that I have a different background and need to do my homework to learn the language better. – Pait Jul 13 '11 at 15:07 The famous KAM tori arose out of HJ considerations. They are Lagrangian torii. They were found by attempting to solve the HJ equation generally, and then finding one can only solve it when certain appropriately irrational frequency conditions hold. They occur in perturbations of integrable systems, or near `typical' linearly stable periodic orbits in a fixed Hamiltonian systems. You can read about them in an Appendix to Arnol'd's Classical Mechanics, and also get some idea from Chris Golé's book 'Symplectic Twist Maps', or from Siegel and Moser's 'Stable and Random Motion'. - As you suspect these generalized solutions and their apparent singularities (=points of the Lagrangian submanifold where condition 2. fails) are unavoidable. First observe that any Lagrangian submanifold contained in $H^{-1}(0)$ must be tangent to the Hamiltonian field $X_H$ (this is the method of characteristics). I assume here that $H^{-1}(0)$ is smoot and $2n-1$ dimensional. Now start with some non-characteristic classical initial data (= an $n-1$ dimensional submanifold in $H^{-1}(0)$ transversal to $\tau^_M$ and transverslat to $X_M$). If you let the initial datum flow with $X_H$ this will swipe out the unique solution in $T^M$. For short times this Lagrangian manifold will be transversal but at some point it can start to bend so that condition 2. fails. The projection to $M$ of points where transversality fails are called caustics in the literature. Here's the classical physics example which you'll find for example in Arnolds books (his PDE course but I think also in his mechanics book): in the particle picture, light particles all move along straight lines with the same speed $c$ in possibly different directions (but they don't interact). An initial data would be given by a surface in the room and a direction field along this surface giving the initial direction of light rays. Initially the light rays don't intersect, but after some times they might start to intersect. The solution S(q) of HJ in this example describes the time after which the wave front arrives at a point q in space. If light rays intersect this function becomes multivalued. By the way I'd be interested in the original source of Lie, could you add that to your question? - Dear Michael, I stated that historical attribution not for having read the original papers of Lie on transformation groups written in the seventies of ninenteenth century, but I learned it from Ch.5 §2.2 "The Geometry of Differential Equations" in "Geometry I, EMS 28" of Alekseevskij, Vinogradov, Lychagin. – Giuseppe Tortorella Jul 9 '11 at 13:41	{"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": 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": 0.9227256178855896, "perplexity": 386.79104889043845}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1409535917663.12/warc/CC-MAIN-20140901014517-00388-ip-10-180-136-8.ec2.internal.warc.gz"}
https://grcs.uwseminars.com/pub/algebra.html	# Algebra This is an online resource for instructors and students. While the material is designed to be taught to strong middle school students, these notes are written for instructors who are invited to guide and discuss topics with their students. ## Fundamentals ### The Integers The first kinds of numbers discovered were the natural numbers: $0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, \dots$ There are (infinitely) many natural numbers, and there are many things that we can do with them. For example, we can add any two natural numbers to obtain another. It does not matter how big the numbers become: their sum is always still a natural number. We may also multiply two natural numbers to obtain another. Again, it does not matter how big or small the numbers are: multiplication is something we can do with any natural numbers. Natural numbers are great for representing “things”. In particular, they're great for representing “some number of things”. But they fall short when we want to represent less than nothing. But you can’t have less than nothing, so why would we ever want that? The desire for a more general kind of number comes from the desire to represent change. If yesterday there were $20$ students in attendance, and today there are $24$ students in attendance, then we can say $4$ more people attended today than did yesterday. But what if tomorrow there will again be $24$ students in attendance? Then how many more students will have attended? Even worse, what if the day after tomorrow, there will be $20$ students in attendance again. Then what is the change in the number of students? It is not that these problems cannot be solved with positive natural numbers. In fact, they certainly can. The number of students could increase by $4$, and we have a number for that. Or the number of students could decrease by $4$. We already have a number for that. So to express a change in a quantity, we must convey two pieces of information: firstly, in what direction is the change; and secondly, by how much the quantity is changed. This seems quite complicated. It would be simpler to introduce a new kind of quantity that could represent these changes more conveniently and more compactly. This, of course, is the motivation for negative numbers. Negative numbers do not “exist” in the real world. We cannot have a negative number of people in a class. But they provide an easy way to talk about change: if positive numbers represent increase, then negative numbers represent decrease. The introduction of negative numbers allows us to expand our set of mathematical objects to the integers, $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$. As with natural numbers, we still have these two useful properties of integers, which we will call closure properties (think of a room withh closed door; if applying these operations inside the room of the integers, you never need to worry about anything outside the room): • The sum of any two integers is again an integer. • The product of any two integers is again an integer. But in addition, we have a third closure property, which enables us to describe changes: • The difference of any two integers is again an integer. #### Exercise 1: Arithmetic with Integers Evaluate each of the following expressions: 1. $1 + (-1) =$ 2. $-1 - (-2) =$ 3. $-3 \times 4 =$ 4. $(-4) \times (-10) =$ ##### Solution 1. $1 + (-1) = \boxed{0}$ 2. $-1 - (-2) = \boxed{1}$ 3. $-3 \times 4 = \boxed{-12}$ 4. $(-4) \times (-10) = \boxed{40}$ ### Variables When we first learn arithmetic, we only have to worry about natural numbers. As we continue to develop new techniques, we find new kinds of numbers — like the integers mentioned above. Let’s take a step back and examine the philosophy behind what numbers really are. We are familiar with everyday objects: things we can see, feel, or otherwise interact with. Mathematical objects are like those, but more abstract. In the mathematical world, you can interact with mathematical objects through statements of fact. A common example of such a statement is equality: $1 + 1 = 2$ is a statement of fact that the mathematical object referred to by $1 + 1$ is in fact the same as the mathematical object referred to by $2$. Numbers are the most common mathematical objects, but as we will see later, they are not the only kind. Typically, we study mathematical objects because of some motivation from a real-world problem. In the real world, we may talk of quantity, location, a process, or any number of other things. To use mathematics to solve these problems, we must translate these real-world concepts into mathematical objects by using a model. Inside the model, we have many mathematical objects. Some of these objects represent the information given to us by the problem. Other objects might represent information we need to figure out — they might not be known from the start! There are three ways we can describe a mathematical object in a simple model: • A literal: for example, just a number, like $123$. • A variable: a letter that we understand the meaning of, but we may or may not know the value of, like $x$. Essentially, we are naming a mathematical object. It can sometimes be useful to name known objects (for example, if they are complicated), but usually we only use variables for unknown objects. • An expression: a combination of literals, variables, and operations, like $x - 123$. For instance, a real world problem is as follows: #### Exercise 2: More Lumber Is Required Yahui and Zhen want to build a wooden treehouse. $100$ planks are required. Yahui has $33$ in her shed, and Zhen has $25$ in his shed. How many additional planks must they buy to complete the treehouse? ##### Solution We can use whole numbers to model this problem. First, let’s give names (variables) to all mathematical objects (numbers) that the problem gives us or requests that we find. Let $y$ be the number of planks that Yahui has, $z$ be the number of planks that Zhen has, $r$ be the total number of planks required, and $x$ be the number of additional planks they must buy. The equation is: $y + z + x = r$ This equation is the governing equation of our model. We are given the values of $y$, $z$, and $r$, and we’re asked to find the value of $x$. (This is not always possible!) As mentioned above, we don’t need to use names for the known values, so we can substitute the literals: $33 + 25 + x = 100$ To find the unknown value we must add to $33 + 25 = 58$ to arrive at $100$, we can use subtraction. That is, $x = 100 - 58 = \boxed{42}$ Another way to understand what we have done in the above step is that $58 + x$ and $100$ refer to the same mathematical object, $100$. Therefore, we can subtract $58$ from both of them, and they will still be different names for the same object. That is, $58 + x - 58 = 100 - 58$. By using the properties of addition that we are familiar with, the left hand side is just another name for $x$ — and so we obtain the result seen above. In this example, all of the variables we used represent specific mathematical objects. Three of them were immediately given to us in the question. The other, $x$, still represented a specific mathematical object, but we had to figure it out. It is not always the case that variables represent specific mathematical objects — sometimes, we can attach a quantifier to a variable, to say that a statement is true of all mathematical objects of a certain type at once. #### Exercise 3: Properties of Whole Number Addition Give a concrete example for each of the following properties of addition of whole numbers: 1. For all integers $a$, $a + 0 = a$. 2. For all integers $a$ and $b$, $a + b = b + a$. 3. For all integers $a$, $b$ and $c$, $(a + b) + c = a + (b + c)$. ##### Solution Solutions may vary. 1. Take $a = 2$. Then $2 + 0 = 2$. 2. Take $a = 3$ and $b = 7$. Then $3 + 7 = 10 = 7 + 3$. 3. Take $a = 1$, $b = 2$ and $c = 3$. Then $(1 + 2) + 3 = 3 + 3 = 6 = 1 + 5 = 1 + (2 + 3)$. Note that these concrete examples are applications of, not justifications for, the properties in question. It can be difficult to give a formal justification for true properties that involve quantifiers. However, if a statement is false, it is often much easier: we can simply give a single concrete example which does not satisfy the statement. #### Exercise 4: Counterexamples Give a counterexample for each of the following incorrect statements about whole numbers: 1. For all integers $a$, $a + a > a$. 2. For all integers $a$ and $b$, $a - b = b - a$. 3. For all integers $a$, $b$ and $c$, $a + b = a + c$. ##### Solution Solutions may vary. 1. Take $a = 0$. Then $a + a = 0 + 0 = 0 \not > 0 = a$. 2. Take $a = 1$ and $b = 0$. Then $a - b = 1 - 0 = 1 \ne -1 = 0 - 1 = b - a$. 3. Take $a = 0$, $b = 0$, and $c = 1$. Then $a + b = 0 + 0 = 0 \ne 1 = 0 + 1 = a + c$. Note that even though each of the statements is false, they all have certain cases where they do hold. In the future, we will see techniques to find out exactly which cases the statement is true in. ## Rational Numbers The motivation behind fractions is similar to that of expanding the whole numbers to the integers. Many things in the real world are divisible into portions. For instance, we can cut a cake into slices. Natural numbers are good at counting wholes, but bad at measuring parts of wholes. To solve this problem, we introduce the concept of splitting a single whole into $n$ equal parts, where $n$ is a positive whole number. (Later, we will allow $n$ to be negative, but it cannot be zero — it is not possible to split something into zero equal parts.) We write this as $\frac{1}{n}$, and call each of the equal parts an $n$th part. (If $n := 10$, then they are called tenths.) We may sometimes require more than one such equal part. If we want $m$ parts where $m$ is any integer, we will write it as $\frac{m}{n}$. All numbers that can be formed from fractions of integers are called rational numbers. We can talk about rational numbers as an extension to the integers, just like how we did with integers above, as an extension to the natural numbers. Each integer is still a rational number: for every integer $n$, and $n = \frac{n}{1}$. There are various simple properties of fraction addition and subtraction, and multiplication, by counting the number of of $n$th parts: 1. For all integers $a$, $b$, and $c$, if $c > 0$, then $\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}$. 2. For all integers $a$, $b$, and $c$, if $c > 0$, then $\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}$. 3. For all integers $a$, $b$, and $c$, if $c > 0$, then $\frac{a}{c} \times b = \frac{ab}{c}$. With the integers, we could not in general perform exact division on any two integers. Fractions cover the case of dividing almost any two integers (almost, since the denominator may not be $0$). But do we still have closure under addition, subtraction, and multiplication? Moreover, do we almost have closure under division (if we disallow a zero divisor)? We only saw some special cases of these operations above. We need to introduce stronger techniques — a general way to do addition, subtraction, and multiplication of fractions, to show that this closure is indeed the case. ### Fraction Multiplication We saw above how to multiply a fraction by an integer. This has the same meaning as multiplying two integers; we can think of it as adding (maybe in the negative direction) repeated copies of the fraction. But it is harder to extend this idea toward multiplying two fractions: what do we mean when we say we want $\frac{1}{2}$ copies of $\frac{1}{3}$? Luckily, there is in fact a single reasonable meaning for this. Recall that $\frac{m}{n}$ means to split a single whole into equal $n$th parts, then take $m$ copies of the $n$th parts. We can replace the single whole with something that is itself an arbitrary rational number: let $q \times \frac{m}{n}$ mean to split $q$ into $n$th parts, and then take $m$ of those. Then, $1 \times \frac{m}{n} = \frac{m}{n}$, as we would expect. How do we split $\frac{a}{c}$ into $n$th parts? We can split each of the $a$ copies of the $b$th parts into $n$ parts, then only take one of every $n$. Each small part is a $b$th part split further into $n$ parts. In a single whole, there would be $c\times n$ equal such parts, so these small parts are in fact $\frac{1}{c\times n}$ each. Therefore, we can justify the following fact (or definition, or a sort) about fraction multiplication: #### Fact 1: Fraction Multiplication Let $a$, $b$, $c$ and $d$ be integers with $c \ne 0 \ne d$. Then $\frac{a}{c} \times \frac{b}{d} = \frac{a \times b}{c \times d}$ ### Equivalence Classes It happens to be the case with fractions that distinct ordered pairs might represent the same quantity. For instance, $\frac{3}{6}$ and $\frac{7}{14}$ are different pairs of numbers, but they represent the same fraction: one half. All the fractions that represent the same particular quantity form a so-called equivalence class. In a sense, this is not very different from $2 - 5$ and $3 - 6$ representing the same change in quantity $-3$. How can we decide whether two fractions represent the same quantity? That is, suppose that $\frac{a}{c}$ and $\frac{b}{d}$ are rational numbers. Are they equal? In the case where the denominator is the same, this is easy to answer: just compare the numerators. Rational numbers with the same denominator are equal if and only if the numerators are equal. #### Fact 2: Equivalence of Fractions With Equal Denominator Let $a$, $b$, and $c$ be integers, and $c \ne 0$. Then $\frac{a}{c} = \frac{b}{c} \iff a = b$ If the denominators are not the same, we need to rewrite the two fractions to have the same denominator. We can do this by first noticing the following fact, which we can obtain from our knowledge of fraction multiplication: #### Fact 3: Common Factor of Numerator and Denominator Let $a$, $b$ and $c$ be integers, and $c \ne 0 \ne b$. Then $\frac{a}{c} = \frac{a}{c} \times 1 = \frac{a}{c} \times \frac{b}{b} = \frac{a \times b}{c \times b}$ This fact allows us to rewrite $\frac{a}{c}$ as $\frac{a\times d}{c\times d}$, and $\frac{b}{d}$ as $\frac{b\times c}{d\times c}$. Now the denominators are the same (remember $c\times d = d\times c$ for any integers $c$ and $d$). So we can simply compare $a\times d$ with $b\times c$! Visually, we are multiplying the top-left with the bottom-right, and the top-right with the bottom-left. This makes a cross shape, so one way to remember this technique is that it is often called “cross-multiplication”. #### Fact 4: Cross-multiplication Let $a$, $b$, $c$ and $d$ be integers, and $c \ne 0 \ne d$. Then $\frac{a}{b} = \frac{c}{d} \iff a\times d = b\times c$ We have seen above the example of rewriting fractions with a common denominator in order to compare them. But another use of fractions with a common denominator is that they are easy to add and subtract. We can apply the same technique: #### Exercise 5: Fraction Addition and Subtraction Rewrite the fractions using a common denominator in order to calculate: 1. $\displaystyle\frac{1}{2} + \frac{1}{3} =$ 2. $\displaystyle\frac{1}{2} - \frac{1}{3} =$ 3. $\displaystyle\frac{1}{3} - \frac{2}{-3} =$ ##### Solution 1. $\displaystyle\frac{1}{2} + \frac{1}{3} = \frac{1\times 3}{2\times 3} + \frac{1\times 2}{3\times 2} = \frac{3}{6} + \frac{2}{6} = \boxed{\frac{5}{6}}$ 2. $\displaystyle\frac{1}{2} - \frac{1}{3} = \frac{1\times 3}{2\times 3} - \frac{1\times 2}{3\times 2} = \frac{3}{6} - \frac{2}{6} = \boxed{\frac{1}{6}}$ 3. $\displaystyle\frac{1}{3} - \frac{2}{-3} = \frac{1}{3} - \frac{2\times(-1)}{-3\times(-1)} = \frac{1}{3} - \frac{-2}{3} = \boxed{\frac{1}{3}}$ In general, we can derive formulas for addition and subtraction of fractions, but you should not memorize them. It is more useful to understand the process of arriving at the formulas. #### Fact 5: Fraction Addition & Subtraction Let $a$, $b$, $c$ and $d$ be integers with $c \ne 0 \ne d$. Then $\frac{a}{c} + \frac{b}{d} = \frac{a \times d}{c \times d} + \frac{b \times c}{d \times c} = \frac{a\times d + b\times c}{c \times d}$ and $\frac{a}{c} - \frac{b}{d} = \frac{a \times d}{c \times d} - \frac{b \times c}{d \times c} = \frac{a\times d - b\times c}{c \times d}$ Notice the resemblance to cross-multiplication. This is not accidental! From the subtraction formula, we see that if two fractions are equal, their difference is zero, and vice versa. ### Simplification Often, given some fraction, we want to find the equivalent fraction with the smallest possible positive integer denominator. This is called the simplest form and is is useful for various reasons: • Smaller positive integer denominators are easier for people to understand. • Two fractions that are equal will have the same simplest form, so fractions in simplest form are easy to compare. Reducing a fraction to simplest form is a matter of finding the largest common factor of both the numerator and denominator, and then dividing both the numerator and denominator by it. #### Exercise 6: Fraction Operations and Simplification Compute each of the following, then reduce it to simplest form. 1. $\displaystyle \frac{3}{8} \times \frac{2}{7} =$ 2. $\displaystyle \frac{5}{9} \times \frac{2}{5} =$ ##### Solution 1. $\displaystyle \frac{3}{8} \times \frac{2}{7} = \frac{6}{56} = \boxed{\frac{3}{28}}$ 2. $\displaystyle \frac{5}{9} \times \frac{2}{5} = \frac{10}{45} = \boxed{\frac{2}{9}}$ #### Exercise 7: A Telescoping Product Compute and reduce to simplest form: $\frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} \times \dots \times \frac{99}{100}$ ##### Solution Each fraction in this product, except for the last one, has a numerator which is the same as the denominator of the following fraction. These will cancel out if we multiply the fractions. For instance, $\frac{1}{2} \times \frac{2}{3} = \frac{1\times 2}{2 \times 3}$, and we can divide $2$ from both the numerator and the denominator to get $\frac{1}{3}$. In this manner, all the numbers except for the first $1$ in the numerator and the last $100$ in the denominator will get cancelled out. So we are left with $\boxed{\frac{1}{100}}$. ## Linear Equations ### Ratios and Rates Frequently, we may know certain quantities not in absolute terms, but only in relative terms. What does this mean? Let’s say you see two weights on the ground, labeled A and B. You might notice that B is twice as hard to lift up as A. Without a scale, it is hard for you to measure the actual weight of the objects, but you might be able to estimate the ratio of their weights. In another example, suppose you are counting cars as they pass by on a highway. You might notice that for every $5$ personal cars you count, you see about one truck. It might be hard for you to estimate how many cars are passing by each minute, since it is hard to guess how long a minute is, but you could estimate the ratio of personal cars to trucks on this highway. We will see a couple of word problems that involve known ratios, and try to determine the absolute quantities using additional information provided to us. #### Exercise 8: Carcross Car Count The community of Carcross, Yukon is quite small, with a population of only 301. Caroline counts the number of cars that passed her house over an hour and noticed that: • There were $15$ cars that passed in total. • All cars were either blue or silver. • Twice as many cars were blue than silver. How many blue cars passed by? How many silver cars? ##### Solution Let $b$ and $s$ be integers representing the number of cars that passed her house. Then our equations are: \begin{aligned} b + s &= 15 \\ b &= 2s \\ \end{aligned} We can substitute the second equation into the first equation, since $2s$ and $b$ refer to the same mathematical object. Thus: \begin{aligned} 2s + s &= 15 \\ 3s &= 15 \\ \textcolor{blue}{\frac{1}{3}} \times 3s &= \textcolor{blue}{\frac{1}{3}} \times 15 \\ s &= 5 \\ \end{aligned} so there were $\boxed{5}$ silver cars. Then we can substitute this back into that second equation $b = 2s$. So $b = 2 \times 5$, and so $b = 10$. Therefore there were $\boxed{10}$ blue cars. #### Exercise 9: Produce Price Sum A supermarket stocks four kinds of produce: apples, oranges, tomatos, and potatos. Apples cost twice as much as oranges, and oranges cost twice as much as tomatos. August bought $1\,\mathrm{kg}$ of each kind of produce, and the total price was $\20$. Can we figure out what was the price of tomatos? If so, what was it? ##### Solution Let $a$, $o$, $t$, and $p$ be rational numbers representing the prices of apples, oranges, tomatos, and potatos, all in dollars per kilogram. Then our equations are \begin{aligned} a &= 2o \\ o &= 2t \\ a + o + t + p &= 20 \\ \end{aligned} If we substitute the first and second equations into the third, we get $4t + 2t + t + p = 20$ and thus $7t + p = 20$. But there is a problem: there are multiple solutions to this! There are even multiple integer solutions; for instance, maybe $t = 2$ and $p = 6$, or $t = 1$ and $p = 13$. So we do not have enough information to figure out the price of tomatos. #### Exercise 10: An Unlikely Sprint? Miran, Gosse, and Brayan participated in a $100\,\mathrm{m}$ sprint. Miran tells you that she won and was twice as fast as Brayan. Gosse agrees that Miran won, and says he was close behind with a time only $20\%$ higher than Miran’s. Brayan says that he came in last with a time $8\,\mathrm{s}$ longer than Gosse’s time. You know, however, that sometimes Miran, Gosse, and Brayan aren’t the most reliable. Is it mathematically possible for all of these accounts to be accurate? If so, do we have enough information to determine what were each of their times? If so, calculate the times. ##### Solution Let $x$ denote Miran’s time, $y$ denote Gosse’s time, and $z$ denote Brayan’s time. Based on what everyone said, the equations are: \begin{aligned} x &= \frac{1}{2} z \\ y &= 1.2 x \\ z &= y + 8\,\mathrm{s} \\ \end{aligned} Substitute the expression for $x$ given by first equation into the second equation, to get $y = 1.2 \times \textcolor{blue}{\frac{1}{2} z} = \frac{3}{5} z$ Now substitute this into the third equation, to get \begin{aligned} z &= \textcolor{blue}{\frac{3}{5} z} + 8\,\mathrm{s} \\ \textcolor{blue}{-\frac{3}{5} z} + z &= \textcolor{blue}{-\frac{3}{5} z} + \frac{3}{5} z + 8\,\mathrm{s} \\ \frac{2}{5} z &= 8\,\mathrm{s} \\ \textcolor{blue}{\frac{5}{2}} \times \frac{2}{5} z &= \textcolor{blue}{\frac{5}{2}} \times 8\,\mathrm{s} \\ z &= 20\,\mathrm{s} \\ \end{aligned} We can now substitute this back into $y = \frac{3}{5} z$ to get $y = 12\,\mathrm{s}$, and into $x = \frac{1}{2} z$ to get $x = 10\,\mathrm{s}$ (very fast indeed, maybe suspiciously so!). We can check that these times match all three of the equations above, so it is mathematically possible and unique. This doesn’t mean that the statements were accurate, but they are not mathematically contradictory. #### Exercise 11: Raccoon Population Growth The number of raccoons in the city of Raccoonville is plotted on the following chart: If the current trend continues, by what year will there be 180 raccoons in Raccoonville? ##### Solution In this problem, we have to figure out the rate of increase of raccoons from the chart. The trend seems to be a straight line with an increase of $10$ raccoons every year. We can assume this trend will continue as the question asks us in that hypothetical. Let $x$ denote the number of years after $2019$. Then the number of raccoons will be $130 + 10x$. We want to solve: \begin{aligned} 130 + 10x &= 180 \\ \textcolor{blue}{-130} + 130 + 10x &= \textcolor{blue}{-130} + 180 \\ 10x &= 50 \\ \frac{10x}{\textcolor{blue}{10}} &= \frac{50}{\textcolor{blue}{10}} \\ x &= 5 \\ \end{aligned} Since this means $5$ years after $2019$, the year that there will be $180$ raccoons in Raccoonville is $2019 + 5 = \boxed{2024}$. ### A General Approach The examples above all have the same general form, where we have a number of equations of the form $ax = b$, where $a$, $b$, and $x$ are rational numbers, and we know $a$ and $b$ (but not $x$). Equations of this form are called “linear equations”. Why are they linear? Intuitively, one reason is that if we draw a line graph of the value of $ax$ as we increase the value of $x$, we will find a straight line: The solution to $ax=b$, if one exists, is simply where this straight line reaches a vertical height of $b$. We saw a general technique to do this if $a\ne 0$: we can multiply both sides by $\frac{1}{a}$ (or equivalently, divide both sides by $a$). Thus the solution is $x = \frac{b}{a}$. But what if $a = 0$? In this case, we cannot divide by $a$, since division by $0$ is meaningless. Instead, we have the flat blue line in the graph. Obviously, this line will never reach any vertical height except $0$! Therefore, there is no solution if $b \ne 0$. If $b = 0$, then we still have no information about $x$: any rational number will do. In this case, there are multiple solutions. ## Real Numbers ### Exponents Recall that a positive exponent represents repeated multiplication, much like how a positive multiplier represents repeated addition. We can express this rule recursively using the following identity: $x^{n+1} = xx^n$ which says that if you increase the exponent by $1$ it is the same as multiply one more copy of the base. #### Exercise 12: Positive Integer Exponents Evaluate each expression. Write your answer as an integer in literal form. 1. $2^4=$ 2. $3^2=$ 3. $10^6=$ ##### Solution 1. $2^4=2\times 2\times 2\times 2=\boxed{16}$ 2. $3^2=3\times 3=\boxed{9}$ 3. $10^6=10\times 10 \times 10\times 10\times 10\times 10=\boxed{1000000}$ There are a variety of facts about positive integer exponents that we can justify using the properties of multiplication. Here are a few. You do not need to memorize these, but it is helpful to understand why they are true. #### Fact 6: Sum of Exponents If $a$ is a rational number, and $n$ and $m$ are positive integers, then $a^m \times a^n = a^{m+n}$. That is: $a^m \times a^n = \underbrace{a\times\dots\times a}_{m\text{ times}} \times \underbrace{a\times\dots\times a}_{n\text{ times}} = \underbrace{a\times\dots\times a}_{m+n\text{ times}} = a^{m+n}$ (Notice how when $n=1$, this is just the recursive rule we discussed above.) #### Fact 7: Product of Exponents If $a$ is a rational number, and $n$ and $m$ are positive integers, then ${(a^m)}^n = a^{mn}$. That is: ${(a^m)}^n = \underbrace{\left(\underbrace{a\times\dots\times a}_{m\text{ times}}\right) \times \dots \times \left(\underbrace{a\times\dots\times a}_{m\text{ times}}\right)}_{n\text{ times}} = \underbrace{a\times\dots\times a}_{mn\text{ times}} = a^{mn}$ It is frequently useful to extend the system of exponents to non-positive numbers, which can be done by applying the recursive rule in the other direction. Thus we can derive that $x^0 = 1$ and that $x^{-1} = \frac{1}{x}$ for all non-zero values of $x$. We can check that this extension retains the sum and product rules of exponents that we mentioned above, which is a useful feature. #### Exercise 13: Negative and Zero Exponents Evaluate each expression. Write your answer in simplest form as a fraction or as an integer literal. 1. ${-1}^{-1}=\boxed{-1}$ 2. $4^{-2}=\boxed{\frac{1}{16}}$ 3. ${999}^0=\boxed{1}$ 4. ${\left(\frac{-17}{4}\right)}^0=\boxed{1}$ A useful application of exponents is in shrinking large numbers to an more humanly understandable format. Indeed, we have a poor conception of how large certain numbers are. In science, it's common to see numbers way too large to count or way too small to visualize. Scientists have developed notation using exponents to make comparing such numbers easier. In scientific notation, a number $x$ is written as $y\times 10^n$, where $y$ is a number with exactly one non-zero decimal digit before the decimal point, and $n$ is a (positive, negative, or zero) exponent. #### Exercise 14: Scientific Notation Express in scientific notation. 1. $1234 = \boxed{1.234\times 10^3}$ 2. $0.000987 = \boxed{9.87\times 10^{-4}}$ A natural question to ask after having defined negative exponents is: what about rational exponents? Could those be useful? In fact, for a positive rational number base, we may sometimes define rational exponents in a way that preserves both the sum and product laws of exponents mentioned above. The only way to do this is to ensure that ${\left(x^{\frac{1}{n}}\right)}^n = x$, that is, $x^{\frac{1}{n}}$ must be the $n$th root of $x$. We can also write that as $\sqrt[n]{x}$. With this definition and the product law, we can define $x^q$ for any positive rational base $x$ and any rational exponent $q$. #### Exercise 15: Fractions, Exponents & Radicals Evaluate each expression. Write your answer in simplest form as a fraction, or as an integer using the place value system. 1. $4^{\frac{1}{2}}=\boxed{2}$ 2. $9^{\frac{3}{2}}=\boxed{27}$ 3. ${\left(\frac{2}{3}\right)}^3=\boxed{\frac{8}{27}}$ 4. $\sqrt{\frac{16}{25}}=\boxed{\frac{4}{5}}$ 5. $\sqrt[4]{\frac{256}{81}}=\boxed{\frac{4}{3}}$ With integer exponents of rational numbers, we are always guaranteed that the result exists and is a rational number (since we compute these exponents by multiplying and dividing rational numbers, which are closed under these operations). As we will see later, when rational exponents are concerned, the result may not exist as a rational number. ### Definition of a Real Number In previous sections, we were careful to only discuss square roots and other rational exponents when we knew that there was in fact a rational number that worked. In general, we cannot assume this is always the case. We can give a proof that no rational number is equal to $\sqrt{3}$. One way to see this is a so-called proof by contradiction. In this kind of argument, we assume that $\sqrt{3}$ is in fact rational. That is, if $\sqrt{3} = \frac{p}{q}$ for some integers $p$ and $q\ne 0$ in simplest form, then $\frac{p^2}{q^2} = 3$, so $p^2 = 3q^2$. We see that $3$ is a factor of the right hand side, so it must also be a factor of the left hand side. But the left hand side is a square, so $p=3m$ for some integer $n$. Then $9k^2 = 3q^2$ so $3m^2 = q^2$. Now $3$ is a factor of the left hand side, so it should also be a factor of the right hand side. But the right hand side is a square, so $q=3n$ for some integer $n$. But then $\frac{p}{q} = \frac{3m}{3n}$ is clearly not in simplest form, so we have reached an absurd state — a contradiction. But of course, this is not possible, so something has gone wrong. Our argument is correct, so it must be our assumption that was wrong. The assumption we made was that $\sqrt{3}$ is a rational number. Many of you will probably be uneasy with how we are talking about $\sqrt{3}$ as if it must exist, when we have already shown that no rational number squares to $3$. We have already complicated things by introducing fractions to the easier world of the integers! If we make the claim that $\sqrt{3}$ should exist as a number, then we run the risk of making things even more complicated and difficult. We don’t, in general, say that everything which doesn’t exist must be a new kind of number (we are happy to say that $\frac{1}{0}$ simply does not exist). There is in fact a good reason, however, to suggest that $\sqrt{3}$ might be a useful number to have. The reason for this is that we can already get very very close to a potential square root of $3$! An example of this is the rational number $\frac{3900231685776981}{2251799813685248} \approx 2.9999999999999997$. In fact, we can get arbitrarily close. A way to visualize this is to see a graph that maps numbers to their squares. We can start by plotting a point on some axes for integer values. The horizontal distance represents the number $x$, which we vary to take on the integer values we want to show. The vertical distance represents the square of that number, $x^2$. Of course, we can also take the square of rational numbers. We can think of this as increasing the precision of our graph by plotting more points, for example, every $0.1$. If we imagine that we continue this process, getting more and more precision, we would expect this curve to become continuous. We can see that it reaches a vertical value of $3$ at some point, and we saw earlier that no such rational number point exists. But the curve suggests that we can define a new kind of number that is on the number line, and we can get close to using rational numbers, but can’t get exactly there. This concept is called a real number. There are many ways to formally define a real number, but it is not necessary to understand such a definition to understand what a real number is. Intuitively, we expect real numbers to plug the holes in continuous lines that we can draw. We can get as close as we want to a real number using rational numbers. In fact we don’t even need to use all rational numbers. All finite decimals are rational numbers, and by adding more decimal points, we can get closer and closer to any rereal number we are interested in. $\sqrt{3} \approx 1.73$, but an even better approximation is $\sqrt{3} \approx 1.732$. We can keep getting more and more precise, but we can never reach the number itself because it is not rational (hence not a decimal). ### Operations on Real Numbers We are able to add, subtract, multiply, and divide (except by zero) real numbers, just as we can with rational numbers. (Remember that we can get as close as we want to a real number using rational numbers. So it makes sense that real numbers behave almost identically to rational numbers!) However, unlike rational numbers, it is not always possible to write real numbers in a canonical simplest form. Instead, we can use algebraic techniques to make expressions look simpler from a human perspective. #### Exercise 16: A Linear Equation using Real Numbers Solve the following equation for real number $x$: $\sqrt{2} x = 2$ ##### Solution We divide both sides by $\sqrt{2}$, yielding: $\frac{\sqrt{2}}{\sqrt{2}} x = \frac{2}{\sqrt{2}}$ Then, simplifying, $x = \frac{2}{\sqrt{2}} = 2\cdot \frac{1}{\sqrt{2}} = 2\cdot \frac{1}{2^{\frac{1}{2}}} = 2^2\cdot 2^{-\frac{1}{2}} = 2^{1 - \frac{1}{2}} = 2^{\frac{1}{2}} = \boxed{\sqrt{2}}$ At this stage, it is helpful to understand some supplementary material on sets. This material is on a seperate page because it is not strictly related to what we are studying right now about algebra, but the notational conveniences of the material will be useful. ## Polynomials ### Linear Equations, Again Recall earlier when we solved linear equations of the form $ax = b$, where $a$ and $b$ are known rational (or real) numbers, and $x$ is an unknown rational (or real) number. The solution is to divide both sides of the equation by $a$, which is valid because $ax$ and $b$ are different names for the same rational (or real) number. This results in the solution $x = \frac{b}{a}$. Note that many equations that may look different are really linear equations after some rearranging. For example, $ax + b = 0$ can be rearranged into the linear equation $ax = -b$. $ax + b = c$ can be rearranged into the linear equation $ax = c - b$. We will say that a linear equation that looks like $ax + b = 0$ is in “standard form”. Let’s now look at a graphical method to solve a linear equation in standard form. What we will do is rewrite the right hand of the equation from $0$ to another variable $y$. We will then draw a graph, similar to what we did earlier when we introduced real numbers. Let us first consider the linear equation $2x - 6 = 0$. Note that the equation $2x - 6 = y$ is more general than $2x - 6 = 0$. If we set $y := 0$, then we get back the original equation $2x - 6 = 0$. Therefore, to solve this equation we can look on the graphical plot for all the values on the line corresponding to $y = 0$. We see that this is where the line intersects with the x-axis. This is called an x-intercept, or root, of the function $y = 2x - 6$. From the plot, we see that the only root is $x = 3$, which corresponds to the only solution to this equation. We will now use the knowledge about plots and roots to solve equations which are not linear. Let us start with a simple example. #### Exercise 17: Roots of $x^2 - 1$ Plot $y = x^2 - 1$, determine its roots, and use this information to solve the equation $x^2 = 1$. ##### Solution Here is our plot: The roots are marked. They are $x = -1$ and $x = 1$, which correspond to the solutions to our equation $x^2 = 1$. #### Exercise 18: Roots of $(x-3)(x+2)$ Without using a plot, determine the roots of $y = (x-3)(x+2)$. These are also solutions to the standard form equation $x^2 - x - 6 = 0$; explain why. ##### Solution The roots occur when $y = 0$, so we want to solve $0 = (x-3)(x+2)$. If the product of two numbers is $0$, that means either the first number is $0$ or the second number is $0$ (or both). Therefore, the set of solutions to the equation $0 = (x-3)(x+2)$ is the union of the set of solutions to $x - 3 = 0$ and the set of solutions to $x + 2 = 0$. If $x - 3 = 0$, this is a simple linear equation, where $x = 3$ is the only solution. If $x + 2 = 0$, this is also a simple linear equation, where $x = -2$ is the only solution. Therefore, the roots are $x \in \{-2, 3\}$. Note that by using the distributive property, we find that $(x-3)(x+2) = (x-3)x + (x-3) \cdot 2 = x^2 - 3x + 2x - 6 = x^2 - x - 6$. So in fact, $x^2 - x - 6 = (x-3)(x+2)$ for all values of $x$. Hence the solutions of the two equations must be the same! Exercise 18 suggests a general approach to solving equations that involve $x$ and $x^2$ might be to decompose it into the product of two components which are both linear. This process is called factoring. An expression of the form $ax^2 + bx + c$, where $a\ne 0$, $b$, and $c$ are known, is called a quadratic polynomial, just as $ax + b$ where $a\ne 0$ and $b$ are known is called a linear polynomial. Suppose we have $(sx - u)(tx - v) = 0$, where $s$, $t$, $u$, and $v$ are known real numbers with $s \ne 0 \ne t$. Then we know the solutions are $x \in \left\{\frac{u}{s}, \frac{v}{t}\right\}$. A quadratic polynomial written this way is easy to solve! Our goal is to take a polynomial in standard form, and convert it into this factored form. It is easier to go backwards. From factored form, we can use distributivity to expand: $(sx - u)(tx - v) = st x^2 - (sv + tu)x + uv$. But what we want to figure out is how to turn standard form into factored form. If we can write a polynomial in standard form to look like $stx^2 - (sv + tu)x + uv$, then we have found the solutions! The standard form is $ax^2 + bx + c$, $a \ne 0$, so we need: $a = st$, $b = - (sv + tu)$, $c = uv$. This is not easy to solve (in fact, it does not have a unique solution), so we need to make some simplifications first. First of all, we need to to fix the fact that the solutions are not unique. We can factor, for example, $2x^2 - 2 = (2x-2)(x+1)$, but we can also factor it as $2x^2 - 2 = (x-1)(2x+2)$. The roots are of course the same, because the equations $x+1 = 0$ and $2x+2 = 0$ have the same solutions. But they do not quite look the same. In order to force the factored form to be unique, we need to enforce that the linear polynomials are monic, that is, they have no leading coefficient (multiplier for the $x$ term). Instead, we will pull out those coefficients into a single multiplier for the entire quadratic polynomial. That is, given $(sx - u)(tx - v)$, we would like to turn this into $a(x-u')(x-v')$, where $u'$ and $v'$ are new coefficients. How do we calculate $a$, $u'$, and $v'$? Let’s focus on the two linear polynomials seperately. We know that $sx - u = s(x - \frac{u}{s})$, by distributivity. Similarly, $tx - v = t(x - \frac{v}{t})$. Therefore: $(sx - u)(tx - v) = s(x - \frac{u}{s})t(x - \frac{v}{t}) = st(x - \frac{u}{s})(x - \frac{v}{t})$ Therefore, we can set $a := st$ and $u' := \frac{u}{s}$ and $v' := \frac{v}{s}$. Note that the expansion of $a(x-u')(x-v')$ into standard form is $ax^2 - a(u' + v')x + au'v'$. This is not actually any different from what we had before, but it looks somewhat closer to what we need! We also now see why we have reused $a$ for the leading coefficient in both forms — in fact, this leading coefficient will be the same going from monic factored form to standard form. Again, we have not really made much progress — the goal is not to go from a factored form to standard form, but actually the opposite! But it turns out the new system of equations is easier to solve. We need: $b = -a(u' + v')$, $c = au'v'$. Another way to write this is $\frac{b}{a} = -(u' + v')$, and $\frac{c}{a} = u'v'$. That is: after dividing by the leading coefficient, we want the constant coefficient to be the product of the solutions, and the $x$ coefficient to be the negative sum of the solutions. We can solve this problem with trial and error. #### Exercise 19: Factoring a Quadratic via Trial and Error Using trial and error, factor $25 + 5x - 6x^2$. ##### Solution In standard form, this is $-6x^2 + 5x + 25$. We can divide by the leading coefficient to obtain $6 (x^2 - \frac{5}{6}x - \frac{25}{6})$. We want two numbers whose sum is $\frac{5}{6}$, and whose product is $-\frac{25}{6}$. One of them will need to be negative! Let’s guess that $u'$ and $v'$ will be rational numbers. Write them as $\frac{m}{p}$ and $\frac{n}{q}$. We would like $mn = -25$ and $pq = 6$. There are essentially two options for $p$ and $q$, such as $1$ and $6$, or $2$ and $3$ (all other options involve negatives or just switching around $p$ and $q$). There are then six options for $m$: $-25, -5, -1, 1, 5, 25$. The value of $n$ corresponding to each of those options would be $1, 5, 25, -25, -5, -1$. Now we just need to guess and check! • If $p=1$, $q=6$, $m=-25$, $n=1$, then the sum is $\frac{-25}{1} + \frac{1}{6} = \frac{-149}{6} \ne \frac{5}{6}$. • If $p=1$, $q=6$, $m=-5$, $n=5$, then the sum is $\frac{-5}{1} + \frac{5}{6} = \frac{-25}{6} \ne \frac{5}{6}$. • If $p=1$, $q=6$, $m=-1$, $n=25$, then the sum is $\frac{-1}{1} + \frac{25}{6} = \frac{19}{6} \ne \frac{5}{6}$. • If $p=1$, $q=6$, $m=1$, $n=-25$, then the sum is $\frac{1}{1} + \frac{-25}{6} = \frac{-19}{6} \ne \frac{5}{6}$. • If $p=1$, $q=6$, $m=5$, $n=-5$, then the sum is $\frac{5}{1} + \frac{-5}{6} = \frac{25}{6} \ne \frac{5}{6}$. • If $p=1$, $q=6$, $m=25$, $n=-1$, then the sum is $\frac{25}{1} + \frac{-1}{6} = \frac{149}{6} \ne \frac{5}{6}$. • If $p=2$, $q=3$, $m=-25$, $n=1$, then the sum is $\frac{-25}{2} + \frac{1}{3} = \frac{-73}{6} \ne \frac{5}{6}$. • If $p=2$, $q=3$, $m=-5$, $n=5$, then the sum is $\frac{-5}{2} + \frac{5}{3} = \frac{-5}{6} \ne \frac{5}{6}$. • If $p=2$, $q=3$, $m=-1$, $n=25$, then the sum is $\frac{-1}{2} + \frac{25}{3} = \frac{47}{6} \ne \frac{5}{6}$. • If $p=2$, $q=3$, $m=1$, $n=-25$, then the sum is $\frac{1}{2} + \frac{-25}{3} = \frac{-47}{6} \ne \frac{5}{6}$. • If $p=2$, $q=3$, $m=5$, $n=-5$, then the sum is $\frac{5}{2} + \frac{-5}{3} = \frac{5}{6}$ — we’re finally done! So the factored form is $-6 (x - \frac{5}{2})(x + \frac{5}{3})$. You might have noticed that this exercise was really tedious. In fact, we will find that there is a more direct way to figure out the numbers we want. Nevertheless, this primitive method can be useful sometimes when the solutions are less difficult to find. We will introduce an additional form, besides standard form and factored form, for a quadratic polynomial. This new form will be called vertex form. The motivation for vertex form is that the graph of any quadratic polynomial will look like a parabola, either opening upwards or downwards based on the leading coefficient. All parabolas have either a minimum or a maximum $y$-value, which is attained at exactly one point. This point is called the vertex. The vertex form of a quadratic polynomial is: $a(x - x_0)^2 + y_0$, where $x_0$ and $y_0$ are real numbers such that $(x_0, y_0)$ are the coordinates of the vertex. #### Exercise 20: Vertex Form to Standard Form Rewrite $4(x + 1)^2 - 9$ in standard form. ##### Solution We expand the square using distributivity: \begin{aligned} 4(x+1)^2 - 9 &= 4(x+1)(x+1) - 9 \\ &= 4(x^2 + 2x + 1) - 9 \\ &= 4x^2 + 8x + 4 - 9 \\ &= 4x^2 + 8x - 5 \end{aligned} which is in standard form. #### Exercise 21: Solving a Quadratic in Vertex Form Solve $4(x + 1)^2 - 9 = 0$. First, add $9$ to both sides: $4(x+1)^2 = 9$ Then, divide both sides by $2$: $(x+1)^2 = \frac{9}{4}$ We know that there are two possibilities for $x+1$, i.e. $x + 1 \in \left\{\frac{-3}{2}, \frac{3}{2}\right\}$ Therefore, the two possibilities for $x$ are $x \in \left\{\frac{-5}{2}, \frac{1}{2}\right\}$ Therefore, we can solve quadratic equations if we put them in vertex form. Based on whether the parabola opens upwards or downwards, and where the vertex is relative to the $x$-axis, the equation will either have $0$, $1$, or $2$ real solutions. Now the question becomes: given a quadratic polynomial in standard form, can we put it in vertex form? Yes, we can! We just need to use distributivity in a clever way. We know that $(x - x_0)^2 = x^2 - 2x_0 x + {x_0}^2$. Therefore, we want to get something which looks like this by manipulating the standard form expression. Let’s start with a concrete example: \begin{aligned} 4x^2 + 8x - 5 &= 4\left(x^2 + 2x\right) - 5 && \text{so we need }x_0 = 1 \\ &= 4\left(x^2 + 2x + \textcolor{blue}{1 - 1}\right) - 5 \\ &= 4\left(x^2 + 2x + 1\right) - 4 - 5 \\ &= 4{(x + 1)}^2 - 9 \end{aligned} So in fact, it is possible to rewrite a standard form quadratic polynomial into vertex form, and this time we did not need any trial and error. In general: \begin{aligned} ax^2 + bx + c &= a\left(x^2 + \frac{b}{a}x\right) + c \\ &= a\left(x^2 + 2\frac{b}{2a}x + \textcolor{blue}{\frac{b^2}{4a^2} - \frac{b^2}{4a^2}}\right) + c \\ &= a\left(x^2 + 2\frac{b}{2a}x + \frac{b^2}{4a^2}\right) - \frac{b^2}{4a} + c \\ &= a{\left(x + \frac{b}{2a}\right)}^2 - \frac{b^2}{4a} + c \\ &= a{\left(x - \frac{-b}{2a}\right)}^2 - \frac{b^2 - 4ac}{4a} \end{aligned} We can furthermore find the roots of this quadratic polynomial in vertex form, using the method above: \begin{aligned} ax^2 + bx + c &= 0 \\ a{\left(x - \frac{-b}{2a}\right)}^2 - \frac{b^2 - 4ac}{4a} &= 0 \\ a{\left(x - \frac{-b}{2a}\right)}^2 &= \frac{b^2 - 4ac}{4a} \\ {\left(x - \frac{-b}{2a}\right)}^2 &= \frac{b^2 - 4ac}{4a^2} \\ x - \frac{-b}{2a} &= \pm \frac{\sqrt{b^2 - 4ac}}{2a} \\ x &= \frac{-b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a} \\ x &= \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \\ \end{aligned} This formula gives us a way to find the solutions of any quadratic polynomial in standard form. We will call it the quadratic formula. ### Polynomials Let us first define two terms that will be useful. A monomial is a whole number power of a variable $x$ multiplied by some coefficient. For example, the following are monomials: • $x$ • $3x^2$ • $7$ • $-\frac{x^{10}}{4}$ The following are not monomials: • $2^x$ • $x + 2$ • $x^{-1}$ A polynomial in a single variable $x$ is an expression that involves the sum of some monomials (maybe just one). All monomials are also polynomials. For example, the following are polynomials: • $0$ • $x + 2$ • $8 + x^2 - 4x^7$ The following are not polynomials: • $\frac{x+1}{x-1}$ • $1 + 3^x$ A polynomial equation in a single variable $x$ is an equation that has polynomials on the left and right hand sides. For example, the following are polynomial equations: • $7x = 3$ • $1 - x - x^2 = \frac{1}{5}$ • $x^2 = x^7$ In particular, all quadratic equations and linear equations (which we saw in the previous few weeks) are also polynomial equations. The degree of a polynomial is the highest exponent (with a non-zero coefficient). For instance, the following polynomials correspond to degrees: • $0x^2 + x$ → 1 • $5$ → 0 • $x^{99} - x^{199}$ → 199 We will define the degree of $0$ to be $-∞$, because there is no term with a a non-zero coefficient. • $0$ → 1 The leading coefficient is the coefficient of the highest exponent term. The constant term is the coefficient of the $x^0$ term, i.e. the monomial which does not depend on $x$ (hence, constant). • $0x^2 + x$ → Leading coefficient: $1$, constant term: $0$ • $5$ → Leading coefficient: $5$, coconstant term: $5$ • $x^{99} - x^{199}$ → Leading coeffcient: $-1$, coconstant term: $0$ ### Factoring Polynomials We have already seen how to solve polynomial equations of degrees $1$ and $2$ (they are linear equations and quadratic equations respectively). To solve polynomial equations with higher degrees, we would ideally want to write the equation in factored form (just like what we did with quadratic equations). There are a few tools we can use to do this. The first tool we will need to learn is called long division. The idea behind this technique is that if we know one factor of a polynomial, we can get the other factor. This is similar to the concept of division for rational numbers that we are familiar with. In fact, the technique is exactly the same, except that we use the exponents of the variables instead of place values. The idea is to consider only the leading coefficient (coefficients corresponding to the highest degree monomial) each time. Let us do some examples of long division. #### Exercise 22: Examples of Polynomial Long Division Use polynomial long division to compute the following quotients. 1. $\frac{x^2 - 1}{x + 1}$ 2. $\frac{x^3 - 27}{x^2 + 3x + 9}$ ##### Solution 1. We write \begin{aligned} x^2 - 1 &= \textcolor{blue}{x(x + 1)} + x^2 - 1 - \textcolor{red}{(x^2 + x)} \\ &= x(x + 1) - x - 1 \\ &= x(x + 1) - \textcolor{blue}{1(x + 1)} - x - 1 - \textcolor{red}{(- x - 1)} \\ &= x(x + 1) - 1(x + 1) \\ &= (x - 1)(x + 1) \\ \end{aligned} and so $\frac{x^2 - 1}{x + 1} = x - 1$. 2. We write \begin{aligned} x^3 - 27 &= \textcolor{blue}{x(x^2 + 3x + 9)} + x^3 - 27 - \textcolor{red}{(x^3 + 3x^2 + 9x)} \\ &= x(x^2 + 3x + 9) - 3x^2 - 9x - 27 \\ &= x(x^2 + 3x + 9) - \textcolor{blue}{3(x^2 + 3x + 9)} - 3x^2 - 9x - 27 - \textcolor{red}{(- 3x^2 - 9x - 27)} \\ &= x(x^2 + 3x + 9) - 3(x^2 + 3x + 9) \\ &= (x - 3)(x^2 + 3x + 9) \\ \end{aligned} and so $\frac{x^3 - 27}{x^2 + 3x + 9} = x - 3$. We now know what to do when we have a factor of the polynomial. How do we figure out The next tool that will be useful is called the remainder theorem. In fact, the remainder theorem is more general (it tells us more) than the version we will look at, but the version we will learn is enough for our purposes. The remainder theorem states that if $p(x)$ is a polynomial, and $a$ is some number, then $p(a) = 0$ if and only if $(x - a)$ is a factor of $p(x)$. This means that if we can guess a root of a polynomial, then we can find at least one factor of it. Let us look at a few examples. #### Exercise 23: Applications of the Remainder Theorem Find a single factor of each polynomial by using the remainder theorem. 1. $x^7 + 1$ 2. $x^4 + x^3 + x^2 + x$ 3. $10x^{10} - 9x - 1$ ##### Solution 1. ${(-1)}^7 + 1 = 0$, so by the remainder theorem, $x + 1$ is a factor. 2. $0^4 + 0^3 + 0^2 + 0 = 0$, so by the remainder theorem, $x$ is a factor. 3. $10\cdot 1^{10} - 9\cdot 1 - 1 = 0$, so by the remainder theorem, $x - 1$ is a factor. By combining the remainder theorem with the technique of long division, we can factor some polynomials for which we can easily guess a root. But we will want yet a third tool to make guessing roots easier, if they are rational. This third tool is called the rational root theorem. It allows us to use trial and error to guess all the possible rational roots; if none of them work, then we know there are no rational roots. (Recall that we did something similar to find the rational roots of a quadratic equation.) The rational root theorem states that if we have a polynomial $p(x)$ with integer coefficients, and the leading coefficient is $a$ while the constant term is $c$, then any rational root $q = \frac{m}{n}$ (in lowest form) must satisfy the following: $m$ is a factor of $c$, and $n$ is a factor of $a$. As a special case, if $a = 1$ (so that t	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 612, "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": 0.995243489742279, "perplexity": 223.40784437596284}, "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/1664030334855.91/warc/CC-MAIN-20220926082131-20220926112131-00760.warc.gz"}
http://www.cut-the-knot.org/Curriculum/Geometry/GeoGebra/TriangleThreeCirclesEllipse.shtml	# Ellipse Touching Sides of Triangle at Midpoints Cõ Gẫng Lên suggested in a post at the CutTheKnotMath facebook page that the six points on the sides of a triangle equidistant from the respective midpoints lie on an ellipse. The statement below goes a step further, not requiring that all six distances be equal, but only for the points on the same side. Assume $D,E,F$ are the midpoints of sides $BC,AC,AB$ of $\Delta ABC$. Points $G,H$ on $BC$; $I,J$ on $AC$, and $K,L$ on $AB$ are such that $DG=DH$, $EI=EJ$, $FK=FL$. There is a unique ellipse through the six points $G,H,I,J,K,L.$ When the points coincide pairwise, the ellipse passes through and is tangent to the midpoints of the $\Delta ABC.$ (The applet below illustrates the configuration.) Proof Assume $D,E,F$ are the midpoints of sides $BC,AC,AB$ of $\Delta ABC$. Points $G,H$ on $BC$; $I,J$ on $AC$, and $K,L$ on $AB$ are such that $DG=DH$, $EI=EJ$, $FK=FL$. There is a unique ellipse through the six points $G,H,I,J,K,L.$ When the points coincide pairwise, the ellipse passes through and is tangent to the midpoints of the $\Delta ABC.$ ### Proof For a proof, observe that affine transformations do not change the configuration: midpoints remain midpoints (of the sides of the triangle and of, say, segment $GH)$ and an ellipse remains ellipse. (You perform an affine transformation by moving any of the vertices of the triangle in the applet above.) It follows that we may choose a triangle to our liking, while the proof will be still valid in the general case. So, let's set up a Cartesian coordinates system, with $B$ at the origin, $A(0,2)$ and $C(2,0).$ Then the midpoints also have simple coordinates, $D(1,0),$ $E(1,1)$, $F(0,1)$. For the other six points we have, $G,H=(1\pm h,0)$, $I,J=(1\pm i,1\mp i),$ $K,L=(0,1\pm j).$ We are going to look for a conic (if such exists) through the six points. A conic is given by a second degree equation: $ax^{2}+2bxy+cy^{2}+2dx+2ey+f=0,$ with only $5$ coefficients independent (because the conic would not change after the equation was divided by one of non-zero coefficients.) Let's substitute the coordinates of points $G,H,I,J,K,L$ into the equation: $a(1\pm h)^{2}+2d(1\pm h)+f=0,$ $c(1\pm i)^{2}+2e(1\pm i)+f=0,$ $a(1\pm j)^{2}+2b(1-j^{2})+c(1\mp j)^{2}+2d(1\pm j)+2e(1\mp j)+f=0.$ Unfold the first equation into two: $a(1+2h+h^{2})+2d(1+h)+f=0,$ and $a(1-2h+h^{2})+2d(1-h)+f=0.$ Solving these for $a$ and $d$ gives $\displaystyle a = -d = \frac{f}{h^{2}-1}.$ Similarly, unfolding the second pair of equations, we obtain $\displaystyle c = -e = \frac{f}{i^{2}-1}.$ Note that $a+d=c+e=0.$ Adding and subtracting the two unfolded equations for $I$ and $J$ gives $a(1+j^{2})+2b(1-j^{2})+c(1+j^{2})+2d+2e+f=0,$ and $aj-cj+dj-ej=0,$ the latter of which holds automatically because, as we found, $a+d=c+e=0.$ From the former we can find the remaining coefficient $b$ in terms of $f.$ The six points $G,H,I,J,K,L$ determine a unique ellipse. The cases we ignored where $h^{2}-1=0,$ or, $i^{2}-1=0,$ or, $j^{2}-1=0,$ correspond to the configurations where some of the points $G,H,I,J,K,L$ coincide with the vertices of the triangle. No ellipse, of cause, can pass through three collinear points. Now, as $h\rightarrow 0$, $H$ and $G$ move toward $D$ and, at the limit, coincide with the latter. The resulting ellipse becomes tangent to $BC$ at $D$. Letting also $i\rightarrow 0$ and $j\rightarrow 0$ leads to an ellipse tangent to the sides of $\Delta ABC$ at the midpoints. Such ellipse is also unique. I do not see how its uniqueness follows from that of the ellipse through six points. But here is an independent proof. Let's again make use of affine transformations. Let's preform a transformation (squashing or stretching) that converts one of the ellipses into a circle. In other words, let's assume without loss of generality that in $\Delta ABC$ the incircle is tangent to the sides at the midpoints $D,E,F.$ Let $S$ be the incenter. Then, on one hand, $DS=ES=FS,$ because these are radii of the incircle. Also, say, $BD=CD,$ making right triangles $BDS$ and $CDS$ equal; and the same is true for the other two pairs of right triangles. On the other hand, $BD=BF$ as two tangents from point $B$ to the same circle. It thus follows that all six right triangles are equal, making $\Delta ABC$ equilateral. The question is thus reduced to whether, besides the incircle, there is an ellipse in an equilateral triangle tangent to the midpoints of the sides. The answer is No because by the same device we would be able to convert that ellipse and the equilateral triangle it is inscribed into into a circle inscribed in another equilateral triangles. But an affine transformation (the inverse one) that maps an equilateral triangle onto an equilateral triangle also maps circles onto circles. This ellipse is known as Steiner's inellipse. Its equation in barycentric coordinates has been derived elsewhere. Note that we chose the hard way to establish the existence of an ellipse tangent to the midpoints of a triangle. An affine transformation of an equilateral triangle onto a given one would map the incircle onto an ellipse immediately resolving the problem of existence. I believe the existence of an ellipse through the six points as discussed above is an independently curious feature. The ellipse tangent to the midpoints has an exciting property that was designated by Dan Kalman as The Most Marvelous Theorem in Mathematics. The theorem is stated in the complex plane where points are identified with complex numbers: The roots of the derivative of the polynomial $P(z)=(z-z_{1})(z-z_{2})(z-z_{3})$ are the foci of the ellipse tangent to the midpoints of $\Delta z_{1}z_{2}z_{3}.$	{"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": 0.924078106880188, "perplexity": 215.68750821861687}, "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/1552912202628.42/warc/CC-MAIN-20190322034516-20190322060516-00543.warc.gz"}
http://maxspywareremover.com/standard-error/what-is-a-standard-error-of-difference.php	Home > Standard Error > What Is A Standard Error Of Difference What Is A Standard Error Of Difference Contents Why is the FBI making such a big deal out Hillary Clinton's private email server? However, different samples drawn from that same population would in general have different values of the sample mean, so there is a distribution of sampled means (with its own mean and Student approximation when σ value is unknown Further information: Student's t-distribution §Confidence intervals In many practical applications, the true value of σ is unknown. To do this, you have available to you a sample of observations $\mathbf{x} = \{x_1, \ldots, x_n \}$ along with some technique to obtain an estimate of $\theta$, $\hat{\theta}(\mathbf{x})$. Standard Error Of Difference Calculator As before, the problem can be solved in terms of the sampling distribution of the difference between means (girls - boys). How are they different and why do you need to measure the standard error? The notation for standard error can be any one of SE, SEM (for standard error of measurement or mean), or SE. Good estimators are consistent which means that they converge to the true parameter value. This approximate formula is for moderate to large sample sizes; the reference gives the exact formulas for any sample size, and can be applied to heavily autocorrelated time series like Wall The sample mean will very rarely be equal to the population mean. The distribution of these 20,000 sample means indicate how far the mean of a sample may be from the true population mean. Sample Mean Difference Formula JSTOR2340569. (Equation 1) ^ James R. My only comment was that, once you've already chosen to introduce the concept of consistency (a technical concept), there's no use in mis-characterizing it in the name of making the answer Standard Error Of Difference Definition Inferential statistics used in the analysis of this type of experiment depend on the sampling distribution of the difference between means. Using this convention, we can write the formula for the variance of the sampling distribution of the difference between means as: Since the standard error of a sampling distribution is the Notice that s x ¯ = s n {\displaystyle {\text{s}}_{\bar {x}}\ ={\frac {s}{\sqrt {n}}}} is only an estimate of the true standard error, σ x ¯ = σ n Now let's look at an application of this formula. Standard Error Of The Difference In Sample Means Calculator Know These 9 Commonly Confused... Standard error of the mean (SEM) This section will focus on the standard error of the mean. A practical result: Decreasing the uncertainty in a mean value estimate by a factor of two requires acquiring four times as many observations in the sample. Standard Error Of Difference Definition School of Psychology, University of Bradford, UK Continue reading... http://link.springer.com/chapter/10.1007%2F978-94-011-7241-7_15 Standard Error of the Difference Between the Means of Two Samples The logic and computational details of this procedure are described in Chapter 9 of Concepts and Applications. Standard Error Of Difference Calculator The mean age was 23.44 years. Standard Error Of The Difference Between Means Calculator ISBN 0-521-81099-X ^ Kenney, J. The standard error of $\hat{\theta}(\mathbf{x})$ (=estimate) is the standard deviation of $\hat{\theta}$ (=random variable). news Thus the probability that the mean of the sample from Species 1 will exceed the mean of the sample from Species 2 by 5 or more is 0.934. doi:10.2307/2340569. Because the age of the runners have a larger standard deviation (9.27 years) than does the age at first marriage (4.72 years), the standard error of the mean is larger for Standard Error Of Difference Between Two Proportions B. The 5 cm can be thought of as a measure of the average of each individual plant height from the mean of the plant heights. The standard error estimated using the sample standard deviation is 2.56. have a peek at these guys The distribution of the mean age in all possible samples is called the sampling distribution of the mean. B. Mean Difference Calculator The standard error (SE) is the standard deviation of the sampling distribution of a statistic,[1] most commonly of the mean. We calculate the mean of each of these samples and now have a sample (usually called a sampling distribution) of means. Two sample variances are 80 or 120 (symmetrical). A difference between means of 0 or higher is a difference of 10/4 = 2.5 standard deviations above the mean of -10. The graphs below show the sampling distribution of the mean for samples of size 4, 9, and 25. The SEM gets smaller as your samples get larger. Standard Error Formula If eight boys and eight girls were sampled, what is the probability that the mean height of the sample of girls would be higher than the mean height of the sample The standard deviation of the age for the 16 runners is 10.23, which is somewhat greater than the true population standard deviation σ = 9.27 years. National Center for Health Statistics typically does not report an estimated mean if its relative standard error exceeds 30%. (NCHS also typically requires at least 30 observations – if not more In lieu of taking many samples one can estimate the standard error from a single sample. http://maxspywareremover.com/standard-error/what-does-the-standard-error-of-the-difference-tell-us.php B. n is the size (number of observations) of the sample. But you can't predict whether the SD from a larger sample will be bigger or smaller than the SD from a small sample. (This is a simplification, not quite true. B. As you collect more data, you'll assess the SD of the population with more precision. Correction for correlation in the sample Expected error in the mean of A for a sample of n data points with sample bias coefficient ρ. doi:10.4103/2229-3485.100662. ^ Isserlis, L. (1918). "On the value of a mean as calculated from a sample". If it is large, it means that you could have obtained a totally different estimate if you had drawn another sample. The confidence interval of 18 to 22 is a quantitative measure of the uncertainty – the possible difference between the true average effect of the drug and the estimate of 20mg/dL. It makes them farther apart. Standard error is instead related to a measurement on a specific sample. Ecology 76(2): 628 – 639. ^ Klein, RJ. "Healthy People 2010 criteria for data suppression" (PDF). If values of the measured quantity A are not statistically independent but have been obtained from known locations in parameter space x, an unbiased estimate of the true standard error of The concept of a sampling distribution is key to understanding the standard error. Because these 16 runners are a sample from the population of 9,732 runners, 37.25 is the sample mean, and 10.23 is the sample standard deviation, s. The mean of these 20,000 samples from the age at first marriage population is 23.44, and the standard deviation of the 20,000 sample means is 1.18. If people are interested in managing an existing finite population that will not change over time, then it is necessary to adjust for the population size; this is called an enumerative The confidence interval is consistent with the P value. First, let's determine the sampling distribution of the difference between means. If symmetrical as variances, they will be asymmetrical as SD. The standard error of the mean (SEM) (i.e., of using the sample mean as a method of estimating the population mean) is the standard deviation of those sample means over all Blackwell Publishing. 81 (1): 75–81. Dobson (4) Author Affiliations 3. We want to know whether the difference between sample means is a real one or whether it could be reasonably attributed to chance, i.e. If numerous samples were taken from each age group and the mean difference computed each time, the mean of these numerous differences between sample means would be 34 - 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": 0.9682376384735107, "perplexity": 419.48827788683633}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891816647.80/warc/CC-MAIN-20180225150214-20180225170214-00305.warc.gz"}
http://mathhelpforum.com/advanced-statistics/113153-jack-jill-waiting-time-question-print.html	# Jack and Jill Waiting Time question. • November 8th 2009, 05:34 AM billym Jack and Jill Waiting Time question. Jack and Jill agree to meet at 1:30. If Jack arrives at a time uniformly distributed between 1:15 and 1:45, and if Jill independently arrives at a time uniformly distributed between 1:30 and 2, 1) What is the probability that Jack arrives first? Straight forward enough. I figure it's 3/4. 2)What is the probability that Jack waits more that 15 minutes. I think I'm stumped by this, maybe not. I set X to be the time after 1:15 that Jack arrives, and Y to be the time after 1:15 that Jill arrives. Am I correct in assuming that this is the way to solve it: $\int_{15}^{45}\int_{0}^{y-15}f_{XY}(x,y)dxdy$	{"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": 1, "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": 0.8111861944198608, "perplexity": 663.1323844705782}, "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-2015-06/segments/1422121934081.85/warc/CC-MAIN-20150124175214-00135-ip-10-180-212-252.ec2.internal.warc.gz"}
http://physics.stackexchange.com/questions/52620/constant-pressure-and-temperature-mixing-of-2-different-ideal-gases-possible-w	Constant pressure and temperature mixing of 2 different ideal gases - possible work and heat? A simple question I hope... Initially, have two separate containers of 2 different ideal gases, 1.) N1, P, T, V1 and 2.) N2, P, T, V2. After mixing, the pressure and temperature are still P and T, but the volume is additive. Assuming isolated system. So I calculated the change in entropy with $\Delta\,S_i = n\,R\,\ln((V1+V2)/Vi)$. I found the change to be positive, which is to be expected. Then when I look at the first law, $d\,U = \delta\,Q - \delta\,W$, I think that as the temperatures did not change then the internal energy of the gases did not change and $d\,U = 0$. Also, I think that there was no work done and so there is no heat transfer, $\delta\,Q$. Is this right? Is the total energy change zero too? - Let's assume that in your setup: 1. The combined system of gas 1 + gas 2 is thermally insulated from the environment (so that there is no heat exchange between the environment and the gases as they mix. 2. The original gas samples are separated by a partition, and to allow them to mix we simply remove the partition and allow them to freely expand into one another to fill the entire container. In this case, one should be careful not to use the first law in differential form (which only holds for quasi-static processes). The first law is still true, however in the form $\Delta U = Q-W$ where $Q$ is the total heat transferred to the combined system, and $W$ is the total work done by the system. You are right that $Q=0$ and $W=0$ in such a situation and therefore $\Delta U = 0$ as you indicate. Cheers! -	{"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": 0.917640209197998, "perplexity": 271.0482783432464}, "config": {"markdown_headings": false, "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-2013-48/segments/1387345769117/warc/CC-MAIN-20131218054929-00096-ip-10-33-133-15.ec2.internal.warc.gz"}
https://myanalyticaldelights.blog/author/yohanceosborne/	# An Integral for the Occasion: Happy New Year! To celebrate the new year 2021, I’ve created an integral for the occasion which reads as follows. 2021 New Years Integral. Let $\alpha,\beta,$ and $\gamma$ be real-valued continuous functions defined in $\mathbb{R}^3$ such that $\displaystyle 1<\gamma(x)<\frac{1+\sqrt{5}}{2},$ $\displaystyle 1<\alpha(x)<\frac{\gamma(x)}{1+\gamma(x)-\gamma(x)^2},$ and $\displaystyle \beta(x):=\gamma(x)+\frac{1}{\alpha(x)}-\frac{1}{\gamma(x)},$ for all $x\in\mathbb{R}^3$. Furthermore, let $f,g:\mathbb{R}^3\to\mathbb{R}$ be functions given by $\displaystyle f(x):=\pi^{-\frac{3}{2}}\exp\left(-\frac{x_1^2}{H^2}-\frac{x_2^2}{a^2p^2}-\frac{x_3^2}{p^2y^2}\right),$ and $\displaystyle g(x):=\pi^{-\frac{3}{2}}\exp\left(-\frac{x_1^2}{4}-\frac{x_2^2}{2019!^2}-\frac{x_3^2}{2041210^2}\right),$ where $H,a,p,y>0$ are constants. If $G$ is a function on $\mathbb{R}^3$ defined by $\displaystyle G(x):=\frac{(\alpha(x)\beta(x)\gamma(x))^2-1}{\alpha(x)\beta(x)\gamma(x)^2+\beta(x)\gamma(x)+1}\text{ }\text{ \emph{for all} }x\in\mathbb{R}^3,$ then the integral $\displaystyle \int_{\mathbb{R}^3}G(x)\log\sqrt[\alpha(x)]{\exp\left({\frac{f(x)}{\gamma(x)}}\right)\sqrt[\beta(x)]{\exp({g(x)})\sqrt[\gamma(x)]{\exp\left({\frac{f(x)}{\gamma(x)}}\right) \sqrt[\alpha(x)]{\exp({g(x)})\sqrt[\beta(x)]{\cdots}}}}}\text{ }\mathrm{d}x$ evaluates to $Happy+2021!$ Proof. Our first step to evaluating the integral is to simplify the nested radical appearing in the integrand. For this, we employ Theorem 6 found on my Nested Radicals page. Theorem 6. Let $(a_n)_{n\in\mathbb{Z}_{\geq 1}}$ and $(b_n)_{n\in\mathbb{Z}_{\geq 1}}$ be periodic sequences of real numbers with periods $T_a,T_b\in \mathbb{Z}_{\geq 1}$ such that $a_n>0$, $a_n=a_{n+T_a}$ and $b_n=b_{n+T_b}$ for all $n\geq 1$. If $\|b_1\|,\|b_2\|,\cdots \|b_{T_b-1}\|,\|b_{T_b}\|>1$. Then $\displaystyle \sqrt[b_1]{a_1\sqrt[b_2]{a_2\sqrt[b_3]{a_3\sqrt[b_4]{a_4\sqrt[b_5]{\cdots}}}}}=\prod_{n=1}^{T_a}a_n^{S(n)}$ where $\displaystyle S(n)=\frac{(b_1\cdots b_{T_b})^{A}}{(b_1\cdots b_{T_b})^{A}-1}\sum_{k=0}^{B-1}\prod_{j=1}^{T_ak+n}b_j^{-1}$ for $n\in\mathbb{Z}_{\geq 1}$, with $A=\frac{l.c.m(T_a,T_b)}{T_b}$ and $B=\frac{l.c.m(T_a,T_b)}{T_a}$. This result can be derived using the idea behind the proof of Theorem 2 found on my Nested Radicals page. For this it suffices to consider the pair of sequences $(\tilde{a}_n)$ and $(\tilde{b}_n)$ given by • $\tilde{a}_n:=a_n$ for $1\leq n \leq T$, $\tilde{b}_n:=b_n$ for $1\leq n\leq T$ with $T=l.c.m(T_a,T_b)$, and • $\tilde{a}_n=\tilde{a}_{n+T}$, and $\tilde{b}_n=\tilde{b}_{n+T}$ for $n\geq 1$. Then one simplifies the resulting expressions to arrive at Theorem 6. For our current problem, we let $(a_n)=(a_n(x))$ and $(b_n)=(b_n(x))$ be periodic sequences of functions defined on $\mathbb{R}^3$ where $\displaystyle a_n=a_{n+2},\text{ }\text{ }b_{n}=b_{n+3}\text{ }\forall n\geq 1$ with $\displaystyle a_1:=\exp\left({\frac{f(x)}{\gamma(x)}}\right),\text{ }a_2:=\exp(g(x)),$ $\displaystyle b_1:=\alpha(x),\text{ }b_2:=\beta(x),\text{ }\text{ and }\text{ }b_3:=\gamma(x).$ It is clear that, for each $x\in\mathbb{R}^3$, the sequence $(a_n(x))$ fits the assumptions of the theorem with $T_a=2$. But for $(b_n)$ we need to check that $\|b_2\|=\|\beta(x)\|>1$ for each $x\in\mathbb{R}^3$ since we already know that $\|b_1\|=\alpha(x)>1$ and $\|b_3\|=\gamma(x)>1$ for all $x\in\mathbb{R}^3$. From the definition of $\beta$ and the assumed bounds on $\alpha$ we find $\displaystyle \beta(x)=\gamma(x)+\frac{1}{\alpha(x)}-\frac{1}{\gamma(x)}>\gamma(x)+\frac{1+\gamma(x)-\gamma(x)^2}{\gamma(x)}-\frac{1}{\gamma(x)}=1$ for all $x\in\mathbb{R}^3$. Noting that $T_a=2$ and $T_b=3$, we find in the context of the theorem that $A=2$ and $B=3$, giving $\displaystyle \sqrt[b_1]{a_1\sqrt[b_2]{a_2\sqrt[b_3]{a_3\sqrt[b_4]{a_4\sqrt[b_5]{\cdots}}}}}=\prod_{n=1}^{2}a_n^{S(n)}$ where $\displaystyle S(n)=\frac{(b_1b_2 b_{3})^2}{(b_1b_2 b_{3})^{2}-1}\sum_{k=0}^{2}\prod_{j=1}^{2k+n}b_j^{-1}$ for $n\in\{1,2\}$. Let’s rewrite $S(1)$ and $S(2)$ in terms of the functions $\alpha,\beta$ and $\gamma$. By periodicity of $(b_n)$ we find $\displaystyle S(1)=\frac{(b_1b_2b_3)^2}{(b_1b_2b_3)^{2}-1}\sum_{k=0}^{2}\prod_{j=1}^{2k+1}b_j^{-1}=\frac{(b_1b_2b_3)^2}{(b_1b_2b_3)^{2}-1}\left(b_1^{-1}+b_1^{-1}b_2^{-1}b_3^{-1}+b_1^{-2}b_2^{-2}b_3^{-1}\right)$ or $\displaystyle S(1)=\frac{\gamma(x)\left(\alpha(x)\beta(x)^2\gamma(x)+\alpha(x)\beta(x)+1\right)}{(\alpha(x)\beta(x)\gamma(x))^2-1},\text{ }\forall x\in\mathbb{R}^3.$ Similarly, we get $\displaystyle S(2)=\frac{(b_1b_2 b_{3})^2}{(b_1b_2 b_{3})^{2}-1}\sum_{k=0}^{2}\prod_{j=1}^{2k+2}b_j^{-1}=\frac{\alpha(x)\beta(x)\gamma(x)^2+\beta(x)\gamma(x)+1}{(\alpha(x)\beta(x)\gamma(x))^2-1},$ for all $x\in\mathbb{R}^3.$ With these, we can simplify the expression for the function $l:\mathbb{R}^3\to\mathbb{R}$ given by $\displaystyle l(x):=\log\sqrt[\alpha(x)]{\exp\left({\frac{f(x)}{\gamma(x)}}\right)\sqrt[\beta(x)]{\exp({g(x)})\sqrt[\gamma(x)]{\exp\left({\frac{f(x)}{\gamma(x)}}\right) \sqrt[\alpha(x)]{\exp({g(x)})\sqrt[\beta(x)]{\cdots}}}}}.$ From our application of Theorem 6, we get for each $x\in\mathbb{R}^3$ $\displaystyle l(x)=\log \prod_{n=1}^{2}a_n^{S(n)}=S(1)\log\left(\exp\left({\frac{f(x)}{\gamma(x)}}\right) \right)+S(2)\log( \exp({g(x)}))$ or $\displaystyle l(x)=\frac{ (\alpha(x)\beta(x)^2\gamma(x)+\alpha(x)\beta(x)+1)f(x)+(\alpha(x)\beta(x)\gamma(x)^2+\beta(x)\gamma(x)+1)g(x)}{(\alpha(x)\beta(x)\gamma(x))^2-1}.$ Recalling the definition of the function $G$, our integrand thus takes the form $\displaystyle G(x)l(x)=\frac{ (\alpha(x)\beta(x)^2\gamma(x)+\alpha(x)\beta(x)+1)f(x)+(\alpha(x)\beta(x)\gamma(x)^2+\beta(x)\gamma(x)+1)g(x)}{\alpha(x)\beta(x)\gamma(x)^2+\beta(x)\gamma(x)+1}$ or $\displaystyle G(x)l(x)=\left(\frac{\alpha(x)\beta(x)^2\gamma(x)+\alpha(x)\beta(x)+1}{\alpha(x)\beta(x)\gamma(x)^2+\beta(x)\gamma(x)+1}\right)f(x)+g(x)\text{ }\forall x\in \mathbb{R}^3.$ Using the expression for $\beta$ we can simplify the coefficient of $f$ as follows: for each $x\in\mathbb{R}^3$ $\displaystyle \frac{\alpha(x)\beta(x)^2\gamma(x)+\alpha(x)\beta(x)+1}{\alpha(x)\beta(x)\gamma(x)^2+\beta(x)\gamma(x)+1}=\frac{\beta(x)^2\gamma(x)+\beta(x)+\frac{1}{\alpha(x)}}{\beta(x)\gamma(x)^2+\frac{\beta(x)\gamma(x)}{\alpha(x)}+\frac{1}{\alpha(x)}}$ $\displaystyle = \frac{\beta(x)^2\gamma(x)+\beta(x)+\beta(x)+\frac{1}{\gamma(x)}-\gamma(x)}{\beta(x)\gamma(x)^2+\beta(x)\gamma(x)\left(\beta(x)+\frac{1}{\gamma(x)}-\gamma(x)\right)+ \beta(x)+\frac{1}{\gamma(x)}-\gamma(x) }$ $\displaystyle =\frac{\beta(x)^2\gamma(x)+2\beta(x)+\frac{1}{\gamma(x)}-\gamma(x)}{\beta(x)^2\gamma(x)+ 2\beta(x)+\frac{1}{\gamma(x)}-\gamma(x) }=1.$ Let $I$ denote the integral that we are seeking to evaluate. With the above, our problem is now simplified massively: evaluate $\displaystyle I=\int_{\mathbb{R}^3}(f(x)+g(x))\mathrm{d}x$ with $\displaystyle f(x)=\pi^{-\frac{3}{2}}\exp\left(-\frac{x_1^2}{H^2}-\frac{x_2^2}{a^2p^2}-\frac{x_3^2}{p^2y^2}\right),$ and $\displaystyle g(x)=\pi^{-\frac{3}{2}}\exp\left(-\frac{x_1^2}{4}-\frac{x_2^2}{2019!^2}-\frac{x_3^2}{2041210^2}\right).$ Apply Fubini’s theorem to find that $\displaystyle \int_{\mathbb{R}^3}f(x)\mathrm{d}x=\pi^{-\frac{3}{2}}\left(\int_{\mathbb{R}}e^{-\frac{x_1^2}{H^2}}\mathrm{d}x_1\right)\left(\int_{\mathbb{R}}e^{-\frac{x_2^2}{(ap^2)}}\mathrm{d}x_2 \right)\left(\int_{\mathbb{R}}e^{-\frac{x_3^2}{(py)^2}}\mathrm{d}x_3\right),$ and $\displaystyle \int_{\mathbb{R}^3}g(x)\mathrm{d}x=\pi^{-\frac{3}{2}}\left(\int_{\mathbb{R}}e^{-\frac{x_1^2}{4}}\mathrm{d}x_1\right)\left(\int_{\mathbb{R}}e^{-\frac{x_2^2}{2019!^2}}\mathrm{d}x_2 \right)\left(\int_{\mathbb{R}}e^{-\frac{x_3^2}{2041210^2}}\mathrm{d}x_3\right).$ The integrals in the above products are all of the form $\displaystyle \int_{\mathbb{R}}e^{-\frac{x^2}{w^2}}\mathrm{d}x$ which evaluates to $\sqrt{\pi}w$ whenever $w>0$. Consequently, $\displaystyle \int_{\mathbb{R}^3}f(x)\mathrm{d}x=Happy$ while $\displaystyle \int_{\mathbb{R}^3}g(x)\mathrm{d}x=2\times 2019!\times 2041210.$ The conclusion then follows once we note that $\displaystyle 2\times 2019!\times 2041210=2!(2021-2)!\times 2041210=2021!\frac{2041210}{\begin{pmatrix} 2021 \\ 2 \end{pmatrix}}$ and $\displaystyle \begin{pmatrix} 2021 \\ 2 \end{pmatrix}=2041210,$ giving $\displaystyle \int_{\mathbb{R}^3}g(x)\mathrm{d}x=2021!$ \\\\ # An Integral for the Occasion: Merry Christmas! This post is dedicated to a proof of the following result. Christmas Determinant Integral. Let $C,M,e,r,i,t,m,a,s,h,y>0$ be given constants such that $M^er^r\neq C^hr^is^tm^as!$ . Then, the determinant integral $\displaystyle I:=\det\int_{0}^{\infty}\exp\begin{pmatrix}\log\left(\frac{\log^2\left(1+\frac{y}{x^2}\right)}{4\pi\log(2)}\right)-C^hr^is^tm^as!w & M^eC^hr^iw \\-s^tm^as!r^rw &\log\left(\frac{\log^2\left(1+\frac{y}{x^2}\right)}{4\pi\log(2)}\right)+M^er^rw\end{pmatrix}\mathrm{d}x$ where $\displaystyle w:=\frac{\log\left(M^er^r\right)-\log\left(C^hr^is^tm^as!\right)}{M^er^r-C^hr^is^tm^as!},$ evaluates to $\displaystyle I=\frac{M^er^ry}{C^hr^is^tm^as!}.$ Proof. Let $A$ be a 2×2 matrix $\displaystyle A=\begin{pmatrix} \tilde{a} & \tilde{b} \\ \tilde{c} & \tilde{d} \end{pmatrix}$ that is diagonalisable, so that there exists an inveritable matrix $P$ $\displaystyle P=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ and a diagonal matrix $D$ $\displaystyle D=\begin{pmatrix} g & 0 \\ 0 & f \end{pmatrix}$ satisfying $A=PDP^{-1}$ (where of course $ad\neq bc$). As such, we have $\displaystyle A=\frac{1}{ad-bc} \begin{pmatrix} adg-bcf & abf-abg \\ cdg-cdf & adf-bcg \end{pmatrix}.$ With the above decomposition, it can be shown that the exponential of $A$ reads $\displaystyle \exp(A)=P\begin{pmatrix}\exp(g) & 0 \\ 0 & \exp(f) \end{pmatrix} P^{-1}$ or $\displaystyle \exp(A)=\frac{1}{\det(P)}\begin{pmatrix} ad\exp(g)-bc\exp(f) & ab\exp(f)-ab\exp(g) \\cd\exp(g)-cd\exp(f) & ad\exp(f)-bc\exp(g) \end{pmatrix}.$ Assuming that $a,b,c,d$ are constants, $\Omega\subset\mathbb{R}^n$ is given, and that $\exp(g)$ and $\exp(f)$ are integrable with respect to Lebesgue measure over $\Omega$, we have $\displaystyle T(\exp(A))=\frac{1}{\det(P)}\begin{pmatrix} \int_{\Omega}(ad\exp(g)-bc\exp(f))\mathrm{d}x & \int_{\Omega}(ab\exp(f)-ab\exp(g))\mathrm{d}x \\ \int_{\Omega}(cd\exp(g)-cd\exp(f))\mathrm{d}x & \int_{\Omega}(ad\exp(f)-bc\exp(g))\mathrm{d}x \end{pmatrix}.$ Here, I’ve let $T(\exp(A))$ denote the integration of the matrix $\exp(A)$ entry-wise. This operation is discussed on my Determinant Integrals page. Now, assume $a,b,c,d> 0$, and suppose $f:=\log\left(\frac{ad}{bc}\exp(g)\right)$. Then, $\displaystyle T(\exp(A))=\frac{1}{\det(P)}\begin{pmatrix} 0 & \int_{\Omega}(ab\exp(f)-ab\exp(g))\mathrm{d}x \\ \int_{\Omega}(cd\exp(g)-cd\exp(f))\mathrm{d}x & \int_{\Omega}(ad\exp(f)-bc\exp(g))\mathrm{d}x\end{pmatrix}$ which implies $\displaystyle \det(T(\exp(A)))=\frac{abcd}{(ad-bc)^2}\left(\int_{\Omega}(\exp(f)-\exp(g))\mathrm{d}x\right)^2$ or rather (1)……. $\displaystyle \det(T(\exp(A)))=\frac{ad}{bc}\left(\int_{\Omega}\exp(g)\mathrm{d}x\right)^2.$ With the choice of $f$ made above, it can be shown that $\displaystyle A=\begin{pmatrix} g-bc\frac{\log(ad)-\log(bc)}{ad-bc} & ab\frac{\log(ad)-\log(bc)}{ad-bc} \\ -cd\frac{\log(ad)-\log(bc)}{ad-bc} & g+ad\frac{\log(ad)-\log(bc)}{ad-bc} \end{pmatrix}.$ Setting $\Omega=(0,\infty)\subset\mathbb{R}$ and $\displaystyle g:=\log\left(\frac{\log^2\left(1+\frac{y}{x^2}\right)}{4\pi\log(2)}\right), \text{ }x>0$ where $y>0$ is an arbitrary constant, it follows from (1) that $\displaystyle \det(T(\exp(A)))=\frac{ad}{bc}\left(\int_{\Omega}\exp(g)\mathrm{d}x\right)^2=\frac{ad}{bc}\left(\int_{\Omega}\frac{\log^2\left(1+\frac{y}{x^2}\right)}{4\pi\log(2)} \mathrm{d}x\right)^2=\frac{ady}{bc}.$ Here, I have used the fact that $\displaystyle \int_{0}^{\infty}\log^2\left(1+\frac{q}{x^2}\right)\mathrm{d}x=4\pi\sqrt{q}\log(2)$ holds for all $q>0$. This follows by using integration by parts and a trigonometric substitution to reduce the integral to the problem of evaluating $\int_0^{\pi/2}\frac{x}{\tan{x}}\mathrm{d}x$. Wolfram Alpha indicates that this latter integral is equal to $\frac{\pi}{2}\log{2}.$ We now have that $A$ takes the form $\displaystyle A(x)=\begin{pmatrix} \log\left(\frac{\log^2\left(1+\frac{y}{x^2}\right)}{4\pi\log(2)}\right) -bc\frac{\log(ad)-\log(bc)}{ad-bc} & ab\frac{\log(ad)-\log(bc)}{ad-bc} \\ -cd\frac{\log(ad)-\log(bc)}{ad-bc} & \log\left(\frac{\log^2\left(1+\frac{y}{x^2}\right)}{4\pi\log(2)}\right)+ad\frac{\log(ad)-\log(bc)}{ad-bc} \end{pmatrix},$ for $x>0$, with $\displaystyle \det(T(\exp(A)))=\frac{ady}{bc},\text{ }y>0.$ Setting $w$ to be the real number given by $\displaystyle w:= \frac{\log(ad)-\log(bc)}{ad-bc},$ the proposed result follows by setting $a=M^e,$ $d=r^r$, $b=C^hr^i,$ and $c=s^tm^as!$, where $C,M,e,r,i,t,m,a,s,h>0$ are arbitrary constants such that $M^er^r\neq C^hr^is^tm^as!$. \\\\ # New Integral Corner Collection: Complex Analysis to the Rescue! My latest addition the Integral Corner page is a collection of neat results that can be found using methods from Complex Analysis. Deriving these results by any other method isn’t particularly clear so I’ve called the collection Complex Analysis to the Rescue! Let’s visit the first few results. The first of the collection asserts that, 1) For $a,b,c \in \mathbb{R}$ satisfying $\|a\|>\|b\|+\|c\|>0$, and given $n \in \mathbb{Z}_{\geq 0}$, there hold $\displaystyle \int_0^{2\pi}\frac{\cos(nx)}{a+b\cos(x)+c\sin(x)}dx=2\pi \lambda (a,b,c,n) \cos(n\phi)$ and $\displaystyle \int_0^{2\pi}\frac{\sin(nx)}{a+b\cos(x)+c\sin(x)}dx=2\pi \lambda (a,b,c,n) \sin(n\phi)$ where $\displaystyle \lambda (a,b,c,n):=\frac{(\text{sgn}(a))^n\left(-\|a\|+\sqrt{a^2-b^2-c^2}\right)^n}{\sqrt{(a^2-b^2-c^2)(b^2+c^2)^n}}$ and $\displaystyle \phi:=\text{arg}(b+ic)\in[-\pi,\pi).$ To derive these results one can rewrite each integral as a complex contour integral over the unit circle $\|z\|=1$ in $\mathbb{C}$ and employ the Residue Theorem after having checked the positions of poles of the integrand relative to the unit disc (which is where the inequality involving $a,b$ and $c$ comes in). A similar approach can be used to tackle the second result in my collection which is a generalisation of the above that tells more about the kind of result to expect when integrating quotients of the form $\displaystyle \frac{a_1\cos(nx)+a_2\sin(nx)+a_3}{a_4\cos(mx)+a_5\sin(mx)+a_6}$ over one period, provided $\|a_6\|>\|a_5\|+\|a_4\|>0$ and $n,m$ are nonnegative integers. It reads 2) For any $a,b,c,d,e,f\in\mathbb{R}$ and $m,n\in\mathbb{Z}_{\geq 0}$ such that $c>\|a\|+\|b\|>0$ and $m\geq 1$, $\displaystyle \int_{0}^{2\pi}\frac{d\cos(nx)+e\sin(nx)+f}{a\cos(mx)+b\sin(mx)+c}\mathrm{d}x=\frac{2\pi }{m\sqrt{c^2-a^2-b^2}}\sum_{k=0}^{m-1}\left(f+\sqrt{d^2+e^2}R\cos\left(\beta-n\alpha_k\right)\right)$ where $\displaystyle R:=\left(\frac{c-\sqrt{c^2-a^2-b^2}}{\sqrt{a^2+b^2}}\right)^{\frac{n}{m}},\text{ } \beta:=\text{arg}(d+ie)\in(-\pi,\pi],$ and, for $k\in\{0,\cdots,m-1\}$, we define $\displaystyle \alpha_k:=\frac{\theta+(2k+1)\pi}{m} \text{ with } \theta=\text{arg}(a+bi) \in (-\pi,\pi].$ Using the first result above in 1), one can easily tackle the following third item of the collection 3) Given $\|a\|>\|b\|+\|c\|>0$ where $a,b,c\in\mathbb{R}$ and $m\in\mathbb{Z}_{\geq 0}$, for all $z\in\mathbb{R}$ there holds $\displaystyle \int_0^{2\pi}\int_0^{2\pi}\frac{(\cos(mx)-\cos(my))(\cos(my)-\cos(mz))}{(a+b\cos(x)+c\sin(x))(a+b\cos(y)+c\sin(y))}\mathrm{d}x\mathrm{d}y=\frac{2\pi^2}{a^2-b^2-c^2}\left(\frac{(b^2+c^2)^m-\left(\|a\|+\sqrt{a^2-b^2-c^2}\right)^{2m}}{\left(\|a\|+\sqrt{a^2-b^2-c^2}\right)^{2m}}\right)$ Consequently, for all $n\in\mathbb{N}$ and $x_{2n+1}\in\mathbb{R}$ we have $\displaystyle \int_0^{2\pi}\int_0^{2\pi}\cdots\int_0^{2\pi}\prod_{j=1}^{2n}\frac{\cos(mx_{j})-\cos(mx_{j+1})}{a+b\cos(x_{j})+c\sin(x_{j})}\mathrm{d}x_1\cdots\mathrm{d}x_{2n-1}\mathrm{d}x_{2n}= \frac{2^n\pi^{2n}}{(a^2-b^2-c^2)^n}\left(\frac{(b^2+c^2)^m-\left(\|a\|+\sqrt{a^2-b^2-c^2}\right)^{2m}}{\left(\|a\|+\sqrt{a^2-b^2-c^2}\right)^{2m}}\right)^n.$ Notice the first result in 3) does not depend on $z\in\mathbb{R}$, which allows us to deduce the second result by Fubini’s Theorem and induction on $n$. The same approach allows one to derive the following 4th result involving an odd number of terms in the product integrand instead. 4) Given $\|a\|>\|b\|+\|c\|>0$ where $a,b,c\in\mathbb{R}$ and $m,n\in\mathbb{Z}_{\geq 0}$, for all $x_{2n+2}\in\mathbb{R}$ there holds $\displaystyle \int_0^{2\pi}\int_0^{2\pi}\cdots\int_0^{2\pi}\prod_{j=1}^{2n+1}\frac{\cos(mx_{j})-\cos(mx_{j+1})}{a+b\cos(x_{j})+c\sin(x_{j})}\mathrm{d}x_1\cdots\mathrm{d}x_{2n}\mathrm{d}x_{2n+1}= \Gamma(a,b,c,n,m)\left(\lambda(a,b,c,m)\cos(m\phi)-\lambda(a,b,c,0)\cos\left(mx_{2n+2}\right)\right)$ where $\displaystyle \Gamma(a,b,c,n,m):=\frac{2^{n+1}\pi^{2n+1}}{(a^2-b^2-c^2)^n}\left(\frac{(b^2+c^2)^m-\left(\|a\|+\sqrt{a^2-b^2-c^2}\right)^{2m}}{\left(\|a\|+\sqrt{a^2-b^2-c^2}\right)^{2m}}\right)^n,$ $\displaystyle \lambda (a,b,c,w):=\frac{(\text{sgn}(a))^w\left(-\|a\|+\sqrt{a^2-b^2-c^2}\right)^w}{\sqrt{(a^2-b^2-c^2)(b^2+c^2)^w}},$ and $\phi:=\text{arg}(b+ic)\in[-\pi,\pi).$ Other problems I hope to consider involve integrands of the form $\displaystyle \frac{f(a_1\cos(nx)+a_2\sin(nx)+a_3)}{a_4\cos(mx)+a_5\sin(mx)+a_6}$ where $f$ is an analytic function locally. # A Picturesque Reduction of An Integral Determinant Equation 1 In this post we discuss example solutions of the following integral determinant equation that was derived on my Determinant Integrals page: (1) …… $\displaystyle \left(\int_{\Omega}\det(A)\mathrm{d}x\right)\left(\int_{\Omega}\det(A^{-1})\mathrm{d}x\right)=1$ Let $\Omega \subset \mathbb{R}^m$ be a bounded open set. Given an invertible matrix $A\in\mathcal{M}_n(\Omega,\textbf{r})$ (see the Determinant Integral page for a description of the set $\mathcal{M}_n(\Omega,\textbf{r})$ if needed) we know that its determinant is integrable and $\displaystyle \text{det}(A)=\frac{1}{\text{det}(A^{-1})}\text{ }\text{ a.e in }\Omega.$ Consequently, if equation (1) holds we have $\displaystyle \left(\int_{\Omega}\text{det}(A)\text{ }\mathrm{d}x\right)\left(\int_{\Omega}\frac{1}{\text{det}(A)}\text{ }\mathrm{d}x\right)=1.$ This leads us to consider examples of bounded open sets $\Omega$ and scalar functions $f$ which satisfy (2) …… $\displaystyle \left(\int_{\Omega}f\text{ }\mathrm{d}x\right)\left(\int_{\Omega}\frac{1}{f}\text{ }\mathrm{d}x\right)=1.$ Clearly, equation (2) is equation (1) in the case when $A$ is a $1\times 1$ matrix, so we’ve massively simplified the problem presented by equation (1). Given an integer $n\geq 2$, notice equation (1) on its own does not have a unique $n\times n$ matrix solution $A$ defined a.e in $\Omega$ if there exists a function $f$ satisfying equation (2). With a function $f$ known to satisfy equation (2), we could construct an uncountable number of matrices $A$ for which $\text{det}(A)=f$ a.e in $\Omega$. For example, consider triangular matrices $A$ whose main diagonals take the form $\text{diag}(1,\cdots,1,f,1,\cdots,1)\in\mathbb{R}^n$. We then have full freedom to determine the possibly non-degenerate off-diagonal entries of $A$, according to whether the matrix is upper or lower triangular. We now present an example solution pair $(f,\Omega)$ to equation (2). Denote by $I[f,\Omega]$ the left-hand side of equation (2): $\displaystyle I[f,\Omega]:= \left(\int_{\Omega}f\text{ }\mathrm{d}x\right)\left(\int_{\Omega}\frac{1}{f}\text{ }\mathrm{d}x\right).$ Let $f(x)=\cos(\alpha x)$ be defined over $\Omega=(a,b)$ with $\alpha \neq 0$ and $a, b\in\mathbb{R}$ such that $a\neq b$. I’ve allowed simply for the condition $a\neq b$ as opposed to the natural condition $a because equation (2) holds even if the limits of integration are interchanged, which allows for the case $a>b$. On the other hand, equation (2) can’t hold if $a=b$ or else we have $0=1$. Nonetheless, for suitable $a,b\in\mathbb{R}$ there holds $\displaystyle I[f,\Omega]=\frac{1}{\alpha^2}\left(\sin(\alpha b)-\sin(\alpha a)\right)\log\left\|\frac{\sec(\alpha b)+\tan(\alpha b)}{\sec(\alpha a)+\tan(\alpha a)}\right\|.$ If we let $z_{\alpha}(a,b)=\left(\sin(\alpha b)-\sin(\alpha a)\right)\log\left\|\frac{\sec(\alpha b)+\tan(\alpha b)}{\sec(\alpha a)+\tan(\alpha a)}\right\|$ equation (2) reads $z_{\alpha}(a,b)=\alpha^2$. Below we plot this equation implicitly in Python over $(a,b)\in\left[0,\frac{\pi}{2}\right]^2$ for $\alpha=1,2,3,4,5,$ and $9$. In Figure 1 all curves of a given colour constitute the solution locus corresponding to one choice of $\alpha$. We describe this correspondence below: • $\alpha=1\to$ Light Blue curves • $\alpha=2\to$ Orange curves • $\alpha=3 \to$ Red curves • $\alpha=4\to$ Blue curves • $\alpha=5\to$ Magenta curves • $\alpha=9\to$ Green curves In essence, for fixed $\alpha$ that generate curves in the plane of positive length, there are infinitely many choices of $\Omega$ that ensure equation (2) holds. In Figure 2 below we extend the plot to $[0,4\pi]^2$ which gives a picturesque pattern that would make for a neat wallpaper! Between figures 1 and 2 we observe closed curves of decreasing diameter for $\alpha=1,2,3,4,$ and $9$, whereas for $\alpha=5$ we see small horizontal/vertical spikes which do not cross the red curves although they appear to be touching prior to zooming in. In Figure 2 we see a pattern repeating for any given coloured curve corresponding to a choice of $\alpha$. This is expected as $z_{\alpha}(a,b)$ is periodic with period $2\pi$ in both its arguments when $\alpha$ is an integer. It would be interesting then to see what family curves we observe when $\alpha$ is not an integer or irrational. In this post I present my argument which proves integral no. 5 under the Radical Integrals section of The Integral Corner. The result in question reads as follows. Radical Integral 5. Let $P$ and $Q$ real satisfy $0< P\leq Q$. Consider the function $\displaystyle a(x):=1+e\gamma(x+1,1)$ for $x\geq 0$ where $\gamma(u,v)$ is the incomplete gamma function with integral representation $\displaystyle \gamma(u,v)=\int_0^{v}t^{u-1}e^{-t}\mathrm{d}t.$ Then, $\displaystyle \int_P^Q\Lambda(x)\log \sqrt[x+1]{a(x)^{x+2}\sqrt[x+2]{a(x)\sqrt[x+3]{a(x)\sqrt[x+4]{a(x)\sqrt[x+5]{\cdots}}}}}\text{ }\mathrm{d}x=\frac{1}{4e}\log\left(H(P,Q)\right)$ with $\displaystyle \Lambda(x):=\int_0^{1}t^{x}e^{-t}\log(t)\text{ }\mathrm{d}t$ and $\displaystyle H(P,Q):=\frac{a(Q)^{2a(Q)^2}}{a(P)^{2a(P)^2}}\exp(a(P)^2-a(Q)^2).$ Proof. If we let $\displaystyle L(x):=\sqrt[x+1]{a(x)^{x+2}\sqrt[x+2]{a(x)\sqrt[x+3]{a(x)\sqrt[x+4]{a(x)\sqrt[x+5]{\cdots}}}}} \text{ }\text{ }(x\geq 0)$ we have for each $x>0$ $\displaystyle L(x)=a(x)^{1+\frac{1}{x+1}+\frac{1}{(x+1)(x+2)}+\frac{1}{(x+1)(x+2)(x+3)}+\cdots}.$ This can be further simplified to $\displaystyle L(x)=a(x)^{1+e\gamma(x+1,1)}=a(x)^{a(x)}$ after establishing $\displaystyle \sum_{k=0}^{\infty}\prod_{j=1}^{k+1}\frac{1}{x+j}=e\gamma(x+1,1)$ for $x\geq 0$. To see this, note that the “lower” incomplete gamma function $\gamma(s,z)$ is a holomorphic function with singularities at points $(s,z)$ where $z=0$ or $s$ is a non-positive integer (check out the Incomplete gamma function Wikipedia page). Moreover, it admits the representation $\displaystyle \gamma(s,z)=z^s\Gamma(s)e^{-z}\sum_{k=0}^{\infty}\frac{z^k}{\Gamma(s+k+1)}.$ Setting $s=x+1$ and $z=1$, we find for each $x>0$ $\displaystyle \gamma(x+1,1)=e^{-1}\Gamma(x+1)\sum_{k=0}^{\infty}\frac{1}{\Gamma(x+k+2)}=e^{-1}\Gamma(x+1)\left(\frac{1}{\Gamma(x+2)}+\frac{1}{\Gamma(x+3)}+\cdots\right)$ Rearranging, we get $\displaystyle \sum_{k=0}^{\infty}\frac{\Gamma(x+1)}{\Gamma(x+k+2)}=e\gamma(x+1,1).$ But notice that, formally, $\prod_{j=1}^{k+1}\frac{1}{x+j}\equiv\frac{\Gamma(x+1)}{\Gamma(x+k+2)}.$ As such, we arrive at the desired identity for the infinite sum, justifying the identity $L(x)=a(x)^{a(x)}$ for $x>0$. Writing out $a$ as $\displaystyle a(x)=1+e\int_0^{1}t^xe^{-t}\mathrm{d}t \text{ }(x\geq 0)$ we see that $a$ is differentiable over $(0,\infty)$ with derivative given by $\displaystyle \frac{da}{dx}=e\int_0^1t^xe^{-t}\log(t)\mathrm{d}t=e\Lambda(x).$ Consequently, $a$ is monotone decreasing over $(0,\infty)$. Therefore, our proposed integral for given $0 can be evaluated as follows. $\displaystyle \int_P^Q\Lambda(x)\log{L(x)}\mathrm{d}x=e^{-1}\int_P^Qa(x)\log(a(x))\frac{da}{dx}\mathrm{d}x=e^{-1}\int_{a(P)}^{a(Q)}v\log(v)\mathrm{d}v=e^{-1}\left[\frac{v^2}{2}\log(v)-\frac{1}{4}v^2\right]_{a(P)}^{a(Q)} =\frac{1}{4e}\left[\log\left((v^{2v^2}\exp(-v^{2})\right)\right]_{a(P)}^{a(Q)}.$ Simplification leads to the stated result: $\displaystyle \int_P^Q\Lambda(x)\log\sqrt[x+1]{a(x)^{x+2}\sqrt[x+2]{a(x)\sqrt[x+3]{a(x)\sqrt[x+4]{a(x)\sqrt[x+5]{\cdots}}}}}\text{ }\mathrm{d}x=\frac{1}{4e}\log\left(H(P,Q)\right)$ with $\displaystyle \Lambda(x):=\int_0^{1}t^{x}e^{-t}\log(t)\text{ }\mathrm{d}t$ and $\displaystyle H(P,Q):=\frac{a(Q)^{2a(Q)^2}}{a(P)^{2a(P)^2}}\exp(a(P)^2-a(Q)^2).$ Featured # Welcome! Below are a few plots of mine for your viewing pleasure. These images depict the evolution of discrete-time dynamical systems that I devised following a coding exercise on Ikeda Maps. While you’re here, feel free to explore • The Integral Corner: a place where I collect some of my favourite integrals, most of which I created for fun since August 2017; • Nested Radicals: an account of my results on evaluating nested radicals through periodicity. This constitutes a short study that I undertook in July 2020; • Determinant Integrals: my introduction to an interplay between matrix determinant and Lebesgue integration. I initiated this work in August 2020 and I look forward to developing it further; • Sobolev Spaces: a discussion on fine properties of functions that are ubiquitous in the theory of partial differential equations. Some of my own work on PDE theory will be discussed here. 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http://edustatistics.org/nathanvan/setup/stat202/lectures/lecture9.html	# Stat 202: Lecture 9 (covers pp. 118-123) Nathan VanHoudnos 10/13/2014 ### Agenda 1. Homework comments 2. Checkpoint #10 results 3. Lecture 9 (covers pp. 118-123) ### Agenda 1. Homework comments 2. Checkpoint #10 results 3. Lecture 9 (covers pp. 118-123) to fill in ### Checkpoint #10 Question 5 Dogs are inbred for such desirable characteristics as blue eye color; but an unfortunate by-product of such inbreeding can be the emergence of characteristics such as deafness. A 1992 study of Dalmatians (by Strain and others, as reported in The Dalmatians Dilemma) found the following: (i) 31% of all Dalmatians have blue eyes. (ii) 38% of all Dalmatians are deaf. (iii) 42% of blue-eyed Dalmatians are deaf. Based on the results of this study is “having blue eyes” independent of “being deaf”? ### Checkpoint #10 Question 5 Write this out: (i) 31% of all Dalmatians have blue eyes. $P(B) = .31$ (ii) 38% of all Dalmatians are deaf. $P(D) = .38$ (iii) 42% of blue-eyed Dalmatians are deaf. $P(D\|B) \text{ or } P(B\|D)$ $P(D\|B) = .42$ ### Checkpoint #10 Question 5 \begin{aligned} P(B) & = .31 & P(D) & = .38 \\ P(D\|B) & = .42 \ \end{aligned} Based on the results of this study is “having blue eyes” independent of “being deaf”? • a) No, since .31 * .38 is not equal to .42. • b) No, since .38 is not equal to .42. • c) No, since .31 is not equal to .42. Write out the symbols… ### Checkpoint #10 Question 5 \begin{aligned} P(B) & = .31 & P(D) & = .38 \\ P(D\|B) & = .42 \ \end{aligned} Based on the results of this study is “having blue eyes” independent of “being deaf”? • a) No, since $$P(B)P(D)$$ is not equal to $$P(D\|B)$$. • b) No, since $$P(D)$$ is not equal to $$P(D\|B)$$. • c) No, since $$P(B)$$ is not equal to $$P(D\|B)$$. ### Checkpoint #10 Question 5 \begin{aligned} P(B) & = .31 & P(D) & = .38 \\ P(D\|B) & = .42 \ \end{aligned} Based on the results of this study is “having blue eyes” independent of “being deaf”? • a) No, since $$P(B)P(D)$$ is not equal to $$P(D\|B)$$. • b) No, since $$P(D)$$ is not equal to $$P(D\|B)$$. • c) No, since $$P(B)$$ is not equal to $$P(D\|B)$$. Independence if and only if: $P(D\|B) = P(D)$ therefore b) is the correct choice. ### Agenda 1. Homework comments 2. Checkpoint #10 results 3. Lecture 9 (covers pp. 118-123) ### Probability Rules! Let $$S$$ be the sample space, $$A$$ any event, $$A^c$$ its complement, and $$B$$ another event. 1. $$0 \le P(A) \le 1$$ 2. $$P(S) = 1$$ 3. $$P(A^c) = 1 - P(A)$$ 4. $$P(A \text{ or } B ) = P(A) + P(B) - P(A \text{ and } B)$$ 5. If and only if $$A$$ and $$B$$ are independent, then $P(A \text{ and } B ) = P(A) \times P(B)$ ### General multiplication rule Recall the definition of conditional probability: $P(A\|B) = \frac{P(A \text{ and } B )}{P(B)}$ Therefore, for all events $$A$$ and $$B$$: $P(A \text{ and } B )= P(A\|B) \times P(B)$ ### Why a generalization? The general rule: $P(A \text{ and } B ) = P(A\|B) \times P(B)$ Recall that $$A$$ is independent of $$B$$ if and only if $P(A\|B) = P(A)$ Therefore, if $$A$$ is independent of $$B$$, \begin{aligned} P(A \text{ and } B ) & = P(A\|B) P(B) \\ & = P(A) P(B) \end{aligned} which is rule #5. ### Probability Rules! Let $$S$$ be the sample space, $$A$$ any event, $$A^c$$ its complement, and $$B$$ another event. 1. $$0 \le P(A) \le 1$$ 2. $$P(S) = 1$$ 3. $$P(A^c) = 1 - P(A)$$ 4. $$P(A \text{ or } B ) = P(A) + P(B) - P(A \text{ and } B)$$ 5. $$P(A \text{ and } B ) = P(A\|B) P(B)$$ 6. Independent: if and only if $$P(A\|B) = P(A)$$. ### General multiplication rule Note that: \begin{aligned} P(A\|B) & = \frac{P(A \text{ and } B )}{P(B)} \\ P(B\|A) & = \frac{P(A \text{ and } B )}{P(A)} \end{aligned} Therefore: $P(A \text{ and } B ) = P(A\|B) P(B) = P(B\|A) P(A)$ Both are correct. ### More than two events: In later courses you will work with objects like this: \begin{aligned} P(X, & \mu, \text{ and } \sigma) \\ & = P(X, \big\{ \mu \text{ and } \sigma \big\} ) \\ & = P(X\|\big\{ \mu \text{ and } \sigma \big\}) P( \big\{ \mu \text{ and } \sigma \big\} ) \\ & = P(X\| \mu, \sigma ) P(\mu\|\sigma)P(\sigma) ) \end{aligned} This chaining of the general multiplication rule is important for: • Hierarchical Linear Models (HLM) • All of Bayesian statistics ### An example In a certain region, one in every thousand people (0.001) of all individuals are infected by the HIV virus that causes AIDS. Tests for presence of the virus are fairly accurate but not perfect. If someone actually has HIV, the probability of testing positive is 0.95. Let $$H$$ denote the event of having HIV, and $$T$$ the event of testing positive. \begin{aligned} P(H) & = ? & P(T) & = ? \\ P(H \text{ and } T) & = ? & P(H \text{ or } T) & = ? \\ P(H\|T) & = ? & P(T\|H) & = ? \end{aligned} ### An example In a certain region, one in every thousand people (0.001) of all individuals are infected by the HIV virus that causes AIDS. Tests for presence of the virus are fairly accurate but not perfect. If someone actually has HIV, the probability of testing positive is 0.95. Let $$H$$ denote the event of having HIV, and $$T$$ the event of testing positive. \begin{aligned} P(H) & = 0.001 & P(T) & = ? \\ P(H \text{ and } T) & = ? & P(H \text{ or } T) & = ? \\ P(H\|T) & = ? & P(T\|H) & = 0.95 \end{aligned} ### An example What is the probability that someone chosen at random tests has HIV and tests positive? \begin{aligned} P(H) & = 0.001 & P(T) & = ? \\ P(H \text{ and } T) & = ? & P(H \text{ or } T) & = ? \\ P(H\|T) & = ? & P(T\|H) & = 0.95 \end{aligned} We need: \begin{aligned} P(H \text{ and } T) & = P(T\|H)P(H) \\ & = (0.95) (0.001) = 0.00095 \end{aligned} implying that approximately 1/10 of 1% of people will have HIV and will test positive for it. ### A further example A sales representative tells his friend that the probability of landing a major contract by the end of the week, resulting in a large commission, is .4. If the commission comes through, the probability that he will indulge in a weekend vacation in Bermuda is .9. Even if the commission doesn't come through, he may still go to Bermuda, but only with probability .3. \begin{aligned} P(C) & = ? & P(V) & = ? \\ P(V\|C) & = ? & P(V\|\text{not } C) & = ? \\ \end{aligned} ### A further example A sales representative tells his friend that the probability of landing a major contract by the end of the week, resulting in a large commission, is .4. If the commission comes through, the probability that he will indulge in a weekend vacation in Bermuda is .9. Even if the commission doesn't come through, he may still go to Bermuda, but only with probability .3. \begin{aligned} P(C) & = 0.40 & P(V) & = ? \\ P(V\|C) & = 0.90 & P(V\|\text{not } C) & = 0.3 \\ \end{aligned} ### Probability Trees “the probability of landing a major contract … is .4” +----0.40 / C / ---< \ not C \ +----0.60 ### Probability Trees “If the commission comes through, the probability [of a] vacation … is .9.” /-----0.90 V / +----0.40-----< / \ C not V / \-------------[ ---< \ /--------------[ not C V \ / +----0.60----< \ not V \--------------[ ### Probability Trees “If the commission comes through, the probability [of a] vacation … is .9.” /-----0.90 V / +----0.40-----< / \ C not V / \-----0.10 ---< \ /--------------[ not C V \ / +----0.60----< \ not V \--------------[ ### Probability Trees “Even if the commission doesn't come through, he may still go … with probability .3.” /-----0.90 V / +----0.40-----< / \ C not V / \-----0.10 ---< \ /------0.30 not C V \ / +----0.60----< \ not V \------0.70 ### Read off conditional probabilities /--P(V\|C) = 0.90 V P(C) / +--0.40--< / \ C not V / \--P(not V\|C) = 0.10 ---< \ /--P(V\|not C) = 0.30 not C V \ / +--0.60--< P(not C) \ not V \--P(not V\|not C) = 0.70 ### A two-way probability table \| V \| not V \| Total ------\|------\|-------\|------ C \| \| \| 0.40 ------\|------\|-------\|------ not C \| \| \| 0.60 ------\|------\|-------\|------ Total \| \| \| 1 From the tree we have • $$P(V\|C)$$, $$P(\text{not } V\|C)$$, • $$P(V\|\text{not } C)$$ and $$P(\text{not } V\|\text{not } C)$$. How do we get $$P(V \text{ and } C)$$ etc. to put in the table? ### E.g. Probability of a vacation? There are two ways to take a vacation, with and without the commission: $P(V) = P(V \text{ and } C) + P(V \text{ and not } C)$ By the general multiplication rule we have: \begin{aligned} P(V \text{ and } C) & = P(V\|C) P(C) \\ P(V \text{ and not } C) & = P(V\|\text{not } C) P(\text{not } C) \end{aligned} Therefore: $P(V) = P(V\|C) P(C) + P(V\|\text{not } C) P(\text{not } C)$ ### Probability of a vacation? $$P(V) = P(V\|C) P(C) + P(V\|\text{not } C) P(\text{not } C)$$ +-0.90 # P(V\|C)P(C) = 0.90 * 0.40 V = 0.36 \| +-0.40-+ \| \| C not V \| +-0.10 < \| +-0.30 not C V \| \| +-0.60-+ \| not V +-0.70 ### Probability of a vacation? $$P(V) = P(V\|C) P(C) + P(V\|\text{not } C) P(\text{not } C)$$ +-0.90 # P(V\|C)P(C) = 0.90 * 0.40 V = 0.36 \| +-0.40-+ \| \| C not V \| +-0.10 < \| +-0.30 # P(V\|not C)P(not C) not C V = 0.60 * 0.30 \| \| = 0.18 +-0.60-+ \| not V +-0.70 ### Probability of a vacation? $$P(V) = P(V\|C) P(C) + P(V\|\text{not } C) P(\text{not } C)$$ +-0.90 # P(V\|C)P(C) = 0.90 * 0.40 V = 0.36 \| +-0.40-+ \| \| C not V \| +-0.10 < \| +-0.30 # P(V\|not C)P(not C) not C V = 0.60 * 0.30 \| \| = 0.18 +-0.60-+ ______________________ \| { Therefore: } not V { P(V) = 0.36 + 0.18 } +-0.70 { = 0.54 } ### A two-way probability table After multiplying out the tree: \| V \| not V \| Total ------\|------\|-------\|------ C \| 0.36 \| \| 0.40 ------\|------\|-------\|------ not C \| 0.18 \| \| 0.60 ------\|------\|-------\|------ Total \| 0.54 \| \| 1 ### A two-way probability table And finding the rest: \| V \| not V \| Total ------\|------\|-------\|------ C \| 0.36 \| 0.04 \| 0.40 ------\|------\|-------\|------ not C \| 0.18 \| 0.42 \| 0.60 ------\|------\|-------\|------ Total \| 0.54 \| 0.46 \| 1 ### Summary thus far.... Two way tables (or Venn Diagrams) • when the problem gives $$P(A \text{ and } B)$$ etc. Probability trees • when the problem gives $$P(A\|B)$$ etc. Can convert back-and-forth between them as needed. ### An exercise Suppose the friend left for a week and came back to the office. When the friend returned, the salesman had left for Bermuda. What is the probability that the salesman received a commission given that he is on vacation in Bermuda? $P(C\|V) = ?$ ### From the probability table ... $P(C\|V) = ?$ \| V \| not V \| Total ------\|------\|-------\|------ C \| 0.36 \| 0.04 \| 0.40 ------\|------\|-------\|------ not C \| 0.18 \| 0.42 \| 0.60 ------\|------\|-------\|------ Total \| 0.54 \| 0.46 \| 1 $P(C\|V) = \frac{P(C \text{ and } V)}{P(V)} = \frac{.36}{.54} = .667$ implying that there is a 67% chance that he received the commission. ### Tree to table is an unsatisfying The probability of landing a major contract by the end of the week is .4. If the commission comes through, the probability that he will vacation in Bermuda is .9. Even if the commission doesn't come through, he may still go to Bermuda, but only with probability .3. \begin{aligned} P(C) & = 0.40 & P(V) & = ? \\ P(V\|C) & = 0.90 & P(V\|\text{not } C) & = 0.3 \\ \end{aligned} What is the probability that the salesman received a commission given that he is on vacation in Bermuda? $P(C\|V) = ?$ ### A better way Recall: \begin{aligned} P(A\|B) & = \frac{P(A \text{ and } B )}{P(B)} \\ P(B\|A) & = \frac{P(A \text{ and } B )}{P(A)} \\ P(A \text{ and } B ) & = P(A\|B) P(B) = P(B\|A) P(A) \end{aligned} Therefore: $P(A\|B) = \frac{P(B\|A) P(A)}{P(B)}$ This is Bayes' Rule. ### Bayes' Rule and Total Probability Bayes' Rule: $P(A\|B) = \frac{P(B\|A) P(A)}{P(B)}$ • Allows you to reverse a conditional probability. Law of Total Probability \begin{aligned} P(B) & = P(B \text{ and } A) + P(B \text{ and not} A) \\ & = P(B\|A)P(A) + P(B\|\text{not }A)P(\text{not A}) \end{aligned} • combine with Bayes' Rule to reverse a probability tree. ### Salesman Reprise \begin{aligned} P(C) & = 0.40 & P(V) & = ? \\ P(V\|C) & = 0.90 & P(V\|\text{not } C) & = 0.3 \\ \end{aligned} What is the probability that the salesman recieved a commission given that he is on vacation in Bermuda? $P(C\|V) = ?$ \begin{aligned} P(C\|V) & = \frac{P(V\|C) P(C)}{P(V)} \\ P(V) & = P(V\|C)P(C) + P(V\|\text{not }C)P(\text{not C}) \end{aligned} ### Salesman Reprise \begin{aligned} P(C) & = 0.40 & P(V) & = ? \\ P(V\|C) & = 0.90 & P(V\|\text{not } C) & = 0.3 \\ \end{aligned} What is the probability that the salesman recieved a commission given that he is on vacation in Bermuda? $P(C\|V) = ?$ \begin{aligned} P(C\|V) & = \frac{ 0.90 * 0.40 }{P(V)} \\ P(V) & = .90 * 0.40 + 0.3 * (1 - 0.40) = 0.667 \end{aligned} ### Salesman Reprise \begin{aligned} P(C) & = 0.40 & P(V) & = ? \\ P(V\|C) & = 0.90 & P(V\|\text{not } C) & = 0.3 \\ \end{aligned} What is the probability that the salesman recieved a commission given that he is on vacation in Bermuda? \begin{aligned} P(C\|V) & = \frac{ 0.90 * 0.40 }{0.54} \\ & = 0.667 \end{aligned} implying that there is a 67% chance that he recieved the comission. ### Summary Recall that Stat 202 will let you solve probability problems your own way. • brute force: (i) make a probability tree, then (ii) make a table, then (iii) find the relevant conditional probability. • elegant: Bayes' Rule, the Law of Total Probability, and other probability rules. ### A spy example Polygraph (lie-detector) tests are often routinely administered to employees or prospective employees in sensitive positions. Lie detector results are “better than chance, but well below perfection.” Typically, the test may conclude someone is a spy 80% of the time when he or she actually is a spy, but 16% of the time the test will conclude someone is a spy when he or she is not. Let us assume that 1 in 1,000, or .001, of the employees in a certain highly classified workplace are actual spies. ### A spy example Test may conclude someone is a spy 80% of the time when he or she actually is a spy, but 16% of the time the test will conclude someone is a spy when he or she is not. Assume that 1 in 1,000, or .001, are actual spies. $P(S) = ?$ $P(S) = 0.001$ $P(D\|S) \text{ or } P(S\|D)?$ $P(D\|S) = 0.80$ $P(D\|\text{not }S) = ?$ $P(D\|\text{not }S) = .16$ ### A spy example $P(S) = 0.001$ $P(D\|S) = 0.80$ $P(D\|\text{not }S) = .16$ If the polygraph detects a spy, are you convinced that the person is actually a spy? $P(S\|D) \text{ or } P(D\|S)?$ $P(S\|D) = \frac{P(D\|S)P(S)}{P(D)}$ $P(D) = P(D\|S)P(S) \\ + P(D\|\text{not }S)P(\text{not }S)$ ### A spy example Law of total probability: \begin{aligned} P(D) & = P(D\|S)P(S) + P(D\|\text{not }S)P(\text{not }S) \\ & = .80 * .001 + .16 * (1 - .001) \\ & = .161 \end{aligned} Bayes' Rule \begin{aligned} P(S\|D) & = \frac{P(D\|S)P(S)}{P(D)} \\ & = \frac{ 0.80 * 0.001 }{ .161} \\ & = 0.005 \end{aligned} ### A spy example If the polygraph detects a spy, are you convinced that the person is actually a spy? $P(S\|D) = 0.005$ implying that about one half of one percent of “detections” are actual spies. Are you convinced? ### $$P(S\|D) \ne P(D\|S) \text{reprise}$$ The order of conditioning matters: $P(S\|D) = 0.005$ implying that about one half of one percent of “detections” are actual spies. $P(D\|S) = 0.80$ implying that 80% of actual spies are dectected by the test. Careful attention is required! ### Bayesian Statistics c. 1701-1761 • Richard Price published Bayes' Rule after Bayes' death. 1749 - 1847 • Set the foundation for Bayesian Statististics	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 2, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9987303018569946, "perplexity": 3680.277738390379}, "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-2017-09/segments/1487501172156.69/warc/CC-MAIN-20170219104612-00170-ip-10-171-10-108.ec2.internal.warc.gz"}
https://planetmath.org/mathitSL2F3	$\mathit{SL}_{2}(F_{3})$ The special linear group over the finite field $\mathbbmss{F}_{3}$ is represented by $\mathit{SL}_{2}(\mathbbmss{F}_{3})$ and consists of the $2\times 2$ invertible matrices with determinant equal to $1$ and whose entries belong to $\mathbbmss{F}_{3}$. Title $\mathit{SL}_{2}(F_{3})$ mathitSL2F3 2013-03-22 14:00:32 2013-03-22 14:00:32 drini (3) drini (3) 9 drini (3) Definition msc 20G40 SL(F3) FiniteField Field Group	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 7, "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": 0.9144585132598877, "perplexity": 1918.6051790289782}, "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-2019-39/segments/1568514572879.28/warc/CC-MAIN-20190916155946-20190916181946-00042.warc.gz"}
http://support.river-valley.com/wiki/index.php?title=FAQ_-_elsarticle.cls	# FAQ - elsarticle.cls ## Preamble and Front matter 1. When I compile the pdf with elsarticle, I always get a blank first page which is numbered 1. The rest of the document starts at page 2. [Answer] 2. I have a problem with the document class elsarticle. In the final document, the first page is empty except for a comma, and the title. The abstract begins in the second page. With elsart.cls I did not encounter this problem. [Answer] 3. I am using elsarticle.cls and having problems with the abstract part. The abstract is missing from the PDF even if I code it using the {abstract} environment. [Answer] 4. The abstract is obtained as two successive lines only instead of a paragraph, even if it is coded in the {abstract} environment. [Answer] 5. I am using the lineno package to add line numbers to the manuscript before submission. This does not number the lines in the abstract. Is there a way around this? [Answer] 6. The title page appeared twice. How can I remove one? [Answer] 7. How can I typeset the title page and the main matter in separate pages? [Answer] 8. I have made numerous attempts to get the email addresses to appear on the title page of my article, but with no luck. [Answer] 9. How should I code a dedication? [Answer] 10. Whenever I try to add one of the options 1p, 3p, or 5p, I get the error" Package keyval Error: centering undefined.. What can be the problem [Answer] 11. I am writing on behalf of the IceCube collaboration, which has around 250 scientists and 39 institutions involved. When I put in all authors and institutions, I get the following error message when trying to compile with latex: ! LaTeX Error: Counter too large.. Could you please help? [Answer] 12. Some Elsevier journals require a Title page with only Title of the manuscript, Authors and Addresses, following with the main matter, including only Title and Abstract, beginning in a new page. I am wondering how can I do this in elsarticle class. [Answer] ## Main matter 1. I am preparing my paper using elsarticle.cls and LaTeX. Can you tell me how to compress lists of at least three consecutive numerical citations that occur together in the text? [Answer] 2. I am using the elsarticle.cls package for my paper submission. It really works very well, but I cannot use the natbib package with the sort&compress option; an option clash error appears when I use it. In such a case, the reference citation appears as [30, 40, 15, 31, 32, 4] rather than [4, 15, 30-32, 40] as does the ordinary article class. [Answer] 3. I am using elsarticle.cls, and I want to present two subfigures side by side, caption them separately and also would like to incorporate a global caption. [Answer] 4. When I use the review option, my tables become distorted due to double spacing. How can I prevent double spacing? Also, is there a way to prevent double spacing for a portion of the document? [Answer] 5. I have coded the bibliography for author–year citations. But still I obtain numbered citations. How can I format the citation in the author–year form? [Answer] 7. How can I create an acknowledgement section? [Answer] 8. When I use \Box, an error message appears — \Box not provided in base LaTeX2e'. [Answer] ## Back matter 1. I just cannot typeset the references and bibliography in harvard style (i.e., author–year format). [Answer] 2. I am using elsarticle-harv.bst' as a bibtex style file that is supposed to produce references in an author–year citation style. However, it produces the references in numerical style. Can you please tell me what went wrong? [Answer] 3. I have a problem with bibliography. The reference list does not appear. Also the ?' only appears where the references are cross-referred to. [Answer] 4. How do I get natbib to generate the references with journal abbreviations rather than full journal names. [Answer] ## General questions 1. I would like to sumbit a paper to Elsevier. Please advise me on how to proceed? [Answer] 2. I would like to know the status of my submited paper. [Answer] 3. I am preparing a paper for Chemical Engineering Science. I have downloaded the required LaTeX package. However, I am confused about which reference style I should use for this journal? [Answer] 4. I am using a template file (elsarticle-template-num). For the abstract part, the paper asks for an MSC code. Where do I find that code for my paper? [Answer] 6. How do I create the same line- and page-breaks as in the final print copy? [Answer] 7. I downloaded the elsarticle class file, put it in the appropriate directory, and when I try to compile the file using the simple template file provided by elsvier, it will not output. In fact, it gets hung up on the following line. I checked the path, and I do have the upsy.fd file installed on my computer, so I really am hung on this one. [Answer] 8. I use Latex at first time. I don't understand what I have do write from the beginning until \begin{frontmatter}. [Answer] 9. I have the following problem when compiling. What to do? [Answer] “C:\Program Files\MiKTeX 2.8\tex\latex\psnfss\upsy.fd”))latex.exe: GUI framework cannot be initialized” ?`	{"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": 0.8605863451957703, "perplexity": 1408.8385662993794}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860121976.98/warc/CC-MAIN-20160428161521-00202-ip-10-239-7-51.ec2.internal.warc.gz"}
http://arxiv.org/abs/hep-th/0312069	hep-th (what is this?) (what is this?) # Title: The M-theory 3-form and E8 gauge theory Abstract: We give a precise formulation of the M-theory 3-form potential C in a fashion applicable to topologically nontrivial situations. In our model the 3-form is related to the Chern-Simons form of an E8 gauge field. This leads to a precise version of the Chern-Simons interaction of 11-dimensional supergravity on manifolds with and without boundary. As an application of the formalism we give a formula for the electric C-field charge, as an integral cohomology class, induced by self-interactions of the 3-form and by gravity. As further applications, we identify the M-theory Chern-Simons term as a cubic refinement of a trilinear form, we clarify the physical nature of Witten's global anomaly for 5-brane partition functions, we clarify the relation of M-theory flux quantization to K-theoretic quantization of RR charge, and we indicate how the formalism can be applied to heterotic M-theory. Comments: 48pp. harvmac b-mode; v2: Several improvements added in response to the referee's comments. Typos fixed Subjects: High Energy Physics - Theory (hep-th) Report number: RUNHETC-2003-34 Cite as: arXiv:hep-th/0312069 (or arXiv:hep-th/0312069v2 for this version) ## Submission history From: Gregory Moore [view email] [v1] Fri, 5 Dec 2003 18:58:12 GMT (47kb) [v2] Tue, 23 Mar 2004 17:12:43 GMT (50kb)	{"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": 0.8434420228004456, "perplexity": 2452.2139924168077}, "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-2016-30/segments/1469257828009.82/warc/CC-MAIN-20160723071028-00082-ip-10-185-27-174.ec2.internal.warc.gz"}
http://mathoverflow.net/revisions/76765/list	A $2n\times 2n$ dimensional Hermitian matrix that can be diagonalized by a symplectic transformation can be viewed as an $n\times n$ matrix with elements consisting of $2\times 2$ blocks of the quaternion real form ${\bar{z}\;-\bar{w}}\choose{w\;\; z}$ so if you choose real $z$ and $w$ you have constructed a real symmetric matrix $M$ that can be diagonalized by a symplectic $S$; is S$. @Federico: this what you need?is the general form for matrices that commute,$MT=TM$, with$T=1_{N}\otimes{0\; 1}\choose{-1\; 0}K$($K$is the operator of complex conjugation); alternatively, one can take matrices that anticommute,$MT=-TM$; then the$2\times 2$blocks have the form${\bar{z}\;\bar{w}}\choose{w\;\; -z}$and again, for a real$M$one would choose real$w,z$. these two choices exhaust the possibilities. In applications to physical systems, the matrix$M$is a Hamiltonian and$T$is the operator of time reversal. Then only commuting matrices,$MT=TM$, are permitted. For a discussion in the physics context, see Section 1.4.2 of Forrester's book, online here: http://www.ms.unimelb.edu.au/~matpjf/b1.ps Post Undeleted by Carlo Beenakker 2 new attempt at an answer The condition is A$2n\times 2n$dimensional Hermitian matrix that the 2n eigenvalues can be diagonalized by a symplectic transformation can be viewed as an$n\times n$matrix with elements consisting of M should come in n twofold degenerate pairs. Then$2\times 2$blocks of the matrix S is both orthogonal quaternion real form${\bar{z}\;-\bar{w}}\choose{w\;\; z}$so if you choose real$z$and$w$you have constructed a real symmetric matrix$M$that can be diagonalized by a symplectic .$S\$; is this what you need?	{"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": 0.842527449131012, "perplexity": 435.73906664557677}, "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/1368706631378/warc/CC-MAIN-20130516121711-00022-ip-10-60-113-184.ec2.internal.warc.gz"}
http://mathoverflow.net/questions/73551/smoothness-along-rays-sufficient-for-global-smoothness	# Smoothness along rays sufficient for global smoothness Hi, Suppose I have a function $f:\mathbb{R}^d \to \mathbb{R}$ and I know that $f$ is smooth ($C^\infty$) along each ray $t \mapsto f(td)$ on $t \in [-\epsilon, \epsilon]$ and all directions $d \in \mathbb{R}^d$. Is smoothness along these rays sufficient for $f$ to be smooth around $0$ as a multivariate function (all partial derivatives exist)? Thanks. - No. Just take a non-smooth function $g \colon S^{d-1} \to \mathbb{R}$ with $g(-p) = -g(p)$ and define $f(v) = \\|v\\|g(v/\\|v\\|)$. This is smooth since on each ray it is just $t \mapsto k t$ for some $k$, but the non-smoothness of $g$ means that $f$ is not smooth. The closest to this that I know of is if you take all smooth curves (not just rays). Then $f$ is smooth. This is due to Jan Boman. – Andrew Stacey Aug 24 '11 at 10:13 That is indeed a nice counterexample. Would you happen to have a reference of this result of Jan Boman? In addition, would smoothness along all real analytic curves be sufficient too? – Bart Aug 24 '11 at 12:15 The result Andrew Stacey refers to is this paper of Boman (MathSciNet link): ams.org/mathscinet-getitem?mr=237728 The article doesn't look like it is available online. In the MathSciNet review the following counterexample is mentioned: there exists a non-continuous function $f$ from $\mathbb{R}^2\to\mathbb{R}$ such that $f\circ u \in C^{\infty}(\mathbb{R},\mathbb{R})$ for every quasianalytic $u$. (Though it is not clear which quasianalytic class the example applies to.) This should give at least a partial answer to your follow-up question. – Willie Wong Aug 24 '11 at 13:12 Thanks Willie. Looking at the Boman paper, demanding smoothness along only analytic curves is indeed not sufficient. It turns out that when $f \circ u$ is real analytic for every real analytic curve $u$, that $f$ is real analytic. See "An Ontology of Directional Regularity Implying Joint Regularity" published in Real Analysis Exchange, available at math.wustl.edu/~sk/joint.pdf . – Bart Aug 24 '11 at 13:35 Apparently, in the published version there is the condition that the k-th partial derivative of $f \circ u$ needs to be smaller than $C k! / r^k$ for some $r>0$. – Bart Aug 24 '11 at 14:19 show 1 more comment	{"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": 0.807005763053894, "perplexity": 520.4350206103924}, "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-2013-48/segments/1387345777253/warc/CC-MAIN-20131218054937-00033-ip-10-33-133-15.ec2.internal.warc.gz"}
https://support.airkit.com/reference/exp	The function EXP take some number n and returns en. This function takes a Number as input. It returns another Number: the numerical constant e taken to the power of the given Number. ### Declaration ``````EXP(n) -> number `````` Parameters n (required, type: `Number`) Any Number; this is what e will be taken to the power of. ### Return Values number (type: `Number`) The numerical estimation of e raised to the nth power, or en. ### Examples The following example calculates a numerical estimation of e1, or simply e. Note that e is an irrational number; the output of this function is an approximation of the value of e rather than an infinite decimal. This should be sufficient for most practical applications, though it can be the source of small rounding errors: ``````EXP(1) -> 2.718281828459045 `````` The EXP function does not require input in whole numbers. The following example calculates a numerical estimation of e0.5 (otherwise known as the square root of e): ``````EXP(0.5) -> 1.6487212707001282 `````` ### Discussion The function EXP only takes the constant e to a given power. To calculate any number to a given power, use the POWER function.	{"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": 0.8094032406806946, "perplexity": 1260.0529660103073}, "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-2022-49/segments/1669446710417.25/warc/CC-MAIN-20221127173917-20221127203917-00248.warc.gz"}
https://www.physicsforums.com/threads/conceptual-understanding-of-derivates-question.768488/	# Conceptual understanding of Derivates Question 1. Sep 1, 2014 ### RJLiberator Question:A bicyclist starts from home and rides back and forth along a straight East/West path. Her (instantaneous) velocity as a function of time is given by v(t), where time t is measured in minutes. Consider: $d_{1} = \int^{30}_{0} v(t) dt$ and $d_{2} = \int^{30}_{0} \|v(t)\| dt$ Choose what it represents from the following: (a) the total distance the bicyclist rode in 30 minutes (b) the bicyclist's average velocity over 30 minutes (c) the bicyclist's distance from the home after 30 minutes (d) none of the above. My thinking: Allright well, the two integrals (d1 and d2) are essentially the same thing other than the absolute value sign. This leads me to believe that d1 is (c) and d2 is (a). This is kind of a logical guess on my part. Can anyone explain what the equation is actually saying? To me, d1 is saying: The area under the curve from 0minutes to 30 minutes is represented by v'(t). This would mean to me that (c) should be the correct answer. While, d2 is saying the absolute value (negative and positive area combined) of v'(t) from 0mintues to 30 minutes. This would be the total distance traveled aka (a). If my conceptual understanding/writing is wrong, please do point it out to me. Is my thinking correct? Thank you all for your help. 2. Sep 1, 2014 ### mal4mac Do you mean, "Choose what each represents from the following"? What's v'(t)? General tip - draw (rough) graphs of v against t and \|v\| against t and remember the general definition of integration = "area under the curve". 3. Sep 1, 2014 ### RJLiberator Yes, I do mean that. Thank you. Hm. The only problem with graphing is, I don't have a function to graph. So this would be a purely conceptual question. I suppose. Maybe I can use a dummy function and try to graph that to help my conceptual understanding. I will try this. 4. Sep 1, 2014 ### HallsofIvy Staff Emeritus Yes, you are correct- (d2) is the total distance ridden, (d1) is the distance from home. You can check that by taking a simple example. Suppose she rides directly away from home at constant velocity v (meters per minute)> 0 for 15 minutes, then rides directly back home with velocity -v for 15 minutes. After 15 minutes she will be 15v meters from home, then she turns around and after another 15 minutes is back home: 0 meters from home. But she rode a total of 30v meters. $$d1= \int_0^{30} v(t)dt= \int_0^{15} v dt+ \int_{15}^{30}(-v)dt= v(15- 0)+ (-v)(30-15)= 15v- 15v= 0$$ $$d2= \int_0^{30} \|v(t)\|dt= \int_0^{30} v dt= 30v$$ Last edited: Sep 1, 2014 5. Sep 1, 2014 ### RJLiberator HallsofIvy, thank you for explaining the conceptual side AND the mathematical side of the problem to me. My hunch of logic has been verified. My kindest regards to you for helping me understand this. 6. Sep 1, 2014 ### RJLiberator One last Calculus question: So I understand that the integral of e^t^2 is not a normal integral and cannot be easily calculated. But in situation where the integral from 0 to sqrt(x) of e^t^2 Is stated and then it asks me to Compute f'(x) Can I accomplish this? My thinking is that, Sure the Integral is not really possible for me to computer, by does f'(x) = the integral? I don't believe so. 7. Sep 1, 2014 ### WWGD Look into the Fundamental Theorem of Calculus, e.g.,http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus . WWGD :What Would Gauss Do? 8. Sep 1, 2014 ### RJLiberator Okay, so I am trying to work on this. I plug the integral into a calculator (for integrals) and receive the answer of: e$^{x}F\sqrt{x}$ I'm just not quite sure what this is telling me. This seems to be telling me the area under the curve from 0 to sqrt(x) of the initial function. I am trying to find f'(x) So I've been supplied (post above) with the fundamental theorem of Calculus. This seems to help, but I'm not putting everything together. So, since f(x) = e^t^2 dt is f'(x) = 2t*e^t^2 Is it that simple? 9. Sep 1, 2014 ### Ray Vickson No. I hope you realize that what you wrote is nonsense: you have x on one side and t on the other. On the other hand, maybe you do not actually mean what you wrote. Originally you had $$f(x) =\int_0^{\sqrt{x}} e^{t^2} \, dt \,$$ although you wrote something that could be interpreted as $(e^t)^2$, which is very different. Use parentheses, like this: e^(t^2). Also: if you are just a beginner at this material, I would recommend that you avoid using the calculator, except for numerics; just do things directly, by hand, and reason it out step-by-step, taking as long as necessary and using as much paper as you need. With practice you will get better and faster---but likely only if you avoid the calculator. 10. Sep 1, 2014 ### mal4mac Suggestion on writing equations in physics forums - look up to the right of the editor and you will see an editing symbol that looks like X2. Click on that to write expressions like et2 Draft saved Draft deleted Similar Discussions: Conceptual understanding of Derivates Question	{"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": 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": 0.8843520879745483, "perplexity": 1187.295418108658}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886106754.4/warc/CC-MAIN-20170820131332-20170820151332-00690.warc.gz"}
