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http://www.onemathematicalcat.org/algebra_book/online_problems/finite_or_inf_rep.htm	Deciding if a Fraction is a Finite or Infinite Repeating Decimal DECIDING IF A FRACTION IS A FINITE OR INFINITE REPEATING DECIMAL • PRACTICE (online exercises and printable worksheets) Want more details, more exercises? Read the full text! RATIONAL and IRRATIONAL NUMBERS The rational numbers are numbers that can be written in the form $\displaystyle\,\frac{a}{b}\,$, where $\,a\,$ and $\,b\,$ are integers, and $\,b\,$ is nonzero. Recall that the integers are: $\,\ldots , -3, -2, -1, 0, 1, 2, 3,\, \ldots\,$ That is, the integers are the whole numbers, together with their opposites. Thus, the rational numbers are ratios of integers. For example, $\,\frac25\,$ and $\,\frac{-7}{4}\,$ are rational numbers. Every real number is either rational, or it isn't. If it isn't rational, then it is said to be irrational. FINITE and INFINITE REPEATING DECIMALS By doing a long division, every rational number can be written as a finite decimal or an infinite repeating decimal. A finite decimal is one that stops, like $\,0.157\,$. An infinite repeating decimal is one that has a specified sequence of digits that repeat, like $\,0.263737373737\ldots = 0.26\overline{37}\,$. Notice that in an infinite repeating decimal, the over-bar indicates the digits that repeat. PRONUNCIATION OF ‘FINITE’ and ‘INFINITE’ Finite is pronounced FIGH-night (FIGH rhymes with ‘eye’; long i). However, infinite is pronounced IN-fi-nit (both short i's). WHICH RATIONAL NUMBERS ARE FINITE DECIMALS, and WHICH ARE INFINITE REPEATING DECIMALS? • start by putting the fraction in simplest form; • then, factor the denominator into primes. • If there are only prime factors of $\,2\,$ and $\,5\,$ in the denominator, then the fraction has a finite decimal name. The following example illustrates the idea: $\displaystyle\frac{9}{60} \ = \ \frac{3}{20} \ = \ \frac{3}{2\cdot2\cdot 5}\cdot\frac{5}{5} \ = \ \frac{15}{100} \ = \ 0.15$ If there are only factors of $\,2\,$ and $\,5\,$ in the denominator, then additional factors can be introduced, as needed, so that there are equal numbers of twos and fives. Then, the denominator is a power of $\,10\,$, which is easy to write in decimal form. When the fraction is in simplest form, then any prime factors other than $\,2\,$ or $\,5\,$ in the denominator will give an infinite repeating decimal. For example: $\displaystyle\frac{1}{6} = \frac{1}{2\cdot 3} = 0.166666\ldots = 0.1\overline{6}$ (bar over just the $6$) $\displaystyle\frac{2}{7} = 0.\overline{285714}$ (bar over the digits $285714$) $\displaystyle\frac{3}{11} = 0.\overline{27}$ (bar over the digits $27$) EXAMPLES: Consider the given fraction. In decimal form, determine if the given fraction is a finite decimal, or an infinite repeating decimal. Fraction: $\displaystyle\frac25$ Fraction: $\displaystyle\frac57$ Master the ideas from this section When you're done practicing, move on to: Deciding if Numbers are Equal or Approximately Equal DO NOT USE YOUR CALCULATOR FOR THESE PROBLEMS. Feel free, however, to use pencil and paper. Consider this fraction: In decimal form, this number is a:	{"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": 0.9039232730865479, "perplexity": 1479.6750168456997}, "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/1534221210105.8/warc/CC-MAIN-20180815122304-20180815142304-00049.warc.gz"}
http://mathhelpforum.com/discrete-math/119066-solved-composition-two-relations-print.html	# [SOLVED] Composition of two relations • December 7th 2009, 04:49 AM sudeepmansh [SOLVED] Composition of two relations If R and S are two relations on a set A. Then which is the correct way to denote the composition of R and S a) R o S or b) S o R One author has suggested (a) and the other has suggested (b). • December 7th 2009, 06:53 AM Plato Quote: Originally Posted by sudeepmansh If R and S are two relations on a set A. Then which is the correct way to denote the composition of R and S a) R o S or b) S o R One author has suggested (a) and the other has suggested (b). I am confused. Which the correct one. Not to be too flippant about, it depends on which author wrote your textbook. Actually, the phrase “the composition of R and S” is much too vague to really answer that question. If you know that $R:A\mapsto B~\&~ S:B\mapsto C$ the domains demand that it be written $S \circ R :A\mapsto C$. On the other hand, if it were $R:A\mapsto A~\&~ S:A\mapsto A$ the vagueness of the phrase would allow for either. • December 7th 2009, 09:21 AM emakarov I agree that it is a matter of convention. When $R\subseteq A\times B$ and $S\subseteq B\times C$, then the composition of $R$ and $S$ is a subset of $A\times C$, and $(x,z)$ is in the composition if for some $y\in B$, $(x,y)\in R$ and $(y,z)\in S$. So to speak, $R$ is "applied" first and $S$ second, even though $R$ and $S$ are not functions and we cannot use the term "applied" in the same way we use it for functions. So far there is no ambiguity. However, with all this, we may agree to denote the composition of $R$ and $S$ by $R\circ S$ or by $S\circ R$. This is just a notation, and as as long as we use it consistently and understand what it means, we can get away with it. To give an illustration, in Genovia it may be an old tradition to denote 2 to the power 3 as $3^2$, while the rest of the world writes $2^3$. This by itself would not make the collaboration between Genovian and American mathematicians impossible because each of them understands what they are talking about. E.g., Genovians have laws like $x^z\times y^z=(x+y)^z$, which are exactly the same as in the rest of the world, only written in a weird way. That said, for functions it is pretty standard to denote $g(f(x))$ as $(g\circ f)(x)$. We say "the composition of $f$ and $g$" because $f$ is applied first, but we write $g\circ f$ to remind ourselves about $g(f(x))$. I would say that anybody who uses a different convention intentionally tries to confuse people. Now, functions are just special kinds of relations, so to keep the notation consistent, the composition of $R\subseteq A\times B$ and $S\subseteq B\times C$ should be denoted by $S\circ R$.	{"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": 33, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8996304869651794, "perplexity": 262.5017886859792}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783398216.41/warc/CC-MAIN-20160624154958-00122-ip-10-164-35-72.ec2.internal.warc.gz"}
http://tex.stackexchange.com/questions/51688/applying-lowercase-to-index-entries?answertab=votes	# Applying \lowercase to index entries I would like to sort the index entries independent of case. Initially I was getting (code provided in MWE) the image on the left. So, that seemed like an easy fix: just apply \lowercase (uncomment the \def in the MWE) but that yields the image on the right: So, how do I get these sorted alphabetically, and get the two entries for zero to be displayed as a sub-entries under a single heading? ## Failed Attempts: 1. I thought this was an explansion issue, so I tried to use \edef\LowerCaseWord{\lowercase{#2}} \index{\LowerCaseWord!#1} but this also yielded results identical to the image on the right. 2. Since a wise member here (who I think no longer wants to be associated with this comment :-)) once alluded that a few carefully placed \expandafters should fix anything I tried this and it also does not change the output: \edef\LowerCaseWord{\lowercase{#2}} \expandafter\index{\LowerCaseWord!#1} ## Code: %\def\UseLowercase{}% Uncomment to use lowercase \documentclass{article} \usepackage{imakeidx} \ifdefined\UseLowercase% Select whether we use lowercase or not \newcommand{\IndexTitle}{Index (lowercase)}% \else \newcommand{\IndexTitle}{Index}% \renewcommand{\lowercase}[1]{#1}% \fi % #1 = indexed term, #2 = word to index this under \par\noindent Indexing: #2 \index{\lowercase{#2}!#1} }% \makeindex[title={\IndexTitle},columns=1] \begin{document} \printindex \end{document} - \lowercase isn't expandable, so no amount of \expandafter can do. But there's something that can be done. Be patient. :) – egreg Apr 12 '12 at 19:22 I'd say that \newcommand*{\AddIndexEntry}[2]{% % #1 = indexed term, #2 = word to index this under \par\noindent \lowercase{\def\temp{#2}}% Indexing: #2% \expandafter\index\expandafter{\temp!#1}% } should be what you need. How does \lowercase works? It sends its argument to a further processor (it's not a macro, so it doesn't its work in TeX's “mouth”); the token list is converted using the \lccode table: each character token that hasn't a zero \lccode is converted to its lowercase correspondent, but symbolic tokens such as \def or \temp are untouched. The token list so obtained is put back in the input as if it had been there from the beginning. There's no expansion during this process: so if TeX finds \lowercase{\def\temp{Xyz}} when it's executing things, then it "waits" a bit, processes the token list as explained, then it processes \def\temp{xyz} and goes along. In case you have more than one index, you can use a modified form: \documentclass{article} \usepackage{imakeidx} % #1 = indexed term, #2 = word to index this under \par\noindent \lowercase{\def\temp{#3}}% Indexing: #3% \if!#1! \expandafter\index\expandafter{\temp!#2}% \else \expandafter\indexopt\expandafter{\temp!#2}{#1} \fi } \newcommand{\indexopt}[2]{\index[#2]{#1}} \makeindex \makeindex[name=Name,title=Title,columns=1] \begin{document} The \indexopt macro takes care of switching the arguments, so that the \expandafter doesn't need to jump over the optional argument to \index. Well, it definitely does what I need, but have to ask: what does it mean to apply a macro like \lowercase to a \def? Don't think I have seen that kind of voodoo magic before. – Peter Grill Apr 12 '12 at 19:34	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8305250406265259, "perplexity": 2675.0585179978134}, "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-2015-22/segments/1432207928501.75/warc/CC-MAIN-20150521113208-00304-ip-10-180-206-219.ec2.internal.warc.gz"}
http://mathhelpforum.com/trigonometry/134971-solving-trig-equations.html	# Math Help - Solving trig equations 1. ## Solving trig equations F(t)=-sin2t + 2cos2t=0 What is the value of t for this equation ? 2. Originally Posted by nyasha F(t)=-sin2t + 2cos2t=0 What is the value of t for this equation ? 2cos(2t) = sin(2t) Or tan(2t) = 2. Do you want to find the value of t in degrees? In that case 2t = arctan(2). Then find t.	{"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": 0.9409199357032776, "perplexity": 3313.2318926640687}, "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-2015-14/segments/1427131302478.63/warc/CC-MAIN-20150323172142-00009-ip-10-168-14-71.ec2.internal.warc.gz"}
http://napitupulu-jon.appspot.com/posts/intro-linear-regression-coursera-statistics.html	# Introduction to Linear Regression \| Source So far we've been looking into the problem where we have single categorical/numerical, relationship between categorical-numerical or categorical-categorical. In this blog, we're going to discuss about two numerical variables. We have a correlation test, want to test the strengthness relationship of both numerical variables, inference, and introduction to linear regression.For those that familliar with machine learning, this blog will make additional flavour from statistics.Why is it call linear regression? Because it's all about regress towards the line. ## Correlation¶ Screenshot taken from Coursera 01:00 The picture above is about the rate between poverty and high school graduate in the US. First we have poverty rate, as response, and we want to see whether it's affected by HS graduate, the explanatory. When looking into two numerical variable, we often observe the linearity, the direction, and whether both have strong correlation.Correlation here means linear association between two numerical variables. Non-linear will be just called association. Correlation is all about linearity, and often called linear association, and it's denoted by R.The explanatory acts as a independent variable(predictor), and the response variable is the dependent variable.We can see the linearity, by looking at the foremention 3 categories: Screenshot taken from Coursera 02:00 The higher the correlation(absolute value), represent the stronger relationship, linear between both variables. Screenshot taken from Coursera 02:28 The direction of the linear decide sign value of R. Screenshot taken from Coursera 03:09 The linear will always between -1 and 1. It will fall perfectly in a line. Whereas 0, as you can see x increases, but nothing's to do with y. Screenshot taken from Coursera 04:30 R is unitless, meaning no matter how you change the unit, or scale, it will always retain its value. Here we have R close to zero, and you can see based on the previous example, it almost show no correlation. Screenshot taken from Coursera 05:11 Even if you flip both axes, R will still be the same. Screenshot taken from Coursera 06:15 When you move even 1 point to corner outliers, R will vary significantly. correlation coefficient is sensitive to outlier. Screenshot taken from Coursera 08:20 Take a look at the example. correlation coefficient is always between -1 and 1. So you can eliminate e. Its direction is negative, so it only betwen b and c. All that's left is strong/weak correlation. Try to imagine the negative space (shaded by orange color). This will help us better intuition. R with 0.1 will almost not give us negative space. ## Residuals¶ Screenshot taken from Coursera 00:44 Residual is the distance between the actual output of y-axis and your hypothesis. This will serve as a basis of predicting the output, in this case poverty. So given grad rate, we want to predict rate in poverty. What we want to do is minimizing the residual overall. Screenshot taken from Coursera 01:44 It will overestimate if the predicted is beyond the actual output, and underestimate if the predicted below the actual output. ## Least Squares Line¶ In this section we want to talk about how we minimize the line in linear regression, by taking least squares (cost function).There's another option where we talk about the distance error as the absolute value, but as we talk before, we want to give higher magnitude to those with longer distance. So we squares all distance, this also give advantage as it's easier to calculate and more common. Screenshot taken from Coursera 01:24 So this would be familliar to those with machine learning experience. We have $\hat{y}$, as our hypothesis, the predicted output. We have intercept, as a bias unit. We have the weight parameter, a slope that can then multiplied by explanatory variable. We have seen this formula, when we're calculating gradient in high school. Recall that y = mx. With m the slope, and we have additional bias unit. Screenshot taken from Coursera 02:44 Here we have b1, as our slope point estimate (b0 will be represented as intercept point estimate). And the calculation of b1 is just swapping from previous formula.So the slope is standard deviation of response variable, times correlation coeffecient for response-explanatory, divided by standard deviaton of the explanatory variable. $$b1 = \frac{s_y}{s_x}R$$ Screenshot taken from Coursera 06:08 Here we simply have all the parameter, and can calculate the slope. Standard deviation is always positive for both response and explanatory (as you recall, it derrive from squared difference). So the sign of the slope is always depends on the sign of the correlation. If you see the image, you can see that it has negative direction. Since we're talking about the percentage rate, 0.62 is the percentage, so we can say that percentage living in porverty is lower(negative sign!) on average by 0.62%. Mind that since this is observational study, pay attention that we're trying to make a correlation and not causal statements. So 'would expect' is the chosen phrase instead of 'will'. And explanatory variable is the one unit meassurement to the predicted response variable. So we have one unit (percentage), so we say as one percent of HS graduate increase, we would expect poverty to be lower on average by 0.62 %. Pay attention to the units of response and explanatory, as both can be different. Screenshot taken from Coursera 07:11 We can replace the formula with $\bar{y}$, which denotes the average of response variables, and $\bar{x}$, which denotes the average of explanatory variable.The intercept will then simply swapping the parameters.You see that in linear regression, it expected that the line is go through the center of the data. Which means, the line is all of the average of data points. Screenshot taken from Coursera 09:28 linear regression is always about the intercept and the slope. Explaining intercept alone is less meaningful. Recall the formula as: $$\hat{y} = \beta0 + \beta1\hat{x}$$ So when the explanatory variable is zero then, $$64.68 = \beta0 \\ \beta0 = -64.68$$ Based on the formula, simplifying to just the intercept, as the picture stated, "States with no HS graduates(assumed zero explanatory for the sake of statement), are expected to have 64.68% of their residents living below the poverty line(response). Screenshot taken from Coursera 10:38 By doing computation software, the table shows the intercept and slope in the estimate column, while the rest of the table will be explained later. So in intercept, explaining only that will provide no useful information. you only setting the x = zero, which will intercept at the y-axis. And intercept only benefit as provide base y-axis(height of the line). While slope, explaning the correlation of x/y axis. As x increase(each unit), what happen to the y-axis(higher/lower?) on average. So least squares line always passes through point average of explanatory and response variable. Using this idea, we can calculate the intercept value as: $$b_0 = \bar{y} - b_1\bar{x}$$ there are different interpretation depending on whether you use numerical/categorical explanatory variable.To interpret intercept when x is numerical " When x = 0, we would expect y to be equal on average to {intercept}". When x is categorical, "We expect the value average of response variable for {reference level} of the {explanatory variable} is {intercept value}". and the slope like before calculated as: $$b_1 = \frac{s_y}{s_x}R$$ To interpret slope, If x is numerical " For each unit increase x, it's expected an increase/decrease by {slope units} for y in average". When x is categorical however, we say, "The value of response variable is predicted to have {slope units} higher/lower between reference level and the other value of explanatory variable. Increase/decrease is depend by sign of the slope. So indicator is the binary explanatory variables. And based on this fact, we can interpret intercept(with level 0) and slope(with level 1). Suppose we're given this example,"The model below predicts GPA based on an indicator variable (0: not premed, 1: premed). Interpret the intercept and slope estimates in context of the data." gpaˆ=3.57−0.01×premed For the intercept, "When premed equals zero, we would expect gpa in average to be 3.57". For the slope, "For each increase of 1 premed, we would expect the gpa on average to be lower on average by 0.01 unit" In [2]: xbar = 70 xs = 2 ybar = 140 ys = 25 R = 0.6 slope = (ys/xs)R intercept = ybar - slopexbar c(intercept,slope) Out[2]: [1] -385.0 7.5 In [6]: xp = 72 yp = 115 predicted =(intercept + slopexp) yp - predicted Out[6]: [1] -40 ## Prediction & Extrapolation¶ Recall that in the prediction, we're going to map x into the line, and infer y based on the point projected on the linear regression. In other words, which y-value that correspond to given x-value that construct point in the linear regression. Screenshot taken from Coursera 10:38 Using intercept and slope from previous example, we can simply plug the explanatory, 82 to the resulting formula. What we get is 13.84. So based we can say on states, given that the state have 82% HS graduation rate, we predict that 13.84% on average in the poverty line. But beware of some extrapolation. Screenshot taken from Coursera 02:49 Extrapolation would means that we want to infer something outside realm of the data. The data that we have are the explanatory in range from 70-95(more or less). But far more outside the realm, we don't know if the intercept consistent with linearity, or maybe in exponential(logistic) manner. Take a look at the red line, intercept with the red line would be wrong. And any exponential line would interpret wrong result outside the realm. So if the problem will need you to predict the poverty rate at 20% HS graduate, you can't do that. The resulting estimate will yield unreliable output. So we can predict for any given value in the explanatory variable, x, the response variable will be: $$\hat{y} = b0 + b1x^*$$ Always remember to not extrapolate response variable beyond the realm of data. REFERENCES: Dr. Mine Çetinkaya-Rundel, Cousera	{"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": 0.8402978181838989, "perplexity": 1655.9537442062938}, "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-2017-30/segments/1500549423812.87/warc/CC-MAIN-20170721222447-20170722002447-00435.warc.gz"}
https://en.formulasearchengine.com/wiki/Infinite_set	# Infinite set {{ safesubst:#invoke:Unsubst\|\|\$N=Refimprove \|date=__DATE__ \|\$B= {{#invoke:Message box\|ambox}} }} In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Some examples are: ## Properties The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set that is directly required by the axioms to be infinite. The existence of any other infinite set can be proved in Zermelo–Fraenkel set theory (ZFC) only by showing that it follows from the existence of the natural numbers. A set is infinite if and only if for every natural number the set has a subset whose cardinality is that natural number. If the axiom of choice holds, then a set is infinite if and only if it includes a countable infinite subset. If a set of sets is infinite or contains an infinite element, then its union is infinite. The powerset of an infinite set is infinite. Any superset of an infinite set is infinite. If an infinite set is partitioned into finitely many subsets, then at least one of them must be infinite. Any set which can be mapped onto an infinite set is infinite. The Cartesian product of an infinite set and a nonempty set is infinite. The Cartesian product of an infinite number of sets each containing at least two elements is either empty or infinite; if the axiom of choice holds, then it is infinite. If an infinite set is a well-ordered set, then it must have a nonempty subset that has no greatest element. In ZF, a set is infinite if and only if the powerset of its powerset is a Dedekind-infinite set, having a proper subset equinumerous to itself.{{ safesubst:#invoke:Unsubst\|\|date=__DATE__ \|\$B= {{#invoke:Category handler\|main}}{{#invoke:Category handler\|main}}[citation needed] }} If the axiom of choice is also true, infinite sets are precisely the Dedekind-infinite sets. If an infinite set is a well-orderable set, then it has many well-orderings which are non-isomorphic. ## History The first known{{ safesubst:#invoke:Unsubst\|\|date=__DATE__ \|\$B= {{#invoke:Category handler\|main}}{{#invoke:Category handler\|main}}[citation needed] }} occurrence of explicitly infinite sets is in Galileo's last book Two New Sciences written while he was under house arrest by the Inquisition.[1] Galileo argues that the set of squares ${\displaystyle \mathbb {S} =\{1,4,9,16,25,\ldots \}}$ is the same size as ${\displaystyle \mathbb {N} =\{1,2,3,4,5,\ldots \}}$ because there is a one-to-one correspondence: ${\displaystyle 1\leftrightarrow 1,2\leftrightarrow 4,3\leftrightarrow 9,4\leftrightarrow 16,5\leftrightarrow 25,\ldots }$ And yet, as he says, ${\displaystyle \mathbb {S} }$ is a proper subset of ${\displaystyle \mathbb {N} }$ and ${\displaystyle \mathbb {S} }$ even gets less dense as the numbers get larger.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 6, "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": 0.9712023735046387, "perplexity": 238.0968449073446}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488273983.63/warc/CC-MAIN-20210621120456-20210621150456-00629.warc.gz"}
https://galoisrepresentations.wordpress.com/2014/11/15/mysterious-formulae/?replytocom=1620	## Mysterious Formulae I’m not one of those mathematicians who is in love with abstraction for its own sake (not that there’s anything wrong with that). I can still be seduced by an explicit example, or even — quell horreur — a definite integral. When I was younger, however, those tendencies were certainly more pronounced than they are now. Still, who can fail to appreciate an identity like the following: $\displaystyle{e^{-2 \pi} \prod_{n=1}^{\infty} (1 - e^{-2 \pi n})^{24} = \frac{\Gamma(1/4)^{24}}{2^{24} \pi^{18}}}.$ But man can not live on identities alone, and ultimately one’s efforts turn in other directions. So it’s always nice when the old and new words coincide, and an identity is revealed to have a deeper meaning. The formula above is a special case of the Chowla-Selberg formula, which is, possible typos in transcription aside, $\displaystyle{\sum_{CM(K)} \log \left( y^{6} \|\Delta(\tau)\| \right) + 6h \log(4 \pi \sqrt{\Delta_K}) = 3 w_K \sum \chi(r) \log \Gamma(r/\Delta_K).}$ Here the notation is as you might guess — $y$ is the imaginary part of $\tau$, which is ranging over the equivalence classes of CM points for a fixed ring of integers in an imaginary quadratic field (there is presumably a version for orders as well). The existence of this identity (and a vague sense that it was related to the Kronecker limit formula) was basically all that I new about this identity, but Tonghai Yang gave a beautiful number theory seminar this week explaining the geometric ideas behind this formula, and some generalizations (the latter being the new work). So, just as in the Gross-Zagier paper on the special values of $j$ at CM points, one now has two proofs of this result which complement each other, one analytic, and one geometric. (I apologize in advance for not being able to attribute all [or really any] of the ideas, Tonghai certainly mentioned many names but I never take notes and this was 5 days ago.) The first remark is that the RHS is essentially the logarithmic derivative of the corresponding Artin L-function. On the other hand, it turns out (non-obviously) that the left hand side can be related to the Faltings height(s) of the corresponding Elliptic curves with CM by $\mathcal{O}_K$. I think this relation was discovered by Colmez in his ’93 Annals paper. The Faltings height has always been a slippery concept to me, and in fact the theory of heights in general has always struck me as being connected to the dark arts. In particular, various definitions depend on certain choices of height function, although they actually don’t depend on that choice in the end. So when actually doing a calculation, it’s always nice if you can magically produce some choice which makes calculation possible. And of course, when making a choice of function on some (tensor power of) $\omega$ over the modular curve, what better choice is there (if one wants to control the zeros and poles) than $\Delta$. (Tonghai mentioned another version of the formula where one instead used certain forms which are Borcherds products — of which $\Delta$ is a highly degenerate example. I had the sense that this formulation was more generalizable to other Shimura varieties, but I never understood Borcherds products so I shall say no more.) Key difficulties in understanding generalizations of these formulas involve ruling out certain vertical components in certain arithmetic divisors on Shimura varieties, which I guess must ultimately be related to understanding the mod-p reduction of these varieties in recalcitrant characteristics (blech). Colmez also formulated a conjectural generalization of the CS-formula, which is what Tonghai was talking about, and on which he (and now he together with his co-authors) have made some progress. The viewpoint in the talk was to re-interpret these identities in terms of arithmetic intersection numbers of arithmetic divisors on Shimura varieties. Of course, this is intimately related to the ideas of Gross-Zagier and its subsequent developments, especially in the work of Kudla, Rapoport, Brunier, Ben Howard, and Tonghai himself (and surely others… see caveat above). In light of this, one can start to see how special values of L-functions and their derivatives might appear. I can’t possibly begin to do this topic justice in a blog post, but I will at least strongly recommend watching Ben Howard talk about this at MSRI in a few weeks (Harris-fest, Tuesday Dec 2 at 11:00). I’ll be there to watch in person, but for those of you playing at home, the video will certainly be posted online. Ben is talking about exactly this problem. Since he is an excellent lecturer, I can safely promise this will be a great talk. Added: Dick Gross emailed me the following (which also gives me the chance to say that Tonghai did indeed mention Greg Anderson during his talk): ************ …if you want to read a nice analytic treatment of the Chowla-Selberg formula, using Kronecker’s first limit formula, you can find it in the last chapter of Weil’s book “Eisenstein and Kronecker”. I found an algebraic proof of C-S when I was a graduate student, using the moduli of abelian varieties with multiplication by an imaginary quadratic field (what we would now call unitary Shimura varieties). Deligne figured out what I was actually doing, and generalized it to prove his wonderful theorem that Hodge cycles on abelian varieties are absolutely Hodge. Greg Anderson formulated a generalization of C-S for the periods of abelian varieties with complex multiplication. This was refined by Colmez, and we know how to prove all the refinements when the CM field is abelian over Q. Tonghai and Ben have been making progress in some non-abelian cases. Dick This entry was posted in Mathematics and tagged , , , , , , , , , , . Bookmark the permalink. ### 2 Responses to Mysterious Formulae 1. Eric Katz says: I’ve been curious about this result as a sort of amateur in that part of number theory. As far as I know, every geometric proof has two parts: 1. Shows that one side of the formula is motivic or constant in families; here motivic means that it depends on some linearized data abstracting being CM; constant in families means for families of a certain CM type 2. Reducing the case of Jacobians of Fermat curves where the formula is evaluated explicitly. Is there a good reason to guess the formula, even in the case of Fermat curves? • Dick Gross says: No reason to guess, but the connection seemed reasonable after David Rohrlich calculated the period lattice of the Fermat curve of exponent N explicitly. The values of the Gamma function at rational arguments (a/N) appear through Euler’s evaluation of Beta function integrals.	{"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": 9, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8483567833900452, "perplexity": 538.1324910406023}, "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-2019-26/segments/1560628000894.72/warc/CC-MAIN-20190627055431-20190627081431-00030.warc.gz"}
http://cage.ugent.be/~kthas/Fun/index.php/noncommutative-f_un-geometry-2.html	# noncommutative F_un geometry (2) Posted by on Oct 16, 2008 in researchNo comments Last time we tried to generalize the Connes-Consani approach to commutative algebraic geometry over the field with one element to the noncommutative world by considering covariant functors which over resp. become visible by a complex (resp. integral) algebra having suitable universal properties. However, we didn’t specify what we meant by a complex noncommutative variety (resp. an integral noncommutative scheme). In particular, we claimed that the -’points’ associated to the functor (here denotes all elements of order of ) were precisely the modular dessins d’enfants of Grothendieck, but didn’t give details. We’ll try to do this now. For algebras over a field we follow the definition, due to Kontsevich and Soibelman, of so called “noncommutative thin schemes”. Actually, the thinness-condition is implicit in both Soule’s-approach as that of Connes and Consani : we do not consider R-points in general, but only those of rings R which are finite and flat over our basering (or field). So, what is a noncommutative thin scheme anyway? Well, its a covariant functor (commuting with finite projective limits) from finite-dimensional (possibly noncommutative) -algebras to sets. Now, the usual dual-space operator gives an anti-equivalence of categories so a thin scheme can also be viewed as a contra-variant functor (commuting with finite direct limits) In particular, we are interested to associated to any {tex]k[/tex]-algebra its representation functor : This may look strange at first sight, but is a finite dimensional algebra and any -dimensional representation of is an algebra map and we take to be the dual coalgebra of this image. Kontsevich and Soibelman proved that every noncommutative thin scheme is representable by a -coalgebra. That is, there exists a unique coalgebra (which they call the coalgebra of ‘distributions’ of ) such that for every finite dimensional -algebra we have In the case of interest to us, that is for the functor the coalgebra of distributions is Kostant’s dual coalgebra . This is the not the full linear dual of but contains only those linear functionals on which factor through a finite dimensional quotient. So? You’ve exchanged an algebra for some coalgebra , but where’s the geometry in all this? Well, let’s look at the commutative case. Suppose is the coordinate ring of a smooth affine variety , then its dual coalgebra looks like the direct sum of all universal (co)algebras of tangent spaces at points . But how do we get the variety out of this? Well, any coalgebra has a coradical (being the sun of all simple subcoalgebras) and in the case just mentioned we have so every point corresponds to a unique simple component of the coradical. In the general case, the coradical of the dual coalgebra is the direct sum of all simple finite dimensional representations of . That is, the direct summands of the coalgebra give us a noncommutative variety whose points are the simple representations, and the remainder of the coalgebra of distributions accounts for infinitesimal information on these points (as do the tangent spaces in the commutative case). In fact, it was a surprise to me that one can describe the dual coalgebra quite explicitly, and that -structures make their appearance quite naturally. See this paper if you’re in for the details on this. That settles the problem of what we mean by the noncommutative variety associated to a complex algebra. But what about the integral case? In the above, we used extensively the theory of Kostant-duality which works only for algebras over fields… Well, not quite. In the case of (or more general, of Dedekind domains) one can repeat Kostant’s proof word for word provided one takes as the definition of the dual -coalgebra of an algebra (which is -torsion free) (over general rings there may be also variants of this duality, as in Street’s book an Quantum groups). Probably lots of people have come up with this, but the only explicit reference I have is to the first paper I’ve ever written. So, also for algebras over we can define a suitable noncommutative integral scheme (the coradical approach accounts only for the maximal ideals rather than all primes, but somehow this is implicit in all approaches as we consider only thin schemes). Fine! So, we can make sense of the noncommutative geometrical objects corresponding to the group-algebras and where is the modular group (the algebras corresponding to the -functor). But, what might be the points of the noncommutative scheme corresponding to ??? Well, let’s continue the path cut out before. “Points” should correspond to finite dimensional “simple representations”. Hence, what are the finite dimensional simple -representations of ? (Or, for that matter, of any group ) Here we come back to Javier’s post on this : a finite dimensional -vectorspace is a finite set. A -representation on this set (of n-elements) is a group-morphism hence it gives a permutation representation of on this set. But then, if finite dimensional -representations of are the finite permutation representations, then the simple ones are the transitive permutation representations. That is, the points of the noncommutative scheme corresponding to are the conjugacy classes of subgroups such that is finite. But these are exactly the modular dessins d’enfants introduced by Grothendieck as I explained a while back elsewhere (see for example this post and others in the same series). Print This Post	{"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": 0.9415494799613953, "perplexity": 592.368983979125}, "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/1531676589557.39/warc/CC-MAIN-20180717031623-20180717051623-00638.warc.gz"}
https://www.sawaal.com/quantitative-aptitude-arithmetic-ability-questions-and-answers/the-simple-interest-accrued-on-an-amount-of-rs-2-500-at-the-end-of-six-years-is-rs1875-what-would-be_9873	22 Q: # The Simple interest accrued on an amount of Rs. 2,500 at the end of six years is Rs.1875. What would be the simple interest accrued on an amount of Rs. 6875 at the same rate and Same period? A) Rs. 4,556.5 B) Rs. 5,025.25 C) Rs.5,245.5 D) None of these Explanation: Q: ΔXYZ is right angled at Y. If cosX = 3/5, then what is the value of cosecZ ? A) 3/4 B) 5/3 C) 4/5 D) 4/3 Explanation: 0 0 Q: If the cost price of 20 books is the same as selling price of 25 books, then the loss percentage is A) 20 B) 25 C) 22 D) 24 Explanation: 0 11 Q: The railway fares of air conditioned sleeper and ordinary sleeper class are in the ratio 4:1. The number of passengers travelled by air conditioned sleeper and ordinary sleeper classes were in the ratio 3:25. If the total collection was Rs. 37,000, how much did air conditioner sleeper passengers pay? A) Rs. 15,000 B) Rs. 10,000 C) Rs. 12,000 D) Rs. 16,000 Explanation: 0 2 Q: While selling a shirt, a shopkeeper gives a discount of 7%. If he gives discount of 9% he earns Rs. 15 less on profit. The market price of the shirt is : A) 712 B) 787 C) 750 D) 697 Explanation: 0 2 Q: A hollow hemispherical bowl is made of silver with its outer radius 8 cm and inner radius 4 cm respectively. The bowl is melted to form a solid right circular cone of radius 8 cm. The height of the cone formed is A) 7 cm B) 9 cm C) 12 cm D) 14 cm Explanation: 0 3 Q: Koushik can do a piece of work in X days and Krishnu can do the same work in Y days. If they work together, then they can do the work in A) (X+Y) days B) 1x+y days C) xyx+y days D) x+yxy days Explanation: 0 2 Q: If a - b = -5 and $a2+b2=73$, then find ab. A) 35 B) 14 C) 50 D) 24 Explanation: 0 0 Q: Which of the following is correct? A) (6x + y)(x - 6y) = 6x^2+ 35xy - 6y^2 B) (6x + y)(x - 6y) = 6x^2- 35xy - 6y^2 C) (6x + y)(x - 6y) = 6x^2- 37xy - 6y^2 D) (6x + y)(x - 6y) = 6x^2+ 37xy - 6y^2 Answer & Explanation Answer: B) (6x + y)(x - 6y) = 6x^2- 35xy - 6y^2 Explanation:	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8757437467575073, "perplexity": 2406.6667895558016}, "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/1573496670559.66/warc/CC-MAIN-20191120134617-20191120162617-00293.warc.gz"}
http://mathhelpforum.com/advanced-statistics/170149-complement-proof-independent-events.html	# Math Help - Complement Proof of independent events 1. ## Complement Proof of independent events If A and B are independent events, show that $\bar{A} \ \ \text{and} \ \ \bar{B}$ are independent. I haven't a clue what to do for this one. 2. Originally Posted by dwsmith If A and B are independent events, show that $\bar{A} \ \ \text{and} \ \ \bar{B}$ are independent. I haven't a clue what to do for this one. I usually write $A^c$ for complements. You can show $P(A^c \cap B^c) = P(A^c) P(B^c)$. $P(A^c \cap B^c)$ $= P[(A \cup B)^c]$ $= 1-P(A \cup B)$ $= 1-P(A)-P(B)+P(A\cap B)$ now use $P(A\cap B) = P(A) P(B)\; and\; P(A^c)=1-P(A)\;and\;P(B^c)=1-P(B)$ and simplify	{"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": 9, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8923864364624023, "perplexity": 497.13081552186435}, "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-2014-41/segments/1410657124356.76/warc/CC-MAIN-20140914011204-00266-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"}
https://www.physicsforums.com/threads/thermal-conduction-between-3-rods.818425/	# Thermal conduction between 3 rods Tags: 1. Jun 10, 2015 ### j3dwards 1. The problem statement, all variables and given/known data Rods of copper, brass and steel are welded together to form a Y-shaped figure. The cross-sectional area of each rod is 2.0 cm2 . The free end of the copper rod is maintained at 100C, and the free ends of the brass and steel rods at 0 C. Assume there is no heat loss from the surface of the rods. The lengths of the rods are: copper, 13 cm; brass, 18 cm; steel, 24 cm. The thermal conductivities are: copper, 385 W m−1 K −1 ; brass, 109 W m−1 K −1 ; steel, 50.2 W m−1 K −1 (a)What is the temperature of the junction point? (b)What is the heat current in each of the three rods? 2. Relevant equations H = kA (TH - TC)/L 3. The attempt at a solution (a) I assumed that the heat flow through all 3 rods was the same: kc (100 - T)/Lc = kb (T - 0.0)/Lb = ks (T - 0.0)/Ls Lb kc (100 - T) = Lc kb]T And with rearranging: T = (100Lbkc)/(Lckb + Lbkc) = 83.0C Is this correct? Can I just assume that heat flow is the same and ignore the steel rod? (b) Do i just used: H = kA (TH - TC)/L again but for each metal? Because H = dQ/dt? Copper: dQ/dt = (385)(2 x 10-4)(100-83)/0.13 = 10.1 Last edited: Jun 10, 2015 2. Jun 10, 2015 ### Hesch I don't think that's a good assumption. I'd switch the thermal circuit to an electric circuit ( temperatures = voltages, thermal conductivity = resistors, heat flow = current ). Use Kirchhoffs current law ( KCL ) to calculate voltage and currents. ( It's only one equation needed ). 3. Jun 11, 2015 ### rude man Presumably, the junction temperature will be somewhere between 0 and 100C. Does it then make sense that, the ends of the steel rod being at different temperatures, that there be no heat flow thru the steel rod? Draft saved Draft deleted Similar Discussions: Thermal conduction between 3 rods	{"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": 0.8178828954696655, "perplexity": 3506.8165754533557}, "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/1502886103891.56/warc/CC-MAIN-20170817170613-20170817190613-00647.warc.gz"}
https://wikivisually.com/wiki/Planck_constant	# Planck constant Value of h Units Ref. 6.62607015×10−34 Js [1] 4.135667696×10−15 eVs [note 2] 2π EPtP Values of ħ (h-bar) Units Ref. 1.054571817×10−34 Js [note 3] 6.582119569×10−16 eVs [note 4] 1 EPtP Values of hc Units Ref. 1.98644568×10−25 Jm [note 5] 1.23984193 eVμm [note 6] 2π EPP Values of ħc (h-bar) Units Ref. 3.16152649×10−26 Jm 0.1973269804 eVμm 1 EPP Plaque at the Humboldt University of Berlin: "Max Planck, discoverer of the elementary quantum of action ${\displaystyle h}$, taught in this building from 1889 to 1928." The Planck constant, or Planck's constant, denoted ${\displaystyle h}$ is a physical constant that is the quantum of electromagnetic action, which relates the energy carried by a photon to its frequency. A photon's energy is equal to its frequency multiplied by the Planck constant; the Planck constant is of fundamental importance in quantum mechanics, and in metrology it is the basis for the definition of the kilogram. The value of Planck constant is ${\displaystyle h=6.626\ 070\ 15\times 10^{-34}\ {\rm {J\cdot s}}}$,[2] as published by 2018 CODATA. The value of Planck constant is defined as exact, with no uncertainty. At the end of the 19th century, physicists were unable to explain why the observed spectrum of black body radiation, which is still considered to have been accurately measured, diverged significantly at higher frequencies from that predicted by existing theories. In 1900, Max Planck empirically derived a formula for the observed spectrum, he assumed that a hypothetical electrically charged oscillator in a cavity that contained black-body radiation could only change its energy in a minimal increment, ${\displaystyle E}$, that was proportional to the frequency of its associated electromagnetic wave.[3] He was able to calculate the proportionality constant, ${\displaystyle h}$, from the experimental measurements, and that constant is named in his honor. In 1905, the value ${\displaystyle E}$ was associated by Albert Einstein with a "quantum" or minimal element of the energy of the electromagnetic wave itself. The light quantum behaved in some respects as an electrically neutral particle, as opposed to an electromagnetic wave, it was eventually called a photon. Max Planck received the 1918 Nobel Prize in Physics "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta." Since energy and mass are equivalent, the Planck constant also relates mass to frequency. ## Origin of the constant Intensity of light emitted from a black body at any given wavelength. Each curve represents behaviour at a different body temperature. Max Planck was the first to explain the shape of these curves. In the last years of the 19th century, Max Planck was investigating the problem of black-body radiation first posed by Kirchhoff some 40 years earlier. Every physical body spontaneously and continuously emits electromagnetic radiation. At low frequencies, Planck's law tends to the Rayleigh–Jeans law, while in the limit of high frequencies (i.e. small wavelengths) it tends to the Wien approximation, but there was no overall expression or explanation for the shape of the observed emission spectrum. Approaching this problem, Planck hypothesized that the equations of motion for light describe a set of harmonic oscillators, one for each possible frequency, he examined how the entropy of the oscillators varied with the temperature of the body, trying to match Wien's law, and was able to derive an approximate mathematical function for the black-body spectrum.[3] To create Planck's law, which correctly predicts blackbody emissions by fitting the observed curves, he multiplied the classical expression by a factor that involves a constant, ${\displaystyle h}$, in both the numerator and the denominator, which subsequently became known as the Planck Constant. The spectral radiance of a body, ${\displaystyle B_{\nu }}$, describes the amount of energy it emits at different radiation frequencies. It is the power emitted per unit area of the body, per unit solid angle of emission, per unit frequency. Planck showed that the spectral radiance of a body for frequency ν at absolute temperature T is given by ${\displaystyle B_{\nu }(\nu ,T)={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{k_{\mathrm {B} }T}}-1}}}$ where ${\displaystyle k_{B}}$ is the Boltzmann constant, ${\displaystyle h}$ is the Planck constant, and ${\displaystyle c}$ is the speed of light in the medium, whether material or vacuum.[4][5][6] The spectral radiance can also be expressed per unit wavelength ${\displaystyle \lambda }$ instead of per unit frequency. In this case, it is given by ${\displaystyle B_{\lambda }(\lambda ,T)={\frac {2hc^{2}}{\lambda ^{5}}}{\frac {1}{e^{\frac {hc}{\lambda k_{\mathrm {B} }T}}-1}},}$ showing how radiated energy emitted at shorter wavelengths increases more rapidly with temperature than energy emitted at longer wavelengths.[7] The law may also be expressed in other terms, such as the number of photons emitted at a certain wavelength, or the energy density in a volume of radiation; the SI units of ${\displaystyle B_{\nu }}$ are W·sr−1·m−2·Hz−1, while those of ${\displaystyle B_{\lambda }}$ are W·sr−1·m−3. Planck soon realized that his solution was not unique. There were several different solutions, each of which gave a different value for the entropy of the oscillators.[3] To save his theory, Planck resorted to using the then-controversial theory of statistical mechanics,[3] which he described as "an act of despair … I was ready to sacrifice any of my previous convictions about physics."[8] One of his new boundary conditions was to interpret UN [the vibrational energy of N oscillators] not as a continuous, infinitely divisible quantity, but as a discrete quantity composed of an integral number of finite equal parts. Let us call each such part the energy element ε; — Planck, On the Law of Distribution of Energy in the Normal Spectrum[3] With this new condition, Planck had imposed the quantization of the energy of the oscillators, "a purely formal assumption … actually I did not think much about it…" in his own words,[9] but one which would revolutionize physics. Applying this new approach to Wien's displacement law showed that the "energy element" must be proportional to the frequency of the oscillator, the first version of what is now sometimes termed the "Planck–Einstein relation": ${\displaystyle E=hf.}$ Planck was able to calculate the value of ${\displaystyle h}$ from experimental data on black-body radiation: his result, 6.55×10−34 J⋅s, is within 1.2% of the currently accepted value.[3] He also made the first determination of the Boltzmann constant ${\displaystyle k_{B}}$ from the same data and theory.[10] The divergence of the theoretical Rayleigh–Jeans (black) curve from the observed Planck curves at different temperatures. ## Development and application The black-body problem was revisited in 1905, when Rayleigh and Jeans (on the one hand) and Einstein (on the other hand) independently proved that classical electromagnetism could never account for the observed spectrum; these proofs are commonly known as the "ultraviolet catastrophe", a name coined by Paul Ehrenfest in 1911. They contributed greatly (along with Einstein's work on the photoelectric effect) in convincing physicists that Planck's postulate of quantized energy levels was more than a mere mathematical formalism; the very first Solvay Conference in 1911 was devoted to "the theory of radiation and quanta".[11] ### Photoelectric effect The photoelectric effect is the emission of electrons (called "photoelectrons") from a surface when light is shone on it, it was first observed by Alexandre Edmond Becquerel in 1839, although credit is usually reserved for Heinrich Hertz,[12] who published the first thorough investigation in 1887. Another particularly thorough investigation was published by Philipp Lenard in 1902.[13] Einstein's 1905 paper[14] discussing the effect in terms of light quanta would earn him the Nobel Prize in 1921,[12] when his predictions had been confirmed by the experimental work of Robert Andrews Millikan;[15] the Nobel committee awarded the prize for his work on the photo-electric effect, rather than relativity, both because of a bias against purely theoretical physics not grounded in discovery or experiment, and dissent amongst its members as to the actual proof that relativity was real.[16][17] Prior to Einstein's paper, electromagnetic radiation such as visible light was considered to behave as a wave: hence the use of the terms "frequency" and "wavelength" to characterise different types of radiation; the energy transferred by a wave in a given time is called its intensity. The light from a theatre spotlight is more intense than the light from a domestic lightbulb; that is to say that the spotlight gives out more energy per unit time and per unit space (and hence consumes more electricity) than the ordinary bulb, even though the colour of the light might be very similar. Other waves, such as sound or the waves crashing against a seafront, also have their own intensity. However, the energy account of the photoelectric effect didn't seem to agree with the wave description of light. The "photoelectrons" emitted as a result of the photoelectric effect have a certain kinetic energy, which can be measured; this kinetic energy (for each photoelectron) is independent of the intensity of the light,[13] but depends linearly on the frequency;[15] and if the frequency is too low (corresponding to a photon energy that is less than the work function of the material), no photoelectrons are emitted at all, unless a plurality of photons, whose energetic sum is greater than the energy of the photoelectrons, acts virtually simultaneously (multiphoton effect).[18] Assuming the frequency is high enough to cause the photoelectric effect, a rise in intensity of the light source causes more photoelectrons to be emitted with the same kinetic energy, rather than the same number of photoelectrons to be emitted with higher kinetic energy.[13] Einstein's explanation for these observations was that light itself is quantized; that the energy of light is not transferred continuously as in a classical wave, but only in small "packets" or quanta; the size of these "packets" of energy, which would later be named photons, was to be the same as Planck's "energy element", giving the modern version of the Planck–Einstein relation: ${\displaystyle E=hf.}$ Einstein's postulate was later proven experimentally: the constant of proportionality between the frequency of incident light ${\displaystyle f}$ and the kinetic energy of photoelectrons ${\displaystyle E}$ was shown to be equal to the Planck constant ${\displaystyle h}$.[15] ### Atomic structure A schematization of the Bohr model of the hydrogen atom. The transition shown from the n = 3 level to the n = 2 level gives rise to visible light of wavelength 656 nm (red), as the model predicts. Niels Bohr introduced the first quantized model of the atom in 1913, in an attempt to overcome a major shortcoming of Rutherford's classical model.[19] In classical electrodynamics, a charge moving in a circle should radiate electromagnetic radiation. If that charge were to be an electron orbiting a nucleus, the radiation would cause it to lose energy and spiral down into the nucleus. Bohr solved this paradox with explicit reference to Planck's work: an electron in a Bohr atom could only have certain defined energies ${\displaystyle E_{n}}$ ${\displaystyle E_{n}=-{\frac {hcR_{\infty }}{n^{2}}},}$ where ${\displaystyle c}$ is the speed of light in vacuum, ${\displaystyle R_{\infty }}$ is an experimentally determined constant (the Rydberg constant) and ${\displaystyle n\in \{1,2,3,...\}}$. Once the electron reached the lowest energy level (${\displaystyle n=1}$), it could not get any closer to the nucleus (lower energy). This approach also allowed Bohr to account for the Rydberg formula, an empirical description of the atomic spectrum of hydrogen, and to account for the value of the Rydberg constant ${\displaystyle R_{\infty }}$ in terms of other fundamental constants. Bohr also introduced the quantity ${\displaystyle {\frac {h}{2\pi }}}$, now known as the reduced Planck constant, as the quantum of angular momentum. At first, Bohr thought that this was the angular momentum of each electron in an atom: this proved incorrect and, despite developments by Sommerfeld and others, an accurate description of the electron angular momentum proved beyond the Bohr model; the correct quantization rules for electrons – in which the energy reduces to the Bohr model equation in the case of the hydrogen atom – were given by Heisenberg's matrix mechanics in 1925 and the Schrödinger wave equation in 1926: the reduced Planck constant remains the fundamental quantum of angular momentum. In modern terms, if ${\displaystyle J}$ is the total angular momentum of a system with rotational invariance, and ${\displaystyle J_{z}}$ the angular momentum measured along any given direction, these quantities can only take on the values {\displaystyle {\begin{aligned}J^{2}=j(j+1)\hbar ^{2},\qquad &j=0,{\tfrac {1}{2}},1,{\tfrac {3}{2}},\ldots ,\\J_{z}=m\hbar ,\qquad \qquad \quad &m=-j,-j+1,\ldots ,j.\end{aligned}}} ### Uncertainty principle The Planck constant also occurs in statements of Werner Heisenberg's uncertainty principle. Given a large number of particles prepared in the same state, the uncertainty in their position, ${\displaystyle \Delta x}$, and the uncertainty in their momentum, ${\displaystyle \Delta p_{x}}$, obey ${\displaystyle \Delta x\,\Delta p_{x}\geq {\frac {\hbar }{2}},}$ where the uncertainty is given as the standard deviation of the measured value from its expected value. There are a number of other such pairs of physically measurable values which obey a similar rule. One example is time vs. energy. The either-or nature of uncertainty forces measurement attempts to choose between trade offs, and given that they are quanta, the trade offs often take the form of either-or (as in Fourier analysis), rather than the compromises and gray areas of time series analysis. In addition to some assumptions underlying the interpretation of certain values in the quantum mechanical formulation, one of the fundamental cornerstones to the entire theory lies in the commutator relationship between the position operator ${\displaystyle {\hat {x}}}$ and the momentum operator ${\displaystyle {\hat {p}}}$: ${\displaystyle [{\hat {p}}_{i},{\hat {x}}_{j}]=-i\hbar \delta _{ij},}$ where ${\displaystyle \delta _{ij}}$ is the Kronecker delta. ## Photon energy The Planck–Einstein relation connects the particular photon energy E with its associated wave frequency f: ${\displaystyle E=hf}$ This energy is extremely small in terms of ordinarily perceived everyday objects. Since the frequency f, wavelength λ, and speed of light c are related by ${\displaystyle f={\frac {c}{\lambda }}}$, the relation can also be expressed as ${\displaystyle E={\frac {hc}{\lambda }}.}$ The de Broglie wavelength λ of the particle is given by ${\displaystyle \lambda ={\frac {h}{p}}}$ where p denotes the linear momentum of a particle, such as a photon, or any other elementary particle. In applications where it is natural to use the angular frequency (i.e. where the frequency is expressed in terms of radians per second instead of cycles per second or hertz) it is often useful to absorb a factor of 2π into the Planck constant. The resulting constant is called the reduced Planck constant, it is equal to the Planck constant divided by 2π, and is denoted ħ (pronounced "h-bar"): ${\displaystyle \hbar ={\frac {h}{2\pi }}.}$ The energy of a photon with angular frequency ω = 2πf is given by ${\displaystyle E=\hbar \omega ,}$ while its linear momentum relates to ${\displaystyle p=\hbar k,}$ where k is an angular wavenumber. In 1923, Louis de Broglie generalized the Planck–Einstein relation by postulating that the Planck constant represents the proportionality between the momentum and the quantum wavelength of not just the photon, but the quantum wavelength of any particle; this was confirmed by experiments soon afterwards. This holds throughout quantum theory, including electrodynamics. Problems can arise when dealing with frequency or the Planck constant because the units of angular measure (cycle or radian) are omitted in SI.[20][21][22] In the language of quantity calculus,[23] the expression for the "value" of the Planck constant, or of a frequency, is the product of a "numerical value" and a "unit of measurement"; when we use the symbol f (or ν) for the value of a frequency it implies the units cycles per second or hertz, but when we use the symbol ω for its value it implies the units radians per second; the numerical values of these two ways of expressing the value of a frequency have a ratio of 2π, but their values are equal. Omitting the units of angular measure "cycle" and "radian" can lead to an error of 2π. A similar state of affairs occurs for the Planck constant. We use the symbol h when we express the value of the Planck constant in J⋅s/cycle, and we use the symbol ħ when we express its value in J⋅s/rad. Since both represent the value of the Planck constant, but in different units, we have h = ħ. Their "values" are equal but, as discussed below, their "numerical values" have a ratio of 2π. In this Wikipedia article the word "value" as used in the tables means "numerical value", and the equations involving the Planck constant and/or frequency actually involve their numerical values using the appropriate implied units; the distinction between "value" and "numerical value" as it applies to frequency and the Planck constant is explained in more detail in this pdf file Link. These two relations are the temporal and spatial component parts of the special relativistic expression using 4-vectors. ${\displaystyle P^{\mu }=\left({\frac {E}{c}},{\vec {p}}\right)=\hbar K^{\mu }=\hbar \left({\frac {\omega }{c}},{\vec {k}}\right)}$ Classical statistical mechanics requires the existence of h (but does not define its value).[24] Eventually, following upon Planck's discovery, it was recognized that physical action cannot take on an arbitrary value. Instead, it must be some integer multiple of a very small quantity, the "quantum of action", now called the reduced Planck constant or the natural unit of action; this is the so-called "old quantum theory" developed by Bohr and Sommerfeld, in which particle trajectories exist but are hidden, but quantum laws constrain them based on their action. This view has been largely replaced by fully modern quantum theory, in which definite trajectories of motion do not even exist, rather, the particle is represented by a wavefunction spread out in space and in time, thus there is no value of the action as classically defined. Related to this is the concept of energy quantization which existed in old quantum theory and also exists in altered form in modern quantum physics. Classical physics cannot explain either quantization of energy or the lack of a classical particle motion. In many cases, such as for monochromatic light or for atoms, quantization of energy also implies that only certain energy levels are allowed, and values in between are forbidden.[25] ## Value The Planck constant has dimensions of physical action; i.e., energy multiplied by time, or momentum multiplied by distance, or angular momentum. In SI units, the Planck constant is expressed in joule-seconds (J⋅s or Nms or kg⋅m2⋅s−1). Implicit in the dimensions of the Planck constant is the fact that the SI unit of frequency, the Hertz, represents one complete cycle, 360 degrees or 2π radians, per second. An angular frequency in radians per second is often more natural in mathematics and physics and many formulas use a reduced Planck constant (pronounced h-bar) ${\displaystyle h=6.626\ 070\ 15\times 10^{-34}\ {\text{J}}{\cdot }{\text{s}}}$ ${\displaystyle \hbar ={{h} \over {2\pi }}=1.054\ 571\ 817...\times 10^{-34}\ {\text{J}}{\cdot }{\text{s}}=6.582\ 119\ 569...\times 10^{-16}\ {\text{eV}}{\cdot }{\text{s}}}$ The above values are recommended by 2018 CODATA. In atomic units, ${\displaystyle h=2\pi {\text{ a.u.}}}$ ${\displaystyle \hbar =\ \ 1{\text{ a.u.}}}$ ### Understanding the 'fixing' of the value of h Since 2019, the numerical value of the Planck constant has been fixed, with infinite significant figures. Under the present definition of the kilogram, which states "the kilogram is defined by taking the fixed numerical value of h to be 6.62607015×10−34 when expressed in the unit J⋅s, which is equal to kg⋅m2⋅s−1, where the metre and the second are defined in terms of speed of light c and duration of hyperfine transition of the ground state of an unperturbed Cesium-133 atom ΔνCs."[26] This implies that mass metrology is now aimed to find the value of one kilogram, and thus it is kilogram which is compensating; every experiment aiming to measure the kilogram (such as the Kibble balance and the X-ray crystal density method), will essentially refine the value of a kilogram. As an illustration of this, suppose the decision of making h to be exact was taken in 2010, when its measured value was 6.62606957×10−34 J⋅s, thus the present definition of kilogram was also enforced. In future, the value of one kilogram must have become refined to ${\displaystyle {6.626\ 070\ 15 \over 6.626\ 069\ 57}\approx \ 1.0\ 000\ 001}$ times the mass of the International Prototype of the Kilogram (IPK), neglecting the metre and second units' share, for sake of simplicity. ## Significance of the value The Planck constant is related to the quantization of light and matter, it can be seen as a subatomic-scale constant. In a unit system adapted to subatomic scales, the electronvolt is the appropriate unit of energy and the petahertz the appropriate unit of frequency. Atomic unit systems are based (in part) on the Planck constant; the physical meaning of the Planck's constant could suggest some basic features of our physical world. The Planck constant is one of the smallest constants used in physics; this reflects the fact that on a scale adapted to humans, where energies are typically of the order of kilojoules and times are typically of the order of seconds or minutes, the Planck constant (the quantum of action) is very small. One can regard the Planck constant to be only relevant to the microscopic scale instead of the macroscopic scale in our everyday experience. Equivalently, the order of the Planck constant reflects the fact that everyday objects and systems are made of a large number of microscopic particles. For example, green light with a wavelength of 555 nanometres (a wavelength that can be perceived by the human eye to be green) has a frequency of 540 THz (540×1012 Hz). Each photon has an energy E = hf = 3.58×10−19 J. That is a very small amount of energy in terms of everyday experience, but everyday experience is not concerned with individual photons any more than with individual atoms or molecules. An amount of light more typical in everyday experience (though much larger than the smallest amount perceivable by the human eye) is the energy of one mole of photons; its energy can be computed by multiplying the photon energy by the Avogadro constant, NA = 6.02214076×1023 mol−1, with the result of 216 kJ/mol, about the food energy in three apples. ## Determination In principle, the Planck constant can be determined by examining the spectrum of a black-body radiator or the kinetic energy of photoelectrons, and this is how its value was first calculated in the early twentieth century. In practice, these are no longer the most accurate methods. Since the value of the Planck constant is fixed now, it is no longer determined or calculated in laboratories; some of the below given practices used to determine Planck constant are now used to determine the value of kilogram.The below given methods except the X-ray crystal density method rely on the theoretical basis of the Josephson effect and the quantum Hall effect. ### Josephson constant The Josephson constant KJ relates the potential difference U generated by the Josephson effect at a "Josephson junction" with the frequency ν of the microwave radiation. The theoretical treatment of Josephson effect suggests very strongly that KJ = 2e/h. ${\displaystyle K_{\rm {J}}={\frac {\nu }{U}}={\frac {2e}{h}}\,}$ The Josephson constant may be measured by comparing the potential difference generated by an array of Josephson junctions with a potential difference which is known in SI volts; the measurement of the potential difference in SI units is done by allowing an electrostatic force to cancel out a measurable gravitational force, in a Kibble balance. Assuming the validity of the theoretical treatment of the Josephson effect, KJ is related to the Planck constant by ${\displaystyle h={\frac {8\alpha }{\mu _{0}c_{0}K_{\rm {J}}^{2}}}.}$ ### Kibble balance A Kibble balance (formerly known as a watt balance)[27] is an instrument for comparing two powers, one of which is measured in SI watts and the other of which is measured in conventional electrical units. From the definition of the conventional watt W90, this gives a measure of the product KJ2RK in SI units, where RK is the von Klitzing constant which appears in the quantum Hall effect. If the theoretical treatments of the Josephson effect and the quantum Hall effect are valid, and in particular assuming that RK = h/e2, the measurement of KJ2RK is a direct determination of the Planck constant. ${\displaystyle h={\frac {4}{K_{\rm {J}}^{2}R_{\rm {K}}}}.}$ ### Magnetic resonance The gyromagnetic ratio γ is the constant of proportionality between the frequency ν of nuclear magnetic resonance (or electron paramagnetic resonance for electrons) and the applied magnetic field B: ν = γB. It is difficult to measure gyromagnetic ratios precisely because of the difficulties in precisely measuring B, but the value for protons in water at 25 °C is known to better than one part per million. The protons are said to be "shielded" from the applied magnetic field by the electrons in the water molecule, the same effect that gives rise to chemical shift in NMR spectroscopy, and this is indicated by a prime on the symbol for the gyromagnetic ratio, γp. The gyromagnetic ratio is related to the shielded proton magnetic moment μp, the spin number I (I = 12 for protons) and the reduced Planck constant. ${\displaystyle \gamma _{\rm {p}}^{\prime }={\frac {\mu _{\rm {p}}^{\prime }}{I\hbar }}={\frac {2\mu _{\rm {p}}^{\prime }}{\hbar }}}$ The ratio of the shielded proton magnetic moment μp to the electron magnetic moment μe can be measured separately and to high precision, as the imprecisely known value of the applied magnetic field cancels itself out in taking the ratio. The value of μe in Bohr magnetons is also known: it is half the electron g-factor ge. Hence ${\displaystyle \mu _{\rm {p}}^{\prime }={\frac {\mu _{\rm {p}}^{\prime }}{\mu _{\rm {e}}}}{\frac {g_{\rm {e}}\mu _{\rm {B}}}{2}}}$ ${\displaystyle \gamma _{\rm {p}}^{\prime }={\frac {\mu _{\rm {p}}^{\prime }}{\mu _{\rm {e}}}}{\frac {g_{\rm {e}}\mu _{\rm {B}}}{\hbar }}.}$ A further complication is that the measurement of γp involves the measurement of an electric current: this is invariably measured in conventional amperes rather than in SI amperes, so a conversion factor is required. The symbol Γp-90 is used for the measured gyromagnetic ratio using conventional electrical units. In addition, there are two methods of measuring the value, a "low-field" method and a "high-field" method, and the conversion factors are different in the two cases. Only the high-field value Γp-90(hi) is of interest in determining the Planck constant. ${\displaystyle \gamma _{\rm {p}}^{\prime }={\frac {K_{\rm {J-90}}R_{\rm {K-90}}}{K_{\rm {J}}R_{\rm {K}}}}\Gamma _{\rm {p-90}}^{\prime }({\rm {hi}})={\frac {K_{\rm {J-90}}R_{\rm {K-90}}e}{2}}\Gamma _{\rm {p-90}}^{\prime }({\rm {hi}})}$ Substitution gives the expression for the Planck constant in terms of Γp-90(hi): ${\displaystyle h={\frac {c_{0}\alpha ^{2}g_{\rm {e}}}{2K_{\rm {J-90}}R_{\rm {K-90}}R_{\infty }\Gamma _{\rm {p-90}}^{\prime }({\rm {hi}})}}{\frac {\mu _{\rm {p}}^{\prime }}{\mu _{\rm {e}}}}.}$ The Faraday constant F is the charge of one mole of electrons, equal to the Avogadro constant NA multiplied by the elementary charge e. It can be determined by careful electrolysis experiments, measuring the amount of silver dissolved from an electrode in a given time and for a given electric current. In practice, it is measured in conventional electrical units, and so given the symbol F90. Substituting the definitions of NA and e, and converting from conventional electrical units to SI units, gives the relation to the Planck constant. ${\displaystyle h={\frac {c_{0}M_{\rm {u}}A_{\rm {r}}({\rm {e}})\alpha ^{2}}{R_{\infty }}}{\frac {1}{K_{\rm {J-90}}R_{\rm {K-90}}F_{90}}}}$ ### X-ray crystal density The X-ray crystal density method is primarily a method for determining the Avogadro constant NA but as the Avogadro constant is related to the Planck constant it also determines a value for h. The principle behind the method is to determine NA as the ratio between the volume of the unit cell of a crystal, measured by X-ray crystallography, and the molar volume of the substance. Crystals of silicon are used, as they are available in high quality and purity by the technology developed for the semiconductor industry; the unit cell volume is calculated from the spacing between two crystal planes referred to as d220. The molar volume Vm(Si) requires a knowledge of the density of the crystal and the atomic weight of the silicon used. The Planck constant is given by ${\displaystyle h={\frac {M_{\rm {u}}A_{\rm {r}}({\rm {e}})c_{0}\alpha ^{2}}{R_{\infty }}}{\frac {{\sqrt {2}}d_{220}^{3}}{V_{\rm {m}}({\rm {Si}})}}.}$ ### Particle accelerator The experimental measurement of the Planck constant in the Large Hadron Collider laboratory was carried out in 2011; the study called PCC using a giant particle accelerator helped to better understand the relationships between the Planck constant and measuring distances in space.[citation needed] ## Notes 1. ^ Set on 20 November 2018, by the CGPM to this exact value. This value took effect on 20 May 2019. 2. ^ exact value; approximated upto 9 decimal places only. 3. ^ 2018 CODATA value; shown upto 9 decimal places only. 4. ^ 2018 CODATA value; shown upto 9 decimal places only. 5. ^ 2018 CODATA value; shown upto 8 decimal places only. 6. ^ 2018 CODATA value; shown upto 8 decimal places only. ## References ### Citations 1. ^ "Resolutions of the 26th CGPM" (PDF). BIPM. 2018-11-16. Retrieved 2018-11-20. 2. ^ 2018, CODATA. "NIST". The NIST Reference on Units, Constants, and Uncertainty. 3. Planck, Max (1901), "Ueber das Gesetz der Energieverteilung im Normalspectrum" (PDF), Ann. Phys., 309 (3): 553–63, Bibcode:1901AnP...309..553P, doi:10.1002/andp.19013090310. English translation: "On the Law of Distribution of Energy in the Normal Spectrum Archived 2008-04-18 at the Wayback Machine"."Archived copy" (PDF). Archived from the original (PDF) on 2011-10-06. Retrieved 2011-10-13.CS1 maint: archived copy as title (link) 4. ^ Planck 1914, pp. 6, 168 5. ^ Chandrasekhar 1960, p. 8 6. ^ Rybicki & Lightman 1979, p. 22 7. ^ Shao, Gaofeng; et al. (2019). "Improved oxidation resistance of high emissivity coatings on fibrous ceramic for reusable space systems". Corrosion Science. 146: 233–246. doi:10.1016/j.corsci.2018.11.006. 8. ^ Kragh, Helge (1 December 2000), Max Planck: the reluctant revolutionary, PhysicsWorld.com 9. ^ Kragh, Helge (1999), Quantum Generations: A History of Physics in the Twentieth Century, Princeton University Press, p. 62, ISBN 978-0-691-09552-3 10. ^ 11. ^ Previous Solvay Conferences on Physics, International Solvay Institutes, archived from the original on 16 December 2008, retrieved 12 December 2008 12. ^ a b See, e.g., Arrhenius, Svante (10 December 1922), Presentation speech of the 1921 Nobel Prize for Physics 13. ^ a b c Lenard, P. (1902), "Ueber die lichtelektrische Wirkung", Ann. Phys., 313 (5): 149–98, Bibcode:1902AnP...313..149L, doi:10.1002/andp.19023130510 14. ^ 15. ^ a b c Millikan, R. A. (1916), "A Direct Photoelectric Determination of Planck's h", Phys. Rev., 7 (3): 355–88, Bibcode:1916PhRv....7..355M, doi:10.1103/PhysRev.7.355 16. ^ Isaacson, Walter (2007-04-10), Einstein: His Life and Universe, ISBN 978-1-4165-3932-2, pp. 309–314. 17. ^ "The Nobel Prize in Physics 1921". Nobelprize.org. Retrieved 2014-04-23. 18. ^ Smith, Richard (1962), "Two Photon Photoelectric Effect", Physical Review, 128 (5): 2225, Bibcode:1962PhRv..128.2225S, doi:10.1103/PhysRev.128.2225.Smith, Richard (1963), "Two-Photon Photoelectric Effect", Physical Review, 130 (6): 2599, Bibcode:1963PhRv..130.2599S, doi:10.1103/PhysRev.130.2599.4. 19. ^ Bohr, Niels (1913), "On the Constitution of Atoms and Molecules", Phil. Mag., 6th Series, 26 (153): 1–25, doi:10.1080/14786441308634993 20. ^ Mohr, J. C.; Phillips, W. D. (2015). "Dimensionless Units in the SI". Metrologia. 52 (1): 40–47. arXiv:1409.2794. Bibcode:2015Metro..52...40M. doi:10.1088/0026-1394/52/1/40. 21. ^ Mills, I. M. (2016). "On the units radian and cycle for the quantity plane angle". Metrologia. 53 (3): 991–997. Bibcode:2016Metro..53..991M. doi:10.1088/0026-1394/53/3/991. 22. ^ Nature (2017) ‘’A Flaw in the SI system,’' Volume 548, Page 135 23. ^ Maxwell J.C. (1873) A Treatise on Electricity and Magnetism, Oxford University Press 24. ^ Giuseppe Morandi; F. Napoli; E. Ercolessi (2001), Statistical mechanics: an intermediate course, p. 84, ISBN 978-981-02-4477-4 25. ^ Einstein, Albert (2003), "Physics and Reality" (PDF), Daedalus, 132 (4): 24, doi:10.1162/001152603771338742, archived from the original (PDF) on 2012-04-15, The question is first: How can one assign a discrete succession of energy value Hσ to a system specified in the sense of classical mechanics (the energy function is a given function of the coordinates qr and the corresponding momenta pr)? The Planck constant h relates the frequency Hσ/h to the energy values Hσ. It is therefore sufficient to give to the system a succession of discrete frequency values. 26. ^ 9th edition, SI BROCHURE. "BIPM" (PDF). BIPM. 27. ^ Materese, Robin (2018-05-14). "Kilogram: The Kibble Balance". NIST. Retrieved 2018-11-13.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 64, "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": 0.9617606997489929, "perplexity": 801.2023955353462}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987833089.90/warc/CC-MAIN-20191023094558-20191023122058-00130.warc.gz"}
http://physics.stackexchange.com/questions/23207/references-for-circular-restricted-3-body-problem	# References for circular restricted 3 body problem? Does anyone know of any good references for the CR3BP -- the circular restricted 3 body problem? Emphasizing on real-life applications, and interpretation of the numerical solutions? Thank you. There are quite a few hits when I googled the topic, but I am wondering if anyone knows of any "Bible" for the subject. - Hi chump, and welcome to Physics Stack Exchange! This would be a better question if you just ask what it is you want to know about the CR3BP, rather than asking for references. People can still provide references to support their answers if they want. – David Z Apr 3 '12 at 21:32 What is the circular restricted 3-body problem? 3 bodies attracting and restricted to move along a hoop? – Ron Maimon Apr 4 '12 at 0:40	{"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": 0.8553138375282288, "perplexity": 644.1677903994276}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00457-ip-10-147-4-33.ec2.internal.warc.gz"}
http://www.gradesaver.com/textbooks/math/calculus/calculus-10th-edition/chapter-1-limits-and-their-properties-1-3-exercises-page-67/58	# Chapter 1 - Limits and Their Properties - 1.3 Exercises: 58 $\lim\limits_{x\to0}\dfrac{[1/(x+4)]-(1/4)}{x}=-\dfrac{1}{16}.$ #### Work Step by Step $f(x)=\dfrac{[1/(x+4)]-(1/4)}{x}=\dfrac{4-(x+4)}{x(4)(x+4)}=-\dfrac{1}{4x+16}=g(x).$ The function $g(x)$ agrees with the function $f(x)$ at all points except $x=0$. Therefore we find the limit as x approaches $0$ of $f(x)$ by substituting the value into $g(x)$. $\lim\limits_{x\to0}\dfrac{[1/(x+4)]-(1/4)}{x}=\lim\limits_{x\to0}\dfrac{-1}{4x+16}=-\dfrac{1}{4(0)+16}=-\dfrac{1}{16}.$ After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.	{"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": 0.8110719919204712, "perplexity": 406.9010610327328}, "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-2018-05/segments/1516084892059.90/warc/CC-MAIN-20180123171440-20180123191440-00737.warc.gz"}
https://link.springer.com/chapter/10.1007/978-3-642-55245-8_16	# On $$\hat{\mathbb{Z}}$$-Zeta Function Conference paper Part of the Contributions in Mathematical and Computational Sciences book series (CMCS, volume 7) ## Abstract We present in this note a definition of zeta function of the field $$\mathbb{Q}$$ which incorporates all p-adic L-functions of Kubota-Leopoldt for all p and also so called Soulé classes of the field $$\mathbb{Q}$$. This zeta function is a measure, which we construct using the action of the absolute Galois group $$G_{\mathbb{Q}}$$ on fundamental groups. Suffix ## Notes ### Acknowledgements These research were started in January 2011 during our visit in Max-Planck-Institut für Mathematik in Bonn. We would like to thank very much MPI for support. We would like also thank to Professor C. Greither for invitation on the conference Iwasawa 2008 in Kloster Irsee. We acknowledge the financial help of the Laboratoire de Dieudonné, which allows us to participate in the meeting Iwasawa 2012 in Heidelberg. ### References 1. Coates, J., Sujatha, R.: Cyclotomic Fields and Zeta Values. Springer Monographs in Mathematics. Springer, Berlin (2006) 2. Deligne, P.: Le groupe fondamental de la droite projective moins trois points. In: Galois Groups over Q. Mathematical Sciences Research Institute Publications, vol. 16, pp. 79–297. Springer, New York (1989)Google Scholar 3. de Shalit, E.: Iwasawa Theory of Elliptic Curves with Complex Multiplication. Perspective in Mathematics, vol. 3. Academic, Boston (1987)Google Scholar 4. Ichimura, H., Sakaguchi, K.: The non-vanishing of a certain Kummer character χ m (after Soulé), and some related topics. In: Galois Representations and Arithmetic Algebraic Geometry. Advanced Studies in Pure Mathematics, vol. 12, pp. 53–64. Kinokuniya Co./North-Holland/Elsevier, Tokyo/Amsterdam/New York (1987)Google Scholar 5. Ihara, Y.: Profinite braid groups, Galois representations and complex multiplications. Ann. Math. 123, 43–106 (1986) 6. Ihara, Y.: Braids, Galois groups, and some arithmetic functions. In: Proceedings of the International Congress of Mathematics, Kyoto, pp. 99–120. Springer (1990)Google Scholar 7. Kubota, T., Leopoldt, H.W.: Eine p-adische Theorie der Zetawerte, I. J. Reine und angew. Math. 214/215, 328–339 (1964)Google Scholar 8. Lang, S.: Cyclotomic Fields I and II. Graduate Texts in Mathematics, vol. 121. Springer, New York (1990)Google Scholar 9. Nakamura, H.: On exterior Galois representations associated with open elliptic curves. J. Math. Sci. Univ. Tokyo 2, 197–231 (1995) 10. Nakamura, H.: On arithmetic monodromy representations of Eisenstein type in fundamental groups of once punctured elliptic curves. Publ. RIMS Kyoto Univ. 49(3), 413–496 (2013) 11. Nakamura, H., Wojtkowiak, Z.: On the explicit formulae for -adic polylogarithms. In: Arithmetic Fundamental Groups and Noncommutative Algebra. Proceedings of Symposia in Pure Mathematics (AMS), vol. 70, pp. 285–294. American Mathematical Society, Providence (2002)Google Scholar 12. Nakamura, H., Wojtkowiak, Z.: Tensor and homotopy criteria for functional equations of l-adic and classical iterated integrals. In: Non-abelian Fundamental Groups and Iwasawa Theory. London Mathematical Society Lecture Note Series, vol. 393, pp. 258–310. Cambridge University Press, Cambridge/New York (2012)Google Scholar 13. Soulé, C.: Éléments Cyclotomiques en K-Théorie. Astérisque 147/148, 225–258 (1987)Google Scholar 14. Wojtkowiak, Z.: On -adic iterated integrals, I analog of Zagier conjecture. Nagoya Math. J. 176, 113–158 (2004) 15. Wojtkowiak, Z.: On -adic iterated integrals, II functional equations and l-adic polylogarithms. Nagoya Math. J. 177, 117–153 (2005) 16. Wojtkowiak, Z.: On -adic Galois periods, relations between coefficients of Galois representations on fundamental groups of a projective line minus a finite number of points. Publ. Math. de Besançon, Algèbra et Théorie des Nombres, 2007–2009, pp. 155–174, Février (2009)Google Scholar	{"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": 0.9207780957221985, "perplexity": 3501.7733292847665}, "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/1518891814290.11/warc/CC-MAIN-20180222200259-20180222220259-00710.warc.gz"}
https://wiki.loliot.net/docs/nn/basics/nn-basics/	# Neural Network Basics ## Neuron(Perceptron)# $z^l_j = \sum_k{\omega^l_{jk}a^{l-1}_k} + b^l_j$ $a^l_j = \sigma\left(z^l_j\right)$ ## Loss function# ### L2 loss function# $Loss \equiv \frac{1}{2} \lVert \mathbf{y} - \mathbf{a}^L \rVert^2 = \frac{1}{2} \sum_i{\left(y_i - a^L_i\right)^2}$ $Loss \geq 0 \quad \left(\mathbf{y}\text{ is the desired output}\right)$ ## Neural Network Training# What we need to look for through neural network training are weights and biases to minimize the consequences of the loss function. When $\mathbf{w}$ is a vector representing weights and biases, $Loss_{next} = Loss + \Delta Loss \approx Loss + \nabla Loss \cdot \Delta \mathbf{w}$ It must be $\nabla Loss \cdot \Delta \mathbf{w} < 0$, because $Loss$ should decrease. Therfore, $\Delta \mathbf{w}$ can be determined as $\Delta \mathbf{w} = - \eta \nabla Loss = - \epsilon \frac{\nabla Loss}{\lVert \nabla Loss \rVert} \quad ( \epsilon > 0)$ $\eta$ is called learning rate and $\epsilon$ is called step. If the step is large, $Loss$ may diverge, and if the step is small, the convergence speed may be slow, so an appropriate value should be determined. If $\Delta \mathbf{w}$ is determined, then $\mathbf{w}_{next}$ can be $\mathbf{w}_{next} = \mathbf{w} + \Delta \mathbf{w}$ ### Stochastic gradient descent# $\nabla Loss = \frac{1}{n}\sum_x{\nabla Loss_x}$ When the number of training inputs is very large, this can take a long time. Stochastic gradient descent works by randomly picking out a small number $m$ of randomly chosen training inputs. $\nabla Loss = \frac{1}{n}\sum_x{\nabla Loss_x} \approx \frac{1}{m}\sum^m_{i=1}{\nabla Loss_{X_i}}$ Those random training inputs $X_1, X_2, ..., X_m$ are called mini-batch. ### Forward-propagation# Forward propagation (or forward pass) refers to the calculation and storage of intermediate variables (including outputs) for a neural network in order from the input layer to the output layer. ### Back-propagation# $z^l_j = \sum_k{\omega^l_{jk}a^{l-1}_k} + b^l_j$ $a^l_j = \sigma\left(z^l_j\right)$ Back-propagation is used to find $\nabla Loss$, because it is difficult for a computer to obtain $\nabla Loss$ by differentiating loss function. Error $\delta^l_j$ of neuron $j$ in layer $l$ is defined as $\delta^l_j \equiv \frac{\partial Loss}{\partial z^l_j}$ Since $z^l_j$ was obtained from forward propagation, If we know $\mathbf{\delta}^{l+1}$, we can get $\delta^l_j$ as below. \begin{aligned} \delta^l_j = \frac{\partial Loss}{\partial z^l_j} & = \sum_i{\frac{\partial Loss}{\partial z^{l+1}_i} \frac{\partial z^{l+1}_i}{\partial z^l_j}} \quad \left( \frac{\partial z^{l+1}_i}{\partial z^l_j} = \omega^{l+1}_{ij} \, \sigma' \left(z^l_j\right) \right)\\ & = \sum_i{\frac{\partial Loss}{\partial z^{l+1}_i} \omega^{l+1}_{ij} \, \sigma' \left(z^l_j\right)} \\ & = \sum_i{\delta^{l+1}_i \omega^{l+1}_{ij} \, \sigma' \left(z^l_j\right)} \end{aligned} If we use L2 loss, since $a^L_j$ was obtained from forward propagation and $\delta^L_j = (a^L_j - y_j) \, \sigma' \left( z^L_j \right)$, we can get the errors like this: $\delta^L_j = (a^L_j - y_j) \, \sigma' \left( z^L_j \right)$ $\delta^{L-1}_j = \sum_i{ \delta^L_i \omega^L_{ij} \, \sigma' \left(z^{L-1}_j\right)} \\ \vdots$ Finally, $\nabla Loss$ can be obtained by using the errors obtained above. $\frac{\partial Loss}{\partial b^l_j} = \frac{\partial Loss}{\partial z^l_j} \frac{\partial z^l_j}{\partial b^l_j} = \delta^l_j$ $\frac{\partial Loss}{\partial \omega^l_{jk}} = \frac{\partial Loss}{\partial z^l_j} \frac{\partial z^l_j}{\partial \omega^l_{jk}} = \delta^l_j a^{l-1}_k$ ### Training# Set initail weights and biases to random and repeat process Forward-propagation -> Back-propagation -> weights and biases update. When it is judged that $Loss$ cannot be made smaller, the final weights and biases are determined. Last updated on	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 40, "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": 0.9856494665145874, "perplexity": 1146.3542559014363}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487610196.46/warc/CC-MAIN-20210613161945-20210613191945-00127.warc.gz"}
https://www.physicsforums.com/threads/what-formula-is-this.152873/	What formula is this? 1. Jan 24, 2007 thenokiaguru great forum, been looking at some past papers and saw this formula in a stats paper - don't know what it is. i want to know as it is used to calculate a question. please help... its put up as an attachment... or... t = x1 - x2 / s (√ 1/n1 + 1/n2) the x's are x bar Attached Files: • form.bmp File size: 79.4 KB Views: 181 Last edited: Jan 24, 2007 2. Jan 25, 2007 drpizza Looked familiar, so I glanced around; it looks like the two-mean hypothesis test, where s_1 = s_2 (i.e. the formula I found for the two-mean hypothesis test was the same as yours, except the s was under the square root as s_1 and s_2 squared.) Here: http://www.duxbury.com/statistics_d/templates/student_resources/0534377556_woodbury/artfinal/Formulas/Formula%205.jpg [Broken] Last edited by a moderator: May 2, 2017	{"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": 0.9026750326156616, "perplexity": 2278.0723654510784}, "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-2018-34/segments/1534221213158.51/warc/CC-MAIN-20180817221817-20180818001817-00140.warc.gz"}
https://learn.saylor.org/mod/page/view.php?id=12962&forceview=1	## Unit 1 Learning Outcomes Upon completion of this course, you will be able to: • Demonstrate an understanding the purpose of the Internet. • Demonstrate an understanding of Web History. • Demonstrate an understanding of Internet Protocols. • Demonstrate an understanding of Hypertext Transfer Protocol. • Demonstrate an understanding of Extensible Markup Language.	{"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": 0.920508861541748, "perplexity": 4641.211724937537}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986673250.23/warc/CC-MAIN-20191017073050-20191017100550-00184.warc.gz"}
http://math.stackexchange.com/questions/36447/bounding-probability-where-markov-chernoff-bounds-seem-to-fail	# Bounding probability where Markov/Chernoff bounds seem to fail This is related to the question I have asked yesterday: Expected value of max/min of random variables. Assume you have $n$ urns and $k$ balls. Each ball is placed uniformly at random in one of the urns. Let $X_i$ denote the number of balls in urn $i$ and let $X = \min\{X_1,\ldots,X_n\}$. I am looking for a $k$ such that $Pr[X < 2\log(n)] < \frac{2}{n}.$ Clearly $Pr[X < 2\log(n)] = Pr[\bigcup_{i=0}^n X_i < 2\log(n)] \leq nPr[X_1 < 2\log(n)]$ Here is where it stops for me. We have to find an upper bound for $Pr[X_1 < 2\log(n)]$ but as far as I am apt in applying Chernoff/Markov bounds, one can only get a lower bound for this kind of expression. Am I missing something? Or is there perhaps another way to solve the problem? - $X_1$ is a binomial random variable with parameters $k$ and $1/n$, so mean $\mu = k/n$. Chernoff should say $P(X_1/k < 1/n - \epsilon) \le e^{-D k}$ where $D = (1/n - \epsilon) \log(1 - \epsilon n) + (1 - 1/n + \epsilon) \log(1 + \epsilon n/(n-1))$. Chernoff bound for lower deviations reads $P(Y\le y)\le a^yE(a^{-Y})$ for every $a\ge1$. Then, as is usual in these deviations bounds, one optimizes over $a\ge1$. If $Y$ is sufficiently integrable and $y<E(Y)$, one knows there exists some $a>1$ such that $a^yE(a^{-Y})<1$, hence the upper bound is not trivial. It can be more convenient to use the equivalent upper bound $P(Y\le y)\le \mathrm{e}^{ty}E(\mathrm{e}^{-tY})$ for every $t\ge0$. In your case, any $k\le 2n\log n$ is hopeless and it seems every $k\ge6.3n\log n$ works. To see this, recall that, for $t\ge0$ and $X_1$ binomial $(k,1/n)$, $$E(\mathrm{e}^{-tX_1})=(1-(1-\mathrm{e}^{-t})/n)^k\le\exp(-(1-\mathrm{e}^{-t})k/n).$$ Using this for $k=cn\log(n)$, one gets $$P(X_1\le2\log n)\le\exp(2t\log(n)-(1-\mathrm{e}^{-t})c\log(n)).$$ This upper bound is less than $1/n^2$ as soon as $$2t\log(n)-(1-\mathrm{e}^{-t})c\log(n)\le-2\log(n),$$ that is, as soon as $c$ is such that there exists $t\ge0$ such that $$2t-(1-\mathrm{e}^{-t})c+2\le0,$$ that is, for every $c\ge c^$, with $$c^*=2\,\inf_{t\ge0}\frac{1+t}{1-\mathrm{e}^{-t}}<6.2924.$$ Note that the OP asked for an upper bound of $2/n$ and this gives an upper bound of $1/n$.	{"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": 0.9981669783592224, "perplexity": 89.40359457695115}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207928965.67/warc/CC-MAIN-20150521113208-00283-ip-10-180-206-219.ec2.internal.warc.gz"}
https://www.dhruvonmath.com/2019/02/25/eigenvectors/	# You Could Have Come Up With Eigenvectors - Here's How In the last post, we developed an intuition for matrices. We found that they are just compact representations of linear maps and that adding and multiplying matrices are just ways of combining the underlying linear maps. In this post, we’re going to dive deeper into the world of linear algebra and cover eigenvectors. Eigenvectors are central to Linear Algebra and help us understand many interesting properties of linear maps including: 1. The effect of applying the linear map repeatedly on an input. 2. How the linear map rotates the space. In fact eigenvectors were first derived to study the axis of rotation of planets! Eigenvectors helped early mathematicians study how the planets rotate. Image Source: Wikipedia. For a more modern example, eigenvectors are at the heart of one of the most important algorithms of all time - the original Page Rank algorithm that powers Google Search. #### Our Goals In this post we’re going to try and derive eigenvectors ourselves. To really create a strong motivation, we’re going to explore basis vectors, matrices in different bases, and matrix diagonalization. So hang in there and wait for the big reveal - I promise it will be really exciting when it all comes together! Everything we’ll be doing is going to be in the 2D space $R^2$ - the standard coordinate plane over real numbers you’re probably already used to. ### Basis Vectors We saw in the last post how we can derive the matrix for a given linear map $f$: $f(x)$ (as we defined it in the previous section) can be represented by the notation $\begin{bmatrix} f(\textcolor{blue}{\begin{bmatrix} 1 \\ 0 \end{bmatrix}}) & f(\textcolor{#228B22}{\begin{bmatrix} 0 \\ 1 \end{bmatrix}}) \end{bmatrix}$ $= \begin{bmatrix} \textcolor{blue}{3} & \textcolor{#228B22}{0} \\ \textcolor{blue}{0} & \textcolor{#228B22}{5} \end{bmatrix}$ This is extremely cool - we can describe the entire function and how it operates on an infinite number of points by a little 4 value table. But why did we choose $\textcolor{blue}{\begin{bmatrix} 1 \\ 0 \end{bmatrix}}$ and $\textcolor{#228B22}{\begin{bmatrix} 0 \\ 1 \end{bmatrix}}$ to define the columns of the matrix? Why not some other pair like $\textcolor{blue}{\begin{bmatrix} 3 \\ 3 \end{bmatrix}}$ and $\textcolor{#228B22}{\begin{bmatrix} 0 \\ 0 \end{bmatrix}}$? Intuitively, we think of $\textcolor{blue}{\begin{bmatrix} 1 \\ 0 \end{bmatrix}}$ and $\textcolor{#228B22}{\begin{bmatrix} 0 \\ 1 \end{bmatrix}}$ as units that we can use to create other vectors. In fact, we can break down every vector in $R^2$ into some combination of these two vectors. We can reach any point in the coordinate plan by combining our two vectors. More formally, when two vectors are able to combine in different ways to create all other vectors in $R^2$, we say that those vectors $span$ the space. The minimum number of vectors you need to span $R^2$ is 2. So when we have 2 vectors that span $R^2$, we call those vectors a basis. $\textcolor{blue}{\begin{bmatrix} 1 \\ 0 \end{bmatrix}}$ and $\textcolor{#228B22}{\begin{bmatrix} 0 \\ 1 \end{bmatrix}}$ are basis vectors for $R^2$. You can think of basis vectors as the minimal building blocks for the space. We can combine them in different amounts to reach all vectors we could care about. We can think of basis vectors as the building blocks of the space - we can combine them to create all possible vectors in the space. Image Source: instructables.com. ### Other Basis Vectors for $R^2$ Now are there other pairs of vectors that also form a basis for $R^2$? $\textcolor{blue}{\begin{bmatrix} 1 \\ 0 \end{bmatrix}}$ and $\textcolor{#228B22}{\begin{bmatrix} -1 \\ 0 \end{bmatrix}}$. Can you combine these vectors to create ${\begin{bmatrix} 2 \\ 3 \end{bmatrix}}$? Clearly you can’t - we don’t have any way to move in the $y$ direction. No combination of these two vectors could possible get us the vector $P$. ### Good Example What about $\textcolor{blue}{\begin{bmatrix} 1 \\ 0 \end{bmatrix}}$ and $\textcolor{#228B22}{\begin{bmatrix} 1 \\ 1 \end{bmatrix}}$? Our new basis vectors. Surprisingly, you can! The below image shows how we can reach our previously unreachable point $P$. Note we can can combine $3$ units of ${\begin{bmatrix} 1 \\ 1 \end{bmatrix}}$ and $-1$ units of ${\begin{bmatrix} 1 \\ 0 \end{bmatrix}}$ to get us the vector $P$. I’ll leave a simple proof of this as an appendix at the end of this post so we can keep moving - but it’s not too complicated so if you’re up for it, give it a go! The main thing we’ve learned here is that: There are multiple valid bases for $R^2$. ### Bases as New Coordinate Axes In many ways, choosing a new basis is like choosing a new set of axes for the coordinate plane. When we when we switch our basis to say $B = \{\textcolor{blue}{\begin{bmatrix} 1 \\ 0 \end{bmatrix}}, \textcolor{#228B22}{\begin{bmatrix} 1 \\ 1 \end{bmatrix}}\}$, our axes just rotate as shown below: As our second basis vector changed from $\textcolor{#228B22}{\begin{bmatrix} 0 \\ 1 \end{bmatrix}}$ to $\textcolor{#228B22}{\begin{bmatrix} 1 \\ 1 \end{bmatrix}}$, our y axis rotates to be in line with $\textcolor{#228B22}{\begin{bmatrix} 1 \\ 1 \end{bmatrix}}$. As a result of this, the same notation for a vector means different things in different bases. In the original basis, ${\begin{bmatrix} \textcolor{blue}{3} \\ \textcolor{#228B22}{4} \end{bmatrix}}$ meant: • The vector you get when you compute $\textcolor{blue}{3 \cdot \begin{bmatrix} 1 \\ 0 \end{bmatrix}} + \textcolor{#228B22}{4 \cdot \begin{bmatrix} 0 \\ 1 \end{bmatrix}}$. • Or just $\textcolor{blue}{3} \cdot$ first basis vector plus $\textcolor{#228B22}{4} \cdot$ second basis vector. In our usual notation, ${\begin{bmatrix} \textcolor{blue}{3} \\ \textcolor{#228B22}{4} \end{bmatrix}}$ means $3$ units of $\textcolor{blue}{\begin{bmatrix} 1 \\ 0 \end{bmatrix}}$ and $4$ units of $\textcolor{#228B22}{\begin{bmatrix} 0 \\ 1 \end{bmatrix}}$ Now when we use a different basis , the meaning of this notation actually changes. For the basis is $B = \{\textcolor{blue}{\begin{bmatrix} 1 \\ 0 \end{bmatrix}}, \textcolor{#228B22}{\begin{bmatrix} 1 \\ 1 \end{bmatrix}}\}$, the vector $\begin{bmatrix} \textcolor{blue}{3} \\ \textcolor{#228B22}{4} \end{bmatrix}_{B}$ means: • The vector you get from: $\textcolor{blue}{3 \cdot \begin{bmatrix} 1 \\ 0 \end{bmatrix}} + \textcolor{#228B22}{4 \cdot \begin{bmatrix} 1 \\ 1 \end{bmatrix}}$. You can see this change below: In the notation of basis $B$, ${\begin{bmatrix} \textcolor{blue}{3} \\ \textcolor{#228B22}{4} \end{bmatrix}}_{B}$ means $3$ units of $\textcolor{blue}{\begin{bmatrix} 1 \\ 0 \end{bmatrix}}$ and $4$ units of $\textcolor{#228B22}{\begin{bmatrix} 1 \\ 1 \end{bmatrix}}$ giving us point $P_{B}$. By changing the underlying axes, we changed the location of $P$ even though it’s still called $(3, 4)$. You can see this below: The point $P$ also changes position when we change the basis. It is still $3$ parts first basis vector, $4$ parts second basis vector. But since the underlying basis vectors have changed, it also changes. So the vectors ${\begin{bmatrix} \textcolor{blue}{3} \\ \textcolor{#228B22}{4} \end{bmatrix}}$ and ${\begin{bmatrix} \textcolor{blue}{3} \\ \textcolor{#228B22}{4} \end{bmatrix}}_{B}$ refer to different actual vectors based on basis $B$. ### Matrix Notation Based on Bases Similarly the same notation also means different things for matrices based on the basis. Earlier, the matrix $F$ for the function $f$ was represented by: $F = \begin{bmatrix} f(\textcolor{blue}{\begin{bmatrix} 1 \\ 0 \end{bmatrix}}) & f(\textcolor{#228B22}{\begin{bmatrix} 0 \\ 1 \end{bmatrix}}) \end{bmatrix}$ When I use the basis $B = \{\textcolor{blue}{\begin{bmatrix} 1 \\ 0 \end{bmatrix}}, \textcolor{#228B22}{\begin{bmatrix} 1 \\ 1 \end{bmatrix}}\}$, the matrix $F_{B}$ in basis $B$ becomes: $F_{B} = \begin{bmatrix} f(\textcolor{blue}{\begin{bmatrix} 1 \\ 0 \end{bmatrix}})_{B} & f(\textcolor{#228B22}{\begin{bmatrix} 1 \\ 1 \end{bmatrix}})_{B} \end{bmatrix}$ More generally, for a basis $B = \{b_1, b_2\}$, the matrix is: $F_{B} = \begin{bmatrix} f(\textcolor{blue}{b_1})_{B} & f(\textcolor{#228B22}{b_2})_{B} \end{bmatrix}$ ### The Power of Diagonals We took this short detour into notation for a very specific reason - rewriting a matrix in a different basis is actually a neat trick that allows us to reconfigure the matrix to make it easier to use. How? Let’s find out with a quick example. Let’s say I have a matrix $F$ (representing a linear function) that I need to apply again and again (say 5 times) on a vector $v$. This would be: $F \cdot F \cdot F \cdot F \cdot F \cdot v$. Usually, calculating this is really cumbersome. Can you imagine doing this 5 times in a row? Yeesh. Image Source: Wikipedia. But let’s imagine for a moment that $F$ was a diagonal matrix (i.e. something like $F = \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix}$). If this were the case, then this multiplication would be EASY. Why? Let’s see what $F \cdot F$ is: $F \cdot F = \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix} \cdot \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix}$ $F \cdot F = \begin{bmatrix} a \cdot a + 0 \cdot 0 & a \cdot 0 + 0 \cdot b \\ 0 \cdot a + b \cdot 0 & 0 \cdot 0 + b \cdot b \end{bmatrix}$ $F \cdot F = \begin{bmatrix} a^2 & 0 \\ 0 & b^2 \end{bmatrix}$ More generally, $F^{n} = \begin{bmatrix} a^n & 0 \\ 0 & b^n \end{bmatrix}$ This is way easier to work with! So how can we get $F$ to be a diagonal matrix? ### Which Basis makes a Matrix Diagonal? Earlier, we saw that choosing a new basis makes us change how we write down the matrix. So can we find a basis $B = \{b_1, b_2\}$ that converts $F$ into a diagonal matrix? From earlier, we know that $F_{B}$, the matrix $F$ in the basis $B$, is written as: $F_B = \begin{bmatrix} f(\textcolor{blue}{b_1})_{B} & f(\textcolor{#228B22}{b_2})_{B} \end{bmatrix}$ For this to be diagonal, we must have: $F_B = \begin{bmatrix} f(\textcolor{blue}{b_1})_{B} & f(\textcolor{#228B22}{b_2})_{B} \end{bmatrix} = {\begin{bmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{bmatrix}}_{B}$ for some $\lambda_1$ and $\lambda_2$ (i.e. the the top-right and bottom-left elements are $0$). This implies: 1. $f(\textcolor{blue}{b_1})_{B} = {\begin{bmatrix}\lambda_1 \\ 0 \end{bmatrix}}_{B}$. 2. $f(\textcolor{#228B22}{b_2})_{B} = {\begin{bmatrix}0 \\ \lambda_2 \end{bmatrix}}_{B}$. Recall our discussion on vector notation in a different basis: Say my basis is $B = \{\textcolor{blue}{\begin{bmatrix} 1 \\ 0 \end{bmatrix}}, \textcolor{#228B22}{\begin{bmatrix} 1 \\ 1 \end{bmatrix}}\}$. Then the vector $\begin{bmatrix} \textcolor{blue}{3} \\ \textcolor{#228B22}{4} \end{bmatrix}_{B}$ means: • The vector you get when you compute: $\textcolor{blue}{3 \cdot \begin{bmatrix} 1 \\ 0 \end{bmatrix}} + \textcolor{#228B22}{4 \cdot \begin{bmatrix} 1 \\ 1 \end{bmatrix}}$. So, we know the following additional information: $f(\textcolor{blue}{b_1}) = {\begin{bmatrix}\lambda_1 \\ 0 \end{bmatrix}}_B = \lambda_1 \cdot \textcolor{blue}{b_1} + 0 \cdot \textcolor{#228B22}{b_2}$ $f(\textcolor{blue}{b_1}) = \mathbf{\lambda_1 \cdot \textcolor{blue}{b_1}}$ Similarly, $f(\textcolor{#228B22}{b_2}) = {\begin{bmatrix}0 \\ \lambda_2 \end{bmatrix}}_B = 0 \cdot \textcolor{blue}{b_1} + \lambda_2 \cdot \textcolor{#228B22}{b_2}$ $f(\textcolor{#228B22}{b_2}) = \mathbf{\lambda_2 \cdot \textcolor{#228B22}{b_2}}$ #### Seeing this Visually What do these vectors look like on our coordinate axis? We saw earlier that choosing a new basis $B = \{b_1, b_2\}$ creates a new coordinate axis for $R^2$ like below: A new basis $B = \{b_1, b_2\}$ gives us new coordinate axis. Let’s plot $f(\textcolor{blue}{b_1})_{B} = {\begin{bmatrix}\lambda_1 \\ 0 \end{bmatrix}}_{B}$: In the graph above, we can see that ${\begin{bmatrix}\lambda_1 \\ 0 \end{bmatrix}}_{B} = \lambda_1 b_1$, so $f(\textcolor{blue}{b_1})_{B} = \lambda_1 b_1$ Similarly, let’s plot $f(\textcolor{#228B22}{b_2})_{B} = {\begin{bmatrix}0 \\ \lambda_2 \end{bmatrix}}_{B}$: From the above, we see clearly that ${\begin{bmatrix}0 \\ \lambda_2 \end{bmatrix}}_{B} = \lambda_2 b_2$, so $f(\textcolor{blue}{b_2})_{B} = \lambda_2 b_2$ #### Rules For Getting a Diagonal So if we can find a basis $B$ formed by $b_1$ and $b_2$ such that: 1. $f(\textcolor{blue}{b_1}) = \lambda_1 \textcolor{blue}{b_1}$ and 2. $f(\textcolor{#228B22}{b_2}) = \lambda_2 \textcolor{#228B22}{b_2}$, then, $F$ can be rewritten as $F_{B}$, where $F_{B} = \begin{bmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{bmatrix}$ A nice diagonal matrix! ### Enter Eigenvectors Is there a special name for the vectors above $b_1$ and $b_2$ that magically let us rewrite a matrix as a diagonal? Yes! These vectors are the eigenvectors of $f$. That’s right - you derived eigenvectors all by yourself. You the real MVP. More formally, we define an eigenvector of $f$ as any non-zero vector $v$ such that: $f(v) = \lambda v$ or $F \cdot v = \lambda v$ The basis formed by the eigenvectors is known as the eigenbasis. Once we switch to using the eigenbasis, our original problem of finding $f\circ f\circ f \circ f \circ f (v)$ becomes: $F_{B} \cdot F_{B} \cdot F_{B} \cdot F_{B} \cdot F_{B} \cdot v_{B}$ $= {\begin{bmatrix} {\lambda_1}^5 & 0 \\ 0 & {\lambda_2}^5 \end{bmatrix}}_{B}$ So. Much. Easier. ### An Example Well this has all been pretty theoretical with abstract vectors like $b$ and $v$ - let’s make this concrete with real vectors and matrices to see it in action. Imagine we had the matrix $F = \begin{bmatrix}2 & 1 \\ 1 & 2 \end{bmatrix}$. Since the goal of this post is not learning how to find eigenvectors, I’m just going to give you the eigenvectors for this matrix. They are: $b_{1} = \begin{bmatrix} 1 \\ -1 \end{bmatrix}$ $b_{2} = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$ The eigenbasis is just $B = \{b_1, b_2\}$. What is $F_{B}$, the matrix $F$ written in the eigenbasis $B$? Since $F_B = \begin{bmatrix} f(\textcolor{blue}{b_1})_{B} & f(\textcolor{#228B22}{b_2})_{B} \end{bmatrix}$, we need to find : • $f(\textcolor{blue}{b_1})_{B}$ and $f(\textcolor{#228B22}{b_2})_{B}$ We’ll break this down by first finding $f(\textcolor{blue}{b_1})$ and $f(\textcolor{#228B22}{b_2})$, and rewrite them in the notation of the eigenbasis $B$ to get $f(\textcolor{blue}{b_1})_{B}$ and $f(\textcolor{#228B22}{b_2})_{B}$. #### Finding $f(\textcolor{blue}{b_1})$ $f(\textcolor{blue}{b_1})$ is: $f(\textcolor{blue}{b_1}) = F\cdot \textcolor{blue}{b_1} = \begin{bmatrix}2 & 1 \\ 1 & 2\end{bmatrix} \cdot \begin{bmatrix} 1 \\ -1 \end{bmatrix}$ $f(\textcolor{blue}{b_1}) = \begin{bmatrix} 1 \\ -1 \end{bmatrix}$ #### Finding $f(\textcolor{#228B22}{b_2})$ Similarly, $f(\textcolor{#228B22}{b_2}) = F\cdot \textcolor{#228B22}{b_2} = \begin{bmatrix}2 & 1 \\ 1 & 2\end{bmatrix} \cdot \begin{bmatrix} 1 \\ 1 \end{bmatrix}$ $f(\textcolor{#228B22}{b_2}) = \begin{bmatrix} 3 \\ 3 \end{bmatrix}$ #### Rewriting the vectors in the basis $B$ We’ve now found $f(b_1)$ and $f(b_2)$. We need to rewrite these vectors in the notation for our new basis $B$. What’s $f(b_1)_{B}$? $f(b_1) = \begin{bmatrix} 1 \\ -1 \end{bmatrix} = \textcolor{blue}{1} \cdot \begin{bmatrix} 1 \\ -1 \end{bmatrix} + \textcolor{#228B22}{0} \cdot \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \textcolor{blue}{1} \cdot b_1 + \textcolor{#228B22}{0} \cdot b_2$ $f(b_1)_{B} = \begin{bmatrix} \textcolor{blue}{1} \\ \textcolor{#228B22}{0} \end{bmatrix}$ Similarly, $f(b_2) = \begin{bmatrix} 3 \\ 3 \end{bmatrix} = \textcolor{blue}{0} \cdot \begin{bmatrix} 1 \\ -1 \end{bmatrix} + \textcolor{#228B22}{3} \cdot \begin{bmatrix}1 \\ 1 \end{bmatrix} = \textcolor{blue}{0} \cdot b_1 + \textcolor{#228B22}{3} \cdot b_2$ $f(b_2)_{B} = \begin{bmatrix} \textcolor{blue}{0} \\ \textcolor{#228B22}{3} \end{bmatrix}$ Putting this all together, $F_B = \begin{bmatrix} f(\textcolor{blue}{b_1})_{B} & f(\textcolor{#228B22}{b_2})_{B} \end{bmatrix}$ $F_B = \begin{bmatrix} 1 & 0 \\ 0 & 3 \end{bmatrix}$ So we get the nice diagonal we wanted! ### Geometric Interpretation of Eigenvectors Eigenvectors also have extremely interesting geometric properties worth understanding. To see this, let’s go back to the definition for an eiegenvector of a linear map $f$ and its matrix $F$. An eigenvector, is a vector $v$ such that: $F \cdot v = \lambda v$ How are $\lambda v$ and $v$ related? $\lambda v$ is just a scaling of $v$ in the same direction - it can’t be rotated in any way. Notice how $\lambda v$ is in the same direction as $v$. Image Source: Wikipedia. In this sense, the eigenvectors of a linear map $f$ show us the axes along which the map simply scales or stretches its inputs. The single best visualization I’ve seen of this is by 3Blue1Brown who has a fantastic youtube channel on visualizing math in general. I’m embedding his video on eigenvectors and their visualizations below as it is the best geometric intuition out there: Source: 3Blue1Brown Like we saw at the beginning of this post, eigenvectors are not just an abstract concept used by eccentric mathematicians in dark rooms - they underpin some of the most useful technology in our lives including Google Search. For the brave, here’s Larry Page and Sergey’s original paper on PageRank, the algorithm that makes it possible for us to type in a few letters on a search box and instantly find every relevant website on the internet. In the next post, we’re going to actually dig through this paper and see how eigenvectors are applied in Google search! Stay tuned. ### Appendix Proof that $\textcolor{blue}{\begin{bmatrix} 1 \\ 0 \end{bmatrix}}$ and $\textcolor{#228B22}{\begin{bmatrix} 1 \\ 1 \end{bmatrix}}$ span $R^2$: 1. We know already that ${\begin{bmatrix} 1 \\ 0 \end{bmatrix}}$ and ${\begin{bmatrix} 0 \\ 1 \end{bmatrix}}$ can be used to reach every coordinate. 2. We can create ${\begin{bmatrix} 0 \\ 1 \end{bmatrix}}$ by computing: • $\textcolor{#228B22}{\begin{bmatrix} 1 \\ 1 \end{bmatrix}} - \textcolor{blue}{\begin{bmatrix} 1 \\ 0 \end{bmatrix}} = {\begin{bmatrix} 0 \\ 1 \end{bmatrix}}$ 3. Thus we can combine our vectors to obtain both ${\begin{bmatrix} 1 \\ 0 \end{bmatrix}}$ and ${\begin{bmatrix} 0 \\ 1 \end{bmatrix}}$. By point 1, this means every vector in $R^2$ is reachable by combining $\textcolor{blue}{\begin{bmatrix} 0 \\ 1 \end{bmatrix}}$ and $\textcolor{#228B22}{\begin{bmatrix} 1 \\ 1 \end{bmatrix}}$.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 190, "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": 0.9113097786903381, "perplexity": 345.12801681015765}, "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/1614178358798.23/warc/CC-MAIN-20210227084805-20210227114805-00144.warc.gz"}
http://www.koreascience.or.kr/article/ArticleFullRecord.jsp?cn=HNSHCY_2009_v31n4_633	UNIQUENESS OF TOEPLITZ OPERATOR IN THE COMPLEX PLANE • Journal title : Honam Mathematical Journal • Volume 31, Issue 4, 2009, pp.633-637 • Publisher : The Honam Mathematical Society • DOI : 10.5831/HMJ.2009.31.4.633 Title & Authors UNIQUENESS OF TOEPLITZ OPERATOR IN THE COMPLEX PLANE Chung, Young-Bok; Abstract We prove using the Szeg$\small{\H{o}}$ kernel and the Garabedian kernel that a Toeplitz operator on the boundary of $\small{C^{\infty}}$ smoothly bounded domain associated to a smooth symbol vanishes only when the symbol vanishes identically. This gives a generalization of previous results on the unit disk to more general domains in the plane. Keywords Szeg$\small{\H{o}}$ kernel;Toeplitz operator;Garabedian kernel; Language English Cited by References 1. S. Bell, Solving the Dirichlet problem in the plane by means of the Cauchy integral, Indiana Univ. Math. J. 39 (1990), no. 4, 1355-1371. 2. Steven R. Bell, The Cauchy transform, potential theory, and conformal mapping, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR MR1228442 (94k:30013) 3. Steve Bell, The Szego projection and the classical objects of potential theory in the plane, Duke Math. J. 64 (1991), no. 1, 1-26. MR MR1131391 (93e:30018) 4. P. R. Garabedian, Schwarz's lemma and the Szego kernel function, Trans. Amer. Math. Soc. 67 (1949), 1-35. 5. Dennis A. Hejhal, Theta functions, kernel functions, and Abelian integrals, American Mathematical Society, Providence, R.I., 1972, Memoirs of the American Mathematical Society, No. 129. 6. N. Kerzman and E. M. Stein, The Cauchy kernel. the Szego kernel, and the Riemann mapping function, Math. Ann. 236 (1978), 85-93. 7. N. Keramen and M. Trummer, Numerical conformal mapping via the Szego kernel, (1986), Numerical conformal mapping, 111-123, Trefethen, ed., North Holland, Amsterdam. 8. Menahem Schiffer, Various types of orthogonalization, Duke Math. J. 17 (1950), 329-366. MR MR0039071 (12,491g) 9. Boo Rim Chce, Hyungwoon Koo, and Young Joo Lee, Zero products of Toeplitz operators with n-harmonic symbols, Integral Equations Operator Theory 57(2007), no. 1, 43-66. MR MR2294274 (2008c:47050) 10. Young Joo Lee. Commuting Toeplitz operators on the Hardy space of the bidisk, J. Math. Anal. Appl. 341 (2008), no. 1, 738-749. MR MR2394121 (2009c:47037) 11. Stefan Bergman, The kernel function and conformal mapping. revised ed., American Mathematical Society, Providence, R.I., 1970, Mathematical Surveys, No. V. MR MR0507701 (58 #22502)	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 3, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9326195120811462, "perplexity": 1169.4160290978784}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549424079.84/warc/CC-MAIN-20170722142728-20170722162728-00135.warc.gz"}
http://bkms.kms.or.kr/journal/view.html?uid=3017	- Current Issue - Ahead of Print Articles - All Issues - Search - Open Access - Information for Authors - Downloads - Guideline - Regulations ㆍPaper Submission ㆍPaper Reviewing ㆍPublication and Distribution - Code of Ethics - For Authors ㆍOnlilne Submission ㆍMy Manuscript - For Reviewers - For Editors Ricci curvature for conjugate and focal points on GRW space-times Bull. Korean Math. Soc. 2001 Vol. 38, No. 2, 285-292 Jeong-Sik Kim and Seon-Bu Kim Chonnam National University, Chonnam National University Abstract : The authors compute the Ricci curvature of the GRW space-time to obtain two conditions for the conjugate points which appear as the Timelike Convergence Condition(TCG) and the Jacobi inequality. Moreover, under such two conditions, we obtain a lower bound of the length of a unit timelike geodesic for focal points emanating from the immersed spacelike hypersurface, the graph over the fiber in the GRW space-time. Keywords : conjugate point, focal point, Timelike Convergence Condition(TCC), Generalized Robertson-Walker(GRW) space-time MSC numbers : 53C20, 53C50 Downloads: Full-text 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": 0.8430542349815369, "perplexity": 3693.2429757830396}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400227524.63/warc/CC-MAIN-20200925150904-20200925180904-00411.warc.gz"}
https://www.physicsforums.com/threads/what-is-the-charge-coulombs-of-a-nanogram-of-electrons.821415/	# What is the charge (Coulombs) of a nanogram of electrons? 1. Jun 30, 2015 ### Luke Cohen 1. The problem statement, all variables and given/known data What is the charge of a nanogram of electrons? This was a test question for me. I didn't know the exact definition of a coulomb, so I guessed about 1. something C. The options were 1.something C, 0.03C, or like 3.64C. Someone care to explain/help? thanks 2. Relevant equations 3. The attempt at a solution 2. Jun 30, 2015 ### Staff: Mentor Under the Relevant Equations section, you should list the mass of an electron and the charge on an electron. Try using that approach and show us what you get... 3. Jun 30, 2015 ### Luke Cohen But I don't wannaaaaaa. Can't you just do it for me and tell me the answer?! :) 4. Jun 30, 2015 ### Staff: Mentor LOL. Nope, that's not how it works around here. Draft saved Draft deleted Similar Discussions: What is the charge (Coulombs) of a nanogram of electrons?	{"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": 0.8135890364646912, "perplexity": 3861.2804466871808}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187825227.80/warc/CC-MAIN-20171022113105-20171022133105-00543.warc.gz"}
https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=136&t=62469&p=239171	## Test 2 Calculate K after a temperature change $\ln K = -\frac{\Delta H^{\circ}}{RT} + \frac{\Delta S^{\circ}}{R}$ Philip Posts: 100 Joined: Sat Sep 07, 2019 12:16 am ### Test 2 Calculate K after a temperature change Given that delta H reaction = 161 kJ/mol and K = 8.43 x 10^-12 for the reaction at 25 degrees C. Calculate K at 125 degrees C. Assume that delta H reaction and delta S reaction remain constant over this temperature range. Idk what I did because somehow I ended up with a new K value of 6.808 x 10^-96 Julia Holsinger_1A Posts: 50 Joined: Tue Feb 26, 2019 12:16 am ### Re: Test 2 Calculate K after a temperature change make sure to set up the equation by first isolating ln(K2). So the equation would be: ln(K2)=(deltaH/R) (1/T1 - 1/T2) + ln(K1). Then plug in your numbers. I have found that when I don't isolate the ln(K2) first I get a strange answer.	{"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": 0.9424256086349487, "perplexity": 2871.142707695333}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141674082.61/warc/CC-MAIN-20201201104718-20201201134718-00614.warc.gz"}
http://math.stackexchange.com/questions/100268/transforming-poisson	# Transforming Poisson Let $N, Y_n,n\in\mathbb{N}$ be independent random variables, with $N \sim P(\lambda), \lambda \lt \infty$ and $\mathbb{P}(Y_n = j)=p_j$, for $j=1,\dots,k$ and all $n$. Set $$N_j = \sum_{n=1}^{N}\mathbb{1}(Y_n=j).$$ Show that $N_1,\dots,N_k$ are independent random variables with $N_j \sim P(\lambda p_j)$ for all $j$. For distribution I used that $N_j \| N \sim Bin(N, p_j)$ and so $$\mathbb{P}(N_j=k)=\sum_{N=k}^{\infty}{{N}\choose{k}}p_j^k(1-p_j)^k\frac{\lambda^N}{N!}e^{-\lambda}=\dots=\frac{(p_j\lambda)^k}{k!}e^{-\lambda p_j}$$ Need help with independence. - Let $T$, $T_1$, $\dots$, $T_k$ be indeterminates. Then the probability generating function of $N$ is $$f(T):={\Bbb E}[T^N]=e^{\lambda T}.$$ Now, if we condition on $N=n$, the joint distribution of the $N_j$'s will be multinomial and given by the coefficients of $(\sum_j p_j T_j)^n$. Therefore, setting $T=\sum_j p_j T_j$ in $f(T)$ will give the joint probability generating function of the $N_j$'s: $${\Bbb E}[\prod_j T_j^{N_j}]= f(\sum_j p_j T_j) = e^{\lambda (\sum_j p_j T_j)}.$$ This equals the product $$\prod_j e^{\lambda p_j T_j},$$ and so the $N_j$'s are independent with each $N_j\sim P(\lambda p_j)$. -	{"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": 0.9878694415092468, "perplexity": 62.014953318206445}, "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-2014-52/segments/1418802775656.66/warc/CC-MAIN-20141217075255-00070-ip-10-231-17-201.ec2.internal.warc.gz"}
http://mathhelpforum.com/advanced-algebra/139205-write-inverse-terms-matrix-print.html	# Write the inverse of A in terms of the matrix A • Apr 14th 2010, 03:56 PM DarK 1 Attachment(s) Write the inverse of A in terms of the matrix A I have no idea where to begin, if someone could help me get started. Also, I'm a little unsure about what to do for the first question as well (part a). • Apr 14th 2010, 04:04 PM dwsmith Matrix multiplication general isn't commutative. $(A-B)(A+B)=A^2+AB-BA-B^2$ If $AB \neq BA$, then $AB-BA$ isn't guaranteed to equal to 0. • Apr 15th 2010, 06:00 AM HallsofIvy You titled this "write the inverse of A in terms of the matrix A" but then didn't ask about (b)! If $A^2+ A- I_n= 0$ then $A(A+ I_n)= (A+ I_n)A= I_n$.	{"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9181904792785645, "perplexity": 898.4307932249495}, "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-2016-44/segments/1476988721067.83/warc/CC-MAIN-20161020183841-00096-ip-10-171-6-4.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/147726/how-to-prove-gausss-digamma-theorem?answertab=active	# How to prove Gauss's Digamma Theorem? Here $\psi(z)$ is digamma function, $\Gamma(z)$ is gamma function. $$\psi(z)=\frac{{\Gamma}'(z)}{\Gamma(z)},$$ For positive integers $m$ and $k$ (with $m < k$), the digamma function may be expressed in terms of elementary functions as: $$\psi\left(\frac{m}{k}\right)=-\gamma-\ln(2k)-\frac{\pi}{2}\cot\left(\frac{m\pi}{k}\right)+2\sum^{[(k-1)/2]}_{n=1}\cos\left(\frac{2\pi nm}{k}\right)\ln\left(\sin \left(\frac{n\pi}{k}\right)\right).$$ How to prove it ? - ## 1 Answer You can look at this, and the references therein. Added: In fact, a quick Google search gives several references for the proof. Also, if the math does not render well, the Planetmath team suggests to switch the view style to HTML with pictures (you can choose at the bottom of the page). - @M Turgeon Thank you very much! I think it's helpful, but I can't find a simple proof. – Daoyi Peng May 22 '12 at 4:21 @DaoyiPeng What would be a simple proof for you? – M Turgeon May 22 '12 at 12:29 The proof is ill formatted! – Pedro Tamaroff May 22 '12 at 23:21 @PeterTamaroff Well, I sent a comment so that someone check and fix it. Meanwhile, it is still possible to look at the source file and figure out what is not being processed. – M Turgeon May 23 '12 at 0:27 @MTurgeon Thanks! – Pedro Tamaroff May 28 '12 at 20:19	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8830975294113159, "perplexity": 634.9410870052616}, "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-2015-27/segments/1435375096706.9/warc/CC-MAIN-20150627031816-00207-ip-10-179-60-89.ec2.internal.warc.gz"}
https://phys.libretexts.org/Courses/University_of_California_Davis/UCD%3A_Biophysics_241_-_Membrane_Biology/Membrane_Phases/The_Ripple_Phase	$$\require{cancel}$$ # The Ripple Phase Lipids consist of hydrophilic polar head groups attached to hydrocarbon chains and arrange themselves in bilayers to make biological membrane structures. At lower temperatures, the bilayer is in a Lβ' 'gel' phase and there is a transition to 'fluid' phaseLα , at higher temperatures due to an increase in mobility of individual lipids in the bilayer. A smectic ripple phase Pβ' is observed in hydrated lipid bilayers between the Lβ' and Lα phase. This phase is characterized by corrugations of the membrane surface with well-defined periodicity with an axis parallel to the mean bilayer plane [1]. The molecular origin of ripple-phase formation is traditionally been associated with the lipid headgroup region and hence lipids can be classified into ripple-forming and non-ripple forming lipids based on their headgroups. One of the lipid families belonging to ripple-forming class is phosphatidylcholines and has been studied in extensive detail [1]. Scheme above shows different physical states adopted by a lipid bilayer in aqueous medium [2] ### Thermodynamics and Existence The existence of the ripple phase at first sight is paradoxical on thermodynamic grounds since it involves an apparent lowering of symmetry (from Lβ' to Pβ' ) on increasing the temperature. Some models suggest that ripples exist because of periodic local spontaneous curvature in the lipid bilayers formed due to electrostatic coupling between water molecules and the polar headgroups or coupling between membrane curvature and molecular tilt. It has also been speculated that ripples form to relieve packing frustrations that arise whenever the relationship between head-group cross sectional area and cross-sectional area of the apolar tails exceeds a certain threshold [1]. However, there is not one conclusive theory to explain ripple phase formation. ### Phase Diagram Depicting Ripple Phase Experimental phase diagram for (1,2-dimyristoyl-sn-glycero-3-phosphocholine) DMPC, plotted as a function of temperature and hydration. Solid lines indicate first order transitions. Arrows indicate directions of increasing tilt in the Lβ′ phase. The rightmost schematic shows, from top to bottom, the forms of the phases Lα, Pβ′ and Lβ' [3] ### Types of Ripple Structures Two different co-existing ripple phases have been reported, one is asymmetric, having a sawtooth profile with alternating thin and thick arms and a periodicity of 13-15 nms and the other one is symmetric and has a wavy sinusoidal structure with twice the periodicity of the asymmetric structure [4]. In phosphatidylcholine bilayers, asymmetric ripple phase formed is more stable which forms at the pretransition temperature upon heating from the gel phase. The metastable ripple phase is formed at the main phase transition upon cooling from the fluid phase and has approximately double the ripple repeat distance as compared to the stable phase [1]. Figure above shows the model of asymmetric (upper) and symmetric (lower) ripple phase with 720 lipids molecules in a bilayer [4]. ### Experiments to Understand Ripple Phases Freeze-fracture electron microscopy (FFEM) has been utilized to understand the structure of ripple phases. Freeze-fracture preparation rapidly freezes the lipid bilayer suspension at certain temperature (cryofixation) which is then fractured. The cold fractured surface is then shadowed with evaporated platinum or gold at an average angle of 45° in a high vacuum evaporator. A second coat of carbon, evaporated perpendicular to the average surface plane is often performed to improve stability of the replica coating. The specimen is returned to room temperature and pressure, then the extremely fragile "pre-shadowed" metal replica of the fracture surface is released from the underlying biological material by careful chemical digestion. The still-floating replica is thoroughly cleaned from all the chemical residues, dried and then viewed in the TEM (Transmission electron microscopy) [6]. T he results obtained through FFES show periodic linear arrays of ripples which change direction by characteristic angles of 60 or 120 degrees reflecting the hexagonal packing in the lipids [1]. Atomic Force Microscopy (AFM) allows for direct visualization of the ripple phase in supported hydrated bilayers and dynamics of formation and disappearance of ripple phases can be studied at pretransition temperatures [1]. Some examples of AFM images depicting Ripple phase are shown below from Kaasgaard et al [1]. In the image descriptions, Λ/2 represents the asymmetric phase and Λ shows the symmetric phase because of twice the periodicity of metastable phase than the stable phase. A phase with periodicity 2Λ has also been observed. Low Angle and Wide Angle X-ray scattering have also been utilized to understand the ripple phase behavior by mapping electron density as shown above [5]. ### Future Research Although many theories, simulations and experiments have been conducted on Ripple phase, the exact parameters affecting its formation are still unknown for different systems. It is still not known how the hydrocarbon chains are oriented in the bilayer in Pβ' phase [8]. The existence of ripple phase still remains enigmatic and determining the detailed molecular structure would seem to be prerequisite to understanding the interactions that are responsible for its formation [7]. ### References 1. Thomas Kaasgaard,* Chad Leidy,* John H. Crowe,y Ole G. Mouritsen,z and Kent Jørgensen* Temperature-Controlled Structure and Kinetics of Ripple Phases in One- and Two-Component Supported Lipid Bilayers, Biophysical Journal Volume 85 July 2003 350–360 . 2. Image from- http://popups.ulg.ac.be/1780-4507/index.php?id=6568 3. Carlson, J. M. and Sethna, J. P. Phys. Rev. A 36, 3359–3374 Oct (1987). 4. Olaf Lenz, Friederike Schmid, Structure of symmetric and asymmetric “ripple” phases in lipid bilayers, Phys. Rev. Lett. 98, 058104, January 2007. 5. Kiyotaka Akabori and John F. Nagle, Structure of the DMPC lipid bilayer ripple phase, Soft Matter, 2015, 11, 918 6. https://en.wikipedia.org/wiki/Electron_microscope 7. Nagle et al., Lipid bilayer structure, Volume 10, Issue 4, 1 August 2000, Pages 474–480 8. Nagle JF, Tristram-Nagle S. Structure of lipid bilayersBiochimica et biophysica acta. 2000;1469(3):159-195.	{"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": 0.830244243144989, "perplexity": 3568.349526150182}, "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-2019-04/segments/1547583659890.6/warc/CC-MAIN-20190118045835-20190118071835-00253.warc.gz"}
https://www.physicsforums.com/threads/separation-vector.60269/	# Separation vector 1. Jan 18, 2005 ### starbaj12 Let c' be the separation vector from a fixed point(x'',y'',z'') to the point (x,y,z) and let c be its length. show that Thnaks for the help 2. Jan 18, 2005 ### cronxeh 3. Jun 25, 2008 ### kkan2243 begin by writing 1/c in terms of cartesian coordinates. c = sqrt[(x - x)^2 + (y - y)^2 + (z - z`)^2] 1/c = ? then differentiate using multiple applications of the chain rule. Remember that the primed terms are constant when differentiating respect to x, y or z. This was the part that confused me at the beginning as I didn't know how to differentiate those. 4. Jun 28, 2008 ### mathwizarddud What is "hat"? 5. Jun 28, 2008 ### tiny-tim ^ "hat" is ^ it means the unit vector in the direction of c'	{"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": 0.9332337379455566, "perplexity": 2737.2574973770693}, "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-2016-50/segments/1480698543170.25/warc/CC-MAIN-20161202170903-00296-ip-10-31-129-80.ec2.internal.warc.gz"}
http://unapologetic.wordpress.com/2012/01/30/amperes-law/?like=1&source=post_flair&_wpnonce=a34a077b61	# The Unapologetic Mathematician ## Ampère’s Law Let’s go back to the way we derived the magnetic version of Gauss’ law. We wrote $\displaystyle B(r)=\nabla\times\left(\frac{\mu_0}{4\pi}\int\limits_{\mathbb{R}^3}\frac{J(s)}{\lvert r-s\rvert}\,d^3s\right)$ Back then, we used this expression to show that the divergence of $B$ vanished automatically, but now let’s see what we can tell about its curl. \displaystyle\begin{aligned}\nabla\times B&=\frac{\mu_0}{4\pi}\nabla\times\nabla\times\left(\int\limits_{\mathbb{R}^3}\frac{J(s)}{\lvert r-s\rvert}\,d^3s\right)\\&=\frac{\mu_0}{4\pi}\left(\nabla\left(\nabla\cdot\int\limits_{\mathbb{R}^3}\frac{J(s)}{\lvert r-s\rvert}\,d^3s\right)-\nabla^2\int\limits_{\mathbb{R}^3}\frac{J(s)}{\lvert r-s\rvert}\,d^3s\right)\end{aligned} Let’s handle the first term first: \displaystyle\begin{aligned}\nabla_r\left(\nabla_r\cdot\int\limits_{\mathbb{R}^3}\frac{J(s)}{\lvert r-s\rvert}\,d^3s\right)&=\nabla_r\int\limits_{\mathbb{R}^3}J(s)\cdot\nabla_r\frac{1}{\lvert r-s\rvert}\,d^3s\\&=-\nabla_r\int\limits_{\mathbb{R}^3}J(s)\cdot\nabla_s\frac{1}{\lvert r-s\rvert}\,d^3s\\&=-\nabla_r\int\limits_{\mathbb{R}^3}\nabla_s\frac{J(s)}{\lvert r-s\rvert}-\frac{1}{\lvert r-s\rvert}\nabla_s\cdot J(s)\,d^3s\\&=-\nabla_r\int\limits_{\mathbb{R}^3}\nabla_s\frac{J(s)}{\lvert r-s\rvert}\,d^3s+\nabla_r\int\limits_{\mathbb{R}^3}\frac{\nabla_s\cdot J(s)}{\lvert r-s\rvert}]\,d^3s\end{aligned} Now the divergence theorem tells us that the first term is $\displaystyle-\nabla_r\int\limits_S\frac{J(s)}{\lvert r-s\rvert}\cdot dS$ where $S=\partial V$ is some closed surface whose interior $V$ contains the support of the whole current distribution $J(s)$. But then the integrand is constantly zero on this surface, so the term is zero. For the other term (and for the moment, no pun intended) we’ll assume that the whole system is in a steady state, so nothing changes with time. The divergence of the current distribution at a point — the amount of charge “moving away from” the point — is the rate at which the charge at that point is decreasing. That is, $\displaystyle\nabla\cdot J=-\frac{\partial\rho}{\partial t}$ But our steady-state assumption says that charge shouldn’t be changing, and thus this term will be taken as zero. So we’re left with: $\displaystyle\nabla\times B(r)=-\frac{\mu_0}{4\pi}\int\limits_{\mathbb{R}^3}J(s)\nabla^2\frac{1}{\lvert r-s\rvert}\,d^3s$ But this is great. We know that the gradient of $\frac{1}{\lvert r\rvert}$ is $\frac{r}{\lvert r\rvert^3}$, and we also know that the divergence of this function is (basically) the “Dirac delta function”. That is: $\displaystyle\nabla^2\frac{1}{\lvert r\vert}=-4\pi\delta(r)$ So in our case we have $\displaystyle\nabla\times B(r)=\frac{\mu_0}{4\pi}\int\limits_{\mathbb{R}^3}J(s)4\pi\delta(r-s)=\mu_0J(r)$ This is Ampère’s law, at least in the case of magnetostatics, where nothing changes in time. January 30, 2012 - 1. [...] we worked out Ampères law in the case of magnetostatics, we used a certain [...] Pingback by Conservation of Charge « The Unapologetic Mathematician \| February 1, 2012 \| Reply 2. [...] magnetism. The third is directly equivalent to Faraday’s law of induction, while the last is Ampère’s law, with Maxwell’s correction. Share this:StumbleUponDiggRedditTwitterLike this:LikeBe the first [...] Pingback by Maxwell’s Equations « The Unapologetic Mathematician \| February 1, 2012 \| Reply 3. [...] we can use Ampère’s law to [...] Pingback by Energy and the Magnetic Field « The Unapologetic Mathematician \| February 14, 2012 \| Reply	{"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": 14, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9831820726394653, "perplexity": 663.086114949226}, "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-2014-10/segments/1394011101613/warc/CC-MAIN-20140305091821-00038-ip-10-183-142-35.ec2.internal.warc.gz"}
https://zbmath.org/?q=an%3A1292.60012	× # zbMATH — the first resource for mathematics Analysis of a generalized Friedman’s urn with multiple drawings. (English) Zbl 1292.60012 Summary: We study a generalized Friedman’s urn model with multiple drawings of white and blue balls. After a drawing, the replacement follows a policy of opposite reinforcement. We give the exact expected value and variance of the number of white balls after a number of draws, and determine the structure of the moments. Moreover, we obtain a strong law of large numbers, and a central limit theorem for the number of white balls. Interestingly, the central limit theorem is obtained combinatorially via the method of moments and probabilistically via martingales. We briefly discuss the merits of each approach. The connection to a few other related urn models is briefly sketched. ##### MSC: 60C05 Combinatorial probability Full Text: ##### References: [1] Arya, S.; Golin, M.; Mehlhorn, K., On the expected depth of random circuits, Combinatorics, Probability and Computing, 8, 209-228, (1999) · Zbl 0941.68001 [2] Chen, M.-R.; Wei, C.-Z., A new urn model, Journal of Applied Probability, 42, 4, 964-976, (2005) · Zbl 1093.60007 [3] Chern, H.-H.; Hwang, H.-K., Phase changes in random $$m$$-ary search trees and generalized quicksort, Random Structures and Algorithms, 19, 316-358, (2001) · Zbl 0990.68052 [4] Freedman, D., Bernard friedman’s urn, The Annals of Mathematical Statistics, 36, 956-970, (1965) · Zbl 0138.12003 [5] Friedman, B., A simple urn model, Communications in Pure and Applied Mathematics, 2, 59-70, (1949) · Zbl 0033.07101 [6] Graham, R.; Knuth, D.; Patashnik, O., Concrete mathematics, (1994), Addison-Wesley Reading [7] Hall, P.; Heyde, C., Martingale limit theory and its application, (1980), Academic Press New York · Zbl 0462.60045 [8] Hill, B.; Lane, D.; Sudderth, W., A strong law for some generalized urn processes, Annals of Probability, 8, 214-226, (1980) · Zbl 0429.60021 [9] Johnson, N.; Kotz, S., Urn models and their applications: an approach to modern discrete probability theory, (1977), Wiley New York · Zbl 0352.60001 [10] Johnson, N.; Kotz, S.; Mahmoud, H., Pólya-type urn models with multiple drawings, Journal of the Iranian Statistical Society, 3, 165-173, (2004) · Zbl 06657086 [11] Kotz, S.; Balakrishnan, N., Advances in urn models during the past two decades, (Advances in Combinatorial Methods and Applications to Probability and Statistics, (1997), Birkhäuser Boston, MA), 203-257 · Zbl 0888.60014 [12] Loève, M., Probability theory I, (1977), Springer New York · Zbl 0359.60001 [13] Mahmoud, H., Pólya urn models, (2008), Chapman-Hall Boca Raton [14] Moler, J.; Plo, F.; Urmeneta, H., A generalized Pólya urn and limit laws for the number of outputs in a family of random circuits, Test, 22, 46-61, (2013) · Zbl 1262.60025 [15] Tsukiji, T.; Mahmoud, H., A limit law for outputs in random circuits, Algorithmica, 31, 403-412, (2001) · Zbl 0989.68107 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": 0.8927628397941589, "perplexity": 3195.817689818095}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038060927.2/warc/CC-MAIN-20210411030031-20210411060031-00471.warc.gz"}
https://proceedings.mlr.press/v75/holden18a.html	# Subpolynomial trace reconstruction for random strings \{and arbitrary deletion probability Nina Holden, Robin Pemantle, Yuval Peres Proceedings of the 31st Conference On Learning Theory, PMLR 75:1799-1840, 2018. #### Abstract The deletion-insertion channel takes as input a bit string ${\bf x}\in \{0,1\}^{n}$, and outputs a string where bits have been deleted and inserted independently at random. The trace reconstruction problem is to recover $\bf x$ from many independent outputs (called “traces”) of the deletion-insertion channel applied to $\bf x$. We show that if $\bf x$ is chosen uniformly at random, then $\exp(O(\log^{1/3} n))$ traces suffice to reconstruct $\bf x$ with high probability. For the deletion channel with deletion probability $q<1/2$ the earlier upper bound was $\exp(O(\log^{1/2} n))$. The case of $q\geq 1/2$ or the case where insertions are allowed has not been previously analysed, and therefore the earlier upper bound was as for worst-case strings, i.e., $\exp(O( n^{1/3}))$. A key ingredient in our proof is a delicate two-step alignment procedure where we estimate the location in each trace corresponding to a given bit of $\bf x$. The alignment is done by viewing the strings as random walks, and comparing the increments in the walk associated with the input string and the trace, respectively.	{"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": 0.9536280632019043, "perplexity": 494.1360280297552}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363229.84/warc/CC-MAIN-20211206012231-20211206042231-00584.warc.gz"}
https://www.coursehero.com/file/5580814/Finite-Semigroups-and-Recognizable-Languages-An-Introduction-1995/	Finite Semigroups and Recognizable Languages An Introduction (1995) Finite Semigroups and Recognizable Languages An Introduction (1995) This preview shows page 1. Sign up to view the full content. This is the end of the preview. Sign up to access the rest of the document. Unformatted text preview: FINITE SEMIGROUPS AND RECOGNIZABLE LANGUAGES: AN INTRODUCTION Bull Research and Development, rue Jean Jaures, 78340 Les Clayes-sous-Bois, FRANCE. Jean-Eric Pin ( ) This paper is an attempt to share with a larger audience some modern developments in the theory of nite automata. It is written for the mathematician who has a background in semigroup theory but knows next to nothing on automata and languages. No proofs are given, but the main results are illustrated by several examples and counterexamples. What is the topic of this theory ? It deals with languages, automata and semigroups, although recent developments have shown interesting connections with model theory in logic, symbolic dynamics and topology. Historically, in their attempt to formalize natural languages, linguists such as Chomsky gave a mathematical de nition of natural concepts such as words, languages or grammars: given a nite set A, a word on A is simply an element of the free monoid on A, and a language is a set of words. But since scientists are fond of classi cations of all sorts, language theory didn't escape to this mania. Chomsky established a rst hierarchy, based on his formal grammars. In this paper, we are interested in the recognizable languages, which form the lower level of the Chomsky hierarchy. A recognizable language can be described in terms of nite automata while, for the higher levels, more powerful machines, ranging from pushdown automata to Turing machines, are required. For this reason, problems on nite automata are often under-estimated, according to the vague \| but totally erroneous \| feeling ( ) 1. Foreword From 1st Sept 1993, LITP, Universite Paris VI, Tour 55-65, 4 Place Jussieu, 75252 Paris Cedex 05, FRANCE. E-mail: [email protected] 2 J.E. Pin that \if a problem has been reduced to a question about nite automata, then it should be easy to solve". Kleene's theorem 23] is usually considered as the foundation of the theory. It shows that the class of recognizable languages (i.e. recognized by nite automata), coincides with the class of rational languages, which are given by rational expressions. Rational expressions can be thought of as a generalization of polynomials involving three operations: union (which plays the role of addition), product and star operation. An important corollary of Kleene's theorem is that rational languages are closed under complement. In the sixties, several classi cation schemes for the rational languages were proposed, based on the number of nested use of a particular operator (star or product, for instance). This led to the natural notions of star height, extended star height, dot-depth and concatenation level. However, the rst natural questions attached to these notions \| \do they de ne strict hierarchies ?", \given a rational language, is there an algorithm for computing its star height, extended star height", etc. ? \| appeared to be extremely di cult. Actually, several of them, like the hierarchy problem for the extended star height, are still open. A break-through was realized by Schutzenberger in the mid sixties 53]. Schutzenberger established the equivalence between nite automata and nite semigroups and showed that a nite monoid, called the syntactic monoid , is canonically attached to each recognizable language. Then he made a non trivial use of this invariant to characterize the languages of extended star height 0, also called star-free languages. Schutzenberger's theorem states that a language is star-free if and only if its syntactic monoid is aperiodic. Two other \syntactic" characterizations were obtained in the early seventies: Simon 57] proved that a language is of concatenation level 1 if and only if its syntactic monoid is J -trivial and Brzozowski-Simon 9] and independently, McNaughton 29] characterized an important subfamily of the languages of dot-depth one, the locally testable languages. These successes settled the power of the semigroup approach, but it was Eilenberg who discovered the appropriate framework to formulate this type of results 17]. Recall that a variety of nite monoids is a class of monoids closed under the taking of submonoids, quotients and nite direct product. Eilenberg's theorem states that varieties of nite monoids are in one to one correspondence with certain classes of recognizable languages, the varieties of languages. For instance, the rational languages correspond to the variety of all nite monoids, the star-free languages correspond to the variety of aperiodic monoids, and the piecewise testable languages correspond to the variety of J -trivial monoids. Numerous similar results have been established during the past fteen years and the theory of nite automata is now intimately related to the theory of nite semigroups. This had a considerable in uence on both theories: for instance algebraic de nitions such as the graph of a semigroup or the Schutzenberger product were motivated by considerations of language theory. The same thing can be said for the systematic study of power semigroups. In the other direction, Straubing's wreath product principle FINITE SEMIGROUPS AND RECOGNIZABLE LANGUAGES 3 has permitted to obtain important new results on recognizable languages. The open question of the decidability of the dot-depth is a good example of a problem that interests both theories (and also formal logic !). The paper is organized as follows. Sections 2 and 3 present the necessary material to understand Kleene's theorem. The equivalence between nite automata and nite semigroups is detailed in section 4. The various hierarchies of rational languages, based on star height, extended star height, dot-depth and concatenation level are introduced in section 5. The syntactic characterization of star-free, piecewise testable and locally testable languages are formulated in sections 6, 7 and 8, respectively. The variety theorem is stated in section 9 and some examples of its application are given in section 10. Other consequences about the hierarchies are analyzed in section 11 and recent developments are reported in section 12. The last section 13 contains the conclusion of this article. The terminology used in the theory of automata originates from various founts. Part of it came from linguistics, some other parts were introduced by physicists or by logicians. This gives sometimes a curious mixture but it is rather convenient in practice. An alphabet is a nite set whose elements are letters . Alphabets are usually denoted by capital letters: A, B , : : : and letters by lower case letters from the beginning of the latin alphabet: a, b, c; : : : A word (over the alphabet A) is a nite sequence (a1 ; a2 ; : : : ; an ) of letters of A; the integer n is the length of the word. In practice, the notation (a1 ; a2 ; : : : ; an ) is shortened to a1 a2 an . The empty word, which is the unique word of length 0, is denoted by 1. Given a letter a, the number of occurrences of a in a word u is denoted by juja . For instance, jabbabja = 2 and jabbabjb = 3. The (concatenation) product of two words u = a1 a2 ap and v = b1 b2 bq is the word uv = a1 a2 ap b1 b2 bq . The product is an associative operation on words. The set of all words on the alphabet A is denoted by A . Equipped with the product of words, it is a monoid, with the empty word as an identity. It is in fact the free monoid on the set A. The set of non-empty words is denoted by A+ ; it is the free semigroup on the set A. A language of A is a set of words over A, that is, a subset of A . The rational operations on languages are the three operations union, product and star, de ned as follows (1) Union : L1 + L2 = fu j u 2 L1 or u 2 L2 g (2) Product : L1 L2 = fu1 u2 j u1 2 L1 and u2 2 L2 g (3) Star : L = fu1 un j n 0 and u1 ; : : : ; un 2 Lg It is also convenient to introduce the operator L+ = LL = fu1 un j n > 0 and u1 ; : : : ; un 2 Lg 2. Rational and recognizable sets 4 J.E. Pin Note that L+ is exactly the subsemigroup of A generated by L, while L is the submonoid of A generated by L. The set of rational languages of A is the smallest set of languages of A containing the nite languages and closed under nite union, nite product and star. For instance, (a + ab) ab + (ba b) denotes a rational language on the alphabet fa; bg. The set of rational languages of A+ is the smallest set of languages of A+ containing the nite languages and closed under nite union, product and plus. It is easy to verify that the rational languages of A+ are exactly the rational languages of A that do not contain the empty word. It may seem a little awkward to have two separate de nitions for the rational languages: one for the free monoid A and another one for the free semigroup A+ . There are actually two parallel theories and although the di erence between them may appear of no great signi cance at rst sight, it turns out to be crucial. The reason is that the algebraic classi cation of rational languages, as given in the forthcoming sections, rests on the notion of varieties of nite monoids (for languages of the free monoid) or varieties of nite semigroups (for languages of the free semigroup). And varieties of nite semigroups cannot be considered as varieties of nite monoids. The simplest example is the variety of nite nilpotent semigroups, which, as we shall see, characterizes the nite or co nite languages of the free semigroup. If one tries, in a naive attempt, to add an identity to convert each nilpotent semigroup into a monoid, the variety of nite monoids obtained in this way is the variety of all nite monoids whose idempotents commute with every element. But this variety of monoids does not characterize the nite-co nite languages of the free monoid. Rational languages are often called regular sets in the literature. However, in the author's opinion, this last term should be avoided for two reasons. First, it interferes with the standard use of this word in semigroup theory. Second, the term rational has a sound mathematical foundation. Indeed one can extend the theory of languages to series with non commutative variables over a commutative P ring or semiring( ) k. Such series can be written as s = u2A (s; u)u, where (s; u) is an element of k. In this context, languages appear naturally as series over the boolean semiring. Now the rational series form the smallest set of series R satisfying the following conditions: (1) Every polynomial is in R, (2) R is a semiring under the usual sum and product of series, P (3) If s is a series in R such that (s; 1) = 0, then s = n 0 sn belongs to R. ( ) A semiring is a set k equipped with an addition and a multiplication. It is a commutative monoid with identity 0 for the addition and a monoid with identity 1 for the multiplication. Multiplication is distributive over addition and 0 satis es 0x = x0 = 0 for every x 2 k. The simplest example of a semiring which is not a ring is the boolean semiring B = f0; 1g de ned by 0 + 0 = 0, 0 + 1 = 1 + 1 = 1 + 0 = 1, 1:1 = 1 and 1: 0 = 0 : 0 = 0 : 1 = 0 . FINITE SEMIGROUPS AND RECOGNIZABLE LANGUAGES 5 Note that if k is a ring, then s = (1 ? s)?1 . In particular, in the one variable case, this de nition coincide with the usual de nition of rational series, which explains the terminology. We shall not detail any further this nice extension of the theory of languages, but we refer the interested reader to 4] for more details. A nite (non deterministic) automaton is a quintuple A = (Q; A; E; I; F ) where Q is a nite set (the set of states ), A is an alphabet, E is a subset of Q A Q, called the set of transitions and I and F are subsets of Q, called the set of initial and nal states, respectively. Two transitions (p; a; q) and (p0 ; a0 ; q0 ) are consecutive if q = p0 . A path in A is a nite sequence of consecutive transitions 3. Finite automata and recognizable sets e0 = (q0 ; a0 ; q1 ); e1 = (q1 ; a1 ; q2 ); : : : ; en?1 = (qn?1 ; an?1 ; qn ) qn?1 an?1 qn ?! The state q0 is the origin of the path, the state qn is its end , and the word x = a0 a1 an?1 is its label . It is convenient to have also, for each state q, an empty path of label 1 from q to q. A path in A is successful if its origin is in I and its end is in F . The language recognized by A is the set, denoted L (A), of the labels of all successful paths of A. A language X is recognizable if there exists a nite automaton A such that X = L (A). Two automata are said to be equivalent if they recognize a0 a1 q0 ?! q1 ?! q2 also denoted the same language. Automata are conveniently represented by labeled graphs, as in the example below. Incoming arrows indicate initial states and outgoing arrows indicate nal states. Example 3.1. Let A = (f1; 2g; fa; bg; E; f1g; f2g) be an automaton, with E = f(1; a; 1); (1; b; 1); (1; a; 2)g. The path (1; a; 1)(1; b; 1)(1; a; 2) is a successful path of label aba. The path (1; a; 1)(1; b; 1)(1; a; 1) has the same label but is unsuccessful since its end is 1. a, b 1 a 2 an a. Figure 3.1. An automaton. The set of words accepted by A is L (A) = A a, the set of all words ending with In the case of the free semigroup, the de nitions are the same, except that we omit the empty paths of label 1. In this case, the language recognized by A is denoted L+ (A). Kleene's theorem states the equivalence between automata and rational expressions. 6 J.E. Pin Theorem 3.1. A language is rational if and only if it is recognizable. In fact, there is one version of Kleene's theorem for the free semigroup and one version for the free monoid. The proof of Kleene's theorem can be found in most books of automata theory 21]. An automaton is deterministic if it has exactly one initial state, usually denoted q0 and if E contains no pair of transitions of the form (q; a; q1 ); (q; a; q2 ) with q1 6= q2 . a q a q 2 q1 In this case, each letter a de nes a partial function from Q to Q, which associates with every state q the unique state q:a, if it exists, such that (q; a; q:a) 2 E . This can be extended into a right action of A on Q by setting, for every q 2 Q, a 2 A and u 2 A : Figure 3.2. The forbidden pattern in a deterministic automaton. q:1 = q n q:(ua) = (q:u):aned if q:u and (q:u):a are de ned unde otherwise One can show that every nite automaton is equivalent to a deterministic one, in the sense that they recognize the same language. States which cannot be reached from the initial state or from which one cannot access to any nal state are clearly useless. This leads to the following de nition. A deterministic automaton A = (Q; A; E; q0 ; F ) is trim if for every state q 2 Q there exist two words u and v such that q0 :u = q and q:v 2 F . It is not di cult to see that every deterministic automaton is equivalent to a trim one. Deterministic automata are partially ordered as follows. Let A = (Q; A; E; q0 ; F ) 0 and A0 = (Q0 ; A; E 0 ; q0 ; F 0 ) be two deterministic automata. Then A A0 if there 0 is a surjective function ' : Q ! Q0 such that '(q0 ) = q0 , '?1 (F 0 ) = F and, for every u 2 A and q 2 Q, '(q:u) = '(q):u. One can show that, amongst the trim deterministic automata recognizing a given recognizable language L, there is a minimal one for this partial order. This automaton is called the minimal automaton of L. Again, there are standard algorithms for minimizing a given nite automaton 21]. FINITE SEMIGROUPS AND RECOGNIZABLE LANGUAGES 7 In this section, we turn to a more algebraic de nition of the recognizable sets, using semigroups in place of automata. Although this de nition is more abstract than the de nition using automata, it is more suitable to handle the ne structure of recognizable sets. Indeed, as illustrated in the next sections, semigroups provide a powerful and systematic tool to classify recognizable sets. We treat the case of the free semigroup. For free monoids, just replace every occurrence of \A+ " by \A " and \semigroup" by \monoid" in the de nitions below. The abstract de nition of recognizable sets is based on the following observation. Let A = (Q; A; E; I; F ) be a nite automaton. To each word u 2 A+ , there corresponds a boolean square matrix of size Card(Q), denoted by (u), and de ned by n 1 if there exists a path from p to q with label u (u)p;q = 0 otherwise It is not di cult to see that is a semigroup morphism from A+ into the multiplicative semigroup of square boolean matrices of size Card(Q). Furthermore, a word u is recognized by A if and only if (u)p;q = 1 for some initial state p and some nal state q. Therefore, a word is recognized by A if and only if (u) 2 fm 2 (A+ ) j mp;q = 1 for some p 2 I and q 2 F g. The semigroup (A+ ) is called the transition semigroup of A, denoted S (A). Example 4.1. Let A = (Q; A; E; I; F ) be the automaton represented below a a 1 b 2 a,b 4. Automata and semigroups Figure 4.1. A non deterministic automaton. Here Q = f1; 2g, A = fa; bg and E = f(1; a; 1); (1; a; 2); (2; a; 2); (2; b; 1); (2; b; 2)g, I = f1g, F = f2g, whence (a) = 1 1 0 1 (ab) = 1 1 1 1 (b) = 0 0 1 1 (ba) = (bb) = (b) (aa) = (a) Thus (A+ ) = f 0 0 ; 1 1 ; 1 1 g. 1 1 0 1 1 1 This leads to the following de nition. A semigroup morphism ' : A+ ! S recognizes a language L A+ if L = '?1 '(L), that is, if u 2 L and '(u) = '(v) implies v 2 L. This is also equivalent to saying that there is a subset P of S ? ? ? 8 J.E. Pin such that L = '?1 (P ). By extension, a semigroup S recognizes a language L if there exists a semigroup morphism ' : A+ ! S that recognizes L. As shown by the previous example, a set recognized by a nite automaton is recognized by the transition semigroup of this automaton. Proposition 4.1. If a nite automaton recognizes a language L, then S (A) recognizes L. The previous computation can be simpli ed if A is deterministic. Indeed, in this case, the transition semigroup of A is naturally embedded into the semigroup of partial functions on Q under composition. Example 4.2. Let A be the deterministic automaton represented below. a 1 b 2 Figure 4.2. A deterministic automaton The transition semigroup S (A) of A contains ve elements which correspond to the words a, b, ab, ba and aa. If one identi es the elements of S (A) with these words, one has the relations aba = a, bab = b and bb = aa. Thus S (A) is the aperiodic Brandt semigroup BA . Here is the transition table of A: 2 a b aa ab ba ? 1 ? ? 1 ? ? 2 1 2 ? 2 Conversely, given a semigroup morphism ' : A+ ! S recognizing a subset X of A+ , one can build a nite automaton recognizing X as follows. Denote by S 1 the monoid equal to S if S has an identity and to S f1g otherwise. Take the right representation of A on S 1 de ned by s:a = s'(a). This de nes a deterministic automaton A = (S 1 ; A; E; f1g; P ), where E = f(s; a; s:a) j s 2 S 1 ; a 2 Ag. FINITE SEMIGROUPS AND RECOGNIZABLE LANGUAGES s a s(a) 9 This automaton recognizes L and thus, the two notions of recognizable sets (by nite automata and by nite semigroups) are equivalent. Example 4.3. Let A = fa; b; cg and let S = f1; a; bg be the three element monoid de ned by a2 = a, b2 = b, ab = b and ba = a. Let ' : A+ ! S be the semigroup morphism de ned by '(a) = a, '(b) = b and '(c) = 1 and let P = fag. Then '?1 (P ) = A ac and the construction above yields the automaton represented in gure 4.4: c 1 a b a,c a a b b,c b Figure 4.3. The transitions of A. Now, Kleene's theorem can be reformulated as follows. + Figure 4.4. The automaton associated with S . Theorem 4.2. Let L be a language of A . The following conditions are equivalent. (1) (2) (3) (4) L is recognized by a nite automaton, L is recognized by a nite deterministic automaton, L is recognized by a nite semigroup, L is rational. Kleene's theorem has important consequences. Corollary 4.3. Recognizable languages are closed under nite boolean operations ( ) , inverse morphisms and morphisms. The trick is that it is easy to prove the last property (closure under morphisms) for rational sets and the other ones for recognizable sets. Here are two examples to illustrate these techniques: ( ) Boolean operations comprise union, intersection, complementation and set di erence. 10 + J.E. Pin + let L = a b + bab be a rational set. Then '(L) = (aba) ca + caabaca is a rational set. Example 4.4. (Closure of recognizable sets under morphism). Let ' : fa; bg ! fa; b; cg be the semigroup morphism de ned by '(a) = aba and '(b) = ca and Example 4.5. (Closure of recognizable sets under complement). Let L be a recognizable set. Then there exists a nite semigroup S , a semigroup morphism ' : A+ ! S and a subset P of S such that L = '?1 (P ). Now A+ n L = '?1 (S n P ) and thus the complement of L is recognizable. The patient reader can, as an exercise, prove the remaining properties by using either semigroups or automata. The impatient reader may consult 16,37]. Let L be a recognizable language of A+ . Amongst the nite semigroups that recognize X , there is a minimal one (with respect to division). This nite semigroup is called the syntactic semigroup of L. It can be de ned directly as the quotient of A+ under the congruence L de ned by u L v if and only if, for every x; y 2 A , xuy 2 L () xvy 2 L. It is also equal to the transition semigroup of the minimal automaton of L. This last property is especially useful for practical computations. It is a good exercise to take a rational expression at random, to compute the minimal automaton of the language represented by this rational expression and then to compute the syntactic semigroup of the language. See examples 6.1 and 7.2 below for outlines of such computations. Kleene's theorem shows that recognizable languages are closed under complementation. Therefore, every recognizable language can be represented by a extended rational expression , that is, a formal expression constructed from the letters by mean of the operations union, product, star and complement. In order to keep concise algebraic notations, we shall denote by Lc the complement of the language L( ) , by 0 the empty language and by u the language fug, for every word u. In particular, the language f1g, containing the empty word, is denoted 1. These notations are coherent with the intuitive formul 1L = L1 = L and 0L = L0 = 0 which hold for every language L. For instance, if A = fa; bg, the expression ?0c(ab + ba)0c c + (aba) c represents the language (A abA A baA ) n (aba) of all words containing the factors ab and ba which are not powers of aba. Thus we have an algebra on A with four operations: +, :, and c . Now a natural attempt to classify recognizable languages is to nd a notion analogous with the degree of a polynomial for these extended rational expressions. It is a remarkable fact that all the hierarchies based on these \extended degrees" suggested so far lead to some extremely di cult problems, most of which are still open. ( ) 5. Early attempts to classify recognizable languages If L is a language of A , the complement of L is A complement is A+ n L n L; if L is a language of A , the + FINITE SEMIGROUPS AND RECOGNIZABLE LANGUAGES 11 The rst proposal concerned the star operation. The star height of an extended rational expression is de ned inductively as follows: (1) The star height of the basic languages is 0. Formally sh(0) = 0 sh(1) = 0 and sh(a) = 0 for every letter a sh(ec ) = sh(e) (2) Union, product and complement do not a ect star height. If e and f are two extended rational expressions, then sh(e + f ) = sh(ef ) = maxfsh(e); sh(f )g sh(e ) = sh(e) + 1 (3) Star increases star height. For each extended rational expression e, Thus the star height counts the number of nested uses of the star operation. For instance (a + bc a ) + (b ab ) (b a + b) is an extended rational expression of star height 3. Now, the extended star height ( ) of a recognizable language L is the minimum of the star heights of the extended rational expressions representing L esh(L) = minfsh(e) j e is an extended rational expression for L g The di culty in computing the extended star height is that a given language can be represented in many di erent ways by an extended rational expression ! The languages of extended star height 0 (or star-free languages ) are characterized by a beautiful theorem of Schutzenberger that will be presented in section 6.1. Schutzenberger's theorem implies the existence of languages of extended star height 1, such as (aa) on the alphabet fag, but, as surprising as it may seem, nobody has been able so far to prove the existence of a language of extended star height greater than 1, although the general feeling is that such languages do exist. In the opposite direction, our knowledge of the languages proven to be of extended star height 1 is rather poor (see 46,51,52] for recent advances on this topic). The star height of a recognizable language is obtained by considering rational expressions instead of extended rational expressions 15]. sh(L) = minfstar height(e) j e is a rational expression for L g That is, one simply removes complement from the list of the basic operations. This time, the corresponding hierarchy was proved to be in nite by Dejean and Schutzenberger 14]. ( ) also called generalized star height 12 J.E. Pin Theorem 5.1. For each n 0, there exists a language of star height n. It is easy to see that the languages of star height 0 are the nite languages, but the e ective characterization of the other levels was left open for several years until Hashiguchi rst settled the problem for star height 1 18] and a few years later for the general case 19]. Theorem 5.2. There is an algorithm to determine the star height of a given recognizable language. Hashiguchi's rst paper is now well understood, although it is still a very di cult result, but volunteers are called to simplify the very long induction proof of the second paper. The second proposal to construct hierarchies was to ignore the star operation (which amounts to working with star-free languages) and to consider the concatenation product or, more precisely, a variation of it, called the marked concatenation product . Given languages L0 , L1 , : : : , Ln and letters a1 , a2 , : : : , an , the product of L0 , : : : Ln marked by a1 , : : : an is the language L0 a1 L1 a2 an Ln . As product is often denoted by a dot, Brzozowski de ned the \dot-depth" of languages of the free semigroup 5]. Later on, Therien (implicitly) and Straubing (explicitly) introduced a similar notion (often called the concatenation level in the literature) for the languages of the free monoid. The languages of dot-depth 0 are the nite or co nite languages, while the languages of concatenation level 0 are A and the empty language 0. Otherwise, the two hierarchies are constructed in the same way and count the number of alternations in the use of the two di erent types of operations: boolean operations and marked product. More precisely, the languages of dot-depth (resp. concatenation level) n + 1 are the nite boolean combinations of marked products of the form L0 a1 L1 a2 a k Lk where L0 , L1 , : : : , Lk are languages of dot-depth (resp. concatenation level) n and a1 , : : : , ak are letters. Note that a language of dot-depth (resp. concatenation level) m is also a language of dot-depth (resp. concatenation level) n for every n m. Brzozowski and Knast 8] have shown that the hierarchy is strict: if A contains at least two letters, then for every n, there exist some languages of dot-depth (resp. level) n + 1 that are not of level n. It is still an outstanding open problem to know whether there is an algorithm to compute the dot-depth (resp. concatenation level) of a given star-free language. The problem has been solved positively, however, for the dot-depth (resp. concatenation level) 1: there is an algorithm to decide whether a language is of dot-depth (resp. concatenation level) 1. These results are detailed in sections 7 and 11. FINITE SEMIGROUPS AND RECOGNIZABLE LANGUAGES 13 The other partial results concerning these hierarchies are brie y reviewed in section 11. Another remarkable fact about these hierarchies is their connections with some hierarchies of formal logic. See the article of W. Thomas in this volume or the survey article 41]. But it is time for us to hark back to Schutzenberger's theorem on star-free sets. 6. Star-free languages The set of star-free subsets of A is the smallest set of subsets of A containing the nite sets and closed under nite boolean operations and product. For instance, A is star-free, since A is the complement of the empty set. More generally, if B is a subset of the alphabet A, the set B is also star-free since B is the complement of the set of words that contain at least one letter of B 0 = A n B . This leads to the following star-free expression B = A n A (A n B )A = (0c (A n B )0c )c = (0c (Ac B )c 0c )c If A = fa; bg, the set (ab) is star-free, since (ab) is the set of words not beginning with b, not nishing by a and containing neither the factor aa, nor the factor bb. This gives the star-free expression (ab) = A n bA ? A a A aaA ? A bbA = b0c + 0c a + 0c aa0c + 0c bb0c c ? Readers may convince themselves that the sets fab; bag and a(ab) b also are star-free but may also wonder whether there exist any non star-free rational sets. In fact, there are some, for instance the sets (aa) and fb; abag , or similar examples that can be derived from the algebraic approach presented below. Let S be a nite semigroup and let s be an element of S . Then the subsemigroup of S generated by s contains a unique idempotent, denoted s! . Recall that a nite semigroup M is aperiodic if and only if, for every x 2 M , x! = x!+1 . This notion is in some sense \orthogonal" to the notion of group. Indeed, one can show that a semigroup is aperiodic if and only if it is H-trivial, or, equivalently, if it contains no non-trivial subgroup. The connection between aperiodic semigroups and star-free sets was established by Schutzenberger 53]. Theorem 6.1. A recognizable subset of A is star-free if and only if its syntactic monoid is aperiodic. There are essentially two techniques to prove this result. The original proof of Schutzenberger 53,37,22], slightly simpli ed in 32], is by induction on the J depth of the syntactic semigroup. The second proof 11,31] makes use of a weak form of the Krohn-Rhodes theorem: every aperiodic nite semigroup divides a wreath product of copies of the monoid U2 = f1; a; bg, given by the multiplication table aa = a, ab = b, ba = b and bb = b. 14 language is star-free. J.E. Pin ( ) Corollary 6.2. There is an algorithm to decide whether a given recognizable Given the minimal automaton A of the language, the algorithm consists to check whether the transition monoid of M is aperiodic. The complexity of this algorithm is analyzed in 10,58]. Example 6.1. Let A = fa; bg and consider the set L = (ab) . Its minimal automaton is represented below: a 1 b 2 The transitions and the relations de ning the syntactic monoid M of L are given in the following tables 1 2 a 2 ? b ? 