https://www.clutchprep.com/physics/practice-problems/41064/a-small-rock-block-with-mass-0-400-kg-is-placed-against-a-compressed-spring-at-t-1	Springs & Elastic Potential Energy Video Lessons Example # Problem: A small rock block with mass 0.400 kg is placed against a compressed spring at the bottom of a 37.0° incline. The compressed spring has 50.0 J of elastic potential energy stored in it. The spring is released and the block moves a distance of 12.0 m along the incline before momentarily coming to rest. How much work does the friction force do on the block during the motion? What is the coefficient of kinetic frinction μk between the block and the incline? 91% (4 ratings) ###### Problem Details A small rock block with mass 0.400 kg is placed against a compressed spring at the bottom of a 37.0° incline. The compressed spring has 50.0 J of elastic potential energy stored in it. The spring is released and the block moves a distance of 12.0 m along the incline before momentarily coming to rest. How much work does the friction force do on the block during the motion? What is the coefficient of kinetic frinction μk between the block and the incline?	{"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": 0.9300209283828735, "perplexity": 400.70499419770994}, "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-2020-45/segments/1603107897022.61/warc/CC-MAIN-20201028073614-20201028103614-00292.warc.gz"}
http://comprensivobovalino.it/jktg/norm-in-latex-overleaf.html	Use dollar symbols around the text. You just need to escape characters that have special meaning: # $% & ~ _ ^ \ { }. The last point is the norm of a vector, the square root from the sum of its components. aa aah aahed aahing aahs aal aalii aaliis aals aardvark aardvarks aardwolf aardwolves aargh aas aasvogel aasvogels aba abaca abacas abaci aback abacus abacuses abaft. There are currently two packages providing international language support, namely, Polyglossia and Babel. I suggest to define new commands (again with amsmath ) in the preamble for an easier workflow. Number sets (prime, natural, integer, rational, real and complex) in LaTeX. x 0 doit être placé sous x -- 0 qui. Number sets such as natural numbers or complex numbers are not provided by default by LaTeX. It's widely used throughout the academic world to publish technical. This online tool is a wrapper script for latexdiff, which is the LaTeX alternative to Track Changes in Microsoft Word. Such basics can be found in introductions like lshort. The main idea of CS is, by exploiting the sparsity nature of the signal (in any domain), we can reconstruct the signal from very fewer samples than required by Shannon-Nyquist sampling theorem. 예를 들어 em이나 px. d3 is the first move in \variation{} Open an example of the skak package in ShareLaTeX. - texasbruce Jan 7 '13 at 20:29. d4 is the last move before analysing a variation, hence 3. I usually add a LaTeX comment that mentions the lipsum package and \lipsum command simply generate filler text. Using prod Edit. Posted: (18 days ago) Beamer - Overleaf, Online LaTeX Editor. Uppercase Greek Letters Binary operation symbols. This tag should be used for questions that concern the backend; questions concerning use of the web service are probably better asked on tex. Open an example in Overleaf. 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See images here : Overleaf v2$\endgroup$– Frobenius Jul 16 '18 at 21:25 \|.$\begingroup@knzhou So, I was lucky to find an online editor : Overleaf (first v1 and later v2). Author and date are the key components in the in-text citation of the APA referencing style. Similar is for limit expressions. Add a New Comment. sourceforge. \left[ \begin {array}{ccc} 9&13&17\\ oalign{\medskip} 14&18&22 \end {array} \right] If you put this code inside a LaTeX displaymath environment, you will get the matrix typeset. References are not included in the 10-page. Um editor de LaTeX online fácil de usar. Formatting Optimization Problems with LaTeX. Prebiotic foods like garlic, onions and avocado help to reduce stress, as well as food high in Omega 3. Starting at step (2-4) there is a single sentence that continues to the end of p60. Para escribir el valor absoluto o la norma en LaTeX es conveniente crear un comando antes que usar la barra vertical "\|". 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The following table shows the whole Greek alphabet along with the commands in a nice table. Citations are an abomination, as are numbering equations; random crashes, corruption of files that contain a lot of graphics etc. \begin {document} View which changes have been added and removed. Similar is for limit expressions. Some of these commands are LATEX macros, while oth- ers belong to plain TEX; no attempt to di eren-. Open an example in ShareLaTeX. x \pod a x (a) Use \equiv to get the congruence symbol (three…. Compressive Sensing is a Signal Processing technique, which gave a break through in 2004. If you want the limits of an integral/sum/product to be specified above and below the symbol in inline math mode, use the \limits command before limits specification. The Möbius function is a more complicated example of a three-part definition. % Specifies … Continue reading "LaTeX - bold vectors and arrow vectors". 1 Beginning a document ndocumentclassfarticleg nusepackagefgraphicx, amssymbg nbeginfdocumentg ntextwidth 6. For example, Authorea has tools to handle the annoying parts of writing like automated reference insert and formatting, and export in different journal formats. Integral\int_ {a}^ {b} x^2 dxinside. Code: [Select all] [Expand/Collapse] [Download] (untitled. Add a New Comment. In a similar manner to the summation symbol, the product symbol can easily be added to a LaTeX document using the \prod notation. Specifically a state diagram describes the behavior of a single object in response to a series of events in a system. LaTeX is a high-quality typesetting system; it includes features designed for the production of technical and scientific documentation. 常用英语词汇6000_英语考试_外语学习_教育专区。常用英语词汇6000. Information used to produce the title is obtained from the following declarations, which should precede the \maketitle command. They'll be productive from day one and be able to pick up small amounts of LaTeX as they go. Overleaf is a collaborative cloud based LaTeX tool which makes the process of writing, editing and publishing scientific papers faster and easier. This is the basic introduction to Matlab. 4A65G69 1995 95-36881 688. The main idea of CS is, by exploiting the sparsity nature of the signal (in any domain), we can reconstruct the signal from very fewer samples than required by Shannon-Nyquist sampling theorem. We appreciate Lindsay being able to svolcano at our inaugural Raleigh FuturePub and share the work Research Square is doing in the scholarly issueing and collaborative survey space. here 1 is subscript of C. Sign in to make your opinion count. For more information about the Encyclopedia, see the Welcome page. d3 is the first move in \variation{} Open an example of the skak package in ShareLaTeX. Reduces page margins to 0. A rendered preview of all letters is shown alongside all commands in a nice table. Foreign symbols. com , or follow them on twitter at @Overleaf. 93 silver badges. If you do want full latex, try Overleaf. tab samples azimuth azimuth_dec_correction citations description dip geologic_classes geologic_types lat lithologies lon method_codes orientation_quality sample site 260 0 This study Archived samples from 1965, 66 expeditions. You can help LaTeX Wiki by expanding it. cursive 'l' Post by abhimanyulad ». Each provides a table for expressions, aligned in rows and columns. Para escribir el valor absoluto o la norma en LaTeX es conveniente crear un comando antes que usar la barra vertical "\|". Step 2: Enter author information. If you have any enquiries about this website or the content on it, please contact: [email protected] 1 DeclareRobustCommand. Eine gute Möglichkeit, Zitate einzufügen und zu formatieren, ist BibTeX (). \end {document}. The sybmol can be compressed to fit on one line (useful for small equations displayed within a text block), or enlarged to make it more readable. The command \variation{} helps to analyse variations of a move. Nobody knows if funeral prices will go up or down in future. Removes page numbers. Sometimes you find yourself wanting to put something over another symbol, e. Here's how to easily switch between a bold vector and an arrow vector. What are Green3, Blue3 or Red3? #2 Brent Longborough, October 8, 2013 at 3:10 p. d4 is the last move before analysing a variation, hence 3. You can use one of the predeﬁned layouts (pseudocode, pascal and c and others), with or without modiﬁcations,. Relational Operators (math mode). Here’s how to use the \$$\\LaTeX \$$ template from Springer for a Computer Science conference publication. In this post, I’m not so much interested in. (Since matrices are large, they are almost always set as displays. The literature argues that the cause of norm death is widespread violation. I will be using OverLeaf to run these written tutorials where we shall unleash a new part of LaTeX every week. As you can see, there are three basic commands and they can be nested to get combined effects. Student t Distributed Linear Value-at-Risk December 2, 2015 by Pawel One of the most underestimated feature of the financial asset distributions is their kurtosis. 75in for more space. They consist of plain text interspersed with some LaTeX commands. The LaTeX for Physicists Header has the following features: Sets font size to 11pt. The sybmol can be compressed to fit on one line (useful for small equations displayed within a text block), or enlarged to make it more readable. Sublime Text. ShareLaTeX is so easy to get started with that you'll be able to invite your non-LaTeX colleagues to contribute directly to your LaTeX documents. tex extension. HTML The icon in HTML, if it is defined as a named mark. Designed with a LinkedIn-like layout, this template reliefs the burden of worrying about customization and rather focuses on increasing productivity by filling out your CV's content in proper sectioned files; that can be later imported to the main. You edit and the compilation is direct and online showing you your probable mistakes. 85 Trachyte 166. This is an information page. d3 is the first move in \variation{} Open an example of the skak package in ShareLaTeX. Embed formulas in your text by surrounding them with dollar signs The equation environment is used to typeset one formula. In combination with a text editor and distributed version control, both reproducibility and simultaneous collaboration are improved by these plain text mark-up languages. at midnight than anyone else. 2 New environments. I think that providing compilable documents makes it lot easier for the reader to test the code to see if it has desired effect. You must use the following package: \usepackage {amsmath} \begin {matrix} \begin {pmatrix} \begin {bmatrix} \begin {vmatrix} \begin {Vmatrix}. Specifically a state diagram describes the behavior of a single object in response to a series of events in a system. Open an example in Overleaf. In LaTeX, I use a \qquad for this. To get exp to appear as a superscript, you type ^{exp}. Our team is passionate about delivering a flawless customer experience as we play matchmaker between great hotels with unsold rooms and the on-the-go people who want to book them. Miscellaneous symbols. The main idea of CS is, by exploiting the sparsity nature of the signal (in any domain), we can reconstruct the signal from very fewer samples than required by Shannon-Nyquist sampling theorem. In a three-part series for Weekend, Fiona Cairns reveals how to make a simplified version of the cake she made for the Duke and Duchess of Cambridge's Royal Wedding and other treats. By using this tool you avoid the command line and having to install Perl. Some preferred LaTeX/Overleaf, while others preferred the R/Knitr/RMarkdown/Sweave ecosystem. The tombstone is sometimes open: (hollow black square). My mental model is that the best we can hope for, the best attainable policy is the projection into the manifold of expertise-informed policies subject to the contraints of pressure from vested interests allowed at the time we chose that policy and available attention. The frequently used left delimiters include (, [ and {, which are obtained by typing (, [and \{respectively. It even does the right thing when something has both a subscript and a superscript. If you want the limits of an integral/sum/product to be specified above and below the symbol in inline math mode, use the \limits command before limits specification. It is a way of bridging the Wikipedia–academia gap by enabling academics, scholars and professionals to contribute expert knowledge to the Wikimedia movement in the familiar academic. Using a 30 day notice letter sample can give you a guide to work with and ensure that your letter of notice says on track. 3 Declare commands within new environment. bonsoir à tous j'aimerai écrire en latex limite quand x tend vers 0, x 0. November 2013 at 18:28. LaTeX template for the CFM 2017 article submission Instructions: Comments in the file below indicate where the following steps have to be performed. Since anti-HIV tests became commercially available in 1985 they have been widely used in diagnostic and transfusion laboratories in the developed world. Languages: English Shqip العربية Bangla Български Català 中文 ( 正體字 , 简化字 (1) , 简化字 (2)) Hrvatski Čeština Dansk Nederlands Esperanto Eesti فارسی Suomi Français Deutsch Ελληνικά ગુજરાતી עברית. sty file that contains these commands. This is a simple step, if you use LaTeX frequently surely you already know this. Thanks for contributing an answer to Stack Overflow! Please be sure to answer the question. See also % command used to denote a comment This article is a stub. As you are aware, there are commands to put a bar or a tilde over a symbol in math mode in LaTeX. com Knowledge base dedicated to Linux and applied mathematics. Basic LaTeX Julie Mitchell This resource was adapted from notes provided by Jerry Marsden 1 Basic Formatting 1. By default Latex justifies all your text so that it lines up on both the left and right margins. improve this answer. When the compiler processes your base file and reaches one of the commands \input or \include, it reads. 1 LaTeX LaTeX is a free document markup system speci cally for technical and scienti c writing. asd - Free ebook download as Text File (. The word command may sound scary. UnicodeMath resembles real mathematical notation the most in comparison to all of the math linear formats, and it is the most concise linear format, though some may prefer editing in the LaTeX input over UnicodeMath since that is widely used in academia. Computerized typesetting. Computerized typesetting. LaTeX symbols in MediaWiki. Bold, italics and underlining Simple text formatting helps to highlight important concepts within a document and make it more readable. In this article, I will walk you through the 5 different LaTeX templates that you will use one day or another and provide a simple overview of how to use them and what they mean. Area between Curves Calculator - eMathHelp. Overleaf es un programa online que nos permite editar y compilar desde nuestro navegador un documento de LaTeX. Relational Operators (math mode). The main idea of CS is, by exploiting the sparsity nature of the signal (in any domain), we can reconstruct the signal from very fewer samples than required by Shannon-Nyquist sampling theorem. The usage is pretty easy, you can basically type the name of the letter and put a backslash in front of it. When two maths elements appear on either side of the sign, it is assumed to be a binary operator, and as such, allocates some space to either side of the sign. Ce document intitulé « LaTeX - Table de caractères » issu de Comment Ça Marche (www. Consider using overleaf. The Bullet Material CV is a minimalist template with a professional feel. with manual sizing (by using two g's) and the unneeded \left. 574 silver badges. These are not guaranteed to work in MathJax but are a good place to start. So LaTeX with its strengths in math typesetting is a very good choice for writers. IEEE Access received an impact factor of 4. -57 Extrusive:Igneous Lava Flow -77. Juli 2014 um 10. Some other constructions. In the preamble of the document include the code: \usepackage{amsmath} Open an example of the amsmath package in Overleaf. Delimiters. Using lists in LaTeX is pretty straightforward and doesn't require you do add any additional packages. In Excel, there are multiple ways to draw this function:. RESPONSE XS90 P0240031 XS90 package £2,999. Literaturverzeichnis erstellen - Schreiben Sie einfach den Titel (Autor, Verlag, etc. First we must take a quick look at LaTeX syntax. The effect should be almost the same with $#^a$ except that a is on the top instead of top right of #. Thanks for contributing an answer to Mathematica Stack Exchange! Please be sure to answer the question. Labels are a necessary part of typesetting as they are efficient pointers to information. Um editor de LaTeX online fácil de usar. Vous pouvez. Boston, MA. cond(A) berechnet die Konditionszahl von A in 2-Norm. Document history. 1 3 ing the instruments that generated some of the data. There is a wealth of guides on how to do this available on. Carol JVF Burns's page of. 폰트 크기 (font size) 문서에서 폰트 크기는 10 pt, 11 pt, 12 pt 따위가 가장 널리 쓰인다. See also % command used to denote a comment This article is a stub. I think I'd lean towards the latter option, just because collectively edited answers tend to see more action and "peer review" than tag wikis: Post edits displayed at the top of the main (meta) page, whereas a tag wiki edit is only (?) visible in the suggested edits. Electrical units are easily written complying to standards using the siunitx package. The \quad command adds in some additional spacing between the definition and the inequalities. Input: two revisions of one LaTeX document. The colour of a second block of text, delimited by { and }, is set to red with the command \color{red}, then a. The template is in accordance with the latest version of the Norms for the printing of thesis/dissertations of the UNICAMP (CCPG Nº 001/2019). To get an expression exp to appear as a subscript, you just type _{exp}. The amsmath package provides commands to typeset matrices with different delimiters. Related MATLAB, Maple, Mathematica, LaTeX News on Phys. Similar is for limit expressions. Das Literaturverzeichnis Nicht erst seit den Rücktritten der Bundesminister Gutenberg und Schavan sollte allen Studierenden klar sein, dass korrekte Zitate und ein vollständiges Literaturverzeichnis für eine wissenschaftliche Arbeit essenziell sind 📚☝️ – egal ob Hausarbeit, Bachelor- oder Masterarbeit. Sublime Text. Open an example in Overleaf. Integral expressions are formed from the use of sub- and superscript, the judicious use of spacing, and simply writing out the differential. This post summarizes symbols used in complex number theory. Lyx is a nice mix of the professional layout and typesetting provided by Latex, with a nice visual interface that provides immediate feedback to the user. New to LaTeX? This tool will help you learn the basics and collaborate with others. August 2007 by tom 42 Comments. In combination with a text editor and distributed version control, both reproducibility and simultaneous collaboration are improved by these plain text mark-up languages. x \bmod a : there will be a short gap between x and mod. This is another way to generate a PDF of your entire write-up. LaTeX symbols have either names (denoted by backslash) or special characters. Compressive Sensing is a Signal Processing technique, which gave a break through in 2004. 825 bronze badges. Phone Number Information; 781-723-2024: Esnaider Mcclone - Shadow Valley Dr, Malden, MA: 781-723-5549: Dorrian Disla - Adams Rd, Malden, MA: 781-723-6157. HotelTonight is a pioneer in mobile commerce. To learn more, see our tips on writing great. These notations describe the limiting behavior of a function in mathematics or classify algorithms in computer science according to their complexity / processing time. dictionary book. Any text in between \begin{flushleft}\end{flushleft} will be aligned with the left-hand margin, but have a ragged right-hand edge. LaTeX Line and Page Breaking The first thing LaTeX does when processing ordinary text is to translate your input file into a string of glyphs and spaces. Uppgifter utan källhänvisning kan ifrågasättas och tas bort utan att det behöver diskuteras på diskussionssidan. cond(A) berechnet die Konditionszahl von A in 2-Norm. Many script-languages use backslash "\" to denote special commands. Step 3: Enter key words. LyX is a graphical interface, nearly WYSIWYG, to the LaTeX word processing package. The segment is called How to LaTeX with OverLeaf. Some examples from the MathJax demos site are reproduced below, as well as the Markdown+TeX source. The Best Latex How To Make A Table Free Download PDF And Video. Whether researchers occasionally turn to Bayesian statistical methods out of convenience or whether they firmly subscribe to the Bayesian paradigm for philosophical reasons: The use of Bayesian statistics in the social sciences is becoming increasingly widespread. By using this tool you avoid the command line and having to install Perl. The math environment is for formulas that appear right in the text. Mucho más que documentos. Hi all - Do you know of any research that compares the typesetting of LaTeX, MS Word, and LibreOffice? I'm especially interested in work that compares the justification algorithms, kerning, and microtypography features, using modern versions of these applications (e. Refer to the external references at the end of this article for more information. \author \date. We've documented and categorized hundreds of macros!. at midnight than anyone else. LaTeX formats mathematics the way it's done in mathematics texts. Step 5: Enter references, e. Hi everyone! I'm working on a proof by cases in Overleaf LaTeX formatter and I'm having trouble formatting the cases within the proof. Da es praktisch unmöglich ist, alle jemals in der Mathematik verwendeten Symbole aufzuführen, werden in dieser Liste nur diejenigen Symbole angegeben, die häufig im Mathematikunterricht oder im Mathematikstudium auftreten. We've documented and categorized hundreds of macros!. View Larger Preview. Posted: (18 days ago) Beamer - Overleaf, Online LaTeX Editor. , are commonly used for inverse hyperbolic trigonometric functions (area hyperbolic functions), even though they are misnomers, since the prefix arc is the abbreviation for arcus, while the prefix ar stands for area. 202544536–dc20 CIP Printed on acid-free paper °c Birkh. Tags: cap, cup, document, intersection, LaTeX, layout, math, maths, SET, sets, union. Add a New Comment. Word 2019 or 2016, LibreOffice 6. Um editor de LaTeX online fácil de usar. Fortunately, there's a tool that can greatly simplify the search for the command for a specific symbol. Some might say that the resulting norm "fences" in the example above are a bit too large and thus threaten to dominate visually the rest of the math stuff. Math symbols deﬁned by LaTeX package «fourier» CapitalGreeklettersdonotchangeshapeinmathalphabets. ISBN -8176-3805-9 (acid-free paper) (pbk. How to use Overleaf with IEEE Collabratec™ - your quick guide to getting started Posted by John on December 15, 2015 NOTE: This article shows screenshots of the integration of IEEE Collabratec™ with the Overleaf v1 platform, but the process is largely the same in the new Overleaf v2 platform. A list of LaTEX Math mode symbols. Overleaf is so easy to get started with that you'll be able to invite your non-LaTeX colleagues to contribute directly to your LaTeX documents. r o u g h g u i d e s. Free essays, homework help, flashcards, research papers, book reports, term papers, history, science, politics. My mental model is that the best we can hope for, the best attainable policy is the projection into the manifold of expertise-informed policies subject to the contraints of pressure from vested interests allowed at the time we chose that policy and available attention. They consist of plain text interspersed with some LaTeX commands. Open an example in Overleaf. Home; Domestic appliances; Large home appliances; Washing machines; Instructions for installation and use WASHER-DRYER Contents. Such basics can be found in introductions like lshort. Letters are printed in italics, with more space left in-between, spaces are ignored. \begin {document} View which changes have been added and removed. What they have in common is that they process the contents of filename. 常用数学符号的 LaTeX 表示方法（以下内容主要摘自“一份不太简短的 LATEX2e 介绍”） 1、指数和下标可以用^和_后加相应字符来实现。比如： 2、平方根（square root）的输入命令为：\sqrt，n 次方根相应地为: \sqrt[n]。方根符号的大小由LATEX自动加以调整。. sty file that contains these commands. The shaft is not flexible below the electrodes, but it is. % Minimal latex example: % Shows how to switch between bold and arrow vectors. The template and relevant files are found at this page: Information for Authors of Computer Science Publications I remember this being pretty confusing the first time I looked at how to do this. We've documented and categorized hundreds of macros!. Other options here include c, for center-aligned, and r for right-aligned. To insert a citation where label is the label of a bibliographic entry in a. Keine Installation notwendig, Zusammenarbeit in Echtzeit, Versionskontrolle, Hunderte von LaTeX-Vorlagen und mehr. 636-46-05-15 VICEPRESIDENTA: CONCEPCIÓN PÉREZ GONZÁLEZ. Integer and sum limits improvement. This keeps the main body of text concise. Greek letters []. @CharlesStewart Thanks! If others find this useful, we could either use it as a tag wiki, or link here from the tag wiki. d3 is the first move in \variation{} Open an example of the skak package in ShareLaTeX. Domicilio Fiscal: C/ Melíes, nº 50, Urbanización Santa María - 08800 - Vila Nova i la Geltrú - BARCELONA. beginner woodwork. TeX has \\int as the integral sign. , there are no licence fees, etc. 85 Trachyte 166. De forma muy simple antes del cuerpo del documento basta añadir las dos instrucciones siguientes: \providecommand{\abs}[1]{\lvert#1\rvert} \providecommand{ orm}[1]{\lVert#1\rVert} Estos comandos se usan de la siguiente manera:. The mathematical symbol is produced using \partial. LaTeX equations always start with \ ( and end with \). Compressive Sensing is a Signal Processing technique, which gave a break through in 2004. This is a simple step, if you use LaTeX frequently surely you already know this. Manhattan: Take the sum of the absolute values of the differences of the coordinates. Open, and a variety of correspondence (postal) tournaments, and (4) officially represents the. As you can see, there are three basic commands and they can be nested to get combined effects. IEEE article templates let you quickly format your article and prepare a draft for peer review. 098 in the 2018 JCR release. stackexchange. latex2exp is an R package that parses and converts LaTeX math formulas to R's plotmath expressions. The differences between these two ways to include files is explained below. Mathematical modes. \begin {document} View which changes have been added and removed. Table 238: fge Math-mode Accents. An online LaTeX editor that's easy to use. The usage is pretty easy, you can basically type the name of the letter and put a backslash in front of it. Open an example in ShareLaTeX. Readbag users suggest that JRCALC Clinical Practice Guidelines 2004 is worth reading. Embed formulas in your text by surrounding them with dollar signs The equation environment is used to typeset one formula. norm(X,p) berechnet die p-Norm des Vektors X. This document is also listed in a special topic for beginners [1]. Formatting Optimization Problems with LaTeX. Open a text editor like Notepad and create a new LaTeX document by typing: \begin {document} \end {document} Type the following between the "begin" and "end" commands to create your bullet point list: Video of the Day. answered May 24 '10 at 4:21. Part 1 \| Part 2 \| Part 3 \| Part 4 \| Part 5. Hyperbolic functions The abbreviations arcsinh, arccosh, etc. To learn more about Overleaf, visit their website: www. Each column ends with an ampersand (&). Hat and underscore are used for superscripts and subscripts. bmatrix Latex matrix pmatrix vmatrix. A state diagram shows the behavior of classes in response to external stimuli. tex) [1] LaTeX Resources for Beginners. This typically indicates Rd problems. This is the basic introduction to Matlab. The calculator will find the principal unit normal vector of the vector-valued function at the given point, with steps shown. The first one is used to write formulas that are part of a text. 9f November19,2018 Thegeneral-purposedrawingpackageTikZcanbeusedtotypesetcommutativediagramsandotherkinds. I made report in LaTeX during my six weeks training. How exactly you format such citations then depends on the citation style that you are being asked to use (e. No installation, real-time collaboration, version control, hundreds of LaTeX templates, and more. Arrow symbols. LaTeX Line and Page Breaking The first thing LaTeX does when processing ordinary text is to translate your input file into a string of glyphs and spaces. Juli 2014 um 10. Add text to the graph that contains an integral expression using LaTeX markup. In a three-part series for Weekend, Fiona Cairns reveals how to make a simplified version of the cake she made for the Duke and Duchess of Cambridge's Royal Wedding and other treats. L a T e X allows two writing modes for mathematical expressions: the inline mode and the display mode. (although it still helps to know how to code maths in Latex). GB GB IT DE Contents English,1 Italiano,13 Deutsch,25 Installation, 2-3 Unpacking and levelling Connecting the electricity and water supplies Technical data Care and maintenance, 4 Cutting off the water and electricity supplies Cleaning the machine Cleaning the detergent dispenser. My view is that as with any service where there is increased competition, prices will stagnate and potentially come down in future. Variable-sized symbols. How do I change my cursor back to vertical? I accidentally pressed a key, and it seemed to have changed my vertical cursor to a horizontal/underscore type cursor!. All results are completely general, numerically sound, and based on general realizations allowing for poles at infinity. Personally I use ShareLaTeX, although I can see how Overleaf would be easier for a non-LaTeX field since you can edit directly in a WYSIWYG format or in the LaTeX format. Amphibian study shows stress increases vulnerability to virus; Mutations in SARS-CoV-2 offer insights into virus evolution. tex) files, or else just remain in the LyX domain (. Similar is for limit expressions. Hey guys, I'm writing a beamer presentation on LaTex and I'm facing a problem I can't seem to solve. LaTeX deals with the + and − signs in two possible ways. The leading scientists behind eLife are committed to rapid, fair, and constructive. LaTeX is a very flexible program for typesetting math, but sometimes figuring out how to get the effect you want can be tricky. Math symbols deﬁned by LaTeX package «fourier» CapitalGreeklettersdonotchangeshapeinmathalphabets. com , or follow them on twitter at @Overleaf. Literaturverzeichnis erstellen - Schreiben Sie einfach den Titel (Autor, Verlag, etc. Use \left\lVert before the expression and \right\rVert after it. Greek letters. If you want different spacing, LaTeX provides the following four commands for use in math mode: \; - a thick space \: - a medium space \, - a thin space. The usage is pretty easy, you can basically type the name of the letter and put a backslash in front of it. Log-like symbols. Writing congruence relations in latex. 1 DeclareRobustCommand. November 2013 by tom 8 Comments. 예를 들어 em이나 px. Fortunately, there are alternative commands that do the same task differently that we can try and there are…. Juli 2014 um 10. Step 2: Enter author information. Um editor de LaTeX online fácil de usar. Permanent Link Edit Delete. It even does the right thing when something has both a subscript and a superscript. Overleaf /LaTex Not sure students need to know too much latex anymore… markdown/r-md is a lot simpler and using it with css and html bits is very flexible. No author: when author information is not available, use the source title to replace the author's position. Letters are printed in italics, with more space left in-between, spaces are ignored. ) The names of certain standard functions and abbreviations are obtained by typing a backlash \ before the name. Bachelor-, Diplom- oder Doktorarbeiten sind in der Regel mit einem enormen Aufwand verbunden. Many script-languages use backslash "\" to denote special commands. This is a simple step, if you use LaTeX frequently surely you already know this. Citations are an abomination, as are numbering equations; random crashes, corruption of files that contain a lot of graphics etc. Once you have loaded \usepackage {amsmath} in your preamble, you can use the following environments in your math environments: If you need to create matrices with different delimiters, you can add them manually to a plain matrix. Sometimes, the output doesn't come out the way some of us might expect or want. the course of thirty-nine editions, Hart's Rules has grown to be the standard work in itsfield,explaining subject by subject each major aspect of punctuation, capitalization, italics, hyphenation. 7 posts • Page 1 of 1. Area between Curves Calculator - eMathHelp. What are LaTeX “environments” While TeX makes direct provision for commands, LaTeX adds a concept of “environment”; environments perform an action on a block (of something or other) rather than than just doing something at one place in your document. aa aah aahed aahing aahs aal aalii aaliis aals aardvark aardvarks aardwolf aardwolves aargh aas aasvogel aasvogels aba abaca abacas abaci aback abacus abacuses abaft. Easy-to-use symbol, keyword, package, style, and formatting reference for LaTeX scientific publishing markup language. 예를 들어 em이나 px. LaTeX forum ⇒ Math & Science ⇒ Norm symbol to correspond argument height Topic is solved Information and discussion about LaTeX's math and science related features (e. To learn more, see our tips on writing great. Right now I have \theoremstyle{case} \newtheorem{case}{Case} at the beginning of my document, and \begin{case} within my proof. com , or follow them on twitter at @Overleaf. texblog because LaTeX matters. Convert Latex equations into beautiful, transparency-correct PNGs. The next chapter will focus on Plain TeX and will explain advanced techniques for programming. Any text in between \begin{flushleft}\end{flushleft} will be aligned with the left-hand margin, but have a ragged right-hand edge. formulas, graphs). Diese Liste mathematischer Symbole zeigt eine Auswahl der gebräuchlichsten Symbole, die in moderner mathematischer Notation innerhalb von Formeln verwendet werden. SaxLove Recommended for you. Integral\int_ {a}^ {b} x^2 dxinside. Languages: English Shqip العربية Bangla Български Català 中文 ( 正體字 , 简化字 (1) , 简化字 (2)) Hrvatski Čeština Dansk Nederlands Esperanto Eesti فارسی Suomi Français Deutsch Ελληνικά ગુજરાતી עברית. Such basics can be found in introductions like lshort. here 1 is subscript of C. Document history. There is a simple way to add "normal text" fragments in. Step 1: Enter abstract title. A Normal distribution with a mean of zero and a standard deviation of 1 is also known as the Standard Normal Distribution (m =0, s =1) as in Figure 1. Plotmath expressions are used to enter mathematical formulas and symbols to be rendered as text, axis labels, etc. The Mac app is finally stable enough. A list of LaTEX Math mode symbols. Creation of vectors is included with a few basic operations. In LaTeX backslash is used to generate a special symbol or a command. LaTeX The LaTeX command that creates the icon. Dictionary - Free ebook download as Text File (. pro woodwork projects. you want to put a small piece of text in a specific type style, you can do it as follows: If you want to put larger amounts of text into these type styles, you can use {{\begin}} and {{\end}} commands; i. Includes index. The main idea of CS is, by exploiting the sparsity nature of the signal (in any domain), we can reconstruct the signal from very fewer samples than required by Shannon-Nyquist sampling theorem. NUESTRA JUNTA DIRECTIVA ESTÁ FORMADA POR: PRESIDENTA: FRANCISCA GIL QUINTANA-- TELF. improve this answer. HOME: Next: Arrow symbols (amssymb) Last: Relation symbols (amssymb) Top: Index Page Index Page. Relation symbols. Consider using overleaf. The amsmath package provides commands \lvert, \rvert, \lVert, \rVert which change size dynamically. Home > Latex > FAQ > Latex - FAQ > LateX Derivatives, Limits, Sums, Products and Integrals. (Refer to bio-contamination data overleaf) Suspension Systems 'Blue Tongue' Aluminium Clean Room Flat Faced Tee Grid & PeakForm Prelude Steel 24mm Tee above. 9f November19,2018 Thegeneral-purposedrawingpackageTikZcanbeusedtotypesetcommutativediagramsandotherkinds. It even does the right thing when something has both a subscript and a superscript. The amsmath The amsmath package comes standard with most L A TEX distributions and is loaded by physics for your convenience. The appropriate LaTeX command is \overset{annotation}{symbol}. Use MathJax to format equations. Computerized typesetting. Type H for immediate help. LaTeX is a fairly high-level language compared to Plain TeX and thus is more limited. Was ist 9+4 ? Oh, die ist schwer. Integer and sum limits improvement. Today I tried to write the solution of a differential equation in LaTeX. Similar is for limit expressions. Miscellaneous symbols. Other options here include c, for center-aligned, and r for right-aligned. This is another way to generate a PDF of your entire write-up. This tag should be used for questions that concern the backend; questions concerning use of the web service are probably better asked on tex. journals) on Instagram: “made it to college! super late spread but more are headin' your way <3 - #bujo #bulletjournal. Part 1 \| Part 2 \| Part 3 \| Part 4 \| Part 5. The amsmath package provides commands \lvert, \rvert, \lVert, \rVert which change size dynamically. A key transformed subproblem involving K„i, x„i, d„i appears in the middle. Newer Post Older Post Home. No extra packages are required to use these symbols. Similar is for limit expressions. 75in for more space. This is the template for LaTeX submissions to eLife. Using lists in LaTeX is pretty straightforward and doesn't require you do add any additional packages. Diese können Sie mit help funcname erfahren. Other tests depend on the binding of a fluorescein or chemiluminescent conjugate, or the visible agglutination of HIV-coated gelatin or latex particles.\begingroup$@knzhou So, I was lucky to find an online editor : Overleaf (first v1 and later v2). In addition to the actual “math mode” environments, wherein math symbols and structures are the norm and text is the exception, you may also want environments in which the content is primarily textual, but which contain logical constructs, such as algorithms, answers, assertions and axioms (and that’s just the A’s!). The displaymath environment is for formulas that appear on their own line. x \mod a : there will be a long gap between x and mod. Den här artikeln behöver källhänvisningar för att kunna verifieras. Here I'd figured I must have been doing something wrong. Easy-to-use symbol, keyword, package, style, and formatting reference for LaTeX scientific publishing markup language. improve this answer. Das Literaturverzeichnis Nicht erst seit den Rücktritten der Bundesminister Gutenberg und Schavan sollte allen Studierenden klar sein, dass korrekte Zitate und ein vollständiges Literaturverzeichnis für eine wissenschaftliche Arbeit essenziell sind 📚☝️ – egal ob Hausarbeit, Bachelor- oder Masterarbeit. Kurz: Es ist praktisch und ich kann von dort aus entscheiden, ob ich länger in Sublime weiterschreibe oder ein PDF oder eine LaTeX-Datei generiere. , there are no licence fees, etc. Overleaf, Online LaTeX Editor. \end {document}. Grade ]?ercentage of f. au/whatson/academic 1497947400 2017 6 20 Tuesday 16:30 1497951000 2017 6 20 Tuesday 17:30. \begin {document} View which changes have been added and removed. You can use Overleaf to write and collaborate online in LaTeX using the template. sty file that contains these commands. Previous ones: Basics and overview Use of mathematical symbols in formulas and equations Many of the examples shown here were adapted from the Wikipedia article Displaying a formula, which is actually about formulas in Math Markup. Using the C++ programming language as an example, one can find nearly every citation for the C++ standards in BibTeX format. cond(A) berechnet die Konditionszahl von A in 2-Norm. Spacing in Math Mode. 098 in the 2018 JCR release. Hi everyone! I'm working on a proof by cases in Overleaf LaTeX formatter and I'm having trouble formatting the cases within the proof. pdf) or read book online for free. In LaTeX backslash is used to generate a special symbol or a command. Select, ver Figura 13. Dictionary - Free ebook download as Text File (. All the versions of this article: < français > Here are few examples to write quickly matrices. Im Bildungs- und Erziehungsauftrag des Gymnasiums wird gefordert, dass das Gymnasium den Schüler „[] auch dazu befähigt, den Anforderungen einer modernen Berufs- und Arbeitswelt gewachsen zu sein" ([4], S. The main things used in it are: Fractions : These can be written as: \frac{x/y} Subscripts: These are wriiten as. Overleaf es un programa online que nos permite editar y compilar desde nuestro navegador un documento de LaTeX. In combination with a text editor and distributed version control, both reproducibility and simultaneous collaboration are improved by these plain text mark-up languages. Compressive Sensing is a Signal Processing technique, which gave a break through in 2004. com JavaScript MIT 17 5 4 2 Updated Apr 23, 2020. tex before continuing with the rest of the base file (the file that contains these statements). Eine gute Möglichkeit, Zitate einzufügen und zu formatieren, ist BibTeX (). The following table shows the whole Greek alphabet along with the commands in a nice table. It's widely used throughout the academic world to publish technical. Tags: document, LaTeX, multiplication, prod, product, symbol. formulas, graphs). For Microsoft Word, and other word processors, you can choose PDF inside of the “File !Save As” menu. No installation, real-time collaboration, version control, hundreds of LaTeX templates, and more. Some other constructions. nach dem Autor oder dem Alphabet) - an einer gewünschten. Embed formulas in your text by surrounding them with dollar signs$ The equation environment is used to typeset one formula. Markdown into LaTeX with Style Post-publication update (13 May, 2017): We are grateful to Vít Novotný, the author/maintainer of the markdown package, for writing to us with some helpful feedback concerning the original article. 2 posts • Page 1 of 1. If you want different spacing, LaTeX provides the following four commands for use in math mode:. \bibliographystyle {bstfilename} To choose a BibTeX bibliographic style file with the extension. HOME: Next: Arrow symbols (amssymb) Last: Relation symbols (amssymb) Top: Index Page Index Page. Quotation Marks and Dashes. LaTeX files usually have a. Overleaf es un programa online que nos permite editar y compilar desde nuestro navegador un documento de LaTeX. Math into LaTeX : an introduction to LaTeX and AMS-LaTeX / George Gr¨atzer p. I'm taking a matrix algebra course this term and thought I'd LaTeX my homework assignments just for practice. » You can assign values to patterns involving Integrate to give results for new classes of integrals. Dictionary - Free ebook download as Text File (. Most of the stock math commands are written for typesetting math or computer science papers for academic journals, so you might need to dig deeper into LaTeX commands to get the vector notation styles that are common in physics textbooks and articles. This puts the annotation in a smaller type size directly above the symbol. Brackets and Norms The frequently used left delimiters include (, [ and {, which are obtained by typing ( , [ and \{ respectively. \frac {d} {dt} \bigg\|_ {t=0} f (t) achieves a similar effect. Vous pouvez. 3 posts • Page 1 of 1. Was ist 9+4 ? Oh, die ist schwer. If you want the limits above and below, place the \limits command after the sum command as follows: $\sum\limits_{k=1}^n k$. This is another way to generate a PDF of your entire write-up. ShareLaTeX is so easy to get started with that you'll be able to invite your non-LaTeX colleagues to contribute directly to your LaTeX documents. To produce a printed document, this string must be broken into lines, and these lines must be broken into pages. Relation symbols. 8, AUGUST 2015 1 How to Use the IEEEtran LATEX Class Michael Shell, Member, IEEE (Invited Paper) Abstract—This article describes how to use the IEEEtran class. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. Im Bildungs- und Erziehungsauftrag des Gymnasiums wird gefordert, dass das Gymnasium den Schüler „[] auch dazu befähigt, den Anforderungen einer modernen Berufs- und Arbeitswelt gewachsen zu sein" ([4], S. Amphibian study shows stress increases vulnerability to virus; Mutations in SARS-CoV-2 offer insights into virus evolution. No extra packages are required to use these symbols. The usage is pretty easy, you can basically type the name of the letter and put a backslash in front of it. Poetica – Get clear feedback, wherever you’re writing. GitHub is home to over 40 million developers working together to host and review code, manage projects, and build software together. Delimiters. Arrow symbols. It even does the right thing when something has both a subscript and a superscript. The command \variation{} helps to analyse variations of a move. \end {document}. \end {document}. The USCF is the national chess organization of the United States. 09 layered on T X v2. For example, one obtains by typing $\cos(\theta + \phi) = \cos \theta \cos \phi - \sin \theta \sin \phi$ The following standard functions are represented by control sequences defined in LaTeX:. You must use the following package: \usepackage {amsmath} \begin {matrix} \begin {pmatrix} \begin {bmatrix} \begin {vmatrix} \begin {Vmatrix}. Embed formulas in your text by surrounding them with dollar signs \$ The equation environment is used to typeset one formula. LaTeX is available as free software. {tikzcd} CommutativediagramswithTikZ Version0. As you see, the way the equations are displayed depends on the delimiter, in this case and . tex file via the \input{filename} command. The version I have is on a compilation of Martin Hannett produced tracks called “And Here Is The Young Man”. In case you are working with LaTeX, there are two very good (free!) sites offering collaboration functionality: Overleaf and ShareLaTeX. edited Feb 23 '15 at 10:57. Previous ones: Basics and overview Use of mathematical symbols in formulas and equations Many of the examples shown here were adapted from the Wikipedia article Displaying a formula, which is actually about formulas in Math Markup. , are commonly used for inverse hyperbolic trigonometric functions (area hyperbolic functions), even though they are misnomers, since the prefix arc is the abbreviation for arcus, while the prefix ar stands for area. chl963vc3mcb4v, v0ib0yqnq7z, a2duuk7lr6774v, tgt7xjmy994nzk, 58yoq5ndnj2at, sz7mfcqo96a9, 74dfdtpva689c, dihas4pxmbql9em, 3c1xb0tdliaz8, wj42qd866ded, fyxr9ovnk7kzks, xn13terxmdky, 81rdrk0mwer4k, 6itdbd6eaen2n, wvjwzh0f55zc4z, x74vpbh2xq1, vm7a6m60f5, 5uhtn8psxdppi, l7vbt4zdf63r, 6yp50ax9uslyfo, 95il4wtczlun, 8zufxzjndu9rmnl, tnmsbdgekbmkqr, 9zkutef0qjey, cwsewu8zpo5toh4, w72b5f9yogc7w6y	{"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": 2, "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": 0.8828639388084412, "perplexity": 4535.0644922643}, "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-2020-29/segments/1593655897027.14/warc/CC-MAIN-20200708124912-20200708154912-00560.warc.gz"}
https://zbmath.org/?q=an:1114.60084	× On multidimensional branching random walks in random environment.(English)Zbl 1114.60084 Summary: We study branching random walks in random i.i.d. environment in $$\mathbb{Z}^d$$, $$d\geq 1$$. For this model, the population size cannot decrease, and a natural ural definition of recurrence is introduced. We prove a dichotomy for recurrence/transience, depending only on the support of the environmental law. We give sufficient conditions for recurrence and for transience. In the recurrent case, we study the asymptotics of the tail of the distribution of the hitting times and prove a shape theorem for the set of lattice sites which are visited up to a large time. MSC: 60K37 Processes in random environments 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses) Full Text: References: [1] Alves, O. S. M., Machado, F. P. and Popov, S. Yu. (2002). The shape theorem for the frog model. Ann. Appl. Probab. 12 533–546. · Zbl 1013.60081 [2] Athreya, K. B. and Ney, P. E. (1972). Branching Processes . Springer, New York. · Zbl 0259.60002 [3] Baillon, J.-B., Clément, P., Greven, A. and den Hollander, F. (1993). A variational approach to branching random walk in random environment. Ann. Probab . 21 290–317. · Zbl 0770.60088 [4] Biggins, J. (1978). The asymptotic shape of the branching random walk. Adv. in Appl. Probab . 10 62–84. JSTOR: · Zbl 0383.60078 [5] Bramson, M. and Griffeath, D. (1980). On the Williams–Bjerknes tumor growth model. II. Math. Proc. Cambridge Philos. Soc. 88 339–357. · Zbl 0459.92013 [6] Comets, F., Menshikov, M. V. and Popov, S. Yu. (1998). One-dimensional branching random walk in random environment: A classification. Markov Process. Related Fields 4 465–477. · Zbl 0938.60081 [7] Dembo, A., Peres, Y. and Zeitouni, O. (1996). Tail estimates for one-dimensional random walk in random environment. Comm. Math. Phys. 181 667–683. · Zbl 0868.60058 [8] Devulder, A. (2005). A branching system of random walks in random environment. Available at http://www.proba.jussieu.fr/mathdoc/textes/PMA-834.pdf. · Zbl 1138.60341 [9] Durrett, R. and Griffeath, D. (1982). Contact processes in several dimensions. Z. Wahrsch. Verw. Gebiete 59 535–552. · Zbl 0483.60089 [10] Engländer, J. (2005). Branching Brownian motion with ‘mild’ Poissonian obstacles. Available at http://arxiv.org/math.PR/0508585. [11] Fayolle, G., Malyshev, V. A. and Menshikov, M. V. (1995). Topics in the Constructive Theory of Countable Markov Chains. Cambridge Univ. Press. · Zbl 0823.60053 [12] Gantert, N. and Müller, S. (2005). The critical branching random walk is transient. Available at http://arxiv.org/math.PR/0510556. · Zbl 1115.60077 [13] Gantert, N. and Zeitouni, O. (1998). Quenched sub-exponential tail estimates for one-dimensional random walk in random environment. Comm. Math. Phys. 194 177–190. · Zbl 0982.60037 [14] Greven, A. and den Hollander, F. (1992). Branching random walk in random environment: Phase transitions for local and global growth rates. Probab. Theory Related Fields 91 195–249. · Zbl 0744.60079 [15] den Hollander, F., Menshikov, M. V. and Popov, S. Yu. (1999). A note on transience versus recurrence for a branching random walk in random environment. J. Statist. Phys. 95 587–614. · Zbl 0933.60089 [16] Kingman, J. F. C. (1973). Subadditive ergodic theory. Ann. Probab. 1 883–909. JSTOR: · Zbl 0311.60018 [17] Lawler, G. F. (1983). A discrete stochastic integral inequality and balanced random walk in a random environment. Duke Math. J. 50 1261–1274. · Zbl 0569.60071 [18] Liggett, T. M. (1985). An improved subadditive ergodic theorem. Ann. Probab. 13 1279–1285. · Zbl 0579.60023 [19] Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York. · Zbl 0559.60078 [20] Machado, F. P. and Popov, S. Yu. (2000). One-dimensional branching random walk in a Markovian random environment. J. Appl. Probab. 37 1157–1163. · Zbl 0995.60070 [21] Machado, F. P. and Popov, S. Yu. (2003). Branching random walk in random environment on trees. Stochastic Process. Appl. 106 95–106. · Zbl 1075.60570 [22] Nagaev, S. V. (1979). Large deviations of sums of independent random variables. Ann. Probab. 7 745–789. · Zbl 0418.60033 [23] Pisztora, A. and Povel, T. (1999). Large deviation principle for random walk in a quenched random environment in the low speed regime. Ann. Probab. 27 1389–1413. · Zbl 0964.60056 [24] Pisztora, A., Povel, T. and Zeitouni, O. (1999). Precise large deviation estimates for a one-dimensional random walk in a random environment. Probab. Theory Related Fields 113 191–219. · Zbl 0922.60059 [25] Sznitman, A.-S. (1999). Slowdown and neutral pockets for a random walk in random environment. Probab. Theory Related Fields 115 287–323. · Zbl 0947.60095 [26] Sznitman, A.-S. (2000). Slowdown estimates and central limit theorem for random walks in random environment. J. Eur. Math. Soc. (JEMS) 2 93–143. · Zbl 0976.60097 [27] Sznitman, A.-S. (2002). An effective criterion for ballistic behavior of random walks in random environment. Probab. Theory Related Fields 122 509–544. · Zbl 0995.60097 [28] Sznitman, A.-S. (2003). On new examples of ballistic random walks in random environment. Ann. Probab. 31 285–322. · Zbl 1017.60104 [29] Sznitman, A.-S. and Zerner, M. (1999). A law of large numbers for random walks in random environment. Ann. Probab. 27 1851–1869. · Zbl 0965.60100 [30] Varadhan, S. R. S. (2003). Large deviations for random walks in a random environment. Comm. Pure Appl. Math. 56 1222–1245. · Zbl 1042.60071 [31] Volkov, S. (2001). Branching random walk in random environment: Fully quenched case. Markov Process. Related Fields 7 349–353. · Zbl 0991.60073 [32] Zeitouni, O. (2004). Random walks in random environment. Lecture Notes in Math. 1837 190–312. Springer, Berlin. · Zbl 1060.60103 [33] Zerner, M. (1998). Lyapounov exponents and quenched large deviations for multidimensional random walk in random environment. Ann. Probab. 26 1446–1476. · Zbl 0937.60095 [34] Zerner, M. (2002). A non-ballistic law of large numbers for random walks in i.i.d. random environment. Electron. Comm. Probab. 7 191–197. · Zbl 1008.60107 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.	{"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": 0.8872588872909546, "perplexity": 2745.087653374088}, "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/1664030337339.70/warc/CC-MAIN-20221002181356-20221002211356-00194.warc.gz"}
https://www.sawaal.com/aptitude-reasoning/verbal-reasoning-mental-ability/series-questions-and-answers.htm?page=2&sort=	# Number Series Questions Q: Find the wrong number in the series. 3, 8, 15, 24, 34, 48, 63 A) 15 B) 24 C) 34 D) 48 Explanation: The difference between consecutive numbers of the given series are respectively 5, 7, 9, 11, 13, and 15. Therefore, 24+11=35 But in your problem it is given as 34.so 34 is wrong number 491 74871 Q: Find the missing number in the given number series? 625, 625, 600, ?, 475, 875 A) 545 B) 700 C) 675 D) 725 Explanation: Here the given number series 625, 625, 600, ?, 475, 875 follows a pattern that 625 625 + (0 x 0) = 625 625 - (5 x 5) = 625 - 25 = 600 600 + (10 x 10) = 600 + 100 = 700 700 - (15 x 15) = 700 - 225 = 475 475 + (20 x 20) = 475 + 400 = 875 Hence, the missing number in the given number series is 700. 126 60965 Q: Find the missing number in the following series? 2, 3, 10, 39, ?, 885 A) 128 B) 156 C) 172 D) 189 Explanation: Take a Look at the below sereies $2×1+12=3$ $3×2+22=10$ $10×3+32=39$ $39×4+42=172$ $172×5+52=885$ 218 57453 Q: 8, 15, 28, 53, ?, 199 A) 101 B) 102 C) 103 D) 104 Explanation: Here the series of the form is 53 x 2 - 4 = 106 - 4 = 102 500 54292 Q: 7, 8, 18, 57, ?, 1165, 6996 A) 228 B) 542 C) 232 D) 415 Explanation: Here 2nd number = (1st number x 1 )+1 3rd number = (2nd number x 2 +2) 4th number = (3rd number x 3 )+3 and so on.. Therefore, 5th number = (4th number x 4) +4 =57 x 4 + 4 =232. 294 50028 Q: Find the wrong number in the series 7, 28, 63, 124, 215, 342, 511 A) 28 B) 124 C) 215 D) 342 Explanation: Here the number follows the given rule But 28 has been given in problem series. so 28 is wrong number. 226 49279 Q: What will be the next term in BKS, DJT, FIU, HHV, __ A) IJX B) IGX C) JGW D) JGU Explanation: Here the first letter of the group is moved with a gap of one letter, the second letter of the group is backwardness and the third letter of the group is forwardness proceeding like this we get the letter group "JGW" 97 47317 Q: Which fraction comes next in the sequence A) 1/27 B) 11/278 C) 9/48 D) 7/123 Explanation: Clearly, the numerators of the fractions in the given sequence follow the series of 2, 3, 5, 7,... and the denominator follows 3, 9, 27, 81, 243, ... Then, the next term in the given sequence is .	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 5, "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": 0.8524905443191528, "perplexity": 1610.4939222849302}, "config": {"markdown_headings": true, "markdown_code": false, "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-2022-40/segments/1664030337338.11/warc/CC-MAIN-20221002150039-20221002180039-00067.warc.gz"}
http://mathoverflow.net/questions/97015/hilton-eckmann-dual-of-the-steenrod-algebra	# Hilton-Eckmann dual of the Steenrod Algebra In essence my question can be stated as follows: fill in the analogy $$\text{cup product} \qquad\qquad \leftrightarrow \qquad \text{Samelson product}$$ $$\updownarrow \qquad\qquad \qquad\qquad \qquad\qquad\updownarrow$$ $$\quad \quad \text{cup}_i \text{-product} \quad \qquad \leftrightarrow \qquad \qquad \quad\qquad\text{?}\qquad \qquad\quad\quad$$ It is known that Samelson products (for a loop space) are Hilton-Eckmann dual to cup products (see e.g., Arkowitz, Martin: Commutators and cup products. Illinois J. Math. 8 1964 571–581. ) Taking this a bit further, the construction of the Steenrod algebra uses the fact that the reduced diagonal $X \to X\wedge X$ is $\Bbb Z/2$-equivariant. It is not hard to show that the Samelson product also has an equivariance. This suggests to me that the graded Lie algebra structure on the homotopy groups of a space can be refined to take this into account. Any ideas? - Your upper $\leftrightarrow$ is Koszul-Quillen duality between commutative and Lie algebras, therefore, since the $\smile_i$-products form an $E_\infty$-algebra, the lower one should be the Koszul-Quillen duality between $E_\infty$-algebras and $L_\infty$-algebras. Have a look at any reference book on rational homotopy theory, where this theory works simplest. – Fernando Muro May 15 '12 at 15:38 @Fernando: In my understanding in the rational case, we can take a simplicial group model for the loop space, say $G$, and form its group ring over $\Bbb Q$ to get a simlpicial Hopf algebra. The degree-wise primitives then give a differential graded Lie algebra, which is a strict version of an $L_\infty$-algebra. However, if we work integrally (or maybe mod 2), are you saying that we get a non-strict $L_\infty$-algebra? Is this written down anywhere? – John Klein May 15 '12 at 17:18 Not really. In fact, the characteristic $>0$ case is not yet well understood or developed, but since you just asked about analogies I offered you the characteristic $0$ analogue ;-) – Fernando Muro May 16 '12 at 7:35 @Fernando, can we take your suggestion a little further and ask whether there are interesting analogues of Steenrod $p$'th powers coming from the $\mathbb{Z}/p$-equivariant homology of the p-th term $L_{\infty}(p)$ of the $L_{\infty}$-operad? – Craig Westerland May 16 '12 at 12:48 Looking for something else I've come up with this paper: Smirnov, V. A. E∞-structures on homotopy groups. (Russian) Mat. Zametki 61 (1997), no. 1, 152--156; translation in Math. Notes 61 (1997), no. 1-2, 127–130. It seems to contain an appropriate notion of $L_\infty$-algebras over $\mathbb{F}_p$. Unforturately the paper is very short and gives no proofs, and I haven't found references where the claims in this paper are proven. – Fernando Muro May 25 '12 at 23:20	{"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": 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": 0.9190407395362854, "perplexity": 505.0155080026783}, "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-2015-11/segments/1424936464303.77/warc/CC-MAIN-20150226074104-00069-ip-10-28-5-156.ec2.internal.warc.gz"}
https://worldwidescience.org/topicpages/f/field+theory+applications.html	#### Sample records for field theory applications 1. Neural fields theory and applications CERN Document Server Graben, Peter; Potthast, Roland; Wright, James 2014-01-01 With this book, the editors present the first comprehensive collection in neural field studies, authored by leading scientists in the field - among them are two of the founding-fathers of neural field theory. Up to now, research results in the field have been disseminated across a number of distinct journals from mathematics, computational neuroscience, biophysics, cognitive science and others. Starting with a tutorial for novices in neural field studies, the book comprises chapters on emergent patterns, their phase transitions and evolution, on stochastic approaches, cortical development, cognition, robotics and computation, large-scale numerical simulations, the coupling of neural fields to the electroencephalogram and phase transitions in anesthesia. The intended readership are students and scientists in applied mathematics, theoretical physics, theoretical biology, and computational neuroscience. Neural field theory and its applications have a long-standing tradition in the mathematical and computational ... 2. Closed superstring field theory and its applications Science.gov (United States) de Lacroix, Corinne; Erbin, Harold; Kashyap, Sitender Pratap; Sen, Ashoke; Verma, Mritunjay 2017-10-01 We review recent developments in the construction of heterotic and type II string field theories and their various applications. These include systematic procedures for determining the shifts in the vacuum expectation values of fields under quantum corrections, computing renormalized masses and S-matrix of the theory around the shifted vacuum and a proof of unitarity of the S-matrix. The S-matrix computed this way is free from all divergences when there are more than 4 noncompact space-time dimensions, but suffers from the usual infrared divergences when the number of noncompact space-time dimensions is 4 or less. 3. Holographic applications of logarithmic conformal field theories NARCIS (Netherlands) Grumiller, D.; Riedler, W.; Rosseel, J.; Zojer, T. 2013-01-01 We review the relations between Jordan cells in various branches of physics, ranging from quantum mechanics to massive gravity theories. Our main focus is on holographic correspondences between critically tuned gravity theories in anti-de Sitter space and logarithmic conformal field theories in 4. Introduction to conformal field theory. With applications to string theory International Nuclear Information System (INIS) Blumenhagen, Ralph; Plauschinn, Erik 2009-01-01 Based on class-tested notes, this text offers an introduction to Conformal Field Theory with a special emphasis on computational techniques of relevance for String Theory. It introduces Conformal Field Theory at a basic level, Kac-Moody algebras, one-loop partition functions, Superconformal Field Theories, Gepner Models and Boundary Conformal Field Theory. Eventually, the concept of orientifold constructions is explained in detail for the example of the bosonic string. In providing many detailed CFT calculations, this book is ideal for students and scientists intending to become acquainted with CFT techniques relevant for string theory but also for students and non-specialists from related fields. (orig.) 5. Higgs effective field theories. Systematics and applications Energy Technology Data Exchange (ETDEWEB) Krause, Claudius G. 2016-07-28 Researchers of the Large Hadron Collider (LHC) at the European Organization for Nuclear Research (CERN) announced on July 4th, 2012, the observation of a new particle. The properties of the particle agree, within the relatively large experimental uncertainties, with the properties of the long-sought Higgs boson. Particle physicists around the globe are now wondering, ''Is it the Standard Model Higgs that we observe; or is it another particle with similar properties?'' We employ effective field theories (EFTs) for a general, model-independent description of the particle. We use a few, minimal assumptions - Standard Model (SM) particle content and a separation of scales to the new physics - which are supported by current experimental results. By construction, effective field theories describe a physical system only at a certain energy scale, in our case at the electroweak-scale v. Effects of new physics from a higher energy-scale, Λ, are described by modified interactions of the light particles. In this thesis, ''Higgs Effective Field Theories - Systematics and Applications'', we discuss effective field theories for the Higgs particle, which is not necessarily the Higgs of the Standard Model. In particular, we focus on a systematic and consistent expansion of the EFT. The systematics depends on the dynamics of the new physics. We distinguish two different consistent expansions. EFTs that describe decoupling new-physics effects and EFTs that describe non-decoupling new-physics effects. We briefly discuss the first case, the SM-EFT. The focus of this thesis, however, is on the non-decoupling EFTs. We argue that the loop expansion is the consistent expansion in the second case. We introduce the concept of chiral dimensions, equivalent to the loop expansion. Using the chiral dimensions, we expand the electroweak chiral Lagrangian up to next-to-leading order, O(f{sup 2}/Λ{sup 2})=O(1/16π{sup 2}). Further, we discuss how different 6. Noether's theorems applications in mechanics and field theory CERN Document Server 2016-01-01 The book provides a detailed exposition of the calculus of variations on fibre bundles and graded manifolds. It presents applications in such area's as non-relativistic mechanics, gauge theory, gravitation theory and topological field theory with emphasis on energy and energy-momentum conservation laws. Within this general context the first and second Noether theorems are treated in the very general setting of reducible degenerate graded Lagrangian theory. 7. Cosmological applications of algebraic quantum field theory in curved spacetimes CERN Document Server Hack, Thomas-Paul 2016-01-01 This book provides a largely self-contained and broadly accessible exposition on two cosmological applications of algebraic quantum field theory (QFT) in curved spacetime: a fundamental analysis of the cosmological evolution according to the Standard Model of Cosmology; and a fundamental study of the perturbations in inflation. The two central sections of the book dealing with these applications are preceded by sections providing a pedagogical introduction to the subject. Introductory material on the construction of linear QFTs on general curved spacetimes with and without gauge symmetry in the algebraic approach, physically meaningful quantum states on general curved spacetimes, and the backreaction of quantum fields in curved spacetimes via the semiclassical Einstein equation is also given. The reader should have a basic understanding of General Relativity and QFT on Minkowski spacetime, but no background in QFT on curved spacetimes or the algebraic approach to QFT is required. 8. Probabilistic theory of mean field games with applications CERN Document Server Carmona, René 2018-01-01 This two-volume book offers a comprehensive treatment of the probabilistic approach to mean field game models and their applications. The book is self-contained in nature and includes original material and applications with explicit examples throughout, including numerical solutions. Volume I of the book is entirely devoted to the theory of mean field games without a common noise. The first half of the volume provides a self-contained introduction to mean field games, starting from concrete illustrations of games with a finite number of players, and ending with ready-for-use solvability results. Readers are provided with the tools necessary for the solution of forward-backward stochastic differential equations of the McKean-Vlasov type at the core of the probabilistic approach. The second half of this volume focuses on the main principles of analysis on the Wasserstein space. It includes Lions' approach to the Wasserstein differential calculus, and the applications of its results to the analysis of stochastic... 9. Application of KBc Subalgebra in String Field Theory Science.gov (United States) Zeze, S. Recently, a classical solution of open cubic string field theory (CSFT) which corresponds to the closed string vacuum is found by Erler and Schnabl. In their work, a very simple subalgebra of open string star algebra --- called K B c subalgebra --- plays a crucial role. In this talk, we demonstrate two applications of the K B c subalgebra. One is evaluation of classical and effective tachyon potential. It turns out that the level expansion in the K B c subalgebra terminates at a certain level, so that analytic evaluation of effective potential is available. The other application is regularization of the identity based solutions. It is demonstrated that the Okawa-Erler-Schnabl type solution naturally includes gauge invariant regularization of identity based solutions. 10. Conformal field theory and its application to strings International Nuclear Information System (INIS) Verlinde, E.P. 1988-01-01 Conformal field theories on Riemann surfaces are considered and the result is applied to study the loop amplitudes for bosonic strings. It is shown that there is a close resemblance between the loop amplitudes for φ 3 -theory and the expressions for string multi-loop amplitudes. The similarity between φ 3 -amplitudes in curved backgrounds and the analytic structure of string amplitudes in backgrounds described by conformal field theories is also pointed out. 60 refs.; 5 figs.; 200 schemes 11. Post Modernity Theory and Its Educational Applications in School Fields Science.gov (United States) El-Baz, Maaly Bent Mohamed Saleh 2017-01-01 This paper aims to identify the fundamental principles on which the post modernity theory is based and to notice this in the field of Education, since this theory deals with two basic rules on which the postmodernist orientation is based, one of them denies on the absolute truth on Ontology level (related to the existence nature), and the other… 12. Field theory CERN Multimedia 1999-11-08 In these lectures I will build up the concept of field theory using the language of Feynman diagrams. As a starting point, field theory in zero spacetime dimensions is used as a vehicle to develop all the necessary techniques: path integral, Feynman diagrams, Schwinger-Dyson equations, asymptotic series, effective action, renormalization etc. The theory is then extended to more dimensions, with emphasis on the combinatorial aspects of the diagrams rather than their particular mathematical structure. The concept of unitarity is used to, finally, arrive at the various Feynman rules in an actual, four-dimensional theory. The concept of gauge-invariance is developed, and the structure of a non-abelian gauge theory is discussed, again on the level of Feynman diagrams and Feynman rules. 13. Bound state quantum field theory application to atoms and ions CERN Document Server Sapirstein, Jonathan 2019-01-01 Two aspects of the book should appeal to a wide audience. One aspect would be the comprehensive coverage on the latest updates and developments this book provides, besides Bethe and Salpeter's handbook on hydrogen and helium, which is still widely regarded as useful. The other aspect would be that a major part of the book uses effective field theory, a way of including quantum electrodynamics (QED) that starts with the familiar Schrödinger equation, and then adds perturbing operators derived in a rather simple manner that incorporates QED. Effective field theory is used in a number of fields including particle physics and nuclear physics, and readership is targeted at these communities too.Additionally, students using this book in conjunction with Peskin's textbook could learn to carry out fairly sophisticated calculations in QED in order to learn the technique, as this book comes with practical calculations.Also included is a very clear exposition of the BetheSalpeter equation, which is simply either ... 14. The application of mean field theory to image motion estimation. Science.gov (United States) Zhang, J; Hanauer, G G 1995-01-01 Previously, Markov random field (MRF) model-based techniques have been proposed for image motion estimation. Since motion estimation is usually an ill-posed problem, various constraints are needed to obtain a unique and stable solution. The main advantage of the MRF approach is its capacity to incorporate such constraints, for instance, motion continuity within an object and motion discontinuity at the boundaries between objects. In the MRF approach, motion estimation is often formulated as an optimization problem, and two frequently used optimization methods are simulated annealing (SA) and iterative-conditional mode (ICM). Although the SA is theoretically optimal in the sense of finding the global optimum, it usually takes many iterations to converge. The ICM, on the other hand, converges quickly, but its results are often unsatisfactory due to its "hard decision" nature. Previously, the authors have applied the mean field theory to image segmentation and image restoration problems. It provides results nearly as good as SA but with much faster convergence. The present paper shows how the mean field theory can be applied to MRF model-based motion estimation. This approach is demonstrated on both synthetic and real-world images, where it produced good motion estimates. 15. Applicability of self-consistent mean-field theory International Nuclear Information System (INIS) Guo Lu; Sakata, Fumihiko; Zhao Enguang 2005-01-01 Within the constrained Hartree-Fock (CHF) theory, an analytic condition is derived to estimate whether a concept of the self-consistent mean field is realized in the level repulsive region. The derived condition states that an iterative calculation of the CHF equation does not converge when the quantum fluctuations coming from two-body residual interaction and quadrupole deformation become larger than a single-particle energy difference between two avoided crossing orbits. By means of numerical calculation, it is shown that the analytic condition works well for a realistic case 16. Test-particle motion in Einstein's unified field theory. I. General theory and application to neutral test particles International Nuclear Information System (INIS) Johnson, C.R. 1985-01-01 We develop a method for finding the exact equations of structure and motion of multipole test particles in Einstein's unified field theory: the theory of the nonsymmetric field. The method is also applicable to Einstein's gravitational theory. Particles are represented by singularities in the field. The method is covariant at each step of the analysis. We also apply the method and find both in Einstein's unified field theory and in Einstein's gravitational theory the equations of structure and motion of neutral pole-dipole test particles possessing no electromagnetic multipole moments. In the case of Einstein's gravitational theory the results are the well-known equations of structure and motion of a neutral pole-dipole test particle in a given background gravitational field. In the case of Einstein's unified field theory the results are the same, providing we identify a certain symmetric second-rank tensor field appearing in Einstein's theory with the metric and gravitational field. We therefore discover not only the equations of structure and motion of a neutral test particle in Einstein's unified field theory, but we also discover what field in Einstein's theory plays the role of metric and gravitational field 17. Algebric generalization of symmetry Dirac bracket. Application to field theory International Nuclear Information System (INIS) Rocha Filho, T.M. da. 1987-01-01 The A set of observable of a physical system with finite e infinite number of degrees of freedom and submitted to certain constraint conditions, is considered. Using jordan algebra structure on A in relation to bymmetric Poisson bracket obtained by Droz-Vincent, a jordan product is obtained on the A/I quocient set with regard to I ideal generated by constraints of second class. It is shown that this product on A/I corresponds to symmetric Dirac bracket. The developed formulation is applied to a system corresponding to harmonic oscillators, non relativistic field, Rarita-Schwinger field and the possibility of its utilization in fermionic string theories is discussed. (M.C.K.) 18. Field theories with subcanonical fields International Nuclear Information System (INIS) Bigi, I.I.Y. 1976-01-01 The properties of quantum field theories with spinor fields of dimension less than the canonical value of 3/2 are studied. As a starting point for the application of common perturbation theory we look for the linear version of these theories. A gange-interaction is introduced and with the aid of power counting the renormalizability of the theory is shown. It follows that in the case of a spinor-field with negative dimension renormalization can only be attained if the interaction has a further symmetry. By this symmetry the theory is determined in an unequivocal way. The gange-interaction introduced in the theory leads to a spontaneous breakdown of scale invariance whereby masses are produced. At the same time the spinor-field operators can now be separated in two orthogonal sections with opposite norm. It is proposed to use the section with negative (positive) norm to describe hadrons (leptons) respectively. (orig./WL) [de 19. Extremes in random fields a theory and its applications CERN Document Server Yakir, Benjamin 2013-01-01 Presents a useful new technique for analyzing the extreme-value behaviour of random fields Modern science typically involves the analysis of increasingly complex data. The extreme values that emerge in the statistical analysis of complex data are often of particular interest. This book focuses on the analytical approximations of the statistical significance of extreme values. Several relatively complex applications of the technique to problems that emerge in practical situations are presented. All the examples are difficult to analyze using classical methods, and as a result, the author pr 20. Probabilistic theory of mean field games with applications I mean field FBSDEs, control, and games CERN Document Server Carmona, René 2018-01-01 This two-volume book offers a comprehensive treatment of the probabilistic approach to mean field game models and their applications. The book is self-contained in nature and includes original material and applications with explicit examples throughout, including numerical solutions. Volume I of the book is entirely devoted to the theory of mean field games without a common noise. The first half of the volume provides a self-contained introduction to mean field games, starting from concrete illustrations of games with a finite number of players, and ending with ready-for-use solvability results. Readers are provided with the tools necessary for the solution of forward-backward stochastic differential equations of the McKean-Vlasov type at the core of the probabilistic approach. The second half of this volume focuses on the main principles of analysis on the Wasserstein space. It includes Lions' approach to the Wasserstein differential calculus, and the applications of its results to the analysis of stochastic... 1. Superspace conformal field theory International Nuclear Information System (INIS) Quella, Thomas 2013-07-01 Conformal sigma models and WZW models on coset superspaces provide important examples of logarithmic conformal field theories. They possess many applications to problems in string and condensed matter theory. We review recent results and developments, including the general construction of WZW models on type I supergroups, the classification of conformal sigma models and their embedding into string theory. 2. Superspace conformal field theory Energy Technology Data Exchange (ETDEWEB) Quella, Thomas [Koeln Univ. (Germany). Inst. fuer Theoretische Physik; Schomerus, Volker [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany) 2013-07-15 Conformal sigma models and WZW models on coset superspaces provide important examples of logarithmic conformal field theories. They possess many applications to problems in string and condensed matter theory. We review recent results and developments, including the general construction of WZW models on type I supergroups, the classification of conformal sigma models and their embedding into string theory. 3. Conformal field theory CERN Document Server Ketov, Sergei V 1995-01-01 Conformal field theory is an elegant and powerful theory in the field of high energy physics and statistics. In fact, it can be said to be one of the greatest achievements in the development of this field. Presented in two dimensions, this book is designed for students who already have a basic knowledge of quantum mechanics, field theory and general relativity. The main idea used throughout the book is that conformal symmetry causes both classical and quantum integrability. Instead of concentrating on the numerous applications of the theory, the author puts forward a discussion of the general 4. Microwave field-efffect transistors theory, design, and application CERN Document Server Pengelly, Raymond 1994-01-01 This book covers the use of devices in microwave circuits and includes such topics as semiconductor theory and transistor performance, CAD considerations, intermodulation, noise figure, signal handling, S-parameter mapping, narrow- and broadband techniques, packaging and thermal considerations. 5. Engineering field theory CERN Document Server 2014-01-01 Engineering Field Theory focuses on the applications of field theory in gravitation, electrostatics, magnetism, electric current flow, conductive heat transfer, fluid flow, and seepage.The manuscript first ponders on electric flux, electrical materials, and flux function. Discussions focus on field intensity at the surface of a conductor, force on a charged surface, atomic properties, doublet and uniform field, flux tube and flux line, line charge and line sink, field of a surface charge, field intensity, flux density, permittivity, and Coulomb's law. The text then takes a look at gravitation 6. Shielding Flowers Developing under Stress: Translating Theory to Field Application Directory of Open Access Journals (Sweden) Noam Chayut 2014-07-01 Full Text Available Developing reproductive organs within a flower are sensitive to environmental stress. A higher incidence of environmental stress during this stage of a crop plants’ developmental cycle will lead to major breaches in food security. Clearly, we need to understand this sensitivity and try and overcome it, by agricultural practices and/or the breeding of more tolerant cultivars. Although passion fruit vines initiate flowers all year round, flower primordia abort during warm summers. This restricts the season of fruit production in regions with warm summers. Previously, using controlled chambers, stages in flower development that are sensitive to heat were identified. Based on genetic analysis and physiological experiments in controlled environments, gibberellin activity appeared to be a possible point of horticultural intervention. Here, we aimed to shield flowers of a commercial cultivar from end of summer conditions, thus allowing fruit production in new seasons. We conducted experiments over three years in different settings, and our findings consistently show that a single application of an inhibitor of gibberellin biosynthesis to vines in mid-August can cause precocious flowering of ~2–4 weeks, leading to earlier fruit production of ~1 month. In this case, knowledge obtained on phenology, environmental constraints and genetic variation, allowed us to reach a practical solution. 7. Effective field theories CERN Document Server Petrov, Alexey A 2016-01-01 This book is a broad-based text intended to help the growing student body interested in topics such as gravitational effective theories, supersymmetric effective theories, applications of effective theory techniques to problems in condensed matter physics (superconductivity) and quantum chromodynamics (such as soft-collinear effective theory). It begins with a review of the use of symmetries to identify the relevant degrees of freedom in a problem, and then presents a variety of methods that can be used to solve physical problems. A detailed discussion of canonical examples of effective field theories with increasing complexity is then conducted. Special cases such as supersymmetry and lattice EFT are discussed, as well as recently-found applications to problems in gravitation and cosmology. An appendix includes various factoids from group theory and other topics that are used throughout the text, in an attempt to make the book self-contained. 8. String theory or field theory? International Nuclear Information System (INIS) Marshakov, A.V. 2002-01-01 The status of string theory is reviewed, and major recent developments - especially those in going beyond perturbation theory in the string theory and quantum field theory frameworks - are analyzed. This analysis helps better understand the role and place of experimental phenomena, it is emphasized that there are some insurmountable problems inherent in it - notably the impossibility to formulate the quantum theory of gravity on its basis - which prevent it from being a fundamental physical theory of the world of microscopic distances. It is this task, the creation of such a theory, which string theory, currently far from completion, is expected to solve. In spite of its somewhat vague current form, string theory has already led to a number of serious results and greatly contributed to progress in the understanding of quantum field theory. It is these developments, which are our concern in this review [ru 9. String theory or field theory? International Nuclear Information System (INIS) Marshakov, Andrei V 2002-01-01 The status of string theory is reviewed, and major recent developments - especially those in going beyond perturbation theory in the string theory and quantum field theory frameworks - are analyzed. This analysis helps better understand the role and place of string theory in the modern picture of the physical world. Even though quantum field theory describes a wide range of experimental phenomena, it is emphasized that there are some insurmountable problems inherent in it - notably the impossibility to formulate the quantum theory of gravity on its basis - which prevent it from being a fundamental physical theory of the world of microscopic distances. It is this task, the creation of such a theory, which string theory, currently far from completion, is expected to solve. In spite of its somewhat vague current form, string theory has already led to a number of serious results and greatly contributed to progress in the understanding of quantum field theory. It is these developments which are our concern in this review. (reviews of topical problems) 10. Quantum field theory of material properties. Its application to models of Rashba spin splitting International Nuclear Information System (INIS) Schober, Giulio Albert Heinrich 2016-01-01 In this thesis, we argue that microscopic field theories - which as such are already scientifically established - have emerged as a new paradigm in materials physics. We hence seek to elaborate on such field theories which underlie modern ab initio calculations, and we apply them to the bismuth tellurohalides (BiTeX with X=I,Br,Cl) as a prototypical class of spin-based materials. For this purpose, we begin by constructing tight-binding models which approximately describe the spin-split conduction bands of BiTeI. Following this, we derive the theory of temperature Green functions systematically from their fundamental equations of motion. This in turn enables us to develop a combined functional renormalization and mean-field approach which is suitable for application to multiband models. For the Rashba model including an attractive, local interaction, this approach yields an unconventional superconducting phase with a singlet gap function and a mixed singlet-triplet order parameter. We further investigate the unusual electromagnetic response of BiTeI, which is caused by the Rashba spin splitting and which includes, in particular, an orbital paramagnetism. Finally, we conclude by summarizing the Functional Approach to electrodynamics of media as a microscopic field theory of electromagnetic material properties which sits in accordance with ab initio physics. 11. Theory of interacting quantum fields International Nuclear Information System (INIS) Rebenko, Alexei L. 2012-01-01 This monograph is devoted to the systematic presentation of foundations of the quantum field theory. Unlike numerous monographs devoted to this topic, a wide range of problems covered in this book are accompanied by their sufficiently clear interpretations and applications. An important significant feature of this monograph is the desire of the author to present mathematical problems of the quantum field theory with regard to new methods of the constructive and Euclidean field theory that appeared in the last thirty years of the 20 th century and are based on the rigorous mathematical apparatus of functional analysis, the theory of operators, and the theory of generalized functions. The monograph is useful for students, post-graduate students, and young scientists who desire to understand not only the formality of construction of the quantum field theory but also its essence and connection with the classical mechanics, relativistic classical field theory, quantum mechanics, group theory, and the theory of path integral formalism. 12. Advances in dynamic and mean field games theory, applications, and numerical methods CERN Document Server Viscolani, Bruno 2017-01-01 This contributed volume considers recent advances in dynamic games and their applications, based on presentations given at the 17th Symposium of the International Society of Dynamic Games, held July 12-15, 2016, in Urbino, Italy. Written by experts in their respective disciplines, these papers cover various aspects of dynamic game theory including mean-field games, stochastic and pursuit-evasion games, and computational methods for dynamic games. Topics covered include Pedestrian flow in crowded environments Models for climate change negotiations Nash Equilibria for dynamic games involving Volterra integral equations Differential games in healthcare markets Linear-quadratic Gaussian dynamic games Aircraft control in wind shear conditions Advances in Dynamic and Mean-Field Games presents state-of-the-art research in a wide spectrum of areas. As such, it serves as a testament to the continued vitality and growth of the field of dynamic games and their applications. It will be of interest to an interdisciplinar... 13. Application of self-consistent field theory to self-assembled bilayer membranes International Nuclear Information System (INIS) Zhang Ping-Wen; Shi An-Chang 2015-01-01 Bilayer membranes self-assembled from amphiphilic molecules such as lipids, surfactants, and block copolymers are ubiquitous in biological and physiochemical systems. The shape and structure of bilayer membranes depend crucially on their mechanical properties such as surface tension, bending moduli, and line tension. Understanding how the molecular properties of the amphiphiles determine the structure and mechanics of the self-assembled bilayers requires a molecularly detailed theoretical framework. The self-consistent field theory provides such a theoretical framework, which is capable of accurately predicting the mechanical parameters of self-assembled bilayer membranes. In this mini review we summarize the formulation of the self-consistent field theory, as exemplified by a model system composed of flexible amphiphilic chains dissolved in hydrophilic polymeric solvents, and its application to the study of self-assembled bilayer membranes. (topical review) 14. The application of Regge calculus to quantum gravity and quantum field theory in a curved background International Nuclear Information System (INIS) Warner, N.P. 1982-01-01 The application of Regge calculus to quantum gravity and quantum field theory in a curved background is discussed. A discrete form of exterior differential calculus is developed, and this is used to obtain Laplacians for p-forms on the Regge manifold. To assess the accuracy of these approximations, the eigenvalues of the discrete Laplacians were calculated for the regular tesselations of S 2 and S 3 . The results indicate that the methods obtained in this paper may be used in curved space-times with an accuracy comparing with that obtained in lattice gauge theories on a flat background. It also becomes evident that Regge calculus provides particularly suitable lattices for Monte-Carlo techniques. (author) 15. Field theory and strings International Nuclear Information System (INIS) Bonara, L.; Cotta-Ramusino, P.; Rinaldi, M. 1987-01-01 It is well-known that type I and heterotic superstring theories have a zero mass spectrum which correspond to the field content of N=1 supergravity theory coupled to supersymmetric Yang-Mills theory in 10-D. The authors study the field theory ''per se'', in the hope that simple consistency requirements will determine the theory completely once one knows the field content inherited from string theory. The simplest consistency requirements are: N=1 supersymmetry; and absence of chiral anomalies. This is what the authors discuss in this paper here leaving undetermined the question of the range of validity of the resulting field theory. As is known, a model of N=1 supergravity (SUGRA) coupled to supersymmetric Yang-Mills (SYM) theory was known in the form given by Chapline and Manton. The coupling of SUGRA to SYM was determined by the definition of the ''field strength'' 3-form H in this paper 16. Quantum field theory International Nuclear Information System (INIS) Ryder, L.H. 1985-01-01 This introduction to the ideas and techniques of quantum field theory presents the material as simply as possible and is designed for graduate research students. After a brief survey of particle physics, the quantum theory of scalar and spinor fields and then of gauge fields, is developed. The emphasis throughout is on functional methods, which have played a large part in modern field theory. The book concludes with a bridge survey of ''topological'' objects in field theory and assumes a knowledge of quantum mechanics and special relativity 17. String field theory International Nuclear Information System (INIS) Kaku, M. 1987-01-01 In this article, the authors summarize the rapid progress in constructing string field theory actions, such as the development of the covariant BRST theory. They also present the newer geometric formulation of string field theory, from which the BRST theory and the older light cone theory can be derived from first principles. This geometric formulation allows us to derive the complete field theory of strings from two geometric principles, in the same way that general relativity and Yang-Mills theory can be derived from two principles based on global and local symmetry. The geometric formalism therefore reduces string field theory to a problem of finding an invariant under a new local gauge group they call the universal string group (USG). Thus, string field theory is the gauge theory of the universal string group in much the same way that Yang-Mills theory is the gauge theory of SU(N). The geometric formulation places superstring theory on the same rigorous group theoretical level as general relativity and gauge theory 18. Algebraic conformal field theory International Nuclear Information System (INIS) Fuchs, J.; Nationaal Inst. voor Kernfysica en Hoge-Energiefysica 1991-11-01 Many conformal field theory features are special versions of structures which are present in arbitrary 2-dimensional quantum field theories. So it makes sense to describe 2-dimensional conformal field theories in context of algebraic theory of superselection sectors. While most of the results of the algebraic theory are rather abstract, conformal field theories offer the possibility to work out many formulae explicitly. In particular, one can construct the full algebra A-bar of global observables and the endomorphisms of A-bar which represent the superselection sectors. Some explicit results are presented for the level 1 so(N) WZW theories; the algebra A-bar is found to be the enveloping algebra of a Lie algebra L-bar which is an extension of the chiral symmetry algebra of the WZW theory. (author). 21 refs., 6 figs 19. Field theory approach to gravitation International Nuclear Information System (INIS) Yilmaz, H. 1978-01-01 A number of authors considered the possibility of formulating a field-theory approach to gravitation with the claim that such an approach would uniquely lead to Einstein's theory of general relativity. In this article it is shown that the field theory approach is more generally applicable and uniqueness cannot be claimed. Theoretical and experimental reasons are given showing that the Einsteinian limit appears to be unviable 20. Field theory and particle physics International Nuclear Information System (INIS) Eboli, O.J.P.; Gomes, M.; Santoro, A. 1990-01-01 This book contains the proceedings of the topics covered during the fifth Jorge Andre Swieca Summer School. The first part of the book collects the material devoted to quantum field theory. There were four courses on methods in Field Theory; H. O. Girotti lectured on constrained dynamics, R. Jackiw on the Schrodinger representation in Field Theory, S.-Y. Pi on the application of this representation to quantum fields in a Robertson-Walker spacetime, and L. Vinet on Berry Connections. There were three courses on Conformal Field Theory: I. Todorov focused on the problem of construction and classification of conformal field theories. Lattice models, two-dimensional S matrices and conformal field theory were looked from the unifying perspective of the Yang-Baxter algebras in the lectures given by M. Karowski. Parasupersymmetric quantum mechanics was discussed in the lectures by L. Vinet. Besides those courses, there was an introduction to string field theory given by G. Horowitz. There were also three seminars: F. Schaposnik reported on recent applications of topological methods in field theory, P. Gerbert gave a seminar on three dimensional gravity and V. Kurak talked on two dimensional parafermionic models. The second part of this proceedings is devoted to phenomenology. There were three courses on Particle Physics: Dan Green lectured on collider physics, E. Predrazzi on strong interactions and G. Cohen-Tanoudji on the use of strings in strong interactions 1. Electron traps in polar liquids. An application of the formalism of the random field theory International Nuclear Information System (INIS) Hilczer, M.; Bartczak, W.M. 1992-01-01 The potential energy surface in a disordered medium is described, using the concepts of the mathematical theory of random fields. The statistics of trapping sites (the regions of an excursion of the random field) is obtained for liquid methanol as a numerical example of the theory. (author). 15 refs, 4 figs 2. Finite discrete field theory International Nuclear Information System (INIS) Souza, Manoelito M. de 1997-01-01 We discuss the physical meaning and the geometric interpretation of implementation in classical field theories. The origin of infinities and other inconsistencies in field theories is traced to fields defined with support on the light cone; a finite and consistent field theory requires a light-cone generator as the field support. Then, we introduce a classical field theory with support on the light cone generators. It results on a description of discrete (point-like) interactions in terms of localized particle-like fields. We find the propagators of these particle-like fields and discuss their physical meaning, properties and consequences. They are conformally invariant, singularity-free, and describing a manifestly covariant (1 + 1)-dimensional dynamics in a (3 = 1) spacetime. Remarkably this conformal symmetry remains even for the propagation of a massive field in four spacetime dimensions. We apply this formalism to Classical electrodynamics and to the General Relativity Theory. The standard formalism with its distributed fields is retrieved in terms of spacetime average of the discrete field. Singularities are the by-products of the averaging process. This new formalism enlighten the meaning and the problem of field theory, and may allow a softer transition to a quantum theory. (author) 3. Geophysical Field Theory International Nuclear Information System (INIS) Eloranta, E. 2003-11-01 The geophysical field theory includes the basic principles of electromagnetism, continuum mechanics, and potential theory upon which the computational modelling of geophysical phenomena is based on. Vector analysis is the main mathematical tool in the field analyses. Electrostatics, stationary electric current, magnetostatics, and electrodynamics form a central part of electromagnetism in geophysical field theory. Potential theory concerns especially gravity, but also electrostatics and magnetostatics. Solid state mechanics and fluid mechanics are central parts in continuum mechanics. Also the theories of elastic waves and rock mechanics belong to geophysical solid state mechanics. The theories of geohydrology and mass transport form one central field theory in geophysical fluid mechanics. Also heat transfer is included in continuum mechanics. (orig.) 4. Nonlocal continuum field theories CERN Document Server 2002-01-01 Nonlocal continuum field theories are concerned with material bodies whose behavior at any interior point depends on the state of all other points in the body -- rather than only on an effective field resulting from these points -- in addition to its own state and the state of some calculable external field. Nonlocal field theory extends classical field theory by describing the responses of points within the medium by functionals rather than functions (the "constitutive relations" of classical field theory). Such considerations are already well known in solid-state physics, where the nonlocal interactions between the atoms are prevalent in determining the properties of the material. The tools developed for crystalline materials, however, do not lend themselves to analyzing amorphous materials, or materials in which imperfections are a major part of the structure. Nonlocal continuum theories, by contrast, can describe these materials faithfully at scales down to the lattice parameter. This book presents a unif... 5. Group theory and general relativity representations of the Lorentz group and their applications to the gravitational field CERN Document Server Carmeli, Moshe 2000-01-01 This is the only book on the subject of group theory and Einstein's theory of gravitation. It contains an extensive discussion on general relativity from the viewpoint of group theory and gauge fields. It also puts together in one volume many scattered, original works, on the use of group theory in general relativity theory.There are twelve chapters in the book. The first six are devoted to rotation and Lorentz groups, and their representations. They include the spinor representation as well as the infinite-dimensional representations. The other six chapters deal with the application of groups 6. An application of random field theory to analysis of electron trapping sites in disordered media International Nuclear Information System (INIS) Hilczer, M.; Bartczak, W.M. 1993-01-01 The potential energy surface in a disordered medium is considered a random field and described using the concepts of the mathematical theory of random fields. The preexisting traps for excess electrons are identified with certain regions of excursion (extreme regions) of the potential field. The theory provides an analytical method of statistical analysis of these regions. Parameters of the cavity-averaged potential field, which are provided by computer simulation of a given medium, serve as input data for the analysis. The statistics of preexisting traps are obtained for liquid methanol as a numerical example of the random field method. 26 refs., 6 figs 7. Hyperfunction quantum field theory International Nuclear Information System (INIS) Nagamachi, S.; Mugibayashi, N. 1976-01-01 The quantum field theory in terms of Fourier hyperfunctions is constructed. The test function space for hyperfunctions does not contain C infinitely functios with compact support. In spite of this defect the support concept of H-valued Fourier hyperfunctions allows to formulate the locality axiom for hyperfunction quantum field theory. (orig.) [de 8. Quantum field theory CERN Document Server Mandl, Franz 2010-01-01 Following on from the successful first (1984) and revised (1993) editions, this extended and revised text is designed as a short and simple introduction to quantum field theory for final year physics students and for postgraduate students beginning research in theoretical and experimental particle physics. The three main objectives of the book are to: Explain the basic physics and formalism of quantum field theory To make the reader proficient in theory calculations using Feynman diagrams To introduce the reader to gauge theories, which play a central role in elementary particle physic 9. Analytical Thermal Field Theory Applicable to Oil Hydraulic Fluid Film Lubrication DEFF Research Database (Denmark) Johansen, Per; Roemer, Daniel Beck; Pedersen, Henrik C. 2014-01-01 An analytical thermal field theory is derived by a perturbation series expansion solution to the energy conservation equation. The theory is valid for small values of the Brinkman number and the modified Peclet number. This condition is sufficiently satisfied for hydraulic oils, whereby...... expansion of the thermal field. The series solution is truncated at first order in order to obtain a closed form approximation. Finally a numerical thermohydrodynamic simulation of a piston-cylinder interface is presented, and the results are used for a comparison with the analytical theory in order... 10. Algebraic quantum field theory International Nuclear Information System (INIS) Foroutan, A. 1996-12-01 The basic assumption that the complete information relevant for a relativistic, local quantum theory is contained in the net structure of the local observables of this theory results first of all in a concise formulation of the algebraic structure of the superselection theory and an intrinsic formulation of charge composition, charge conjugation and the statistics of an algebraic quantum field theory. In a next step, the locality of massive particles together with their spectral properties are wed for the formulation of a selection criterion which opens the access to the massive, non-abelian quantum gauge theories. The role of the electric charge as a superselection rule results in the introduction of charge classes which in term lead to a set of quantum states with optimum localization properties. Finally, the asymptotic observables of quantum electrodynamics are investigated within the framework of algebraic quantum field theory. (author) 11. Closed string field theory International Nuclear Information System (INIS) Strominger, A. 1987-01-01 A gauge invariant cubic action describing bosonic closed string field theory is constructed. The gauge symmetries include local spacetime diffeomorphisms. The conventional closed string spectrum and trilinear couplings are reproduced after spontaneous symmetry breaking. The action S is constructed from the usual ''open string'' field of ghost number minus one half. It is given by the associator of the string field product which is non-vanishing because of associativity anomalies. S does not describe open string propagation because open string states associate and can thereby be shifted away. A field theory of closed and open strings can be obtained by adding to S the cubic open string action. (orig.) 12. Lectures on matrix field theory CERN Document Server 2017-01-01 These lecture notes provide a systematic introduction to matrix models of quantum field theories with non-commutative and fuzzy geometries. The book initially focuses on the matrix formulation of non-commutative and fuzzy spaces, followed by a description of the non-perturbative treatment of the corresponding field theories. As an example, the phase structure of non-commutative phi-four theory is treated in great detail, with a separate chapter on the multitrace approach. The last chapter offers a general introduction to non-commutative gauge theories, while two appendices round out the text. Primarily written as a self-study guide for postgraduate students – with the aim of pedagogically introducing them to key analytical and numerical tools, as well as useful physical models in applications – these lecture notes will also benefit experienced researchers by providing a reference guide to the fundamentals of non-commutative field theory with an emphasis on matrix models and fuzzy geometries. 13. Interpolating string field theories International Nuclear Information System (INIS) Zwiebach, B. 1992-01-01 This paper reports that a minimal area problem imposing different length conditions on open and closed curves is shown to define a one-parameter family of covariant open-closed quantum string field theories. These interpolate from a recently proposed factorizable open-closed theory up to an extended version of Witten's open string field theory capable of incorporating on shell closed strings. The string diagrams of the latter define a new decomposition of the moduli spaces of Riemann surfaces with punctures and boundaries based on quadratic differentials with both first order and second order poles 14. Application of the random field theory in PET imaging - injection dose optimization Czech Academy of Sciences Publication Activity Database Dvořák, Jiří; Boldyš, Jiří; Skopalová, M.; Bělohlávek, O. 2013-01-01 Roč. 49, č. 2 (2013), s. 280-300 ISSN 0023-5954 R&D Projects: GA MŠk 1M0572 Institutional support: RVO:67985556 Keywords : random field theory * Euler characteristic * PET imaging * PET image quality Subject RIV: BD - Theory of Information Impact factor: 0.563, year: 2013 http://library.utia.cas.cz/separaty/2013/ZOI/boldys-0397176.pdf 15. 2D fractional supersymmetry for rational conformal field theory: application for third-integer spin states International Nuclear Information System (INIS) Perez, A.; Simon, P. 1996-01-01 A 2D fractional supersymmetry theory is algebraically constructed. The Lagrangian is derived using an adapted superspace including, in addition to a scalar field, two fields with spins 1/3,2/3. This theory turns out to be a rational conformal field theory. The symmetry of this model goes beyond the super-Virasoro algebra and connects these third-integer spin states. Besides the stress-momentum tensor, we obtain a supercurrent of spin 4/3. Cubic relations are involved in order to close the algebra; the basic algebra is no longer a Lie or a super-Lie algebra. The central charge of this model is found to be 5/3. Finally, we analyze the form that a local invariant action should take. (orig.) 16. Axiomatic conformal field theory International Nuclear Information System (INIS) Gaberdiel, M.R.; Goddard, P. 2000-01-01 A new rigourous approach to conformal field theory is presented. The basic objects are families of complex-valued amplitudes, which define a meromorphic conformal field theory (or chiral algebra) and which lead naturally to the definition of topological vector spaces, between which vertex operators act as continuous operators. In fact, in order to develop the theory, Moebius invariance rather than full conformal invariance is required but it is shown that every Moebius theory can be extended to a conformal theory by the construction of a Virasoro field. In this approach, a representation of a conformal field theory is naturally defined in terms of a family of amplitudes with appropriate analytic properties. It is shown that these amplitudes can also be derived from a suitable collection of states in the meromorphic theory. Zhu's algebra then appears naturally as the algebra of conditions which states defining highest weight representations must satisfy. The relationship of the representations of Zhu's algebra to the classification of highest weight representations is explained. (orig.) 17. Quantum theory of fields CERN Document Server Wentzel, Gregor 1949-01-01 A prominent figure in twentieth-century physics, Gregor Wentzel made major contributions to the development of quantum field theory, first in Europe and later at the University of Chicago. His Quantum Theory of Fields offers a knowledgeable view of the original literature of elementary quantum mechanics and helps make these works accessible to interested readers.An introductory volume rather than an all-inclusive account, the text opens with an examination of general principles, without specification of the field equations of the Lagrange function. The following chapters deal with particular 18. Theoretical physics. Field theory International Nuclear Information System (INIS) Landau, L.; Lifchitz, E. 2004-01-01 This book is the fifth French edition of the famous course written by Landau/Lifchitz and devoted to both the theory of electromagnetic fields and the gravity theory. The talk of the theory of electromagnetic fields is based on special relativity and relates to only the electrodynamics in vacuum and that of pointwise electric charges. On the basis of the fundamental notions of the principle of relativity and of relativistic mechanics, and by using variational principles, the authors develop the fundamental equations of the electromagnetic field, the wave equation and the processes of emission and propagation of light. The theory of gravitational fields, i.e. the general theory of relativity, is exposed in the last five chapters. The fundamentals of the tensor calculus and all that is related to it are progressively introduced just when needed (electromagnetic field tensor, energy-impulse tensor, or curve tensor...). The worldwide reputation of this book is generally allotted to clearness, to the simplicity and the rigorous logic of the demonstrations. (A.C.) 19. Application of Stochastic Unsaturated Flow Theory, Numerical Simulations, and Comparisons to Field Observations DEFF Research Database (Denmark) Jensen, Karsten Høgh; Mantoglou, Aristotelis 1992-01-01 A stochastic unsaturated flow theory and a numerical simulation model have been coupled in order to estimate the large-scale mean behavior of an unsaturated flow system in a spatially variable soil. On the basis of the theoretical developments of Mantoglou and Gelhar (1987a, b, c), the theory...... unsaturated flow equation representing the mean system behavior is solved using a finite difference numerical solution technique. The effective parameters are evaluated from the stochastic theory formulas before entering them into the numerical solution for each iteration. The stochastic model is applied...... to a field site in Denmark, where information is available on the spatial variability of soil parameters and variables. Numerical simulations have been carried out, and predictions of the mean behavior and the variance of the capillary tension head and the soil moisture content have been compared to field... 20. Introduction to gauge field theory International Nuclear Information System (INIS) Bailin, David; Love, Alexander 1986-01-01 The book is intended as an introduction to gauge field theory for the postgraduate student of theoretical particle physics. The topics discussed in the book include: path integrals, classical and quantum field theory, scattering amplitudes, feynman rules, renormalisation, gauge field theories, spontaneous symmetry breaking, grand unified theory, and field theories at finite temperature. (UK) 1. Gauge field theory International Nuclear Information System (INIS) Aref'eva, I.Ya.; Slavnov, A.A. 1981-01-01 This lecture is devoted to the discussion of gauge field theory permitting from the single point of view to describe all the interactions of elementary particles. The authors used electrodynamics and the Einstein theory of gravity to search for a renormgroup fixing a form of Lagrangian. It is shown that the gauge invariance added with the requirement of the minimum number of arbitraries in Lagrangian fixes unambigously the form of the electromagnetic interaction. The generalization of this construction for more complicate charge spaces results in the Yang-Mills theory. The interaction form in this theory is fixed with the relativity principle in the charge space. A quantum scheme of the Yang-Mills fields through the explicit separation of true dynamic variables is suggested. A comfortable relativistically invariant diagram technique for the calculation of a producing potential for the Green functions is described. The Ward generalized identities have been obtained and a procedure of the elimination of ultraviolet and infrared divergencies has been accomplished. Within the framework of QCD (quantum-chromodynamic) the phenomenon of the asymptotic freedom being the most successful prediction of the gauge theory of strong interactions was described. Working methods with QCD outside the framework of the perturbation theory have been described from a coupling constant. QCD is represented as a single theory possessing both the asymptotical freedom and the freedom retaining quarks [ru 2. Theory of electromagnetic fields CERN Document Server Wolski, Andrzej 2011-01-01 We discuss the theory of electromagnetic fields, with an emphasis on aspects relevant to radiofrequency systems in particle accelerators. We begin by reviewing Maxwell's equations and their physical significance. We show that in free space, there are solutions to Maxwell's equations representing the propagation of electromagnetic fields as waves. We introduce electromagnetic potentials, and show how they can be used to simplify the calculation of the fields in the presence of sources. We derive Poynting's theorem, which leads to expressions for the energy density and energy flux in an electromagnetic field. We discuss the properties of electromagnetic waves in cavities, waveguides and transmission lines. 3. Applications of Canonical transformations and nontrivial vacuum solutions to flavor mixing and critical phenomena in quantum field theory Energy Technology Data Exchange (ETDEWEB) Mishchenko, Yuriy [North Carolina State Univ., Raleigh, NC (United States) 2004-12-01 MISHCHENKO, YURIY. Applications of Canonical Transformations and Nontrivial Vacuum Solutions to flavor mixing and critical phenomena in Quantum Field Theory. (Under the direction of Chueng-Ryong Ji.) In this dissertation we consider two recent applications of Bogoliubov Transformation to the phenomenology of quantum mixing and the theory of critical phenomena. In recent years quantum mixing got in the focus of the searches for New Physics due to its unparalleled sensitivity to SM parameters and indications of neutrino mixing. It was recently suggested that Bogoliubov Transformation may be important in proper definition of the flavor states that otherwise results in problems in perturbative treatment. As first part of this dissertation we investigate this conjecture and develop a complete formulation of such a mixing field theory involving introduction of general formalism, analysis of space-time conversion and phenomenological implications. As second part of this dissertati 4. Applications of the renormalization group approach to problems in quantum field theory International Nuclear Information System (INIS) Renken, R.L. 1985-01-01 The presence of fluctuations at many scales of length complicates theories of quantum fields. However, interest is often focused on the low-energy consequences of a theory rather than the short distance fluctuations. In the renormalization-group approach, one takes advantage of this by constructing an effective theory with identical low-energy behavior, but without short distance fluctuations. Three problems of this type are studied here. In chapter 1, an effective lagrangian is used to compute the low-energy consequences of theories of technicolor. Corrections to weak-interaction parameters are found to be small, but conceivably measurable. In chapter 2, the renormalization group approach is applied to second order phase transitions in lattice gauge theories such as the deconfining transition in the U(1) theory. A practical procedure for studying the critical behavior based on Monte Carlo renormalization group methods is described in detail; no numerical results are presented. Chapter 3 addresses the problem of computing the low-energy behavior of atoms directly from Schrodinger's equation. A straightforward approach is described, but is found to be impractical 5. Euclidean quantum field theory International Nuclear Information System (INIS) Jaffe, A. 1985-01-01 In four seminal papers, written from 1963 to 1968, Kurt Symanzik laid the foundations for his euclidean quantum field theory program (EQFT). His original goal was to use EQFT as a tool to approach the existence question for interacting quantum fields. In 1968, when other methods appeared better suited for the existence question, Symanzik abandoned this heroic attempt and redirected his research toward different questions. (orig./HSI) 6. Quantum Field Theory CERN Document Server Zeidler, Eberhard This is the first volume of a modern introduction to quantum field theory which addresses both mathematicians and physicists ranging from advanced undergraduate students to professional scientists. The book tries to bridge the existing gap between the different languages used by mathematicians and physicists. For students of mathematics it is shown that detailed knowledge of the physical background helps to motivate the mathematical subjects and to discover interesting interrelationships between quite different mathematical topics. For students of physics, fairly advanced mathematics is presented, which is beyond the usual curriculum in physics. It is the author's goal to present the state of the art of realizing Einstein's dream of a unified theory for the four fundamental forces in the universe (gravitational, electromagnetic, strong, and weak interaction). From the reviews: "… Quantum field theory is one of the great intellectual edifices in the history of human thought. … This volume differs from othe... 7. Microcontinuum field theories CERN Document Server Eringen, A Cemal 1999-01-01 Microcontinuum field theories constitute an extension of classical field theories -- of elastic bodies, deformations, electromagnetism, and the like -- to microscopic spaces and short time scales. Material bodies are here viewed as collections of large numbers of deformable particles, much as each volume element of a fluid in statistical mechanics is viewed as consisting of a large number of small particles for which statistical laws are valid. Classical continuum theories are valid when the characteristic length associated with external forces or stimuli is much larger than any internal scale of the body under consideration. When the characteristic lengths are comparable, however, the response of the individual constituents becomes important, for example, in considering the fluid or elastic properties of blood, porous media, polymers, liquid crystals, slurries, and composite materials. This volume is concerned with the kinematics of microcontinua. It begins with a discussion of strain, stress tensors, balanc... 8. Covariant density functional theory beyond mean field and applications for nuclei far from stability International Nuclear Information System (INIS) Ring, P 2010-01-01 Density functional theory provides a very powerful tool for a unified microscopic description of nuclei all over the periodic table. It is not only successful in reproducing bulk properties of nuclear ground states such as binding energies, radii, or deformation parameters, but it also allows the investigation of collective phenomena, such as giant resonances and rotational excitations. However, it is based on the mean field concept and therefore it has its limits. We discuss here two methods based based on covariant density functional theory going beyond the mean field concept, (i) models with an energy dependent self energy allowing the coupling to complex configurations and a quantitative description of the width of giant resonances and (ii) methods of configuration mixing between Slater determinants with different deformation and orientation providing are very successful description of transitional nuclei and quantum phase transitions. 9. Affine field theories International Nuclear Information System (INIS) 1989-01-01 The author constructs a non-Abelian field theory by gauging a Kac-Moody algebra, obtaining an infinite tower of interacting vector fields and associated ghosts, that obey slightly modified Feynman rules. She discusses the spontaneous symmetry breaking of such theory via the Higgs mechanism. If the Higgs particle lies in the Cartan subalgebra of the Kac-Moody algebra, the previously massless vectors acquire a mass spectrum that is linear in the Kac-Moody index and has additional fine structure depending on the associated Lie algebra. She proceeds to show that there is no obstacle in implementing the affine extension of supersymmetric Yang-Mills theories. The result is valid in four, six and ten space-time dimensions. Then the affine extension of supergravity is investigated. She discusses only the loop algebra since the affine extension of the super-Poincare algebra appears inconsistent. The construction of the affine supergravity theory is carried out by the group manifold method and leads to an action describing infinite towers of spin 2 and spin 3/2 fields that interact subject to the symmetries of the loop algebra. The equations of motion satisfy the usual consistency check. Finally, she postulates a theory in which both the vector and scalar fields lie in the loop algebra of SO(3). This theory has an expanded soliton sector, and corresponding to the original 't Hooft-Polyakov solitonic solutions she now finds an infinite family of exact, special solutions of the new equations. She also proposes a perturbation method for obtaining an arbitrary solution of those equations for each level of the affine index 10. Slave Boson Theory of Orbital Differentiation with Crystal Field Effects: Application to UO2 Science.gov (United States) Lanatà, Nicola; Yao, Yongxin; Deng, Xiaoyu; Dobrosavljević, Vladimir; Kotliar, Gabriel 2017-03-01 We derive an exact operatorial reformulation of the rotational invariant slave boson method, and we apply it to describe the orbital differentiation in strongly correlated electron systems starting from first principles. The approach enables us to treat strong electron correlations, spin-orbit coupling, and crystal field splittings on the same footing by exploiting the gauge invariance of the mean-field equations. We apply our theory to the archetypical nuclear fuel UO2 and show that the ground state of this system displays a pronounced orbital differentiation within the 5 f manifold, with Mott-localized Γ8 and extended Γ7 electrons. 11. Slave Boson Theory of Orbital Differentiation with Crystal Field Effects: Application to UO_{2}. Science.gov (United States) Lanatà, Nicola; Yao, Yongxin; Deng, Xiaoyu; Dobrosavljević, Vladimir; Kotliar, Gabriel 2017-03-24 We derive an exact operatorial reformulation of the rotational invariant slave boson method, and we apply it to describe the orbital differentiation in strongly correlated electron systems starting from first principles. The approach enables us to treat strong electron correlations, spin-orbit coupling, and crystal field splittings on the same footing by exploiting the gauge invariance of the mean-field equations. We apply our theory to the archetypical nuclear fuel UO_{2} and show that the ground state of this system displays a pronounced orbital differentiation within the 5f manifold, with Mott-localized Γ_{8} and extended Γ_{7} electrons. 12. Bucharest PhD Training School : Modern Aspects of Quantum Field Theory and Applications CERN Document Server 2015-01-01 Bucharest 2015 – Modern Aspects of Quantum Field Theory is part of the CERN – SEENET-MTP PhD Training Program, which consists of a number of seminars in theoretical high energy Physics. This is the second seminar organized by this Program. Here are some photos from this event held in Bucharest between 8-14 November 2015. The previous seminar was organized in Belgrade, under the name Belgrade 2015 - Supergravity. 13. Probabilistic theory of mean field games with applications II mean field games with common noise and master equations CERN Document Server Carmona, René 2018-01-01 This two-volume book offers a comprehensive treatment of the probabilistic approach to mean field game models and their applications. The book is self-contained in nature and includes original material and applications with explicit examples throughout, including numerical solutions. Volume II tackles the analysis of mean field games in which the players are affected by a common source of noise. The first part of the volume introduces and studies the concepts of weak and strong equilibria, and establishes general solvability results. The second part is devoted to the study of the master equation, a partial differential equation satisfied by the value function of the game over the space of probability measures. Existence of viscosity and classical solutions are proven and used to study asymptotics of games with finitely many players. Together, both Volume I and Volume II will greatly benefit mathematical graduate students and researchers interested in mean field games. The authors provide a detailed road map t... 14. Neural field theory of synaptic metaplasticity with applications to theta burst stimulation. Science.gov (United States) Fung, P K; Robinson, P A 2014-01-07 Transcranial magnetic stimulation (TMS) is characterized by strong nonlinear plasticity effects. Experimental results that highlight such nonlinearity include continuous and intermittent theta-burst stimulations (cTBS and iTBS, respectively), where depression is induced in the continuous case, but insertion of an off period of around 8s for every 2s of stimulation changes the induced plasticity to potentiation in the intermittent case. Another nonlinearity is that cTBS and iTBS exhibit dosage dependency, where doubling of the stimulation duration changes the direction of induced plasticity. Guided by previous experimental results, this study postulates on the characteristics of metaplasticity and formulates a physiological system-level plasticity theory to predict TMS experiments. In this theory, plasticity signaling induces plasticity in NMDA receptors to modulate further plasticity signals, and is followed by a signal transduction delayed plasticity expression. Since this plasticity in NMDA receptor affects subsequent plasticity induction, it is a form of metaplasticity. Incorporating this metaplasticity into a recent neural field theory of calcium dependent plasticity gives a physiological basis for the theory of Bienenstock, Cooper, Munro (1982), where postsynaptic intracellular calcium level becomes the measure of temporal averaged postsynaptic activity, and converges to the plasticity threshold to give homeostatic effects. Simulations of TMS protocol responses show that intracellular calcium oscillations around the threshold predicts the aforementioned nonlinearities in TMS-induced plasticity, as well as the interpersonal TBS response polarity found experimentally, where the same protocol may induce opposite plasticity effect for different subjects. Thereby, recommendations for future experiments and TMS protocol optimizations are made. Input selectivity via spatially extended, mean field neural dynamics is also explored. © 2013 Elsevier Ltd. All rights 15. Theory and applications of internal photoemission in the MOS system at low electric fields Science.gov (United States) Przewlocki, Henryk M. 2001-08-01 A new theory is presented of the photoelectric phenomena, which take place in UV illuminated MOS structures, in the presence of weak electric fields (\|E\|photoelectric measurement methods of the MOS system parameters. Two of such methods are shortly presented. The first is the measurement method of the φMS factor of the MOS system, which has already been fully verified experimentally and has been shown to be the most accurate of the existing methods of this parameter determination. The second is the method to determine trapping properties of the dielectric in the MOS system, which is currently being optimized and verified experimentally. 16. Introduction to string field theory International Nuclear Information System (INIS) Horowitz, G.T. 1989-01-01 A light cone gauge superstring field theory is constructed. The BRST approach is described discussing generalizations to yield gauge invariant free superstring field theory and interacting theory for superstrings. The interaction term is explicitly expressed in terms of first quantized oscillators. A purily cubic action for superstring field theory is also derived. (author) 17. On the application of the field-redefinition theorem to the heterotic superstring theory Science.gov (United States) Pollock, M. D. 2015-05-01 The ten-dimensional effective action which defines the heterotic superstring theory at low energy is constructed by hypothesis in such a way that the resulting classical equation of motion for the space-time metric simultaneously implies the vanishing of the beta-function for the N = 1 supersymmetric non-linear sigma-model on the world sheet. At four-loop order it was found by Grisaru and Zanon (see also Freeman et al.) that the effective Lagrangian so constructed differs in the numerical coefficient of the term from that obtained directly from the four-point gravitational scattering amplitude. The two expressions can be related via a metric field redefinition , activation of which, however, results in the appearance of ghosts at higher gravitational order , n > 4, as shown by Lawrence. Here, we prove, after reduction of to the physical dimensionality D = 4, that the corresponding field redefinition yields the identity g' ij = g ij , signified by L 3/ R = 0, in a Friedmann space-time generated by a perfect-fluid source characterized by adiabatic index γ ≡ 1 + p/ ρ, where p is the pressure and ρ is the energy density, if, and only if, κ 6 ρ 3 γ 2( γ - 1) = 0. That is, the theory remains free of ghosts in Minkowski space ρ = 0, in a maximally symmetric space-time γ = 0, or in a dust Universe γ = 1. Further aspects of ghost freedom and dimensional reduction, especially to D = 4, are discussed. 18. Superstring field theory International Nuclear Information System (INIS) Green, M.B. 1984-01-01 Superstring field theories are formulated in terms of light-cone-gauge superfields that are functionals of string coordinates chi(sigma) and theta(sigma). The formalism used preserves only the manifest SU(4) symmetry that corresponds to rotations among six of the eight transverse directions. In type I theories, which have one ten-dimensional supersymmetry and describe both open and closed strings, there are five interaction terms of two basic kinds. One kind is a breaking or joining interaction, which is a string generalization of a cubic Yang-Mills coupling. It is relevant to both the three open-string vertex and the open-string to closed-string transition vertex. The other kind is an exchange or crossing-over interaction, which is a string generalization of a cubic gravitational coupling. All the interactions can be uniquely determined by requiring continuity of the coordinates chi(sigma) and theta(sigma) (which implies local conservation of the conjugate momenta) and by imposing the global supersymmetry algebra. Specific local operators are identified for each of the two kinds of interactions. In type II theories, which have two ten-dimensional supersymmetries and contain closed strings only, the entire interaction hamiltonian consists of a single cubic vertex. The higher-order contact terms of the N=8 supergravity theory that arises in the low-energy limit give an effective description of the exchange of massive string modes. (orig.) 19. A superstring field theory for supergravity Science.gov (United States) Reid-Edwards, R. A.; Riccombeni, D. A. 2017-09-01 A covariant closed superstring field theory, equivalent to classical tendimensional Type II supergravity, is presented. The defining conformal field theory is the ambitwistor string worldsheet theory of Mason and Skinner. This theory is known to reproduce the scattering amplitudes of Cachazo, He and Yuan in which the scattering equations play an important role and the string field theory naturally incorporates these results. We investigate the operator formalism description of the ambitwsitor string and propose an action for the string field theory of the bosonic and supersymmetric theories. The correct linearised gauge symmetries and spacetime actions are explicitly reproduced and evidence is given that the action is correct to all orders. The focus is on the NeveuSchwarz sector and the explicit description of tree level perturbation theory about flat spacetime. Application of the string field theory to general supergravity backgrounds and the inclusion of the Ramond sector are briefly discussed. 20. Beyond mean field theory: statistical field theory for neural networks. Science.gov (United States) Buice, Michael A; Chow, Carson C 2013-03-01 Mean field theories have been a stalwart for studying the dynamics of networks of coupled neurons. They are convenient because they are relatively simple and possible to analyze. However, classical mean field theory neglects the effects of fluctuations and correlations due to single neuron effects. Here, we consider various possible approaches for going beyond mean field theory and incorporating correlation effects. Statistical field theory methods, in particular the Doi-Peliti-Janssen formalism, are particularly useful in this regard. 1. Media Accountability Online in Israel. An application of Bourdieu’s field theory Directory of Open Access Journals (Sweden) Ronja Kniep 2015-12-01 Full Text Available Due to structural changes in journalism, such as deregulation, privatisation and the influence of new technologies, it has become increasingly important to study media accountability (MA. By applying Bourdieu’s theory of social fields, this paper proposes a new approach to do so: MA is defined as a function of both journalistic autonomy and influence in the media field. Here, online communication potentially widens the scope of action for media’s transparency, responsiveness as well as the articulation of media criticism by a variety of actors. In Israel, media criticism is driven by the agent’s struggle for interpretive authority over public discourse in a politically polarized society. Semi-structured interviews with Israeli journalists, media activists and experts suggest that journalistic agents who have yet to earn credibility and reputation exploit online communication to its full potential, while agents in the field of power tend to dismiss online criticism. The influence of the audience’s media criticism is not solely dependent on the technical ability of connecting and hearing the voices of the masses; it has to be in combination with symbolic or political capital. However, the demand for media’s social responsibility is also related to being more careful and less critical, which is very evident in Israel. Thus, it is important to critically reflect on what happens when media accountability practices become more efficient and a stronger sense for “being watched” develops. 2. Thermal Field Theory in Equilibrium OpenAIRE Andersen, Jens O. 2000-01-01 In this talk, I review recent developments in equilibrium thermal field theory. Screened perturbation theory and hard-thermal-loop perturbation theory are discussed. A self-consistent $\\Phi$-derivable approach is also briefly reviewed. 3. Class field theory CERN Document Server Artin, Emil 2009-01-01 This classic book, originally published in 1968, is based on notes of a year-long seminar the authors ran at Princeton University. The primary goal of the book was to give a rather complete presentation of algebraic aspects of global class field theory, and the authors accomplished this goal spectacularly: for more than 40 years since its first publication, the book has served as an ultimate source for many generations of mathematicians. In this revised edition, two mathematical additions complementing the exposition in the original text are made. The new edition also contains several new foot 4. Higgs Effective Field Theories CERN Document Server 2016-01-01 The main focus of this meeting is to present new theoretical advancements related to effective field theories, evaluate the impact of initial results from the LHC Run2, and discuss proposals for data interpretation/presentation during Run2. A crucial role of the meeting is to bring together theorists from different backgrounds and with different viewpoints and to extend bridges towards the experimental community. To this end, we would like to achieve a good balance between senior and junior speakers, enhancing the visibility of younger scientists while keeping some overview talks. 5. Finite volume method for self-consistent field theory of polymers: Material conservation and application Science.gov (United States) Yong, Daeseong; Kim, Jaeup U. 2017-12-01 For the purpose of checking material conservation of various numerical algorithms used in the self-consistent-field theory (SCFT) of polymeric systems, we develop an algebraic method using matrix and bra-ket notation, which traces the Hermiticity of the product of the volume and evolution matrices. Algebraic tests for material conservation reveal that the popular pseudospectral method in the Cartesian grid conserves material perfectly, while the finite-volume method (FVM) is the proper tool when real-space SCFT with the Crank-Nicolson method is adopted in orthogonal coordinate systems. We also find that alternating direction implicit methods combined with the FVM exhibit small mass errors in the SCFT calculation. By introducing fractional cells in the FVM formulation, accurate SCFT calculations are performed for systems with irregular geometries and the results are consistent with previous experimental and theoretical works. 6. Studies in quantum field theory International Nuclear Information System (INIS) Bender, C.M.; Mandula, J.E.; Shrauner, J.E. 1982-01-01 Washington University is currently conducting research in many areas of high energy theoretical and mathematical physics. These areas include: strong-coupling approximation; classical solutions of non-Abelian gauge theories; mean-field approximation in quantum field theory; path integral and coherent state representations in quantum field theory; lattice gauge calculations; the nature of perturbation theory in large orders; quark condensation in QCD; chiral symmetry breaking; the l/N expansion in quantum field theory; effective potential and action in quantum field theories, including QCD 7. Quantum field theory International Nuclear Information System (INIS) Mancini, F. 1986-01-01 Theoretical physicists, coming from different countries, working on different areas, gathered at Positano: the Proceedings contain all the lectures delivered as well as contributed papers. Many areas of physics are represented, elementary particles in high energy physics, quantum relativity, quantum geometry, condensed matter physics, statistical mechanics; but all works are concerned with the use of the methods of quantum field theory. The first motivation of the meeting was to pay homage to a great physicist and a great friend; it was also an occasion in which theoretical physicists got together to discuss and to compare results in different fields. The meeting was very intimate; the relaxed atmosphere allowed constructive discussions and contributed to a positive exchange of ideas. (orig.) 8. Statistical field theory with constraints: Application to critical Casimir forces in the canonical ensemble. Science.gov (United States) Gross, Markus; Gambassi, Andrea; Dietrich, S 2017-08-01 The effect of imposing a constraint on a fluctuating scalar order parameter field in a system of finite volume is studied within statistical field theory. The canonical ensemble, corresponding to a fixed total integrated order parameter (e.g., the total number of particles), is obtained as a special case of the theory. A perturbative expansion is developed which allows one to systematically determine the constraint-induced finite-volume corrections to the free energy and to correlation functions. In particular, we focus on the Landau-Ginzburg model in a film geometry (i.e., in a rectangular parallelepiped with a small aspect ratio) with periodic, Dirichlet, or Neumann boundary conditions in the transverse direction and periodic boundary conditions in the remaining, lateral directions. Within the expansion in terms of ε=4-d, where d is the spatial dimension of the bulk, the finite-size contribution to the free energy of the confined system and the associated critical Casimir force are calculated to leading order in ε and are compared to the corresponding expressions for an unconstrained (grand canonical) system. The constraint restricts the fluctuations within the system and it accordingly modifies the residual finite-size free energy. The resulting critical Casimir force is shown to depend on whether it is defined by assuming a fixed transverse area or a fixed total volume. In the former case, the constraint is typically found to significantly enhance the attractive character of the force as compared to the grand canonical case. In contrast to the grand canonical Casimir force, which, for supercritical temperatures, vanishes in the limit of thick films, in the canonical case with fixed transverse area the critical Casimir force attains for thick films a negative value for all boundary conditions studied here. Typically, the dependence of the critical Casimir force both on the temperaturelike and on the fieldlike scaling variables is different in the two ensembles. 9. Digestible quantum field theory CERN Document Server Smilga, Andrei 2017-01-01 This book gives an intermediate level treatment of quantum field theory, appropriate to a reader with a first degree in physics and a working knowledge of special relativity and quantum mechanics. It aims to give the reader some understanding of what QFT is all about, without delving deep into actual calculations of Feynman diagrams or similar. The author serves up a seven‐course menu, which begins with a brief introductory Aperitif. This is followed by the Hors d'oeuvres, which set the scene with a broad survey of the Universe, its theoretical description, and how the ideas of QFT developed during the last century. In the next course, the Art of Cooking, the author recaps on some basic facts of analytical mechanics, relativity, quantum mechanics and also presents some nutritious “extras” in mathematics (group theory at the elementary level) and in physics (theory of scattering). After these preparations, the reader should have a good appetite for the Entrées ‐ the central par t of the book where the... 10. Control theory in physics and other fields of science concepts, tools and applications CERN Document Server Schulz, Michael 2006-01-01 This book covers systematically and in a simple language the mathematical and physical foundations of controlling deterministic and stochastic evolutionary processes in systems with a high degree of complexity. Strong emphasis is placed on concepts, methods and techniques for modelling, assessment and the solution or estimation of control problems in an attempt to understand the large variability of these problems in several branches of physics, chemistry and biology as well as in technology and economics. The main focus of the book is on a clear physical and mathematical understanding of the dynamics and kinetics behind several kinds of control problems and their relation to self-organizing principles in complex systems. The book is a modern introduction and a helpful tool for researchers, engineers as well as post-docs and graduate students interested in an application oriented control theory and related topics. 11. Application of optimal control theory to laser heating of a plasma in a solenoidal magnetic field International Nuclear Information System (INIS) Neal, R.D. 1975-01-01 Laser heating of a plasma column confined by a solenoidal magnetic field is studied via modern optimal control techniques. A two-temperature, constant pressure model is used for the plasma so that the temperature and density are functions of time and location along the plasma column. They are assumed to be uniform in the radial direction so that refraction of the laser beam does not occur. The laser intensity used as input to the column at one end is taken as the control variable and plasma losses are neglected. The localized behavior of the plasma heating dynamics is first studied and conventional optimal control theory applied. The distributed parameter optimal control problem is next considered with minimum time to reach a specified final ion temperature criterion as the objective. Since the laser intensity can only be directly controlled at the input end of the plasma column, a boundary control situation results. The problem is unique in that the control is the boundary value of one of the state variables. The necessary conditions are developed and the problem solved numerically for typical plasma parameters. The problem of maximizing the space-time integral of neutron production rate in the plasma is considered for a constant distributed control problem where the laser intensity is assumed fixed at maximum and the external magnetic field is taken as a control variable 12. Study of the convergence of the nuclear field theory and its application on the lead isotopes International Nuclear Information System (INIS) Scoccola, N.N. 1985-01-01 It is shown that highly satisfactory results can be obtained not only in schematic problems (four particles in a degenerate j-shell), but in realistic ones (low lying 204 Pb spectrum), provided second order diagrams and/or diagonalization procedures are used. In both cases energies and two-body transfer amplitudes are calculated and compared with exact and other approximate results. In the second part, the electromagnetic emission of the giant quadrupole resonance (GQR) in 208 Pb after its excitation by inelastic scattering of 17 O to 380 MeV is studied. As the GQR is unstable with respect to the decay to compound nucleous, the reaction mechanism is carefully analized. A formalism is proposed in which the emission probability is factorized in three independent contributions: one due to the electromagnetic field, another to the nuclear reaction and the third to the nuclear structure. The last one is carefully studied in the lowest order of the nuclear field theory, taking into account the mixture of the different isospin states. The results are consistent with the upper experimental limit of the ratio between the transition populating the 3 - (2.62 MeV) state and the one that populates the ground state. However, they failed to reproduce the strong dipole transition to the 3 - (4.97 MeV) state. (Author) [es 13. Correlation functions in finite temperature field theories: formalism and applications to quark-gluon plasma International Nuclear Information System (INIS) Gelis, Francois 1998-12-01 The general framework of this work is thermal field theory, and more precisely the perturbative calculation of thermal Green's functions. In a first part, I consider the problems closely related to the formalism itself. After two introductory chapters devoted to set up the framework and the notations used afterwards, a chapter is dedicated to a clarification of certain aspects of the justification of the Feynman rules of the real time formalism. Then, I consider in the chapter 4 the problem of cutting rules in the real time formalisms. In particular, after solving a controversy on this subject, I generalize these cutting rules to the 'retarded-advanced' version of this formalism. Finally, the last problem considered in this part is that of the pion decay into two photons in a thermal bath. I show that the discrepancies found in the literature are due to peculiarities of the analytical properties of the thermal Green's functions. The second part deals with the calculations of the photons or dilepton (virtual photon) production rate by a quark gluon plasma. The framework of this study is the effective theory based on the resummation of hard thermal loops. The first aspects of this study is related to the production of virtual photons, where we show that important contributions arise at two loops, completing the result already known at one loop. In the case of real photon production, we show that extremely strong collinear singularities make two loop contributions dominant compared to one loop ones. In both cases, the importance of two loop contributions can be interpreted as weaknesses of the hard thermal loop approximation. (author) 14. Topics in quantum field theory International Nuclear Information System (INIS) Svaiter, N.F. 2006-11-01 This paper presents some important aspects on quantum field theory, covering the following aspects: the triumph and limitations of the quantum field theory; the field theory in curved spaces - Hawking and Unruh-Davies effects; the problem of divergent theory of the zero-point; the problem of the spinning detector and the Trocheries-Takeno vacuum; the field theory at finite temperature - symmetry breaking and phase transition; the problem of the summability of the perturbative series and the perturbative expansion for the strong coupling; quantized fields in presence of classical macroscopic structures; the Parisi-Wu stochastic quantization method 15. Fractional Stochastic Field Theory Science.gov (United States) Honkonen, Juha 2018-02-01 Models describing evolution of physical, chemical, biological, social and financial processes are often formulated as differential equations with the understanding that they are large-scale equations for averages of quantities describing intrinsically random processes. Explicit account of randomness may lead to significant changes in the asymptotic behaviour (anomalous scaling) in such models especially in low spatial dimensions, which in many cases may be captured with the use of the renormalization group. Anomalous scaling and memory effects may also be introduced with the use of fractional derivatives and fractional noise. Construction of renormalized stochastic field theory with fractional derivatives and fractional noise in the underlying stochastic differential equations and master equations and the interplay between fluctuation-induced and built-in anomalous scaling behaviour is reviewed and discussed. 16. Chameleon field theories International Nuclear Information System (INIS) Khoury, Justin 2013-01-01 Chameleons are light scalar fields with remarkable properties. Through the interplay of self-interactions and coupling to matter, chameleon particles have a mass that depends on the ambient matter density. The manifestation of the fifth force mediated by chameleons therefore depends sensitively on their environment, which makes for a rich phenomenology. In this paper, we review two recent results on chameleon phenomenology. The first result a pair of no-go theorems limiting the cosmological impact of chameleons and their generalizations: (i) the range of the chameleon force at cosmological density today can be at most ∼Mpc; (ii) the conformal factor relating Einstein- and Jordan-frame scale factors is essentially constant over the last Hubble time. These theorems imply that chameleons have negligible effect on the linear growth of structure, and cannot account for the observed cosmic acceleration except as some form of dark energy. The second result pertains to the quantum stability of chameleon theories. We show how requiring that quantum corrections be small, so as to allow reliable predictions of fifth forces, leads to an upper bound of m −3 ) 1/3 eV for gravitational strength coupling, whereas fifth force experiments place a lower bound of m > 0.0042 eV. An improvement of less than a factor of 2 in the range of fifth force experiments could test all classical chameleon field theories whose quantum corrections are well-controlled and couple to matter with nearly gravitational strength regardless of the specific form of the chameleon potential. (paper) 17. An introduction to conformal field theory International Nuclear Information System (INIS) Zuber, J.B. 1995-01-01 The aim of these lectures is to present an introduction at a fairly elementary level to recent developments in two dimensional field theory, namely in conformal field theory. We shall see the importance of new structures related to infinite dimensional algebras: current algebras and Virasoro algebra. These topics will find physically relevant applications in the lectures by Shankar and Ian Affeck. (author) 18. Quantum field theory of fluids. Science.gov (United States) Gripaios, Ben; Sutherland, Dave 2015-02-20 The quantum theory of fields is largely based on studying perturbations around noninteracting, or free, field theories, which correspond to a collection of quantum-mechanical harmonic oscillators. The quantum theory of an ordinary fluid is "freer", in the sense that the noninteracting theory also contains an infinite collection of quantum-mechanical free particles, corresponding to vortex modes. By computing a variety of correlation functions at tree and loop level, we give evidence that a quantum perfect fluid can be consistently formulated as a low-energy, effective field theory. We speculate that the quantum behavior is radically different from both classical fluids and quantum fields. 19. Relativistic quantum mechanics and field theory CERN Document Server Gross, Franz 1999-01-01 An accessible, comprehensive reference to modern quantum mechanics and field theory.In surveying available books on advanced quantum mechanics and field theory, Franz Gross determined that while established books were outdated, newer titles tended to focus on recent developments and disregard the basics. Relativistic Quantum Mechanics and Field Theory fills this striking gap in the field. With a strong emphasis on applications to practical problems as well as calculations, Dr. Gross provides complete, up-to-date coverage of both elementary and advanced topics essential for a well-rounded understanding of the field. 20. Worked examples in engineering field theory CERN Document Server 1976-01-01 Worked Examples in Engineering Field Theory is a product of a lecture course given by the author to first-year students in the Department of Engineering in the University of Leicester. The book presents a summary of field theory together with a large number of worked examples and solutions to all problems given in the author's other book, Engineering Field Theory. The 14 chapters of this book are organized into two parts. Part I focuses on the concept of flux including electric flux. This part also tackles the application of the theory in gravitation, ideal fluid flow, and magnetism. Part II d 1. Advanced number theory with applications CERN Document Server Mollin, Richard A 2009-01-01 Algebraic Number Theory and Quadratic Fields Algebraic Number Fields The Gaussian Field Euclidean Quadratic Fields Applications of Unique Factorization Ideals The Arithmetic of Ideals in Quadratic Fields Dedekind Domains Application to Factoring Binary Quadratic Forms Basics Composition and the Form Class Group Applications via Ambiguity Genus Representation Equivalence Modulo p Diophantine Approximation Algebraic and Transcendental Numbers Transcendence Minkowski's Convex Body Theorem Arithmetic Functions The Euler-Maclaurin Summation Formula Average Orders The Riemann zeta-functionIntroduction to p-Adic AnalysisSolving Modulo pn Introduction to Valuations Non-Archimedean vs. Archimedean Valuations Representation of p-Adic NumbersDirichlet: Characters, Density, and Primes in Progression Dirichlet Characters Dirichlet's L-Function and Theorem Dirichlet DensityApplications to Diophantine Equations Lucas-Lehmer Theory Generalized Ramanujan-Nagell Equations Bachet's Equation The Fermat Equation Catalan and the A... 2. Naturality in conformal field theory International Nuclear Information System (INIS) Moore, G.; Seiberg, N. 1989-01-01 We discuss constraints on the operator product coefficients in diagonal and nondiagonal rational conformal field theories. Nondiagonal modular invariants always arise from automorphisms of the fusion rule algebra or from extensions of the chiral algebra. Moreover, when the chiral algebra has been maximally extended a strong form of the naturality principle of field theory can be proven for rational conformal field theory: operator product coefficients vanish if and only if the corresponding fusion rules vanish; that is, if and only if the vanishing can be understood in terms of a symmetry. We illustrate these ideas with several examples. We also generalize our ideas about rational conformal field theories to a larger class of theories: 'quasi-rational conformal field theories' and we explore some of their properties. (orig.) 3. An Empirical Study on the Application of Theme Theory in the Field of Writing Pedagogy Science.gov (United States) Jingxia, Liu; Li, Liu 2013-01-01 English writing instruction is an important part in college English pedagogy. Traditional way of teaching English writing lays much emphasis on word, grammar and sentence rather than the level of discourse. Under the traditional way, the students have difficulties to yield well-organized and coherent compositions. Theme Theory provides a… 4. Conformal field theories and tensor categories. Proceedings Energy Technology Data Exchange (ETDEWEB) Bai, Chengming [Nankai Univ., Tianjin (China). Chern Institute of Mathematics; Fuchs, Juergen [Karlstad Univ. (Sweden). Theoretical Physics; Huang, Yi-Zhi [Rutgers Univ., Piscataway, NJ (United States). Dept. of Mathematics; Kong, Liang [Tsinghua Univ., Beijing (China). Inst. for Advanced Study; Runkel, Ingo; Schweigert, Christoph (eds.) [Hamburg Univ. (Germany). Dept. of Mathematics 2014-08-01 First book devoted completely to the mathematics of conformal field theories, tensor categories and their applications. Contributors include both mathematicians and physicists. Some long expository articles are especially suitable for beginners. The present volume is a collection of seven papers that are either based on the talks presented at the workshop ''Conformal field theories and tensor categories'' held June 13 to June 17, 2011 at the Beijing International Center for Mathematical Research, Peking University, or are extensions of the material presented in the talks at the workshop. These papers present new developments beyond rational conformal field theories and modular tensor categories and new applications in mathematics and physics. The topics covered include tensor categories from representation categories of Hopf algebras, applications of conformal field theories and tensor categories to topological phases and gapped systems, logarithmic conformal field theories and the corresponding non-semisimple tensor categories, and new developments in the representation theory of vertex operator algebras. Some of the papers contain detailed introductory material that is helpful for graduate students and researchers looking for an introduction to these research directions. The papers also discuss exciting recent developments in the area of conformal field theories, tensor categories and their applications and will be extremely useful for researchers working in these areas. 5. Topological quantum field theory and four manifolds CERN Document Server Marino, Marcos 2005-01-01 The present book is the first of its kind in dealing with topological quantum field theories and their applications to topological aspects of four manifolds. It is not only unique for this reason but also because it contains sufficient introductory material that it can be read by mathematicians and theoretical physicists. On the one hand, it contains a chapter dealing with topological aspects of four manifolds, on the other hand it provides a full introduction to supersymmetry. The book constitutes an essential tool for researchers interested in the basics of topological quantum field theory, since these theories are introduced in detail from a general point of view. In addition, the book describes Donaldson theory and Seiberg-Witten theory, and provides all the details that have led to the connection between these theories using topological quantum field theory. It provides a full account of Witten’s magic formula relating Donaldson and Seiberg-Witten invariants. Furthermore, the book presents some of the ... 6. Mean Field Theory of a Coupled Heisenberg Model and Its Application to an Organic Antiferromagnet with Magnetic Anions Science.gov (United States) Ito, Kazuhiro; Shimahara, Hiroshi 2016-02-01 We examine the mean field theory of a uniaxial coupled Heisenberg antiferromagnet with two subsystems, one of which consists of strongly interacting small spins and the other consists of weakly interacting large spins. We reanalyze the experimental data of specific heat and magnetic susceptibility obtained by previous authors for the organic compound λ-(BETS)2FeCl4 at low temperatures, where BETS stands for bis(ethylenedithio)tetraselenafulvalene. The model parameters for this compound are evaluated, where the applicability of the theory is checked. As a result, it is found that J1 ≫ J12 ≫ J2, where J1, J2, and J12 denote the exchange coupling constant between π spins, that between 3d spins, and that between π and 3d spins, respectively. At the low-temperature limit, both sublattice magnetizations of the 3d and π spins are saturated, and the present model is reduced to the Schottky model, which successfully explains experimental observations in previous studies. As temperature increases, fluctuations of 3d spins increase, while π spins remain almost saturated. Near the critical temperature, both spins fluctuate significantly, and thus the mean field approximation breaks down. It is revealed that the magnetic anisotropy, which may be crucial to the antiferromagnetic long-range order, originates from J12 rather than from J2 and that the angle between the magnetic easy-axis and the crystal c-axis is approximately 26-27° in the present effective model. 7. Class field theory from theory to practice CERN Document Server Gras, Georges 2003-01-01 Global class field theory is a major achievement of algebraic number theory, based on the functorial properties of the reciprocity map and the existence theorem. The author works out the consequences and the practical use of these results by giving detailed studies and illustrations of classical subjects (classes, idèles, ray class fields, symbols, reciprocity laws, Hasse's principles, the Grunwald-Wang theorem, Hilbert's towers,...). He also proves some new or less-known results (reflection theorem, structure of the abelian closure of a number field) and lays emphasis on the invariant (/cal T) p, of abelian p-ramification, which is related to important Galois cohomology properties and p-adic conjectures. This book, intermediary between the classical literature published in the sixties and the recent computational literature, gives much material in an elementary way, and is suitable for students, researchers, and all who are fascinated by this theory. In the corrected 2nd printing 2005, the author improves s... 8. Broken symmetries in field theory NARCIS (Netherlands) Kok, Mark Okker de 2008-01-01 The thesis discusses the role of symmetries in Quantum Field Theory. Quantum Field Theory is the mathematical framework to describe the physics of elementary particles. A symmetry here means a transformation under which the model at hand is invariant. Three types of symmetry are distinguished: 1. 9. Renormalization and effective field theory CERN Document Server Costello, Kevin 2011-01-01 This book tells mathematicians about an amazing subject invented by physicists and it tells physicists how a master mathematician must proceed in order to understand it. Physicists who know quantum field theory can learn the powerful methodology of mathematical structure, while mathematicians can position themselves to use the magical ideas of quantum field theory in "mathematics" itself. The retelling of the tale mathematically by Kevin Costello is a beautiful tour de force. --Dennis Sullivan This book is quite a remarkable contribution. It should make perturbative quantum field theory accessible to mathematicians. There is a lot of insight in the way the author uses the renormalization group and effective field theory to analyze perturbative renormalization; this may serve as a springboard to a wider use of those topics, hopefully to an eventual nonperturbative understanding. --Edward Witten Quantum field theory has had a profound influence on mathematics, and on geometry in particular. However, the notorio... 10. Introductory lectures on quantum field theory International Nuclear Information System (INIS) Alvarez-Gaume, L.; Vasquez-Mozo, M.A. 2011-01-01 In these lectures we present a few topics in quantum field theory in detail. Some of them are conceptual and some more practical. They have been selected because they appear frequently in current applications to particle physics and string theory. (author) 11. Perturbation series at large orders in quantum mechanics and field theories: application to the problem of resummation International Nuclear Information System (INIS) Zinn-Justin, J.; Freie Univ. Berlin 1981-01-01 In this review I present a method to estimate the large order behavior of perturbation theory in quantum mechanics and field theory. The basic idea, due to Lipatov, is to relate the large order behavior to (in general complex) instanton contributions to the path integral representation of Green's functions. I explain the method first in the case of a simple integral and of the anharmonic oscillator and recover the results of Bender and Wu. I apply it then to the PHI 4 field theory. I study general potentials and boson field theories. I show, following Parisi, how the method can be generalized to theories with fermions. Finally I outline the implications of these results for the summability of the series. In particular I explain a method to sum divergent series based on a Borel transformation. In a last section I compare the larger order behavior predictions to actual series calculation. I present also some numerical examples of series summation. (orig.) 12. Acoustic array systems theory, implementation, and application CERN Document Server Bai, Mingsian R; Benesty, Jacob 2013-01-01 Presents a unified framework of far-field and near-field array techniques for noise source identification and sound field visualization, from theory to application. Acoustic Array Systems: Theory, Implementation, and Application provides an overview of microphone array technology with applications in noise source identification and sound field visualization. In the comprehensive treatment of microphone arrays, the topics covered include an introduction to the theory, far-field and near-field array signal processing algorithms, practical implementations, and common applic 13. Unified field theory from the classical wave equation: Preliminary application to atomic and nuclear structure Science.gov (United States) Múnera, Héctor A. 2016-07-01 It is postulated that there exists a fundamental energy-like fluid, which occupies the flat three-dimensional Euclidean space that contains our universe, and obeys the two basic laws of classical physics: conservation of linear momentum, and conservation of total energy; the fluid is described by the classical wave equation (CWE), which was Schrödinger's first candidate to develop his quantum theory. Novel solutions for the CWE discovered twenty years ago are nonharmonic, inherently quantized, and universal in the sense of scale invariance, thus leading to quantization at all scales of the universe, from galactic clusters to the sub-quark world, and yielding a unified Lorentz-invariant quantum theory ab initio. Quingal solutions are isomorphic under both neo-Galilean and Lorentz transformations, and exhibit nother remarkable property: intrinsic unstability for large values of ℓ (a quantum number), thus limiting the size of each system at a given scale. Unstability and scale-invariance together lead to nested structures observed in our solar system; unstability may explain the small number of rows in the chemical periodic table, and nuclear unstability of nuclides beyond lead and bismuth. Quingal functions lend mathematical basis for Boscovich's unified force (which is compatible with many pieces of evidence collected over the past century), and also yield a simple geometrical solution for the classical three-body problem, which is a useful model for electronic orbits in simple diatomic molecules. A testable prediction for the helicoidal-type force is suggested. 14. An application of modular inclusion to quantum field theory in curved space-time International Nuclear Information System (INIS) Summers, S.J.; Verch, R. 1993-09-01 Applying recent results by Borchers connecting geometric modular action, modular inclusion and the spectrum condition, earlier results by Kay and Wald concerning the temperature of physically significant states of the linear Hermitean scalar field propagating in the background of a space-time with a bifurcate Killing horizon are generalized. (orig.) 15. Semiclassical methods in field theories International Nuclear Information System (INIS) Ventura, I. 1978-10-01 A new scheme is proposed for semi-classical quantization in field theory - the expansion about the charge (EAC) - which is developed within the canonical formalism. This method is suitable for quantizing theories that are invariant under global gauge transformations. It is used in the treatment of the non relativistic logarithmic theory that was proposed by Bialynicki-Birula and Mycielski - a theory we can formulate in any number of spatial dimensions. The non linear Schroedinger equation is also quantized by means of the EAC. The classical logarithmic theories - both, the non relativistic and the relativistic one - are studied in detail. It is shown that the Bohr-Sommerfeld quantization rule(BSQR) in field theory is, in many cases, equivalent to charge quantization. This rule is then applied to the massive Thirring Model and the logarithmic theories. The BSQR can be see as a simplified and non local version of the EAC [pt 16. Austerity and geometric structure of field theories International Nuclear Information System (INIS) Kheyfets, A. 1986-01-01 The relation between the austerity idea and the geometric structure of the three basic field theories - electrodynamics, Yang-Mills theory, and general relativity - is studied. One of the most significant manifestations of the austerity idea in field theories is thought to be expressed by the boundary of a boundary principle (BBP). The BBP says that almost all content of the field theories can be deduced from the topological identity of delta dot produced with delta = 0 used twice, at the 1-2-3-dimensional level (providing the homogeneous field equations), and at the 2-3-4-dimensional level (providing the conservation laws for the source currents). There are some difficulties in this line of thought due to the apparent lack of universality in application of the BBP to the three basic modern field theories above. This dissertation: (a) analyzes the difficulties by means of algebraic topology, integration theory, and modern differential geometry based on the concepts of principal bundles and Ehresmann connections: (b) extends the BBP to the unified Kaluza-Klein theory; (c) reformulates the inhomogeneous field equations and the BBP in terms of E. Cartan moment of rotation, in the way universal for the three theories and compatible with the original austerity idea; and (d) underlines the important role of the soldering structure on spacetime, and indicates that the future development of the austerity idea would involve the generalized theories 17. Lectures on quantum field theory CERN Document Server Das, Ashok 2008-01-01 This book consists of the lectures for a two-semester course on quantum field theory, and as such is presented in a quite informal and personal manner. The course starts with relativistic one-particle systems, and develops the basics of quantum field theory with an analysis of the representations of the Poincaré group. Canonical quantization is carried out for scalar, fermion, Abelian and non-Abelian gauge theories. Covariant quantization of gauge theories is also carried out with a detailed description of the BRST symmetry. The Higgs phenomenon and the standard model of electroweak interactio 18. Introduction to quantum field theory International Nuclear Information System (INIS) Kazakov, D.I. 1988-01-01 The lectures appear to be a continuation to the introduction to elementary principles of the quantum field theory. The work is aimed at constructing the formalism of standard particle interaction model. Efforts are made to exceed the limits of the standard model in the quantum field theory context. Grand unification models including strong and electrical weak interactions, supersymmetric generalizations of the standard model and grand unification theories and, finally, supergravitation theories including gravitation interaction to the universal scheme, are considered. 3 refs.; 19 figs.; 2 tabs 19. Bayesian theory and applications CERN Document Server Dellaportas, Petros; Polson, Nicholas G; Stephens, David A 2013-01-01 The development of hierarchical models and Markov chain Monte Carlo (MCMC) techniques forms one of the most profound advances in Bayesian analysis since the 1970s and provides the basis for advances in virtually all areas of applied and theoretical Bayesian statistics. This volume guides the reader along a statistical journey that begins with the basic structure of Bayesian theory, and then provides details on most of the past and present advances in this field. The book has a unique format. There is an explanatory chapter devoted to each conceptual advance followed by journal-style chapters that provide applications or further advances on the concept. Thus, the volume is both a textbook and a compendium of papers covering a vast range of topics. It is appropriate for a well-informed novice interested in understanding the basic approach, methods and recent applications. Because of its advanced chapters and recent work, it is also appropriate for a more mature reader interested in recent applications and devel... 20. A landscape of field theories Energy Technology Data Exchange (ETDEWEB) Maxfield, Travis [Enrico Fermi Institute, University of Chicago,Chicago, IL 60637 (United States); Robbins, Daniel [George P. and Cynthia W. Mitchell Institute for Fundamental Physics and Astronomy,Texas A& M University,College Station, TX 77843-4242 (United States); Sethi, Savdeep [Enrico Fermi Institute, University of Chicago,Chicago, IL 60637 (United States) 2016-11-28 Studying a quantum field theory involves a choice of space-time manifold and a choice of background for any global symmetries of the theory. We argue that many more choices are possible when specifying the background. In the context of branes in string theory, the additional data corresponds to a choice of supergravity tensor fluxes. We propose the existence of a landscape of field theory backgrounds, characterized by the space-time metric, global symmetry background and a choice of tensor fluxes. As evidence for this landscape, we study the supersymmetric six-dimensional (2,0) theory compactified to two dimensions. Different choices of metric and flux give rise to distinct two-dimensional theories, which can preserve differing amounts of supersymmetry. 1. Embedding classical fields in quantum field theories International Nuclear Information System (INIS) Blaha, S. 1978-01-01 We describe a procedure for quantizing a classical field theory which is the field-theoretica analog of Sudarshan's method for embedding a classical-mechanical system in a quantum-mechanical system. The essence of the difference between our quantization procedure and Fock-space quantization lies in the choice of vacuum states. The key to our choice of vacuum is the procedure we outline for constructing Lagrangians which have gradient terms linear in the field varialbes from classical Lagrangians which have gradient terms which are quadratic in field variables. We apply this procedure to model electrodynamic field theories, Yang-Mills theories, and a vierbein model of gravity. In the case of electrodynamics models we find a formalism with a close similarity to the coherent-soft-photon-state formalism of QED. In addition, photons propagate to t = + infinity via retarded propagators. We also show how to construct a quantum field for action-at-a-distance electrodynamics. In the Yang-Mills case we show that a previously suggested model for quark confinement necessarily has gluons with principle-value propagation which allows the model to be unitary despite the presence of higher-order-derivative field equations. In the vierbein-gravity model we show that our quantization procedure allows us to treat the classical and quantum parts of the metric field in a unified manner. We find a new perturbation scheme for quantum gravity as a result 2. Topological field theories and duality International Nuclear Information System (INIS) Stephany, J.; Universidad Simon Bolivar, Caracas 1996-05-01 Topologically non trivial effects appearing in the discussion of duality transformations in higher genus manifold are discussed in a simple example, and their relation with the properties of Topological Field Theories is established. (author). 16 refs 3. Renormalization in classical field theory International Nuclear Information System (INIS) Corbo, Guido 2010-01-01 We discuss simple examples in which renormalization is required in classical field theory. The presentation is accessible to undergraduate students with a knowledge of the basic notions of classical electromagnetism. (letters and comments) 4. Games, theory and applications CERN Document Server Thomas, L C 2011-01-01 Anyone with a knowledge of basic mathematics will find this an accessible and informative introduction to game theory. It opens with the theory of two-person zero-sum games, two-person non-zero sum games, and n-person games, at a level between nonmathematical introductory books and technical mathematical game theory books. Succeeding sections focus on a variety of applications - including introductory explanations of gaming and meta games - that offer nonspecialists information about new areas of game theory at a comprehensible level. Numerous exercises appear with full solutions, in addition 5. Finite-temperature field theory International Nuclear Information System (INIS) Kapusta, J.I.; Landshoff, P.V. 1989-01-01 Particle number is not conserved in relativistic theories although both lepton and baryon number are. Therefore when discussing the thermodynamics of a quantum field theory one uses the grand canonical formalism. The entropy S is maximised, keeping fixed the ensemble averages E and N of energy and lepton number. Two lagrange multipliers are introduced. (author) 6. Supersymmetric field theories and generalized cohomology OpenAIRE Teichner, Peter; Stolz, Stephan 2011-01-01 This survey discusses our results and conjectures concerning supersymmetric field theories and their relationship to cohomology theories. A careful definition of supersymmetric Euclidean field theories is given, refining Segal's axioms for conformal field theories. We state and give an outline of the proof of various results relating field theories to cohomology theories. 7. Octonionic methods in field theory International Nuclear Information System (INIS) Duendarer, A.R. 1987-01-01 Some applications of octonion algebra and octonionic analysis to group theory and higher dimensional field theories are presented. To this end an eight dimensional covariant treatment of the octonion algebra is needed. The existing formulations which are covariant only in seven dimensions are reviewed. In this work the eight dimensional formulation is developed through the introduction of fourth rank tensors f abcd and f' abcd in eight dimensions that generalize the octonionic structure constants. The seven octonion units e α are generalized to an 8-vector e a and two second rank tensors e ab and e' ab . Higher rank tensors associated with e α are also introduced. Chirality and duality properties of the structure tensors, f,f' and the octonionic tensors e a , e ab , etc. are discussed and various new identities relating these quantities are derived. New vector products for two, three and four octonions are introduced and their duality properties with respect to the eight-dimensional Levi-Civita tensor as well as their orthogonality properties are studied 8. Introduction to classical and quantum field theory International Nuclear Information System (INIS) Ng, Tai-Kai 2009-01-01 This is the first introductory textbook on quantum field theory to be written from the point of view of condensed matter physics. As such, it presents the basic concepts and techniques of statistical field theory, clearly explaining how and why they are integrated into modern quantum (and classical) field theory, and includes the latest developments. Written by an expert in the field, with a broad experience in teaching and training, it manages to present such substantial topics as phases and phase transitions or solitons and instantons in an accessible and concise way. Divided into three parts, the first part covers fundamental physics and the mathematics background needed by students in order to enter the field, while the second part introduces more advanced concepts and techniques. Part III discusses applications of quantum field theory to a few basic problems. The emphasis here lies on how modern concepts of quantum field theory are embedded in these approaches, and also on the limitations of standard quantum field theory techniques in facing, 'real' physics problems. Throughout there are numerous end-of-chapter problems, and a free solutions manual is available for lecturers. (orig.) 9. Dimensional continuation in field theory International Nuclear Information System (INIS) Lee, T. 1988-01-01 The continuation of space-time dimension to an arbitrary complex number is discussed. The ultra-violet and infra-red divergences are simply regularized by analytically continuing to some proper dimension n. Combined with functional integral quantization, it provides a simple and elegant description of quantum field theory. Two well known field theories are discussed. Scalar field theory and quantum electrodynamics. In the scalar theory, the focus is on the operator product expansion. It is showed that a renormalization scheme (minimal subtraction) clearly defines the operator product expansion. In the quantum electrodynamics, it is shown that BRS symmetry can simplify the renormalization process. Composite operators are the renormalized and renormalized stress-energy tensor is formed 10. [Studies in quantum field theory International Nuclear Information System (INIS) 1990-01-01 During the period 4/1/89--3/31/90 the theoretical physics group supported by Department of Energy Contract No. AC02-78ER04915.A015 and consisting of Professors Bender and Shrauner, Associate Professor Papanicolaou, Assistant Professor Ogilvie, and Senior Research Associate Visser has made progress in many areas of theoretical and mathematical physics. Professors Bender and Shrauner, Associate Professor Papanicolaou, Assistant Professor Ogilvie, and Research Associate Visser are currently conducting research in many areas of high energy theoretical and mathematical physics. These areas include: strong-coupling approximation; classical solutions of non-Abelian gauge theories; mean-field approximation in quantum field theory; path integral and coherent state representations in quantum field theory; lattice gauge calculations; the nature of perturbation theory in large order; quark condensation in QCD; chiral symmetry breaking; the 1/N expansion in quantum field theory; effective potential and action in quantum field theories, including OCD; studies of the early universe and inflation, and quantum gravity 11. Field theory of strings International Nuclear Information System (INIS) Ramond, P. 1987-01-01 We review the construction of the free equations of motion for open and closed strings in 26 dimensions, using the methods of the Florida Group. Differing from previous treatments, we argue that the constraint L 0 -anti L 0 =0 should not be imposed on all the fields of the closed string in the gauge invariant formalism; we show that it can be incorporated in the gauge invariant formalism at the price of being unable to extract the equations of motion from a Langrangian. We then describe our purely algebraic method to introduce interactions, which works equally well for open and closed strings. Quartic interactions are absent except in the Physical Gauge. Finally, we speculate on the role of the measure of the open string path functional. (orig.) 12. Field theory of strings International Nuclear Information System (INIS) Ramond, P. 1986-01-01 We review the construction of the free equations of motion for open and closed strings in 26 dimensions, using the methods of the Florida Group. Differing from previous treatments, we argue that the constraint L 0 - L 0 -bar = 0 should not be imposed on all the fields of the closed string in the gauge invariant formalism: we show that it can be incorporated in the invariant formalism at the price of being unable to extract the equations of motion from a Lagrangian. We then describe our purely algebraic method to introduce interactions, which works equally well for open and closed strings. Quartic interactions are absent except in the Physical Gauge. Finally, we speculate on the role of the measure of the open string path functional. 20 refs 13. Probability theory and applications CERN Document Server Hsu, Elton P 1999-01-01 This volume, with contributions by leading experts in the field, is a collection of lecture notes of the six minicourses given at the IAS/Park City Summer Mathematics Institute. It introduces advanced graduates and researchers in probability theory to several of the currently active research areas in the field. Each course is self-contained with references and contains basic materials and recent results. Topics include interacting particle systems, percolation theory, analysis on path and loop spaces, and mathematical finance. The volume gives a balanced overview of the current status of probability theory. An extensive bibliography for further study and research is included. This unique collection presents several important areas of current research and a valuable survey reflecting the diversity of the field. 14. Classical field theory with fermions International Nuclear Information System (INIS) Borsanyi, Sz.; Hindmarsh, M. 2009-01-01 Classical field theory simulations have been essential for our understanding of non-equilibrium phenomena in particle physics. In this talk we discuss the possible extension of the bosonic classical field theory simulations to include fermions. In principle we use the inhomogeneous mean field approximation as introduced by Aarts and Smit. But in practice we turn from their deterministic technique to a stochastic approach. We represent the fermion field as an ensemble of pairs of spinor fields, dubbed male and female. These c-number fields solve the classical Dirac equation. Our improved algorithm enables the extension of the originally 1+1 dimensional analyses and is suitable for large-scale inhomogeneous settings, like defect networks. 15. Wavelet theory and its applications Energy Technology Data Exchange (ETDEWEB) Faber, V.; Bradley, JJ.; Brislawn, C.; Dougherty, R.; Hawrylycz, M. 1996-07-01 This is the final report of a three-year, Laboratory-Directed Research and Development (LDRD) project at the Los Alamos National Laboratory (LANL). We investigated the theory of wavelet transforms and their relation to Laboratory applications. The investigators have had considerable success in the past applying wavelet techniques to the numerical solution of optimal control problems for distributed- parameter systems, nonlinear signal estimation, and compression of digital imagery and multidimensional data. Wavelet theory involves ideas from the fields of harmonic analysis, numerical linear algebra, digital signal processing, approximation theory, and numerical analysis, and the new computational tools arising from wavelet theory are proving to be ideal for many Laboratory applications. 10 refs. 16. Conformal techniques in string theory and string field theory International Nuclear Information System (INIS) Giddings, S.B. 1987-01-01 The application of some conformal and Riemann surface techniques to string theory and string field theory is described. First a brief review of Riemann surface techniques and of the Polyakov approach to string theory is presented. This is followed by a discussion of some features of string field theory and of its Feynman rules. Specifically, it is shown that the Feynman diagrams for Witten's string field theory respect modular invariance, and in particular give a triangulation of moduli space. The Polyakov formalism is then used to derive the Feynman rules that should follow from this theory upon gauge-fixing. It should also be possible to apply this derivation to deduce the Feynman rules for other gauge-fixed string field theories. Following this, Riemann surface techniques are turned to the problem of proving the equivalence of the Polyakov and light-cone formalisms. It is first shown that the light-cone diagrams triangulate moduli space. Then the Polyakov measure is worked out for these diagrams, and shown to equal that deduced from the light-cone gauge fixed formalism. Also presented is a short description of the comparison of physical states in the two formalisms. The equivalence of the two formalisms in particular constitutes a proof of the unitarity of the Polyakov framework for the closed bosonic string 17. Application of effective field theory on nuclear matter and neutron matter; Anwendung effektiver Feldtheorie auf Kernmaterie und Neutronenmaterie Energy Technology Data Exchange (ETDEWEB) Saviankou, Pavel 2009-05-15 In the thesis the effective field theory in NLO and NNLO order is applied. The order NLO still knows no three-particle forces. The theory yields however already in this order the saturation behaviour of nuclear matter. This is due to the fact that in the NLO order the scattering phases are qualitatively correctly reproduced, especially the scattering phases {sup 1}S{sub 0} and {sup 3}S{sub 1} are for energies above 200 MeV negative, which is in all potentials by a so called hard core represented. In the NNLO orde three-particle forces occur, which lead to a larger improvement of the saturation curve, however the saturation point lies still at too high densities. A correction of the low-energy constants by scarcely three percent of the value in the vacuum generates however a saturation curve, which reproduces the empirical binding energy per particle, the density and the compressibility of nuclear matter. About the equation of state of neutron matter is empirically few known. At small densities of neutron matter (k{sub f}<1 fm{sup -1}) the NLO and NNLO orders scarcely differ, but indeed from the free Fermi gas. For applications in finite nuclei a simplified parametrization of the nucleon-nucleon interactions was developed, which reproduces both the known scattering phases with an NLO-comparable accuracy and the empirical saturation behaviour. [German] In der Arbeit wird die Effektive Feldtheorie in der Ordnung NLO und NNLO angewandt. Die Ordnung NLO kennt noch keine Dreiteilchenkraefte. Die Theorie liefert jedoch bereits in dieser Ordnung das Saettigungsverhalten von Kernmaterie. Dies liegt daran, dass bereits in der Ordnung NLO die Streuphasen qualitativ korrekt reproduziert werden, insbesondere sind die Streuphasen {sup 1}S{sub 0} und {sup 3}S{sub 1} fuer Energien oberhalb 200 MeV negativ, was in allen Potentialen durch einen sogenannten ''hard core'' dargestellt wird. In der Ordnung NNLO treten Dreiteilchenkraefte auf, die zu einer grossen 18. Electromagnetic field theories for engineering CERN Document Server Salam, Md Abdus 2014-01-01 A four year Electrical and Electronic engineering curriculum normally contains two modules of electromagnetic field theories during the first two years. However, some curricula do not have enough slots to accommodate the two modules. This book, Electromagnetic Field Theories, is designed for Electrical and Electronic engineering undergraduate students to provide fundamental knowledge of electromagnetic fields and waves in a structured manner. A comprehensive fundamental knowledge of electric and magnetic fields is required to understand the working principles of generators, motors and transformers. This knowledge is also necessary to analyze transmission lines, substations, insulator flashover mechanism, transient phenomena, etc. Recently, academics and researches are working for sending electrical power to a remote area by designing a suitable antenna. In this case, the knowledge of electromagnetic fields is considered as important tool. 19. Generalized field theory of gravitation International Nuclear Information System (INIS) Yilmaz, H. 1976-01-01 It is shown that if, on empirical grounds, one rules out the existence of cosmic fields of Dicke-Brans (scalar) and Will Nordvedt (vector, tensor) type, then the most general experimentally viable and theoretically reasonable theory of gravitation seems to be a LAMBDA-dependent generalization of Einstein and Yilmez theories, which reduces to the former for LAMBDA=0 and to the latter for LAMBDA=1 20. Towards the mathematics of quantum field theory CERN Document Server Paugam, Frédéric 2014-01-01 The aim of this book is to introduce mathematicians (and, in particular, graduate students) to the mathematical methods of theoretical and experimental quantum field theory, with an emphasis on coordinate-free presentations of the mathematical objects in play. This should in turn promote interaction between mathematicians and physicists by supplying a common and flexible language for the good of both communities, even if the mathematical one is the primary target. This reference work provides a coherent and complete mathematical toolbox for classical and quantum field theory, based on categorical and homotopical methods, representing an original contribution to the literature. The first part of the book introduces the mathematical methods needed to work with the physicists' spaces of fields, including parameterized and functional differential geometry, functorial analysis, and the homotopical geometric theory of non-linear partial differential equations, with applications to general gauge theories. The second... 1. Renormalization of topological field theory International Nuclear Information System (INIS) Birmingham, D.; Rakowski, M.; Thompson, G. 1988-11-01 One loop corrections to topological field theory in three and four dimensions are presented. By regularizing determinants, we compute the effective action and β-function in four dimensional topological Yang-Mills theory and find that the BRST symmetry is preserved. Moreover, the minima of the effective action still correspond to instanton configurations. In three dimensions, an analysis of the Chern-Simons theory shows that the topological nature of the theory is also preserved to this order. In addition, we find that this theory possesses an extra supersymmetry when quantized in the Landau gauge. Using dimensional regularization, we then study the Ward identities of the extended BRST symmetry in the three dimensional topological Yang-Mills-Higgs model. (author). 22 refs 2. Gauge and supergauge field theories International Nuclear Information System (INIS) Slavnov, A. 1977-01-01 The most actual problems concerning gauge fields are reviwed. Theoretical investigations conducted since as early as 1954 are enclosed. Present status of gauge theories is summarized, including intermediate vector mesons, heavy leptons, weak interactions of hadrons, V-A structure, universal interaction, infrared divergences in perturbation theory, particle-like solutions in gauge theories, spontaneous symmetry breaking. Special emphasis is placed on strong interactions, or more precisely, on the alleged unobservability of ''color'' objects (quark confinement). Problems dealing with the supersymmetric theories invariant under gauge transformations and spontaneous breaking of supersymmetry are also discussed. Gauge theories are concluded to provide self-consistent apparatus for weak and electromagnetic interactions. As to strong interactions such models are still to be discovered 3. Embedded mean-field theory. Science.gov (United States) Fornace, Mark E; Lee, Joonho; Miyamoto, Kaito; Manby, Frederick R; Miller, Thomas F 2015-02-10 We introduce embedded mean-field theory (EMFT), an approach that flexibly allows for the embedding of one mean-field theory in another without the need to specify or fix the number of particles in each subsystem. EMFT is simple, is well-defined without recourse to parameters, and inherits the simple gradient theory of the parent mean-field theories. In this paper, we report extensive benchmarking of EMFT for the case where the subsystems are treated using different levels of Kohn-Sham theory, using PBE or B3LYP/6-31G* in the high-level subsystem and LDA/STO-3G in the low-level subsystem; we also investigate different levels of density fitting in the two subsystems. Over a wide range of chemical problems, we find EMFT to perform accurately and stably, smoothly converging to the high-level of theory as the active subsystem becomes larger. In most cases, the performance is at least as good as that of ONIOM, but the advantages of EMFT are highlighted by examples that involve partitions across multiple bonds or through aromatic systems and by examples that involve more complicated electronic structure. EMFT is simple and parameter free, and based on the tests provided here, it offers an appealing new approach to a multiscale electronic structure. 4. Geometry of lattice field theory International Nuclear Information System (INIS) Honan, T.J. 1986-01-01 Using some tools of algebraic topology, a general formalism for lattice field theory is presented. The lattice is taken to be a simplicial complex that is also a manifold and is referred to as a simplicial manifold. The fields on this lattice are cochains, that are called lattice forms to emphasize the connections with differential forms in the continuum. This connection provides a new bridge between lattice and continuum field theory. A metric can be put onto this simplicial manifold by assigning lengths to every link or I-simplex of the lattice. Regge calculus is a way of defining general relativity on this lattice. A geometric discussion of Regge calculus is presented. The Regge action, which is a discrete form of the Hilbert action, is derived from the Hilbert action using distribution valued forms. This is a new derivation that emphasizes the underlying geometry. Kramers-Wannier duality in statistical mechanics is discussed in this general setting. Nonlinear field theories, which include gauge theories and nonlinear sigma models are discussed in the continuum and then are put onto a lattice. The main new result here is the generalization to curved spacetime, which consists of making the theory compatible with Regge calculus 5. Bosonic colored group field theory Energy Technology Data Exchange (ETDEWEB) Ben Geloun, Joseph [Universite Paris XI, Laboratoire de Physique Theorique, Orsay Cedex (France); University of Abomey-Calavi, Cotonou (BJ). International Chair in Mathematical Physics and Applications (ICMPA-UNESCO Chair); Universite Cheikh Anta Diop, Departement de Mathematiques et Informatique, Faculte des Sciences et Techniques, Dakar (Senegal); Magnen, Jacques [Ecole Polytechnique, Centre de Physique Theorique, Palaiseau Cedex (France); Rivasseau, Vincent [Universite Paris XI, Laboratoire de Physique Theorique, Orsay Cedex (France) 2010-12-15 Bosonic colored group field theory is considered. Focusing first on dimension four, namely the colored Ooguri group field model, the main properties of Feynman graphs are studied. This leads to a theorem on optimal perturbative bounds of Feynman amplitudes in the ''ultraspin'' (large spin) limit. The results are generalized in any dimension. Finally, integrating out two colors we write a new representation, which could be useful for the constructive analysis of this type of models. (orig.) 6. Gravitation and bilocal field theory International Nuclear Information System (INIS) Vollendorf, F. 1975-01-01 The starting point is the conjecture that a field theory of elementary particles can be constructed only in a bilocal version. Thus the 4-dimensional space time has to be replaced by the 8-dimensional manifold R 8 of all ordered pairs of space time events. With special reference to the Schwarzschild metric it is shown that the embedding of the time space into the manifold R 8 yields a description of the gravitational field. (orig.) [de 7. Computers for lattice field theories International Nuclear Information System (INIS) Iwasaki, Y. 1994-01-01 Parallel computers dedicated to lattice field theories are reviewed with emphasis on the three recent projects, the Teraflops project in the US, the CP-PACS project in Japan and the 0.5-Teraflops project in the US. Some new commercial parallel computers are also discussed. Recent development of semiconductor technologies is briefly surveyed in relation to possible approaches toward Teraflops computers. (orig.) 8. Dimensional analysis in field theory International Nuclear Information System (INIS) Stevenson, P.M. 1981-01-01 Dimensional Transmutation (the breakdown of scale invariance in field theories) is reconciled with the commonsense notions of Dimensional Analysis. This makes possible a discussion of the meaning of the Renormalisation Group equations, completely divorced from the technicalities of renormalisation. As illustrations, I describe some very farmiliar QCD results in these terms 9. Informetrics theory, methods and applications CERN Document Server Qiu, Junping; Yang, Siluo; Dong, Ke 2017-01-01 This book provides an accessible introduction to the history, theory and techniques of informetrics. Divided into 14 chapters, it develops the content system of informetrics from the theory, methods and applications; systematically analyzes the six basic laws and the theory basis of informetrics and presents quantitative analysis methods such as citation analysis and computer-aided analysis. It also discusses applications in information resource management, information and library science, science of science, scientific evaluation and the forecast field. Lastly, it describes a new development in informetrics- webometrics. Providing a comprehensive overview of the complex issues in today's environment, this book is a valuable resource for all researchers, students and practitioners in library and information science. 10. Neuronal coupling by endogenous electric fields: Cable theory and applications to coincidence detector neurons in the auditory brainstem OpenAIRE Goldwyn, Joshua H.; Rinzel, John 2015-01-01 The ongoing activity of neurons generates a spatially- and time-varying field of extracellular voltage ($V_e$). This $V_e$ field reflects population-level neural activity, but does it modulate neural dynamics and the function of neural circuits? We provide a cable theory framework to study how a bundle of model neurons generates $V_e$ and how this $V_e$ feeds back and influences membrane potential ($V_m$). We find that these "ephaptic interactions" are small but not negligible. The model neur... 11. Text Mining Applications and Theory CERN Document Server Berry, Michael W 2010-01-01 Text Mining: Applications and Theory presents the state-of-the-art algorithms for text mining from both the academic and industrial perspectives. The contributors span several countries and scientific domains: universities, industrial corporations, and government laboratories, and demonstrate the use of techniques from machine learning, knowledge discovery, natural language processing and information retrieval to design computational models for automated text analysis and mining. This volume demonstrates how advancements in the fields of applied mathematics, computer science, machine learning 12. Hydrodynamics, fields and constants in gravitational theory International Nuclear Information System (INIS) Stanyukovich, K.P.; Mel'nikov, V.N. 1983-01-01 Results of original inveatigations into problems of standard gravitation theory and its generalizations are presented. The main attention is paid to the application of methods of continuous media techniques in the gravitation theory; to the specification of the gravitation role in phenomena of macro- and microworld, accurate solutions in the case, when the medium is the matter, assigned by hydrodynamic energy-momentum tensor; and to accurate solutions for the case when the medium is the field. GRT generalizations are analyzed, such as the new cosmologic hypothesis which is based on the gravitation vacuum theory. Investigations are performed into the quantization of cosmological models, effects of spontaneous symmetry violation and particle production in cosmology. Graeity theory with fundamental Higgs field is suggested in the framework of which in the atomic unit number one can explain possible variations of the effective gravitational bonds, and in the gravitation bond, variations of masses of all particles 13. Double field theory: a pedagogical review International Nuclear Information System (INIS) Aldazabal, Gerardo; Marqués, Diego; Núñez, Carmen 2013-01-01 Double field theory (DFT) is a proposal to incorporate T-duality, a distinctive symmetry of string theory, as a symmetry of a field theory defined on a double configuration space. The aim of this review is to provide a pedagogical presentation of DFT and its applications. We first introduce some basic ideas on T-duality and supergravity in order to proceed to the construction of generalized diffeomorphisms and an invariant action on the double space. Steps towards the construction of a geometry on the double space are discussed. We then address generalized Scherk–Schwarz compactifications of DFT and their connection to gauged supergravity and flux compactifications. We also discuss U-duality extensions and present a brief parcours on worldsheet approaches to DFT. Finally, we provide a summary of other developments and applications that are not discussed in detail in the review. (topical review) 14. Growing up with field theory International Nuclear Information System (INIS) Vajskopf, V.F. 1982-01-01 The article deals with the history of the development of quantum electrodynamics since the date of publishing the work by P.A.M. Dirac ''The Quantum Theory of the Emission and Absorption of Radiation''. Classic ''before-Dirac'' electrodynamics related with the names of Maxwell, Lorenz, Hertz, is outlined. Work of Bohr and Rosenfeld is shown to clarify the physical sense of quantized field and to reveal the existence of uncertainties between the strengths of different fields. The article points to the significance of the article ''Quantum theory of radiation'' by E. Fermi which clearly describes the Dirac theory of radiation, relativistic wave equation and fundamentals of quantum electrodynamics. Shown is work on elimination of troubles related with the existence of states with negative kinetic energy or with negative mass. Hypothesis on the Dirac filled-in vacuum led to understanding of the existence of antiparticles and two unknown till then fundamental processes - pair production and annihilation. Ways of fighting against the infinite quantities in quantum electrodynamics are considered. Renormalization of the theory overcame all the infinities and gave a pattern for calculation of any processes of electron interactions with electromagnetic field to any desired accuracy 15. Cutkosky rules for superstring field theory International Nuclear Information System (INIS) Pius, Roji; Sen, Ashoke 2016-01-01 Superstring field theory expresses the perturbative S-matrix of superstring theory as a sum of Feynman diagrams each of which is manifestly free from ultraviolet divergences. The interaction vertices fall off exponentially for large space-like external momenta making the ultraviolet finiteness property manifest, but blow up exponentially for large time-like external momenta making it impossible to take the integration contours for loop energies to lie along the real axis. This forces us to carry out the integrals over the loop energies by choosing appropriate contours in the complex plane whose ends go to infinity along the imaginary axis but which take complicated form in the interior navigating around the various poles of the propagators. We consider the general class of quantum field theories with this property and prove Cutkosky rules for the amplitudes to all orders in perturbation theory. Besides having applications to string field theory, these results also give an alternative derivation of Cutkosky rules in ordinary quantum field theories. 16. Cutkosky rules for superstring field theory Energy Technology Data Exchange (ETDEWEB) Pius, Roji [Perimeter Institute for Theoretical Physics,Waterloo, ON N2L 2Y5 (Canada); Sen, Ashoke [Harish-Chandra Research Institute,Chhatnag Road, Jhusi, Allahabad 211019 (India) 2016-10-06 Superstring field theory expresses the perturbative S-matrix of superstring theory as a sum of Feynman diagrams each of which is manifestly free from ultraviolet divergences. The interaction vertices fall off exponentially for large space-like external momenta making the ultraviolet finiteness property manifest, but blow up exponentially for large time-like external momenta making it impossible to take the integration contours for loop energies to lie along the real axis. This forces us to carry out the integrals over the loop energies by choosing appropriate contours in the complex plane whose ends go to infinity along the imaginary axis but which take complicated form in the interior navigating around the various poles of the propagators. We consider the general class of quantum field theories with this property and prove Cutkosky rules for the amplitudes to all orders in perturbation theory. Besides having applications to string field theory, these results also give an alternative derivation of Cutkosky rules in ordinary quantum field theories. 17. Effective field theory for triaxially deformed nuclei Energy Technology Data Exchange (ETDEWEB) Chen, Q.B. [Technische Universitaet Muechen, Physik-Department, Garching (Germany); Peking University, State Key Laboratory of Nuclear Physics and Technology, School of Physics, Beijing (China); Kaiser, N. [Technische Universitaet Muechen, Physik-Department, Garching (Germany); Meissner, Ulf G. [Universitaet Bonn, Helmholtz-Institut fuer Strahlen- und Kernphysik and Bethe Center for Theoretical Physics, Bonn (Germany); Institute for Advanced Simulation, Institut fuer Kernphysik, Juelich Center for Hadron Physics and JARA-HPC, Forschungszentrum Juelich, Juelich (Germany); Meng, J. [Peking University, State Key Laboratory of Nuclear Physics and Technology, School of Physics, Beijing (China); Beihang University, School of Physics and Nuclear Energy Engineering, Beijing (China); University of Stellenbosch, Department of Physics, Stellenbosch (South Africa) 2017-10-15 Effective field theory is generalized to investigate the rotational motion of triaxially deformed even-even nuclei. The Hamiltonian for the triaxial rotor is obtained up to next-to-leading order within the effective field theory formalism. Its applicability is examined by comparing with a five-dimensional rotor-vibrator Hamiltonian for the description of the energy spectra of the ground state and γ band in Ru isotopes. It is found that by taking into account the next-to-leading order corrections, the ground state band in the whole spin region and the γ band in the low spin region are well described. The deviations for high-spin states in the γ bands point towards the importance of including vibrational degrees of freedom in the effective field theory formulation. (orig.) 18. Exceptional field theory: SL(5) International Nuclear Information System (INIS) Musaev, Edvard T. 2016-01-01 In this work the exceptional field theory formulation of supergravity with SL(5) gauge group is considered. This group appears as a U-duality group of D=7 maximal supergravity. In the formalism presented the hidden global duality group is promoted into a gauge group of a theory in dimensions 7+number of extended directions. This work is a continuation of the series of works for E 8,7,6 ,SO(5,5) and SL(3)×SL(2) duality groups. 19. Perturbative coherence in field theory International Nuclear Information System (INIS) Aldrovandi, R.; Kraenkel, R.A. 1987-01-01 A general condition for coherent quantization by perturbative methods is given, because the basic field equations of a fild theory are not always derivable from a Lagrangian. It's seen that non-lagrangian models way have well defined vertices, provided they satisfy what they call the 'coherence condition', which is less stringent than the condition for the existence of a Lagrangian. They note that Lagrangian theories are perturbatively coherent, in the sense that they have well defined vertices, and that they satisfy automatically that condition. (G.D.F.) [pt 20. Einstein's theory of unified fields CERN Document Server Tonnelat, Marie Antoinette 2014-01-01 First published in1966, here is presented a comprehensive overview of one of the most elusive scientific speculations by the pre-eminent genius of the 20th century. The theory is viewed by some scientists with deep suspicion, by others with optimism, but all agree that it represents an extreme challenge. As the author herself affirms, this work is not intended to be a complete treatise or 'didactic exposition' of the theory of unified fields, but rather a tool for further study, both by students and professional physicists. Dealing with all the major areas of research whic 1. A course in field theory CERN Document Server Baal, Pierre Van 2014-01-01 ""… a pleasant novelty that manages the impossible: a full course in field theory from a derivation of the Dirac equation to the standard electroweak theory in less than 200 pages. Moreover, the final chapter consists of a careful selection of assorted problems, which are original and either anticipate or detail some of the topics discussed in the bulk of the chapters. Instead of building a treatise out of a collection of lecture notes, the author took the complementary approach and constructed a course out of a number of well-known and classic treatises. The result is fresh and useful. … the 2. Introduction to quantum field theory CERN Document Server Chang, Shau-Jin 1990-01-01 This book presents in a short volume the basics of quantum field theory and many body physics. The first part introduces the perturbative techniques without sophisticated apparatus and applies them to numerous problems including quantum electrodynamics (renormalization), Fermi and Bose gases, the Brueckner theory of nuclear system, liquid Helium and classical systems with noise. The material is clear, illustrative and the important points are stressed to help the reader get the understanding of what is crucial without overwhelming him with unnecessary detours or comments. The material in the s 3. Multifractals theory and applications CERN Document Server Harte, David 2001-01-01 Although multifractals are rooted in probability, much of the related literature comes from the physics and mathematics arena. Multifractals: Theory and Applications pulls together ideas from both these areas using a language that makes them accessible and useful to statistical scientists. It provides a framework, in particular, for the evaluation of statistical properties of estimates of the Renyi fractal dimensions.The first section provides introductory material and different definitions of a multifractal measure. The author then examines some of the various constructions for describing multifractal measures. Building from the theory of large deviations, he focuses on constructions based on lattice coverings, covering by point-centered spheres, and cascades processes. The final section presents estimators of Renyi dimensions of integer order two and greater and discusses their properties. It also explores various applications of dimension estimation and provides a detailed case study of spatial point patte... 4. The Global Approach to Quantum Field Theory International Nuclear Information System (INIS) Folacci, Antoine; Jensen, Bruce 2003-01-01 is rather difficult to read because of its great breadth. From the start he is faithful to his own view of field theory by developing a powerful formalism which permits him to discuss broad general features common to all field theories. He demands a considerable effort from the reader to penetrate his formalism, and a reading of Appendix A which presents the basics of super-analysis is a prerequisite. To keep the reader on course, DeWitt offers a series of exercises on applications of global formalism in Part 8, nearly 200 pages worth. The exercises are to be worked in parallel with reading the text, starting from the beginning. Before concluding, some criticisms. DeWitt has anticipated some criticism himself in the Preface, where he warns the reader that 'this book is in no sense a reference book on quantum field theory and its application to particle physics. The selection of topics is idiosyncratic. But the reviewers should add a few more remarks: (1) There are very few references. Of course, this is because the work is largely original. Even where the work of other researchers is presented, it has mostly been transformed by the DeWittian point of view. (2) There are very few diagrams, which sometimes hinders the exposition. In summary, in our opinion, this is one of the best books dealing with quantum field theory existing today. It will be of great interest for graduate and postgraduate students as well as workers in the domains of quantum field theory in flat and in curved spacetime and string theories. But we believe that the reader must have previously studied standard textbooks on quantum field theory and general relativity. Even with this preparation, it is by no means an easy book to read. However, the reward is to be able to share the deep and unique vision of the quantum theory of fields and its formalism by one of its greatest expositors. (book review) 5. Plasmonics theory and applications CERN Document Server Shahbazyan, Tigran V 2014-01-01 This contributed volume summarizes recent theoretical developments in plasmonics and its applications in physics, chemistry, materials science, engineering, and medicine. It focuses on recent advances in several major areas of plasmonics including plasmon-enhanced spectroscopies, light scattering, many-body effects, nonlinear optics, and ultrafast dynamics. The theoretical and computational methods used in these investigations include electromagnetic calculations, density functional theory calculations, and nonequilibrium electron dynamics calculations. The book presents a comprehensive overview of these methods as well as their applications to various current problems of interest. 6. Graphs Theory and Applications CERN Document Server Fournier, Jean-Claude 2008-01-01 This book provides a pedagogical and comprehensive introduction to graph theory and its applications. It contains all the standard basic material and develops significant topics and applications, such as: colorings and the timetabling problem, matchings and the optimal assignment problem, and Hamiltonian cycles and the traveling salesman problem, to name but a few. Exercises at various levels are given at the end of each chapter, and a final chapter presents a few general problems with hints for solutions, thus providing the reader with the opportunity to test and refine their knowledge on the 7. Weak gravity conjecture and effective field theory Science.gov (United States) Saraswat, Prashant 2017-01-01 The weak gravity conjecture (WGC) is a proposed constraint on theories with gauge fields and gravity, requiring the existence of light charged particles and/or imposing an upper bound on the field theory cutoff Λ . If taken as a consistency requirement for effective field theories (EFTs), it rules out possibilities for model building including some models of inflation. I demonstrate simple models which satisfy all forms of the WGC, but which through Higgsing of the original gauge fields produce low-energy EFTs with gauge forces that badly violate the WGC. These models illustrate specific loopholes in arguments that motivate the WGC from a bottom-up perspective; for example the arguments based on magnetic monopoles are evaded when the magnetic confinement that occurs in a Higgs phase is accounted for. This indicates that the WGC should not be taken as a veto on EFTs, even if it turns out to be a robust property of UV quantum gravity theories. However, if the latter is true, then parametric violation of the WGC at low energy comes at the cost of nonminimal field content in the UV. I propose that only a very weak constraint is applicable to EFTs, Λ ≲(log 1/g )-1 /2Mpl , where g is the gauge coupling, motivated by entropy bounds. Remarkably, EFTs produced by Higgsing a theory that satisfies the WGC can saturate but not violate this bound. 8. Neuronal coupling by endogenous electric fields: cable theory and applications to coincidence detector neurons in the auditory brain stem. Science.gov (United States) Goldwyn, Joshua H; Rinzel, John 2016-04-01 The ongoing activity of neurons generates a spatially and time-varying field of extracellular voltage (Ve). This Ve field reflects population-level neural activity, but does it modulate neural dynamics and the function of neural circuits? We provide a cable theory framework to study how a bundle of model neurons generates Ve and how this Ve feeds back and influences membrane potential (Vm). We find that these "ephaptic interactions" are small but not negligible. The model neural population can generate Ve with millivolt-scale amplitude, and this Ve perturbs the Vm of "nearby" cables and effectively increases their electrotonic length. After using passive cable theory to systematically study ephaptic coupling, we explore a test case: the medial superior olive (MSO) in the auditory brain stem. The MSO is a possible locus of ephaptic interactions: sounds evoke large (millivolt scale)Vein vivo in this nucleus. The Ve response is thought to be generated by MSO neurons that perform a known neuronal computation with submillisecond temporal precision (coincidence detection to encode sound source location). Using a biophysically based model of MSO neurons, we find millivolt-scale ephaptic interactions consistent with the passive cable theory results. These subtle membrane potential perturbations induce changes in spike initiation threshold, spike time synchrony, and time difference sensitivity. These results suggest that ephaptic coupling may influence MSO function. Copyright © 2016 the American Physiological Society. 9. Euler-Poincare reduction for discrete field theories International Nuclear Information System (INIS) Vankerschaver, Joris 2007-01-01 In this note, we develop a theory of Euler-Poincare reduction for discrete Lagrangian field theories. We introduce the concept of Euler-Poincare equations for discrete field theories, as well as a natural extension of the Moser-Veselov scheme, and show that both are equivalent. The resulting discrete field equations are interpreted in terms of discrete differential geometry. An application to the theory of discrete harmonic mappings is also briefly discussed 10. Variational methods for field theories International Nuclear Information System (INIS) Ben-Menahem, S. 1986-09-01 Four field theory models are studied: Periodic Quantum Electrodynamics (PQED) in (2 + 1) dimensions, free scalar field theory in (1 + 1) dimensions, the Quantum XY model in (1 + 1) dimensions, and the (1 + 1) dimensional Ising model in a transverse magnetic field. The last three parts deal exclusively with variational methods; the PQED part involves mainly the path-integral approach. The PQED calculation results in a better understanding of the connection between electric confinement through monopole screening, and confinement through tunneling between degenerate vacua. This includes a better quantitative agreement for the string tensions in the two approaches. Free field theory is used as a laboratory for a new variational blocking-truncation approximation, in which the high-frequency modes in a block are truncated to wave functions that depend on the slower background modes (Boron-Oppenheimer approximation). This ''adiabatic truncation'' method gives very accurate results for ground-state energy density and correlation functions. Various adiabatic schemes, with one variable kept per site and then two variables per site, are used. For the XY model, several trial wave functions for the ground state are explored, with an emphasis on the periodic Gaussian. A connection is established with the vortex Coulomb gas of the Euclidean path integral approach. The approximations used are taken from the realms of statistical mechanics (mean field approximation, transfer-matrix methods) and of quantum mechanics (iterative blocking schemes). In developing blocking schemes based on continuous variables, problems due to the periodicity of the model were solved. Our results exhibit an order-disorder phase transition. The transfer-matrix method is used to find a good (non-blocking) trial ground state for the Ising model in a transverse magnetic field in (1 + 1) dimensions 11. Variational methods for field theories Energy Technology Data Exchange (ETDEWEB) Ben-Menahem, S. 1986-09-01 Four field theory models are studied: Periodic Quantum Electrodynamics (PQED) in (2 + 1) dimensions, free scalar field theory in (1 + 1) dimensions, the Quantum XY model in (1 + 1) dimensions, and the (1 + 1) dimensional Ising model in a transverse magnetic field. The last three parts deal exclusively with variational methods; the PQED part involves mainly the path-integral approach. The PQED calculation results in a better understanding of the connection between electric confinement through monopole screening, and confinement through tunneling between degenerate vacua. This includes a better quantitative agreement for the string tensions in the two approaches. Free field theory is used as a laboratory for a new variational blocking-truncation approximation, in which the high-frequency modes in a block are truncated to wave functions that depend on the slower background modes (Boron-Oppenheimer approximation). This ''adiabatic truncation'' method gives very accurate results for ground-state energy density and correlation functions. Various adiabatic schemes, with one variable kept per site and then two variables per site, are used. For the XY model, several trial wave functions for the ground state are explored, with an emphasis on the periodic Gaussian. A connection is established with the vortex Coulomb gas of the Euclidean path integral approach. The approximations used are taken from the realms of statistical mechanics (mean field approximation, transfer-matrix methods) and of quantum mechanics (iterative blocking schemes). In developing blocking schemes based on continuous variables, problems due to the periodicity of the model were solved. Our results exhibit an order-disorder phase transition. The transfer-matrix method is used to find a good (non-blocking) trial ground state for the Ising model in a transverse magnetic field in (1 + 1) dimensions. 12. Types of two-dimensional N = 4 superconformal field theories Superconformal field theory; free field realization; string theory; AdS-CFT correspon- dence. PACS Nos 11.25.Hf; 11.25.-w; 11.30.Ly; 11.30.Pb. Conformal symmetries in two space-time dimensions have been very extensively studied owing to their applications both in string theory and two-dimensional statistical systems. 13. Theory of field reversed configurations International Nuclear Information System (INIS) Steinhauer, L.C. 1990-01-01 This final report surveys the results of work conducted on the theory of field reversed configurations. This project has spanned ten years, beginning in early 1980. During this period, Spectra Technology was one of the leading contributors to the advances in understanding FRC. The report is organized into technical topic areas, FRC formation, equilibrium, stability, and transport. Included as an appendix are papers published in archival journals that were generated in the course of this report. 33 refs 14. Renormalization and Interaction in Quantum Field Theory International Nuclear Information System (INIS) RATSIMBARISON, H.M. 2008-01-01 This thesis works on renormalization in quantum field theory (QFT), in order to show the relevance of some mathematical structures as C-algebraic and probabilistic structures. Our work begins with a study of the path integral formalism and the Kreimer-Connes approach in perturbative renormalization, which allows to situate the statistical nature of QFT and to appreciate the ultra-violet divergence problem of its partition function. This study is followed by an emphasis of the presence of convolution products in non perturbative renormalisation, through the construction of the Wilson effective action and the Legendre effective action. Thanks to these constructions and the definition of effective theories according J. Polchinski, the non perturbative renormalization shows in particular the general approach of regularization procedure. We begin the following chapter with a C-algebraic approach of the scale dependence of physical theories by showing the existence of a hierarchy of commutative spaces of states and its compatibility with the fiber bundle formulation of classical field theory. Our Hierarchy also allows us to modelize the notion of states and particles. Finally, we develop a probabilistic construction of interacting theories starting from simple model, a Bernoulli random processes. We end with some arguments on the applicability of our construction -such as the independence between the free and interacting terms and the possibility to introduce a symmetry group wich will select the type of interactions in quantum field theory. [fr 15. Dynamic random walks theory and applications CERN Document Server 2006-01-01 The aim of this book is to report on the progress realized in probability theory in the field of dynamic random walks and to present applications in computer science, mathematical physics and finance. Each chapter contains didactical material as well as more advanced technical sections. Few appendices will help refreshing memories (if necessary!).· New probabilistic model, new results in probability theory· Original applications in computer science· Applications in mathematical physics· Applications in finance 16. Electricity markets theories and applications CERN Document Server Lin, Jeremy 2017-01-01 Electricity Markets: Theories and Applications offers students and practitioners a clear understanding of the fundamental concepts of the economic theories, particularly microeconomic theories, as well as information on some advanced optimization methods of electricity markets. The authors--noted experts in the field--cover the basic drivers for the transformation of the electricity industry in both the United States and around the world and discuss the fundamentals of power system operation, electricity market design and structures, and electricity market operations. The text also explores advanced topics of power system operations and electricity market design and structure including zonal versus nodal pricing, market performance and market power issues, transmission pricing, and the emerging problems electricity markets face in smart grid and micro-grid environments. The authors also examine system planning under the context of electricity market regime. They explain the new ways to solve problems with t... 17. On incompleteness of classical field theory OpenAIRE Sardanashvily, G. 2009-01-01 Classical field theory is adequately formulated as Lagrangian theory on fibre bundles and graded manifolds. One however observes that non-trivial higher stage Noether identities and gauge symmetries of a generic reducible degenerate Lagrangian field theory fail to be defined. Therefore, such a field theory can not be quantized. 18. Holography for field theory solitons Science.gov (United States) Domokos, Sophia K.; Royston, Andrew B. 2017-07-01 We extend a well-known D-brane construction of the AdS/dCFT correspondence to non-abelian defects. We focus on the bulk side of the correspondence and show that there exists a regime of parameters in which the low-energy description consists of two approximately decoupled sectors. The two sectors are gravity in the ambient spacetime, and a six-dimensional supersymmetric Yang-Mills theory. The Yang-Mills theory is defined on a rigid AdS4 × S 2 background and admits sixteen supersymmetries. We also consider a one-parameter deformation that gives rise to a family of Yang-Mills theories on asymptotically AdS4 × S 2 spacetimes, which are invariant under eight supersymmetries. With future holographic applications in mind, we analyze the vacuum structure and perturbative spectrum of the Yang-Mills theory on AdS4 × S 2, as well as systems of BPS equations for finite-energy solitons. Finally, we demonstrate that the classical Yang-Mills theory has a consistent truncation on the two-sphere, resulting in maximally supersymmetric Yang-Mills on AdS4. 19. Generalized filtering of laser fields in optimal control theory: application to symmetry filtering of quantum gate operations International Nuclear Information System (INIS) Schroeder, Markus; Brown, Alex 2009-01-01 We present a modified version of a previously published algorithm (Gollub et al 2008 Phys. Rev. Lett.101 073002) for obtaining an optimized laser field with more general restrictions on the search space of the optimal field. The modification leads to enforcement of the constraints on the optimal field while maintaining good convergence behaviour in most cases. We demonstrate the general applicability of the algorithm by imposing constraints on the temporal symmetry of the optimal fields. The temporal symmetry is used to reduce the number of transitions that have to be optimized for quantum gate operations that involve inversion (NOT gate) or partial inversion (Hadamard gate) of the qubits in a three-dimensional model of ammonia. 20. Braided quantum field theories and their symmetries International Nuclear Information System (INIS) Sasai, Yuya; Sasakura, Naoki 2007-01-01 Braided quantum field theories, proposed by Oeckl, can provide a framework for quantum field theories that possess Hopf algebra symmetries. In quantum field theories, symmetries lead to non-perturbative relations among correlation functions. We study Hopf algebra symmetries and such relations in the context of braided quantum field theories. We give the four algebraic conditions among Hopf algebra symmetries and braided quantum field theories that are required for the relations to hold. As concrete examples, we apply our analysis to the Poincare symmetries of two examples of noncommutative field theories. One is the effective quantum field theory of three-dimensional quantum gravity coupled to spinless particles formulated by Freidel and Livine, and the other is noncommutative field theory on the Moyal plane. We also comment on quantum field theory in κ-Minkowski spacetime. (author) 1. Causality Constraints in Conformal Field Theory CERN Multimedia CERN. Geneva 2015-01-01 Causality places nontrivial constraints on QFT in Lorentzian signature, for example fixing the signs of certain terms in the low energy Lagrangian. In d-dimensional conformal field theory, we show how such constraints are encoded in crossing symmetry of Euclidean correlators, and derive analogous constraints directly from the conformal bootstrap (analytically). The bootstrap setup is a Lorentzian four-point function corresponding to propagation through a shockwave. Crossing symmetry fixes the signs of certain log terms that appear in the conformal block expansion, which constrains the interactions of low-lying operators. As an application, we use the bootstrap to rederive the well known sign constraint on the (∂φ)4 coupling in effective field theory, from a dual CFT. We also find constraints on theories with higher spin conserved currents. Our analysis is restricted to scalar correlators, but we argue that similar methods should also impose nontrivial constraints on the interactions of spinni... 2. Classification of networks of automata by dynamical mean field theory International Nuclear Information System (INIS) Burda, Z.; Jurkiewicz, J.; Flyvbjerg, H. 1990-01-01 Dynamical mean field theory is used to classify the 2 24 =65,536 different networks of binary automata on a square lattice with nearest neighbour interactions. Application of mean field theory gives 700 different mean field classes, which fall in seven classes of different asymptotic dynamics characterized by fixed points and two-cycles. (orig.) 3. Progress on The GEMS (Gravity Electro-Magnetism-Strong) Theory of Field Unification and Its Application to Space Problems International Nuclear Information System (INIS) Brandenburg, J. E. 2008-01-01 Progress on the GEMS (Gravity Electro-Magnetism-Strong), theory is presented as well as its application to space problems. The GEMS theory is now validated through the Standard Model of physics. Derivation of the value of the Gravitation constant based on the observed variation of α with energy: results in the formula G congruent with (ℎ/2π)c/M ηc 2 exp(-1/(1.61α)), where α is the fine structure constant,(ℎ/2π), is Planck's constant, c, is the speed of light, and M ηc is the mass of the η cc Charmonium meson that is shown to be identical to that derived from the GEM postulates. Covariant formulation of the GEM theory is now possible through definition of the spacetime metric tensor as a portion of the EM stress tensor normalized by its own trace: g ab = 4(F c a F cb )/(F ab F ab ), it is found that this results in a massless ground state vacuum and a Newtonian gravitation potential φ = 1/2 E 2 /B 2 . It is also found that a Lorentz or flat-space metric is recovered in the limit of a full spectrum ZPF 4. Large-signal model of the bilayer graphene field-effect transistor targeting radio-frequency applications: Theory versus experiment Energy Technology Data Exchange (ETDEWEB) Pasadas, Francisco, E-mail: [email protected]; Jiménez, David [Departament d' Enginyeria Electrònica, Escola d' Enginyeria, Universitat Autònoma de Barcelona, 08193 Bellaterra (Spain) 2015-12-28 Bilayer graphene is a promising material for radio-frequency transistors because its energy gap might result in a better current saturation than the monolayer graphene. Because the great deal of interest in this technology, especially for flexible radio-frequency applications, gaining control of it requires the formulation of appropriate models for the drain current, charge, and capacitance. In this work, we have developed them for a dual-gated bilayer graphene field-effect transistor. A drift-diffusion mechanism for the carrier transport has been considered coupled with an appropriate field-effect model taking into account the electronic properties of the bilayer graphene. Extrinsic resistances have been included considering the formation of a Schottky barrier at the metal-bilayer graphene interface. The proposed model has been benchmarked against experimental prototype transistors, discussing the main figures of merit targeting radio-frequency applications. 5. Group theory and lattice gauge fields International Nuclear Information System (INIS) Creutz, M. 1988-09-01 Lattice gauge theory, formulated in terms of invariant integrals over group elements on lattice bonds, benefits from many group theoretical notions. Gauge invariance provides an enormous symmetry and powerful constraints on expectation values. Strong coupling expansions require invariant integrals over polynomials in group elements, all of which can be evaluated by symmetry considerations. Numerical simulations involve random walks over the group. These walks automatically generate the invariant group measure, avoiding explicit parameterization. A recently proposed overrelaxation algorithm is particularly efficient at exploring the group manifold. These and other applications of group theory to lattice gauge fields are reviewed in this talk. 17 refs 6. Inverse bootstrapping conformal field theories Science.gov (United States) Li, Wenliang 2018-01-01 We propose a novel approach to study conformal field theories (CFTs) in general dimensions. In the conformal bootstrap program, one usually searches for consistent CFT data that satisfy crossing symmetry. In the new method, we reverse the logic and interpret manifestly crossing-symmetric functions as generating functions of conformal data. Physical CFTs can be obtained by scanning the space of crossing-symmetric functions. By truncating the fusion rules, we are able to concentrate on the low-lying operators and derive some approximate relations for their conformal data. It turns out that the free scalar theory, the 2d minimal model CFTs, the ϕ 4 Wilson-Fisher CFT, the Lee-Yang CFTs and the Ising CFTs are consistent with the universal relations from the minimal fusion rule ϕ 1 × ϕ 1 = I + ϕ 2 + T , where ϕ 1 , ϕ 2 are scalar operators, I is the identity operator and T is the stress tensor. 7. About Applications of the Fixed Point Theory Directory of Open Access Journals (Sweden) Bucur Amelia 2017-06-01 Full Text Available The fixed point theory is essential to various theoretical and applied fields, such as variational and linear inequalities, the approximation theory, nonlinear analysis, integral and differential equations and inclusions, the dynamic systems theory, mathematics of fractals, mathematical economics (game theory, equilibrium problems, and optimisation problems and mathematical modelling. This paper presents a few benchmarks regarding the applications of the fixed point theory. This paper also debates if the results of the fixed point theory can be applied to the mathematical modelling of quality. 8. Regularization of quantum field theories International Nuclear Information System (INIS) Rayski, J. 1985-01-01 General idea of regularization and renormalization in quantum field theory is presented. It is postulated that it is possible not to go to infinity with the auxiliary masses of regularization but to attach to them a certain physical meaning, but it is equivalent with a violation of unitarity of the operator of evolution in time. It may be achieved in two different ways: it might be simply assumed that only the direction but not the length of the state vector possesses a physical meaning and that not all possible physical events are predictable. 3 refs., 1 fig. (author) 9. Quantum field theory of point particles and strings CERN Document Server Hatfield, Brian 1992-01-01 The purpose of this book is to introduce string theory without assuming any background in quantum field theory. Part I of this book follows the development of quantum field theory for point particles, while Part II introduces strings. All of the tools and concepts that are needed to quantize strings are developed first for point particles. Thus, Part I presents the main framework of quantum field theory and provides for a coherent development of the generalization and application of quantum field theory for point particles to strings.Part II emphasizes the quantization of the bosonic string. 10. The Global Approach to Quantum Field Theory Energy Technology Data Exchange (ETDEWEB) Folacci, Antoine; Jensen, Bruce [Faculte des Sciences, Universite de Corse (France); Department of Mathematics, University of Southampton (United Kingdom) 2003-12-12 be noted that DeWitt's book is rather difficult to read because of its great breadth. From the start he is faithful to his own view of field theory by developing a powerful formalism which permits him to discuss broad general features common to all field theories. He demands a considerable effort from the reader to penetrate his formalism, and a reading of Appendix A which presents the basics of super-analysis is a prerequisite. To keep the reader on course, DeWitt offers a series of exercises on applications of global formalism in Part 8, nearly 200 pages worth. The exercises are to be worked in parallel with reading the text, starting from the beginning. Before concluding, some criticisms. DeWitt has anticipated some criticism himself in the Preface, where he warns the reader that 'this book is in no sense a reference book on quantum field theory and its application to particle physics. The selection of topics is idiosyncratic. (book review)[abstract truncated 11. Twisted conformal field theories and Morita equivalence Energy Technology Data Exchange (ETDEWEB) Marotta, Vincenzo [Dipartimento di Scienze Fisiche, Universita di Napoli ' Federico II' and INFN, Sezione di Napoli, Compl. universitario M. Sant' Angelo, Via Cinthia, 80126 Napoli (Italy); Naddeo, Adele [CNISM, Unita di Ricerca di Salerno and Dipartimento di Fisica ' E.R. Caianiello' , Universita degli Studi di Salerno, Via Salvador Allende, 84081 Baronissi (Italy); Dipartimento di Scienze Fisiche, Universita di Napoli ' Federico II' , Compl. universitario M. Sant' Angelo, Via Cinthia, 80126 Napoli (Italy)], E-mail: [email protected] 2009-04-01 The Morita equivalence for field theories on noncommutative two-tori is analysed in detail for rational values of the noncommutativity parameter {theta} (in appropriate units): an isomorphism is established between an Abelian noncommutative field theory (NCFT) and a non-Abelian theory of twisted fields on ordinary space. We focus on a particular conformal field theory (CFT), the one obtained by means of the m-reduction procedure [V. Marotta, J. Phys. A 26 (1993) 3481; V. Marotta, Mod. Phys. Lett. A 13 (1998) 853; V. Marotta, Nucl. Phys. B 527 (1998) 717; V. Marotta, A. Sciarrino, Mod. Phys. Lett. A 13 (1998) 2863], and show that it is the Morita equivalent of a NCFT. Finally, the whole m-reduction procedure is shown to be the image in the ordinary space of the Morita duality. An application to the physics of a quantum Hall fluid at Jain fillings {nu}=m/(2pm+1) is explicitly discussed in order to further elucidate such a correspondence and to clarify its role in the physics of strongly correlated systems. A new picture emerges, which is very different from the existing relationships between noncommutativity and many body systems [A.P. Polychronakos, arXiv: 0706.1095]. 12. Topics in low-dimensional field theory Energy Technology Data Exchange (ETDEWEB) Crescimanno, M.J. 1991-04-30 Conformal field theory is a natural tool for understanding two- dimensional critical systems. This work presents results in the lagrangian approach to conformal field theory. The first sections are chiefly about a particular class of field theories called coset constructions and the last part is an exposition of the connection between two-dimensional conformal theory and a three-dimensional gauge theory whose lagrangian is the Chern-Simons density. 13. Number theory arising from finite fields analytic and probabilistic theory CERN Document Server Knopfmacher, John 2001-01-01 ""Number Theory Arising from Finite Fields: Analytic and Probabilistic Theory"" offers a discussion of the advances and developments in the field of number theory arising from finite fields. It emphasizes mean-value theorems of multiplicative functions, the theory of additive formulations, and the normal distribution of values from additive functions. The work explores calculations from classical stages to emerging discoveries in alternative abstract prime number theorems. 14. Perturbative study in quantum field theory at finite temperature, application to lepton pair production from a quark-gluon plasma International Nuclear Information System (INIS) Altherr, T. 1989-12-01 The main topic of this thesis is a perturbative study of Quantum Field Theory at Finite Temperature. The real-time formalism is used throughout this work. We show the cancellation of infrared and mass singularities in the case of the first order QCD corrections to lepton pair production from a quark-gluon plasma. Two methods of calculation are presented and give the same finite result in the limit of vanishing quark mass. These finite terms are analysed and give small corrections in the region of interest for ultra-relativistic heavy ions collisions, except for a threshold factor. Specific techniques for finite temperature calculations are explicited in the case of the fermionic self-energy in QED [fr 15. Vertex operator algebras and conformal field theory International Nuclear Information System (INIS) Huang, Y.Z. 1992-01-01 This paper discusses conformal field theory, an important physical theory, describing both two-dimensional critical phenomena in condensed matter physics and classical motions of strings in string theory. The study of conformal field theory will deepen the understanding of these theories and will help to understand string theory conceptually. Besides its importance in physics, the beautiful and rich mathematical structure of conformal field theory has interested many mathematicians. New relations between different branches of mathematics, such as representations of infinite-dimensional Lie algebras and Lie groups, Riemann surfaces and algebraic curves, the Monster sporadic group, modular functions and modular forms, elliptic genera and elliptic cohomology, Calabi-Yau manifolds, tensor categories, and knot theory, are revealed in the study of conformal field theory. It is therefore believed that the study of the mathematics involved in conformal field theory will ultimately lead to new mathematical structures which would be important to both mathematics and physics 16. On quantum field theory in gravitational background International Nuclear Information System (INIS) Haag, R.; Narnhofer, H.; Stein, U. 1984-02-01 We discuss Quantum Fields on Riemannian space-time. A principle of local definitness is introduced which is needed beyond equations of motion and commutation relations to fix the theory uniquely. It also allows to formulate local stability. In application to a region with a time-like Killing vector field and horizons it yields the value of the Hawking temperature. The concept of vacuum and particles in a non stationary metric is treated in the example of the Robertson-Walker metric and some remarks on detectors in non inertial motion are added. (orig.) 17. Light front field theory: an advanced primer International Nuclear Information System (INIS) Martinovic, L. 2007-01-01 We present an elementary introduction to quantum field theory formulated in terms of Dirac's light front variables. In addition to general principles and methods, a few more specific topics and approaches based on the author's work will be discussed. Most of the discussion deals with massive two-dimensional models formulated in a finite spatial volume starting with a detailed comparison between quantization of massive free fields in the usual field theory and the light front (LF) quantization. We discuss basic properties such as relativistic invariance and causality. After the LF treatment of the soluble Federbush model, a LF approach to spontaneous symmetry breaking is explained and a simple gauge theory - the massive Schwinger model in various gauges is studied. A LF version of bosonization and the massive Thirring model are also discussed. A special chapter is devoted to the method of discretized light cone quantization and its application to calculations of the properties of quantum solitons. The problem of LF zero modes is illustrated with the example of the two/dimensional Yukawa model. Hamiltonian perturbation theory in the LF formulation is derived and applied to a few simple processes to demonstrate its advantages. As a byproduct, it is shown that the LF theory cannot be obtained as a 'light-like' limit of the usual field theory quantized on a initial space-like surface. A simple LF formulation of the Higgs mechanism is then given Since our intention was to provide a treatment of the light front quantization accessible to postgradual students, an effort was made to discuss most of the topics pedagogically and number of technical details and derivations are contained in the appendices (Author) 18. Quantum Field Theory in (0 + 1) Dimensions Science.gov (United States) Boozer, A. D. 2007-01-01 We show that many of the key ideas of quantum field theory can be illustrated simply and straightforwardly by using toy models in (0 + 1) dimensions. Because quantum field theory in (0 + 1) dimensions is equivalent to quantum mechanics, these models allow us to use techniques from quantum mechanics to gain insight into quantum field theory. In… 19. Application of the nuclear field theory to monopole interactions which include all the vertices of a general force International Nuclear Information System (INIS) Bes, D.R.; Dussel, G.G.; Liotta, R.J.; Sofia, H.M.; Broglia, R.A. 1976-01-01 The field treatment is applied to the monopole pairing and monopole particle-hole interactions in a two-level model. All the vertices of realistic interactions appear, and the problems treated here have most of the complexities of real nuclei. Yet, the model remains sufficiently simple, so that a close comparison with the results of a (conventional) treatment in which only the fermion degrees of freedom are considered is possible. The applicability to actual physical situations appears to be feasible, both for schematic or realistic forces. The advantage of including the exchange components of the interaction in the construction of the phonon is discussed. (Auth.) 20. Gaussian processes and constructive scalar field theory International Nuclear Information System (INIS) Benfatto, G.; Nicolo, F. 1981-01-01 The last years have seen a very deep progress of constructive euclidean field theory, with many implications in the area of the random fields theory. The authors discuss an approach to super-renormalizable scalar field theories, which puts in particular evidence the connections with the theory of the Gaussian processes associated to the elliptic operators. The paper consists of two parts. Part I treats some problems in the theory of Gaussian processes which arise in the approach to the PHI 3 4 theory. Part II is devoted to the discussion of the ultraviolet stability in the PHI 3 4 theory. (Auth.) 1. Effective Field Theory on Manifolds with Boundary Science.gov (United States) Albert, Benjamin I. In the monograph Renormalization and Effective Field Theory, Costello made two major advances in rigorous quantum field theory. Firstly, he gave an inductive position space renormalization procedure for constructing an effective field theory that is based on heat kernel regularization of the propagator. Secondly, he gave a rigorous formulation of quantum gauge theory within effective field theory that makes use of the BV formalism. In this work, we extend Costello's renormalization procedure to a class of manifolds with boundary and make preliminary steps towards extending his formulation of gauge theory to manifolds with boundary. In addition, we reorganize the presentation of the preexisting material, filling in details and strengthening the results. 