1 1 Figure 6.1. The minimal automaton of (ab) . aa ? ? ab 1 ? ba ? 2 a2 = b2 = 0 aba = a bab = b Since a2 = a3 , b2 = b3 , (ab)2 = ab and (ba)2 = ba, M is aperiodic and thus L is star-free. Consider now the set L0 = (aa) . Its minimal automaton is represented below: a 1 a 2 Figure 6.2. The minimal automaton of (aa) . ( ) A recognizable set can be given either by a nite automaton, by a nite semigroup or by a rational expression since there are standard algorithms to pass from one representation to the other. FINITE SEMIGROUPS AND RECOGNIZABLE LANGUAGES 15 The transitions and the relations de ning the syntactic monoid M 0 of L0 are given in the following tables 1 1 2 a3 = a a 2 1 b=0 b ? ? aa 1 2 Thus M 0 is not aperiodic and hence L0 is not star-free. Recall that the languages of concatenation level 0 of A are A and 0. According to the general de nition, the languages of concatenation level 1 are the nite boolean combinations of the languages of the form A a1 A a2 A A ak A , where k 0 and ai 2 A. The languages of this form are called piecewise testable . Intuitively, such a language can be recognized by an automaton that one could call a Hydra automaton . a 1 a2 a 3 a 4 a 5 1 2 3 ... 4 an 7. Piecewise testable languages Finite Memory Figure 7.1. A Hydra automaton with four heads. Such an automaton has a nite number h of heads, each of which can read a letter of the input word. The heads are ordered, so that together they permit to read a subword (in the sense of a subsequence of non necessarily consecutive letters) of the input word. The automaton computes in this way the set of all subwords of length h of the input word. This set is then compared to the nite collection of sets of words contained in the memory. If it occurs in the memory, the word is accepted, otherwise it is rejected. For instance, for the language (A aA bA aA \ A bA bA aA ) n (A aA bA bA A bA bA bA ), the memory would contain the collection of all sets of words of length 3 containing aba and bba but containing neither abb nor bbb. Piecewise testable languages are characterized by a deep result of I. Simon 57]. 16 J.E. Pin Theorem 7.1. A language of A is piecewise testable if and only if its syntactic monoid is J -trivial, or, equivalently, if it satis es the equations x! = x!+1 and (xy)! = (yx)! . Corollary 7.2. There is an algorithm to decide whether a given star-free language is of concatenation level 1. Given the minimal automaton A of the language, the algorithm consists in checking whether the transition monoid of M is J -trivial. Actually, this condition can be directly checked on A in polynomial time 10,58]. There exist several proofs of Simon's theorem 2,57,69,58]. The central argument of Simon's original proof 57] is a careful study of the combinatorics of the subword relation. Stern's proof 58] borrows some ideas from model theory. The proof of Straubing and Therien 69] is the only one that avoids totally combinatorics on words. In the spirit of the proof of Schutzenberger, it works by induction on the cardinality of the syntactic monoid. The proof of Almeida 2] is based on implicit operations (see the papers of J. Almeida and P. Weil in this volume for more details). Example 7.1. Let A = fa; b; cg and let L = A abA . The minimal automaton of L is represented below b, c 1 c a a 2 b a, b, c 3 Figure 7.2. The minimal automaton of L. The transitions and the relations de ning the syntactic monoid M of L are given in the following tables a2 = a 1 1 2 3 ab = 0 a 2 2 3 ac = c b2 = b b 1 3 3 bc = b c 1 1 3 ca = a ab 3 3 3 cb = c ba 2 3 3 c2 = c The J -class structure of M is represented in the following diagram. FINITE SEMIGROUPS AND RECOGNIZABLE LANGUAGES 1 17 a ba 0 c b Figure 7.3. The J -classes of M . In particular, a J c and thus M is not J -trivial. Therefore L is not piecewise testable. Example 7.2. Consider now the language L0 = A abA on the alphabet A = fa; bg. Then the minimal automaton of L0 is obtained from that of L by erasing the transitions with label c. b 1 a a 2 b a, b 3 Figure 7.4. The minimal automaton of L0 . The transitions and the relations de ning the syntactic monoid M 0 of L0 are given in the following tables 1 1 a 2 b 1 ab 3 ba 2 2 2 3 3 3 3 3 3 3 3 a2 = a ab = 0 b2 = b The J -class structure of M 0 is represented in the following diagram. 18 J.E. Pin 1 a ba 0 b Thus M 0 is J -trivial and L0 is piecewise testable. In fact L0 = A aA bA . Simon's theorem also has some nice consequences of pure semigroup theory. An ordered monoid is a monoid equipped with a stable order relation. An ordered monoid (M; ) is called 1-ordered if, for every x 2 M , x 1. A nite 1-ordered monoid is always J -trivial. Indeed, if u J v, there exist x; y; z; t 2 M such that u = xvy and v = zyt. Now x 1, y 1 and thus u = xvy v and similarly, v u whence u = v. The converse is not true: there exist nite J -trivial monoids which cannot be 1-ordered. Example 7.3. Let M be the monoid with zero presented on fa; b; cg by the relations aa = ac = ba = bb = ca = cb = cc = 0. Thus M = f1; a; b; c; ab; bc; abc; 0g and M is J -trivial. However, M is not a 1-ordered monoid. Otherwise, one would have on the one hand, b 1, whence abc ac = 0 and on the other hand, 0 1, whence 0 = 0:abc 1:abc = abc, a contradiction since abc 6= 0. However, Straubing and Therien 69] proved that 1-ordered monoids generate all the nite J -trivial monoids in the following sense. monoid. Figure 7.5. The J -classes of M 0 . Theorem 7.3. A monoid is J -trivial if and only if it is a quotient of a 1-ordered Actually, it is not di cult to establish that this result is equivalent to Simon's theorem. But Straubing and Therien also gave an ingenious direct proof of their result by induction on the cardinality of the monoid. This gives in turn a proof of Simon's theorem. Straubing 63] also observed the following connection with semigroups of relations. relations on a nite set. Theorem 7.4. A monoid is J -trivial if and only if it divides a monoid of re exive FINITE SEMIGROUPS AND RECOGNIZABLE LANGUAGES 19 A language of A+ is locally testable if it is a boolean combination of languages of the form uA , A v or A wA where u; v; w 2 A+ . For instance, if A = fa; bg, the language (ab)+ is locally testable since (ab)+ = (aA \ A b) n (A aaA A bbA ). These languages occur naturally in the study of the languages of dot-depth one. Actually they form the rst level of a natural subhierarchy of the languages of dot-depth one (see 36] for more details). Locally testable languages also have a natural interpretation in terms of automata. They are recognized by scanners . A scanner is a machine equipped with a nite memory and a window of size n to scan the input word. a 1 a2 a 3 a 4 a 5 ... an 8. Locally testable languages Finite Memory The window can also be moved beyond the rst and last letter of the word, so that the pre xes and su xes of length < n can be read. For instance, if n = 3, and u = abbaaabab, the di erent positions of the window are represented on the following diagrams: Figure 8.1. A scanner. a bbaaabab ab baaabab abb aaabab a bba aabab abbaaaba b At the end of the scan, the scanner memorizes the pre xes and the su xes of length < n and the set of factors of length n of the input word, but does not count the multiplicities. That is, if a factor occurs several times, it is memorized just once. This information is then compared to a collection of permitted sets of pre xes, su xes and factors contained in the memory. The word is accepted or rejected, according to the result of this test. The algebraic characterization of locally testable languages is slightly more involved than for star-free or piecewise testable languages. Recall that a nite semigroup S is said to have a property locally , if, for every idempotent e of S , the subsemigroup eSe = fese j s 2 S g has the property. In particular, a semigroup is locally trivial if, for every idempotent e of S , eSe = e and is locally idempotent and commutative if, for every idempotent e of S , eSe is idempotent and commutative. Equivalently, S is locally idempotent and commutative if, for every e; s; t 2 S such that e = e2 , (ese)2 = (ese) and (ese)(ete) = (ete)(ese). The following result was proved independently by Brzozowski and Simon 9] and by McNaughton 29]. 20 J.E. Pin + its syntactic semigroup is locally idempotent and commutative. Theorem 8.1. A recognizable language of A is locally testable if and only if This result, or more precisely the proof of this result, had a strong in uence on pure semigroup theory. The reason is that Theorem 8.1 can be divided into two separate statements. Proposition 8.2. A recognizable language of A is locally testable if and only + if its syntactic semigroup divides a semidirect product of a semilattice by a locally trivial semigroup. Proposition 8.3. A semigroup divides a semidirect product of a semilattice by a locally trivial semigroup if and only if it is locally idempotent and commutative. The proof of Proposition 8.2 is relatively easy, but Proposition 8.3 is much more di cult and relies on an interesting property. Given a semigroup S , form a graph G(S ) as follows: the vertices are the idempotents of S and the edges from e to f are the elements of the form esf . Then one can show that a semigroup divides a semidirect product of a semilattice by a locally trivial semigroup if and only if its graph is locally idempotent and commutative in the following sense: if p and q are loops around the same vertex, then p = p2 and pq = qp. We shall encounter another condition on graphs in Theorem 11.1. This type of graph conditions is now well understood, although numerous problems are still pending. The graph of a semigroup is a special instance of a derived category and is deeply connected with the study of the semidirect product (see Straubing 68] and Tilson 71]). In 1974, the syntactic characterizations of the star-free, piecewise testable and locally testable languages had already established the power of the semigroup approach. However, these theorems were still isolated. In 1976, Eilenberg presented in his book a uni ed framework for these three results. The cornerstone of this approach is the concept of variety. Recall that a variety of nite semigroups (or pseudovariety ) is a class of semigroups V such that: (1) if S 2 V and if T is a subsemigroup of S , then T 2 V, (2) if S 2 V and if T is a quotient of S , then T 2 V, Q (3) if (Si )i2I is a nite family of semigroups of V, then i2I Si is also in V. Varieties of nite monoids are de ned in the same way. Condition (3) can be replaced by the conjunction of conditions (3.a) and (3.b): (3.a) the trivial semigroup 1 belongs to V, (3.b) if S1 and S2 are semigroups of V, then S1 S2 is also in V. Indeed, condition (3.a) is obtained by taking I = ; in (3). 9. Varieties, another approach to recognizable languages. FINITE SEMIGROUPS AND RECOGNIZABLE LANGUAGES 21 Recall that a semigroup T divides a semigroup S if T is a quotient of a subsemigroup of S . Division is a transitive relation on semigroups and thus conditions (1) and (2) can be replaced by condition (10 ) (10 ) if S 2 V and if T divides S , then T 2 V. Given a class C of semigroups, the intersection of all varieties containing C is still a variety, called the variety generated by C , and denoted by hCi. In a more constructive way, hCi is the class of all semigroups that divide a nite product of semigroups of C . (1) The class M of all nite monoids forms a variety of nite monoids. (2) The smallest variety of nite monoids is the trivial variety, denoted by I, consisting only of the monoid 1. (3) The class Com of all nite commutative monoids form a variety of nite monoids. (4) The class J1 of all nite idempotent and commutative monoids (or semilattices) forms a variety of nite monoids. (5) The class A of all nite aperiodic monoids forms a variety of nite monoids. (6) The class J of all nite J -trivial monoids forms a variety of nite monoids. (7) The class of LI of all nite locally trivial semigroups forms a variety of nite semigroups. (8) The class LJ1 of all nite locally idempotent and commutative semigroups forms a variety of nite semigroups. Equations are a convenient way to de ne varieties. For instance, the variety of nite commutative semigroups is de ned by the equation xy = yx, the variety of aperiodic semigroups is de ned by the equation x! = x!+1 . Of course, x! = x!+1 is not an equation in the usual sense, since ! is not a xed integer: : : However, one can give a rigorous meaning to this \pseudoequation". Since J. Almeida and P. Weil present this topic in great detail in this volume, we refer the reader to their article for more information. For our purpose, it su ces to remember that equations (or pseudoequations) give an elegant description of the varieties of nite semigroups, but are sometimes very di cult to determine. We shall now extend this purely algebraic approach to recognizable languages. If V is a variety of semigroups, we denote by V (A+ ) the set of recognizable languages of A+ whose syntactic semigroup belongs to V. This is also the set of languages of A+ recognized by a semigroup of V. A +-class of recognizable languages is a correspondence which associates with every nite alphabet A, a set C (A+ ) of recognizable languages of A+ . Similarly, a -class of recognizable languages is a correspondence which associates with every nite alphabet A, a set C (A ) of recognizable languages of A . In particular, the correspondence V ! V associates with every variety of semigroups a +-class of recognizable languages. Eilenberg gave a combinatorial description of the classes of languages that occur in this way. Example 9.1. 22 J.E. Pin If X is a language of A+ and if u 2 A , the left quotient (resp. right quotient ) of X by u is the language u?1 X = fv 2 A+ j uv 2 X g (resp. Xu?1 = fv 2 A+ j vu 2 X g) Left and right quotients are de ned similarly for languages of A by replacing A+ by A in the de nition. A +-variety is a class of recognizable languages such that (1) for every alphabet A, V (A+ ) is closed under nite boolean operations ( nite union and complement), (2) for every semigroup morphism ' : A+ ! B + , X 2 V (B + ) implies '?1 (X ) 2 V (A+ ), (3) If X 2 V (A+ ) and u 2 A+ , then u?1 X 2 V (A+ ) and Xu?1 2 V (A+ ). Similarly, a -variety is a class of recognizable languages such that (1) for every alphabet A, V (A ) is closed under nite boolean operations, (2) for every monoid morphism ' : A ! B , X 2 V (B ) implies '?1 (X ) 2 V (A ), (3) If X 2 V (A ) and u 2 A , then u?1 X 2 V (A ) and Xu?1 2 V (A ). We are ready to state Eilenberg's theorem. Theorem 9.1. The correspondence V ! V de nes a bijection between the varieties of semigroups and the +-varieties. The variety of nite semigroups corresponding to a given +-variety is the variety of semigroups generated by the syntactic semigroups of all the languages L 2 V (A+ ), for every nite alphabet A. There is, of course, a similar statement for the varieties. Theorem 9.2. The correspondence V ! V de nes a bijection between the varieties of monoids and the -varieties. Varieties of nite semigroups or monoids are usually denoted by boldface letters and the corresponding varieties of languages are denoted by the corresponding cursive letters. We already know four instances of Eilenberg's variety theorem. (1) By Kleene's theorem, the -variety corresponding to M is the -variety of rational languages. (2) By Schutzenberger's theorem, the -variety corresponding to A is the variety of star-free languages. (3) By Simon's theorem, the -variety corresponding to J is the -variety of piecewise testable languages. (4) By Theorem 8.1, the +-variety corresponding to LJ1 is the +-variety of locally testable languages. FINITE SEMIGROUPS AND RECOGNIZABLE LANGUAGES 23 To clear up any possible misunderstanding, note that the four theorems mentioned above (Kleene, Schutzenberger, etc.) are not corollaries of the variety theorem. For instance, the variety theorem indicates that the languages corresponding to the nite aperiodic monoids form a -variety; it doesn't say that this -variety is the variety of star-free languages: : : Actually, it is often a di cult problem to nd an explicit description of the -variety of languages corresponding to a given variety of nite monoids, or, conversely, to nd the variety of nite monoids corresponding to a given -variety. However, the variety theorem provided a new direction to classify recognizable languages. Systematic searches for the variety of monoids (resp. languages) corresponding to a given variety of languages (resp. monoids) were soon undertaken. A partial account of these results is given into the next section. 10. Bestiary We review in this section a few examples of correspondence between varieties of nite monoids (or semigroups) and varieties of languages. A boolean algebra is a set of languages containing the empty language and closed under nite union, nite intersection and complement. Let us start our visit of the zoo with the subvarieties of the variety Com of all nite commutative monoids: the variety Acom of commutative aperiodic monoids, the variety Gcom of commutative groups, the variety J1 of idempotent and commutative monoids (or semilattices) and the trivial variety I. Proposition 10.1. For every alphabet A, I (A ) = f0; A g. Proposition 10.2. For every alphabet A, J 1 (A ) is the boolean algebra generated by the languages of the form A aA where a is a letter. Equivalently, J 1 (A ) is the boolean algebra generated by the languages of the form B where B is a subset of A. Proposition 10.3. For every alphabet A, G com(A ) is the boolean algebra generated by the languages of the form L(a; k; n) = fu 2 A j juja k mod ng where a 2 A and 0 k < n. Proposition 10.4. For every alphabet A, Acom(A ) is the boolean algebra generated by the languages of the form where a 2 A and k 0. L(a; k) = fu 2 A+ j juja = kg 24 J.E. Pin Proposition 10.5. For every alphabet A, C om(A ) is the boolean algebra generated by the languages of the form L(a; k) = fu 2 A+ j juja = kg or L(a; k; n) = fu 2 A+ j juja k mod ng where a 2 A and 0 k < n. Consider now the variety LI of all locally trivial semigroups and its subvarieties Lr I, L I and Nil. A nite semigroup S belongs to LI if and only if, for every e 2 E (S ) and every s 2 S , ese = e. The asymmetrical versions of this condition de ne the varieties Lr I and L I. Thus Lr I (resp. L I) is the variety of all nite semigroups S such that se = e (resp. es = e). Equivalently, a semigroup belongs to LI (resp. Lr I, L I) if it is a nilpotent extension of a rectangular band (resp. a right rectangular band, a left rectangular band). Finally Nil is the variety of nilpotent semigroups, de ned by the condition es = se = e for every e 2 E (S ) and every s 2 S . Recall that a subset F of a set E is co nite if its complement in E is nite. languages of A . + Proposition 10.6. For every alphabet A, N il(A ) is the set of nite or co nite + of languages of the form A X Y (resp. XA subsets of A+ . Proposition 10.7. For every alphabet A, Lr I (A ) (resp. L I (A )) is the set + + + Y ), where X and Y are nite Proposition 10.8. For every alphabet A, LI (A ) is the set of languages of the form XA Y Z , where X , Y and Z are nite subsets of A+ . Note that the previous characterizations do not make use of the complement, although the sets N il(A+ ), Lr I (A+ ), L I (A+ ) and LI (A+ ) are closed under complement. Actually, the following characterizations hold. Proposition 10.9. For every alphabet A, (1) Lr I (A ) is the boolean algebra generated by the languages of the form A u, where u 2 A , (2) L` I (A ) is the boolean algebra generated by the languages of the form uA , where u 2 A , (3) LI (A ) is the boolean algebra generated by the languages of the form uA or A u, where u 2 A . + + + + + + It would be to long to state in full detail all known results on varieties of languages. Let us just mention that the languages corresponding to the following varieties of nite semigroups or monoids are known: all varieties of bands ( 45] for the FINITE SEMIGROUPS AND RECOGNIZABLE LANGUAGES 25 lower levels and 56] for the general case), the varieties of R-trivial (resp. Ltrivial) monoids 17,7,37], the varieties of p-groups (resp. nilpotent groups) 17], the varieties of solvable groups 60], the varieties of monoids whose groups are commutative 54,26], nilpotent 17], solvable 60], the variety of monoids with commuting idempotents 27], the variety of J -trivial monoids with commuting idempotents 3], the variety of monoids whose regular J -classes are rectangular bands 55], the variety of block groups (see the author's article \BG = PG, a success story" in this volume) and many others which follow in particular from the general results given in section 12. As the variety approach proved to be successful in many di erent situations, it was expected to shed some new light on the di cult problems mentioned in section 5. The reality is more contrasted. In brief, varieties do not seem to be helpful for the star height, it is so far the most successful approach for the dot-depth and the concatenation levels and, with regard to the extended star height, it seems to be a useful tool, but probably nothing more. Let us comment on this judgment in more details. Varieties do not seem to be helpful for the star height, simply because the languages of a given star height are not closed under inverse morphisms between free monoids and thus, do not form a variety of languages. However, the notion of syntactic semigroup arises in the proof of Hashiguchi's theorems. Schutzenberger's theorem shows that the languages of extended star height 0 form a variety. However, it seems unlikely that a similar result holds for the languages of extended star height 1. Indeed, one can show 33] that every nite monoid divides the syntactic monoid of a language of the form L , where L is nite. It follows that if the languages of extended star height 1 form a variety of languages, then this variety is the variety of all rational languages. In particular, this would imply that every recognizable language is of extended star height 0 or 1. Varieties are much more useful in the study of the concatenation product. We have already seen the syntactic characterization of the languages of concatenation level 1. There is a similar result for the languages of dot-depth one. It is easy to see from the general de nition that a language of A+ is of \dot-depth one" if it is a boolean combination of languages of the form 11. Back to the early attempts u0 A u1 A u2 A u k ? 1 A uk where k 0 and ui 2 A . The syntactic characterization of these languages was settled by Knast 24,25]. Theorem 11.1. A language of A is of dot-depth one if and only if the graph of + its syntactic semigroup satis es the following condition : if e and f are two vertices, p and r edges from e to f , and q and s edges from f to e, then (pq)! ps(rs)! = (pq)! (rs)! . 26 J.E. Pin p, r e q, s f More generally, one can show that the languages of dot-depth n form a +-variety of languages. The corresponding variety of nite semigroups is usually denoted by Bn . Similarly, the languages of concatenation level n form a -variety of languages and the corresponding variety of nite monoids is denoted Vn . The two hierarchies are strict 8]. 0, there exists a language of dot-depth n + 1 which is not of dot-depth n and a language of concatenation level n + 1 which is not of concatenation level n. An important connection between the two hierarchies was found by Straubing 67]. Given a variety of nite monoids V and a variety of nite semigroups W, denote by V W the variety of nite semigroups generated by the semidirect products S T with S 2 V and T 2 W such that the action of T on S is right unitary. Theorem 11.2. For every n Theorem 11.3. For every n > 0, one has Bn = Vn LI and Vn = Bn \ M. In particular B1 = J LI. It follows also, thanks to e deep result of Straubing 67] that Bn is decidable if and only if Vn is decidable. However, it is still an open problem to know whether the varieties Vn are decidable for n 2. The case n = 2 is especially frustrating, but although several partial results have been obtained 44,68,72,70,74,13], the general case remains open. FINITE SEMIGROUPS AND RECOGNIZABLE LANGUAGES 27 12. Recent developments We shall not discuss in detail the numerous developments of the theory since Eilenberg's variety theorem, but we shall indicate the main trends. A quick glance at the known examples shows that the combinatorial description of a variety of languages follow most often the following pattern: the variety is described as the smallest variety closed under a given class of operations, such as boolean operations, product, etc. Varieties of semigroups are also often de ned with the help of operators: join, semidirect products, Malcev products, etc. In view of Eilenberg's theorem, one may expect some relationship between the operators on languages (of combinatorial nature) and the operators on semigroups (of algebraic nature). V ??????????????????! W ? ? ? ? y V Operation on semigroups Operation on languages ? ? ? ? y ?????????????????! W In the late seventies, several results of this type were established, in particular by H. Straubing. We rst consider the marked product. One of the most useful tools for studying this product is the Schutzenberger product of n monoids, which was originally de ned by Schutzenberger for two monoids 53], and extended by Straubing 64] for any number of monoids. Given a monoid M , the set of subsets of M , denoted P (M ), is a semiring under union as addition and the product of subsets as multiplication, de ned, for all X; Y M by XY = fxy j x 2 X and y 2 Y g. Let M1 ; : : : ; Mn be monoids. We denote by M the product monoid M1 Mn , k the semiring P (M ) and by Mn (k) the semiring of square matrices of size n with entries in k. The Schutzenberger product of M1 ; : : : ; Mn , denoted }n (M1 ; : : : ; Mn ) is the submonoid of the multiplicative monoid Mn (k) composed of all the matrices P satisfying the three following conditions: (1) If i > j , Pi;j = 0 (2) If 1 i n, Pi;i = f(1; : : : ; 1; si ; 1; : : : ; 1)g for some si 2 Si (3) If 1 i j n, Pi;j 1 1 Mi Mj 1 1. Condition (1) indicates that the matrices of the Schutzenberger product are upper triangular, condition (2) enables to identify the diagonal coe cient Pi;i with an element si of Mi and condition (3) shows that if i < j , Pi;j can be identi ed with a subset of Mi Mj . With this convention, a matrix of }3 (M1 ; M2 ; M3 ) will 28 have the form 1 J.E. Pin 0s P P 1 @ 0 s ; P ;; A 12 2 13 with si 2 Mi , P1;2 M1 M2 , P1;3 M1 Notice that the Schutzenberger product is not associative, in the sense that in general the monoids }2 (M1 ; }2 (M2 ; M3 )), }3 (M1 ; M2 ; M3 ) and }2 (}2 (M1 ; M2 ); M3 ) are pairwise distinct. The following result shows that the Schutzenberger product is the algebraic operation on monoids that corresponds to the marked product. 0 0 s3 M2 M3 and P2;3 M2 M3 . 23 Proposition 12.1. Let L ; L ; : : : ; Ln be languages of A recognized by monoids 0 1 M0 ; M1 ; : : : ; Mn and let a1 ; : : : ; an be letters of A. Then the marked product L0 a1 L1 an Ln is recognized by the monoid }n+1 (M0 ; M1 ; : : : ; Mn ). This result was extended to varieties by Reutenauer 50] for n = 1 and by the author 36] in the general case (see also 73] for a simpler proof). Let V0 , : : : , Vn be varieties of nite monoids and let }n+1 (V0 ; V1 ; : : : ; Vn ) be the variety of nite monoids generated by the Schutzenberger products of the form }n+1 (M0 ; M1 ; : : : ; Mn ) with M0 2 V0 , M1 2 V1 , : : : , Mn 2 Vn . Theorem 12.2. Let V be the -variety corresponding to the variety of nite monoids }n+1 (V0 ; V1 ; : : : ; Vn ). Then, for all alphabet A, V (A ) is the boolean algebra generated by all the marked products of the form L0 a1 L1 an Ln where L0 2 V 0 (A ),: : : , Ln 2 V n (A ) and a1 ; : : : ; an 2 A. If V0 = V1S= : : : = Vn = V, the variety }n+1 (V; V; : : : ; V) is denoted }n+1 V and }V = n>0 }n V denotes the union of all }n V. Given a -variety of languages V , the extension of V under marked product is the -variety V 0 such that, for every alphabet A, V 0 (A ) is the boolean algebra generated by the marked products of the form L0 a1 L1 an Ln where L0 ; L1 ; : : : ; Ln 2 V (A ) and a1 ; : : : ; an 2 A. The closure of V under marked product is the smallest -variety V such that, for every alphabet A, V (A ) contains V (A ) and all the marked products of the form L0 a1 L1 an Ln where L0 ; L1 ; : : : ; Ln 2 V (A ) and a1 ; : : : ; an 2 A. The -variety corresponding to }V is described in the following theorem. marked product. Theorem 12.3. Let V be a monoid variety and let V be the corresponding -variety. Then the -variety corresponding to }V is the extension of V under Corollary 12.4. A -variety is closed under marked product if and only if the corresponding variety of monoids V satis es V = }V. FINITE SEMIGROUPS AND RECOGNIZABLE LANGUAGES 29 The Schutzenberger product has a remarkable algebraic property 64,39]. Let M1 , : : : , Mn be monoids and let be the monoid morphism from }n (M1 ; : : : ; Mn ) into M1 Mn that maps a matrix onto its diagonal. Theorem 12.5. For every idempotent e of M is in the variety B . 1 1 Mn , the semigroup ?1 (e) Given a variety of nite semigroups V, a nite monoid M is called a V-extension of a nite monoid N if there exists a surjective morphism ' : M ! N such that, for every idempotent e of N , '?1 (e) 2 V. Theorem 12.5 shows that the Schutzenberger product of n nite monoids is a B1 -extension of their product. Given a variety of nite monoids W, the Malcev product V M W is the variety of nite monoids generated by all the V-extensions of monoids of W. This gives the following relation between the Vn . Theorem 12.6. For every n 0, Vn is contained in B +1 1 M Vn . It is conjectured that Vn+1 = B1 M Vn for every n. If this conjecture were true, it would reduce the decidability of the dot-depth to a problem on the Malcev products of the form B1 M V. Malcev products actually play an important role in the study of the marked product. For instance, Straubing has established the important following result, which gives support to the previous conjecture. Theorem 12.7. Let V be a monoid variety and let V be the corresponding -variety. Then the -variety corresponding to A V is the closure of V under marked product. M Example 12.1. Let H be a variety of nite groups (for instance, the variety of all nite commutative groups, nilpotent groups, solvable groups, etc.). Denote by H the variety of all monoids whose subgroups (that is, H-classes containing an idempotent) belong to H. One can show that A M H = H. Therefore, the corresponding -variety is closed under marked product. The marked product L = L0 a1 L1 an Ln of n languages L0 , L1 , : : : , Ln is unambiguous if every word u of L admits a unique factorization of the form u0 a1 u1 an un with u0 2 L0 , u1 2 L1 , : : : , un 2 Ln . The following result was established in 35,46] as a generalization of a former result of Schutzenberger 55]. Theorem 12.8. Let V be a monoid variety and let V be the corresponding -variety. Then the -variety corresponding to LI V is the closure of V under unambiguous marked product. M 30 J.E. Pin The extension of a given -variety is also characterized in 46]. Other variations of the marked product have been considered 55,35,49]. They lead to some interesting algebraic constructions. Another operation on semigroups has a natural counterpart in terms of languages. Given a variety of nite monoids V, denote by PV the variety of nite monoids generated by all the monoids of the form P (M ), for M 2 V. A monoid morphism ' : B ! A is length preserving if it maps a letter of B onto a letter of A. Given a -variety of languages V , the extension of V under length preserving morphisms is the smallest -variety V 0 such that, for every alphabet A, V 0 (A ) contains the languages of the form '(L) where L 2 V (B ) and ' : B ! A is a length preserving morphism. The closure of V under length preserving morphisms is the smallest -variety V containing V such that, for every length preserving morphism ' : B ! A , L 2 V (A ) implies '?1 (L) 2 V (B ). We can now state the result found independently by Reutenauer 50] and Straubing 62]. Theorem 12.9. Let V be a monoid variety and let V be the corresponding -variety. Then the -variety corresponding to PV is the extension of V under length preserving morphisms. Corollary 12.10. A -variety is closed under length preserving morphisms if and only if the corresponding variety of monoids V satis es V = PV. These results motivated the systematic study of the varieties of the form PV, which is not yet achieved. See the survey article 38] for the known results prior to 1986 and the book of J. Almeida 1] for the more recent results. The Schutzenberger product and the power monoid are actually particular cases of a general construction which gives the monoid counterpart of a given operation on languages 42,43,40]. This general construction works for most operations on languages, with the notable exception of the star operation, but its presentation would take us to far a eld. We conclude this section by a few results on the semidirect product of two varieties. We have already de ned the semidirect product V W of a variety of nite monoids V and a variety of nite semigroups W. One can de ne similarly the semidirect product of two varieties of nite monoids or of two varieties of nite semigroups. For instance, if V and W are two varieties of nite monoids, V W is the variety of nite monoids generated by the semidirect products M N with M 2 V and N 2 W such that the action of N on M is unitary. This variety is also generated by the wreath products M N with M 2 V and N 2 W. Straubing has given a general construction to describe the languages recognized by the wreath product of two nite monoids. Let M 2 V and N 2 W be two nite monoids and let : A ! M N be a monoid morphism. We denote by FINITE SEMIGROUPS AND RECOGNIZABLE LANGUAGES 31 : M N ! N the monoid morphism de ned by (f; n) = n and we put ' = . Thus ' is a monoid morphism from A into N . Let B = N A and : A ! B be the map (which is not a morphism!) de ned by (a1 a2 an ) = (1; a1 )('(a1 ); a2 ) ('(a1 a2 an1 ); an ) Then Straubing's \wreath product principle" can be stated as follows. is recognized by M and where X Theorem 12.11. If a language L is recognized by : A ! M N , then L is a nite boolean combination of languages of the form X \ ? (Y ), where Y B A is recognized by N . 1 Conversely, the nite boolean combinations of languages of the form X \ ?1 (Y ) are not necessarily recognized by M N , but are certainly recognized by a monoid of the variety V W. Therefore, a careful study of the languages of the form ?1 (Y ) usually su ces to give a combinatorial description of the languages corresponding to V W. A similar wreath product principle holds when V or W are varieties of nite semigroups. Examples of application of this technique include Proposition 8.2 and the proof of Schutzenberger's theorem based on the fact that every nite aperiodic monoid divides a wreath product of copies of U2 . Straubing also has successfully used this principle to describe the variety of languages corresponding to solvable groups (solvable groups are wreath products of commutative groups) and in his proof of the equality Bn = Vn LI. We have centered our presentation around the notion of variety and voluntarily left out several aspects of the theory which are developed extensively in other articles of this volume: H. Straubing, D. Therien and W. Thomas survey the connections with formal logic and boolean circuits, J. Almeida and P. Weil present the implicit operations, D. Perrin and the author treat the theory of automata on in nite words, J. Rhodes states a general conjecture on Malcev products, the topological aspects are mentioned in the author's account of the success story BG = PG, S.W. Margolis and J. Meakin cover the extensions of automata theory to inverse monoids, M. Sapir demarcates the border between decidable and undecidable and H. Short shows that automata are also useful in group theory. Some other extensions are not covered at all in this volume, in particular the connections with the variable length codes, the rational and recognizable sets on arbitrary monoids and the extension of the theory to power series and algebras. We hope that the reading of the articles of this volume will convince the reader that the algebraic theory of automata is a recent but ourishing subject. It is intimately related to the theory of nite semigroups and certainly one of the most convincing applications of this theory. 13. Conclusion 32 J.E. Pin I would like to thank Pascal Weil, Marc Zeitoun and Monica Mangold for many useful suggestions. Acknowledgements References 1] J. Almeida, Semigrupos Finitos e Algebra Universal, Publicacoes do Instituto de Matematica e Estat stica da Universidade de Sa~ Paulo, (1992) o 2] J. Almeida, Implicit operations on nite J -trivial semigroups and a conjecture of I. Simon, J. Pure Appl. Algebra 69, (1990) 205{218. 3] C. J. Ash, T.E. Hall and J.E. 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Pin, Languages and inverse semigroups, 11th ICALP, Lecture Notes in Computer Science 199, Springer, Berlin (1985) 285{299. 28] S. W. Margolis and J.E. Pin, Inverse semigroups and varieties of nite semigroups, Journal of Algebra 110 (1987) 306{323. 29] R. McNaughton, Algebraic decision procedures for local testability, Math. Syst. Theor. 8, (1974) 60{76. 30] R. McNaughton and S. Pappert, Counter-free Automata, MIT Press, (1971). 31] A. R. Meyer, A note on star-free events, J. Assoc. Comput. Mach. 16, (1969) 220{225. 32] D. Perrin, Automata, Chapter 1 in Handbook of Theoretical Computer Science (J. Van Leeuwen, ed.), Vol B: Formal Models and Semantics, Elsevier (1990). 33] J.-E. Pin, Sur le mono de de L lorsque L est un langage ni. Theor. Comput. Sci. 7 (1978) 211{215. 34] J.-E. Pin, Varietes de langages et mono de des parties, Semigroup Forum 20 (1980) 11{47. 35] J.-E. Pin, Proprietes syntactiques du produit non ambigu. 7th ICALP, Lecture Notes in Computer Science 85 (1980) 483{499. 36] J.-E. Pin, Hierarchies de concatenation, RAIRO Informatique Theorique 18, (1984) 23{46. 37] J.-E. Pin, 1984 Varietes de langages formels, Masson, Paris; English translation: 1986, Varieties of formal languages, Plenum, New-York. 38] J.-E. Pin, Power semigroups and related varieties of nite semigroups, Semigroups and Their Applications, ed. S. M. Goberstein and P. M. Higgins, D. Reidel (1986) 139{152. 34 J.E. Pin 39] J.-E. Pin, A property of the Schutzenberger product, Semigroup Forum 35, (1987) 53{62. 40] J.-E. Pin, Relational morphisms, transductions and operations on languages, in J.E. Pin (ed.), Formal properties of nite automata and applications, Lect. Notes in Comp. Sci. 386, Springer, (1989) 120{137. 41] J.-E. Pin, Logic, Semigroups and Automata on Words, Proceedings of the Spring School on Logic and Computer Science, ed. S. Grigorie , to appear. 42] J.-E. 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Sezinando, The varieties of languages corresponding to the varieties of nite band monoids, Semigroup Forum 44 (1992) 283{305. 57] I. Simon, Piecewise testable events, Proc. 2nd GI Conf., Lect. Notes in Comp. Sci. 33, Springer, Berlin, (1975) 214{222. 58] J. Stern, Characterization of some classes of regular events, Theoret. Comp. Sci. 35, (1985) 17{42. FINITE SEMIGROUPS AND RECOGNIZABLE LANGUAGES 35 59] J. Stern, Complexity of some problems from the theory of automata, Inform. and Control 66, (1985) 63{176. 60] H. Straubing, Families of recognizable sets corresponding to certain varieties of nite monoids, J. Pure Appl. Algebra, 15 (1979) 305{318. 61] H. Straubing, Aperiodic homomorphisms and the concatenation product of recognizable sets, J. Pure Appl. Algebra, 15 (1979) 319{327. 62] H. Straubing, Recognizable sets and power sets of nite semigroups, Semigroup Forum, 18 (1979) 331{340. 63] H. Straubing, On nite J -trivial monoids, Semigroup Forum 19, (1980) 107{110. 64] H. 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Weil, Closure of varieties of languages under products with counter, Journal of Computer and System Sciences 45 (1992) 316{339. 74] P. Weil, Some results on the dot-depth hierarchy, Semigroup Forum 46 (1993) 352{370. ... View Full Document Ask a homework question - tutors are online	{"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": 0.9067734479904175, "perplexity": 1231.8682460628795}, "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-2017-04/segments/1484560280801.0/warc/CC-MAIN-20170116095120-00380-ip-10-171-10-70.ec2.internal.warc.gz"}
https://www.sarthaks.com/1079714/f-x-sin-x-3cos-x-ismaximum-when-x-a-3-b-4-c-6-d-0?show=1079721	# f(x) = sin x+ √3cos x is maximum when x = A.π/3 B.π/4 C.π/6 D. 0 8 views closed f(x) = sin x + $\sqrt{3}$ cos x is maximum when x = A. $\frac{\pi}{3}$ B. $\frac{\pi}{4}$ C. $\frac{\pi}{6}$ D. 0 +1 vote by (11.2k points) selected by Option : (C) f(x) = sin x + $\sqrt{3}$cos x Differentiating f(x) with respect to x, we get f'(x) = cos x - $\sqrt{3}$sin x Differentiating f’(x) with respect to x, we get f''(x) = - sin x - $\sqrt{3}$cos x For maxima at x = c, f’(c) = 0 and f’’(c) < 0 f’(x) = 0 ⇒ tan x = $\frac{1}{\sqrt 3}$ or, x = $\frac{\pi}{6}$ or $\frac{7\pi}{6}$ f"($\frac{\pi}{6}$) = - 2 < 0 and f" ($\frac{7\pi}{6}$) = 2 > 0 Hence, x = $\frac{\pi}{6}$ is a point of maxima for f(x).	{"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": 0.9494889974594116, "perplexity": 4531.366780092182}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991812.46/warc/CC-MAIN-20210515004936-20210515034936-00251.warc.gz"}
https://planetmath.org/PolynomialEquationOfOddDegree	polynomial equation of odd degree Theorem. The equation $\displaystyle a_{0}x^{n}+a_{1}x^{n-1}+\cdots+a_{n-1}x+a_{n}=0$ (1) with odd degree $n$ and real coefficients $a_{i}$ ($a_{0}\neq 0$) has at least one real root $x$. Proof. Denote by $f(x)$ the left hand side of (1). We can write $f(x)=a_{0}x^{n}[1+g(x)]$ where $\displaystyle g(x):=\frac{a_{1}}{x}\!+\cdots\!+\!\frac{a_{n-1}}{x^{n-1}}\!+\!% \frac{a_{n}}{x^{n}}$. But we have $\displaystyle\lim_{\|x\|\to\infty}g(x)=0$ because $\lim_{\|x\|\to\infty}\frac{a_{i}}{x^{i}}=0$ for all $i=1,\,...,\,n$. Thus there exists an $M>0$ such that $\|g(x)\|<1\,\,\mbox{for}\,\,\|x\|\geqq M.$ Accordingly $1+g(\pm M)>0$ and $\mbox{sign}f(\pm M)=(\mbox{sign}a_{0})(\mbox{sign}(\pm M))^{n}\cdot 1=(\mbox{% sign}a_{0})(\pm 1)$ since $n$ is odd. Therefore the real polynomial function $f$ has opposite signs in the end points of the interval$[-M,\,M]$. Thus the continuity of $f$ guarantees, according to Bolzano’s theorem, at least one zero $x$ of $f$ in that interval. So (1) has at least one real root $x$. Title polynomial equation of odd degree PolynomialEquationOfOddDegree 2013-03-22 15:39:19 2013-03-22 15:39:19 pahio (2872) pahio (2872) 7 pahio (2872) Theorem msc 26A15 msc 26A09 msc 12D10 msc 26C05 AlgebraicEquation ExampleOfSolvingACubicEquation	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 22, "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": 0.9869291186332703, "perplexity": 2739.771329301969}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610704799711.94/warc/CC-MAIN-20210126073722-20210126103722-00507.warc.gz"}
http://scholar.cnki.net/WebPress/brief.aspx?dbcode=SJMF	作者：Ismaël Soudères 来源：[J].Algebraic & Geometric Topology, 2018, Vol.18 (2), pp.635-685MSP 摘要：We prove the Beilinson--Soul\'e vanishingconjecture formotives attached to the moduli spaces $\mathcal{M}_{0,n}$ of curves of genus$0$ with$n$ marked points. As part of the proof, we also show that thesemotives aremixed Tate. As a consequence of Levine's work, we thus obtain awel... 作者：Kohei Tanaka 来源：[J].Algebraic & Geometric Topology, 2018, Vol.18 (2), pp.779-796MSP 摘要：This paper presents a discrete analog of topological complexityfor finite spaces using purely combinatorial terms. We demonstratethat this coincides with the genuine topological complexity ofthe original finite space. Furthermore, we study therelationship withsimplicial complexit... 作者：Shengkui Ye 来源：[J].Algebraic & Geometric Topology, 2018, Vol.18 (2), pp.1195-1204MSP 摘要：Let $M^{r}$ be a connected orientable manifold with the Eulercharacteristic$\chi(M)\not \equiv 0\operatorname{mod}6$. Denote by$\operatorname{SAut}(F_{n})$the unique subgroup of index two in the automorphism group of a freegroup.Then any group action of $\operatorname{SAut}(F_{n}... 作者：Andrew Blumberg , Michael Hill 来源：[J].Algebraic & Geometric Topology, 2018, Vol.18 (2), pp.723-766MSP 摘要：For a genuine'' equivariant commutative ring spectrum$R$,$\pi_0(R)$admits a rich algebraic structure known as a Tambarafunctor. This algebraic structure mirrors the structure on$R$arising from the existence of multiplicative norm maps. Motivated bythe surprising fact that Bo... 作者：Jens Hornbostel 来源：[J].Algebraic & Geometric Topology, 2018, Vol.18 (2), pp.1257-1258MSP 摘要：We correct a claim concerningmotivic$S^1$--deloopings. 作者：Bernhard Hanke , Peter Quast 来源：[J].Algebraic & Geometric Topology, 2018, Vol.18 (2), pp.877-895MSP 摘要：$\Gamma\mkern-1.5mu$--structures are weak forms of multiplications on closedoriented manifolds. As wasshown by Hopf the rational cohomologyalgebras ofmanifolds admitting$\Gamma\mkern-1.5mu$--structures are free over odd-degreegenerators.We prove that this condition is also suffi... 作者：Melissa Zhang 来源：[J].Algebraic & Geometric Topology, 2018, Vol.18 (2), pp.1147-1194MSP 摘要：For a$2\!$--periodic link$\tilde L$in the thickened annulus and itsquotient link$L$, we exhibit a spectral sequence with\[E^1 \cong \operatorname{AKh}(\tilde L) \otimes_{\mathbb{F}} \mathbb{F}[\theta, \theta^{-1}]\rightrightarrows E^\infty \cong \operatorname{AKh}(L)\otimes_{... 作者：Simon Gritschacher 来源：[J].Algebraic & Geometric Topology, 2018, Vol.18 (2), pp.1205-1249MSP 摘要：We study commutative complex$K$--theory, a generalised cohomologytheory built from spaces of ordered commuting tuples in the unitarygroups. We show that the spectrum for commutative complex$K$--theory isstably equivalent to the$k\mkern-1.5mu u$--group ring of$B\mkern-2mu U(1)... 作者：Nancy Guelman , Cristóbal Rivas 来源：[J].Algebraic & Geometric Topology, 2018, Vol.18 (2), pp.1067-1076MSP 摘要：Weshow that if $G$ is a solvable group acting on the lineand if there is $T\in G$ having no fixed points, then there is a Radonmeasure $\mu$ on the line quasi-invariant under~$G\!$. In fact, our methodallows for the same conclusion for $G$ inside a class of groups that isclosed u... 作者：Matthew Young 来源：[J].Algebraic & Geometric Topology, 2018, Vol.18 (2), pp.975-1039MSP 摘要：We introduce a relative version of the $2\mskip-1.5mu$--Segal simplicial spacesdefined by Dyckerhoff and Kapranov, and G\'{a}lvez-Carrillo, Kock andTonks. Examples of relative $2\mskip-1.5mu$--Segal spaces include the categorifiedunoriented cyclic nerve, real pseudoholomorphic po...	{"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": 0.9414829015731812, "perplexity": 2263.2141355764675}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125945222.55/warc/CC-MAIN-20180421125711-20180421145711-00051.warc.gz"}
http://wire.waynflete.org/edible-opener/	# Edible opener Students in Cathy Douglas’s and Lisa Kramer’s Calculus Accelerated classes are in the thick of a unit on the disk method and volumes by rotation. In today’s “opener” (an exercise used to get the math brain engaged), students were asked to find the volume of a Hostess cupcake by (a) a traditional formula for a truncated cone and (b) “by finding a function that can be rotated about an axis to create the shape, then using calculus to find the volume.” (!) The class compared the two results to see if the volumes were identical. Students were also asked to determine whether their findings were consistent with the volume listed on the cupcake package. Their results showed about 10 additional cubic centimeters over the package’s listed 45 grams (the gram was originally defined as the mass of one cubic centimeter of water at its maximum density at 4 degrees Celsius). “The creamy spherical filling seemed to be less dense,” says Cathy. “One of my students estimated the cubic centimeters of the creamy filling and found that it accounted for the difference in our volumes and the package’s unit weight.” Needless to say, the subject matter was consumed at the end of the class…	{"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": 0.832181990146637, "perplexity": 1215.8727307269412}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247481624.10/warc/CC-MAIN-20190217051250-20190217073250-00572.warc.gz"}
https://worldwidescience.org/topicpages/g/generalized+conformational+energy.html	#### Sample records for generalized conformational energy 1. General Conformity Science.gov (United States) The General Conformity requirements ensure that the actions taken by federal agencies in nonattainment and maintenance areas do not interfere with a state’s plans to meet national standards for air quality. 2. Conformal General Relativity CERN Document Server Pervushin, V 2001-01-01 The inflation-free solution of problems of the modern cosmology (horizon, cosmic initial data, Planck era, arrow of time, singularity,homogeneity, and so on) is considered in the conformal-invariant unified theory given in the space with geometry of similarity where we can measure only the conformal-invariant ratio of all quantities. Conformal General Relativity is defined as the $SU_c(3)\\times SU(2)\\times U(1)$-Standard Model where the dimensional parameter in the Higgs potential is replaced by a dilaton scalar field described by the negative Penrose-Chernikov-Tagirov action. Spontaneous SU(2) symmetry breaking is made on the level of the conformal-invariant angle of the dilaton-Higgs mixing, and it allows us to keep the structure of Einstein's theory with the equivalence principle. We show that the lowest order of the linearized equations of motion solves the problems mentioned above and describes the Cold Universe Scenario with the constant temperature T and z-history of all masses with respect to an obser... 3. 40 CFR 52.2133 - General conformity. Science.gov (United States) 2010-07-01 ... 40 Protection of Environment 4 2010-07-01 2010-07-01 false General conformity. 52.2133 Section 52...) APPROVAL AND PROMULGATION OF IMPLEMENTATION PLANS (CONTINUED) South Carolina § 52.2133 General conformity. The General Conformity regulations adopted into the South Carolina State Implementation Plan... 4. 40 CFR 52.938 - General conformity. Science.gov (United States) 2010-07-01 ... 40 Protection of Environment 3 2010-07-01 2010-07-01 false General conformity. 52.938 Section 52...) APPROVAL AND PROMULGATION OF IMPLEMENTATION PLANS Kentucky § 52.938 General conformity. The General Conformity regulations were submitted on November 10, 1995, and adopted into the Kentucky... 5. Conformal Gravity: Dark Matter and Dark Energy Directory of Open Access Journals (Sweden) Robert K. Nesbet 2013-01-01 Full Text Available This short review examines recent progress in understanding dark matter, dark energy, and galactic halos using theory that departs minimally from standard particle physics and cosmology. Strict conformal symmetry (local Weyl scaling covariance, postulated for all elementary massless fields, retains standard fermion and gauge boson theory but modifies Einstein–Hilbert general relativity and the Higgs scalar field model, with no new physical fields. Subgalactic phenomenology is retained. Without invoking dark matter, conformal gravity and a conformal Higgs model fit empirical data on galactic rotational velocities, galactic halos, and Hubble expansion including dark energy. 6. Conformal methods in general relativity CERN Document Server Valiente Kroon, Juan A 2016-01-01 This book offers a systematic exposition of conformal methods and how they can be used to study the global properties of solutions to the equations of Einstein's theory of gravity. It shows that combining these ideas with differential geometry can elucidate the existence and stability of the basic solutions of the theory. Introducing the differential geometric, spinorial and PDE background required to gain a deep understanding of conformal methods, this text provides an accessible account of key results in mathematical relativity over the last thirty years, including the stability of de Sitter and Minkowski spacetimes. For graduate students and researchers, this self-contained account includes useful visual models to help the reader grasp abstract concepts and a list of further reading, making this the perfect reference companion on the topic. 7. Conformal collider physics: Energy and charge correlations CERN Document Server Hofman, Diego M 2008-01-01 We study observables in a conformal field theory which are very closely related to the ones used to describe hadronic events at colliders. We focus on the correlation functions of the energies deposited on calorimeters placed at a large distance from the collision. We consider initial states produced by an operator insertion and we study some general properties of the energy correlation functions for conformal field theories. We argue that the small angle singularities of energy correlation functions are controlled by the twist of non-local light-ray operators with a definite spin. We relate the charge two point function to a particular moment of the parton distribution functions appearing in deep inelastic scattering. The one point energy correlation functions are characterized by a few numbers. For ${\\cal N}=1$ superconformal theories the one point function for states created by the R-current or the stress tensor are determined by the two parameters $a$ and $c$ characterizing the conformal anomaly. Demandin... 8. Conformal general relativity contains the quantum CERN Document Server Bonal, R; Cardenas, R 2000-01-01 Based on the de Broglie-Bohm relativistic quantum theory of motion we show that the conformal formulation of general relativity, being linked with a Weyl-integrable geometry, may implicitly contain the quantum effects of matter. In this context the Mach's principle is discussed. 9. Generalized Orbifold Construction for Conformal Nets CERN Document Server Bischoff, Marcel 2016-01-01 Let $\\mathcal{B}$ be a conformal net. We give the notion of a proper action of a finite hypergroup acting by vacuum preserving unital completely positive (so-called stochastic) maps, which generalizes the proper actions of finite groups. Taking fixed points under such an action gives a finite index subnet $\\mathcal{B}^K$ of $\\mathcal{B}$, which generalizes the $G$-orbifold. Conversely, we show that if $\\mathcal{A}\\subset \\mathcal{B}$ is a finite inclusion of conformal nets, then $\\mathcal{A}$ is a generalized orbifold $\\mathcal{A}=\\mathcal{B}^K$ of the conformal net $\\mathcal{B}$ by a unique finite hypergroup $K$. There is a Galois correspondence between intermediate nets $\\mathcal{B}^K\\subset \\mathcal{A} \\subset \\mathcal{B}$ and subhypergroups $L\\subset K$ given by $\\mathcal{A}=\\mathcal{B}^L$. In this case, the fixed point of $\\mathcal{B}^K\\subset \\mathcal{A}$ is the generalized orbifold by the hypergroup of double cosets $L\\backslash K/ L$. If $\\mathcal{A}\\subset \\mathcal{B}$ is an finite index inclusion of... 10. Willmore energy estimates in conformal Berger spheres Energy Technology Data Exchange (ETDEWEB) Barros, Manuel, E-mail: [email protected] [Departamento de Geometria y Topologia, Facultad de Ciencias Universidad de Granada, 1807 Granada (Spain); Ferrandez, Angel, E-mail: [email protected] [Departamento de Matematicas, Universidad de Murcia Campus de Espinardo, 30100 Murcia (Spain) 2011-07-15 Highlights: > The Willmore energy is computed in a wide class of surfaces. > Isoperimetric inequalities for the Willmore energy of Hopf tori are obtained. > The best possible lower bound is achieved on isoareal Hopf tori. - Abstract: We obtain isoperimetric inequalities for the Willmore energy of Hopf tori in a wide class of conformal structures on the three sphere. This class includes, on the one hand, the family of conformal Berger spheres and, on the other hand, a one parameter family of Lorentzian conformal structures. This allows us to give the best possible lower bound of Willmore energies concerning isoareal Hopf tori. 11. Slice Energy and Conformal Frames in Theories of Gravitation CERN Document Server Cotsakis, S 2004-01-01 We examine and compare the behaviour of the scalar field slice energy in different classes of theories of gravity, in particular higher-order and scalar-tensor theories. We find a universal formula for the energy and compare the resulting conservation laws with those known in general relativity. This leads to a comparison between the inflaton, the dilaton and other forms of scalar fields present in these generalized theories. It also shows that all such conformally-related, generalized theories of gravitation allow for the energy on a slice to be invariably defined and its fundamental properties be insensitive to conformal transformations. 12. On gravitational energy in conformal teleparallel gravity Science.gov (United States) da Silva, J. G.; Ulhoa, S. C. 2017-07-01 The paper deals with the definition of gravitational energy in conformal teleparallel gravity. The total energy is defined by means of the field equations which allow a local conservation law. Then such an expression is analyzed for a homogeneous and isotropic Universe. This model is implemented by the Friedmann-Robertson-Walker (FRW) line element. The energy of the Universe in the absence of matter is identified with the dark energy, however it can be expanded for curved models defining such an energy as the difference between the total energy and the energy of the perfect fluid which is the matter field in the FRW model. 13. Simple Space-Time Symmetries: Generalizing Conformal Field Theory CERN Document Server Mack, G; Mack, Gerhard; Riese, Mathias de 2004-01-01 We study simple space-time symmetry groups G which act on a space-time manifold M=G/H which admits a G-invariant global causal structure. We classify pairs (G,M) which share the following additional properties of conformal field theory: 1) The stability subgroup H of a point in M is the identity component of a parabolic subgroup of G, implying factorization H=MAN, where M generalizes Lorentz transformations, A dilatations, and N special conformal transformations. 2) special conformal transformations in N act trivially on tangent vectors to the space-time manifold M. The allowed simple Lie groups G are the universal coverings of SU(m,m), SO(2,D), Sp(l,R), SO(4n) and E_7(-25) and H are particular maximal parabolic subgroups. All these groups G admit positive energy representations. It will also be shown that the classical conformal groups SO(2,D) are the only allowed groups which possess a time reflection automorphism; in all other cases space-time has an intrinsic chiral structure. 14. 75 FR 17253 - Revisions to the General Conformity Regulations Science.gov (United States) 2010-04-05 ... Protection Agency 40 CFR Parts 51 and 93 Revisions to the General Conformity Regulations; Final Rule #0;#0... PROTECTION AGENCY 40 CFR Parts 51 and 93 RIN 2060-AH93 Revisions to the General Conformity Regulations AGENCY... or Federal implementation plan (SIP, TIP, or FIP) for attaining clean air (General... 15. General Information for Transportation and Conformity Science.gov (United States) Transportation conformity is required by the Clean Air Act section 176(c) (42 U.S.C. 7506(c)) to ensure that federal funding and approval are given to highway and transit projects that are consistent with SIP. 16. On Useful Conformal Tranformations In General Relativity CERN Document Server Carneiro, D F; De Lima, A G; Shapiro, I L 2004-01-01 Local conformal transformations are known as a useful tool in various applications of the gravitational theory, especially in cosmology. We describe some new aspects of these transformations, in particular using them for derivation of Einstein equations for the cosmological and Schwarzschild metrics. Furthermore, the conformal transformation is applied for the dimensional reduction of the Gauss-Bonnet topological invariant in $d=4$ to the spaces of lower dimensions. 17. On useful conformal tranformations in general relativity Science.gov (United States) Carneiro, D. F.; Freiras, E. A.; Gonçalves, B.; de Lima, A. G.; Shapiro, I. 2004-12-01 Local conformal transformations are known as a useful tool in various applications of the gravitational theory, especially in cosmology. We describe some new aspects of these transformations, in particular using them for derivation of Einstein equations for the cosmological and Schwarzschild metrics. Furthermore, the conformal transformation is applied for the dimensional reduction of the Gauss-Bonnet topological invariant in $d=4$ to the spaces of lower dimensions. 18. Energy of knots and conformal geometry CERN Document Server O'Hara, Jun 2003-01-01 Energy of knots is a theory that was introduced to create a "canonical configuration" of a knot - a beautiful knot which represents its knot type. This book introduces several kinds of energies, and studies the problem of whether or not there is a "canonical configuration" of a knot in each knot type. It also considers this problems in the context of conformal geometry. The energies presented in the book are defined geometrically. They measure the complexity of embeddings and have applications to physical knotting and unknotting through numerical experiments. Contents: In Search of the "Optima 19. Conformal Invariance and Conserved Quantities of General Holonomic Systems Institute of Scientific and Technical Information of China (English) CAI Jian-Le 2008-01-01 Conformal invarianee and conserved quantities of general holonomic systems are studied. A one-parameter infinitesimal transformation group and its infinitesimal transformation vector of generators are described.The definition of conformal invariance and determining equation for the system are provided.The conformal factor expression is deduced from conformal invariance and Lie symmetry.The necessary and sufficient condition,that conformal invariance of the system would be Lie symmetry,is obtained under the infinitesimal one-parameter transformation group. The corresponding conserved quantity is derived with the aid of a structure equation.Lastly,an example is given to demonstrate the application of the result. 20. Basic Information About the General Conformity Rule Science.gov (United States) These regulations ensure that federal activities or actions don't cause new violations to the NAAQS and ensure that NAAQS attainment is not delayed. This page has general information about how and where these regulations apply. 1. Conformational Nonequilibrium Enzyme Kinetics: Generalized Michaelis-Menten Equation. Science.gov (United States) Piephoff, D Evan; Wu, Jianlan; Cao, Jianshu 2017-08-03 In a conformational nonequilibrium steady state (cNESS), enzyme turnover is modulated by the underlying conformational dynamics. On the basis of a discrete kinetic network model, we use an integrated probability flux balance method to derive the cNESS turnover rate for a conformation-modulated enzymatic reaction. The traditional Michaelis-Menten (MM) rate equation is extended to a generalized form, which includes non-MM corrections induced by conformational population currents within combined cyclic kinetic loops. When conformational detailed balance is satisfied, the turnover rate reduces to the MM functional form, explaining its general validity. For the first time, a one-to-one correspondence is established between non-MM terms and combined cyclic loops with unbalanced conformational currents. Cooperativity resulting from nonequilibrium conformational dynamics can be achieved in enzymatic reactions, and we provide a novel, rigorous means of predicting and characterizing such behavior. Our generalized MM equation affords a systematic approach for exploring cNESS enzyme kinetics. 2. PDB ligand conformational energies calculated quantum-mechanically. Science.gov (United States) Sitzmann, Markus; Weidlich, Iwona E; Filippov, Igor V; Liao, Chenzhong; Peach, Megan L; Ihlenfeldt, Wolf-Dietrich; Karki, Rajeshri G; Borodina, Yulia V; Cachau, Raul E; Nicklaus, Marc C 2012-03-26 (RSCC). We repeated these calculations with the solvent model IEFPCM, which yielded energy differences that were generally somewhat lower than the corresponding vacuum results but did not produce a qualitatively different picture. Torsional sampling around the crystal conformation at the molecular mechanics level using the MMFF94s force field typically led to an increase in energy. © 2012 American Chemical Society 3. Energy flow in non-equilibrium conformal field theory Science.gov (United States) Bernard, Denis; Doyon, Benjamin 2012-09-01 We study the energy current and its fluctuations in quantum gapless 1d systems far from equilibrium modeled by conformal field theory, where two separated halves are prepared at distinct temperatures and glued together at a point contact. We prove that these systems converge towards steady states, and give a general description of such non-equilibrium steady states in terms of quantum field theory data. We compute the large deviation function, also called the full counting statistics, of energy transfer through the contact. These are universal and satisfy fluctuation relations. We provide a simple representation of these quantum fluctuations in terms of classical Poisson processes whose intensities are proportional to Boltzmann weights. 4. Generalized Wilson-Fisher critical points from the conformal OPE CERN Document Server Gliozzi, Ferdinando; Petkou, Anastasios C; Wen, Congkao 2016-01-01 We study possible smooth deformations of Generalized Free Conformal Field Theories in arbitrary dimensions by exploiting the singularity structure of the conformal blocks dictated by the null states. We derive in this way, at the first non trivial order in the $\\epsilon$-expansion, the anomalous dimensions of an infinite class of scalar local operators, without using the equations of motion. In the cases where other computational methods apply, the results agree. 5. Conformal anomaly of generalized form factors and finite loop integrals CERN Document Server Chicherin, Dmitry 2017-01-01 We reveal a new mechanism of conformal symmetry breaking at Born level. It occurs in generalized form factors with several local operators and an on-shell state of massless particles. The effect is due to hidden singularities on collinear configurations of the momenta. This conformal anomaly is different from the holomorphic anomaly of amplitudes. We present a number of examples in four and six dimensions. We find an application of the new conformal anomaly to finite loop momentum integrals with one or more massless legs. The collinear region around a massless leg creates a contact anomaly, made visible by the loop integration. The anomalous conformal Ward identity for an $\\ell-$loop integral is a 2nd-order differential equation whose right-hand side is an $(\\ell-1)-$loop integral. We show several examples, in particular the four-dimensional scalar double box. 6. 76 FR 77182 - Approval and Promulgation of Air Quality Implementation Plans; Virginia; General Conformity... Science.gov (United States) 2011-12-12 ... Conformity Requirements for Federal Agencies Applicable to Federal Actions AGENCY: Environmental Protection... adopted by Virginia for the purpose of incorporating Federal general conformity requirements revisions... approving Virginia's general conformity SIP revision and if that provision may be severed from the... 7. 76 FR 77150 - Approval and Promulgation of Air Quality Implementation Plans; Virginia; General Conformity... Science.gov (United States) 2011-12-12 ... Conformity Requirements for Federal Agencies Applicable to Federal Actions AGENCY: Environmental Protection... regulation adopted by Virginia to incorporate revisions to Federal general conformity requirements... state general conformity requirements rule for Federal agencies applicable to Federal actions... 8. Singular conformally invariant trilinear forms and generalized Rankin Cohen operators CERN Document Server Jean-Louis, Clerc 2011-01-01 The most singular residues of the standard meromorphic family of trilinear conformally invariant forms on $\\mathcal C^\\infty_c(\\mathbb R^d)$ are computed. Their expression involves covariant bidifferential operators (generalized Rankin Cohen operators), for which new formul\\ae \\ are obtained. The main tool is a Bernstein-Sato identity for the kernel of the forms. 9. A phenomenological relationship between molecular geometry change and conformational energy change Science.gov (United States) Bodi, Andras; Bjornsson, Ragnar; Arnason, Ingvar 2010-08-01 A linear correlation is established between the change in the axial/equatorial conformational energy difference and the change in the molecular geometry transformation during conformational inversion in substituted six-membered ring systems, namely in the 1-substituted cyclohexane/silacyclohexane, cyclohexane/ N-substituted piperidine and 1-substituted silacyclohexane/ P-substituted phosphorinane compound families, and for the analogous gauche/anti conformational isomerism in 1-substituted propanes/1-silapropanes. The nuclear repulsion energy parameterizes the molecular geometry, and changes in the conformational energy between the related compound families are linearly correlated with the changes in the nuclear repulsion energy difference based on DFT (B3LYP, M06-2X), G3B3, and CBS-QB3 calculations. This correlation reproduces the sometimes remarkable contrast between the conformational behavior of analogous compounds, e.g., the lack of a general equatorial preference in silacyclohexanes. 10. Vacuum energy sequestering and conformal symmetry Science.gov (United States) Ben-Dayan, Ido; Richter, Robert; Ruehle, Fabian; Westphal, Alexander 2016-05-01 In a series of recent papers Kaloper and Padilla proposed a mechanism to sequester standard model vacuum contributions to the cosmological constant. We study the consequences of embedding their proposal into a fully local quantum theory. In the original work, the bare cosmological constant Λ and a scaling parameter λ are introduced as global fields. We find that in the local case the resulting Lagrangian is that of a spontaneously broken conformal field theory where λ plays the role of the dilaton. A vanishing or a small cosmological constant is thus a consequence of the underlying conformal field theory structure. 11. Conformal Gravity with the most general ELKO Matter OpenAIRE Fabbri, Luca 2011-01-01 Recently we have constructed the conformal gravity with metric and torsion, finding the gravitational field equations that give the conservation laws and trace condition; in the present paper we apply this theory to the case of ELKO matter field, proving that their spin and energy densities once the matter field equations are considered imply the validity of the conservation laws and trace condition mentioned above. 12. Conformally-modified gravity and vacuum energy CERN Document Server Henke, Christian 2016-01-01 The paper deals with a modified theory of gravity and the cosmological consequences. Instead of concerning the field equations directly, we modify a conformally-related and equivalent equation, such that a spontaneous symmetry breaking at Planck scale occurs in the trace equation. As the consequence the cosmological constant problem is solved. 13. Anisotropic scaling and generalized conformal invariance at Lifshitz points Science.gov (United States) Henkel, Malte; Pleimling, Michel 2002-08-01 A new variant of the Wolff cluster algorithm is proposed for simulating systems with competing interactions. This method is used in a high-precision study of the Lifshitz point of the 3D ANNNI model. At the Lifshitz point, several critical exponents are found and the anisotropic scaling of the correlators is verified. The functional form of the two-point correlators is shown to be consistent with the predictions of generalized conformal invariance. 14. Prosocial Conformity: Prosocial Norms Generalize Across Behavior and Empathy. Science.gov (United States) Nook, Erik C; Ong, Desmond C; Morelli, Sylvia A; Mitchell, Jason P; Zaki, Jamil 2016-08-01 Generosity is contagious: People imitate others' prosocial behaviors. However, research on such prosocial conformity focuses on cases in which people merely reproduce others' positive actions. Hence, we know little about the breadth of prosocial conformity. Can prosocial conformity cross behavior types or even jump from behavior to affect? Five studies address these questions. In Studies 1 to 3, participants decided how much to donate to charities before learning that others donated generously or stingily. Participants who observed generous donations donated more than those who observed stingy donations (Studies 1 and 2). Crucially, this generalized across behaviors: Participants who observed generous donations later wrote more supportive notes to another participant (Study 3). In Studies 4 and 5, participants observed empathic or non-empathic group responses to vignettes. Group empathy ratings not only shifted participants' own empathic feelings (Study 4), but they also influenced participants' donations to a homeless shelter (Study 5). These findings reveal the remarkable breadth of prosocial conformity. © 2016 by the Society for Personality and Social Psychology, Inc. 15. 78 FR 57335 - Approval and Promulgation of Implementation Plans; State of Missouri; Conformity of General... Science.gov (United States) 2013-09-18 ... AGENCY 40 CFR Part 52 Approval and Promulgation of Implementation Plans; State of Missouri; Conformity of... conformity rule in its entirety to bring it into compliance with the Federal general conformity rule which was updated in the Federal Register on April 5, 2010. General conformity regulations prohibit... 16. Energy flux positivity and unitarity in conformal field theories NARCIS (Netherlands) Kulaxizi, M.; Parnachev, A. 2011-01-01 We show that in most conformal field theories the condition of the energy flux positivity, proposed by Hofman and Maldacena, is equivalent to the absence of ghosts. At finite temperature and large energy and momenta, the two-point functions of the stress energy tensor develop lightlike poles. The re 17. Gauge formulation of general relativity using conformal and spin symmetries. Science.gov (United States) Wang, Charles H-T 2008-05-28 The gauge symmetry inherent in Maxwell's electromagnetics has a profound impact on modern physics. Following the successful quantization of electromagnetics and other higher order gauge field theories, the gauge principle has been applied in various forms to quantize gravity. A notable development in this direction is loop quantum gravity based on the spin-gauge treatment. This paper considers a further incorporation of the conformal gauge symmetry in canonical general relativity. This is a new conformal decomposition in that it is applied to simplify recently formulated parameter-free construction of spin-gauge variables for gravity. The resulting framework preserves many main features of the existing canonical framework for loop quantum gravity regarding the spin network representation and Thiemann's regularization. However, the Barbero-Immirzi parameter is converted into the conformal factor as a canonical variable. It behaves like a scalar field but is somehow non-dynamical since the Hamiltonian constraint does not depend on its momentum. The essential steps of the mathematical derivation of this parameter-free framework for the spin-gauge variables of gravity are spelled out. The implications for the loop quantum gravity programme are briefly discussed. 18. Enhanced conformational sampling technique provides an energy landscape view of large-scale protein conformational transitions. Science.gov (United States) Shao, Qiang 2016-10-26 Large-scale conformational changes in proteins are important for their functions. Tracking the conformational change in real time at the level of a single protein molecule, however, remains a great challenge. In this article, we present a novel in silico approach with the combination of normal mode analysis and integrated-tempering-sampling molecular simulation (NMA-ITS) to give quantitative data for exploring the conformational transition pathway in multi-dimensional energy landscapes starting only from the knowledge of the two endpoint structures of the protein. The open-to-closed transitions of three proteins, including nCaM, AdK, and HIV-1 PR, were investigated using NMA-ITS simulations. The three proteins have varied structural flexibilities and domain communications in their respective conformational changes. The transition state structure in the conformational change of nCaM and the associated free-energy barrier are in agreement with those measured in a standard explicit-solvent REMD simulation. The experimentally measured transition intermediate structures of the intrinsically flexible AdK are captured by the conformational transition pathway measured here. The dominant transition pathways between the closed and fully open states of HIV-1 PR are very similar to those observed in recent REMD simulations. Finally, the evaluated relaxation times of the conformational transitions of three proteins are roughly at the same level as reported experimental data. Therefore, the NMA-ITS method is applicable for a variety of cases, providing both qualitative and quantitative insights into the conformational changes associated with the real functions of proteins. 19. Generalized conformal realizations of Kac-Moody algebras Science.gov (United States) Palmkvist, Jakob 2009-01-01 We present a construction which associates an infinite sequence of Kac-Moody algebras, labeled by a positive integer n, to one single Jordan algebra. For n =1, this reduces to the well known Kantor-Koecher-Tits construction. Our generalization utilizes a new relation between different generalized Jordan triple systems, together with their known connections to Jordan and Lie algebras. Applied to the Jordan algebra of Hermitian 3×3 matrices over the division algebras R, C, H, O, the construction gives the exceptional Lie algebras f4, e6, e7, e8 for n =2. Moreover, we obtain their infinite-dimensional extensions for n ≥3. In the case of 2×2 matrices, the resulting Lie algebras are of the form so(p +n,q+n) and the concomitant nonlinear realization generalizes the conformal transformations in a spacetime of signature (p,q). 20. Anisotropic scaling and generalized conformal invariance at Lifshitz points CERN Document Server Pleimling, M; Pleimling, Michel; Henkel, Malte 2001-01-01 The behaviour of the 3D axial next-nearest neighbour Ising (ANNNI) model at the uniaxial Lifshitz point is studied using Monte Carlo techniques. A new variant of the Wolff cluster algorithm permits the analysis of systems far larger than in previous studies. The Lifshitz point critical exponents are $\\alpha=0.18(2)$, $\\beta=0.238(5)$ and $\\gamma=1.36(3)$. Data for the spin-spin correlation function are shown to be consistent with the explicit scaling function derived from the assumption of local scale invariance, which is a generalization of conformal invariance to the anisotropic scaling {\\em at} the Lifshitz point. 1. Conformal invariance and generalized Hojman conserved quantities of mechanico-electrical systems Institute of Scientific and Technical Information of China (English) Li Yuan-Cheng; Xia Li-Li; Wang Xiao-Ming 2009-01-01 This paper studies conformal invariance and generalized Hojman conserved quantities of mechanico-electrical systems. The definition and the determining equation of conformal invariance for mechanico-electrical systems are provided. The conformal factor expression is deduced from conformal invariance and Lie symmetry under the infinitesimal singleparameter transformation group. The generalized Hojman conserved quantities from the conformal invariance of the system are given. An example is given to illustrate the application of the result. 2. General Relativity and Energy Science.gov (United States) Jackson, A. T. 1973-01-01 Reviews theoretical and experimental fundamentals of Einstein's theory of general relativity. Indicates that recent development of the theory of the continually expanding universe may lead to revision of the space-time continuum of the finite and unbounded universe. (CC) 3. Wormholes admitting conformal Killing vectors and supported by generalized Chaplygin gas Energy Technology Data Exchange (ETDEWEB) Kuhfittig, Peter K.F. [Milwaukee School of Engineering, Department of Mathematics, Milwaukee, WI (United States) 2015-08-15 When Morris and Thorne first proposed that traversable wormholes may be actual physical objects, they concentrated on the geometry by specifying the shape and redshift functions. This mathematical approach necessarily raises questions regarding the determination of the required stress-energy tensor. This paper discusses a natural way to obtain a complete wormhole solution by assuming that the wormhole (1) is supported by generalized Chaplygin gas and (2) admits conformal Killing vectors. (orig.) 4. Conformal symmetry wormholes and the null energy condition CERN Document Server Kuhfittig, Peter K F 2016-01-01 In this paper we seek a relationship between the assumption of conformal symmetry and the exotic matter needed to hold a wormhole open. By starting with a Morris-Thorne wormhole having a constant energy density, it is shown that the conformal factor provides the extra degree of freedom sufficient to account for the exotic matter. The same holds for Morris-Thorne wormholes in a noncommutative-geometry setting. Applied to thin shells, there would exist a radius that results in a wormhole with positive surface density and negative surface pressure and which violates the null energy condition on the thin shell. 5. Slice Energy in Higher Order Gravity Theories and Conformal Transformations CERN Document Server Cotsakis, S 2004-01-01 We show that there is a generic transport of energy between the scalar field generated by the conformal transformation of higher order gravity theories and the matter component. We give precise relations of this exchange and show that, unless we are in a stationary spacetime, slice energy is not generically conserved. These results translate into statements about the relative behaviour of ordinary matter, dark matter and dark energy in the context of higher order gravity. 6. Conformal invariance, dark energy, and CMB non-gaussianity Energy Technology Data Exchange (ETDEWEB) Antoniadis, Ignatios [Department of Physics, CERN, Theory Division CH-1211 Geneva 23 (Switzerland); Mazur, Pawel O. [Department of Physics and Astronomy, University of South Carolina Columbia SC 29208 (United States); Mottola, Emil, E-mail: [email protected], E-mail: [email protected], E-mail: [email protected] [Theoretical Division, MS B285 Los Alamos National Laboratory Los Alamos, NM 87545 (United States) 2012-09-01 In addition to simple scale invariance, a universe dominated by dark energy naturally gives rise to correlation functions possessing full conformal invariance. This is due to the mathematical isomorphism between the conformal group of certain three dimensional slices of de Sitter space and the de Sitter isometry group SO(4,1). In the standard homogeneous, isotropic cosmological model in which primordial density perturbations are generated during a long vacuum energy dominated de Sitter phase, the embedding of flat spatial R{sup 3} sections in de Sitter space induces a conformal invariant perturbation spectrum and definite prediction for the shape of the non-Gaussian CMB bispectrum. In the case in which the density fluctuations are generated instead on the de Sitter horizon, conformal invariance of the S{sup 2} horizon embedding implies a different but also quite definite prediction for the angular correlations of CMB non-Gaussianity on the sky. Each of these forms for the bispectrum is intrinsic to the symmetries of de Sitter space, and in that sense, independent of specific model assumptions. Each is different from the predictions of single field slow roll inflation models, which rely on the breaking of de Sitter invariance. We propose a quantum origin for the CMB fluctuations in the scalar gravitational sector from the conformal anomaly that could give rise to these non-Gaussianities without a slow roll inflaton field, and argue that conformal invariance also leads to the expectation for the relation n{sub S}−1 = n{sub T} between the spectral indices of the scalar and tensor power spectrum. Confirmation of this prediction or detection of non-Gaussian correlations in the CMB of one of the bispectral shape functions predicted by conformal invariance can be used both to establish the physical origins of primordial density fluctuations, and distinguish between different dynamical models of cosmological vacuum dark energy. 