2. Logarithmic conformal field theory: beyond an introduction Science.gov (United States) Creutzig, Thomas; Ridout, David 2013-12-01 of the underlying chiral algebra and the modular data pertaining to the characters of the representations. Each of the archetypal logarithmic conformal field theories is studied here by first determining its irreducible spectrum, which turns out to be continuous, as well as a selection of natural reducible, but indecomposable, modules. This is followed by a detailed description of how to obtain character formulae for each irreducible, a derivation of the action of the modular group on the characters, and an application of the Verlinde formula to compute the Grothendieck fusion rules. In each case, the (genuine) fusion rules are known, so comparisons can be made and favourable conclusions drawn. In addition, each example admits an infinite set of simple currents, hence extended symmetry algebras may be constructed and a series of bulk modular invariants computed. The spectrum of such an extended theory is typically discrete and this is how the triplet model \\mathfrak {W} (1,2) arises, for example. Moreover, simple current technology admits a derivation of the extended algebra fusion rules from those of its continuous parent theory. Finally, each example is concluded by a brief description of the computation of some bulk correlators, a discussion of the structure of the bulk state space, and remarks concerning more advanced developments and generalizations. The final part gives a very short account of the theory of staggered modules, the (simplest class of) representations that are responsible for the logarithmic singularities that distinguish logarithmic theories from their rational cousins. These modules are discussed in a generality suitable to encompass all the examples met in this review and some of the very basic structure theory is proven. Then, the important quantities known as logarithmic couplings are reviewed for Virasoro staggered modules and their role as fundamentally important parameters, akin to the three-point constants of rational conformal field 3. Negative power spectra in quantum field theory International Nuclear Information System (INIS) Hsiang, Jen-Tsung; Wu, Chun-Hsien; Ford, L.H. 2011-01-01 We consider the spatial power spectra associated with fluctuations of quadratic operators in field theory, such as quantum stress tensor components. We show that the power spectrum can be negative, in contrast to most fluctuation phenomena where the Wiener-Khinchin theorem requires a positive power spectrum. We show why the usual argument for positivity fails in this case, and discuss the physical interpretation of negative power spectra. Possible applications to cosmology are discussed. -- Highlights: → Wiener-Khinchin theorem usually implies a positive power spectrum of fluctuations. → We show this is not always the case in quantum field theory. → Quantum stress tensor fluctuations can have a negative power spectrum. → Negative power interchanges correlations and anticorrelations. 4. PPP mode’s applications motivation in the field of water conservancy project - based on the “money service” theory of Milton Friedman Science.gov (United States) Chen, Zurong; Feng, Jingchun; Wang, Yuting; Xue, Song 2017-06-01 We study on PPP mode’s applications motivation in the field of water conservancy project, on the basis of analyzing Friedman’s “money service” theory, for the disadvantages of traditional investment mode in water conservancy project field. By analyzing the way of government and social capital spending money in PPP projects, we get conclusion that both of which are the way of “spending their own money to do their own thing”, which fully reflects that the two sides are a win-win partnership in PPP mode. From the application motivation, PPP mode can not only compensate for the lack of local funds, improve the investment efficiency of the government, but also promote marketization and the supply-side structural reforms. 5. N=1 field theory duality from M theory International Nuclear Information System (INIS) Schmaltz, M.; Sundrum, R. 1998-01-01 We investigate Seiberg close-quote s N=1 field theory duality for four-dimensional supersymmetric QCD with the M-theory 5-brane. We find that the M-theory configuration for the magnetic dual theory arises via a smooth deformation of the M-theory configuration for the electric theory. The creation of Dirichlet 4-branes as Neveu-Schwarz 5-branes are passed through each other in type IIA string theory is given an elegant derivation from M theory. copyright 1998 The American Physical Society 6. Quantum Field Theory A Modern Perspective CERN Document Server Parameswaran Nair, V 2005-01-01 Quantum field theory, which started with Paul Dirac’s work shortly after the discovery of quantum mechanics, has produced an impressive and important array of results. Quantum electrodynamics, with its extremely accurate and well-tested predictions, and the standard model of electroweak and chromodynamic (nuclear) forces are examples of successful theories. Field theory has also been applied to a variety of phenomena in condensed matter physics, including superconductivity, superfluidity and the quantum Hall effect. The concept of the renormalization group has given us a new perspective on field theory in general and on critical phenomena in particular. At this stage, a strong case can be made that quantum field theory is the mathematical and intellectual framework for describing and understanding all physical phenomena, except possibly for a quantum theory of gravity. Quantum Field Theory: A Modern Perspective presents Professor Nair’s view of certain topics in field theory loosely knit together as it gr... 7. Quantum field theory of universe International Nuclear Information System (INIS) Hosoya, Akio; Morikawa, Masahiro. 1988-08-01 As is well-known, the wave function of universe dictated by the Wheeler-DeWitt equation has a difficulty in its probabilistic interpretation. In order to overcome this difficulty, we explore a theoretical possibility of the second quantization of universe, following the same passage historically taken for the Klein-Gordon particles and the Nambu-Goto strings. It turns out that multiple production of universes is an inevitable consequence even if the initial state is nothing. The problematical interpretation of wave function of universe is circumvented by introducing an internal comoving model detector, which is an analogue of the DeWitt-Unruh detector in the quantum field theory in curved space-time. (author) 8. Families and degenerations of conformal field theories Energy Technology Data Exchange (ETDEWEB) Roggenkamp, D. 2004-09-01 In this work, moduli spaces of conformal field theories are investigated. In the first part, moduli spaces corresponding to current-current deformation of conformal field theories are constructed explicitly. For WZW models, they are described in detail, and sigma model realizations of the deformed WZW models are presented. The second part is devoted to the study of boundaries of moduli spaces of conformal field theories. For this purpose a notion of convergence of families of conformal field theories is introduced, which admits certain degenerated conformal field theories to occur as limits. To such a degeneration of conformal field theories, a degeneration of metric spaces together with additional geometric structures can be associated, which give rise to a geometric interpretation. Boundaries of moduli spaces of toroidal conformal field theories, orbifolds thereof and WZW models are analyzed. Furthermore, also the limit of the discrete family of Virasoro minimal models is investigated. (orig.) 9. Dependence logic theory and applications CERN Document Server Kontinen, Juha; Väänänen, Jouko; Vollmer, Heribert 2016-01-01 In this volume, different aspects of logics for dependence and independence are discussed, including both the logical and computational aspects of dependence logic, and also applications in a number of areas, such as statistics, social choice theory, databases, and computer security. The contributing authors represent leading experts in this relatively new field, each of whom was invited to write a chapter based on talks given at seminars held at the Schloss Dagstuhl Leibniz Center for Informatics in Wadern, Germany (in February 2013 and June 2015) and an Academy Colloquium at the Royal Netherlands Academy of Arts and Sciences (March 2014). Altogether, these chapters provide the most up-to-date look at this developing and highly interdisciplinary field and will be of interest to a broad group of logicians, mathematicians, statisticians, philosophers, and scientists. Topics covered include a comprehensive survey of many propositional, modal, and first-order variants of dependence logic; new results concerning ... 10. Evolution operator equation: Integration with algebraic and finite difference methods. Applications to physical problems in classical and quantum mechanics and quantum field theory Energy Technology Data Exchange (ETDEWEB) Dattoli, Giuseppe; Torre, Amalia [ENEA, Centro Ricerche Frascati, Rome (Italy). Dipt. Innovazione; Ottaviani, Pier Luigi [ENEA, Centro Ricerche Bologna (Italy); Vasquez, Luis [Madris, Univ. Complutense (Spain). Dept. de Matemateca Aplicado 1997-10-01 The finite-difference based integration method for evolution-line equations is discussed in detail and framed within the general context of the evolution operator picture. Exact analytical methods are described to solve evolution-like equations in a quite general physical context. The numerical technique based on the factorization formulae of exponential operator is then illustrated and applied to the evolution-operator in both classical and quantum framework. Finally, the general view to the finite differencing schemes is provided, displaying the wide range of applications from the classical Newton equation of motion to the quantum field theory. 11. BRST field theory of relativistic particles International Nuclear Information System (INIS) Holten, J.W. van 1992-01-01 A generalization of BRST field theory is presented, based on wave operators for the fields constructed out of, but different from the BRST operator. The authors discuss their quantization, gauge fixing and the derivation of propagators. It is shown, that the generalized theories are relevant to relativistic particle theories in the Brink-Di Vecchia-Howe-Polyakov (BDHP) formulation, and argue that the same phenomenon holds in string theories. In particular it is shown, that the naive BRST formulation of the BDHP theory leads to trivial quantum field theories with vanishing correlation functions. (author). 22 refs 12. Large N Field Theories, String Theory and Gravity CERN Document Server Aharony, O; Maldacena, J M; Ooguri, H; Oz, Y 2000-01-01 We review the holographic correspondence between field theories and string/M theory, focusing on the relation between compactifications of string/M theory on Anti-de Sitter spaces and conformal field theories. We review the background for this correspondence and discuss its motivations and the evidence for its correctness. We describe the main results that have been derived from the correspondence in the regime that the field theory is approximated by classical or semiclassical gravity. We focus on the case of the N=4 supersymmetric gauge theory in four dimensions, but we discuss also field theories in other dimensions, conformal and non-conformal, with or without supersymmetry, and in particular the relation to QCD. We also discuss some implications for black hole physics. 13. Stochastic Gravity: Theory and Applications Directory of Open Access Journals (Sweden) Hu Bei Lok 2008-05-01 Full Text Available Whereas semiclassical gravity is based on the semiclassical Einstein equation with sources given by the expectation value of the stress-energy tensor of quantum fields, stochastic semiclassical gravity is based on the Einstein–Langevin equation, which has, in addition, sources due to the noise kernel. The noise kernel is the vacuum expectation value of the (operator-valued stress-energy bitensor, which describes the fluctuations of quantum-matter fields in curved spacetimes. A new improved criterion for the validity of semiclassical gravity may also be formulated from the viewpoint of this theory. In the first part of this review we describe the fundamentals of this new theory via two approaches: the axiomatic and the functional. The axiomatic approach is useful to see the structure of the theory from the framework of semiclassical gravity, showing the link from the mean value of the stress-energy tensor to the correlation functions. The functional approach uses the Feynman–Vernon influence functional and the Schwinger–Keldysh closed-time-path effective action methods. In the second part, we describe three applications of stochastic gravity. First, we consider metric perturbations in a Minkowski spacetime, compute the two-point correlation functions of these perturbations and prove that Minkowski spacetime is a stable solution of semiclassical gravity. Second, we discuss structure formation from the stochastic-gravity viewpoint, which can go beyond the standard treatment by incorporating the full quantum effect of the inflaton fluctuations. Third, using the Einstein–Langevin equation, we discuss the backreaction of Hawking radiation and the behavior of metric fluctuations for both the quasi-equilibrium condition of a black-hole in a box and the fully nonequilibrium condition of an evaporating black hole spacetime. Finally, we briefly discuss the theoretical structure of stochastic gravity in relation to quantum gravity and point out 14. On magnetohydrodynamic gauge field theory Science.gov (United States) Webb, G. M.; Anco, S. C. 2017-06-01 Clebsch potential gauge field theory for magnetohydrodynamics is developed based in part on the theory of Calkin (1963 Can. J. Phys. 41 2241-51). It is shown how the polarization vector {P} in Calkin’s approach naturally arises from the Lagrange multiplier constraint equation for Faraday’s equation for the magnetic induction {B} , or alternatively from the magnetic vector potential form of Faraday’s equation. Gauss’s equation, (divergence of {B} is zero) is incorporated in the variational principle by means of a Lagrange multiplier constraint. Noether’s theorem coupled with the gauge symmetries is used to derive the conservation laws for (a) magnetic helicity, (b) cross helicity, (c) fluid helicity for non-magnetized fluids, and (d) a class of conservation laws associated with curl and divergence equations which applies to Faraday’s equation and Gauss’s equation. The magnetic helicity conservation law is due to a gauge symmetry in MHD and not due to a fluid relabelling symmetry. The analysis is carried out for the general case of a non-barotropic gas in which the gas pressure and internal energy density depend on both the entropy S and the gas density ρ. The cross helicity and fluid helicity conservation laws in the non-barotropic case are nonlocal conservation laws that reduce to local conservation laws for the case of a barotropic gas. The connections between gauge symmetries, Clebsch potentials and Casimirs are developed. It is shown that the gauge symmetry functionals in the work of Henyey (1982 Phys. Rev. A 26 480-3) satisfy the Casimir determining equations. 15. A philosophical approach to quantum field theory CERN Document Server Öttinger, Hans Christian 2015-01-01 This text presents an intuitive and robust mathematical image of fundamental particle physics based on a novel approach to quantum field theory, which is guided by four carefully motivated metaphysical postulates. In particular, the book explores a dissipative approach to quantum field theory, which is illustrated for scalar field theory and quantum electrodynamics, and proposes an attractive explanation of the Planck scale in quantum gravity. Offering a radically new perspective on this topic, the book focuses on the conceptual foundations of quantum field theory and ontological questions. It also suggests a new stochastic simulation technique in quantum field theory which is complementary to existing ones. Encouraging rigor in a field containing many mathematical subtleties and pitfalls this text is a helpful companion for students of physics and philosophers interested in quantum field theory, and it allows readers to gain an intuitive rather than a formal understanding. 16. A philosophical approach to quantum field theory CERN Document Server Öttinger, Hans Christian 2017-01-01 This text presents an intuitive and robust mathematical image of fundamental particle physics based on a novel approach to quantum field theory, which is guided by four carefully motivated metaphysical postulates. In particular, the book explores a dissipative approach to quantum field theory, which is illustrated for scalar field theory and quantum electrodynamics, and proposes an attractive explanation of the Planck scale in quantum gravity. Offering a radically new perspective on this topic, the book focuses on the conceptual foundations of quantum field theory and ontological questions. It also suggests a new stochastic simulation technique in quantum field theory which is complementary to existing ones. Encouraging rigor in a field containing many mathematical subtleties and pitfalls this text is a helpful companion for students of physics and philosophers interested in quantum field theory, and it allows readers to gain an intuitive rather than a formal understanding. 17. Generalized locally Toeplitz sequences theory and applications CERN Document Server Garoni, Carlo 2017-01-01 Based on their research experience, the authors propose a reference textbook in two volumes on the theory of generalized locally Toeplitz sequences and their applications. This first volume focuses on the univariate version of the theory and the related applications in the unidimensional setting, while the second volume, which addresses the multivariate case, is mainly devoted to concrete PDE applications. This book systematically develops the theory of generalized locally Toeplitz (GLT) sequences and presents some of its main applications, with a particular focus on the numerical discretization of differential equations (DEs). It is the first book to address the relatively new field of GLT sequences, which occur in numerous scientific applications and are especially dominant in the context of DE discretizations. Written for applied mathematicians, engineers, physicists, and scientists who (perhaps unknowingly) encounter GLT sequences in their research, it is also of interest to those working in the fields of... 18. Particles, fields and quantum theory International Nuclear Information System (INIS) Bongaarts, P.J.M. 1982-01-01 The author gives an introduction to the development of gauge theories of the fundamental interactions. Starting from classical mechanics and quantum mechanics the development of quantum electrodynamics and non-abelian gauge theories is described. (HSI) 19. Toward a gauge field theory of gravity. Science.gov (United States) Yilmaz, H. Joint use of two differential identities (Bianchi and Freud) permits a gauge field theory of gravity in which the gravitational energy is localizable. The theory is compatible with quantum mechanics and is experimentally viable. 20. Towards weakly constrained double field theory Directory of Open Access Journals (Sweden) Kanghoon Lee 2016-08-01 Full Text Available We show that it is possible to construct a well-defined effective field theory incorporating string winding modes without using strong constraint in double field theory. We show that X-ray (Radon transform on a torus is well-suited for describing weakly constrained double fields, and any weakly constrained fields are represented as a sum of strongly constrained fields. Using inverse X-ray transform we define a novel binary operation which is compatible with the level matching constraint. Based on this formalism, we construct a consistent gauge transform and gauge invariant action without using strong constraint. We then discuss the relation of our result to the closed string field theory. Our construction suggests that there exists an effective field theory description for massless sector of closed string field theory on a torus in an associative truncation. 1. Gauge field theories: various mathematical approaches OpenAIRE Jordan, François; Serge, Lazzarini; Thierry, Masson 2014-01-01 This paper presents relevant modern mathematical formulations for (classical) gauge field theories, namely, ordinary differential geometry, noncommutative geometry, and transitive Lie algebroids. They provide rigorous frameworks to describe Yang-Mills-Higgs theories or gravitation theories, and each of them improves the paradigm of gauge field theories. A brief comparison between them is carried out, essentially due to the various notions of connection. However they reveal a compelling common... 2. Stochastic Gravity: Theory and Applications Directory of Open Access Journals (Sweden) Hu Bei Lok 2004-01-01 Full Text Available Whereas semiclassical gravity is based on the semiclassical Einstein equation with sources given by the expectation value of the stress-energy tensor of quantum fields, stochastic semiclassical gravity is based on the Einstein-Langevin equation, which has in addition sources due to the noise kernel. The noise kernel is the vacuum expectation value of the (operator-valued stress-energy bi-tensor which describes the fluctuations of quantum matter fields in curved spacetimes. In the first part, we describe the fundamentals of this new theory via two approaches: the axiomatic and the functional. The axiomatic approach is useful to see the structure of the theory from the framework of semiclassical gravity, showing the link from the mean value of the stress-energy tensor to their correlation functions. The functional approach uses the Feynman-Vernon influence functional and the Schwinger-Keldysh closed-time-path effective action methods which are convenient for computations. It also brings out the open systems concepts and the statistical and stochastic contents of the theory such as dissipation, fluctuations, noise, and decoherence. We then focus on the properties of the stress-energy bi-tensor. We obtain a general expression for the noise kernel of a quantum field defined at two distinct points in an arbitrary curved spacetime as products of covariant derivatives of the quantum field's Green function. In the second part, we describe three applications of stochastic gravity theory. First, we consider metric perturbations in a Minkowski spacetime. We offer an analytical solution of the Einstein-Langevin equation and compute the two-point correlation functions for the linearized Einstein tensor and for the metric perturbations. Second, we discuss structure formation from the stochastic gravity viewpoint, which can go beyond the standard treatment by incorporating the full quantum effect of the inflaton fluctuations. Third, we discuss the backreaction 3. Quantum field theory in gravitational background International Nuclear Information System (INIS) Narnhofer, H. 1986-01-01 The author suggests ignoring the influence of the quantum field on the gravitation as the first step to combine quantum field theory and gravitation theory, but to consider the gravitational field as fixed and thus study quantum field theory on a manifold. This subject evoked interest when thermal radiation of a black hole was predicted. The author concentrates on the free quantum field and can split the problem into two steps: the Weyl-algebra of the free field and the Wightman functional on the tangent space 4. The general theory of quantized fields in the 1950s International Nuclear Information System (INIS) Wightman, A.S. 1989-01-01 This review describes developments in theoretical particle physics in the 1950s which were important in the race to develop a putative general theory of quantized fields, especially ideas that offered a mathematically rigorous theory. Basic theoretical concepts then available included the Hamiltonian formulation of quantum dynamics, canonical quantization, perturbative renormalization theory and the theory of distributions. Following a description of various important theoretical contributions of this era, the review ends with a summary of the most important contributions of axiomatic field theory to concrete physics applications. (UK) 5. Analytic aspects of rational conformal field theories International Nuclear Information System (INIS) Kiritsis, E.B.; Lawrence Berkeley Lab., CA 1990-01-01 The problem of deriving linear differential equations for correlation functions of Rational Conformal Field Theories is considered. Techniques from the theory of fuchsian differential equations are used to show that knowledge of the central charge, dimensions of primary fields and fusion rules are enough to fix the differential equations for one- and two-point functions on the tours. Any other correlation function can be calculated along similar lines. The results settle the issue of 'exact solution' of rational conformal field theories. (orig.) 6. Nonequilibrium molecular dynamics theory, algorithms and applications CERN Document Server Todd, Billy D 2017-01-01 Written by two specialists with over twenty-five years of experience in the field, this valuable text presents a wide range of topics within the growing field of nonequilibrium molecular dynamics (NEMD). It introduces theories which are fundamental to the field - namely, nonequilibrium statistical mechanics and nonequilibrium thermodynamics - and provides state-of-the-art algorithms and advice for designing reliable NEMD code, as well as examining applications for both atomic and molecular fluids. It discusses homogenous and inhomogenous flows and pays considerable attention to highly confined fluids, such as nanofluidics. In addition to statistical mechanics and thermodynamics, the book covers the themes of temperature and thermodynamic fluxes and their computation, the theory and algorithms for homogenous shear and elongational flows, response theory and its applications, heat and mass transport algorithms, applications in molecular rheology, highly confined fluids (nanofluidics), the phenomenon of slip and... 7. Nonlinear boundary value problems in quantum field theory International Nuclear Information System (INIS) 1989-01-01 We discuss the general structure of a quantum field theory which is free in the interior of a bounded set B of R n . It is shown how to recover the field theory in the interior of B from a certain quantum field theory on the boundary. With an application to string theory in mind, we discuss the example where B equals an interval and the boundary value problem is given in terms of a euclidean functional integral with a P(var phi) interaction restricted to the boundary. copyright 1989 Academic Press, Inc 8. Alternative approaches to maximally supersymmetric field theories International Nuclear Information System (INIS) Broedel, Johannes 2010-01-01 The central objective of this work is the exploration and application of alternative possibilities to describe maximally supersymmetric field theories in four dimensions: N=4 super Yang-Mills theory and N=8 supergravity. While twistor string theory has been proven very useful in the context of N=4 SYM, no analogous formulation for N=8 supergravity is available. In addition to the part describing N=4 SYM theory, twistor string theory contains vertex operators corresponding to the states of N=4 conformal supergravity. Those vertex operators have to be altered in order to describe (non-conformal) Einstein supergravity. A modified version of the known open twistor string theory, including a term which breaks the conformal symmetry for the gravitational vertex operators, has been proposed recently. In a first part of the thesis structural aspects and consistency of the modified theory are discussed. Unfortunately, the majority of amplitudes can not be constructed, which can be traced back to the fact that the dimension of the moduli space of algebraic curves in twistor space is reduced in an inconsistent manner. The issue of a possible finiteness of N=8 supergravity is closely related to the question of the existence of valid counterterms in the perturbation expansion of the theory. In particular, the coefficient in front of the so-called R 4 counterterm candidate has been shown to vanish by explicit calculation. This behavior points into the direction of a symmetry not taken into account, for which the hidden on-shell E 7(7) symmetry is the prime candidate. The validity of the so-called double-soft scalar limit relation is a necessary condition for a theory exhibiting E 7(7) symmetry. By calculating the double-soft scalar limit for amplitudes derived from an N=8 supergravity action modified by an additional R 4 counterterm, one can test for possible constraints originating in the E 7(7) symmetry. In a second part of the thesis, the appropriate amplitudes are calculated 9. Strings - Links between conformal field theory, gauge theory and gravity International Nuclear Information System (INIS) Troost, J. 2009-05-01 String theory is a candidate framework for unifying the gauge theories of interacting elementary particles with a quantum theory of gravity. The last years we have made considerable progress in understanding non-perturbative aspects of string theory, and in bringing string theory closer to experiment, via the search for the Standard Model within string theory, but also via phenomenological models inspired by the physics of strings. Despite these advances, many deep problems remain, amongst which a non-perturbative definition of string theory, a better understanding of holography, and the cosmological constant problem. My research has concentrated on various theoretical aspects of quantum theories of gravity, including holography, black holes physics and cosmology. In this Habilitation thesis I have laid bare many more links between conformal field theory, gauge theory and gravity. Most contributions were motivated by string theory, like the analysis of supersymmetry preserving states in compactified gauge theories and their relation to affine algebras, time-dependent aspects of the holographic map between quantum gravity in anti-de-Sitter space and conformal field theories in the bulk, the direct quantization of strings on black hole backgrounds, the embedding of the no-boundary proposal for a wave-function of the universe in string theory, a non-rational Verlinde formula and the construction of non-geometric solutions to supergravity 10. Singularity theory and N = 2 superconformal field theories International Nuclear Information System (INIS) Warner, N.P. 1989-01-01 The N = 2 superconformal field theories that appear at the fixed points of the renormalization group flows of Landau-Ginsburg models are discussed. Some of the techniques of singularity theory are employed to deduce properties of these superconformal theories. These ideas are then used to deduce the relationship between Calabi-Yau compactifications and tensored discrete series models. The chiral rings of general N = 2 superconformal theories are also described. 14 refs 11. Advanced electromagnetism foundations, theory and applications CERN Document Server Barrett, Terence W 1995-01-01 Advanced Electromagnetism: Foundations, Theory and Applications treats what is conventionally called electromagnetism or Maxwell's theory within the context of gauge theory or Yang-Mills theory. A major theme of this book is that fields are not stand-alone entities but are defined by their boundary conditions. The book has practical relevance to efficient antenna design, the understanding of forces and stresses in high energy pulses, ring laser gyros, high speed computer logic elements, efficient transfer of power, parametric conversion, and many other devices and systems. Conventional electro 12. Algebraic quantum field theory, perturbation theory, and the loop expansion International Nuclear Information System (INIS) Duetsch, M.; Fredenhagen, K. 2001-01-01 The perturbative treatment of quantum field theory is formulated within the framework of algebraic quantum field theory. We show that the algebra of interacting fields is additive, i.e. fully determined by its subalgebras associated to arbitrary small subregions of Minkowski space. We also give an algebraic formulation of the loop expansion by introducing a projective system A (n) of observables ''up to n loops'', where A (0) is the Poisson algebra of the classical field theory. Finally we give a local algebraic formulation for two cases of the quantum action principle and compare it with the usual formulation in terms of Green's functions. (orig.) 13. Operator algebras and conformal field theory International Nuclear Information System (INIS) Gabbiani, F.; Froehlich, J. 1993-01-01 We define and study two-dimensional, chiral conformal field theory by the methods of algebraic field theory. We start by characterizing the vacuum sectors of such theories and show that, under very general hypotheses, their algebras of local observables are isomorphic to the unique hyperfinite type III 1 factor. The conformal net determined by the algebras of local observables is proven to satisfy Haag duality. The representation of the Moebius group (and presumably of the entire Virasoro algebra) on the vacuum sector of a conformal field theory is uniquely determined by the Tomita-Takesaki modular operators associated with its vacuum state and its conformal net. We then develop the theory of Mebius covariant representations of a conformal net, using methods of Doplicher, Haag and Roberts. We apply our results to the representation theory of loop groups. Our analysis is motivated by the desire to find a 'background-independent' formulation of conformal field theories. (orig.) 14. Fundamental number theory with applications CERN Document Server Mollin, Richard A 2008-01-01 An update of the most accessible introductory number theory text available, Fundamental Number Theory with Applications, Second Edition presents a mathematically rigorous yet easy-to-follow treatment of the fundamentals and applications of the subject. The substantial amount of reorganizing makes this edition clearer and more elementary in its coverage. New to the Second Edition Removal of all advanced material to be even more accessible in scope New fundamental material, including partition theory, generating functions, and combinatorial number theory Expa 15. Classical gravity and quantum matter fields in unified field theory Science.gov (United States) von Borzeszkowski, Horst-Heino; Treder, Hans-Jürgen 1996-01-01 The Einstein-Schrödinger purely affine field theory of the non-symmetric field provides canonical field equations without constraints. These equations imply the Heisenberg-Pauli commutation rules of quantum field theory. In the Schrödinger gauging of the Einstein field coordinatesU {/kl i }=Γ{/kl i }-δ{/l i }Γ{/km m }, this unified geometric field theory becomes a model of the coupling between a quantized Maxwellian field in a medium and classical gravity. Therefore, independently of the question as to the physical truth of this model, its analysis performed in the present paper demonstrates that, in the framework of a quantized unified field theory, gravity can appear as a genuinely classical field. 16. Applications of polyfold theory I CERN Document Server Hofer, H; Zehnder, E 2017-01-01 In this paper the authors start with the construction of the symplectic field theory (SFT). As a general theory of symplectic invariants, SFT has been outlined in Introduction to symplectic field theory (2000), by Y. Eliashberg, A. Givental and H. Hofer who have predicted its formal properties. The actual construction of SFT is a hard analytical problem which will be overcome be means of the polyfold theory due to the present authors. The current paper addresses a significant amount of the arising issues and the general theory will be completed in part II of this paper. To illustrate the polyfold theory the authors use the results of the present paper to describe an alternative construction of the Gromov-Witten invariants for general compact symplectic manifolds. 17. Singularity Theory and its Applications CERN Document Server Stewart, Ian; Mond, David; Montaldi, James 1991-01-01 A workshop on Singularities, Bifuraction and Dynamics was held at Warwick in July 1989, as part of a year-long symposium on Singularity Theory and its applications. The proceedings fall into two halves: Volume I mainly on connections with algebraic geometry and volume II on connections with dynamical systems theory, bifurcation theory and applications in the sciences. The papers are original research, stimulated by the symposium and workshop: All have been refereed and none will appear elsewhere. The main topic of volume II is new methods for the study of bifurcations in nonlinear dynamical systems, and applications of these. 18. Mathematical aspects of quantum field theories CERN Document Server Strobl, Thomas 2015-01-01 Despite its long history and stunning experimental successes, the mathematical foundation of perturbative quantum field theory is still a subject of ongoing research. This book aims at presenting some of the most recent advances in the field, and at reflecting the diversity of approaches and tools invented and currently employed. Both leading experts and comparative newcomers to the field present their latest findings, helping readers to gain a better understanding of not only quantum but also classical field theories. Though the book offers a valuable resource for mathematicians and physicists alike, the focus is more on mathematical developments. This volume consists of four parts: The first Part covers local aspects of perturbative quantum field theory, with an emphasis on the axiomatization of the algebra behind the operator product expansion. The second Part highlights Chern-Simons gauge theories, while the third examines (semi-)classical field theories. In closing, Part 4 addresses factorization homolo... 19. Lectures on classical and quantum theory of fields CERN Document Server Arodz, Henryk 2017-01-01 This textbook addresses graduate students starting to specialize in theoretical physics. It provides didactic introductions to the main topics in the theory of fields, while taking into account the contemporary view of the subject. The student will find concise explanations of basic notions essential for applications of the theory of fields as well as for frontier research in theoretical physics. One third of the book is devoted to classical fields. Each chapter contains exercises of varying degree of difficulty with hints or solutions, plus summaries and worked examples as useful. It aims to deliver a unique combination of classical and quantum field theory in one compact course. 20. New results in topological field theory and Abelian gauge theory International Nuclear Information System (INIS) Thompson, G. 1995-10-01 These are the lecture notes of a set of lectures delivered at the 1995 Trieste summer school in June. I review some recent work on duality in four dimensional Maxwell theory on arbitrary four manifolds, as well as a new set of topological invariants known as the Seiberg-Witten invariants. Much of the necessary background material is given, including a crash course in topological field theory, cohomology of manifolds, topological gauge theory and the rudiments of four manifold theory. My main hope is to wet the readers appetite, so that he or she will wish to read the original works and perhaps to enter this field. (author). 41 refs, 5 figs 1. Theory and computation of the matrix elements of the full interaction of the electromagnetic field with an atomic state: Application to the Rydberg and the continuous spectrum International Nuclear Information System (INIS) Komninos, Yannis; Mercouris, Theodoros; Nicolaides, Cleanthes A. 2002-01-01 We develop practical formulas for the calculation of the matrix elements of the interaction of the electromagnetic field with an atomic state, beyond the long-wavelength approximation. The atom-plus-field Hamiltonian is chosen to have the multipolar form, containing the electric, paramagnetic, and diamagnetic operators. The final workable expressions include the interactions to all orders and are derived by first expanding the fields in partial waves. The electric-field operator reaches a constant value as the radial variable becomes large, contrary to the result of the electric-dipole approximation (EDA) where the value of the corresponding operator increases indefinitely. Applications are given for Rydberg states of hydrogen up to n=50 and for free-free transitions in a Coulomb potential. Such matrix elements are relevant to a number of real and virtual processes occurring during laser-atom interactions. The computation is done numerically, using a combination of analytic with numerical techniques. By comparing the results of the EDA with those of the exact treatment, it is shown that the former is inadequate in such cases. This finding has repercussions on the theory and understanding of the physics of quantum systems in high-lying Rydberg levels and wave packets or in scattering states 2. Introduction to algebraic quantum field theory International Nuclear Information System (INIS) Horuzhy, S.S. 1990-01-01 This volume presents a systematic introduction to the algebraic approach to quantum field theory. The structure of the contents corresponds to the way the subject has advanced. It is shown how the algebraic approach has developed from the purely axiomatic theory of observables via superselection rules into the dynamical formalism of fields and observables. Chapter one discusses axioms and their consequences -many of which are now classical theorems- and deals, in general, with the axiomatic theory of local observable algebras. The absence of field concepts makes this theory incomplete and, in chapter two, superselection rules are shown to be the key to the reconstruction of fields from observables. Chapter three deals with the algebras of Wightman fields, first unbounded operator algebras, then Von Neumann field algebras (with a special section on wedge region algebras) and finally local algebras of free and generalised free fields. (author). 447 refs.; 4 figs 3. Using field theory in hadron physics International Nuclear Information System (INIS) Abarbanel, H.D.I. 1978-03-01 Topics are covered on the connection of field theory and hadron physics. The renormalization group and infrared and ultraviolet limits of field theory, in particular quantum chromodynamics, spontaneous mass generation, color confinement, instantons, and the vacuum state in quantum chromodynamics are treated. 21 references 4. Calculations in perturbative string field theory International Nuclear Information System (INIS) Thorn, C.B. 1987-01-01 The author discusses methods for evaluating the Feynman diagrams of string field theory, with particular emphasis on Witten's version of open string field theory. It is explained in some detail how the rules states by Giddings and Martinec for relating a given diagram to a Polyakov path integral emerge from the Feynman rules 5. Using field theory in hadron physics Energy Technology Data Exchange (ETDEWEB) Abarbanel, H.D.I. 1978-03-01 Topics are covered on the connection of field theory and hadron physics. The renormalization group and infrared and ultraviolet limits of field theory, in particular quantum chromodynamics, spontaneous mass generation, color confinement, instantons, and the vacuum state in quantum chromodynamics are treated. 21 references. (JFP) 6. Semiclassical Quantization of Classical Field Theories NARCIS (Netherlands) Cattaneo, A.; Mnev, P.; Reshetikhin, N.; Calaque, D.; Strobi, Th. 2015-01-01 Abstract These lectures are an introduction to formal semiclassical quantization of classical field theory. First we develop the Hamiltonian formalism for classical field theories on space time with boundary. It does not have to be a cylinder as in the usual Hamiltonian framework. Then we outline 7. Neoclassical Theory and Its Applications Energy Technology Data Exchange (ETDEWEB) Shaing, Ker-Chung [Univ. of Wisconsin, Madison, WI (United States) 2015-11-20 The grant entitled Neoclassical Theory and Its Applications started on January 15 2001 and ended on April 14 2015. The main goal of the project is to develop neoclassical theory to understand tokamak physics, and employ it to model current experimental observations and future thermonuclear fusion reactors. The PI had published more than 50 papers in refereed journals during the funding period. 8. On the interplay between string theory and field theory International Nuclear Information System (INIS) Brunner, I. 1998-01-01 In this thesis, we have discussed various aspects of branes in string theory and M-theory. In chapter 2 we were able to construct six-dimensional chiral interacting eld theories from Hanany-Witten like brane setups. The field theory requirement that the anomalies cancel was reproduced by RR-charge conservation in the brane setup. The data of the Hanany-Witten setup, which consists of brane positions, was mapped to instanton data. The orbifold construction can be extended to D and E type singularities. In chapter 3 we discussed a matrix conjecture, which claims that M-theory in the light cone gauge is described by the quantum mechanics of D0 branes. Toroidal compactifications of M-theory have a description in terms of super Yang-Mills theory an the dual torus. For more than three compactified dimensions, more degrees of freedom have to be added. In some sense, the philosophy in this chapter is orthogonal to the previous chapter: Here, we want to get M-theory results from eld theory considerations, whereas in the previous chapter we obtained eld theory results by embedding the theories in string theory. Our main focus was on the compactification on T 6 , which leads to complications. Here, the Matrix model is again given by an eleven dimensional theory, not by a lower dimensional field theory. Other problems and possible resolutions of Matrix theory are discussed at the end of chapter 3. In the last chapter we considered M- and F-theory compactifications on Calabi-Yau fourfolds. After explaining some basics of fourfolds, we showed that the web of fourfolds is connected by singular transitions. The two manifolds which are connected by the transition are different resolutions of the same singular manifold. The resolution of the singularities can lead to a certain type of divisors, which lead to non-perturbative superpotentials, when branes wrap them. The vacua connected by the transitions can be physically very different. (orig.) 9. Quantum Field Theory in a Semiotic Perspective CERN Document Server Günter Dosch, Hans; Sieroka, Norman 2005-01-01 Viewing physical theories as symbolic constructions came to the fore in the middle of the nineteenth century with the emancipation of the classical theory of the electromagnetic field from mechanics; most notably this happened through the work of Helmholtz, Hertz, Poincaré, and later Weyl. The epistemological problems that nourished this development are today highlighted within quantum field theory. The present essay starts off with a concise and non-technical outline of the firmly based aspects of relativistic quantum field theory, i.e. the very successful description of subnuclear phenomena. The particular methods, by which these different aspects have to be accessed, then get described as distinct facets of quantum field theory. The authors show how these different facets vary with respect to the relation between quantum fields and associated particles. Thus, by emphasising the respective role of various basic concepts involved, the authors claim that only a very general epistemic approach can properly ac... 10. Introduction to field theory of strings International Nuclear Information System (INIS) Kikkawa, K. 1987-01-01 The field theory of bosonic string is reviewed. First, theory is treated in a light-cone gauge. After a brief survey of the first quantized theory of free string, the second quantization is discussed. All possible interactions of strings are introduced based on a smoothness condition of work sheets swept out by strings. Perturbation theory is developed. Finally a possible way to the manifest covariant formalism is discussed 11. Variational Wigner-Kirkwood approach to relativistic mean field theory International Nuclear Information System (INIS) Del Estal, M.; Centelles, M.; Vinas, X. 1997-01-01 The recently developed variational Wigner-Kirkwood approach is extended to the relativistic mean field theory for finite nuclei. A numerical application to the calculation of the surface energy coefficient in semi-infinite nuclear matter is presented. The new method is contrasted with the standard density functional theory and the fully quantal approach. copyright 1997 The American Physical Society 12. Constrained variational calculus for higher order classical field theories International Nuclear Information System (INIS) Campos, Cedric M; De Leon, Manuel; De Diego, David MartIn 2010-01-01 We develop an intrinsic geometrical setting for higher order constrained field theories. As a main tool we use an appropriate generalization of the classical Skinner-Rusk formalism. Some examples of applications are studied, in particular to the geometrical description of optimal control theory for partial differential equations. 13. Schrodinger representation in renormalizable quantum field theory International Nuclear Information System (INIS) Symanzik, K. 1983-01-01 The problem of the Schrodinger representation arose from work on the Nambu-Goto Ansatz for integration over surfaces. Going beyond semiclassical approximation leads to two problems of nonrenormalizibility and of whether Dirichlet boundary conditions can be imposed on a ''Euclidean'' quantum field theory. The Schrodinger representation is constructed in a way where the principles of general renormalization theory can be refered to. The Schrodinger function of surface terms is studied, as well as behaviour at the boundary. The Schrodinger equation is derived. Completeness, unitarity, and computation of expectation values are considered. Extensions of these methods into other Bose field theories such as Fermi fields and Marjorana fields is straightforward 14. Local algebras in Euclidean quantum field theory International Nuclear Information System (INIS) Guerra, Francesco. 1975-06-01 The general structure of the local observable algebras of Euclidean quantum field theory is described, considering the very simple examples of the free scalar field, the vector meson field, and the electromagnetic field. The role of Markov properties, and the relations between Euclidean theory and Hamiltonian theory in Minkowski space-time are especially emphasized. No conflict appears between covariance (in the Euclidean sense) and locality (in the Markov sense) on one hand and positive definiteness of the metric on the other hand [fr 15. Random measures, theory and applications CERN Document Server Kallenberg, Olav 2017-01-01 Offering the first comprehensive treatment of the theory of random measures, this book has a very broad scope, ranging from basic properties of Poisson and related processes to the modern theories of convergence, stationarity, Palm measures, conditioning, and compensation. The three large final chapters focus on applications within the areas of stochastic geometry, excursion theory, and branching processes. Although this theory plays a fundamental role in most areas of modern probability, much of it, including the most basic material, has previously been available only in scores of journal articles. The book is primarily directed towards researchers and advanced graduate students in stochastic processes and related areas. 16. Mathematical aspects of quantum field theory CERN Document Server de Faria, Edson 2010-01-01 Over the last century quantum field theory has made a significant impact on the formulation and solution of mathematical problems and inspired powerful advances in pure mathematics. However, most accounts are written by physicists, and mathematicians struggle to find clear definitions and statements of the concepts involved. This graduate-level introduction presents the basic ideas and tools from quantum field theory to a mathematical audience. Topics include classical and quantum mechanics, classical field theory, quantization of classical fields, perturbative quantum field theory, renormalization, and the standard model. The material is also accessible to physicists seeking a better understanding of the mathematical background, providing the necessary tools from differential geometry on such topics as connections and gauge fields, vector and spinor bundles, symmetries and group representations. 17. Structural aspects of quantum field theory and noncommutative geometry CERN Document Server Grensing, Gerhard 2013-01-01 This book is devoted to the subject of quantum field theory. It is divided into two volumes. The first can serve as a textbook on the main techniques and results of quantum field theory, while the second treats more recent developments, in particular the subject of quantum groups and noncommutative geometry, and their interrelation. The first volume is directed at graduate students who want to learn the basic facts about quantum field theory. It begins with a gentle introduction to classical field theory, including the standard model of particle physics, general relativity, and also supergravity. The transition to quantized fields is performed with path integral techniques, by means of which the one-loop renormalization of a self-interacting scalar quantum field, of quantum electrodynamics, and the asymptotic freedom of quantum chromodynamics is treated. In the last part of the first volume, the application of path integral methods to systems of quantum statistical mechanics is covered. The book ends with a r... 18. Quantum scattering from classical field theory International Nuclear Information System (INIS) Gould, T.M.; Poppitz, E.R. 1995-01-01 We show that scattering amplitudes between initial wave packet states and certain coherent final states can be computed in a systematic weak coupling expansion about classical solutions satisfying initial-value conditions. The initial-value conditions are such as to make the solution of the classical field equations amenable to numerical methods. We propose a practical procedure for computing classical solutions which contribute to high energy two-particle scattering amplitudes. We consider in this regard the implications of a recent numerical simulation in classical SU(2) Yang-Mills theory for multiparticle scattering in quantum gauge theories and speculate on its generalization to electroweak theory. We also generalize our results to the case of complex trajectories and discuss the prospects for finding a solution to the resulting complex boundary value problem, which would allow the application of our method to any wave packet to coherent state transition. Finally, we discuss the relevance of these results to the issues of baryon number violation and multiparticle scattering at high energies. ((orig.)) 19. Supersymmetry and Duality in Field Theory and String Theory CERN Document Server Kiritsis, Elias B 1999-01-01 This is a set of lectures given at the 99' Cargese Summer School, "Particle Physics : Ideas and Recent Developments". They contain a pedestrian exposition of recent theoretical progress in non-perturbative field theory and string theory based on ideas of duality. 20. Introduction to conformal field theory and string theory International Nuclear Information System (INIS) Dixon, L.J. 1989-12-01 These lectures are meant to provide a brief introduction to conformal field theory (CFT) and string theory for those with no prior exposure to the subjects. There are many excellent reviews already available, and most of these go in to much more detail than I will be able to here. 52 refs., 11 figs 1. Introduction to conformal field theory and string theory Energy Technology Data Exchange (ETDEWEB) Dixon, L.J. 1989-12-01 These lectures are meant to provide a brief introduction to conformal field theory (CFT) and string theory for those with no prior exposure to the subjects. There are many excellent reviews already available, and most of these go in to much more detail than I will be able to here. 52 refs., 11 figs. 2. Application of the weak-field asymptotic theory to the analysis of tunneling ionization of linear molecules DEFF Research Database (Denmark) Madsen, Lars Bojer; Tolstikhin, Oleg I.; Morishita, Toru 2012-01-01 Hartree-Fock wave functions for the diatomics, and a Hartree-Fock quantum chemistry wave function for CO2. The structure factors are expanded in terms of standard functions and the associated structure coefficients, allowing the determination of the ionization rate for any orientation of the molecule...... with respect to the field, are tabulated. Our results, which are exact in the weak-field limit for H2+ and, in addition, under the Hartree-Fock approximation for the diatomics, are compared with results from the recent literature.... 3. String field theory in curved space International Nuclear Information System (INIS) Kikkawa, Keiji; Maeno, Masahiro; Sawada, Shiro 1988-01-01 The purely cubic action in the string field theory is shown to provide a set of equations of motion for background fields which agree to those obtained by the vanishing condition of β-functions in the non-linear sigma model. Using the sigma model as an auxiliary tool, a systematic method for solving the string field theory in curved space is proposed. (author) 4. Classical field theory on electrodynamics, non-Abelian gauge theories and gravitation CERN Document Server Scheck, Florian 2012-01-01 The book describes Maxwell's equations first in their integral, directly testable form, then moves on to their local formulation. The first two chapters cover all essential properties of Maxwell's equations, including their symmetries and their covariance in a modern notation. Chapter 3 is devoted to Maxwell theory as a classical field theory and to solutions of the wave equation. Chapter 4 deals with important applications of Maxwell theory. It includes topical subjects such as metamaterials with negative refraction index and solutions of Helmholtz' equation in paraxial approximation relevant for the description of laser beams. Chapter 5 describes non-Abelian gauge theories from a classical, geometric point of view, in analogy to Maxwell theory as a prototype, and culminates in an application to the U(2) theory relevant for electroweak interactions. The last chapter 6 gives a concise summary of semi-Riemannian geometry as the framework for the classical field theory of gravitation. The chapter concludes wit... 5. A Conceptual Application of Attachment Theory and Research to the Social Work Student-Field Instructor Supervisory Relationship Science.gov (United States) Bennett, Susanne; Saks, Loretta Vitale 2006-01-01 This article conceptualizes an attachment-based model of the student-field instructor relationship, based on empirical research concerning internal working models of attachment, which continue into adulthood and serve as templates for life-long relating. Supportive relationships within a noncritical context are salient for effective supervision;… 6. Representations of classical groups on the lattice and its application to the field theory on discrete space-time OpenAIRE Lorente, M. 2003-01-01 We explore the mathematical consequences of the assumption of a discrete space-time. The fundamental laws of physics have to be translated into the language of discrete mathematics. We find integral transformations that leave the lattice of any dimension invariant and apply these transformations to field equations. 7. Light-front quantization of field theory Energy Technology Data Exchange (ETDEWEB) Srivastava, Prem P. [Universidade do Estado, Rio de Janeiro, RJ (Brazil). Inst. de Fisica]\|[Centro Brasileiro de Pesquisas Fisicas (CBPF), Rio de Janeiro, RJ (Brazil) 1996-07-01 Some basic topics in Light-Front (LF) quantized field theory are reviewed. Poincare algebra and the LF spin operator are discussed. The local scalar field theory of the conventional framework is shown to correspond to a non-local Hamiltonian theory on the LF in view of the constraint equations on the phase space, which relate the bosonic condensates to the non-zero modes. This new ingredient is useful to describe the spontaneous symmetry breaking on the LF. The instability of the symmetric phase in two dimensional scalar theory when the coupling constant grows is shown in the LF theory renormalized to one loop order. Chern-Simons gauge theory, regarded to describe excitations with fractional statistics, is quantized in the light-cone gauge and a simple LF Hamiltonian obtained which may allow us to construct renormalized theory of anyons. (author). 20 refs. 8. Field theory of relativistic strings: I. Trees International Nuclear Information System (INIS) Kaku, M.; Kikkawa, K. 1985-01-01 The authors present an entirely new kind of field theory, a field theory quantized not at space-time points, but quantized along an extended set of multilocal points on a string. This represents a significant departure from the usual quantum field theory, whose free theory represents a definite set of elementary particles, because the field theory on relativistic strings can accommodate an infinite set of linearly rising Regge trajectories. In this paper, the authors (1) present canonical quantization and the Green's function of the free string, (2) introduce three-string interactions, (3) resolve the question of multiple counting, (4) complete the counting arguments for all N-point trees, and (5) introduce four-string interactions which yield a Yang-Mills structure when the zero-slope limit is taken 9. A Field Theory with Curvature and Anticurvature Directory of Open Access Journals (Sweden) M. I. Wanas 2014-01-01 Full Text Available The present work is an attempt to construct a unified field theory in a space with curvature and anticurvature, the PAP-space. The theory is derived from an action principle and a Lagrangian density using a symmetric linear parameterized connection. Three different methods are used to explore physical contents of the theory obtained. Poisson’s equations for both material and charge distributions are obtained, as special cases, from the field equations of the theory. The theory is a pure geometric one in the sense that material distribution, charge distribution, gravitational and electromagnetic potentials, and other physical quantities are defined in terms of pure geometric objects of the structure used. In the case of pure gravity in free space, the spherical symmetric solution of the field equations gives the Schwarzschild exterior field. The weak equivalence principle is respected only in the case of pure gravity in free space; otherwise it is violated. 10. Mass corrections in string theory and lattice field theory International Nuclear Information System (INIS) Del Debbio, Luigi; Kerrane, Eoin; Russo, Rodolfo 2009-01-01 Kaluza-Klein (KK) compactifications of higher-dimensional Yang-Mills theories contain a number of 4-dimensional scalars corresponding to the internal components of the gauge field. While at tree level the scalar zero modes are massless, it is well known that quantum corrections make them massive. We compute these radiative corrections at 1 loop in an effective field theory framework, using the background field method and proper Schwinger-time regularization. In order to clarify the proper treatment of the sum over KK modes in the effective field theory approach, we consider the same problem in two different UV completions of Yang-Mills: string theory and lattice field theory. In both cases, when the compactification radius R is much bigger than the scale of the UV completion (R>>√(α ' ), a), we recover a mass renormalization that is independent of the UV scale and agrees with the one derived in the effective field theory approach. These results support the idea that the value of the mass corrections is, in this regime, universal for any UV completion that respects locality and gauge invariance. The string analysis suggests that this property holds also at higher loops. The lattice analysis suggests that the mass of the adjoint scalars appearing in N=2, 4 super Yang-Mills is highly suppressed, even if the lattice regularization breaks all supersymmetries explicitly. This is due to an interplay between the higher-dimensional gauge invariance and the degeneracy of bosonic and fermionic degrees of freedom. 11. Fermion boson metamorphosis in field theory International Nuclear Information System (INIS) Ha, Y.K. 1982-01-01 In two-dimensional field theories many features are especially transparent if the Fermi fields are represented by non-local expressions of the Bose fields. Such a procedure is known as boson representation. Bilinear quantities appear in the Lagrangian of a fermion theory transform, however, as simple local expressions of the bosons so that the resulting theory may be written as a theory of bosons. Conversely, a theory of bosons may be transformed into an equivalent theory of fermions. Together they provide a basis for generating many interesting equivalences between theories of different types. In the present work a consistent scheme for constructing a canonical Fermi field in terms of a real scalar field is developed and such a procedure is valid and consistent with the tenets of quantum field theory is verified. A boson formulation offers a unifying theme in understanding the structure of many theories. This is illustrated by the boson formulation of a multifermion theory with chiral and internal symmetries. The nature of dynamical generation of mass when the theory undergoes boson transmutation and the preservation of continuous chiral symmetry in the massive case are examined. The dynamics of the system depends to a great extent on the specific number of fermions and different models of the same system can have very different properties. Many unusual symmetries of the fermion theory, such as hidden symmetry, duality and triality symmetries, are only manifest in the boson formulation. The underlying connections between some models with U(N) internal symmetry and another class of fermion models built with Majorana fermions which have O(2N) internal symmetry are uncovered 12. Elements of the theory of Markov processes and their applications CERN Document Server Bharucha-Reid, A T 2010-01-01 This graduate-level text and reference in probability, with numerous applications to several fields of science, presents nonmeasure-theoretic introduction to theory of Markov processes. The work also covers mathematical models based on the theory, employed in various applied fields. Prerequisites are a knowledge of elementary probability theory, mathematical statistics, and analysis. Appendixes. Bibliographies. 1960 edition. 13. Abelian gauge theories with tensor gauge fields International Nuclear Information System (INIS) Kapuscik, E. 1984-01-01 Gauge fields of arbitrary tensor type are introduced. In curved space-time the gravitational field serves as a bridge joining different gauge fields. The theory of second order tensor gauge field is developed on the basis of close analogy to Maxwell electrodynamics. The notion of tensor current is introduced and an experimental test of its detection is proposed. The main result consists in a coupled set of field equations representing a generalization of Maxwell theory in which the Einstein equivalence principle is not satisfied. (author) 14. Quantum field theory in curved space-times: with an application to the reduced model of deSitter universe International Nuclear Information System (INIS) Peter, I. J. 1995-06-01 The work deals with space-times with fixed background metric. The topics were arranged in a straight course, the first chapter collects basic facts on Lorentzian manifolds as time-orientability, causal structure, ... Further free neutral scalar fields and spinor fields described by the Klein-Gordon equation resp. the Dirac equation are dealt with. Having in mind the construction of the Weyl algebra and the Fermi algebra in the second chapter, it was put emphasis on the structure of the spaces of solutions of these equations: In the first case the space of solutions is a symplectic vector space in a canonical manner, in the second case a Hilbert space. It was made some effort to stay as general as possible. Most of the material in the second chapter already exists for several years, but it is largely scattered over various journal articles. In the third chapter the construction of a vacuum on the special example of deSitter universe is described. A close investigation of a recent work by J. Bros and U. Moschella made it possible to refine a result concerning temperature felt by an accelerated observer in deSitter space. The last part of this thesis is concerned with vacua for spinor fields on the two-dimensional deSitter universe. A procedure introduced by R. Haag, H. Narnhofer and U. Stein for four dimensional space-times does not seem to work in two dimensions. (author) 15. Effective theories of single field inflation when heavy fields matter CERN Document Server Achucarro, Ana; Hardeman, Sjoerd; Palma, Gonzalo A; Patil, Subodh P 2012-01-01 We compute the low energy effective field theory (EFT) expansion for single-field inflationary models that descend from a parent theory containing multiple other scalar fields. By assuming that all other degrees of freedom in the parent theory are sufficiently massive relative to the inflaton, it is possible to derive an EFT valid to arbitrary order in perturbations, provided certain generalized adiabaticity conditions are respected. These conditions permit a consistent low energy EFT description even when the inflaton deviates off its adiabatic minimum along its slowly rolling trajectory. By generalizing the formalism that identifies the adiabatic mode with the Goldstone boson of this spontaneously broken time translational symmetry prior to the integration of the heavy fields, we show that this invariance of the parent theory dictates the entire non-perturbative structure of the descendent EFT. The couplings of this theory can be written entirely in terms of the reduced speed of sound of adiabatic perturbat... 16. [Studies in quantum field theory]: Progress report International Nuclear Information System (INIS) Polmar, S.K. 1988-01-01 The theoretical physics group at Washington University has been devoted to the solution of problems in theoretical and mathematical physics. All of the personnel on this task have a similar approach to their research in that they apply sophisticated analytical and numerical techniques to problems primarily in quantum field theory. Specifically, this group has worked on quantum chromodynamics, classical Yang-Mills fields, chiral symmetry breaking condensates, lattice field theory, strong-coupling approximations, perturbation theory in large order, nonlinear waves, 1/N expansions, quantum solitons, phase transitions, nuclear potentials, and early universe calculations 17. Effective field theory for NN interactions International Nuclear Information System (INIS) Tran Duy Khuong; Vo Hanh Phuc 2003-01-01 The effective field theory of NN interactions is formulated and the power counting appropriate to this case is reviewed. It is more subtle than in most effective field theories since in the limit that the S-wave NN scattering lengths go to infinity. It is governed by nontrivial fixed point. The leading two body terms in the effective field theory for nucleon self interactions are scale invariant and invariant under Wigner SU(4) spin-isospin symmetry in this limit. Higher body terms with no derivatives (i.e. three and four body terms) are automatically invariant under Wigner symmetry. (author) 18. Clifford algebra in finite quantum field theories International Nuclear Information System (INIS) Moser, M. 1997-12-01 We consider the most general power counting renormalizable and gauge invariant Lagrangean density L invariant with respect to some non-Abelian, compact, and semisimple gauge group G. The particle content of this quantum field theory consists of gauge vector bosons, real scalar bosons, fermions, and ghost fields. We assume that the ultimate grand unified theory needs no cutoff. This yields so-called finiteness conditions, resulting from the demand for finite physical quantities calculated by the bare Lagrangean. In lower loop order, necessary conditions for finiteness are thus vanishing beta functions for dimensionless couplings. The complexity of the finiteness conditions for a general quantum field theory makes the discussion of non-supersymmetric theories rather cumbersome. Recently, the F = 1 class of finite quantum field theories has been proposed embracing all supersymmetric theories. A special type of F = 1 theories proposed turns out to have Yukawa couplings which are equivalent to generators of a Clifford algebra representation. These algebraic structures are remarkable all the more than in the context of a well-known conjecture which states that finiteness is maybe related to global symmetries (such as supersymmetry) of the Lagrangean density. We can prove that supersymmetric theories can never be of this Clifford-type. It turns out that these Clifford algebra representations found recently are a consequence of certain invariances of the finiteness conditions resulting from a vanishing of the renormalization group β-function for the Yukawa couplings. We are able to exclude almost all such Clifford-like theories. (author) 19. Bayreuther festspiele as a field for application of Peter Berger’s and Thomas Luckmann’s Theory of social construction of reality Directory of Open Access Journals (Sweden) Jeremić-Molnar Dragana 2006-01-01 Full Text Available The Stage festival in Bauyeuth (Bayreuther Festspiele, established in 1876. by German composer Richard Wagner (1813-1883, is, even nowadays, a complex and unique phenomenon which continually attracts the attention of scholars from various (mainly humanistic and social scientific fields. In many different methodological approaches to Bayreuther Festspiele, including those made by social scientists, one can not find the application of the sociological theory of Peter Berger and Thomas Luckmann. However, one has to bare in mind the important fact that Richard Wagner founded his completely innovative festive institution mainly in order to carry out and to spread his regenerative Weltanschauung - already formulated in his numerous theoretical writings and incorporated into his musical dramas. The fact that Wagner’s Weltanschauung was based on the idea of changing the reality of everyday life by constructing the new reality, is of equal importance. Considering all this, it becomes appropriate to explain Wagner’s motivation for establishing the stage festival, as well as his idea of festival, from the standpoint of the theory of social construction of reality. 20. Polynomial field theories and nonintegrability International Nuclear Information System (INIS) Euler, N.; Steeb, W.H.; Cyrus, K. 1990-01-01 The nonintegrability of the nonlinear field equation v ηξ = v 3 is studied with the help of the Painleve test. The condition at the resonance is discussed in detail. Particular solutions are given. (orig.) 1. Workshop on Thermal Field Theory to Neural Networks CERN Document Server Veneziano, Gabriele; Aurenche, Patrick 1996-01-01 Tanguy Altherr was a Fellow in the Theory Division at CERN, on leave from LAPP (CNRS) Annecy. At the time of his accidental death in July 1994, he was only 31.A meeting was organized at CERN, covering the various aspects of his scientific interests: thermal field theory and its applications to hot or dense media, neural networks and its applications to high energy data analysis. Speakers were among his closest collaborators and friends. 2. Graph theory with applications CERN Document Server Vasudev, C 2006-01-01 Salient Features Over 1500 problems are used to illustrate concepts, related to different topics, and introduce applications. Over 1000 exercises in the text with many different types of questions posed. Precise mathematical language is used without excessive formalism and abstraction. Care has been taken to balance the mix of notation and words in mathematical statements. Problem sets are stated clearly and unambiguously, and all are carefully graded for various levels of difficulty. This text has been carefully designed for flexible use. 3. Metric quantum field theory: A preliminary look International Nuclear Information System (INIS) Watson, W.N. 1988-01-01 Spacetime coordinates are involved in uncertainty relations; spacetime itself appears to exhibit curvature. Could the continua associated with field variables exhibit curvature? This question, as well as the ideas that (a) difficulties with quantum theories of gravitation may be due to their formulation in an incorrect analogy with other quantum field theories, (b) spacetime variables should not be any more basic than others for describing physical phenomena, and (c) if field continua do not exhibit curvature, the reasons would be of interest, motivated the formulation of a theory of variable curvature and torsion in the electromagnetic four-potential's reciprocal space. Curvature and torsion equation completely analogous to those for a gauge theory of gravitation (the Einstein-Cartan-Sciama-Kibble theory) are assumed for this continuum. The interaction-Hamiltonian density of this theory, to a first approximation, implies that in addition to the Maxwell-Dirac field interaction of ordinary quantum electrodynamics, there should also be an interaction between Dirac-field vector and pseudovector currents unmediated by photons, as well as other interactions involving two or three Dirac-field currents interacting with the Maxwell field at single spacetime events. Calculations expressing Bhabha-scattering cross sections for incident beams with parallel spins differ from those of unmodified quantum electrodynamics by terms of first order in the gravitational constant of the theory, but the corresponding cross section for unpolarized incident beams differs from that of the unmodified theory only by terms of higher order in that constant. Undesirable features of the present theory include its nonrenormalizability, the obscurity of the meaning of its inverse field operator, and its being based on electrodynamics rather than electroweak dynamics 4. Lectures on classical and quantum theory of fields Energy Technology Data Exchange (ETDEWEB) Arodz, Henryk; Hadasz, Leszek [Jagiellonian Univ., Krakow (Poland). Inst. Physics 2010-07-01 This textbook on classical and quantum theory of fields addresses graduate students starting to specialize in theoretical physics. It provides didactic introductions to the main topics in the theory of fields, while taking into account the contemporary view of the subject. The student will find concise explanations of basic notions essential for applications of the theory of fields as well as for frontier research in theoretical physics. One third of the book is devoted to classical fields. Each chapter contains exercises of varying degree of difficulty with hints or solutions, plus summaries and worked examples as useful. The textbook is based on lectures delivered to students of theoretical physics at Jagiellonian University. It aims to deliver a unique combination of classical and quantum field theory in one compact course. (orig.) 5. Rheology v.3 theory and applications CERN Document Server Eirich, Frederick 1960-01-01 Rheology: Theory and Applications, Volume 3 is a collection of articles contributed by experts in the field of rheology - the science of deformation and flow. This volume is composed of specialized chapters on the application of normal coordinate analysis to the theory of high polymers; principles of rheometry; and the rheology of cross-linked plastics, poly electrolytes, latexes, inks, pastes, and clay. Also included are a series of technological articles on lubrication, spinning, molding, extrusion, and adhesion and a survey of the general features of industrial rheology. Materials scientist 6. Solitons and their interactions in classical field theory International Nuclear Information System (INIS) Belova, T.I.; Kudryavtsev, A.E. 1997-01-01 Effects of nonlinearity in the classical field theory for non-integrated systems are considered, such as soliton scattering, soliton bound states, the fractal nature of resonant structures, kink scattering by inhomogeneities, and domain bladder collapse. The results are presented in both (1 + 1) and higher dimensions. Both neutral and charged scalar fields are considered. Possible applications areas for the nonlinearity effects are discussed 7. Global integrability of field theories. Proceedings International Nuclear Information System (INIS) Calmet, J.; Seiler, W.M.; Tucker, R.W. 2006-01-01 The GIFT 2006 workshop covers topics related to the Global Integration of Field Theories. These topics span several domains of science including Mathematics, Physics and Computer Science. It is indeed an interdisciplinary event and this feature is well illustrated by the diversity of papers presented at the workshop. Physics is our main target. A simple approach would be to state that we investigate systems of partial differential equations since it is widely believed that they provide a fair description of our world. The questions whether this world is Einsteinian or not, is described by String Theory or not are not however on our agenda. At this stage we have defined what we mean with field theories. To assess what global integrability means we surf on the two other domains of our interest. Mathematics delivers the main methodologies and tools to achieve our goal. It is a trivial remark to say that there exists several approaches to investigate the concept of integrability. Only selected ones are to be found in these proceedings. We do not try to define precisely what global integrability means. Instead, we only suggest two tracks. The first one is by analogy with the design of algorithms, in Computer Algebra or Computer Science, to solve systems of differential equations. The case of ODEs is rather well understood since a constructive methodology exists. Although many experts claim that numerous results do exist to solve systems of PDEs, no constructive decision method exists. This is our first track. The second track follows directly since the real world is described by systems of PDEs, which are mainly non-linear ones. To be able to decide in such a case of the existence of solutions would increase immediately the scope of new technologies applicable to indus trial problems. It is this latter remark that led to the European NEST project with the same name. The GIFT project aims at making progresses in the investigation of field theories through the use of very 8. The conceptual framework of quantum field theory CERN Document Server Duncan, Anthony 2012-01-01 The book attempts to provide an introduction to quantum field theory emphasizing conceptual issues frequently neglected in more "utilitarian" treatments of the subject. The book is divided into four parts, entitled respectively "Origins", "Dynamics", "Symmetries", and "Scales". The emphasis is conceptual - the aim is to build the theory up systematically from some clearly stated foundational concepts - and therefore to a large extent anti-historical, but two historical Chapters ("Origins") are included to situate quantum field theory in the larger context of modern physical theories. The three remaining sections of the book follow a step by step reconstruction of this framework beginning with just a few basic assumptions: relativistic invariance, the basic principles of quantum mechanics, and the prohibition of physical action at a distance embodied in the clustering principle. The "Dynamics" section of the book lays out the basic structure of quantum field theory arising from the sequential insertion of quan... 9. Factorization algebras in quantum field theory CERN Document Server Costello, Kevin 2017-01-01 Factorization algebras are local-to-global objects that play a role in classical and quantum field theory which is similar to the role of sheaves in geometry: they conveniently organize complicated information. Their local structure encompasses examples like associative and vertex algebras; in these examples, their global structure encompasses Hochschild homology and conformal blocks. In this first volume, the authors develop the theory of factorization algebras in depth, but with a focus upon examples exhibiting their use in field theory, such as the recovery of a vertex algebra from a chiral conformal field theory and a quantum group from Abelian Chern-Simons theory. Expositions of the relevant background in homological algebra, sheaves and functional analysis are also included, thus making this book ideal for researchers and graduates working at the interface between mathematics and physics. 10. Magnetic fields, special relativity and potential theory elementary electromagnetic theory CERN Document Server Chirgwin, B H; Kilmister, C W 1972-01-01 Magnetic Fields, Special Relativity and Potential Theory is an introduction to electromagnetism, special relativity, and potential theory, with emphasis on the magnetic field of steady currents (magnetostatics). Topics covered range from the origin of the magnetic field and the magnetostatic scalar potential to magnetization, electromagnetic induction and magnetic energy, and the displacement current and Maxwell's equations. This volume is comprised of five chapters and begins with an overview of magnetostatics, followed by a chapter on the methods of solving potential problems drawn from elec 11. The Mie Theory Basics and Applications CERN Document Server Wriedt, Thomas 2012-01-01 This book presents in a concise way the Mie theory and its current applications. It begins with an overview of current theories, computational methods, experimental techniques, and applications of optics of small particles. There is also some biographic information on Gustav Mie, who published his famous paper on the colour of Gold colloids in 1908. The Mie solution for the light scattering of small spherical particles set the basis for more advanced scattering theories and today there are many methods to calculate light scattering and absorption for practically any shape and composition of particles. The optics of small particles is of interest in industrial, atmospheric, astronomic and other research. The book covers the latest developments in divers fields in scattering theory such as plasmon resonance, multiple scattering and optical force. 12. Thermo field dynamics: a quantum field theory at finite temperature International Nuclear Information System (INIS) Mancini, F.; Marinaro, M.; Matsumoto, H. 1988-01-01 A brief review of the theory of thermo field dynamics (TFD) is presented. TFD is introduced and developed by Umezawa and his coworkers at finite temperature. The most significant concept in TFD is that of a thermal vacuum which satisfies some conditions denoted as thermal state conditions. The TFD permits to reformulate theories at finite temperature. There is no need in an additional principle to determine particle distributions at T ≠ 0. Temperature and other macroscopic parameters are introduced in the definition of the vacuum state. All operator formalisms used in quantum field theory at T=0 are preserved, although the field degrees of freedom are doubled. 8 refs 13. Repeated sprints, high-intensity interval training, small-sided games: theory and application to field sports. Science.gov (United States) Hoffmann, James J; Reed, Jacob P; Leiting, Keith; Chiang, Chieh-Ying; Stone, Michael H 2014-03-01 Due to the broad spectrum of physical characteristics necessary for success in field sports, numerous training modalities have been used develop physical preparedness. Sports like rugby, basketball, lacrosse, and others require athletes to be not only strong and powerful but also aerobically fit and able to recover from high-intensity intermittent exercise. This provides coaches and sport scientists with a complex range of variables to consider when developing training programs. This can often lead to confusion and the misuse of training modalities, particularly in the development of aerobic and anaerobic conditioning. This review outlines the benefits and general adaptations to 3 commonly used and effective conditioning methods: high-intensity interval training, repeated-sprint training, and small-sided games. The goals and outcomes of these training methods are discussed, and practical implementations strategies for coaches and sport scientists are provided. 14. String theory inspired deformations of quantum field theories Science.gov (United States) Chiou, Dah-Wei In this dissertation, some extensions on field theories with deformations inspired by string theory are explored and their implications are investigated. These are: (i) noncommutative dipole field theory (DFT) and unitarity; (ii) three dimensional super Yang-Mills theory and mini-twistor string theory; (iii) massive super Yang-Mills theory and twistor string theory; and (iv) a deformation of twistor space and N = 4 super Yang-Mills theory with a chiral mass term. The DFT with fixed spacetime vectors ("dipole-vectors") is formulated for gauge theory coupled with a scalar field of adjoint charge. The argument for the violation of unitarity in field theories on a noncommutative spacetime is extended to the case of DFT: with a timelike dipole vector, 1-loop amplitudes are shown not to obey the optical theorem and thus violate unitarity. Likewise, a simple 0 + 1D quantum mechanical system with nonlocal potential of finite extent in time also gives violation of unitarity. Associated with D = 3 super Yang-Mills theory, the topological B-model is constructed for the twistor string theory, of which the target space is the (super-)mini-twistor space. As the D = 4 twistor space can be considered as a fibration over D = 3 mini-twistor space, the dimensional reduction from D = 4 to D = 3 is conducted to obtain the scattering amplitudes for D = 3 super Yang-Mills theory. The result shows that, analogous to the D = 4 case, the twistor transformed D = 3 amplitudes are supported on holomorphic curves in the (super-)mini-twistor space. Another alternative twistor description---Berkovits's open string theory---is also analyzed. By the prescription which interrelates Witten's B-model and Berkovits's open string theory, the dimensional reduction can be made for Berkovits's model as well, in which the enhanced R-symmetry Spin(7) is recognized, whereas only the subgroup SU(4) is manifest in the B-model. The extension of the twistor string theory by adding mass terms is then proposed and 15. Functional analysis theory and applications CERN Document Server Edwards, RE 2011-01-01 ""The book contains an enormous amount of information - mathematical, bibliographical and historical - interwoven with some outstanding heuristic discussions."" - Mathematical Reviews.In this massive graduate-level study, Emeritus Professor Edwards (Australian National University, Canberra) presents a balanced account of both the abstract theory and the applications of linear functional analysis. Written for readers with a basic knowledge of set theory, general topology, and vector spaces, the book includes an abundance of carefully chosen illustrative examples and excellent exercises at the 16. Lectures on interacting string field theory International Nuclear Information System (INIS) Jevicki, A. 1986-09-01 We give a detailed review of the current formulations of interacting string field theory. The historical development of the subject is taken beginning with the old dual resonance model theory. The light cone approach is reviewed in some detail with emphasis on conformal mapping techniques. Witten's covariant approach is presented. The main body of the lectures concentrates on developing the operator formulation of Witten's theory. 38 refs., 22 figs., 5 tabs 17. Dynamical symmetry breaking in quantum field theories CERN Document Server 1993-01-01 The phenomenon of dynamical symmetry breaking (DSB) in quantum field theory is discussed in a detailed and comprehensive way. The deep connection between this phenomenon in condensed matter physics and particle physics is emphasized. The realizations of DSB in such realistic theories as quantum chromodynamics and electroweak theory are considered. Issues intimately connected with DSB such as critical phenomenona and effective lagrangian approach are also discussed. 18. Recent progress in reggeon field theory International Nuclear Information System (INIS) Sugar, R.L. 1977-01-01 The present status of the pomeron theory in the reggeon field theory is summarized. For α 0 ( 0 -a bare intercept, αsub(oc) - a certain critical value) the theory is in a very good shape. It appears to satisfy both S and t-channel unitarity, and to avoid all of the decreases which plagued the simple pole model of the pomeron. For α 0 >αsub(oc) the situation is less clear 19. Pure field theories and MACSYMA algorithms Science.gov (United States) Ament, W. S. 1977-01-01 A pure field theory attempts to describe physical phenomena through singularity-free solutions of field equations resulting from an action principle. The physics goes into forming the action principle and interpreting specific results. Algorithms for the intervening mathematical steps are sketched. Vacuum general relativity is a pure field theory, serving as model and providing checks for generalizations. The fields of general relativity are the 10 components of a symmetric Riemannian metric tensor; those of the Einstein-Straus generalization are the 16 components of a nonsymmetric. Algebraic properties are exploited in top level MACSYMA commands toward performing some of the algorithms of that generalization. The light cone for the theory as left by Einstein and Straus is found and simplifications of that theory are discussed. 20. Axion topological field theory of topological superconductors Science.gov (United States) Qi, Xiao-Liang; Witten, Edward; Zhang, Shou-Cheng 2013-04-01 Topological superconductors are gapped superconductors with gapless and topologically robust quasiparticles propagating on the boundary. In this paper, we present a topological field theory description of three-dimensional time-reversal invariant topological superconductors. In our theory the topological superconductor is characterized by a topological coupling between the electromagnetic field and the superconducting phase fluctuation, which has the same form as the coupling of “axions” with an Abelian gauge field. As a physical consequence of our theory, we predict the level crossing induced by the crossing of special “chiral” vortex lines, which can be realized by considering s-wave superconductors in proximity with the topological superconductor. Our theory can also be generalized to the coupling with a gravitational field. 1. An introduction to relativistic quantum field theory CERN Document Server Schweber, Silvan S 1961-01-01 Complete, systematic, and self-contained, this text introduces modern quantum field theory. "Combines thorough knowledge with a high degree of didactic ability and a delightful style." - Mathematical Reviews. 1961 edition. 2. Mathematical game theory and applications CERN Document Server 2014-01-01 An authoritative and quantitative approach to modern game theory with applications from diverse areas including economics, political science, military science, and finance. Explores areas which are not covered in current game theory texts, including a thorough examination of zero-sum game.Provides introductory material to game theory, including bargaining, parlour games, sport, networking games and dynamic games.Explores Bargaining models, discussing new result such as resource distributions, buyer-seller instructions and reputation in bargaining models.Theoretical results are presented along 3. Quantum field theory with infinite component local fields as an alternative to the string theories International Nuclear Information System (INIS) Krasnikov, N.V. 1987-05-01 We show that the introduction of the infinite component local fields with higher order derivatives in the interaction makes the theory completely ultraviolet finite. For the γ 5 -anomalous theories the introduction of the infinite component field makes the theory renormalizable or superrenormalizable. (orig.) 4. The conceptual basis of Quantum Field Theory NARCIS (Netherlands) Hooft, G. 't 2005-01-01 Relativistic Quantum Field Theory is a mathematical scheme to describe the sub-atomic particles and forces. The basic starting point is that the axioms of Special Relativity on the one hand and those of Quantum Mechanics on the other, should be combined into one theory. The fundamental 5. Renormalizability of effective scalar field theory CERN Document Server Ball, R D 1994-01-01 We present a comprehensive discussion of the consistency of the effective quantum field theory of a single $Z_2$ symmetric scalar field. The theory is constructed from a bare Euclidean action which at a scale much greater than the particle's mass is constrained only by the most basic requirements; stability, finiteness, analyticity, naturalness, and global symmetry. We prove to all orders in perturbation theory the boundedness, convergence, and universality of the theory at low energy scales, and thus that the theory is perturbatively renormalizable in the sense that to a certain precision over a range of such scales it depends only on a finite number of parameters. We then demonstrate that the effective theory has a well defined unitary and causal analytic S--matrix at all energy scales. We also show that redundant terms in the Lagrangian may be systematically eliminated by field redefinitions without changing the S--matrix, and discuss the extent to which effective field theory and analytic S--matrix theory... 6. Klein Topological Field Theories from Group Representations Directory of Open Access Journals (Sweden) Sergey A. Loktev 2011-07-01 Full Text Available We show that any complex (respectively real representation of finite group naturally generates a open-closed (respectively Klein topological field theory over complex numbers. We relate the 1-point correlator for the projective plane in this theory with the Frobenius-Schur indicator on the representation. We relate any complex simple Klein TFT to a real division ring. 7. Electromagnetic Field Theory A Collection of Problems CERN Document Server Mrozynski, Gerd 2013-01-01 After a brief introduction into the theory of electromagnetic fields and the definition of the field quantities the book teaches the analytical solution methods of Maxwell’s equations by means of several characteristic examples. The focus is on static and stationary electric and magnetic fields, quasi stationary fields, and electromagnetic waves. For a deeper understanding, the many depicted field patterns are very helpful. The book offers a collection of problems and solutions which enable the reader to understand and to apply Maxwell’s theory for a broad class of problems including classical static problems right up to waveguide eigenvalue problems. Content Maxwell’s Equations - Electrostatic Fields - Stationary Current Distributions – Magnetic Field of Stationary Currents – Quasi Stationary Fields: Eddy Currents - Electromagnetic Waves Target Groups Advanced Graduate Students in Electrical Engineering, Physics, and related Courses Engineers and Physicists Authors Professor Dr.-Ing. Gerd Mrozynski... 8. Field theories with multiple fermionic excitations International Nuclear Information System (INIS) Crawford, J.P. 1978-01-01 The reason for the existence of the muon has been an enigma since its discovery. Since that time there has been a continuing proliferation of elementary particles. It is proposed that this proliferation of leptons and quarks is comprehensible if there are only four fundamental particles, the leptons ν/sub e/ and e - , and the quarks u and d. All other leptons and quarks are imagined to be excited states of these four fundamental entities. Attention is restricted to the charged leptons and the electromagnetic interactions only. A detailed study of a field theory in which there is only one fundamental charged fermionic field having two (or more) excitations is made. When the electromagnetic interactions are introduced and the theory is second quantized, under certain conditions this theory reproduces the S matrix obtained from usual OED. In this case no electromagnetic transitions are allowed. A leptonic charge operator is defined and a superselection rule for this leptonic charge is found. Unfortunately, the mass spectrum cannot be obtained. This theory has many renormalizable generalizations including non-abelian gauge theories, Yukawa-type theories, and Fermi-type theories. Under certain circumstances the Yukawa- and Fermi-type theories are finite in perturbation theory. It is concluded that there are no fundamental objections to having fermionic fields with more than one excitation 9. Simple recursion relations for general field theories International Nuclear Information System (INIS) Cheung, Clifford; Shen, Chia-Hsien; Trnka, Jaroslav 2015-01-01 On-shell methods offer an alternative definition of quantum field theory at tree-level, replacing Feynman diagrams with recursion relations and interaction vertices with a handful of seed scattering amplitudes. In this paper we determine the simplest recursion relations needed to construct a general four-dimensional quantum field theory of massless particles. For this purpose we define a covering space of recursion relations which naturally generalizes all existing constructions, including those of BCFW and Risager. The validity of each recursion relation hinges on the large momentum behavior of an n-point scattering amplitude under an m-line momentum shift, which we determine solely from dimensional analysis, Lorentz invariance, and locality. We show that all amplitudes in a renormalizable theory are 5-line constructible. Amplitudes are 3-line constructible if an external particle carries spin or if the scalars in the theory carry equal charge under a global or gauge symmetry. Remarkably, this implies the 3-line constructibility of all gauge theories with fermions and complex scalars in arbitrary representations, all supersymmetric theories, and the standard model. Moreover, all amplitudes in non-renormalizable theories without derivative interactions are constructible; with derivative interactions, a subset of amplitudes is constructible. We illustrate our results with examples from both renormalizable and non-renormalizable theories. Our study demonstrates both the power and limitations of recursion relations as a self-contained formulation of quantum field theory. 10. Phase-field-crystal dynamics for binary systems: Derivation from dynamical density functional theory, amplitude equation formalism, and applications to alloy heterostructures. Science.gov (United States) Huang, Zhi-Feng; Elder, K R; Provatas, Nikolas 2010-08-01 The dynamics of phase field crystal (PFC) modeling is derived from dynamical density functional theory (DDFT), for both single-component and binary systems. The derivation is based on a truncation up to the three-point direct correlation functions in DDFT, and the lowest order approximation using scale analysis. The complete amplitude equation formalism for binary PFC is developed to describe the coupled dynamics of slowly varying complex amplitudes of structural profile, zeroth-mode average atomic density, and system concentration field. Effects of noise (corresponding to stochastic amplitude equations) and species-dependent atomic mobilities are also incorporated in this formalism. Results of a sample application to the study of surface segregation and interface intermixing in alloy heterostructures and strained layer growth are presented, showing the effects of different atomic sizes and mobilities of alloy components. A phenomenon of composition overshooting at the interface is found, which can be connected to the surface segregation and enrichment of one of the atomic components observed in recent experiments of alloying heterostructures. 11. Classical theory of electric and magnetic fields CERN Document Server Good, Roland H 1971-01-01 Classical Theory of Electric and Magnetic Fields is a textbook on the principles of electricity and magnetism. This book discusses mathematical techniques, calculations, with examples of physical reasoning, that are generally applied in theoretical physics. This text reviews the classical theory of electric and magnetic fields, Maxwell's Equations, Lorentz Force, and Faraday's Law of Induction. The book also focuses on electrostatics and the general methods for solving electrostatic problems concerning images, inversion, complex variable, or separation of variables. The text also explains ma 12. Best matching theory & applications CERN Document Server 2017-01-01 Mismatch or best match? This book demonstrates that best matching of individual entities to each other is essential to ensure smooth conduct and successful competitiveness in any distributed system, natural and artificial. Interactions must be optimized through best matching in planning and scheduling, enterprise network design, transportation and construction planning, recruitment, problem solving, selective assembly, team formation, sensor network design, and more. Fundamentals of best matching in distributed and collaborative systems are explained by providing: § Methodical analysis of various multidimensional best matching processes § Comprehensive taxonomy, comparing different best matching problems and processes § Systematic identification of systems’ hierarchy, nature of interactions, and distribution of decision-making and control functions § Practical formulation of solutions based on a library of best matching algorithms and protocols, ready for direct applications and apps development. Design... 13. Unified-field theory: yesterday, today, tomorrow International Nuclear Information System (INIS) Bergman, P.G. 1982-01-01 Beginning with the expounding of Einstein understanding of advantages and disadvantages of general relativity theory, the authors proceed to consideration of what the complete unified theory have to be according to Einstein. The four theories which can be considered as ''unified'', namely weyl and Calutsa ones, worked out a half of century ago, and twistor twisting and supersymmetry theories, nowadays attracting attention, are briefly described and discussed. The authors come to a conclusion that achievements in elementary-particle physics have to affect any future theory, that this theory has to explain the principle contradictions between classical and quantum field theories, and that finally it can lead to change of the modern space-time model as a four-dimensional variety 14. Quantum field theory in a semiotic perspective International Nuclear Information System (INIS) Dosch, H.G. 2005-01-01 Viewing physical theories as symbolic constructions came to the fore in the middle of the nineteenth century with the emancipation of the classical theory of the electromagnetic field from mechanics; most notably this happened through the work of Helmholtz, Hertz, Poincare, and later Weyl. The epistemological problems that nourished this development are today highlighted within quantum field theory. The present essay starts off with a concise and non-technical outline of the firmly based aspects of relativistic quantum field theory, i.e. the very successful description of subnuclear phenomena. The particular methods, by which these different aspects have to be accessed, then get described as distinct facets of quantum field theory. The authors show how these different facets vary with respect to the relation between quantum fields and associated particles. Thus, by emphasising the respective role of various basic concepts involved, the authors claim that only a very general epistemic approach can properly account for this diversity - an account they trace back to the philosophical writings of the aforementioned physicists and mathematicians. Finally, what they call their semiotic perspective on quantum field theory gets related to recent discussions within the philosophy of science and turns out to act as a counterbalance to, for instance, structural realism. (orig.) 15. Quantum field theory in a semiotic perspective Energy Technology Data Exchange (ETDEWEB) Dosch, H.G. [Heidelberg Univ. (Germany). Inst. fuer Theoretische Physik; Mueller, V.F. [Technische Univ. Kaiserslautern (Germany). Fachbereich Physik; Sieroka, N. [Zurich Univ. (Switzerland) 2005-07-01 Viewing physical theories as symbolic constructions came to the fore in the middle of the nineteenth century with the emancipation of the classical theory of the electromagnetic field from mechanics; most notably this happened through the work of Helmholtz, Hertz, Poincare, and later Weyl. The epistemological problems that nourished this development are today highlighted within quantum field theory. The present essay starts off with a concise and non-technical outline of the firmly based aspects of relativistic quantum field theory, i.e. the very successful description of subnuclear phenomena. The particular methods, by which these different aspects have to be accessed, then get described as distinct facets of quantum field theory. The authors show how these different facets vary with respect to the relation between quantum fields and associated particles. Thus, by emphasising the respective role of various basic concepts involved, the authors claim that only a very general epistemic approach can properly account for this diversity - an account they trace back to the philosophical writings of the aforementioned physicists and mathematicians. Finally, what they call their semiotic perspective on quantum field theory gets related to recent discussions within the philosophy of science and turns out to act as a counterbalance to, for instance, structural realism. (orig.) 16. Superstring field theory equivalence: Ramond sector International Nuclear Information System (INIS) Kroyter, Michael 2009-01-01 We prove that the finite gauge transformation of the Ramond sector of the modified cubic superstring field theory is ill-defined due to collisions of picture changing operators. Despite this problem we study to what extent could a bijective classical correspondence between this theory and the (presumably consistent) non-polynomial theory exist. We find that the classical equivalence between these two theories can almost be extended to the Ramond sector: We construct mappings between the string fields (NS and Ramond, including Chan-Paton factors and the various GSO sectors) of the two theories that send solutions to solutions in a way that respects the linearized gauge symmetries in both sides and keeps the action of the solutions invariant. The perturbative spectrum around equivalent solutions is also isomorphic. The problem with the cubic theory implies that the correspondence of the linearized gauge symmetries cannot be extended to a correspondence of the finite gauge symmetries. Hence, our equivalence is only formal, since it relates a consistent theory to an inconsistent one. Nonetheless, we believe that the fact that the equivalence formally works suggests that a consistent modification of the cubic theory exists. We construct a theory that can be considered as a first step towards a consistent RNS cubic theory. 17. Quantum groups, quantum categories and quantum field theory CERN Document Server Fröhlich, Jürg 1993-01-01 This book reviews recent results on low-dimensional quantum field theories and their connection with quantum group theory and the theory of braided, balanced tensor categories. It presents detailed, mathematically precise introductions to these subjects and then continues with new results. Among the main results are a detailed analysis of the representation theory of U (sl ), for q a primitive root of unity, and a semi-simple quotient thereof, a classfication of braided tensor categories generated by an object of q-dimension less than two, and an application of these results to the theory of sectors in algebraic quantum field theory. This clarifies the notion of "quantized symmetries" in quantum fieldtheory. The reader is expected to be familiar with basic notions and resultsin algebra. The book is intended for research mathematicians, mathematical physicists and graduate students. 18. Magnetic Catalysis in Graphene Effective Field Theory. Science.gov (United States) DeTar, Carleton; Winterowd, Christopher; Zafeiropoulos, Savvas 2016-12-23 We report on the first calculation of magnetic catalysis at zero temperature in a fully nonperturbative simulation of the graphene effective field theory. Using lattice gauge theory, a nonperturbative analysis of the theory of strongly interacting, massless, (2+1)-dimensional Dirac fermions in the presence of an external magnetic field is performed. We show that in the zero-temperature limit, a nonzero value for the chiral condensate is obtained which signals the spontaneous breaking of chiral symmetry. This result implies a nonzero value for the dynamical mass of the Dirac quasiparticle. 19. Supersymmetric gauge theories, quantization of Mflat, and conformal field theory International Nuclear Information System (INIS) Teschner, J.; Vartanov, G.S. 2013-02-01 We propose a derivation of the correspondence between certain gauge theories with N=2 supersymmetry and conformal field theory discovered by Alday, Gaiotto and Tachikawa in the spirit of Seiberg-Witten theory. Based on certain results from the literature we argue that the quantum theory of the moduli spaces of flat SL(2,R)-connections represents a nonperturbative ''skeleton'' of the gauge theory, protected by supersymmetry. It follows that instanton partition functions can be characterized as solutions to a Riemann-Hilbert type problem. In order to solve it, we describe the quantization of the moduli spaces of flat connections explicitly in terms of two natural sets of Darboux coordinates. The kernel describing the relation between the two pictures represents the solution to the Riemann Hilbert problem, and is naturally identified with the Liouville conformal blocks. 20. Random light beams theory and applications CERN Document Server Korotkova, Olga 2013-01-01 Random Light Beams: Theory and Applications contemplates the potential in harnessing random light. This book discusses light matter interactions, and concentrates on the various phenomena associated with beam-like fields. It explores natural and man-made light fields and gives an overview of recently introduced families of random light beams. It outlines mathematical tools for analysis, suggests schemes for realization, and discusses possible applications. The book introduces the essential concepts needed for a deeper understanding of the subject, discusses various classes of deterministic par 1. Infrared problems in field perturbation theory International Nuclear Information System (INIS) David, Francois. 1982-12-01 The work presented mainly covers questions related to the presence of ''infrared'' divergences in perturbation expansions of the Green functions of certain massless field theories. It is important to determine the mathematical status of perturbation expansions in field theory in order to define the region in which they are valid. Renormalization and the symmetry of a theory are important factors in infrared problems. The main object of this thesis resides in the mathematical techniques employed: integral representations of the Feynman amplitudes; methods for desingularization, regularization and dimensional renormalization. Nonlinear two dimensional space-time sigma models describing Goldstone's low energy boson dynamics associated with a breaking of continuous symmetry are studied. Random surface models are then investigated followed by infrared divergences in super-renormalizable theories. Finally, nonperturbation effects in massless theories are studied by expanding the two-dimensional nonlinear sigma model in 1/N [fr 2. Proceedings of the 5. Jorge Andre Swieca Summer School Field Theory and Particle Physics International Nuclear Information System (INIS) Eboli, O.J.P.; Gomes, M.; Santoro, A. 1989-01-01 Lectures on quantum field theories and particle physics are presented. The part of quantum field theories contains: constrained dynamics; Schroedinger representation in field theory; application of this representation to quantum fields in a Robertson-Walker space-time; Berry connection; problem of construction and classification of conformal field theories; lattice models; two-dimensional S matrices and conformal field theory for unifying perspective of Yang-Baxter algebras; parasupersymmetric quantum mechanics; introduction to string field theory; three dimensional gravity and two-dimensional parafermionic model. The part of particle physics contains: collider physics; strong interactions and use of strings in strong interactions. (M.C.K.) 3. Towards field theory in spaces with multivolume junctions CERN Document Server Fomin, P I 2002-01-01 We consider a spacetime formed by several pieces with common timelike boundary which plays the role of a junction between them. We establish junction conditions for fields of various spins and derive the resulting laws of wave propagation through the junction, which turn out to be quite similar for fields of all spins. As an application, we consider the case of multivolume junctions in four-dimensional spacetime that may arise in the context of the theory of quantum creation of a closed universe on the background of a big mother universe. The theory developed can also be applied to braneworld models and to the superstring theory. 4. Theory of semigroups and applications CERN Document Server Sinha, Kalyan B 2017-01-01 The book presents major topics in semigroups, such as operator theory, partial differential equations, harmonic analysis, probability and statistics and classical and quantum mechanics, and applications. Along with a systematic development of the subject, the book emphasises on the explorations of the contact areas and interfaces, supported by the presentations of explicit computations, wherever feasible. Designed into seven chapters and three appendixes, the book targets to the graduate and senior undergraduate students of mathematics, as well as researchers in the respective areas. The book envisages the pre-requisites of a good understanding of real analysis with elements of the theory of measures and integration, and a first course in functional analysis and in the theory of operators. Chapters 4 through 6 contain advanced topics, which have many interesting applications such as the Feynman–Kac formula, the central limit theorem and the construction of Markov semigroups. Many examples have been given in... 5. Sinusoids theory and technological applications CERN Document Server Kythe, Prem K 2014-01-01 A Complete Treatment of Current Research Topics in Fourier Transforms and Sinusoids Sinusoids: Theory and Technological Applications explains how sinusoids and Fourier transforms are used in a variety of application areas, including signal processing, GPS, optics, x-ray crystallography, radioastronomy, poetry and music as sound waves, and the medical sciences. With more than 200 illustrations, the book discusses electromagnetic force and sychrotron radiation comprising all kinds of waves, including gamma rays, x-rays, UV rays, visible light rays, infrared, microwaves, and radio waves. It also covers topics of common interest, such as quasars, pulsars, the Big Bang theory, Olbers' paradox, black holes, Mars mission, and SETI.The book begins by describing sinusoids-which are periodic sine or cosine functions-using well-known examples from wave theory, including traveling and standing waves, continuous musical rhythms, and the human liver. It next discusses the Fourier series and transform in both continuous and... 6. Gravitation Field Dynamics in Jeans Theory 2016-01-27 Jan 27, 2016 ... Closed system of time equations for nonrelativistic gravitation field and hydrodynamic medium was obtained by taking into account binary correlations of the field, which is the generalization of Jeans theory. Distribution function of the systemwas built on the basis of the Bogolyubov reduced description ... 7. Field theory of polar continua International Nuclear Information System (INIS) Heinz, C. 1988-01-01 A Lagrangian density in the polar space X 1+3+3 depending of the potentials and their derivativs and of the fluxes is introduced. The potentials are then the mechanical and electromagnetic potentials, the potentials of gravity and in the polar space X 1+3+3 the components of affine connection. The fluxes are essentially the tangential motors of the mechanical and electromagnetic world-lines multiplied with the density of mass and electric charge. The Hamilton principle gives, with the in variational calculus usual integrations by part, here done via the theorem of Gauss, the equations of motion and the field equations. The conditions of integrability for these equations are discussed. (author) 8. Mean-field magnetohydrodynamics and dynamo theory CERN Document Server Krause, F 2013-01-01 Mean-Field Magnetohydrodynamics and Dynamo Theory provides a systematic introduction to mean-field magnetohydrodynamics and the dynamo theory, along with the results achieved. Topics covered include turbulence and large-scale structures; general properties of the turbulent electromotive force; homogeneity, isotropy, and mirror symmetry of turbulent fields; and turbulent electromotive force in the case of non-vanishing mean flow. The turbulent electromotive force in the case of rotational mean motion is also considered. This book is comprised of 17 chapters and opens with an overview of the gen 9. Circuit complexity in quantum field theory Science.gov (United States) Jefferson, Robert A.; Myers, Robert C. 2017-10-01 Motivated by recent studies of holographic complexity, we examine the question of circuit complexity in quantum field theory. We provide a quantum circuit model for the preparation of Gaussian states, in particular the ground state, in a free scalar field theory for general dimensions. Applying the geometric approach of Nielsen to this quantum circuit model, the complexity of the state becomes the length of the shortest geodesic in the space of circuits. We compare the complexity of the ground state of the free scalar field to the analogous results from holographic complexity, and find some surprising similarities. 10. Dark Matter, Elko Fields and Weinberg's Quantum Field Theory Formalism Science.gov (United States) 2012-02-01 The Elko quantum field was introduced by Ahluwalia and Grumiller, who proposed it as a candidate for dark matter. We study the Elko field in Wemberg's formalism for quantum field theory. We prove that if one takes the symmetry group to be the full Pomcaré group then the Elko field is not a quantum field in the sense of Weinberg. This confirms results of Ahluwalia, Lee and Schritt, who showed using a different approach that the Elko field does not transform covariantly under rotations and hence has a preferred axis. 11. Coadjoint orbits and conformal field theory Energy Technology Data Exchange (ETDEWEB) Taylor, IV, Washington [Univ. of California, Berkeley, CA (United States) 1993-08-01 This thesis is primarily a study of certain aspects of the geometric and algebraic structure of coadjoint orbit representations of infinite-dimensional Lie groups. The goal of this work is to use coadjoint orbit representations to construct conformal field theories, in a fashion analogous to the free-field constructions of conformal field theories. The new results which are presented in this thesis are as follows: First, an explicit set of formulae are derived giving an algebraic realization of coadjoint orbit representations in terms of differential operators acting on a polynomial Fock space. These representations are equivalent to dual Verma module representations. Next, intertwiners are explicitly constructed which allow the construction of resolutions for irreducible representations using these Fock space realizations. Finally, vertex operators between these irreducible representations are explicitly constructed as chain maps between the resolutions; these vertex operators allow the construction of rational conformal field theories according to an algebraic prescription. 12. Applications of the complex-mass renormalization scheme in effective field theory; Anwendungen des Komplexe-Masse-Renormierungsschemas in effektiver Feldtheorie Energy Technology Data Exchange (ETDEWEB) Bauer, Torsten 2012-07-11 In the first part of the this doctoral thesis the perturbative unitarity in the complex-mass scheme (CMS) is analysed. To that end a procedure for calculating cutting rules for loop integrals containing propagators with finite widths is presented. A toy-model Lagrangian describing the interaction of a heavy vector boson with a light fermion is used to demonstrate that the CMS respects unitarity order by order in perturbation theory provided that the renormalized coupling constant remains real. The second part of the thesis deals with various applications of the CMS to chiral effective field theory (EFT). In particular, mass and width of the delta resonance, elastic electromagnetic form factors of the Roper resonance, form factors of the nucleon-to-Roper transition, pion-nucleon scattering, and pion photo- and electroproduction for center-of-mass energies in the region of the Roper mass are calculated. By choosing appropriate renormalization conditions, a consistent chiral power counting scheme for EFT with resonant degrees of freedom can be established. This allows for a systematic investigation of the above processes in terms of an expansion in small quantities. The obtained results can be applied to the extrapolation of corresponding simulations in the context of lattice QCD to the physical value of the pion mass. Therefore, in addition to the Q{sup 2} dependence of the form factors, also the pion-mass dependence of the magnetic moment and electromagnetic radii of the Roper resonance is explored. Both a partial wave decomposition and a multipole expansion are performed for pion-nucleon scattering and pion photo- and electroproduction, respectively. In this connection the P11 partial wave as well as the M{sub 1-} and S{sub 1-} multipoles are fitted via non-linear regression to empirical data. 13. The space-time operator product expansion in string theory duals of field theories International Nuclear Information System (INIS) Aharony, Ofer; Komargodski, Zohar 2008-01-01 We study the operator product expansion (OPE) limit of correlation functions in field theories which possess string theory duals, from the point of view of the string worldsheet. We show how the interesting ('single-trace') terms in the OPE of the field theory arise in this limit from the OPE of the worldsheet theory of the string dual, using a dominant saddle point which appears in computations of worldsheet correlation functions in the space-time OPE limit. The worldsheet OPE generically contains only non-physical operators, but all the non-physical contributions are resummed by the saddle point to a contribution similar to that of a physical operator, which exactly matches the field theory expectations. We verify that the OPE limit of the worldsheet theory does not have any other contributions to the OPE limit of space-time correlation functions. Our discussion is completely general and applies to any local field theory (conformal at high energies) that has a weakly coupled string theory dual (with arbitrary curvature). As a first application, we compare our results to a proposal of R. Gopakumar for the string theory dual of free gauge theories 14. Tephra: field, theory and application Science.gov (United States) Pouget, Solene In this work we briefly introduced the current state of the art for plume dynamics and plume modelling (chapters 1 and 2). From these, it was found that several questions remained unanswered. One of them what about adding some quantitative methodology to tephra identification when using geochemistry. Using discontinuous two tephra layers discovered at Burney Spring Mountain, northern California, this aspect was explored. Stratigraphic relationships suggest that they are two distinct tephras. Binary plots and standard similarity coefficients of electron probe microanalysis data have been supplemented with principal component analysis in log-ratio transformed data to correlate the two tephra layers to known regional tephras. Using principal component analysis, we are furthermore able to bound our uncertainty in the correlation of the two tephra layers (chapter 3). After removal of outliers, within the 95% prediction interval, we can say that one tephra layer is likely the Rockland tephra, aged 565-610 ka, and the second layer is likely from Mt Mazama, the Trego Hot Springs tephra, aged ~29 ka. Using cluster analysis on several vectors of chemical elements another quantitative methodology was explored (chapter 4). It was found that in most cases, geochemical analysis of a tephra layer will be assign to a single cluster, however in some cases the analysis are spread over several clusters. This spreading is a direct result of mixing and reworking happening in the tephra layer. The dynamics of volcanic plumes were also investigated. We introduce a new method to estimate mass eruption rate (MER) and mass loading from the growth of a volcanic umbrella cloud or downwind plume using satellite images, or photographs where ground-based observations are available with a gravity current model (chapter 5). The results show a more fully characterised MER as a function of time than do the results given by pre-existing methods, and allow a faster, remote assessment of the mass eruption rate, even for volcanoes that are difficult to study. A new gravity current model for umbrella cloud was tested which allows to transition from one regime to another against measured data from several eruptions (chapter 6). Once the model was proved to be accurate, the different variables were tested to observed their impact on the spreading of the umbrella cloud. As a result it was found that the evolution of the radius changes not only in power-law with time but also indicates transitions in regimes. Chapter 7 is an end-to-end framework to probabilistic forecasting of volcanic ash transport and improved eruption source parameters. In summary, this dissertation demonstrates four main contributions to volcanology: 1. The importance of bringing quantitative methods to tephra identification and how these methods can help in characterization of tephra. 2. The importance of the spread of volcanic cloud in the atmosphere as a gravity current. Particularly for prediction of the ash dispersal since the spreading as a gravity current happens over large distances from the volcano and even upwind. But also because it can help in getting a first and fast estimation of the mass eruption rate of an eruption which can be followed with time. 3. The importance of studying the structures and features on volcanic plume as they can reveal information about the dynamics of spreading and improve the estimation of regime transitions. 4. The need for the different communities working on tephra to communicate and understand each others approach fo better collaboration and multi-approach work. 15. Distributed hash table theory, platforms and applications CERN Document Server Zhang, Hao; Xie, Haiyong; Yu, Nenghai 2013-01-01 This SpringerBrief summarizes the development of Distributed Hash Table in both academic and industrial fields. It covers the main theory, platforms and applications of this key part in distributed systems and applications, especially in large-scale distributed environments. The authors teach the principles of several popular DHT platforms that can solve practical problems such as load balance, multiple replicas, consistency and latency. They also propose DHT-based applications including multicast, anycast, distributed file systems, search, storage, content delivery network, file sharing and c 16. Knots, topology and quantum field theories International Nuclear Information System (INIS) Lusanna, L. 1989-01-01 The title of the workshop, Knots, Topology and Quantum Field Theory, accurate reflected the topics discussed. There have been important developments in mathematical and quantum field theory in the past few years, which had a large impact on physicist thinking. It is historically unusual and pleasing that these developments are taking place as a result of an intense interaction between mathematical physicists and mathematician. On the one hand, topological concepts and methods are playing an increasingly important lead to novel mathematical concepts: for instance, the study of quantum groups open a new chapter in the deformation theory of Lie algebras. These developments at present will lead to new insights into the theory of elementary particles and their interactions. In essence, the talks dealt with three, broadly defined areas of theoretical physics. One was topological quantum field theories, the other the problem of quantum groups and the third one certain aspects of more traditional field theories, such as, for instance, quantum gravity. These topics, however, are interrelated and the general theme of the workshop defies rigid classification; this was evident from the cross references to be found in almo all the talks 17. On the unitary transformation between non-quasifree and quasifree state spaces and its application to quantum field theory on curved spacetimes International Nuclear Information System (INIS) Gottschalk, Hanno; Hack, Thomas-Paul 2009-12-01 Using -calculus on the dual of the Borchers-Uhlmann algebra endowed with a combinatorial co-product, we develop a method to calculate a unitary transformation relating the GNS representations of a non-quasifree and a quasifree state of the free hermitian scalar field. The motivation for such an analysis and a further result is the fact that a unitary transformation of this kind arises naturally in scattering theory on non-stationary backgrounds. Indeed, employing the perturbation theory of the Yang-Feldman equations with a free CCR field in a quasifree state as an initial condition and making use of extended Feynman graphs, we are able to calculate the Wightman functions of the interacting and outgoing fields in a φ p -theory on arbitrary curved spacetimes. A further examination then reveals two major features of the aforementioned theory: firstly, the interacting Wightman functions fulfil the basic axioms of hermiticity, invariance, spectrality (on stationary spacetimes), perturbative positivity, and locality. Secondly, the outgoing field is free and fulfils the CCR, but is in general not in a quasifree state in the case of a non-stationary spacetime. In order to obtain a sensible particle picture for the outgoing field and, hence, a description of the scattering process in terms of particles (in asymptotically flat spacetimes), it is thus necessary to compute a unitary transformation of the abovementioned type. (orig.) 18. Experimental signature of scaling violation implied by field theories International Nuclear Information System (INIS) Tung, W. 1975-01-01 Renormalizable field theories are found to predict a surprisingly specific pattern of scaling violation in deep inelastic scattering. Comparison with experiments is discussed. The feasibility of distinguishing asymptotically free field theories from conventional field theories is evaluated 19. Nonequilibrium statistical field theory for classical particles: Basic kinetic theory. Science.gov (United States) Viermann, Celia; Fabis, Felix; Kozlikin, Elena; Lilow, Robert; Bartelmann, Matthias 2015-06-01 Recently Mazenko and Das and Mazenko [Phys. Rev. E 81, 061102 (2010); J. Stat. Phys. 149, 643 (2012); J. Stat. Phys. 152, 159 (2013); Phys. Rev. E 83, 041125 (2011)] introduced a nonequilibrium field-theoretical approach to describe the statistical properties of a classical particle ensemble starting from the microscopic equations of motion of each individual particle. We use this theory to investigate the transition from those microscopic degrees of freedom to the evolution equations of the macroscopic observables of the ensemble. For the free theory, we recover the continuity and Jeans equations of a collisionless gas. For a theory containing two-particle interactions in a canonical perturbation series, we find the macroscopic evolution equations to be described by the Born-Bogoliubov-Green-Kirkwood-Yvon hierarchy with a truncation criterion depending on the order in perturbation theory. This establishes a direct link between the classical and the field-theoretical approaches to kinetic theory that might serve as a starting point to investigate kinetic theory beyond the classical limits. 20. Further studies in aesthetic field theory International Nuclear Information System (INIS) 1984-01-01 We study different facets of Aesthetic Field Theory. First, we have found within the complex version of the theory a bounded particle system which has more structure than what we have previously observed. The particle is built from planar 3 maxima--minima confluence regions. The confluence region closes in 3 spatial dimensions, so once again we have a ''topological'' particle. If we characterize bound stats by the number of large magnitude regions in close proximity, then the simplest interpretation of what we are seeing is that of a 3 particle bound system. Secondly, again within the framework of complex Aesthetic Field Theory, but using a more symmetric system of equations, we observe a confluence type topological particle spontaneously arising out of the vacuum (creation effect). The particle again has a loop shape. The extended particles thus far found in 4 dimensional Aesthetic Field Theory have always had problems with the spreading of the particle system as time went on. Thirdly, we found a bounded confluence particle system, which in addition to confinement and non attenuation shows the desirable property of not spreading in time. In this case, we work exclusively with real fields. The particle system has a dipole looking shape. We also studied complex null Aesthetic Field Theory in 8 dimensions having a 4 direct-sum 4 structure. We were not able to find a bound to our particle system here 1. Massive deformations of Type IIA theory within double field theory Science.gov (United States) Çatal-Özer, Aybike 2018-02-01 We obtain massive deformations of Type IIA supergravity theory through duality twisted reductions of Double Field Theory (DFT) of massless Type II strings. The mass deformation is induced through the reduction of the DFT of the RR sector. Such reductions are determined by a twist element belonging to Spin+(10, 10), which is the duality group of the DFT of the RR sector. We determine the form of the twists and give particular examples of twists matrices, for which a massive deformation of Type IIA theory can be obtained. In one of the cases, requirement of gauge invariance of the RR sector implies that the dilaton field must pick up a linear dependence on one of the dual coordinates. In another case, the choice of the twist matrix violates the weak and the strong constraints explicitly in the internal doubled space. 2. Recent Developments in D=2 String Field Theory OpenAIRE Kaku, Michio 1994-01-01 In this review article, we review the recent developments in constructing string field theories that have been proposed, all of which correctly reproduce the correlation functions of two-dimensional string theory. These include: (a) free fermion field theory (b) collective string field theory (c) temporal gauge string field theory (d) non-polynomial string field theory. We analyze discrete states, the $w(\\infty)$ symmetry, and correlation functions in terms of these different string field the... 3. A non-linear field theory International Nuclear Information System (INIS) Skyrme, T.H.R. 1994-01-01 A unified field theory of mesons and their particle sources is proposed and considered in its classical aspects. The theory has static solutions of a singular nature, but finite energy, characterized by spin directions; the number of such entities is a rigorously conserved constant of motion; they interact with an external meson field through a derivative-type coupling with the spins, akin to the formalism of strong-coupling meson theory. There is a conserved current identifiable with isobaric spin, and another that may be related to hypercharge. The postulates include one constant of the dimensions of length, and another that is conjecture necessarily to have the value (h/2π)c, or perhaps 1/2(h/2π)c, in the quantized theory. (author). 5 refs 4. Field theory of the Eulerian perfect fluid Science.gov (United States) Ariki, Taketo; Morales, Pablo A. 2018-01-01 The Eulerian perfect-fluid theory is reformulated from its action principle in a pure field-theoretic manner. Conservation of the convective current is no longer imposed by Lin’s constraints, but rather adopted as the central idea of the theory. Our formulation, for the first time, successfully reduces redundant degrees of freedom promoting one half of the Clebsch variables to true dynamical fields. Interactions on these fields allow for the exchange of the convective current of quantities such as mass and charge, which are uniformly understood as the breaking of the underlying symmetry of the force-free fluid. The Clebsch fields play the essential role of exchanging angular momentum with the force field producing vorticity. 5. A general field-covariant formulation of quantum field theory International Nuclear Information System (INIS) Anselmi, Damiano 2013-01-01 In all nontrivial cases renormalization, as it is usually formulated, is not a change of integration variables in the functional integral, plus parameter redefinitions, but a set of replacements, of actions and/or field variables and parameters. Because of this, we cannot write simple identities relating bare and renormalized generating functionals, or generating functionals before and after nonlinear changes of field variables. In this paper we investigate this issue and work out a general field-covariant approach to quantum field theory, which allows us to treat all perturbative changes of field variables, including the relation between bare and renormalized fields, as true changes of variables in the functional integral, under which the functionals Z and W=lnZ behave as scalars. We investigate the relation between composite fields and changes of field variables, and we show that, if J are the sources coupled to the elementary fields, all changes of field variables can be expressed as J-dependent redefinitions of the sources L coupled to the composite fields. We also work out the relation between the renormalization of variable-changes and the renormalization of composite fields. Using our transformation rules it is possible to derive the renormalization of a theory in a new variable frame from the renormalization in the old variable frame, without having to calculate it anew. We define several approaches, useful for different purposes, in particular a linear approach where all variable changes are described as linear source redefinitions. We include a number of explicit examples. (orig.) 6. Neutrix calculus and finite quantum field theory International Nuclear Information System (INIS) Ng, Y Jack; Dam, H van 2005-01-01 In general, quantum field theories (QFT) require regularizations and infinite renormalizations due to ultraviolet divergences in their loop calculations. Furthermore, perturbation series in theories like quantum electrodynamics are not convergent series, but are asymptotic series. We apply neutrix calculus, developed in connection with asymptotic series and divergent integrals, to QFT, obtaining finite renormalizations. While none of the physically measurable results in renormalizable QFT is changed, quantum gravity is rendered more manageable in the neutrix framework. (letter to the editor) 7. Quantum field theory in generalised Snyder spaces International Nuclear Information System (INIS) Meljanac, S.; Meljanac, D.; Mignemi, S.; Štrajn, R. 2017-01-01 We discuss the generalisation of the Snyder model that includes all possible deformations of the Heisenberg algebra compatible with Lorentz invariance and investigate its properties. We calculate perturbatively the law of addition of momenta and the star product in the general case. We also undertake the construction of a scalar field theory on these noncommutative spaces showing that the free theory is equivalent to the commutative one, like in other models of noncommutative QFT. 8. Staircase Models from Affine Toda Field Theory CERN Document Server Dorey, P; Dorey, Patrick; Ravanini, Francesco 1993-01-01 We propose a class of purely elastic scattering theories generalising the staircase model of Al. B. Zamolodchikov, based on the affine Toda field theories for simply-laced Lie algebras g=A,D,E at suitable complex values of their coupling constants. Considering their Thermodynamic Bethe Ansatz equations, we give analytic arguments in support of a conjectured renormalisation group flow visiting the neighbourhood of each W_g minimal model in turn. 9. Quantum field theory in generalised Snyder spaces Energy Technology Data Exchange (ETDEWEB) Meljanac, S.; Meljanac, D. [Rudjer Bošković Institute, Bijenička cesta 54, 10002 Zagreb (Croatia); Mignemi, S., E-mail: [email protected] [Dipartimento di Matematica e Informatica, Università di Cagliari, viale Merello 92, 09123 Cagliari (Italy); INFN, Sezione di Cagliari, Cittadella Universitaria, 09042 Monserrato (Italy); Štrajn, R. [Dipartimento di Matematica e Informatica, Università di Cagliari, viale Merello 92, 09123 Cagliari (Italy); INFN, Sezione di Cagliari, Cittadella Universitaria, 09042 Monserrato (Italy) 2017-05-10 We discuss the generalisation of the Snyder model that includes all possible deformations of the Heisenberg algebra compatible with Lorentz invariance and investigate its properties. We calculate perturbatively the law of addition of momenta and the star product in the general case. We also undertake the construction of a scalar field theory on these noncommutative spaces showing that the free theory is equivalent to the commutative one, like in other models of noncommutative QFT. 10. Magnetic monopoles in field theory and cosmology. Science.gov (United States) Rajantie, Arttu 2012-12-28 The existence of magnetic monopoles is predicted by many theories of particle physics beyond the standard model. However, in spite of extensive searches, there is no experimental or observational sign of them. I review the role of magnetic monopoles in quantum field theory and discuss their implications for particle physics and cosmology. I also highlight their differences and similarities with monopoles found in frustrated magnetic systems. 11. Catastrophe theory and its application status in mechanical engineering Directory of Open Access Journals (Sweden) Jinge LIU Full Text Available Catastrophe theory is a kind of mathematical method which aims to apply and interpret the discontinuous phenomenon. Since its emergence, it has been widely used to explain a variety of emergent phenomena in the fields of natural science, social science, management science and some other science and technology fields. Firstly, this paper introduces the theory of catastrophe in several aspects, such as its generation, radical principle, basic characteristics and development. Secondly, it summarizes the main applications of catastrophe theory in the field of mechanical engineering, focusing on the research progress of catastrophe theory in revealing catastrophe of rotor vibration state, analyzing friction and wear failure, predicting metal fracture, and so on. Finally, it advises that later development of catastrophe theory should pay more attention to the combination of itself with other traditional nonlinear theories and methods. This paper provides a beneficial reference to guide the application of catastrophe theory in mechanical engineering and related fields for later research. 12. Adsorption refrigeration technology theory and application CERN Document Server Wang, Ruzhu; Wu, Jingyi 2014-01-01 Gives readers a detailed understanding of adsorption refrigeration technology, with a focus on practical applications and environmental concerns Systematically covering the technology of adsorption refrigeration, this book provides readers with a technical understanding of the topic as well as detailed information on the state-of-the-art from leading researchers in the field. Introducing readers to background on the development of adsorption refrigeration, the authors also cover the development of adsorbents, various thermodynamic theories, the design of adsorption systems and adsorption refri 13. Wireless network security theories and applications CERN Document Server Chen, Lei; Zhang, Zihong 2013-01-01 Wireless Network Security Theories and Applications discusses the relevant security technologies, vulnerabilities, and potential threats, and introduces the corresponding security standards and protocols, as well as provides solutions to security concerns. Authors of each chapter in this book, mostly top researchers in relevant research fields in the U.S. and China, presented their research findings and results about the security of the following types of wireless networks: Wireless Cellular Networks, Wireless Local Area Networks (WLANs), Wireless Metropolitan Area Networks (WMANs), Bluetooth 14. Effective field theory for magnetic compactifications Science.gov (United States) Buchmuller, Wilfried; Dierigl, Markus; Dudas, Emilian; Schweizer, Julian 2017-04-01 Magnetic flux plays an important role in compactifications of field and string theories in two ways, it generates a multiplicity of chiral fermion zero modes and it can break supersymmetry. We derive the complete four-dimensional effective action for N = 1 supersymmetric Abelian and non-Abelian gauge theories in six dimensions compactified on a torus with flux. The effective action contains the tower of charged states and it accounts for the mass spectrum of bosonic and fermionic fields as well as their level-dependent interactions. This allows us to compute quantum corrections to the mass and couplings of Wilson lines. We find that the one-loop corrections vanish, contrary to the case without flux. This can be traced back to the spontaneous breaking of symmetries of the six-dimensional theory by the background gauge field, with the Wilson lines as Goldstone bosons. 15. Effective field theory for magnetic compactifications Energy Technology Data Exchange (ETDEWEB) Buchmuller, Wilfried; Dierigl, Markus; Schweizer Julian [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany); Dudas, Emilian [Univ. Paris-Saclay, Palaiseau (France). Ecole Polytechnique 2016-12-15 Magnetic flux plays an important role in compactifications of field and string theories in two ways, it generates a multiplicity of chiral fermion zero modes and it can break supersymmetry. We derive the complete four-dimensional effective action for N=1 supersymmetric Abelian and non-Abelian gauge theories in six dimensions compactified on a torus with flux. The effective action contains the tower of charged states and it accounts for the mass spectrum of bosonic and fermionic fields as well as their level-dependent interactions. This allows us to compute quantum corrections to the mass and couplings of Wilson lines. We find that the one-loop corrections vanish, contrary to the case without flux. This can be traced back to the spontaneous breaking of symmetries of the six-dimensional theory by the background gauge field, with the Wilson lines as Goldstone bosons. 16. Effective field theory for magnetic compactifications Energy Technology Data Exchange (ETDEWEB) Buchmuller, Wilfried; Dierigl, Markus [Deutsches Elektronen-Synchrotron DESY,22607 Hamburg (Germany); Dudas, Emilian [Centre de Physique Théorique, École Polytechnique, CNRS, Université Paris-Saclay,F-91128 Palaiseau (France); Schweizer, Julian [Deutsches Elektronen-Synchrotron DESY,22607 Hamburg (Germany) 2017-04-10 Magnetic flux plays an important role in compactifications of field and string theories in two ways, it generates a multiplicity of chiral fermion zero modes and it can break supersymmetry. We derive the complete four-dimensional effective action for N=1 supersymmetric Abelian and non-Abelian gauge theories in six dimensions compactified on a torus with flux. The effective action contains the tower of charged states and it accounts for the mass spectrum of bosonic and fermionic fields as well as their level-dependent interactions. This allows us to compute quantum corrections to the mass and couplings of Wilson lines. We find that the one-loop corrections vanish, contrary to the case without flux. This can be traced back to the spontaneous breaking of symmetries of the six-dimensional theory by the background gauge field, with the Wilson lines as Goldstone bosons. 17. Group theory for chemists fundamental theory and applications CERN Document Server Molloy, K C 2010-01-01 The basics of group theory and its applications to themes such as the analysis of vibrational spectra and molecular orbital theory are essential knowledge for the undergraduate student of inorganic chemistry. The second edition of Group Theory for Chemists uses diagrams and problem-solving to help students test and improve their understanding, including a new section on the application of group theory to electronic spectroscopy.Part one covers the essentials of symmetry and group theory, including symmetry, point groups and representations. Part two deals with the application of group theory t 18. Classical field theory on electrodynamics, non-abelian gauge theories and gravitation CERN Document Server Scheck, Florian 2018-01-01 Scheck’s successful textbook presents a comprehensive treatment, ideally suited for a one-semester course. The textbook describes Maxwell's equations first in their integral, directly testable form, then moves on to their local formulation. The first two chapters cover all essential properties of Maxwell's equations, including their symmetries and their covariance in a modern notation. Chapter 3 is devoted to Maxwell's theory as a classical field theory and to solutions of the wave equation. Chapter 4 deals with important applications of Maxwell's theory. It includes topical subjects such as metamaterials with negative refraction index and solutions of Helmholtz' equation in paraxial approximation relevant for the description of laser beams. Chapter 5 describes non-Abelian gauge theories from a classical, geometric point of view, in analogy to Maxwell's theory as a prototype, and culminates in an application to the U(2) theory relevant for electroweak interactions. The last chapter 6 gives a concise summary... 19. Conformal field theory with gauge symmetry CERN Document Server Ueno, Kenji 2008-01-01 This book presents a systematic approach to conformal field theory with gauge symmetry from the point of view of complex algebraic geometry. After presenting the basic facts of the theory of compact Riemann surfaces and the representation theory of affine Lie algebras in Chapters 1 and 2, conformal blocks for pointed Riemann surfaces with coordinates are constructed in Chapter 3. In Chapter 4 the sheaf of conformal blocks associated to a family of pointed Riemann surfaces with coordinates is constructed, and in Chapter 5 it is shown that this sheaf supports a projective flat connection-one of 20. Reggeon field theory for large Pomeron loops International Nuclear Information System (INIS) Altinoluk, Tolga; Kovner, Alex; Levin, Eugene; Lublinsky, Michael 2014-01-01 We analyze the range of applicability of the high energy Reggeon Field Theory H RFT derived in http://dx.doi.org/10.1088/1126-6708/2009/03/109. We show that this theory is valid as long as at any intermediate value of rapidity η throughout the evolution at least one of the colliding objects is dilute. Importantly, at some values of η the dilute object could be the projectile, while at others it could be the target, so that H RFT does not reduce to either H JIMWLK or H KLWMIJ . When both objects are dense, corrections to the evolution not accounted for in http://dx.doi.org/10.1088/1126-6708/2009/03/109 become important. The same limitation applies to other approaches to high energy evolution available today, such as for example (http://dx.doi.org/10.1103/PhysRevD.78.054019; http://dx.doi.org/10.1103/PhysRevD.78.054020 and http://dx.doi.org/10.1016/S0370-2693(00)00571-2; http://dx.doi.org/10.1140/epjc/s2003-01565-9; http://dx.doi.org/10.1016/j.physletb.2005.10.054). We also show that, in its regime of applicability H RFT can be simplified. We derive the simpler version of H RFT and in the large N c limit rewrite it in terms of the Reggeon creation and annihilation operators. The resulting H RFT is explicitly self dual and provides the generalization of the Pomeron calculus developed in (http://dx.doi.org/10.1016/S0370-2693(00)00571-2; http://dx.doi.org/10.1140/epjc/s2003-01565-9; http://dx.doi.org/10.1016/j.physletb.2005.10.054) by including higher Reggeons in the evolution. It is applicable for description of ‘large’ Pomeron loops, namely Reggeon graphs where all the splittings occur close in rapidity to one dilute object (projectile), while all the merging close to the other one (target). Additionally we derive, in the same regime expressions for single and double inclusive gluon production (where the gluons are not separated by a large rapidity interval) in terms of the Reggeon degrees of freedom 1. Supergauge Field Theory of Covariant Heterotic Strings OpenAIRE Michio, KAKU; Physics Department, Osaka University : Physics Department, City College of the City University of New York 1986-01-01 We present the gauge covariant second quantized field theory for free heterotic strings, which is leading candidate for a unified theory of all known particles. Our action is invariant under the semi-direct product of the super Virasoro and the Kac-Moody E_8×E_8 or Spin(32)/Z_2 group. We derive the covariant action by path integrals in the same way that Feynman originally derived the Schrodinger equation. By adding an infinite number of auxiliary fields, we can also make the action explicitly... 2. Field theory a path integral approach CERN Document Server Das, Ashok 2006-01-01 This unique book describes quantum field theory completely within the context of path integrals. With its utility in a variety of fields in physics, the subject matter is primarily developed within the context of quantum mechanics before going into specialized areas.Adding new material keenly requested by readers, this second edition is an important expansion of the popular first edition. Two extra chapters cover path integral quantization of gauge theories and anomalies, and a new section extends the supersymmetry chapter, where singular potentials in supersymmetric systems are described. 3. A geometric formulation of exceptional field theory Energy Technology Data Exchange (ETDEWEB) Bosque, Pascal du [Arnold Sommerfeld Center for Theoretical Physics,Department für Physik, Ludwig-Maximilians-Universität München,Theresienstraße 37, 80333 München (Germany); Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, Föhringer Ring 6, 80805 München (Germany); Hassler, Falk [Department of Physics and Astronomy, University of North Carolina, Phillips Hall, CB #3255, 120 E. Cameron Ave., Chapel Hill, NC 27599-3255 (United States); City University of New York, The Graduate Center, 365 Fifth Avenue, New York, NY 10016 (United States); Department of Physics, Columbia University, Pupin Hall, 550 West 120th St., New York, NY 10027 (United States); Lüst, Dieter [Arnold Sommerfeld Center for Theoretical Physics,Department für Physik, Ludwig-Maximilians-Universität München,Theresienstraße 37, 80333 München (Germany); Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, Föhringer Ring 6, 80805 München (Germany); Malek, Emanuel [Arnold Sommerfeld Center for Theoretical Physics,Department für Physik, Ludwig-Maximilians-Universität München,Theresienstraße 37, 80333 München (Germany) 2017-03-01 We formulate the full bosonic SL(5) exceptional field theory in a coordinate-invariant manner. Thereby we interpret the 10-dimensional extended space as a manifold with SL(5)×ℝ{sup +}-structure. We show that the algebra of generalised diffeomorphisms closes subject to a set of closure constraints which are reminiscent of the quadratic and linear constraints of maximal seven-dimensional gauged supergravities, as well as the section condition. We construct an action for the full bosonic SL(5) exceptional field theory, even when the SL(5)×ℝ{sup +}-structure is not locally flat. 4. Statistical field theory of futures commodity prices Science.gov (United States) Baaquie, Belal E.; Yu, Miao 2018-02-01 The statistical theory of commodity prices has been formulated by Baaquie (2013). Further empirical studies of single (Baaquie et al., 2015) and multiple commodity prices (Baaquie et al., 2016) have provided strong evidence in support the primary assumptions of the statistical formulation. In this paper, the model for spot prices (Baaquie, 2013) is extended to model futures commodity prices using a statistical field theory of futures commodity prices. The futures prices are modeled as a two dimensional statistical field and a nonlinear Lagrangian is postulated. Empirical studies provide clear evidence in support of the model, with many nontrivial features of the model finding unexpected support from market data. 5. From topological quantum field theories to supersymmetric gauge theories International Nuclear Information System (INIS) Bossard, G. 2007-10-01 This thesis contains 2 parts based on scientific contributions that have led to 2 series of publications. The first one concerns the introduction of vector symmetry in cohomological theories, through a generalization of the so-called Baulieu-Singer equation. Together with the topological BRST (Becchi-Rouet-Stora-Tyutin) operator, this symmetry gives an off-shell closed sub-sector of supersymmetry that permits to determine the action uniquely. The second part proposes a methodology for re-normalizing supersymmetric Yang-Mills theory without assuming a regularization scheme which is both supersymmetry and gauge invariance preserving. The renormalization prescription is derived thanks to the definition of 2 consistent Slavnov-Taylor operators for supersymmetry and gauge invariance, whose construction requires the introduction of the so-called shadow fields. We demonstrate the renormalizability of supersymmetric Yang-Mills theories. We give a fully consistent, regularization scheme independent, proof of the vanishing of the β function and of the anomalous dimensions of the one half BPS operators in maximally supersymmetric Yang-Mills theory. After a short introduction, in chapter two, we give a review of the cohomological Yang-Mills theory in eight dimensions. We then study its dimensional reductions in seven and six dimensions. The last chapter gives quite independent results, about a geometrical interpretation of the shadow fields, an unpublished work about topological gravity in four dimensions, an extension of the shadow formalism to superconformal invariance, and finally the solution of the constraints in a twisted superspace. (author) 6. Uncertainty quantification theory, implementation, and applications CERN Document Server Smith, Ralph C 2014-01-01 The field of uncertainty quantification is evolving rapidly because of increasing emphasis on models that require quantified uncertainties for large-scale applications, novel algorithm development, and new computational architectures that facilitate implementation of these algorithms. Uncertainty Quantification: Theory, Implementation, and Applications provides readers with the basic concepts, theory, and algorithms necessary to quantify input and response uncertainties for simulation models arising in a broad range of disciplines. The book begins with a detailed discussion of applications where uncertainty quantification is critical for both scientific understanding and policy. It then covers concepts from probability and statistics, parameter selection techniques, frequentist and Bayesian model calibration, propagation of uncertainties, quantification of model discrepancy, surrogate model construction, and local and global sensitivity analysis. The author maintains a complementary web page where readers ca... 7. Effective field theory and the quark model International Nuclear Information System (INIS) Durand, Loyal; Ha, Phuoc; Jaczko, Gregory 2001-01-01 We analyze the connections between the quark model (QM) and the description of hadrons in the low-momentum limit of heavy-baryon effective field theory in QCD. By using a three-flavor-index representation for the effective baryon fields, we show that the 'nonrelativistic' constituent QM for baryon masses and moments is completely equivalent through O(m s ) to a parametrization of the relativistic field theory in a general spin-flavor basis. The flavor and spin variables can be identified with those of effective valence quarks. Conversely, the spin-flavor description clarifies the structure and dynamical interpretation of the chiral expansion in effective field theory, and provides a direct connection between the field theory and the semirelativistic models for hadrons used in successful dynamical calculations. This allows dynamical information to be incorporated directly into the chiral expansion. We find, for example, that the striking success of the additive QM for baryon magnetic moments is a consequence of the relative smallness of the non-additive spin-dependent corrections 8. On space of integrable quantum field theories Directory of Open Access Journals (Sweden) F.A. Smirnov 2017-02-01 Full Text Available We study deformations of 2D Integrable Quantum Field Theories (IQFT which preserve integrability (the existence of infinitely many local integrals of motion. The IQFT are understood as “effective field theories”, with finite ultraviolet cutoff. We show that for any such IQFT there are infinitely many integrable deformations generated by scalar local fields Xs, which are in one-to-one correspondence with the local integrals of motion; moreover, the scalars Xs are built from the components of the associated conserved currents in a universal way. The first of these scalars, X1, coincides with the composite field (TT¯ built from the components of the energy–momentum tensor. The deformations of quantum field theories generated by X1 are “solvable” in a certain sense, even if the original theory is not integrable. In a massive IQFT the deformations Xs are identified with the deformations of the corresponding factorizable S-matrix via the CDD factor. The situation is illustrated by explicit construction of the form factors of the operators Xs in sine-Gordon theory. We also make some remarks on the problem of UV completeness of such integrable deformations. 9. Singular traces theory and applications CERN Document Server Sukochev, Fedor; Zanin, Dmitriy 2012-01-01 This text is the first complete study and monograph dedicated to singular traces. For mathematical readers the text offers, due to Nigel Kalton's contribution, a complete theory of traces on symmetrically normed ideals of compact operators. For mathematical physicists and other users of Connes' noncommutative geometry the text offers a complete reference to Dixmier traces and the deeper mathematical features of singular traces. An application section explores the consequences of these features, which previously were not discussed in general texts on noncommutative geometry. 10. Asymptotic functions and their application in quantum theory International Nuclear Information System (INIS) Khristov, Kh.Ya.; Damyanov, B.P. 1979-01-01 An asymptotic function introduced as a limit for a certain class of successions has been determined. The basic properties of the functions are given: continuity, differentiability, integrability. The fields of application of the asymptotic functions in the quantum field theory are presented. The shortcomings and potentialities of further development of the theory are enumerated 11. Wavelets theory, algorithms, and applications CERN Document Server Montefusco, Laura 2014-01-01 Wavelets: Theory, Algorithms, and Applications is the fifth volume in the highly respected series, WAVELET ANALYSIS AND ITS APPLICATIONS. This volume shows why wavelet analysis has become a tool of choice infields ranging from image compression, to signal detection and analysis in electrical engineering and geophysics, to analysis of turbulent or intermittent processes. The 28 papers comprising this volume are organized into seven subject areas: multiresolution analysis, wavelet transforms, tools for time-frequency analysis, wavelets and fractals, numerical methods and algorithms, and applicat 12. Diatomic interaction potential theory applications CERN Document Server Goodisman, Jerry 2013-01-01 Diatomic Interaction Potential Theory, Volume 2: Applications discusses the variety of applicable theoretical material and approaches in the calculations for diatomic systems in their ground states. The volume covers the descriptions and illustrations of modern calculations. Chapter I discusses the calculation of the interaction potential for large and small values of the internuclear distance R (separated and united atom limits). Chapter II covers the methods used for intermediate values of R, which in principle means any values of R. The Hartree-Fock and configuration interaction schemes des 13. The nonlinearity of the scalar field in a relativistic mean-field theory of the nucleus International Nuclear Information System (INIS) Reinhard, P.G. 1987-10-01 The form of the nonlinear selfcoupling of the scalar meson field in a nuclear relativistic mean-field theory is investigated. The conventional ansatz is shown to produce instabilities in critical applications. A modified selfcoupling is proposed which guarantees stability under all conditions. (orig.) 14. On the general theory of quantized fields International Nuclear Information System (INIS) Fredenhagen, K. 1991-10-01 In my lecture I describe the present stage of the general theory of quantized fields on the example of 5 subjects. They are ordered in the direction from large to small distances. The first one is the by now classical problem of the structure of superselection sectors. It involves the behavior of the theory at spacelike infinity and is directly connected with particle statistics and internal symmetries. It has become popular in recent years by the discovery of a lot of nontrivial models in 2d conformal-field theory, by connections to integrable models and critical behavior in statistical mechanics and by the relations to the Jones' theory of subfactors in von Neumann algebras and to the corresponding geometrical objects (braids, knots, 3d manifolds, ...). At large timelike distances the by far most important feature of quantum field theory is the particle structure. This will be the second subject of my lecture. It follows the technically most involved part which is concerned with the behavior at finite distances. Two aspets, nuclearity which emphasizes the finite density of states in phase space, and the modular structure which relies on the infinite number of degrees of freedom present even locally, and their mutual relations will be treated. The next point, involving the structure at infinitesimal distances, is the connection between the Haag-Kastler framework of algebras of local and the framework of Wightman fields. Finally, problems in approaches to quantum gravity will be discussed, as far as they are accessible by the methods of the general theory of quantized fields. (orig.) 15. On the History of Unified Field Theories Directory of Open Access Journals (Sweden) Goenner Hubert F.M. 2004-01-01 Full Text Available This article is intended to give a review of the history of the classical aspects of unified field theories in the 20th century. It includes brief technical descriptions of the theories suggested, short biographical notes concerning the scientists involved, and an extensive bibliography. The present first installment covers the time span between 1914 and 1933, i.e., when Einstein was living and working in Berlin - with occasional digressions into other periods. Thus, the main theme is the unification of the electromagnetic and gravitational fields augmented by short-lived attempts to include the matter field described by Schrödinger's or Dirac's equations. While my focus lies on the conceptual development of the field, by also paying attention to the interaction of various schools of mathematicians with the research done by physicists, some prosopocraphical remarks are included. 16. Maslow's theory and its application to librarianship OpenAIRE Sridhar, M. S. 1981-01-01 Explains the basis for Maslow’s theory, enumerates Maslow’s hierarchy of needs, describes implications of the theory and finally presents application of Maslow’s theory to librarianship with suitable examples and illustrations. 17. The path integral method in quantum field theory International Nuclear Information System (INIS) Burden, C.J. 1990-01-01 Richard Feynman is reputed to have once the that in his whole life he had only ever had two really clever ideas. As it has turned out, these two ideas, the path integral formulation of quantum mechanics and the diagrammatic representation of perturbation theory have become the cornerstone of modern quantum field theory and particle physics. The path integral, first hinted at by Dirac in the thirties but developed fully by Feynman in the late 40's provides us with an alternative, though equivalent, description of quantum mechanics to canonical quantization. It was not until the seventies that it found broad application in quantum field theory (QFT), leading of gauge theories. It has also lead to the invention of non-perturbative techniques such as lattice gauge theory and has revealed an intimate connection between QFT and statistical mechanics. This paper, the author develops the basic of QFT form the path integral formalism and introduce the idea of Feynman diagrams for calculating perturbation expansions. I will concentrate mainly on the example of φ 4 theory since it is probably the simplest example of an interacting field theory, though the methods generalize readily to more sophisticated theories 18. Symmetry analysis for anisotropic field theories International Nuclear Information System (INIS) Parra, Lorena; Vergara, J. David 2012-01-01 The purpose of this paper is to study with the help of Noether's theorem the symmetries of anisotropic actions for arbitrary fields which generally depend on higher order spatial derivatives, and to find the corresponding current densities and the Noether charges. We study in particular scale invariance and consider the cases of higher derivative extensions of the scalar field, electrodynamics and Chern-Simons theory. 19. Anomalies in Witten's NSR superstring field theory International Nuclear Information System (INIS) Aref'eva, I.Ya.; Medvedev, P.B. 1988-01-01 The action of Witten's NSR superstring field theory if shown to depend on the regularization being choosen to define its value on non-smooth states that are generated by supertransformation. The necessity of additional regularization originates from the appearance of products of picture-changing operators in coincident points. Two different regularization are described, one corresponding to Witten's scheme and the other to the scheme based on the notion of truncated fields 20. Proceedings of quantum field theory, quantum mechanics, and quantum optics International Nuclear Information System (INIS) Dodonov, V.V.; Man; ko, V.I. 1991-01-01 This book contains papers presented at the XVIII International Colloquium on Group Theoretical Methods in Physics held in Moscow on June 4-9, 1990. Topics covered include; applications of algebraic methods in quantum field theory, quantum mechanics, quantum optics, spectrum generating groups, quantum algebras, symmetries of equations, quantum physics, coherent states, group representations and space groups 1. Grassmann methods in lattice field theory and statistical mechanics International Nuclear Information System (INIS) Bilgici, E.; Gattringer, C.; Huber, P. 2006-01-01 Full text: In two dimensions models of loops can be represented as simple Grassmann integrals. In our work we explore the generalization of these techniques to lattice field theories and statistical mechanic systems in three and four dimensions. We discuss possible strategies and applications for representations of loop and surface models as Grassmann integrals. (author) 2. Compositional Data Analysis Theory and Applications CERN Document Server Pawlowsky-Glahn, Vera 2011-01-01 This book presents the state-of-the-art in compositional data analysis and will feature a collection of papers covering theory, applications to various fields of science and software. Areas covered will range from geology, biology, environmental sciences, forensic sciences, medicine and hydrology. Key features:Provides the state-of-the-art text in compositional data analysisCovers a variety of subject areas, from geology to medicineWritten by leading researchers in the fieldIs supported by a website featuring R code 3. Integrable structures in quantum field theory International Nuclear Information System (INIS) Negro, Stefano 2016-01-01 This review was born as notes for a lecture given at the Young Researchers Integrability School (YRIS) school on integrability in Durham, in the summer of 2015. It deals with a beautiful method, developed in the mid-nineties by Bazhanov, Lukyanov and Zamolodchikov and, as such, called BLZ. This method can be interpreted as a field theory version of the quantum inverse scattering, also known as the algebraic Bethe ansatz. Starting with the case of conformal field theories (CFTs) we show how to build the field theory analogues of commuting transfer T matrices and Baxter Q -operators of integrable lattice models. These objects contain the complete information of the integrable structure of the theory, viz. the integrals of motion, and can be used, as we will show, to derive the thermodynamic Bethe ansatz and nonlinear integral equations. This same method can be easily extended to the description of integrable structures of certain particular massive deformations of CFTs; these, in turn, can be described as quantum group reductions of the quantum sine-Gordon model and it is an easy step to include this last theory in the framework of BLZ approach. Finally we show an interesting and surprising connection of the BLZ structures with classical objects emerging from the study of classical integrable models via the inverse scattering transform method. This connection goes under the name of ODE/IM correspondence and we will present it for the specific case of quantum sine-Gordon model only. (topical review) 4. Dual field theories of quantum computation International Nuclear Information System (INIS) Vanchurin, Vitaly 2016-01-01 Given two quantum states of N q-bits we are interested to find the shortest quantum circuit consisting of only one- and two- q-bit gates that would transfer one state into another. We call it the quantum maze problem for the reasons described in the paper. We argue that in a large N limit the quantum maze problem is equivalent to the problem of finding a semiclassical trajectory of some lattice field theory (the dual theory) on an N+1 dimensional space-time with geometrically flat, but topologically compact spatial slices. The spatial fundamental domain is an N dimensional hyper-rhombohedron, and the temporal direction describes transitions from an arbitrary initial state to an arbitrary target state and so the initial and final dual field theory conditions are described by these two quantum computational states. We first consider a complex Klein-Gordon field theory and argue that it can only be used to study the shortest quantum circuits which do not involve generators composed of tensor products of multiple Pauli Z matrices. Since such situation is not generic we call it the Z-problem. On the dual field theory side the Z-problem corresponds to massless excitations of the phase (Goldstone modes) that we attempt to fix using Higgs mechanism. The simplest dual theory which does not suffer from the massless excitation (or from the Z-problem) is the Abelian-Higgs model which we argue can be used for finding the shortest quantum circuits. Since every trajectory of the field theory is mapped directly to a quantum circuit, the shortest quantum circuits are identified with semiclassical trajectories. We also discuss the complexity of an actual algorithm that uses a dual theory prospective for solving the quantum maze problem and compare it with a geometric approach. We argue that it might be possible to solve the problem in sub-exponential time in 2 N , but for that we must consider the Klein-Gordon theory on curved spatial geometry and/or more complicated (than N 5. String amplitudes: from field theories to number theory CERN Multimedia CERN. Geneva 2017-01-01 In a variety of recent developments, scattering amplitudes hint at new symmetries of and unexpected connections between physical theories which are otherwise invisible in their conventional description via Feynman diagrams or Lagrangians. Yet, many of these hidden structures are conveniently accessible to string theory where gauge interactions and gravity arise as the low-energy excitations of open and closed strings. In this talk, I will give an intuitive picture of gravity as a double copy of gauge interactions and extend the web of relations to scalar field theories including chiral Lagrangians for Goldstone bosons. The string corrections to gauge and gravity amplitudes beyond their point-particle limit exhibit elegant mathematical structures and offer a convenient laboratory to explore modern number-theoretic concepts in a simple context. As a common theme with Feynman integrals, string amplitudes introduce a variety of periods and special functions including multiple zeta values and polylogarithms, orga... 6. Correlation functions in finite temperature field theories: formalism and applications to quark-gluon plasma; Fonctions de correlations en theorie des champs a temperature finie: aspects formels et applications au plasma de quarks et de gluons Energy Technology Data Exchange (ETDEWEB) Gelis, Francois [Savoie Univ., 73 - Chambery (France) 1998-12-01 The general framework of this work is thermal field theory, and more precisely the perturbative calculation of thermal Greens functions. In a first part, I consider the problems closely related to the formalism itself. After two introductory chapters devoted to set up the framework and the notations used afterwards, a chapter is dedicated to a clarification of certain aspects of the justification of the Feynman rules of the real time formalism. Then, I consider in the chapter 4 the problem of cutting rules in the real time formalisms. In particular, after solving a controversy on this subject, I generalize these cutting rules to the retarded-advanced version of this formalism. Finally, the last problem considered in this part is that of the pion decay into two photons in a thermal bath. I show that the discrepancies found in the literature are due to peculiarities of the analytical properties of the thermal Greens functions. The second part deals with the calculations of the photons or dilepton (virtual photon) production rate by a quark gluon plasma. The framework of this study is the effective theory based on the resummation of hard thermal loops. The first aspects of this study is related to the production of virtual photons, where we show that important contributions arise at two loops, completing the result already known at one loop. In the case of real photon production, we show that extremely strong collinear singularities make two loop contributions dominant compared to one loop ones. In both cases, the importance of two loop contributions can be interpreted as weaknesses of the hard thermal loop approximation. (author) 366 refs., 109 figs. 7. Cross Sections From Scalar Field Theory Science.gov (United States) Norbury, John W.; Dick, Frank; Norman, Ryan B.; Nasto, Rachel 2008-01-01 A one pion exchange scalar model is used to calculate differential and total cross sections for pion production through nucleon- nucleon collisions. The collisions involve intermediate delta particle production and decay to nucleons and a pion. The model provides the basic theoretical framework for scalar field theory and can be applied to particle production processes where the effects of spin can be neglected. 8. Gravitational descendants in symplectic field theory NARCIS (Netherlands) Fabert, O. 2011-01-01 It was pointed out by Y. Eliashberg in his ICM 2006 plenary talk that the rich algebraic formalism of symplectic field theory leads to a natural appearance of quantum and classical integrable systems, at least in the case when the contact manifold is the prequantization space of a symplectic 9. Quantum field theory with soliton conservation laws CERN Document Server Schrör, B 1978-01-01 Field theories with soliton conservation laws are the most promising candidates for explicitly constructable models. The author exemplifies in the case of the massive Thirring model how the old S matrix bootstrap idea, supplemented with a soliton factorization property, may be used as a systematic starting point for the construction of the S matrix, form factors and (hopefully) correlation functions. (34 refs). 10. Covariant field theory of closed superstrings International Nuclear Information System (INIS) Siopsis, G. 1989-01-01 The authors construct covariant field theories of both type-II and heterotic strings. Toroidal compactification is also considered. The interaction vertices are based on Witten's vertex representing three strings interacting at the mid-point. For closed strings, the authors thus obtain a bilocal interaction 11. Fusion rules in conformal field theory International Nuclear Information System (INIS) Fuchs, J. 1993-06-01 Several aspects of fusion rings and fusion rule algebras, and of their manifestations in two-dimensional (conformal) field theory, are described: diagonalization and the connection with modular invariance; the presentation in terms of quotients of polynomial rings; fusion graphs; various strategies that allow for a partial classification; and the role of the fusion rules in the conformal bootstrap programme. (orig.) 12. The quantum symmetry of rational field theories International Nuclear Information System (INIS) Fuchs, J. 1993-12-01 The quantum symmetry of a rational quantum field theory is a finite-dimensional multi-matrix algebra. Its representation category, which determines the fusion rules and braid group representations of superselection sectors, is a braided monoidal C-category. Various properties of such algebraic structures are described, and some ideas concerning the classification programme are outlined. (orig.) 13. Reconstructing bidimensional scalar field theory models International Nuclear Information System (INIS) Flores, Gabriel H.; Svaiter, N.F. 2001-07-01 In this paper we review how to reconstruct scalar field theories in two dimensional spacetime starting from solvable Scrodinger equations. Theree different Schrodinger potentials are analyzed. We obtained two new models starting from the Morse and Scarf II hyperbolic potencials, the U (θ) θ 2 In 2 (θ 2 ) model and U (θ) = θ 2 cos 2 (In(θ 2 )) model respectively. (author) 14. General relativity invariance and string field theory International Nuclear Information System (INIS) Aref'eva, I.Ya.; Volovich, I.V. 1987-04-01 The general covariance principle in the string field theory is considered. The algebraic properties of the string Lie derivative are discussed. The string vielbein and spin connection are introduced and an action invariant under general co-ordinate transformation is proposed. (author). 18 refs 15. Construction of topological field theories using BV NARCIS (Netherlands) Jonghe, F. de; Vandoren, S. 1993-01-01 We discuss in detail the construction of topological field theories us- ing the Batalin–Vilkovisky (BV) quantisation scheme. By carefully examining the dependence of the antibracket on an external metric, we show that differentiating with respect to the metric and the BRST charge do not commute 16. Wilson lines in quantum field theory CERN Document Server Cherednikov, Igor O; Veken, Frederik F van der 2014-01-01 The objective of this book is to get the reader acquainted with theoretical and mathematical foundations of the concept of Wilson loops in the context of modern quantum field theory. It teaches how to perform independently with some elementary calculations on Wilson lines, and shows the recent development of the subject in different important areas of research. 17. Translationally invariant self-consistent field theories International Nuclear Information System (INIS) Shakin, C.M.; Weiss, M.S. 1977-01-01 We present a self-consistent field theory which is translationally invariant. The equations obtained go over to the usual Hartree-Fock equations in the limit of large particle number. In addition to deriving the dynamic equations for the self-consistent amplitudes we discuss the calculation of form factors and various other observables 18. Causality and analyticity in quantum fields theory International Nuclear Information System (INIS) Iagolnitzer, D. 1992-01-01 This is a presentation of results on the causal and analytical structure of Green functions and on the collision amplitudes in fields theories, for massive particles of one type, with a positive mass and a zero spin value. (A.B.) 19. Asymptotic mass degeneracies in conformal field theories International Nuclear Information System (INIS) Kani, I.; Vafa, C. 1990-01-01 By applying a method of Hardy and Ramanujan to characters of rational conformal field theories, we find an asymptotic expansion for degeneracy of states in the limit of large mass which is exact for strings propagating in more than two uncompactified space-time dimensions. Moreover we explore how the rationality of the conformal theory is reflected in the degeneracy of states. We also consider the one loop partition function for strings, restricted to physical states, for arbitrary (irrational) conformal theories, and obtain an asymptotic expansion for it in the limit that the torus degenerates. This expansion depends only on the spectrum of (physical and unphysical) relevant operators in the theory. We see how rationality is consistent with the smoothness of mass degeneracies as a function of moduli. (orig.) 20. Superconformal quantum field theories in string. Gauge theory dualities Energy Technology Data Exchange (ETDEWEB) Wiegandt, Konstantin 2012-08-14 In this thesis aspects of superconformal field theories that are of interest in the so-called AdS/CFT correspondence are investigated. The AdS/CFT correspondence states a duality between string theories living on Anti-de Sitter space and superconformal quantum field theories in Minkowski space. In the context of the AdS/CFT correspondence the so-called Wilson loop/amplitude duality was discovered, stating the equality of the finite parts of n-gluon MHV amplitudes and n-sided lightlike polygonal Wilson loops in N=4 supersymmetric Yang-Mills (SYM) theory. It is the subject of the first part of this thesis to investigate the Wilson loop side of a possible similar duality in N=6 superconformal Chern-Simons matter (ABJM) theory. The main result is, that the expectation value of n-sided lightlike polygonal Wilson loops vanishes at one-loop order and at two-loop order is identical in its functional form to the Wilson loop in N=4 SYM theory at one-loop order. Furthermore, an anomalous conformal Ward identity for Wilson loops in Chern-Simons theory is derived. Related developments and symmetries of amplitudes and correlators in ABJM theory are discussed as well. In the second part of this thesis we calculate three-point functions of two protected operators and one twist-two operator with arbitrary even spin j in N=4 SYM theory. In order to carry out the calculations, the indices of the spin j operator are projected to the light-cone and the correlator is evaluated in a soft-limit where the momentum coming in at the spin j operator becomes zero. This limit largely simplifies the perturbative calculation, since all three-point diagrams effectively reduce to two-point diagrams and the dependence on the one-loop mixing matrix drops out completely. The result is in agreement with the analysis of the operator product expansion of four-point functions of half-BPS operators by Dolan and Osborn in 2004. 1. A periodic table of effective field theories Energy Technology Data Exchange (ETDEWEB) Cheung, Clifford [Walter Burke Institute for Theoretical Physics,California Institute of Technology,Pasadena, CA (United States); Kampf, Karol; Novotny, Jiri [Institute of Particle and Nuclear Physics,Faculty of Mathematics and Physics, Charles University,Prague (Czech Republic); Shen, Chia-Hsien [Walter Burke Institute for Theoretical Physics,California Institute of Technology,Pasadena, CA (United States); Trnka, Jaroslav [Center for Quantum Mathematics and Physics (QMAP),Department of Physics, University of California,Davis, CA (United States) 2017-02-06 We systematically explore the space of scalar effective field theories (EFTs) consistent with a Lorentz invariant and local S-matrix. To do so we define an EFT classification based on four parameters characterizing 1) the number of derivatives per interaction, 2) the soft properties of amplitudes, 3) the leading valency of the interactions, and 4) the spacetime dimension. Carving out the allowed space of EFTs, we prove that exceptional EFTs like the non-linear sigma model, Dirac-Born-Infeld theory, and the special Galileon lie precisely on the boundary of allowed theory space. Using on-shell momentum shifts and recursion relations, we prove that EFTs with arbitrarily soft behavior are forbidden and EFTs with leading valency much greater than the spacetime dimension cannot have enhanced soft behavior. We then enumerate all single scalar EFTs in d<6 and verify that they correspond to known theories in the literature. Our results suggest that the exceptional theories are the natural EFT analogs of gauge theory and gravity because they are one-parameter theories whose interactions are strictly dictated by properties of the S-matrix. 2. Noncommutative gravity and quantum field theory on noncummutative curved spacetimes International Nuclear Information System (INIS) Schenkel, Alexander 2011-01-01 quantum field theory at short distances, i.e. in the ultraviolet. In the third part we develop elements of a more powerful, albeit more abstract, mathematical approach to noncommutative gravity. The goal is to better understand global aspects of homomorphisms between and connections on noncommutative vector bundles, which are fundamental objects in the mathematical description of noncommutative gravity. We prove that all homomorphisms and connections of the deformed theory can be obtained by applying a quantization isomorphism to undeformed homomorphisms and connections. The extension of homomorphisms and connections to tensor products of modules is clarified, and as a consequence we are able to add tensor fields of arbitrary type to the noncommutative gravity theory of Wess et al. As a nontrivial application of the new mathematical formalism we extend our studies of exact noncommutative gravity solutions to more general deformations. 3. Noncommutative gravity and quantum field theory on noncummutative curved spacetimes Energy Technology Data Exchange (ETDEWEB) Schenkel, Alexander 2011-10-24 noncommutative quantum field theory at short distances, i.e. in the ultraviolet. In the third part we develop elements of a more powerful, albeit more abstract, mathematical approach to noncommutative gravity. The goal is to better understand global aspects of homomorphisms between and connections on noncommutative vector bundles, which are fundamental objects in the mathematical description of noncommutative gravity. We prove that all homomorphisms and connections of the deformed theory can be obtained by applying a quantization isomorphism to undeformed homomorphisms and connections. The extension of homomorphisms and connections to tensor products of modules is clarified, and as a consequence we are able to add tensor fields of arbitrary type to the noncommutative gravity theory of Wess et al. As a nontrivial application of the new mathematical formalism we extend our studies of exact noncommutative gravity solutions to more general deformations. 4. Quantum tunneling and field electron emission theories CERN Document Server Liang, Shi-Dong 2013-01-01 Quantum tunneling is an essential issue in quantum physics. Especially, the rapid development of nanotechnology in recent years promises a lot of applications in condensed matter physics, surface science and nanodevices, which are growing interests in fundamental issues, computational techniques and potential applications of quantum tunneling. The book involves two relevant topics. One is quantum tunneling theory in condensed matter physics, including the basic concepts and methods, especially for recent developments in mesoscopic physics and computational formulation. The second part is the f 5. A symplectic framework for field theories International Nuclear Information System (INIS) Kijowski, J.; Tulczyjew, W.M. 1979-01-01 These notes are concerned with the formulation of a new conceptual framework for classical field theories. Although the formulation is based on fairly advanced concepts of symplectic geometry these notes cannot be viewed as a reformulation of known structures in more rigorous and elegant torns. Our intention is rather to communicate to theoretical physicists a set of new physical ideas. We have chosen for this purpose the language of local coordinates which is more elementary and more widely known than the abstract language of modern differntial geometry. Our emphasis is directed more to physical intentions than to mathematical vigour. We start with a symplectic analysis of staties. Both discrete and continuous systems are considered on a largely intuitive level. The notion of reciprocity and potentiality of the theory is discussed. Chapter II is a presentation of particle dynamics together with more rigorous definitions of the geometric structure. Lagrangian-Submanifolds and their generating function 3 are defined and the time evolution of particle states is studied. Chapter II form the main part of these notes. Here we describe the construction of canonical momenta and discuss the field dynamics in finite domains of space-time. We also establish the relation between our symplectic framework and the geometric formulation of the calculus of variations of multiple integrals. In the following chapter we give a few examples of field theories selected to illustrate various features of the new approach. A new formulation of the theory of gravity consists of using the affine connection in space-time as the field configuration. In the past section we present an analysis of hydrodynamics within our framework which reveals a formal analogy with electrodynamics. The discovery of potentials for hydrodynamics and the subsequent formulation of a variational principle provides an excellent example for the fruitfulness of the new approach to field theory. A short review of 6. Management applications of discontinuity theory Science.gov (United States) Angeler, David G.; Allen, Craig R.; Barichievy, Chris; Eason, Tarsha; Garmestani, Ahjond S.; Graham, Nicholas A.J.; Granholm, Dean; Gunderson, Lance H.; Knutson, Melinda; Nash, Kirsty L.; Nelson, R. John; Nystrom, Magnus; Spanbauer, Trisha; Stow, Craig A.; Sundstrom, Shana M. 2015-01-01 Human impacts on the environment are multifaceted and can occur across distinct spatiotemporal scales. Ecological responses to environmental change are therefore difficult to predict, and entail large degrees of uncertainty. Such uncertainty requires robust tools for management to sustain ecosystem goods and services and maintain resilient ecosystems.We propose an approach based on discontinuity theory that accounts for patterns and processes at distinct spatial and temporal scales, an inherent property of ecological systems. Discontinuity theory has not been applied in natural resource management and could therefore improve ecosystem management because it explicitly accounts for ecological complexity.Synthesis and applications. We highlight the application of discontinuity approaches for meeting management goals. Specifically, discontinuity approaches have significant potential to measure and thus understand the resilience of ecosystems, to objectively identify critical scales of space and time in ecological systems at which human impact might be most severe, to provide warning indicators of regime change, to help predict and understand biological invasions and extinctions and to focus monitoring efforts. Discontinuity theory can complement current approaches, providing a broader paradigm for ecological management and conservation. 7. Quantum field theory in topology changing spacetimes International Nuclear Information System (INIS) Bauer, W. 2007-03-01 The goal of this diploma thesis is to present an overview of how to reduce the problem of topology change of general spacetimes to the investigation of elementary cobordisms. In the following we investigate the possibility to construct quantum fields on elementary cobordisms, in particular we discuss the trousers topology. Trying to avoid the problems occuring at spacetimes with instant topology change we use a model for simulating topology change. We construct the algebra of observables for a free scalar field with the algebraic approach to quantum field theory. Therefore we determine a fundamental solution of the eld equation. (orig.) 8. Continuous and distributed systems theory and applications CERN Document Server 2014-01-01 In this volume, the authors close the gap between abstract mathematical approaches, such as abstract algebra, number theory, nonlinear functional analysis, partial differential equations, methods of nonlinear and multi-valued analysis, on the one hand, and practical applications in nonlinear mechanics, decision making theory and control theory on the other. Readers will also benefit from the presentation of modern mathematical modeling methods for the numerical solution of complicated engineering problems in hydromechanics, geophysics and mechanics of continua. This compilation will be of interest to mathematicians and engineers working at the interface of these field. It presents selected works of the open seminar series of Lomonosov Moscow State University and the National Technical University of Ukraine “Kyiv Polytechnic Institute”. The authors come from Germany, Italy, Spain, Russia, Ukraine, and the USA. 9. Scattering of decuplet baryons in chiral effective field theory Energy Technology Data Exchange (ETDEWEB) Haidenbauer, J. [Institut fuer Kernphysik, Institute for Advanced Simulation and Juelich Center for Hadron Physics, Juelich (Germany); Petschauer, S.; Kaiser, N.; Weise, W. [Technische Universitaet Muenchen, Physik Department, Garching (Germany); Meissner, Ulf G. [Institut fuer Kernphysik, Institute for Advanced Simulation and Juelich Center for Hadron Physics, Juelich (Germany); Universitaet Bonn, Helmholtz-Institut fuer Strahlen- und Kernphysik and Bethe Center for Theoretical Physics, Bonn (Germany) 2017-11-15 A formalism for treating the scattering of decuplet baryons in chiral effective field theory is developed. The minimal Lagrangian and potentials in leading-order SU(3) chiral effective field theory for the interactions of octet baryons (B) and decuplet baryons (D) for the transitions BB → BB, BB <-> DB, DB → DB, BB <-> DD, DB <-> DD, and DD → DD are provided. As an application of the formalism we compare with results from lattice QCD simulations for ΩΩ and NΩ scattering. Implications of our results pertinent to the quest for dibaryons are discussed. (orig.) 10. Symmetry aspects of nonholonomic field theories Energy Technology Data Exchange (ETDEWEB) Vankerschaver, Joris [Control and Dynamical Systems, California Institute of Technology, MC 107-81, Pasadena, CA 91125 (United States); Diego, David MartIn de [Instituto de Matematicas y Fisica Fundamental, Consejo Superior de Investigaciones CientIficas, Serrano 123, 28006 Madrid (Spain) 2008-01-25 The developments in this paper are concerned with nonholonomic field theories in the presence of symmetries. Having previously treated the case of vertical symmetries, we now deal with the case where the symmetry action can also have a horizontal component. As a first step in this direction, we derive a new and convenient form of the field equations of a nonholonomic field theory. Nonholonomic symmetries are then introduced as symmetry generators whose virtual work is zero along the constraint submanifold, and we show that for every such symmetry, there exists a so-called momentum equation, describing the evolution of the associated component of the momentum map. Keeping up with the underlying geometric philosophy, a small modification of the derivation of the momentum lemma allows us to also treat generalized nonholonomic symmetries, which are vector fields along a projection. Such symmetries arise for example in practical examples of nonholonomic field theories such as the Cosserat rod, for which we recover both energy conservation (a previously known result) and a modified conservation law associated with spatial translations. 11. Mean fields and self consistent normal ordering of lattice spin and gauge field theories International Nuclear Information System (INIS) Ruehl, W. 1986-01-01 Classical Heisenberg spin models on lattices possess mean field theories that are well defined real field theories on finite lattices. These mean field theories can be self consistently normal ordered. This leads to a considerable improvement over standard mean field theory. This concept is carried over to lattice gauge theories. We construct first an appropriate real mean field theory. The equations determining the Gaussian kernel necessary for self-consistent normal ordering of this mean field theory are derived. (orig.) 12. Interaction vertices in reduced string field theories International Nuclear Information System (INIS) Embacher, F. 1989-01-01 In contrast to previous expectations, covariant overlap vertices are not always suitable for gauge-covariant formulations of bosonic string field theory with a reduced supplementary field content. This is demonstrated for the version of the theory suggested by Neveu, Schwarz and West. The method to construct the interaction, as formulated by Neveu and West, fails at one level higher than these authors have considered. The condition for a general vertex to describe formally a local gauge-invariant interaction is derived. The solution for the action functional and the gauge transformation law is exhibited for all fields at once, to the first order in the coupling constant. However, all these vertices seem to be unphysical. 21 refs. (Author) 13. Extending Gurwitsch's field theory of consciousness. Science.gov (United States) Yoshimi, Jeff; Vinson, David W 2015-07-01 Aron Gurwitsch's theory of the structure and dynamics of consciousness has much to offer contemporary theorizing about consciousness and its basis in the embodied brain. On Gurwitsch's account, as we develop it, the field of consciousness has a variable sized focus or "theme" of attention surrounded by a structured periphery of inattentional contents. As the field evolves, its contents change their status, sometimes smoothly, sometimes abruptly. Inner thoughts, a sense of one's body, and the physical environment are dominant field contents. These ideas can be linked with (and help unify) contemporary theories about the neural correlates of consciousness, inattention, the small world structure of the brain, meta-stable dynamics, embodied cognition, and predictive coding in the brain. Published by Elsevier Inc. 14. Quantum field theory the why, what and how CERN Document Server 2016-01-01 This book describes, in clear terms, the Why, What and the How of Quantum Field Theory. The raison d'etre of QFT is explained by starting from the dynamics of a relativistic particle and demonstrating how it leads to the notion of quantum fields. Non-perturbative aspects and the Wilsonian interpretation of field theory are emphasized right from the start. Several interesting topics such as the Schwinger effect, Davies-Unruh effect, Casimir effect and spontaneous symmetry breaking introduce the reader to the elegance and breadth of applicability of field theoretical concepts. Complementing the conceptual aspects, the book also develops all the relevant mathematical techniques in detail, leading e.g., to the computation of anomalous magnetic moment of the electron and the two-loop renormalisation of the self-interacting scalar field. It contains nearly a hundred problems, of varying degrees of difficulty, making it suitable for both self-study and classroom use. 15. A Generalized Field Theory: Charged Spherical Symmetric Solution Science.gov (United States) Wanas, M. I. 1985-06-01 Three solutions with spherical symmetry are obtained for the field equations of the generalized field theory established recently by Mikhail and Wanas. The solutions found are in agreement with classical known results. The solution representing a generalized field, outside a spherical symmetric charged body, is found to have an extra term compared with the Reissner-Nordström metric. The space used for application is of type FIGI, so the solutions obtained correspond to a field in a matter-free space. A brief comparison between the solutions obtained and those given by other field theories is given. Two methods have been used to get physical results: the first is the type analysis, and the second is the comparison with classical known results by writing down the metric of the associated Riemannian space. 16. Cosmological field theory for observational astronomers International Nuclear Information System (INIS) Zel'Dovich, Y.B. 1987-01-01 Theories of the very early Universe that use scalar fields (i.e., the so-called inflationary models of the Universe) have now come into wide use. The inflationary universe approach may perhaps solve some of the most difficult enigmas about the Universe as a whole. The inflationary universe forms a good bridge between the quantum theory of the birth of the Universe (which is still in the initial stages of development) and the standard hot Big Bang theory (which is well established, at least qualitatively). Therefore, an understanding of the basic ideas of inflation is a must for astronomers interested in the broad picture of the science. Astronomers are mathematically oriented enough (via celestial mechanics, electromagnetic theory, magnetohydrodynamics, nuclear reactions,etc.) that there is no negative attitude towards formulae in general. What the astronomer lacks is a knowledge of recent developments in particle physics and field theory. The astronomer should not be blamed for this, because these branches of physics are developing in a very peculiar fashion: some subfields of it are progressing comparatively slowly, with experimental verifications at each and every step, while other subfields progress rapidly 17. Copula Theory and Its Applications CERN Document Server Jaworski, Piotr; Hardle, Wolfgang Karl; Rychlik, Tomasz 2010-01-01 Copulas are mathematical objects that fully capture the dependence structure among random variables and hence offer great flexibility in building multivariate stochastic models. Since their introduction in the early 50's, copulas have gained considerable popularity in several fields of applied mathematics, such as finance, insurance and reliability theory. Today, they represent a well-recognized tool for market and credit models, aggregation of risks, portfolio selection, etc. This book is divided into two main parts: Part I - 'Surveys' contains 11 chapters that provide an up-to-date account o 18. Relating the archetypes of logarithmic conformal field theory International Nuclear Information System (INIS) Creutzig, Thomas; Ridout, David 2013-01-01 Logarithmic conformal field theory is a rich and vibrant area of modern mathematical physics with well-known applications to both condensed matter theory and string theory. Our limited understanding of these theories is based upon detailed studies of various examples that one may regard as archetypal. These include the c=−2 triplet model, the Wess–Zumino–Witten model on SL(2;R) at level k=−1/2 , and its supergroup analogue on GL(1\|1). Here, the latter model is studied algebraically through representation theory, fusion and modular invariance, facilitating a subsequent investigation of its cosets and extended algebras. The results show that the archetypes of logarithmic conformal field theory are in fact all very closely related, as are many other examples including, in particular, the SL(2\|1) models at levels 1 and −1/2 . The conclusion is then that the archetypal examples of logarithmic conformal field theory are practically all the same, so we should not expect that their features are in any way generic. Further archetypal examples must be sought 19. Relating the archetypes of logarithmic conformal field theory Energy Technology Data Exchange (ETDEWEB) Creutzig, Thomas, E-mail: [email protected] [Department of Physics and Astronomy, University of North Carolina, Phillips Hall, CB 3255, Chapel Hill, NC 27599-3255 (United States); Fachbereich Mathematik, Technische Universität Darmstadt, Schloßgartenstraße 7, 64289 Darmstadt (Germany); Ridout, David, E-mail: [email protected] [Department of Theoretical Physics, Research School of Physics and Engineering, Australian National University, Canberra, ACT 0200 (Australia); Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200 (Australia) 2013-07-21 Logarithmic conformal field theory is a rich and vibrant area of modern mathematical physics with well-known applications to both condensed matter theory and string theory. Our limited understanding of these theories is based upon detailed studies of various examples that one may regard as archetypal. These include the c=−2 triplet model, the Wess–Zumino–Witten model on SL(2;R) at level k=−1/2 , and its supergroup analogue on GL(1\|1). Here, the latter model is studied algebraically through representation theory, fusion and modular invariance, facilitating a subsequent investigation of its cosets and extended algebras. The results show that the archetypes of logarithmic conformal field theory are in fact all very closely related, as are many other examples including, in particular, the SL(2\|1) models at levels 1 and −1/2 . The conclusion is then that the archetypal examples of logarithmic conformal field theory are practically all the same, so we should not expect that their features are in any way generic. Further archetypal examples must be sought. 20. Noncommutative analysis, operator theory and applications CERN Document Server Cipriani, Fabio; Colombo, Fabrizio; Guido, Daniele; Sabadini, Irene; Sauvageot, Jean-Luc 2016-01-01 This book illustrates several aspects of the current research activity in operator theory, operator algebras and applications in various areas of mathematics and mathematical physics. It is addressed to specialists but also to graduate students in several fields including global analysis, Schur analysis, complex analysis, C-algebras, noncommutative geometry, operator algebras, operator theory and their applications. Contributors: F. Arici, S. Bernstein, V. Bolotnikov, J. Bourgain, P. Cerejeiras, F. Cipriani, F. Colombo, F. D'Andrea, G. Dell'Antonio, M. Elin, U. Franz, D. Guido, T. Isola, A. Kula, L.E. Labuschagne, G. Landi, W.A. Majewski, I. Sabadini, J.-L. Sauvageot, D. Shoikhet, A. Skalski, H. de Snoo, D. C. Struppa, N. Vieira, D.V. Voiculescu, and H. Woracek. 1. Social signals: from theory to applications. Science.gov (United States) Poggi, Isabella; D'Errico, Francesca; Vinciarelli, Alessandro 2012-10-01 The Special Issue Editorial introduces the research milieu in which Social Signal Processing originates, by merging computer scientists and social scientists and giving rise to this field in parallel with Human-Computer Interaction, Affective Computing, and Embodied Conversational Agents, all similarly characterized by high interdisciplinarity, stress on multimodality of communication, and the continuous loop from theory to simulation and application. Some frameworks of the cognitive and social processes underlying social signals are identified as reference points (Theory of Mind and Intersubjectivity, mirror neurons, and the ontogenesis and phylogenesis of communication), while three dichotomies (automatic vs. controlled, individualistic vs. intersubjective, and meaning vs. influence) are singled out as leads to navigate within the theoretical and applicative studies presented in the Special Issue. 2. BRST quantization of topological field theories International Nuclear Information System (INIS) Birmingham, D.; Rakowski, M.; Thompson, G. 1988-07-01 We consider in detail the construction of a variety of topological quantum field theories through BRST quantization. In particular, we show that supersymmetric quantum mechanics on an arbitrary Riemannian manifold can be obtained as the BRST quantization of a purely bosonic theory. The introduction of a new local symmetry allows for the possibility of different gauge choices, and we show how this freedom can simplify the evaluation of the Witten index in certain cases. Topological sigma models are also constructed via the same mechanism. In three dimensions, we consider a Yang-Mills-Higgs model related to the four dimensional TQFT of Witten. (author). 24 refs 3. Thermo field theory versus imaginary time formalism International Nuclear Information System (INIS) Fujimoto, Y.; Nishino, H.; Grigjanis, R. 1983-11-01 We calculate a two-loop diagram at finite temperature to compare Thermo Field Theory (=Th.F.Th.) with the conventional imaginary time formalism (=Im.T.F.). The summation over the Matsubara frequency in Im.T.F. is carried out at two-loop level, and the result is shown to coincide with that of Th.F.Th. We confirm that in Im.T.F. the temperature dependent divergences cancel out at least in the calculation of effective potential of phi 4 theory, as in Th.F.Th. (author) 4. Recursion equations in gauge field theories International Nuclear Information System (INIS) Migdal, A.A. 1975-01-01 An approximate recursive equation describing scale transformation of the effective action of a gauging field has been formulated. The equation becomes exact in the two-dimensional space-time. In the four-dimensional theory it reproduces the asymptotic freedom with an accuracy of 30% in β-function coefficients. In the region of strong coupling β-function remains negative, that leads to an asymptotic ''prison'' in the infrared range. Some possible generalizations and appendices to the colour quark-gluon gauging theory are being discussed 5. Twistors and supertwistors for exceptional field theory Energy Technology Data Exchange (ETDEWEB) Cederwall, Martin [Dept. of Fundamental Physics, Chalmers University of Technology, Gothenburg, SE 412 96 (Sweden) 2015-12-18 As a means of examining the section condition and its possible solutions and relaxations, we perform twistor transforms related to versions of exceptional field theory with Minkowski signature. The spinor parametrisation of the momenta naturally solves simultaneously both the mass-shell condition and the (weak) section condition. It is shown that the incidence relations for multi-particle twistors force them to share a common section, but not to be orthogonal. The supersymmetric extension contains additional scalar fermionic variables shown to be kappa-symmetry invariants. We speculate on some implications, among them a possible relation to higher spin theory. 6. Biometrics Theory, Methods, and Applications CERN Document Server Boulgouris, N V; Micheli-Tzanakou, Evangelia 2009-01-01 An in-depth examination of the cutting edge of biometrics. This book fills a gap in the literature by detailing the recent advances and emerging theories, methods, and applications of biometric systems in a variety of infrastructures. Edited by a panel of experts, it provides comprehensive coverage of:. Multilinear discriminant analysis for biometric signal recognition;. Biometric identity authentication techniques based on neural networks;. Multimodal biometrics and design of classifiers for biometric fusion;. Feature selection and facial aging modeling for face recognition;. Geometrical and 7. Quantile regression theory and applications CERN Document Server Davino, Cristina; Vistocco, Domenico 2013-01-01 A guide to the implementation and interpretation of Quantile Regression models This book explores the theory and numerous applications of quantile regression, offering empirical data analysis as well as the software tools to implement the methods. The main focus of this book is to provide the reader with a comprehensivedescription of the main issues concerning quantile regression; these include basic modeling, geometrical interpretation, estimation and inference for quantile regression, as well as issues on validity of the model, diagnostic tools. Each methodological aspect is explored and 8. Surface chemistry theory and applications CERN Document Server Bikerman, J J 2013-01-01 Surface Chemistry Theory and Applications focuses on liquid-gas, liquid-liquid, solid-gas, solid-liquid, and solid-solid surfaces. The book first offers information on liquid-gas surfaces, including surface tension, measurement of surface tension, rate of capillarity rise, capillary attraction, bubble pressure and pore size, and surface tension and temperature. The text then ponders on liquid-liquid and solid-gas surfaces. Discussions focus on surface energy of solids, surface roughness and cleanness, adsorption of gases and vapors, adsorption hysteresis, interfacial tension, and interfacial t 9. Self-consistent normal ordering of gauge field theories International Nuclear Information System (INIS) Ruehl, W. 1987-01-01 Mean-field theories with a real action of unconstrained fields can be self-consistently normal ordered. This leads to a considerable improvement over standard mean-field theory. This concept is applied to lattice gauge theories. First an appropriate real action mean-field theory is constructed. The equations determining the Gaussian kernel necessary for self-consistent normal ordering of this mean-field theory are derived. (author). 4 refs 10. Protective relaying theory and applications CERN Document Server Elmore, Walter A 2003-01-01 Targeting the latest microprocessor technologies for more sophisticated applications in the field of power system short circuit detection, this revised and updated source imparts fundamental concepts and breakthrough science for the isolation of faulty equipment and minimization of damage in power system apparatus. The Second Edition clearly describes key procedures, devices, and elements crucial to the protection and control of power system function and stability. It includes chapters and expertise from the most knowledgeable experts in the field of protective relaying, and describes micropro 11. Effective field theory for halo nuclei International Nuclear Information System (INIS) Hagen, Philipp Robert 2014-01-01 We investigate properties of two- and three-body halo systems using effective field theory. If the two-particle scattering length a in such a system is large compared to the typical range of the interaction R, low-energy observables in the strong and the electromagnetic sector can be calculated in halo EFT in a controlled expansion in R/ vertical stroke a vertical stroke. Here we focus on universal properties and stay at leading order in the expansion. Motivated by the existence of the P-wave halo nucleus 6 He, we first set up an EFT framework for a general three-body system with resonant two-particle P-wave interactions. Based on a Lagrangian description, we identify the area in the effective range parameter space where the two-particle sector of our model is renormalizable. However, we argue that for such parameters, there are two two-body bound states: a physical one and an additional deeper-bound and non-normalizable state that limits the range of applicability of our theory. With regard to the three-body sector, we then classify all angular-momentum and parity channels that display asymptotic discrete scale invariance and thus require renormalization via a cut-off dependent three-body force. In the unitary limit an Efimov effect occurs. However, this effect is purely mathematical, since, due to causality bounds, the unitary limit for P-wave interactions can not be realized in nature. Away from the unitary limit, the three-body binding energy spectrum displays an approximate Efimov effect but lies below the unphysical, deep two-body bound state and is thus unphysical. Finally, we discuss possible modifications in our halo EFT approach with P-wave interactions that might provide a suitable way to describe physical three-body bound states. We then set up a halo EFT formalism for two-neutron halo nuclei with resonant two-particle S-wave interactions. Introducing external currents via minimal coupling, we calculate observables and universal correlations for such 12. Automata theory and its applications CERN Document Server 2001-01-01 The theory of finite automata on finite stings, infinite strings, and trees has had a dis tinguished history. First, automata were introduced to represent idealized switching circuits augmented by unit delays. This was the period of Shannon, McCullouch and Pitts, and Howard Aiken, ending about 1950. Then in the 1950s there was the work of Kleene on representable events, of Myhill and Nerode on finite coset congruence relations on strings, of Rabin and Scott on power set automata. In the 1960s, there was the work of Btichi on automata on infinite strings and the second order theory of one successor, then Rabin's 1968 result on automata on infinite trees and the second order theory of two successors. The latter was a mystery until the introduction of forgetful determinacy games by Gurevich and Harrington in 1982. Each of these developments has successful and prospective applications in computer science. They should all be part of every computer scientist's toolbox. Suppose that we take a computer scientist's ... 13. Quantum field theory and critical phenomena CERN Document Server Zinn-Justin, Jean 1996-01-01 Over the last twenty years quantum field theory has become not only the framework for the discussion of all fundamental interactions except gravity, but also for the understanding of second-order phase transitions in statistical mechanics. This advanced text is based on graduate courses and summer schools given by the author over a number of years. It approaches the subject in terms of path and functional intergrals, adopting a Euclidean metric and using the language of partition and correlation functions. Renormalization and the renormalization group are examined, as are critical phenomena and the role of instantons. Changes for this edition 1. Extensive revision to eliminate a few bugs that had survived the second edition and (mainly) to improve the pedagogical presentation, as a result of experience gathered by lecturing. 2. Additional new topics; holomorphic or coherent state path integral; functional integral and representation of the field theory S-matrix in the holomorphic formalis; non-relativistic li... 14. Propositional systems in local field theories International Nuclear Information System (INIS) Banai, M. 1980-07-01 The authors investigate propositional systems for local field theories, which reflect intrinsically the uncertainties of measurements made on the physical system, and satisfy the isotony and local commutativity postulates of Haag and Kastler. The spacetime covariance can be implemented in natural way in these propositional systems. New techniques are introduced to obtain these propositional systems: the lattice-valued logics. The decomposition of the complete orthomodular lattice-valued logics shows that these logics are more general than the usual two-valued ones and that in these logics there is enough structure to characterize the classical and quantum, non relativistic and relativistic local field theories in a natural way. The Hilbert modules give the natural inner product ''spaces'' (modules) for the realization of the lattice-valued logics. (author) 15. Propositional systems in local field theories Energy Technology Data Exchange (ETDEWEB) Banai, M. 1981-03-01 We investigate propositional systems for local field theories, which reflect intrinsically the uncertainties of measurements made on the physical system, and satisfy the isotony and local commutativity postulates of Haag and Kastler. The space-time covariance can be implemented in a natural way in these propositional systems. New techniques are introduced to obtain these propositional systems: the lattice-valued logics. The decomposition of the complete orthomodular lattice-valued logics shows that these logics are more general than the usual two-valued ones and that in these there is enough structure to characterize the classical and quantum, nonrelativistic and relativistic local field theories in a natural way. The Hilbert modules give the natural inner product ''spaces'' (modules) for the realization of the lattice-valued logics. 16. Magnetic fields and density functional theory International Nuclear Information System (INIS) Salsbury, Freddie Jr. 1999-01-01 A major focus of this dissertation is the development of functionals for the magnetic susceptibility and the chemical shielding within the context of magnetic field density functional theory (BDFT). These functionals depend on the electron density in the absence of the field, which is unlike any other treatment of these responses. There have been several advances made within this theory. The first of which is the development of local density functionals for chemical shieldings and magnetic susceptibilities. There are the first such functionals ever proposed. These parameters have been studied by constructing functionals for the current density and then using the Biot-Savart equations to obtain the responses. In order to examine the advantages and disadvantages of the local functionals, they were tested numerically on some small molecules 17. New ideas about unified field theory International Nuclear Information System (INIS) Gleiser, M. 1986-01-01 An outline of the physical concepts evolution is given from the ancient philosophers to the present time. With qualitative explanations about the meaning of the theories that is the milestones of these concepts evolution, it mentions the ideas which lead the studies to the conception of a unified field theory. Chronologically, it has brief information about the ideas of Laplace (mechanical determinism), Maxwell (the field concept), Einsten (the space-time structure), Heisenberg and Schroedinger (the quantum mechanics), Dirac (the relativistic quantum and the antiparticles), Gell-Mann (the quarks), Weinberg-Salam (Weak interactions and eletromagnetic unification), H. Georgi and S. Glashon (strong interactions plus Weinberg-Salam), Kaluza-Klein (a fifth space-time coordinate), and Zumino-Weiss (supersymmetry and supergravity). (G.D.F.) [pt 18. Discrete Curvature Theories and Applications KAUST Repository Sun, Xiang 2016-08-25 Discrete Di erential Geometry (DDG) concerns discrete counterparts of notions and methods in di erential geometry. This thesis deals with a core subject in DDG, discrete curvature theories on various types of polyhedral surfaces that are practically important for free-form architecture, sunlight-redirecting shading systems, and face recognition. Modeled as polyhedral surfaces, the shapes of free-form structures may have to satisfy di erent geometric or physical constraints. We study a combination of geometry and physics { the discrete surfaces that can stand on their own, as well as having proper shapes for the manufacture. These proper shapes, known as circular and conical meshes, are closely related to discrete principal curvatures. We study curvature theories that make such surfaces possible. Shading systems of freeform building skins are new types of energy-saving structures that can re-direct the sunlight. From these systems, discrete line congruences across polyhedral surfaces can be abstracted. We develop a new curvature theory for polyhedral surfaces equipped with normal congruences { a particular type of congruences de ned by linear interpolation of vertex normals. The main results are a discussion of various de nitions of normality, a detailed study of the geometry of such congruences, and a concept of curvatures and shape operators associated with the faces of a triangle mesh. These curvatures are compatible with both normal congruences and the Steiner formula. In addition to architecture, we consider the role of discrete curvatures in face recognition. We use geometric measure theory to introduce the notion of asymptotic cones associated with a singular subspace of a Riemannian manifold, which is an extension of the classical notion of asymptotic directions. We get a simple expression of these cones for polyhedral surfaces, as well as convergence and approximation theorems. We use the asymptotic cones as facial descriptors and demonstrate the 19. Consistency relations in effective field theory Energy Technology Data Exchange (ETDEWEB) Munshi, Dipak; Regan, Donough, E-mail: [email protected], E-mail: [email protected] [Astronomy Centre, School of Mathematical and Physical Sciences, University of Sussex, Brighton BN1 9QH (United Kingdom) 2017-06-01 The consistency relations in large scale structure relate the lower-order correlation functions with their higher-order counterparts. They are direct outcome of the underlying symmetries of a dynamical system and can be tested using data from future surveys such as Euclid. Using techniques from standard perturbation theory (SPT), previous studies of consistency relation have concentrated on continuity-momentum (Euler)-Poisson system of an ideal fluid. We investigate the consistency relations in effective field theory (EFT) which adjusts the SPT predictions to account for the departure from the ideal fluid description on small scales. We provide detailed results for the 3D density contrast δ as well as the scaled divergence of velocity θ-bar . Assuming a ΛCDM background cosmology, we find the correction to SPT results becomes important at k ∼> 0.05 h/Mpc and that the suppression from EFT to SPT results that scales as square of the wave number k , can reach 40% of the total at k ≈ 0.25 h/Mpc at z = 0. We have also investigated whether effective field theory corrections to models of primordial non-Gaussianity can alter the squeezed limit behaviour, finding the results to be rather insensitive to these counterterms. In addition, we present the EFT corrections to the squeezed limit of the bispectrum in redshift space which may be of interest for tests of theories of modified gravity. 20. New framework for gauge field theories International Nuclear Information System (INIS) Blaha, S. 1979-01-01 Gauge theories are formulated within the framework of a generalization of quantum field theory. In particular, models of electrodynamics and of Yang-Mills theories, we discuss a model of the strong interaction with higher-order derivatives and quark confinement and a renormalizable model of pure quantum gravity with Einstein Lagrangian. In the case of electrodynamics it is shown that two models are possible: one with predictions which are identical to QED and one which is a quantum action-at-a-distance model of electrodynamics. In the case of Yang-Mills theories a model is constructed which is identical in predictions to any conventional model, or a quantum action-at-a-distance model. In the second case it is possible to eliminate all loops of Yang-Mills particles (in all gauges) in a manner consistent with unitarity. A variation of Yang-Mills models exists in this formulation which has higher-order derivative field equations. It is unitary and has positive probabilities. It can be used to construct a model of the strong interactions which has a linear potential and manifest quark confinement. Finally, it is shown how to construct an action-at-a-distance model of pure quantum gravity (whose classical limit is the dynamics of the Einstein Lagrangian) coupled to an external classical source. The model is trivially renormalizable. (author) 1. Why are tensor field theories asymptotically free? Science.gov (United States) Rivasseau, V. 2015-09-01 In this pedagogic letter we explain the combinatorics underlying the generic asymptotic freedom of tensor field theories. We focus on simple combinatorial models with a 1/p2 propagator and quartic interactions and on the comparison between the intermediate field representations of the vector, matrix and tensor cases. The transition from asymptotic freedom (tensor case) to asymptotic safety (matrix case) is related to the crossing symmetry of the matrix vertex, whereas in the vector case, the lack of asymptotic freedom (“Landau ghost”), as in the ordinary scalar φ^44 case, is simply due to the absence of any wave function renormalization at one loop. 2. Un-reduction in field theory. Science.gov (United States) Arnaudon, Alexis; López, Marco Castrillón; Holm, Darryl D 2018-01-01 The un-reduction procedure introduced previously in the context of classical mechanics is extended to covariant field theory. The new covariant un-reduction procedure is applied to the problem of shape matching of images which depend on more than one independent variable (for instance, time and an additional labelling parameter). Other possibilities are also explored: nonlinear [Formula: see text]-models and the hyperbolic flows of curves. 3. Two problems in thermal field theory F can be calculated perturbatively as a sum of vacuum ... F / F id eal d c b a. Figure 4. Results of the screened perturbative expansion for the free energy as a func- tion of the coupling constant in scalar field theory [8]. (a) and (b): first ... for the pressure of a SU(3) Yang–Mills gas just by introducing a mass in the propagator. 4. Special relativity and classical field theory CERN Document Server Susskind, Leonard 2017-01-01 Physicist Leonard Susskind and data engineer Art Friedman are back. This time, they introduce readers to Einstein's special relativity and Maxwell's classical field theory. Using their typical brand of real math, enlightening drawings, and humor, Susskind and Friedman walk us through the complexities of waves, forces, and particles by exploring special relativity and electromagnetism. It's a must-read for both devotees of the series and any armchair physicist who wants to improve their knowledge of physics' deepest truths. 5. Multibrane solutions in open string field theory Czech Academy of Sciences Publication Activity Database Murata, Masaki; Schnabl, Martin 2012-01-01 Roč. 2012, č. 7 (2012), 1-26 ISSN 1126-6708 R&D Projects: GA MŠk(CZ) LH11106 Grant - others:EUROHORC and ESF(XE) EYI/07/E010 Institutional research plan: CEZ:AV0Z10100502 Keywords : string field theory tachyon condensation Subject RIV: BF - Elementary Particles and High Energy Physics Impact factor: 5.618, year: 2012 http://link.springer.com/article/10.1007%2FJHEP07%282012%29063 6. Topics on field theories at finite temperature International Nuclear Information System (INIS) Eboli, O.J.P. 1985-01-01 The dynamics of a first order phase transition through the study of the decay rate of the false vacuum in the high temperature limit are analysed. An alternative approach to obtain the phase diagram of a field theory which is based on the study of the free energy of topological defects, is developed the behavior of coupling constants with the help of the Dyson-Schwinger equations at finite temperature, is evaluated. (author) [pt 7. Numerical studies of gauge field theories International Nuclear Information System (INIS) Creutz, M. 1981-06-01 Monte Carlo simulation of statistical systems is a well established technique of the condensed matter physicist. In the last few years, particle theorists have rediscovered this method and are having a marvelous time applying it to quantized gauge field theories. The main result has been strong numerical evidence that the standard SU(3) non-Abelian gauge theory of the strong interaction is capable of simultaneously confining quarks into the physical hadrons and exhibiting asymptotic freedom, the phenomenon of quark interactions being small at short distances. In four dimensions, confinement is a non-perturbative phenomenon. Essentially all models of confinement tie widely separated quarks together with strings of gauge field flux. This gives rise to a linear potential at long distances. A Monte Carlo program generates a sequence of field configuration by a series of random changes of the fields. The algorithm is so constructed that ultimately the probability density for finding any given configuration is proportional to the Boltzmann weighting. We bring our lattices into thermal equilibrium with a heat bath at a temperature specified by the coupling constant. Thus we do computer experiments with four-dimensional crystals stored in a computer memory. As the entire field configuration is stored, we have access to any correlation function desired. These lectures describe the kinds of experiments being done and the implications of these results for strong interaction physics 8. Heavy Quarks, QCD, and Effective Field Theory Energy Technology Data Exchange (ETDEWEB) Thomas Mehen 2012-10-09 The research supported by this OJI award is in the area of heavy quark and quarkonium production, especially the application Soft-Collinear E ective Theory (SCET) to the hadronic production of quarkonia. SCET is an e ffective theory which allows one to derive factorization theorems and perform all order resummations for QCD processes. Factorization theorems allow one to separate the various scales entering a QCD process, and in particular, separate perturbative scales from nonperturbative scales. The perturbative physics can then be calculated using QCD perturbation theory. Universal functions with precise fi eld theoretic de nitions describe the nonperturbative physics. In addition, higher order perturbative QCD corrections that are enhanced by large logarithms can be resummed using the renormalization group equations of SCET. The applies SCET to the physics of heavy quarks, heavy quarkonium, and similar particles. 9. Nonlinear analysis approximation theory, optimization and applications CERN Document Server 2014-01-01 Many of our daily-life problems can be written in the form of an optimization problem. Therefore, solution methods are needed to solve such problems. Due to the complexity of the problems, it is not always easy to find the exact solution. However, approximate solutions can be found. The theory of the best approximation is applicable in a variety of problems arising in nonlinear functional analysis and optimization. This book highlights interesting aspects of nonlinear analysis and optimization together with many applications in the areas of physical and social sciences including engineering. It is immensely helpful for young graduates and researchers who are pursuing research in this field, as it provides abundant research resources for researchers and post-doctoral fellows. This will be a valuable addition to the library of anyone who works in the field of applied mathematics, economics and engineering. 10. Tachyon condensation in superstring field theory International Nuclear Information System (INIS) Berkovits, Nathan; Sen, Ashoke; Zwiebach, Barton 2000-01-01 It has been conjectured that at the stationary point of the tachyon potential for the D-brane-anti-D-brane pair or for the non-BPS D-brane of superstring theories, the negative energy density cancels the brane tensions. We study this conjecture using a Wess-Zumino-Witten-like open superstring field theory free of contact term divergences and recently shown to give 60% of the vacuum energy by condensation of the tachyon field alone. While the action is non-polynomial, the multiscalar tachyon potential to any fixed level involves only a finite number of interactions. We compute this potential to level three, obtaining 85% of the expected vacuum energy, a result consistent with convergence that can also be viewed as a successful test of the string field theory. The resulting effective tachyon potential is bounded below and has two degenerate global minima. We calculate the energy density of the kink solution interpolating between these minima finding good agreement with the tension of the D-brane of one lower dimension 11. Regularity Theory for Mean-Field Game Systems KAUST Repository Gomes, Diogo A. 2016-09-14 Beginning with a concise introduction to the theory of mean-field games (MFGs), this book presents the key elements of the regularity theory for MFGs. It then introduces a series of techniques for well-posedness in the context of mean-field problems, including stationary and time-dependent MFGs, subquadratic and superquadratic MFG formulations, and distinct classes of mean-field couplings. It also explores stationary and time-dependent MFGs through a series of a-priori estimates for solutions of the Hamilton-Jacobi and Fokker-Planck equation. It shows sophisticated a-priori systems derived using a range of analytical techniques, and builds on previous results to explain classical solutions. The final chapter discusses the potential applications, models and natural extensions of MFGs. As MFGs connect common problems in pure mathematics, engineering, economics and data management, this book is a valuable resource for researchers and graduate students in these fields. 12. Regularity theory for mean-field game systems CERN Document Server Gomes, Diogo A; Voskanyan, Vardan 2016-01-01 Beginning with a concise introduction to the theory of mean-field games (MFGs), this book presents the key elements of the regularity theory for MFGs. It then introduces a series of techniques for well-posedness in the context of mean-field problems, including stationary and time-dependent MFGs, subquadratic and superquadratic MFG formulations, and distinct classes of mean-field couplings. It also explores stationary and time-dependent MFGs through a series of a-priori estimates for solutions of the Hamilton-Jacobi and Fokker-Planck equation. It shows sophisticated a-priori systems derived using a range of analytical techniques, and builds on previous results to explain classical solutions. The final chapter discusses the potential applications, models and natural extensions of MFGs. As MFGs connect common problems in pure mathematics, engineering, economics and data management, this book is a valuable resource for researchers and graduate students in these fields. 13. Lattice theory special topics and applications CERN Document Server Wehrung, Friedrich George Grätzer's Lattice Theory: Foundation is his third book on lattice theory (General Lattice Theory, 1978, second edition, 1998). In 2009, Grätzer considered updating the second edition to reflect some exciting and deep developments. He soon realized that to lay the foundation, to survey the contemporary field, to pose research problems, would require more than one volume and more than one person. So Lattice Theory: Foundation provided the foundation. Now we complete this project with Lattice Theory: Special Topics and Applications, written by a distinguished group of experts, to cover some of the vast areas not in Foundation. This first volume is divided into three parts. Part I. Topology and Lattices includes two chapters by Klaus Keimel, Jimmie Lawson and Ales Pultr, Jiri Sichler. Part II. Special Classes of Finite Lattices comprises four chapters by Gabor Czedli, George Grätzer and Joseph P. S. Kung. Part III. Congruence Lattices of Infinite Lattices and Beyond includes four chapters by Friedrich W... 14. Topics in string theory and quantum field theory Science.gov (United States) Giombi, Simone In this dissertation we study several topics in string theory and quantum field theory, which we collect into three main parts. The first part contains some studies in the context of twistor string theory. Witten proposed that the perturbative expansion of N = 4 super Yang-Mills theory has a dual formulation in terms of a topological string theory on the supertwistor space CP3\|4 . We discuss extensions of this construction in two directions. First, we make some preliminary considerations on the possibility of having a similar twistor approach to perturbative gravity. Then we extend the construction to theories with lower supersymmetry by taking orbifolds in the fermionic directions of CP3\|4 . We consider N = 1 and N = 2 superconformal quiver gauge theories as specific examples. In the second part of the dissertation we study worldline methods in curved space. In particular, we use the N = 2 spinning particle to describe antisymmetric tensors of arbitrary rank propagating in a curved background. The path integral quantization of the N = 2 particle produces a novel and compact representation of the one loop effective action for generic differential p-forms, including the vector field as a special example. We study both the massless and massive case, and show that the worldline representation of the one loop effective action can be used to efficiently study various quantum effects for antisymmetric tensor fields of arbitrary rank in arbitrary dimension. In the last and final part we study some topics in the context of the AdS/CFT correspondence. We start by investigating the recently discovered description of half-BPS supergravity backgrounds in terms of one-dimensional free fermions. We study a generalization of this construction obtained by considering free fermions at non-zero temperature. The ADM mass of the corresponding supergravity background is shown to agree with the fermion thermal energy, and we propose a way to qualitatively match the entropy in the two 15. Undergraduate Lecture Notes in Topological Quantum Field Theory OpenAIRE Ivancevic, Vladimir G.; Ivancevic, Tijana T. 2008-01-01 These third-year lecture notes are designed for a 1-semester course in topological quantum field theory (TQFT). Assumed background in mathematics and physics are only standard second-year subjects: multivariable calculus, introduction to quantum mechanics and basic electromagnetism. Keywords: quantum mechanics/field theory, path integral, Hodge decomposition, Chern-Simons and Yang-Mills gauge theories, conformal field theory 16. Particle versus field structure in conformal quantum field theories International Nuclear Information System (INIS) Schroer, Bert 2000-06-01 I show that a particle structure in conformal field theory is incompatible with interactions. As a substitute one has particle-like excitations whose interpolating fields have in addition to their canonical dimension an anomalous contribution. The spectra of anomalous dimension is given in terms of the Lorentz invariant quadratic invariant (compact mass operator) of a conformal generator R μ with pure discrete spectrum. The perturbative reading of R o as a Hamiltonian in its own right, associated with an action in a functional integral setting naturally leads to the Ad S formulation. The formal service role of Ad S in order to access C QFT by a standard perturbative formalism (without being forced to understand first massive theories and then taking their scale-invariant limit) vastly increases the realm of conventionally accessible 4-dim. C QFT beyond those for which one had to use Lagrangians with supersymmetry in order to have a vanishing Beta-function. (author) 17. On the background independence of string field theory International Nuclear Information System (INIS) Sen, A. 1990-01-01 Given a solution Ψ cl of the classical equations of motion in either closed or open string field theory formulated around a given conformal field theory background, we can construct a new operator Q B in the corresponding two-dimensional field theory such that (Q B ) 2 =0. It is shown that in the limit when the background field Ψ cl is weak, Q B can be identified with the BRST charge of a new local conformal field theory. This indicates that the string field theories formulated around these two different conformal field theories are actually the same theory, and that these two conformal field theories may be regarded as different classical solutions of this string field theory. (orig.) 18. An application of Anthony Giddens' structuration theory African Journals Online (AJOL) The aim of this article is to discuss the structuration theory of Anthony Giddens with regard to its applicability to translation studies. Key concepts of Giddens' sociological theory as agent, agency, structure, system and structuration will be explored in terms of their applicability to translation. In this article, structuration theory ... 19. Theory of singlet-ground-state magnetism. Application to field-induced transitions in CsFeCl3 and CsFeBr3 DEFF Research Database (Denmark) Lindgård, P.-A.; Schmid, B. 1993-01-01 In the singlet ground-state systems CsFeCl3 and CsFeBr3 a large single-ion anisotropy causes a singlet ground state and a doubly degenerate doublet as the first excited states of the Fe2+ ion. In addition the magneteic interaction is anisotropic being much larger along the z axis than perpendicular...... to it. Therefore, these quasi-one-dimensional magnetic model systems are ideal to demonstrate unique correlation effects. Within the framework of the correlation theory we derive the expressions for the excitation spectrum. When a magnetic field is applied parallel to the z axis both substances have... 20. (Non-)decoupled supersymmetric field theories International Nuclear Information System (INIS) Pietro, Lorenzo Di; Dine, Michael; Komargodski, Zohar 2014-01-01 We study some consequences of coupling supersymmetric theories to (super)gravity. To linear order, the couplings are determined by the energy-momentum supermultiplet. At higher orders, the couplings are determined by contact terms in correlation functions of the energy-momentum supermultiplet. We focus on the couplings of one particular field in the supergravity multiplet, the auxiliary field M. We discuss its linear and quadratic (seagull) couplings in various supersymmetric theories. In analogy to the local renormalization group formalism (http://dx.doi.org/10.1016/0370-2693(89)90729-6; http://dx.doi.org/10.1016/0550-3213(90)90584-Z; http://dx.doi.org/10.1016/0550-3213(91)80030-P), we provide a prescription for how to fix the quadratic couplings. They generally arise at two-loops in perturbation theory. We check our prescription by explicitly computing these couplings in several examples such as mass-deformed N=4 and in the Coulomb phase of some theories. These couplings affect the Lagrangians of rigid supersymmetric theories in curved space. In addition, our analysis leads to a transparent derivation of the phenomenon known as Anomaly Mediation. In contrast to previous approaches, we obtain both the gaugino and scalar masses of Anomaly Mediation by relying just on classical, minimal supergravity and a manifestly local and supersymmetric Wilsonian point of view. Our discussion naturally incorporates the connection between Anomaly Mediation and supersymmetric AdS 4 Lagrangians. This note can be read without prior familiarity with Anomaly Mediated Supersymmetry Breaking (AMSB) 1. Motion of small bodies in classical field theory International Nuclear Information System (INIS) Gralla, Samuel E. 2010-01-01 I show how prior work with R. Wald on geodesic motion in general relativity can be generalized to classical field theories of a metric and other tensor fields on four-dimensional spacetime that (1) are second-order and (2) follow from a diffeomorphism-covariant Lagrangian. The approach is to consider a one-parameter-family of solutions to the field equations satisfying certain assumptions designed to reflect the existence of a body whose size, mass, and various charges are simultaneously scaled to zero. (That such solutions exist places a further restriction on the class of theories to which our results apply.) Assumptions are made only on the spacetime region outside of the body, so that the results apply independent of the body's composition (and, e.g., black holes are allowed). The worldline 'left behind' by the shrinking, disappearing body is interpreted as its lowest-order motion. An equation for this worldline follows from the 'Bianchi identity' for the theory, without use of any properties of the field equations beyond their being second-order. The form of the force law for a theory therefore depends only on the ranks of its various tensor fields; the detailed properties of the field equations are relevant only for determining the charges for a particular body (which are the ''monopoles'' of its exterior fields in a suitable limiting sense). I explicitly derive the force law (and mass-evolution law) in the case of scalar and vector fields, and give the recipe in the higher-rank case. Note that the vector force law is quite complicated, simplifying to the Lorentz force law only in the presence of the Maxwell gauge symmetry. Example applications of the results are the motion of 'chameleon' bodies beyond the Newtonian limit, and the motion of bodies in (classical) non-Abelian gauge theory. I also make some comments on the role that scaling plays in the appearance of universality in the motion of bodies. 2. Fuzzy neural network theory and application CERN Document Server Liu, Puyin 2004-01-01 This book systematically synthesizes research achievements in the field of fuzzy neural networks in recent years. It also provides a comprehensive presentation of the developments in fuzzy neural networks, with regard to theory as well as their application to system modeling and image restoration. Special emphasis is placed on the fundamental concepts and architecture analysis of fuzzy neural networks. The book is unique in treating all kinds of fuzzy neural networks and their learning algorithms and universal approximations, and employing simulation examples which are carefully designed to he 3. Modeling and Optimization : Theory and Applications Conference CERN Document Server Terlaky, Tamás 2015-01-01 This volume contains a selection of contributions that were presented at the Modeling and Optimization: Theory and Applications Conference (MOPTA) held at Lehigh University in Bethlehem, Pennsylvania, USA on August 13-15, 2014. The conference brought together a diverse group of researchers and practitioners, working on both theoretical and practical aspects of continuous or discrete optimization. Topics presented included algorithms for solving convex, network, mixed-integer, nonlinear, and global optimization problems, and addressed the application of deterministic and stochastic optimization techniques in energy, finance, logistics, analytics, healthcare, and other important fields. The contributions contained in this volume represent a sample of these topics and applications and illustrate the broad diversity of ideas discussed at the meeting. 4. Modeling and Optimization : Theory and Applications Conference CERN Document Server Terlaky, Tamás 2017-01-01 This volume contains a selection of contributions that were presented at the Modeling and Optimization: Theory and Applications Conference (MOPTA) held at Lehigh University in Bethlehem, Pennsylvania, USA on August 17-19, 2016. The conference brought together a diverse group of researchers and practitioners, working on both theoretical and practical aspects of continuous or discrete optimization. Topics presented included algorithms for solving convex, network, mixed-integer, nonlinear, and global optimization problems, and addressed the application of deterministic and stochastic optimization techniques in energy, finance, logistics, analytics, health, and other important fields. The contributions contained in this volume represent a sample of these topics and applications and illustrate the broad diversity of ideas discussed at the meeting. 5. Constraints on Interacting Scalars in 2T Field Theory and No Scale Models in 1T Field Theory CERN Document Server Bars, Itzhak 2010-01-01 In this paper I determine the general form of the physical and mathematical restrictions that arise on the interactions of gravity and scalar fields in the 2T field theory setting, in d+2 dimensions, as well as in the emerging shadows in d dimensions. These constraints on scalar fields follow from an underlying Sp(2,R) gauge symmetry in phase space. Determining these general constraints provides a basis for the construction of 2T supergravity, as well as physical applications in 1T-field theory, that are discussed briefly here, and more detail elsewhere. In particular, no scale models that lead to a vanishing cosmological constant at the classical level emerge naturally in this setting. 6. Triboluminescence theory, synthesis, and application CERN Document Server Okoli, Okenwa; Fontenot, Ross; Hollerman, William 2016-01-01 This book expounds on progress made over the last 35 years in the theory, synthesis, and application of triboluminescence for creating smart structures. It presents in detail the research into utilization of the triboluminescent properties of certain crystals as new sensor systems for smart engineering structures, as well as triboluminescence-based sensor systems that have the potential to enable wireless, in-situ, real time and distributed (WIRD) structural health monitoring of composite structures. The sensor component of any structural health monitoring (SHM) technology — measures the effects of the external load/event and provides the necessary inputs for appropriate preventive/corrective action to be taken in a smart structure — sits at the heart of such a system. This volume explores advances in materials properties and structural behavior underlying creation of smart composite structures and sensor systems for structural health monitoring of critical engineering structures, such as bridges, aircraf... 7. Conformal Field Theory, Automorphic Forms and Related Topics CERN Document Server Weissauer, Rainer; CFT 2011 2014-01-01 This book, part of the series Contributions in Mathematical and Computational Sciences, reviews recent developments in the theory of vertex operator algebras (VOAs) and their applications to mathematics and physics. The mathematical theory of VOAs originated from the famous monstrous moonshine conjectures of J.H. Conway and S.P. Norton, which predicted a deep relationship between the characters of the largest simple finite sporadic group, the Monster, and the theory of modular forms inspired by the observations of J. MacKay and J. Thompson. The contributions are based on lectures delivered at the 2011 conference on Conformal Field Theory, Automorphic Forms and Related Topics, organized by the editors as part of a special program offered at Heidelberg University that summer under the sponsorship of the MAThematics Center Heidelberg (MATCH). 