7. Conformal invariance and particle aspects in general relativity CERN Document Server Salehi, H; Darabi, F 2000-01-01 We study the breakdown of conformal symmetry in a conformally invariantgravitational model. The symmetry breaking is introduced by defining apreferred conformal frame in terms of the large scale characteristics of theuniverse. In this context we show that a local change of the preferredconformal frame results in a Hamilton-Jacobi equation describing a particlewith adjustable mass. In this equation the dynamical characteristics of theparticle substantially depends on the applied conformal factor and localgeometry. Relevant interpretations of the results are also discussed. 8. Conformally invariant gauge conditions in electromagnetism and general relativity Energy Technology Data Exchange (ETDEWEB) Esposito, Giampiero; Stornaiolo, Cosimo 2000-06-01 The construction of conformally invariant gauge conditions for Maxwell and Einstein theories on a manifold M is found to involve two basic ingredients. First, covariant derivatives of a linear gauge (e.g. Lorenz or de Donder), completely contracted with the tensor field representing the metric on the vector bundle of the theory. Second, the addition of a compensating term, obtained by covariant differentiation of a suitable tensor field built from the geometric data of the problem. The existence theorem for such a gauge in gravitational theory is here proved when the manifold M is endowed with a m-dimensional positive-definite metric g. An application to a generally covariant integral formulation of the Einstein equations is also outlined. 9. 40 CFR 51.858 - Criteria for determining conformity of general Federal actions. Science.gov (United States) 2010-07-01 ... 40 Protection of Environment 2 2010-07-01 2010-07-01 false Criteria for determining conformity of... Determining Conformity of General Federal Actions to State or Federal Implementation Plans § 51.858 Criteria for determining conformity of general Federal actions. Link to an amendment published at 75 FR... 10. 40 CFR 51.859 - Procedures for conformity determinations of general Federal actions. Science.gov (United States) 2010-07-01 ... 40 Protection of Environment 2 2010-07-01 2010-07-01 false Procedures for conformity... IMPLEMENTATION PLANS Determining Conformity of General Federal Actions to State or Federal Implementation Plans § 51.859 Procedures for conformity determinations of general Federal actions. Link to an... 11. Investigation on the low energy conformational surface of tabun to probe the role of its different conformers on biological activity Science.gov (United States) Paukku, Yuliya; Michalkova, Andrea; Majumdar, D.; Leszczynski, Jerzy 2006-05-01 Conformational studies have been carried out on the two different enantiomers of tabun at the density functional and second order Møller-Plesset perturbation levels of theory to generate low energy potential energy surfaces in the gas phase as well as in aqueous environment. The structures of the low energy conformers together with their molecular electrostatic potential surfaces have been compared with those of the non-aged acetylcholinesterase-tabun complex to locate the active conformer of this molecule. 12. Accurate calculation of conformational free energy differences in explicit water: the confinement-solvation free energy approach. Science.gov (United States) Esque, Jeremy; Cecchini, Marco 2015-04-23 The calculation of the free energy of conformation is key to understanding the function of biomolecules and has attracted significant interest in recent years. Here, we present an improvement of the confinement method that was designed for use in the context of explicit solvent MD simulations. The development involves an additional step in which the solvation free energy of the harmonically restrained conformers is accurately determined by multistage free energy perturbation simulations. As a test-case application, the newly introduced confinement/solvation free energy (CSF) approach was used to compute differences in free energy between conformers of the alanine dipeptide in explicit water. The results are in excellent agreement with reference calculations based on both converged molecular dynamics and umbrella sampling. To illustrate the general applicability of the method, conformational equilibria of met-enkephalin (5 aa) and deca-alanine (10 aa) in solution were also analyzed. In both cases, smoothly converged free-energy results were obtained in agreement with equilibrium sampling or literature calculations. These results demonstrate that the CSF method may provide conformational free-energy differences of biomolecules with small statistical errors (below 0.5 kcal/mol) and at a moderate computational cost even with a full representation of the solvent. 13. Conformal invariance and conserved quantities of general holonomic systems in phase space Institute of Scientific and Technical Information of China (English) Xia Li-Li; Cai Jian-Le; Li Yuan-Cheng 2009-01-01 This paper studies the conformed invariance and conserved quantities of general holonomic systems in phase space.The definition and the determining equation of conformed invariance for general holonomic systems in phase space are provided.The conformal factor expression is deduced from conformeal invariance and Lie symmetry.The relationship between the conformed invaxiance and the Lie symmetry is discussed,and the necessary and sufficient condition that the conformal invaxiance would be the Lie symmetry of the system under the infinitesimal single-parameter transformation group is deduced.The conserved quantities of the system axe given.An example is given to illustrate the application of the result. 14. An introduction to conformal geometry and tractor calculus, with a view to applications in general relativity CERN Document Server Curry, Sean 2014-01-01 The following are expanded lecture notes for the course of eight one hour lectures given by the second author at the 2014 summer school Asymptotic Analysis in General Relativity held in Grenoble by the Institut Fourier. The first four lectures deal with conformal geometry and the conformal tractor calculus, taking as primary motivation the search for conformally invariant tensors and diffrerential operators. The final four lectures apply the conformal tractor calculus to the study of conformally compactified geometries, motivated by the conformal treatment of infinity in general relativity. 15. Hawking-Hayward quasi-local energy under conformal transformations CERN Document Server Prain, Angus; Faraoni, Valerio; Lapierre-Léonard, Marianne 2015-01-01 We derive a formula describing the transformation of the Hawking-Hayward quasi-local energy under a conformal rescaling of the spacetime metric. A known formula for the transformation of the Misner-Sharp-Hernandez mass is recovered as a special case. 16. Generalized BRST symmetry for arbitrary spin conformal field theory Energy Technology Data Exchange (ETDEWEB) Upadhyay, Sudhaker, E-mail: [email protected] [Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016 (India); Mandal, Bhabani Prasad, E-mail: [email protected] [Department of Physics, Banaras Hindu University, Varanasi 221005 (India) 2015-05-11 We develop the finite field-dependent BRST (FFBRST) transformation for arbitrary spin-s conformal field theories. We discuss the novel features of the FFBRST transformation in these systems. To illustrate the results we consider the spin-1 and spin-2 conformal field theories in two examples. Within the formalism we found that FFBRST transformation connects the generating functionals of spin-1 and spin-2 conformal field theories in linear and non-linear gauges. Further, the conformal field theories in the framework of FFBRST transformation are also analyzed in Batalin–Vilkovisky (BV) formulation to establish the results. 17. Wormhole supported by dark energy admitting conformal motion Science.gov (United States) Bhar, Piyali; Rahaman, Farook; Manna, Tuhina; Banerjee, Ayan 2016-12-01 In this article, we study the possibility of sustaining static and spherically symmetric traversable wormhole geometries admitting conformal motion in Einstein gravity, which presents a more systematic approach to search a relation between matter and geometry. In wormhole physics, the presence of exotic matter is a fundamental ingredient and we show that this exotic source can be dark energy type which support the existence of wormhole spacetimes. In this work we model a wormhole supported by dark energy which admits conformal motion. We also discuss the possibility of the detection of wormholes in the outer regions of galactic halos by means of gravitational lensing. Studies of the total gravitational energy for the exotic matter inside a static wormhole configuration are also performed. 18. Wormhole supported by dark energy admitting conformal motion Energy Technology Data Exchange (ETDEWEB) Bhar, Piyali [Government General Degree College, Singur, Department of Mathematics, Hooghly, West Bengal (India); Rahaman, Farook; Banerjee, Ayan [Jadavpur University, Department of Mathematics, Kolkata, West Bengal (India); Manna, Tuhina [St. Xavier' s College, Department of Mathematics and Statistics (Commerce Evening), Kolkata, West Bengal (India) 2016-12-15 In this article, we study the possibility of sustaining static and spherically symmetric traversable wormhole geometries admitting conformal motion in Einstein gravity, which presents a more systematic approach to search a relation between matter and geometry. In wormhole physics, the presence of exotic matter is a fundamental ingredient and we show that this exotic source can be dark energy type which support the existence of wormhole spacetimes. In this work we model a wormhole supported by dark energy which admits conformal motion. We also discuss the possibility of the detection of wormholes in the outer regions of galactic halos by means of gravitational lensing. Studies of the total gravitational energy for the exotic matter inside a static wormhole configuration are also performed. (orig.) 19. A robust force field based method for calculating conformational energies of charged drug-like molecules DEFF Research Database (Denmark) Pøhlsgaard, Jacob; Harpsøe, Kasper; Jørgensen, Flemming Steen 2012-01-01 The binding affinity of a drug like molecule depends among other things on the availability of the bioactive conformation. If the bioactive conformation has a significantly higher energy than the global minimum energy conformation, the molecule is unlikely to bind to its target. Determination of ...... compounds generated by conformational analysis with modified electrostatics are good approximations of the conformational distributions predicted by experimental data and in simulated annealing performed in explicit solvent.......The binding affinity of a drug like molecule depends among other things on the availability of the bioactive conformation. If the bioactive conformation has a significantly higher energy than the global minimum energy conformation, the molecule is unlikely to bind to its target. Determination...... of the global minimum energy conformation and calculation of conformational penalties of binding are prerequisites for prediction of reliable binding affinities. Here, we present a simple and computationally efficient procedure to estimate the global energy minimum for a wide variety of structurally diverse... 20. 40 CFR 1033.201 - General requirements for obtaining a certificate of conformity. Science.gov (United States) 2010-07-01 ... certificate of conformity. 1033.201 Section 1033.201 Protection of Environment ENVIRONMENTAL PROTECTION AGENCY....201 General requirements for obtaining a certificate of conformity. Certification is the process by... certificate of conformity for freshly manufactured locomotives. Anyone meeting the definition... 1. 77 FR 47621 - Appalachian Gateway Project; Notice of Availability of Draft General Conformity Analysis Science.gov (United States) 2012-08-09 ... Conformity Analysis In accordance with the National Environmental Policy Act of 1969, the Clean Air Act and... prepared this draft General Conformity Determination (GCD) for the Appalachian Gateway Project (Project) to... the Project will achieve conformity in Pennsylvania with the use of Pennsylvania Department... 2. 40 CFR 93.158 - Criteria for determining conformity of general Federal actions. Science.gov (United States) 2010-07-01 ... 40 Protection of Environment 20 2010-07-01 2010-07-01 false Criteria for determining conformity of... (CONTINUED) AIR PROGRAMS (CONTINUED) DETERMINING CONFORMITY OF FEDERAL ACTIONS TO STATE OR FEDERAL IMPLEMENTATION PLANS Determining Conformity of General Federal Actions to State or Federal Implementation... 3. 40 CFR 93.159 - Procedures for conformity determinations of general Federal actions. Science.gov (United States) 2010-07-01 ... 40 Protection of Environment 20 2010-07-01 2010-07-01 false Procedures for conformity... PROTECTION AGENCY (CONTINUED) AIR PROGRAMS (CONTINUED) DETERMINING CONFORMITY OF FEDERAL ACTIONS TO STATE OR FEDERAL IMPLEMENTATION PLANS Determining Conformity of General Federal Actions to State or... 4. The energy profiles of atomic conformational transition intermediates of adenylate kinase. Science.gov (United States) Feng, Yaping; Yang, Lei; Kloczkowski, Andrzej; Jernigan, Robert L 2009-11-15 The elastic network interpolation (ENI) (Kim et al., Biophys J 2002;83:1620-1630) is a computationally efficient and physically realistic method to generate conformational transition intermediates between two forms of a given protein. However it can be asked whether these calculated conformations provide good representatives for these intermediates. In this study, we use ENI to generate conformational transition intermediates between the open form and the closed form of adenylate kinase (AK). Based on C(alpha)-only intermediates, we construct atomic intermediates by grafting all the atoms of known AK structures onto the C(alpha) atoms and then perform CHARMM energy minimization to remove steric conflicts and optimize these intermediate structures. We compare the energy profiles for all intermediates from both the CHARMM force-field and from knowledge-based energy functions. We find that the CHARMM energies can successfully capture the two energy minima representing the open AK and closed AK forms, while the energies computed from the knowledge-based energy functions can detect the local energy minimum representing the closed AK form and show some general features of the transition pathway with a somewhat similar energy profile as the CHARMM energies. The combinatorial extension structural alignment (Shindyalov et al., 1998;11:739-747) and the k-means clustering algorithm are then used to show that known PDB structures closely resemble computed intermediates along the transition pathway. 5. Energy Flux Positivity and Unitarity in Conformal Field Theories Science.gov (United States) Kulaxizi, Manuela; Parnachev, Andrei 2011-01-01 We show that in most conformal field theories the condition of the energy flux positivity, proposed by Hofman and Maldacena, is equivalent to the absence of ghosts. At finite temperature and large energy and momenta, the two-point functions of the stress energy tensor develop lightlike poles. The residues of the poles can be computed, as long as the only spin-two conserved current, which appears in the stress energy tensor operator-product expansion and acquires a nonvanishing expectation value at finite temperature, is the stress energy tensor. The condition for the residues to stay positive and the theory to remain ghost-free is equivalent to the condition of positivity of energy flux. 6. Energy flux positivity and unitarity in conformal field theories. Science.gov (United States) Kulaxizi, Manuela; Parnachev, Andrei 2011-01-07 We show that in most conformal field theories the condition of the energy flux positivity, proposed by Hofman and Maldacena, is equivalent to the absence of ghosts. At finite temperature and large energy and momenta, the two-point functions of the stress energy tensor develop lightlike poles. The residues of the poles can be computed, as long as the only spin-two conserved current, which appears in the stress energy tensor operator-product expansion and acquires a nonvanishing expectation value at finite temperature, is the stress energy tensor. The condition for the residues to stay positive and the theory to remain ghost-free is equivalent to the condition of positivity of energy flux. 7. Generally covariant vs. gauge structure for conformal field theories Energy Technology Data Exchange (ETDEWEB) Campigotto, M., E-mail: [email protected] [Dipartimento di Fisica, University of Torino, Via P. Giuria 1, 10125, Torino (Italy); Istituto Nazionale di Fisica Nucleare (INFN), Via P. Giuria 1, 10125, Torino (Italy); Fatibene, L. [Dipartimento di Matematica, University of Torino, Via C. Alberto 10, 10123, Torino (Italy); Istituto Nazionale di Fisica Nucleare (INFN), Via P. Giuria 1, 10125, Torino (Italy) 2015-11-15 We introduce the natural lift of spacetime diffeomorphisms for conformal gravity and discuss the physical equivalence between the natural and gauge natural structure of the theory. Accordingly, we argue that conformal transformations must be introduced as gauge transformations (affecting fields but not spacetime point) and then discuss special structures implied by the splitting of the conformal group. -- Highlights: •Both a natural and a gauge natural structure for conformal gravity are defined. •Global properties and natural lift of spacetime transformations are described. •The possible definitions of physical state are considered and discussed. •The gauge natural theory has less physical states than the corresponding natural one. •The dynamics forces to prefer the gauge natural structure over the natural one. 8. Conformation of antifreeze glycoproteins as determined from conformational energy calculations and fully assigned proton NMR spectra Energy Technology Data Exchange (ETDEWEB) Bush, C.A.; Rao, B.N.N. 1986-05-01 The /sup 1/H NMR spectra of AFGP's ranging in molecular weight from 2600 to 30,000 Daltons isolated from several different species of polar fish have been measured. The spectrum of AFGP 1-4 from Pagothenia borchgrevinki with an average of 30 repeating subunits has a single resonance for each proton of the glycotripeptide repeating unit, (ala-(gal-(..beta..-1..-->..3) galNAc-(..cap alpha..--O-)thr-ala)/sub n/. Its /sup 1/H NMR spectrum including resonances of the amide protons has been completely assigned. Coupling constants and nuclear Overhauser enhancements (n.O.e.) between protons on distant residues imply conformational order. The 2600 dalton molecular weight glycopeptides (AFGP-8) have pro in place of ala at certain specific points in the sequence and AFGP-8R of Eleginus gracilis has arg in place of one thr. The resonances of pro and arg were assigned by decoupling. The resonances of the carboxy and amino terminals have distinct chemical shifts and were assigned in AFGP-8 of Boreogadus saida by titration. n.O.e. between ..cap alpha..--protons and amide protons of the adjacent residue (sequential n.O.e.) were used in assignments of additional resonances and to assign the distinctive resonances of thr followed by pro. Conformational energy calculations on the repeating glycotripeptide subunit of AFGP show that the ..cap alpha..--glucosidic linkage has a fixed conformation while the ..beta..--linkage is less rigid. A conformational model for AFGP 1-4, which is based on the calculations has the peptide in an extended left-handed helix with three residues per turn similar to polyproline II. The model is consistent with CD data, amide proton coupling constants, temperature dependence of amide proton chemical shifts. 9. CALCULATION OF CONFORMATIONAL ENTROPY AND FREE ENERGY OF POLYSILANE CHAIN Institute of Scientific and Technical Information of China (English) Meng-bo Luo; Ying-cai Chen; Jian-hua Huang; Jian-min Xu 2001-01-01 The conformational entropy S and free energy F were calculated by exact enumeration of polysilane chain up to 23 segments with excluded volume (EV) and long-range van der Waals (VW) interaction. A nonlinear relation between SEV+VW and chain length n was found though SEV was found to vary linearly with n. We found that the second-order transition temperature of polysilane chain with VW interaction increases with the increase of chain length, while that of polysilane chain without VW interaction is chain length independent. Moreover, the free energies FEV+VW and FEV are both linearly related with n, and FEV+VW＜FEV for all temperatures. 10. Assignment of Side-Chain Conformation Using Adiabatic Energy Mapping, Free Energy Perturbation, and Molecular Dynamic Simulations DEFF Research Database (Denmark) Frimurer, Thomas M.; Günther, Peter H.; Sørensen, Morten Dahl 1999-01-01 adiabatic mapping, conformational change, essentialdynamics, free energy simulations, Kunitz type inhibitor ga3(VI)......adiabatic mapping, conformational change, essentialdynamics, free energy simulations, Kunitz type inhibitor ga3(VI)... 11. 75 FR 20591 - AES Sparrows Point LNG, LLC and Mid-Atlantic Express, LLC; Notice of Final General Conformity... Science.gov (United States) 2010-04-20 ... Energy Regulatory Commission AES Sparrows Point LNG, LLC and Mid-Atlantic Express, LLC; Notice of Final General Conformity Determination for Pennsylvania for the Proposed Sparrows Point LNG Terminal and... liquefied natural gas (LNG) import terminal and natural gas pipeline proposed by AES Sparrows Point LNG,... 12. Stable phantom-energy wormholes admitting conformal motions CERN Document Server Kuhfittig, Peter K F 2016-01-01 It has been argued that wormholes are as good a prediction of Einstein's theory as black holes but the theoretical construction requires a reverse strategy, specifying the desired geometric properties of the wormhole and leaving open the determination of the stress-energy tensor. We begin by confirming an earlier result by the author showing that a complete wormhole solution can be obtained by adopting the equation of state $p=\\omega\\rho$ and assuming that the wormhole admits a one-parameter group of conformal motions. The main purpose of this paper is to use the assumption of conformal symmetry to show that the wormhole is stable to linearized radial perturbations whenever $-1.5<\\omega <-1$. 13. Stable phantom-energy wormholes admitting conformal motions Science.gov (United States) Kuhfittig, Peter K. F. It has been argued that wormholes are as good a prediction of Einstein’s theory as black holes but the theoretical construction requires a reverse strategy, specifying the desired geometric properties of the wormhole and leaving open the determination of the stress-energy tensor. We begin by confirming an earlier result by the author showing that a complete wormhole solution can be obtained by adopting the equation of state p = ωρ and assuming that the wormhole admits a one-parameter group of conformal motions. The main purpose of this paper is to use the assumption of conformal symmetry to show that the wormhole is stable to linearized radial perturbations whenever ‑ 1.5 < ω < ‑1. 14. [Conformal radiotherapy of prostatic cancer: a general review]. Science.gov (United States) Chauvet, B; Oozeer, R; Bey, P; Pontvert, D; Bolla, M 1999-01-01 Recent progress in radiotherapeutic management of localized prostate cancer is reviewed. Clinical aspects--including dose-effect beyond 70 Gy, relative role of conformal radiation therapy techniques and of early hormonal treatment--are discussed as well as technical components--including patient immobilization, organ motion, prostate contouring, beam arrangement, 3-D treatment planning and portal imaging. The local control and biological relapse-free survival rates appear to be improved by high dose conformal radiotherapy from 20 to 30% for patients with intermediate and high risk of relapse. A benefit of overall survival is expected but not yet demonstrated. Late reactions, especially the rectal toxicity, remain moderate despite the dose escalation. However, conformal radiotherapy demands a high precision at all steps of the procedure. 15. Conformational energy calculations and proton nuclear overhauser enhancements reveal a unique conformation for blood group A oligosaccharides Energy Technology Data Exchange (ETDEWEB) Bush, C.A.; Yan, Z.Y.; Rao, B.N.N. 1986-10-01 The H NMR spectra of a series of blood group A active oligosaccharides containing from four to ten sugar residues have been completely assigned, and quantitative nuclear Overhauser enhancements (NOE) have been measured between protons separated by known distances within the pyranoside ring. The observation of NOE between anomeric protons and those of the aglycon sugar as well as small effects between protons of distant rings suggests that the oligosaccharides have well-defined conformations. Conformational energy calculations were carried out on a trisaccharide, Fuc( -1 2)(GalNAc( -1 3))-GalUS -O-me, which models the nonreducing terminal fragments of the blood group A oligosaccharides. The results of calculations with three different potential energy functions which have been widely used in peptides and carbohydrates gave several minimum energy conformations. In NOE calculations from conformational models, the rotational correlation time was adjusted to fit T1's and intra-ring NOE. Comparison of calculated maps of NOE as a function of glycosidic dihedral angles showed that only a small region of conformational space was consistent with experimental data on a blood group A tetrasaccharide alditol. This conformation occurs at an energy minimum in all three energy calculations. Temperature dependence of the NOE implies that the oligosaccharides adopt single rigid conformations which do not change with temperature. 16. Predictive Sampling of Rare Conformational Events in Aqueous Solution: Designing a Generalized Orthogonal Space Tempering Method. Science.gov (United States) Lu, Chao; Li, Xubin; Wu, Dongsheng; Zheng, Lianqing; Yang, Wei 2016-01-12 In aqueous solution, solute conformational transitions are governed by intimate interplays of the fluctuations of solute-solute, solute-water, and water-water interactions. To promote molecular fluctuations to enhance sampling of essential conformational changes, a common strategy is to construct an expanded Hamiltonian through a series of Hamiltonian perturbations and thereby broaden the distribution of certain interactions of focus. Due to a lack of active sampling of configuration response to Hamiltonian transitions, it is challenging for common expanded Hamiltonian methods to robustly explore solvent mediated rare conformational events. The orthogonal space sampling (OSS) scheme, as exemplified by the orthogonal space random walk and orthogonal space tempering methods, provides a general framework for synchronous acceleration of slow configuration responses. To more effectively sample conformational transitions in aqueous solution, in this work, we devised a generalized orthogonal space tempering (gOST) algorithm. Specifically, in the Hamiltonian perturbation part, a solvent-accessible-surface-area-dependent term is introduced to implicitly perturb near-solute water-water fluctuations; more importantly in the orthogonal space response part, the generalized force order parameter is generalized as a two-dimension order parameter set, in which essential solute-solvent and solute-solute components are separately treated. The gOST algorithm is evaluated through a molecular dynamics simulation study on the explicitly solvated deca-alanine (Ala10) peptide. On the basis of a fully automated sampling protocol, the gOST simulation enabled repetitive folding and unfolding of the solvated peptide within a single continuous trajectory and allowed for detailed constructions of Ala10 folding/unfolding free energy surfaces. The gOST result reveals that solvent cooperative fluctuations play a pivotal role in Ala10 folding/unfolding transitions. In addition, our assessment 17. Positive Energy Conditions in 4D Conformal Field Theory CERN Document Server Farnsworth, Kara; Prilepina, Valentina 2015-01-01 We argue that all consistent 4D quantum field theories obey a spacetime-averaged weak energy inequality $\\langle T^{00} \\rangle \\ge -C/L^4$, where $L$ is the size of the smearing region, and $C$ is a positive constant that depends on the theory. If this condition is violated, the theory has states that are indistinguishable from states of negative total energy by any local measurement, and we expect instabilities or other inconsistencies. We apply this condition to 4D conformal field theories, and find that it places constraints on the OPE coefficients of the theory. The constraints we find are weaker than the "conformal collider" constraints of Hofman and Maldacena. We speculate that there may be theories that violate the Hofman-Maldacena bounds, but satisfy our bounds. In 3D CFTs, the only constraint we find is equivalent to the positivity of 2-point function of the energy-momentum tensor, which follows from unitarity. Our calculations are performed using momentum-space Wightman functions, which are remarka... 18. Positive energy conditions in 4D conformal field theory Science.gov (United States) Farnsworth, Kara; Luty, Markus A.; Prilepina, Valentina 2016-10-01 We argue that all consistent 4D quantum field theories obey a spacetime-averaged weak energy inequality ≥ - C/L 4, where L is the size of the smearing region, and C is a positive constant that depends on the theory. If this condition is violated, the theory has states that are indistinguishable from states of negative total energy by any local measurement, and we expect instabilities or other inconsistencies. We apply this condition to 4D conformal field theories, and find that it places constraints on the OPE coefficients of the theory. The constraints we find are weaker than the "conformal collider" constraints of Hofman and Maldacena. In 3D CFTs, the only constraint we find is equivalent to the positivity of 2-point function of the energy-momentum tensor, which follows from unitarity. Our calculations are performed using momentum-space Wightman functions, which are remarkably simple functions of momenta, and may be of interest in their own right. 19. Theoretical study of the conformation and energy of supercoiled DNA Energy Technology Data Exchange (ETDEWEB) Hunt, N. G. [Lawrence Berkeley Lab., CA (United States). Structural Biology Div.; California Univ., Berkeley, CA (United States). Dept. of Physics 1992-01-01 The two sugar-phosphate backbones of the DNA molecule wind about each other in helical paths. For circular DNA molecules, or for linear pieces of DNA with the ends anchored, the two strands have a well-defined linking number, Lk. If Lk differs from the equilibrium linking number Lk{sub 0}, the molecule is supercoiled. The linking difference {Delta}Lk = Lk-Lk{sub 0} is partitioned between torsional deformation of the DNA, or twist ({Delta}Tw), and a winding of the DNA axis about itself known as writhe (Wr). In this dissertation, the conformation and energy of supercoiled DNA are examined by treating DNA as an elastic cylinder. Finite-length and entropic effects are ignored, and all extensive quantities are treated as linear densities. Two classes of conformation are considered: the plectonemic or interwound form, in which the axis of the DNA double helix winds about itself in a double superhelix, and the toroidal shape in which the axis is wrapped around a torus. Minimum energy conformation are found. For biologically relevant values of specific linking differences, the plectonemic DNA, the superhelical pitch angle {alpha} is in the range 45{degree} < {alpha} {le} 90{degree}. For low values of specific linking difference {vert_bar}{sigma}{vert_bar} ({sigma} = {Delta}Lk/Lk{sub 0}), most linking difference is in writhe. As {vert_bar}{sigma}{vert_bar} increases, a greater proportion of linking difference is in twist. Interaction between DNA strands is treated first as a hard-body excluded volume and then as a screened electrostatic repulsion. Ionic strength is found to have a large effect, resulting in significantly greater torsional stress in supercoiled DNA at low ionic strength. 20. Binary cluster collision dynamics and minimum energy conformations Energy Technology Data Exchange (ETDEWEB) 2013-10-15 The collision dynamics of one Ag or Cu atom impinging on a Au{sub 12} cluster is investigated by means of DFT molecular dynamics. Our results show that the experimentally confirmed 2D to 3D transition of Au{sub 12}→Au{sub 13} is mostly preserved by the resulting planar Au{sub 12}Ag and Au{sub 12}Cu minimum energy clusters, which is quite remarkable in view of the excess energy, well larger than the 2D–3D potential barrier height. The process is accompanied by a large s−d hybridization and charge transfer from Au to Ag or Cu. The dynamics of the collision process mainly yields fusion of projectile and target, however scattering and cluster fragmentation also occur for large energies and large impact parameters. While Ag projectiles favor fragmentation, Cu favors scattering due to its smaller mass. The projectile size does not play a major role in favoring the fragmentation or scattering channels. By comparing our collision results with those obtained by an unbiased minimum energy search of 4483 Au{sub 12}Ag and 4483 Au{sub 12}Cu configurations obtained phenomenologically, we find that there is an extra bonus: without increase of computer time collisions yield the planar lower energy structures that are not feasible to obtain using semi-classical potentials. In fact, we conclude that phenomenological potentials do not even provide adequate seeds for the search of global energy minima for planar structures. Since the fabrication of nanoclusters is mainly achieved by synthesis or laser ablation, the set of local minima configurations we provide here, and their distribution as a function of energy, are more relevant than the global minimum to analyze experimental results obtained at finite temperatures, and is consistent with the dynamical coexistence of 2D and 3D liquid Au clusters conformations obtained previously. 1. Exclusion Statistics in Conformal Field Theory -- generalized fermions and spinons for level-1 WZW theories OpenAIRE 1998-01-01 We systematically study the exclusion statistics for quasi-particles for Conformal Field Theory spectra by employing a method based on recursion relations for truncated spectra. Our examples include generalized fermions in c 2. General Conformity Training Module 2.5: Proactive Role for Federal Agencies Science.gov (United States) Module 2.5 explains how taking a proactive role will allow a federal agency to more effectively participate in newly promulgated programs under the General Conformity Regulations, such as the emission reduction credits and the emission budgets programs. 3. Conformational disorder in energy transfer: beyond Förster theory. Science.gov (United States) Nelson, Tammie; Fernandez-Alberti, Sebastian; Roitberg, Adrian E; Tretiak, Sergei 2013-06-21 Energy transfer in donor-acceptor chromophore pairs, where the absorption of each species is well separated while donor emission and acceptor absorption overlap, can be understood through a Förster resonance energy transfer model. The picture is more complex for organic conjugated polymers, where the total absorption spectrum can be described as a sum of the individual contributions from each subunit (chromophore), whose absorption is not well separated. Although excitations in these systems tend to be well localized, traditional donors and acceptors cannot be defined and energy transfer can occur through various pathways where each subunit (chromophore) is capable of playing either role. In addition, fast torsional motions between individual monomers can break conjugation and lead to reordering of excited state energy levels. Fast torsional fluctuations occur on the same timescale as electronic transitions leading to multiple trivial unavoided crossings between excited states during dynamics. We use the non-adiabatic excited state molecular dynamics (NA-ESMD) approach to simulate energy transfer between two poly-phenylene vinylene (PPV) oligomers composed of 3-rings and 4-rings, respectively, separated by varying distances. The change in the spatial localization of the transient electronic transition density, initially localized on the donors, is used to determine the transfer rate. Our analysis shows that evolution of the intramolecular transition density can be decomposed into contributions from multiple transfer pathways. Here we present a detailed analysis of ensemble dynamics as well as a few representative trajectories which demonstrate the intertwined role of electronic and conformational processes. Our study reveals the complex nature of energy transfer in organic conjugated polymer systems and emphasizes the caution that must be taken in performing such an analysis when a single simple unidirectional pathway is unlikely. 4. Enhanced conformational sampling to visualize a free-energy landscape of protein complex formation. Science.gov (United States) Iida, Shinji; Nakamura, Haruki; Higo, Junichi 2016-06-15 We introduce various, recently developed, generalized ensemble methods, which are useful to sample various molecular configurations emerging in the process of protein-protein or protein-ligand binding. The methods introduced here are those that have been or will be applied to biomolecular binding, where the biomolecules are treated as flexible molecules expressed by an all-atom model in an explicit solvent. Sampling produces an ensemble of conformations (snapshots) that are thermodynamically probable at room temperature. Then, projection of those conformations to an abstract low-dimensional space generates a free-energy landscape. As an example, we show a landscape of homo-dimer formation of an endothelin-1-like molecule computed using a generalized ensemble method. The lowest free-energy cluster at room temperature coincided precisely with the experimentally determined complex structure. Two minor clusters were also found in the landscape, which were largely different from the native complex form. Although those clusters were isolated at room temperature, with rising temperature a pathway emerged linking the lowest and second-lowest free-energy clusters, and a further temperature increment connected all the clusters. This exemplifies that the generalized ensemble method is a powerful tool for computing the free-energy landscape, by which one can discuss the thermodynamic stability of clusters and the temperature dependence of the cluster networks. 5. Generalized Wick theorems in conformal field theory and the Borcherds identity CERN Document Server Takagi, Taichiro 2016-01-01 As the missing counterpart of the well-known generalized Wick theorem for interacting fields in two dimensional conformal field theory, we present a new formula for the operator product expansion of a normally ordered operator and a single operator on its right hand. Quite similar to the original Wick theorem for the opposite order operator product, it expresses the contraction i.e. the singular part of the operator product expansion as a contour integral of only two terms, each of which is a product of a contraction and a single operator. We discuss the relationship between these formulas and the Borcherds identity satisfied by the quantum fields associated with the theory of vertex algebras. A derivation of these formulas by an analytic method is also presented. The validity of our new formula is illustrated by a few examples including the Sugawara construction of the energy momentum tensor for the quantized currents of affine Lie algebras. 6. Comparative structural and vibrational study of the four lowest energy conformers of serotonin Science.gov (United States) 2017-02-01 A computational investigation of all possible lowest energy conformers of serotonin was carried out at the B3LYP/6-311 ++G* level. Out of the 14 possible lowest energy conformers, the first 4 conformers were investigated thoroughly for the optimized geometries, fundamental frequencies, the potential energy distributions, APT and natural charges, natural bond orbital (NBO) analysis, MEP, Contour map, total density array, HOMO, LUMO energies. The second third and fourth conformers are energetically at higher temperatures of 78, 94 and 312 K respectively with respect to the first one. Bond angles and bond lengths do not show significant variations while the dihedral angles vary significantly in going from one conformer to the other. Some of the vibrational modes of the indole moiety are conformation dependent to some extent whereas most of the normal modes of vibration of amino-ethyl side chain vary significantly in going from one conformer to conformer. The MEP for the four conformers suggested that the sites of the maximum positive and negative ESP change on changing the conformation. The charges at some atomic sites also change significantly from conformer to conformer. 7. A Calculus for Conformal Hypersurfaces and new higher Willmore energy functionals CERN Document Server Gover, A Rod 2016-01-01 The invariant theory for conformal hypersurfaces is studied by treating these as the conformal infinity of a conformally compact manifold: For a given conformal hypersurface embedding, a distinguished ambient metric is found (within its conformal class) by solving a singular version of the Yamabe problem. Using existence results for asymptotic solutions to this problem, we develop the details of how to proliferate conformal hypersurface invariants. In addition we show how to compute the the solution's asymptotics. We also develop a calculus of conformal hypersurface invariant differential operators and in particular, describe how to compute extrinsically coupled analogues of conformal Laplacian powers. Our methods also enable the study of integrated conformal hypersurface invariants and their functional variations. As a main application we develop new higher dimensional analogues of the Willmore energy for embedded surfaces. This complements recent progress on the existence and construction of such functional... 8. Numerical conformal mapping via a boundary integral equation with the adjoint generalized Neumann kernel OpenAIRE Nasser, Mohamed M. S.; Murid, Ali H. M.; Sangawi, Ali W. K. 2013-01-01 This paper presents a new uniquely solvable boundary integral equation for computing the conformal mapping, its derivative and its inverse from bounded multiply connected regions onto the five classical canonical slit regions. The integral equation is derived by reformulating the conformal mapping as an adjoint Riemann-Hilbert problem. From the adjoint Riemann-Hilbert problem, we derive a boundary integral equation with the adjoint generalized Neumann kernel for the derivative of the boundary... 9. Rotational Spectroscopy of the Lowest Energy Conformer of 2-Cyanobutane. Science.gov (United States) Müller, Holger S P; Zingsheim, Oliver; Wehres, Nadine; Grabow, Jens-Uwe; Lewen, Frank; Schlemmer, Stephan 2017-09-28 Isopropyl cyanide was recently detected in space as the first branched alkyl compound. Its abundance with respect to n-propyl cyanide in the Galactic center source Sagittarius B2(N2) is about 0.4. Astrochemical model calculations suggest that for the heavier homologue butyl cyanide the branched isomers dominate over the unbranched n-butyl cyanide and that 2-cyanobutane is the most abundant isomer. We have studied the rotational spectrum of 2-cyanobutane between 2 and 24 GHz using Fourier transform microwave spectroscopy and between 36 and 402 GHz employing (sub)millimeter absorption spectroscopy. Transitions of the lowest energy conformer were identified easily. Its rotational spectrum is very rich, and the quantum numbers J and Ka reach values of 111 and 73, respectively. This wealth of data yielded rotational and centrifugal distortion parameters up to tenth order, diagonal and one off-diagonal (14)N nuclear quadrupole coupling parameters, and one nuclear spin-rotation coupling parameter. We have also carried out quantum chemical calculations in part to facilitate the assignments. The molecule 2-cyanobutane was not found in the present ALMA data of Sagittarius B2(N2), but it may be found in the more sensitive data that have been completed very recently in the ALMA Cycle 4. 10. Triangulating Nucleic Acid Conformations Using Multicolor Surface Energy Transfer. Science.gov (United States) Riskowski, Ryan A; Armstrong, Rachel E; Greenbaum, Nancy L; Strouse, Geoffrey F 2016-02-23 Optical ruler methods employing multiple fluorescent labels offer great potential for correlating distances among several sites, but are generally limited to interlabel distances under 10 nm and suffer from complications due to spectral overlap. Here we demonstrate a multicolor surface energy transfer (McSET) technique able to triangulate multiple points on a biopolymer, allowing for analysis of global structure in complex biomolecules. McSET couples the competitive energy transfer pathways of Förster Resonance Energy Transfer (FRET) with gold-nanoparticle mediated Surface Energy Transfer (SET) in order to correlate systematically labeled points on the structure at distances greater than 10 nm and with reduced spectral overlap. To demonstrate the McSET method, the structures of a linear B-DNA and a more complex folded RNA ribozyme were analyzed within the McSET mathematical framework. The improved multicolor optical ruler method takes advantage of the broad spectral range and distances achievable when using a gold nanoparticle as the lowest energy acceptor. The ability to report distance information simultaneously across multiple length scales, short-range (10-50 Å), mid-range (50-150 Å), and long-range (150-350 Å), distinguishes this approach from other multicolor energy transfer methods. 11. Stalking Higher Energy Conformers on the Potential Energy Surface of Charged Species. Science.gov (United States) Brites, Vincent; Cimas, Alvaro; Spezia, Riccardo; Sieffert, Nicolas; Lisy, James M; Gaigeot, Marie-Pierre 2015-03-10 Combined theoretical DFT-MD and RRKM methodologies and experimental spectroscopic infrared predissociation (IRPD) strategies to map potential energy surfaces (PES) of complex ionic clusters are presented, providing lowest and high energy conformers, thresholds to isomerization, and cluster formation pathways. We believe this association not only represents a significant advance in the field of mapping minima and transition states on the PES but also directly measures dynamical pathways for the formation of structural conformers and isomers. Pathways are unraveled over picosecond (DFT-MD) and microsecond (RRKM) time scales while changing the amount of internal energy is experimentally achieved by changing the loss channel for the IRPD measurements, thus directly probing different kinetic and isomerization pathways. Demonstration is provided for Li(+)(H2O)3,4 ionic clusters. Nonstatistical formation of these ionic clusters by both direct and cascade processes, involving isomerization processes that can lead to trapping of high energy conformers along the paths due to evaporative cooling, has been unraveled. 12. General Randic matrix and general Randi'c energy Directory of Open Access Journals (Sweden) Ran Gu; 2014-09-01 Full Text Available Let $G$ be a simple graph with vertex set $V(G = {v_1, v_2,ldots , v_n}$ and $d_i$ the degree of its vertex $v_i$, $i = 1, 2, cdots, n$. Inspired by the Randi'c matrix and the general Randi'c index of a graph, we introduce the concept of general Randi'c matrix $textbf{R}_alpha$ of $G$, which is defined by $(textbf{R}_alpha_{i,j}=(d_id_j^alpha$ if $v_i$ and $v_j$ are adjacent, and zero otherwise. Similarly, the general Randi'{c} eigenvalues are the eigenvalues of the general Randi'{c} matrix, the greatest general Randi'{c} eigenvalue is the general Randi'{c} spectral radius of $G$, and the general Randi'{c} energy is the sum of the absolute values of the general Randi'{c} eigenvalues. In this paper, we prove some properties of the general Randi'c matrix and obtain lower and upper bounds for general Randi'{c} energy, also, we get some lower bounds for general Randi'{c} spectral radius of a connected graph. Moreover, we give a new sharp upper bound for the general Randi'{c} energy when $alpha=-1/2$.[2mm] noindent{bf Keywords:} general Randi'c matrix, general Randi'c energy, eigenvalues, spectral radius. 13. Energy levels and quantum states of [Leu]enkephalin conformations based on theoretical and experimental investigations Energy Technology Data Exchange (ETDEWEB) Abdali, Salim; Jensen, Morten O; Bohr, Henrik [Quantum Protein Centre (QUP), Department of Physics, Bldg. 309, Technical University of Denmark, DK-2800 Kgs. Lyngby (Denmark) 2003-05-14 This paper describes a theoretical and experimental study of [Leu]enkephalin conformations with respect to the quantum states of the atomic structure of the peptide. Results from vibrational absorption measurements and quantum calculations are used to outline a quantum picture and to assign vibrational modes to the different conformations. The energy landscape of the conformations is reported as a function of a Hamming distance in Ramachandran space. Molecular dynamics simulations reveal a pronounced stability of the so-called single-bend low-energy conformation, which supports the derived quantum picture of this peptide. 14. Energy levels and quantum states of [Leu]enkephalin conformations based on theoretical and experimental investigations DEFF Research Database (Denmark) Abdali, Salim; Jensen, Morten Østergaard; Bohr, Henrik 2003-01-01 This paper describes a theoretical and experimental study of [Leu]enkephalin conformations with respect to the quantum estates of the atomic structure of the peptide. Results from vibrational absorption measurements and quantum calculations are used to outline a quantum picture and to assign...... vibrational modes to the different conformations. The energy landscape of the conformations is reported as a function of a Hamming distance in Ramachandran space. Molecular dynamics simulations reveal a pronounced stability of the so-called single-bend low-energy conformation, which supports the derived...... quantum picture of this peptide.... 15. Conformal transformations and conformal invariance in gravitation CERN Document Server Dabrowski, Mariusz P; Blaschke, David B 2008-01-01 Conformal transformations are frequently used tools in order to study relations between various theories of gravity and Einstein relativity. Because of that, in this paper we discuss the rules of conformal transformations for geometric quantities in general relativity. In particular, we discuss the conformal transformations of the matter energy-momentum tensor. We thoroughly discuss the latter and show the subtlety of the conservation law (i.e., the geometrical Bianchi identity) imposed in one of the conformal frames in reference to the other. The subtlety refers to the fact that conformal transformation creates'' an extra matter term composed of the conformal factor which enters the conservation law. In an extreme case of the flat original spacetime the matter is created'' due to work done by the conformal transformation to bend the spacetime which was originally flat. We also discuss how to construct the conformally invariant gravity which, in the simplest version, is a special case of the Brans-Dicke t... 16. Peptoid conformational free energy landscapes from implicit-solvent molecular simulations in AMBER. Science.gov (United States) Voelz, Vincent A; Dill, Ken A; Chorny, Ilya 2011-01-01 To test the accuracy of existing AMBER force field models in predicting peptoid conformation and dynamics, we simulated a set of model peptoid molecules recently examined by Butterfoss et al. (JACS 2009, 131, 16798-16807) using QM methods as well as three peptoid sequences with experimentally determined structures. We found that AMBER force fields, when used with a Generalized Born/Surface Area (GBSA) implicit solvation model, could accurately reproduce the peptoid torsional landscape as well as the major conformers of known peptoid structures. Enhanced sampling by replica exchange molecular dynamics (REMD) using temperatures from 300 to 800 K was used to sample over cis-trans isomerization barriers. Compared to (Nrch)5 and cyclo-octasarcosyl, the free energy of N-(2-nitro-3-hydroxyl phenyl)glycine-N-(phenyl)glycine has the most "foldable" free energy landscape, due to deep trans-amide minima dictated by N-aryl sidechains. For peptoids with (S)-N (1-phenylethyl) (Nspe) side chains, we observe a discrepancy in backbone dihedral propensities between molecular simulations and QM calculations, which may be due to force field effects or the inability to capture n --> n* interactions. For these residues, an empirical phi-angle biasing potential can "rescue" the backbone propensities seen in QM. This approach can serve as a general strategy for addressing force fields without resorting to a complete reparameterization. Overall, this study demonstrates the utility of implicit-solvent REMD simulations for efficient sampling to predict peptoid conformational landscapes, providing a potential tool for first-principles design of sequences with specific folding properties. 17. Free Energy-Based Conformational Search Algorithm Using the Movable Type Sampling Method. Science.gov (United States) Pan, Li-Li; Zheng, Zheng; Wang, Ting; Merz, Kenneth M 2015-12-08 In this article, we extend the movable type (MT) sampling method to molecular conformational searches (MT-CS) on the free energy surface of the molecule in question. Differing from traditional systematic and stochastic searching algorithms, this method uses Boltzmann energy information to facilitate the selection of the best conformations. The generated ensembles provided good coverage of the available conformational space including available crystal structures. Furthermore, our approach directly provides the solvation free energies and the relative gas and aqueous phase free energies for all generated conformers. The method is validated by a thorough analysis of thrombin ligands as well as against structures extracted from both the Protein Data Bank (PDB) and the Cambridge Structural Database (CSD). An in-depth comparison between OMEGA and MT-CS is presented to illustrate the differences between the two conformational searching strategies, i.e., energy-based versus free energy-based searching. These studies demonstrate that our MT-based ligand conformational search algorithm is a powerful approach to delineate the conformational ensembles of molecular species on free energy surfaces. 18. 78 FR 57267 - Approval and Promulgation of Implementation Plans; State of Missouri; Conformity of General... Science.gov (United States) 2013-09-18 ... (44 U.S.C. 3501 et seq.); is certified as not having a significant economic impact on a substantial... revision amends rule 10 CSR 10-6.300 Conformity of General Federal Actions to State Implementation Plans... affect the stringency of the SIP or adversely impact air quality. II. Have the requirements for approval... 19. On a Generalization of GKO Coset Construction of Conformal Field Theories CERN Document Server Kumar, Dushyant 2015-01-01 We introduce a generalization of Goddard-Kent-Olive (GKO) coset construction of two dimensional conformal field theories based on a choice of a scaled affine subalgebra $\\hat{\\mathfrak{h}}^s$ of a given affine Lie algebra $\\hat{\\mathfrak{h}}$. We study some aspects of the construction through the example of Ising CFT as a generalized GKO coset of $\\text{su(2)}_1$ with a scaling factor $s=2$. 20. Elucidation of the conformational free energy landscape in H.pylori LuxS and its implications to catalysis Directory of Open Access Journals (Sweden) Bhattacharyya Moitrayee 2010-08-01 Full Text Available Abstract Background One of the major challenges in understanding enzyme catalysis is to identify the different conformations and their populations at detailed molecular level in response to ligand binding/environment. A detail description of the ligand induced conformational changes provides meaningful insights into the mechanism of action of enzymes and thus its function. Results In this study, we have explored the ligand induced conformational changes in H.pylori LuxS and the associated mechanistic features. LuxS, a dimeric protein, produces the precursor (4,5-dihydroxy-2,3-pentanedione for autoinducer-2 production which is a signalling molecule for bacterial quorum sensing. We have performed molecular dynamics simulations on H.pylori LuxS in its various ligand bound forms and analyzed the simulation trajectories using various techniques including the structure network analysis, free energy evaluation and water dynamics at the active site. The results bring out the mechanistic details such as co-operativity and asymmetry between the two subunits, subtle changes in the conformation as a response to the binding of active and inactive forms of ligands and the population distribution of different conformations in equilibrium. These investigations have enabled us to probe the free energy landscape and identify the corresponding conformations in terms of network parameters. In addition, we have also elucidated the variations in the dynamics of water co-ordination to the Zn2+ ion in LuxS and its relation to the rigidity at the active sites. Conclusions In this article, we provide details of a novel method for the identification of conformational changes in the different ligand bound states of the protein, evaluation of ligand-induced free energy changes and the biological relevance of our results in the context of LuxS structure-function. The methodology outlined here is highly generalized to illuminate the linkage between structure and function in 1. Experimental conformational energy maps of proteins and peptides. Science.gov (United States) Balaji, Govardhan A; Nagendra, H G; Balaji, Vitukudi N; Rao, Shashidhar N 2017-06-01 We have presented an extensive analysis of the peptide backbone dihedral angles in the PDB structures and computed experimental Ramachandran plots for their distributions seen under a various constraints on X-ray resolution, representativeness at different sequence identity percentages, and hydrogen bonding distances. These experimental distributions have been converted into isoenergy contour plots using the approach employed previously by F. M. Pohl. This has led to the identification of energetically favored minima in the Ramachandran (ϕ, ψ) plots in which global minima are predominantly observed either in the right-handed α-helical or the polyproline II regions. Further, we have identified low energy pathways for transitions between various minima in the (ϕ,ψ) plots. We have compared and presented the experimental plots with published theoretical plots obtained from both molecular mechanics and quantum mechanical approaches. In addition, we have developed and employed a root mean square deviation (RMSD) metric for isoenergy contours in various ranges, as a measure (in kcal.mol(-1) ) to compare any two plots and determine the extent of correlation and similarity between their isoenergy contours. In general, we observe a greater degree of compatibility with experimental plots for energy maps obtained from molecular mechanics methods compared to most quantum mechanical methods. The experimental energy plots we have investigated could be helpful in refining protein structures obtained from X-ray, NMR, and electron microscopy and in refining force field parameters to enable simulations of peptide and protein structures that have higher degree of consistency with experiments. Proteins 2017; 85:979-1001. © 2017 Wiley Periodicals, Inc. © 2017 Wiley Periodicals, Inc. 2. Protein Conformational Change Based on a Two-dimensional Generalized Langevin Equation Institute of Scientific and Technical Information of China (English) Ying-xi Wang; Shuang-mu Linguang; Nan-rong Zhao; Yi-jing Yan 2011-01-01 A two-dimensional generalized Langevin equation is proposed to describe the protein conformational change,compatible to the electron transfer process governed by atomic packing density model.We assume a fractional Gaussian noise and a white noise through bond and through space coordinates respectively,and introduce the coupling effect coming from both fluctuations and equilibrium variances.The general expressions for autocorrelation functions of distance fluctuation and fluorescence lifetime variation are derived,based on which the exact conformational change dynamics can be evaluated with the aid of numerical Laplace inversion technique.We explicitly elaborate the short time and long time approximations.The relationship between the two-dimensional description and the one-dimensional theory is also discussed. 3. NIR Laser Radiation Induced Conformational Changes and Tunneling Lifetimes of High-Energy Conformers of Amino Acids in Low-Temperature Matrices Science.gov (United States) Bazso, Gabor; Najbauer, Eszter E.; Magyarfalvi, Gabor; Tarczay, Gyorgy 2013-06-01 We review our recent results on combined matrix isolation FT-IR and NIR laser irradiation studies on glycine alanine, and cysteine. The OH and the NH stretching overtones of the low-energy conformers of these amino acids deposited in Ar, Kr, Xe, and N_{2} matrices were irradiated. At the expense of the irradiated conformer, other conformers were enriched and new, high-energy, formerly unobserved conformers were formed in the matrices. This enabled the separation and unambiguous assignment of the vibrational transitions of the different conformers. The main conversion paths and their efficiencies are described qualitatively showing that there are significant differences in different matrices. It was shown that the high-energy conformer decays in the matrix by H-atom tunneling. The lifetimes of the high-energy conformers in different matrices were measured. Based on our results we conclude that some theoretically predicted low-energy conformers of amino acids are likely even absent in low-energy matrices due to fast H-atom tunneling. G. Bazso, G. Magyarfalvi, G. Tarczay J. Mol. Struct. 1025 (Light-Induced Processes in Cryogenic Matrices Special Issue) 33-42 (2012). G. Bazso, G. Magyarfalvi, G. Tarczay J. Phys. Chem. A 116 (43) 10539-10547 (2012). G. Bazso, E. E. Najbauer, G. Magyarfalvi, G. Tarczay J. Phys. Chem. A in press, DOI: 10.1021/jp400196b. E. E. Najbauer, G. Bazso, G. Magyarfalvi, G. Tarczay in preparation. 4. Conformal generally covariant quantum field theory. The scalar field and its Wick products Energy Technology Data Exchange (ETDEWEB) Pinamonti, N. [Hamburg Univ. (Germany). 2. Inst. fuer Theoretische Physik 2008-06-15 In this paper we generalize the construction of generally covariant quantum theories given in [R. Brunetti, K. Fredenhagen, R. Verch, Commun. Math. Phys. 237, 31 (2003)] to encompass the conformal covariant case. After introducing the abstract framework, we discuss the massless conformally coupled Klein Gordon field theory, showing that its quantization corresponds to a functor between two certain categories. At the abstract level, the ordinary fields, could be thought as natural transformations in the sense of category theory. We show that, the Wick monomials without derivatives (Wick powers), can be interpreted as fields in this generalized sense, provided a non trivial choice of the renormalization constants is given. A careful analysis shows that the transformation law of Wick powers is characterized by a weight, and it turns out that the sum of fields with different weights breaks the conformal covariance. At this point there is a difference between the previously given picture due to the presence of a bigger group of covariance. It is furthermore shown that the construction does not depend upon the scale {mu} appearing in the Hadamard parametrix, used to regularize the fields. Finally, we briefly discuss some further examples of more involved fields. (orig.) 5. Solar energy education. Renewable energy activities for general science Energy Technology Data Exchange (ETDEWEB) 1985-01-01 Renewable energy topics are integrated with the study of general science. The literature is provided in the form of a teaching manual and includes such topics as passive solar homes, siting a home for solar energy, and wind power for the home. Other energy topics are explored through library research activities. (BCS) 6. Characterization of low-energy conformational domains for Met-enkephalin. Science.gov (United States) Perez, J J; Villar, H O; Loew, G H 1992-04-01 An extensive exploration of the conformational hypersurface of Met-enkephalin has been carried out, in order to characterize different low-energy conformational domains accessible to this pentapeptide. The search strategy used consisted of two steps. First, systematic nested rotations were performed using the ECEPP potential. Ninety-two low-energy structures were found and minimized using the CHARMm potential. High and low-temperature molecular dynamics trajectories were then computed for the lowest energy structures in an interative fashion until no lower energy conformers could be found. The same search strategy was used in these studies simulating three different environments, a distance-dependent dielectric epsilon = r, and two constant dielectrics epsilon = 10 and epsilon = 80. The lowest energy structure found in a distance-dependent dielectric is a Gly-Gly beta-II'-type turn. All other structures found for epsilon = r within 10 kcal/mol of this lowest energy structure are also bends. In the more polar environments, the density of conformational states is significantly larger compared to the apolar media. Moreover, fewer hydrogen bonds are formed in the more polar environments, which increases the flexibility of the peptide and results in less structured conformers. Comparisons are made with previous calculations and experimental results. 7. Assessing Energy-Dependent Protein Conformational Changes in the TonB System. Science.gov (United States) Larsen, Ray A 2017-01-01 Changes in conformation can alter a protein's vulnerability to proteolysis. Thus, in vivo differential proteinase sensitivity provides a means for identifying conformational changes that mark discrete states in the activity cycle of a protein. The ability to detect a specific conformational state allows for experiments to address specific protein-protein interactions and other physiological components that potentially contribute to the function of the protein. This chapter presents the application of this technique to the TonB-dependent energy transduction system of Gram-negative bacteria, a strategy that has refined our understanding of how the TonB protein is coupled to the ion electrochemical gradient of the cytoplasmic membrane. 8. Conformational Control of Energy Transfer: A Mechanism for Biocompatible Nanocrystal-Based Sensors OpenAIRE Kay, Euan R; Lee, Jungmin; Nocera, Daniel; Bawendi, Moungi G 2012-01-01 Fold-up fluorophore: A new paradigm for designing self-referencing fluorescent nanosensors is demonstrated by interfacing a pH-triggered molecular conformational switch with quantum dots. Analytedependent, large-amplitude conformational motion controls the distance between the nanocrystal energy donor and an organic FRET acceptor. The result is a fluorescence signal capable of reporting pH values from individual endosomes in living cells. 9. Scalar vacuum structure in general relativity and alternative theories. Conformal continuations CERN Document Server Bronnikov, K A 2001-01-01 We discuss the global properties of static, spherically symmetric configurations of a self-gravitating real scalar field $\\phi$ in general relativity (GR), scalar-tensor theories (STT) and high-order gravity ($L=f(R)$) in various dimensions. In GR, for fields with arbitrary potentials $V(\\phi)$, not necessarily positive-definite, it is shown that the list of all possible types of space-time causal structure in the models under consideration is the same as the one for $\\phi = const$. In particular, there are no regular black holes with any asymptotics. These features are extended to STT and $f(R)$ theories, connected with GR by conformal mappings, unless there is a conformal continuation, i.e., a case when a singularity in a solution of GR maps to a regular surface in an alternative theory, and the solution is continued through such a surface. This effect is exemplified by exact solutions in GR with a massless conformal scalar field, considered as a special STT. Necessary conditions for the existence of a conf... 10. The analytic structure of conformal blocks and the generalized Wilson-Fisher fixed points Science.gov (United States) Gliozzi, Ferdinando; Guerrieri, Andrea L.; Petkou, Anastasios C.; Wen, Congkao 2017-04-01 We describe in detail the method used in our previous work arXiv:1611.10344 https://arxiv.org/abs/1611.10344 to study the Wilson-Fisher critical points nearby generalized free CFTs, exploiting the analytic structure of conformal blocks as functions of the conformal dimension of the exchanged operator. Our method is equivalent to the mechanism of conformal multiplet recombination set up by null states. We compute, to the first non-trivial order in the ɛ-expansion, the anomalous dimensions and the OPE coefficients of infinite classes of scalar local operators using just CFT data. We study single-scalar and O( N)-invariant theories, as well as theories with multiple deformations. When available we agree with older results, but we also produce a wealth of new ones. Unitarity and crossing symmetry are not used in our approach and we are able to apply our method to non-unitary theories as well. Some implications of our results for the study of the non-unitary theories containing partially conserved higher-spin currents are briefly mentioned. 11. Octonionic M-theory and /D=11 generalized conformal and superconformal algebras Science.gov (United States) Lukierski, Jerzy; Toppan, Francesco 2003-08-01 Following [Phys. Lett. B 539 (2002) 266] we further apply the octonionic structure to supersymmetric D=11 M-theory. We consider the octonionic 2n+1×2n+1 Dirac matrices describing the sequence of Clifford algebras with signatures (9+n,n) (n=0,1,2,…) and derive the identities following from the octonionic multiplication table. The case n=1 (4×4 octonion-valued matrices) is used for the description of the D=11 octonionic M-superalgebra with 52 real bosonic charges; the n=2 case (8×8 octonion-valued matrices) for the D=11 conformal M-algebra with 232 real bosonic charges. The octonionic structure is described explicitly for n=1 by the relations between the 528 Abelian O(10,1) tensorial charges Zμ, Zμν, Zμ…μ5 of the M-superalgebra. For n=2 we obtain 2080 real non-Abelian bosonic tensorial charges Zμν, Zμ1μ2μ3, Zμ1…μ6 which, suitably constrained describe the generalized D=11 octonionic conformal algebra. Further, we consider the supersymmetric extension of this octonionic conformal algebra which can be described as D=11 octonionic superconformal algebra with a total number of 64 real fermionic and 239 real bosonic generators. 12. Generalized dark energy interactions with multiple fluids CERN Document Server van de Bruck, Carsten; Mimoso, José P; Nunes, Nelson J 2016-01-01 In the search for an explanation for the current acceleration of the Universe, scalar fields are the most simple and useful tools to build models of dark energy. This field, however, must in principle couple with the rest of the world and not necessarily in the same way to different particles or fluids. We provide the most complete dynamical system analysis to date, consisting of a canonical scalar field conformally and disformally coupled to both dust and radiation. We perform a detailed study of the existence and stability conditions of the systems and comment on constraints imposed on the disformal coupling from Big-Bang Nucleosynthesis and given current limits on the variation of the fine-structure constant. 13. Generalized dark energy interactions with multiple fluids Science.gov (United States) van de Bruck, Carsten; Mifsud, Jurgen; Mimoso, José P.; Nunes, Nelson J. 2016-11-01 In the search for an explanation for the current acceleration of the Universe, scalar fields are the most simple and useful tools to build models of dark energy. This field, however, must in principle couple with the rest of the world and not necessarily in the same way to different particles or fluids. We provide the most complete dynamical system analysis to date, consisting of a canonical scalar field conformally and disformally coupled to both dust and radiation. We perform a detailed study of the existence and stability conditions of the systems and comment on constraints imposed on the disformal coupling from Big-Bang Nucleosynthesis and given current limits on the variation of the fine-structure constant. 14. Anisotropic Generalized Ghost Pilgrim Dark Energy Model in General Relativity Science.gov (United States) Santhi, M. Vijaya; Rao, V. U. M.; Aditya, Y. 2017-02-01 A spatially homogeneous and anisotropic locally rotationally symmetric (LRS) Bianchi type- I Universe filled with matter and generalized ghost pilgrim dark energy (GGPDE) has been studied in general theory of relativity. To obtain determinate solution of the field equations we have used scalar expansion proportional to the shear scalar which leads to a relation between the metric potentials. Some well-known cosmological parameters (equation of state (EoS) parameter ( ω Λ), deceleration parameter ( q) and squared speed of sound {vs2}) and planes (ω _{Λ }-dot {ω }_{Λ } and statefinder) are constructed for obtained model. The discussion and significance of these parameters is totally done through pilgrim dark energy parameter ( β) and cosmic time ( t). 15. Simple implementation of general dark energy models Energy Technology Data Exchange (ETDEWEB) Bloomfield, Jolyon K. [MIT Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, 77 Massachusetts Ave #37241, Cambridge, MA, 02139 (United States); Pearson, Jonathan A., E-mail: [email protected], E-mail: [email protected] [Centre for Particle Theory, Department of Mathematical Sciences, Durham University, South Road, Durham, DH1 3LE (United Kingdom) 2014-03-01 We present a formalism for the numerical implementation of general theories of dark energy, combining the computational simplicity of the equation of state for perturbations approach with the generality of the effective field theory approach. An effective fluid description is employed, based on a general action describing single-scalar field models. The formalism is developed from first principles, and constructed keeping the goal of a simple implementation into CAMB in mind. Benefits of this approach include its straightforward implementation, the generality of the underlying theory, the fact that the evolved variables are physical quantities, and that model-independent phenomenological descriptions may be straightforwardly investigated. We hope this formulation will provide a powerful tool for the comparison of theoretical models of dark energy with observational data. 16. Potential Energy Surface-Based Automatic Deduction of Conformational Transition Networks and Its Application on Quantum Mechanical Landscapes of d-Glucose Conformers. Science.gov (United States) Satoh, Hiroko; Oda, Tomohiro; Nakakoji, Kumiyo; Uno, Takeaki; Tanaka, Hiroaki; Iwata, Satoru; Ohno, Koichi 2016-11-08 This paper describes our approach that is built upon the potential energy surface (PES)-based conformational analysis. This approach automatically deduces a conformational transition network, called a conformational reaction route map (r-map), by using the Scaled Hypersphere Search of the Anharmonic Downward Distortion Following method (SHS-ADDF). The PES-based conformational search has been achieved by using large ADDF, which makes it possible to trace only low transition state (TS) barriers while restraining bond lengths and structures with high free energy. It automatically performs sampling the minima and TS structures by simply taking into account the mathematical feature of PES without requiring any a priori specification of variable internal coordinates. An obtained r-map is composed of equilibrium (EQ) conformers connected by reaction routes via TS conformers, where all of the reaction routes are already confirmed during the process of the deduction using the intrinsic reaction coordinate (IRC) method. The postcalculation analysis of the deduced r-map is interactively carried out using the RMapViewer software we have developed. This paper presents computational details of the PES-based conformational analysis and its application to d-glucose. The calculations have been performed for an isolated glucose molecule in the gas phase at the RHF/6-31G level. The obtained conformational r-map for α-d-glucose is composed of 201 EQ and 435 TS conformers and that for β-d-glucose is composed of 202 EQ and 371 TS conformers. For the postcalculation analysis of the conformational r-maps by using the RMapViewer software program we have found multiple minimum energy paths (MEPs) between global minima of (1)C4 and (4)C1 chair conformations. The analysis using RMapViewer allows us to confirm the thermodynamic and kinetic predominance of (4)C1 conformations; that is, the potential energy of the global minimum of (4)C1 is lower than that of (1)C4 (thermodynamic predominance 17. The first experimental observation of the higher-energy trans conformer of trifluoroacetic acid Science.gov (United States) Apóstolo, R. F. G.; Bazsó, Gábor; Bento, R. R. F.; Tarczay, G.; Fausto, R. 2016-12-01 We report here the first experimental observation of the higher-energy conformer of trifluoroacetic acid (trans-TFA). The new conformer was generated by selective narrowband near-infrared vibrational excitation of the lower-energy cis-TFA conformer isolated in cryogenic matrices (Ar, Kr, N2) and shown to spontaneously decay to this latter form in the various matrix media, by tunneling. The decay rates in the different matrices were measured and compared with those of the trans conformers of other carboxylic acids in similar experimental conditions. The experimental studies received support from quantum chemistry calculations undertaken at various levels of approximation, which allowed a detailed characterization of the relevant regions of the potential energy surface of the molecule and the detailed assignment of the infrared spectra of the two conformers in the various matrices. Noteworthly, in contrast to cis-TFA that has its trifluoromethyl group eclipsed with the Cdbnd O bond of the carboxylic moiety, trans-TFA has the trifluoromethyl group eclipsed with the Csbnd O bond. This unusual structure of trans-TFA results from the fact that the relative orientation of the CF3 and COOH groups in this geometry facilitates the establishment of an intramolecular hydrogen-bond-like interaction between the OH group and the closely located in-plane fluorine atom of the CF3 moiety. 18. Conformity to the surviving sepsis campaign international guidelines among physicians in a general intensive care unit in Nairobi. Science.gov (United States) Mung'ayi, V; Karuga, R 2010-08-01 There are emerging therapies for managing septic critically-ill patients. There is little data from the developing world on their usage. To determine the conformity rate for resuscitation and management bundles for septic patients amongst physicians in a general intensive care unit. Cross sectional observational study. The general intensive care unit, Aga Khan University Hospital,Nairobi. Admitting physicians from all specialties in the general intensive care unit. The physicians had high conformity rates of 92% and 96% for the fluid resuscitation and use of va so pressors respectively for the initial resuscitation bundle. They had moderate conformity rates for blood cultures prior to administering antibiotics (57%) and administration of antibiotics within first hour of recognition of septic shock (54%). There was high conformity rate to the glucose control policy (81%), use of protective lung strategy in acute lung injury/Acute respiratory distress syndrome, venous thromboembolism prophylaxis (100%) and stress ulcer prophylaxis (100%) in the management bundle. Conformity was moderate for use of sedation, analgesia and muscle relaxant policy (69%), continuous renal replacement therapies (54%) and low for steroid policy (35%), administration ofdrotrecogin alfa (0%) and selective digestive decontamination (15%). There is varying conformity to the international sepsis guidelines among physicians caring for patients in our general ICU. Since increased conformity would improve survival and reduce morbidity, there is need for sustained education and guideline based performance improvement. 19. 40 CFR 1039.201 - What are the general requirements for obtaining a certificate of conformity? Science.gov (United States) 2010-07-01 ... obtaining a certificate of conformity? 1039.201 Section 1039.201 Protection of Environment ENVIRONMENTAL... obtaining a certificate of conformity? (a) You must send us a separate application for a certificate of conformity for each engine family. A certificate of conformity is valid from the indicated effective... 20. 40 CFR 1042.201 - General requirements for obtaining a certificate of conformity. Science.gov (United States) 2010-07-01 ... certificate of conformity. 1042.201 Section 1042.201 Protection of Environment ENVIRONMENTAL PROTECTION AGENCY... of conformity. (a) You must send us a separate application for a certificate of conformity for each engine family. A certificate of conformity is valid starting with the indicated effective date, but it... 1. 40 CFR 1048.201 - What are the general requirements for obtaining a certificate of conformity? Science.gov (United States) 2010-07-01 ... obtaining a certificate of conformity? 1048.201 Section 1048.201 Protection of Environment ENVIRONMENTAL... certificate of conformity? (a) You must send us a separate application for a certificate of conformity for each engine family. A certificate of conformity is valid starting with the indicated effective... 2. 40 CFR 1045.201 - What are the general requirements for obtaining a certificate of conformity? Science.gov (United States) 2010-07-01 ... obtaining a certificate of conformity? 