8. An invitation to quantum field theory International Nuclear Information System (INIS) Alvarez-Gaume, Luis; Vazquez-Mozo, Miguel A. 2012-01-01 This book provides an introduction to Quantum Field Theory (QFT) at an elementary level - with only special relativity, electromagnetism and quantum mechanics as prerequisites. For this fresh approach to teaching QFT, based on numerous lectures and courses given by the authors, a representative sample of topics has been selected containing some of the more innovative, challenging or subtle concepts. They are presented with a minimum of technical details, the discussion of the main ideas being more important than the presentation of the typically very technical mathematical details necessary to obtain the final results. Special attention is given to the realization of symmetries in particle physics: global and local symmetries, explicit, spontaneously broken, and anomalous continuous symmetries, as well as discrete symmetries. Beyond providing an overview of the standard model of the strong, weak and electromagnetic interactions and the current understanding of the origin of mass, the text enumerates the general features of renormalization theory as well as providing a cursory description of effective field theories and the problem of naturalness in physics. Among the more advanced topics the reader will find are an outline of the first principles derivation of the CPT theorem and the spin-statistics connection. As indicated by the title, the main aim of this text is to motivate the reader to study QFT by providing a self-contained and approachable introduction to the most exciting and challenging aspects of this successful theoretical framework. (orig.) 9. An invitation to quantum field theory Energy Technology Data Exchange (ETDEWEB) Alvarez-Gaume, Luis [CERN, Geneva (Switzerland). Physics Dept.; Vazquez-Mozo, Miguel A. [Salamanca Univ. (Spain). Dept. de Fisica Fundamental 2012-07-01 This book provides an introduction to Quantum Field Theory (QFT) at an elementary level - with only special relativity, electromagnetism and quantum mechanics as prerequisites. For this fresh approach to teaching QFT, based on numerous lectures and courses given by the authors, a representative sample of topics has been selected containing some of the more innovative, challenging or subtle concepts. They are presented with a minimum of technical details, the discussion of the main ideas being more important than the presentation of the typically very technical mathematical details necessary to obtain the final results. Special attention is given to the realization of symmetries in particle physics: global and local symmetries, explicit, spontaneously broken, and anomalous continuous symmetries, as well as discrete symmetries. Beyond providing an overview of the standard model of the strong, weak and electromagnetic interactions and the current understanding of the origin of mass, the text enumerates the general features of renormalization theory as well as providing a cursory description of effective field theories and the problem of naturalness in physics. Among the more advanced topics the reader will find are an outline of the first principles derivation of the CPT theorem and the spin-statistics connection. As indicated by the title, the main aim of this text is to motivate the reader to study QFT by providing a self-contained and approachable introduction to the most exciting and challenging aspects of this successful theoretical framework. (orig.) 10. Monoidal categories and topological field theory CERN Document Server 2017-01-01 This monograph is devoted to monoidal categories and their connections with 3-dimensional topological field theories. Starting with basic definitions, it proceeds to the forefront of current research. Part 1 introduces monoidal categories and several of their classes, including rigid, pivotal, spherical, fusion, braided, and modular categories. It then presents deep theorems of Müger on the center of a pivotal fusion category. These theorems are proved in Part 2 using the theory of Hopf monads. In Part 3 the authors define the notion of a topological quantum field theory (TQFT) and construct a Turaev-Viro-type 3-dimensional state sum TQFT from a spherical fusion category. Lastly, in Part 4 this construction is extended to 3-manifolds with colored ribbon graphs, yielding a so-called graph TQFT (and, consequently, a 3-2-1 extended TQFT). The authors then prove the main result of the monograph: the state sum graph TQFT derived from any spherical fusion category is isomorphic to the Reshetikhin-Turaev surgery gr... 11. Introduction to soliton theory applications to mechanics CERN Document Server Munteanu, Ligia 2005-01-01 This monograph provides the application of soliton theory to solve certain problems selected from the fields of mechanics. The work is based of the authors' research, and on some specified, significant results existing in the literature. The present monograph is not a simple translation of its predecessor appeared in Publishing House of the Romanian Academy in 2002. Improvements outline the way in which the soliton theory is applied to solve some engineering problems. The book addresses concrete resolution methods of certain problems such as the motion of thin elastic rod, vibrations of initial deformed thin elastic rod, the coupled pendulum oscillations, dynamics of left ventricle, transient flow of blood in arteries, the subharmonic waves generation in a piezoelectric plate with Cantor-like structure, and some problems related to Tzitzeica surfaces. This comprehensive study enables the readers to make connections between the soliton physical phenomenon and some partical, engineering problems. 12. Relativistic mean field theory for unstable nuclei International Nuclear Information System (INIS) Toki, Hiroshi 2000-01-01 We discuss the properties of unstable nuclei in the framework of the relativistic mean field (RMF) theory. We take the RMF theory as a phenomenological theory with several parameters, whose form is constrained by the successful microscopic theory (RBHF), and whose values are extracted from the experimental values of unstable nuclei. We find the outcome with the newly obtained parameter sets (TM1 and TMA) is promising in comparison with various experimental data. We calculate systematically the ground state properties of even-even nuclei up to the drip lines; about 2000 nuclei. We find that the neutron magic shells (N=82, 128) at the standard magic numbers stay at the same numbers even far from the stability line and hence provide the feature of the r-process nuclei. However, many proton magic numbers disappear at the neutron numbers far away from the magic numbers due to the deformations. We discuss how to describe giant resonances for the case of the non-linear coupling terms for the sigma and omega mesons in the relativistic RPA. We mention also the importance of the relativistic effect on the spin observables as the Gamow-Teller strength and the longitudinal and transverse spin responses. (author) 13. GNSS remote sensing theory, methods and applications CERN Document Server Jin, Shuanggen; Xie, Feiqin 2014-01-01 This book presents the theory and methods of GNSS remote sensing as well as its applications in the atmosphere, oceans, land and hydrology. It contains detailed theory and study cases to help the reader put the material into practice. 14. Game Theory: An Application to Tanners and 'Pomo' Wholesalers in ... African Journals Online (AJOL) Game Theory: An Application to Tanners and 'Pomo' Wholesalers in Hides Marketing Competition in Nigeria. ... Agricultural economics is an applied social science so game theory was applied practically from a field survey to determine the level of competition between tanners and 'pomo' wholesalers in Nigeria. All the ... 15. Bare coupling constants and asymptotic behaviour in reggeon field theory International Nuclear Information System (INIS) Baig, M. 1983-01-01 A relation between the values of bare coupling constants and the asymptotic behaviour of the reggeon field theory (RFT) is discussed. It is shown how the numerical values of bare coupling constants fix the starting point of renormalization group trajectories which, in turn, determine the asymptotic behaviour of the RFT. Applications to a pure pomeron theory and a pomeron plus f-pole model are discussed. Some nontrivial phenomenological information concerning the values of bare triple-Regge pomeron-f-pole coupling constants is obtained 16. Yang-Baxter algebra - Integrable systems - Conformal quantum field theories International Nuclear Information System (INIS) Karowski, M. 1989-01-01 This series of lectures is based on investigations [1,2] of finite-size corrections for the six-vertex model by means of Bethe ansatz methods. In addition a review on applications of Yang-Baxter algebras and an introduction to the theory of integrable systems and the algebraic Bethe ansatz is presented. A Θ-vacuum like angle appearing in the RSOS-models is discussed. The continuum limit in the critical case of these statistical models is performed to obtain the minimal models of conformal quantum field theory. (author) 17. Theory of field-reversed configurations International Nuclear Information System (INIS) Steinhauer, L.C. 1993-01-01 This report summarizes results from the theoretical program on field reversed configurations (FRC) at STI Optronics. The program, which has spanned the last 13 years, has included analytical as well as computational components. It has led to published papers on every major topic of FRC theory. The report is outlined to summarize results from each of these topic areas: formation, equilibrium, stability, and confinement. Also briefly described are Steinhauer's activities as Compact Toroid Theory Listening Post. Appendix A is a brief listing of the major advances achieved in this program. Attached at the back of this report is a collection of technical papers in archival journals that resulted from work in this program. The discussion within each subsection is given chronologically in order to give a historical sense of the evolution of understanding of FRC physics 18. Working Group Report: Lattice Field Theory Energy Technology Data Exchange (ETDEWEB) Blum, T.; et al., 2013-10-22 This is the report of the Computing Frontier working group on Lattice Field Theory prepared for the proceedings of the 2013 Community Summer Study ("Snowmass"). We present the future computing needs and plans of the U.S. lattice gauge theory community and argue that continued support of the U.S. (and worldwide) lattice-QCD effort is essential to fully capitalize on the enormous investment in the high-energy physics experimental program. We first summarize the dramatic progress of numerical lattice-QCD simulations in the past decade, with some emphasis on calculations carried out under the auspices of the U.S. Lattice-QCD Collaboration, and describe a broad program of lattice-QCD calculations that will be relevant for future experiments at the intensity and energy frontiers. We then present details of the computational hardware and software resources needed to undertake these calculations. 19. A Cahn-Hilliard-type phase-field theory for species diffusion coupled with large elastic deformations: Application to phase-separating Li-ion electrode materials Science.gov (United States) Di Leo, Claudio V.; Rejovitzky, Elisha; Anand, Lallit 2014-10-01 We formulate a unified framework of balance laws and thermodynamically-consistent constitutive equations which couple Cahn-Hilliard-type species diffusion with large elastic deformations of a body. The traditional Cahn-Hilliard theory, which is based on the species concentration c and its spatial gradient ∇c, leads to a partial differential equation for the concentration which involves fourth-order spatial derivatives in c; this necessitates use of basis functions in finite-element solution procedures that are piecewise smooth and globally C1-continuous. In order to use standard C0-continuous finite-elements to implement our phase-field model, we use a split-method to reduce the fourth-order equation into two second-order partial differential equations (pdes). These two pdes, when taken together with the pde representing the balance of forces, represent the three governing pdes for chemo-mechanically coupled problems. These are amenable to finite-element solution methods which employ standard C0-continuous finite-element basis functions. We have numerically implemented our theory by writing a user-element subroutine for the widely used finite-element program Abaqus/Standard. We use this numerically implemented theory to first study the diffusion-only problem of spinodal decomposition in the absence of any mechanical deformation. Next, we use our fully coupled theory and numerical-implementation to study the combined effects of diffusion and stress on the lithiation of a representative spheroidal-shaped particle of a phase-separating electrode material. 20. Vortex operators in gauge field theories International Nuclear Information System (INIS) Polchinski, J.G. 1980-01-01 We study several related aspects of the t Hooft vortex operator. The first chapter reviews the current picture of the vacuum of quantum chromodynamics, the idea of dual field theories, and the idea of the vortex operator. The second chapter deals with the Abelian vortex operator written in terms of elementary fields and with the calculation of its Green's functions. The Dirac veto problem appears in a new guise. We present a two dimensional solvable model of a Dirac string. This leads us to a new solution of the veto problem; we discuss its extension to four dimensions. We then show how the Green's functions can be expressed more neatly in terms of Wu and Yang's geometrical idea of sections. In the third chapter we discuss the dependence of the Green's functions of the Wilson and t Hooft operators on the nature of the vacuum. In the fourth chapter we consider systems which have fields in the fundamental representation, so that there are no vortex operators. When these fields enter only weakly into the dynamics, as is the case in QCD and in real superconductors, we would expect to be able to define a vortex-like operator. We show that any such operator can no longer be local looplike, but must have commutators at long range. We can still find an operator with useful properties, its cluster property, though more complicated than that of the usual vortex operator, still appears to distinguish Higgs, confining and perturbative phases. To test this, we consider a U(1) lattice gauge theory with two matter fields, one singly charged (fundamental) and one doubly charged (adjoint) 1. The Supersymmetric Effective Field Theory of Inflation Energy Technology Data Exchange (ETDEWEB) Delacrétaz, Luca V.; Gorbenko, Victor [Stanford Institute for Theoretical Physics, Stanford University,Stanford, CA 94306 (United States); Senatore, Leonardo [Stanford Institute for Theoretical Physics, Stanford University,Stanford, CA 94306 (United States); Kavli Institute for Particle Astrophysics and Cosmology, Stanford University and SLAC,Menlo Park, CA 94025 (United States) 2017-03-10 We construct the Supersymmetric Effective Field Theory of Inflation, that is the most general theory of inflationary fluctuations when time-translations and supersymmetry are spontaneously broken. The non-linear realization of these invariances allows us to define a complete SUGRA multiplet containing the graviton, the gravitino, the Goldstone of time translations and the Goldstino, with no auxiliary fields. Going to a unitary gauge where only the graviton and the gravitino are present, we write the most general Lagrangian built out of the fluctuations of these fields, invariant under time-dependent spatial diffeomorphisms, but softly-breaking time diffeomorphisms and gauged SUSY. With a suitable Stückelberg transformation, we introduce the Goldstone boson of time translation and the Goldstino of SUSY. No additional dynamical light field is needed. In the high energy limit, larger than the inflationary Hubble scale for the Goldstino, these fields decouple from the graviton and the gravitino, greatly simplifying the analysis in this regime. We study the phenomenology of this Lagrangian. The Goldstino can have a non-relativistic dispersion relation. Gravitino and Goldstino affect the primordial curvature perturbations at loop level. The UV modes running in the loops generate three-point functions which are degenerate with the ones coming from operators already present in the absence of supersymmetry. Their size is potentially as large as corresponding to f{sub NL}{sup equil.,orthog.}∼1 or, for particular operators, even ≫1. The non-degenerate contribution from modes of order H is estimated to be very small. 2. Quantitative graph theory mathematical foundations and applications CERN Document Server Dehmer, Matthias 2014-01-01 The first book devoted exclusively to quantitative graph theory, Quantitative Graph Theory: Mathematical Foundations and Applications presents and demonstrates existing and novel methods for analyzing graphs quantitatively. Incorporating interdisciplinary knowledge from graph theory, information theory, measurement theory, and statistical techniques, this book covers a wide range of quantitative-graph theoretical concepts and methods, including those pertaining to real and random graphs such as:Comparative approaches (graph similarity or distance)Graph measures to characterize graphs quantitat 3. Towers of algebras in rational conformal field theories International Nuclear Information System (INIS) Gomez, C.; Sierra, G. 1991-01-01 This paper reports on Jones fundamental construction applied to rational conformal field theories. The Jones algebra which emerges in this application is realized in terms of duality operations. The generators of the algebra are an open version of Verlinde's operators. The polynomial equations appear in this context as sufficient conditions for the existence of Jones algebra. The ADE classification of modular invariant partition functions is put in correspondence with Jones classification of subfactors 4. Bookshelf (The Quantum Theory of Fields, La lumiere des neutrinos) International Nuclear Information System (INIS) Anon. 1995-01-01 The Quantum Theory of Fields Volume 1: Foundations by Steven Weinberg, Cambridge University Press, 1995: Steven Weinberg is celebrated for his many contributions to quantum field theory and its applications to elementary particle physics - most notably, for developing the electroweak theory, the unified model of the electromagnetic and weak forces that forms part of the Standard Model that has explained essentially all accelerator data on the behaviour of elementary particles. This is the culmination of the developments in quantum field theory that started in the early days of quantum mechanics and came to maturity with the development of quantum electrodynamics in the late 1940s. Quantum field theory is the basic theoretical framework for research in particle physics as well as in many areas of condensed matter physics. No wonder the community has been waiting with anticipation for Weinberg's exposition of the subject in the form of a two-volume textbook - the more so since, despite the existence of many textbooks, few of them are written with the insight and detail that are needed for a thorough understanding. The community will not be disappointed, at least on the basis of this first volume - Volume 2 is due to appear next year. Volume 1 is 600 pages of meticulous exposition of the fundamentals of the subject, starting from a historical introduction and the canonical formulation of quantum field theory to modern path integral methods applied to the quantization of electrodynamics and a first discussion of renormaiization. In addition to a superb treatment of all the conventional topics there are numerous sections covering areas that are not normally emphasized, such as the subject of field redefinitions, higher-rank tensor fields and an unusually clear and thorough treatment of infrared effects. This is only the basics - Volume 2 promises to develop the subjects at the cutting edge of modern research such as Yang-Mills theory, the renormalization group 5. Basics of thermal field theory a tutorial on perturbative computations CERN Document Server Laine, Mikko 2016-01-01 This book presents thermal field theory techniques, which can be applied in both cosmology and the theoretical description of the QCD plasma generated in heavy-ion collision experiments. It focuses on gauge interactions (whether weak or strong), which are essential in both contexts. As well as the many differences in the physics questions posed and in the microscopic forces playing a central role, the authors also explain the similarities and the techniques, such as the resummations, that are needed for developing a formally consistent perturbative expansion. The formalism is developed step by step, starting from quantum mechanics; introducing scalar, fermionic and gauge fields; describing the issues of infrared divergences; resummations and effective field theories; and incorporating systems with finite chemical potentials. With this machinery in place, the important class of real-time (dynamic) observables is treated in some detail. This is followed by an overview of a number of applications, ranging from t... 6. Coarse grainings and irreversibility in quantum field theory International Nuclear Information System (INIS) Anastopoulos, C. 1997-01-01 In this paper we are interested in studying coarse graining in field theories using the language of quantum open systems. Motivated by the ideas of Hu and Calzetta on correlation histories we employ the Zwanzig projection technique to obtain evolution equations for relevant observables in self-interacting scalar field theories. Our coarse-graining operation consists in concentrating solely on the evolution of the correlation functions of degree less than n, a treatment which corresponds to the familiar truncation of the BBKGY hierarchy at the nth level. We derive the equations governing the evolution of mean-field and two-point functions thus identifying the terms corresponding to dissipation and noise. We discuss possible applications of our formalism, the emergence of classical behavior, and the connection to the decoherent histories framework. copyright 1997 The American Physical Society 7. Mathematical analysis, approximation theory and their applications CERN Document Server Gupta, Vijay 2016-01-01 Designed for graduate students, researchers, and engineers in mathematics, optimization, and economics, this self-contained volume presents theory, methods, and applications in mathematical analysis and approximation theory. Specific topics include: approximation of functions by linear positive operators with applications to computer aided geometric design, numerical analysis, optimization theory, and solutions of differential equations. Recent and significant developments in approximation theory, special functions and q-calculus along with their applications to mathematics, engineering, and social sciences are discussed and analyzed. Each chapter enriches the understanding of current research problems and theories in pure and applied research. 8. Field theory approach to quantum hall effect International Nuclear Information System (INIS) Cabo, A.; Chaichian, M. 1990-07-01 The Fradkin's formulation of statistical field theory is applied to the Coulomb interacting electron gas in a magnetic field. The electrons are confined to a plane in normal 3D-space and also interact with the physical 3D-electromagnetic field. The magnetic translation group (MTG) Ward identities are derived. Using them it is shown that the exact electron propagator is diagonalized in the basis of the wave functions of the free electron in a magnetic field whenever the MTG is unbroken. The general tensor structure of the polarization operator is obtained and used to show that the Chern-Simons action always describes the Hall effect properties of the system. A general proof of the Streda formula for the Hall conductivity is presented. It follows that the coefficient of the Chern-Simons terms in the long-wavelength approximation is exactly given by this relation. Such a formula, expressing the Hall conductivity as a simple derivative, in combination with diagonal form of the full propagator allows to obtain a simple expressions for the filling factor and the Hall conductivity. Indeed, these results, after assuming that the chemical potential lies in a gap of the density of states, lead to the conclusion that the Hall conductivity is given without corrections by σ xy = νe 2 /h where ν is the filling factor. In addition it follows that the filling factor is independent of the magnetic field if the chemical potential remains in the gap. (author). 21 ref, 1 fig 9. Theory of microemulsions in a gravitational field Science.gov (United States) Jeng, J. F.; Miller, Clarence A. 1989-01-01 A theory of microemulsions developed previously is extended to include the effect of a gravitational field. It predicts variation with position of drop size, drop volume fraction, and area per molecule in the surfactant films within a microemulsion phase. Variation in volume fraction is greatest and occurs in such a way that oil content increases with increasing elevation, as has been found experimentally. Large composition variations are predicted within a middle phase microemulsion near optimal conditions because inversion from the water-continuous to the oil-continuous arrangement occurs with increasing elevation. Generally speaking, gravity reduces solubilization within microemulsions and promotes separation of excess phases. 10. A matrix model from string field theory Directory of Open Access Journals (Sweden) Syoji Zeze 2016-09-01 Full Text Available We demonstrate that a Hermitian matrix model can be derived from level truncated open string field theory with Chan-Paton factors. The Hermitian matrix is coupled with a scalar and U(N vectors which are responsible for the D-brane at the tachyon vacuum. Effective potential for the scalar is evaluated both for finite and large N. Increase of potential height is observed in both cases. The large $N$ matrix integral is identified with a system of N ZZ branes and a ghost FZZT brane. 11. The Effective Field Theory of nonsingular cosmology International Nuclear Information System (INIS) Cai, Yong; Wan, Youping; Li, Hai-Guang; Qiu, Taotao; Piao, Yun-Song 2017-01-01 In this paper, we explore the nonsingular cosmology within the framework of the Effective Field Theory (EFT) of cosmological perturbations. Due to the recently proved no-go theorem, any nonsingular cosmological models based on the cubic Galileon suffer from pathologies. We show how the EFT could help us clarify the origin of the no-go theorem, and offer us solutions to break the no-go. Particularly, we point out that the gradient instability can be removed by using some spatial derivative operators in EFT. Based on the EFT description, we obtain a realistic healthy nonsingular cosmological model, and show the perturbation spectrum can be consistent with the observations. 12. Purely cubic action for string field theory Science.gov (United States) Horowitz, G. T.; Lykken, J.; Rohm, R.; Strominger, A. 1986-01-01 It is shown that Witten's (1986) open-bosonic-string field-theory action and a closed-string analog can be written as a purely cubic interaction term. The conventional form of the action arises by expansion around particular solutions of the classical equations of motion. The explicit background dependence of the conventional action via the Becchi-Rouet-Stora-Tyutin operator is eliminated in the cubic formulation. A closed-form expression is found for the full nonlinear gauge-transformation law. 13. Exact integrability in quantum field theory International Nuclear Information System (INIS) Thacker, H.B. 1980-08-01 The treatment of exactly integrable systems in various branches of two-dimensional classical and quantum physics has recently been placed in a unified framework by the development of the quantum inverse method. This method consolidates a broad range of developments in classical nonlinear wave (soliton) physics, statistical mechanics, and quantum field theory. The essential technique for analyzing exactly integrable quantum systems was invested by Bethe in 1931. The quantum-mechanical extension of the inverse scattering method and its relationship to the methods associated with Bethe's ansatz are examined here 14. Field theory approaches to new media practices DEFF Research Database (Denmark) Hartley, Jannie Møller; Willig, Ida; Waltorp, Karen 2015-01-01 In this article introducing the theme of the special issue we argue that studies of new media practices might benefit from especially Pierre Bourdieu’s research on cultural production. We introduce some of the literature, which deals with the use of digital media, and which have taken steps...... on more studies within a field theory framework, as the ability of the comprehensive theoretical work and the ideas of a reflexive sociology is able to trigger the good questions, more than it claims to offer a complete and self-sufficient sociology of media and inherent here also new media.... 15. Spectral theory and nonlinear analysis with applications to spatial ecology CERN Document Server Cano-Casanova, S; Mora-Corral , C 2005-01-01 This volume details some of the latest advances in spectral theory and nonlinear analysis through various cutting-edge theories on algebraic multiplicities, global bifurcation theory, non-linear Schrödinger equations, non-linear boundary value problems, large solutions, metasolutions, dynamical systems, and applications to spatial ecology. The main scope of the book is bringing together a series of topics that have evolved separately during the last decades around the common denominator of spectral theory and nonlinear analysis - from the most abstract developments up to the most concrete applications to population dynamics and socio-biology - in an effort to fill the existing gaps between these fields. 16. A simple proof of orientability in colored group field theory. Science.gov (United States) Caravelli, Francesco 2012-01-01 Group field theory is an emerging field at the boundary between Quantum Gravity, Statistical Mechanics and Quantum Field Theory and provides a path integral for the gluing of n-simplices. Colored group field theory has been introduced in order to improve the renormalizability of the theory and associates colors to the faces of the simplices. The theory of crystallizations is instead a field at the boundary between graph theory and combinatorial topology and deals with n-simplices as colored graphs. Several techniques have been introduced in order to study the topology of the pseudo-manifold associated to the colored graph. Although of the similarity between colored group field theory and the theory of crystallizations, the connection between the two fields has never been made explicit. In this short note we use results from the theory of crystallizations to prove that color in group field theories guarantees orientability of the piecewise linear pseudo-manifolds associated to each graph generated perturbatively. Colored group field theories generate orientable pseudo-manifolds. The origin of orientability is the presence of two interaction vertices in the action of colored group field theories. In order to obtain the result, we made the connection between the theory of crystallizations and colored group field theory. 17. Quantum field theory lectures of Sidney Coleman CERN Document Server Derbes, David; Griffiths, David; Hill, Brian; Sohn, Richard; Ting, Yuan-Sen 2018-01-01 Sidney Coleman was a physicist's physicist. He is largely unknown outside of the theoretical physics community, and known only by reputation to the younger generation. He was an unusually effective teacher, famed for his wit, his insight and his encyclopedic knowledge of the field to which he made many important contributions. There are many first-rate quantum field theory books (the ancient Bjorken and Drell, the more modern Itzykson and Zuber, the now-standard Peskin and Schroder, and the recent Zee), but the immediacy of Prof. Coleman's approach and his ability to present an argument simply without sacrificing rigor makes his book easy to read and ideal for the student. Part of the motivation in producing this book is to pass on the work of this outstanding physicist to later generations, a record of his teaching that he was too busy to leave himself. 18. Advanced concepts in particle and field theory CERN Document Server Hübsch, Tristan 2015-01-01 Uniting the usually distinct areas of particle physics and quantum field theory, gravity and general relativity, this expansive and comprehensive textbook of fundamental and theoretical physics describes the quest to consolidate the basic building blocks of nature, by journeying through contemporary discoveries in the field, and analysing elementary particles and their interactions. Designed for advanced undergraduates and graduate students and abounding in worked examples and detailed derivations, as well as including historical anecdotes and philosophical and methodological perspectives, this textbook provides students with a unified understanding of all matter at the fundamental level. Topics range from gauge principles, particle decay and scattering cross-sections, the Higgs mechanism and mass generation, to spacetime geometries and supersymmetry. By combining historically separate areas of study and presenting them in a logically consistent manner, students will appreciate the underlying similarities and... 19. Field theory approaches to new media practices DEFF Research Database (Denmark) Willig, Ida; Waltorp, Karen; Hartley, Jannie Møller 2015-01-01 This special issue of MedieKultur specifically addresses new media practices and asks how field theory approaches can help us understand how culture is (prod)used via various digital platforms. In this article introducing the theme of the special issue, we argue that studies of new media practices...... could benefit particularly from Pierre Bourdieu’s research on cultural production. We introduce some of the literature that concerns digital media use and has been significant for field theory’s development in this context. We then present the four thematic articles in this issue and the articles...... of a reflexive sociology are capable of prompting important questions without necessarily claiming to offer a complete and self-sufficient sociology of media, including new media.... 20. Histories and observables in covariant field theory Science.gov (United States) Paugam, Frédéric 2011-09-01 Motivated by DeWitt's viewpoint of covariant field theory, we define a general notion of a non-local classical observable that applies to many physical Lagrangian systems (with bosonic and fermionic variables), by using methods that are now standard in algebraic geometry. We review the methods of local functional calculus, as they are presented by Beilinson and Drinfeld, and relate them to our construction. We partially explain the relation of these with Vinogradov's secondary calculus. The methods present here are all necessary to understand mathematically properly, and with simple notions, the full renormalization of the standard model, based on functional integral methods. Our approach is close in spirit to non-perturbative methods since we work with actual functions on spaces of fields, and not only formal power series. This article can be seen as an introduction to well-grounded classical physical mathematics, and as a good starting point to study quantum physical mathematics, which make frequent use of non-local functionals, like for example in the computation of Wilson's effective action. We finish by describing briefly a coordinate-free approach to the classical Batalin-Vilkovisky formalism for general gauge theories, in the language of homotopical geometry.	{"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": 0.8130489587783813, "perplexity": 983.3647510646471}, "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-2018-22/segments/1526794866938.68/warc/CC-MAIN-20180525024404-20180525044404-00538.warc.gz"}
https://www.physicsforums.com/threads/pendulum-maximum-displacement.85042/	# Pendulum- Maximum displacement? • Thread starter yourmom98 • Start date • #1 yourmom98 42 0 what is the maximum displacement of a pendulum i don't know what it is. is it the distance from the central point to the end of the arm? and how do i solve it if i am given a periodic Sin function? ## Answers and Replies • #2 Homework Helper 43,021 971 The "maximum displacement" is usually the distance, measured along the arc, from the lowest point of the pendulum to the highest. If you are given a function of the form s= A sin(ωt), then the displacement is the amplitude, A. • Last Post Replies 26 Views 471 • Last Post Replies 10 Views 673 • Last Post Replies 1 Views 1K • Last Post Replies 4 Views 638 • Last Post Replies 2 Views 532 • Last Post Replies 4 Views 288 • Last Post Replies 23 Views 506 • Last Post Replies 14 Views 664 • Last Post Replies 15 Views 153 • Last Post Replies 7 Views 996	{"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": 0.8148371577262878, "perplexity": 2666.5338822272065}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500303.56/warc/CC-MAIN-20230206015710-20230206045710-00789.warc.gz"}
http://alexkritchevsky.com/2019/12/22/many-worlds.html	A possible way to get the Born Rule in Many Worlds [December 22, 2019] The Many-Worlds Interpretation (MWI) of quantum mechanics is probably roughly correct. There is no reason to think that the rules of atomic phenomena would stop applying at larger scales when an experimenter becomes entangled with their experiment (kooky interjections about consciousness notwithstanding…). However, MWI has the problem that it does not easily explain why quantum randomness leads to the probabilities that we observe. The Born Rule says that if a system is in a state $\alpha \\| 0 \> + \beta \\| 1 \>$, upon ‘measurement’ (in which we entangle with one or the other outcome), we measure the eigenvalue associated with the state $\\| 0 \>$ with probability The Born Rule is normally included as an additional postulate in MWI, and this is somewhat unsatisfying. Or at least, it is apparently difficult to justify, given that I’ve read a bunch of attempts, each of which talks about how there haven’t been any other satisfactory attempts. I think it would be unobjectionable to say that there is not a consensus on how to motivate the Born Rule from MWI without any other assumptions. Anyway here’s an argument that I find somewhat compelling? See what you think. 1. A classical coin First let’s think about classical probability, but write it in a notation suggestive of quantum mechanics. Suppose we’re flipping a biased coin that gets heads with probability $P[H] = p$ and $P[T] = q$. Let’s call its states $\\| H \>$ and $\\| T \>$, so the results of a coin flip are written as $p \\| H \> + q \\| T \>$ with $p + q= 1$. Upon $n$ iterations of classical coin-flipping we end up in state Where $\\| H^k T^{n-k} \>$ means a state in which we have observed $k$ heads and $n-k$ tails (in any order). Now suppose this whole experiment is being performed by a poor experimenter who’s, like, locked in a box or something. The experimenter does the experiment, writes down what they think the probability of heads is, and then transmits that to us, outside of the box. So the only value we end up seeing is the value of their measurement of $P[H] = p$, which we’ll call $\hat{P}[H]$. The best estimate that the experimenter can give, of course, is their observed frequency $\frac{k}{n}$, so we might say that the resulting system’s states are identified by the probability perceived by the experimenter: If you let $n$ get very large, the system with $\hat{P}[H] = p$ will end up having the highest-magnitude amplitude, and so we expect to end up in a ‘universe’ where the measurement of the probability $p$ converges on the true value of $p$. This is easily seen, because for large $n$ the binomial distribution $B(n, p, q)$ converges to a normal distribution $\mathcal{N}(np, npq)$ with mean $np$. So, asymptotically, the state $\\| \hat{P}[H] = \frac{np}{n} = p \>$ becomes increasingly high-amplitude relative to all of the others. This is a way of phrasing the law of large numbers. I think this is as good an explanation as any as to what probability ‘is’. Instead of trying to figure out what it means for us to experience an infinite number of events and observe a probability, let’s just let an experimenter who’s locked in a box figure it out for us, and then just have them send us their results! Unsurprisingly, the experimenter does a good job of recovering classical probability. 2. A quantum coin Now let’s try it for a quantum coin (okay, a qubit). The individual experiment runs are now given by $\alpha \\| 0 \> + \beta \\| 1 \>$ where $\alpha, \beta$ are probability amplitudes with $\\| \alpha \\|^2 + \\| \beta \\|^2 = 1$. Note that normalizing these to sum to 1 doesn’t predetermine what the experienced probabilities are, and as we will see the normalization isn’t necessary. As before we generate a state that’s something like: Where are things going to go differently? A potential problem is that each of the measurement results that comprise a $\\| P = \frac{k}{n} \>$ macrostate could have different phases, and there is no reason to think that they will add up neatly – there could be interference between different ways of getting the same result. I’m not totally sure this is reasonable, but it leads to an interesting result, so let’s assume it is. Consider running the experiment twice, but letting each $\\| 0 \>$ state have a different have a different phase $\alpha_j = \\| \alpha \\| e^{i \theta_j}$. (We can ignore the $\beta$ phase without loss of generality by treating it as an overall coefficient to the entire wave function) The state we generate will be: This is no longer a clean binomial distribution. Writing $a = \\| \alpha \\|$ and $b = \\| \beta \\|$ for clarity, the two-iteration wave function is: And $ab (e^{i \theta_1} + e^{i \theta_2}) \\| 0^1 1^1 \>$ only has the same magnitude as $2ab \\| 0^1 1^1 \>$ when $\theta_1 = \theta_2$. 3. Random Walks in Phase Space Now let’s consider what this looks like as $n \ra \infty$. For a state with $k$ $\alpha\\| 0 \>$ terms, we end up with a sum of exponentials with $k$ phases in them: Here $S_{k,n}$ is the set of $k$-element subsets of $n$ elements. For instance if $k=2, n=3$: Our wave function for $n$ iterations of the experiment is given by The classical version of this is a binomial distribution because $E_{k, n}$ is replaced with ${n \choose k}$. The quantum version observes some cancellation. We want to know: as $n \ra \infty$, what value of $k$ dominates? We don’t know anything the phases themselves, so we’ll treat them as classical independent random variables. This means that $\bb{E}[e^{i \theta}] = 0$ and therefore $\bb{E}[E_{k, n}] = 0$ for all $k$. But the expected magnitude is not 0. The sum of all of these random vectors forms a random walk in the complex plane, and the expected amplitude of a random walk is given by $\bb{E}[ \\| E_{1, n} \\|^2 ] = n$. Briefly: this comes from the fact that This means that the magnitude of the $k=1$ term for our quantum coin is proportional to $\sqrt{n}$, rather than the classical value of $n$. For $k > 1$, the same argument applies (it’s still basically a random walk), except that there are ${ n \choose k }$ terms in the sum, so in every case we get an expected amplitude $\bb{E} [ \\| E_{k, n} \\|^2 ] = { n \choose k }$. 4. The Born Rule These don’t tell us the constant of proportionality, since $\bb{E}[ \\| E_{k, n} \\|^2] \neq \bb{E}[ \\| E_{k, n} \\|]^2$, but fortunately we only need to compute the value of $k$ at the peak, and we can find that using $\\| \psi \\|^2$, which is easy to work with: This is a binomial distribution $B(n, a^2, b^2) = B(n, \\|\alpha\\|^2, \\| \beta \\|^2)$, which asymptotically looks like a normal distribution $\mathcal{N}(n \\| \alpha \\|^2, n \\| \alpha \\|^2 \\| \beta \\|^2)$ with maximum $k = n \\| \alpha \\|^2$, which means that the highest-amplitude state measures is: Thus we conclude that the observed probability of measuring $\\| 0 \>$ when interacting with a system in state $\alpha \\| 0 \> + \beta \\| 1 \>$ is centered around $\\| \alpha \\|^2$, as reported by an experimenter in a box who runs the measurement many times, which is what we probably are anyway. And that’s the Born Rule. Ultimately this seems to be because different ways of seeing the same result interfere with each other, suppressing the amplitudes of seeing less uniform results by a factor of the square root of their multiplicity. (Note that this argument should still work if $\\|\alpha \\|^2 + \\| \beta \\|^2 \neq 1$; the resulting asymptotic normal distribution will end up having mean $\frac{n \\| \alpha \\|^2}{\\| \alpha \\|^2 + \\| \beta \\|^2}$.) So that’s interesting. I find the argument that “random walks in phase space might lead to a peak amplitude that matches the Born Rule” to be suspiciously clean, and therefore compelling, but I don’t any confidence that I’ve correctly identified what might actually lead to the random interference in this experiment. Is it the experimental apparatus interfering with itself? Is it hidden degrees of freedom in the experiment itself? Or maybe it’s all of reality, from the point of view of an observer trying to make sense of all historical evidence for the Born Rule. And it’s unclear to me how carefully isolated an experiment would have to be for different orderings of its results to interfere with each other. Presumably the answer is “a lot”, but what if it isn’t? Suffice to say I would love to know a) what’s wrong with this argument (maybe it’s circular, but I haven’t figured out how), or b) if it exists in the literature somewhere, cause I haven’t found anything, although admittedly I didn’t look very hard. I can think of some strange implications of this argument but I don’t want to get ahead of myself. I should go to graduate school.	{"extraction_info": {"found_math": true, "script_math_tex": 70, "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": 0.9374213218688965, "perplexity": 253.12365123256083}, "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/1581875149238.97/warc/CC-MAIN-20200229114448-20200229144448-00242.warc.gz"}
https://dsp.stackexchange.com/questions/43227/rnn-as-posterior-probability-estimation-in-speech-recognition-with-htk	# RNN as posterior probability estimation in speech recognition with HTK I'm new to speech recognition and deep learning and in a learning phase. I'm trying to follow this paper to learn how to use RNN as posterior probability estimation in an HTK environment. The paper proposes RNN-HMM hybrid system, so for the HMM part I need to use HTK platform. The problem is that I couldn't even start from anywhere. I have a sample code which uses HMM to recognize digits, but I'm unable to solve at which part should I insert RNN to the code. If there are any ideas, I would be glad. The system in the paper is shown as follows: I have codes in python environment and HMM is applied using HTK. After converting data to MFCC format, I should use RNN, but after using RNN, which steps of HMM should I apply to generate RNN-HMM acoustic system.	{"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": 0.8090977072715759, "perplexity": 1164.9659827231972}, "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-2020-45/segments/1603107882581.13/warc/CC-MAIN-20201024110118-20201024140118-00356.warc.gz"}
https://deepai.org/publication/a-reductions-approach-to-fair-classification	# A Reductions Approach to Fair Classification We present a systematic approach for achieving fairness in a binary classification setting. While we focus on two well-known quantitative definitions of fairness, our approach encompasses many other previously studied definitions as special cases. Our approach works by reducing fair classification to a sequence of cost-sensitive classification problems, whose solutions yield a randomized classifier with the lowest (empirical) error subject to the desired constraints. We introduce two reductions that work for any representation of the cost-sensitive classifier and compare favorably to prior baselines on a variety of data sets, while overcoming several of their disadvantages. ## Authors • 53 publications • 10 publications • 23 publications • 47 publications • 28 publications • ### Wasserstein Fair Classification We propose an approach to fair classification that enforces independence... 07/28/2019 ∙ by Ray Jiang, et al. ∙ 7 • ### Classification with Fairness Constraints: A Meta-Algorithm with Provable Guarantees Developing classification algorithms that are fair with respect to sensi... 06/15/2018 ∙ by L. 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In response to this scrutiny—and driven by ongoing debates and collaborations with lawyers, policy-makers, social scientists, and others (e.g., Barocas & Selbst, 2016)—machine learning researchers have begun to turn their attention to the topic of “fairness in machine learning,” and, in particular, to the design of fair classification and regression algorithms. In this paper we study the task of binary classification subject to fairness constraints with respect to a pre-defined protected attribute, such as race or sex. Previous work in this area can be divided into two broad groups of approaches. The first group of approaches incorporate specific quantitative definitions of fairness into existing machine learning methods, often by relaxing the desired definitions of fairness, and only enforcing weaker constraints, such as lack of correlation (e.g., Woodworth et al., 2017; Zafar et al., 2017; Johnson et al., 2016; Kamishima et al., 2011; Donini et al., 2018). The resulting fairness guarantees typically only hold under strong distributional assumptions, and the approaches are tied to specific families of classifiers, such as SVMs. The second group of approaches eliminate the restriction to specific classifier families and treat the underlying classification method as a “black box,” while implementing a wrapper that either works by pre-processing the data or post-processing the classifier’s predictions (e.g., Kamiran & Calders, 2012; Feldman et al., 2015; Hardt et al., 2016; Calmon et al., 2017). Existing pre-processing approaches are specific to particular definitions of fairness and typically seek to come up with a single transformed data set that will work across all learning algorithms, which, in practice, leads to classifiers that still exhibit substantial unfairness (see our evaluation in Section 4). In contrast, post-processing allows a wider range of fairness definitions and results in provable fairness guarantees. However, it is not guaranteed to find the most accurate fair classifier, and requires test-time access to the protected attribute, which might not be available. We present a general-purpose approach that has the key advantage of this second group of approaches—i.e., the underlying classification method is treated as a black box—but without the noted disadvantages. Our approach encompasses a wide range of fairness definitions, is guaranteed to yield the most accurate fair classifier, and does not require test-time access to the protected attribute. Specifically, our approach allows any definition of fairness that can be formalized via linear inequalities on conditional moments, such as demographic parity or equalized odds (see Section 2.1). We show how binary classification subject to these constraints can be reduced to a sequence of cost-sensitive classification problems. We require only black-box access to a cost-sensitive classification algorithm, which does not need to have any knowledge of the desired definition of fairness or protected attribute. We show that the solutions to our sequence of cost-sensitive classification problems yield a randomized classifier with the lowest (empirical) error subject to the desired fairness constraints. Corbett-Davies et al. (2017) and Menon & Williamson (2018) begin with a similar goal to ours, but they analyze the Bayes optimal classifier under fairness constraints in the limit of infinite data. In contrast, our focus is algorithmic, our approach applies to any classifier family, and we obtain finite-sample guarantees. Dwork et al. (2018) also begin with a similar goal to ours. Their approach partitions the training examples into subsets according to protected attribute values and then leverages transfer learning to jointly learn from these separate data sets. Our approach avoids partitioning the data and assumes access only to a classification algorithm rather than a transfer learning algorithm. A preliminary version of this paper appeared at the FAT/ML workshop (Agarwal et al., 2017), and led to extensions with more general optimization objectives (Alabi et al., 2018) and combinatorial protected attributes (Kearns et al., 2018). In the next section, we formalize our problem. While we focus on two well-known quantitative definitions of fairness, our approach also encompasses many other previously studied definitions of fairness as special cases. In Section 3, we describe our reductions approach to fair classification and its guarantees in detail. The experimental study in Section 4 shows that our reductions compare favorably to three baselines, while overcoming some of their disadvantages and also offering the flexibility of picking a suitable accuracy–fairness tradeoff. Our results demonstrate the utility of having a general-purpose approach for combining machine learning methods and quantitative fairness definitions. ## 2 Problem Formulation We consider a binary classification setting where the training examples consist of triples , where is a feature vector, is a protected attribute, and is a label. The feature vector can either contain the protected attribute as one of the features or contain other features that are arbitrarily indicative of . For example, if the classification task is to predict whether or not someone will default on a loan, each training example might correspond to a person, where represents their demographics, income level, past payment history, and loan amount; represents their race; and represents whether or not they defaulted on that loan. Note that might contain their race as one of the features or, for example, contain their zipcode—a feature that is often correlated with race. Our goal is to learn an accurate classifier from some set (i.e., family) of classifiers , such as linear threshold rules, decision trees, or neural nets, while satisfying some definition of fairness. Note that the classifiers in do not explicitly depend on . ### 2.1 Fairness Definitions We focus on two well-known quantitative definitions of fairness that have been considered in previous work on fair classification; however, our approach also encompasses many other previously studied definitions of fairness as special cases, as we explain at the end of this section. The first definition—demographic (or statistical) parity—can be thought of as a stronger version of the US Equal Employment Opportunity Commission’s “four-fifths rule,” which requires that the “selection rate for any race, sex, or ethnic group [must be at least] four-fifths (4/5) (or eighty percent) of the rate for the group with the highest rate.”111See the Uniform Guidelines on Employment Selection Procedures, 29 C.F.R. §1607.4(D) (2015). ###### Definition 1 (Demographic parity—DP). A classifier satisfies demographic parity under a distribution over if its prediction is statistically independent of the protected attribute —that is, if for all , . Because , this is equivalent to for all . The second definition—equalized odds—was recently proposed by Hardt et al. (2016) to remedy two previously noted flaws with demographic parity (Dwork et al., 2012). First, demographic parity permits a classifier which accurately classifies data points with one value , such as the value with the most data, but makes random predictions for data points with as long as the probabilities of match. Second, demographic parity rules out perfect classifiers whenever is correlated with . In contrast, equalized odds suffers from neither of these flaws. ###### Definition 2 (Equalized odds—EO). A classifier satisfies equalized odds under a distribution over if its prediction is conditionally independent of the protected attribute given the label —that is, if for all , , and . Because , this is equivalent to for all , . We now show how each definition can be viewed as a special case of a general set of linear constraints of the form Mμ(h)≤c, (1) where matrix and vector describe the linear constraints, each indexed by , and is a vector of conditional moments of the form where and is an event defined with respect to . Crucially, depends on , while cannot depend on in any way. ###### Example 1 (Dp). In a binary classification setting, demographic parity can be expressed as a set of equality constraints, each of the form . Letting , for all , , and , where refers to the event encompassing all points in the sample space, each equality constraint can be expressed as .222Note that . Finally, because each such constraint can be equivalently expressed as a pair of inequality constraints of the form μa(h)−μ⋆(h) ≤0 −μa(h)+μ⋆(h) ≤0, demographic parity can be expressed as equation (1), where , , , , , and . Expressing each equality constraint as a pair of inequality constraints allows us to control the extent to which each constraint is enforced by positing for some (or all) . ###### Example 2 (Eo). In a binary classification setting, equalized odds can be expressed as a set of equality constraints, each of the form . Letting , for all , , and , each equality constraint can be equivalently expressed as μ(a,y)(h)−μ(⋆,y)(h) ≤0 −μ(a,y)(h)+μ(⋆,y)(h) ≤0. As a result, equalized odds can be expressed as equation (1), where , , , , , and . Again, we can posit for some (or all) to allow small violations of some (or all) of the constraints. Although we omit the details, we note that many other previously studied definitions of fairness can also be expressed as equation (1). For example, equality of opportunity (Hardt et al., 2016) (also known as balance for the positive class; Kleinberg et al., 2017), balance for the negative class (Kleinberg et al., 2017), error-rate balance (Chouldechova, 2017), overall accuracy equality (Berk et al., 2017), and treatment equality (Berk et al., 2017) can all be expressed as equation (1); in contrast, calibration (Kleinberg et al., 2017) and predictive parity (Chouldechova, 2017) cannot because to do so would require the event to depend on . We note that our approach can also be used to satisfy multiple definitions of fairness, though if these definitions are mutually contradictory, e.g., as described by Kleinberg et al. (2017), then our guarantees become vacuous. ### 2.2 Fair Classification In a standard (binary) classification setting, the goal is to learn the classifier with the minimum classification error: . However, because our goal is to learn the most accurate classifier while satisfying fairness constraints, as formalized above, we instead seek to find the solution to the constrained optimization problem333We consider misclassification error for concreteness, but all the results in this paper apply to any error of the form , where . minh∈Herr(h)subject toMμ(h)≤c. (2) Furthermore, rather than just considering classifiers in the set , we can enlarge the space of possible classifiers by considering randomized classifiers that can be obtained via a distribution over . By considering randomized classifiers, we can achieve better accuracy–fairness tradeoffs than would otherwise be possible. A randomized classifier makes a prediction by first sampling a classifier from and then using to make the prediction. The resulting classification error is and the conditional moments are (see Appendix A for the derivation). Thus we seek to solve minQ∈Δerr(Q)subject toMμ(Q)≤c, (3) where is the set of all distributions over . In practice, we do not know the true distribution over and only have access to a data set of training examples . We therefore replace and in equation (3) with their empirical versions and . Because of the sampling error in , we also allow errors in satisfying the constraints by setting for all , where . After these modifications, we need to solve the empirical version of equation (3): minQ∈Δˆerr(Q)subject toMˆμ(Q)≤ˆc. (4) ## 3 Reductions Approach We now show how the problem (4) can be reduced to a sequence of cost-sensitive classification problems. We further show that the solutions to our sequence of cost-sensitive classification problems yield a randomized classifier with the lowest (empirical) error subject to the desired constraints. ### 3.1 Cost-sensitive Classification We assume access to a cost-sensitive classification algorithm for the set . The input to such an algorithm is a data set of training examples , where and denote the losses—costs in this setting—for predicting the labels or , respectively, for . The algorithm outputs argminh∈Hn∑i=1h(Xi)C1i+(1−h(Xi))C0i. (5) This abstraction allows us to specify different costs for different training examples, which is essential for incorporating fairness constraints. Moreover, efficient cost-sensitive classification algorithms are readily available for several common classifier representations (e.g., Beygelzimer et al., 2005; Langford & Beygelzimer, 2005; Fan et al., 1999). In particular, equation (5) is equivalent to a weighted classification problem, where the input consists of labeled examples with and , and the goal is to minimize the weighted classification error . This is equivalent to equation (5) if we set and . ### 3.2 Reduction To derive our fair classification algorithm, we rewrite equation (4) as a saddle point problem. We begin by introducing a Lagrange multiplier for each of the constraints, summarized as , and form the Lagrangian L(Q,λ) =ˆerr(Q)+λ⊤(Mˆμ(Q)−ˆc). Thus, equation (4) is equivalent to For computational and statistical reasons, we impose an additional constraint on the norm of and seek to simultaneously find the solution to the constrained version of (6) as well as its dual, obtained by switching min and max: Because is linear in and and the domains of and are convex and compact, both problems have solutions (which we denote by and ) and the minimum value of (P) and the maximum value of (D) are equal and coincide with . Thus, is the saddle point of (Corollary 37.6.2 and Lemma 36.2 of Rockafellar, 1970). We find the saddle point by using the standard scheme of Freund & Schapire (1996), developed for the equivalent problem of solving for an equilibrium in a zero-sum game. From game-theoretic perspective, the saddle point can be viewed as an equilibrium of a game between two players: the -player choosing and the -player choosing . The Lagrangian specifies how much the -player has to pay to the -player after they make their choices. At the saddle point, neither player wants to deviate from their choice. Our algorithm finds an approximate equilibrium in which neither player can gain more than by changing their choice (where is an input to the algorithm). Such an approximate equilibrium corresponds to a -approximate saddle point of the Lagrangian, which is a pair , where L(ˆQ,ˆλ) ≤L(Q,ˆλ)+ν for all Q∈Δ, L(ˆQ,ˆλ) ≥L(ˆQ,λ)−ν for all λ∈R\|K\|+, ∥λ∥1≤B. We proceed iteratively by running a no-regret algorithm for the -player, while executing the best response of the -player. Following Freund & Schapire (1996), the average play of both players converges to the saddle point. We run the exponentiated gradient algorithm (Kivinen & Warmuth, 1997) for the -player and terminate as soon as the suboptimality of the average play falls below the pre-specified accuracy . The best response of the -player can always be chosen to put all of the mass on one of the candidate classifiers , and can be implemented by a single call to a cost-sensitive classification algorithm for the set . Algorithm 1 fully implements this scheme, except for the functions and , which correspond to the best-response algorithms of the two players. (We need the best response of the -player to evaluate whether the suboptimality of the current average play has fallen below .) The two best response functions can be calculated as follows. #### \textscBestλ(Q): the best response of the λ-player. The best response of the -player for a given is any maximizer of over all valid s. In our setting, it can always be chosen to be either or put all of the mass on the most violated constraint. Letting and letting denote the vector of the standard basis, returns {0if ˆγ(Q)≤ˆc,Bek∗otherwise, where k∗=argmaxk[ˆγk(Q)−ˆck]. #### \textscBesth(λ): the best response of the Q-player. Here, the best response minimizes over all s in the simplex. Because is linear in , the minimizer can always be chosen to put all of the mass on a single classifier . We show how to obtain the classifier constituting the best response via a reduction to cost-sensitive classification. Letting be the empirical event probabilities, the Lagrangian for which puts all of the mass on a single is then L(h,λ)=ˆerr(h)+λ⊤(Mˆμ(h)−ˆc) =ˆE[1{h(X)≠Y}]−λ⊤ˆc+∑k,jMk,jλkˆμj(h) =−λ⊤ˆc+ˆE[1{h(X)≠Y}] +∑k,jMk,jλkpjˆE[gj(X,A,Y,h(X))1{(X,A,Y)∈Ej}]. Assuming a data set of training examples , the minimization of over then corresponds to cost-sensitive classification on with costs444For general error, , the costs and contain, respectively, the terms and instead of and . C0i =1{Yi≠0} +∑k,jMk,jλkpjgj(Xi,Ai,Yi,0)1{(Xi,Ai,Yi)∈Ej} C1i =1{Yi≠1} +∑k,jMk,jλkpjgj(Xi,Ai,Yi,1)1{(Xi,Ai,Yi)∈Ej}. ###### Theorem 1. Letting , Algorithm 1 satisfies the inequality νt≤Blog(\|K\|+1)ηt+ηρ2B. Thus, for , Algorithm 1 will return a -approximate saddle point of in at most iterations. This theorem, proved in Appendix B, bounds the suboptimality of the average play , which is equal to its suboptimality as a saddle point. The right-hand side of the bound is optimized by , leading to the bound . This bound decreases with the number of iterations and grows very slowly with the number of constraints . The quantity is a problem-specific constant that bounds how much any single classifier can violate the desired set of fairness constraints. Finally, is the bound on the -norm of , which we introduced to enable this specific algorithmic scheme. In general, larger values of will bring the problem (P) closer to (6), and thus also to (4), but at the cost of needing more iterations to reach any given suboptimality. In particular, as we derive in the theorem, achieving suboptimality may need up to iterations. ###### Example 3 (Dp). Using the matrix for demographic parity as described in Section 2, the cost-sensitive reduction for a vector of Lagrange multipliers uses costs C0i=1{Yi≠0},C1i=1{Yi≠1}+λAipAi−∑a∈Aλa, where and , effectively replacing two non-negative Lagrange multipliers by a single multiplier, which can be either positive or negative. Because for all , . Furthermore, because all empirical moments are bounded in , we can assume , which yields the bound . Thus, Algorithm 1 terminates in at most iterations. ###### Example 4 (Eo). For equalized odds, the cost-sensitive reduction for a vector of Lagrange multipliers uses costs C0i =1{Yi≠0}, C1i =1{Yi≠1}+λ(Ai,Yi)p(Ai,Yi)−∑a∈Aλ(a,Yi)p(⋆,Yi), where , , and . If we again assume , then we obtain the bound . Thus, Algorithm 1 terminates in at most iterations. ### 3.3 Error Analysis Our ultimate goal, as formalized in equation (3), is to minimize the classification error while satisfying fairness constraints under a true but unknown distribution over . In the process of deriving Algorithm 1, we introduced three different sources of error. First, we replaced the true classification error and true moments with their empirical versions. Second, we introduced a bound on the magnitude of . Finally, we only run the optimization algorithm for a fixed number of iterations, until it reaches suboptimality level . The first source of error, due to the use of empirical rather than true quantities, is unavoidable and constitutes the underlying statistical error. The other two sources of error, the bound and the suboptimality level , stem from the optimization algorithm and can be driven arbitrarily small at the cost of additional iterations. In this section, we show how the statistical error and the optimization error affect the true accuracy and the fairness of the randomized classifier returned by Algorithm 1—in other words, how well Algorithm 1 solves our original problem (3). To bound the statistical error, we use the Rademacher complexity of the classifier family , which we denote by , where is the number of training examples. We assume that for some and . We note that in the vast majority of classifier families, including norm-bounded linear functions (see Theorem 1 of Kakade et al., 2009 ), neural networks (see Theorem 18 of Bartlett & Mendelson, 2002), and classifier families with bounded VC dimension (see Lemma 4 and Theorem 6 of Bartlett & Mendelson, 2002). Recall that in our empirical optimization problem we assume that , where are error bounds that account for the discrepancy between and . In our analysis, we assume that these error bounds have been set in accordance with the Rademacher complexity of . ###### Assumption 1. There exists and such that and , where is the number of data points that fall in , nj\coloneqq∣∣{i:(Xi,Ai,Yi)∈Ej}∣∣. The optimization error can be bounded via a careful analysis of the Lagrangian and the optimality conditions of (P) and (D). Combining the three different sources of error yields the following bound, which we prove in Appendix C. ###### Theorem 2. Let Assumption 1 hold for , where . Let be any -approximate saddle point of , let minimize subject to , and let . Then, with probability at least , the distribution satisfies err(ˆQ) ≤err(Q⋆)+2ν+˜O(n−α), γk(ˆQ) ≤ck+1+2νB+∑j∈J\|Mk,j\|˜O(n−αj) for all k, where suppresses polynomial dependence on . If for all , then, for all , γk(ˆQ)≤ck+1+2νB+∑j∈J\|Mk,j\|˜O((np⋆j)−α). In other words, the solution returned by Algorithm 1 achieves the lowest feasible classification error on the true distribution up to the optimization error, which grows linearly with , and the statistical error, which grows as . Therefore, if we want to guarantee that the optimization error does not dominate the statistical error, we should set . The fairness constraints on the true distribution are satisfied up to the optimization error and up to the statistical error. Because the statistical error depends on the moments, and the error in estimating the moments grows as , we can set to guarantee that the optimization error does not dominate the statistical error. Combining this reasoning with the learning rate setting of Theorem 1 yields the following theorem (proved in Appendix C). ###### Theorem 3. Let . Let Assumption 1 hold for , where . Let minimize subject to . Then Algorithm 1 with , and terminates in iterations and returns , which with probability at least satisfies err(ˆQ) ≤err(Q⋆)+˜O(n−α), γk(ˆQ) ≤ck+∑j∈J\|Mk,j\|˜O(n−αj) for all k. ###### Example 5 (Dp). If denotes the number of training examples with , then Assumption 1 states that we should set and Theorem 3 then shows that for a suitable setting of , , , and , Algorithm 1 will return a randomized classifier with the lowest feasible classification error up to while also approximately satisfying the fairness constraints ∣∣E[h(X)\|A=a]−E[h(X)]∣∣≤˜O(n−αa)for all a, where is with respect to as well as . ###### Example 6 (Eo). Similarly, if denotes the number of examples with and and denotes the number of examples with , then Assumption 1 states that we should set and Theorem 3 then shows that for a suitable setting of , , , and , Algorithm 1 will return a randomized classifier with the lowest feasible classification error up to while also approximately satisfying the fairness constraints ∣∣E[h(X)\|A=a,Y=y]−E[h(X)\|Y=y]∣∣≤˜O(n−α(a,y)) for all , . Again, includes randomness under the true distribution over as well as . ### 3.4 Grid Search In some situations, it is preferable to select a deterministic classifier, even if that means a lower accuracy or a modest violation of the fairness constraints. A set of candidate classifiers can be obtained from the saddle point . Specifically, because is a minimizer of and is linear in , the distribution puts non-zero mass only on classifiers that are the -player’s best responses to . If we knew , we could retrieve one such best response via the reduction to cost-sensitive learning introduced in Section 3.2. We can compute using Algorithm 1, but when the number of constraints is very small, as is the case for demographic parity or equalized odds with a binary protected attribute, it is also reasonable to consider a grid of values , calculate the best response for each value, and then select the value with the desired tradeoff between accuracy and fairness. ###### Example 7 (Dp). When the protected attribute is binary, e.g., , then the grid search can in fact be conducted in a single dimension. The reduction formally takes two real-valued arguments and , and then adjusts the costs for predicting by the amounts δa=λapa−λa−λa′andδa′=λa′pa′−λa−λa′, respectively, on the training examples with and . These adjustments satisfy , so instead of searching over and , we can carry out the grid search over alone and apply the adjustment to the protected attribute value . With three attribute values, e.g., , we similarly have , so it suffices to conduct grid search in two dimensions rather than three. ###### Example 8 (Eo). If , we obtain the adjustment δ(a,y)=λ(a,y)p(a,y)−λ(a,y)+λ(a′,y)p(⋆,y) for an example with protected attribute value and label , and similarly for protected attribute value . In this case, separately for each , the adjustments satisfy p(a,y)δ(a,y)+p(a′,y)δ(a′,y)=0, so it suffices to do the grid search over and and set the parameters for to . ## 4 Experimental Results We now examine how our exponentiated-gradient reduction performs at the task of binary classification subject to either demographic parity or equalized odds. We provide an evaluation of our grid-search reduction in Appendix D. We compared our reduction with the score-based post-processing algorithm of Hardt et al. (2016), which takes as its input any classifier, (i.e., a standard classifier without any fairness constraints) and derives a monotone transformation of the classifier’s output to remove any disparity with respect to the training examples. This post-processing algorithm works with both demographic parity and equalized odds, as well as with binary and non-binary protected attributes. For demographic parity, we also compared our reduction with the reweighting and relabeling approaches of Kamiran & Calders (2012). Reweighting can be applied to both binary and non-binary protected attributes and operates by changing importance weights on each example with the goal of removing any statistical dependence between the protected attribute and label.666Although reweighting was developed for demographic parity, the weights that it induces are achievable by our grid search, albeit the grid search for equalized odds rather than demographic parity. Relabeling was developed for binary protected attributes. First, a classifier is trained on the original data (without considering fairness). The training examples close to the decision boundary are then relabeled to remove all disparity while minimally affecting accuracy. The final classifier is then trained on the relabeled data. As the base classifiers for our reductions, we used the weighted classification implementations of logistic regression and gradient-boosted decision trees in scikit-learn (Pedregosa et al., 2011). In addition to the three baselines described above, we also compared our reductions to the “unconstrained” classifiers trained to optimize accuracy only. We used four data sets, randomly splitting each one into training examples (75%) and test examples (25%): • [nosep] • The adult income data set (Lichman, 2013) (48,842 examples). Here the task is to predict whether someone makes more than \$50k per year, with gender as the protected attribute. To examine the performance for non-binary protected attributes, we also conducted another experiment with the same data, using both gender and race (binarized into white and non-white) as the protected attribute. Relabeling, which requires binary protected attributes, was therefore not applicable here. • ProPublica’s COMPAS recidivism data (7,918 examples). The task is to predict recidivism from someone’s criminal history, jail and prison time, demographics, and COMPAS risk scores, with race as the protected attribute (restricted to white and black defendants). • Law School Admissions Council’s National Longitudinal Bar Passage Study (Wightman, 1998) (20,649 examples). Here the task is to predict someone’s eventual passage of the bar exam, with race (restricted to white and black only) as the protected attribute. • The Dutch census data set (Dutch Central Bureau for Statistics, 2001) (60,420 examples). Here the task is to predict whether or not someone has a prestigious occupation, with gender as the protected attribute. While all the evaluated algorithms require access to the protected attribute at training time, only the post-processing algorithm requires access to at test time. For a fair comparison, we included in the feature vector , so all algorithms had access to it at both the training time and test time. We used the test examples to measure the classification error for each approach, as well as the violation of the desired fairness constraints, i.e., and for demographic parity and equalized odds, respectively. We ran our reduction across a wide range of tradeoffs between the classification error and fairness constraints. We considered and for each value ran Algorithm 1 with across all . As expected, the returned randomized classifiers tracked the training Pareto frontier (see Figure 2 in Appendix D). In Figure 1, we evaluate these classifiers alongside the baselines on the test data. For all the data sets, the range of classification errors is much smaller than the range of constraint violations. Almost all the approaches were able to substantially reduce or remove disparity without much impact on classifier accuracy. One exception was the Dutch census data set, where the classification error increased the most in relative terms. Our reduction generally dominated or matched the baselines. The relabeling approach frequently yielded solutions that were not Pareto optimal. Reweighting yielded solutions on the Pareto frontier, but often with substantial disparity. As expected, post-processing yielded disparities that were statistically indistinguishable from zero, but the resulting classification error was sometimes higher than achieved by our reduction under a statistically indistinguishable disparity. In addition, and unlike the post-processing algorithm, our reduction can achieve any desired accuracy–fairness tradeoff, allows a wider range of fairness definitions, and does not require access to the protected attribute at test time. Our grid-search reduction, evaluated in Appendix D, sometimes failed to achieve the lowest disparities on the training data, but its performance on the test data very closely matched that of our exponentiated-gradient reduction. However, if the protected attribute is non-binary, then grid search is not feasible. For instance, for the version of the adult income data set where the protected attribute takes on four values, the grid search would need to span three dimensions for demographic parity and six dimensions for equalized odds, both of which are prohibitively costly. ## 5 Conclusion We presented two reductions for achieving fairness in a binary classification setting. Our reductions work for any classifier representation, encompass many definitions of fairness, satisfy provable guarantees, and work well in practice. Our reductions optimize the tradeoff between accuracy and any (single) definition of fairness given training-time access to protected attributes. Achieving fairness when training-time access to protected attributes is unavailable remains an open problem for future research, as does the navigation of tradeoffs between accuracy and multiple fairness definitions. ## Acknowledgements We would like to thank Aaron Roth, Sam Corbett-Davies, and Emma Pierson for helpful discussions. ## References • Agarwal et al. (2017) Agarwal, A., Beygelzimer, A., Dudík, M., and Langford, J. A reductions approach to fair classiﬁcation. In Fairness, Accountability, and Transparency in Machine Learning (FATML), 2017. • Alabi et al. (2018) Alabi, D., Immorlica, N., and Kalai, A. T. Unleashing linear optimizers for group-fair learning and optimization. In Proceedings of the 31st Annual Conference on Learning Theory (COLT), 2018. • Barocas & Selbst (2016) Barocas, S. and Selbst, A. D. Big data’s disparate impact. California Law Review, 104:671–732, 2016. • Bartlett & Mendelson (2002) Bartlett, P. L. and Mendelson, S. 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Online learning and online convex optimization. Foundations and Trends in Machine Learning, 4(2):107–194, 2012. • Wightman (1998) Wightman, L. LSAC National Longitudinal Bar Passage Study, 1998. • Woodworth et al. (2017) Woodworth, B. E., Gunasekar, S., Ohannessian, M. I., and Srebro, N. Learning non-discriminatory predictors. In Proceedings of the 30th Conference on Learning Theory (COLT), pp. 1920–1953, 2017. • Zafar et al. (2017) Zafar, M. B., Valera, I., Rodriguez, M. G., and Gummadi, K. P. Fairness constraints: Mechanisms for fair classification. In Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS) , pp. 962–970, 2017. ## Appendix A Error and Fairness for Randomized Classifiers Let denote the distribution over triples . The accuracy of a classifier is measured by 0-1 error, , which for a randomized classifier becomes err(Q)\coloneqqP(X,A,Y)∼D,h∼Q[h(X)≠Y]=∑h∈HQ(h)err(h). The fairness constraints on a classifier are . Recall that . For a randomized classifier we define its moment as μj(Q)\coloneqqE(X,A,Y)∼D,h∼Q[gj(X,A,Y,h(X))∣∣Ej]=∑h∈HQ(h)μj(h), where the last equality follows because is independent of the choice of . ## Appendix B Proof of Theorem 1 The proof follows immediately from the analysis of Freund & Schapire (1996) applied to the Exponentiated Gradient (EG) algorithm (Kivinen & Warmuth, 1997), which in our specific case is also equivalent to Hedge (Freund & Schapire, 1997). Let and . We associate any with the that is equal to on coordinates through and puts the remaining mass on the coordinate . Consider a run of Algorithm 1. For each , let be the associated element of . Let and let be equal to on coordinates through and put zero on the coordinate . Thus, for any and the associated , we have, for all , λ⊤rt=(λ′)⊤r′t, (7) and, in particular, λ⊤t(Mˆ	{"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": 0.8228172063827515, "perplexity": 1585.2439617203968}, "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/1631780056392.79/warc/CC-MAIN-20210918093220-20210918123220-00505.warc.gz"}
https://www.gradesaver.com/textbooks/math/algebra/introductory-algebra-for-college-students-7th-edition/chapter-7-section-7-1-rational-expressions-and-their-simplification-exercise-set-page-491/3	## Introductory Algebra for College Students (7th Edition) $x=8$ A rational expression is undefined when the value of its denominator is zero. Write and solve an expression where the denominator is zero. $x-8=0$ Add 8 to both sides. $x-8+8=0+8$ Simplify. $x=8$	{"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": 0.980858325958252, "perplexity": 517.4313179634551}, "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-2018-30/segments/1531676592636.68/warc/CC-MAIN-20180721145209-20180721165209-00369.warc.gz"}
https://slideplayer.com/slide/3062426/	# Lecture 3 Universal TM. Code of a DTM Consider a one-tape DTM M = (Q, Σ, Γ, δ, s). It can be encoded as follows: First, encode each state, each direction, ## Presentation on theme: "Lecture 3 Universal TM. Code of a DTM Consider a one-tape DTM M = (Q, Σ, Γ, δ, s). It can be encoded as follows: First, encode each state, each direction,"— Presentation transcript: Lecture 3 Universal TM Code of a DTM Consider a one-tape DTM M = (Q, Σ, Γ, δ, s). It can be encoded as follows: First, encode each state, each direction, and each symbol into a natural number (code(B) = 0, code(R) =1, code(L) = 2, code(s)=3, code(h)=4,... ). Then encode each transition δ(q, a) = (p, b, D) into a string 0 10 10 10 10 qapbD The code of M is obtained by combining all codes code i of transitions together: 111code 1 11code 2 11∙∙∙11code m 111. Remark: Each TM has many codes. All codes of TMs form a Turing-decidable language. Universal DTM One can design a three-tape DTM M* which behaves as follows: On input, M* first decodes M on the second tape and then simulates M on the output tape. Clearly, L(M) = { \| x ε L(M)}. Thus, Theorem 1. { \| x ε L(M)} is Turing- acceptable. Next, we prove Theorem 2. { \| x ε L(M)} isn't Turing-decidable. To do so, we consider A = { M \| M accepts M} and prove Lemma. A isn't Turing-decidable. Barber cuts his own hair Class 1: {Barber \| he can cut his own hair} Class 2: {Barber \| he cannot cut own hair} Question: Is there a barber who cuts hair of everybody in class 2, but not cut hair of anybody in class 1. Answer: No!!! Proof. Suppose such a barber exists. If he cuts his own hair, then he is in class 1 and hence he cannot cut his own hair, a contradiction. If he cannot cut his own hair, then he belongs to class 2 and hence he can cut his own hair, a contradiction. This argument is called diagonalization. hair barber Example. There exists an irrational number. Proof. Consider all rational numbers in (0,1). They are countable, a1, a2, …. Now, we construct a number such that its i-th digit is different from the i-th digit of ai. Then this number is not rational. a1 a2 digits Proof. For contradiction, suppose that A is accepted by a one-tape DTM M’. We look at M’ on input M’. If M’ accepts M’, then M’ is in A, which means that M’ rejects M’, a contradiction. If M’ rejects M’, then M’ isn’t in A which means that M’ accepts M’, a contradiction. Many-one reduction Consider two sets A c Σ and B c Γ. If there exists a Turing-computable total function f : Σ → Γ* such that x ε A iff f(x) ε B, then we say that A is many-one reducible to B, and write A ≤ m B. A = { M \| M accepts M} B = { \| M accepts x} Claim. A ≤ m B. Proof. Define f(M) =. M ε A iff M accepts M iff ε B Theorem. A ≤ m B, B ≤ m C imply A ≤ m C. (This means that ≤ m is a partial ordering.) Theorem. If A ≤ m B and B is Turing- decidable, then A is Turing-decidable. By this theorem, { \| M accepts x} isn’t Turing-decidable. Complete in r. e. An r. e. set A is complete in r. e. if for every r. e. set B, B ≤ m A. Halting problem Theorem. K = { \| M accepts x } is complete in r. e.. Proof. (1) K is a r. e. set. (2) For any r. e. set A, there exists a DTM M A such that A = L(M A ). For every input x of M A, define f(x) =. Then x ε A iff f(x) ε K. Halting problem Theorem. K = { \| M accepts x } is complete in r. e.. Proof. (1) K is a r. e. set. (2) For any r. e. set A, there exists a DTM M A such that A = L(M A ). For every input x of M A, define f(x) =. Then x ε A iff f(x) ε K. Nonempty Nonempty = {M \| L(M) ≠ Φ } is complete in r. e. Proof. (1) Nonempty is a r. e. set. Construct a DTM M* as follows: For each M, we may try every input of M, one by one. If M accepts an input, then M is accepted by M. (2) K ≤ m Nonempty. Suppose M’ is a DTM accepting every input. For each input of K, we define f( ) = M where M is a DTM working as follows: on an input y, Step 1. M simulates M on input x. If M accepts x, then go to Step 2. Step 2. M simulates M’ on input y Therefore, ε K => M accepts x => M accepts every input y => f( ) = M ε Nonempty not in K => M doesn’t halt on x => M doesn’t halt on y => L(M ) = Φ => f( ) not in Nonempty r. e. –hard A set B is r. e.-hard if for every r. e. set A, A ≤ m B Remark Every complete set is r. e.-hard. However, not every r. e.-hard set is complete. Every r. e.-hard set is not recursive. All = {M \| M accepts all inputs} All is r. e. hard. All is not r. e. All is not complete. All = {M \| M accepts all inputs} All is r. e. hard. All is not r. e. All is not complete. r. e. property A subset P of TM codes is called a r. e. property if M ε P and L(M’) = L(M) imply M’ ε P. e.g., Nonempty, Empty, All are r. e. properties. Question: Give an example which is a subsets of TM codes, but not a r. e. property. Nontrivial A r. e. property is trivial if either it is empty or it contains all r. e. set. Rice Theorem 1 Every nontrivial r. e. property is not recursive. Proof Let P be a nontrivial r. e. property. For contradiction. Suppose P is a recursive set. So is its complement. Note that either P or its complement P does not contains the empty set. Without loss of generality, assume that P does not contains the empty set. Since P is nontrivial, P contains a nonempty r. e. set A. Let Ma be a TM accepting A, i.e., A=L(Ma). We want to prove K ≤m P. For each input of K, we define f( ) = M where M is a DTM working as follows. For each input y of M, it first goes to Step 1. Step 1. M simulates M on input x of M. If M accepts x, then go to Step 2. Step 2. M simulates Ma on y. If Ma accepts y, then M accepts y. Therefore, if ε K then L(M ) = L(Ma) = A ε P, and if not in K, then L(M ) = Φ not in P Since K is not recursive and K ≤ m P, we obtain a contradiction. Recursive = {M \| L(M) is recursive} is not recursive. RE = {M \| L(M) is r. e.} is trivial. Question: Is K an r. e. property? Is every r. e. property complete? Is it true that for any r. e. property, either it or its complement is complete? Rice Theorem 2 A r. e. property P is r. e. iff the following three conditions hold: (1)If A ε P and A c B for some r. e. set B, then B ε P. (2) If A is an infinite set in P, then A has a finite subset in P. (3) The set of finite languages in P is enumerable, in the sense that there is a TM that generates the (possibly) infinite string code1#code2# …, where code i is a code for the ith finite languages in P. The code for the finite language {w 1, w 2, …, w n } is [w 1,w 2,…,w n]. In other words, there exists an r. e. set B that is a subset of codes of finite languages in P such that for every finite language F in P, B contains at least one code of F. Examples All is not r. e. because All does not satisfy condition (2). The complement of ALL is not r. e. because it does not satisfy condition (1). Empty is not r. e. because it does not satisfy (1) Nonempty is r. e. because it satisfies (1), (2) and (3). Undecidable Problems Given TMs M and M’, is it true that L(M)=L(M’)? This problem is undecidable, i.e., A = { \| L(M) = L(M’)} is not recursive. Proof. Empty ≤ m A. Let M o be a fixed TM such that L(M o ) = Φ. Define f(M) =. Then, M ε Empty iff ε A. Let A and B be two nonempty proper subsets of Σ. If A B and B A are recursive, then A ≤ m B. Proof. Let y ε B and z ε B. Define y if x ε A B f(x) = z if x ε B A x, otherwise Research Problem For a DFA M=(Q, Σ, δ, s, F), L(M) = L(M) where M = (Q, Σ, δ, s, Q-F). Given a DTM M, could we have an algorithm to compute a DTM M* such that L(M) = L(M) when L(M) is regular, and M will not halt when L(M) is not regular? Proof of Hierarchy Theorems Diagonalization Proof of Hierarchy Theorems Space-constructible function s(n) is fully space-constructible if there exists a DTM M such that for sufficiently large n and any input x with \|x\|=n, Space M (x) = s(n). Space Hierarchy If s 2 (n) is a fully space-constructible function, s 1 (n)/s 2 (n) → 0 as n → infinity, s 1 (n) > log n, then DSPACE(s 2 (n)) DSPACE(s 1 (n)) ≠ Φ Download ppt "Lecture 3 Universal TM. Code of a DTM Consider a one-tape DTM M = (Q, Σ, Γ, δ, s). It can be encoded as follows: First, encode each state, each direction," Similar presentations	{"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": 0.9143061637878418, "perplexity": 1639.3576660874942}, "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/1620243989690.55/warc/CC-MAIN-20210516044552-20210516074552-00360.warc.gz"}
https://www.physicsforums.com/threads/tricky-mean-life-question-potassium.63074/	# Tricky mean life question (potassium) 1. Feb 7, 2005 ### JamesJames Natural potassiu, has an atomic weight of 39.089 and contains 0.0118 atomic percent of the isotope 19 K 40, which has 2 decay modes: 19 K 40 -> 20 Ca 40 + beta particle + neutrino (nu bar) 19 K 40 + e- -> 18 Ar 40 * + beta particle + neutrino (nu no bar) where 18 Ar 40 * means an excited state of 18 Ar 40. In this case, this excited state decays to the fround state by emitting a single gamma ray. The total intensity of beta particles emitted is 2.7*10^4 kg^-1 . s^-1 of natural potassium and on average there are 12 gamma rays emitted to every 100 beta particles emitted. Estimate the mean life of 19 K 40. If someone can set up the question for me, I will take it from there. I am very confused right now as to how the intensity can be used to compute the lifetime. Can someone please explain this using equations? James 2. Feb 7, 2005 ### JamesJames Please guys..any equations you can write to help me would be greatly appreciated. James 3. Feb 8, 2005 ### JamesJames Come on guys...someone msut be able to help me...please JAmes 4. Feb 8, 2005 ### HallsofIvy Staff Emeritus I'm no expert on this but you are told how many, on average, beta particles are emitted per second. Since each Argon atom has to emit a beta particle when it disintegrates, that gives you the the average number of Argon atoms that disintegrate per second (atoms/second) which should be the reciprocal of the average lifetime (seconds/atom). Any suggestions, Doc Al? 5. Feb 8, 2005 ### Gamma I agree with HallsofIvy. Problem can be solved with some mathematics. However, since there are two modes of decay, which one do we use to find the mean life? For the first mode of decay, one needs to consider the number of beta particles emitted per second.(mean life is the reciprocal of that as HallsofIvy said) regards. Similar Discussions: Tricky mean life question (potassium)	{"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": 0.942287802696228, "perplexity": 1588.6316798404173}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886105108.31/warc/CC-MAIN-20170818194744-20170818214744-00607.warc.gz"}
https://www.physicsforums.com/threads/finding-general-solution-of-radical-equation.639250/	# Finding general solution of Radical Equation • #1 I_am_learning 686 16 Before trying to find out the general solution of a radical equation; I would first like to know if it can be found? For example I have a equation of the form $\text{A1}+\text{A2} x + \text{A3}\sqrt{\text{B1}+\text{B2} x+\text{B3} x^{\frac{3}{2}}+\text{B4}\sqrt{x}+\text{B5} x^2}+ \text{A4}\sqrt{\text{C1}+\text{C2} x+\text{C3} x^{\frac{3}{2}}+\text{C4}\sqrt{x}+\text{C5} x^2}=0$ Can I find x in terms of the Constants A1,A2 etc? What is the general view on deciding whether a general solution to radical equation exist or not? I tried searching, but couldn't find out the answer regarding radical equation. For polynomial equation though, I learned that a general solution doesn't exist for polynomials of degree 5 or higher. http://en.wikipedia.org/wiki/Abel–Ruffini_theorem • #2 Homework Helper 43,021 971 1. Isolate one of the square roots. 2. Square both sides which will leave a single square root. 3. Isolate that square root. 4. Now you will have an equation involving powers of x of 2 3/2, 1, and 1/2. Let y= x1/2 so that you have a polynomial involving y4, y3, y2, and y. (There cannot be a "general solution for a polynomial equation" of degee greater than 5 involving only powers and roots of the coefficients because they may have solution that cannot be written in terms of roots.) • #3 I_am_learning 686 16 1. Isolate one of the square roots. 2. Square both sides which will leave a single square root. 3. Isolate that square root and Square both sides 4. Now you will have an equation involving powers of x of 2 3/2, 1, and 1/2. Let y= x1/2 so that you have a polynomial involving y4, y3, y2, and y. (There cannot be a "general solution for a polynomial equation" of degee greater than 5 involving only powers and roots of the coefficients because they may have solution that cannot be written in terms of roots.) (I made a little amendments). Thanks. 1.$\text{A1}+\text{A2} x + \text{A3}\sqrt{\text{B1}+\text{B2} x+\text{B3} x^{\frac{3}{2}}+\text{B4}\sqrt{x}+\text{B5} x^2}= \text{A4}\sqrt{\text{C1}+\text{C2} x+\text{C3} x^{\frac{3}{2}}+\text{C4}\sqrt{x}+\text{C5} x^2}$ 2.$\left(\text{A1}+\text{A2} x + \text{A3}\sqrt{\text{B1}+\text{B2} x+\text{B3} x^{\frac{3}{2}}+\text{B4}\sqrt{x}+\text{B5} x^2}\right)^2= \left(\text{A4}\sqrt{\text{C1}+\text{C2} x+\text{C3} x^{\frac{3}{2}}+\text{C4}\sqrt{x}+\text{C5} x^2}\right)^2$ $\left(\text{A1}^2+\text{A3}^2 \text{B1}+\text{A3}^2 \text{B4} \sqrt{x}+2 \text{A1} \text{A2} x+\text{A3}^2 \text{B2} x+\text{A3}^2 \text{B3} x^{3/2}+\text{A2}^2 x^2+\text{A3}^2 \text{B5} x^2+\text{A3} (2 \text{A1}+2 \text{A2} x) \sqrt{\text{B1}+\text{B4} \sqrt{x}+\text{B2} x+\text{B3} x^{3/2}+\text{B5} x^2}\right)=\text{A4}^2 \text{C1}+\text{A4}^2 \text{C4} \sqrt{x}+\text{A4}^2 \text{C2} x+\text{A4}^2 \text{C3} x^{3/2}+\text{A4}^2 \text{C5} x^2$ 3.$\text{A3} (2 \text{A1}+2 \text{A2} x) \sqrt{\text{B1}+\text{B4} \sqrt{x}+\text{B2} x+\text{B3} x^{3/2}+\text{B5} x^2}=\left(\text{A4}^2 \text{C1}+\text{A4}^2 \text{C4} \sqrt{x}+\text{A4}^2 \text{C2} x+\text{A4}^2 \text{C3} x^{3/2}+\text{A4}^2 \text{C5} x^2\right)-\left(\text{A1}^2+\text{A3}^2 \text{B1}+\text{A3}^2 \text{B4} \sqrt{x}+2 \text{A1} \text{A2} x+\text{A3}^2 \text{B2} x+\text{A3}^2 \text{B3} x^{3/2}+\text{A2}^2 x^2+\text{A3}^2 \text{B5} x^2\right)$ $\left(\text{A3} (2 \text{A1}+2 \text{A2} x) \sqrt{\text{B1}+\text{B4} \sqrt{x}+\text{B2} x+\text{B3} x^{3/2}+\text{B5} x^2}\right)^2=\left(\left(\text{A4}^2 \text{C1}+\text{A4}^2 \text{C4} \sqrt{x}+\text{A4}^2 \text{C2} x+\text{A4}^2 \text{C3} x^{3/2}+\text{A4}^2 \text{C5} x^2\right)-\left(\text{A1}^2+\text{A3}^2 \text{B1}+\text{A3}^2 \text{B4} \sqrt{x}+2 \text{A1} \text{A2} x+\text{A3}^2 \text{B2} x+\text{A3}^2 \text{B3} x^{3/2}+\text{A2}^2 x^2+\text{A3}^2 \text{B5} x^2\right)\right)^2$ 4.$4 \text{A1}^2 \text{A3}^2 \text{B1}+4 \text{A1}^2 \text{A3}^2 \text{B4} \sqrt{x}+8 \text{A1} \text{A2} \text{A3}^2 \text{B1} x+4 \text{A1}^2 \text{A3}^2 \text{B2} x+4 \text{A1}^2 \text{A3}^2 \text{B3} x^{3/2}+8 \text{A1} \text{A2} \text{A3}^2 \text{B4} x^{3/2}+4 \text{A2}^2 \text{A3}^2 \text{B1} x^2+8 \text{A1} \text{A2} \text{A3}^2 \text{B2} x^2+4 \text{A1}^2 \text{A3}^2 \text{B5} x^2+8 \text{A1} \text{A2} \text{A3}^2 \text{B3} x^{5/2}+4 \text{A2}^2 \text{A3}^2 \text{B4} x^{5/2}+4 \text{A2}^2 \text{A3}^2 \text{B2} x^3+8 \text{A1} \text{A2} \text{A3}^2 \text{B5} x^3+4 \text{A2}^2 \text{A3}^2 \text{B3} x^{7/2}+4 \text{A2}^2 \text{A3}^2 \text{B5} x^4=\text{A1}^4+2 \text{A1}^2 \text{A3}^2 \text{B1}+\text{A3}^4 \text{B1}^2-2 \text{A1}^2 \text{A4}^2 \text{C1}-2 \text{A3}^2 \text{A4}^2 \text{B1} \text{C1}+\text{A4}^4 \text{C1}^2+2 \text{A1}^2 \text{A3}^2 \text{B4} \sqrt{x}+2 \text{A3}^4 \text{B1} \text{B4} \sqrt{x}-2 \text{A3}^2 \text{A4}^2 \text{B4} \text{C1} \sqrt{x}-2 \text{A1}^2 \text{A4}^2 \text{C4} \sqrt{x}-2 \text{A3}^2 \text{A4}^2 \text{B1} \text{C4} \sqrt{x}+2 \text{A4}^4 \text{C1} \text{C4} \sqrt{x}+4 \text{A1}^3 \text{A2} x+4 \text{A1} \text{A2} \text{A3}^2 \text{B1} x+2 \text{A1}^2 \text{A3}^2 \text{B2} x+2 \text{A3}^4 \text{B1} \text{B2} x+\text{A3}^4 \text{B4}^2 x-4 \text{A1} \text{A2} \text{A4}^2 \text{C1} x-2 \text{A3}^2 \text{A4}^2 \text{B2} \text{C1} x-2 \text{A1}^2 \text{A4}^2 \text{C2} x-2 \text{A3}^2 \text{A4}^2 \text{B1} \text{C2} x+2 \text{A4}^4 \text{C1} \text{C2} x-2 \text{A3}^2 \text{A4}^2 \text{B4} \text{C4} x+\text{A4}^4 \text{C4}^2 x+2 \text{A1}^2 \text{A3}^2 \text{B3} x^{3/2}+2 \text{A3}^4 \text{B1} \text{B3} x^{3/2}+4 \text{A1} \text{A2} \text{A3}^2 \text{B4} x^{3/2}+2 \text{A3}^4 \text{B2} \text{B4} x^{3/2}-2 \text{A3}^2 \text{A4}^2 \text{B3} \text{C1} x^{3/2}-2 \text{A3}^2 \text{A4}^2 \text{B4} \text{C2} x^{3/2}-2 \text{A1}^2 \text{A4}^2 \text{C3} x^{3/2}-2 \text{A3}^2 \text{A4}^2 \text{B1} \text{C3} x^{3/2}+2 \text{A4}^4 \text{C1} \text{C3} x^{3/2}-4 \text{A1} \text{A2} \text{A4}^2 \text{C4} x^{3/2}-2 \text{A3}^2 \text{A4}^2 \text{B2} \text{C4} x^{3/2}+2 \text{A4}^4 \text{C2} \text{C4} x^{3/2}+6 \text{A1}^2 \text{A2}^2 x^2+2 \text{A2}^2 \text{A3}^2 \text{B1} x^2+4 \text{A1} \text{A2} \text{A3}^2 \text{B2} x^2+\text{A3}^4 \text{B2}^2 x^2+2 \text{A3}^4 \text{B3} \text{B4} x^2+2 \text{A1}^2 \text{A3}^2 \text{B5} x^2+2 \text{A3}^4 \text{B1} \text{B5} x^2-2 \text{A2}^2 \text{A4}^2 \text{C1} x^2-2 \text{A3}^2 \text{A4}^2 \text{B5} \text{C1} x^2-4 \text{A1} \text{A2} \text{A4}^2 \text{C2} x^2-2 \text{A3}^2 \text{A4}^2 \text{B2} \text{C2} x^2+\text{A4}^4 \text{C2}^2 x^2-2 \text{A3}^2 \text{A4}^2 \text{B4} \text{C3} x^2-2 \text{A3}^2 \text{A4}^2 \text{B3} \text{C4} x^2+2 \text{A4}^4 \text{C3} \text{C4} x^2-2 \text{A1}^2 \text{A4}^2 \text{C5} x^2-2 \text{A3}^2 \text{A4}^2 \text{B1} \text{C5} x^2+2 \text{A4}^4 \text{C1} \text{C5} x^2+4 \text{A1} \text{A2} \text{A3}^2 \text{B3} x^{5/2}+2 \text{A3}^4 \text{B2} \text{B3} x^{5/2}+2 \text{A2}^2 \text{A3}^2 \text{B4} x^{5/2}+2 \text{A3}^4 \text{B4} \text{B5} x^{5/2}-2 \text{A3}^2 \text{A4}^2 \text{B3} \text{C2} x^{5/2}-4 \text{A1} \text{A2} \text{A4}^2 \text{C3} x^{5/2}-2 \text{A3}^2 \text{A4}^2 \text{B2} \text{C3} x^{5/2}+2 \text{A4}^4 \text{C2} \text{C3} x^{5/2}-2 \text{A2}^2 \text{A4}^2 \text{C4} x^{5/2}-2 \text{A3}^2 \text{A4}^2 \text{B5} \text{C4} x^{5/2}-2 \text{A3}^2 \text{A4}^2 \text{B4} \text{C5} x^{5/2}+2 \text{A4}^4 \text{C4} \text{C5} x^{5/2}+4 \text{A1} \text{A2}^3 x^3+2 \text{A2}^2 \text{A3}^2 \text{B2} x^3+\text{A3}^4 \text{B3}^2 x^3+4 \text{A1} \text{A2} \text{A3}^2 \text{B5} x^3+2 \text{A3}^4 \text{B2} \text{B5} x^3-2 \text{A2}^2 \text{A4}^2 \text{C2} x^3-2 \text{A3}^2 \text{A4}^2 \text{B5} \text{C2} x^3-2 \text{A3}^2 \text{A4}^2 \text{B3} \text{C3} x^3+\text{A4}^4 \text{C3}^2 x^3-4 \text{A1} \text{A2} \text{A4}^2 \text{C5} x^3-2 \text{A3}^2 \text{A4}^2 \text{B2} \text{C5} x^3+2 \text{A4}^4 \text{C2} \text{C5} x^3+2 \text{A2}^2 \text{A3}^2 \text{B3} x^{7/2}+2 \text{A3}^4 \text{B3} \text{B5} x^{7/2}-2 \text{A2}^2 \text{A4}^2 \text{C3} x^{7/2}-2 \text{A3}^2 \text{A4}^2 \text{B5} \text{C3} x^{7/2}-2 \text{A3}^2 \text{A4}^2 \text{B3} \text{C5} x^{7/2}+2 \text{A4}^4 \text{C3} \text{C5} x^{7/2}+\text{A2}^4 x^4+2 \text{A2}^2 \text{A3}^2 \text{B5} x^4+\text{A3}^4 \text{B5}^2 x^4-2 \text{A2}^2 \text{A4}^2 \text{C5} x^4-2 \text{A3}^2 \text{A4}^2 \text{B5} \text{C5} x^4+\text{A4}^4 \text{C5}^2 x^4$ But now there are $x^4,x^{\frac{7}{2}},x^3,x^{\frac{5}{2}},x^2,x^{\frac{3}{2}},x,\sqrt{x}$ If I replace x = y^2 then I will have polynomial of degree 8. So it appears the equation won't have general solution. (Atleast not in terms of roots and powers as you said). But I wonder in what form I might get the solution, if at all. • Last Post Replies 1 Views 420 • Last Post Replies 5 Views 140 • Last Post Replies 4 Views 603 • Last Post Replies 44 Views 2K • Last Post Replies 0 Views 500 • Last Post Replies 4 Views 371 • Last Post Replies 5 Views 850 • Last Post Replies 5 Views 1K • Last Post Replies 4 Views 1K • Last Post Replies 10 Views 2K	{"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": 0.8776156306266785, "perplexity": 780.0742243405955}, "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/1674764501555.34/warc/CC-MAIN-20230209081052-20230209111052-00458.warc.gz"}
https://www.intechopen.com/chapters/60249	Open access peer-reviewed chapter # Ultrafast Intramolecular Proton Transfer Reaction of 1,2- Dihydroxyanthraquinone in the Excited State By Sebok Lee, Myungsam Jen, Kooknam Jeon, Jaebeom Lee, Joonwoo Kim and Yoonsoo Pang Submitted: December 6th 2017Reviewed: February 21st 2018Published: March 27th 2018 DOI: 10.5772/intechopen.75783 ## Abstract 1,2-Dihydroxyanthraquinone (alizarin) shows an ultrafast intramolecular proton transfer in the excited states between the adjacent hydroxyl and carbonyl groups. Due to the ground and electronic structure of locally excited and proton-transferred tautomers, alizarin shows dual emission bands with strong Stokes shifts. The energy barriers between the locally excited (LE) and proton-transferred (PT) tautomers in the excited state are strongly dependent on the solvent polarity and thus alizarin shows complicated photophysical properties including solvent and excitation dependences. The excited-state intramolecular proton transfer (ESIPT) of alizarin was monitored in time-resolved stimulated Raman spectroscopic investigation, where the instantaneous structural changes of anthraquinone backbone in 70~80 fs were captured. Two major vibrational modes of alizarin, ν(C=C) and ν(C=O) represent the proton transfer reaction in the excited state, which then leads to the vibrational relaxation of the product and the restructuring of solvent molecules. Ultrafast changes in solvent vibrational modes of dimethyl sulfoxide (DMSO) were also investigated for the solvation dynamics including hydrogen bond breaking and reformation. ### Keywords • excited-state intramolecular proton transfer • tautomerization • femtosecond stimulated Raman • transient absorption ## 1. Introduction Proton transfer occurring either intramolecularly or intermolecularly is one of the fundamental chemical reactions and has been of great interest in chemistry, biology, and related disciplines [1, 2, 3, 4, 5]. Molecules with the excited-state intramolecular proton transfer (ESIPT) often show large Stokes shifts, which is beneficial in many photonic applications due to small self-absorption [6, 7]. The ESIPT reactions have been extensively studied by time-resolved spectroscopic methods, where the ultrafast laser pulses initiate the chemical reaction in the excited state [6, 7, 8, 9, 10]. Femtosecond transient absorption technique was used as the time-resolved electronic probe in monitoring ultrafast proton transfer reactions in the time scales of ~30 fs [8, 10]. Recently, a much faster ESIPT of ~13 fs in 10-hydroxybenzo[h]quinolone has been reported by a fluorescence upconversion technique [9]. 1,2-Dihydroxyanthraquinone (alizarin) is one of natural red pigments which forms an intramolecular hydrogen bond between a carbonyl and a hydroxyl group in the ground and excited states [11, 12, 13, 14, 15, 16, 17, 18]. Upon photoexcitation, a proton transfer from the hydroxyl to the carbonyl group occurs and the dual emission bands of the locally excited (LE) and proton-transferred (PT) tautomers have been reported [11, 12, 13, 14]. The scheme of electronic structure of alizarin in LE and PT tautomers is shown in Figure 1. Since the barrier between LE and PT tautomers of alizarin in the excited state is tunable by changing solvent polarity [19, 20], the proton transfer reaction dynamics between the LE and PT tautomers might also be controlled by this factor. The emission band of the PT tautomer dominates in nonpolar aprotic solvents while the dual emission bands both from the LE and PT tautomers appear in polar aprotic solvents with the inhibition of the ESIPT reaction [21, 22]. According to the density functional theory (DFT)/time-dependent DFT (TDDFT) simulation results, the LE tautomer of alizarin is more stable (4.7–4.8 kcal/mol) than the PT tautomer in ground state, while the LE tautomer becomes less stable in the excited state with tunable energy barriers between the LE and PT [23]. For example, the energy barrier from the LE to PT tautomer in the exited state was estimated as 1.30 kcal/mol in benzene (nonpolar aprotic) and 3.19 kcal/mol in ethanol (polar aprotic). The excited state lifetime of the alizarin PT tautomer was observed as 60–80 ps in several time-resolved spectroscopic investigations on alizarin, but the exact proton transfer dynamics of alizarin was not clearly obtained due to strong excited state absorption and emission signals, and complicated excited state dynamics including vibrational relaxations, solvations, etc., [20, 24, 25, 26]. However, the ultrafast ESIPT reactions (45–120 fs) of several anthraquinone derivatives including 1-hydroxyanthraquinone and 1-chloroacetylaminoanthraquinone have been measured by time-resolved fluorescent measurements [11, 15, 27, 28]. Femtosecond stimulated Raman spectroscopy (FSRS) with both high temporal (<50 fs) and spectral (<10 cm−1) resolutions was introduced recently for the study of excited state dynamics and reaction mechanisms [29, 30] and has been widely used to study the photo-induced population and structural dynamics in many chemical and biological systems [31, 32, 33, 34]. In this chapter, the ESIPT reaction and excited state dynamics of alizarin will be overviewed by using experimental results of steady-state absorption and emission, femtosecond transient absorption, and femtosecond stimulated Raman measurement. The excited state dynamics of alizarin was examined by changing the solvent polarity and the evidence for the ultrafast proton transfer reaction and subsequent structural changes in the product state were inspected by the time-dependent skeletal vibrational modes of alizarin. ## 2. Experimental details ### 2.1. Chemical preparation Alizarin (Sigma-Aldrich, St. Louis, MO), dimethyl sulfoxide (DMSO, Daejung Chemicals and Metals, Siheung, Korea), ethanol (Duksan Pure Chemicals, Ansan, Korea), and other chemicals were used as received. Alizarin hardly dissolves in water but dissolves in most organic solvents, so alizarin solutions (33–50 μM) were prepared in ethanol and DMSO for the steady-state absorption and emission, and transient absorption measurements. A 2 mm cell with a stirring magnet was used for transient absorption measurements to avoid photo-damage from the laser pulses. The DMSO solutions of alizarin up to 20-mM concentrations in a 0.5 mm flow cell recirculated by a peristaltic pump were used for stimulated Raman measurements. ### 2.2. Steady-state absorption and emission measurements The absorption spectra were recorded by a UV/Vis spectrometer (S-3100, Scinco, Seoul, Korea) and the emission spectra were obtained by a time-resolved fluorescence setup based on a time-correlated single photon-counting module (Picoharp 300, PicoQuant, Berlin, Germany), a picosecond diode laser (λex = 405 nm; P-C-405, PicoQuant), a monochromator (Cornerstone 260, Newport Corp., Irvine, CA), and a photomultiplier tube detector (PMA 192, PicoQuant). ### 2.3. Transient absorption spectroscopy A femtosecond transient absorption setup based on a Ti:sapphire regenerative amplifier (LIBRA-USP-HE, Coherent Inc., Santa Clara, CA) was used for transient absorption measurements [35, 36]. The pump pulses at 403 nm were generated by sum-harmonic generation (SHG) in a BBO crystal (θ = 29.2°, Eksma Optics, Vilnius, Lithuania) and compressed in a prism-pair compressor. The whitelight supercontinuum probe pulses (450–1000 nm) generated in a sapphire window were tightly focused to the sample with the pump and detected with a fiber-based spectrometer (QE65Pro, Ocean Optics, Largo, FL). Transient absorption spectra and kinetics were analyzed in a global fit analysis by using a software package Glotaran [37]. ### 2.4. Femtosecond stimulated Raman spectroscopy A femtosecond stimulated Raman setup based on the Ti:sapphire regenerative amplifier (LIBRA-USP-HE) was used for time-resolved Raman measurements. A narrowband picosecond pulses (802 nm, 0.6 nm, 1.2 ps) generated by a home-built grating filter (1200 gr/mm) was used for the Raman pump, and a broadband (850–1000 nm) whitelight continuum generated in a YAG window (Newlight Photonics, Toronto, ON) was used for the Raman probe. The Raman probe filtered with long pass filters (FEL0850, Thorlabs Inc., Newton, NJ; 830 DCLP, Omega Optical Inc., Brattleboro, VT) was combined at the sample with the Raman pump and the actinic pump at 403 nm generated from SHG. The Raman pump was modulated at 500 Hz by an optical chopper (MC2000, Thorlabs Inc.) and the Raman probe was recorded at 1 kHz shot-to-shot level by a spectrograph (Triax 320, Horiba Jobin Yvon GmbH, Bensheim, Germany) and a CCD detector (PIXIS 100, Princeton Instruments, Trento, NJ). The optical time delay between the actinic pump and the Raman pump/probe pair was controlled by a motorized stage (MFN25PP, Newport Inc.) and a controller (ESP300, Newport Inc.). The Raman pump of 350 nJ pulse energy and the actinic pump of 750 nJ pulse energy were used in a typical FSRS measurement. ### 2.5. Computational details DFT simulations for the Raman vibrational modes of alizarin were conducted by the Gaussian 09 software package (Gaussian Inc., Wallingford, CT), and the B3LYP/6-31G(d,p) level of theory with the optimized geometries from previous TDDFT results was used [23, 38]. The scaling factors for the vibrational frequencies obtained in previous reports were used to visualize the Raman spectra of alizarin both in ground and excited electronic states with arbitrary bandwidths of 10–15 cm−1 [39]. ## 3. Steady-state absorption and emission spectra of alizarin In the ground state, the LE tautomer exists lower than the PT tautomer in energy and the energy barrier between two tautomers is too high for the proton transfer in the ground state to be observed [19]. On the other hand, the LE tautomer in the excited state which can be approached by photoexcitation exists higher in energy than the PT tautomer, and the tautomerization to the PT tautomer can occur depending on the barrier height separating two tautomers [11, 12, 13, 14, 16, 27, 40]. The absorption and emission spectra of alizarin in n-heptane, ethanol, and DMSO solution are shown in Figure 2(a). The absorption spectrum of alizarin in n-heptane appears as several vibronic bands at ~405, 425, and 450 nm and the absorption bands of alizarin in both ethanol and DMSO are inhomogeneously broadened and red-shifted by 20–30 nm from the bands in n-heptane. The emission bands of alizarin in n-heptane centered at 610 and 660 nm show large Stokes’ shifts from the absorption band representing the intramolecular proton transfer in the excited state. The emission spectra of alizarin in ethanol and DMSO show increased emission in the range of 500–600 nm in addition to main emission bands at 620 and 670 nm, which is interpreted as the emission signal originating from the LE state in the excited state. Figure 2(b) shows the dependence of the excitation wavelength in the emission spectra of alizarin in ethanol. Alizarin shows two emission bands in ethanol solution. One centered at 535 nm from the LE tautomer appears strongly with 485 nm excitation, while the other centered at 620 nm from the PT tautomer becomes the main band with 405-nm excitation. The excess energy in the 405-nm excitation may facilitate the ESIPT by overcoming the energy barrier of the LE-PT tautomerization. Concentration and wavelength dependences in the emission spectra of alizarin can be the evidence for the existence of the energy barrier between the LE and PT tautomers [19, 20, 41]. To further investigate the intramolecular proton transfer of alizarin and the solvent dependence, the steady-state absorption and emission spectra of alizarin in binary mixtures of ethanol and water were measured with 405-nm excitation as shown in Figure 3. The absorption spectra of alizarin show a slight increase in absorbance with the addition of water to ethanol up to 50% without any spectral change. However, the emission spectra of alizarin show a strong solvent dependence. The PT emission bands at 615 and 670 nm decrease as the fraction of water increases up to 50% while the LE emission band at 530 nm increases. The isosbestic point between the LE and PT emission bands is clearly observed at 560 nm, which clearly supports the transition between the LE and PT tautomers. In addition, a decrease of overall quantum yield of alizarin with the addition of water may represent a nonradiative rate constant of the LE tautomer is much smaller than that of the PT tautomer. Recently, the effect of water on the ESIPT reaction of alizarin was further investigated by the simulations based on the time-dependent density functional theory [42]. It has been noted that the strong intramolecular hydrogen bonding of alizarin between the carbonyl and hydroxyl group may facilitate the ESIPT reaction in the excited state. Furthermore, the inhibition of the ESIPT process by water molecules by forming hydrogen bonds with the carbonyl or hydroxyl groups of alizarin was proposed, which weakens the intramolecular hydrogen bonding associated with the ESIPT process and thus increases the energy barrier between the LE and PT tautomer [42]. We have used time-resolved electronic (femtosecond transient absorption) and vibrational spectroscopy (FSRS) to further study the detailed kinetics and mechanism of the ESIPT reaction of alizarin in the excited state. ## 4. Excited state intramolecular proton transfer of alizarin ### 4.1. Femtosecond transient absorption results Transient absorption results of alizarin in ethanol and in a binary mixture of ethanol:water = 1:1 with 403-nm excitation are shown in Figure 4. Within 10 ps time delay, the excited state absorption (ESA) band centered at 510 nm and the stimulated emission (SE) band in the 570–750 nm range are observed in both ethanol and ethanol-water mixture. A broad and weak ESA band in the 500–550 nm range is left after 1 ns time delay for the ethanol-water mixture, while all the excited state population of alizarin in ethanol solution decays to the ground state by the same time. The global fit results for alizarin in ethanol are summarized by two kinetic components of 8.3 and 87 ps whose evolution associated difference spectra (EADS) are shown in Figure 4(a). The 8.3 ps component with slightly broader absorption band (450–580 nm) but without emission signal represents the vibrationally hot PT tautomer, and the 87 ps component with both absorption and emission (580–750 nm) signals represents the relaxed PT tautomer in the excited state, which is consistent with previous results [20, 24, 25, 26]. The excited state dynamics of alizarin in ethanol-water mixture is somewhat complex. Instead of performing the global fit analysis of the whole transient absorption data, we analyzed the absorption (<580 nm) and emission (>580 nm) part of the data separately in the global fit analysis. We found three kinetic components of 7.6, 31.8, and 890 ps from the absorption part and two components of 15.7 and 540 ps from the emission part of the data. The kinetic components of 7.6 and 31.8 ps in the absorption part are tentatively assigned as the vibrationally hot and relaxed PT tautomers of alizarin, respectively, by inferring from the results of ethanol solution. However, the 7.6 ps component may include the decay of the LE state, as the blocking of the proton transfer reaction was observed with the addition of water from results of the steady-state emission spectra. The 15.7 ps lifetime of the first emission component of the data is much shorter than the lifetime of 31.8 ps component in the absorption part, thus this component may also represent the emission signal of both the LE and PT tautomer which cannot be separated in all the analysis we have done. In addition to the fast kinetic components for the LE and PT tautomers, a long-lived component (540 or 890 ps) appeared as a very broad absorption band in 500–600 nm. It is noted that the shortened lifetime of the PT tautomer (87 → 31.8 ps) with the addition of water to ethanol observed in transient absorption measurements is consistent to the reduced quantum yield and the increased nonradiative rate constant of alizarin observed in the steady-state emission measurements. As suggested by the recent theoretical study [42], water molecules may form hydrogen bonds with the carbonyl and hydroxyl groups of alizarin and impede the intramolecular proton transfer reaction of alizarin. Thus the long decay component in the transient absorption of alizarin in ethanol-water mixture may be considered as the “trapped” state of alizarin with water molecules. Further details on the solute-solvent interaction and resulting ESIPT kinetics can be investigated by FSRS, where time-resolved structural changes of solute and solvent molecules can be monitored. It has been proposed that faster components of 300–400 fs time constant generally observed from the transient absorption signals of alizarin in the wavelengths (570–585 nm) where the strong ESA and SE signals cancel out, may represent the kinetics for the vibrational relaxation in the LE tautomer and the ESIPT to the PT tautomer [41, 43]. We also observed these fast components universally in the transient absorption results of alizarin in ethanol, methanol, DMSO, and ethanol-water mixture (examples are shown in Figure 4(c) and (d)), but did not show any dependence on the solvent polarity. Since the ESA, SE, and the ground-state bleaching signals of two tautomers of alizarin in transient absorption measurements are overlapped in wavelength and time, it seems to be very difficult to separate the kinetic components of the vibrational relaxation of the LE and PT tautomers, the ESIPT, etc. We conclude that the transient absorption measurements may be inadequate for the correct analysis of the ESIPT process, a further investigation by FSRS was performed to obtain the population and structural dynamics of alizarin upon photoexcitation. ### 4.2. Femtosecond stimulated Raman results #### 4.2.1. FSRS details Alizarin is soluble in ethanol and DMSO but the Raman spectrum of ethanol overlaps that of alizarin in many spectral regions. Then small changes in the vibrational modes of alizarin might not be observed in ethanol solution due to strong Raman modes of ethanol. The Raman bands of DMSO, however, can be separable from the Raman modes of alizarin. Thus femtosecond stimulated Raman measurements of alizarin were done with DMSO solution. From the analysis of transient absorption result of alizarin in DMSO solution, two kinetic components were obtained [44]. Two components of 1.1 and 83.3 ps represent the vibrational relaxation in the PT tautomer and the lifetime of PT tautomer in the excited state, respectively. Although a fast (~600 fs) kinetic component was observed at 590 nm where the strong ESA and SE signals cancel out, it is not clear whether this component represents the ESIPT dynamics of alizarin. The Raman intensity of the FSRS, often called the Raman gain can be evaluated by Eq. (1): RamanGain=IR.PumpONIBkgIR.PumpOFFIBkgE1 where IR.PumpONand IR.PumpOFFrepresent the intensity of Raman probe with and without the Raman pump, respectively, and IBkgrepresents the dark signal of the CCD detector. The Raman probe of 600,000 pulses (60 accumulations of 10-sec acquisition) was averaged in a typical Raman gain measurement with the half of them focused together with the Raman pump to the sample, to obtain a Raman gain signal in a signal-to-noise level of 2 × 10−5 (or 0.002%) at a specific time delay with the actinic pump pulses. Time-resolved stimulated Raman spectra of alizarin in DMSO at multiple time delays were obtained at time delays of −1 to 100 ps and the ground state spectrum measured at −10 ps time delay, for example, was subtracted from each stimulated Raman spectrum to obtain the difference stimulated Raman spectra shown in Figure 5(a). A small portion of transient absorption signal, for example, the ESA and SE can be obtained together with stimulated Raman gain signals at most time delays, thus a polynomial background subtraction was performed to remove the transient absorption signal. #### 4.2.2. The population and structural dynamics of the ESIPT Major Raman bands of the ground electronic state of alizarin in the 1500–1800 cm−1 range shown in Figure 5(a) were assigned as the ring ν(C=C) at 1573 and 1594 cm−1, and ν(C=O) at 1634 and 1661 cm−1 mainly according to the DFT simulation results. One ν(C=O) at 1661 cm−1 is assigned to the isolated carbonyl at C10 position and the other at 1634 cm−1 is the carbonyl at the site of the ESIPT and adjacent to a hydroxyl group [44]. Another Raman band at 1191 cm−1 is assigned as δ(CH) and δ(OH). In the excited stimulated Raman spectra of alizarin, several Raman bands at 1162, 1555, and 1632 cm−1 appeared in 50–100 fs after the photoexcitation and showed a decay after 20 ps or so. According to the TDDFT simulation results [23, 44], we tentatively assign the 1162 cm−1 band as the δ(CH) and δ(OH) of the PT tautomer, and the 1555 and 1632 cm−1 as ν(C=C) and ν(C=O) bands also in the PT tautomer of alizarin. To obtain the details of the excited state dynamics and the ESIPT from the stimulated Raman bands of alizarin, the experimental data were fit with a low-order polynomial background and several Gaussian functions for Raman bands. The population dynamics of ν(C=C) and ν(C=O) bands at 1555 and 1632 cm−1 shown in Figure 5(b) shows a ubiquitous sharp rise in 70–80 fs and a slow decay into the ground state which is compatible to the PT tautomer’s lifetime of 83.3 ps as shown in the transient absorption results. The structural dynamics of ν(C=C) and ν(C=O) modes of the PT tautomer are shown in Figure 5(c) and (d) as the time-dependent changes in the peak position and the bandwidth. The peak shift of the solvent vibrational mode, δ(CH3) of DMSO at 1421 cm−1 represents the instrument response function of FSRS measurements for comparison with the excited dynamics of alizarin Raman bands. It is interesting to note that a strong blue-shift (1540 → 1553 cm−1) and a decrease of bandwidth (28 → 24 cm−1) of ν(C=C) band all occur in an ultrafast time scale of ~150 fs after photoexcitation. On the other hand, the ν(C=O) band shows a strong red-shift (1645 → 1630 cm−1) in the same time delay of 100–150 fs although this vibrational band appears too broad for the bandwidth analysis. As well represented in Figure 5(b–d), the population growth and the structural changes of ν(C=C) and ν(C=O) Raman bands of the PT tautomer of alizarin are interpreted as the ESIPT process from the LE to the PT tautomer. Since no Raman band of the LE tautomer has been identified from the FSRS results, this could also be included for the dynamics of ν(C=C) and ν(C=O) bands at 1555 and 1632 cm−1. The vibrational relaxation along an electronic potential surface would generally result in slight blue-shifts in the strongly coupled vibrational modes due to the anharmonicity of the electric potential surface. However, the ν(C=C) and ν(C=O) bands showed strong (~15 cm−1) peak shifts either in increasing and decreasing bandwidth, respectively, during the ultrafast period of the population growth for the PT tautomer. This cannot be explained by any type of relaxation inside the same potential surface but has to be understood as the nuclear rearrangements for the intramolecular proton transfer reaction. Therefore, we conclude that the intramolecular proton transfer reaction of alizarin in the excited state occurs in ultrafast time scale of 70–80 fs. Another interesting fact is the strong and opposite peak shifts observed for ν(C=C) and ν(C=O) bands during the ESIPT reaction. We could imagine a transition state for the ESIPT reaction of alizarin as a six-membered ring formed by intramolecular hydrogen bonding between carbonyl group and hydroxyl group. We propose that the strong and opposite peak shifts of the ν(C=C) and ν(C=O) band directly represent the changes in the resonance structure of the alizarin backbone which is composed of multiple C=C and C-C bonds and a C=O. The details of the ESIPT reaction mechanism of alizarin need be confirmed by thorough theoreticalinvestigations, which is beyond of the scope of this chapter. The reaction mechanism of many ESIPT reactions and the existence of transition states have been recently reported by several theoretical works based on TDDFT and several transition states of six-membered ring between carbonyl and hydroxyl groups were represented for 1,8-dihydroxy-2-naphthaldehyde [7, 45, 46]. Although a separate transition state of the ESIPT reaction was not resolved from the FSRS results, we also propose the reaction may occur via the transition state of a new hydrogen-bond six-membered ring attached to the anthraquinone backbone. As shown in Figure 5(b–d), two more kinetic components other than the population decay of the PT tautomer were identified. A slight blue-shift (1553 → 1557 cm−1) and an increased bandwidth (24 → 26 cm−1) of ν(C=C) mode observed in 3–10 ps and another slight blue-shift (1630 → 1636 cm−1) of ν(C=O) mode shown in 20–30 ps represent the vibrational relaxation in the product potential surface of the PT conformer. There were no further changes in peak position and bandwidth of ν(C=C) and ν(C=O) modes during the population decay of the PT tautomer, which also supports the assignment of the vibrational relaxation in the PT potential surface. ## 5. Solvation dynamics The intramolecular proton transfer reaction of alizarin in the excited state was evidenced by the population and structural dynamics of two major vibrational modes of ν(C=C) and ν(C=O). Ultrafast ESIPT reaction of alizarin in the excited state can also be observed indirectly by the changes in the solvent vibrational spectrum such as the instantaneous disruption of the solvation shells and the formation of new solvation. Figure 6(a) and (b) show the difference stimulated Raman spectra of the solvent DMSO, especially ν(S=O) at 1044 cm−1 with alizarin concentrations of 20 and 1 mM, respectively. Solvent DMSO is known to form hydrogen bonds in solution between S=O and C-H groups, and also a polymeric structure is formed at low temperature [47]. The Raman band of ν(S=O) is composed of multiple subbands including the symmetric (1026 cm−1) and antisymmetric stretching (1042 cm−1) of dimer, the symmetric stretching (1058 cm−1) of monomer, etc., [47, 48]. In the DMSO solutions of alizarin in 0–15 mM concentration range, the ν(S=O) band of DMSO shows strong peak shifts (1042 → 1024 cm−1), which represents changes in the hydrogen bonding network of DMSO [44]. The δ(CH3) band of DMSO shows no major spectral changes upon the alizarin concentration, thus the solvation of alizarin with DMSO mainly occurs via hydrogen bonds with the sulfoxide group of DMSO [44]. As shown in Figure 6(a) and (b), a sharp dispersive pattern in the ν(S=O) band appears instantly with the actinic pulse for both 20 and 1 mM concentrations of alizarin. The Raman intensity for the symmetric stretching of monomer around 1060 cm−1 decreases and the symmetric stretching of dimer around 1025 cm−1 increases, which clearly shows the instantaneous changes in hydrogen bonding of DMSO molecules. As clearly seen with the 1-mM solution case, this dispersive pattern disappears very quickly as the actinic pulse leaves the solution. In other words, a disruption in the hydrogen bonding of DMSO molecules created by ultrafast laser pulses is removed quickly by reforming hydrogen bonds between DMSO molecules. It seems that the hydrogen bond reformation occurs much faster than the instrument response function of FSRS (~100 fs). Considering from the ultrafast dynamics of the dispersive signals of ν(S=O) band, the nonpolar solvation effect may be understood as the origin of this sharp dispersive signal [49, 50, 51]. We have also observed a similar dispersive background signal in the δ(CH3) band of DMSO [44]. On the other hand, the 20-mM alizarin results showed clearly distinct dynamics for the ν(S=O) band as shown in Figure 6(a). The initial dispersive Raman signals were almost removed in about 100 fs then another type of dispersive Raman signals appeared, which is composed of a small bleaching with almost the same spectral shape as the ground state ν(S=O) band and a much broader positive signal around 990 cm−1. Figure 7 clearly shows this dispersive Raman pattern in the ν(S=O) of DMSO appearing 100 fs after the actinic pump, where the initial dispersive Raman signals of the ν(S=O) obtained with 1-mM alizarin solution were subtracted from the results with 20-mM alizarin solution. The bleaching of the ground state ν(S=O) Raman band may result from the local heating due to the vibrational cooling of solute molecules, then the recovery of the bleaching signals by the local cooling would take several tens of picoseconds [51, 52, 53]. The decay of the second dispersive Raman signals at 990 and 1043 cm−1 is compatible to the local cooling time but a huge frequency difference (~50 cm−1) cannot be explained by the local heating of solvent DMSO. In a control experiment to measure temperature-dependent Raman spectra for the ν(S=O) of DMSO, the spectral changes of less than 5 cm−1 were observed with temperature increase of 40–50 °C. Therefore, we conclude that the second dispersive Raman signals of ν(S=O) appearing at 100 fs time delay and between 990 and 1043 cm−1 cannot be explained as the local heating of solvent molecules pumped by vibrationally cooled solute molecules and the changes in the solvation shells of DMSO molecules due to the ESIPT reaction of alizarin molecules from the LE to the PT tautomers must be considered. Different from the initial dispersive signals in the ν(S=O) and δ(CH3) bands of DMSO, the second dispersive signals in the ν(S=O) must be considered as originating from the polar or dielectric type of solvation [51]. The dispersive Raman signal of the ν(S=O) band at 1044 cm−1 showed a growth with a 60 ± 30 fs time constant and a decay with a 4.9 ± 1.5 ps time constant, which clearly shows that the hydrogen-bonding network of DMSO was created by the ESIPT reaction of alizarin and decayed as the vibrational relaxation of the product along the potential surface of the PT tautomer in the excited state. In this work, we showed that the ν(S=O) band of solvent DMSO can be used to determine the ultrafast ESIPT reaction and the subsequent vibrational relaxation in the reaction product. The breaking and reforming of hydrogen-bonding network of DMSO can be successfully observed by the ν(S=O) band of DMSO thus this method can also be applied to many chemical reactions occurring in the photoinduced excited states. ## 6. Conclusion In this chapter, the ESIPT reaction and the excited state dynamics of alizarin were explored by time-resolved electronic and vibrational spectroscopy with the femtosecond time-resolution. The dependence on solvent polarity and excitation wavelength was observed in the steady-state emission spectra of alizarin, where the barrier height between the LE and PT tautomers in the excited state may exist and be controlled by the solvent polarity. The transient absorption results of alizarin in ethanol and ethanol-water mixture were so complicated and overlapping, so the ESIPT rate constant from the LE to the PT tautomers was not separable from the vibrational relaxation and population decay of both tautomers. Instead, the ESIPT of alizarin in the excited state was clearly observed in femtosecond stimulated Raman measurements. The population and structural dynamics of two major vibrational modes of ν(C=C) and ν(C=O) clearly showed the dynamics of the ESIPT rate to the PT tautomer, the vibrational relaxation and the population decay of the product PT tautomer. The vibrational signature of the LE tautomer was not observed in FSRS, but the reaction mechanism of the ESIPT including a transition state of a newly formed six-membered ring composed of the carbonyl and hydroxyl groups was estimated by the strong and opposite peak shifts of ν(C=C) and ν(C=O) seen in the stimulated Raman spectra of alizarin during the reaction. From the population growth and structural transformation into the PT tautomer, we concluded that the ESIPT of alizarin occurs in an ultrafast time scale of 70–80 fs. During the ESIPT reaction of alizarin, solvent DMSO molecules showed ultrafast structural changes involving hydrogen bonds with solute molecules. When the solute concentration is very low, DMSO shows a dispersive Raman signal in the ν(S=O) and δ(CH3) modes only with the actinic pump. The instantaneous disruption and reformation of hydrogen bonds may suggest a nonpolar type of solvation between solvent molecules. On the other hand, complicated dispersive Raman signals in the ν(S=O) mode of DMSO were observed with a concentrated (20 mM) solution of alizarin. After the same instantaneous solvent responses completed in 100 fs, the second dispersive Raman pattern with a bleaching of the ground state spectrum appeared and decayed with 60 fs and 5 ps time scales. This also represents the disruption of hydrogen bonds of DMSO molecules, more specifically between solute molecules and in the polar or dielectric solvation shells. Interestingly, the dynamics for the ultrafast proton transfer reaction and the vibrational relaxation in the product state was measured by the solvation signals of solvent DMSO. ## Acknowledgments This work was supported by Basic Science Research Program funded by the Ministry of Education (2017R1A1D1B03027870, 2014R1A1A2058409) and by the International Cooperation Program (2016K2A9A1A01951845), through the National Research Foundation of Korea (NRF). The GIST Research Institute (GRI) in 2018 and the PLSI supercomputing resources of the Korea Institute of Science and Technology Information also supported this research. ## Conflict of interest The authors declare no conflict of interest. chapter PDF Citations in RIS format Citations in bibtex format ## More © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ## How to cite and reference ### Cite this chapter Copy to clipboard Sebok Lee, Myungsam Jen, Kooknam Jeon, Jaebeom Lee, Joonwoo Kim and Yoonsoo Pang (March 27th 2018). Ultrafast Intramolecular Proton Transfer Reaction of 1,2- Dihydroxyanthraquinone in the Excited State, Photochemistry and Photophysics - Fundamentals to Applications, Satyen Saha and Sankalan Mondal, IntechOpen, DOI: 10.5772/intechopen.75783. Available from: ### Related Content Next chapter #### Tailoring the Photophysical Signatures of BODIPY Dyes: Toward Fluorescence Standards across the Visible Spectral Region By Rebeca Sola Llano, Edurne Avellanal Zaballa, Jorge Bañuelos, César Fernando Azael Gómez Durán, José Luis Belmonte Vázquez, Eduardo Peña Cabrera and Iñigo López Arbeloa #### Photophysics, Photochemical and Substitution Reactions Edited by Satyen Saha First chapter #### Photophysical Properties of 4-(Dicyanomethylene)-2-Methyl-6-(4-Dimethylaminostyryl)-4H-Pyran (DCM) and Optical Sensing Applications By Ravi Kumar Kanaparthi, Satyen Saha, Manjeev Singh and Akhila M We are IntechOpen, the world's leading publisher of Open Access books. Built by scientists, for scientists. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals. We share our knowledge and peer-reveiwed research papers with libraries, scientific and engineering societies, and also work with corporate R&D departments and government entities.	{"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": 0.8911291360855103, "perplexity": 3807.9014550741026}, "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/1642320303956.14/warc/CC-MAIN-20220123015212-20220123045212-00196.warc.gz"}
https://www.palmsens.com/knowledgebase-article/potentiostat/the-equations-behind-the-potentiostat/	# The equations behind the potentiostat This article explains the working of a potentiostat more in depth, using Faraday’s law and the Nernst equation. For the basics of the potentiostat, please read the potentiostat article. A potentiostat controls the potential of the working electrode and measures the current flowing through it. Why not just two electrodes? One of the reasons is that we cannot measure the potential of the working electrode against a fixed point when we just have two electrodes. Imagine a two-electrode system that consists of the already mentioned working electrode and the electrode, which potential should be our fixed reference point, the reference electrode. We apply a certain potential between these electrodes and an electrochemical reaction happens at the working electrode, but since the circuit needs to be closed and current needs to flow, a reaction that is inverse to the reaction at the working electrode must occur, that is if oxidation occurs at the working electrode, a reduction must take place at the reference electrode. If a current flows at a constant potential, an electrochemical reaction must happen according to Faraday’s law: This equation says that the charge Q flowing through an electrode is proportional to the amount n of a species that took or gave z electrons at the electrode. F is the Faraday constant and represents the charge of 1 mol electrons. The current I is the charge Q per time t flowing through the electrode: The equations 3.1 and 3.2 combination shows that the current I flowing is connected to the reaction happening at the electrode via the amount n: ## Nernst equation Imagine now that the current is flowing at the reference electrode. At this electrode a species’ amount of n is converted. This conversion leads to a change of the surface or the concentration of the solution surrounding the electrode. The Nernst equation shows a clear correlation between the potential E of an electrode and its surrounding: E0 is the standard potential of the redox couple Red and Ox. R is the gas constant and T the temperature. The activity of the oxidized and reduced form of the species aOx and aRed in the surrounding solution is not always easy to predict. This often leads to a simplification of the equation: The two activity coefficients fOx and fRed are included in the resulting potential E0’, which is called the formal potential. Since it contains parameters that depend on the environment, such as temperature and activity coefficients, E0’ cannot be listed but needs to be determined for each experiment, if necessary. Most experiments in analytical chemistry are performed at room temperature (295 K). This makes another simplification possible. Out of convenience also the ln will be transferred to the log. For practical application equation 3.6 is the most used form of the Nernst equation. For many applications one can assume that E0 is roughly the same as E0’, because both of the activity coefficients are close to one. In this form (equation 3.6) the correlation between the surrounding of an electrode and its potential is visible more easily. As mentioned before all the simplifications at equation 3.4 were performed: The change of the solution surrounding the reference electrode, due to a flowing current, leads to a change of the potential that is supposed to be our fixed reference point. But we cannot limit the current flow through the reference electrode (RE), because all limitations should be caused by the process that we want to investigate, that is the process at the working electrode (WE). ## Using a third electrode To create a fixed reference point, we use a third electrode. At this counter electrode (CE), also known as the auxiliary electrode, the counter-reaction to the working electrode’s reactions takes place. The current is flowing between the working and the counter electrode. The potential is controlled between the working and reference electrode (see Figure 3.1). The potential between the counter and reference electrode is adjusted in such a way that the current flowing through the working electrode at a certain potential between working and reference electrode is satisfied. There are limits for the potential a potentiostat can apply between RE and WE (DC potential range) and CE and WE (compliance voltage). Since you control the potential between RE and WE it is easy to stay within the limits of the DC potential range. The CE has to be bigger than the WE, because the compliance voltage cannot be controlled by the user. A bigger surface at the same potential leads to a higher current and the CE should provide enough current without running into the compliance voltage. A rule of thumb suggests that the CE should be 100 times bigger than the WE. For many experiments this may not be necessary, but for a good practice you should ensure that the CE is big enough so that it does not limit the current flowing at the WE. Usually the distance between CE and WE is big enough so the reactions of the two electrodes do not influence each other, and the counter-reaction can be ignored, but sometimes, in small volumes for example, it can be helpful to know which reaction happens at the counter electrode. The potential is applied between reference and working electrode, while the current flows through working and counter electrode. This way a constant reference point for the potential is maintained.	{"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": 0.882846474647522, "perplexity": 548.8727610818358}, "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-2023-14/segments/1679296948932.75/warc/CC-MAIN-20230329023546-20230329053546-00238.warc.gz"}
https://math.stackexchange.com/questions/3013566/buffons-needle-expected-number-of-intersections-pmf-when-l-d	# Buffon's needle: expected number of intersections & pmf when $l > d$ Earlier results have shown that when $$l < d$$, the expected number of crossings of a needle of length $$l$$ with vertical lines spaced $$d$$ apart is $$\frac{2l}{\pi d}$$, which is also the expression for the probability that a needle intersects a line. I'm looking for an intuitive explanation for why that is the case (is that even the case...?) when the needle is longer ie. $$l > d$$ (consider $$l = 3, d = 1$$ for example). This does not match the expression for the probability that a needle intersects a line when $$l > d$$; rather, it matches the expression for the probability that a needle intersects a line when $$l < d$$. Is this just because the possible numbers of crossings are no longer restricted to $$0$$ and $$1$$ (ie. the $$0$$ term cancels out when computing the expected value)? And, how would one find the PMF of the number of crossings when $$l > d$$ (for a simpler case such as $$l = 3, d = 1$$)? The possible values for the numbers of crossings are $$0, 1, 2, 3, 4$$ if I'm not mistaken. But I don't know where to go from there. edit: still looking for the PMF! • Possibly helpful: cs.umb.edu/~eb/piday/whypi.pdf – Ethan Bolker Nov 25 '18 at 23:38 • @EthanBolker I guess so! Still only deals with the situation when $l = d$. was really hoping for some hard-hitting intuition when $l > d$, but maybe it's just not intuitive and that's all there is to it. – 0k33 Nov 25 '18 at 23:44 • I think that discussion covers the case you are interested in since it explains (or at least asserts) that the crossing number in fact depends on the ratio of $d$ to $l$. – Ethan Bolker Nov 26 '18 at 0:02 • When $l \lt d$, the probability of crossing is equal to the expected number of crossings for precisely the reason you give: the number of crossings can only be $0$ or $1$. When $l\gt d$ then the $\frac{2l}{\pi d}$ gives the number of expected crossings but not the (smaller) probability of of at least one crossing - the expression is obviously not the probability when $\frac{l}{d}l \gt \frac{\pi}{2}$ since the expression will be greater than $1$ – Henry Nov 26 '18 at 0:36 • @Henry Yes, of course. – Ethan Bolker Nov 26 '18 at 0:42 It is clear that we can rescale the problem and take, wlog, $$d=1$$ and $$l/d=r$$. Therefore we can take the lines to be the vertical lines at $$x \in \mathbb Z$$. Consider the needle placed with one end at $$(s,0)$$ and forming an angle $$\alpha$$ wrt the $$x$$ axis: we can sketch the following scheme Considering the simmetry of the problem, we can limit to the I and II quadrants. Also, the variable $$s$$ will be limited to the range $$\left[ {0,1} \right)$$. However, there is a symmetry around $$s=1/2$$, so we will reduce our analysis to $$0 \le s < 1/2$$, considering $$s$$ and $$1-s$$ to be equivalent. The circle with center in $$(s,0)$$ and radius $$r$$ encompasses the abscissas $$s-r \le x \le s+r$$. The set of lines that the needle can cross are those given by $$x = n\quad \left\| {\;\left\lceil {s - r} \right\rceil \le n \le \left\lfloor {s + r} \right\rfloor } \right.$$ It is convenient to extend the values of $$n$$ by two additional elements at the extremes, and define a set of boundary values for $$x$$ and for the angle $$\alpha$$ defined as follows $$\left\{ \matrix{ N = \left\{ {n\quad \left\| {\;\left\lceil {s - r} \right\rceil - 1 \le n \le \left\lfloor {s + r} \right\rfloor + 1} \right.} \right\} \hfill \cr X = \left\{ {x(n)} \right\} = \left\{ {\left( {s - r} \right),\;\left\lceil {s - r} \right\rceil ,\;\left\lceil {s - r} \right\rceil + 1,\; \cdots ,\;0, \;1, \cdots ,\left\lfloor {s + r} \right\rfloor ,\left( {s + r} \right)} \right\} \hfill \cr A = \left\{ {\alpha (n) = \arccos \left( {{{x(n) - s} \over r}} \right)} \right\} = \left\{ {\pi ,\;\arccos \left( {{{\left\lceil {s - r} \right\rceil - s} \over r}} \right),\; \cdots ,\; \arccos \left( {{{\left\lfloor {s + r} \right\rfloor - s} \over r}} \right),\;0} \right\} \hfill \cr} \right.$$ where the set $$A$$ is in non-increasing order, contrary to the others. In this way, the arc corresponding to $$q$$ intersections will be individuated by the values of $$x$$ such that $$\bbox[lightyellow] { x \in \left( {\left( { - q, - q + 1} \right] \cup \left[ {q,q + 1} \right)} \right) \cap \left[ {s - r,\;s + r} \right] } \tag{1}$$ so that we have in general two arcs, except - at $$q=0$$ in which case we have just one range; - (possibly) at the extremes , where the range could be void or of null measure, depending on the values of $$r$$ and $$s$$. In an another perspective, by the above we are assigning a value $$q$$ to the intervals delimited by the points in $$X$$, and correspondingly to the arcs delimited by the angles in $$A$$. Thus we are constructing a measure of the angle $$Ang(q,s;r)$$ as the sum of one or two angles. The position $$s$$ and the angle $$\alpha$$ are supposed independent and uniformly distributed, thus the probability of having $$N$$ intersections is given by \bbox[lightyellow] { \eqalign{ & dP(q,\,s;\;r) = dP(q,\,1 - s;\;r)\quad \left\| \matrix{ \;0 \le s < 1/2 \hfill \cr \;0 < r \hfill \cr \;0 \le q \in Z \hfill \cr} \right. = \cr & = {1 \over \pi }{{ds} \over {1/2}}\left( {\alpha \left( { - q} \right) - \alpha \left( { - q + 1} \right) + \alpha \left( q \right) - \alpha \left( {q + 1} \right)} \right) \cr} } \tag{2} After that, since \eqalign{ & \int {\arccos \left( {{{n - s} \over r}} \right)ds} = - r\int {\arccos \left( {{{n - s} \over r}} \right)d\left( {{{n - s} \over r}} \right)} = \cr & = r\left( {\sqrt {1 - \left( {{{n - s} \over r}} \right)^{\,2} } - \left( {{{n - s} \over r}} \right)\arccos \left( {{{n - s} \over r}} \right)} \right) \cr} we can integrate the above for $$0 \le s < 1/2$$, with due consideration for the variation in $$s$$ of the intervals: the $$n$$ indicated above may vary $$\pm 1$$ at varying $$s$$, which will require to split the integral. • This is fantastic. The diagram is incredibly illustrative. Thank you for such a thorough explanation! – 0k33 Nov 27 '18 at 3:32 • @Ok33: actually, in my previous version, I missed some important details: sorry. I re-casted the answer to deal more precisely with them. – G Cab Nov 28 '18 at 15:20 • @G Cab thank you so much for the follow up! – 0k33 Nov 29 '18 at 23:58 • @Ok33 : the argument was interesting for me as well, but it is a pleasure to help people so nice to leave a thank! – G Cab Nov 30 '18 at 0:32	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 55, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9899411201477051, "perplexity": 348.70203314683476}, "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-2019-22/segments/1558232255536.6/warc/CC-MAIN-20190520021654-20190520043654-00366.warc.gz"}
http://www.cs.cornell.edu/Info/People/raman/phd-thesis/html/node23.html	Refining the quasi-prefix form [Next] [Up] [Previous] Next: Constructing high-level representations Up: Representing mathematical content Previous: Math object encapsulates ## Refining the quasi-prefix form In s:high-level-models, we mentioned that all objects in our document model are linked. This is true of the objects appearing in the quasi-prefix representation. Each node in the tree is linked to its parent, as well as to its previous and next siblings. Math attributes have their parent link set to the object being attributed. We refine the quasi-prefix form by adding the following subtypes. This makes recognizing and handling complex mathematical content cleaner. We first introduce object math subformula, which is used to capture subexpressions appearing within the [tex2html_wrap5306] and [tex2html_wrap5308] of La)TeX. Object math subformula can be thought of as being the math equivalent of object text block described in s:high-level-models. It has the following structure: • Attribute: Visual attributes. • Content: The mathematical content represented as a math object. Object math subformula can be intuitively thought of as a dummy object that encapsulates an expression. We need object math subformula to represent expressions of the form: [displaymath5302] [displaymath5303] In representing each of the above examples, object math subformula is essential in capturing the expression to which the overbrace/underbrace applies. To enable recognition of written mathematics, tokens have to be appropriately classified. Our classification of tokens when processing written mathematics is inspired by appendix F of the TeX Book, [Knu84]. The symbols divide naturally into groups based on their mathematical class (Ord, Op, Bin, Rel, Open, Close, or Punct), [tex2html_wrap5310] We introduce subtypes of object math object to correspond to each token type: • Ordinary: TeX ord. Letters, numbers and some miscellaneous symbols. • Big operator: TeX Op. The large operators that typically appear as unary operators, e.g., [tex2html_wrap5312], [tex2html_wrap5314], [tex2html_wrap5316]. • Binary operator: TeX Bin. The binary operators, e.g., +, [tex2html_wrap5320]. • Relational operator: TeX Rel, e.g., <, [tex2html_wrap5324]. We subdivide the TeX Rel class into relational and arrow operators. • Arrow operators: Arrows such as [tex2html_wrap5326], [tex2html_wrap5328]. • Mathematical function: Plain TeX and LaTeX define [tex2html_wrap5330] etc. as macros. We introduce an object type, mathematical function to represent these. • Open delimiter: TeX Open, e.g., [tex2html_wrap5332], [tex2html_wrap5334]. • Close delimiter: TeX Close, e.g., [tex2html_wrap5336], [tex2html_wrap5338]. • Math punctuation : TeX Punct -punctuation marks. Written mathematical notation uses juxtaposition as an infix operator. Juxtaposition, as in [tex2html_wrap5340], mostly denotes multiplication, but can mean function application in certain contexts -[tex2html_wrap5342]. We introduce a new operator to represent juxtaposition, and to define it precisely, we also assert that all mathematical variables are single letters. Thus, [tex2html_wrap5344] is represented as the juxtaposition of three ordinary objects. This assertion is not specific to our internal representation, rather, it specifies the concrete syntax used in the electronic markup and reflects the choice made in the design of TeX. We do allow mathematical variables made up of more than one character, but these should be clearly marked up as such, e.g., as [tex2html_wrap5346], by using `\mbox` as in `\$\mbox{cab}=cab\$`. The classification of a math object is defined using the following command: (define-math-classification token classification) In certain special cases, the predefined classification shown above can be modified. A good example of this is recognizing a mathematical text that consistently uses the letters [tex2html_wrap5348], [tex2html_wrap5350] and [tex2html_wrap5352] to denote functions. Using the predefined classification, the recognizer would treat [tex2html_wrap5354] as object ordinary, leading to [tex2html_wrap5356] being represented as the juxtaposition of two objects, namely, [tex2html_wrap5358] and [tex2html_wrap5360]. Declaring [tex2html_wrap5362] to be a mathematical function by executing (define-math-classification f mathematical-function-name) results in occurrences of [tex2html_wrap5364] being treated as a function. Hence, [tex2html_wrap5366] is correctly recognized as a function application. Note that the correct interpretation of such notation is more important for browsing than for speaking the expression. [Next] [Up] [Previous] Next: Constructing high-level representations Up: Representing mathematical content Previous: Math object encapsulates TV Raman Thu Mar 9 20:10:41 EST 1995	{"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": 0.9827204346656799, "perplexity": 4931.617588170709}, "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/1368699798457/warc/CC-MAIN-20130516102318-00037-ip-10-60-113-184.ec2.internal.warc.gz"}
http://cms.math.ca/cmb/kw/nilpotent	location: Publications → journals Search results Search: All articles in the CMB digital archive with keyword nilpotent Expand all Collapse all Results 1 - 8 of 8 1. CMB 2014 (vol 57 pp. 884) Xu, Yong; Zhang, Xinjian $m$-embedded Subgroups and $p$-nilpotency of Finite Groups Let $A$ be a subgroup of a finite group $G$ and $\Sigma : G_0\leq G_1\leq\cdots \leq G_n$ some subgroup series of $G$. Suppose that for each pair $(K,H)$ such that $K$ is a maximal subgroup of $H$ and $G_{i-1}\leq K \lt H\leq G_i$, for some $i$, either $A\cap H = A\cap K$ or $AH = AK$. Then $A$ is said to be $\Sigma$-embedded in $G$; $A$ is said to be $m$-embedded in $G$ if $G$ has a subnormal subgroup $T$ and a $\{1\leq G\}$-embedded subgroup $C$ in $G$ such that $G = AT$ and $T\cap A\leq C\leq A$. In this article, some sufficient conditions for a finite group $G$ to be $p$-nilpotent are given whenever all subgroups with order $p^{k}$ of a Sylow $p$-subgroup of $G$ are $m$-embedded for a given positive integer $k$. Keywords:finite group, $p$-nilpotent group, $m$-embedded subgroupCategories:20D10, 20D15 2. CMB 2013 (vol 57 pp. 125) Mlaiki, Nabil M. Camina Triples In this paper, we study Camina triples. Camina triples are a generalization of Camina pairs. Camina pairs were first introduced in 1978 by A .R. Camina. Camina's work was inspired by the study of Frobenius groups. We show that if $(G,N,M)$ is a Camina triple, then either $G/N$ is a $p$-group, or $M$ is abelian, or $M$ has a non-trivial nilpotent or Frobenius quotient. Keywords:Camina triples, Camina pairs, nilpotent groups, vanishing off subgroup, irreducible characters, solvable groupsCategory:20D15 3. CMB 2012 (vol 56 pp. 606) Mazorchuk, Volodymyr; Zhao, Kaiming Characterization of Simple Highest Weight Modules We prove that for simple complex finite dimensional Lie algebras, affine Kac-Moody Lie algebras, the Virasoro algebra and the Heisenberg-Virasoro algebra, simple highest weight modules are characterized by the property that all positive root elements act on these modules locally nilpotently. We also show that this is not the case for higher rank Virasoro and for Heisenberg algebras. Keywords:Lie algebra, highest weight module, triangular decomposition, locally nilpotent actionCategories:17B20, 17B65, 17B66, 17B68 4. CMB 2011 (vol 55 pp. 579) Ndogmo, J. C. Casimir Operators and Nilpotent Radicals It is shown that a Lie algebra having a nilpotent radical has a fundamental set of invariants consisting of Casimir operators. A different proof is given in the well known special case of an abelian radical. A result relating the number of invariants to the dimension of the Cartan subalgebra is also established. Keywords:nilpotent radical, Casimir operators, algebraic Lie algebras, Cartan subalgebras, number of invariantsCategories:16W25, 17B45, 16S30 5. CMB 2009 (vol 52 pp. 535) Daigle, Daniel; Kaliman, Shulim A Note on Locally Nilpotent Derivations\\ and Variables of $k[X,Y,Z]$ We strengthen certain results concerning actions of $(\Comp,+)$ on $\Comp^{3}$ and embeddings of $\Comp^{2}$ in $\Comp^{3}$, and show that these results are in fact valid over any field of characteristic zero. Keywords:locally nilpotent derivations, group actions, polynomial automorphisms, variable, affine spaceCategories:14R10, 14R20, 14R25, 13N15 6. CMB 2004 (vol 47 pp. 343) Drensky, Vesselin; Hammoudi, Lakhdar Combinatorics of Words and Semigroup Algebras Which Are Sums of Locally Nilpotent Subalgebras We construct new examples of non-nil algebras with any number of generators, which are direct sums of two locally nilpotent subalgebras. Like all previously known examples, our examples are contracted semigroup algebras and the underlying semigroups are unions of locally nilpotent subsemigroups. In our constructions we make more transparent than in the past the close relationship between the considered problem and combinatorics of words. Keywords:locally nilpotent rings,, nil rings, locally nilpotent semigroups,, semigroup algebras, monomial algebras, infinite wordsCategories:16N40, 16S15, 20M05, 20M25, 68R15 7. CMB 2001 (vol 44 pp. 266) Cencelj, M.; Dranishnikov, A. N. Extension of Maps to Nilpotent Spaces We show that every compactum has cohomological dimension $1$ with respect to a finitely generated nilpotent group $G$ whenever it has cohomological dimension $1$ with respect to the abelianization of $G$. This is applied to the extension theory to obtain a cohomological dimension theory condition for a finite-dimensional compactum $X$ for extendability of every map from a closed subset of $X$ into a nilpotent $\CW$-complex $M$ with finitely generated homotopy groups over all of $X$. Keywords:cohomological dimension, extension of maps, nilpotent group, nilpotent spaceCategories:55M10, 55S36, 54C20, 54F45 8. CMB 1999 (vol 42 pp. 335) Kim, Goansu; Tang, C. Y. Cyclic Subgroup Separability of HNN-Extensions with Cyclic Associated Subgroups We derive a necessary and sufficient condition for HNN-extensions of cyclic subgroup separable groups with cyclic associated subgroups to be cyclic subgroup separable. Applying this, we explicitly characterize the residual finiteness and the cyclic subgroup separability of HNN-extensions of abelian groups with cyclic associated subgroups. We also consider these residual properties of HNN-extensions of nilpotent groups with cyclic associated subgroups. Keywords:HNN-extension, nilpotent groups, cyclic subgroup separable $(\pi_c)$, residually finiteCategories:20E26, 20E06, 20F10 top of page \| contact us \| privacy \| site map \|	{"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": 0.843544065952301, "perplexity": 919.6325822227919}, "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/1448398464396.48/warc/CC-MAIN-20151124205424-00219-ip-10-71-132-137.ec2.internal.warc.gz"}
https://artofproblemsolving.com/wiki/index.php?title=Ceva%27s_Theorem&diff=next&oldid=18121	Difference between revisions of "Ceva's Theorem" Ceva's Theorem is a criterion for the concurrence of cevians in a triangle. Statement Let be a triangle, and let be points on lines , respectively. Lines concur iff (if and only if) , where lengths are directed. (Note that the cevians do not necessarily lie within the triangle, although they do in this diagram.) Proof We will use the notation to denote the area of a triangle with vertices . First, suppose meet at a point . We note that triangles have the same altitude to line , but bases and . It follows that . The same is true for triangles , so . Similarly, and , so . Now, suppose satisfy Ceva's criterion, and suppose intersect at . Suppose the line intersects line at . We have proven that must satisfy Ceva's criterion. This means that , so , and line concurrs with and . Trigonometric Form The trigonometric form of Ceva's Theorem (Trig Ceva) states that cevians concur if and only if Proof First, suppose concur at a point . We note that , and similarly, . It follows that . Here, sign is irrelevant, as we may interpret the sines of directed angles mod to be either positive or negative. The converse follows by an argument almost identical to that used for the first form of Ceva's Theorem. Examples 1. Suppose AB, AC, and BC have lengths 13, 14, and 15. If and , find BD and DC. If and , then , and . From this, we find and . 2. The concurrence of the altitudes of a triangle at the orthocenter and the concurrence of the perpendicual bisectors of a triangle at the circumcenter can both be proven by Ceva's Theorem (the latter is a little harder). Furthermore, the existance of the centroid can be shown by Ceva, and the existance of the incenter can be shown using trig Ceva. However, there are more elegant methods for proving each of these results, and in any case, any result obtained by classic Ceva's Theorem can be proven using ratios of areas. 3. The existance of isotonic conjugates can be shown by classic Ceva, and the existance of isogonal conjugates can be shown by trig Ceva.	{"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": 0.9836283326148987, "perplexity": 679.2083853727665}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107867463.6/warc/CC-MAIN-20201019232613-20201020022613-00434.warc.gz"}
https://www.scm.com/highlights/surface-activation-of-mao/	# Surface activation of MAO In a recent study, the interaction of various methylaluminoxane (MAO) models on the (110) MgCl2 surface were studied with dispersion corrected DFT (revPBE-D3) in ADF. MAO activates the metallocene catalyst, but the structure(s) of MAO are unknown and depend on the reaction conditions (see also this highlight). The use of a support affects the equilibrium between species that could potentially be active in polymerization and those that are dormant. The work was carried out to get a better understanding of how the support lowers the Al:catalyst ratio necessary for good polymer activities. As the calculations in this paper show, the active state of MAO is stabilized by the (110) MgCl2 surface. An ETS-NOCV analyses revealed that the active MAO species interacted with the surface via surface to adsorbate donation (Cl to Al) as well as via sizable interactions involving charge donation to the surface Mg atoms from MAO methyl groups or O atoms. Insight in the bonding interactions of the surface, the catalyst and co-catalyst can help to understand the complexities of the heterogeneously catalyzed olefin polymerization reaction. Bonding interactions between MAO and MgCl2 support	{"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": 0.822845458984375, "perplexity": 4041.7581277762733}, "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/1669446710155.67/warc/CC-MAIN-20221127005113-20221127035113-00429.warc.gz"}
http://math.stackexchange.com/questions/9960/complex-inequality-up-1u-vp-1v-leq-c-p-u-vup-1vp-1	# Complex inequality $\|\|u\|^{p-1}u - \|v\|^{p-1}v\|\leq c_p \|u-v\|(\|u\|^{p-1}+\|v\|^{p-1})$ How does one show for complex numbers u and v, and for p>1 that \begin{equation} \|\|u\|^{p-1}u - \|v\|^{p-1}v\|\leq c_p \|u-v\|(\|u\|^{p-1}+\|v\|^{p-1}), \end{equation} where $c_p$ is some constant dependent on p. My intuition is to use some version of the mean value theorem with $F(u) = \|u\|^{p-1}u$, but I'm not sure how to make this work for complex-valued functions. Plus there seems to be an issue with the fact that $F$ may not smooth near the origin. For context, this shows up in Terry Tao's book Nonlinear Dispersive Equations: Local and Global Analysis on pg. 136, where it is stated without proof as an "elementary estimate". - Suppose without loss of generality that $\|u\| \geq \|v\| > 0$. Then you can divide the equation through by $\|v\|^p$ and your task it to prove $\|\|w\|^{p-1}w - 1\| \leq c_p\|w - 1\|(\|w\|^{p-1} + 1)$, where $w = u/v$. Note that $$\|\|w\|^{p-1}w - 1\| = \|\|w\|^{p-1}w - \|w\|^{p-1} + \|w\|^{p-1} - 1\|$$ $$\leq \|\|w\|^{p-1}w - \|w\|^{p-1}\| + \|\|w\|^{p-1} - 1\|$$ Note the first term is $\|w\|^{p-1}\|w - 1\|$ is automatically bounded by your right hand side. So you're left trying to show that $\|\|w\|^{p-1} - 1\|$ is bounded by your right hand side. For this it suffices to show that $$\|\|w\|^{p-1} - 1\| \leq c_p\|\|w\| - 1\|\| (\|w\|^{p-1} + 1)$$ Since $\|w\| \geq 1$ by the assumption that $\|u\| \geq \|v\|$, it suffices to show that for all real $r \geq 1$ one has $$r^{p-1} - 1 \leq c_p(r - 1)(r^{p-1} + 1)$$ Now use the mean value theorem as you originally wanted to. Without loss, you can assume $\|u\|\le\|v\|$ and replace $u$ by $uv$ to reduce the problem to the situation $v=1$ and $\|u\|\le1$. Unless you need $c_p$ explicitly, then it's clear, yes?	{"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": 0.9767878651618958, "perplexity": 86.04309301102441}, "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-2016-30/segments/1469257824217.36/warc/CC-MAIN-20160723071024-00062-ip-10-185-27-174.ec2.internal.warc.gz"}
https://scipost.org/SciPostPhys.3.5.034	## Entanglement Entropy in Lifshitz Theories Temple He, Javier M. Magan, Stefan Vandoren SciPost Phys. 3, 034 (2017) · published 16 November 2017 ### Abstract We discuss and compute entanglement entropy (EE) in (1+1)-dimensional free Lifshitz scalar field theories with arbitrary dynamical exponents. We consider both the subinterval and periodic sublattices in the discretized theory as subsystems. In both cases, we are able to analytically demonstrate that the EE grows linearly as a function of the dynamical exponent. Furthermore, for the subinterval case, we determine that as the dynamical exponent increases, there is a crossover from an area law to a volume law. Lastly, we deform Lifshitz field theories with certain relevant operators and show that the EE decreases from the ultraviolet to the infrared fixed point, giving evidence for a possible $c$-theorem for deformed Lifshitz theories. ### Ontology / Topics See full Ontology or Topics database. ### Authors / Affiliations: mappings to Contributors and Organizations See all Organizations. Funders for the research work leading to this publication	{"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": 0.8784164786338806, "perplexity": 2674.973061034539}, "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-2022-05/segments/1642320300253.51/warc/CC-MAIN-20220117000754-20220117030754-00682.warc.gz"}
https://gmatclub.com/forum/set-x-consists-of-100-numbers-the-average-arithmetic-mean-103536.html	GMAT Question of the Day - Daily to your Mailbox; hard ones only It is currently 18 Jan 2019, 00:08 ### GMAT Club Daily Prep #### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email. Customized for You we will pick new questions that match your level based on your Timer History Track Your Progress every week, we’ll send you an estimated GMAT score based on your performance Practice Pays we will pick new questions that match your level based on your Timer History ## Events & Promotions ###### Events & Promotions in January PrevNext SuMoTuWeThFrSa 303112345 6789101112 13141516171819 20212223242526 272829303112 Open Detailed Calendar • ### Free GMAT Strategy Webinar January 19, 2019 January 19, 2019 07:00 AM PST 09:00 AM PST Aiming to score 760+? Attend this FREE session to learn how to Define your GMAT Strategy, Create your Study Plan and Master the Core Skills to excel on the GMAT. • ### FREE Quant Workshop by e-GMAT! January 20, 2019 January 20, 2019 07:00 AM PST 07:00 AM PST Get personalized insights on how to achieve your Target Quant Score. # Set X consists of 100 numbers. The average (arithmetic mean) new topic post reply Question banks Downloads My Bookmarks Reviews Important topics Author Message TAGS: ### Hide Tags Retired Moderator Status: The last round Joined: 18 Jun 2009 Posts: 1220 Concentration: Strategy, General Management GMAT 1: 680 Q48 V34 Set X consists of 100 numbers. The average (arithmetic mean) of set X [#permalink] ### Show Tags Updated on: 24 Sep 2015, 02:52 3 13 00:00 Difficulty: 35% (medium) Question Stats: 63% (00:45) correct 37% (00:54) wrong based on 502 sessions ### HideShow timer Statistics Set X consists of 100 numbers. The average (arithmetic mean) of set X is 10, and the standard deviation is 4.6. Which of the following two numbers, when added to set X, will decrease the set’s standard deviation by the greatest amount? A. -100 and -100 B. -10 and -10 C. 0 and 0 D. 0 and 20 E. 10 and 10 _________________ Originally posted by Hussain15 on 30 May 2010, 05:49. Last edited by Bunuel on 24 Sep 2015, 02:52, edited 1 time in total. Renamed the topic, edited the question and added the OA. ##### Most Helpful Expert Reply Math Expert Joined: 02 Sep 2009 Posts: 52266 Re: Set X consists of 100 numbers. The average (arithmetic mean) of set X [#permalink] ### Show Tags 04 Jun 2010, 05:49 4 3 Hussain15 wrote: Hussain15 wrote: I have another doubt: If we will add 10 & 10 to set X, then the arithematic mean will not change i.e it will still be 10. Hence the standar deviation will not change & will remain the same. But our requirement is that SD will decrease in maximum. What am I missing here? I think the red portion in my statement above is not correct. SD will come down. "Standard deviation shows how much variation there is from the mean. A low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data are spread out over a large range of values." So when we add numbers, which are far from the mean we are stretching the set making SD bigger and when we add numbers which are close to the mean we are shrinking the set making SD smaller. According to the above adding two numbers which are closest to the mean will shrink the set most, thus decreasing SD by the greatest amount. Closest to the mean are 10 and 10 (actually these numbers equal to the mean) thus adding them will shrink the set most, thus decreasing SD by the greatest amount. Answer: E. amitjash wrote: I have a doubt for this explaination. The question says "will decrease the set’s standard deviation by the GREATEST amount" Now if SD is +ve then adding a negetive number will reduce the standard deviation correct?? and if SD is -ve then adding the positive number will reduce the standard deviation. the value of the added will depend on the number of data given. Please correct me if i am wrong. SD is always $$\geq{0}$$. SD is 0 only when the list contains all identical elements (or which is same only 1 element). For more on this issue please check Standard Deviation chapter of Math Book (link in my signature) and the following two topics for practice: ps-questions-about-standard-deviation-85897.html ds-questions-about-standard-deviation-85896.html _________________ ##### General Discussion Intern Joined: 27 Aug 2009 Posts: 40 Schools: London Business School Re: Set X consists of 100 numbers. The average (arithmetic mean) of set X [#permalink] ### Show Tags 30 May 2010, 06:22 4 Hussain15 wrote: Set X consists of 100 numbers. The average (arithmetic mean) of set X is 10, and the standard deviation is 4.6. Which of the following two numbers, when added to set X, will decrease the set’s standard deviation by the greatest amount? A. -100 & -100 B. -10 & -10 C. 0 & 0 D. 0 & 20 E. 10 & 10 Answer E Standard Deviation is deviation from the mean. If all the numbers in a set are equal to mean, then the standard deviation will be zero. Therefore, in the given set with mean of 10, if we add 10 & 10, then the standard deviation will reduce. All other numbers will increase the standard deviation. Please note that B is not the answer. Mean is +10 and, therefore, -10 is 20 points away from +10 (its not equal to the mean). Therefore, -10 will increase the standard deviation of the given set. _________________ Salaries are low in recession. So, working for kudos now. Manager Joined: 04 Apr 2010 Posts: 86 Schools: UCLA Anderson Re: Set X consists of 100 numbers. The average (arithmetic mean) of set X [#permalink] ### Show Tags 02 Jun 2010, 16:48 1 Clearly E, as the mean of the set is 10. Adding the mean as an extra value in the set will decrease the standard deviation. _________________ If this post was helpful, please give Kudos. Manager Joined: 16 Mar 2010 Posts: 143 Re: Set X consists of 100 numbers. The average (arithmetic mean) of set X [#permalink] ### Show Tags 04 Jun 2010, 02:41 I have a doubt for this explaination. The question says "will decrease the set’s standard deviation by the GREATEST amount" Now if SD is +ve then adding a negetive number will reduce the standard deviation correct?? and if SD is -ve then adding the positive number will reduce the standard deviation. the value of the added will depend on the number of data given. Please correct me if i am wrong. Retired Moderator Status: The last round Joined: 18 Jun 2009 Posts: 1220 Concentration: Strategy, General Management GMAT 1: 680 Q48 V34 Re: Set X consists of 100 numbers. The average (arithmetic mean) of set X [#permalink] ### Show Tags 04 Jun 2010, 05:04 I have another doubt: If we will add 10 & 10 to set X, then the arithematic mean will not change i.e it will still be 10. Hence the standar deviation will not change & will remain the same. But our requirement is that SD will decrease in maximum. What am I missing here? _________________ Retired Moderator Status: The last round Joined: 18 Jun 2009 Posts: 1220 Concentration: Strategy, General Management GMAT 1: 680 Q48 V34 Re: Set X consists of 100 numbers. The average (arithmetic mean) of set X [#permalink] ### Show Tags 04 Jun 2010, 05:14 Hussain15 wrote: I have another doubt: If we will add 10 & 10 to set X, then the arithematic mean will not change i.e it will still be 10. Hence the standar deviation will not change & will remain the same. But our requirement is that SD will decrease in maximum. What am I missing here? I think the red portion in my statement above is not correct. SD will come down. _________________ Director Joined: 24 Aug 2007 Posts: 741 WE 1: 3.5 yrs IT WE 2: 2.5 yrs Retail chain Re: Set X consists of 100 numbers. The average (arithmetic mean) of set X [#permalink] ### Show Tags 04 Jun 2010, 05:43 Yes, this part is incorrect. There will be no change in the average if you add (10, 10), but the difference will be zero, when we calculate for SD. So, SD will be <4.6. E is correct, which brings the miminum sum after subtracting and squaring the differences for SD calculations. Hussain15 wrote: Hussain15 wrote: I have another doubt: If we will add 10 & 10 to set X, then the arithematic mean will not change i.e it will still be 10. Hence the standar deviation will not change & will remain the same. But our requirement is that SD will decrease in maximum. What am I missing here? I think the red portion in my statement above is not correct. SD will come down. _________________ Want to improve your CR: http://gmatclub.com/forum/cr-methods-an-approach-to-find-the-best-answers-93146.html Tricky Quant problems: http://gmatclub.com/forum/50-tricky-questions-92834.html Important Grammer Fundamentals: http://gmatclub.com/forum/key-fundamentals-of-grammer-our-crucial-learnings-on-sc-93659.html Manager Joined: 04 Apr 2010 Posts: 86 Schools: UCLA Anderson Re: Set X consists of 100 numbers. The average (arithmetic mean) of set X [#permalink] ### Show Tags 04 Jun 2010, 09:48 1 You can prove it yourself with a couple quick calculations. Try calculating the SD for the set {1,2,3} and then calculate it for the set {1,2,2,3} Notice how it decreases. _________________ If this post was helpful, please give Kudos. Math Expert Joined: 02 Sep 2009 Posts: 52266 Re: Standard Devi [#permalink] ### Show Tags 24 Oct 2010, 13:10 5 1 shrive555 wrote: Set X consists of 100 numbers. The average (arithmetic mean) of set X is 10, and the standard deviation is 4.6. Which of the following two numbers, when added to set X, will decrease the set’s standard deviation by the greatest amount? A. -100 and -100 B. -10 and -10 C. 0 and 0 D. 0 and 20 E. 10 and 10 "Standard deviation shows how much variation there is from the mean. A low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data are spread out over a large range of values." So when we add numbers, which are far from the mean we are stretching the set making SD bigger and when we add numbers which are close to the mean we are shrinking the set making SD smaller. According to the above adding two numbers which are closest to the mean will shrink the set most, thus decreasing SD by the greatest amount. Closest to the mean are 10 and 10 (actually these numbers equal to the mean) thus adding them will shrink the set most, thus decreasing SD by the greatest amount. Answer: E. For more on this issue please check Standard Deviation chapter of Math Book (link in my signature) and the following two topics for practice: ps-questions-about-standard-deviation-85897.html ds-questions-about-standard-deviation-85896.html Hope it helps. _________________ Math Expert Joined: 02 Sep 2009 Posts: 52266 Re: Set X consists of 100 numbers. The average (arithmetic mean) [#permalink] ### Show Tags 08 Dec 2013, 23:33 Bunuel wrote: gmatgambler wrote: Set X consists of 100 numbers. The average (arithmetic mean) of set X is 10, and the standard deviation is 4.6. Which of the following two numbers, when added to set X, will decrease the set’s standard deviation by the greatest amount? A)-100 and -100 B)-10 and -10 C)0 and 0 D)0 and 20 E)10 and 10 Merging similar topics. Please refer to the solutions above. Theory on Statistics and Sets problems: math-standard-deviation-87905.html All DS Statistics and Sets problems to practice: search.php?search_id=tag&tag_id=34 All PS Statistics and Sets problems to practice: search.php?search_id=tag&tag_id=55 Similar questions to practice: a-certain-list-has-an-average-of-6-and-a-standard-deviation-97473.html a-certain-list-of-100-data-has-an-average-arithmetic-mean-87743.html a-certain-list-of-200-test-scores-has-an-average-131448.html new-ds-set-150653-60.html#p1211907 ps-questions-about-standard-deviation-85897.html lately-many-questions-were-asked-about-the-standard-85896.html Hope this helps. _________________ Intern Joined: 29 May 2014 Posts: 17 Concentration: Finance, Strategy GPA: 3.8 Re: Set X consists of 100 numbers. The average (arithmetic mean) [#permalink] ### Show Tags 14 Jun 2014, 04:51 How I think about problems associated with means and standard deviations is with scattered data points. Since this question is regarding standard deviations (and how to reduce it), a quantity calculated to indicate the extent of deviation for the group -- the greater the deviation from the mean the greater the standard deviation. Therefore what data points will help to reduce deviation from the mean of set X [10]. If you add two points with the value of the mean their deviation will be 0, which is the smallest deviation you can add to the set of numbers. Manager Joined: 20 Apr 2014 Posts: 89 Re: Set X consists of 100 numbers. The average (arithmetic mean) of set X [#permalink] ### Show Tags 24 Sep 2015, 01:33 i reviewed math of Gmat club and found that 10,10 will not decrease the SD Manager Joined: 17 Jun 2015 Posts: 208 GMAT 1: 540 Q39 V26 GMAT 2: 680 Q46 V37 Re: Set X consists of 100 numbers. The average (arithmetic mean) [#permalink] ### Show Tags 25 Dec 2015, 12:01 HIgh SD = Wide distribution of data in the set Low SD = Closely associated data in the set The intention is to reduce the SD That implies the newer entrants into the set need to near the mean Option A would Increase the SD the most Option E would not affect as much. Hence E _________________ Fais de ta vie un rêve et d'un rêve une réalité Target Test Prep Representative Status: Founder & CEO Affiliations: Target Test Prep Joined: 14 Oct 2015 Posts: 4548 Location: United States (CA) Re: Set X consists of 100 numbers. The average (arithmetic mean) [#permalink] ### Show Tags 10 Apr 2018, 16:06 Hussain15 wrote: Set X consists of 100 numbers. The average (arithmetic mean) of set X is 10, and the standard deviation is 4.6. Which of the following two numbers, when added to set X, will decrease the set’s standard deviation by the greatest amount? A. -100 and -100 B. -10 and -10 C. 0 and 0 D. 0 and 20 E. 10 and 10 The standard deviation is a measure of the spread of data values around the mean. If data values are close to the mean, the standard deviation is small, and if data values are further away from the mean, the standard deviation is larger. Thus, in order to decrease the standard deviation, we want to find two values that are as close to the mean as possible. Since the mean is 10, the two values of 10 and 10 will decrease the standard deviation by the greatest amount. Answer: E _________________ Scott Woodbury-Stewart Founder and CEO GMAT Quant Self-Study Course 500+ lessons 3000+ practice problems 800+ HD solutions Study Buddy Forum Moderator Joined: 04 Sep 2016 Posts: 1298 Location: India WE: Engineering (Other) Re: Set X consists of 100 numbers. The average (arithmetic mean) [#permalink] ### Show Tags 21 Apr 2018, 22:03 niks18 Bunuel chetan2u Quote: Set X consists of 100 numbers. The average (arithmetic mean) of set X is 10, and the standard deviation is 4.6. Which of the following two numbers, when added to set X, will decrease the set’s standard deviation by the greatest amount? A. -100 and -100 B. -10 and -10 C. 0 and 0 D. 0 and 20 E. 10 and 10 If one of the answer option is -10 and 10, will it mean same as (C)? _________________ It's the journey that brings us happiness not the destination. Retired Moderator Joined: 25 Feb 2013 Posts: 1220 Location: India GPA: 3.82 Re: Set X consists of 100 numbers. The average (arithmetic mean) [#permalink] ### Show Tags 21 Apr 2018, 23:04 adkikani wrote: niks18 Bunuel chetan2u Quote: Set X consists of 100 numbers. The average (arithmetic mean) of set X is 10, and the standard deviation is 4.6. Which of the following two numbers, when added to set X, will decrease the set’s standard deviation by the greatest amount? A. -100 and -100 B. -10 and -10 C. 0 and 0 D. 0 and 20 E. 10 and 10 If one of the answer option is -10 and 10, will it mean same as (C)? hi adkikani Can you elaborate your query? If Option is not present and instead of Option E, your option of -10 and 10 is given then in my opinion none of the answer choice will decease the SD but will increase the SD. So the question will become irrelevant. Between Option C i.e. 0 & 0 and your option -10 & 10, 0 & 0 will have a less incremental impact on SD than -10 & 10 because both 0 & 0 and -10 & 10 will only reduce the average of the set but when you subtract each element of the set with the reduced average and then square it to get the variance, then -10 & 10 will give you higher variance than 0 & 0, resulting in higher SD SC Moderator Joined: 23 Sep 2015 Posts: 1611 Re: Set X consists of 100 numbers. The average (arithmetic mean) [#permalink] ### Show Tags 03 Dec 2018, 05:56 Official Explanation: Standard deviation is a tricky subject. Here’s a GMAT blog that explains absolutely everything you need to know about the standard deviation. First we have to understand what the mean is. The mean is just the plain old average --- add up all the terms on the list, and divide by the number of terms on the list. Adding -100 and -100 would lower the mean the most, but that’s not what the question is asking. The standard deviation is something much more complicated. There's a technical definition, which I explain in that GMAT blog, but the more informal definition is fine for this problem. The informal definition --- standard deviation is sort of an average of the deviations from the mean. You see, every term has its own distance from the mean, and we call that a “deviation from the mean” ---- for these purposes, we count it as positive whether it’s greater than or less than the mean. In this problem, the mean = 10. A value of 13 would have a deviation from the mean of 3. A value of 7 would also have a deviation from the mean of 3. Each of those terms, 13 and 7, are a distance of 3 from the mean, so in terms of the standard deviation, they contribute the same thing. Very high or very low numbers, far away from the mean, would have large deviations from the mean. We find the deviation from the mean of every number on the list --- all 100 numbers in this problem ---- and the standard deviation is sort of an average of all 100 of those deviations from the mean. (Again, this is not the precise definition, it’s not a strict average, but for this problem it’s close enough.) The question gives a numerical value for the standard deviation, 4.6, but that's just a distractor. We don't need that. The question asks us to add two numbers that lower the standard deviation the most. Well, the standard deviation is a kind of average, and if we want to lower any average, we have to add new terms that make contributions as small as possible. What's the smallest possible deviation a number could have from the mean? If we added a new term with a value of 10, that term equals the mean, so its deviation from the mean—its distance from the mean—is zero. If the new term is anything above or below 10, it will have a deviation from the mean greater than zero. Therefore, the lowest possible deviation from the mean a term can have is zero, and this is possible only if the term equals the mean of 10. Thus, if we two terms, both equal to the mean, we will be adding two terms with the lowest possible deviation from the mean, and that will lower the standard deviation, the average across all deviations from the mean, as much as possible. Answer = E _________________ Thanks! Do give some kudos. Simple strategy: “Once you’ve eliminated the impossible, whatever remains, however improbable, must be the truth.” Want to improve your Score: GMAT Ninja YouTube! Series 1\| GMAT Ninja YouTube! Series 2 \| How to Improve GMAT Quant from Q49 to a Perfect Q51 My Notes: Reading comprehension \| Critical Reasoning \| Absolute Phrases \| Subjunctive Mood Intern Joined: 03 Dec 2018 Posts: 7 Set X consists of 100 numbers. The average (arithmetic mean) [#permalink] ### Show Tags 07 Dec 2018, 03:24 Hussain15 wrote: Set X consists of 100 numbers. The average (arithmetic mean) of set X is 10, and the standard deviation is 4.6. Which of the following two numbers, when added to set X, will decrease the set’s standard deviation by the greatest amount? A. -100 and -100 B. -10 and -10 C. 0 and 0 D. 0 and 20 E. 10 and 10 if the question had asked "...will decrease the set’s standard deviation by the LEAST amount . the answer would be (0,20) & (0,0) & (-10,10) right? cause : -100, -100 will make numerator very large , thus increasing the SD. only (0,0) & (0,20) & (-10,-10) will add 100,100 to the numerator ( least among all) for increase by greatest , answer would be -100,-100 for increase by least , answer = cant say Set X consists of 100 numbers. The average (arithmetic mean) &nbs [#permalink] 07 Dec 2018, 03:24 Display posts from previous: Sort by # Set X consists of 100 numbers. The average (arithmetic mean) new topic post reply Question banks Downloads My Bookmarks Reviews Important topics Powered by phpBB © phpBB Group \| Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.	{"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": 0.8627052903175354, "perplexity": 1654.4822391536102}, "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-2019-04/segments/1547583659944.3/warc/CC-MAIN-20190118070121-20190118092121-00471.warc.gz"}
https://www.physicsforums.com/threads/coulombs-law.87298/	# Coulomb's law 1. Sep 2, 2005 ### mystry4 Can someone explain this law to me in simple terms? I know that it is the electrostatic force between 2 charged objects in relation to the quantity and inversly related to the square of distance F=K q1 q2 / d^2 ..but what if you have atoms that are spearated by a certain distance and have extra electrons? What happens to the force if you double the charges? Thank you. 2. Sep 2, 2005 ### codyg1985 $$q_1$$ and $$q_2$$ represents the net charge of the atoms. For example, if you have $$Fe^2^+$$ and $$Cl^-$$, then $$q_1$$ and $$q_2$$ would be 2 and 1, respectively. The second number would be positive because it accepts a negative charge by default, making the answer a positive number. 3. Sep 3, 2005 ### mystry4 so, the formula I typed above is correct and could be used with 2 oxygen atoms, 3 cm apart and each have 2 extra electrons ? Wouldn't I get a -2 charge? (-2)(-2) / 3cm ?? 4. Sep 3, 2005 ### teclo well, atoms for the most part are neutral. if you had two oxygen atoms that somehow gained electrons and each had a negative 2 charge that were placed next to eachother, they'd fly apart. negative * negative = positive -- repulsive force. negative * posative = negative -- attractive force. doubling the charges of each one -- look at the equation. F = (1/4piepsilon-zero)(q1)(q2)/d^2 say each charge is e (charge of an electron), and you double each one (2e), the magnitude of the force will increase by a factor of 4 (22). dealing with atoms is a bit tricky, but to simplify it enough to say that there is a uniform sphereical charge distribution at the location of each ion would work. two negative ions placed near eachother in a closed system would accelerate away from eachother on the line that they create.	{"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": 0.9510542750358582, "perplexity": 752.1726864861579}, "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-2016-44/segments/1476988717954.1/warc/CC-MAIN-20161020183837-00564-ip-10-171-6-4.ec2.internal.warc.gz"}
https://www.gradesaver.com/textbooks/math/algebra/intermediate-algebra-for-college-students-7th-edition/chapter-1-review-exercises-page-98/26	## Intermediate Algebra for College Students (7th Edition) $-\dfrac{1}{10}$ Make the fractions similar using their LCD of 10 as denominator to obtain: $=-(\frac{3\cdot2}{5\cdot2}) - (-\frac{1\cdot5}{2\cdot5}) \\=-\frac{6}{10} - (-\frac{5}{10})$ RECALL: $a-b = a+(-b)$ Use the rule above to obtain: $=-\dfrac{6}{10} + \dfrac{5}{10} \\=\dfrac{-6+5}{10} \\=\dfrac{-1}{10}$	{"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": 0.9647720456123352, "perplexity": 2276.671197793061}, "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-2018-34/segments/1534221219197.89/warc/CC-MAIN-20180821230258-20180822010258-00311.warc.gz"}
https://mitchellkember.com/sph4u/thin-film-interference.html	## Thin-film interference Thin-film interference occur when light waves reflect on the upper and lower boundaries of a thin film. This causes interference and it is responsible for the rainbow of colours that you see in soap bubbles and in puddles that have a thin layer of oil. When the light strikes the thin film, it is both reflected and refracted. The refracted ray reflects off the bottom and refracts through the top to come out parallel with the original reflected ray: When a wave reflects is in a fast medium and reflects on the interface to a slower medium, it inverts. This means the first wave inverts when it reflects, but the second wave does not. All other things being equal, they should be out of phase. Of course, other things aren’t equal because the second wave travels an extra distance. For small angles of incidence, this extra distance is about 2t where t is the thickness of the film. There are three interesting cases for the thickness of the film. When t ≪ lambda, the extra distance is so small that the interference is destructive for all colours. When t=1//4lambda, the two waves are in phase. Why? If we don’t consider the extra distance, the waves are out of phase, meaning a phase delay of 1//2lambda; when we do consider it, we have 1//2lambda + 2t = lambda. Since they are in phase, they interfere constructively—all other colours (with different values of lambda) are blocked because the interference is destructive. When t=1//2lambda, the waves are out of phase and therefore destructive. This means that only that colour is blocked—the others are not. phase delay type of interference The lambda colour other colours t ≪ lambda small destructive blocked blocked t = 1/4lambda lambda constructive reflected blocked t = 1/2lambda 3/2lambda destructive blocked reflected	{"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": 0.8020318746566772, "perplexity": 971.188365024544}, "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/1544376825349.51/warc/CC-MAIN-20181214022947-20181214044447-00343.warc.gz"}
https://undergroundmathematics.org/vector-geometry/r7982/solution	Review question # How fast do these connected particles move? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource Ref: R7982 ## Solution The diagram shows two particles $A$ and $B$, connected by a light inextensible string which passes over a smooth fixed peg. The system is held with the string taut and with $A$ and $B$ each at a height of $\quantity{0.09}{m}$ above a fixed horizontal plane; it is then released from rest. When $B$ reaches the plane is becomes stationary. Calculate 1. the tension in the string while both particles are in motion, [Take $g$ to be $\quantity{10}{m\,s^{-2}}$.] Let the acceleration of the particles be $a$, measured upwards on $A$ and downwards on $B$, and let the tension in the string be $T$. The following diagram shows the forces acting on the two particles. Applying Newton’s second law, $F=ma$, to each of the particles and using $g=10$ we have \begin{align} T-8 &= 0.8a \\ 12-T &= 1.2a. \end{align} Adding gives us $4=2a$, $a=2$. Substituting this into one of the equations gives $T=\quantity{9.6}{N}.$ Calculate 1. the speed of the particles when $B$ reaches the plane, We know that $B$ starts at rest and travels $\quantity{0.09}{m}$. We know its acceleration and want to find its final velocity, so we use the equation of motion \begin{align} v^2 &= u^2+2as. \\ v^2 &= 0+2\times2\times 0.09 = 0.36 \\ v &= 0.6 \end{align} The particles are moving at $\quantity{0.6}{m\,s^{-1}}$ when $B$ reaches the plane. Calculate 1. the maximum height above the plane attained by $A$, assuming that $A$ does not reach the height of the fixed peg. Assuming that neither the peg nor the string interfere with its motion, $A$ will now be under the influence only of gravity. We know its initial velocity is $\quantity{0.6}{m\,s^{-1}}$ upwards and its acceleration is $g$ downwards. We want to know its upwards displacement, so we’ll use the same equation of motion as before. \begin{align} v^2 &= u^2+2as \\ 0 &= 0.6^2+2(-10)s \\ s &= \frac{0.36}{20} = 0.018 \end{align} This is how far it rises after the string becomes slack but this happens when $A$ is $2\times\quantity{0.09}{m}$ above the plane, so the maximum height above the plane is $2\times0.09+0.018=\quantity{0.198}{m}.$	{"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": 0.9999110698699951, "perplexity": 370.5669533056185}, "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-2018-34/segments/1534221217970.87/warc/CC-MAIN-20180821053629-20180821073629-00719.warc.gz"}
https://math.stackexchange.com/questions/1745646/find-a-vector-v-such-that-v-mathbbct-cdot-v	# Find a vector $v$ such that $V=\mathbb{C}[T]\cdot v.$ Let $V=\mathbb{C}^2$ and let $\alpha:V\to V$ be the $\mathbb{C}$-linear map given by \begin{equation} \begin{pmatrix} a\\ b \end{pmatrix} \longmapsto \begin{pmatrix} 2 & 3\\ -3 & 8 \end{pmatrix} \cdot \begin{pmatrix} a\\ b \end{pmatrix}. \end{equation} Consider $V$ as a $\mathbb{C}[x]$-vector space with scalar multiplication $$\left(\sum_{i=0}^nc_ix^i\right)\cdot v\doteq\sum_{i=0}^nc_i\alpha^i(v),$$ where $\alpha^i$ denoted the $i$-times composition of $\alpha$. Find a vector $v\in V$ such that $V=\mathbb{C}[x]\cdot v.$ I'm not sure where to start. I suppose the problem is asking for a generator for $V$ as a $\mathbb{C}[x]$-vector space, but where I get confused is that $V=\mathbb{C}^2$ has rank $2$. How do I find a single generator? Any help/ hints are greatly appreciated. • The polynomial ring $\;\Bbb C[x]\;$ is not a division ring, much less a field. I think the intention here is to give $\;V\;$ as module over $\;\Bbb C[x]\;$, not a vector space, which is a term usually reserved when working over fields or at least division rings. . – DonAntonio Apr 16 '16 at 22:39 • I believe the right-hand side term in the definition of your scalar multiplication should $\sum_{i=0}^nc_i\alpha^i(v)$. – Arnaud D. Apr 17 '16 at 9:27 • @ArnaudD. Yes, it should. My mistake. – Jfemdl Apr 18 '16 at 0:18 One way to do this is to take a vector $v$ such that $v$ and $\alpha(v)$ are linearly independent. Indeed, in this case they form a basis of $V$, so that for any $w\in V$ there exist unique $c_0,c_1\in \mathbb{C}$ such that $w=c_0v+c_1\alpha(v)$. Thus $$w=c_0v+c_1\alpha(v)=(c_0+c_1x)\cdot v\in \mathbb{C}[x]\cdot v,$$which shows that $V=\mathbb{C}[x]\cdot v$. Now finding a vector $v$ such that $v$ and $\alpha(v)$ are linearly independent is quite easy; it suffices to take a vector that is not an eigenvector, so almost any vector will do the job.	{"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": 0.986409068107605, "perplexity": 68.42720899024326}, "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/1575540529006.88/warc/CC-MAIN-20191210205200-20191210233200-00164.warc.gz"}
https://collegemathteaching.wordpress.com/category/number-theory/	# College Math Teaching ## September 23, 2016 ### Carmichael Numbers: “not quite” primes… Filed under: algebra, elementary number theory, number theory, recreational mathematics — collegemathteaching @ 9:49 pm We had a fun mathematics seminar yesterday. Andrew Shallue gave a talk about the Carmichael numbers and gave a glimpse into his research. Along the way he mentioned the work of another mathematician…one that I met during my ultramarathon/marathon walking adventures! Talk about a small world.. So, to kick start my brain cells, I’ll say a few words about these. First of all, prime numbers are very important in encryption schemes and it is a great benefit to be able to find them. However, for very large numbers, it can be difficult to determine whether a number is prime or not. So one can take short cuts in determining whether a number is likely prime or not: one can say “ok, prime numbers have property P and if this number doesn’t have property P, it is not a prime. But if it DOES have property P, we hare X percent sure that it really is a prime. If this said property is relatively “easy” to implement (via a computer), we might be able to live with the small amount of errors that our test generates. One such test is to see if this given number satisfies “Fermat’s Little Theorem” which is as follows: Let $a$ be a positive integer and $p$ be a prime, and suppose $a \neq kp$, that is $a \neq 0 (mod p)$ Then $a^{p-1} = 1 (mod p)$ If you forgotten how this works, recall that $Z_p$ is a field if $p$ is a prime, so $a \in Z_p, a \neq 0 (mod p)$ means that the set $\{a, 2a, 3a, ...(p-1)a \}$ consists of $\{1, 2, 3, ...(p-1) \}$. So take the product $(a)(2a)(3a)...((p-1)a)) = 1(2)(3)..(p-1)a^{p-1} = 1(2)(3)...(p-1) (mod p)$. Now note that we are working in a field, so we can cancel the $(1)(2)...(p-1)$ factor on both sides to get $a^{p-1} = 1 (mod p)$. So one way to check to see if a number $q$ might be a prime is to check all $a^{q-1}$ for all $a \leq q$ and see if $a^{q-1} = 1 mod q$. Now this is NOT a perfect check; there are non-prime numbers for which $a^{q-1} = 1 mod q$ for all $a \leq q$; these are called the Carmichael numbers. The 3 smallest such numbers are 561, 41041 and 825265. The talk was about much more than this, but this was interesting. ## February 5, 2016 ### More fun with selective sums of divergent series Just a reminder: if $\sum_{k=1}^{\infty} a_k$ is a series and $c_1, c_2, ...c_n ,,$ is some sequence consisting of 0’s and 1’s then a selective sum of the series is $\sum_{k=1}^{\infty} c_k a_k$. The selective sum concept is discussed in the MAA book Real Infinite Series (MAA Textbooks) by Bonar and Khoury (2006) and I was introduced to the concept by Ferdinands’s article Selective Sums of an Infinite Series in the June 2015 edition of Mathematics Magazine (Vol. 88, 179-185). There is much of interest there, especially if one considers convergent series or alternating series. This post will be about divergent series of positive terms for which $lim_{n \rightarrow \infty} a_n = 0$ and $a_{n+1} < a_n$ for all $n$. The first fun result is this one: any selected $x > 0$ is a selective sum of such a series. The proof of this isn’t that bad. Since $lim_{n \rightarrow \infty} a_n = 0$ we can find a smallest $n$ such that $a_n \leq x$. Clearly if $a_n = x$ we are done: our selective sum has $c_n = 1$ and the rest of the $c_k = 0$. If not, set $n_1 = n$ and note that because the series diverges, there is a largest $m_1$ so that $\sum_{k=n_1}^{m_1} a_k \leq x$. Now if $\sum_{k=n_1}^{m_1} a_k = x$ we are done, else let $\epsilon_1 = x - \sum_{k=n_1}^{m_1} a_k$ and note $\epsilon_1 < a_{m_1+1}$. Now because the $a_k$ tend to zero, there is some first $n_2$ so that $a_{n_2} \leq \epsilon_1$. If this is equality then the required sum is $a_{n_2} + \sum_{k=n_1}^{m_1} a_k$, else we can find the largest $m_2$ so that $\sum_{k=n_1}^{m_1} a_k + \sum_{k=n_2}^{m_2} a_k \leq x$ This procedure can be continued indefinitely. So if we label $\sum_{k=n_j}^{m_{j}} a_k = s_j$ we see that $s_1 + s_2 + ...s_{n} = t_{n}$ form an increasing, bounded sequence which converges to the least upper bound of its range, and it isn’t hard to see that the least upper bound is $x$ because $x-t_{n} =\epsilon_n < a_{m_n+1}$ So now that we can obtain any positive real number as the selective sum of such a series, what can we say about the set of all selective sums for which almost all of the $c_k = 0$ (that is, all but a finite number of the $c_k$ are zero). Answer: the set of all such selective sums are dense in the real line, and this isn’t that hard to see, given our above construction. Let $(a,b)$ be any open interval in the real line and let $a < x < b$. Then one can find some $N$ such that for all $n > N$ we have $x - a_n > a$. Now consider our construction and choose $m$ large enough such that $x - t_m > x - a_n > a$. Then the $t_m$ represents the finite selected sum that lies in the interval $(a,b)$. We can be even more specific if we now look at a specific series, such as the harmonic series $\sum_{k=1}^{\infty} \frac{1}{k}$. We know that the set of finite selected sums forms a dense subset of the real line. But it turns out that the set of select sums is the rationals. I’ll give a slightly different proof than one finds in Bonar and Khoury. First we prove that every rational in $(0,1]$ is a finite select sum. Clearly 1 is a finite select sum. Otherwise: Given $\frac{p}{q}$ we can find the minimum $n$ so that $\frac{1}{n} \leq \frac{p}{q} < \frac{1}{n-1}$. If $\frac{p}{q} = \frac{1}{n}$ we are done. Otherwise: the strict inequality shows that $pn-p < q$ which means $pn-q < p$. Then note $\frac{p}{q} - \frac{1}{n} = \frac{pn-q}{qn}$ and this fraction has a strictly smaller numerator than $p$. So we can repeat our process with this new rational number. And this process must eventually terminate because the numerators generated from this process form a strictly decreasing sequence of positive integers. The process can only terminate when the new faction has a numerator of 1. Hence the original fraction is some sum of fractions with numerator 1. Now if the rational number $r$ in question is greater than one, one finds $n_1$ so that $\sum^{n_1}_{k=1} \frac{1}{k} \leq r$ but $\sum^{n_1+1}_{k=1} \frac{1}{k} > r$. Then write $r-\sum^{n_1+1}_{k=1} \frac{1}{k}$ and note that its magnitude is less than $\frac{1}{n_1+1}$. We then use the procedure for numbers in $(0,1)$ noting that our starting point excludes the previously used terms of the harmonic series. There is more we can do, but I’ll stop here for now. ## January 20, 2016 ### Congratulations to the Central Missouri State Mathematics Department Filed under: advanced mathematics, editorial, number theory — Tags: — collegemathteaching @ 10:43 pm The largest known prime has been discovered by mathematicians at Central Missouri State University. For what it is worth, it is: $2^{74,207,281} -1$. Now if you want to be depressed, go to the Smithsonian Facebook page and read the comment. The Dunning-Kruger effect is real. Let’s just say that in our era, our phones are smarter than our people. 🙂 ## October 25, 2013 ### Why the sequence cos(n) diverges We are in the sequences section of our Freshman calculus class. One of the homework problems was to find whether the sequence $a_n = cos(\frac{n}{2})$ converged or diverged. This sequence diverges, but it isn’t easy for a freshman to see. I’ll discuss this problem and how one might go about explaining it to a motivated student. To make things a bit simpler, I’ll discuss the sequence $a_n = cos(n)$ instead. Of course $cos(x)$ is periodic with a fundamental region $[0, 2\pi]$ so we will work with that region. Now we notice the following: $n (mod 2 \pi)$ is a group with the usual operation of addition. By $n (mod 2 \pi)$, I mean the set $n + k2\pi$ where $k \in \{..-2, -1, 0, 1, 2, 3,...\}$; one can think of the analogue of modular arithmetic, or one might see the elements of the group $\{ r\| r \in [0, 2 \pi), r = n - k 2\pi \}$. Of course, to get additive inverses, we need to include the negative integers, but ultimately that won’t matter. Example: $1, 2, 3, 4, 5, 6$ are just equal to themselves $mod 2 \pi.$ $7 = 7 - 2\pi (mod 2\pi), 13 = 13 - 4 \pi (mod 2\pi)$, etc. So, I’ll denote the representative of $n (mod 2\pi)$ by $[n]$. Now if $n \ne m$ then $[n] \ne [m]$; for if $[n]=[m]$ then there would be integers $j, k$ so that $n + j2\pi = m +k2\pi$ which would imply that $\|m-n\|$ is a multiple of $\pi$. Therefore there are an infinite number of $[n]$ in $[0, 2\pi]$ which means that the set $\{[n]\}$ has a limit point in the compact set $[0, 2\pi]$ which means that given any positive integer $m$ there is some interval of width $\frac{2\pi}{m}$ that contains two distinct $[i], [j]$ (say, $j$ greater than $i$.) This means that $[j-i] \in (0, \frac{2\pi}{m})$ so there is some integers $k_2, k_3,$ so that $k_2[j-i] \in (\frac{2\pi}{m}, \frac{22\pi}{m}), k_3[j-i] \in (\frac{22\pi}{m}, \frac{32\pi}{m})$, etc. Therefore there is some multiple of $[j-i]$ in every interval of width $\frac{2\pi}{m}$. But $m$ was an arbitrary positive integer; this means that the $[n]$ are dense in $[0,2\pi]$. It follows that $cos([n]) = cos(n)$ is dense in $[-1,1]$ and hence $a_n = cos(n)$ cannot converge as a sequence. Frankly, I think that this is a bit tough for most Freshman calculus classes (outside of, say those at MIT, Harvard, Cal Tech, etc.). ## May 16, 2013 ### Big Breakthrough in Number Theory: progress toward the twin primes conjecture. Filed under: advanced mathematics, number theory — Tags: , , — collegemathteaching @ 7:26 pm It is a long standing conjecture in number theory that there exists an infinite number of twin primes: twin primes are prime integers that differ by 2. Example: 3 and 5, 11 and 13, 17 and 19 are examples of twin prime pairs. Very large twin primes have been found: $(2,003,663,613 \times 2^{2195,000}) - 1$ and $(2,003,663,613 \times 2^{2195,000}) + 1$. But, up to now: We don’t know if this pairing “stops” at some point (is there a largest pair?) In fact, up to recently, we had no statement of the following form: given a finite integer $M$ there exists an infinite number or pairs of primes $p, q$ such that $p - q \le M$ (assuming that $p$ is the greater of the pair). Well, now we do. The Annals of Mathematics (the top ranked mathematics journal in the world) has accepted a paper that shows the infinite pairs statement is true, if $M = 70,000,000$: The twin prime conjecture says that there is an infinite number of such twin pairs. Some attribute the conjecture to the Greek mathematician Euclid of Alexandria, which would make it one of the oldest open problems in mathematics. The problem has eluded all attempts to find a solution so far. A major milestone was reached in 2005 when Goldston and two colleagues showed that there is an infinite number of prime pairs that differ by no more than 16. But there was a catch. “They were assuming a conjecture that no one knows how to prove,” says Dorian Goldfeld, a number theorist at Columbia University in New York. The new result, from Yitang Zhang of the University of New Hampshire in Durham, finds that there are infinitely many pairs of primes that are less than 70 million units apart without relying on unproven conjectures. Although 70 million seems like a very large number, the existence of any finite bound, no matter how large, means that that the gaps between consecutive numbers don’t keep growing forever. The jump from 2 to 70 million is nothing compared with the jump from 70 million to infinity. “If this is right, I’m absolutely astounded,” says Goldfeld. Zhang presented his research on 13 May to an audience of a few dozen at Harvard University in Cambridge, Massachusetts, and the fact that the work seems to use standard mathematical techniques led some to question whether Zhang could really have succeeded where others failed. But a referee report from the Annals of Mathematics, to which Zhang submitted his paper, suggests he has. “The main results are of the first rank,” states the report, a copy of which Zhang provided to Nature. “The author has succeeded to prove a landmark theorem in the distribution of prime numbers. … We are very happy to strongly recommend acceptance of the paper for publication in the Annals.” Hey, 70 million is a LOT less than “infinity”. 🙂 ## January 17, 2013 ### Enigma Machines: some of the elementary math Note: this type of cipher is really an element of the group $S_{26}$, the symmetric group on 26 letters. Never allowing a letter to go to itself reduced the possibilites to products of cycles that covered all of the letters. ## August 8, 2011 ### MathFest Day Three (Lexington 2011) I left after the second large lecture and didn’t get a chance to blog about them before now. But what I saw was very good. The early lecture was by Lauren Ancel Meyers (Texas-Austin) on Mathematical Approaches to Infectious Disease and Control This is one of those talks where I wish I had access to the slides; they were very useful. She started out by giving a brief review of the classical SIR model of the spread of a disease which uses the mass action principle (from science) that says that the rate of of change of those infected with a disease is proportional to the product of those who are susceptible to the disease and those who can transmit the disease: $\frac{dI}{dt}=\beta S I$. (this actually came from chemistry). Of course, those who are infected either recover or die; this action reduces the number infected. Of course, the number of susceptible also drop. This leads to a system of differential equations. The basic reproduction number is significant: $= R_0 = \frac{\beta S}{\nu + \delta}$ where $\nu$ is the recovery rate and $\delta$ is the death rate. Note: if $R_0 < 1$ then the disease will die off; if it is greater than 1 we have a pandemic. We can reduce this by reducing $S$ (vaccination or quarantine), increasing recovery or, yes, increasing the death rate (as we do with livestock; remember the massive poultry slaughters to stop the spread of flu). Of course, this model assumes that the infected organisms contact others at random and have equal probabilities of spreading, that the virus doesn’t evolve, etc. So this model had to be improved on; methods from percolation theory were developed. So many factors had to be taken into account such as: how much vaccine is there to spread? How far along is the outbreak? (at first children get it; then adults). How severe is the consequences? (we don’t want the virus to evolve to a more dangerous, more resistant form). Note that the graph model of transmission is dynamic; it can actually change with time. Of special interest: one can recover the rate of infections of the various strains (and the strains vary from season to season) by looking at the number of times flu related words were searched for on Google. The graph overlap (search rate versus reported cases) was stunning; the only exception is when a scare occurred; then the word search rate lead the actual cases, but that happened only once (2009). Note also that predictions of what will happen get better with a shorter time window (not a surprise). There was much more in the talk; for example the role of the location of the providers of vaccines was discussed (what is the optimal way to spread out the availability of a given vaccine?) Manjur Bhargava, Lecture III First, he noted that in the case where $f(x,y)$ was cubic, that there is always a rational change of variable to put the curve into the following form: $y^2 = x^3 + Ax + B$ where $A, B$ are integers that have the following property: if $p$ is any prime where $p^4$ divides $A$ then $p^6$ does NOT divide $B$. So this curve can be denoted as $E_{A,B}$. Also, there are two “generic” cases of curves depending on whether the cubic in $x$ has only one real root or three real roots. This is a catalog of elliptical algebraic curves of the form $y^2 = x^3 + ax + b$ taken from here. The everywhere smooth curves are considered; the ones with a disconnected graph are said to have “an egg”; those are the ones in which the cubic in $x$ has three real roots. In the connected case, the cubic has only one; remember that these are genus one curves; we are seeing a slice of a torus in 4-space (a space with two complex dimensions) in the plane. Also recall that the rational points on the curve may be finite or infinite. It turns out that the rational points (both coordinates rational) have a group structure (this is called the “divisor class group” in algebraic geometry). This group has a structure that can be understood by a simple geometric construction in the plane, though checking that the operation is associative can be very tedious. I’ll give a description of the group operation and provide an elementary example: First, note that if $(x,y)$ is a point on an elliptical curve, then so is $(x, -y)$ (note: the $y^2$ on the left hand side of the defining equation). That is important. Also note that we will restrict ourselves to smooth curves (that have a well defined tangent line). The elements of our group will be the rational points of the curve (if any?) along with the point at infinity. If $P = (x_1, y_1)$ I will denote $(x_1, -y_1) = P'$. The operation: if $P, Q$ are rational points on the curve, construct the line $l$ with equation $y = m(x-x_1)+ y_1$ Substitute this into $y^2 = x^3 + Ax + B$ and note that we now have a cubic equation in $x$ that has two rational solutions; hence there must be a third rational solution $x_r$. Associated to that $x$ value is two $y$ values (possibly double if the $y$ value is zero). Call that point on the curve $R$ then define $P + Q = R'$ where $R'$ is the reflection of $R$ about the $x$ axis. Note the following: that this operation commutes is immediate. If one adds a point to itself, one uses the tangent line as the line through two points; note that such a line might not hit the curve a third time. If such a line is vertical (parallel to the $y$ axis) the result is said to be “0” (the point at infinity); if the line is not vertical but still misses the rest of the curve, it is counted three times; that is: $P + P = P'$. Here are the situations: Of course, $\infty$ is the group identity. Associativity is difficult to check directly (elementary algebra but very tedious; perhaps 3-4 pages of it?). Since the group is Abelian, if the group is finite it must be isomorphic to $\oplus_{i = 1}^r Z_i \oplus \frac{Z}{n_1 Z} \oplus \frac{Z}{n_2 Z}....\frac{Z}{n_k Z}$ where the second part is the torsion part and the number of infinite cyclic factors is the rank. The rank turns out to be the geometric rank; that is, the minimum number of points required to obtain all of the rational points (infinite number) of the curve. Let $T$ be the torsion subgroup; Mazur proved that $\|T\|\le 16$. Let’s look at an example of a subgroup of such a curve: let the curve be given by $y^2 = X^3 + 1$ It is easy to see that $(0,1), (0, -1), (2, 3), (2, -3), (-1, 0)$ are all rational points. Let’s see how these work: $(-1, 0) + (-1, 0) = 0$ so this point has order 2. But there is also some interesting behavior: note that $\frac{d}{dx} (y^2) = \frac{d}{dx}(x^3 + 1)$ which implies that $\frac{dy}{dx} = \frac{3x^2}{2y}$ So the tangent line through $(0, 1)$ and $(0, -1)$ are both horizontal; that means that both of these points have order 3. Note also that $(2, 3) + (2,3) = (0,1)$ as the tangent line runs through the point $(0, -1)$. Similarly $(2, 3) + (0, -1) = (2, -3)$ So, we can see that $(2,3), (2, -3)$ have order 6, $(0, 1), (0, -1)$ have order 3 and $(-1, 0)$ has order 2. So there is an isomorphism $\theta$ where $\theta(2,3) = 1, \theta(2,-3) = 5, \theta(0, 1) = 2, \theta(0, -1) = 4, \theta(-1, 0) = 3$ where the integers are mod 6. So, we’ve shown a finite Abelian subgroup of the group of rationals of this curve. It turns out that these are the only rational points; here all we get is the torsion group. This curve has rank zero (not obvious). Note: the group of rationals for $y^2 = x^3 + 2x + 3$ is isomorphic to $Z \oplus \frac{Z}{2Z}$ though this isn’t obvious. The generator of the $Z$ term is $(3,6)$ and $(-1,0)$ generates the the torsion term. History note Some of this was tackled by computers many years ago (Birch, Swinnerton-Dyer). Because computers were so limited in those days, the code had to be very efficient and therefore people had to do quite a bit of work prior to putting it into code; evidently this lead to progress. The speaker joked that such progress might not have been so quickly today due to better computers! If one looks at $y^2 = x^3 + Ax + B mod p$ where $p$ is prime, we should have about $p$ points on the curve. So we’d expect that $\frac{N_p}{p} \approx 1$. If there are a lot of rational points on the curve, most of these points would correspond to $mod p$ points. So there is a conjecture by Birch, Swinnerton-Dyer: $\prod_{p \le X} \frac{N_p}{p} \approx c (log(X))^r$ where $r$ is the rank. Yes, this is hard; win one million US dollars if you prove it. 🙂 Back to the curves: there are ways of assigning “heights” to these curves; some include: $H(E_{(A,B)}) = max(4\|A\|^3, 27B^2)$ or $\Delta(E_{(A,B)} -4A^3 - 27B^2$ Given this ordering, what are average sizes of ranks? Katz-Sarnak: half have rank 0, half have rank 1. It was known that average ranks are bounded; previous results had the bound at 2.3, 2, 1.79, assuming that the Generalized Riemann Hypothesis and the Birch, Swinnerton-Dyer conjecture were asssumed. The speaker and his students got some results without making these large assumptions: Result 1: when $E/Q$ is ordered by height, the average rank is less than 1. Result 2: A positive portion (10 percent, at least) have rank 0. Result 3: at least 80 percent have rank 0 or 1. Corollary: the BSD is true for a positive proportion of elliptic curves; The speaker (with his student) proved results 1, 2, and 3 and then worked backwards on the existing “BSD true implies X” results to show that BSD was true for a positive proportion of the elliptic curves. ## August 6, 2011 ### MathFest Day 2 (2011: Lexington, KY) I went to the three “big” talks in the morning. Dawn Lott’s talk was about applied mathematics and its relation to the study of brain aneurysms; in particular the aneurysm model was discussed (partial differential equations with a time coordinate and stresses in the radial, circumference and latitudinal directions were modeled). There was also modeling of the clipping procedure (where the base of the aneurysm was clipped with a metal clip); various clipping strategies were investigated (straight across? diagonal?). One interesting aspect was that the model of the aneurysm was discussed; what shape gave the best results? Note: this is one procedure that was being modeled: Next, Bhargava gave his second talk (on rational points on algebraic curves) It was excellent. In the previous lecture, we saw that a quadratic curve either has an infinite number of rational points or zero rational points. Things are different with a cubic curve. For example, $y^2 = x^3 - 3x$ has exactly one rational point (namely (0,0) ) but $y^2 = x^3-2x$ has an infinite number! It turns out that the number of rational points an algebraic curve has is related to the genus of the graph of the curve in $C^2$ (where one uses complex values for both variables). The surface is a punctured multi-holed torus of genus $g$ with the punctures being “at infinity”. The genus is as follows: 0 if the degree is 1 or 2, 1 if the degree is 3, and greater than 1 if the degree is 4 or higher. So what about the number of rational points: 0 or finite if the genus is zero finite if the genus is strictly greater than 1 (Falting’s Theorem; 1983) indeterminate if the genus is 1. Hence much work is done in this area. No general algorithm is known to make the determination if the curve is cubic (and therefore of genus 1) Note: the set of rational points has a group structure. Note: a rational cubic has a rational change of variable which changes the curve to elliptic form: Weierstrauss form: $y^2 = x^3 + Ax + B$ where $A, B$ are integers. Hence this is the form that is studied. Sometimes the rational points can be found in the following way (example: $y^2 = x^3 + 2x + 3$: note: this curve is symmetric about the $x$ axis. $(-1, 0)$ is a rational point. So is $(3, 6)$. This line intersects the curve in a third point; this line and the cubic form a cubic in $x$ with two rational roots; hence the third must be rational. So we get a third rational point. Then we use $(3, -6)$ to obtain another line and still another rational point; we keep adding rational points in this manner. This requires proof, but eventually we get all of the rational points in this manner. The minimum number of “starting points” that we need to find the rational points is called the “rank” of the curve. Our curve is of rank 1 since we really needed only $(3, 6)$ (which, after reflecting, yields a line and a third rational point). Mordell’s Theorem: every cubic is of finite rank, though it is unknown (as of this time) what the maximum rank is (maximum known example: rank 28), what an expected size would be, or even if “most” are rank 0 or rank 1. Note: rank 0 means only a finite number of rational points. Smaller talks I enjoyed many of the short talks. Of note: there was a number theory talk by Jay Schiffman in which different conjectures of the following type were presented: if $S$ is some sequence of positive integers and we look at the series of partial sums, or partial products (plus or minus some set number), what can we say about the number of primes that we obtain? Example: Consider the Euclid product of primes (used to show that there is no largest prime number) $E(1) = 2 + 1 = 3, E(2) = 23 + 1 = 7, E(3) = 235 + 1 = 31, E(4) = 2357 + 1 = 211$ etc. It is unknown if there is a largest prime in the sequence $E(1), E(2), E(3)....$. Another good talk was given by Charlie Smith. It was about the proofs of the irrationality of various famous numbers; it was shown that many of the proofs follow a similar pattern and use a series of 3 techniques/facts that the presenter called “rabbits”. I might talk about this in a later post. Another interesting talk was given by Jack Mealy. It was about a type of “hyper-hyperbolic” geometry called a “Snell geometry”. Basically one sets up the plane and then puts in a smooth closed boundary curve (say, a line or a sphere). One then declares that the geodesics are those that result from a straight lines…that stay straight until they hit the boundary; they then obey the Snell’s law from physics with respect to the normal of the boundary surface; the two rays joined together from the geodesic in the new geometry. One can do this with, say, a concentric series of circles. If one arranges the density coefficient in the correct manner, one’s density (in terms of area) can be made to increase as one goes outward; this can lead to interesting area properties of triangles. ## August 5, 2011 ### Blogging MathFest, 2011 (Lexington, KY) Filed under: advanced mathematics, algebraic curves, conference, elementary number theory, number theory — collegemathteaching @ 1:50 am I started the day by attending three large lectures: Laura DeMarco, University of Illinois at Chicago who spoke about dynamical systems (that result from complex polynomials; for example if $f: C \rightarrow C$ is a function of the complex plane, one can talk about the orbit of a point $z \in C, z, f(z), f(f(z)) = f^{(2)}(z), f(f(f(z))) = f^{(3)}(z)....$. One can then talk about sets of points $w, w\in C$ and $sup\|f^{(n)}\| < \infty$ This is called the Filled Julia Set. Ed Burger of Williams (a graduate school classmate of mine who made good) gave the second; he talked about Fibonacci numbers and their relation to irrational ratios (which can be obtained by continued fractions) and various theorems which say that natural numbers can be written uniquely as specified sums of such gadgets. Lastly Manjul Bhargava of Princeton (who is already full professor though he is less than half my age; he was an Andrew Wiles student) gave a delightful lecture on algebraic curves. What I noted: all three of these mathematicians are successful enough to be arrogant (especially the third). They could have blown us all away. Yes, they took the time and care to give presentations that actually taught us something. Of the three, I was the most intrigued by the last one, so I’ll comment on the mathematics. You’ve probably heard that a Pythagorean triple is a triple of integers $a, b, c$ such that $a^2 + b^2 = c^2$. For now, we’ll limit ourselves to primitive triples; that is, we’ll assume that $a, b, c$ have no common factor. You might have heard that any Pythagorean triple is of the form: $a = m^2 - n^2, b = 2mn, c = m^2 + n^2$ for $m, n$ integers. It is true that $a, b, c$ being defined that way leads to a Pythagorean triple, but why do ALL Pythagorean triples come in this form? One way to see this is to look at an algebraic curve; in this case, the curve corresponding to $x^2 + y^2 = 1$. Why? Start with $a^2 + b^2 = c^2$ and divide both sides by $c^2$ to obtain $((\frac{a}{c})^2 + (\frac{b}{c})^2 = 1$ One then notes that one is now reduced to looking to rational solutions to $x^2 + y^2 = 1$ (a rational solution to this can be put in the $((\frac{a}{c})^2 + (\frac{b}{c})^2 = 1$ form by getting a common denominator). We now wish to find all rational points (both coordinates rational) on the circle; clearly $(-1,0)$ is one of them. Easy claim: if $(u, v)$ is such a rational point, then the line from $(-1,0)$ to $(u, v)$ has rational slope. Not quite as easy claim: if a line running through $(-1, 0)$ has rational slope $s < \infty$ then the line intersects the circle in a rational point. Verification: such a line has equation $y = s(x+1)$ and intersects the circle in a point whose $x$ value satisfies $x^2 + s^2(x+1)^2-1 = 0$. This is a quadratic that has rational coefficients and root $x = -1$ hence the second root must also be rational. Let’s calculate the second root by doing division: $\frac{(s^2 + 1)x^2 +2s^2x + s^2-1}{x+1} = (s^2+1)x + s^2 - 1$. So the point of intersection has $x = \frac{1-s^2}{s^2 + 1}$ latex and $y = s(\frac{1-s^2}{s^2 + 1} + 1) = \frac{2s}{s^2 + 1}$. Both are rational. Therefore, there is a one to one correspondence between rational slopes and rational points on the circle and all are of the form $(\frac{1-s^2}{s^2 +1}, \frac{2s}{s^2 + 1})$. Note: we obtain $(-1,0)$ by letting $s$ go to infinity; use L’Hopital’s rule on the first coordinate). So if we have any Pythagorean triple $(a,b,c)$ then $\frac{a}{c} = \frac{1-s^2}{s^2 + 1}, \frac{b}{c} = \frac{2s}{s^2 + 1}.$ But $s$ is rational hence we write $s = \frac{p}{q}$ where $p, q$ are relatively prime integers. Just a bit of easy algebra reveals $\frac{a}{c} = \frac{q^2 -p^2}{p^2 + q^2}, \frac{b}{c} = \frac{2pq}{p^2 + q^2}$ which gives us $a = q^2 - p^2, b =2pq, c = q^2 + p^2$ as required. The point: the algebraic curve motivated the proof that all Pythagorean triples are of that form. Note: we can extract even more: if $f(x,y) = 0$ latex is any quadratic rational curve (i. e., $f(x,y) = a_1 x^2 + a_2 x + a_3 + a_4 y^2 + a_5 y + a_6 xy$, all coefficients rational, and $(u, v)$ is any rational point and there is a line through $(u,v)$ of rational slope $s$ which intersects the curve in a second point (the quadratic nature forbids more than 2 points), the second point must also be rational. This follows by obtaining a quadratic in $x$ by substituting $y = s(x - u) + v$ and obtaining a quadratic with rational coefficients that has one rational root. Of course, it might be the case that there is no rational point to choose for $(u, v)$. In fact, that is the case for $x^2 + y^2 = 3.$ Why? Suppose there is a rational point on this curve $x = \frac{p}{q}, y = \frac{a}{b}$ with both fractions in lowest terms. We obtain $(pb)^2 + (aq)^2 = 3(qb)^2$ Now let’s work Mod 4 (hint from the talk): note that in $mod 4, 2^2 = 0, 3^2 = 1$ therefore the sum of two squares can only be 0, 1 or 2. The right hand side is either 3 or 0; equality means that both sides are zero. This means that $pb, aq$ are both even and therefore $3(qb)^2$ is divisible by 4 therefore either $q$ is even or $b$ is even. Suppose $b$ is odd. Then $q$ is even and because $pb$ is even, $p$ is even. This contradicts the fact that $p, q$ are relatively prime. If $q$ is odd, then because $aq$ is even, $a$ is even. This contradicts the fact that $a, b$ are relatively prime. So both $q, b$ are even which means that $p, a$ are odd. Write $q = 2^I m, b = 2^J n$ where $m, n$ are odd (possibly 1). Then $(p^2)(2^{2J})n^2 + (a^2)(2^{2I}) m^2 = 2^{2J + 2I}3 m^2n^2$. Now if $J = I$ we obtain $(pn)^2 + (am)^2$ on the left hand side (sum of two odd numbers squared) which must be 2 mod 4. The right hand side is still only 3 or 0; this is impossible. Now if, say, $J \ge I$ then we get $(pn)^2 2^{2(J-I)} + (am)^2 = 2^{2J} 3 (mn)^2$ which means that the odd number $(am)^2$ is the difference of two even numbers. That too is impossible. Hence $x^2 + y^2 = 3$ contains no rational coordinates; that circle manages to miss that dense set. The point of all of this is that algebraic curves can yield significant information about number theory. Photos This is the German Enigma Coding machine (with plug board) at the NSA booth. This is another view of the Enigma	{"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": 290, "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": 0.8965559005737305, "perplexity": 271.5868361229141}, "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-2018-05/segments/1516084890187.52/warc/CC-MAIN-20180121040927-20180121060927-00098.warc.gz"}
http://www.gradesaver.com/textbooks/math/algebra/intermediate-algebra-6th-edition/chapter-5-section-5-1-exponents-and-scientific-notation-exercise-set-page-262/70	## Intermediate Algebra (6th Edition) $\frac{15}{16}$ We are given the expression $1^{-3}-4^{-2}$ $1^{-3}-4^{-2}=\frac{1}{1^{3}}-\frac{1}{4^{2}}=1-\frac{1}{16}=\frac{16}{16}-\frac{1}{16}=\frac{15}{16}$ In general, $a^{-n}=\frac{1}{a^{n}}$ (where a is a nonzero real number and n is a positive integer).	{"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": 0.9188057780265808, "perplexity": 267.1876034289652}, "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/1510934806353.62/warc/CC-MAIN-20171121113222-20171121133222-00537.warc.gz"}
https://tex.stackexchange.com/questions/63232/why-can-words-with-hyphen-char-not-be-hyphenated/63234	# Why can words with hyphen char not be hyphenated? Assume I have a word like Baden-Württemberg. TeX can't hyphenate any of these two word parts. Why? why isn't it something like this: where the small ticks indicate possible hyphenation points. A technical (TeXnical) explanation is welcome. BTW: I am not asking how to circumvent this (by using the babel shorthand "= for example). • Because so decided Knuth. – egreg Jul 13 '12 at 19:35 • @egreg but why? I mean, what parts of TeX makes decide to disable hyphenation? Is there an explicit "if word contains hyphen char, then exit" somewhere in the hyphenation routine? – topskip Jul 13 '12 at 19:37 • See the posting Switch meaning of hyphenation commands for a LuaLaTeX-based solution, which changes all instances of - on the fly to "= if they (the instances of -) are sandwiched between two letters. Thus, Baden-Württemberg, branchen-üblich, and Gesäß-Muskulatur (and many others!) can automatically be hyphenated at many additional places. :-) – Mico Aug 7 '16 at 19:53 The TeXbook, page 454, last but one double dangerous bend paragraph If a trial word l1 … ln has been found by this process, hyphenation will still be abandoned unless n ≥ λ + ρ, where λ = max(1,\|\lefthyphenmin\|) and ρ = max(1,\|\righthyphenmin\|). (Plain TeX takes λ = 2 and ρ = 3.) Furthermore, the items immediately following the trial word must consist of zero or more characters, ligatures, and implicit kerns, followed immediately by either glue or an explicit kern or a penalty item or a whatsit or an item of vertical mode material from \mark, \insert, or \vadjust. Thus, a box or rule or math formula or discretionary following too closely upon the trial word will inhibit hyphenation. (Since TeX inserts empty discretionaries after explicit hyphens, these rules imply that already-hyphenated compound words will not be further hyphenated by the algorithm.) An explicit hyphen is a character whose character code matches the font's \hyphenchar value or a ligature that ends with such a character (that's why also -- or --- inhibit hyphenation). Indeed, if you try the following example, you'll see that TeX hyphenates the compound word: \documentclass{article} \usepackage[T1]{fontenc} \usepackage[ngerman]{babel} \begin{document} \hyphenchar\font=\string"7F \end{document} The result is In Ba- den-Würt- tem- berg The T1 encoded fonts have in position 0x7F a character which is identical to the normal hyphen. Changing the \hyphenchar to denote this slot, the normal hyphen does not inhibit hyphenation any more. • 8 consecutive lines with hyphenated words in the TeXbook (the mentioned paragraph). That should be worth a Knuth-error-award-cheque. – topskip Jul 13 '12 at 19:55 • @PatrickGundlach Read at the bottom of page 451. :) – egreg Jul 13 '12 at 19:59 • This solution works fine for me if I just load fontenc with the T1 encoding so that it uses cm-super. However, if I load lmodern as well, for example, it doesn't seem to work. Looking at the encoding files, both seem to put hyphen.alt in that spot. Does anybody know why it doesn't work with lmodern? – cfr Nov 26 '13 at 23:07 • @egreg Am I right that changing \hyphenchar ist not really a practical »solution«? One would need to change it for all fonts and fonts series in use to get uniform behaviour for a document... (and still a "= may be needed in Baden-Württemberg so there isn't much gained when typing, anyway) – clemens Feb 27 '14 at 11:49 • @cgnieder Yes, you're right. IIRC, there should a package that can hook the choice of \hyphenchar at every font loading; but also hyphenation patterns should be modified to have possible breaks at - (the usual hyphen). – egreg Feb 27 '14 at 11:55 The \hyphenchar\font=\string"7F seems not the correct work around for this problem, since it is not font independent. A better way would be to set the \defaulthyphenchar=127 which seems font independent. Also hyphenation which are defined in acronyms will be correct too. BUT there are still issues when having hyphens in the text like: Baden-W\"urttemberg \gls{BP}-Test If you look at "den-Würt-", it is still not split correctly into two lines. Also the last example "BP-Test" isn't split. The only work around I have found is to use "= instead of - in the text which works for now. But I would appreciate a better solution... Here is the MWE: \documentclass{article} \usepackage[T1]{fontenc} \usepackage[ngerman]{babel} %\usepackage{lmodern} \usepackage{mathptmx} \usepackage[ nonumberlist, acronym, nopostdot, section] {glossaries} \newacronym{BP}{BP}{Borderline-Pers\"onlichkeitsst\"orung} \defaulthyphenchar=127 \begin{document}	{"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": 0.914587676525116, "perplexity": 2991.934960187395}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583662690.13/warc/CC-MAIN-20190119054606-20190119080606-00515.warc.gz"}
https://cs.stackexchange.com/tags/real-numbers/hot	People who code: we want your input. Take the Survey # Tag Info 37 Sage is an open source computer algebra system. Let's see if it can handle your basic example: sage: sqrt(3) * (4/sqrt(3) - sqrt(3)) 1 What is happening under the hood? Sage is storing everything as a symbolic expression, which it is able to manipulate and simplify using some basic rules. Here is another example: sage: 1 + exp(pi*i) 0 So sage can also ... 22 There is no way to represent all real numbers without errors if each number is to have a finite representation. There are uncountably many real numbers but only countably many finite strings of 1's and 0's that you could use to represent them with. 20 It all depends what you want to do. For example, what you show is a great way of representing rational numbers. But it still can't represent something like $\pi$ or $e$ perfectly. In fact, many languages such as Haskell and Scheme have built in support for rational numbers, storing them in the form $\frac{a}{b}$ where $a,b$ are integers. The main reason ... 18 Computer algebra is a huge area, probably at least a semester-long university-level course to get most of the basics. However, I think we can cover some of the flavour of it here. Your case is the easy case, because it is entirely within the language of algebraic numbers (i.e. roots of polynomials), and manipulating polynomials is really the foundation of ... 9 Any machine model in which a machine can be described by a string over a fixed alphabet can only compute countably many things. Since there are uncountably many real numbers, all of these machine models fail to compute almost all real numbers. In fact, the alphabet need not be fixed. It is enough that the alphabet is taken from some countable set of finite ... 8 A real number cannot be input into a Turing machine, since it is an infinite object. There are various models for providing a Turing machine with oracle access to real numbers, but these are by definition computable. You could imagine a Turing machine given a ZFC definition of a real number, and in that case it's undecidable. For example, given a Turing ... 8 Yes. There are. There is the real-RAM/BSS model mentioned in the other answer. The model has some issues and AFAIK there is not much research activity about it. Arguably, it is not a realistic model of computation. The more active notion of real computability is that of higher type computation model. The basic idea is that you define complexity for higher ... 8 The model you describe is known as the Blum-Shub-Smale (BSS) model (also Real RAM model) and indeed used to define complexity classes. Some interesting problems in this domain are the classes $P_R$, $NP_R$, and of course the question of whether $P_R$ = $NP_R$. By $P_R$ we mean the problem is polynomially decidable, $NP_R$ is the problem is polynomially ... 8 Yes, Rice's theorem for reals holds in every reasonable version of computable reals. I will first prove a certain theorem and a corollary, and explain what it has to do with computability later. Theorem: Suppose $p : \mathbb{R} \to \{0,1\}$ is a map and $a, b \in \mathbb{R}$ two reals such that $p(a) = 0$ and $p(b) = 1$. Then there exists a Cauchy sequence ... 7 Your idea does not work because a number represented in base $b$ with mantissa $m$ and exponent $e$ is the rational number $b \cdot m^{-e}$, thus your representation works precisely for rational numbers and no others. You cannot represent $\sqrt{2}$ for instance. There is a whole branch of computable mathematics which deals with exact real arithmetic. Many ... 7 There are many effective Rational Number implementations but one that has been proposed many times and can even handle some irrationals quite well is Continued Fractions. Quote from Continued Fractions by Darren C. Collins: Theorem 5-1. - The continued fraction expression of a real number is finite if and only if the real number is rational. Quote ... 7 Type-2 Turing machines are not more powerful than ordinary Turing machines in the sense that any map $\mathbb{N} \to \mathbb{N}$ that can be computed by a type-2 machine can also be computed by an ordinary machine. To see this, suppose a type-2 Turing machine $T$ computes a function $f : \mathbb{N} \to \mathbb{N}$. We can convert $T$ to an ordinary machine ... 6 Let's assume that the numbers $a_1,\ldots,a_n$ are integers, so that the problem is in NP for any fixed $f$. We construct a polynomial $f$ so that the problem is NP-complete, by reduction from vertex cover in cubic graphs ($3$-regular graphs). Let the instance of cubic vertex cover consist of a cubic graph $G=(V,E)$ and an integer $m$, and let $\|V\| = n$. ... 6 Here is the question: You are given a list of length $n+1$ which contains the numbers $1,\ldots,n$, one of them appearing twice (and the rest appearing once). Find the number which appears twice. The sum of numbers from $1$ to $n$ is $\frac{n(n+1)}{2}$, so if you subtract that from the sum of the list you get the number appearing twice. 5 There are a number of "exact real" suggestions in the comments (e.g. continued fractions, linear fractional transformations, etc). The typical catch is that while you can compute answers to a formula, equality is often undecidable. However, if you're just interested in algebraic numbers, then you're in luck: The theory of real closed fields is complete, o-... 5 In a nutshell: Printing a random non-computable real is a meaningless task, for precise technical reasons. The meaningful problem is to print non-computable numbers precisely identified by some unique property. But these cannot be printed by any program precisely because they are not computable. Using randomness in the hope of printing by chance the ... 5 Turing machines, in the classical sense, decide languages of finite strings over finite alphabets. Your logical language has uncountably many constant symbols so you can't even write down all the formulas as strings, let alone ask a Turing machine to decide things about the collection of formulas. 5 The continued fraction algorithm is easy enough to implement. The first step is to compute the continued fraction of the input $x = [c_0;c_1,\ldots]$. You start with $x_0 = x$, and use the recurrence $c_i = \lfloor x_i \rfloor$, $x_{i+1} = 1/(x_i - c_i)$. You stop when $x_i - c_i$ is "small enough". The next step is to compute the convergent of the continued ... 5 We can compute $$f(\theta) = -N\log (2\pi)-\frac{1}{2}\sum_{i=1}^N (\langle x_i,\theta \rangle -y_i)^2.$$ Expanding this out, we get that $f(\theta)$ is some quadratic form: $$f(\theta) = \theta' A \theta + v'\theta + C,$$ where $A$ is symmetric. The next step is to get rid of the linear term. Let $\theta = \psi + \epsilon$. Then $$\theta' A \theta + \... 4 If you use the operators \{+,-,\times,/\} (i.e., you don't included the power operator), then all of your problems are likely decidable. Testing equality with zero For instance, let's consider L = \mathbb{Z} \cup \{\pi\}. Then you can treat \pi as a formal symbol, so that each leaf is a polynomial in \mathbb{Z}[\pi] (e.g., the integer 5 is the ... 4 This is a rather tricky question! As you seem to understand, the real issue is the presence of \hat{}. It is intimately related to a well known conjecture: Schanuel's conjecture, which states that, essentially, there are no non-trivial algebraic relationships between \pi and e. The (expected) positive answer to this conjecture would give you a ... 4 The set of higher-order primitive recursive reals is essentially the class of functions \mathbb{N}\rightarrow\mathbb{N} which can be represented by a term \mathrm{Nat}\rightarrow\mathrm{Nat} in Gödel's system T. Since every such function is total, and every well-typed term in the system can be enumerated effectively, there is a relatively easy proof by ... 4 There are several issues with your question but perhaps I can clarify some issues. First off you assume f(1) = 1.999... and also that no x \in \mathbb{N} exists such that f(x) = 2 but that's a contradiction in terms because 1.999... = 2 and thus f(1) = 2. Why does 1.999... = 2? Well there's an easy answer but not fulfilling answer and a more ... 3 Consider the following reasonable definition for a Turing machine computing an irrational number in [0,1]. A Turing machine computes an irrational r \in [0,1] if, on input n, it outputs the first n digits (after the decimal) of the binary representation of r. One can think of many extensions of this definition for probabilistic Turing machines. ... 3 You are looking for an optimal 1-dimensional k-means algorithm. The k-means objective function for partitioning the data x_1, \ldots, x_n into k sets S = \{S_1, \ldots, S_k\}.$$ \sum\limits_{i=1}^k \sum\limits_{x \in S_i}\lVert x - \mu_i \rVert^2 where $\mu_i$ is the mean of $S_i$ [1]. You can apply a dynamic programming algorithm to the ... 3 It's pretty clearly an algorithm according to my definition in the linked question. I think your real question is "what is the problem, if any, for which this algorithm is correct?" The answer would be something like "given some stuff and a number of iterations, output a value satisfying some condition with an accuracy related to the iteration count." 3 Original formulation (Originally, the OP was interested in the minimum absolute difference rather than the minimum non-zero absolute difference.) You are asking two different questions. For the first, consider the restricted version in which all numbers are rational, and the decision variant in which the problem is to decide whether your expression is at ... 3 Check this out! http://coq.io/opam/coq-markov.8.5.0.html. A library for Markov's inequality built on mathematical probability theory. 3 You can run any standard quantum algorithm on a real-amplitude quantum computer with one additional qubit and only constant-factor slowdown (or perhaps linear-factor slowdown considering loss of parallelism) by replacing each $a{+}bi$ in your unitary matrices by $\big(\begin{smallmatrix}a&-b\\b&a\end{smallmatrix}\big)$. Likewise, you can simulate a ... 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http://math.stackexchange.com/questions/7926/if-v-is-a-vector-space-that-what-is-v-mathbbn/7930	# If $V$ is a vector space, that what is $V^\mathbb{N}$? Notation question (I believe the notes that I'm reading uses pretty common notation): Let $V$ be a vector space over a field $K$. What is $V^\mathbb{N}$? Is it a vector space of infinite dimensions? What are the elements of $V^\mathbb{N}$? Are they infinite length tuples? - BTW I'd have preferred to ask this question on chat.math.stackexchange.com but it's not setup yet. – user2468 Oct 26 '10 at 15:47 In my opinion, this notation is ambiguous (edit: but maybe I'm wrong). It could mean two things: it could mean the space of functions $\mathbb{N} \to V$, which is a vector space in the obvious way. This is the countable product in the category of vector spaces. This vector space is very large; in particular, it has uncountable dimension and it is not spanned by the functions $e_i$ which are equal to $1$ at $i$ and equal to $0$ elsewhere. Or it could mean the space of functions $\mathbb{N} \to V$ which are zero except at finitely many places, which is also a vector space in the obvious way, but countable-dimensional, and this is the vector space spanned by the $e_i$. This is the countable coproduct in the category of vector spaces. (Note that it is not necessary to distinguish between finite products and finite coproducts because they are the same.) If I had to guess, I would guess that it means the former. Can you give a link to the notes? It might be possible to figure out which one is meant from context. - Unfortunately the notes are under protected access within my campus :[ (edit:oops return) Thanks. I check both function spaces and infer which is consistent with the notes. – user2468 Oct 26 '10 at 15:42 I would guess the former too. Most of the time I see the notation $V^\mathbb{N}$ it means "the space of sequences taking values in $V$". – Willie Wong Oct 26 '10 at 15:44 @Qiaochu: I don't think $V^{\mathbb{N}}$ is used for the coproduct, though I have seen $V^{(\mathbb{N})}$ for the elements of $V^{\mathbb{N}}$ of finite support... – Arturo Magidin Oct 26 '10 at 15:59 @Qiaochu: Could you elaborate on why the former has uncountable dimension? – Rahul Oct 26 '10 at 18:25 @Arturo: Thanks for the link. I'm not very well-versed in infinite-dimensional vector spaces, so it will take me a while to digest that proof. @Qiaochu: Ah, I see; I was forgetting that the span is composed of linear combinations, which by definition involve only finitely many vectors. – Rahul Oct 26 '10 at 23:04 It would be the vector space consisting of an infinite product V × V × ... of copies of V. More generally: for two sets A and B, AB often denotes the set of functions f : B → A. (The only exception I'm aware of is when the sets A and B happen to be ordinals, in set theory; then a different notion of exponentiation is often used.) How are these two concepts related? Well: as a vector space, V is also a set of vectors. And so a function f : ℕ → V is essentially a block-vector description of a vector v = (f(0), f(1), ...) ∈ V × V × ... . - surely in the beginning of the second paragraph you meant $f:B\to A$. – Willie Wong Oct 26 '10 at 15:48 @Willie Wong: that's right, silly mistake. – Niel de Beaudrap Oct 26 '10 at 15:50 I'd think of it as the set of sequences of elements of $V$, i.e. $$V^\mathbb{N} = \prod_{n=1}^\infty V.$$ But I don't think there is a really nice description. In the case where $V$ is finite-dimensional, it'd just be $K^\mathbb{N}$ in a clever (and non-canonically isomorphic) disguise, but if $V$ were to be infinite-dimensional, $V^\mathbb{N}$ could be rather ugly. E.g. $\ell_1^\mathbb{N}$: sequences of absolutely summable sequences. Then a sequence of sequences would converge to a sequence iff it converged coordinatewise. Complicated and very confusing. Following Munkres' "Topology", I prefer to use $V^\infty$ for set of sequences in $V$ for which only finitely many elements are nonzero, i.e. $$V^\infty = \bigoplus_{n=1}^\infty V = \left\lbrace (v_n) \in V^\mathbb{N} \;{\large\mid}\; \text{v_n = 0 from some index onwards}\right\rbrace.$$ -	{"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": 0.9523640871047974, "perplexity": 208.5267945939571}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438043062635.98/warc/CC-MAIN-20150728002422-00135-ip-10-236-191-2.ec2.internal.warc.gz"}
https://jd2718.org/2007/04/02/puzzle-one-over-one-plus-methods-of-solution/	This is the place to offer methods of solutions for the puzzle below: What is the value of $\frac{1}{1+\frac{1}{1 + \frac{1}{1 + \dotsb}}}$ ? A month ago Dave Marain ran a discussion of $\sqrt{2 + \sqrt{2 + \sqrt{2 + \dotsb}}}$ over here. I know this one has been done many many times before. But he generated a nice discussion. There are also places for: What makes these two problems so similar? April 3, 2007 pm30 9:48 pm 9:48 pm $\sqrt{2 + \sqrt{2 + \sqrt{2 + \dotsb}}}$ First examine what this ‘object’ is. It looks a like a sequence, or better, the limit of the sequence. The sequence can be described by: $a_1 = b \hspace{1cm}\mathrm{(initial value)}$ $a_2 = \sqrt{2+a_1}$ $a_3 = \sqrt{2+a_2} = \sqrt{2+ \sqrt{2+a_1}}$ In general: $a_{n+1}= \sqrt{2+ a_{n}}$ If the limit exists, then $a_{n}$ and $a_{n+1}$ have the same limit. Thus, $a_{n}=a_{n+1}=B$, where $B$ is the limit of the sequence. Pluggin this into the equation $a_{n+1}= \sqrt{2+ a_{n}}$ yields: $B = \sqrt{2 + B}$ $B^2-B-2=0$ Solving for B yields (B>0): $B = 2$. Of course, one has to prove the several steps. April 3, 2007 pm30 10:00 pm 10:00 pm Correction for my post above: If the limit exists, then $a_{n}$ and $a_{n+1}$ have the same limit. This can be written as $\rm{lim} a_{n} = B$ and $\rm{lim} a_{n+1} = B$ I forgot to write $\rm{lim}$ in my former post. ———– Now, I think I know how this sequence was generated: $2 = \sqrt{2+2}$ Plugging this into the right hand side yields: $2 = \sqrt{2+ \sqrt{2+2}}$ and so on… April 3, 2007 pm30 10:20 pm 10:20 pm I just figured out how such a sequence looks for 3 (this is fun): $3 = \sqrt{9}= \sqrt{6+3}$ $= \sqrt{ 6+ \sqrt{6+3} }$ $= \sqrt{ 6+ \sqrt{ 6 + \sqrt{ 6 + 3 }}}$ $latex = \sqrt{ 6 + \sqrt{ 6 + \sqrt{ 6 + \dotsb}}}$ April 3, 2007 pm30 11:17 pm 11:17 pm $\frac{1}{1+\frac{1}{1 + \frac{1}{1 + \dotsb}}}$ This can be analyzed in a similar way. The ‘object’ is the limit of a sequence: $a_{1} = b (\mathrm{initial value})$ $a_{2} = 1+ \frac{1}{a_{1}}$ $a_{3} = 1 + \frac{1}{a_{2}} = 1 + \frac{1}{1+ \frac{1}{a_{1}}}$ In general: $a_{n+1} = 1 + \frac{1}{a_{n}}$ If we assume that the sequence $a_{n}$ converges, then $a_{n}$ and $a_{n+1}$ have the same limit. This yields $B = 1 + \frac{1}{B}$, where B is the limit of the sequence. It follows: $B^2-B-1 = 0$ Solving for B yields (B>0): $B = \frac{1}{2}(\sqrt{5}+1)$ $\frac{1}{2} \left(\sqrt{5}+1 \right) = \frac{1}{2} \left(\sqrt{5}+1 \right) + 1 - 1$ $= 1+ \frac{1}{2} \left(\sqrt{5}+1 \right) -1$ $= 1+ \frac{\sqrt{5}-1}{2}$ $= 1+ \frac{5-1}{2(\sqrt{5}+1)}$ $= 1+ \frac{4}{2(\sqrt{5}+1)}$ $= 1+ \frac{2}{\sqrt{5}+1}$ $= 1+ \frac{1}{\frac{1}{2}(\sqrt{5}+1)}$ The tactic is obvious, in the first step a 1 is added because we want to have the 1 + 1/(1+/…) term. The remaining task is to manipulate the term after the 1 in such a way, that $\frac{1}{2} \left(\sqrt{5}+1 \right)$ appears again. April 3, 2007 pm30 11:26 pm 11:26 pm I just noticed that I calculated the value of $1+\frac{1}{1+\frac{1}{1 + \frac{1}{1 + \dotsb}}}$ $\frac{1}{1+\frac{1}{1 + \frac{1}{1 + \dotsb}}}$ Thus, the limit of $\frac{1}{1+\frac{1}{1 + \frac{1}{1 + \dotsb}}}$ is $\frac{1}{2}(\sqrt{5}-1)$	{"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": 44, "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": 0.9350156784057617, "perplexity": 880.1249367995355}, "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-2021-17/segments/1618038087714.38/warc/CC-MAIN-20210415160727-20210415190727-00475.warc.gz"}
https://www.physicsforums.com/threads/rlc-circuit-with-damping-questions.142374/	# Homework Help: RLC Circuit with damping questions 1. Nov 5, 2006 ### totallydesperate I have an RLC Circuit. I've changed the resistors 3 times to give me a case of overdamping, underdamping, and critical damping. In taking my data (this was a long long time ago) I apparently missed taking measurements of current. I'm trying to solve the three differential equations to make a "predicted results" graph. To do this, I need initial conditions V0, which I have, and I0, which I should have. Is there any way besides redoing my experiment to determine what the current at t=0 would be? If so, what other information do you need from me to explain it to me? 2. Nov 5, 2006 ### ultimateguy If you know the value of the resistances and the voltage, then technically you should be able to calculate the current using the Kirchoff law. Even so, keep in mind that calculated values are often a bit different from measured values due to uncertainties. If you're using a calculated value for current when you were supposed to use a measured one, it kind of misses the point.	{"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": 0.9426530599594116, "perplexity": 550.3337252503062}, "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-2018-30/segments/1531676592875.98/warc/CC-MAIN-20180722002753-20180722022753-00371.warc.gz"}