1045.201 Section 1045.201 Protection of Environment ENVIRONMENTAL... obtaining a certificate of conformity? Engine manufacturers must certify their engines with respect to the... conformity: (a) You must send us a separate application for a certificate of conformity for each... 3. Conformational search by potential energy annealing: Algorithm and application to cyclosporin A Science.gov (United States) van Schaik, René C.; van Gunsteren, Wilfred F.; Berendsen, Herman J. C. 1992-04-01 A major problem in modelling (biological) macromolecules is the search for low-energy conformations. The complexity of a conformational search problem increases exponentially with the number of degrees of freedom which means that a systematic search can only be performed for very small structures. Here we introduce a new method (PEACS) which has a far better performance than conventional search methods. To show the advantages of PEACS we applied it to the refinement of Cyclosporin A and compared the results with normal molecular dynamics (MD) refinement. The structures obtained with PEACS were lower in energy and agreed with the NMR parameters much better than those obtained with MD. From the results it is further clear that PEACS samples a much larger part of the available conformational space than MD does. 4. A New Conformal Theory of Semi-Classical Quantum General Relativity Directory of Open Access Journals (Sweden) Suhendro I. 2007-10-01 Full Text Available We consider a new four-dimensional formulation of semi-classical quantum general relativity in which the classical space-time manifold, whose intrinsic geometric properties give rise to the effects of gravitation, is allowed to evolve microscopically by means of a conformal function which is assumed to depend on some quantum mechanical wave function. As a result, the theory presented here produces a unified field theory of gravitation and (microscopic electromagnetism in a somewhat simple, effective manner. In the process, it is seen that electromagnetism is actually an emergent quantum field originating in some kind of stochastic smooth extension (evolution of the gravitational field in the general theory of relativity. 5. Dynamic energy landscapes of riboswitches help interpret conformational rearrangements and function. Directory of Open Access Journals (Sweden) Giulio Quarta Full Text Available Riboswitches are RNAs that modulate gene expression by ligand-induced conformational changes. However, the way in which sequence dictates alternative folding pathways of gene regulation remains unclear. In this study, we compute energy landscapes, which describe the accessible secondary structures for a range of sequence lengths, to analyze the transcriptional process as a given sequence elongates to full length. In line with experimental evidence, we find that most riboswitch landscapes can be characterized by three broad classes as a function of sequence length in terms of the distribution and barrier type of the conformational clusters: low-barrier landscape with an ensemble of different conformations in equilibrium before encountering a substrate; barrier-free landscape in which a direct, dominant "downhill" pathway to the minimum free energy structure is apparent; and a barrier-dominated landscape with two isolated conformational states, each associated with a different biological function. Sharing concepts with the "new view" of protein folding energy landscapes, we term the three sequence ranges above as the sensing, downhill folding, and functional windows, respectively. We find that these energy landscape patterns are conserved in various riboswitch classes, though the order of the windows may vary. In fact, the order of the three windows suggests either kinetic or thermodynamic control of ligand binding. These findings help understand riboswitch structure/function relationships and open new avenues to riboswitch design. 6. Conformational energy range of ligands in protein crystal structures: The difficult quest for accurate understanding. Science.gov (United States) Peach, Megan L; Cachau, Raul E; Nicklaus, Marc C 2017-02-24 In this review, we address a fundamental question: What is the range of conformational energies seen in ligands in protein-ligand crystal structures? This value is important biophysically, for better understanding the protein-ligand binding process; and practically, for providing a parameter to be used in many computational drug design methods such as docking and pharmacophore searches. We synthesize a selection of previously reported conflicting results from computational studies of this issue and conclude that high ligand conformational energies really are present in some crystal structures. The main source of disagreement between different analyses appears to be due to divergent treatments of electrostatics and solvation. At the same time, however, for many ligands, a high conformational energy is in error, due to either crystal structure inaccuracies or incorrect determination of the reference state. Aside from simple chemistry mistakes, we argue that crystal structure error may mainly be because of the heuristic weighting of ligand stereochemical restraints relative to the fit of the structure to the electron density. This problem cannot be fixed with improvements to electron density fitting or with simple ligand geometry checks, though better metrics are needed for evaluating ligand and binding site chemistry in addition to geometry during structure refinement. The ultimate solution for accurately determining ligand conformational energies lies in ultrahigh-resolution crystal structures that can be refined without restraints. 7. Calculation of relative free energies for ligand-protein binding, solvation, and conformational transitions using the GROMOS software. Science.gov (United States) Riniker, Sereina; Christ, Clara D; Hansen, Halvor S; Hünenberger, Philippe H; Oostenbrink, Chris; Steiner, Denise; van Gunsteren, Wilfred F 2011-11-24 The calculation of the relative free energies of ligand-protein binding, of solvation for different compounds, and of different conformational states of a polypeptide is of considerable interest in the design or selection of potential enzyme inhibitors. Since such processes in aqueous solution generally comprise energetic and entropic contributions from many molecular configurations, adequate sampling of the relevant parts of configurational space is required and can be achieved through molecular dynamics simulations. Various techniques to obtain converged ensemble averages and their implementation in the GROMOS software for biomolecular simulation are discussed, and examples of their application to biomolecules in aqueous solution are given. 8. Four-Node Generalized Conforming Membrane Elements with Drilling DOFs Using Quadrilateral Area Coordinate Methods Directory of Open Access Journals (Sweden) Xiao-Ming Chen 2015-01-01 Full Text Available Two 4-node generalized conforming quadrilateral membrane elements with drilling DOF, named QAC4θ and QAC4θM, were successfully developed. Two kinds of quadrilateral area coordinates are used together in the assumed displacement fields of the new elements, so that the related formulations are quite straightforward and will keep the order of the Cartesian coordinates unchangeable while the mesh is distorted. The drilling DOF is defined as the additional rigid rotation at the element nodes to avoid improper constraint. Both elements can pass the strict patch test and exhibit better performance than other similar models. In particular, they are both free of trapezoidal locking in MacNeal’s beam test and insensitive to various mesh distortions. 9. A general approach to visualize protein binding and DNA conformation without protein labelling. Science.gov (United States) Song, Dan; Graham, Thomas G W; Loparo, Joseph J 2016-01-01 Single-molecule manipulation methods, such as magnetic tweezers and flow stretching, generally use the measurement of changes in DNA extension as a proxy for examining interactions between a DNA-binding protein and its substrate. These approaches are unable to directly measure protein-DNA association without fluorescently labelling the protein, which can be challenging. Here we address this limitation by developing a new approach that visualizes unlabelled protein binding on DNA with changes in DNA conformation in a relatively high-throughput manner. Protein binding to DNA molecules sparsely labelled with Cy3 results in an increase in fluorescence intensity due to protein-induced fluorescence enhancement (PIFE), whereas DNA length is monitored under flow of buffer through a microfluidic flow cell. Given that our assay uses unlabelled protein, it is not limited to the low protein concentrations normally required for single-molecule fluorescence imaging and should be broadly applicable to studying protein-DNA interactions. 10. 77 FR 59100 - Approval and Promulgation of Implementation Plans; Alabama: General and Transportation Conformity... Science.gov (United States) 2012-09-26 ... Transportation Conformity & New Source Review Prevention of Significant Deterioration for Fine Particulate Matter... transportation conformity regulations. EPA is approving portions of Alabama's May 2, 2011, SIP revision because... transportation conformity regulations into the SIP. Alabama's May 2, 2011, SIP revision includes changes to... 11. Parallel cascade selection molecular dynamics for efficient conformational sampling and free energy calculation of proteins Science.gov (United States) Kitao, Akio; Harada, Ryuhei; Nishihara, Yasutaka; Tran, Duy Phuoc 2016-12-01 Parallel Cascade Selection Molecular Dynamics (PaCS-MD) was proposed as an efficient conformational sampling method to investigate conformational transition pathway of proteins. In PaCS-MD, cycles of (i) selection of initial structures for multiple independent MD simulations and (ii) conformational sampling by independent MD simulations are repeated until the convergence of the sampling. The selection is conducted so that protein conformation gradually approaches a target. The selection of snapshots is a key to enhance conformational changes by increasing the probability of rare event occurrence. Since the procedure of PaCS-MD is simple, no modification of MD programs is required; the selections of initial structures and the restart of the next cycle in the MD simulations can be handled with relatively simple scripts with straightforward implementation. Trajectories generated by PaCS-MD were further analyzed by the Markov state model (MSM), which enables calculation of free energy landscape. The combination of PaCS-MD and MSM is reported in this work. 12. Mapping transiently formed and sparsely populated conformations on a complex energy landscape Science.gov (United States) Wang, Yong; Papaleo, Elena; Lindorff-Larsen, Kresten 2016-01-01 Determining the structures, kinetics, thermodynamics and mechanisms that underlie conformational exchange processes in proteins remains extremely difficult. Only in favourable cases is it possible to provide atomic-level descriptions of sparsely populated and transiently formed alternative conformations. Here we benchmark the ability of enhanced-sampling molecular dynamics simulations to determine the free energy landscape of the L99A cavity mutant of T4 lysozyme. We find that the simulations capture key properties previously measured by NMR relaxation dispersion methods including the structure of a minor conformation, the kinetics and thermodynamics of conformational exchange, and the effect of mutations. We discover a new tunnel that involves the transient exposure towards the solvent of an internal cavity, and show it to be relevant for ligand escape. Together, our results provide a comprehensive view of the structural landscape of a protein, and point forward to studies of conformational exchange in systems that are less characterized experimentally. DOI: http://dx.doi.org/10.7554/eLife.17505.001 PMID:27552057 13. Toward accurate prediction of pKa values for internal protein residues: the importance of conformational relaxation and desolvation energy. Science.gov (United States) Wallace, Jason A; Wang, Yuhang; Shi, Chuanyin; Pastoor, Kevin J; Nguyen, Bao-Linh; Xia, Kai; Shen, Jana K 2011-12-01 Proton uptake or release controls many important biological processes, such as energy transduction, virus replication, and catalysis. Accurate pK(a) prediction informs about proton pathways, thereby revealing detailed acid-base mechanisms. Physics-based methods in the framework of molecular dynamics simulations not only offer pK(a) predictions but also inform about the physical origins of pK(a) shifts and provide details of ionization-induced conformational relaxation and large-scale transitions. One such method is the recently developed continuous constant pH molecular dynamics (CPHMD) method, which has been shown to be an accurate and robust pK(a) prediction tool for naturally occurring titratable residues. To further examine the accuracy and limitations of CPHMD, we blindly predicted the pK(a) values for 87 titratable residues introduced in various hydrophobic regions of staphylococcal nuclease and variants. The predictions gave a root-mean-square deviation of 1.69 pK units from experiment, and there were only two pK(a)'s with errors greater than 3.5 pK units. Analysis of the conformational fluctuation of titrating side-chains in the context of the errors of calculated pK(a) values indicate that explicit treatment of conformational flexibility and the associated dielectric relaxation gives CPHMD a distinct advantage. Analysis of the sources of errors suggests that more accurate pK(a) predictions can be obtained for the most deeply buried residues by improving the accuracy in calculating desolvation energies. Furthermore, it is found that the generalized Born implicit-solvent model underlying the current CPHMD implementation slightly distorts the local conformational environment such that the inclusion of an explicit-solvent representation may offer improvement of accuracy. 14. Teaching Energy to a General Audience Science.gov (United States) 2010-02-01 A new, interdisciplinary course entitled Energy!'' has been developed by faculty in the physics and chemistry departments to meet the university's science and technology general education requirement. This course now enrolls over 400 students each semester in a single lecture where faculty from both departments co-teach throughout the term. Topics include the fundamentals of energy, fossil fuels, global climate change, nuclear energy, and renewable energy sources. The students represent an impressive range of majors (science, engineering, business, humanities, etc.) and comprise freshmen to seniors. To effectively teach this diverse audience and increase classroom engagement, in-class clickers'' are used with guided questions to teach concepts, which are then explicitly reinforced with online LON-CAPAfootnotetextFree open-source distributed learning content management and assessment system (www.lon-capa.org) homework. This online system enables immediate feedback in a structured manner, where students can practice randomized versions of problems for homework, quizzes, and exams. The course is already in high demand after only two semesters, in part because it is particularly relevant to students given the challenging energy and climate issues facing the nation and world. ) 15. General solar energy information user study Energy Technology Data Exchange (ETDEWEB) Belew, W.W.; Wood, B.L.; Marle, T.L.; Reinhardt, C.L. 1981-03-01 This report describes the results of a series of telephone interviews with groups of users of information on general solar energy. These results, part of a larger study on many different solar technologies, identify types of information each group needed and the best ways to get information to each group. The report is 1 of 10 discussing study results. The overall study provides baseline data about information needs in the solar community. An earlier study identified the information user groups in the solar community and the priority (to accelerate solar energy commercialization) of getting information to each group. In the current study only high-priority groups were examined. Results from 13 groups of respondents are analyzed in this report: Loan Officers, Real Estate Appraisers, Tax Assessors, Insurers, Lawyers, Utility Representatives, Public Interest Group Representatives, Information and Agricultural Representatives, Public Interest Group Representatives, Information and Agricultural Specialists at State Cooperative Extension Service Offices, and State Energy Office Representatives. The data will be used as input to the determination of information products and services the Solar Energy Research Institute, the Solar Energy Information Data Bank Network, and the entire information outreach community should be preparing and disseminating. 16. Elucidating molecular motion through structural and dynamic filters of energy-minimized conformer ensembles. Science.gov (United States) Emani, Prashant S; Bardaro, Michael F; Huang, Wei; Aragon, Sergio; Varani, Gabriele; Drobny, Gary P 2014-02-20 Complex RNA structures are constructed from helical segments connected by flexible loops that move spontaneously and in response to binding of small molecule ligands and proteins. Understanding the conformational variability of RNA requires the characterization of the coupled time evolution of interconnected flexible domains. To elucidate the collective molecular motions and explore the conformational landscape of the HIV-1 TAR RNA, we describe a new methodology that utilizes energy-minimized structures generated by the program "Fragment Assembly of RNA with Full-Atom Refinement (FARFAR)". We apply structural filters in the form of experimental residual dipolar couplings (RDCs) to select a subset of discrete energy-minimized conformers and carry out principal component analyses (PCA) to corroborate the choice of the filtered subset. We use this subset of structures to calculate solution T1 and T(1ρ) relaxation times for (13)C spins in multiple residues in different domains of the molecule using two simulation protocols that we previously published. We match the experimental T1 times to within 2% and the T(1ρ) times to within less than 10% for helical residues. These results introduce a protocol to construct viable dynamic trajectories for RNA molecules that accord well with experimental NMR data and support the notion that the motions of the helical portions of this small RNA can be described by a relatively small number of discrete conformations exchanging over time scales longer than 1 μs. 17. The effect of tensile stress on the conformational free energy landscape of disulfide bonds. Directory of Open Access Journals (Sweden) Full Text Available Disulfide bridges are no longer considered to merely stabilize protein structure, but are increasingly recognized to play a functional role in many regulatory biomolecular processes. Recent studies have uncovered that the redox activity of native disulfides depends on their C-C-S-S dihedrals, χ2 and χ'2. Moreover, the interplay of chemical reactivity and mechanical stress of disulfide switches has been recently elucidated using force-clamp spectroscopy and computer simulation. The χ2 and χ'2 angles have been found to change from conformations that are open to nucleophilic attack to sterically hindered, so-called closed states upon exerting tensile stress. In view of the growing evidence of the importance of C-C-S-S dihedrals in tuning the reactivity of disulfides, here we present a systematic study of the conformational diversity of disulfides as a function of tensile stress. With the help of force-clamp metadynamics simulations, we show that tensile stress brings about a large stabilization of the closed conformers, thereby giving rise to drastic changes in the conformational free energy landscape of disulfides. Statistical analysis shows that native TDi, DO and interchain Ig protein disulfides prefer open conformations, whereas the intrachain disulfide bridges in Ig proteins favor closed conformations. Correlating mechanical stress with the distance between the two a-carbons of the disulfide moiety reveals that the strain of intrachain Ig protein disulfides corresponds to a mechanical activation of about 100 pN. Such mechanical activation leads to a severalfold increase of the rate of the elementary redox S(N2 reaction step. All these findings constitute a step forward towards achieving a full understanding of functional disulfides. 18. FEARCF a multidimensional free energy method for investigating conformational landscapes and chemical reaction mechanisms Institute of Scientific and Technical Information of China (English) NAIDOO Kevin J. 2012-01-01 The development and implementation of a computational method able to produce free energies in multiple dimensions,descriptively named the free energies from adaptive reaction coordinate forces (FEARCF) method is described in this paper.While the method can be used to calculate free energies of association,conformation and reactivity here it is shown in the context of chemical reaction landscapes.A reaction free energy surface for the Claisen rearrangement of chorismate to prephenate is used as an illustration of the method's efficient convergence.FEARCF simulations are shown to achieve fiat histograms for complex multidimensional free energy volumes.The sampling efficiency by which it produces multidimensional free energies is demonstrated on the complex puckering of a pyranose ring,that is described by a three dimensional W(θ1,θ2,θ3) potential of mean force. 19. Millimeter and submillimeter wave spectroscopy of higher energy conformers of 1,2-propanediol Science.gov (United States) Zakharenko, O.; Bossa, J.-B.; Lewen, F.; Schlemmer, S.; Müller, H. S. P. 2017-03-01 We have performed a study of the millimeter/submillimeter wave spectrum of four higher energy conformers of 1,2-propanediol. The present analysis of rotational transitions carried out in the frequency range 38-400 GHz represents a significant extension of previous microwave work. The new data were combined with previously-measured microwave transitions and fitted using a Watson's S-reduced Hamiltonian. The final fits were within experimental accuracy, and included spectroscopic parameters up to sixth order of angular momentum, for the ground states of the four higher energy conformers following previously studied ones: g‧Ga, gG‧g‧, aGg‧ and g‧Gg. The present analysis provides reliable frequency predictions for astrophysical detection of 1,2-propanediol by radio telescope arrays at millimeter wavelengths. 20. Transportation Conformity Science.gov (United States) This section provides information on: current laws, regulations and guidance, policy and technical guidance, project-level conformity, general information, contacts and training, adequacy review of SIP submissions 1. Energy flow in the cryptophyte PE545 antenna is directed by bilin pigment conformation. Science.gov (United States) Curutchet, Carles; Novoderezhkin, Vladimir I; Kongsted, Jacob; Muñoz-Losa, Aurora; van Grondelle, Rienk; Scholes, Gregory D; Mennucci, Benedetta 2013-04-25 Structure-based calculations are combined with quantitative modeling of spectra and energy transfer dynamics to detemine the energy transfer scheme of the PE545 principal light-harvesting antenna of the cryptomonad Rhodomonas CS24. We use a recently developed quantum-mechanics/molecular mechanics (QM/MM) method that allows us to account for pigment-protein interactions at atomic detail in site energies, transition dipole moments, and electronic couplings. In addition, conformational flexibility of the pigment-protein complex is accounted for through molecular dynamics (MD) simulations. We find that conformational disorder largely smoothes the large energetic differences predicted from the crystal structure between the pseudosymmetric pairs PEB50/61C-PEB50/61D and PEB82C-PEB82D. Moreover, we find that, in contrast to chlorophyll-based photosynthetic complexes, pigment composition and conformation play a major role in defining the energy ladder in the PE545 complex, rather than specific pigment-protein interactions. This is explained by the remarkable conformational flexibility of the eight bilin pigments in PE545, characterized by a quasi-linear arrangement of four pyrrole units. The MD-QM/MM site energies allow us to reproduce the main features of the spectra, and minor adjustments of the energies of the three red-most pigments DBV19A, DBV19B, and PEB82D allow us to model the spectra of PE545 with a similar quality compared to our original model (model E from Novoderezhkin et al. Biophys. J.2010, 99, 344), which was extracted from the spectral and kinetic fit. Moreover, the fit of the transient absorption kinetics is even better in the new structure-based model. The largest difference between our previous and present results is that the MD-QM/MM calculations predict a much smaller gap between the PEB50/61C and PEB50/61D sites, in better accord with chemical intuition. We conclude that the current adjusted MD-QM/MM energies are more reliable in order to explore the 2. The generalized Erlangen program and setting a geometry for four- dimensional conformal fields Energy Technology Data Exchange (ETDEWEB) Neeman, Y. [Tel Aviv Univ. (Israel). Sackler Faculty of Exact Sciences]\|[Texas Univ., Austin, TX (United States). Center for Particle Physics; Hehl, F.W.; Mielke, E.W. [Koeln Univ. (Germany). Inst. fuer Theoretische Physik 1993-10-22 This is the text of a talk at the International Symposium on Mathematical Physics towards the XXI Century held in March 1993 at Beersheva, Israel. In the first part we attempt to summarize XXth Century Physics, in the light of Kelvins 1900 speech Dark Clouds over XIXth Century Physics. Contrary to what is usually said, Kelvin predicted that the clouds (relativity and quantum mechanics) would revolutionize physics and that one hundred years might be needed to harmonize them with classical physics. Quantum Gravity can be considered as a leftover from Kelvins program -- so are the problems with the interpretation of quantum mechanics. At the end of the XXth Century, the Standard Model is the new panoramic synthesis, drawn in gauge-geometric lines -- realizing the Erlangen program beyond F. Kleins expectations. The hierarchy problem and the smallness of the cosmological constant are our clouds, generations and the Higgs sector are to us what radioactivity was in 1900. In the second part we describe Metric-Affine spacetimes. We construct the Noether machinery and provide expressions for the conserved energy and hypermomentum. Superimposing conformal invariance over the affine structure induces the Virasoro-like infinite constraining algebra of diffeomorphisms, applied with constant parameters and opening the possibility of a 4DCFT, similar to 2DCFT. 3. Elucidating Hyperconjugation from Electronegativity to Predict Drug Conformational Energy in a High Throughput Manner. Science.gov (United States) Liu, Zhaomin; Pottel, Joshua; Shahamat, Moeed; Tomberg, Anna; Labute, Paul; Moitessier, Nicolas 2016-04-25 Computational chemists use structure-based drug design and molecular dynamics of drug/protein complexes which require an accurate description of the conformational space of drugs. Organic chemists use qualitative chemical principles such as the effect of electronegativity on hyperconjugation, the impact of steric clashes on stereochemical outcome of reactions, and the consequence of resonance on the shape of molecules to rationalize experimental observations. While computational chemists speak about electron densities and molecular orbitals, organic chemists speak about partial charges and localized molecular orbitals. Attempts to reconcile these two parallel approaches such as programs for natural bond orbitals and intrinsic atomic orbitals computing Lewis structures-like orbitals and reaction mechanism have appeared. In the past, we have shown that encoding and quantifying chemistry knowledge and qualitative principles can lead to predictive methods. In the same vein, we thought to understand the conformational behaviors of molecules and to encode this knowledge back into a molecular mechanics tool computing conformational potential energy and to develop an alternative to atom types and training of force fields on large sets of molecules. Herein, we describe a conceptually new approach to model torsion energies based on fundamental chemistry principles. To demonstrate our approach, torsional energy parameters were derived on-the-fly from atomic properties. When the torsional energy terms implemented in GAFF, Parm@Frosst, and MMFF94 were substituted by our method, the accuracy of these force fields to reproduce MP2-derived torsional energy profiles and their transferability to a variety of functional groups and drug fragments were overall improved. In addition, our method did not rely on atom types and consequently did not suffer from poor automated atom type assignments. 4. A Low Energy Consumption DOA Estimation Approach for Conformal Array in Ultra-Wideband Directory of Open Access Journals (Sweden) Liangtian Wan 2013-12-01 Full Text Available Most direction-of-arrival (DOA estimation approaches for conformal array suffer from high computational complexity, which cause high energy loss for the direction finding system. Thus, a low energy consumption DOA estimation algorithm for conformal array antenna is proposed in this paper. The arbitrary baseline direction finding algorithm is extended to estimate DOA for a conformal array in ultra-wideband. The rotation comparison method is adopted to solve the ambiguity of direction finding. The virtual baseline approach is used to construct the virtual elements. Theoretically, the virtual elements can be extended in the space flexibility. Four elements (both actual and virtual elements can be used to obtain a group of solutions. The space angle estimation can be obtained by using sub-array divided technique and matrix inversion method. The stability of the proposed algorithm can be guaranteed by averaging the angles obtained by different sub-arrays. Finally, the simulation results verify the effectiveness of the proposed method with high DOA estimation accuracy and relatively low computational complexity. 5. Conformational Explosion: Understanding the Complexity of the Para-Dialkylbenzene Potential Energy Surfaces Science.gov (United States) Mishra, Piyush; Hewett, Daniel M.; Zwier, Timothy S. 2017-06-01 This talk focuses on the single-conformation spectroscopy of small-chain para-dialkylbenzenes. This work builds on previous studies from our group on long-chain n-alkylbenzenes that identified the first folded structure in octylbenzene. The dialkylbenzenes are representative of a class of molecules that are common components of coal and aviation fuel and are known to be present in vehicle exhaust. We bring the molecules para-diethylbenzene, para-dipropylbenzene and para-dibutylbenzene into the gas phase and cool the molecules in a supersonic expansion. The jet-cooled molecules are then interrogated using laser-induced fluorescence excitation, fluorescence dip IR spectroscopy (FDIRS) and dispersed fluorescence. The LIF spectra in the S_{0}-S_{1} origin region show dramatic increases in the number of resolved transitions with increasing length of alkyl chains, reflecting an explosion in the number of unique low-energy conformations formed when two independent alkyl chains are present. Since the barriers to isomerization of the alkyl chain are similar in size, this results in an 'egg carton' shape to the potential energy surface. We use a combination of electronic frequency shift and alkyl CH stretch infrared spectra to generate a consistent set of conformational assignments. 6. Coarse-grained free energy functions for studying protein conformational changes: a double-well network model. Science.gov (United States) Chu, Jhih-Wei; Voth, Gregory A 2007-12-01 In this work, a double-well network model (DWNM) is presented for generating a coarse-grained free energy function that can be used to study the transition between reference conformational states of a protein molecule. Compared to earlier work that uses a single, multidimensional double-well potential to connect two conformational states, the DWNM uses a set of interconnected double-well potentials for this purpose. The DWNM free energy function has multiple intermediate states and saddle points, and is hence a "rough" free energy landscape. In this implementation of the DWNM, the free energy function is reduced to an elastic-network model representation near the two reference states. The effects of free energy function roughness on the reaction pathways of protein conformational change is demonstrated by applying the DWNM to the conformational changes of two protein systems: the coil-to-helix transition of the DB-loop in G-actin and the open-to-closed transition of adenylate kinase. In both systems, the rough free energy function of the DWNM leads to the identification of distinct minimum free energy paths connecting two conformational states. These results indicate that while the elastic-network model captures the low-frequency vibrational motions of a protein, the roughness in the free energy function introduced by the DWNM can be used to characterize the transition mechanism between protein conformations. 7. 77 FR 64183 - Notice of Availability of a Final General Conformity Determination for the California High-Speed... Science.gov (United States) 2012-10-18 ... California High-Speed Train System Merced to Fresno Section AGENCY: Federal Railroad Administration (FRA... Section of the California High-Speed Train (HST) System on September 18, 2012. FRA is the lead Federal... General Conformity requirements. The California High Speed Rail Authority (Authority), as the... 8. The Relationship between Alcohol Use and Peer Pressure Susceptibility, Peer Popularity and General Conformity in Northern Irish School Children Science.gov (United States) McKay, Michael T.; Cole, Jon C. 2012-01-01 This cross-sectional study investigated the bivariate and more fully controlled (with socio-demographic measures) relationship between self-reported drinking behaviour and peer pressure susceptibility, desire for peer popularity and general conformity in a sample of 11-16-year-old school children in Northern Ireland. Self-reported drinking… 9. The Relationship between Alcohol Use and Peer Pressure Susceptibility, Peer Popularity and General Conformity in Northern Irish School Children Science.gov (United States) McKay, Michael T.; Cole, Jon C. 2012-01-01 This cross-sectional study investigated the bivariate and more fully controlled (with socio-demographic measures) relationship between self-reported drinking behaviour and peer pressure susceptibility, desire for peer popularity and general conformity in a sample of 11-16-year-old school children in Northern Ireland. Self-reported drinking… 10. Semiexperimental equilibrium structure of the lower energy conformer of glycidol by the mixed estimation method. Science.gov (United States) Demaison, Jean; Craig, Norman C; Conrad, Andrew R; Tubergen, Michael J; Rudolph, Heinz Dieter 2012-09-13 Rotational constants were determined for (18)O-substituted isotopologues of the lower energy conformer of glycidol, which has an intramolecular inner hydrogen bond from the hydroxyl group to the oxirane ring oxygen. Rotational constants were previously determined for the (13)C and the OD species. These rotational constants have been corrected with the rovibrational constants calculated from an ab initio cubic force field. The derived semiexperimental equilibrium rotational constants have been supplemented by carefully chosen structural parameters, including those for hydrogen atoms, from medium level ab initio calculations. The combined data have been used in a weighted least-squares fit to determine an equilibrium structure for the glycidol H-bond inner conformer. This work shows that the mixed estimation method allows us to determine a complete and reliable equilibrium structure for large molecules, even when the rotational constants of a number of isotopologues are unavailable. 11. Cation-π Interactions in Serotonin: Conformational, Electronic Distribution, and Energy Decomposition Analysis. Science.gov (United States) Pratuangdejkul, Jaturong; Jaudon, Pascale; Ducrocq, Claire; Nosoongnoen, Wichit; Guerin, Georges-Alexandre; Conti, Marc; Loric, Sylvain; Launay, Jean-Marie; Manivet, Philippe 2006-05-01 An adiabatic conformational analysis of serotonin (5-hydroxytryptamine, 5-HT) using quantum chemistry led to six stable conformers that can be either +gauche (Gp), -gauche (Gm), and anti (At) depending upon the value taken by ethylamine side chain and 5-hydroxyl group dihedral angles φ1, φ2, and φ4, respectively. Further vibrational frequency analysis of the GmGp, GmGm, and GmAt conformers with the 5-hydroxyl group in the anti position revealed an additional red-shifted N-H stretch mode band in GmGp and GmGm that is absent in GmAt. This band corresponds to the 5-HT side-chain N-H bond involved in an intramolecular nonbonded interaction with the 5-hydroxy indole ring. The influence of this nonbonded interaction on the electronic distribution was assessed by analysis of the spin-spin coupling constants of GmGp and GmGm that show a marked increase for C2-C3 and C8-C9 bonds in GmGm and GmGp, respectively, with a decrease of their double bond character and an increase of their length. The Atoms in Molecules (AIM), Natural Bond Orbital (NBO), and fluorescence and CD spectra (TDDFT method) analyses confirmed the existence in GmGp and GmGm of a through-space charge-transfer between the HOMO and the HOMO-1 π-orbital of the indole ring and the LUMO σ* N-H antibonding orbital of the ammonium group. The strength of the cation-π interaction was determined by calculating binding energies of the NH4(+)/5-hydroxyindole complexes extracted from stable conformers. The energy decomposition analysis indicated that cationic-π interactions in the GmGp and GmGm conformers are governed by the electrostatic term with significant contributions from polarization and charge transfer. The lower stability of the GmGm over the GmGp comes from a higher exchange repulsion and a weaker polarization contributions. Our results provide insight into the nature of intramolecular forces that influence the conformational properties of 5-HT. 12. Narrowband NIR-Induced In Situ Generation of the High-Energy Trans Conformer of Trichloroacetic Acid Isolated in Solid Nitrogen and its Spontaneous Decay by Tunneling to the Low-Energy Cis Conformer Directory of Open Access Journals (Sweden) R. F. G. Apóstolo 2015-12-01 Full Text Available The monomeric form of trichloroacetic acid (CCl3COOH; TCA was isolated in a cryogenic nitrogen matrix (15 K and the higher energy trans conformer (O=C–O–H dihedral: 180° was generated in situ by narrowband near-infrared selective excitation the 1st OH stretching overtone of the low-energy cis conformer (O=C–O–H dihedral: 0°. The spontaneous decay, by tunneling, of the generated high-energy conformer into the cis form was then evaluated and compared with those observed previously for the trans conformers of acetic and formic acids in identical experimental conditions. The much faster decay of the high-energy conformer of TCA compared to both formic and acetic acids (by ~35 and ca. 25 times, respectively was found to correlate well with the lower energy barrier for the trans→cis isomerization in the studied compound. The experimental studies received support from quantum chemistry calculations undertaken at the DFT(B3LYP/cc-pVDZ level of approximation, which allowed a detailed characterization of the potential energy surface of the molecule and the detailed assignment of the infrared spectra of the two conformers. 13. Connecting free energy surfaces in implicit and explicit solvent: an efficient method to compute conformational and solvation free energies. Science.gov (United States) Deng, Nanjie; Zhang, Bin W; Levy, Ronald M 2015-06-09 The ability to accurately model solvent effects on free energy surfaces is important for understanding many biophysical processes including protein folding and misfolding, allosteric transitions, and protein–ligand binding. Although all-atom simulations in explicit solvent can provide an accurate model for biomolecules in solution, explicit solvent simulations are hampered by the slow equilibration on rugged landscapes containing multiple basins separated by barriers. In many cases, implicit solvent models can be used to significantly speed up the conformational sampling; however, implicit solvent simulations do not fully capture the effects of a molecular solvent, and this can lead to loss of accuracy in the estimated free energies. Here we introduce a new approach to compute free energy changes in which the molecular details of explicit solvent simulations are retained while also taking advantage of the speed of the implicit solvent simulations. In this approach, the slow equilibration in explicit solvent, due to the long waiting times before barrier crossing, is avoided by using a thermodynamic cycle which connects the free energy basins in implicit solvent and explicit solvent using a localized decoupling scheme. We test this method by computing conformational free energy differences and solvation free energies of the model system alanine dipeptide in water. The free energy changes between basins in explicit solvent calculated using fully explicit solvent paths agree with the corresponding free energy differences obtained using the implicit/explicit thermodynamic cycle to within 0.3 kcal/mol out of ∼3 kcal/mol at only ∼8% of the computational cost. We note that WHAM methods can be used to further improve the efficiency and accuracy of the implicit/explicit thermodynamic cycle. 14. Conformal "Thin-Sandwich" Data for the Initail-Value Problem of General Relativity CERN Document Server York, J W 1999-01-01 The initial-value problem is posed by giving a conformal three-metric on each of two nearby spacelike hypersurfaces, their proper-time separation up to a multiplier to be determined, and the mean (extrinsic) curvature of one slice. The resulting equations have the {\\it same} elliptic form as does the one-hypersurface formulation. The metrical roots of this form are revealed by a conformal thin sandwich'' viewpoint coupled with the transformation properties of the lapse function. 15. Experimental and computational study of crystalline formic acid composed of the higher-energy conformer. Science.gov (United States) Hakala, Mikko; Marushkevich, Kseniya; Khriachtchev, Leonid; Hämäläinen, Keijo; Räsänen, Markku 2011-02-07 Crystalline formic acid (FA) is studied experimentally and by first-principles simulations in order to identify a bulk solid structure composed of the higher-energy (cis) conformer. In the experiments, deuterated FA (HCOOD) was deposited in a Ne matrix and transformed to the cis conformer by vibrational excitation of the ground state (trans) form. Evaporation of the Ne host above 13 K prepared FA in a bulk solid state mainly composed of cis-FA. Infrared absorption spectroscopy at 4.3 K shows that the obtained solid differs from that composed of trans-FA molecules and that the state persists up to the annealing temperature of at least 110 K. The first-principles simulations reveal various energetically stable periodic chain structures containing cis-FA conformers. These chain structures contain either purely cis or both cis and trans forms. The vibrational frequencies of the calculated structures were compared to the experiment and a tentative assignment is given for a novel solid composed of cis-FA. 16. Visualizing potential energy curves and conformations on ultra high-resolution display walls. Science.gov (United States) Kirschner, Karl N; Reith, Dirk; Jato, Oliver; Hinkenjann, André 2015-11-01 In this contribution, we examine how visualization on an ultra high-resolution display wall can augment force-field research in the field of molecular modeling. Accurate force fields are essential for producing reliable simulations, and subsequently important for several fields of applications (e.g. rational drug design and biomolecular modeling). We discuss how using HORNET, a recently constructed specific ultra high-resolution tiled display wall, enhances the visual analytics that are necessary for conformational-based interpretation of the raw data from molecular calculations. Simultaneously viewing multiple potential energy graphs and conformation overlays leads to an enhanced way of evaluating force fields and in their optimization. Consequently, we have integrated visual analytics into our existing Wolf2Pack workflow. We applied this workflow component to analyze how major AMBER force fields (Parm14SB, Gaff, Lipid14, Glycam06j) perform at reproducing the quantum mechanics relative energies and geometries of saturated hydrocarbons. Included in this comparison are the 1996 OPLS force field and our newly developed ExTrM force field. While we focus on atomistic force fields the ideas presented herein are generalizable to other research areas, particularly those that involve numerous representations of large data amounts and whose simultaneous visualization enhances the analysis. 17. Energy management and conservation at General Motors Energy Technology Data Exchange (ETDEWEB) Kelly, R.L. 1982-07-01 An energy conservation plan on a corporate level, some results and potential benefits, two areas for future savings and a national energy policy, as revealed at the 1982 National Industrial Electric Conference are described. Phases of the program are Administrative Controls, Engineering Solutions, and Financial Controls. Heating, ventilation and air conditioning, the largest energy users in the corporation, comprise 27.6% of the total energy used. Suggested engineering solutions cover product specifications, process changes, heat recovery applications, materials conservation, improved equipment control, and facility changes. Computerized Facility Monitoring and Control Systems (FMC) automatically start and stop energy consuming equipment for maximum conservation. Conservation centers around energy accountability- knowing where, how much, and how wisely it is being used, and its cost. 18. 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 19. A coarse-grained generalized second law for holographic conformal field theories Science.gov (United States) Bunting, William; Fu, Zicao; Marolf, Donald 2016-03-01 We consider the universal sector of a d\\gt 2 dimensional large-N strongly interacting holographic CFT on a black hole spacetime background B. When our CFT d is coupled to dynamical Einstein-Hilbert gravity with Newton constant G d , the combined system can be shown to satisfy a version of the thermodynamic generalized second law (GSL) at leading order in G d . The quantity {S}{CFT}+\\frac{A({H}B,{perturbed})}{4{G}d} is non-decreasing, where A({H}B,{perturbed}) is the (time-dependent) area of the new event horizon in the coupled theory. Our S CFT is the notion of (coarse-grained) CFT entropy outside the black hole given by causal holographic information—a quantity in turn defined in the AdS{}d+1 dual by the renormalized area {A}{ren}({H}{{bulk}}) of a corresponding bulk causal horizon. A corollary is that the fine-grained GSL must hold for finite processes taken as a whole, though local decreases of the fine-grained generalized entropy are not obviously forbidden. Another corollary, given by setting {G}d=0, states that no finite process taken as a whole can increase the renormalized free energy F={E}{out}-{{TS}}{CFT}-{{Ω }}J, with T,{{Ω }} constants set by {H}B. This latter corollary constitutes a 2nd law for appropriate non-compact AdS event horizons. 20. The effect of beam energy on the quality of IMRT plans for prostate conformal radiotherapy. Science.gov (United States) de Boer, Steven F; Kumek, Yunus; Jaggernauth, Wainwright; Podgorsak, Matthew B 2007-04-01 Three dimensional conformal radiation therapy (3DCRT) for prostate cancer is most commonly delivered with high-energy photons, typically in the range of 10-21 MV. With the advent of Intensity Modulated Radiation Therapy (IMRT), an increase in the number of monitor units (MU) relative to 3DCRT has lead to a concern about secondary malignancies. This risk becomes more relevant at higher photon energies where there is a greater neutron contribution. Subsequently, the majority of IMRT prostate treatments being delivered today are with 6-10 MV photons where neutron production is negligible. However, the absolute risk is small [Hall, E. J. Intensity Modulated Radiation Therapy, Protons, and the Risk of Second Cancers. Int J Radiat Oncol Bio Phys 65, 1-7 (2006); Kry, F. S., Salehpour, M., Followill, D. S., Stovall, M., Kuban, D. A., White, R. A., and Rosen, I. I. The Calculated Risk of Fatal Secondary Malignancies From Intensity Modulated Radiation Therapy. Int J Radiat Oncol Bio Phys 62, 1195-1203 (2005).] and therefore it has been suggested that the use of an 18MV IMRT may achieve better target coverage and normal tissue sparing such that this benefit outweighs the risks. This paper investigates whether 18MV IMRT offer better target coverage and normal tissue sparing. Computed Tomography (CT) image sets of ten prostate cancer patients were acquired and two separate IMRT plans were created for each patient. One plan used 6 MV beams, and the other used 18 MV, both in a coplanar, non-opposed beam geometry. Beam arrangements and optimization constraints were the same for all plans. This work includes a comparison and discussion of the total integral dose, neutron dose conformity index, and total number of MU for plans generated with both energies. 1. A reoptimized GROMOS force field for hexopyranose-based carbohydrates accounting for the relative free energies of ring conformers, anomers, epimers, hydroxymethyl rotamers, and glycosidic linkage conformers. Science.gov (United States) Hansen, Halvor S; Hünenberger, Philippe H 2011-04-30 This article presents a reoptimization of the GROMOS 53A6 force field for hexopyranose-based carbohydrates (nearly equivalent to 45A4 for pure carbohydrate systems) into a new version 56A(CARBO) (nearly equivalent to 53A6 for non-carbohydrate systems). This reoptimization was found necessary to repair a number of shortcomings of the 53A6 (45A4) parameter set and to extend the scope of the force field to properties that had not been included previously into the parameterization procedure. The new 56A(CARBO) force field is characterized by: (i) the formulation of systematic build-up rules for the automatic generation of force-field topologies over a large class of compounds including (but not restricted to) unfunctionalized polyhexopyranoses with arbritrary connectivities; (ii) the systematic use of enhanced sampling methods for inclusion of experimental thermodynamic data concerning slow or unphysical processes into the parameterization procedure; and (iii) an extensive validation against available experimental data in solution and, to a limited extent, theoretical (quantum-mechanical) data in the gas phase. At present, the 56A(CARBO) force field is restricted to compounds of the elements C, O, and H presenting single bonds only, no oxygen functions other than alcohol, ether, hemiacetal, or acetal, and no cyclic segments other than six-membered rings (separated by at least one intermediate atom). After calibration, this force field is shown to reproduce well the relative free energies of ring conformers, anomers, epimers, hydroxymethyl rotamers, and glycosidic linkage conformers. As a result, the 56A(CARBO) force field should be suitable for: (i) the characterization of the dynamics of pyranose ring conformational transitions (in simulations on the microsecond timescale); (ii) the investigation of systems where alternative ring conformations become significantly populated; (iii) the investigation of anomerization or epimerization in terms of free-energy differences 2. Energy management and conservation at General Motors Energy Technology Data Exchange (ETDEWEB) Kelly, R.L. 1982-07-01 An energy conservation plan on a corporate level, some of the results and potential benefits, two specific areas for future savings and a national energy policy are described. It is from a paper presented during the 1982 National Industrial Electric Conference, sponsored by The Electrification Council and held in Lexington, Ky. The formal energy conservation program has evolved into three phases: Phase I - Administrative Controls; Phase II - Engineering Solutions; Phase III - Financial Controls. All GM plants worldwide have completed a Five Year Energy Conservation Plan for the period 1981 through 1985. A summary of the Plans from 217 locations reveals that the potential exists to save another 39.1 billion Btu by 1985 at a cost of /572 million. The payback for such projects averages three years. 3. LEAR (Low Energy Antiproton Ring), general view. CERN Multimedia 1990-01-01 When the Antiproton Project was launched in the late 1970s, it was recognized that in addition to the primary purpose of high-energy proton-antiproton collisions in the SPS, there was interesting physics to be done with low-energy antiprotons. In 1982, LEAR was ready to receive antiprotons from the Antiproton Accumulator (AA), via the PS. A year later, delivery of antiprotons to the experiments began, at momenta as low as 100 MeV/c (kinetic energy 5.3 MeV), in an "Ultra-Slow Extraction" mode, dispensing some E9 antiprotons over times counted in hours. For such an achievement, stochastic and electron cooling had to be brought to high levels of perfection. 4. The Activation of c-Src Tyrosine Kinase: Conformational Transition Pathway and Free Energy Landscape. Science.gov (United States) Fajer, Mikolai; Meng, Yilin; Roux, Benoît 2016-10-28 Tyrosine kinases are important cellular signaling allosteric enzymes that regulate cell growth, proliferation, metabolism, differentiation, and migration. Their activity must be tightly controlled, and malfunction can lead to a variety of diseases, particularly cancer. The nonreceptor tyrosine kinase c-Src, a prototypical model system and a representative member of the Src-family, functions as complex multidomain allosteric molecular switches comprising SH2 and SH3 domains modulating the activity of the catalytic domain. The broad picture of self-inhibition of c-Src via the SH2 and SH3 regulatory domains is well characterized from a structural point of view, but a detailed molecular mechanism understanding is nonetheless still lacking. Here, we use advanced computational methods based on all-atom molecular dynamics simulations with explicit solvent to advance our understanding of kinase activation. To elucidate the mechanism of regulation and self-inhibition, we have computed the pathway and the free energy landscapes for the "inactive-to-active" conformational transition of c-Src for different configurations of the SH2 and SH3 domains. Using the isolated c-Src catalytic domain as a baseline for comparison, it is observed that the SH2 and SH3 domains, depending upon their bound orientation, promote either the inactive or active state of the catalytic domain. The regulatory structural information from the SH2-SH3 tandem is allosterically transmitted via the N-terminal linker of the catalytic domain. Analysis of the conformational transition pathways also illustrates the importance of the conserved tryptophan 260 in activating c-Src, and reveals a series of concerted events during the activation process. 5. Computing conformational free energy differences in explicit solvent: An efficient thermodynamic cycle using an auxiliary potential and a free energy functional constructed from the end points. Science.gov (United States) Harris, Robert C; Deng, Nanjie; Levy, Ronald M; Ishizuka, Ryosuke; Matubayasi, Nobuyuki 2016-12-23 Many biomolecules undergo conformational changes associated with allostery or ligand binding. Observing these changes in computer simulations is difficult if their timescales are long. These calculations can be accelerated by observing the transition on an auxiliary free energy surface with a simpler Hamiltonian and connecting this free energy surface to the target free energy surface with free energy calculations. Here, we show that the free energy legs of the cycle can be replaced with energy representation (ER) density functional approximations. We compute: (1) The conformational free energy changes for alanine dipeptide transitioning from the right-handed free energy basin to the left-handed basin and (2) the free energy difference between the open and closed conformations of β-cyclodextrin, a "host" molecule that serves as a model for molecular recognition in host-guest binding. β-cyclodextrin contains 147 atoms compared to 22 atoms for alanine dipeptide, making β-cyclodextrin a large molecule for which to compute solvation free energies by free energy perturbation or integration methods and the largest system for which the ER method has been compared to exact free energy methods. The ER method replaced the 28 simulations to compute each coupling free energy with two endpoint simulations, reducing the computational time for the alanine dipeptide calculation by about 70% and for the β-cyclodextrin by > 95%. The method works even when the distribution of conformations on the auxiliary free energy surface differs substantially from that on the target free energy surface, although some degree of overlap between the two surfaces is required. © 2016 Wiley Periodicals, Inc. 6. Integrated Hamiltonian sampling: a simple and versatile method for free energy simulations and conformational sampling. Science.gov (United States) Mori, Toshifumi; Hamers, Robert J; Pedersen, Joel A; Cui, Qiang 2014-07-17 Motivated by specific applications and the recent work of Gao and co-workers on integrated tempering sampling (ITS), we have developed a novel sampling approach referred to as integrated Hamiltonian sampling (IHS). IHS is straightforward to implement and complementary to existing methods for free energy simulation and enhanced configurational sampling. The method carries out sampling using an effective Hamiltonian constructed by integrating the Boltzmann distributions of a series of Hamiltonians. By judiciously selecting the weights of the different Hamiltonians, one achieves rapid transitions among the energy landscapes that underlie different Hamiltonians and therefore an efficient sampling of important regions of the conformational space. Along this line, IHS shares similar motivations as the enveloping distribution sampling (EDS) approach of van Gunsteren and co-workers, although the ways that distributions of different Hamiltonians are integrated are rather different in IHS and EDS. Specifically, we report efficient ways for determining the weights using a combination of histogram flattening and weighted histogram analysis approaches, which make it straightforward to include many end-state and intermediate Hamiltonians in IHS so as to enhance its flexibility. Using several relatively simple condensed phase examples, we illustrate the implementation and application of IHS as well as potential developments for the near future. The relation of IHS to several related sampling methods such as Hamiltonian replica exchange molecular dynamics and λ-dynamics is also briefly discussed. 7. Universal geometrical factor of protein conformations as a consequence of energy minimization CERN Document Server Wu, Ming-Chya; Ma, Wen-Jong; Kouza, Maksim; Hu, Chin-Kun; 10.1209/0295-5075/96/68005 2012-01-01 The biological activity and functional specificity of proteins depend on their native three-dimensional structures determined by inter- and intra-molecular interactions. In this paper, we investigate the geometrical factor of protein conformation as a consequence of energy minimization in protein folding. Folding simulations of 10 polypeptides with chain length ranging from 183 to 548 residues manifest that the dimensionless ratio (V/(A)) of the van der Waals volume V to the surface area A and average atomic radius of the folded structures, calculated with atomic radii setting used in SMMP [Eisenmenger F., et. al., Comput. Phys. Commun., 138 (2001) 192], approach 0.49 quickly during the course of energy minimization. A large scale analysis of protein structures show that the ratio for real and well-designed proteins is universal and equal to 0.491\\pm0.005. The fractional composition of hydrophobic and hydrophilic residues does not affect the ratio substantially. The ratio also holds for intrinsically disorde... 8. Octonionic M-theory and D=11 Generalized Conformal and Superconformal Algebras CERN Document Server Lukierski, J 2003-01-01 Following [1] we further apply the octonionic structure to supersymmetric D=11 $M$-theory. We consider the octonionic $2^{n+1} \\times 2^{n+1}$ Dirac matrices describing the sequence of Clifford algebras with signatures ($9+n,n$) ($n=0,1,2, ...$) and derive the identities following from the octonionic multiplication table. The case $n=1$ ($4\\times 4$ octonion-valued matrices) is used for the description of the D=11 octonionic $M$-superalgebra with 52 real bosonic charges; the $n=2$ case ($8 \\times 8$ octonion-valued matrices) for the D=11 conformal $M$-algebra with 232 real bosonic charges. The octonionic structure is described explicitly for $n=1$ by the relations between the 512 Abelian O(10,1) tensorial charges $Z_\\mu$, $Z_{\\mu\ 9. The clinical potential of high energy, intensity and energy modulated electron beams optimized by simulated annealing for conformal radiation therapy Science.gov (United States) Salter, Bill Jean, Jr. Purpose. The advent of new, so called IVth Generation, external beam radiation therapy treatment machines (e.g. Scanditronix' MM50 Racetrack Microtron) has raised the question of how the capabilities of these new machines might be exploited to produce extremely conformal dose distributions. Such machines possess the ability to produce electron energies as high as 50 MeV and, due to their scanned beam delivery of electron treatments, to modulate intensity and even energy, within a broad field. Materials and methods. Two patients with 'challenging' tumor geometries were selected from the patient archives of the Cancer Therapy and Research Center (CTRC), in San Antonio Texas. The treatment scheme that was tested allowed for twelve, energy and intensity modulated beams, equi-spaced about the patient-only intensity was modulated for the photon treatment. The elementary beams, incident from any of the twelve allowed directions, were assumed parallel, and the elementary electron beams were modeled by elementary beam data. The optimal arrangement of elementary beam energies and/or intensities was optimized by Szu-Hartley Fast Simulated Annealing Optimization. Optimized treatment plans were determined for each patient using both the high energy, intensity and energy modulated electron (HIEME) modality, and the 6 MV photon modality. The 'quality' of rival plans were scored using three different, popular objective functions which included Root Mean Square (RMS), Maximize Dose Subject to Dose and Volume Limitations (MDVL - Morrill et. al.), and Probability of Uncomplicated Tumor Control (PUTC) methods. The scores of the two optimized treatments (i.e. HIEME and intensity modulated photons) were compared to the score of the conventional plan with which the patient was actually treated. Results. The first patient evaluated presented a deeply located target volume, partially surrounding the spinal cord. A healthy right kidney was immediately adjacent to the tumor volume, separated 10. Paul Scherrer Institute Scientific Report 1998. Volume V: General Energy Energy Technology Data Exchange (ETDEWEB) Daum, C.; Leuenberger, J. [eds. 1999-08-01 In view of its mission to contribute towards the development of a globally more sustainable energy supply system, the General Energy Department is focusing on four topical areas: advancing technologies for the use of renewable energies; investigating options for chemical and electrochemical energy storage on various time scales; developing highly efficient converters for the low emission use of fossil and renewable fuels, including both combustion devices and fuel cells; analyzing the consequences of energy use, and advancing scenarios for the development of the energy supply system. Progress in 1998 in these topical areas is described in this report. A list of scientific publications in 1998 is also provided. (author) figs., tabs., refs. 11. Combined experimental powder X-ray diffraction and DFT data to obtain the lowest energy molecular conformation of friedelin Energy Technology Data Exchange (ETDEWEB) Oliveira, Djalma Menezes de; Mussel, Wagner da Nova; Duarte, Lucienir Pains; Silva, Gracia Divina de Fatima; Duarte, Helio Anderson; Gomes, Elionai Cassiana de Lima [Universidade Federal de Minas Gerais (UFMG), Belo Horizonte, MG (Brazil). Dept. de Quimica; Guimaraes, Luciana [Universidade Federal de Sao Joao Del-Rei (UFSJ), MG (Brazil). Dept. de Ciencias Naturais; Vieira Filho, Sidney A., E-mail: [email protected] [Universidade Federal de Ouro Preto (UFOP), MG (Brazil). Dept. de Farmacia 2012-07-01 Friedelin molecular conformers were obtained by Density Functional Theory (DFT) and by ab initio structure determination from powder X-ray diffraction. Their conformers with the five rings in chair-chair-chair-boat-boat, and with all rings in chair, are energy degenerated in gas-phase according to DFT results. The powder diffraction data reveals that rings A, B and C of friedelin are in chair, and rings D and E in boat-boat, conformation. The high correlation values among powder diffraction data, DFT and reported single crystal data indicate that the use of conventional X-ray diffractometer can be applied in routine laboratory analysis in the absence of a single-crystal diffractometer. (author) 12. Combined experimental powder X-ray diffraction and DFT data to obtain the lowest energy molecular conformation of friedelin Directory of Open Access Journals (Sweden) Djalma Menezes de Oliveira 2012-01-01 Full Text Available Friedelin molecular conformers were obtained by Density Functional Theory (DFT and by ab initio structure determination from powder X-ray diffraction. Their conformers with the five rings in chair-chair-chair-boat-boat, and with all rings in chair, are energy degenerated in gas-phase according to DFT results. The powder diffraction data reveals that rings A, B and C of friedelin are in chair, and rings D and E in boat-boat, conformation. The high correlation values among powder diffraction data, DFT and reported single-crystal data indicate that the use of conventional X-ray diffractometer can be applied in routine laboratory analysis in the absence of a single-crystal diffractometer. 13. Generalized ghost dark energy in Brans-Dicke theory CERN Document Server Sheykhi, A; Yosefi, Y 2013-01-01 It was argued that the vacuum energy of the Veneziano ghost field of QCD, in a time-dependent background, can be written in the general form,$H + O(H^2)$, where$H$is the Hubble parameter. Based on this, a phenomenological dark energy model whose energy density is of the form$\\rho=\\alpha H+\\beta H^{2}$was recently proposed to explain the dark energy dominated universe. In this paper, we investigate this generalized ghost dark energy model in the setup of Brans-Dicke cosmology. We study the cosmological implications of this model. In particular, we obtain the equation of state and the deceleration parameters and a differential equation governing the evolution of this dark energy model. It is shown that the equation of state parameter of the new ghost dark energy can cross the phantom line ($w_D=-1$) provided the parameters of the model are chosen suitably. 14. Commercial Building Partnership General Merchandise Energy Savings Overview Energy Technology Data Exchange (ETDEWEB) None 2013-03-01 The Commercial Building Partnership (CBP) paired selected commercial building owners and operators with representatives of DOE, national laboratories and private sector exports to explore energy efficiency measures across general merchandise commercial buildings. 15. General Merchandise 50% Energy Savings Technical Support Document Energy Technology Data Exchange (ETDEWEB) Hale, E.; Leach, M.; Hirsch, A.; Torcellini, P. 2009-09-01 This report documents technical analysis for medium-box general merchandise stores aimed at providing design guidance that achieves whole-building energy savings of at least 50% over ASHRAE Standard 90.1-2004. 16. Chinese hotel general managers' perspectives on energy-saving practices Science.gov (United States) Zhu, Yidan As hotels' concern about sustainability and budget-control is growing steadily, energy-saving issues have become one of the important management concerns hospitality industry face. By executing proper energy-saving practices, previous scholars believed that hotel operation costs can decrease dramatically. Moreover, they believed that conducting energy-saving practices may eventually help the hotel to gain other benefits such as an improved reputation and stronger competitive advantage. The energy-saving issue also has become a critical management problem for the hotel industry in China. Previous research has not investigated energy-saving in China's hotel segment. To achieve a better understanding of the importance of energy-saving, this document attempts to present some insights into China's energy-saving practices in the tourist accommodations sector. Results of the study show the Chinese general managers' attitudes toward energy-saving issues and the differences among the diverse hotel managers who responded to the study. Study results indicate that in China, most of the hotels' energy bills decrease due to the implementation of energy-saving equipments. General managers of hotels in operation for a shorter period of time are typically responsible for making decisions about energy-saving issues; older hotels are used to choosing corporate level concerning to this issue. Larger Chinese hotels generally have official energy-saving usage training sessions for employees, but smaller Chinese hotels sometimes overlook the importance of employee training. The study also found that for the Chinese hospitality industry, energy-saving practices related to electricity are the most efficient and common way to save energy, but older hotels also should pay attention to other ways of saving energy such as water conservation or heating/cooling system. 17. Conformal Infinity Directory of Open Access Journals (Sweden) Frauendiener Jörg 2000-08-01 Full Text Available The notion of conformal infinity has a long history within the research in Einstein's theory of gravity. Today, conformal infinity'' is related with almost all other branches of research in general relativity, from quantisation procedures to abstract mathematical issues to numerical applications. This review article attempts to show how this concept gradually and inevitably evolved out of physical issues, namely the need to understand gravitational radiation and isolated systems within the theory of gravitation and how it lends itself very naturally to solve radiation problems in numerical relativity. The fundamental concept of null-infinity is introduced. Friedrich's regular conformal field equations are presented and various initial value problems for them are discussed. Finally, it is shown that the conformal field equations provide a very powerful method within numerical relativity to study global problems such as gravitational wave propagation and detection. 18. Conformal Infinity Science.gov (United States) Frauendiener, Jörg 2004-12-01 The notion of conformal infinity has a long history within the research in Einstein's theory of gravity. Today, "conformal infinity" is related to almost all other branches of research in general relativity, from quantisation procedures to abstract mathematical issues to numerical applications. This review article attempts to show how this concept gradually and inevitably evolved from physical issues, namely the need to understand gravitational radiation and isolated systems within the theory of gravitation, and how it lends itself very naturally to the solution of radiation problems in numerical relativity. The fundamental concept of null-infinity is introduced. Friedrich's regular conformal field equations are presented and various initial value problems for them are discussed. Finally, it is shown that the conformal field equations provide a very powerful method within numerical relativity to study global problems such as gravitational wave propagation and detection. 19. Conformal Infinity Directory of Open Access Journals (Sweden) Frauendiener Jörg 2004-01-01 Full Text Available The notion of conformal infinity has a long history within the research in Einstein's theory of gravity. Today, 'conformal infinity' is related to almost all other branches of research in general relativity, from quantisation procedures to abstract mathematical issues to numerical applications. This review article attempts to show how this concept gradually and inevitably evolved from physical issues, namely the need to understand gravitational radiation and isolated systems within the theory of gravitation, and how it lends itself very naturally to the solution of radiation problems in numerical relativity. The fundamental concept of null-infinity is introduced. Friedrich's regular conformal field equations are presented and various initial value problems for them are discussed. Finally, it is shown that the conformal field equations provide a very powerful method within numerical relativity to study global problems such as gravitational wave propagation and detection. 20. Full QM Calculation of RNA Energy Using Electrostatically Embedded Generalized Molecular Fractionation with Conjugate Caps Method. Science.gov (United States) Jin, Xinsheng; Zhang, John Z H; He, Xiao 2017-03-30 In this study, the electrostatically embedded generalized molecular fractionation with conjugate caps (concaps) method (EE-GMFCC) was employed for efficient linear-scaling quantum mechanical (QM) calculation of total energies of RNAs. In the EE-GMFCC approach, the total energy of RNA is calculated by taking a proper combination of the QM energy of each nucleotide-centric fragment with large caps or small caps (termed EE-GMFCC-LC and EE-GMFCC-SC, respectively) deducted by the energies of concaps. The two-body QM interaction energy between non-neighboring ribonucleotides which are spatially in close contact are also taken into account for the energy calculation. Numerical studies were carried out to calculate the total energies of a number of RNAs using the EE-GMFCC-LC and EE-GMFCC-SC methods at levels of the Hartree-Fock (HF) method, density functional theory (DFT), and second-order many-body perturbation theory (MP2), respectively. The results show that the efficiency of the EE-GMFCC-SC method is about 3 times faster than the EE-GMFCC-LC method with minimal accuracy sacrifice. The EE-GMFCC-SC method is also applied for relative energy calculations of 20 different conformers of two RNA systems using HF and DFT, respectively. Both single-point and relative energy calculations demonstrate that the EE-GMFCC method has deviations from the full system results of only a few kcal/mol. 1. Determination of the electron-detachment energies of 2'-deoxyguanosine 5'-monophosphate anion: influence of the conformation. Science.gov (United States) Rubio, Mercedes; Roca-Sanjuán, Daniel; Serrano-Andrés, Luis; Merchán, Manuela 2009-02-26 The vertical electron-detachment energies (VDEs) of the singly charged 2'-deoxyguanosine 5'-monophosphate anion (dGMP-) are determined by using the multiconfigurational second-order perturbation CASPT2 method at the MP2 ground-state equilibrium geometry of relevant conformers. The origin of the unique low-energy band in the gas phase photoelectron spectrum of dGMP-, with maximum at around 5.05 eV, is unambiguously assigned to electron detachment from the highest occupied molecular orbital of pi-character belonging to guanine fragment of a syn conformation. The presence of a short H-bond linking the 2-amino and phosphate groups, the guanine moiety acting as proton donor, is precisely responsible for the pronounced decrease of the computed VDE with respect to that obtained in other conformations. As a whole, the present research supports the nucleobase as the site with the lowest ionization potential in negatively charged (deprotonated) nucleotides at the most stable conformations as well as for B-DNA-like type arrangements, in agreement with experimental evidence. 2. Control systems are General Motors' biggest energy saver Energy Technology Data Exchange (ETDEWEB) 1979-06-01 In 1978, General Motors Corp. used almost 3% less energy than it did in 1972, even though production had increased about 25%. Most of the savings are the result of improved technology and design changes in buildings, equipment, and processes. Computerized energy management control systems are now in operation or being installed in 78 GM buildings. 3. Development of the Model of the Generalized Quintom Dark Energy Institute of Scientific and Technical Information of China (English) WANG Wei; GUI Yuan-Xing; SHAO Ying 2006-01-01 @@ We consider a generalized quintom (GQ) dark energy modelfor changing the equal weight of the negative-kinetic scalar field (phantom) and the normal scalar field (quintessence) in quintom dark energy. Though the phantomdominated scaling solution is a stable late-time attractor, the early evolution of GQ is different from that of the quintom model and the adjustability of the dark energy state equation in the model is improved. 4. Available Potential Energy and the Maintenance of the General Circulation OpenAIRE Lorenz, Edward N. 2011-01-01 The available potential energy of the atmosphere may be defined as the difference between the total potential energy and the minimum total potential energy which could result from any adiabatic redistribution of mass. It vanishes if the density stratification is horizontal and statically stable everywhere, and is positive otherwise. It is measured approximately by a weighted vertical average of the horizontal variance of temperature. In magnitude it is generally about ten times the total kine... 5. Dissecting CFT Correlators and String Amplitudes. Conformal Blocks and On-Shell Recursion for General Tensor Fields Energy Technology Data Exchange (ETDEWEB) Hansen, Tobias 2015-07-15 This thesis covers two main topics: the tensorial structure of quantum field theory correlators in general spacetime dimensions and a method for computing string theory scattering amplitudes directly in target space. In the first part tensor structures in generic bosonic CFT correlators and scattering amplitudes are studied. To this end arbitrary irreducible tensor representations of SO(d) (traceless mixed-symmetry tensors) are encoded in group invariant polynomials, by contracting with sets of commuting and anticommuting polarization vectors which implement the index symmetries of the tensors. The tensor structures appearing in CFT{sub d} correlators can then be inferred by studying these polynomials in a d + 2 dimensional embedding space. It is shown with an example how these correlators can be used to compute general conformal blocks describing the exchange of mixed-symmetry tensors in four-point functions, which are crucial for advancing the conformal bootstrap program to correlators of operators with spin. Bosonic string theory lends itself as an ideal example for applying the same methods to scattering amplitudes, due to its particle spectrum of arbitrary mixed-symmetry tensors. This allows in principle the definition of on-shell recursion relations for string theory amplitudes. A further chapter introduces a different target space definition of string scattering amplitudes. As in the case of on-shell recursion relations, the amplitudes are expressed in terms of their residues via BCFW shifts. The new idea here is that the residues are determined by use of the monodromy relations for open string theory, avoiding the infinite sums over the spectrum arising in on-shell recursion relations. Several checks of the method are presented, including a derivation of the Koba-Nielsen amplitude in the bosonic string. It is argued that this method provides a target space definition of the complete S-matrix of string theory at tree-level in a at background in terms of a 6. A coarse-grained generalized second law for holographic conformal field theories CERN Document Server Bunting, William; Marolf, Donald 2015-01-01 We consider the universal sector of a$d$-dimensional large-$N$strongly-interacting holographic CFT on a black hole spacetime background$B$. When our CFT$_d$is coupled to dynamical Einstein-Hilbert gravity with Newton constant$G_{d}$, the combined system can be shown to satisfy a version of the thermodynamic Generalized Second Law (GSL) at leading order in$G_{d}$. The quantity$S_{CFT} + \\frac{A(H_{B, \\text{perturbed}})}{4G_{d}}$is non-decreasing, where$A(H_{B, \\text{perturbed}})$is the (time-dependent) area of the new event horizon in the coupled theory. Our$S_{CFT}$is the notion of (coarse-grained) CFT entropy outside the black hole given by causal holographic information -- a quantity in turn defined in the AdS$_{d+1}$dual by the renormalized area$A_{ren}(H_{\\rm bulk})$of a corresponding bulk causal horizon. A corollary is that the fine-grained GSL must hold for finite processes taken as a whole, though local decreases of the fine-grained generalized entropy are not obviously forbidden. Anothe... 7. 76 FR 11437 - Application To Export Electric Energy; Societe Generale Energy Corp. Science.gov (United States) 2011-03-02 ... Application To Export Electric Energy; Societe Generale Energy Corp. AGENCY: Office of Electricity Delivery.... (SGEC) has applied for authority to transmit electric energy from the United States to Canada pursuant... application from the SGEC for authority to transmit electric energy from the United States to Canada as... 8. Finding low-energy conformations of lattice protein models by quantum annealing CERN Document Server Perdomo-Ortiz, Alejandro; Drew-Brook, Marshall; Rose, Geordie; Aspuru-Guzik, Alán 2012-01-01 Lattice protein folding models are a cornerstone of computational biophysics. Although these models are a coarse grained representation, they provide useful insight into the energy landscape of natural proteins. Finding low-energy three-dimensional structures is an intractable problem even in the simplest model, the Hydrophobic-Polar (HP) model. Exhaustive search of all possible global minima is limited to sequences in the tens of amino acids. Description of protein-like properties are more accurately described by generalized models, such as the one proposed by Miyazawa and Jernigan (MJ), which explicitly take into account the unique interactions among all 20 amino acids. There is theoretical and experimental evidence of the advantage of solving classical optimization problems using quantum annealing over its classical analogue (simulated annealing). In this report, we present a benchmark implementation of quantum annealing for a biophysical problem (six different experiments up to 81 superconducting quantum ... 9. General relativistic models for rotating magnetized neutron stars in conformally flat space-time Science.gov (United States) Pili, A. G.; Bucciantini, N.; Del Zanna, L. 2017-09-01 The extraordinary energetic activity of magnetars is usually explained in terms of dissipation of a huge internal magnetic field of the order of 1015-16 G. How such a strong magnetic field can originate during the formation of a neutron star (NS) is still subject of active research. An important role can be played by fast rotation: if magnetars are born as millisecond rotators dynamo mechanisms may efficiently amplify the magnetic field inherited from the progenitor star during the collapse. In this case, the combination of rapid rotation and strong magnetic field determine the right physical condition not only for the development of a powerful jet-driven explosion, manifesting as a gamma-ray burst, but also for a copious gravitational waves emission. Strong magnetic fields are indeed able to induce substantial quadrupolar deformations in the star. In this paper, we analyse the joint effect of rotation and magnetization on the structure of a polytropic and axisymmetric NS, within the ideal magneto-hydrodynamic regime. We will consider either purely toroidal or purely poloidal magnetic field geometries. Through the sampling of a large parameter space, we generalize previous results in literature, inferring new quantitative relations that allow for a parametrization of the induced deformation, that takes into account also the effects due to the stellar compactness and the current distribution. Finally, in the case of purely poloidal field, we also discuss how different prescription on the surface charge distribution (a gauge freedom) modify the properties of the surrounding electrosphere and its physical implications. 