https://www.peterkrautzberger.org/0104/	# Rapid idempotent ultrafilters Welcome back to the second (and final) part on why strongly summable ultrafilters are rapid. Why the new title? Well, first, I needed another one (I’ve had too many posts with “Part X” in them, I think). Second, after I proved the results I mentioned last time, I quickly found an additional, somewhat more general observation regarding idempotents. Here’s the problem with this post though. I want to give the argument. But you know how it is in mathematics: standing on the shoulders of the huddled masses giants and all that. All work relies on established results. In the first post, I gave a lot of “unnecessary” proofs to motivate the story behind the result. In this post, I’ll get to the actual proof and I’m torn, I’m not sure how much to quote and how much to prove. So bear with me and leave comments whenever I screw up. ### Strongly summables are rapid. Why could this be true? On the one hand, because we already established partial results last time. On the other hand there’s an old result attributed to Pierre Matet which, for now, I can only state in a obfuscated fashion. Theorem (Matet 87) If is a strongly summable ultrafilter, then there exists a function “” such that is rapid. So you see, strongly summable ultrafilters imply the existence of rapid ultrafilters – via a very simple -function. What’s , you’re asking? Ah yes, I should talk about ### \max to the Max. For FS-sets, there is a natural notion of maximum. If you have a bunch of elements summed up, then it makes sense to call the largest summand the maximum. So intuitively, the maximum should simply map each element in the FS-set to the largest generator involved in producing it. This might not appear very well defined but bear with me. Consider an FS-set, let’s keep calling it . If we’re lucky, there is a unique way to write each as a sum of (or should I write ? indices are hard…). For example, the sequence has this property while fails to have this property (quite badly, I suppose). It turns out that there’s an easy property to ensure this: the sequence just has to grow quickly. Proposition (folklore? can be found in Blass, Hindman 87) If for all , then In other words, each element in has a unique representation. [Edit May 22, 2012: modified attribution] The proof is an easy induction on , using the growth factor to argue that the maximal element of and must be equal. • If , then follows immediately from the growth assumption. • Now assume inductively that we’ve proved the claim for smaller sets. • Let be the maximum of . • Wlog, . • Then we must also have – otherwise a contradiction to the assumed equality of both sums. • But if both and contain , we also have . • Using our induction hypothesis, we have , hence . So if we have unique representations of elements in , we can define a function ### Growth, growth, growth and more growth Unfortunately, we need to tweak this a little bit more. Remember that FS-sets are all about sums. Very often will have . So assuming growth, each have a unique representation in terms of the . It’s natural to assume that there’s some kind of connection between the representations of . For one thing, we know that if , then . The growth as above does not guarantee the reverse, though (just consider ), and the reverse often simplifies things. Fortunately, all we have to do is improve the growth! Proposition (Hindman, Blass 87) If , then if and only if and . The proof is much like the earlier proof. • Again, we’re doing an induction on . • If , then or must be empty and the growth condition does the rest. • For the inductive step, let . • Due to the growth condition, must be in . • But it must also lie in . • Else . • But it can’t be in both and . • Else . • Wlog • Th . • Applying our induction hypothesis to , we get and . • This in turn gives us (as ) and – as desired. Why all this trouble? Well, there are many uses for this. For what’s coming below, it simplifies an important calculation. In general, it is extremely important since it allows us to switch from the addition of numbers to the union of disjoint, finite sets. (I don’t know about you, but I find the union operation on disjoint sets much easier to comprehend.) Corollary If has growth as above, then if we ever have (and we will), then, assuming , we have . In particular, if , then . ### Strongly summable ultrafitlers are rapid – the proof. Anyway, let’s get back to where we started. First, we should make a connection to strongly summable ultrafilters. Lemma (Blass, Hindman 87) If is strongly summable, then has a base of FS-sets whose sequences satisfy the growth condition (the stronger one with factor , of course). This is a great lemma (though maybe not a true lemma) and the reason why I spend so much time above talking about growth conditions – it comes in handy in many situations and really tells us something about strongly summable ultrafilters and the sets they contain. The proof, however, is weird so I’ll skip it (unless you insist in the comments). And now it makes sense to state the initial theorem. Theorem (Matet, 87 / Blass, Hindman 87) Let be a strongly summable ultrafilter and with growth (or just unique representations); fix the -function for as above. Then is a rapid P-point. [Edit on May 21, 2012: I rephrased the theorem to improve clarity – thanks to the comment-by-email who suggested it!] You can skip the proof if you like because it’s not important to us (and I’m cheating a little on the important part, rapidity). But I find the argument appealing and since I’ve had to go through all the trouble to introduce the growth condition and so forth, I think I might as well include this, too. It’s a typical proof for strongly summable ultrafilters – just write down a good partition and let it do the work for you. • Fix . • We will prove that is either constant on a set in or it is finite-to-one. • Pick any with . • Let be the minimum function analogous to . • Now partition into and • Our strongly sumable ultrafilter will give us (with the usual growth condition) included in one of these two parts. • If is included in the first part, then is bounded on . (In particular, is constant on a set in .) • Consider and pick any other . • Due to the growth condition, we have and reversely . • In particular, . • So is bounded on and we find a • If is included in the second part, then is finite-to-one on . • Consider some point in the image of , say . • If for some we have , then by our assumption. • But how many can there be with ? At most -many! • The will have pairwise distinct minima. Why? • Remember that for we naturally have . • By the growth condition of the , we know that and are sums of disjoint sets of ’s. • In particular, their minima will differ! • Therefore, is finite-to-one. • And now the cheating: the last argument shows that is at most size . To be able to make any finite-to-one function an -to-one function is, in fact, equivalent to being a rapid ultrafilter. It’s a nice exercise, but feel free to insist in the comments. This theorem is the reason I originally (back in 2010, in my conversations with Jana at BLAST) thought there’s a chance that all strongly summable ultrafilters are rapid. First, is a finite-to-one function. It’s an old, probably folklore result (cf. Miller, 1980) that the finite-to-one image of a rapid ultrafilter is again rapid. Now the reverse is not true but our function is so easy that it’s possible to prove this. [Edit May 22, 2012: modified attribution] Theorem (Krautzberger (yep, this is it)) If is strongly summable, then is rapid. Here’s the gist: the trick is simple: speed up functions by and let that sped-up function be dominated in the rapid image. Then we pick an FS-set in our strongly summable ultrafilter that witnesses this domination, in particular, it’s generating sequence will dominate that sped-up function. Finally, just as in our initial observations in the first post, the FS-set will still grow fast enough to dominate the original function. • By Matet’s theorem pick such that is rapid. • Now pick any . • We may assume that is strictly monotone (that’s all the functions we need to dominate). • By Matet’s theorem we can find a set that dominates . • Now fix such that ; for simplicity, we can assume that the also satisfy the growth condition. • Then dominates . • Let . We’ll show that . • Pick the maximal . • So , i.e., we only need to find out how large is. • Of course, . • Now , so is greater or equal to the -th element of . • Since the dominates , this gives us . • By ’s monotonicity, , • But the set contains exactly -many elements, i.e., less than -many elements – precisely as desired. Whew, ok. That’s done. ### Jana asked one more question After I got around to writing my argument up properly after the conference, Jana asked me whether there are could be other rapid idempotent ultrafilters. In particular, could there be so-called minimal idempotents which are rapid? This, again, sounded rather drastic to me. Minimal idempotents have extremely rich algebraic properties, in particular, any set in them is central and thus all versions of the Central Sets Theorem hold for such sets (as opposed to FS-sets where no FS-set with the growth condition satisfies even the simplest Central Sets Theorem). But, of course, by now I was skeptical of my own skepticism. To understand this question, we have to go back to strongly summable ultafilters for a second. Due to Matet’s result, we know that the existence of stongly summable ultrafilters imply the existence of rapid P-points. In particular, the existence of strongly summable ultrafilters cannot be proved using ZFC alone. But speaking of P-points and rapidity, it is a famous open problems whether there is a model with neither P-points nor Q-points. We can achieve a model without P-points and a model without Q-points, but incidentally not both. (As a taste of the problem: the continuum must at least be in such a model.) On top of that, there exists a model without rapid ultrafilters (hence without Q-points), but disturbingly, afaik, nobody has a model without Q-points but with rapid ultrafilters! In other words, all known models without Q-points are without rapid ultrafilters (but with P-points). Also, as a consequence of the big open question, any known model without P-points has Q-points, hence rapid ultrafilters. What I’m trying to say is that Jana’s question leads to a whole bunch of interesting and classical open problems. So it was very much worth thinking about. If there are other rapid idempotents, how do we get them? It turns out we can get the possibly strongest positive answer to this question. Theorem (Krautzberger (yippie, another micro-contribution)) If there exists a rapid ultrafilter, then there exist rapid idempotent ultrafilters. In fact, then there exists a whole closed left ideal of rapid ultrafilters, in particular there are minimal idempotents which are rapid. As it turns out this follows easily from two well-known results on rapid ultrafilters which give us the following: Proposition If is rapid, any ultrafilter, then is rapid. • Since is rapid, the tensor product is rapid (this can be found in Miller, 1980). • Also, the finite-to-one image of a rapid ultrafilter is rapid (again, see Miller, 1980). • But and addition is a finite-to-one map. • Hence is rapid. Then the proof of the theorem is as follows: • Let be a rapid ultrafilter. • Then is a closed left ideal containing only rapid ultrafilters. • This is a closed left ideal since is a continuous map. • If is any ultrafilter, then is rapid. • Every closed left ideal contains (by compactness) a minimal left ideal which in turn contains a minimal idempotent (that’s one way of defining them, actually). This theorem seems very strong to me. If I have one rapid, I have an entire closed left ideal of rapid ultrafilters – that’s one of the crucial structures in Algebra in the Stone–Čech compactification! If you happen to have strongly summable ultrafilters, this gives an even nicer observation. You see, the definition of minimal idempotent can be given in terms of minimality in a certain partial order on the idempotents, namely There are two obvious related orders, iff , and (guess how it’s defined). It’s an easy exercise (really), that minimality in either partial order is minimality in all others. An old result is that strongly summable ultrafilters are right maximal (in fact, strongly right maximal: has only one solution, ). This means that assuming we have strongly summable ultrafilters, then we have a “full spectrum” of rapid idempotents – from right maximal all the way to (right) minimal. Well, and that’s all folks. I hope you enjoyed my little experiment as much as I have. I’ll certainly write a follow-up post when I’ve decided where to put a fixed copy of this. Please let me know if you find any errors in the proof and, more importantly, if you think I should clarify certain parts.	{"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": 0.9877647757530212, "perplexity": 882.1626330280505}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676593051.79/warc/CC-MAIN-20180722061341-20180722081341-00092.warc.gz"}
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By the classical theory of Tannakian duality we know that every $k$-linear rigid abelian tensor category ($k$ a field) which has a fibre functor to $\mathrm{Vec}_k$ (finite dim. vector spaces over ... 379 views ### Non-trivial topological line bundles over cartesian product of manifolds not coming from a pullback Let $X$ and $Y$ be connected smooth manifolds. Let $L$ be a topological real line bundle over $X\times Y$. Then we know that the isomorphism class of such a line bundle is determined by its first ... 6k views ### Applications for p-Sylow subgroups theorem I have searched for such a question and didn't find it. I recently had a presentation in which I introduced $p$-Sylow subgroups and proved Sylow's theorems. I will have another one soon, concerning ... 1k views ### Errata database? Some authors do a really great job by collecting errors and comments to their books and putting a list on their websites. I wonder if there is some (perhaps wiki-style) website where errata are ... 48k views ### Fourier vs Laplace transforms In solving a linear system, when would I use a Fourier transform versus a Laplace transform? I am not a mathematician, so the little intuition I have tells me that it could be related to the boundary ... 3k views ### Why is Lie's Third Theorem difficult? Recall the following classical theorem of Cartan (!): Theorem (Lie III): Any finite-dimensional Lie algebra over $\mathbb R$ is the Lie algebra of some analytic Lie group. Similarly, one can propose ... 1k views ### What is the trace in the Chern-Simons action? Warning: This is a very stupid question regarding a basic misunderstanding that I have. I realize that the question is very elementary, but I guess asking stupid questions is better than remaining ... 2k views ### Is the radical of an irreducible ideal irreducible? I originally posted this to math.stackexchange.com here. I got a partial answer, but I now suspect that the complete answer is much harder than I thought, so I'm posting it here. Fix a commutative ... 3k views ### Tannakian Formalism The Tannakian formalism says you can recover a complex algebraic group from its category of finite dimensional representations, the tensor structure, and the forgetful functor to Vect. Intuitively, ... 523 views ### What are the limits of the Erdős-Rankin method for covering intervals by arithmetic progressions? To construct gaps between primes which are marginally larger than average, Erdős and Rankin covered an interval $[1,y]$ with arithmetic progressions with prime differences. A nice short exposition is ... 1k views ### Crystalline cohomology via the syntomic site Hello, Let $k$ be a field of characteristic $p > 0$, and let $Y$ be a $k$-scheme. Consider the sites $Y_{syn}$ and $(Y/W_n)_{cris}$ (where $W_n$ are the Witt vectors of $k$ of length $n$), of $Y$ ... 1k views 97 views ### quasi-projective and separated as topological properties Let $X$ be a non-reduced noetherian scheme over $\mathbb{Z}$ or $\mathbb{C}$. Assume that $X^{red}$ is quasi-projective and separated, does the same hold for $X$ ? (By the way, projective implies a ... 140 views ### Modular factorization of Dedekind zeta functions It is well known that for abelian number fields, the factorization of its Dedekind zeta function goes like this: $$\zeta_K=\zeta\prod_\chi L(s,\chi)$$ with the Dirichlet characters distinct and ... 742 views ### Catalan numbers as sums of squares of numbers in the rows of the Catalan triangle - is there a combinatorial explanation? This question arose from an answer to my recent question How many traces are there on Temperley-Lieb, Fuss-Catalan, Iwahori-Hecke, Birman-Wenzl-Murakami-Kauffman, ... algebras? What I need from that ... 96 views ### Asymptotics of a Bivariate Generating Function I have the following generating function, $$G(x,y)=\sum_{n,k \geq 0}a(n,k)x^ny^k = \frac{(y^2-y)x+1}{(y-y^3)x^2-(y+1)x+1}$$ and I am interested in obtaining an asymptotic for the sequence $a(n,k)$ ... 366 views I have three questions related to the theory of modular forms and it was frequently asked to me by my collegues and even my invited teacher in our seminars of the number theory at the faculty of ... 85 views ### If $C$ has all geometric realizations of simplicial objects, what other colimits does it have? Let $C$ be an $\infty$-category. Suppose that every diagram $\Delta^{\mathit{op}} \to C$ has a colimit. Is there any characterization of small categories $I$ such that every diagram $I \to C$ has a ... 47 views ### roots in a root system which have nonzero coefficients with respect to each simple root If we consider crystallographic root systems, then for each $k$ such that $n \leq k \leq d-1$ where $d$ is the Coxeter number, it seems to be the case that there is exactly one root of height $k$ with ... 196 views ### Counting function for prime pair with bounded gaps between them [duplicate] I'll start by noting that I am not at all an expert on number theory. However I do use it in a proof and would like your assistance if possible. Yitang Zhang breakthrough result established that ... 508 views ### Minimum value of $\|p(1)\|^2+\|p(2)\|^2 +…+ \|p(n+3)\|^2$ over all monic polynomials $p$ Let $n$ be a positive integer. Determine the smallest possible value of $\|p(1)\|^2+\|p(2)\|^2 +...+ \|p(n+3)\|^2$ over all monic polynomials $p$ of degree $n$. This question was proposed (problem ... Given a (finite) abelian group $G = \langle S \mid R \rangle$, has the problem of counting the number of elements which can be expressed as a word (in $S$) of length $\leq k$ been studied? If so, ...	{"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": 0.939540445804596, "perplexity": 483.2739315578799}, "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-2015-35/segments/1440644064019.39/warc/CC-MAIN-20150827025424-00160-ip-10-171-96-226.ec2.internal.warc.gz"}
https://www.expii.com/t/horizontal-asymptotes-of-rational-functions-9859	Expii # Horizontal Asymptotes of Rational Functions - Expii Learn how to visualize and find the horizontal asymptotes of a rational function. A horizontal asymptote refers to "end behavior like a constant (flat line with zero slope)," which happens when the degree of the numerator is no more than the degree of the denominator.	{"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": 0.9407891035079956, "perplexity": 316.0575212362992}, "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/1593657149205.56/warc/CC-MAIN-20200714051924-20200714081924-00598.warc.gz"}
https://hal-amu.archives-ouvertes.fr/hal-01840679	Power spectrum of the cosmic infrared background at 60 and 100 μ m with IRAS - Archive ouverte HAL Access content directly Journal Articles Astronomy and Astrophysics - A&A Year : 2002 ## Power spectrum of the cosmic infrared background at 60 and 100 μ m with IRAS M.-A. Miville-Deschênes Guilaine Lagache J.-L. Puget • Function : Author #### Abstract Based on a power spectrum analysis of the IRAS ISSA maps, we present the first detection of the Cosmic far-Infrared Background (CIB) fluctuations at 60 and 100 µm. The power spectrum of 12 low cirrus emission regions is characterized by a power excess at spatial frequencies higher than k ∼ 0.02 arcmin −1. Most of this excess is due to noise and to nearby point sources with a flux stronger than 1 Jy. But we show that when these contributions are carefully removed, there is still a power excess that is the signature of the CIB fluctuations. The power spectrum of the CIB at 60 and 100 µm is compatible with a Poissonian distribution, at spatial frequencies between 0.025 and 0.2 arcmin −1. The fluctuation level is ∼1.6 × 10 3 Jy 2 /sr and ∼5.8 × 10 3 Jy 2 /sr at 60 and 100 µm respectively. The levels of the fluctuations are used in a larger framework, with other observationnal data, to constrain the evolution of IR galaxies (Lagache et al. 2002). The detections reported here, coupled with the level of the fluctuations at 170 µm, give strong constraints on the evolution of the IR luminosity function. The combined results at 60, 100 and 170 µm for the CIB and its fluctuations allows, on the CIB at 60 µm, to put a firm upper limit of 0.27 MJy/sr and to give an estimate of 0.18 MJy/sr. ### Dates and versions hal-01840679 , version 1 (16-07-2018) ### Identifiers • HAL Id : hal-01840679 , version 1 • DOI : ### Cite M.-A. Miville-Deschênes, Guilaine Lagache, J.-L. Puget. Power spectrum of the cosmic infrared background at 60 and 100 μ m with IRAS. Astronomy and Astrophysics - A&A, 2002, 393 (3), pp.749 - 756. ⟨10.1051/0004-6361:20020929⟩. ⟨hal-01840679⟩ ### Export BibTeX TEI Dublin Core DC Terms EndNote Datacite 52 View	{"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": 0.8282087445259094, "perplexity": 4216.301322403465}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945473.69/warc/CC-MAIN-20230326142035-20230326172035-00102.warc.gz"}
http://www.checkmyknowledge.com/a-level/physics/training/a2/thermal-energy	###### Boyle's law Select the correct statement of Boyle's law. Select the correct answer. For a constant mass of gas at a constant temperature, the pressure exerted by the gas is inversely proportional to the volume it occupies. ###### Charles's law Select the correct statement of Charles's law. Select the correct statement. For a constant mass of gas at a constant pressure, the volume occupied by the gas is proportional to its absolute temperature. ###### Absolute zero Select the correct definition of absolute zero temperature. Select the correct answer. Absolute zero is the temperature at which the pressure of an ideal gas becomes zero. This is 0К on the Kelvin temperature scale, which is equivalent to -273 °C. ###### Absolute temperature Select the correct definition of absolute temperature. Select the correct answer. Absolute temperature is the Kelvin or thermodynamic temperature scale with zero at -273 °C, the temperature at which the pressure of an ideal gas becomes zero. ###### Specific heat capacity How long will a 1.5 kW kettle take to raise the temperature of 1kg of water from 20° C to 100° C? The specific heat capacity of water is $$4200{J\over kg\times K}$$. Select the correct answer. $$\Delta \theta=100-20 = 80K$$ $$\Delta E = c m \Delta T = 4200180=336000J$$ $$\Delta t=E/P=336000/1500=224s$$ ###### RMS speed Find the r.m.s. speed of the five molecules of atmospheric oxygen. Velocities of the molecules are: v1 = 500 m/s, v2 = 496 m/s, v3 = 508 m/s, v4 = 503 m/s, v5 = 499 m/s. Select the correct answer. The root-mean-square speed $$c=\sqrt{v^2_1+v^2_2+v^2_3+v^2_4+v^2_5\over 5}\approx501.217m/s$$. ###### RMS speed - 2 What is the root-mean-square speed of a balloon filled with dioxygen at 20° C? The mass of a dioxygen miolecule is $$5.312\times10^{-26} kg$$. Select the correct answer. $$T = 20+273 =293K$$ $${m<c^2>\over2}={3\over2}kT$$ Root-mean-square speed is $$\sqrt{<c^2>}=\sqrt{3kT \over m}\approx477.865m/s$$ ###### Pressure law Select the correct definition of pressure law. Select the correct option. For a constant mass of gas at a constant volume, the pressure exerted by the gas is proportional to its absolute temperature. ###### Equation of state Select the correct expression for the equation of state for an ideal gas. Select the correct option. The equation of state for an ideal gas is $$pV=nRT$$. It relates the pressure, volume and temperature of an ideal gas. n is the number of moles of the gas; and R is the Universal gas constant. $$R = 8.31{J\over kg\times mol}$$. ###### Ideal gas Which of the features listed are wrong for an ideal gas? Select only wrong options. An ideal gas would have the following properties: 1 The molecules have zero size. 2 The molecules are identical. 3 The molecules collide with each other and the walls of their containers without any loss of energy, collisions take zero time. 4 The molecules exert no forces on each other, except during collisions. 5 There are enough molecules for statistical laws to be applied.	{"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": 0.9750916957855225, "perplexity": 1272.0252813413613}, "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-2017-26/segments/1498128321961.50/warc/CC-MAIN-20170627235941-20170628015941-00592.warc.gz"}
https://math.stackexchange.com/questions/1742848/what-is-condition-for-second-degree-equation-to-represent-a-pair-of-straight-lin	# What is condition for second degree equation to represent a pair of straight lines? According to my text the necessary and sufficient condition for a general equation of second degree i.e. $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ to represent a pair of straight lines is that 1) the discriminant $abc + 2fgh - af^2 - bg^2 - cf^2 = 0$ and 2) $h^2 \ge ab, g^2 \ge ca$ and $f^2 \ge bc$. I was able to prove part 1) but I am not able to work out why part 2) should always be satisfied. Can someone help prove part 2) and what happens if part 1) is true but part 2) is false? I think $h^2 \ge ab$ must always be true because then angle between two lines represented by the equation of second degree will be not defined as $tan \theta = {2 \sqrt{h^2-ab} \over a+b}$ where $\theta$ is angle between two lines. • With $h^2<ab$, the trinomial formed by the quadratic terms is positive (or negative) definite, hence the conic is of the ellipse type. – Yves Daoust Apr 14 '16 at 20:38 If the conic represents a pair of lines, then it can be written as $(px+qy+r)(p'x+q'y+r')$. Therefore, we get $ab = pp'qq'$ and $h = \frac{pq'+p'q}{2}$. 1) If both $pq'$ and $p'q$ are greater than equal to zero then apply AM-GM to $pq'$ and $p'q$ to get $$\frac{pq'+ p'q}{2} \geq \sqrt{pq' \cdot p'q}$$ Squaring both sides, we get $$h^2 = {\bigg(\frac{pq'+ p'q}{2}\bigg)}^2 \geq pp'qq' = ab$$ 2) If exactly one of $pq'$ and $p'q$ is less than zero then $ab \leq 0$ and therefore is always less than equal to $h^2 \geq 0$. 3) If both $pq'$ and $p'q$ are less than zero then apply AM-GM to $-pq'$ and $-p'q$ to get $$\frac{-pq'+ (-p'q)}{2} \geq \sqrt{-pq' \cdot -p'q} = \sqrt{pp'qq'}$$ Squaring both sides, we get $$h^2 = {\bigg(\frac{-pq'+ (-p'q)}{2}\bigg)}^2 \geq pp'qq' = ab$$ The other two conditions come out in a similar manner. • thanks that was pretty easy.! – Matt Apr 14 '16 at 21:04 • I realized that we can't quite apply AM-GM since $a$ and $b$ need not be positive in general. But in that case, $ab \leq 0$ and is hence less than equal to the square $h^2$. – Seven Apr 14 '16 at 21:11 • may you please show how can we prove any one of three by A.M G.M inequality? – Matt Apr 15 '16 at 17:11 • @Raghav I have elaborated on the answer. Hope that helps. – Seven Apr 15 '16 at 20:36 Hint: The classification of conics is done through the quadratic form in $\mathbf R^3$ associated to the matrix $$\begin{pmatrix} a&h&g\\h&b&f\\g&f&c \end{pmatrix}.$$ The conic splits into two lines if and only if the quadratic form has signature $(1,1)$.	{"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": 0.8925583958625793, "perplexity": 130.62256375443528}, "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/1563195529481.73/warc/CC-MAIN-20190723172209-20190723194209-00539.warc.gz"}
https://engineering.stackexchange.com/questions/5913/how-do-you-find-this-vx-in-the-given-circuit	# How do you find “this” Vx in the given circuit? So I was told that $V_x$ is equal to 10V. I tried reasoning out why $V_x$ was 10V and this is what I got: -You can treat the Vx as an open circuit which means current does not flow through the resistor (meaning it's a short circuit) and thus from KVL we get $V_x$ = 10V However, I am still not convinced that $V_x$= 10V. A better explanation would be greatly appreciated. • Welcome to Engineering SE. You are asked to calculate $V_{th}$ for the circuit. So the first step is to calculate the open circuit voltage across $V_{AB}$. Since no current is flowing across the Circuit $V_{AB}$ is same as $V_x$ is same as 10V. Here is a good link Thévenin's theorem or Thevenin's Equivalent – Mahendra Gunawardena Oct 26 '15 at 2:08 You start by making A-B an open circuit. This implies that the current entering or leaving node A from the load is 0 Amps. This implies that the current in the two 1 $\Omega$ resistors have the same current flowing through them. The open circuit across A-B also implies that 0 current is flowing into or out of Node B. This means 0 current is flowing through $\frac{5}{11}\Omega$ resistor, so there is 0 voltage drop across that resistor, so $V_x$ is also 10 V. This leaves you with 10 V at $V_x$ and $0.2V_x = 2 V$ across the voltage controlled voltage source. This means the current flowing in the loop with the two 1 $\Omega$ resistors is $\frac{2 V}{2\Omega} = 1 A$. This also means the voltage between the voltage controlled voltage source and the bottom 1 $\Omega$ resistor is $12 V$. With this information you can calculate the voltage drop across the lower 1 $\Omega$ resistor to be $1V$, which gives you a $V_{th}$ of $11 V$. You are right in that $V_x$ can't be 10 V. Fortunately, 10V for $V_x$ simplifies the circuit considerably, making it particularly easy to analyze. Since $V_x$ is the same voltage as the supply on the left, both ends of the 5/11 Ω resistor are at the same voltage. That means the current thru it is zero. That means the current thru anything directly in series with is must also be zero. Therefore, the current thru the 5/11 Ω resistor, the 10 V power supply, and the 1 Ω resistor at right are all 0. The voltage across any resistor with 0 current thru it must also be 0. We therefore know that point A is at 0 V. Taking the loop with the 2 V supply and the two 1 Ω resistors in isolation, we can easily see that 1 A must be circulating, as indicated by your curved arrow. 1 A thru a 1 Ω resistor causes a 1 V drop, so the right end of this loop must be 1 V higher than the left. Putting this back into the rest of the circuit that is known to carry 0 current just doesn't work. According to previous logic, point A must be at 0 V, but adding the +2 V of the supply and -1 V of the resistor to the 10 V at $V_x$ yields 11 V at A, which is a contradiction. Clearly something is wrong somewhere. • This: "That means the current thru anything directly in series with is must also be zero." doesn't imply this: "the 1 Ω resistor at right are all 0". Why? Because you have a voltage controlled voltage voltage source and the loop there. So your second answer of 11 V is correct. – Eric Oct 26 '15 at 16:21 When calculating $V_{th}$ , $V_{AB}$ is open circuited as in the diagram. As a result there is no current flow across $\frac{5}{11} \Omega$. Thus $V_x = 10V$ References: • But there is current in loop I. – Eric Oct 26 '15 at 16:53	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8426347970962524, "perplexity": 382.8504785139403}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347419593.76/warc/CC-MAIN-20200601180335-20200601210335-00293.warc.gz"}
https://math.stackexchange.com/questions/2537849/is-the-borel-sigma-algebra-over-reals-a-complete-lattice	# Is the borel $\sigma$-algebra over reals a complete lattice? I've read in some forum answer that "the Borel sigma-algebra on the real numbers $$\mathscr{B}(\mathbb{R})$$ is not a complete lattice" and I was wondering why and hope you can help. Def. a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). Def. The Borel $$\sigma$$-algebra on the reals is the smallest $$\sigma$$-algebra that contains all the open sets One way for the above statement to hold would be if $$[-\infty,\infty] \notin \mathscr{B}(\mathbb{R})$$ -is this the case and why? I would have guessed that $$[-\infty,\infty] \in \mathscr{B}(\mathbb{R})$$; because $$\mathscr{B}(\mathbb{R})$$ is closed under countable union and all open and closed sets exist in it, thus, $$(0,1),[1,2]\in \mathscr{B}(\mathbb{R})\Rightarrow [-\infty,1),[1,\infty]\in \mathscr{B}(\mathbb{R}) \Rightarrow [-\infty,\infty]\in \mathscr{B}(\mathbb{R})$$. PS: I hope the tags are correct. Note that every singleton is a Borel set in the case of $\Bbb R$. So for the Borel sets to form a complete lattice, any collection of singletons must have a join. • There are $\mathfrak{c}$ many Borel sets (does this depend on choice?) and $2^\mathfrak{c}$ many subsets of $\mathbb{R}$ so counting gives us non-Borel sets already. Nov 26 '17 at 12:32 • @Henno: If $\Bbb R$ is a countable union of countable sets, then every set is Borel. Nov 26 '17 at 12:54 • So this could be formulated as: Every singleton set of $\mathbb{R}$ exists in $\mathscr{B}(\mathbb{R})$. In order for $\mathscr{B}(\mathbb{R})$ to form a complete lattice, any collection of singletons must have a join. This is not true for a set containing all but one of the singletons, e.g. $\{\{r\} \mid r \in \mathbb{R}\setminus \{0\}\} \notin \mathscr{B}(\mathbb{R})$. Nov 30 '17 at 13:56	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 7, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8248457908630371, "perplexity": 166.30425417668386}, "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/1642320303385.49/warc/CC-MAIN-20220121131830-20220121161830-00404.warc.gz"}
https://www.mail-archive.com/search?l=lyx-users%40lists.lyx.org&q=date:20100125&o=newest	### How to determine where I am on a page. I have hacked up my own class file. It does 95% of what I set out to achieve. However, I am stuck with the final bit of polish. I have an element that I don't want to appear at the bottom of a page. Therefore, I need to test to find out how far down the page I am, and if more than 75% of the page ### Re: How to determine where I am on a page. Philip Stubbs wrote: I have an element that I don't want to appear at the bottom of a page. Therefore, I need to test to find out how far down the page I am, and if more than 75% of the page is used, insert a \newpage. One possibility: ### Re: How to determine where I am on a page. On 01/25/2010 04:52 AM, Jürgen Spitzmüller wrote: % Conditional pagebreak \def\condbreak#1{% \vskip 0pt plus #1\pagebreak[3]\vskip 0pt plus -#1\relax} As a related question: What does \relax do, and when does one need it? rh ### Re: Change tracking On 01/23/2010 04:59 AM, E. Kaplan wrote: It seems that inserting notes into a Lyx file while tracking changes modifies the color of some parts of the text in the pdf output to blue, misleading the reader to believe that the blue text is new insertions, while it is not. This is true of both ### Re: How to determine where I am on a page. rgheck wrote: As a related question: What does \relax do, and when does one need it? http://en.wikibooks.org/wiki/TeX/relax Note, though, that the \condbreak macro is not my doing. I've picked it up eventually on comp.text.tex. Jürgen ### Re: How to determine where I am on a page. On 01/25/2010 07:59 AM, Jürgen Spitzmüller wrote: rgheck wrote: As a related question: What does \relax do, and when does one need it? http://en.wikibooks.org/wiki/TeX/relax Thanks. I thought it must be something like that. rh ### Don't understand LaTeX Error Hello, I'm working with Lyx 1.6.5 and JabRef 2.5 When I want to view the following LaTeX Source as PDF (pdflatex), I get at the end of the first LaTeX run 4 LaTeX errors: LaTeX Source: \subsubsection{Die Familien Busch vor der Hardt und Flender\label{sub:Die-Familien-Busch}% \footnote{Aus ### Re: Don't understand LaTeX Error On 01/25/2010 10:37 AM, Matthias Schmidt wrote: Hello, I'm working with Lyx 1.6.5 and JabRef 2.5 When I want to view the following LaTeX Source as PDF (pdflatex), I get at the end of the first LaTeX run 4 LaTeX errors: I think the problem is that you have put a footnote inside a ### Re: Clear double page and start at left page Finally managed to solve the problem using the textpos package. In your preamble, put: % absolute to use whole page, showboxes draws line around each box, useful for debugging \usepackage[absolute,showboxes]{textpos} % sets up a grid, useful for debugging \usepackage[texcoord,colorgrid]{eso-pic} ### lyx2lyx failure I am trying to open a lyx file produced by a conversion script and Lyx (1.6.5) fails claiming lyx2lyx could not convert it. The lyx file is in lyx format 2.16 (or so it claims in its header). Launching lyx2lyx from the console gives the following error: Warning: An error ocurred in 225, ### Index of descriptions Dear list, I was wondering if any of you has had to do this before, and has maybe programmed a script or something to automate it: I need to look for all descriptions in a file and insert an index entry with the described item as text. Thanks, Manolo ### photographs: what format to use for b/w printing via lyx? Does anyone have any useful input on what photo format gives the best quality results on b/w printing via lyx? Are there any optimal values, or things to worry about? TIA Richard ### Re: Don't understand LaTeX Error yes, I didn't think about the table of contents, ok. But I would like to understand, what is happening there: I get the LaTeX error only with \citet{GieslerG1990} but not with \cite{GieslerG1990}. Why doesn't Lyx accept this one citation style but the other one is ok? am Montag, 25. Januar 2010 ### RE: photographs: what format to use for b/w printing via lyx? Hi Richard, What kinds of photos are they? For example, are they screenshots that you've converted to black and white? Or are they text on a white background? Or are they grayscale images or are they true black and white (two-tone images)? Are there any pertinent details that you wish to ### Re: Document with PDFs Is this not the right list for this kind of errors? Should I check bugzilla? Can someone send it to the developer list? My wife told me that today she closed all the footnotes and was able to compile the document. I haven't been able to verify this, but I'll try later tonight. Best regards. ### Re: Index of descriptions On 01/25/2010 11:56 AM, Manolo Martínez wrote: Dear list, I was wondering if any of you has had to do this before, and has maybe programmed a script or something to automate it: I need to look for all descriptions in a file and insert an index entry with the described item as text. ### RE: photographs: what format to use for b/w printing via lyx? Thanks for that encouraging reply, Rob. Can I ask a quick supplementary? I have colour jpgs, not particularly high res but neither are they poor. And I'm making a pdf for submission to a printer, to be printed in b/w. The jpgs are atthe moment much bigger than I need: I'll have to scale them to ### Re: Index of descriptions But that does not save me from going description by description adding the glossary entry manually, does it? Manolo rgheck escribió: On 01/25/2010 11:56 AM, Manolo Martínez wrote: Dear list, I was wondering if any of you has had to do this before, and has maybe programmed a script or ### Re: Index of descriptions On 01/25/2010 02:37 PM, Manolo Martínez wrote: But that does not save me from going description by description adding the glossary entry manually, does it? You could redefine the description environment in such a way that it handled this for you, I think. rh Manolo rgheck escribió: On ### RE: photographs: what format to use for b/w printing via lyx? Hi Richard, Can I ask a quick supplementary? Of course. The jpgs are at the moment much bigger than I need: I'll have to scale them to about 40%. Should I process the jpgs into monochrome tiffs? Or would the conversion cost me definition? By all means, process the files. If possible, do ### Instead of A B .. in Appendix Appendix A Appendix B Hi I would like to have, instead of the standard numbering in the Appendix (A, B, ...) a numbering like Appendix A: HERE COMES MY TITLE Appendix A: HERE COMES MY SECOND TITLE . . . How can I achieve this? Thanks, Rainer -- NEW GERMAN FAX NUMBER!!! Rainer M. Krug, PhD (Conservation ### Re: Instead of A B .. in Appendix Appendix A Appendix B On Monday 25 January 2010 15:24:26 Rainer M Krug wrote: Hi I would like to have, instead of the standard numbering in the Appendix (A, B, ...) a numbering like Appendix A: HERE COMES MY TITLE Appendix A: HERE COMES MY SECOND TITLE . . . How can I achieve this? Thanks, Rainer ### Genealogy symbols Hello, I need genealogy symbols in a lyx-document. I installed the genealogy package with the MiKTeX package manager, reconfigured lyx, started lyx again - but I don't find the symbols (insert/special character/symbols). Please, how can I use the genealogy symbols in my document? -- Mit ### RE: photographs: what format to use for b/w printing via lyx? Thanks, Rob. That was very clear and helpful! Richard On Mon, 2010-01-25 at 13:12 -0700, Rob Oakes wrote: Hi Richard, Can I ask a quick supplementary? Of course. The jpgs are at the moment much bigger than I need: I'll have to scale them to about 40%. Should I process the jpgs ### Re: Instead of A B .. in Appendix Appendix A Appendix B On Mon, Jan 25, 2010 at 11:12 PM, Steve Litt [email protected]: On Monday 25 January 2010 15:24:26 Rainer M Krug wrote: Hi I would like to have, instead of the standard numbering in the Appendix (A, B, ...) a numbering like Appendix A: HERE COMES MY TITLE Appendix A: ### Re: Instead of A B .. in Appendix Appendix A Appendix B On Monday 25 January 2010 16:53:30 Rainer M Krug wrote: On Mon, Jan 25, 2010 at 11:12 PM, Steve Litt [email protected]: On Monday 25 January 2010 15:24:26 Rainer M Krug wrote: Hi I would like to have, instead of the standard numbering in the Appendix (A, B, ...) a ### Re: Instead of A B .. in Appendix Appendix A Appendix B On Tue, Jan 26, 2010 at 12:22 AM, Steve Litt [email protected]: On Monday 25 January 2010 16:53:30 Rainer M Krug wrote: On Mon, Jan 25, 2010 at 11:12 PM, Steve Litt [email protected]: On Monday 25 January 2010 15:24:26 Rainer M Krug wrote: Hi I would ### Re: Index of descriptions OK, thanks, I'll try that. M rgheck escribió: On 01/25/2010 02:37 PM, Manolo Martínez wrote: But that does not save me from going description by description adding the glossary entry manually, does it? You could redefine the description environment in such a way that it handled this for ### How to determine where I am on a page. I have hacked up my own class file. It does 95% of what I set out to achieve. However, I am stuck with the final bit of polish. I have an element that I don't want to appear at the bottom of a page. Therefore, I need to test to find out how far down the page I am, and if more than 75% of the page ### Re: How to determine where I am on a page. Philip Stubbs wrote: I have an element that I don't want to appear at the bottom of a page. Therefore, I need to test to find out how far down the page I am, and if more than 75% of the page is used, insert a \newpage. One possibility: ### Re: How to determine where I am on a page. On 01/25/2010 04:52 AM, Jürgen Spitzmüller wrote: % Conditional pagebreak \def\condbreak#1{% \vskip 0pt plus #1\pagebreak[3]\vskip 0pt plus -#1\relax} As a related question: What does \relax do, and when does one need it? rh ### Re: Change tracking On 01/23/2010 04:59 AM, E. Kaplan wrote: It seems that inserting notes into a Lyx file while tracking changes modifies the color of some parts of the text in the pdf output to blue, misleading the reader to believe that the blue text is new insertions, while it is not. This is true of both ### Re: How to determine where I am on a page. rgheck wrote: As a related question: What does \relax do, and when does one need it? http://en.wikibooks.org/wiki/TeX/relax Note, though, that the \condbreak macro is not my doing. I've picked it up eventually on comp.text.tex. Jürgen ### Re: How to determine where I am on a page. On 01/25/2010 07:59 AM, Jürgen Spitzmüller wrote: rgheck wrote: As a related question: What does \relax do, and when does one need it? http://en.wikibooks.org/wiki/TeX/relax Thanks. I thought it must be something like that. rh ### Don't understand LaTeX Error Hello, I'm working with Lyx 1.6.5 and JabRef 2.5 When I want to view the following LaTeX Source as PDF (pdflatex), I get at the end of the first LaTeX run 4 LaTeX errors: LaTeX Source: \subsubsection{Die Familien Busch vor der Hardt und Flender\label{sub:Die-Familien-Busch}% \footnote{Aus ### Re: Don't understand LaTeX Error On 01/25/2010 10:37 AM, Matthias Schmidt wrote: Hello, I'm working with Lyx 1.6.5 and JabRef 2.5 When I want to view the following LaTeX Source as PDF (pdflatex), I get at the end of the first LaTeX run 4 LaTeX errors: I think the problem is that you have put a footnote inside a ### Re: Clear double page and start at left page Finally managed to solve the problem using the textpos package. In your preamble, put: % absolute to use whole page, showboxes draws line around each box, useful for debugging \usepackage[absolute,showboxes]{textpos} % sets up a grid, useful for debugging \usepackage[texcoord,colorgrid]{eso-pic} ### lyx2lyx failure I am trying to open a lyx file produced by a conversion script and Lyx (1.6.5) fails claiming lyx2lyx could not convert it. The lyx file is in lyx format 2.16 (or so it claims in its header). Launching lyx2lyx from the console gives the following error: Warning: An error ocurred in 225, ### Index of descriptions Dear list, I was wondering if any of you has had to do this before, and has maybe programmed a script or something to automate it: I need to look for all descriptions in a file and insert an index entry with the described item as text. Thanks, Manolo ### photographs: what format to use for b/w printing via lyx? Does anyone have any useful input on what photo format gives the best quality results on b/w printing via lyx? Are there any optimal values, or things to worry about? TIA Richard ### Re: Don't understand LaTeX Error yes, I didn't think about the table of contents, ok. But I would like to understand, what is happening there: I get the LaTeX error only with \citet{GieslerG1990} but not with \cite{GieslerG1990}. Why doesn't Lyx accept this one citation style but the other one is ok? am Montag, 25. Januar 2010 ### RE: photographs: what format to use for b/w printing via lyx? Hi Richard, What kinds of photos are they? For example, are they screenshots that you've converted to black and white? Or are they text on a white background? Or are they grayscale images or are they true black and white (two-tone images)? Are there any pertinent details that you wish to ### Re: Document with PDFs Is this not the right list for this kind of errors? Should I check bugzilla? Can someone send it to the developer list? My wife told me that today she closed all the footnotes and was able to compile the document. I haven't been able to verify this, but I'll try later tonight. Best regards. ### Re: Index of descriptions On 01/25/2010 11:56 AM, Manolo Martínez wrote: Dear list, I was wondering if any of you has had to do this before, and has maybe programmed a script or something to automate it: I need to look for all descriptions in a file and insert an index entry with the described item as text. ### RE: photographs: what format to use for b/w printing via lyx? Thanks for that encouraging reply, Rob. Can I ask a quick supplementary? I have colour jpgs, not particularly high res but neither are they poor. And I'm making a pdf for submission to a printer, to be printed in b/w. The jpgs are atthe moment much bigger than I need: I'll have to scale them to ### Re: Index of descriptions But that does not save me from going description by description adding the glossary entry manually, does it? Manolo rgheck escribió: On 01/25/2010 11:56 AM, Manolo Martínez wrote: Dear list, I was wondering if any of you has had to do this before, and has maybe programmed a script or ### Re: Index of descriptions On 01/25/2010 02:37 PM, Manolo Martínez wrote: But that does not save me from going description by description adding the glossary entry manually, does it? You could redefine the description environment in such a way that it handled this for you, I think. rh Manolo rgheck escribió: On ### RE: photographs: what format to use for b/w printing via lyx? Hi Richard, Can I ask a quick supplementary? Of course. The jpgs are at the moment much bigger than I need: I'll have to scale them to about 40%. Should I process the jpgs into monochrome tiffs? Or would the conversion cost me definition? By all means, process the files. If possible, do ### Instead of A B .. in Appendix Appendix A Appendix B Hi I would like to have, instead of the standard numbering in the Appendix (A, B, ...) a numbering like Appendix A: HERE COMES MY TITLE Appendix A: HERE COMES MY SECOND TITLE . . . How can I achieve this? Thanks, Rainer -- NEW GERMAN FAX NUMBER!!! Rainer M. Krug, PhD (Conservation ### Re: Instead of A B .. in Appendix Appendix A Appendix B On Monday 25 January 2010 15:24:26 Rainer M Krug wrote: Hi I would like to have, instead of the standard numbering in the Appendix (A, B, ...) a numbering like Appendix A: HERE COMES MY TITLE Appendix A: HERE COMES MY SECOND TITLE . . . How can I achieve this? Thanks, Rainer ### Genealogy symbols Hello, I need genealogy symbols in a lyx-document. I installed the genealogy package with the MiKTeX package manager, reconfigured lyx, started lyx again - but I don't find the symbols (insert/special character/symbols). Please, how can I use the genealogy symbols in my document? -- Mit ### RE: photographs: what format to use for b/w printing via lyx? Thanks, Rob. That was very clear and helpful! Richard On Mon, 2010-01-25 at 13:12 -0700, Rob Oakes wrote: Hi Richard, Can I ask a quick supplementary? Of course. The jpgs are at the moment much bigger than I need: I'll have to scale them to about 40%. Should I process the jpgs ### Re: Instead of A B .. in Appendix Appendix A Appendix B On Mon, Jan 25, 2010 at 11:12 PM, Steve Litt [email protected]: On Monday 25 January 2010 15:24:26 Rainer M Krug wrote: Hi I would like to have, instead of the standard numbering in the Appendix (A, B, ...) a numbering like Appendix A: HERE COMES MY TITLE Appendix A: ### Re: Instead of A B .. in Appendix Appendix A Appendix B On Monday 25 January 2010 16:53:30 Rainer M Krug wrote: On Mon, Jan 25, 2010 at 11:12 PM, Steve Litt [email protected]: On Monday 25 January 2010 15:24:26 Rainer M Krug wrote: Hi I would like to have, instead of the standard numbering in the Appendix (A, B, ...) a ### Re: Instead of A B .. in Appendix Appendix A Appendix B On Tue, Jan 26, 2010 at 12:22 AM, Steve Litt [email protected]: On Monday 25 January 2010 16:53:30 Rainer M Krug wrote: On Mon, Jan 25, 2010 at 11:12 PM, Steve Litt [email protected]: On Monday 25 January 2010 15:24:26 Rainer M Krug wrote: Hi I would ### Re: Index of descriptions OK, thanks, I'll try that. M rgheck escribió: On 01/25/2010 02:37 PM, Manolo Martínez wrote: But that does not save me from going description by description adding the glossary entry manually, does it? You could redefine the description environment in such a way that it handled this for ### How to determine where I am on a page. I have hacked up my own class file. It does 95% of what I set out to achieve. However, I am stuck with the final bit of polish. I have an element that I don't want to appear at the bottom of a page. Therefore, I need to test to find out how far down the page I am, and if more than 75% of the page ### Re: How to determine where I am on a page. Philip Stubbs wrote: > I have an element that I don't want to appear at the bottom of a page. > Therefore, I need to test to find out how far down the page I am, and > if more than 75% of the page is used, insert a \newpage. One possibility: ### Re: How to determine where I am on a page. On 01/25/2010 04:52 AM, Jürgen Spitzmüller wrote: % Conditional pagebreak \def\condbreak#1{% \vskip 0pt plus #1\pagebreak[3]\vskip 0pt plus -#1\relax} As a related question: What does \relax do, and when does one need it? rh ### Re: Change tracking On 01/23/2010 04:59 AM, E. Kaplan wrote: It seems that inserting notes into a Lyx file while tracking changes modifies the color of some parts of the text in the pdf output to blue, misleading the reader to believe that the blue text is new insertions, while it is not. This is true of both ### Re: How to determine where I am on a page. rgheck wrote: > As a related question: What does \relax do, and when does one need it? http://en.wikibooks.org/wiki/TeX/relax Note, though, that the \condbreak macro is not my doing. I've picked it up eventually on comp.text.tex. Jürgen ### Re: How to determine where I am on a page. On 01/25/2010 07:59 AM, Jürgen Spitzmüller wrote: rgheck wrote: As a related question: What does \relax do, and when does one need it? http://en.wikibooks.org/wiki/TeX/relax Thanks. I thought it must be something like that. rh ### Don't understand LaTeX Error Hello, I'm working with Lyx 1.6.5 and JabRef 2.5 When I want to view the following LaTeX Source as PDF (pdflatex), I get at the end of the first LaTeX run 4 LaTeX errors: LaTeX Source: \subsubsection{Die Familien Busch vor der Hardt und Flender\label{sub:Die-Familien-Busch}% \footnote{Aus ### Re: Don't understand LaTeX Error On 01/25/2010 10:37 AM, Matthias Schmidt wrote: Hello, I'm working with Lyx 1.6.5 and JabRef 2.5 When I want to view the following LaTeX Source as PDF (pdflatex), I get at the end of the first LaTeX run 4 LaTeX errors: I think the problem is that you have put a footnote inside a ### Re: Clear double page and start at left page Finally managed to solve the problem using the textpos package. In your preamble, put: % absolute to use whole page, showboxes draws line around each box, useful for debugging \usepackage[absolute,showboxes]{textpos} % sets up a grid, useful for debugging \usepackage[texcoord,colorgrid]{eso-pic} ### lyx2lyx failure I am trying to open a lyx file produced by a conversion script and Lyx (1.6.5) fails claiming lyx2lyx could not convert it. The lyx file is in lyx format 2.16 (or so it claims in its header). Launching lyx2lyx from the console gives the following error: Warning: An error ocurred in 225, ### Index of descriptions Dear list, I was wondering if any of you has had to do this before, and has maybe programmed a script or something to automate it: I need to look for all descriptions in a file and insert an index entry with the described item as text. Thanks, Manolo ### photographs: what format to use for b/w printing via lyx? Does anyone have any useful input on what photo format gives the best quality results on b/w printing via lyx? Are there any optimal values, or things to worry about? TIA Richard ### Re: Don't understand LaTeX Error yes, I didn't think about the table of contents, ok. But I would like to understand, what is happening there: I get the LaTeX error only with "\citet{GieslerG1990}" but not with "\cite{GieslerG1990}". Why doesn't Lyx accept this one citation style but the other one is ok? am Montag, 25. Januar ### RE: photographs: what format to use for b/w printing via lyx? Hi Richard, What kinds of photos are they? For example, are they screenshots that you've converted to black and white? Or are they text on a white background? Or are they grayscale images or are they true black and white (two-tone images)? Are there any pertinent details that you wish to ### Re: Document with PDFs Is this not the right list for this kind of errors? Should I check bugzilla? Can someone send it to the developer list? My wife told me that today she closed all the footnotes and was able to compile the document. I haven't been able to verify this, but I'll try later tonight. Best regards. ### Re: Index of descriptions On 01/25/2010 11:56 AM, Manolo Martínez wrote: Dear list, I was wondering if any of you has had to do this before, and has maybe programmed a script or something to automate it: I need to look for all descriptions in a file and insert an index entry with the described item as text. ### RE: photographs: what format to use for b/w printing via lyx? Thanks for that encouraging reply, Rob. Can I ask a quick supplementary? I have colour jpgs, not particularly high res but neither are they poor. And I'm making a pdf for submission to a printer, to be printed in b/w. The jpgs are atthe moment much bigger than I need: I'll have to scale them to ### Re: Index of descriptions But that does not save me from going description by description adding the glossary entry manually, does it? Manolo rgheck escribió: On 01/25/2010 11:56 AM, Manolo Martínez wrote: Dear list, I was wondering if any of you has had to do this before, and has maybe programmed a script or ### Re: Index of descriptions On 01/25/2010 02:37 PM, Manolo Martínez wrote: But that does not save me from going description by description adding the glossary entry manually, does it? You could redefine the description environment in such a way that it handled this for you, I think. rh Manolo rgheck escribió: On ### RE: photographs: what format to use for b/w printing via lyx? Hi Richard, << Can I ask a quick supplementary? >> Of course. << The jpgs are at the moment much bigger than I need: I'll have to scale them to about 40%. Should I process the jpgs into monochrome tiffs? Or would the conversion cost me definition? >> By all means, process the files. If ### Instead of "A" "B" .. in Appendix "Appendix A" "Appendix B" Hi I would like to have, instead of the standard numbering in the Appendix ("A", "B", ...) a numbering like Appendix A: HERE COMES MY TITLE Appendix A: HERE COMES MY SECOND TITLE . . . How can I achieve this? Thanks, Rainer -- NEW GERMAN FAX NUMBER!!! Rainer M. Krug, PhD (Conservation ### Re: Instead of "A" "B" .. in Appendix "Appendix A" "Appendix B" On Monday 25 January 2010 15:24:26 Rainer M Krug wrote: > Hi > > I would like to have, instead of the standard numbering in the Appendix > ("A", "B", ...) a numbering like > > Appendix A: HERE COMES MY TITLE > Appendix A: HERE COMES MY SECOND TITLE > . > . > . > > How can I achieve this? > > ### Genealogy symbols Hello, I need genealogy symbols in a lyx-document. I installed the genealogy package with the MiKTeX package manager, reconfigured lyx, started lyx again - but I don't find the symbols (insert/special character/symbols). Please, how can I use the genealogy symbols in my document? -- Mit ### RE: photographs: what format to use for b/w printing via lyx? Thanks, Rob. That was very clear and helpful! Richard On Mon, 2010-01-25 at 13:12 -0700, Rob Oakes wrote: > Hi Richard, > > << Can I ask a quick supplementary? >> > > Of course. > > << The jpgs are at the moment much bigger than I need: I'll have to scale > them to about 40%. Should I ### Re: Instead of "A" "B" .. in Appendix "Appendix A" "Appendix B" On Mon, Jan 25, 2010 at 11:12 PM, Steve Litt wrote: > On Monday 25 January 2010 15:24:26 Rainer M Krug wrote: > > Hi > > > > I would like to have, instead of the standard numbering in the Appendix > > ("A", "B", ...) a numbering like > > > > Appendix A: HERE COMES MY ### Re: Instead of "A" "B" .. in Appendix "Appendix A" "Appendix B" On Monday 25 January 2010 16:53:30 Rainer M Krug wrote: > On Mon, Jan 25, 2010 at 11:12 PM, Steve Litt wrote: > > On Monday 25 January 2010 15:24:26 Rainer M Krug wrote: > > > Hi > > > > > > I would like to have, instead of the standard numbering in the Appendix > > > ### Re: Instead of "A" "B" .. in Appendix "Appendix A" "Appendix B" On Tue, Jan 26, 2010 at 12:22 AM, Steve Litt wrote: > On Monday 25 January 2010 16:53:30 Rainer M Krug wrote: > > On Mon, Jan 25, 2010 at 11:12 PM, Steve Litt > wrote: > > > On Monday 25 January 2010 15:24:26 Rainer M Krug wrote: > > > > Hi > ### Re: Index of descriptions OK, thanks, I'll try that. M rgheck escribió: On 01/25/2010 02:37 PM, Manolo Martínez wrote: But that does not save me from going description by description adding the glossary entry manually, does it? You could redefine the description environment in such a way that it handled this for	{"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": 0.8430993556976318, "perplexity": 2802.0049184879645}, "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-2021-25/segments/1623488539480.67/warc/CC-MAIN-20210623134306-20210623164306-00465.warc.gz"}
http://prof.clontz.org/classes/2017/06/ma227/standards/s08/	# MA 227 Standard S08 Transformations of Variables At the end of the course, each student should be able to… • S08: TransVar. Compute and apply a transformation of variables. ## S08: Transformations of Variables • A transformation of variables is a function $$\vect T(u,v)=\<x,y\>$$ that converts vectors $$\<u,v\>$$ in the $$uv$$ plane into vectors $$\<x,y\>$$ in the $$xy$$ plane. • A transformation is called affine if it preserves parallelograms. • All affine transformations are of the form $$\vect T(u,v)=\<a_1u+b_1v+c_1,a_2u+b_2v+c_2\>$$. • To find a transformation from the unit square in the $$uv$$ plane to a parallelogram, the values of $$a,b,c$$ may be calculated by setting each of $$\vect T(0,0),\vect T(1,0), \vect T(1,1), \vect T(0,1)$$ to each of its corners. • To find a transformation from the unit triangle in the $$uv$$ plane to a parallelogram, the values of $$a,b,c$$ may be calculated by setting each of $$\vect T(0,0),\vect T(1,0), \vect T(1,1)$$ to each of its corners. • Affine transformations scale areas by a factor of $$\detTwo{a_1}{b_1}{a_2}{b_2}$$ (where this value is negative when the transformation reflects orientation). This generalizes to the Jacobian $$\frac{\partial\vect T}{\partial\<u,v\>}=\detTwo {\frac{\partial x}{\partial u}}{\frac{\partial x}{\partial v}} {\frac{\partial y}{\partial u}}{\frac{\partial y}{\partial v}}$$ for more arbitrary transformations. • It follows that if the transformation $$\vect T(u,v)$$ transforms the region $$G$$ in the $$uv$$ plane into the region $$R$$ in the $$xy$$ plane, then $$\iint_R f(x,y)\,dA=\iint_G f(\vect T(u,v))\|\frac{\partial\vect T}{\partial\<u,v\>}\|\,dA$$. • Similarly for 3D transformations $$\vect T(u,v,w)=\<x,y,z\>$$ sending $$H$$ to $$D$$, it may be shown that $$\iiint_D f(x,y,z)\,dV=\iiint_H f(\vect T(u,v,w))\|\frac{\partial\vect T}{\partial\<u,v,w\>}\|\,dV$$ where $$\frac{\partial\vect T}{\partial\<u,v,w\>}=\detThree {\frac{\partial x}{\partial u}}{\frac{\partial x}{\partial v}} {\frac{\partial x}{\partial w}} {\frac{\partial y}{\partial u}}{\frac{\partial y}{\partial v}} {\frac{\partial y}{\partial w}} {\frac{\partial z}{\partial u}}{\frac{\partial z}{\partial v}} {\frac{\partial z}{\partial w}}$$. ### Textbook References • University Calculus: Early Transcendentals (3rd Ed) • 14.8 ### Practice Exercises Let the unit square have vertices $$\<0,0\>$$, $$\<1,0\>$$, $$\<1,1\>$$, and $$\<0,1\>$$. Let the unit triangle have vertices $$\<0,0\>$$, $$\<1,0\>$$, and $$\<1,1\>$$. 1. Find a transformation from either the unit square or unit triangle in the $$uv$$ plane into the given region $$R$$ in the $$xy$$ plane. • $$R$$: the parallelogram bounded by $$y=3x+1$$, $$y=3x-3$$, $$y=x-3$$ $$y=x+1$$ • $$R$$: the triangle bounded by $$y=x$$, $$y=2x$$, $$y=6-x$$ • $$R$$: the square with vertices $$\<2,1\>$$, $$\<-2,3\>$$, $$\<0,7\>$$, $$\<4,5\>$$ • $$R$$: the triangle with vertices $$\<0,-2\>$$ $$\<-1,1\>$$, $$\<1,3\>$$ 2. Evaluate the double integral with variables $$x,y$$ using the given transformation from the $$uv$$ plane. • $$\iint_R (2y-4x)\,dA$$, $$\vect{T}(u,v)=\<u+v,2u-v+3\>$$ from the unit square into the parallelogram $$R$$ with vertices $$\<0,3\>$$, $$\<1,5\>$$, $$\<2,4\>$$, $$\<1,2\>$$. • $$\iint_R (x+y)(x-y-2)\,dA$$, $$\vect{T}(u,v)=\<4-u-v,v-u+2\>$$ from unit triangle into the triangle $$R$$ with vertices $$\<4,2\>$$, $$\<3,1\>$$, $$\<2,2\>$$. • $$\iint_R (x+y)e^{x^2-y^2}\,dA$$, $$\vect{T}(u,v)=\<u+2v,u-2v\>$$ from unit square into the rectangle $$R$$ bounded by $$y=x$$, $$y=x-4$$, $$y=-x$$, $$y=2-x$$. • $$\iint_R \cos(e^x)\,dA$$, $$\vect{T}(u,v)=\<\ln (u+v+1),v\>$$ from unit triangle into the region $$R$$ bounded by $$y=0$$, $$y=e^x-2$$, $$y=\frac{e^x-1}{2}$$. ### Solutions 1. Find the transformation (solutions are not unique): • $$\vect T(u,v)=\<2u+2v-2,2u+6v-5\>$$ • $$\vect T(u,v)=\<3u-v,3u+v\>$$ • $$\vect T(u,v)=\<4u+2v-2,-2u+4v+3\>$$ • $$\vect T(u,v)=\<u+v-1,-3u+5v+1\>$$ 2. Evaluate the integral: • $$9$$ • $$\frac{3}{2}$$ • $$\frac{1}{8}(e^8-9)$$ • $$-\frac{1}{2}\cos(3)+\cos(2)-\frac{1}{2}\cos(1)$$	{"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": 0.9883013367652893, "perplexity": 338.13172973783577}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891813622.87/warc/CC-MAIN-20180221123439-20180221143439-00679.warc.gz"}
https://planetmath.org/SolutionsOf1xx2x3y2	# solutions of $1+x+x^{2}+x^{3}=y^{2}$ This article shows that the only solutions in integers to the equation $1+x+x^{2}+x^{3}=y^{2}$ are the obvious trivial solutions $x=0,\pm 1$ together with $x=7,y=20$. This result was known to Fermat. First, note that the equation is $(1+x)(1+x^{2})=y^{2}$, so we have immediately that $x\geq-1$. So, noting the solutions for $x=0,\pm 1$, assume in what follows that $x>1$. Let $d=\gcd(1+x,1+x^{2})$. Then $x\equiv-1\pmod{d}$, so that $1+x^{2}\equiv 2\pmod{d}$. But $d\mid 1+x^{2}$ so that $2\equiv 0\pmod{d}$ and $d$ is either $1$ or $2$. If $d=1$, so that $1+x$ and $1+x^{2}$ are coprime, then $1+x$ and $1+x^{2}$ must both be squares. But $1+x^{2}$ is not a square for $x>1$. Thus the $\gcd$ of $1+x$ and $1+x^{2}$ is $2$, so each must be twice a square, say $\displaystyle 1+x=2r^{2}$ $\displaystyle 1+x^{2}=2s^{2}$ and then $(r^{2})^{2}+(r^{2}-1)^{2}=\left(\frac{1+x}{2}\right)^{2}+\left(\frac{-1+x}{2}% \right)^{2}=\frac{1+x^{2}}{2}=s^{2}$ so that $r^{2},r^{2}-1,s$ form a primitive Pythagorean triple (note that $r>1$ since $x>1$). Recall that if $(a,b,c)$ is a primitive Pythagorean triple, then precisely one of $a$ and $b$ is even, and we can choose coprime integers $p,q$ such that $a=p^{2}-q^{2},b=2pq,c=p^{2}+q^{2}$ or $a=2pq,b=p^{2}-q^{2},c=p^{2}+q^{2}$ depending on the parity of $a$. Suppose first that $r$ is odd. Then $\displaystyle r^{2}=p^{2}-q^{2}$ $\displaystyle r^{2}-1=2pq$ $\displaystyle s=p^{2}+q^{2}$ Then $1=r^{2}-(r^{2}-1)=(p-q)^{2}-2q^{2}$. Now, note that $p-q$ must be a square, say $p-q=t^{2}$, since $\gcd(p-q,p+q)=1$ and $(p-q)(p+q)$ is a square. Then $t^{4}=(p-q)^{2}=2q^{2}+1=(q^{2}+1)^{2}-q^{4}$ so that $t^{4}+q^{4}=(q^{2}+1)^{2}$ But we know that the sum of two fourth powers can be a square (http://planetmath.org/ExampleOfFermatsLastTheorem) only for the trivial case where all are zero. So $r$ cannot be odd. So suppose that $r$ is even. Then $\displaystyle r^{2}=2pq$ $\displaystyle r^{2}-1=p^{2}-q^{2}$ $\displaystyle s=p^{2}+q^{2}$ From the second of these formulas, we see that $p$ must be even (consider both sides $\pmod{4}$), say $p=2t^{2}$. Now, $\displaystyle(p+q-1)(p+q+1)=(p+q)^{2}-1=p^{2}+2pq+q^{2}-1=\\ \displaystyle p^{2}+2pq+q^{2}-(r^{2}-(r^{2}-1))=p^{2}+2pq+q^{2}-2pq+p^{2}-q^{2% }=2p^{2}=8t^{4}$ Since $p$ and $q$ have opposite parity, $p+q\pm 1$ are even, so that $2t^{4}=\frac{p+q-1}{2}\cdot\left(\frac{p+q-1}{2}+1\right)=u(u+1)$ Thus one of $u,u+1$ is a fourth power and the other is twice a fourth power, say $b^{4}$ and $2c^{4}$. Now, $\displaystyle u=b^{4}\Rightarrow u+1=2c^{4}\Rightarrow b^{4}-2c^{4}=-1$ $\displaystyle u=2c^{4}\Rightarrow u+1=b^{4}\Rightarrow b^{4}-2c^{4}=1$ so that $b^{4}-2c^{4}=\pm 1$. If $b^{4}-2c^{4}=1$, then $((c^{2})^{2}+1)^{2}=(c^{2})^{4}+b^{4}$ and again we have a square being the sum of two fourth powers. So this case is impossible. If $b^{4}-2c^{4}=-1$, write $e=c^{2}$, then $(e^{2}-1)^{2}=e^{4}-b^{4}$. It follows (see here (http://planetmath.org/X4Y4z2HasNoSolutionsInPositiveIntegers)) that either $b=0$ or $e^{2}-1=0$. If $b=0$, we get the impossibility $e^{4}-2e^{2}+1=e^{4}$, while if $(e^{2}-1)^{2}=0$, then $e^{4}=b^{4},e=\pm 1$ and so $b=\pm 1$. Then $\displaystyle\frac{p+q-1}{2}=b^{4}=1$, so that $p+q=3$. Thus $r^{2}=4$, so $r=2$ and, finally, $1+x=2r^{2}=8$, so that $x=7$. Thus, the only nontrivial solution to the equation given is $1+7+49+343=400$ Title solutions of $1+x+x^{2}+x^{3}=y^{2}$ SolutionsOf1xx2x3y2 2013-03-22 17:05:09 2013-03-22 17:05:09 rm50 (10146) rm50 (10146) 10 rm50 (10146) Theorem msc 11F80 msc 14H52 msc 11D41	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 90, "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": 0.998707115650177, "perplexity": 84.49028968761993}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610704835583.91/warc/CC-MAIN-20210128005448-20210128035448-00637.warc.gz"}
https://rupress.org/jcb/article-standard/111/1/79/13902/Anchorage-of-secretion-competent-dense-granules-on	The ultrastructural changes in electropermeabilized bovine platelets that accompany the Ca2(+)-induced secretion of serotonin were investigated in ultra-thin sections of chemically fixed cells. Such preparations permitted us to study both the localization of and the structures associated with serotonin-containing dense granules. Localization of dense granules within cells was examined by measuring the shortest distances between the granular membranes and the plasma membrane. About 40% of total granules were located close to the plasma membrane at an average distance of 10.8 +/- 1.6 nm. 71% of the total number of granules were localized at a similar average distance of 12.5 +/- 2.7 nm in intact platelets. The percentage of granules apposed to the plasma membrane corresponded closely to the percentage of total serotonin that was maximally secreted after stimulation of the permeabilized (38 +/- 4.9%) and the intact platelets (72 +/- 3.6%). Furthermore, the percentage of granules anchored to the membrane, but not of those in other regions of permeabilized cells, decreased markedly when cells were stimulated for 30 s by extracellularly added Ca2+. The decrease in the numbers of granules in the vicinity of the plasma membrane corresponded to approximately 22% of the total number of dense granules that were used for measurements of the distances between the two membranes and corresponded roughly to the overall decrease (15%) in the average number of the granules per cell. Most dense granules were found to be associated with meshwork structures of microfilaments. Upon secretory stimulation, nonfilamentous, amorphous structures found between the plasma membrane and the apposed granules formed a bridge-like structure that connected both membranes without any obvious accompanying changes in the microfilament structures. These results suggest that the dense granules that are susceptible to secretory stimulation are anchored to the plasma membrane before stimulation, and that the formation of the bridge-like structure may participate in the Ca2(+)-regulated exocytosis. This content is only available as a PDF.	{"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": 0.9144551753997803, "perplexity": 2958.2119953226984}, "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/1674764501066.53/warc/CC-MAIN-20230209014102-20230209044102-00804.warc.gz"}
https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=148&t=29178&p=90049	## 15.37C $\frac{d[R]}{dt}=-k[R]; \ln [R]=-kt + \ln [R]_{0}; t_{\frac{1}{2}}=\frac{0.693}{k}$ Wilson Yeh 1L Posts: 42 Joined: Fri Sep 29, 2017 7:06 am ### 15.37C Sulfuryl chloride, SO2Cl2, decomposes by first-order kinetics, and k = 2.81 x 10-3 min-1 at a certain temperature. (a) Determine the half-life for the reaction. (b) Determine the time needed for the concentration of SO2Cl2 to decrease to 10% of its initial concentration. (c) If 14.0 g of SO2Cl2 is sealed in a 2500.-L reaction vessel and heated to the specifi ed temperature, what mass will remain after 1.5 h? For part C, is the 2500.-L irrelevant? I managed to solve for the final answer without ever taking the volume into account. Am I doing something wrong? Janine Chan 2K Posts: 71 Joined: Fri Sep 29, 2017 7:04 am ### Re: 15.37C Nope you're good, if you divide both the initial and final mass by liters to find the concentrations of each, the liters will cancel out when you plug into [A]t=[A]0e-ktanyway. Sue Xu 2K Posts: 58 Joined: Fri Sep 29, 2017 7:06 am ### Re: 15.37C It is possible to solve the question without the volume, but you can still calculate the concentration with the mass and the volume first and then calculate the final mass. Anna Okabe Posts: 30 Joined: Fri Sep 29, 2017 7:06 am ### Re: 15.37C I think it would be safer to use the volume and use the molarity every time, so that you can avoid making mistakes in the future. Emma Miltenberger 2I Posts: 51 Joined: Thu Jul 27, 2017 3:00 am ### Re: 15.37C In this case, the volume cancels out. However, it is good practice to use the volume to solve for molarity and use molarity in the equation to ensure you do not make an arithmetic mistake on a test. Michelle Lu 1F Posts: 50 Joined: Thu Jul 27, 2017 3:01 am Been upvoted: 1 time ### Re: 15.37C In this particular situation, the volumes end up cancelling out, so if you were able to recognize this was going to occur, then you could have left it out of the picture. However, common safe practice involves solving for the concentration using the mass given and the volume 2500. L, and this could lead to less errors.	{"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": 1, "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": 0.9013078808784485, "perplexity": 1802.3237546822281}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371637684.76/warc/CC-MAIN-20200406133533-20200406164033-00284.warc.gz"}
https://robotics.stackexchange.com/questions/21523/how-to-obtain-the-state-space-representation-of-a-computed-torque-control-system	# How to obtain the state space representation of a computed torque control system? Assuming I have a system: $$a \, \ddot{x}_1 + b \, \ddot{x}_2 + c \,\dot{x_1} + d \,\dot{x_2} + e = u_1 \\ f \, \ddot{x}_1 + g \, \ddot{x}_2 + h \,\dot{x_1} + i \,\dot{x_2} + j = u_2$$ How would I write this system in state space representation: $$\dot{x} = Ax+Bu$$ $$y = Cx + Du$$ Usually, I would isolate for $$\ddot{x_1}$$ or $$\ddot{x_2}$$, but in this case, they are functions of each other. Would the only way be to rewrite the dynamics in matrix form after setting new state variables? • The fact they are coupled makes no difference. You can use your standard methodology. :), set one for x2 and x1 and then solve. Dec 20, 2020 at 9:07 First, isolate the second order from the other terms: $$a \, \ddot{x}_1 + b \, \ddot{x}_2 =- c \,\dot{x_1} - d \,\dot{x_2} - e + u_1 \\ f \, \ddot{x}_1 + g \, \ddot{x}_2 = - h \,\dot{x_1} - i \,\dot{x_2} - j + u_2$$ Then, put it in a matrix form: $$\left[\begin{matrix}a&b\\f&g\end{matrix}\right]\left[\begin{matrix}\ddot{x_1}\\\ddot{x_2}\end{matrix}\right]=\left[\begin{matrix}- c \,\dot{x_1} - d \,\dot{x_2} - e + u_1\\- h \,\dot{x_1} - i \,\dot{x_2} - j + u_2\end{matrix}\right].$$ Supposing $$\left[\begin{smallmatrix}a&b\\f&g\end{smallmatrix}\right]$$ is invertible, we can left multiply both sides by $$\left[\begin{smallmatrix}a&b\\f&g\end{smallmatrix}\right]^{-1}$$: $$\left[\begin{matrix}a&b\\f&g\end{matrix}\right]^{-1}\left[\begin{matrix}a&b\\f&g\end{matrix}\right]\left[\begin{matrix}\ddot{x_1}\\\ddot{x_2}\end{matrix}\right]=\left[\begin{matrix}a&b\\f&g\end{matrix}\right]^{-1}\left[\begin{matrix}- c \,\dot{x_1} - d \,\dot{x_2} - e + u_1\\- h \,\dot{x_1} - i \,\dot{x_2} - j + u_2\end{matrix}\right].$$ Thus, we get: $$\left[\begin{matrix}\ddot{x_1}\\\ddot{x_2}\end{matrix}\right] = {1\over ag-bf} \left[\begin{matrix}g&-b\\-f&a\end{matrix}\right]\left[\begin{matrix}- c \,\dot{x_1} - d \,\dot{x_2} - e + u_1\\- h \,\dot{x_1} - i \,\dot{x_2} - j + u_2\end{matrix}\right],$$ $$\left[\begin{matrix}\ddot{x_1}\\\ddot{x_2}\end{matrix}\right] = {1\over ag-bf} \left[\begin{matrix}(bh-gc)\dot{x_1}+(bi-gd)\dot{x_2}+(bj-je)\\ (fc-ah)\dot{x_1}+(fd-ai)\dot{x_2} + (fe-aj)\end{matrix}\right] +{1\over ag-bf} \left[\begin{matrix} g&-b\\ -f&a \end{matrix}\right] \left[\begin{matrix} u_1\\ u_2 \end{matrix}\right].$$ Using auxiliary variables: $$\left[\begin{matrix}\ddot{x_1}\\\ddot{x_2}\end{matrix}\right] = \left[\begin{matrix}\alpha\dot{x_1}+\beta\dot{x_2}+\gamma\\ \delta\dot{x_1}+\epsilon\dot{x_2} + \zeta\end{matrix}\right] + \left[\begin{matrix} \eta&\theta\\ \iota&\kappa \end{matrix}\right] \left[\begin{matrix} u_1\\ u_2 \end{matrix}\right].$$ Here, $$\gamma$$ and $$\zeta$$ can represent either uncontrolled inputs or disturbances. As uncontrolled inputs: $$\left[\begin{matrix}\ddot{x_1}\\\ddot{x_2}\end{matrix}\right] = \left[\begin{matrix}\alpha\dot{x_1}+\beta\dot{x_2}\\ \delta\dot{x_1}+\epsilon\dot{x_2}\end{matrix}\right] + \left[\begin{matrix} \eta&\theta&\gamma\\ \iota&\kappa&\zeta \end{matrix}\right] \left[\begin{matrix} u_1\\ u_2\\ 1 \end{matrix}\right].$$ As disturbances: $$\left[\begin{matrix}\ddot{x_1}\\\ddot{x_2}\end{matrix}\right] = \left[\begin{matrix}\alpha\dot{x_1}+\beta\dot{x_2}\\ \delta\dot{x_1}+\epsilon\dot{x_2}\end{matrix}\right] + \left[\begin{matrix} \eta&\theta\\ \iota&\kappa \end{matrix}\right] \left[\begin{matrix} u_1\\ u_2 \end{matrix}\right]+ \left[\begin{matrix} \gamma\\ \zeta \end{matrix}\right].$$ Creating the state $$s=[s_1\;s_2\;s_3\;s_4]^T=[x_1\;\dot{x_1}\;x_2\;\dot{x_2}]^T$$: $$\dot{s} = \dot{\left[\begin{matrix}s_1\\ s_2\\ s_3\\ s_4\end{matrix}\right]} = \left[\begin{matrix} 0&1&0&0\\ 0&\alpha&0&\beta\\ 0&0&0&1\\ 0&\delta&0&\epsilon \end{matrix}\right] \left[\begin{matrix}s_1\\ s_2\\ s_3\\ s_4\end{matrix}\right] + \left[\begin{matrix} 0&0\\ \eta&\theta\\ 0&0\\ \iota&\kappa \end{matrix}\right] \left[\begin{matrix} u_1\\ u_2 \end{matrix}\right] + \left[\begin{matrix} 0\\ \gamma\\ 0\\ \zeta \end{matrix}\right].$$ Depending on your system, you can design a controller that rejects those disturbances; the presence of integrators in the system can help remove steady-state errors. • Thank you. Is this generally the way computed torque control is done? I am wondering if it would be possible to transform this into a second order state space by representing x1 and x2 in a single vector q = [x1;x2]? Dec 20, 2020 at 17:23 • In youtube.com/watch?v=MV-xBPP3H2k, you can see that a combination of feedfoward and PID can be used to control using the torque as input. Dec 21, 2020 at 16:30 \begin{align} a \, \ddot{x}_1 + b \, \ddot{x}_2 + c \,\dot{x_1} + d \,\dot{x_2} + e &= u_1 \tag{1} \\ f \, \ddot{x}_1 + g \, \ddot{x}_2 + h \,\dot{x_1} + i \,\dot{x_2} + j &= u_2 \tag{2} \end{align} Let's write Eq(1) without $$\ddot{x}_2$$, hence: \begin{align} a \, \ddot{x}_1 + b \, \Big[\frac{u_2-f \, \ddot{x}_1 - h \,\dot{x}_1 - i \,\dot{x}_2 - j}{g}\Big] + c \,\dot{x}_1 + d \,\dot{x}_2 + e &= u_1 \\ \left[a-\frac{bf}{g} \right] \, \ddot{x}_1 + \, \frac{b}{g}u_2 - \frac{bh}{g} \,\dot{x}_1 - \frac{bi}{g} \,\dot{x}_2 - \frac{b}{g}j + c \,\dot{x}_1 + d \,\dot{x}_2 + e &= u_1 \\ % \left[a-\frac{bf}{g} \right] \, \ddot{x}_1 = u_1 - \, \frac{b}{g}u_2 + \frac{bh}{g} \,\dot{x}_1 + \frac{bi}{g} \,\dot{x}_2 + \frac{b}{g}j - c \,\dot{x}_1 - d \,\dot{x}_2 - &e \\ % \ddot{x}_1 = \left[\frac{g}{(ag-bf)} \right] \left[u_1 - \, \frac{b}{g}u_2 + \frac{bh}{g} \,\dot{x}_1 + \frac{bi}{g} \,\dot{x}_2 + \frac{b}{g}j - c \,\dot{x}_1 - d \,\dot{x}_2 - e\right] \tag{3} \end{align} Now we do same procedure which is eliminating $$\ddot{x}_1$$ from Eq(2) hence, \begin{align} f \, \Big[\frac{u_1- b \, \ddot{x}_2 - c \,\dot{x_1} - d \,\dot{x_2} - e}{a}\Big] + g \, \ddot{x}_2 + h \,\dot{x_1} + i \,\dot{x_2} + j &= u_2 \\ % \Big[ g - \frac{fb}{a}\Big] \ddot{x}_2 + \frac{f}{a} u_1 - \frac{fc}{a} \,\dot{x_1} - \frac{fd}{a} \,\dot{x_2} - \frac{fe}{a} + h \,\dot{x_1} + i \,\dot{x_2} + j &= u_2 \\ % \Big[ g - \frac{fb}{a}\Big] \ddot{x}_2 = u_2 - \frac{f}{a} u_1 + \frac{fc}{a} \,\dot{x_1} + \frac{fd}{a} \,\dot{x_2} + \frac{fe}{a} - h \,\dot{x_1} - i \,\dot{x_2} - j \\ % \ddot{x}_2 = \left[\frac{a}{(ag-bf)} \right] \left[ u_2 - \frac{f}{a} u_1 + \frac{fc}{a} \,\dot{x_1} + \frac{fd}{a} \,\dot{x_2} + \frac{fe}{a} - h \,\dot{x_1} - i \,\dot{x_2} - j \right] \tag{4} \end{align} Now let $$y_1=x_1, y_2=\dot{x}_1, y_3=x_2, y_4=\dot{x}_2$$, hence: \begin{align} \dot{y}_1 &= y_2 \\ \dot{y}_2 &= \left[\frac{g}{(ag-bf)} \right] \left[u_1 - \, \frac{b}{g}u_2 + \Big[\frac{bh-cg}{g}\Big] \,y_2 + \Big[\frac{bi-dg}{g} \Big] \,y_4 + \frac{bj-eg}{g}\right] \\ \dot{y}_3 &= y_4 \\ \dot{y}_4 &= \left[\frac{a}{(ag-bf)} \right] \left[ u_2 - \frac{f}{a} u_1 + \Big[\frac{fc-ha}{a} \,\Big] y_2 + \Big[\frac{fd-ia}{a}\Big] \,y_4 + \frac{fe-aj}{a} \right] \end{align} where $$ag\neq bf$$. You can proceed from here. Before the question can be answered with mathematical terms, a bit meta knowledge may help to understand the thought system. The goal of creating a state space formula for robotics problems is a typical application of the matlab software. The students are educated how to use a commercial software package for describing real world problems. With this pre-knowledge in mind it is much easier to identify the correct answer. It was given in the internet already under the URL https://in.mathworks.com/help/control/getstart/linear-lti-models.html The website explains, how to use the famous mathematical software for creating a state space model for a torgue control. What is provided too, is how to create the differential equations and transform them into the matrix notation. • Students are educated on such easy equations to write them out by hand and learn the method, then use software. I don’t know of any good university that would teach students to write a state space model in Matlab, this isn’t a helpful answer on the method on how to write in state space methodology, coming from a poor basis to suggest that universities teach only to use commercial software, particularly for such a simple example. Dec 20, 2020 at 9:04	{"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": 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": 27, "wp-katex-eq": 0, "align": 4, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.999848484992981, "perplexity": 2132.4689751337414}, "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-2023-14/segments/1679296948932.75/warc/CC-MAIN-20230329023546-20230329053546-00048.warc.gz"}
https://www.physicsforums.com/threads/the-set-of-ring-automorphisms-is-an-abstract-group-under-composition.162011/	# Homework Help: The set of ring automorphisms is an abstract group under composition 1. Mar 22, 2007 ### catcherintherye 1. The problem statement, all variables and given/known data Aut(R) denotes the set of ring automorphisms of a ring R Show formally that Aut(R) is a group under composition. 2. Relevant equations 3. The attempt at a solution I Have a very similar question to which I have the solution viz Aut(G) denotes the set of group automorphisms of a Group G, show that Aut(G) is a group under composition. Proof Let a,b: G -> G be automorphisms then $$a\circ b: G \rightarrow g is also an auto$$ $$(a\circ b)(xy) = a(b(xy))=a(b(x)b(y))= a(b(x))a(b(y))$$ $$=(a\circ b)(x)(a\circ b)(y)$$ so $$a\circb$$ is a homomorphism it is also bijective since a,b are bijective $$\circ: aut(G)\times aut(G) \rightarrow aut(G)$$ is automatically associative (because comp of mappings is associative) As identity in Aut(G) $$take Id_g:G \rightarrow G$$ finally inverses Let a: G --> G be an auto then a^-1:G -->G is atleast a mapping and bijective need only show $$a^-1(xy) = a^-1(x)a^-1(y)$$ let $$x,y \in G$$ choose $$c,d \in G$$ : a(c)=x, a(d)=y $$a^-1(xy) = a^-1(a(c)a(d))=a^-1(a(cd)) = cd= a^-1(x)a^-1(y)$$ q.e.d .......the proof for rings is essentially the same right? The only thing that concerns me is the last part(above) since we used the fact that every element has it's inverse in a group but we don't have that in a ring.... Last edited: Mar 22, 2007 2. Mar 22, 2007 ### matt grime Where have you used that fact? 3. Mar 22, 2007 ### catcherintherye ok so i haven't and these two proofs are essentially the same? 4. Mar 22, 2007 ### HallsofIvy You did use a-1 where a is automorphism. You say you know Are you sure of that definition of Aut(G)? Homomorphism, in general, do not have inverses. 5. Mar 22, 2007 ### catcherintherye no i've made a mistake it's supposed to be the group of isomorphisms	{"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": 0.9814321994781494, "perplexity": 1875.408790259568}, "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-2018-22/segments/1526794866870.92/warc/CC-MAIN-20180524205512-20180524225512-00601.warc.gz"}
https://www.tenebre.xyz/math-geometry-triangles.html	## Triangles ### Area of a Triangle There are numerous ways to determine the area of a triangle. Each calculation method depends on what information is already known about the triangle in question. Given the height and width of a right triangle, the area is simply: $$A=\frac{1}{2}ab$$ Given the height and and base edge length of any triangle, the area is similarly: $$A=\frac{1}{2}bh$$ Given one angle and is adjacent lengths, the area is: $$A=\frac{1}{2}bc\,sin\,α$$ Given one side and its two adjacent angles, the area is: $$A=\frac{a^2\,sin\,α\,sin\,β}{2\,sin(α+β)}$$ Given all three side lengths, we use Heron's formula (aka Hero's formula): $$A=\sqrt{s(s-a)(s-b)(s-c)}$$ Where $$s$$ is the semiperimeter: $$s=\frac{1}{2}(a+b+c)$$ Note that a circle inscribed within any triangle has the property: $$r=\frac{A}{s}$$	{"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": 0.8626473546028137, "perplexity": 568.5107548817107}, "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/1679296945182.12/warc/CC-MAIN-20230323163125-20230323193125-00226.warc.gz"}
https://www.thphys.uni-heidelberg.de/~cosmo/dokuwiki/doku.php/doran:cmb	The Theory of the Cosmic Microwave Background Topics In this course, we cover the theory of the cosmic microwave background fluctuations. In particular: • The expanding universe • Stokes parameters, spin harmonics and E and B modes • Cosmological perturbations • Gauge freedom and gauge transformations • The Boltzmann equation for photons • The scattering terms of Boltzmann's equation for photons • The multipole expansion • The line of sight strategy of solving the multipole hierachy • The multipole spectrum of temperature and polarization fluctuations • Finding initial conditions for the fluctuations • The tight coupling limit before recombination • Analytic estimates • If time permits: secondary anisotropies Lecture Notes You can download the lecture notes new from 02-05-2010 (without a chapter on secondary anisotropies yet). Further Literature • The Cosmic Microwave Background, by Ruth Durrer (2008) ISBN-10: 0521847044 • Principles of Physical Cosmology by P.J.E Peebles (1993) ISBN-10: 0691019339 • Fluctuations in the Cosmic Microwave Background, Matthias Zaldarriaga's Thesis • Gauge Invariant Cosmological Perturbations, J. M. Bardeen (1980), Phys.Rev.D22:1882-1905,1980 Credit Points Master students may acquire 4 credit points subject to a written exam at the end of the course.	{"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": 0.8792506456375122, "perplexity": 4814.93816804474}, "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-2019-30/segments/1563195529276.65/warc/CC-MAIN-20190723105707-20190723131707-00013.warc.gz"}
https://socratic.org/questions/what-is-the-slope-of-the-line-perpendicular-to-y-5-3x-3	Algebra Topics # What is the slope of the line perpendicular to y=-5/3x -3 ? Feb 4, 2016 The slope of a line perpendicular to a line with slope $m$ is $- \frac{1}{m}$. The line we are looking for, therefore, has a slope of $\frac{3}{5}$. #### Explanation: Standard form of a line is: $y = m x + b$ where $m$ is the slope and $b$ is the y-intercept. For another line perpendicular, the slope will be $- \frac{1}{m}$. In this case, that is $- \frac{1}{- \frac{5}{3}}$ = $\frac{3}{5}$. ##### Impact of this question 726 views around the world	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 9, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8169571161270142, "perplexity": 335.9361587746539}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496671053.31/warc/CC-MAIN-20191121231600-20191122015600-00020.warc.gz"}
https://byjus.com/trapezoid-formula/	# Trapezoid formula In Geometry, a trapezium is one of the types of a quadrilateral with one pair of parallel sides (called bases) and one pair of non-parallel sides (called legs). The area of a trapezoid is the number of unit squares that can be fit into the trapezoid. It is generally measured using square units, such as cm2, m2, and so on. In this article, we are going to learn the area of trapezoid formula and examples in detail. ## Area of Trapezoid Formula To find the area of trapezoid, it is enough to know the bases (parallel sides) of trapezoid and the perpendicular distance between them. If “a” and “b” are the parallel sides of trapezoid and “h” is the perpendicular distance between, then the area of trapezoid formula is given by: Area of Trapezoid = (½) [(a+b)h] square units. ### Examples on Area of Trapezoid Example 1: Find the area of trapezoid if the bases of the trapezoid are 6 cm and 7 cm and the perpendicular distance between them is 8 cm. Solution: Given: Let the parallel sides of a trapezium be “a” and “b”. Hence, a = 6 cm and b = 7 cm. Also, given that the perpendicular distance between the parallel sides is 8 cm. (i.e) h = 8 cm. We know that the area of trapezoid formula is: A = (½) [(a+b)h] square units Now, substitute the values in the formula, we get A = (½) [(6+7)8] square units A = (6+7)4 cm2 A = (13)4 cm2 A = 52 cm2 Therefore, the area of trapezoid is 52 cm2. Example 2: Find the height of the trapezoid, if the sum of the parallel sides is 25 m and the area is 75 m2. Solution: Given: a+b = 25 m Area, A = 75 m2. To find: h We know that the formula for area of trapezoid, A = (½) [(a+b)h] square units Substituting the values in the formula, we get 75 = (½)[25h] 75/25 = (½)h 3 = h/2 h = 6. Therefore, the height of the trapezoid is 6 m. Stay tuned to BYJU’S – The Learning App to learn more formulas. ## Frequently Asked Questions on Area of Trapezoid Formula ### What is the area of trapezoid? The area of trapezoid is the number of unit squares that fit into the trapezoid shape. In other words, the region occupied by the trapezoid shape is called the area of trapezoid. It is measured in square units. ### What is the area of trapezoid formula? If “a” and “b” are the parallel sides of trapezoid, and “h” is the perpendicular distance between them, then the area of trapezoid formula is (½) [(a+b)h] square units. ### Can we find the area of the trapezoid, if the bases and height are known? Yes, we can find the area of trapezoid if the bases and height are known.	{"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": 0.9180575609207153, "perplexity": 650.708375798142}, "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-2022-21/segments/1652662541747.38/warc/CC-MAIN-20220521205757-20220521235757-00489.warc.gz"}
http://www.thescinewsreporter.com/2018/04/astronomers-observe-strange-quantum.html	### Astronomers Observe Strange Quantum Distortion in Empty Space for the First Time Ever Vacuum birefringence is a weird quantum phenomenon that has only ever been observed on an atomic scale. It occurs when a neutron star is surrounded by a magnetic field so powerful and intense, so that it gives rise to a region in empty space where matter randomly appears and vanishes. This polarization of light in a vacuum due to strong magnetic fields was first thought to be possible in the 1930s by physicists Werner Heisenberg and Hans Heinrich Euler as a product of the theory of quantum electrodynamics (QED). The theory describes how light and matter interact. Now, for the first time ever, this strange quantum effect has been observed by a team of scientists from INAF Milan (Italy) and from the University of Zielona Gora (Poland). Using the European Southern Observatory’s (ESO) Very Large Telescope (VLT), a research team led by Roberto Mignani observed neutron star RX J1856.5-375, which is about 400 light-years from Earth. Neutron stars are rather dim, yet they are 10 times more massive than our sun. As such, they have extremely strong magnetic fields permeating their surface and surroundings. Vacuums are supposedly empty spaces (according to Einstein and Newton, at least) where light can pass through uninhibited or unchanged. But, according to QED, space is full of virtual particles continually popping in and out of existence. Very strong magnetic fields, like those surrounding neutron stars, can modify such spaces as vacuums. Using the FORS2 instrument on the VLT, the researchers were able to observe the neutron star with just visible light, pushing the limits of existing telescope technology. ## Better Telescopes Studying VLT data on the star, the researchers saw linear polarization occurring at a significant degree of around 16%. This is very likely due to vacuum birefringence in the area surrounding RX J1856.5-375. “The high linear [polarization] that we measured with the VLT can’t be easily explained by our models unless the vacuum birefringence effects predicted by QED are included. [Polarization] measurements with the next generation of telescopes, such as ESO’s European Extremely Large Telescope (EELT), could play a crucial role in testing QED predictions of vacuum birefringence effects around many more neutron stars” said Mignani. Given the limited technology used, Mignani believes that future telescopes can discover more about similar strange quantum effects by studying other neutron stars. “This measurement, made for the first time now in visible light, also paves the way to similar measurements to be carried out at X-ray wavelengths,” researcher Kinwah Wu said.	{"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": 0.8249076008796692, "perplexity": 1282.8533752625744}, "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/1563195529406.97/warc/CC-MAIN-20190723130306-20190723152306-00557.warc.gz"}
http://www.contrib.andrew.cmu.edu/~ryanod/index.php?p=2285&replytocom=22868	# Some tips • You might try using analysis of Boolean functions whenever you’re faced with a problems involving Boolean strings in which both the uniform probability distribution and the Hamming graph structure play a role. More generally, the tools may still apply when studying functions on (or subsets of) product probability spaces. • If you’re mainly interested in unbiased functions, or subsets of volume $\frac12$, use the representation $f : \{-1,1\}^n \to \{-1,1\}$. If you’re mainly interested in subsets of small volume, use the representation $f : \{-1,1\}^n \to \{0,1\}$. • As for the domain, if you’re interested in the operation of adding two strings (modulo $2$), use $\mathbb{F}_2^n$. Otherwise use $\{-1,1\}^n$. • If you have a conjecture about Boolean functions: • Test it on dictators, majority, parity, tribes (and maybe recursive majority of $3$). If it’s true for these functions, it’s probably true. • Try to prove it by induction on $n$. • Try to prove it in the special case of functions on Gaussian space. • Try not to prove any bound on Boolean functions $f : \{-1,1\}^n \to \{-1,1\}$ that involves the parameter $n$. • Analytically, the only multivariate polynomials we really know how to control are degree-$1$ polynomials. Try to reduce to this case if you can. • Hypercontractivity is useful in two ways: (i) It lets you show that low-degree functions of independent random variables behave “reasonably”. (ii) It implies that the noisy hypercube graph is a small-set expander. • Almost any result about functions on the hypercube extends to the case of the $p$-biased cube, and more generally, to the case of functions on products of discrete probability spaces in which every outcome has probability at least $p$ — possibly with a dependence on $p$, though. • Every Boolean function consists of a junta part and Gaussian part. ### 5 comments to Some tips • Ravi Boppana Thanks for the tips! In the third tip (about the domain), should $\{ -1, 1 \}$ be $\{ -1, 1 \}^n$? Can you say a word about the rationale for the fifth tip (about the parameter $n$)? • Ryan O'Donnell You’re right about the third tip, thanks! Regarding the 5th… well of course it’s not ironclad, but one of the themes in the area is to try to work in “$\{-1,1\}^\infty$” if possible. I guess I’m mainly thinking about theorems like hypercontractivity and the like (ones proven by induction) where one might be tempted to prove weaker versions with a dependence on $n$, but where it’s possible to prove something independent of $n$. • Ravi Boppana Thanks for the explanation. A million thanks for the entire book! • Luca Is there an “n-free” variant (generalization?) of KKL that has more or less the same applications? • Luca: Yes, take a look at the “KKL Edge-Isoperimetric Theorem” in Chapter 9.6, which gives what I view as the “correct” version of the KKL theorem.	{"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": 0.8858998417854309, "perplexity": 704.9663650579224}, "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/1518891812841.74/warc/CC-MAIN-20180219211247-20180219231247-00778.warc.gz"}
http://www.apniphysics.com/tag/fecl3-solution/	Introduction Quinke’s Method In this video, I have discussed the calculation of the magnetic susceptibility by using the Quinke’s experiment. What are the important points for the measurement of…	{"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": 0.87486332654953, "perplexity": 713.6268206991093}, "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-2019-18/segments/1555578584186.40/warc/CC-MAIN-20190423015050-20190423041050-00248.warc.gz"}
https://indico.cern.ch/event/260380/	TH String Theory Seminar # Double-Detector Correlations in N=4 SYM ## by Stefan Hohenegger Europe/Zurich 4-3-006 - TH Conference Room (CERN) ### 4-3-006 - TH Conference Room #### CERN 100 Show room on map Description In this talk I will consider double-correlations of energy-, charge- and scalar-flow operators in four-dimensional N=4 Super Yang-Mills theory. These operators are constructed from the stress-energy tensor, R-currents, and scalar operators of the theory respectively and can physically be interpreted as detectors. Their correlations can be related to weighted cross sections, which can be calculated perturbatively using amplitudes. In this talk I will describe an efficient way to compute such correlations using superconformal symmetry, which is not only applicable in perturbation theory but can also be extended to the strong coupling regime. Organised by J. Drummond	{"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": 0.8351523280143738, "perplexity": 1986.7993492096668}, "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-2016-30/segments/1469257827781.76/warc/CC-MAIN-20160723071027-00072-ip-10-185-27-174.ec2.internal.warc.gz"}
http://planetmath.org/euclideanspaceasamanifold	# Euclidean space as a manifold Let $\mathbbmss{E}^{n}$ be $n$-dimensional Euclidean space, and let $(\mathbbmss{V},\langle\cdot,\cdot\rangle)$ be the corresponding $n$-dimensional inner product space of translation isometries. Alternatively, we can consider Euclidean space as an inner product space that has forgotten which point is its origin. Forgetting even more information, we have the structure of $\mathbbmss{E}^{n}$ as a differential manifold. We can obtain an atlas with just one coordinate chart, a Cartesian coordinate system $(x^{1},\ldots,x^{n})$ which gives us a bijection between $\mathbbmss{E}^{n}$ and $\mathbbmss{R}^{n}$. The tangent bundle is trivial, with $\operatorname{T}\mathbbmss{E}^{n}\cong\mathbbmss{E}^{n}\times\mathbbmss{V}.$ Equivalently, every tangent space $\operatorname{T}_{p}\mathbbmss{E}^{n},\;p\in\mathbbmss{E}^{n}$. is isomorphic to $\mathbbmss{V}$. We can retain a bit more structure, and consider $\mathbbmss{E}^{n}$ as a Riemannian manifold by equipping it with the metric tensor $\displaystyle g$ $\displaystyle=$ $\displaystyle dx^{1}\otimes dx^{1}+\cdots+dx^{n}\otimes dx^{n}$ $\displaystyle=$ $\displaystyle\delta_{ij}dx^{i}\otimes dx^{j}.$ We can also describe $g$ in a coordinate-free fashion as $g(u,v)=\langle u,v\rangle,\quad u,v\in\mathbbmss{V}.$ ## Properties 1. 1. Geodesics are straight lines in $\mathbbmss{R}^{n}$. 2. 2. The Christoffel symbols vanish identically. 3. 3. The Riemann curvature tensor vanish identically. Conversely, we can characterize Eucldiean space as a connected, complete Riemannian manifold with vanishing curvature and trivial fundamental group. Title Euclidean space as a manifold EuclideanSpaceAsAManifold 2013-03-22 15:29:48 2013-03-22 15:29:48 matte (1858) matte (1858) 9 matte (1858) Definition msc 53B21 msc 53B20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 20, "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": 0.9939942955970764, "perplexity": 455.57514287471133}, "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-2018-13/segments/1521257647600.49/warc/CC-MAIN-20180321082653-20180321102653-00783.warc.gz"}
https://brilliant.org/practice/translation/	Geometry # Translation Point $$(2, 1)$$ is moved to point $$(10,11)$$ by $T: (x, y) \rightarrow (x+a, y+b).$ What is $$a+b$$? If we translate the point $$(9, 3)$$ by $$(13, 2),$$ what would be the new point? The curve $$y=2x^2+3x-9$$ undergoes a parallel translation by $$7$$ units in the positive direction of the $$x$$-axis and $$-3$$ units in the positive direction of the $$y$$-axis. If the equation of the resulting curve is $$y=ax^2+bx+c,$$ what is $$a+b+c?$$ Consider a translation of axes $T: (x, y) \to (x+a, y+b)$ such that the coordinates of a point $$P=(7, 4)$$ with respect to the new system of coordinates after translation are $$P'=(10,-11).$$ What is the value of $$a+b?$$ If the point $$(m,n)$$ is shifted by $$6$$ in the positive $$x$$ direction and by $$-12$$ in the positive $$y$$ direction, then it coincides with the center of the circle $x^2+y^2-6x+2y-13=0.$ What is the value of $$m+n?$$ ×	{"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": 0.8103324174880981, "perplexity": 60.52425101769519}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583700734.43/warc/CC-MAIN-20190120062400-20190120084400-00448.warc.gz"}
https://www.vedantu.com/question-answer/two-coherent-monochromatic-light-beams-of-class-12-physics-cbse-5f5d32f98f2fe24918455b02	Question # Two coherent monochromatic light beams of intensities I and 4I are superposed. The maximum and minimum possible resulting intensities are :A. 5I and 0B. 5I and 3IC. 9I and ID. 9I and 3I Hint: Take ${{I}_{1}}$ and ${{I}_{2}}$ as the two intensities of two coherent monochromatic light beams, then find out the maximum and minimum intensity by constructing a formula with the help of those two assumed variable. If ${{I}_{1}}$ and ${{I}_{2}}$ are two intensities of two coherent monochromatic light beams, then ${{I}_{\max }}$=($\sqrt{{I_1}}$ + $\sqrt{{I_2}}$)$^{2}$ ${{I}_{\min }}$=($\sqrt{{I_1}}$ - $\sqrt{{I_2}}$)$^{2}$ Now substituting the value of ${{I}_{1}}$ and ${{I}_{2}}$ with I and 4I, Therefore, maximum intensity ${{I}_{\max }}$=($\sqrt{I}$+$\sqrt{4I}$)$^{2}$ On solving it comes, ${{I}_{\max }}$=9I Therefore, minimum intensity, ${{I}_{\min }}$=($\sqrt{I}$-$\sqrt{4I}$) $^{2}$ On solving it comes, ${{I}_{\min }}$=I Therefore, Option C is the correct option. Maximum intensity is =($\sqrt{I}$+$\sqrt{4I}$)$^{2}$ Minimum intensity is=($\sqrt{I}$-$\sqrt{4I}$)$^{2}$	{"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": 0.9083420634269714, "perplexity": 2009.5292218204038}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178357641.32/warc/CC-MAIN-20210226115116-20210226145116-00634.warc.gz"}
https://www.physicsforums.com/threads/using-remainder-theorem-to-find-remainder.900226/	# Using Remainder Theorem to find remainder Tags: 1. Jan 13, 2017 ### Schaus 1. The problem statement, all variables and given/known data (y4 - 5y2 + 2y - 15) / (3y - √(2)) The answer says (2√(2)/3)-(1301/81) 2. Relevant equations 3. The attempt at a solution Using synthetic division 3y - √(2) = 0 3y = √(2) y = √(2)/3 View attachment 111547 My final answer that I keep getting is (2√(2)/3)-21. I can't seem to get (2√(2)/3)-(1301/81). I was just wondering if someone could show me where I'm going wrong? Sorry writing this out was very difficult so I hope you can still understand it. Last edited: Jan 13, 2017 2. Jan 13, 2017 ### Staff: Mentor I don't actually can follow your calculations. If I compare my calculation with your result, then I think you simply lost some denominators. I get $\frac{4}{81}-\frac{10}{9}-15+\frac{2\sqrt{2}}{3}$ whereas you seem to have added only the nominators $4-10-15+\frac{2\sqrt{2}}{3}$. I once gave an example on PF here: https://www.physicsforums.com/threa...r-a-polynomial-over-z-z3.889140/#post-5595083 Maybe it helps. 3. Jan 13, 2017 ### Schaus Sorry, I should have just posted this to begin with. 4. Jan 13, 2017 ### Ray Vickson Your long-division process is all wrong. Let $p(y) = y^4 - 5y^2 + 2y - 15$ and $d(y) = 3y - \sqrt{2}$. You want to find a "quotient" $q(y)$ and a number $r$ (the "remainder") that give you $p(y) = q(y) d(y) + r.$ The question is asking you to find $r$. Step 1: see how many times the leading term of $d(y)$ will go into the leading term of $p(y)$; that is the number of times $3y$ goes into $y^4$. The answer in this case is $y^4/(3y) = (1/3)y^3$, so that is the first term in your quotient $q(y)$. Now we have $$p(y) - \frac{1}{3} y^3 d(y) = \frac{\sqrt{2}}{3} y^3 -5 y^2 + 2y - 15 \equiv p_1(y).$$ However, all that would be doing it the hard way. The easy way would be to go ahead and use the "remainder theorem". Step 2: see how many times the leading term of $d(y)$ goes into the leading term of $p_1(y)$. The answer is $(\sqrt{2}/3)/(3 y) = (\sqrt{2}/9) y^2$, so that is the next term in your quotient $q(y)$. We have $$p_2(y) \equiv p_1(y) - \frac{\sqrt{2}}{9} y^2 d(y) = -\frac{43}{9} y^2 + 2y - 15.$$ So, up to now we have $$p(y) = \left( \frac{1}{3} y^3 + \frac{\sqrt{2}}{9} y^2 \right) d(y) + p_2(y).$$ Keep going like that; the next term in the quotient $q(y)$ will be the number of times the leading term of $d(y)$ goes into the leading term of $p_2(y)$, so is the number of times $3y$ goes into $-(43/9) y^2$, etc., etc. The remainder will be what you have left when the process comes to an end. However, all that would be doing it the hard way; I included the material just because you attempted to do it, but did it all wrong. The easy way would be to just use the so-called "remainder theorem". 5. Jan 14, 2017 ### Schaus I was using synthetic division which worked on every other question but I'll try it the way you're suggesting it. 6. Jan 14, 2017 ### Schaus I got the same result as the answer. Thanks for your help! 7. Jan 14, 2017 ### PetSounds It looks to me like you simply made a couple multiplication errors when working with fractions. I worked the problem with synthetic division and got the right answer. Last edited: Jan 14, 2017 Have something to add? Draft saved Draft deleted Similar Discussions: Using Remainder Theorem to find remainder	{"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": 0.8934352397918701, "perplexity": 478.35945243320805}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886105297.38/warc/CC-MAIN-20170819031734-20170819051734-00373.warc.gz"}
http://outdoor.geocities.jp/yokosuka0904_wa/	# index.htm @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ iRXJv`j́AQPNNPO_ސ쌧EI[LOijv`j ƂĔ܂B{EOYnŃEI[LO̕yړIƂĉłB @ @ @ @ @ @ <Counter> @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ ̈ē IV̈ē EI[LÖē iCƕj ʐ^W @ @ @ @ @ (ߑjY){R`݂I NW(H) (H) ₢킹(H) @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ u{R` -n݂vIEEEB @Qs{łB EI[N}i[Ċy܂傤 @ @ ͕PTNxV݂ꂽuU̐iɍvv ĉƂĕ\܂BiQPTj@ڍׂ͉B @ @ @ @ @ @ @ @ @ @ @ @@̍s͑SĉȊO̕\ȂŎQł܂B @ @ @ @ @ @ @ @ @ @ @tԂ܂łɒڂzB҂Ă܂B @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @@(2014.9.9ԑ̐XƊ @xêȂeXg[oh) @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ ܂Pyvo܂B @ @ @ @ @ gcYlsƈꏏɋLOʐ^ @ @ @@ꏏɕFBPl܂B @ @ @ @ @ @ @@Vio邱Ƃ҂āB @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ RXJEI[LO @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ ǁF 239-0802@{snx4-11-23@쓇 @ {fڋL̒쌠̓RXJEI[LOɋA܂B @ @ @ @ @ TEL/FAXF046-842-6117 @ @ @ i{fړeʐ^Ɋւē]ڂ֎~܂B) @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @	{"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": 1.0000097751617432, "perplexity": 7.579453406495737}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860122501.26/warc/CC-MAIN-20160428161522-00001-ip-10-239-7-51.ec2.internal.warc.gz"}
http://daylateanddollarshort.com/bloog/what-more-i-know-about-hyperbolic-tetrahedra/	## What more I know about hyperbolic tetrahedra I have updated my note, “Hedronometric Formulas for a Hyperbolic Tetrahedron” (PDF), with a brand new formula for the volume of an arbitrary tetrahedron in terms of its face and pseudo-face areas. (See Section 8.3.) The formula isn’t the monolithic and symmetric counterpart to Derevnin-Mednykh I’ve been seeking, but it’s a start. It’s complicated enough that I won’t attempt to render it here. The Open Question: As one might expect, the formula involves an integral. One of the limits of integration is the subject of a Conjecture. Again, the notion is too complicated to describe here, but the gist is that I believe that, by appropriately assigning names to the faces (and pseudo-faces), we guarantee that the lower limit is simply one-quarter of a particular pseudo-face area. (If I’m mistaken, then that limit is a less-obvious root of a trigonometric equation.) Numerical experiments in Mathematica suggest that the conjecture is true, but I don’t have even non-constructive proof. (Nevermind that the conjecture wouldn’t really be helpful without a practical way to determine what an “appropriate assignment” of names would be.) When (if?) a properly symmetric formula is finally discovered, the order of face names won’t matter at all; but for now, it makes for an irksome little wrinkle in the formula. Posted 16 February, 2013 by in Hedronometry, Open Question	{"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": 0.8783136606216431, "perplexity": 872.0407334246408}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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-2019-22/segments/1558232257100.22/warc/CC-MAIN-20190523043611-20190523065611-00198.warc.gz"}
http://mathoverflow.net/feeds/question/23002	Analogue of Shimura curves in the symplectic case? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T07:38:27Z http://mathoverflow.net/feeds/question/23002 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23002/analogue-of-shimura-curves-in-the-symplectic-case Analogue of Shimura curves in the symplectic case? Kevin Buzzard 2010-04-29T16:13:19Z 2010-05-20T23:07:03Z <p>My understanding is this: one can attach 2-d Galois representations to classical modular eigenforms because one can look in the etale cohomology of modular curves. For Hilbert modular forms the naive analogy breaks down, because the middle cohomology of the Hilbert modular varieties will (conjecturally at least, and possibly this is known in this case) be built up from tensor products of the 2-dimensional Galois representations attached to the automorphic representations contributing to the cohomology, and one can't unravel the prodands from the product.</p> <p>One way of resolving this problem is to instead use Shimura curves. By Jacquet-Langlands, cuspidal automorphic forms on GL_2 over a totally real field (of odd degree over Q say) biject with cuspidal automorphic forms on a quaternion algebra ramified at all but one infinite place. And we have the happy coincidence that the associated algebraic group satisfies Deligne's axioms for a Shimura variety, and we can again look in the cohomology of a curve to construct the Galois representation.</p> <p>This trick relies on two things, one local and one global: the local thing is that GL_2(R) has an inner form which is compact mod centre, and the global thing is that the quaternion algebra satisfies Deligne's axioms for a Shimura variety.</p> <p>Now let's try and generalise all of this to the symplectic case, so G=GSp_{2g} with g>1. If the base field is Q then Weissauer and others constructed the Galois representations attached to a Siegel modular form in the case g=2 by looking in the etale cohomology of a Siegel modular 3-fold. Now what about if the base is bigger? Can one pull off the same trick?</p> <p>Local question: does GSp_4(R) have an inner form which is compact mod centre?</p> <p>Global question: if so, does GSp_4(F) (F totally real) sometimes have an inner form which is compact mod centre at all but one infinite place, and for which Deligne's axioms hold? If so, might one hope to see the Galois representations attached to Hilbert-Siegel modular forms over F here?</p> <p>[Edit: the local question is solved below by Hansen. I thought that the papers he linked to would deal with the global question too, but now I suspect they don't. I've put a 150-point bounty on for the global question.]</p> <p>[Edit: because of bounty daftness I can now no longer accept any answer for this question.]</p> http://mathoverflow.net/questions/23002/analogue-of-shimura-curves-in-the-symplectic-case/23008#23008 Answer by David Hansen for Analogue of Shimura curves in the symplectic case? David Hansen 2010-04-29T16:58:38Z 2010-04-29T16:58:38Z <p>Local question: Yes, \$GU2(\mathbb{H})\$, where \$\mathbb{H}\$ is the Hamilton quaternions.</p> <p>Global question: Not an answer, but perhaps useful to you - There are two quite relevant papers of Claus Sorenson which can be found <a href="http://www.math.princeton.edu/~csorense/gal_final.pdf" rel="nofollow">here</a> and <a href="http://www.math.princeton.edu/~csorense/lower.pdf" rel="nofollow">here</a>. The first paper constructs the Galois representations attached to Hilbert-Siegel modular forms over totally real fields. The second paper concerns level-lowering for GSp4 - along the way, he proves a Jacquet-Langlands transfer to an inner form of GSp4 compact at <em>all</em> infinite places and split at all finite places (at least when F has even degree and pi satisfies the usual conditions).</p>	{"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": 0.934715747833252, "perplexity": 533.9463494945994}, "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-2013-20/segments/1368705749262/warc/CC-MAIN-20130516120229-00071-ip-10-60-113-184.ec2.internal.warc.gz"}
https://mathspace.co/textbooks/syllabuses/Syllabus-446/topics/Topic-8206/subtopics/Subtopic-107657/?activeTab=theory	# Features of linear relationships Lesson You will have covered the concepts here before, either in previous years or even this year. But I'll summaries the key features of linear equations below so that you have a handy reference point all in one place. ## Slope or Slope The slope of a line (also called the slope) that passes through two known points, say $\left(x_1,y_1\right)$(x1,y1) and $\left(x_2,y_2\right)$(x2,y2) on the cartesian plane can be found easily. Slope is a measure of steepness. It is the ratio of a line's rising (or falling) to its running. If, over a distance of $8$8 metres, a driveway rises $2$2 metres, then its slope is said to be the ratio $\frac{2}{8}=\frac{1}{4}=0.25$28=14=0.25. It is also defined as the tangent of the angle of rise as shown in this simple diagram. Consider the following example of a line passing through two points $\left(-3,7\right)$(3,7) and $\left(5,9\right)$(5,9) as shown here: Looking at the two $y$y values of the two points, the rise is clearly $2$2. We could either use a formula for rise which might look like $Rise=y_2-y_1=9-7=2$Rise=y2y1=97=2 or simply notice that there is a gap of $2$2 between the two values. Looking at the two $x$x values we could again either use the formula $Run=x_2-x_1=5-\left(-3\right)=8$Run=x2x1=5(3)=8 or simply notice that the gap between $-3$3 and $5$5 is $8$8 We also realise that the line is rising and this means that the slope is positive. The slope, often denoted by the letter $m$m is simply the ratio given by: $m=\frac{y_2-y_1}{x_2-x_1}=\frac{2}{8}=0.25$m=y2y1x2x1=28=0.25 From the fact that $\tan\theta=0.25$tanθ=0.25 we can use a scientific calculator to show that $\theta=\tan^{-1}\left(0.25\right)=14^\circ2'$θ=tan1(0.25)=14°2, which gives some sense to the steepness of the rise. Note that if the line is falling, the line's slope will be negative. In such cases the acute angle the line makes with the $x$x axis will be shown on the calculator as a negative angle. Adding $180^\circ$180° to this will reveal the obtuse angle of inclination the line makes with the axis. For example if the slope was given by $m=-0.25$m=0.25, then $\theta=180^\circ+\tan^{-1}\left(-0.25\right)$θ=180°+tan1(0.25), which simplifies to $\theta=165^\circ58'$θ=165°58 Suppose we consider the line given by $5x-2y=20$5x2y=20. By putting $x=0$x=0 we see that $y=-10$y=10 (note the negative sign here). Also, by putting $y=0$y=0, we find that $x=4$x=4. This means that the $x$x and $y$y intercepts are $4$4 and $-10$10 respectively. The situation is shown here: Note that the rise and run can be determined from the $x$x and $y$y intercepts. The positive slope of the line shown is given as $m=\frac{10}{4}=2.5$m=104=2.5 #### Worked Examples ##### Question 1 What is the slope of the line shown in the graph given that point A(3,3) and point B(6,5) both line on the line. ##### Question 2 What is the slope of the line going through A and B? ## Finding the Equation The line with equation $y=mx+b$y=mx+b has a slope $m$m and a y intercept $b$b . It is important to observe that this form of the line shows $y$y explicitly as a function of $x$x with $m$m and $b$b as constants, different values of $x$x will determine different values of $y$y . For example, the line, say $L_1$L1, given by $y=3x+3$y=3x+3 has a slope of $3$3 and a $y$y intercept of $3$3. The $y$y intercept can be determined by noting that at $x=0$x=0$y=3$y=3 The line $L_2$L2 given in general form as $2x+y-8=0$2x+y8=0 can be rearranged to $y=-2x+8$y=2x+8 and the slope $-2$2 and y intercept $8$8 can be easily determined. The line $L_3$L3 given by $5x+4y-29=0$5x+4y29=0 can be rearranged to $4y=29-5x$4y=295x and then to $y=\frac{29}{4}-\frac{5}{4}x$y=29454x with slope $m=-\frac{5}{4}$m=54 and $y$y intercept $b=7.25$b=7.25 We will now go through some of the skills in finding lines, intersections and midpoints by considering a number of questions relating to the lines $L_1,L_2$L1,L2 and $L_3$L3. As we answer the questions, check the sketch below to confirm your understanding of each answer. ##### Question 3 Is the point $A\left(1,6\right)$A(1,6) on $L_1$L1? By substituting $\left(1,6\right)$(1,6) into $y=3x+3$y=3x+3 we see that $6=3\times1+3$6=3×1+3. This is true and so the given point is on $L_1$L1. ##### Question 4 Find $P$P, the $x$x intercept of $L_2$L2. Since $L_2$L2 is given by $2x+y-8=0$2x+y8=0, the $x$x intercept is found by putting $y=0$y=0. Then $2x-8=0$2x8=0 and solving for $x$x, we see that $x=4$x=4. The point of intercept is thus $P\left(4,0\right)$P(4,0). ##### Question 5 Find the equation of the line $L_4$L4, which passes through $P$P and $M$M. With $P\left(4,0\right)$P(4,0) and $M\left(5,1\right)$M(5,1), we have two methods to find $L_4$L4. Both methods require finding the slope of the line given by $m=\frac{y_2-y_1}{x_2-x_1}=\frac{1-0}{5-4}=1$m=y2y1x2x1=1054=1 Then method 1 makes use of the point slope form of the line. Specifically we know that the equation we are looking for must have the form $y=1x+b$y=1x+b. Since $M\left(5,1\right)$M(5,1) is on this line, it must satisfy it. Thus we can write $1=1\times5+b$1=1×5+b and so with a little thought, $b$b must be $-4$4. the equation of $L_4$L4 must be $y=x-4$y=x4. The second method makes use of the point slope formula $y-y_1=m\left(x-x_1\right)$yy1=m(xx1). We know that the slope $m=1$m=1 and choosing one of the known points on the line, say $M\left(5,1\right)$M(5,1), we can determine the equation of $L_4$L4 as $y-5=1\left(x-4\right)$y5=1(x4), and this simplifies once again to $y=x-4$y=x4 ##### Question 6 A line passes through the point $A\left(-2,-9\right)$A(2,9) and has a slope of $-2$2. Using the point-slope formula, express the equation of the line in slope intercept form. ##### Question 7 A line passes through the point $\left(3,-5\right)$(3,5) and $\left(-7,2\right)$(7,2) a) Find the slope of the line b) Find the equation of the line by substituting the slope and one point into $y-y_1=m\left(x-x_1\right)$yy1=m(xx1) ##### Question 8 a) Find the equation, in general form, of the line that passes through $A\left(-12,-2\right)$A(12,2) and $B\left(-10,-7\right)$B(10,7) b) Find the $x$x-coordinate of the point of intersection of the line that goes through $A$A and $B$B, and the line $y=x-2$y=x2 c) Hence find the $y$y-coordinate of the point of intersection ## Intercepts of Horizontal and Vertical Lines ### Horizontal Horizontal lines are lines that follow the horizon. They look like this... Imagine now horizontal lines on the Cartesian plane. Horizontal lines are parallel to the $x$x axis, and as you move along a horizontal line, the $x$x value will change but the $y$y value will remain the same. A horizontal line will: • only have a $y$y intercept • have an equation of the form $y=b$y=b (every point on the line has a $y$y value of $b$b) • have a $y$y intercept = $b$b, no $x$x intercept ### Vertical Vertical Lines are lines that go up and down (they are perpendicular to horizontal lines). Imagine now vertical lines on the Cartesian plane. Vertical lines are parallel to the $y$y axis, and as you move along a vertical line, the $y$y value will change but the $x$x value will remain the same. A vertical line will: • only have an $x$x intercept • have an equation of the form $x=b$x=b (every point on the line has an $x$x value of $b$b) • have an $x$x intercept = $b$b, no $y$y intercept ## Parallel and perpendicular lines ### Parallel lines These occur when we have 2 lines that NEVER cross each other and have no points in common. For this to happen the two lines need to have exactly the same slope. If they have different slopes they will cross. Parallel lines occur often in the real world. Consider the line $y=x$y=x, with slope=1. What would happen if we shifted every point on the line $2$2 units upwards? We would get a new line that is parallel to $y=x$y=x, but with every point having a $y$y value that is two greater: $y=x+2$y=x+2 So parallel lines are just shifts of one another. Parallel lines on the Cartesian Plane have the same slope (slope). ### Perpendicular Lines The Leaning Tower of Pisa Perpendicular is the word used to describe when one object meets another at exactly 90°. So perpendicular lines are simply lines that cross each other at exactly 90°. To see how important the idea of perpendicular really is just think about your floor, walls and roof. If a builder does not take care to make the walls perpendicular to the floor and ceiling you'll end up with an unstable house. The leaning tower of Pisa is a famous example of perpendicular angles gone wrong! Prior to restoration work performed between 1990 and 2001, the tower leaned at an angle of 5.5°, but the tower now leans at about 3.99°. That means the acute angle made by the tower and the ground is 86.01°. Perpendicular lines on the Cartesian plane will have one point of intersection, and at that point of intersection the angle between them will be 90°. ## Intersections and concurrent lines Because lines extend forever in both directions, unless they are parallel they will intersect somewhere. Now when 3 or more lines all pass through the same point we give those lines a special name: they are called concurrent lines. The point of intersection is called the "point of concurrency", labelled point P below. . ### Intersections of two lines Where two lines intersect, they share a common point. The $x$x and $y$y values of this point satisfy the equations of both lines. #### Example If one line has equation $y=2x+3$y=2x+3 and another has equation $y=x+6$y=x+6 then the point of intersection is where both the $y$y's are the same value. If they are the same value, then we can say that: (at the point of intersection) $2x+3$2x+3 is equal to $x+6$x+6 $2x+3=x+6$2x+3=x+6 We can then solve for the $x$x value at the point of intersection. $2x-x=6-3$2xx=63 $x=3$x=3 Now that we have $x$x, we can find the $y$y value at the point of intersection. Which equation should we substitute back into? Well since the point is common to both lines, you can choose either equation. $y=x+6$y=x+6 $y=3+6$y=3+6 $y=9$y=9 So these lines cross at the point $\left(3,9\right)$(3,9). ##### Question 9 Consider the following linear equations: $y=2x+2$y=2x+2 and $y=-2x+2$y=2x+2. a) What are the intercepts of the line $y=2x+2$y=2x+2? b) What are the intercepts of the line $y=-2x+2$y=2x+2? c) Plot the lines of the two equations on the same graph. d) State the values of $x$x and $y$y that satisfy both equations. #### Worked Examples ##### Question 10 Examine the graph attached and assess: 1. the slope of the line. 2. the $y$y-intercept of the line. 3. the $x$x-intercept of the line ##### Question 11 Consider the graph of the linear function shown. 1. What is the slope of the line? 2. What is the $y$y-intercept? 3. What is the $x$x-intercept? 4. What is the equation of the line? 5. What is the zero of the function? ## Zeros The zeros of a function are the values of that function that make it equal to zero. For example for the line $y=2x+1$y=2x+1, the zero is the value of $x$x, that makes the whole function ($2x+1$2x+1) equal to zero. So we set $2x+1=0$2x+1=0 and solve for $x$x. $2x+1=0$2x+1=0 $2x=-1$2x=1 $x=-\frac{1}{2}$x=12 Does that process look familiar? It should. It's exactly the same process we use when we are finding the $x$x intercepts. This means that the phrase zero of a function, (and also sometimes root of a function) is actually asking for the $x$x-intercepts. ### Outcomes #### 10P.LR2.05 Graph lines by hand, using a variety of techniques	{"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": 0.9098852872848511, "perplexity": 761.8103618114194}, "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-2021-49/segments/1637964362287.26/warc/CC-MAIN-20211202175510-20211202205510-00502.warc.gz"}
http://math.stackexchange.com/questions/315845/does-the-series-sum-fracxn1xn-converge-uniformly-on-x-in-0-1	# Does the series $\sum \frac{x^n}{1+x^n}$ converge uniformly on $x\in [0,1)$? Does the series $\sum \frac{x^n}{1+x^n}$ converge uniformly on $x\in [0,1)$? I have no idea where to start. Could somebody give me a hint ? Edit: Could I use something like this ? $$\left\|\frac{x^n}{1+x^n}\right\|\leq x^n$$ Because $x<1$, the geometric series $\sum x^n$ converges. Therefore by the M-test we get that the series converges uniformly. - It does not work. In the M-test, you need to find an upper bound which does not depend on $x$. Here it only gives uniform convergence on $[0,a]$ for any $a<1$. – 1015 Feb 27 '13 at 13:33 @julien Hm... schade... Any better ideas ? ^^ – Kasper Feb 27 '13 at 13:36 @DavidMitra So your line of reasoning is: If the series was uniform convergent, it would have a continuous limit function $f$. But then it would also be continuous in a point very close to $1$, but as this is not true, we get a contradiction, right ? – Kasper Feb 27 '13 at 13:48 @Kasper You don't even need to talk about the limit $f$. Just work on the Cauchy criterion. See my edit. – 1015 Feb 27 '13 at 13:49 Better: The terms $x^n\over 1+x^n$ do not converge to the zero function uniformly on $[0,1)$. Thus the series does not converge uniformly on $[0,1)$. – David Mitra Feb 27 '13 at 13:53 The partial sums $S_N(x)=\sum_{n=0}^N \frac{x^n}{1+x^n}$ converge pointwise on $[0,1)$ to $S(x)=\sum_{n\geq 0} \frac{x^n}{1+x^n}$ by comparison with the geometric series $\sum x^n$. Now $$S(x)-S_N(x)=\sum_{n\geq N+1} \frac{x^n}{1+x^n}\geq \frac{x^{N+1}}{1+x^{N+1}}$$ for all $x\in[0,1)$. So, letting $x$ tend to $1$, we get $$\sup_{[0,1)}S-S_N\geq \frac{1}{2}$$ for all $N$. Hence the convergence to $S$ is not uniform (which by definition is $\sup_{[0,1)}\|S-S_N\|\longrightarrow 0$ as $N\rightarrow +\infty$.) Note: you don't even need $S$, you could simply do it with the Cauchy criterion, if you use the fact that $C([0,1),\mathbb{R})$ is complete when equipped with the uniform norm. Then $$\|S_M(x)-S_N(x)\|\geq \frac{x^M}{1+x^M}$$ for all $M>N$ and all $x\in[0,1)$. Hence $$\sup_{[0,1)}\|S_M-S_N\|\geq \frac{1}{2}$$ and the sequence is not uniformly Cauchy. - I don't really understand. What do you mean with $S_M$ and $S_n$ without the $(x)$ ? – Kasper Feb 27 '13 at 14:01 @Kasper $\\|f\\|_\infty=\sup_{[0,1)}\|f\|=\sup_{x\in[0,1)}\|f(x)\|$. – 1015 Feb 27 '13 at 14:03 aah okay. But why must $\sup_{[0,1)}\|S_M-S_N\|\geq \frac{1}{2}$ go to zero ? – Kasper Feb 27 '13 at 14:04 Okay, some brain malfunction, I understand it now. Thanks for your helping me out :) – Kasper Feb 27 '13 at 14:19 By the way, is the prove of @sbr, also correct ? – Kasper Feb 27 '13 at 14:19 By definition the serie $\displaystyle\sum\frac{x^n}{1+x^n}$ converges uniformly on $[0,1)$ if $$\lim_n\sup_{x\in[0,1)} \sum_{k=n+1}^{\infty}\frac{x^k}{1+x^k}=0.$$ We have $$\sum_{k=n+1}^{\infty}\frac{x^k}{1+x^k}\geq \sum_{k=n+1}^{\infty}\frac{x^k}{2}=\frac{1}{2}\frac{x^{n+1}}{1-x},$$ So it's clear that $\displaystyle\sup_{x\in[0,1)} \frac{x^{n+1}}{1-x}=+\infty,$ and hence the condition of uniform convergence is not verified. - I don't know of the definition you gave. The cauchy criterion I know of is: The sequence of partial sums of a series $\sum g_k$ of functions is uniformly Cauchy on a set $S$ if and only if the series satisfies the Cauchy criterion: $\forall \epsilon >0 \exists N \forall x\in S:(n\geq m>N \implies \|\sum_{k=m}^ng_k(x)\|<\epsilon$ – Kasper Feb 27 '13 at 14:07 Oh wait this implies $\|g_n(x)\|<\epsilon$ for all $x\in S$, which implies $\sup\{\|g_n(x)\|:x\in S\}\leq \epsilon$ .... aah. right ? – Kasper Feb 27 '13 at 14:14 aaaah. things begin to make sense. So then you $\forall \epsilon >0\exists N\forall n>N :\|\sup\{\|g_n(x)\|:x\in S\}-0\|<\epsilon$ Wich proves $\limsup\{\|g_n(x)\|:x\in S\} = 0$ – Kasper Feb 27 '13 at 14:17 @Kasper The serie converges uniformly if the partial sum $S_n$ converge with the uniform norm to the sum $S$ i.e. $\|\|S_n-S\|\|_{\infty}=\sup_x\|S_n(x)-S(x)\|\rightarrow 0$. – Sami Ben Romdhane Feb 27 '13 at 14:56	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9733179211616516, "perplexity": 253.2993508069763}, "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-2014-35/segments/1409535919886.18/warc/CC-MAIN-20140909043920-00116-ip-10-180-136-8.ec2.internal.warc.gz"}
https://brilliant.org/discussions/thread/solving-cyclic-inequalities/	# Solving cyclic inequalities I understand that in case of symmetric inequalities, you can assume $$a \geq b \geq c$$ without loss of generality because you can replace for example $$a$$ with $$b$$ but still the inequality remains unchanged. But in case of cyclic inequalities can we assume $$a = max(a,b,c)$$ without loss of generality. If yes please explain the reason of WLOG in this case as well. Example:- [RMO 2017 P6] Let $$x,y,z$$ be real numbers, each greater than $$1$$. Prove that $$\dfrac{x+1}{y+1}+\dfrac{y+1}{z+1}+\dfrac{z+1}{x+1} \leq \dfrac{x-1}{y-1}+\dfrac{y-1}{z-1}+\dfrac{z-1}{x-1}$$ The official solution uses $$a = max(a,b,c)$$. $$#rmo$$ #inequalities $$#wlog$$ Note by Santu Paul 4 weeks, 1 day ago MarkdownAppears as italics or _italics_ italics bold or __bold__ bold - bulleted- list • bulleted • list 1. numbered2. list 1. numbered 2. list Note: you must add a full line of space before and after lists for them to show up correctly paragraph 1paragraph 2 paragraph 1 paragraph 2 [example link](https://brilliant.org)example link > This is a quote This is a quote # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" MathAppears as Remember to wrap math in $$...$$ or $...$ to ensure proper formatting. 2 \times 3 $$2 \times 3$$ 2^{34} $$2^{34}$$ a_{i-1} $$a_{i-1}$$ \frac{2}{3} $$\frac{2}{3}$$ \sqrt{2} $$\sqrt{2}$$ \sum_{i=1}^3 $$\sum_{i=1}^3$$ \sin \theta $$\sin \theta$$ \boxed{123} $$\boxed{123}$$ Sort by: Since the inequality (or equality) is cyclic, you can get new solutions by "rotating around" the numbers of a solution. For example, if $$(5,2,9)$$ is a solution, then $$(2,9,5)$$ and $$(9,5,2)$$ are solutions as well, but $$(5,9,2)$$ isn't a solution. This works because cyclic (in)equalities only use some operation (in your example $$\frac {a+1}{b+1}$$ for some $$a,b$$) of all pairs of two "consecutive" variables, but since all "consecutive" variable pairs ($$(x,y)$$, $$(y,z)$$ and $$(z,x)$$) are used in the (in)equality, the order of the variables matters, but the "rotation" doesn't and that's why you can set $$a = \text{max} (a,b,c)$$. - 4 weeks ago Imagine all 3 (or any number of) variables arranged in a circle. The cyclic (in)equality involves some operation of all pairs of variables that are next to each other in this circle. You can't change the order of the elements in the circle and you also can't make a mirror image, but you can rotate the variables around and WLOG rotate one specific value (in your example the macimum value) so that it becomes $$a$$. - 4 weeks ago I kind of get the geometric intuition but could you please be a bit more descriptive, especially in the algebraic one. Thank you. - 3 weeks, 6 days ago If a cyclic equality has a solution $$(x,y,z)_1 = (a,b,c)$$, then – since the equality is cyclic – you can get another solution as $$(x,y,z)_2 = (b,c,a)$$. I think you might understand this intuitively, but to prove it algebraically we have to define what exactly a cyclic equality is. My first thought is the definition A cyclic equality is an equality $$f(x,y,z) = 0$$ such that you can substitute $$(a,b,c) = (g(x,y),g(y,z),g(z,x))$$ for some $$g(r,s)$$ into the equation and get a symmetric equation. In some cases you might have to make multiple substitutions. Derived from the case of your inequality, we can show that $$\frac {x+1}{y+1} + \frac {y+1}{z+1} + \frac {z+1}{x+1} = 0$$ is cyclic (by my definition) because the function $$g(r,s) = \frac {r+1}{s+1}$$ and its corresponding substitution $$(a,b,c) = (g(x,y),g(y,z),g(z,x))$$ gives the symmetric equality $$a + b + c = 0$$. $$\frac {x-y}{y-z} + \frac {y-z}{z-x} + \frac {z-x}{x-y} = 0$$ then we would have to use the function $$g_1(r,s) = r-s$$ and get $$\frac ab + \frac bc + \frac ca = 0$$ Since this isn't symmetric, we have to use another substitution $$(d,e,f) = (g_2(d,e),g_2(e,f),g_2(f,d))$$ for $$g_2(r,s) = \frac rs$$. This brings us to the symmetric equality $$d + e + f = 0$$ Another possibility is to define a cyclic equality as an equality where we can get a new solution from a known one $$(x,y,z)_1 = (a,b,c)$$ as $$(x,y,z)_2 = (b,c,a)$$. Then, this fact you asked about is the defining property and therefore doesn't require any proof. Which definition do you like the most, or do you have any other ideas? - 3 weeks, 3 days ago	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.968036413192749, "perplexity": 741.4483096359421}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912205163.72/warc/CC-MAIN-20190326115319-20190326141319-00270.warc.gz"}

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