10. Kerr-Taub-NUT General Frame, Energy, and Momentum in Teleparallel Equivalent of General Relativity Directory of Open Access Journals (Sweden) Gamal G. L. Nashed 2012-01-01 Full Text Available A new exact solution describing a general stationary and axisymmetric object of the gravitational field in the framework of teleparallel equivalent of general relativity (TEGR is derived. The solution is characterized by three parameters “the gravitational mass M, the rotation a, and the NUT L.” The vierbein field is axially symmetric, and the associated metric gives the Kerr-Taub-NUT spacetime. Calculation of the total energy using two different methods, the gravitational energy momentum and the Riemannian connection 1-form Γα̃β, is carried out. It is shown that the two methods give the same results of energy and momentum. The value of energy is shown to depend on the mass M and the NUT parameter L. If L is vanishing, then the total energy reduced to the energy of Kerr black hole. 11. How Do DFT-DCP, DFT-NL, and DFT-D3 Compare for the Description of London-Dispersion Effects in Conformers and General Thermochemistry? Science.gov (United States) Goerigk, Lars 2014-03-11 The dispersion-core-potential corrected B3LYP-DCP method (Torres and DiLabio J. Phys. Chem. Lett. 2012, 3, 1738) is for the first time thoroughly assessed and compared with the B3LYP-NL (Hujo and Grimme J. Chem. Theory Comput. 2011, 7, 3866) and B3LYP-D3 (Grimme et al. J. Comput. Chem. 2011, 32, 1456) methods for a broad range of chemical problems that particularly shed light on intramolecular London-dispersion effects in conformers and general thermochemistry. The analysis is based on a compilation of 473 reference cases, the majority of which are taken from the GMTKN30 database (Goerigk and Grimme J. Chem. Theory Comput. 2010, 6, 107; 2011, 7, 291). The results confirm previous findings that B3LYP-DCP indeed predicts very good binding energies for noncovalently bound complexes, particularly with small basis sets. However, problems are identified for the description of intramolecular effects in some conformers and chemical reactions, for which B3LYP-DCP sometimes gives results similar or worse than uncorrected B3LYP. Surprisingly large errors for total atomization energies reveal an unwanted influence of the DCPs on the short-range electronic structure of the investigated systems. However, a recently modified carbon potential for B3LYP-DCP (DiLabio et al. Theor. Chem. Acc. 2013, 132, 1389) was additionally tested that seems to solve most of those problems and provides improved results. An overall comparison between all tested methods shows that B3LYP-NL is the most robust and accurate approach, closely followed by B3LYP-D3. This is also true when small basis sets of double-ζ quality are applied for which those methods have not been parametrized. However, binding energies of noncovalently bound complexes can be more strongly influenced by basis-set superposition-error effects than for B3LYP-DCP. Finally, it is noted that the DFT-D3 and DFT-NL schemes are readily applicable to a large range of chemical elements and they are therefore particularly recommended for 12. How universal are hydrogen bond correlations? A density functional study of intramolecular hydrogen bonding in low-energy conformers of α-amino acids Science.gov (United States) Ramaniah, Lavanya M.; Kamal, C.; Kshirsagar, Rohidas J.; Chakrabarti, Aparna; Banerjee, Arup 2013-10-01 Hydrogen bonding is one of the most important and ubiquitous interactions present in Nature. Several studies have attempted to characterise and understand the nature of this very basic interaction. These include both experimental and theoretical investigations of different types of chemical compounds, as well as systems subjected to high pressure. The O-H..O bond is of course the best studied hydrogen bond, and most studies have concentrated on intermolecular hydrogen bonding in solids and liquids. In this paper, we analyse and characterise normal hydrogen bonding of the general type, D-H...A, in intramolecular hydrogen bonding interactions. Using a first-principles density functional theory approach, we investigate low energy conformers of the twenty α-amino acids. Within these conformers, several different types of intramolecular hydrogen bonds are identified. The hydrogen bond within a given conformer occurs between two molecular groups, either both within the backbone itself, or one in the backbone and one in the side chain. In a few conformers, more than one (type of) hydrogen bond is seen to occur. Interestingly, the strength of the hydrogen bonds in the amino acids spans quite a large range, from weak to strong. The signature of hydrogen bonding in these molecules, as reflected in their theoretical vibrational spectra, is analysed. With the new first-principles data from 51 hydrogen bonds, various parameters relating to the hydrogen bond, such as hydrogen bond length, hydrogen bond angle, bond length and vibrational frequencies are studied. Interestingly, the correlation between these parameters in these bonds is found to be in consonance with those obtained in earlier experimental studies of normal hydrogen bonds on vastly different systems. Our study provides some of the most detailed first-principles support, and the first involving vibrational frequencies, for the universality of hydrogen bond correlations in materials. 13. Paul Scherrer Institute Scientific Report 1999. Volume V: General Energy Energy Technology Data Exchange (ETDEWEB) Daum, Christina; Leuenberger, Jakob [eds. 2000-07-01 Strengthening of international collaborations represented a strategic goal of the General Energy Research Department for 1999. For the Fifth Framework Program of the European Union, we participated in consortia and in the successful preparation of several proposals. National networks with partners from academia and Industry have been formed in two topical areas of central interest in the context of sustainability, i.e. 'Ecoefficient energy use and material cycles' and 'Sustainable transportation' on the other hand. Research at PSI comprises all aspects of human energy use, with the ultimate goal of promoting development towards a sustainable energy supply system. In the General Energy Research Department, technologies are being advanced for the utilization of renewable energy sources, low-loss energy storage, efficient conversion, and low emission energy use. Experimental and model-based assessment of these emissions forms the basis of a comprehensive assessment of economic, ecological and environmental consequences, for both present and future energy supply systems. The research program of the department is centered around 1) development, use, and characterisation of catalysts for energy technologies in many different fields, like e.g. the partial oxidation of methanol for hydrogen production, the processing of methane by catalytic combustion and reforming; 2) use of concentrated solar radiation to induce chemical conversions, thereby producing energy carriers; 3) development of efficient, less polluting combustion engines and burners by advancing the detailed understanding of reaction mechanisms and combustion pathways; 4) research and development of low temperature polymer electrolyte fuel cells, novel batteries and capacitors, with applications envisaged for electric vehicles, photovoltaics and on-site load leveling; 5) experimental and model based research concerning transportation and chemistry of atmospheric trace gases related to 14. Extended Theories of Gravity with Generalized Energy Conditions CERN Document Server Mimoso, José P; Capozziello, Salvatore 2014-01-01 We address the problem of the energy conditions in modified gravity taking into account the additional degrees of freedom related to scalar fields and curvature invariants. The latter are usually interpreted as generalized {\\it geometrical fluids} that differ in meaning with respect to the matter fluids generally considered as sources of the field equations. In extended gravity theories the curvature terms are encapsulated in a tensor$H^{ab}$and a coupling$g(\\Psi^i)$that can be recast as effective Einstein field equations, with corrections to the energy-momentum tensor of matter. The formal validity of standard energy inequalities does not assure basic requirements such as the attractive nature of gravity, so we argue that the energy conditions have to be considered in a wider sense. 15. Cosmological constraints on the generalized holographic dark energy CERN Document Server Lu, Jianbo; Wu, Yabo; Wang, Tianqiang 2012-01-01 We use the Markov ChainMonte Carlo method to investigate global constraints on the generalized holographic (GH) dark energy with flat and non-flat universe from the current observed data: the Union2 dataset of type supernovae Ia (SNIa), high-redshift Gamma-Ray Bursts (GRBs), the observational Hubble data (OHD), the cluster X-ray gas mass fraction, the baryon acoustic oscillation (BAO), and the cosmic microwave background (CMB) data. The most stringent constraints on the GH model parameter are obtained. In addition, it is found that the equation of state for this generalized holographic dark energy can cross over the phantom boundary wde =-1. 16. High energy collisions of particles inside ergosphere: general approach CERN Document Server Zaslavskii, O B 2013-01-01 We show that recent observation made in Grib and Pavlov, arXiv:1301.0698 for the Kerr black hole is valid in the general case of rotating axially symmetric metric. Namely, collision of two particles in the ergosphere leads to indefinite growth of the energy in the centre of mass frame, provided the angular momentum of one of two particles is negative and increases without limit for a fixed energy at infinity. General approach enabled us to elucidate, why the role of the ergosphere in this process is crucial. 17. Paul Scherrer Institut Scientific Report 2003. Volume V: General Energy Energy Technology Data Exchange (ETDEWEB) Wokaun, Alexander; Daum, Christina (eds.) 2004-03-01 Research at PSI comprises all aspects of human energy use, with the ultimate goal of promoting development towards a sustainable energy supply system. In the General Energy Research Department, technologies are being advanced for the utilization of renewable energy sources, low-loss energy storage, efficient conversion, and low emission energy use. Experimental and model-based assessment of these emissions forms the basis of a comprehensive assessment of economic, ecological and environmental consequences, for both present and future energy supply systems. The research program of the department is centered around 1) development, use, and characterisation of catalysts for energy technologies in many different fields, like e.g. the partial oxidation of methanol for hydrogen production, the processing of methane by catalytic combustion and reforming; 2) use of concentrated solar radiation to induce chemical conversions, thereby producing energy carriers; 3) development of efficient, less polluting combustion engines and burners by advancing the detailed understanding of reaction mechanisms and combustion pathways; 4) research and development of low temperature polymer electrolyte fuel cells, novel batteries and capacitors, with applications envisaged for electric vehicles, photovoltaics and on-site load leveling; 5) experimental and model based research concerning transportation and chemistry of atmospheric trace gases related to anthropogenic energy transformations. Progress in 2000 in these topical areas is described in this report. A list of scientific publications in 2002 is also provided. 18. Paul Scherrer Institut Scientific Report 2001. Volume V: General Energy Energy Technology Data Exchange (ETDEWEB) Wokaun, Alexander; Daum, Christina (eds.) 2002-03-01 Research at PSI comprises all aspects of human energy use, with the ultimate goal of promoting development towards a sustainable energy supply system. In the General Energy Research Department, technologies are being advanced for the utilization of renewable energy sources, low-loss energy storage, efficient conversion, and low emission energy use. Experimental and model-based assessment of these emissions forms the basis of a comprehensive assessment of economic, ecological and environmental consequences, for both present and future energy supply systems. The research program of the department is centered around 1) development, use, and characterisation of catalysts for energy technologies in many different fields, like e.g. the partial oxidation of methanol for hydrogen production, the processing of methane by catalytic combustion and reforming; 2) use of concentrated solar radiation to induce chemical conversions, thereby producing energy carriers; 3) development of efficient, less polluting combustion engines and burners by advancing the detailed understanding of reaction mechanisms and combustion pathways; 4) research and development of low temperature polymer electrolyte fuel cells, novel batteries and capacitors, with applications envisaged for electric vehicles, photovoltaics and on-site load leveling; 5) experimental and model based research concerning transportation and chemistry of atmospheric trace gases related to anthropogenic energy transformations. Progress in 2000 in these topical areas is described in this report. A list of scientific publications in 2001 is also provided. 19. Paul Scherrer Institute Scientific Report 2000. Volume V: General Energy Energy Technology Data Exchange (ETDEWEB) Daum, Christina; Leuenberger, Jakob [eds. 2001-03-01 Research at PSI comprises all aspects of human energy use, with the ultimate goal of promoting development towards a sustainable energy supply system. In the General Energy Research Department, technologies are being advanced for the utilization of renewable energy sources, low-loss energy storage, efficient conversion, and low emission energy use. Experimental and model-based assessment of these emissions forms the basis of a comprehensive assessment of economic, ecological and environmental consequences, for both present and future energy supply systems. The research program of the department is centered around (1) development, use, and characterisation of catalysts for energy technologies in many different fields, like e.g. the partial oxidation of methanol for hydrogen production, the processing of methane by catalytic combustion and reforming; (2) use of concentrated solar radiation to induce chemical conversions, thereby producing energy carriers; (3) development of efficient, less polluting combustion engines and burners by advancing the detailed understanding of reaction mechanisms and combustion pathways; (4) research and development of low temperature polymer electrolyte fuel cells, novel batteries and capacitors, with applications envisaged for electric vehicles, photovoltaics and on-site load leveling; (5) experimental and model based research concerning transportation and chemistry of atmospheric trace gases related to anthropogenic energy transformations. Progress in 2000 in these topical areas is described in this report. A list of scientific publications in 2000 is also provided. 20. General business model patterns for Local Energy Management concepts Directory of Open Access Journals (Sweden) Emanuele eFacchinetti 2016-03-01 Full Text Available The transition towards a more sustainable global energy system, significantly relying on renewable energies and decentralized energy systems, requires a deep reorganization of the energy sector. The way how energy services are generated, delivered and traded is expected to be very different in the coming years. Business model innovation is recognized as a key driver for the successful implementation of the energy turnaround. This work contributes to this topic by introducing a heuristic methodology easing the identification of general business model patterns best suited for Local Energy Management concepts such as Energy Hubs. A conceptual framework characterizing the Local Energy Management business model solution space is developed. Three reference business model patterns providing orientation across the defined solution space are identified, analyzed and compared. Through a market review a number of successfully implemented innovative business models have been analyzed and allocated within the defined solution space. The outcomes of this work offer to potential stakeholders a starting point and guidelines for the business model innovation process, as well as insights for policy makers on challenges and opportunities related to Local Energy Management concepts. 1. Paul Scherrer Institut Scientific Report 2004. Volume V: General Energy Energy Technology Data Exchange (ETDEWEB) Wokaun, Alexander; Daum, Christina (eds.) 2005-03-01 Research at PSI comprises all aspects of human energy use, with the ultimate goal of promoting development towards a sustainable energy supply system. In the General Energy Research Department, technologies are being advanced for the utilization of renewable energy sources, low-loss energy storage, efficient conversion, and low emission energy use. Experimental and model-based assessment of these emissions forms the basis of a comprehensive assessment of economic, ecological and environmental consequences, for both present and future energy supply systems. The research program of the department is centered around 1) development, use, and characterisation of catalysts for energy technologies in many different fields, like e.g. the partial oxidation of methanol for hydrogen production, the processing of methane by catalytic combustion and reforming; 2) use of concentrated solar radiation to induce chemical conversions, thereby producing energy carriers; 3) development of efficient, less polluting combustion engines and burners by advancing the detailed understanding of reaction mechanisms and combustion pathways; 4) research and development of low temperature polymer electrolyte fuel cells, novel batteries and capacitors, with applications envisaged for electric vehicles, photovoltaics and on-site load leveling; 5) experimental and model based research concerning transportation and chemistry of atmospheric trace gases related to anthropogenic energy transformations. Progress in 2000 in these topical areas is described in this report. A list of scientific publications in 2002 is also provided. 2. Paul Scherrer Institut Scientific Report 2002. Volume V: General Energy Energy Technology Data Exchange (ETDEWEB) Wokaun, Alexander; Daum, Christina (eds.) 2003-03-01 Research at PSI comprises all aspects of human energy use, with the ultimate goal of promoting development towards a sustainable energy supply system. In the General Energy Research Department, technologies are being advanced for the utilization of renewable energy sources, low-loss energy storage, efficient conversion, and low emission energy use. Experimental and model-based assessment of these emissions forms the basis of a comprehensive assessment of economic, ecological and environmental consequences, for both present and future energy supply systems. The research program of the department is centered around 1) development, use, and characterisation of catalysts for energy technologies in many different fields, like e.g. the partial oxidation of methanol for hydrogen production, the processing of methane by catalytic combustion and reforming; 2) use of concentrated solar radiation to induce chemical conversions, thereby producing energy carriers; 3) development of efficient, less polluting combustion engines and burners by advancing the detailed understanding of reaction mechanisms and combustion pathways; 4) research and development of low temperature polymer electrolyte fuel cells, novel batteries and capacitors, with applications envisaged for electric vehicles, photovoltaics and on-site load leveling; 5) experimental and model based research concerning transportation and chemistry of atmospheric trace gases related to anthropogenic energy transformations. Progress in 2000 in these topical areas is described in this report. A list of scientific publications in 2002 is also provided. 3. Paul Scherrer Institute Scientific Report 2000. Volume V: General Energy Energy Technology Data Exchange (ETDEWEB) Daum, Christina; Leuenberger, Jakob [eds. 2001-07-01 Research at PSI comprises all aspects of human energy use, with the ultimate goal of promoting development towards a sustainable energy supply system. In the General Energy Research Department, technologies are being advanced for the utilization of renewable energy sources, low-loss energy storage, efficient conversion, and low emission energy use. Experimental and model-based assessment of these emissions forms the basis of a comprehensive assessment of economic, ecological and environmental consequences, for both present and future energy supply systems. The research program of the department is centered around 1) development, use, and characterisation of catalysts for energy technologies in many different fields, like e.g. the partial oxidation of methanol for hydrogen production, the processing of methane by catalytic combustion and reforming; 2) use of concentrated solar radiation to induce chemical conversions, thereby producing energy carriers; 3) development of efficient, less polluting combustion engines and burners by advancing the detailed understanding of reaction mechanisms and combustion pathways; 4) research and development of low temperature polymer electrolyte fuel cells, novel batteries and capacitors, with applications envisaged for electric vehicles, photovoltaics and on-site load leveling; 5) experimental and model based research concerning transportation and chemistry of atmospheric trace gases related to anthropogenic energy transformations. Progress in 2000 in these topical areas is described in this report. A list of scientific publications in 2000 is also provided. 4. Generalized Lorentz invariance with an invariant energy scale CERN Document Server Magueijo, J; Magueijo, Joao; Smolin, Lee 2003-01-01 The hypothesis that the Lorentz transformations may be modified at Planck scale energies is further explored. We present a general formalism for theories which preserve the relativity of inertial frames with a non-linear action of the Lorentz transformations on momentum space. Several examples are discussed in which the speed of light varies with energy and elementary particles have a maximum momenta and/or energy. Energy and momentum conservation are suitably generalized and a proposal is made for how the new transformation laws apply to composite systems. We then use these results to explain the ultra high energy cosmic ray anomaly and we find a form of the theory that explains the anomaly, and leads also to a maximum momentum and a speed of light that diverges with energy. We finally propose that the spatial coordinates be identified as the generators of translation in Minkowski spacetime. In some examples this leads to a commutative geometry, but with an energy dependent Planck constant. 5. Exploring transition pathway and free-energy profile of large-scale protein conformational change by combining normal mode analysis and umbrella sampling molecular dynamics. Science.gov (United States) Wang, Jinan; Shao, Qiang; Xu, Zhijian; Liu, Yingtao; Yang, Zhuo; Cossins, Benjamin P; Jiang, Hualiang; Chen, Kaixian; Shi, Jiye; Zhu, Weiliang 2014-01-09 Large-scale conformational changes of proteins are usually associated with the binding of ligands. Because the conformational changes are often related to the biological functions of proteins, understanding the molecular mechanisms of these motions and the effects of ligand binding becomes very necessary. In the present study, we use the combination of normal-mode analysis and umbrella sampling molecular dynamics simulation to delineate the atomically detailed conformational transition pathways and the associated free-energy landscapes for three well-known protein systems, viz., adenylate kinase (AdK), calmodulin (CaM), and p38α kinase in the absence and presence of respective ligands. For each protein under study, the transient conformations along the conformational transition pathway and thermodynamic observables are in agreement with experimentally and computationally determined ones. The calculated free-energy profiles reveal that AdK and CaM are intrinsically flexible in structures without obvious energy barrier, and their ligand binding shifts the equilibrium from the ligand-free to ligand-bound conformation (population shift mechanism). In contrast, the ligand binding to p38α leads to a large change in free-energy barrier (ΔΔG ≈ 7 kcal/mol), promoting the transition from DFG-in to DFG-out conformation (induced fit mechanism). Moreover, the effect of the protonation of D168 on the conformational change of p38α is also studied, which reduces the free-energy difference between the two functional states of p38α and thus further facilitates the conformational interconversion. Therefore, the present study suggests that the detailed mechanism of ligand binding and the associated conformational transition is not uniform for all kinds of proteins but correlated to their respective biological functions. 6. Conformational selection through electrostatics: Free energy simulations of GTP and GDP binding to archaeal initiation factor 2. Science.gov (United States) Satpati, Priyadarshi; Simonson, Thomas 2012-05-01 Archaeal Initiation Factor 2 is a GTPase involved in protein biosynthesis. In its GTP-bound, "ON" conformation, it binds an initiator tRNA and carries it to the ribosome. In its GDP-bound, "OFF" conformation, it dissociates from tRNA. To understand the specific binding of GTP and GDP and their dependence on the conformational state, molecular dynamics free energy simulations were performed. The ON state specificity was predicted to be weak, with a GTP/GDP binding free energy difference of -1 kcal/mol, favoring GTP. The OFF state specificity is larger, 4 kcal/mol, favoring GDP. The overall effects result from a competition among many interactions in several complexes. To interpret them, we use a simpler, dielectric continuum model. Several effects are robust with respect to the model details. Both nucleotides have a net negative charge, so that removing them from solvent into the binding pocket carries a desolvation penalty, which is large for the ON state, and strongly disfavors GTP binding compared to GDP. Short-range interactions between the additional GTP phosphate group and ionized sidechains in the binding pocket offset most, but not all of the desolvation penalty; more distant groups also contribute significantly, and the switch 1 loop only slightly. The desolvation penalty is lower for the more open, wetter OFF state, and the GTP/GDP difference much smaller. Short-range interactions in the binding pocket and with more distant groups again make a significant contribution. Overall, the simulations help explain how conformational selection is achieved with a single phosphate group. Copyright © 2012 Wiley Periodicals, Inc. 7. The quantum mechanics based on a general kinetic energy CERN Document Server Wei, Yuchuan 2016-01-01 In this paper, we introduce the Schrodinger equation with a general kinetic energy operator. The conservation law is proved and the probability continuity equation is deducted in a general sense. Examples with a Hermitian kinetic energy operator include the standard Schrodinger equation, the relativistic Schrodinger equation, the fractional Schrodinger equation, the Dirac equation, and the deformed Schrodinger equation. We reveal that the Klein-Gordon equation has a hidden non-Hermitian kinetic energy operator. The probability continuity equation with sources indicates that there exists a different way of probability transportation, which is probability teleportation. An average formula is deducted from the relativistic Schrodinger equation, the Dirac equation, and the K-G equation. 8. Generalized energy conditions in Extended Theories of Gravity CERN Document Server Capozziello, Salvatore; Mimoso, José P 2014-01-01 Theories of physics can be considered viable if the initial value problem and the energy conditions are formulated self-consistently. The former allow a uniquely determined dynamical evolution of the system, and the latter guarantee that causality is preserved and that "plausible" physical sources have been considered. In this work, we consider the further degrees of freedom related to curvature invariants and scalar fields in Extended Theories of Gravity (ETG). These new degrees of freedom can be recast as effective perfect fluids that carry different meanings with respect to the standard matter fluids generally adopted as sources of the field equations. It is thus somewhat misleading to apply the standard general relativistic energy conditions to this effective energy-momentum, as the latter contains the matter content and a geometrical quantity, which arises from the ETG considered. Here, we explore this subtlety, extending on previous work, in particular, to cases with the contracted Bianchi identities wi... 9. Generalized Ensemble Sampling of Enzyme Reaction Free Energy Pathways Science.gov (United States) Wu, Dongsheng; Fajer, Mikolai I.; Cao, Liaoran; Cheng, Xiaolin; Yang, Wei 2016-01-01 Free energy path sampling plays an essential role in computational understanding of chemical reactions, particularly those occurring in enzymatic environments. Among a variety of molecular dynamics simulation approaches, the generalized ensemble sampling strategy is uniquely attractive for the fact that it not only can enhance the sampling of rare chemical events but also can naturally ensure consistent exploration of environmental degrees of freedom. In this review, we plan to provide a tutorial-like tour on an emerging topic: generalized ensemble sampling of enzyme reaction free energy path. The discussion is largely focused on our own studies, particularly ones based on the metadynamics free energy sampling method and the on-the-path random walk path sampling method. We hope that this mini presentation will provide interested practitioners some meaningful guidance for future algorithm formulation and application study. PMID:27498634 10. Generalized trends in the formation energies of perovskite oxides DEFF Research Database (Denmark) Zeng, Zhenhua; Calle-Vallejo, Federico; Mogensen, Mogens Bjerg 2013-01-01 Generalized trends in the formation energies of several families of perovskite oxides (ABO3) and plausible explanations to their existence are provided in this study through a combination of DFT calculations, solid-state physics analyses and simple physical/chemical descriptors. The studied...... systematically on the oxidation state of the A-site cation; and (IV) the trends in formation energies of perovskites with elements from different periods at the B site depend on the oxidation state of A-site cations. Since the energetics of perovskites is shown to be the superposition of the individual...... contributions of their constituent oxides, the trends can be rationalized in terms of A–O and B–O interactions in the ionic crystal. These findings reveal the existence of general systematic trends in the formation energies of perovskites and provide further insight into the role of ion–ion interactions... 11. Cosmological General Relativity With Scale Factor and Dark Energy CERN Document Server Oliveira, Firmin J 2014-01-01 In this paper the four-dimensional space-velocity Cosmological General Relativity of Carmeli is developed by a general solution to the Einstein field equations. The metric is given in the Tolman form and the vacuum mass density is included in the energy-momentum tensor. The scale factor redshift equation is obtained, forming the basis for deriving the various redshift-distance relations of cosmological analysis. A linear equation of state dependent on the scale factor is assumed to account for the effects of an evolving dark energy in the expansion of the universe. Modeling simulations are provided for a few combinations of mass density, vacuum density and state parameter values over a sample of high redshift SNe Ia data. Also, the Carmeli cosmological model is derived as a special case of the general solution. 12. General-equilibrium approach to energy/environmental economic analysis Energy Technology Data Exchange (ETDEWEB) Groncki, P J 1978-08-01 This paper presents a brief critique of the use of fixed-coefficient input-output models for use in energy/environmental modeling systems, a shortcoming of input-output models that has been often been noted. Then, given the existence of aggregate, general-equilibrium, variable-coefficient growth models, a methodology is presented for using this information to adjust a recent disaggregated input-output table. This methodology takes into account all of the general-equilibrium aspects of the aggregate model in making the changes in the disaggregate model. The use of various weighting schemes and the implicit technological change biases they embody are examined. The methodology is being tested on historical tables for the United States, and preliminary results are discussed. This methodology's ability to fully capture the general-equilibrium nature of the economy should enhance the usefulness of input-output models in energy/environmental modeling systems. 13. A big bounce, slow-roll inflation and dark energy from conformal gravity CERN Document Server Gegenberg, Jack; Seahra, Sanjeev S 2016-01-01 We examine the cosmological sector of a gauge theory of gravity based on the SO(4,2) conformal group of Minkowski space. We allow for conventional matter coupled to the spacetime metric as well as matter coupled to the field that gauges special conformal transformations. An effective cosmological constant is generated dynamically via solution of the equations of motion, and this allows us to recover the late time acceleration of the universe. Furthermore, gravitational fields sourced by ordinary cosmological matter (i.e. dust and radiation) are significantly weakened in the very early universe, which has the effect of replacing the big bang with a big bounce. Finally, we find that this bounce is followed by a period of nearly-exponential slow roll inflation that can last long enough to explain the large scale homogeneity of the cosmic microwave background. 14. Extracting black-hole's rotational energy: the generalized Penrose process CERN Document Server Lasota, J -P; Abramowicz, M; Tchekhovskoy, A; Narayan, R 2014-01-01 In the case involving particles the necessary and sufficient condition for the Penrose process to extract energy from a rotating black hole is absorption of particles with negative energies and angular momenta. No torque at the black hole horizon occurs. In this article we consider the case of arbitrary fields or matter described by an unspecified, general energy-momentum tensor and show that the necessary and sufficient condition for extraction of black-hole's rotational energy is analogous to that in mechanical Penrose process: absorption of negative energy and negative angular momentum. We also show that a necessary condition for the Penrose process to occur is for the Noether current (the conserved energy-momentum density vector) to be spacelike or past-directed (timelike or null) on some part of the horizon. In the particle case our general criterion for the occurrence of a Penrose process reproduces the standard result. In the case of relativistic jet-producing "magnetically arrested disks" we show that... 15. High Energy Physics Signatures from Inflation and Conformal Symmetry of de Sitter CERN Document Server Kehagias, Alex 2015-01-01 During inflation, the geometry of spacetime is described by a (quasi-)de Sitter phase. Inflationary observables are determined by the underlying (softly broken) de Sitter isometry group SO(1, 4) which acts like a conformal group on R^3: when the fluctuations are on super-Hubble scales, the correlators of the scalar fields are constrained by conformal invariance. Heavy fields with mass m larger than the Hubble rate H correspond to operators with imaginary dimensions in the dual Euclidean three-dimensional conformal field theory. By making use of the dS/CFT correspondence we show that, besides the Boltzmann suppression expected from the thermal properties of de Sitter space, the generic effect of heavy fields in the inflationary correlators of the light fields is to introduce power-law suppressed corrections of the form O(H^2/m^2). This can be seen, for instance, at the level of the four-point correlator for which we provide the correction due to a massive scalar field exchange. 16. 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. 17. Energy-Dependent Fission Q Values Generalized for All Actinides Energy Technology Data Exchange (ETDEWEB) Vogt, R 2008-09-25 We generalize Madland's parameterization of the energy release in fission to obtain the dependence of the fission Q values on incident neutron energy, E{sub n}, for all major and minor actinides. These Q(E{sub n}) parameterizations are included in the ENDL2008 release. This paper describes calculations of energy-dependent fission Q values based on parameterizations of the prompt energy release in fission [1], developed by Madland [1] to describe the prompt energy release in neutron-induced fission of {sup 235}U, {sup 238}U, and {sup 239}Pu. The energy release is then related to the energy deposited during fission so that experimentally measurable quantities can be used to obtain the Q values. A discussion of these specific parameterizations and their implementation in the processing code for Monte Carlo neutron transport, MCFGEN, [2] is described in Ref. [3]. We extend this model to describe Q(E) for all actinides, major and minor, in the Evaluated Nuclear Data Library (ENDL) 2008 release, ENDL2008. 18. Generalized Energy-Dependent Q Values for Fission Energy Technology Data Exchange (ETDEWEB) Vogt, R 2010-03-31 We extend Madland's parameterization of the energy release in fission to obtain the dependence of the fission Q value for major and minor actinides on the incident neutron energies in the range 0 {le} E{sub n} {le} 20 MeV. Our parameterization is based on the actinide evaluations recommended for the ENDF/B-VII.1 release. This paper describes the calculation of energydependent fission Q values based on the calculation of the prompt energy release in fission by Madland. This calculation was adopted for use in the LLNL ENDL database and then generalized to obtain the prompt fission energy release for all actinides. Here the calculation is further generalized to the total energy release in fission. There are several stages in a fission event, depending on the time scale. Neutrons and gammas may be emitted at any time during the fission event.While our discussion here is focussed on compound nucleus creation by an incident neutron, similar parameterizations could be obtained for incident gammas or spontaneous fission. 19. Oscillatory Universe, dark energy equation of state and general relativity CERN Document Server Ghosh, Partha Pratim; Usmani, A A; Mukhopadhyay, Utpal 2012-01-01 The concept of oscillatory Universe appears to be realistic and buried in the dynamic dark energy equation of state. We explore its evolutionary history under the frame work of general relativity. We observe that oscillations do not go unnoticed with such an equation of state and that their effects persist later on in cosmic evolution. The classical' general relativity seems to retain the past history of oscillatory Universe in the form of increasing scale factor as the classical thermodynamics retains this history in the form of increasing cosmological entropy. 20. Disulphide trapping of an in vivo energy-dependent conformation of Escherichia coli TonB protein. Science.gov (United States) Ghosh, Joydeep; Postle, Kathleen 2005-01-01 In Escherichia coli, the TonB system transduces the protonmotive force (pmf) of the cytoplasmic membrane to support a variety of transport events across the outer membrane. Cytoplasmic membrane proteins ExbB and ExbD appear to harvest pmf and transduce it to TonB. Experimental evidence suggests that TonB shuttles to the outer membrane, apparently to deliver conformationally stored potential energy to outer membrane transporters. In the most recent model, discharged TonB is then recycled to the cytoplasmic membrane to be re-energized by the energy coupling proteins, ExbB/D. It has been suggested that the carboxy-terminal 75 amino acids of active TonB could be represented by the rigid, strand-exchanged, dimeric crystal structure of the corresponding fragment. In contrast, recent genetic studies of alanine substitutions have suggested instead that in vivo the carboxy-terminus of intact TonB is dynamic and flexible. The biochemical studies presented here confirm and extend those results by demonstrating that individual cys substitution at aromatic residues in one monomeric subunit can form spontaneous dimers in vivo with the identical residue in the other monomeric subunit. Two energized TonBs appear to form a single cluster of 8-10 aromatic amino acids, including those found at opposite ends of the crystal structure. The aromatic cluster requires both the amino-terminal energy coupling domain of TonB, and ExbB/D (and cross-talk analogues TolQ/R) for in vivo formation. The large aromatic cluster is detected in cytoplasmic membrane-, but not outer membrane-associated TonB. Consistent with those observations, the aromatic cluster can form in the first half of the energy transduction cycle, before release of conformationally stored potential energy to ligand-loaded outer membrane transporters. The model that emerges is one in which, after input of pmf mediated through ExbB/D and the TonB transmembrane domain, the TonB carboxy-terminus can form a meta-stable high-energy 1. Limits of the energy-momentum tensor in general relativity CERN Document Server Paiva, F M; Hall, G S; MacCallum, M A H; Paiva, Filipe M.; Reboucas, Marcelo J.; Hall, Graham S.; Callum, Malcolm A.H. Mac 1998-01-01 A limiting diagram for the Segre classification of the energy-momentum tensor is obtained and discussed in connection with a Penrose specialization diagram for the Segre types. A generalization of the coordinate-free approach to limits of Paiva et al. to include non-vacuum space-times is made. Geroch's work on limits of space-times is also extended. The same argument also justifies part of the procedure for classification of a given spacetime using Cartan scalars. 2. The Generalized Conversion Factor in Einstein's Mass-Energy Equation Directory of Open Access Journals (Sweden) Sharma A. 2008-07-01 Full Text Available Einstein’s September 1905 paper is origin of light energy-mass inter conversion equa- tion ( L = mc 2 and Einstein speculated E = mc 2 from it by simply replacing L by E . From its critical analysis it follows that L = mc 2 is only true under special or ideal conditions. Under general cases the result is L / mc 2 ( E / mc 2 . Conse- quently an alternate equation E = Ac 2 M has been suggested, which implies that energy emitted on annihilation of mass can be equal, less and more than predicted by E = mc 2 . The total kinetic energy of fission fragments of U 235 or Pu 239 is found experimentally 20–60 MeV less than Q -value predicted by mc 2 . The mass of parti- cle Ds (2317 discovered at SLAC, is more than current estimates. In many reactions including chemical reactions E = mc 2 is not confirmed yet, but regarded as true. It implies the conversion factor than c 2 is possible. These phenomena can be explained with help of generalized mass-energy equation E = Ac 2 M . 3. The Generalized Conversion Factor in Einstein's Mass-Energy Equation Directory of Open Access Journals (Sweden) Ajay Sharma 2008-07-01 Full Text Available Einstein's September 1905 paper is origin of light energy-mass inter conversion equation ($L = Delta mc^{2}$and Einstein speculated$E = Delta mc^{2}$from it by simply replacing$L$by$E$. From its critical analysis it follows that$L = Delta mc^{2}$is only true under special or ideal conditions. Under general cases the result is$L propto Delta mc^{2}$($E propto Delta mc^{2}$. Consequently an alternate equation$Delta E = A ub c^{2}Delta M$has been suggested, which implies that energy emitted on annihilation of mass can be equal, less and more than predicted by$Delta E = Delta mc^{2}$. The total kinetic energy of fission fragments of U-235 or Pu-239 is found experimentally 20-60 MeV less than Q-value predicted by$Delta mc^{2}$. The mass of particle Ds (2317 discovered at SLAC, is more than current estimates. In many reactions including chemical reactions$E = Delta mc^{2}$is not confirmed yet, but regarded as true. It implies the conversion factor than$c^{2}$is possible. These phenomena can be explained with help of generalized mass-energy equation$Delta E = A ub c^{2}Delta M$. 4. Generalized trends in the formation energies of perovskite oxides. Science.gov (United States) Zeng, ZhenHua; Calle-Vallejo, Federico; Mogensen, Mogens B; Rossmeisl, Jan 2013-05-28 Generalized trends in the formation energies of several families of perovskite oxides (ABO3) and plausible explanations to their existence are provided in this study through a combination of DFT calculations, solid-state physics analyses and simple physical/chemical descriptors. The studied elements at the A site of perovskites comprise rare-earth, alkaline-earth and alkaline metals, whereas 3d and 5d metals were studied at the B site. We also include ReO3-type compounds, which have the same crystal structure of cubic ABO3 perovskites except without A-site elements. From the observations we extract the following four conclusions for the perovskites studied in the present paper: for a given cation at the B site, (I) perovskites with cations of identical oxidation state at the A site possess close formation energies; and (II) perovskites with cations of different oxidation states at the A site usually have quite different but ordered formation energies. On the other hand, for a given A-site cation, (III) the formation energies of perovskites vary linearly with respect to the atomic number of the elements at the B site within the same period of the periodic table, and the slopes depend systematically on the oxidation state of the A-site cation; and (IV) the trends in formation energies of perovskites with elements from different periods at the B site depend on the oxidation state of A-site cations. Since the energetics of perovskites is shown to be the superposition of the individual contributions of their constituent oxides, the trends can be rationalized in terms of A-O and B-O interactions in the ionic crystal. These findings reveal the existence of general systematic trends in the formation energies of perovskites and provide further insight into the role of ion-ion interactions in the properties of ternary compounds. 5. GENERALIZED ENERGY CONSERVATION AND UNSTABLE PERTURBATION PROPERTY IN BAROTROPIC VORTEX Institute of Scientific and Technical Information of China (English) HUANG Hong; ZHANG Ming 2006-01-01 Based on a barotropic vortex model, generalized energy-conserving equation was derived and two necessary conditions of basic flow destabilization are gained. These conditions correspond to generalized barotropic instability and super speed instability. They are instabilities of vortex and gravity inertial wave respectively. In order to relate to practical situation, a barotropic vortex was analyzed, the basic flow of which is similar to lower level basic wind field of tropical cyclones and the maximum wind radius of which is 500 km.The results show that generalized barotropic instability depending upon the radial gradient of relative vorticity can appear in this vortex. It can be concluded that unstable vortex Rossby wave may appear in barotropic vortex. 6. A general end point free energy calculation method based on microscopic configurational space coarse-graining CERN Document Server Tian, Pu 2015-01-01 Free energy is arguably the most important thermodynamic property for physical systems. Despite the fact that free energy is a state function, presently available rigorous methodologies, such as those based on thermodynamic integration (TI) or non-equilibrium work (NEW) analysis, involve energetic calculations on path(s) connecting the starting and the end macrostates. Meanwhile, presently widely utilized approximate end-point free energy methods lack rigorous treatment of conformational variation within end macrostates, and are consequently not sufficiently reliable. Here we present an alternative and rigorous end point free energy calculation formulation based on microscopic configurational space coarse graining, where the configurational space of a high dimensional system is divided into a large number of sufficiently fine and uniform elements, which were termed conformers. It was found that change of free energy is essentially decided by change of the number of conformers, with an error term that accounts... 7. New potentials for conformal mechanics CERN Document Server Papadopoulos, G 2012-01-01 We show that V=\\alpha x^2+\\beta x^{-2} arises as a potential of 1-dimensional conformal theories. This class of conformal models includes the DFF model \\alpha=0 and the harmonic oscillator \\beta=0. The construction is based on a different embedding of the conformal symmetry group into the time re-parameterizations from that of the DFF model and its generalizations. Depending on the range of the couplings$\\alpha, \\beta$, these models can have a ground state and a well-defined energy spectrum, and exhibit either a$SL(2,\\bR)$or a SO(3) conformal symmetry. The latter group can also be embedded in Diff(S^1). We also present several generalizations of these models which include the Calogero models with harmonic oscillator couplings and non-linear models with suitable metric and potential couplings. In addition, we give the conditions on the couplings for a class of gaugetheories to admit a SL(2,\\bR) or SO(3) conformal symmetry. We present examples of such systems with general gauge groups and global symmetries t... 8. A generalized recipe to construct elementary or multi-step reaction paths via a stochastic formulation: Application to the conformational change in noble gas clusters Energy Technology Data Exchange (ETDEWEB) Talukder, Srijeeta; Sen, Shrabani [Department of Chemistry, University of Calcutta, 92 A P C Road, Kolkata 700 009 (India); Sharma, Rahul [Department of Chemistry, St. Xavier’s College, 30 Mother Teresa Sarani, Kolkata 700 016 (India); Banik, Suman K., E-mail: [email protected] [Department of Chemistry, Bose Institute, 93/1 A P C Road, Kolkata 700 009 (India); Chaudhury, Pinaki, E-mail: [email protected] [Department of Chemistry, University of Calcutta, 92 A P C Road, Kolkata 700 009 (India) 2014-03-18 Highlights: • We demonstrate a general strategy to map out reaction paths irrespective of the number of kinetic steps involved. • The objective function proposed does not need the information of gradient norm and eigenvalue of Hessian matrix explicitly. • A stochastic optimizer Simulated Annealing is used in searching reaction path. • The strategy is applied in mapping out the path for conformational changes in pure Ar clusters and Ar{sub N}Xe mixed clusters. - abstract: In this paper we demonstrate a general strategy to map out reaction paths irrespective of the number of kinetic steps required to bring about the change. i.e., whether the transformation takes place in a single step or in multiple steps with the appearance of intermediates. The objective function proposed is unique and works equally well for a concerted or a multiple step pathway. As the objective function proposed does not explicitly involves the calculation of the gradient of the potential energy function or the eigenvalues of the Hessian Matrix during the iterative process, the calculation is computationally economical. To map out the reaction path, we cast the entire problem as one of optimization and the solution is done with the use of the stochastic optimizer Simulated Annealing. The formalism is tested on Argon clusters (Ar{sub N}) and Argon clusters singly doped with Xenon (Ar{sub N-1}Xe). The size of the systems for which the method is applied ranges from N=7-25, where N is the total number of atoms in the cluster. We also test the results obtained by us by comparing with an established gradient only method. Moreover to demonstrate that our strategy can overcome the standard problems of drag method, we apply our strategy to a two dimensional LEPS + harmonic oscillator Potential to locate the TS, in which standard drag method has been seen to encounter problems. 9. Conformational characteristics of dimeric subunits of RNA from energy minimization studies. Mixed sugar-puckered ApG, ApU, CpG, and CpU. Science.gov (United States) Thiyagarajan, P; Ponnuswamy, P K 1981-09-01 Following the procedure described in the preceding article, the low energy conformations located for the four dimeric subunits of RNA, ApG, ApU, CpG, and CpU are presented. The A-RNA type and Watson-Crick type helical conformations and a number of different kinds of loop promoting ones were identified as low energy in all the units. The 3E-3E and 3E-2E pucker sequences are found to be more or less equally preferred; the 2E-2E sequence is occasionally preferred, while the 2E-3E is highly prohibited in all the units. A conformation similar to the one observed in the drug-dinucleoside monophosphate complex crystals becomes a low energy case only for the CpG unit. The low energy conformations obtained for the four model units were used to assess the stability of the conformational states of the dinucleotide segments in the four crystal models of the tRNAPhe molecule. Information on the occurrence of the less preferred sugar-pucker sequences in the various loop regions in the tRNAPhe molecule has been obtained. A detailed comparison of the conformational characteristics of DNA and RNA subunits at the dimeric level is presented on the basis of the results. 10. Generalized Ghost Dark Energy with Non-Linear Interaction CERN Document Server Ebrahimi, E; Mehrabi, A; Movahed, S M S 2016-01-01 In this paper we investigate ghost dark energy model in the presence of non-linear interaction between dark energy and dark matter. The functional form of dark energy density in the generalized ghost dark energy (GGDE) model is$\\rho_D\\equiv f(H, H^2)$with coefficient of$H^2$represented by$\\zeta$and the model contains three free parameters as$\\Omega_D, \\zeta$and$b^2$(the coupling coefficient of interactions). We propose three kinds of non-linear interaction terms and discuss the behavior of equation of state, deceleration and dark energy density parameters of the model. We also find the squared sound speed and search for signs of stability of the model. To compare the interacting GGDE model with observational data sets, we use more recent observational outcomes, namely SNIa, gamma-ray bursts, baryonic acoustic oscillation and the most relevant CMB parameters including, the position of acoustic peaks, shift parameters and redshift to recombination. For GGDE with the first non-linear interaction, the j... 11. Wang-Landau molecular dynamics technique to search for low-energy conformational space of proteins CERN Document Server Nagasima, Takehiro; Mitsui, Takashi; Nishikawa, Ken-Ichi 2007-01-01 Multicanonical molecular dynamics (MD) is a powerful technique for sampling conformations on rugged potential surfaces such as protein. However, it is notoriously difficult to estimate the multicanonical temperature effectively. Wang and Landau developed a convenient method for estimating the density of states based on a multicanonical Monte Carlo method. In their method, the density of states is calculated autonomously during a simulation. In this paper we develop a set of techniques to effectively apply the Wang-Landau method to MD simulations. In the multicanonical MD, the estimation of the derivative of the density of states is critical. In order to estimate it accurately, we devise two original improvements. First, the correction for the density of states is made smooth by using the Gaussian distribution obtained by a short canonical simulation. Second, an approximation is applied to the derivative, which is based on the Gaussian distribution and the multiple weighted histogram technique. A test of this ... 12. 18 CFR 153.21 - Conformity with requirements. Science.gov (United States) 2010-04-01 ... 18 Conservation of Power and Water Resources 1 2010-04-01 2010-04-01 false Conformity with requirements. 153.21 Section 153.21 Conservation of Power and Water Resources FEDERAL ENERGY REGULATORY... Requirements § 153.21 Conformity with requirements. (a) General Rule. Applications under subparts B and C... 13. Conformational Dynamics and Binding Free Energies of Inhibitors of BACE-1: From the Perspective of Protonation Equilibria. Directory of Open Access Journals (Sweden) M Olivia Kim 2015-10-01 Full Text Available BACE-1 is the β-secretase responsible for the initial amyloidogenesis in Alzheimer's disease, catalyzing hydrolytic cleavage of substrate in a pH-sensitive manner. The catalytic mechanism of BACE-1 requires water-mediated proton transfer from aspartyl dyad to the substrate, as well as structural flexibility in the flap region. Thus, the coupling of protonation and conformational equilibria is essential to a full in silico characterization of BACE-1. In this work, we perform constant pH replica exchange molecular dynamics simulations on both apo BACE-1 and five BACE-1-inhibitor complexes to examine the effect of pH on dynamics and inhibitor binding properties of BACE-1. In our simulations, we find that solution pH controls the conformational flexibility of apo BACE-1, whereas bound inhibitors largely limit the motions of the holo enzyme at all levels of pH. The microscopic pKa values of titratable residues in BACE-1 including its aspartyl dyad are computed and compared between apo and inhibitor-bound states. Changes in protonation between the apo and holo forms suggest a thermodynamic linkage between binding of inhibitors and protons localized at the dyad. Utilizing our recently developed computational protocol applying the binding polynomial formalism to the constant pH molecular dynamics (CpHMD framework, we are able to obtain the pH-dependent binding free energy profiles for various BACE-1-inhibitor complexes. Our results highlight the importance of correctly addressing the binding-induced protonation changes in protein-ligand systems where binding accompanies a net proton transfer. This work comprises the first application of our CpHMD-based free energy computational method to protein-ligand complexes and illustrates the value of CpHMD as an all-purpose tool for obtaining pH-dependent dynamics and binding free energies of biological systems. 14. Generalized holographic Ricci dark energy and generalized second law of thermodynamics in Bianchi Type I universe CERN Document Server Li, En-Kun; Geng, Jin-Ling; Duan, Peng-Fei 2016-01-01 Generalized second law of thermodynamics in the Bianchi type I universe with the generalized holographic Ricci dark energy model is studied in this paper. The behavior of dark energy's equation of state parameter indicates that it is matter-like in the early time of the universe but phantom-like in the future. By analysing the evolution of the deviations of state parameter and the total pressure of the universe, we find that for an anisotropic Bianchi type I universe, it transits from a high anisotropy stage to a more homogeneous stage in the near past. Using the normal entropy given by Gibbs' law of thermodynamics, it is proved that the generalized second law of thermodynamics does not always satisfied throughout the history of the universe when we assume the universe is enclosed by the generalized Ricci scalar radius$R_{gr}. It becomes invalid in the near past to the future, and the formation of the galaxies will be helpful in explaining such phenomenon, for that the galaxies's formation is an entropy inc... 15. Energy density and spatial curvature in general relativity Energy Technology Data Exchange (ETDEWEB) Frankel, T.; Galloway, G.J. 1981-04-01 Positive energy density tends to limit the size of space. This effect is studied within several contexts. We obtain sufficient conditions (which involve the energy density in a crucial way) for the compactness of spatial hypersurfaces in space-time. We then obtain some results concerning static or, more generally, stationary space-times. The Schwarzchild solution puts an upper bound on the size of a static spherically symmetric fluid with density bounded from below. We derive a result of roughly the same nature which, however, requires no symmetry and allows for rotation. Also, we show that static or rotating universes with L = 0 require that the density along some spatial geodesic must fall off rapidly with distance from a point. 16. Dissipative generalized Chaplygin gas as phantom dark energy Energy Technology Data Exchange (ETDEWEB) Cruz, Norman [Departamento de Fisica, Facultad de Ciencia, Universidad de Santiago, Casilla 307, Santiago (Chile)]. E-mail: [email protected]; Lepe, Samuel [Instituto de Fisica, Facultad de Ciencias Basicas y Matematicas, Pontificia Universidad Catolica de Valparaiso, Avenida Brasil 2950, Valparaiso (Chile)]. E-mail: [email protected]; Pena, Francisco [Departamento de Ciencias Fisicas, Facultad de Ingenieria, Ciencias y Administracion, Universidad de la Frontera, Avda. Francisco Salazar 01145, Casilla 54-D, Temuco (Chile)]. E-mail: [email protected] 2007-03-15 The generalized Chaplygin gas, characterized by the equation of state p=-A/{rho}{sup {alpha}}, has been considered as a model for dark energy due to its dark-energy-like evolution at late times. When dissipative processes are taken into account, within the framework of the standard Eckart theory of relativistic irreversible thermodynamics, cosmological analytical solutions are found. Using the truncated causal version of the Israel-Stewart formalism, a suitable model was constructed which crosses the w=-1 barrier. The future-singularities encountered in both approaches are of a new type, and not included in the classification presented by Nojiri and Odintsov [S. Nojiri, S.D. Odintsov, Phys. Rev. D 72 (2005) 023003]. 17. Energies and 2'-Hydroxyl Group Orientations of RNA Backbone Conformations. Benchmark CCSD(T)/CBS Database, Electronic Analysis, and Assessment of DFT Methods and MD Simulations. Science.gov (United States) Mládek, Arnošt; Banáš, Pavel; Jurečka, Petr; Otyepka, Michal; Zgarbová, Marie; Šponer, Jiří 2014-01-14 Sugar-phosphate backbone is an electronically complex molecular segment imparting RNA molecules high flexibility and architectonic heterogeneity necessary for their biological functions. The structural variability of RNA molecules is amplified by the presence of the 2'-hydroxyl group, capable of forming multitude of intra- and intermolecular interactions. Bioinformatics studies based on X-ray structure database revealed that RNA backbone samples at least 46 substates known as rotameric families. The present study provides a comprehensive analysis of RNA backbone conformational preferences and 2'-hydroxyl group orientations. First, we create a benchmark database of estimated CCSD(T)/CBS relative energies of all rotameric families and test performance of dispersion-corrected DFT-D3 methods and molecular mechanics in vacuum and in continuum solvent. The performance of the DFT-D3 methods is in general quite satisfactory. The B-LYP-D3 method provides the best trade-off between accuracy and computational demands. B3-LYP-D3 slightly outperforms the new PW6B95-D3 and MPW1B95-D3 and is the second most accurate density functional of the study. The best agreement with CCSD(T)/CBS is provided by DSD-B-LYP-D3 double-hybrid functional, although its large-scale applications may be limited by high computational costs. Molecular mechanics does not reproduce the fine energy differences between the RNA backbone substates. We also demonstrate that the differences in the magnitude of the hyperconjugation effect do not correlate with the energy ranking of the backbone conformations. Further, we investigated the 2'-hydroxyl group orientation preferences. For all families, we conducted a QM and MM hydroxyl group rigid scan in gas phase and solvent. We then carried out set of explicit solvent MD simulations of folded RNAs and analyze 2'-hydroxyl group orientations of different backbone families in MD. The solvent energy profiles determined primarily by the sugar pucker match well with the 18. Generalized average local ionization energy and its representations in terms of Dyson and energy orbitals. Science.gov (United States) Kohut, Sviataslau V; Cuevas-Saavedra, Rogelio; Staroverov, Viktor N 2016-08-21 Ryabinkin and Staroverov [J. Chem. Phys. 141, 084107 (2014)] extended the concept of average local ionization energy (ALIE) to correlated wavefunctions by defining the generalized ALIE as Ī(r)=-∑jλj\|fj(r)\|(2)/ρ(r), where λj are the eigenvalues of the generalized Fock operator and fj(r) are the corresponding eigenfunctions (energy orbitals). Here we show that one can equivalently express the generalized ALIE as Ī(r)=∑kIk\|dk(r)\|(2)/ρ(r), where Ik are single-electron removal energies and dk(r) are the corresponding Dyson orbitals. The two expressions for Ī(r) emphasize different physical interpretations of this quantity; their equivalence enables one to calculate the ALIE at any level of ab initio theory without generating the computationally expensive Dyson orbitals. 19. Towards understanding the free and receptor bound conformation of neuropeptide Y by fluorescence resonance energy transfer studies. Science.gov (United States) Haack, Michael; Beck-Sickinger, Annette G 2009-06-01 Despite a considerable sequence identity of the three mammalian hormones of the neuropeptide Y family, namely neuropeptide Y, peptide YY and pancreatic polypeptide, their structure in solution is described to be different. A so-called pancreatic polypeptide-fold has been identified for pancreatic polypeptide, whereas the structure of the N-terminal segment of neuropeptide Y is unknown. This element is important for the binding of neuropeptide Y to two of its relevant receptors, Y(1) and Y(5), but not to the Y(2) receptor subtype. In this study now, three doubly fluorescent-labeled analogs of neuropeptide Y have been synthesized that still bind to the Y(5) receptor with high affinity to investigate the conformation in solution and, for the first time, to probe the conformational changes upon binding of the ligand to its receptor in cell membrane preparations. The results obtained from the fluorescence resonance energy transfer investigations clearly show considerable differences in transfer efficiency that depend both on the solvent as well as on the peptide concentration. However, the studies do not support a pancreatic polypeptide-like folding of neuropeptide Y in the presence of membranes that express the human Y(5) receptor subtype. 20. Some invariant solutions for non-conformal perfect fluid plates in 5-flat form in general relativity Indian Academy of Sciences (India) Mukesh Kumar; Y K Gupta 2010-06-01 A set of six invariant solutions for non-conformal perfect fluid plates in 5-flat form is obtained using one-parametric Lie group of transformations. Out of the six solutions so obtained, three are in implicit form while the remaining three could be expressed explicitly. Each solution describes an accelerating fluid distribution and is new as far as authors are aware. 1. Gravitational Energy-Momentum and Conservation of Energy-Momentum in General Relativity Science.gov (United States) Wu, Zhao-Yan 2016-06-01 Based on a general variational principle, Einstein-Hilbert action and sound facts from geometry, it is shown that the long existing pseudotensor, non-localizability problem of gravitational energy-momentum is a result of mistaking different geometrical, physical objects as one and the same. It is also pointed out that in a curved spacetime, the sum vector of matter energy-momentum over a finite hyper-surface can not be defined. In curvilinear coordinate systems conservation of matter energy-momentum is not the continuity equations for its components. Conservation of matter energy-momentum is the vanishing of the covariant divergence of its density-flux tensor field. Introducing gravitational energy-momentum to save the law of conservation of energy-momentum is unnecessary and improper. After reasonably defining “change of a particle's energy-momentum”, we show that gravitational field does not exchange energy-momentum with particles. And it does not exchange energy-momentum with matter fields either. Therefore, the gravitational field does not carry energy-momentum, it is not a force field and gravity is not a natural force. 2. Conformational Transitions Science.gov (United States) Czerminski, Ryszard; Roitberg, Adrian; Choi, Chyung; Ulitsky, Alexander; Elber, Ron 1991-10-01 on the number of the direct paths between the minima. The influence on the distribution of the barriers and the minima energies is less significant. Calculation of reaction paths in large molecular systems requires new computational techniques. We employed our newly developed reaction path algorithm (SPW) for the study of the B to Z transition in DNA. The SPW (Self Penalty Walk) algorithm is explained in detail. A complex reaction coordinate (the B to Z transition in DNA) is calculated and analyzed. The calculated reaction path is for six basepairs DNA (including all 376 atoms). The path consists of 180° flips of two basepairs from the B DNA conformation to the Z DNA conformation. 3. Anisotropic conjugated polymer chain conformation tailors the energy migration in nanofibers CERN Document Server Camposeo, Andrea; Moffa, Maria; Fasano, Vito; Altamura, Davide; Giannini, Cinzia; Pisignano, Dario; Scholes, Gregory D 2016-01-01 Conjugated polymers are complex multi-chromophore systems, with emission properties strongly dependent on the electronic energy transfer through active sub-units. Although the packing of the conjugated chains in the solid state is known to be a key factor to tailor the electronic energy transfer and the resulting optical properties, most of the current solution-based processing methods do not allow for effectively controlling the molecular order, thus making the full unveiling of energy transfer mechanisms very complex. Here we report on conjugated polymer fibers with tailored internal molecular order, leading to a significant enhancement of the emission quantum yield. Steady state and femtosecond time-resolved polarized spectroscopies evidence that excitation is directed toward those chromophores oriented along the fiber axis, on a typical timescale of picoseconds. These aligned and more extended chromophores, resulting from the high stretching rate and electric field applied during the fiber spinning proces... 4. Rotational Spectroscopy of the Low Energy Conformer of 2-METHYLBUTYRONITRILE and Search for it Toward Sagittarius B2(N2) Science.gov (United States) Müller, Holger S. P.; Wehres, Nadine; Zingsheim, Oliver; Lewen, Frank; Schlemmer, Stephan; Grabow, Jens-Uwe; Garrod, Robin T.; Belloche, Arnaud; Menten, Karl M. 2017-06-01 Quite recently, some of us detected iso-propyl cyanide as the first branched alkyl molecule in space. The identification was made in an ALMA Cycle 0 and 1 molecular line survey of Sagittarius B2(N) at 3 mm. The branched isomer was only slightly less abundant than its straight-chain isomer with a ratio of about 2:5. While initial chemical models favored the branched isomer somewhat, more recent models are able to reproduce the observed ratio. Moreover, the models predicted that among the next longer butyl cyanides (BuCNs) 2-methylbutyronitrile (2-MBN) should be more abundant than both n-BuCN and 3-MBN by factors of around 2, with t-BuCN being almost negligible. With the rotational spectra of t- and n-BuCN studied, we investigated those of 2-MBN and 3-MBN betwen ˜40 and ˜400 GHz by conventional absorption spectroscopy and by chirped-pulse and resonator Fourier transform microwave (FTMW) spectroscopy. The analyses were guided by quantum-chemical calculations. A. Belloche, R. T. Garrod, H. S. P. Müller, and K. M. Menten, Science 345 1584. R. T. Garrod, A. Belloche, H. S. P. Müller, and K. M. Menten, Astron. Astrophys., in press; doi: 10.1051/0004-6361/201630254. With the rotational spectra of t- and n-BuCN studied, we investigated those of 2-MBN and 3-MBN betwen ˜40 and ˜400 GHz by conventional absorption spectroscopy and by chirped-pulse and resonator Fourier transform microwave (FTMW) spectroscopy. The analyses were guided by quantum-chemical calculations. Here we report the analysis of the low-energy conformer of 2-MBN and a search for it in our current ALMA data. Two additional conformers are higher by ˜250 and ˜280 cm^{-1}. The low-energy conformer displays a very rich rotational spectrum because of its great asymmetry (κ ≈ 0.14) and large a- and b-dipole moment components. Accurate ^{14}N quadrupole coupling parameters were obtained from the FTMW spectral recordings. 5. EL JUICIO DE APARIENCIA DE BUEN DERECHO FRENTE A LA IMPARCIALIDAD DEL JUEZ QUE DECRETA MEDIDAS CAUTELARES INNOMINADAS CONFORME EL CODIGO GENERAL DEL PROCESO EN COLOMBIA OpenAIRE Laguado Serrano, Cristian Eduardo; Vargas Buitrago, Jordan Aquiles 2015-01-01 La presente tesis denominada “El juicio de apariencia de buen derecho frente a la imparcialidad del juez que decreta medidas cautelares innominadas conforme el código general del proceso en Colombia”, plantea que a través la reforma del código de procedimiento civil buscando lograr un código general del proceso, al implementar las denominadas medidas cautelares innominadas se establecieron una serie de requisitos que a simple vista son claros y precisos; no obstante, al llevar ... 6. Predicting free energy contributions to the conformational stability of folded proteins from the residue sequence with radial basis function networks Energy Technology Data Exchange (ETDEWEB) Casadio, R.; Fariselli, P.; Vivarelli, F. [Univ. of Bologna (Italy); Compiani, M. [Univ. of Camerino (Italy) 1995-12-31 Radial basis function neural networks are trained on a data base comprising 38 globular proteins of well resolved crystallographic structure and the corresponding free energy contributions to the overall protein stability (as computed partially from crystallographic analysis and partially with multiple regression from experimental thermodynamic data by Ponnuswamy and Gromiha (1994)). Starting from the residue sequence and using as input code the percentage of each residue and the total residue number of the protein, it is found with a cross-validation method that neural networks can optimally predict the free energy contributions due to hydrogen bonds, hydrophobic interactions and the unfolded state. Terms due to electrostatic and disulfide bonding free energies are poorly predicted. This is so also when other input codes, including the percentage of secondary structure type of the protein and/or residue-pair information are used. Furthermore, trained on the computed and/or experimental {Delta}G values of the data base, neural networks predict a conformational stability ranging from about 10 to 20 kcal mol{sup -1} rather independently of the residue sequence, with an average error per protein of about 9 kcal mol{sup -1}. 7. Predicting free energy contributions to the conformational stability of folded proteins from the residue sequence with radial basis function networks. Science.gov (United States) Casadio, R; Compiani, M; Fariselli, P; Vivarelli, F 1995-01-01 Radial basis function neural networks are trained on a data base comprising 38 globular proteins of well resolved crystallographic structure and the corresponding free energy contributions to the overall protein stability (as computed partially from chrystallographic analysis and partially with multiple regression from experimental thermodynamic data by Ponnuswamy and Gromiha (1994)). Starting from the residue sequence and using as input code the percentage of each residue and the total residue number of the protein, it is found with a cross-validation method that neural networks can optimally predict the free energy contributions due to hydrogen bonds, hydrophobic interactions and the unfolded state. Terms due to electrostatic and disulfide bonding free energies are poorly predicted. This is so also when other input codes, including the percentage of secondary structure type of the protein and/or residue-pair information are used. Furthermore, trained on the computed and/or experimental delta G values of the data base, neural networks predict a conformational stability ranging from about 10 to 20 kcal mol-1 rather independently of the residue sequence, with an average error per protein of about 9 kcal mol-1. 8. Supergravitational conformal Galileons Science.gov (United States) Deen, Rehan; Ovrut, Burt 2017-08-01 The worldvolume actions of 3+1 dimensional bosonic branes embedded in a five-dimensional bulk space can lead to important effective field theories, such as the DBI conformal Galileons, and may, when the Null Energy Condition is violated, play an essential role in cosmological theories of the early universe. These include Galileon Genesis and "bouncing" cosmology, where a pre-Big Bang contracting phase bounces smoothly to the presently observed expanding universe. Perhaps the most natural arena for such branes to arise is within the context of superstring and M -theory vacua. Here, not only are branes required for the consistency of the theory, but, in many cases, the exact spectrum of particle physics occurs at low energy. However, such theories have the additional constraint that they must be N = 1 supersymmetric. This motivates us to compute the worldvolume actions of N = 1 supersymmetric three-branes, first in flat superspace and then to generalize them to N = 1 supergravitation. In this paper, for simplicity, we begin the process, not within the context of a superstring vacuum but, rather, for the conformal Galileons arising on a co-dimension one brane embedded in a maximally symmetric AdS 5 bulk space. We proceed to N = 1 supersymmetrize the associated worldvolume theory and then generalize the results to N = 1 supergravity, opening the door to possible new cosmological scenarios 9. Conformal piezoelectric energy harvesting and storage from motions of the heart, lung, and diaphragm. Science.gov (United States) Dagdeviren, Canan; Yang, Byung Duk; Su, Yewang; Tran, Phat L; Joe, Pauline; Anderson, Eric; Xia, Jing; Doraiswamy, Vijay; Dehdashti, Behrooz; Feng, Xue; Lu, Bingwei; Poston, Robert; Khalpey, Zain; Ghaffari, Roozbeh; Huang, Yonggang; Slepian, Marvin J; Rogers, John A 2014-02-01 Here, we report advanced materials and devices that enable high-efficiency mechanical-to-electrical energy conversion from the natural contractile and relaxation motions of the heart, lung, and diaphragm, demonstrated in several different animal models, each of which has organs with sizes that approach human scales. A cointegrated collection of such energy-harvesting elements with rectifiers and microbatteries provides an entire flexible system, capable of viable integration with the beating heart via medical sutures and operation with efficiencies of ∼2%. Additional experiments, computational models, and results in multilayer configurations capture the key behaviors, illuminate essential design aspects, and offer sufficient power outputs for operation of pacemakers, with or without battery assist. 10. Probing bulk defect energy bands using generalized charge pumping method Science.gov (United States) Masuduzzaman, Muhammad; Weir, Bonnie; Alam, Muhammad Ashraful 2012-04-01 The multifrequency charge pumping (CP) technique has long been used to probe the density of defects at the substrate-oxide interface, as well as in the bulk of the oxide of MOS transistors. However, profiling the energy levels of the defects has been more difficult due to the narrow scanning range of the voltage of a typical CP signal, and the uncertainty associated with the defect capture cross-section. In this paper, we discuss a generalized CP method that can identify defect energy bands within a bulk oxide, without requiring separate characterization of the defect capture cross-section. We use the new technique to characterize defects in both fresh and stressed samples of various dielectric compositions. By quantifying the way defects are generated as a function of time, we gain insight into the nature of defect generation in a particular gate dielectric. We also discuss the relative merits of voltage, time, and other variables of CP to probe bulk defect density, and compare the technique with related characterization approaches. 11. Laplacian-based generalized gradient approximations for the exchange energy CERN Document Server Cancio, Antonio C 2013-01-01 It is well known that in the gradient expansion approximation to density functional theory (DFT) the gradient and Laplacian of the density make interchangeable contributions to the exchange correlation (XC) energy. This is an arbitrary "gauge" freedom for building DFT models, normally used to eliminate the Laplacian from the generalized gradient approximation (GGA) level of DFT development. We explore the implications of keeping the Laplacian at this level of DFT, to develop a model that fits the known behavior of the XC hole, which can only be described as a system average in conventional GGA. We generate a family of exchange models that obey the same constraints as conventional GGA's, but which in addition have a finite-valued potential at the atomic nucleus unlike GGA's. These are tested against exact densities and exchange potentials for small atoms, and for constraints chosen to reproduce the SOGGA and the APBE variants of the GGA. The model reliably reproduces exchange energies of closed shell atoms, on... 12. Incidence angle modifiers. A general approach for energy calculations Energy Technology Data Exchange (ETDEWEB) Carvalho, Maria Joao; Horta, Pedro; Mendes, Joao Farinha [INETI - Inst. Nacional de Engenharia Tecnologia, Inovacao, IP, Lisboa (Portugal); Collares Pereira, Manuel; Carbajal, Wildor Maldonado [AO SOL, Energias Renovaveis, S.A., Samora Correia (Portugal) 2008-07-01 The calculation of the energy (power) delivered by a given solar collector, requires special care in the consideration of the way it handles the incoming solar radiation. Some collectors, e.g. flat plate types, are easy to characterize from an optical point of view, given their rotational symmetry with respect to the incident angle on the entrance aperture. This in contrast with collectors possessing a 2D (or cylindrical) symmetry, such as collectors using evacuated tubes or CPC collectors, requiring the incident radiation to be decomposed and treated in two orthogonal planes. Analyses of incidence angle modifier (IAM) along these lines were done in the past for parabolic through, evacuated tube (ETC) or compound parabolic concentrator (CPC) collectors. The present paper addresses a general approach to IAM calculation, treating in a general, equivalent and systematic way all collector types. This approach will allow the proper handling of the solar radiation available to each collector type, subdivided in its different components, folding that with the optical effects present in the solar collector and enabling more accurate comparisons between different collector types, in terms of long term performance calculation. (orig.) 13. Feasibility of using intermediate x-ray energies for highly conformal extracranial radiotherapy Energy Technology Data Exchange (ETDEWEB) Dong, Peng; Yu, Victoria; Nguyen, Dan; Demarco, John; Low, Daniel A.; Sheng, Ke, E-mail: [email protected] [Department of Radiation Oncology, University of California Los Angeles, California 90095 (United States); Woods, Kaley; Boucher, Salime [RadiaBeam Technologies, Santa Monica, California 90404 (United States) 2014-04-15 Purpose: To investigate the feasibility of using intermediate energy 2 MV x-rays for extracranial robotic intensity modulated radiation therapy. Methods: Two megavolts flattening filter free x-rays were simulated using the Monte Carlo code MCNP (v4c). A convolution/superposition dose calculation program was tuned to match the Monte Carlo calculation. The modeled 2 MV x-rays and actual 6 MV flattened x-rays from existing Varian Linacs were used in integrated beam orientation and fluence optimization for a head and neck, a liver, a lung, and a partial breast treatment. A column generation algorithm was used for the intensity modulation and beam orientation optimization. Identical optimization parameters were applied in three different planning modes for each site: 2, 6 MV, and dual energy 2/6 MV. Results: Excellent agreement was observed between the convolution/superposition and the Monte Carlo calculated percent depth dose profiles. For the patient plans, overall, the 2/6 MV x-ray plans had the best dosimetry followed by 2 MV only and 6 MV only plans. Between the two single energy plans, the PTV coverage was equivalent but 2 MV x-rays improved organs-at-risk sparing. For the head and neck case, the 2MV plan reduced lips, mandible, tongue, oral cavity, brain, larynx, left and right parotid gland mean doses by 14%, 8%, 4%, 14%, 24%, 6%, 30% and 16%, respectively. For the liver case, the 2 MV plan reduced the liver and body mean doses by 17% and 18%, respectively. For the lung case, lung V20, V10, and V5 were reduced by 13%, 25%, and 30%, respectively. V10 of heart with 2 MV plan was reduced by 59%. For the partial breast treatment, the 2 MV plan reduced the mean dose to the ipsilateral and contralateral lungs by 27% and 47%, respectively. The mean body dose was reduced by 16%. Conclusions: The authors showed the feasibility of using flattening filter free 2 MV x-rays for extracranial treatments as evidenced by equivalent or superior dosimetry compared to 6 MV plans 14. Near-IR laser generation of a high-energy conformer of L-alanine and the mechanism of its decay in a low-temperature nitrogen matrix. Science.gov (United States) Nunes, Cláudio M; Lapinski, Leszek; Fausto, Rui; Reva, Igor 2013-03-28 Monomers of L-alanine (ALA) were isolated in cryogenic nitrogen matrices at 14 K. Two conformers were identified for the compound trapped from the gas-phase into the solid nitrogen environment. The potential energy surface (PES) of ALA was theoretically calculated at the MP2 and QCISD levels. Twelve minima were located on this PES. Seven low-energy conformers fall within the 0-10 kJ mol(-1) range and should be appreciably populated in the equilibrium gas phase prior to deposition. Observation of only two forms in the matrices is explained in terms of calculated barriers to conformational rearrangements. All conformers with the O=C-O-H moiety in the cis orientation are separated by low barriers and collapse to the most stable form I during deposition of the matrix onto the low-temperature substrate. The second observed form II has the O=C-O-H group in the trans orientation. The remaining trans forms have very high relative energies (between 24 and 30 kJ mol(-1)) and are not populated. The high-energy trans form VI, that differs from I only by rotation of the OH group, was found to be separated from other conformers by barriers that are high enough to open a perspective for its stabilization in a matrix. The form VI was photoproduced in situ by narrow-band near-infrared irradiation of the samples at 6935-6910 cm(-1), where the first overtone of the OH stretching vibration in form I appears. The photogenerated form VI decays in N2 matrices back to conformer I with a characteristic decay time of ∼15 min. The mechanism of the VI → I relaxation is rationalized in terms of the proton tunneling. 15. Conformational unfolding in the N-terminal region of ribonuclease A detected by nonradiative energy transfer. Science.gov (United States) McWherter, C A; Haas, E; Leed, A R; Scheraga, H A 1986-04-22 Unfolding in the N-terminal region of RNase A was studied by the nonradiative energy-transfer technique. RNase A was labeled with a nonfluorescent acceptor (2,4-dinitrophenyl) on the alpha-amino group and a fluorescent donor (ethylenediamine monoamide of 2-naphthoxyacetic acid) on a carboxyl group in the vicinity of residue 50 (75% at Glu-49 and 25% at Asp-53). The distribution of donor labeling sites does not affect the results of this study since they are close in both the sequence and the three-dimensional structure. The sites of labeling were determined by peptide mapping. The derivatives possessed full enzymatic activity and underwent reversible thermal transitions. However, there were some quantitative differences in the thermodynamic parameters. When the carboxyl groups were masked, there was a 5 degrees C lowering of the melting temperature at pH 2 and 4, and no significant change in delta H(Tm). Labeling of the alpha-amino group had no effect on the melting temperature or delta H(Tm) at pH 2 but did result in a dramatic decrease in delta H(Tm) of the unfolding reaction at pH 4. The melting temperature did not change appreciably at pH 4, indicating that an enthalpy/entropy compensation had occurred. The efficiencies of energy transfer determined with both fluorescence intensity and lifetime measurements were in reasonably good agreement. The transfer efficiency dropped from about 60% under folding conditions to roughly 20% when the derivatives were unfolded with disulfide bonds intact and was further reduced to 5% when the disulfide bonds were reduced. The interprobe separation distance was estimated to be 35 +/- 2 A under folding conditions. The contribution to the interprobe distance resulting from the finite size of the probes was treated by using simple geometric considerations and a rotational isomeric state model of the donor probe linkage. With this model, the estimated average interprobe distance of 36 A is in excellent agreement with the 16. Conformal Nets II: Conformal Blocks Science.gov (United States) Bartels, Arthur; Douglas, Christopher L.; Henriques, André 2017-03-01 Conformal nets provide a mathematical formalism for conformal field theory. Associated to a conformal net with finite index, we give a construction of the bundle of conformal blocks', a representation of the mapping class groupoid of closed topological surfaces into the category of finite-dimensional projective Hilbert spaces. We also construct infinite-dimensional spaces of conformal blocks for topological surfaces with smooth boundary. We prove that the conformal blocks satisfy a factorization formula for gluing surfaces along circles, and an analogous formula for gluing surfaces along intervals. We use this interval factorization property to give a new proof of the modularity of the category of representations of a conformal net. 17. Conformal Nets II: Conformal Blocks Science.gov (United States) Bartels, Arthur; Douglas, Christopher L.; Henriques, André 2017-08-01 Conformal nets provide a mathematical formalism for conformal field theory. Associated to a conformal net with finite index, we give a construction of the bundle of conformal blocks', a representation of the mapping class groupoid of closed topological surfaces into the category of finite-dimensional projective Hilbert spaces. We also construct infinite-dimensional spaces of conformal blocks for topological surfaces with smooth boundary. We prove that the conformal blocks satisfy a factorization formula for gluing surfaces along circles, and an analogous formula for gluing surfaces along intervals. We use this interval factorization property to give a new proof of the modularity of the category of representations of a conformal net. 18. Energy loss of ions in a magnetized plasma: conformity between linear response and binary collision treatments. Science.gov (United States) Nersisyan, H B; Zwicknagel, G; Toepffer, C 2003-02-01 The energy loss of a heavy ion moving in a magnetized electron plasma is considered within the linear response (LR) and binary collision (BC) treatments with the purpose to look for a connection between these two models. These two complementary approaches yield close results if no magnetic field is present, but there develop discrepancies with growing magnetic field at ion velocities that are lower than, or comparable with, the thermal velocity of the electrons. We show that this is a peculiarity of the Coulomb interaction which requires cutoff procedures to account for its singularity at the origin and its infinite range. The cutoff procedures in the LR and BC treatments are different as the order of integrations in velocity and in ordinary (Fourier) spaces is reversed in both treatments. While BC involves a velocity average of Coulomb logarithms, there appear in LR Coulomb logarithms of velocity averaged cutoffs. The discrepancies between LR and BC vanish, except for small contributions of collective modes, for smoothened potentials that require no cutoffs. This is shown explicitly with the help of an improved BC in which the velocity transfer is treated up to second order in the interaction in Fourier space. 19. Exact solutions of the general equilibrium shape equations in a general power model of free energy for DNA structures Science.gov (United States) Yavari, Morteza 2014-02-01 The aim of this paper is to generalize the results of the Feoli's formalism (A. Feoli et al., Nucl. Phys. B 705, 577 (2005)) for DNA structures. The exact solutions of the general equilibrium shape equations for a general power model of free energy are investigated using the Feoli's formalism. The free energy of B- to Z-DNA transition is also calculated in this formalism. 20. On conformally related -waves Indian Academy of Sciences (India) Varsha Daftardar-Gejji 2001-05-01 Brinkmann [1] has shown that conformally related distinct Ricci flat solutions are -waves. Brinkmann's result has been generalized to include the conformally invariant source terms. It has been shown that [4] ifg_{ik}$and$\\overline{g}_{ik}$($=^{-2}g_{ik}$, : a scalar function), are distinct metrics having the same Einstein tensor,$G_{ik}=\\overline{G}_{ik}$, then both represent (generalized)$pp$-waves and$_{i}$is a null convariantly constant vector of$g_{ik}$. Thus$pp$-waves are the only candidates which yield conformally related nontrivial solutions of$G_{ik}=T_{ik}=\\overline{G}_{ik}$, with$T_{ik}$being conformally invariant source. In this paper the functional form of the conformal factor for the conformally related$pp$-waves/generalized$pp$-waves has been obtained. It has been shown that the most general$pp$-wave, conformally related to${\\rm d}s^{2}=-2{\\rm d}u[{\\rm d}v-m{\\rm d}y+H{\\rm d}u]+P^{-2}[{\\rm d}y^{2}+{\\rm d}z^{2}]$, turns out to the$(au+b)^{-2}{\\rm d}s^{2}$, where , are constants. Only in the special case when$m=0$,$H=1$, and$P=P(y,z)$, the conformal factor is$(au+b)^{-2}$or$(a(u+v)+b)^{-2}$. 1. Conformal Relativity: Theory and Observations CERN Document Server Pervushin, V; Zorin, A 2005-01-01 Theoretical and observational arguments are listed in favor of a new principle of relativity of units of measurements as the basis of a conformal-invariant unification of General Relativity and Standard Model by replacement of all masses with a scalar (dilaton) field. The relative units mean conformal observables: the coordinate distance, conformal time, running masses, and constant temperature. They reveal to us a motion of a universe along its hypersurface in the field space of events like a motion of a relativistic particle in the Minkowski space, where the postulate of the vacuum as a state with minimal energy leads to arrow of the geometric time. In relative units, the unified theory describes the Cold Universe Scenario, where the role of the conformal dark energy is played by a free minimal coupling scalar field in agreement with the most recent distance-redshift data from type Ia supernovae. In this Scenario, the evolution of the Universe begins with the effect of intensive creation of primordial W-Z-b... 2. A generalized model for estimating the energy density of invertebrates Science.gov (United States) James, Daniel A.; Csargo, Isak J.; Von Eschen, Aaron; Thul, Megan D.; Baker, James M.; Hayer, Cari-Ann; Howell, Jessica; Krause, Jacob; Letvin, Alex; Chipps, Steven R. 2012-01-01 Invertebrate energy density (ED) values are traditionally measured using bomb calorimetry. However, many researchers rely on a few published literature sources to obtain ED values because of time and sampling constraints on measuring ED with bomb calorimetry. Literature values often do not account for spatial or temporal variability associated with invertebrate ED. Thus, these values can be unreliable for use in models and other ecological applications. We evaluated the generality of the relationship between invertebrate ED and proportion of dry-to-wet mass (pDM). We then developed and tested a regression model to predict ED from pDM based on a taxonomically, spatially, and temporally diverse sample of invertebrates representing 28 orders in aquatic (freshwater, estuarine, and marine) and terrestrial (temperate and arid) habitats from 4 continents and 2 oceans. Samples included invertebrates collected in all seasons over the last 19 y. Evaluation of these data revealed a significant relationship between ED and pDM (r2 = 0.96, p calorimetry approaches. This model should prove useful for a wide range of ecological studies because it is unaffected by taxonomic, seasonal, or spatial variability. 3. DFT Conformation and Energies of Amylose Fragments at Atomic Resolution Part I: Syn Forms of Alpha-Maltotetraose Science.gov (United States) DFT optimization studies of ninety syn '-maltotetraose (DP-4) amylose fragments have been carried out at the B3LYP/6-311++G level of theory. The DP-4 fragments studied include V-helix, tightly bent conformations, a boat, and a 1C4 conformer. The standard hydroxymethyl rotamers (gg, gt, tg) were ... 4. DFT studies of the conformation and relative energies of alpha-maltotetraose (DP-4): An amylose fragment at atomic resolution Science.gov (United States) DFT optimization studies of more than one hundred conformations of a-maltotetraose have been carried out at the B3LYP/6-311++G level of theory. The DP-4 fragments of predominately 4C1 chair residues include tightly bent forms, helix, band-flips, kinks, boat, and some 1C4 conformers. The three do... 5. The Augmenting Effects of Desolvation and Conformational Energy Terms on the Predictions of Docking Programs against mPGES-1. Directory of Open Access Journals (Sweden) Ashish Gupta Full Text Available In this study we introduce a rescoring method to improve the accuracy of docking programs against mPGES-1. The rescoring method developed is a result of extensive computational study in which different scoring functions and molecular descriptors were combined to develop consensus and rescoring methods. 127 mPGES-1 inhibitors were collected from literature and were segregated into training and external test sets. Docking of the 27 training set compounds was carried out using default settings in AutoDock Vina, AutoDock, DOCK6 and GOLD programs. The programs showed low to moderate correlation with the experimental activities. In order to introduce the contributions of desolvation penalty and conformation energy of the inhibitors various molecular descriptors were calculated. Later, rescoring method was developed as empirical sum of normalised values of docking scores, LogP and Nrotb. The results clearly indicated that LogP and Nrotb recuperate the predictions of these docking programs. Further the efficiency of the rescoring method was validated using 100 test set compounds. The accurate prediction of binding affinities for analogues of the same compounds is a major challenge for many of the existing docking programs; in the present study the high correlation obtained for experimental and predicted pIC50 values for the test set compounds validates the efficiency of the scoring method. 6. The Augmenting Effects of Desolvation and Conformational Energy Terms on the Predictions of Docking Programs against mPGES-1 Science.gov (United States) Gupta, Ashish; Chaudhary, Neha; Kakularam, Kumar Reddy; Pallu, Reddanna; Polamarasetty, Aparoy 2015-01-01 In this study we introduce a rescoring method to improve the accuracy of docking programs against mPGES-1. The rescoring method developed is a result of extensive computational study in which different scoring functions and molecular descriptors were combined to develop consensus and rescoring methods. 127 mPGES-1 inhibitors were collected from literature and were segregated into training and external test sets. Docking of the 27 training set compounds was carried out using default settings in AutoDock Vina, AutoDock, DOCK6 and GOLD programs. The programs showed low to moderate correlation with the experimental activities. In order to introduce the contributions of desolvation penalty and conformation energy of the inhibitors various molecular descriptors were calculated. Later, rescoring method was developed as empirical sum of normalised values of docking scores, LogP and Nrotb. The results clearly indicated that LogP and Nrotb recuperate the predictions of these docking programs. Further the efficiency of the rescoring method was validated using 100 test set compounds. The accurate prediction of binding affinities for analogues of the same compounds is a major challenge for many of the existing docking programs; in the present study the high correlation obtained for experimental and predicted pIC50 values for the test set compounds validates the efficiency of the scoring method. PMID:26305898 7. Decipher the mechanisms of protein conformational changes induced by nucleotide binding through free-energy landscape analysis: ATP binding to Hsp70. Directory of Open Access Journals (Sweden) Adrien Nicolaï Full Text Available ATP regulates the function of many proteins in the cell by transducing its binding and hydrolysis energies into protein conformational changes by mechanisms which are challenging to identify at the atomic scale. Based on molecular dynamics (MD simulations, a method is proposed to analyze the structural changes induced by ATP binding to a protein by computing the effective free-energy landscape (FEL of a subset of its coordinates along its amino-acid sequence. The method is applied to characterize the mechanism by which the binding of ATP to the nucleotide-binding domain (NBD of Hsp70 propagates a signal to its substrate-binding domain (SBD. Unbiased MD simulations were performed for Hsp70-DnaK chaperone in nucleotide-free, ADP-bound and ATP-bound states. The simulations revealed that the SBD does not interact with the NBD for DnaK in its nucleotide-free and ADP-bound states whereas the docking of the SBD was found in the ATP-bound state. The docked state induced by ATP binding found in MD is an intermediate state between the initial nucleotide-free and final ATP-bound states of Hsp70. The analysis of the FEL projected along the amino-acid sequence permitted to identify a subset of 27 protein internal coordinates corresponding to a network of 91 key residues involved in the conformational change induced by ATP binding. Among the 91 residues, 26 are identified for the first time, whereas the others were shown relevant for the allosteric communication of Hsp70 s in several experiments and bioinformatics analysis. The FEL analysis revealed also the origin of the ATP-induced structural modifications of the SBD recently measured by Electron Paramagnetic Resonance. The pathway between the nucleotide-free and the intermediate state of DnaK was extracted by applying principal component analysis to the subset of internal coordinates describing the transition. The methodology proposed is general and could be applied to analyze allosteric communication in 8. Conformal house DEFF Research Database (Denmark) Ryttov, Thomas Aaby; Sannino, Francesco 2010-01-01 fixed point. As a consistency check we recover the previously investigated bounds of the conformal windows when restricting to a single matter representation. The earlier conformal windows can be imagined to be part now of the new conformal house. We predict the nonperturbative anomalous dimensions...... at the infrared fixed points. We further investigate the effects of adding mass terms to the condensates on the conformal house chiral dynamics and construct the simplest instanton induced effective Lagrangian terms... 9. Annual report 2005 General Direction of the Energy and raw materials; Rapport annuel 2005 Direction Generale de L'Energie et des Matieres Premieres Energy Technology Data Exchange (ETDEWEB) NONE 2005-07-01 This 2005 annual report of the DGEMP (General Direction of the Energy and the raw Materials), takes stock on the energy bill and accounting of the France. The first part presents the electric power, natural gas and raw materials market in France. The second part is devoted to the diversification of the energy resources with a special attention to the renewable energies and the nuclear energy. The third part discusses the energy and raw materials prices and the last part presents the international cooperation in the energy domain. (A.L.B.) 10. 77 FR 4203 - Energy Conservation Program: Test Procedures for General Service Fluorescent Lamps, General... Science.gov (United States) 2012-01-27 ... Fluorescent Lamps, General Service Incandescent Lamps, and Incandescent Reflector Lamps AGENCY: Office of... the test procedures for general service fluorescent lamps (GSFLs), general service incandescent lamps (GSILs), and incandescent reflector lamps (IRLs). That proposed rulemaking serves as the basis for... 11. Molecular Mechanism and Energy Basis of Conformational Diversity of Antibody SPE7 Revealed by Molecular Dynamics Simulation and Principal Component Analysis Science.gov (United States) Chen, Jianzhong; Wang, Jinan; Zhu, Weiliang 2016-11-01 More and more researchers are interested in and focused on how a limited repertoire of antibodies can bind and correspondingly protect against an almost limitless diversity of invading antigens. In this work, a series of 200-ns molecular dynamics (MD) simulations followed by principal component (PC) analysis and free energy calculations were performed to probe potential mechanism of conformational diversity of antibody SPE7. The results show that the motion direction of loops H3 and L3 is different relative to each other, implying that a big structural difference exists between these two loops. The calculated energy landscapes suggest that the changes in the backbone angles ψ and φ of H-Y101 and H-Y105 provide significant contributions to the conformational diversity of SPE7. The dihedral angle analyses based on MD trajectories show that the side-chain conformational changes of several key residues H-W33, H-Y105, L-Y34 and L-W93 around binding site of SPE7 play a key role in the conformational diversity of SPE7, which gives a reasonable explanation for potential mechanism of cross-reactivity of single antibody toward multiple antigens. 12. GM energy management: organization and results. [General Motors Energy Technology Data Exchange (ETDEWEB) DeKoker, N. 1975-01-01 Energy conservation is not new to industry. The effective and efficient use of labor and materials including energy has been an important tool in cost control for many years. Today, an even greater emphasis must be placed on conserving energy while at the same time stimulating the longer term development of existing as well as new energy resources. Energy conservation in manufacturing can involve product material and/or finish specification changes, process changes or elimination, improved control of equipment, the installation of heat recovery devices and the consolidation of operations. GM's organization to meet these energy management challenges and some of the specific measures taken to improve the efficiency of manufacturing operations are presented. 13. Conformational sampling techniques. Science.gov (United States) Hatfield, Marcus P D; Lovas, Sándor 2014-01-01 The potential energy hyper-surface of a protein relates the potential energy of the protein to its conformational space. This surface is useful in determining the native conformation of a protein or in examining a statistical-mechanical ensemble of structures (canonical ensemble). In determining the potential energy hyper-surface of a protein three aspects must be considered; reducing the degrees of freedom, a method to determine the energy of each conformation and a method to sample the conformational space. For reducing the degrees of freedom the choice of solvent, coarse graining, constraining degrees of freedom and periodic boundary conditions are discussed. The use of quantum mechanics versus molecular mechanics and the choice of force fields are also discussed, as well as the sampling of the conformational space through deterministic and heuristic approaches. Deterministic methods include knowledge-based statistical methods, rotamer libraries, homology modeling, the build-up method, self-consistent electrostatic field, deformation methods, tree-based elimination and eigenvector following routines. The heuristic methods include Monte Carlo chain growing, energy minimizations, metropolis monte carlo and molecular dynamics. In addition, various methods to enhance the conformational search including the deformation or smoothing of the surface, scaling of system parameters, and multi copy searching are also discussed. 14. A conformal model of gravitons CERN Document Server Donoghue, John F 2016-01-01 In the description of general covariance, the vierbein and the Lorentz connection can be treated as independent fundamental fields. With the usual gauge Lagrangian, the Lorentz connection is characterized by an asymptotically free running coupling. When running from high energy, the coupling gets large at a scale which can be called the Planck mass. If the Lorentz connection is confined at that scale, the low energy theory can have the Einstein Lagrangian induced at low energy through dimensional transmutation. However, in general there will be new divergences in such a theory and the Lagrangian basis should be expanded. I construct a conformally invariant model with a larger basis size which potentially may have the same property. 15. 76 FR 56661 - Energy Conservation Program: Test Procedures for General Service Fluorescent Lamps, General... Science.gov (United States) 2011-09-14 ... energy efficiency, energy use, or estimated annual operating cost of a covered product during a... test procedures and offer the public an opportunity to present oral and written comments on them. (42 U...''), 10 CFR 430.23 (Test procedures for the measurement of energy and water consumption''), 10 CFR... 16. Vacuum polarization energy for general backgrounds in one space dimension Science.gov (United States) Weigel, H. 2017-03-01 For field theories in one time and one space dimensions we propose an efficient method to compute the vacuum polarization energy of static field configurations that do not allow a decomposition into symmetric and anti-symmetric channels. The method also applies to scenarios in which the masses of the quantum fluctuations at positive and negative spatial infinity are different. As an example we compute the vacuum polarization energy of the kink soliton in the ϕ6 model. We link the dependence of this energy on the position of the soliton to the different masses. 17. Vertical Ionization Energies of ÃŽÂ±-L-Amino Acids as a Function of Their Conformation: an Ab Initio Study Directory of Open Access Journals (Sweden) Georges Dive 2004-11-01 Full Text Available Abstract: Vertical ionization energies (IE as a function of the conformation are determined at the quantum chemistry level for eighteen ÃŽÂ±-L-amino acids. Geometry optimization of the neutrals are performed within the Density Functional Theory (DFT framework using the hybrid method B3LYP and the 6-31G*(5d basis set. Few comparisons are made with wave-function-based ab initio correlated methods like MP2, QCISD or CCSD. For each amino acid, several conformations are considered that lie in the range 10-15 kJ/mol by reference to the more stable one. Their IE are calculated using the Outer-Valence-Green's-Functions (OVGF method at the neutrals' geometry. Few comparisons are made with MP2 and QCISD IE. It turns out that the OVGF results are satisfactory but an uncertainty relative to the most stable conformer at the B3LYP level persists. Moreover, the value of the IE can largely depend on the conformation due to the fact that the ionized molecular orbitals (MO can change a lot as a function of the nuclear structure. 18. Energy conservation: Policies, programs, and general studies. Citations from the NTIS data base Science.gov (United States) Hundemann, A. S. 1980-08-01 National policies, programs, and general studies or ways to conserve energy are presented. Topic areas cover such subjects as electric load management, effects of price and taxation on energy conservation, public attitudes and behavior toward energy saving, energy savings through reduction in hot water consumption, and telecommunications substitutability for travel. 19. General engineering ethics and multiple stress of atomic energy engineering Energy Technology Data Exchange (ETDEWEB) Takeda, Kunihiko [Shibaura Inst. of Tech., Tokyo (Japan) 1999-08-01 The factors, by which the modern engineering ethics has been profoundly affected, were classified to three categories, namely mental blow, the destruction of human function and environment damage. The role of atomic energy engineering in the ethic field has been shown in the first place. It is pointed out that it has brought about the mental blow by the elucidation of universal truth and discipline and the functional disorder by the power supply. However, the direct effect of radiation to the human kinds is only a part of the stresses comparing to the accumulation of the social stress which should be taken into account of by the possibility of disaster and the suspicion of the atomic energy politics. An increase in the multiple stresses as well as the restriction of criticism will place obstacles on the promotion of atomic energy. (author) 20. DGP cosmological model with generalized Ricci dark energy Energy Technology Data Exchange (ETDEWEB) Aguilera, Yeremy [Universidad de Santiago, Departamento de Matematicas y Ciencia de la Computacion, Santiago (Chile); Avelino, Arturo [Harvard-Smithsonian Center for Astrophysics, Cambridge, MA (United States); Cruz, Norman [Universidad de Santiago, Departamento de Fisica, Facultad de Ciencia, Santiago (Chile); Lepe, Samuel [Pontificia Universidad Catolica de Valparaiso, Facultad de Ciencias, Instituto de Fisica, Valparaiso (Chile); Pena, Francisco [Universidad de La Frontera, Departamento de Ciencias Fisicas, Facultad de Ingenieria y Ciencias, Temuco (Chile) 2014-11-15 The brane-world model proposed by Dvali, Gabadadze and Porrati (DGP) leads to an accelerated universe without cosmological constant or other form of dark energy for the positive branch (element of = +1). For the negative branch (element of = -1) we have investigated the behavior of a model with an holographic Ricci-like dark energy and dark matter, where the IR cutoff takes the form αH{sup 2} + βH, H being the Hubble parameter and α, β positive constants of the model. We perform an analytical study of the model in the late-time dark energy dominated epoch, where we obtain a solution for r{sub c}H(z), where r{sub c} is the leakage scale of gravity into the bulk, and conditions for the negative branch on the holographic parameters α and β, in order to hold the conditions of weak energy and accelerated universe. On the other hand, we compare the model versus the late-time cosmological data using the latest type Ia supernova sample of the Joint Light-curve Analysis (JLA), in order to constrain the holographic parameters in the negative branch, as well as r{sub c}H{sub 0} in the positive branch, where H{sub 0} is the Hubble constant. We find that the model has a good fit to the data and that the most likely values for (r{sub c}H{sub 0}, α, β) lie in the permitted region found from an analytical solution in a dark energy dominated universe. We give a justification to use a holographic cutoff in 4D for the dark energy in the 5-dimensional DGP model. Finally, using the Bayesian Information Criterion we find that this model is disfavored compared with the flat ΛCDM model. (orig.) 1. Stress-energy-momentum of affine-metric gravity generalized Komar superpotential CERN Document Server Giachetta, G 1995-01-01 In case of the Einstein's gravitation theory and its first order Palatini reformulation, the stress-energy-momentum of gravity has been proved to reduce to the Komar superpotential. We generalize this result to the affine-metric theory of gravity in case of general connections and arbitrary Lagrangian densities invariant under general covariant transformations. In this case, the stress-energy-momentum of gravity comes to the generalized Komar superpotential depending on a Lagrangian density in a precise way. 2. Quantum massive conformal gravity Energy Technology Data Exchange (ETDEWEB) Faria, F.F. [Universidade Estadual do Piaui, Centro de Ciencias da Natureza, Teresina, PI (Brazil) 2016-04-15 We first find the linear approximation of the second plus fourth order derivative massive conformal gravity action. Then we reduce the linearized action to separated second order derivative terms, which allows us to quantize the theory by using the standard first order canonical quantization method. It is shown that quantum massive conformal gravity is renormalizable but has ghost states. A possible decoupling of these ghost states at high energies is discussed. (orig.) 3. Workers’ Conformism Directory of Open Access Journals (Sweden) Nikolay Ivantchev 2013-10-01 Full Text Available Conformism was studied among 46 workers with different kinds of occupations by means of two modified scales measuring conformity by Santor, Messervey, and Kusumakar (2000 – scale for perceived peer pressure and scale for conformism in antisocial situations. The hypothesis of the study that workers’ conformism is expressed in a medium degree was confirmed partly. More than a half of the workers conform in a medium degree for taking risk, and for the use of alcohol and drugs, and for sexual relationships. More than a half of the respondents conform in a small degree for anti-social activities (like a theft. The workers were more inclined to conform for risk taking (10.9%, then – for the use of alcohol, drugs and for sexual relationships (8.7%, and in the lowest degree – for anti-social activities (6.5%. The workers who were inclined for the use of alcohol and drugs tended also to conform for anti-social activities. 4. A CONFORMATIONAL ELASTICITY THEORY Institute of Scientific and Technical Information of China (English) 1998-01-01 A new statistical theory based on the rotational isomeric state model describing the chain conformational free energy has been proposed. This theory can be used to predict different tensions of rubber elongation for chemically different polymers, and the energy term during the elongation of natural rubber coincides with the experimental one. 5. DNA – A General Energy System Simulation Tool DEFF Research Database (Denmark) Elmegaard, Brian; Houbak, Niels 2005-01-01 operation. The program decides at runtime to apply the DAE solver if the system contains differential equations. This makes it easy to extend an existing steady state model to simulate dynamic operation of the plant. The use of the program is illustrated by examples of gas turbine models. The paper also......The paper reviews the development of the energy system simulation tool DNA (Dynamic Network Analysis). DNA has been developed since 1989 to be able to handle models of any kind of energy system based on the control volume approach, usually systems of lumped parameter components. DNA has proven...... to be a useful tool in the analysis and optimization of several types of thermal systems: Steam turbines, gas turbines, fuels cells, gasification, refrigeration and heat pumps for both conventional fossil fuels and different types of biomass. DNA is applicable for models of both steady state and dynamic... 6. Energy, momentum, and center of mass in general relativity CERN Document Server Wang, Mu-Tao 2016-01-01 These notions in the title are of fundamental importance in any branch of physics. However, there have been great difficulties in finding physically acceptable definitions of them in general relativity since Einstein's time. I shall explain these difficulties and progresses that have been made. In particular, I shall introduce new definitions of center of mass and angular momentum at both the quasi-local and total levels, which are derived from first principles in general relativity and by the method of geometric analysis. With these new definitions, the classical formula p=mv is shown to be consistent with Einstein's field equation for the first time. This paper is based on joint work [14][15] with Po-Ning Chen and Shing-Tung Yau. 7. Conformationally averaged vertical detachment energy of finite size NO3(-)·nH2O clusters: a route connecting few to many. Science.gov (United States) Pathak, Arup Kumar; Samanta, Alok Kumar; Maity, Dilip Kumar 2011-04-07 We report conformationally averaged VDEs (VDE(w)(n)) for different sizes of NO(3)(-)·nH(2)O clusters calculated by using uncorrelated HF, correlated hybrid density functional (B3LYP, BHHLYP) and correlated ab intio (MP2 and CCSD(T)) theory. It is observed that the VDE(w)(n) at the B3LYP/6-311++G(d,p), B3LYP/Aug-cc-Pvtz and CCSD(T)/6-311++G(d,p) levels is very close to the experimentally measured VDE. It is shown that the use of calculated results of the conformationally averaged VDE for small-sized solvated negatively-charged clusters and a microscopic theory-based general expression for the same provides a route to obtain the VDE for a wide range of cluster sizes, including bulk. 8. High energy light scattering in the generalized eikonal approximation. Science.gov (United States) Chen, T W 1989-10-01 The generalized eikonal approximation method is applied to the study of light scattering by a dielectric medium. In this method, the propagation of light inside the medium is assumed to be rectilinear, as in the usual eikonal method, but with a parameterized propagator which is used to include the edge effect and ray optics behavior at the limit of very short wavelengths. The resulting formulas for the intensity and extinction efficiency factor are compared numerically and shown to agree excellently with the exact results for a homogeneous dielectric sphere. 9. A Generalization of Electromagnetic Fluctuation-Induced Casimir Energy Directory of Open Access Journals (Sweden) Yi Zheng 2015-01-01 Full Text Available Intermolecular forces responsible for adhesion and cohesion can be classified according to their origins; interactions between charges, ions, random dipole—random dipole (Keesom, random dipole—induced dipole (Debye are due to electrostatic effects; covalent bonding, London dispersion forces between fluctuating dipoles, and Lewis acid-base interactions are due to quantum mechanical effects; pressure and osmotic forces are of entropic origin. Of all these interactions, the London dispersion interaction is universal and exists between all types of atoms as well as macroscopic objects. The dispersion force between macroscopic objects is called Casimir/van der Waals force. It results from alteration of the quantum and thermal fluctuations of the electrodynamic field due to the presence of interfaces and plays a significant role in the interaction between macroscopic objects at micrometer and nanometer length scales. This paper discusses how fluctuational electrodynamics can be used to determine the Casimir energy/pressure between planar multilayer objects. Though it is confirmation of the famous work of Dzyaloshinskii, Lifshitz, and Pitaevskii (DLP, we have solved the problem without having to use methods from quantum field theory that DLP resorted to. Because of this new approach, we have been able to clarify the contributions of propagating and evanescent waves to Casimir energy/pressure in dissipative media. 10. A Generalized Framework for Energy Conservation in Wireless Sensor Network Directory of Open Access Journals (Sweden) V. S. Anita Sofia 2011-01-01 Full Text Available A Wireless Sensor Networks (WSN consists of spatially distributed autonomous sensors to cooperatively monitor physical or environmental conditions, such as temperature, sound, vibration, pressure, motion or pollutants. WSN contains a large number of nodes with a limited energy supply. A wireless sensor network consists of nodes that can communicate with each other via wireless links. Sensors are be remotely deployed in large numbers and operates autonomously in unattended environments. One way to support efficient communication between sensors is to organize the network into several groups, called clusters, with each cluster electing one node as the head of cluster To support scalability, nodes are often grouped into disjoint and mostly non-overlapping clusters. This paper deals about the frame work for energy conservation of a Wireless sensor network. The frame work is developed such a way that the nodes are to be clustered, electing the cluster head, performing intra cluster transmission and from the cluster head the information is transmitted to the base station. 11. 40 CFR 93.154 - Conformity analysis. Science.gov (United States) 2010-07-01 ... 40 Protection of Environment 20 2010-07-01 2010-07-01 false Conformity analysis. 93.154 Section 93...) DETERMINING CONFORMITY OF FEDERAL ACTIONS TO STATE OR FEDERAL IMPLEMENTATION PLANS Determining Conformity of General Federal Actions to State or Federal Implementation Plans § 93.154 Conformity analysis. Any... 12. Students' Misunderstandings about the Energy Conservation Principle: A General View to Studies in Literature Science.gov (United States) Tatar, Erdal; Oktay, Munir 2007-01-01 This paper serves to review previously reported studies on students' misunderstandings about the energy conservation principle (the first law of thermodynamics). Generally, studies in literature highlighted students' misunderstandings about the energy conservation principle stem from preliminaries about energy concept in daily life. Since prior… 13. Paul Scherrer Institut Scientific Report 2001. Volume V: General Energy Energy Technology Data Exchange (ETDEWEB) Wokaun, A.; Daum, C. (eds.) 2002-03-01 Major advances in 'Energy and Materials Cycles' have been achieved in the removal of heavy metals from the solid residues of municipal waste incineration. It has been conclusively shown that the oxidation/reduction conditions established during the thermal treatment of filter ash have a decisive influence on the evaporation of groups of heavy metals. With respect to biomass gasification, studies have been carried out with respect to the best way of extracting pure hydrogen from the low calorific value gas that is typically obtained from a biomass gasifier. The overarching goal of the laboratory 'High Temperature Solar Technology' is the use of solar energy for the production of solar fuels, or for the reduction of CO{sub 2} emissions in large scale industrial processes that are conventionally carried out with the use of fossil fuels. In a short-term project targeted at the solar production of lime, highly encouraging results (98% degree of calcination, adjustable reactivity of the lime) have been obtained in a 10 kW prototype reactor. Hybrid processes, in which the calorific value of fossil fuels is upgraded by solar energy, represent the medium-term strategy. In this context, the successful operation of the SYNMET reactor, in which zinc oxide is reacted with methane to produce zinc and synthesis gas, represents an important milestone. The physical sciences group has come up with a novel scheme in which sulfides, rather than oxides, are used as starting materials. Copper sulfide Cu{sub 2}S has been identified as a promising raw material, from which metallic copper would be produced in a solar reduction step. For the use of a catalytic combustor upstream of the main burning chamber of the gas turbine, it is crucial to know the stream wise distance over the catalyst where homogeneous ignition is initiated. The combustion-group working at this concept has made great advances in matching the observed ignition distances with theory. In addition, the 14. Paul Scherrer Institut Scientific Report 2001. Volume V: General Energy Energy Technology Data Exchange (ETDEWEB) Wokaun, A.; Daum, C. (eds.) 2002-03-01 Major advances in 'Energy and Materials Cycles' have been achieved in the removal of heavy metals from the solid residues of municipal waste incineration. It has been conclusively shown that the oxidation/reduction conditions established during the thermal treatment of filter ash have a decisive influence on the evaporation of groups of heavy metals. With respect to biomass gasification, studies have been carried out with respect to the best way of extracting pure hydrogen from the low calorific value gas that is typically obtained from a biomass gasifier. The overarching goal of the laboratory 'High Temperature Solar Technology' is the use of solar energy for the production of solar fuels, or for the reduction of CO{sub 2} emissions in large scale industrial processes that are conventionally carried out with the use of fossil fuels. In a short-term project targeted at the solar production of lime, highly encouraging results (98% degree of calcination, adjustable reactivity of the lime) have been obtained in a 10 kW prototype reactor. Hybrid processes, in which the calorific value of fossil fuels is upgraded by solar energy, represent the medium-term strategy. In this context, the successful operation of the SYNMET reactor, in which zinc oxide is reacted with methane to produce zinc and synthesis gas, represents an important milestone. The physical sciences group has come up with a novel scheme in which sulfides, rather than oxides, are used as starting materials. Copper sulfide Cu{sub 2}S has been identified as a promising raw material, from which metallic copper would be produced in a solar reduction step. For the use of a catalytic combustor upstream of the main burning chamber of the gas turbine, it is crucial to know the stream wise distance over the catalyst where homogeneous ignition is initiated. The combustion-group working at this concept has made great advances in matching the observed ignition distances with theory. In addition, the 15. Frequent Questions about General Conformity Science.gov (United States) These regulations ensure that federal activities or actions don't cause new violations to the NAAQS and ensure that NAAQS attainment is not delayed. This page has information about other agency representatives or stakeholders 16. 群体工作中用势艺术之探讨%Discussion on Art of Conforming to General Drift During Mass Sports Work Institute of Scientific and Technical Information of China (English) 2013-01-01 Based on the successful examples of conforming to general drift in Chinese history ,this article focus on the reality of jiangsu mass sports .From leadership and methodology perspective , the author discuss mass sports in nowadays ,in order to improve ability and level of mass sports worker to know things ,analyze this situation and solve problems .% 本文源于对江苏群体工作的现实关注，从领导学和方法论的视角出发，在探究中国历史中“用势艺术”成功范例的基础上，试图对当今群体工作作相应的梳理与探讨，以期提高广大群体工作者看待事物、分析情况、解决问题的能力和层次。 17. Charged Dilaton, Energy, Momentum and Angular-Momentum in Teleparallel Theory Equivalent to General Relativity CERN Document Server Nashed, Gamal Gergess Lamee 2008-01-01 We apply the energy-momentum tensor to calculate energy, momentum and angular-momentum of two different tetrad fields. This tensor is coordinate independent of the gravitational field established in the Hamiltonian structure of the teleparallel equivalent of general relativity (TEGR). The spacetime of these tetrad fields is the charged dilaton. Our results show that the energy associated with one of these tetrad fields is consistent, while the other one does not show this consistency. Therefore, we use the regularized expression of the gravitational energy-momentum tensor of the TEGR. We investigate the energy within the external event horizon using the definition of the gravitational energy-momentum. 18. Non-linear Realizations of Conformal Symmetry and Effective Field Theory for the Pseudo-Conformal Universe CERN Document Server Hinterbichler, Kurt; Khoury, Justin 2012-01-01 The pseudo-conformal scenario is an alternative to inflation in which the early universe is described by an approximate conformal field theory on flat, Minkowski space. Some fields acquire a time-dependent expectation value, which breaks the flat space so(4,2) conformal algebra to its so(4,1) de Sitter subalgebra. As a result, weight-0 fields acquire a scale invariant spectrum of perturbations. The scenario is very general, and its essential features are determined by the symmetry breaking pattern, irrespective of the details of the underlying microphysics. In this paper, we apply the well-known coset technique to derive the most general effective lagrangian describing the Goldstone field and matter fields, consistent with the assumed symmetries. The resulting action captures the low energy dynamics of any pseudo-conformal realization, including the U(1)-invariant quartic model and the Galilean Genesis scenario. We also derive this lagrangian using an alternative method of curvature invariants, consisting of ... 19. M(o)ller Energy Complexes of Monopoles and Textures in General Relativity and Teleparallel Gravity Institute of Scientific and Technical Information of China (English) Melis Aygün; Ihsan Yllmaz 2007-01-01 The energy problem of monopole and texture spacetimes is investigated in the context of two different approaches of gravity such as general relativity and teleparallel gravity.In this connection,firstly the energies for monopoles and textures are evaluated by using the Moiler energy-momentum prescription in different approximations.It is obtained that energy distributions of M?ller definition give the same results for these topological defects (monopole and texture)in general relativity(GR)and teleparallel gravity(TG).The results strengthen the importance of the M?ller energy-momentum definitions in giyen spacetimes and the viewpoint of Lessner that M(o)ller energy-momentum complex is a powerful concept for energy and momentum. 20. Nuclear matter symmetry energy from generalized polarizabilities: dependences on momentum, isospin, density and temperature CERN Document Server Braghin, F L 2004-01-01 Symmetry energy terms from macroscopic mass formulae are investigated as generalized polarizabilities of nuclear matter. Besides the neutron-proton (n-p) symmetry energy the spin dependent symmetry energies and a scalar one are also defined. They depend on the nuclear densities ($\\rho$), neutron-proton asymmetry ($b$), temperature ($T$) and exchanged energy and momentum ($q$). Based on a standard expression for the generalized polarizabilities, a differential equation is proposed to constrain the dependence of the symmetry energy on the n-p asymmetry and on the density. Some solutions are discussed. The q-dependence (zero frequence) of the symmetry energy coefficients with Skyrme-type forces is investigated in the four channels of the particle-hole interaction. Spin dependent symmetry energies are also investigated indicating much stronger differences in behavior with$qfor each Skyrme force than the results for the neutron-proton one. 1. Annual report 2001. General direction of energy and raw materials; Rapport annuel 2001. Direction generale de l'energie et des matieres premieres Energy Technology Data Exchange (ETDEWEB) NONE 2001-07-01 This report summarizes the 2001 activity of the French general direction of energy and raw materials (DGEMP) of the ministry of finances and industry: 1 - security of energy supplies: a recurrent problem; 2001, a transition year for nuclear energy worldwide; petroleum refining in font of the 2005 dead-line; the OPEC and the upset of the oil market; the pluri-annual planning of power production investments; renewable energies: a reconfirmed priority; 2 - the opening of markets: the opening of French electricity and gas markets; the international development of Electricite de France (EdF) and of Gaz de France (GdF); electricity and gas industries: first branch agreements; 3 - the present-day topics: 2001, the year of objective contracts; AREVA, the future to be prepared; the new IRSN; the agreements on climate and the energy policy; the mastery of domestic energy consumptions; the safety of hydroelectric dams; Technip-Coflexip: the birth of a para-petroleum industry giant; the cleansing of the mining activity in French Guyana; the future of workmen of Lorraine basin coal mines; 4 - 2001 at a glance: highlights; main legislative and regulatory texts; 5 - DGEMP: November 2001 reorganization and new organization chart; energy and raw materials publications; www.industrie.gouv.fr/energie. (J.S.) 2. Conformational analysis of six- and twelve-membered ring compounds by molecular dynamics DEFF Research Database (Denmark) Christensen, I T; Jørgensen, Flemming Steen 1997-01-01 A molecular dynamics (MD)-based conformational analysis has been performed on a number of cycloalkanes in order to demonstrate the reliability and generality of MD as a tool for conformational analysis. MD simulations on cyclohexane and a series of methyl-substituted cyclohexanes were performed...... provided 19 out of the 20 most stable conformations found in the MM2 force field. Finally, the general performance of the MD method for conformational analysis is discussed........ A series of methyl-substituted 1,3-dioxanes were investigated at 1000 K, and the number of chair-chair interconversions could be quantitatively correlated to the experimentally determined ring inversion barrier. Similarly, the distribution of sampled minimum-energy conformations correlated with the energy... 3. Office of Inspector General audit report on the U.S. Department of Energys aircraft activities Energy Technology Data Exchange (ETDEWEB) NONE 1999-01-01 On October 19, 1998, the Office of Inspector General (OIG) was asked to undertake a review of the Department of Energys aircraft activities. It was also requested that they report back within 90 days. The OIG has gathered information concerning the number of aircraft, the level of utilization, and the cost of the Departments aircraft operations. They have also briefly summarized four issues that, in their judgment, may require management attention. 4. Variations in energy spectra and water-to-material stopping-power ratios in three-dimensional conformal and intensity-modulated photon fields. Science.gov (United States) Jang, Si Young; Liu, H Helen; Mohan, Radhe; Siebers, Jeffrey V 2007-04-01 Because of complex dose distributions and dose gradients that are created in three-dimensional conformal radiotherapy (3D-CRT) and intensity-modulated radiation therapy (IMRT), photon- and electron-energy spectra might change significantly with spatial locations and doses. This study examined variations in photon- and electron-energy spectra in 3D-CRT and IMRT photon fields. The effects of spectral variations on water-to-material stopping-power ratios used in Monte Carlo treatment planning systems and the responses of energy-dependent dosimeters, such as thermoluminescent dosimeters (TLDs) and radiographic films were further studied. The EGSnrc Monte Carlo code was used to simulate megavoltage 3D-CRT and IMRT photon fields. The photon- and electron-energy spectra were calculated in 3D water phantoms and anthropomorphic phantoms based on the fluence scored in voxel grids. We then obtained the water-to-material stopping-power ratios in the local voxels using the Spencer-Attix cavity theory. Changes in the responses of films and TLDs were estimated based on the calculated local energy spectra and published data on the dosimeter energy dependency. Results showed that the photon-energy spectra strongly depended on spatial positions and doses in both the 3D-CRT and IMRT fields. The relative fraction of low-energy photons (stopping-power ratio over the range of calculated dose for both 3D-CRT and IMRT was negligible (< 1.0%) for ICRU tissue, cortical bone, and soft bone and less than 3.6% for dry air and lung. Because of spectral softening at low doses, radiographic films in the phantoms could over-respond to dose by more than 30%, whereas the over-response of TLDs was less than 10%. Thus, spatial variations of the photon- and electron-energy spectra should be considered as important factors in 3D-CRT and IMRT dosimetry. 5. Pontential energies and potential-energy tensors for subsystems: general properties CERN Document Server Caimmi, R 2016-01-01 With regard to generic two-component systems, the theory of first variations of global quantities is reviewed and explicit expressions are inferred for subsystem potential energies and potential-energy tensors. Performing a conceptual experiment, a physical interpretation of subsystem potential energies and potential-energy tensors is discussed. Subsystem tidal radii are defined by requiring an unbound component in absence of the other one. To this respect, a few guidance examples are presented as: (i) an embedding and an embedded homogeneous sphere; (ii) an embedding and an embedded truncated, singular isothermal sphere where related centres are sufficiently distant; (iii) a homogeneous sphere and a Roche system i.e. a mass point surrounded by a vanishing atmosphere. The results are discussed and compared with the findings of earlier investigations. 6. An Analysis of Metaphors Used by Students to Describe Energy in an Interdisciplinary General Science Course Science.gov (United States) Lancor, Rachael 2015-01-01 The meaning of the term energy varies widely in scientific and colloquial discourse. Teasing apart the different connotations of the term can be especially challenging for non-science majors. In this study, undergraduate students taking an interdisciplinary, general science course (n?=?49) were asked to explain the role of energy in five contexts:… 7. Wormhole geometries in fourth-order conformal Weyl gravity Science.gov (United States) Varieschi, Gabriele U.; Ault, Kellie L. 2016-04-01 We present an analysis of the classic wormhole geometries based on conformal Weyl gravity, rather than standard general relativity. The main characteristics of the resulting traversable wormholes remains the same as in the seminal study by Morris and Thorne, namely, that effective super-luminal motion is a viable consequence of the metric. Improving on previous work on the subject, we show that for particular choices of the shape and redshift functions the wormhole metric in the context of conformal gravity does not violate the main energy conditions at or near the wormhole throat. Some exotic matter might still be needed at the junction between our solutions and flat spacetime, but we demonstrate that the averaged null energy condition (as evaluated along radial null geodesics) is satisfied for a particular set of wormhole geometries. Therefore, if fourth-order conformal Weyl gravity is a correct extension of general relativity, traversable wormholes might become a realistic solution for interstellar travel. 8. Stress-energy tensor correlators in N-dimensional hot flat spaces via the generalized zeta-function method Science.gov (United States) Cho, H. T.; Hu, B. L. 2012-09-01 We calculate the expectation values of the stress-energy bitensor defined at two different spacetime points x, x‧ of a massless, minimally coupled scalar field with respect to a quantum state at finite temperature T in a flat N-dimensional spacetime by means of the generalized zeta-function method. These correlators, also known as the noise kernels, give the fluctuations of energy and momentum density of a quantum field which are essential for the investigation of the physical effects of negative energy density in certain spacetimes or quantum states. They also act as the sources of the Einstein-Langevin equations in stochastic gravity which one can solve for the dynamics of metric fluctuations as in spacetime foams. In terms of constitutions these correlators are one rung above (in the sense of the correlation—BBGKY or Schwinger-Dyson—hierarchies) the mean (vacuum and thermal expectation) values of the stress-energy tensor which drive the semiclassical Einstein equation in semiclassical gravity. The low- and the high-temperature expansions of these correlators are also given here: at low temperatures, the leading order temperature dependence goes like TN while at high temperatures they have a T2 dependence with the subleading terms exponentially suppressed by e-T. We also discuss the singular behavior of the correlators in the x‧ → x coincident limit as was done before for massless conformal quantum fields. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker’s 75th birthday devoted to ‘Applications of zeta functions and other spectral functions in mathematics and physics’. 9. Spacetime Conformal Fluctuations and Quantum Dephasing Science.gov (United States) Bonifacio, Paolo M. 2009-06-01 Any quantum system interacting with a complex environment undergoes decoherence. Empty space is filled with vacuum energy due to matter fields in their ground state and represents an underlying environment that any quantum particle has to cope with. In particular quantum gravity vacuum fluctuations should represent a universal source of decoherence. To study this problem we employ a stochastic approach that models spacetime fluctuations close to the Planck scale by means of a classical, randomly fluctuating metric (random gravity framework). We enrich the classical scheme for metric perturbations over a curved background by also including matter fields and metric conformal fluctuations. We show in general that a conformally modulated metric induces dephasing as a result of an effective nonlinear newtonian potential obtained in the appropriate nonrelativistic limit of a minimally coupled Klein-Gordon field. The special case of vacuum fluctuations is considered and a quantitative estimate of the expected effect deduced. Secondly we address the question of how conformal fluctuations could physically arise. By applying the random gravity framework we first show that standard GR seems to forbid spontaneous conformal metric modulations. Finally we argue that a different result follows within scalar-tensor theories of gravity such as e.g. Brans-Dicke theory. In this case a conformal modulation of the metric arises naturally as a result of the fluctuations in the Brans-Dicke field and quantum dephasing of a test particle is expected to occur. For large negative values of the coupling parameter the conformal fluctuations may also contribute to alleviate the well known problem of the large zero point energy due to quantum matter fields. 10. Gravitational radiation fields in teleparallel equivalent of general relativity and their energies Institute of Scientific and Technical Information of China (English) Gamal G.L. Nashed 2010-01-01 We derive two new retarded solutions in the teleparallel theory equivalent to general relativity (TEGR). One of these solutions gives a divergent energy. Therefore, we use the regularized expression of the gravitational energy-momentum tensor, which is a coordinate dependent. A detailed analysis of the loss of the mass of Bondi space-time is carried out using the flux of the gravitational energy-momentum. 11. Gravitational collapse with standard and dark energy in the teleparallel equivalent of general relativity Institute of Scientific and Technical Information of China (English) Gamal G.L.Nashed 2012-01-01 A perfect fluid with self-similarity of the second kind is studied within the framework of the teleparallel equivalent of general relativity (TEGR).A spacetime which is not asymptotically flat is derived.The energy conditions of this spacetime are studied.It is shown that after some time the strong energy condition is not enough to satisfy showing a transition from standard matter to dark energy.The singularities of this solution are discussed. 12. Molecular mechanics conformational analysis of tylosin Science.gov (United States) Ivanov, Petko M. 1998-01-01 The conformations of the 16-membered macrolide antibiotic tylosin were studied with molecular mechanics (AMBER∗ force field) including modelling of the effect of the solvent on the conformational preferences (GB/SA). A Monte Carlo conformational search procedure was used for finding the most probable low-energy conformations. The present study provides complementary data to recently reported analysis of the conformations of tylosin based on NMR techniques. A search for the low-energy conformations of protynolide, a 16-membered lactone containing the same aglycone as tylosin, was also carried out, and the results were compared with the observed conformation in the crystal as well as with the most probable conformations of the macrocyclic ring of tylosin. The dependence of the results on force field was also studied by utilizing the MM3 force field. Some particular conformations were computed with the semiempirical molecular orbital methods AM1 and PM3. 13. On Energy and Momentum of the Friedman and Some More General Universes CERN Document Server Garecki, Janusz 2016-01-01 Recently some authors concluded that the energy and momentum of the Fiedman universes, flat and closed, are equal to zero locally and globally (flat universes) or only globally (closed universes). The similar conclusion was also done for more general only homogeneous universes (Kasner and Bianchi type I). Such conclusions originated from coordinate dependent calculations performed only in comoving Cartesian coordinates by using the so-called {\\it energy-momentum complexes}. By using new coordinate independent expressions on energy and momentum one can show that the Friedman and more general universes {\\it needn't be energetic nonentity}. 14. Energy Savings Forecast of Solid-State Lighting in General Illumination Applications Energy Technology Data Exchange (ETDEWEB) none, 2014-08-29 With declining production costs and increasing technical capabilities, LED adoption has recently gained momentum in general illumination applications. This is a positive development for our energy infrastructure, as LEDs use significantly less electricity per lumen produced than many traditional lighting technologies. The U.S. Department of Energy’s Energy Savings Forecast of Solid-State Lighting in General Illumination Applications examines the expected market penetration and resulting energy savings of light-emitting diode, or LED, lamps and luminaires from today through 2030. 15. Ligand-induced conformational changes: Improved predictions of ligand binding conformations and affinities DEFF Research Database (Denmark) Frimurer, T.M.; Peters, Günther H.J.; Iversen, L.F. 2003-01-01 A computational docking strategy using multiple conformations of the target protein is discussed and evaluated. A series of low molecular weight, competitive, nonpeptide protein tyrosine phosphatase inhibitors are considered for which the x-ray crystallographic structures in complex with protein...... tyrosine phosphatase 1 B (PTP1B) are known. To obtain a quantitative measure of the impact of conformational changes induced by the inhibitors, these were docked to the active site region of various structures of PTP1B using the docking program FlexX. Firstly, the inhibitors were docked to a PTP1B crystal...... predicted binding energy and a correct docking mode. Thirdly, to improve the predictability of the docking procedure in the general case, where only a single target protein structure is known, we evaluate an approach which takes possible protein side-chain conformational changes into account. Here, side... 16. Free Energy and the Generalized Optimality Equations for Sequential Decision Making CERN Document Server Ortega, Pedro A 2012-01-01 The free energy functional has recently been proposed as a variational principle for bounded rational decision-making, since it instantiates a natural trade-off between utility gains and information processing costs that can be axiomatically derived. Here we apply the free energy principle to general decision trees that include both adversarial and stochastic environments. We derive generalized sequential optimality equations that not only include the Bellman optimality equations as a limit case, but also lead to well-known decision-rules such as Expectimax, Minimax and Expectiminimax. We show how these decision-rules can be derived from a single free energy principle that assigns a resource parameter to each node in the decision tree. These resource parameters express a concrete computational cost that can be measured as the amount of samples that are needed from the distribution that belongs to each node. The free energy principle therefore provides the normative basis for generalized optimality equations t... 17. Generalized Scaling of Urban Heat Island Effect and Its Applications for Energy Consumption and Renewable Energy Directory of Open Access Journals (Sweden) T.-W. Lee 2014-01-01 Full Text Available In previous work from this laboratory, it has been found that the urban heat island intensity (UHI can be scaled with the urban length scale and the wind speed, through the time-dependent energy balance. The heating of the urban surfaces during the daytime sets the initial temperature, and this overheating is dissipated during the night-time through mean convection motion over the urban surface. This may appear to be in contrast to the classical work by Oke (1973. However, in this work, we show that if the population density is used in converting the population data into urbanized area, then a good agreement with the current theory is found. An additional parameter is the “urban flow parameter,” which depends on the urban building characteristics and affects the horizontal convection of heat due to wind. This scaling can be used to estimate the UHI intensity in any cities and therefore predict the required energy consumption during summer months. In addition, all urbanized surfaces are expected to exhibit this scaling, so that increase in the surface temperature in large energy-consumption or energy-producing facilities (e.g., solar electric or thermal power plants can be estimated. 18. Conformal Coating of Three-Dimensional Nanostructures via Atomic Layer Deposition for Development of Advanced Energy Storage Devices and Plasmonic Transparent Conductors Science.gov (United States) Malek, Gary A. Due to the prodigious amount of electrical energy consumed throughout the world, there exists a great demand for new and improved methods of generating electrical energy in a clean and renewable manner as well as finding more effective ways to store it. This enormous task is of great interest to scientists and engineers, and much headway is being made by utilizing three-dimensional (3D) nanostructured materials. This work explores the application of two types of 3D nanostructured materials toward fabrication of advanced electrical energy storage and conversion devices. The first nanostructured material consists of vertically aligned carbon nanofibers. This three-dimensional structure is opaque, electrically conducting, and contains active sites along the outside of each fiber that are conducive to chemical reactions. Therefore, they make the perfect 3D conducting nanostructured substrate for advanced energy storage devices. In this work, the details for transforming vertically aligned carbon nanofiber arrays into core-shell structures via atomic layer deposition as well as into a mesoporous manganese oxide coated supercapacitor electrode are given. Another unique type of three-dimensional nanostructured substrate is nanotextured glass, which is transparent but non-conducting. Therefore, it can be converted to a 3D transparent conductor for possible application in photovoltaics if it can be conformally coated with a conducting material. This work details that transformation as well as the addition of plasmonic gold nanoparticles to complete the transition to a 3D plasmonic transparent conductor. 19. Spherically symmetric conformal gravity and "gravitational bubbles" CERN Document Server Berezin, V A; Eroshenko, Yu N 2016-01-01 The general structure of the spherically symmetric solutions in the Weyl conformal gravity is described. The corresponding Bach equation are derived for the special type of metrics, which can be considered as the representative of the general class. The complete set of the pure vacuum solutions is found. It consists of two classes. The first one contains the solutions with constant two-dimensional curvature scalar of our specific metrics, and the representatives are the famous Robertson-Walker metrics. One of them we called the "gravitational bubbles", which is compact and with zero Weyl tensor. The second class is more general, with varying curvature scalar. We found its representative as the one-parameter family. It appears that it can be conformally covered by the thee-parameter Mannheim-Kazanas solution. We also investigated the general structure of the energy-momentum tensor in the spherical conformal gravity and constructed the vectorial equation that reveals clearly the same features of non-vacuum solu... 20. Generalization of radiative jet energy loss to non-zero magnetic mass Energy Technology Data Exchange (ETDEWEB) Djordjevic, Magdalena, E-mail: [email protected] [Institute of Physics Belgrade, University of Belgrade (Serbia); Djordjevic, Marko [Faculty of Biology, University of Belgrade (Serbia) 2012-03-19 Reliable predictions for jet quenching in ultra-relativistic heavy ion collisions require accurate computation of radiative energy loss. While all available energy loss formalisms assume zero magnetic mass - in accordance with the one-loop perturbative calculations - different non-perturbative approaches report a non-zero magnetic mass at RHIC and LHC. We here generalize a recently developed energy loss formalism in a realistic finite size QCD medium, to consistently include a possibility for existence of non-zero magnetic screening. We also present how the inclusion of finite magnetic mass changes the energy loss results. Our analysis suggests a fundamental constraint on magnetic to electric mass ratio. 1. Recursion Relations for Conformal Blocks CERN Document Server Penedones, João; Yamazaki, Masahito 2016-09-12 In the context of conformal field theories in general space-time dimension, we find all the possible singularities of the conformal blocks as functions of the scaling dimension\\Delta$of the exchanged operator. In particular, we argue, using representation theory of parabolic Verma modules, that in odd spacetime dimension the singularities are only simple poles. We discuss how to use this information to write recursion relations that determine the conformal blocks. We first recover the recursion relation introduced in 1307.6856 for conformal blocks of external scalar operators. We then generalize this recursion relation for the conformal blocks associated to the four point function of three scalar and one vector operator. Finally we specialize to the case in which the vector operator is a conserved current. 2. NEW PRINCIPLES OF POWER AND ENERGY RATE OF INCREMENTAL RATE TYPE FOR GENERALIZED CONTINUUM FIELD THEORIES Institute of Scientific and Technical Information of China (English) 戴天民 2001-01-01 The aim of this paper is to establish new principles of power and energy rate of incremental type in generalized continuum mechanics. By combining new principles of virtual velocity and virtual angular velocity as well as of virtual stress and virtual couple stress with cross terms of incremental rate type a new principle of power and energy rate of incremental rate type with cross terms for micropolar continuum field theories is presented and from it all corresponding equations of motion and boundary conditions as well as power and energy rate equations of incremental rate type for micropolar and nonlocal micropolar continua with the help of generalized Piola's theorems in all and without any additional requirement are derived. Complete results for micromorphic continua could be similarly derived. The derived results in the present paper are believed to be new. They could be used to establish corresponding finite element methods of incremental rate type for generalized continuum mechanics. 3. Generalized energy balance and reciprocity relations for thin-film optics Energy Technology Data Exchange (ETDEWEB) Dupertuis, M.A.; Proctor, M. [Institut de Micro- et Optoelectronique, Lausanne (Switzerland); Acklin, B. [AT& T Bell Labs., Holmdel, NJ (United States) 1994-03-01 Energy balance and reciprocity relations are studied for harmonic inhomogeneous plane waves that are incident upon a stack of continuous absorbing dielectric media that are macroscopically characterized by their electric and magnetic permittivities and their conductivities. New cross terms between parallel electric and parallel magnetic modes are identified in the fully generalized Poynting vector. The symmetry and the relations between the general Fresnel coefficients are investigated in the context of energy balance at the interface. The contributions of the so-called mixed Poynting vector are discussed in detail. In particular a new transfer matrix is introduced for energy fluxes in thin-film optics based on the Poynting and mixed Poynting vectors. Finally, the study of reciprocity relations leads to a generalization of a theorem of reversibility for conducting and dielectric media. 16 refs. 4. Energy-Momentum of the Friedmann Models in General Relativity and Teleparallel Theory of Gravity CERN Document Server Sharif, M 2008-01-01 This paper is devoted to the evaluation of the energy-momentum density components for the Friedmann models. For this purpose, we have used M${\\o}$ller's pseudotensor prescription in General Relativity and a certain energy-momentum density developed from his teleparallel formulation. It is shown that the energy density of the closed Friedmann universe vanishes on the spherical shell at the radius$\\rho=2\\sqrt{3}$. This coincides with the earlier results available in the literature. We also discuss the energy of the flat and open models. A comparison shows a partial consistency between the M${\\o}ller's pseudotensor for General Relativity and teleparallel theory. Further, it is shown that the results are independent of the free dimensionless coupling constant of the teleparallel gravity. 5. General Relativistic Energy Conditions The Hubble expansion in the epoch of galaxy formation CERN Document Server Visser, M 1997-01-01 The energy conditions of Einstein gravity (classical general relativity) are designed to extract as much information as possible from classical general relativity without enforcing a particular equation of state for the stress-energy. This systematic avoidance of the need to specify a particular equation of state is particularly useful in a cosmological setting --- since the equation of state for the cosmological fluid in a Friedmann-Robertson-Walker type universe is extremely uncertain. I shall show that the energy conditions provide simple and robust bounds on the behaviour of both the density and look-back time as a function of red-shift. I shall show that current observations suggest that the so-called strong energy condition (SEC) is violated sometime between the epoch of galaxy formation and the present. This implies that no possible combination of normal'' matter is capable of fitting the observational data. 6. Fake Conformal Symmetry in Unimodular Gravity CERN Document Server Oda, Ichiro 2016-01-01 We study Weyl symmetry (local conformal symmetry) in unimodular gravity. It is shown that the Noether currents for both Weyl symmetry and global scale symmetry, identically vanish as in the conformally invariant scalar-tensor gravity. We clearly explain why in the class of conformally invariant gravitational theories, the Noether currents vanish by starting with the conformally invariant scalar-tensor gravity. Moreover, we comment on both classical and quantum-mechanical equivalences among Einstein's general relativity, the conformally invariant scalar-tensor gravity and the Weyl-transverse (WTDiff) gravity. Finally, we discuss the Weyl current in the conformally invariant scalar action and see that it is also vanishing. 7. Fake conformal symmetry in unimodular gravity Science.gov (United States) Oda, Ichiro 2016-08-01 We study Weyl symmetry (local conformal symmetry) in unimodular gravity. It is shown that the Noether currents for both Weyl symmetry and global scale symmetry vanish exactly as in conformally invariant scalar-tensor gravity. We clearly explain why in the class of conformally invariant gravitational theories, the Noether currents vanish by starting with conformally invariant scalar-tensor gravity. Moreover, we comment on both classical and quantum-mechanical equivalences in Einstein's general relativity, conformally invariant scalar-tensor gravity, and the Weyl-transverse gravity. Finally, we discuss the Weyl current in the conformally invariant scalar action and see that it is also vanishing. 8. Proving the Achronal Averaged Null Energy Condition from the Generalized Second Law CERN Document Server Wall, Aron C 2009-01-01 A null line is a complete achronal null geodesic. It is proven that for any quantum fields minimally coupled to semiclassical Einstein gravity, the averaged null energy condition (ANEC) on null lines is a consequence of the generalized second law of thermodynamics for causal horizons. Auxiliary assumptions include CPT and the existence of a suitable renormalization scheme for the generalized entropy. Although the ANEC can be violated on general geodesics in curved spacetimes, as long as the ANEC holds on null lines there exist theorems showing that semiclassical gravity should satisfy positivity of energy, topological censorship, and should not admit closed timelike curves. It is pointed out that these theorems fail once the linearized graviton field is quantized, because then the renormalized shear squared term in the Raychaudhuri equation can be negative. A "shear-inclusive" generalization of the ANEC is proposed to remedy this, and is proven under an additional assumption about perturbations to horizons in... 9. CONTINUOUS-ENERGY MONTE CARLO METHODS FOR CALCULATING GENERALIZED RESPONSE SENSITIVITIES USING TSUNAMI-3D Energy Technology Data Exchange (ETDEWEB) Perfetti, Christopher M [ORNL; Rearden, Bradley T [ORNL 2014-01-01 This work introduces a new approach for calculating sensitivity coefficients for generalized neutronic responses to nuclear data uncertainties using continuous-energy Monte Carlo methods. The approach presented in this paper, known as the GEAR-MC method, allows for the calculation of generalized sensitivity coefficients for multiple responses in a single Monte Carlo calculation with no nuclear data perturbations or knowledge of nuclear covariance data. The theory behind the GEAR-MC method is presented here, and proof of principle is demonstrated by using the GEAR-MC method to calculate sensitivity coefficients for responses in several 3D, continuous-energy Monte Carlo applications. 10. Energy Savings Forecast of Solid-State Lighting in General Illumination Applications Energy Technology Data Exchange (ETDEWEB) Penning, Julie [Navigant Consulting Inc., Washington, DC (United States); Stober, Kelsey [Navigant Consulting Inc., Washington, DC (United States); Taylor, Victor [Navigant Consulting Inc., Washington, DC (United States); Yamada, Mary [Navigant Consulting Inc., Washington, DC (United States) 2016-09-01 The DOE report, Energy Savings Forecast of Solid-State Lighting in General Illumination Applications, is a biannual report which models the adoption of LEDs in the U.S. general-lighting market, along with associated energy savings, based on the full potential DOE has determined to be technically feasible over time. This version of the report uses an updated 2016 U.S. lighting-market model that is more finely calibrated and granular than previous models, and extends the forecast period to 2035 from the 2030 limit that was used in previous editions. 11. General Navier–Stokes-like momentum and mass-energy equations Energy Technology Data Exchange (ETDEWEB) Monreal, Jorge, E-mail: [email protected] 2015-03-15 A new system of general Navier–Stokes-like equations is proposed to model electromagnetic flow utilizing analogues of hydrodynamic conservation equations. Such equations are intended to provide a different perspective and, potentially, a better understanding of electromagnetic mass, energy and momentum behaviour. Under such a new framework additional insights into electromagnetism could be gained. To that end, we propose a system of momentum and mass-energy conservation equations coupled through both momentum density and velocity vectors. 12. 40 CFR 51.854 - Conformity analysis. Science.gov (United States) 2010-07-01 ... 40 Protection of Environment 2 2010-07-01 2010-07-01 false Conformity analysis. 51.854 Section 51... FOR PREPARATION, ADOPTION, AND SUBMITTAL OF IMPLEMENTATION PLANS Determining Conformity of General Federal Actions to State or Federal Implementation Plans § 51.854 Conformity analysis. Link to... 13. Hybridization of General Cargo Ships to meet the Required Energy Efficiency Design Index OpenAIRE Øverleir, Magnus Anders 2015-01-01 In this thesis a hybrid propulsion system is proposed for a general cargo ship with the aim to meet the required Energy Efficiency Design Index (EEDI). The study has investigated how a hybrid propulsion system will influence the ship s EEDI value and fuel economy. The central problem is the coming challenge for the general cargo segment meeting the required efficiency value. Especially small vessels (3 000-15 000 DWT) with high speed will have troubles complying with the stricter regulations.... 14. A general theory of evolution based on energy efficiency: its implications for diseases. Science.gov (United States) Yun, Anthony J; Lee, Patrick Y; Doux, John D; Conley, Buford R 2006-01-01 We propose a general theory of evolution based on energy efficiency. Life represents an emergent property of energy. The earth receives energy from cosmic sources such as the sun. Biologic life can be characterized by the conversion of available energy into complex systems. Direct energy converters such as photosynthetic microorganisms and plants transform light energy into high-energy phosphate bonds that fuel biochemical work. Indirect converters such as herbivores and carnivores predominantly feed off the food chain supplied by these direct converters. Improving energy efficiency confers competitive advantage in the contest among organisms for energy. We introduce a term, return on energy (ROE), as a measure of energy efficiency. We define ROE as a ratio of the amount of energy acquired by a system to the amount of energy consumed to generate that gain. Life-death cycling represents a tactic to sample the environment for innovations that allow increases in ROE to develop over generations rather than an individual lifespan. However, the variation-selection strategem of Darwinian evolution may define a particular tactic rather than an overarching biological paradigm. A theory of evolution based on competition for energy and driven by improvements in ROE both encompasses prior notions of evolution and portends post-Darwinian mechanisms. Such processes may involve the exchange of non-genetic traits that improve ROE, as exemplified by cognitive adaptations or memes. Under these circumstances, indefinite persistence may become favored over life-death cycling, as increases in ROE may then occur more efficiently within a single lifespan rather than over multiple generations. The key to this transition may involve novel methods to address the promotion of health and cognitive plasticity. We describe the implications of this theory for human diseases. 15. Conformational plasticity and dynamics in the generic protein folding catalyst SlyD unraveled by single-molecule FRET. Science.gov (United States) Kahra, Dana; Kovermann, Michael; Löw, Christian; Hirschfeld, Verena; Haupt, Caroline; Balbach, Jochen; Hübner, Christian Gerhard 2011-08-26 The relation between conformational dynamics and chemistry in enzyme catalysis recently has received increasing attention. While, in the past, the mechanochemical coupling was mainly attributed to molecular motors, nowadays, it seems that this linkage is far more general. Single-molecule fluorescence methods are perfectly suited to directly evidence conformational flexibility and dynamics. By labeling the enzyme SlyD, a member of peptidyl-prolyl cis-trans isomerases of the FK506 binding protein type with an inserted chaperone domain, with donor and acceptor fluorophores for single-molecule fluorescence resonance energy transfer, we directly monitor conformational flexibility and conformational dynamics between the chaperone domain and the FK506 binding protein domain. We find a broad distribution of distances between the labels with two main maxima, which we attribute to an open conformation and to a closed conformation of the enzyme. Correlation analysis demonstrates that the conformations exchange on a rate in the 100 Hz range. With the aid from Monte Carlo simulations, we show that there must be conformational flexibility beyond the two main conformational states. Interestingly, neither the conformational distribution nor the dynamics is significantly altered upon binding of substrates or other known binding partners. Based on these experimental findings, we propose a model where the conformational dynamics is used to search the conformation enabling the chemical step, which also explains the remarkable substrate promiscuity connected with a high efficiency of this class of peptidyl-prolyl cis-trans isomerases. 16. Thermal corrections to the Casimir energy in a general weak gravitational field Science.gov (United States) Nazari, Borzoo 2016-12-01 We calculate finite temperature corrections to the energy of the Casimir effect of a two conducting parallel plates in a general weak gravitational field. After solving the Klein-Gordon equation inside the apparatus, mode frequencies inside the apparatus are obtained in terms of the parameters of the weak background. Using Matsubara’s approach to quantum statistical mechanics gravity-induced thermal corrections of the energy density are obtained. Well-known weak static and stationary gravitational fields are analyzed and it is found that in the low temperature limit the energy of the system increases compared to that in the zero temperature case. 17. The Generalized Energy Equation and Instability in the Two-layer Barotropic Vortex Institute of Scientific and Technical Information of China (English) 2007-01-01 The linear two-layer barotropic primitive equations in cylindrical coordinates are used to derive a generalized energy equation, which is subsequently applied to explain the instability of the spiral wave in the model. In the two-layer model, there are not only the generalized barotropic instability and the super highspeed instability, but also some other new instabilities, which fall into the range of the Kelvin-Helmholtz instability and the generalized baroclinic instability, when the upper and lower basic flows are different.They are perhaps the mechanisms of the generation of spiral cloud bands in tropical cyclones as well. 18. How to be smart and energy efficient: a general discussion on thermochromic windows. Science.gov (United States) Long, Linshuang; Ye, Hong 2014-09-19 A window is a unique element in a building because of its simultaneous properties of being "opaque" to inclement weather yet transparent to the observer. However, these unique features make the window an element that can reduce the energy efficiency of buildings. A thermochromic window is a type of smart window whose solar radiation properties vary with temperature. It is thought that the solar radiation gain of a room can be intelligently regulated through the use of thermochromic windows, resulting in lower energy consumption than with standard windows. Materials scientists have made many efforts to improve the performance of thermochromic materials. Despite these efforts, fundamental problems continue to confront us. How should a "smart" window behave? Is a "smart" window really the best candidate for energy-efficient applications? What is the relationship between smartness and energy performance? To answer these questions, a general discussion of smartness and energy performance is provided. 19. An Analysis of Metaphors Used by Students to Describe Energy in an Interdisciplinary General Science Course Science.gov (United States) Lancor, Rachael 2015-04-01 The meaning of the term energy varies widely in scientific and colloquial discourse. Teasing apart the different connotations of the term can be especially challenging for non-science majors. In this study, undergraduate students taking an interdisciplinary, general science course (n = 49) were asked to explain the role of energy in five contexts: radiation, transportation, generating electricity, earthquakes, and the big bang theory. The responses were qualitatively analyzed under the framework of conceptual metaphor theory. This study presents evidence that non-science major students spontaneously use metaphorical language that is consistent with the conceptual metaphors of energy previously identified in the discourse of students in introductory physics, biology, and chemistry courses. Furthermore, most students used multiple coherent metaphors to explain the role of energy in these complex topics. This demonstrates that these conceptual metaphors for energy have broader applicability than just traditional scientific contexts. Implications for this work as a formative assessment tool in instruction will also be discussed. 20. Generalizing the McClelland bounds for total {pi}-electron energy Energy Technology Data Exchange (ETDEWEB) Gutman, I. [Univ. of Kragujevac (Czechoslovakia). Faculty of Science; Indulal, G. [St. Aloysius Coll., Edathua, Alappuzha (India). Dept. of Mathematics; Todeschini, R. [Univ. of Milano (Italy). Dept. of Environmental Science 2008-05-15 In 1971 McClelland obtained lower and upper bounds for the total {pi}-electron energy. We now formulate the generalized version of these bounds, applicable to the energy-like expression E{sub X}={sigma}{sub i=1}{sup n} vertical stroke x{sub i}-x vertical stroke, where x{sub 1},x{sub 2},.., x{sub n} are any real numbers, and x is their arithmetic mean. In particular, if x{sub 1},x{sub 2},..,x{sub n} are the eigenvalues of the adjacency, Laplacian, or distance matrix of some graph G, then E{sub X} is the graph energy, Laplacian energy, or distance energy, respectively, of G. (orig.) 1. Conformation-mediated Förster resonance energy transfer (FRET) in blue-emitting polyvinylpyrrolidone (PVP)-passivated zinc oxide (ZnO) nanoparticles. Science.gov (United States) Kurt, Hasan; Alpaslan, Ece; Yildiz, Burçin; Taralp, Alpay; Ow-Yang, Cleva W 2017-02-15 Homopolymers, such as polyvinylpyrrolidone (PVP), are commonly used to passivate the surface of blue-light emitting ZnO nanoparticles during colloid nucleation and growth. However, although PVP is known to auto-fluoresce at 400nm, which is near the absorption edge of ZnO, the impact of PVP adsorption characteristics on the surface of ZnO and the surface-related photophysics of PVP-capped ZnO nanoparticles is not well understood. To investigate, we have synthesized ZnO nanoparticles in solvents containing PVP of 3 concentrations-0.5, 0.7, and 0.11gmL(-1). Using time-domain NMR, we show that the adsorbed polymer conformation differs with polymer concentration-head-to-tail under low concentration (e.g., 0.05gmL(-1)) and looping, then train-like, with increasing concentration (e.g., 0.07gmL(-1) and 0.11gmL(-1), respectively). When the surface-adsorbed PVP is entrained, the surface states of ZnO are passivated and radiative emission from surface trap states is suppressed, allowing emission to be dominated by exciton transitions in the UV (ca. 310nm). Moreover, the reduced proximity between the PVP molecule and the ZnO gives rise to increased efficiency of energy transfer between the exciton emission of ZnO and the HOMO-LUMO absorption of PVP (ca. 400nm). As a result, light emission in the blue is enhanced in the PVP-capped ZnO nanoparticles. We thus show that the emission properties of ZnO can be tuned by controlling the adsorbed PVP conformation on the ZnO surface via the PVP concentration in the ZnO precipitation medium. 2. Intramolecular ex vivo Fluorescence Resonance Energy Transfer (FRET of Dihydropyridine Receptor (DHPR β1a Subunit Reveals Conformational Change Induced by RYR1 in Mouse Skeletal Myotubes. Directory of Open Access Journals (Sweden) Dipankar Bhattacharya Full Text Available The dihydropyridine receptor (DHPR β1a subunit is essential for skeletal muscle excitation-contraction coupling, but the structural organization of β1a as part of the macromolecular DHPR-ryanodine receptor type I (RyR1 complex is still debatable. We used fluorescence resonance energy transfer (FRET to probe proximity relationships within the β1a subunit in cultured skeletal myotubes lacking or expressing RyR1. The fluorescein biarsenical reagent FlAsH was used as the FRET acceptor, which exhibits fluorescence upon binding to specific tetracysteine motifs, and enhanced cyan fluorescent protein (CFP was used as the FRET donor. Ten β1a reporter constructs were generated by inserting the CCPGCC FlAsH binding motif into five positions probing the five domains of β1a with either carboxyl or amino terminal fused CFP. FRET efficiency was largest when CCPGCC was positioned next to CFP, and significant intramolecular FRET was observed for all constructs suggesting that in situ the β1a subunit has a relatively compact conformation in which the carboxyl and amino termini are not extended. Comparison of the FRET efficiency in wild type to that in dyspedic (lacking RyR1 myotubes revealed that in only one construct (H458 CCPGCC β1a -CFP FRET efficiency was specifically altered by the presence of RyR1. The present study reveals that the C-terminal of the β1a subunit changes conformation in the presence of RyR1 consistent with an interaction between the C-terminal of β1a and RyR1 in resting myotubes. 3. Fluorescence Resonance Energy Transfer-based Structural Analysis of the Dihydropyridine Receptor α1S Subunit Reveals Conformational Differences Induced by Binding of the β1a Subunit Science.gov (United States) Mahalingam, Mohana; Perez, Claudio F.; Fessenden, James D. 2016-01-01 The skeletal muscle dihydropyridine receptor α1S subunit plays a key role in skeletal muscle excitation-contraction coupling by sensing membrane voltage changes and then triggering intracellular calcium release. The cytoplasmic loops connecting four homologous α1S structural domains have diverse functions, but their structural arrangement is poorly understood. Here, we used a novel FRET-based method to characterize the relative proximity of these intracellular loops in α1S subunits expressed in intact cells. In dysgenic myotubes, energy transfer was observed from an N-terminal-fused YFP to a FRET acceptor, ReAsH (resorufin arsenical hairpin binder), targeted to each α1S intracellular loop, with the highest FRET efficiencies measured to the α1S II-III loop and C-terminal tail. However, in HEK-293T cells, FRET efficiencies from the α1S N terminus to the II-III and III-IV loops and the C-terminal tail were significantly lower, thus suggesting that these loop structures are influenced by the cellular microenvironment. The addition of the β1a dihydropyridine receptor subunit enhanced FRET to the II-III loop, thus indicating that β1a binding directly affects II-III loop conformation. This specific structural change required the C-terminal 36 amino acids of β1a, which are essential to support EC coupling. Direct FRET measurements between α1S and β1a confirmed that both wild type and truncated β1a bind similarly to α1S. These results provide new insights into the role of muscle-specific proteins on the structural arrangement of α1S intracellular loops and point to a new conformational effect of the β1a subunit in supporting skeletal muscle excitation-contraction coupling. PMID:27129199 4. Fluorescence Resonance Energy Transfer-based Structural Analysis of the Dihydropyridine Receptor α1S Subunit Reveals Conformational Differences Induced by Binding of the β1a Subunit. Science.gov (United States) Mahalingam, Mohana; Perez, Claudio F; Fessenden, James D 2016-06-24 The skeletal muscle dihydropyridine receptor α1S subunit plays a key role in skeletal muscle excitation-contraction coupling by sensing membrane voltage changes and then triggering intracellular calcium release. The cytoplasmic loops connecting four homologous α1S structural domains have diverse functions, but their structural arrangement is poorly understood. Here, we used a novel FRET-based method to characterize the relative proximity of these intracellular loops in α1S subunits expressed in intact cells. In dysgenic myotubes, energy transfer was observed from an N-terminal-fused YFP to a FRET acceptor, ReAsH (resorufin arsenical hairpin binder), targeted to each α1S intracellular loop, with the highest FRET efficiencies measured to the α1S II-III loop and C-terminal tail. However, in HEK-293T cells, FRET efficiencies from the α1S N terminus to the II-III and III-IV loops and the C-terminal tail were significantly lower, thus suggesting that these loop structures are influenced by the cellular microenvironment. The addition of the β1a dihydropyridine receptor subunit enhanced FRET to the II-III loop, thus indicating that β1a binding directly affects II-III loop conformation. This specific structural change required the C-terminal 36 amino acids of β1a, which are essential to support EC coupling. Direct FRET measurements between α1S and β1a confirmed that both wild type and truncated β1a bind similarly to α1S These results provide new insights into the role of muscle-specific proteins on the structural arrangement of α1S intracellular loops and point to a new conformational effect of the β1a subunit in supporting skeletal muscle excitation-contraction coupling. © 2016 by The American Society for Biochemistry and Molecular Biology, Inc. 5. Wormhole geometries in fourth-order conformal Weyl gravity CERN Document Server Varieschi, Gabriele U 2015-01-01 We present an analysis of the classic wormhole geometries based on conformal Weyl gravity, rather than standard general relativity. The main characteristics of the resulting traversable wormholes remain the same as in the seminal study by Morris and Thorne, namely, that effective super-luminal motion is a viable consequence of the metric. Improving on previous work on the subject, we show that for particular choices of the shape and redshift functions, the wormhole metric in the context of conformal gravity does not violate the main energy conditions, as was the case of the original solutions. In particular, the resulting geometry does not require the use of exotic matter at or near the wormhole throat. Therefore, if fourth-order conformal Weyl gravity is a correct extension of general relativity, traversable wormholes might become a realistic solution for interstellar travel. 6. Conformal Gravity and the Alcubierre Warp Drive Metric CERN Document Server Varieschi, Gabriele U 2012-01-01 We present an analysis of the classic Alcubierre metric based on conformal gravity, rather than standard general relativity. The main characteristics of the resulting warp drive remain the same as in the original study by Alcubierre, namely that effective super-luminal motion is a viable outcome of the metric. We show that for particular choices of the shaping function, the Alcubierre metric in the context of conformal gravity does not violate the weak energy condition, as was the case of the original solution. In particular, the resulting warp drive does not require the use of exotic matter. Therefore, if conformal gravity is a correct extension of general relativity, super-luminal motion via an Alcubierre metric might be a realistic solution, thus allowing faster-than-light interstellar travel. 7. Helix 3 acts as a conformational hinge in Class A GPCR activation: An analysis of interhelical interaction energies in crystal structures. Science.gov (United States) Lans, Isaias; Dalton, James A R; Giraldo, Jesús 2015-12-01 A collection of crystal structures of rhodopsin, β2-adrenergic and adenosine A2A receptors in active, intermediate and inactive states were selected for structural and energetic analyses to identify the changes involved in the activation/deactivation of Class A GPCRs. A set of helix interactions exclusive to either inactive or active/intermediate states were identified. The analysis of these interactions distinguished some local conformational changes involved in receptor activation, in particular, a packing between the intracellular domains of transmembrane helices H3 and H7 and a separation between those of H2 and H6. Also, differential movements of the extracellular and intracellular domains of these helices are apparent. Moreover, a segment of residues in helix H3, including residues L/I3.40 to L3.43, is identified as a key component of the activation mechanism, acting as a conformational hinge between extracellular and intracellular regions. Remarkably, the influence on the activation process of some glutamic and aspartic acidic residues and, as a consequence, the influence of variations on local pH is highlighted. Structural hypotheses that arose from the analysis of rhodopsin, β2-adrenergic and adenosine A2A receptors were tested on the active and inactive M2 muscarinic acetylcholine receptor structures and further discussed in the context of the new mechanistic insights provided by the recently determined active and inactive crystal structures of the μ-opioid receptor. Overall, the structural and energetic analyses of the interhelical interactions present in this collection of Class A GPCRs suggests the existence of a common general activation mechanism featuring a chemical space useful for drug discovery exploration. 8. Energy Conservation: Policies, Programs, and General Studies. 1979-July, 1980 (Citations from the NTIS Data Base). Science.gov (United States) Hundemann, Audrey S. The 135 abstracts presented pertain to national policies, programs, and general strategies for conserving energy. In addition to the abstract, each citation lists the title, author, sponsoring agency, subject categories, number of pages, date, descriptors, identifiers, and ordering information for each document. Topics covered in this compilation… 9. Energy density in general relativity a possible role of cosmological constant CERN Document Server Ray, S; Ray, Saibal; Bhadra, Sumana 2004-01-01 We consider a static spherically symmetric charged anisotropic fluid source of finite physical radius (\\sim 10^{-16} cm) by introducing a scalar variable \\Lambda dependent on the radial coordinate r under general relativity. From the solution sets a possible role of the cosmological constant is investigated which indicates the dependency of energy density of electron on the variable \\Lambda. 10. Non-existence of global solutions to generalized dissipative Klein-Gordon equations with positive energy Directory of Open Access Journals (Sweden) Maxim Olegovich Korpusov 2012-07-01 Full Text Available In this article the initial-boundary-value problem for generalized dissipative high-order equation of Klein-Gordon type is considered. We continue our study of nonlinear hyperbolic equations and systems with arbitrary positive energy. The modified concavity method by Levine is used for proving blow-up of solutions. 11. Deriving Internal Energy by Virtue of Generalized Feynman-Hellmann Theorem for Mixed States Institute of Scientific and Technical Information of China (English) FAN Hong-Yi; JIANG Zhong-Hua 2005-01-01 We show how to directly use the generalized Feynman-Hellmann theorem, which is suitable for mixed state ensemble average, to derive the internal energy of Hamiltonian systems. A concrete example, which is a two coupled harminic oscillators, is used for elucidating our approach. 12. Conformal supermultiplets without superpartners CERN Document Server Jarvis, Peter 2011-01-01 We consider polynomial deformations of Lie superalgebras and their representations. For the class A(n-1,0) ~ sl(n/1), we identify families of superalgebras of quadratic and cubic type, consistent with Jacobi identities. For such deformed superalgebras we point out the possibility of zero step supermultiplets, carried on a single, irreducible representation of the even (Lie) subalgebra. For the conformal group SU(2,2) in 1+3-dimensional spacetime, such irreducible (unitary) representations correspond to standard conformal fields (j_1,j_2;d), where (j_1,j_2) is the spin and d the conformal dimension; in the massless class j_1 j_2=0, and d=j_1+j_2+1. We show that these repesentations are zero step supermultiplets for the superalgebra SU_(2)(2,2/1), the quadratic deformation of conformal supersymmetry SU(2,2/1). We propose to elevate SU_(2)(2,2/1) to a symmetry of the S-matrix. Under this scenario, low-energy standard model matter fields (leptons, quarks, Higgs scalars and gauge fields) descended from such confor... 13. P-loop conformation governed crizotinib resistance in G2032R-mutated ROS1 tyrosine kinase: clues from free energy landscape. Directory of Open Access Journals (Sweden) Huiyong Sun 2014-07-01 Full Text Available Tyrosine kinases are regarded as excellent targets for chemical drug therapy of carcinomas. However, under strong purifying selection, drug resistance usually occurs in the cancer cells within a short term. Many cases of drug resistance have been found to be associated with secondary mutations in drug target, which lead to the attenuated drug-target interactions. For example, recently, an acquired secondary mutation, G2032R, has been detected in the drug target, ROS1 tyrosine kinase, from a crizotinib-resistant patient, who responded poorly to crizotinib within a very short therapeutic term. It was supposed that the mutation was located at the solvent front and might hinder the drug binding. However, a different fact could be uncovered by the simulations reported in this study. Here, free energy surfaces were characterized by the drug-target distance and the phosphate-binding loop (P-loop conformational change of the crizotinib-ROS1 complex through advanced molecular dynamics techniques, and it was revealed that the more rigid P-loop region in the G2032R-mutated ROS1 was primarily responsible for the crizotinib resistance, which on one hand, impaired the binding of crizotinib directly, and on the other hand, shortened the residence time induced by the flattened free energy surface. Therefore, both of the binding affinity and the drug residence time should be emphasized in rational drug design to overcome the kinase resistance. 14. P-loop conformation governed crizotinib resistance in G2032R-mutated ROS1 tyrosine kinase: clues from free energy landscape. Science.gov (United States) Sun, Huiyong; Li, Youyong; Tian, Sheng; Wang, Junmei; Hou, Tingjun 2014-07-01 Tyrosine kinases are regarded as excellent targets for chemical drug therapy of carcinomas. However, under strong purifying selection, drug resistance usually occurs in the cancer cells within a short term. Many cases of drug resistance have been found to be associated with secondary mutations in drug target, which lead to the attenuated drug-target interactions. For example, recently, an acquired secondary mutation, G2032R, has been detected in the drug target, ROS1 tyrosine kinase, from a crizotinib-resistant patient, who responded poorly to crizotinib within a very short therapeutic term. It was supposed that the mutation was located at the solvent front and might hinder the drug binding. However, a different fact could be uncovered by the simulations reported in this study. Here, free energy surfaces were characterized by the drug-target distance and the phosphate-binding loop (P-loop) conformational change of the crizotinib-ROS1 complex through advanced molecular dynamics techniques, and it was revealed that the more rigid P-loop region in the G2032R-mutated ROS1 was primarily responsible for the crizotinib resistance, which on one hand, impaired the binding of crizotinib directly, and on the other hand, shortened the residence time induced by the flattened free energy surface. Therefore, both of the binding affinity and the drug residence time should be emphasized in rational drug design to overcome the kinase resistance. 15. Spacetime Conformal Fluctuations and Quantum Dephasing CERN Document Server Bonifacio, Paolo M Any quantum system interacting with a complex environment undergoes decoherence. Empty space is filled with vacuum energy due to matter fields in their ground state and represents an underlying environment that any quantum particle has to cope with. In particular quantum gravity vacuum fluctuations should represent a universal source of decoherence. To study this problem we employ a stochastic approach that models spacetime fluctuations close to the Planck scale by means of a classical, randomly fluctuating metric (random gravity framework). We enrich the classical scheme for metric perturbations over a curved background by also including matter fields and metric conformal fluctuations. We show in general that a conformally modulated metric induces dephasing as a result of an effective nonlinear newtonian potential obtained in the appropriate nonrelativistic limit of a minimally coupled Klein-Gordon field. The special case of vacuum fluctuations is considered and a quantitative estimate of the expected effect... 16. A chord error conforming tool path B-spline fitting method for NC machining based on energy minimization and LSPIA Directory of Open Access Journals (Sweden) Shanshan He 2015-10-01 Full Text Available Piecewise linear (G01-based tool paths generated by CAM systems lack G1 and G2 continuity. The discontinuity causes vibration and unnecessary hesitation during machining. To ensure efficient high-speed machining, a method to improve the continuity of the tool paths is required, such as B-spline fitting that approximates G01 paths with B-spline curves. Conventional B-spline fitting approaches cannot be directly used for tool path B-spline fitting, because they have shortages such as numerical instability, lack of chord error constraint, and lack of assurance of a usable result. Progressive and Iterative Approximation for Least Squares (LSPIA is an efficient method for data fitting that solves the numerical instability problem. However, it does not consider chord errors and needs more work to ensure ironclad results for commercial applications. In this paper, we use LSPIA method incorporating Energy term (ELSPIA to avoid the numerical instability, and lower chord errors by using stretching energy term. We implement several algorithm improvements, including (1 an improved technique for initial control point determination over Dominant Point Method, (2 an algorithm that updates foot point parameters as needed, (3 analysis of the degrees of freedom of control points to insert new control points only when needed, (4 chord error refinement using a similar ELSPIA method with the above enhancements. The proposed approach can generate a shape-preserving B-spline curve. Experiments with data analysis and machining tests are presented for verification of quality and efficiency. Comparisons with other known solutions are included to evaluate the worthiness of the proposed solution. 17. Energy distributions of Bianchi type-VI h Universe in general relativity and teleparallel gravity Science.gov (United States) Özkurt, Şeref; Aygün, Sezg&idot; n. 2017-04-01 In this paper, we have investigated the energy and momentum density distributions for the inhomogeneous generalizations of homogeneous Bianchi type-VI h metric with Einstein, Bergmann-Thomson, Landau-Lifshitz, Papapetrou, Tolman and Møller prescriptions in general relativity (GR) and teleparallel gravity (TG). We have found exactly the same results for Einstein, Bergmann-Thomson and Landau-Lifshitz energy-momentum distributions in Bianchi type-VI h metric for different gravitation theories. The energy-momentum distributions of the Bianchi type- VI h metric are found to be zero for h = -1 in GR and TG. However, our results agree with Tripathy et al, Tryon, Rosen and Aygün et al. 18. Cloud-radiative effects on implied oceanic energy transport as simulated by atmospheric general circulation models Science.gov (United States) Gleckler, P. J.; Randall, D. A.; Boer, G.; Colman, R.; Dix, M.; Galin, V.; Helfand, M.; Kiehl, J.; Kitoh, A.; Lau, W. 1995-01-01 This paper summarizes the ocean surface net energy flux simulated by fifteen atmospheric general circulation models constrained by realistically-varying sea surface temperatures and sea ice as part of the Atmospheric Model Intercomparison Project. In general, the simulated energy fluxes are within the very large observational uncertainties. However, the annual mean oceanic meridional heat transport that would be required to balance the simulated surface fluxes is shown to be critically sensitive to the radiative effects of clouds, to the extent that even the sign of the Southern Hemisphere ocean heat transport can be affected by the errors in simulated cloud-radiation interactions. It is suggested that improved treatment of cloud radiative effects should help in the development of coupled atmosphere-ocean general circulation models. 19. Generalization of radiative jet energy loss to non-zero magnetic mass CERN Document Server Djordjevic, Magdalena 2011-01-01 Reliable predictions for jet quenching in ultra-relativistic heavy ion collisions require accurate computation of radiative energy loss. With this goal, an energy loss formalism in a realistic finite size dynamical QCD medium was recently developed. While this formalism assumes zero magnetic mass - in accordance with the one-loop perturbative calculations - different non-perturbative approaches report a non-zero magnetic mass at RHIC and LHC. We here generalize the energy loss to consistently include a possibility for existence of non-zero magnetic screening. We also present how the inclusion of finite magnetic mass changes the energy loss results. Our analysis indicates a fundamental constraint on magnetic to electric mass ratio. 20. Moller Energy-Momentum Complex in General Relativity for Higher Dimensional Universes Institute of Scientific and Technical Information of China (English) M. Aygün; S. Aygün; (I). Yilmaz; H. Baysal; (I). Tarhan 2007-01-01 Using the Moller energy-momentum definition in general relativity (GR) we calculate the total energy-momentum distribution associated with (n + 2)-dimensional homogeneous and isotropic model of the universe. It is found that total energy of Moller is vanishing in (n+ 2) dimensions everywhere but n-momentum components of Moller in (n + 2) dimensions are different from zero. Also, we evaluate the static Einstein Universe, FRW universe and de Sitter universe in four dimensions by using (n + 2)-type metric, then calculate the Moller energy-momentum distribution of these spacetimes. However, our results are consistent with the results of Banerjee and Sen, Xulu, Radinschi, Vargas, Cooperstock-Israelit, A ygin et al., Rosen, and Johri et al. in four dimensions. 1. Conformal Patterson-Walker metrics CERN Document Server Hammerl, Matthias; Šilhan, Josef; Taghavi-Chabert, Arman; Žádník, Vojtěch 2016-01-01 The classical Patterson-Walker construction of a split-signature (pseudo-)Riemannian structure from a given torsion-free affine connection is generalized to a construction of a split-signature conformal structure from a given projective class of connections. A characterization of the induced structures is obtained. We achieve a complete description of Einstein metrics in the conformal class formed by the Patterson-Walker metric. Finally, we describe all symmetries of the conformal Patterson-Walker metric. In both cases we obtain descriptions in terms of geometric data on the original structure. 2. Controlling the Conformational Energy of a Phenyl Group by Tuning the Strength of a Nonclassical CH···O Hydrogen Bond: The Case of 5-Phenyl-1,3-dioxane. Science.gov (United States) Bailey, William F; Lambert, Kyle M; Stempel, Zachary D; Wiberg, Kenneth B; Mercado, Brandon Q 2016-12-16 Anancomeric 5-phenyl-1,3-dioxanes provide a unique opportunity to study factors that control conformation. Whereas one might expect an axial phenyl group at C(5) of 1,3-dioxane to adopt a conformation similar to that in axial phenylcyclohexane, a series of studies including X-ray crystallography, NOE measurements, and DFT calculations demonstrate that the phenyl prefers to lie over the dioxane ring in order to position an ortho-hydrogen to participate in a stabilizing, nonclassical CH···O hydrogen bond with a ring oxygen of the dioxane. Acid-catalyzed equilibration of a series of anancomeric 2-tert-butyl-5-aryl-1,3-dioxane isomers demonstrates that remote substituents on the phenyl ring affect the conformational energy of a 5-aryl-1,3-dioxane: electron-withdrawing substituents decrease the conformational energy of the aryl group, while electron-donating substituents increase the conformational energy of the group. This effect is correlated in a very linear way to Hammett substituent parameters. In short, the strength of the CH···O hydrogen bond may be tuned in a predictable way in response to the electron-withdrawing or electron-donating ability of substituents positioned remotely on the aryl ring. This effect may be profound: a 3,5-bis-CF3 phenyl group at C(5) in 1,3-dioxane displays a pronounced preference for the axial orientation. The results are relevant to broader conformational issues involving heterocyclic systems bearing aryl substituents. 3. Conformational changes in glycine tri- and hexapeptide DEFF Research Database (Denmark) Yakubovich, Alexander V.; Solov'yov, Ilia; Solov'yov, Andrey V. 2006-01-01 conformations and calculated the energy barriers for transitions between them. Using a thermodynamic approach, we have estimated the times of the characteristic transitions between these conformations. The results of our calculations have been compared with those obtained by other theoretical methods...... also investigated the influence of the secondary structure of polypeptide chains on the formation of the potential energy landscape. This analysis has been performed for the sheet and the helix conformations of chains of six amino acids.... 4. Strategies to Save 50% Site Energy in Grocery and General Merchandise Stores Energy Technology Data Exchange (ETDEWEB) Hirsch, A.; Hale, E.; Leach, M. 2011-03-01 This paper summarizes the methodology and main results of two recently published Technical Support Documents. These reports explore the feasibility of designing general merchandise and grocery stores that use half the energy of a minimally code-compliant building, as measured on a whole-building basis. We used an optimization algorithm to trace out a minimum cost curve and identify designs that satisfy the 50% energy savings goal. We started from baseline building energy use and progressed to more energy-efficient designs by sequentially adding energy design measures (EDMs). Certain EDMs figured prominently in reaching the 50% energy savings goal for both building types: (1) reduced lighting power density; (2) optimized area fraction and construction of view glass or skylights, or both, as part of a daylighting system tuned to 46.5 fc (500 lux); (3) reduced infiltration with a main entrance vestibule or an envelope air barrier, or both; and (4) energy recovery ventilators, especially in humid and cold climates. In grocery stores, the most effective EDM, which was chosen for all climates, was replacing baseline medium-temperature refrigerated cases with high-efficiency models that have doors. 5. Paul Scherrer Institut annual report 1996. Annex V: PSI general energy technology newsletter 1996 Energy Technology Data Exchange (ETDEWEB) Daum, C.; Leuenberger, J. [eds. 1997-06-01 Surveying the results of General Energy Research in 1996, three major trends can be identified. First, in areas where research results have reached an advanced stage, decisive steps have been taken to promote a transfer towards industrial realization; examples include biomass gasification, advanced battery concepts, and combustion research. Second, in projects with longer term orientation, several options are being evaluated by exploratory studies, e.g. in solar chemistry and reaction analysis. Third, in line with the strategic planning of our institute, the development and characterization of materials for energy research has received increased attention. (author) figs., tabs., refs. 6. Generalized Chou-Yang Model and Meson-Proton Elastic Scattering at High Energies Science.gov (United States) Saleem, Mohammad; Aleem, Fazal-E.; Rashid, Haris The various characteristics of meson-proton elastic scattering at high energies are explained by using the generalized Chou-Yang model which takes into consideration the anisotropic scattering of objects constituting pions(kaons) and protons. A new parametrization of the proton form factor consistent with the recent experimental data is proposed. It is then shown that all the data for meson-proton elastic scattering at 200 and 250 GeV/c are in agreement with theoretical computations. The physical picture of generalized Chou-Yang model which is based on multiple scattering theory is given in detail. 7. Generalized Chou-Yang model and meson-proton elastic scattering at high energies Energy Technology Data Exchange (ETDEWEB) Saleem, M.; Aleem, F.E.; Rashid, H. 1989-01-01 The various characteristics of meson-proton elastic scattering at high energies are explained by using the generalized Chou-Yang model which takes into consideration the anisotropic scattering of objects constituting pions(kaons) and protons. A new parametrization of the proton form factor consistent with the recent experimental data is proposed. It is then shown that all the data for meson-proton elastic scattering at 200 and 250 GeV/c are in agreement with theoretical computations. The physical picture of generalized Chou-Yang model which is based on multiple scattering theory is given in detail. 8. Escherichia coli Phosphoenolpyruvate Dependent Phosphotransferase System. NMR Studies of the Conformation of HPr and P-HPr and the Mechanism of Energy Coupling NARCIS (Netherlands) Dooijewaard, G.; Roossien, F.F.; Robillard, G.T. 1979-01-01 1H and 31P nuclear magnetic resonance investigations of the phosphoprotein intermediate P-HPr and the parent molecule HPr of the E. coli phosphoenolpyruvate dependent phosphotransferase system (PTS) show that HPr can exist in two conformations. These conformations influence the protonation state of 9. DFT Conformation and Energies of Amylose Fragments at Atomic Resolution Part 2: “Band-flip” and “Kink” Forms of Alpha-Maltotetraose Science.gov (United States) In Part 2 of this series of DFT optimization studies of '-maltotetraose, we present results at the B3LYP/6-311++G** level of theory for conformations denoted “band-flips” and “kinks”. Recent experimental X-ray studies have found examples of amylose fragments with conformations distorted from the us... 10. Gas-phase hydrogen/deuterium exchange of 5'- and 3'-mononucleotides in a quadrupole ion trap: exploring the role of conformation and system energy. Science.gov (United States) Chipuk, Joseph E; Brodbelt, Jennifer S 2007-04-01 Gas-phase hydrogen/deuterium (H/D) exchange reactions for deprotonated 2'-deoxy-5'-monophosphate and 2'-deoxy-3'-monophosphate nucleotides with D(2)O were performed in a quadrupole ion trap mass spectrometer. To augment these experiments, molecular modeling was also conducted to identify likely deprotonation sites and potential gas-phase conformations of the anions. A majority of the 5'-monophosphates exchanged extensively with several of the compounds completely incorporating deuterium in place of their labile hydrogen atoms. In contrast, most of the 3'-monophosphate isomers exchanged relatively few hydrogen atoms, even though the rate of the first two exchanges was greater than observed for the 5'-monophosphates. Mononucleotides that failed to incorporate more than two deuterium atoms under default reaction conditions were often found to exchange more extensively when reactions were performed under higher energy conditions. Integration of the experimental and theoretical results supports the use of a relay exchange mechanism and suggests that the exchange behavior depends highly on the identity and orientation of the nucleobase and the position and flexibility of the deprotonated phosphate moiety. These observations also highlight the importance of the distance between the various participating groups in addition to their gas-phase acidity and basicity. 11. Palatini wormholes and energy conditions from the prism of General Relativity CERN Document Server Bejarano, C; Olmo, Gonzalo J; Rubiera-Garcia, D 2016-01-01 Wormholes are hypothetical shortcuts in spacetime that in General Relativity unavoidably violate all of the pointwise energy conditions. In this paper, we consider several wormhole spacetimes that, as opposed to the standard \\emph{designer} procedure frequently employed in the literature, arise directly from gravitational actions including additional terms resulting from contractions of the Ricci tensor with the metric, and which are formulated assuming independence between metric and connection (Palatini approach). We reinterpret such wormhole solutions under the prism of General Relativity and study the matter sources that thread them. We discuss the size of violation of the energy conditions in different cases, and how this is related to the same spacetimes when viewed from the modified gravity side. 12. Fluorescence resonance energy transfer biosensors that detect Ran conformational changes and a Ran x GDP-importin-beta -RanBP1 complex in vitro and in intact cells. Science.gov (United States) Plafker, Kendra; Macara, Ian G 2002-08-16 The Ran GTPase plays a central role in nucleocytoplasmic transport. Association of Ran x GTP with transport carriers (karyopherins) triggers the loading/unloading of export or import cargo, respectively. The C-terminal tail of Ran x GTP is deployed in an extended conformation when associated with a Ran binding domain or importins. To monitor tail orientation, a Ran-GFP fusion was labeled with the fluorophore Alexa546. Fluorescence resonance energy transfer (FRET) occurs efficiently between the green fluorescent protein (GFP) and Alexa546 for Ran x GDP and Ran x GTP, suggesting that the tail is tethered in both states. However, Ran x GTP complexes with importin-beta, RanBP1, and Crm1 all show reduced FRET consistent with tail extension. Displacement of the C-terminal tail of Ran by karyopherins may be a general mechanism to facilitate RanBP1 binding. A Ran x GDP-RanBP1-importin-beta complex also displayed a low FRET signal. To detect this complex in vivo, a bipartite biosensor consisting of Ran-Alexa546 plus GST-GFP-RanBP1, was co-injected into the cytoplasm of cells. The Ran redistributed predominantly to the nucleus, and RanBP1 remained cytoplasmic. Nonetheless, a robust cytoplasmic FRET signal was detectable, which suggests that a significant fraction of cytoplasmic Ran.GDP may exist in a ternary complex with RanBP1 and importins. 13. Deformed Potential Energy of 263Db in a Generalized Liquid Drop Model Institute of Scientific and Technical Information of China (English) 陈宝秋; 马中玉; 赵耀林 2003-01-01 The macroscopic deformed potential energy for super-heavy nuclei 263 Db,which governs the entrance and alpha decay channels,is determined within a generalized liquid drop model(GLDM).A quasi-molecular shape is as sumed in the GLDM,which includes volume-,surface-,and Coulomb-energies,proximity effects,mass asymmetry,and an accurate nuclear radius.The microscopic single particle energies are derived from a shell model in an axially deformed Woods-Saxon potential with a quasi-molecular shape.The shell correction is calculated by the Strutinsky method.The total deformed potential energy of a nucleus can be calculated by the macro-microscopic method as the summation of the liquid-drop energy and the Strutinsky shell correction.The theory is applied to predict the deformed potential energy of the experiment 22Ne + 241Am → 263Db* → 259Db + 4n,which was performed on the Heavy Ion Accelerator in Lanzhou.It is found that the neck in the quasi-molecular shape is responsible for the deep valley of the fusion barrier due to the shell corrections.In the cold fusion path,the double-hump fusion barrier is predicted by the shell correction and complete fusion events may occur. 14. Stereoelectronic effects dictate molecular conformation and biological function of heterocyclic amides. Science.gov (United States) Reid, Robert C; Yau, Mei-Kwan; Singh, Ranee; Lim, Junxian; Fairlie, David P 2014-08-27 Heterocycles adjacent to amides can have important influences on molecular conformation due to stereoelectronic effects exerted by the heteroatom. This was shown for imidazole- and thiazole-amides by comparing low energy conformations (ab initio MP2 and DFT calculations), charge distribution, dipole moments, and known crystal structures which support a general principle. Switching a heteroatom from nitrogen to sulfur altered the amide conformation, producing different three-dimensional electrostatic surfaces. Differences were attributed to different dipole and orbital alignments and spectacularly translated into opposing agonist vs antagonist functions in modulating a G-protein coupled receptor for inflammatory protein complement C3a on human macrophages. Influences of the heteroatom were confirmed by locking the amide conformation using fused bicyclic rings. These findings show that stereoelectronic effects of heterocycles modulate molecular conformation and can impart strikingly different biological properties. 15. Conformally-related Einstein-Langevin equations for metric fluctuations in stochastic gravity CERN Document Server Satin, Seema; Hu, Bei Lok 2016-01-01 For a conformally-coupled scalar field we obtain the conformally-related Einstein-Langevin equations, using appropriate transformations for all the quantities in the equations between two conformally-related spacetimes. In particular, we analyze the transformations of the influence action, the stress energy tensor, the noise kernel and the dissipation kernel. In due course the fluctuation-dissipation relation is also discussed. The analysis in this paper thereby facilitates a general solution to the Einstein-Langevin equation once the solution of the equation in a simpler, conformally-related spacetime is known. For example, from the Minkowski solution of Martin and Verdaguer, those of the Einstein-Langevin equations in conformally-flat spacetimes, especially for spatially-flat Friedmann-Robertson-Walker models, can be readily obtained. 16. Conformally related Einstein-Langevin equations for metric fluctuations in stochastic gravity Science.gov (United States) Satin, Seema; Cho, H. T.; Hu, Bei Lok 2016-09-01 For a conformally coupled scalar field we obtain the conformally related Einstein-Langevin equations, using appropriate transformations for all the quantities in the equations between two conformally related spacetimes. In particular, we analyze the transformations of the influence action, the stress energy tensor, the noise kernel and the dissipation kernel. In due course the fluctuation-dissipation relation is also discussed. The analysis in this paper thereby facilitates a general solution to the Einstein-Langevin equation once the solution of the equation in a simpler, conformally related spacetime is known. For example, from the Minkowski solution of Martín and Verdaguer, those of the Einstein-Langevin equations in conformally flat spacetimes, especially for spatially flat Friedmann-Robertson-Walker models, can be readily obtained. 17. Cosmological isotropic matter-energy generalizations of Schwarzschild and Kerr metrics CERN Document Server Arik, Metin 2016-01-01 We present a time dependent isotropic fluid solution around a Schwarzschild black hole. We offer the solutions and discuss the effects on the field equations and the horizon. We derive the energy density, pressure and the equation of state parameter. In the second part, we generalize the rotating black hole solution to an expanding universe. We derive from the proposed metric the special solutions of the field equations for the dust approximation and the dark energy solution. We show that the presence of a rotating black hole does not modify the scale factorb(t)=t^{2/3}$law for dust, nor$b(t)=e^{\\lambda\\hspace{1mm}t}$and$p=-\\rho$for dark energy. 18. Energy and angular momentum of general 4-dimensional stationary axi-symmetric spacetime in teleparallel geometry CERN Document Server Nashed, Gamal Gergess Lamee 2008-01-01 We derive an exact general axi-symmetric solution of the coupled gravitational and electromagnetic fields in the tetrad theory of gravitation. The solution is characterized by four parameters$M$(mass),$Q$(charge),$a$(rotation) and$L$(NUT). We then, calculate the total exterior energy using the energy-momentum complex given by M{\\o}ller in the framework of Weitzenb$\\ddot{o}$ck geometry. We show that the energy contained in a sphere is shared by its interior as well as exterior. We also calculate the components of the spatial momentum to evaluate the angular momentum distribution. We show that the only non-vanishing components of the angular momentum is in the Z direction. 19. Intercomparison and interpretation of surface energy fluxes in atmospheric general circulation models Science.gov (United States) Randall, D. A.; Cess, R. D.; Blanchet, J. P.; Boer, G. J.; Dazlich, D. A.; Del Genio, A. D.; Deque, M.; Dymnikov, V.; Galin, V.; Ghan, S. J. 1992-01-01 Responses of the surface energy budgets and hydrologic cycles of 19 atmospheric general circulation models to an imposed, globally uniform sea surface temperature perturbation of 4 K were analyzed. The responses of the simulated surface energy budgets are extremely diverse and are closely linked to the responses of the simulated hydrologic cycles. The response of the net surface energy flux is not controlled by cloud effects; instead, it is determined primarily by the response of the latent heat flux. The prescribed warming of the oceans leads to major increases in the atmospheric water vapor content and the rates of evaporation and precipitation. The increased water vapor amount drastically increases the downwelling IR radiation at the earth's surface, but the amount of the change varies dramatically from one model to another. 20. Modeling conformational ensembles of slow functional motions in Pin1-WW. Directory of Open Access Journals (Sweden) Faruck Morcos Full Text Available Protein-protein interactions are often mediated by flexible loops that experience conformational dynamics on the microsecond to millisecond time scales. NMR relaxation studies can map these dynamics. However, defining the network of inter-converting conformers that underlie the relaxation data remains generally challenging. Here, we combine NMR relaxation experiments with simulation to visualize networks of inter-converting conformers. We demonstrate our approach with the apo Pin1-WW domain, for which NMR has revealed conformational dynamics of a flexible loop in the millisecond range. We sample and cluster the free energy landscape using Markov State Models (MSM with major and minor exchange states with high correlation with the NMR relaxation data and low NOE violations. These MSM are hierarchical ensembles of slowly interconverting, metastable macrostates and rapidly interconverting microstates. We found a low population state that consists primarily of holo-like conformations and is a "hub" visited by most pathways between macrostates. These results suggest that conformational equilibria between holo-like and alternative conformers pre-exist in the intrinsic dynamics of apo Pin1-WW. Analysis using MutInf, a mutual information method for quantifying correlated motions, reveals that WW dynamics not only play a role in substrate recognition, but also may help couple the substrate binding site on the WW domain to the one on the catalytic domain. Our work represents an important step towards building networks of inter-converting conformational states and is generally applicable. 1. f(R in Holographic and Agegraphic Dark Energy Models and the Generalized Uncertainty Principle Directory of Open Access Journals (Sweden) Barun Majumder 2013-01-01 Full Text Available We studied a unified approach with the holographic, new agegraphic, and f(R dark energy model to construct the form of f(R which in general is responsible for the curvature driven explanation of the very early inflation along with presently observed late time acceleration. We considered the generalized uncertainty principle in our approach which incorporated the corrections in the entropy-area relation and thereby modified the energy densities for the cosmological dark energy models considered. We found that holographic and new agegraphic f(R gravity models can behave like phantom or quintessence models in the spatially flat FRW universe. We also found a distinct term in the form of f(R which goes as R 3 / 2 due to the consideration of the GUP modified energy densities. Although the presence of this term in the action can be important in explaining the early inflationary scenario, Capozziello et al. recently showed that f(R ~ R 3 / 2 leads to an accelerated expansion, that is, a negative value for the deceleration parameter q which fits well with SNeIa and WMAP data. 2. Implications of conformal invariance in momentum space Energy Technology Data Exchange (ETDEWEB) Bzowski, Adam [Institute for Theoretical Physics,K.U. Leuven, Celestijnenlaan 200D, 3000 Leuven (Belgium); McFadden, Paul [Perimeter Institute for Theoretical Physics,31 Caroline St. N. Waterloo, N2L 2Y5 Ontario (Canada); Skenderis, Kostas [Mathematical Sciences, University of Southampton,Highfield, SO17 1BJ Southampton (United Kingdom) 2014-03-25 We present a comprehensive analysis of the implications of conformal invariance for 3-point functions of the stress-energy tensor, conserved currents and scalar operators in general dimension and in momentum space. Our starting point is a novel and very effective decomposition of tensor correlators which reduces their computation to that of a number of scalar form factors. For example, the most general 3-point function of a conserved and traceless stress-energy tensor is determined by only five form factors. Dilatations and special conformal Ward identities then impose additional conditions on these form factors. The special conformal Ward identities become a set of first and second order differential equations, whose general solution is given in terms of integrals involving a product of three Bessel functions (‘triple-K integrals’). All in all, the correlators are completely determined up to a number of constants, in agreement with well-known position space results. In odd dimensions 3-point functions are finite without renormalisation while in even dimensions non-trivial renormalisation in required. In this paper we restrict ourselves to odd dimensions. A comprehensive analysis of renormalisation will be discussed elsewhere. This paper contains two parts that can be read independently of each other. In the first part, we explain the method that leads to the solution for the correlators in terms of triple-K integrals while the second part contains a self-contained presentation of all results. Readers interested only in results may directly consult the second part of the paper. 3. Conformal Coating of Cobalt-Nickel Layered Double Hydroxides Nanoflakes on Carbon Fibers for High-performance Electrochemical Energy Storage Supercapacitor Devices KAUST Repository Warsi, Muhammad Farooq 2014-07-01 High specific capacitance coupled with the ease of large scale production is two desirable characteristics of a potential pseudo-supercapacitor material. In the current study, the uniform and conformal coating of nickel-cobalt layered double hydroxides (CoNi0.5LDH,) nanoflakes on fibrous carbon (FC) cloth has been achieved through cost-effective and scalable chemical precipitation method, followed by a simple heat treatment step. The conformally coated CoNi0.5LDH/FC electrode showed 1.5 times greater specific capacitance compared to the electrodes prepared by conventional non-conformal (drop casting) method of depositing CoNi0.5LDH powder on the carbon microfibers (1938 Fg-1 vs 1292 Fg-1). Further comparison of conformally and non-conformally coated CoNi0.5LDH electrodes showed the rate capability of 79%: 43% capacity retention at 50 Ag-1 and cycling stability 4.6%: 27.9% loss after 3000 cycles respectively. The superior performance of the conformally coated CoNi0.5LDH is mainly due to the reduced internal resistance and fast ionic mobility between electrodes as compared to non-conformally coated electrodes which is evidenced by EIS and CV studies. © 2014 Elsevier Ltd. 4. Conformal isoparametric hypersurfaces with two distinct conformal principal curvatures in conformal space Institute of Scientific and Technical Information of China (English) 2010-01-01 The conformal geometry of regular hypersurfaces in the conformal space is studied.We classify all the conformal isoparametric hypersurfaces with two distinct conformal principal curvatures in the conformal space up to conformal equivalence. 5. Deformed Potential Energy of Super Heavy Element Z = 120 in a Generalized Liquid Drop Model Institute of Scientific and Technical Information of China (English) CHEN Bao-Qiu; MA Zhong-Yu; ZHU Zhi-Yuan; SONG Hong-Qiu; ZHAO Yao-Lin 2005-01-01 @@ The macroscopic deformed potential energy for super-heavy elements Z = 120 is determined within a generalized liquid drop model (GLDM). The shell correction is calculated with the Strutinsky method and the microscopic single particle energies are derived from the shell model in an axially deformed Woods-Saxon potential with the same quasi-molecular shape. The total potential energy of a nucleus is calculated by the macro-microscopic method as the summation of the liquid-drop energy and the Strutinsky shell correction. The theory is adopted to describe the deformed potential energies in a set of cold reactions. The neck in the quasi-molecular shape is responsible to the deep valley of the fusion barrier due to shell corrections. In the cold fusion path, the doublehump fusion barrier is predicted by the shell correction and complete fusion events may occur. The results show that some of projectile-target combinations in the entrance channel, such as 50Ca+252Fm→ 302120* and 58Fe+244pu→ 302120, favour the fusion reaction, which can be considered as candidates for the synthesis of super heavy nuclei Z = 120 and the former might be the best cold fusion reaction to produce the nucleus 302120among them. 6. Conformal invariant saturation CERN Document Server Navelet, H 2002-01-01 We show that, in onium-onium scattering at (very) high energy, a transition to saturation happens due to quantum fluctuations of QCD dipoles. This transition starts when the order alpha^2 correction of the dipole loop is compensated by its faster energy evolution, leading to a negative interference with the tree level amplitude. After a derivation of the the one-loop dipole contribution using conformal invariance of the elastic 4-gluon amplitude in high energy QCD, we obtain an exact expression of the saturation line in the plane (Y,L) where Y is the total rapidity and L, the logarithm of the onium scale ratio. It shows universal features implying the Balitskyi - Fadin - Kuraev - Lipatov (BFKL) evolution kernel and the square of the QCD triple Pomeron vertex. For large L, only the higher BFKL Eigenvalue contributes, leading to a saturation depending on leading log perturbative QCD characteristics. For initial onium scales of same order, however, it involves an unlimited summation over all conformal BFKL Eigen... 7. Hot Conformal Gauge Theories CERN Document Server Mojaza, Matin; Sannino, Francesco 2010-01-01 We compute the nonzero temperature free energy up to the order g^6 \\ln(1/g) in the coupling constant for vector like SU(N) gauge theories featuring matter transforming according to different representations of the underlying gauge group. The number of matter fields, i.e. flavors, is arranged in such a way that the theory develops a perturbative stable infrared fixed point at zero temperature. Due to large distance conformality we trade the coupling constant with its fixed point value and define a reduced free energy which depends only on the number of flavors, colors and matter representation. We show that the reduced free energy changes sign, at the second, fifth and sixth order in the coupling, when decreasing the number of flavors from the upper end of the conformal window. If the change in sign is interpreted as signal of an instability of the system then we infer a critical number of flavors. Surprisingly this number, if computed to the order g^2, agrees with previous predictions for the lower boundary o... 8. Impacts of the EISA 2007 Energy Efficiency Standard on General Service Lamps Energy Technology Data Exchange (ETDEWEB) Kantner, Colleen L. [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Alstone, Andrea L. [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Ganeshalingam, Mohan [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Gerke, Brian F. [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Hosbach, Robert [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States) 2017-01-20 The Energy Policy and Conservation Act of 1975, as amended by the Energy Independence and Security Act of 2007 (EISA 2007), requires that, effective beginning January 1, 2020, the Secretary of Energy shall prohibit the sale of any general service lamp (GSL) that does not meet a minimum efficacy standard of 45 lumens per watt. This is referred to as the EISA 2007 backstop. The U.S. Department of Energy recently revised the definition of the term GSL to include certain lamps that were either previously excluded or not explicitly mentioned in the EISA 2007 definition. For this subset of GSLs, we assess the impacts of the EISA 2007 backstop on national energy consumption, carbon dioxide emissions, and consumer expenditures. To estimate these impacts, we projected the energy use, purchase price, and operating cost of representative lamps purchased during a 30-year analysis period, 2020-2049, for cases in which the EISA 2007 backstop does and does not take effect; the impacts of the backstop are then given by the difference between the two cases. In developing the projection model, we also performed the most comprehensive assessment to date of usage patterns and lifetime distributions for the analyzed lamp types in the United States. There is substantial uncertainty in the estimated impacts, which arises from uncertainty in the speed and extent of the market conversion to solid state lighting technology that would occur in the absence of the EISA 2007 backstop. In our central estimate we find that the EISA 2007 backstop results in significant energy savings of 27 quads and consumer net present value of$120 billion (at a seven percent discount rate) for lamps shipped between 2020 and 2049, and carbon dioxide emissions reduction of 540 million metric tons by 2030 for those GSLs not explicitly included in the EISA 2007 definition of a GSL. 9. Impact of the EISA 2007 Energy Efficiency Standard on General Service Lamps Energy Technology Data Exchange (ETDEWEB) Kantner, Colleen L.S. [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Alstone, Andrea L. [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Ganeshalingam, Mohan [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Gerke, Brian F. [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Hosbach, Robert [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States) 2017-06-20 The Energy Policy and Conservation Act of 1975, as amended by the Energy Independence and Security Act of 2007 (EISA 2007), requires that, effective beginning January 1, 2020, the Secretary of Energy shall prohibit the sale of any general service lamp (GSL) that does not meet a minimum efficacy standard of 45 lumens per watt. This is referred to as the EISA 2007 backstop. The U.S. Department of Energy recently revised the definition of the term GSL to include certain lamps that were either previously excluded or not explicitly mentioned in the EISA 2007 definition. For this subset of GSLs, we assess the impacts of the EISA 2007 backstop on national energy consumption, carbon dioxide emissions, and consumer expenditures. To estimate these impacts, we projected the energy use, purchase price, and operating cost of representative lamps purchased during a 30-year analysis period, 2020-2049, for cases in which the EISA 2007 backstop does and does not take effect; the impacts of the backstop are then given by the difference between the two cases. In developing the projection model, we also performed the most comprehensive assessment to date of usage patterns and lifetime distributions for the analyzed lamp types in the United States. There is substantial uncertainty in the estimated impacts, which arises from uncertainty in the speed and extent of the market conversion to solid state lighting technology that would occur in the absence of the EISA 2007 backstop. In our central estimate we find that the EISA 2007 backstop results in significant energy savings of 27 quads and consumer net present value of $120 billion (at a seven percent discount rate) for lamps shipped between 2020 and 2049, and carbon dioxide emissions reduction of 540 million metric tons by 2030 for those GSLs not explicitly included in the EISA 2007 definition of a GSL. 10. Nonlocal gravity: Conformally flat spacetimes CERN Document Server Bini, Donato 2016-01-01 The field equations of the recent nonlocal generalization of Einstein's theory of gravitation are presented in a form that is reminiscent of general relativity. The implications of the nonlocal field equations are studied in the case of conformally flat spacetimes. Even in this simple case, the field equations are intractable. Therefore, to gain insight into the nature of these equations, we investigate the structure of nonlocal gravity in two-dimensional spacetimes. While any smooth 2D spacetime is conformally flat and satisfies Einstein's field equations, only a subset containing either a Killing vector or a homothetic Killing vector can satisfy the field equations of nonlocal gravity. 11. Generalization of classical mechanics for nuclear motions on nonadiabatically coupled potential energy surfaces in chemical reactions. Science.gov (United States) Takatsuka, Kazuo 2007-10-18 Classical trajectory study of nuclear motion on the Born-Oppenheimer potential energy surfaces is now one of the standard methods of chemical dynamics. In particular, this approach is inevitable in the studies of large molecular systems. However, as soon as more than a single potential energy surface is involved due to nonadiabatic coupling, such a naive application of classical mechanics loses its theoretical foundation. This is a classic and fundamental issue in the foundation of chemistry. To cope with this problem, we propose a generalization of classical mechanics that provides a path even in cases where multiple potential energy surfaces are involved in a single event and the Born-Oppenheimer approximation breaks down. This generalization is made by diagonalization of the matrix representation of nuclear forces in nonadiabatic dynamics, which is derived from a mixed quantum-classical representation of the electron-nucleus entangled Hamiltonian [Takatsuka, K. J. Chem. Phys. 2006, 124, 064111]. A manifestation of quantum fluctuation on a classical subsystem that directly contacts with a quantum subsystem is discussed. We also show that the Hamiltonian thus represented gives a theoretical foundation to examine the validity of the so-called semiclassical Ehrenfest theory (or mean-field theory) for electron quantum wavepacket dynamics, and indeed, it is pointed out that the electronic Hamiltonian to be used in this theory should be slightly modified. 12. High energy conformers of M(+)(APE)(H2O)(0-1)Ar(0-1) clusters revealed by combined IR-PD and DFT-MD anharmonic vibrational spectroscopy. Science.gov (United States) Brites, V; Nicely, A L; Sieffert, N; Gaigeot, M-P; Lisy, J M 2014-07-14 IR-PD vibrational spectroscopy and DFT-based molecular dynamics simulations are combined in order to unravel the structures of M(+)(APE)(H2O)0-1 ionic clusters (M = Na, K), where APE (2-amino-1-phenyl ethanol) is commonly used as an analogue for the noradrenaline neurotransmitter. The strength of the synergy between experiments and simulations presented here is that DFT-MD provides anharmonic vibrational spectra that unambiguously help assign the ionic clusters structures. Depending on the interacting cation, we have found that the lowest energy conformers of K(+)(APE)(H2O)0-1 clusters are formed, while the lowest energy conformers of Na(+)(APE)(H2O)0-1 clusters can only be observed through water loss channel (i.e. without argon tagged to the clusters). Trapping of higher energy conformers is observed when the argon loss channel is recorded in the experiment. This has been rationalized by transition state energies. The dynamical anharmonic vibrational spectra unambiguously provide the prominent OH stretch due to the OH···NH2 H-bond, within 10 cm(-1) of the experiment, hence reproducing the 240-300 cm(-1) red-shift (depending on the interacting cation) from bare neutral APE. When this H-bond is not present, the dynamical anharmonic spectra provide the water O-H stretches as well as the rotational motion of the water molecule at finite temperature, as observed in the experiment. 13. Spatial separation of molecular conformers and clusters. Science.gov (United States) Horke, Daniel; Trippel, Sebastian; Chang, Yuan-Pin; Stern, Stephan; Mullins, Terry; Kierspel, Thomas; Küpper, Jochen 2014-01-09 Gas-phase molecular physics and physical chemistry experiments commonly use supersonic expansions through pulsed valves for the production of cold molecular beams. However, these beams often contain multiple conformers and clusters, even at low rotational temperatures. We present an experimental methodology that allows the spatial separation of these constituent parts of a molecular beam expansion. Using an electric deflector the beam is separated by its mass-to-dipole moment ratio, analogous to a bender or an electric sector mass spectrometer spatially dispersing charged molecules on the basis of their mass-to-charge ratio. This deflector exploits the Stark effect in an inhomogeneous electric field and allows the separation of individual species of polar neutral molecules and clusters. It furthermore allows the selection of the coldest part of a molecular beam, as low-energy rotational quantum states generally experience the largest deflection. Different structural isomers (conformers) of a species can be separated due to the different arrangement of functional groups, which leads to distinct dipole moments. These are exploited by the electrostatic deflector for the production of a conformationally pure sample from a molecular beam. Similarly, specific cluster stoichiometries can be selected, as the mass and dipole moment of a given cluster depends on the degree of solvation around the parent molecule. This allows experiments on specific cluster sizes and structures, enabling the systematic study of solvation of neutral molecules. 14. RECOVERY AND ENERGY SAVINGS OF ALUMINUM CAN BEVERAGE CONSUMED IN GENERAL AND VOCATIONAL TECHNICAL HIGH SCHOOLS Directory of Open Access Journals (Sweden) Mert ZORAĞA 2012-01-01 Full Text Available In commitments of Kyoto protocol principles, 100% recyclable features aluminum is one of most current metal. In this protocol, Turkey is not contractor to develop policies to prevent climate change to apply, to take measures to increase energy efficiency and savings, to limit greenhouse gas emissions. Aluminum production from used aluminum requires 95% less energy than production from raw material and recycled aluminum put in the production reduces flue gases pollutant emissions at rate of 99%. Between 2004-2005 and 2009-2010 academic year education is estimated that every one of 5 and 10 students were consumed average 1 aluminum can beverage each day to take into account habits of general and vocational high school students. In case of recovery of 50% this cans will save approximately 4.7 and 13.1 million kWh electrical energy, in the case of 75% recovery will save between 7.2 and 19.9 million kWh electrical energy, in the case of 100% will save the 9.4 and 25 million kWh electrical energy than the same amount of aluminum in the primary method (from ore in our country. In the same conditions is estimated that realization of an efficient recycling project will provide between 5.2 and 20 million kWh of electrical energy savings in the 2010 -2011 academic year education. In this study, anymore it turned into a habit of recovery of packaging waste application in most countries as the name “Blue Angels Project” to place in our country has been trying to bring clarity to issues. 15. On the generalized eigenvalue method for energies and matrix elements in lattice field theory CERN Document Server Blossier, Benoit; von Hippel, Georg; Mendes, Tereza; Sommer, Rainer 2009-01-01 We discuss the generalized eigenvalue problem for computing energies and matrix elements in lattice gauge theory, including effective theories such as HQET. It is analyzed how the extracted effective energies and matrix elements converge when the time separations are made large. This suggests a particularly efficient application of the method for which we can prove that corrections vanish asymptotically as$\\exp(-(E_{N+1}-E_n) t)$. The gap$E_{N+1}-E_n$can be made large by increasing the number$N$of interpolating fields in the correlation matrix. We also show how excited state matrix elements can be extracted such that contaminations from all other states disappear exponentially in time. As a demonstration we present numerical results for the extraction of ground state and excited B-meson masses and decay constants in static approximation and to order$1/m_b$in HQET. 16. Energy and momentum of general spherically symmetric frames on the regularizing teleparallelism Institute of Scientific and Technical Information of China (English) Gamal G. L. Nashed 2012-01-01 In the context of the covariant teleparallel framework,we use the 2-form translational momentum to compute the total energy of two general spherically symmetric frames.The first one is characterized by an arbitrary function H(γ),which preserves the spherical symmetry and reproduces all the previous solutions,while the other one is characterized by a parameter ξ which ensures the vanishing of the axial of trace of the torsion.We calculate the total energy by using two procedures,i.e.,when the Weitzenb?ck connection Γαβ is trivial,and show how H(r) and ξ play the role of an inertia that leads the total energy to be unphysical.Therefore,we take into account Γαβ and show that although the spacetimes we use contain an arbitrary function and one parameter,they have no effect on the form of the total energy and momentum as it should be. 17. A generalized free energy perturbation theory accounting for end states with differing configuration space volume. Science.gov (United States) Ullmann, R Thomas; Ullmann, G Matthias 2011-01-27 We present a generalized free energy perturbation theory that is inspired by Monte Carlo techniques and based on a microstate description of a transformation between two states of a physical system. It is shown that the present free energy perturbation theory stated by the Zwanzig equation follows as a special case of our theory. Our method uses a stochastic mapping of the end states that associates a given microstate from one ensemble with a microstate from the adjacent ensemble according to a probability distribution. In contrast, previous free energy perturbation methods use a static, deterministic mapping that associates fixed pairs of microstates from the two ensembles. The advantages of our approach are that end states of differing configuration space volume can be treated easily also in the case of discrete configuration spaces and that the method does not require the potentially cumbersome search for an optimal deterministic mapping. The application of our theory is illustrated by some example problems. We discuss practical applications for which our findings could be relevant and point out perspectives for further development of the free energy perturbation theory. 18. Quasi-Local Energy-Momentum and Angular Momentum in General Relativity Directory of Open Access Journals (Sweden) Szabados László B. 2009-06-01 Full Text Available The present status of the quasi-local mass, energy-momentum and angular-momentum constructions in general relativity is reviewed. First, the general ideas, concepts, and strategies, as well as the necessary tools to construct and analyze the quasi-local quantities, are recalled. Then, the various specific constructions and their properties (both successes and deficiencies are discussed. Finally, some of the (actual and potential applications of the quasi-local concepts and specific constructions are briefly mentioned.This review is based on talks given at the Erwin Schrödinger Institute, Vienna in July 1997, at the Universität Tübingen in May 1998, and at the National Center for Theoretical Sciences in Hsinchu, Taiwan and at the National Central University, Chungli, Taiwan, in July 2000. 19. Energy distributions of Bianchi type-$VI_{h}$Universe in general relativity and teleparallel gravity Indian Academy of Sciences (India) SEREF ÖZKURT; SEZGIN AYGÜN 2017-04-01 In this paper, we have investigated the energy and momentum density distributions for the inhomogeneous generalizations of homogeneous Bianchi type-$VI_{h}$metric with Einstein, Bergmann–Thomson, Landau–Lifshitz,Papapetrou, Tolman and M$\\phi$ller prescriptions in general relativity (GR) and teleparallel gravity (TG). We have found exactly the same results for Einstein, Bergmann–Thomson and Landau–Lifshitz energy–momentum distributions in Bianchi type-$VI_{h}$metric for different gravitation theories. The energy–momentum distributions of the Bianchi type-$VI_{h}$metric are found to be zero for$h$= −1 in GR and TG. However, our results agree with Tripathy et al, Tryon, Rosen and Aygün et al. 20. Quartet-metric general relativity: scalar graviton, dark matter, and dark energy Energy Technology Data Exchange (ETDEWEB) Pirogov, Yury F. [SRC Institute for High Energy Physics of NRC Kurchatov Institute, Protvino (Russian Federation) 2016-04-15 General relativity extended through a dynamical scalar quartet is proposed as a theory of the scalar-vector-tensor gravity, generically describing the unified gravitational dark matter (DM) and dark energy (DE). The implementation in the weak-field limit of the Higgs mechanism for the extended gravity, with a redefinition of metric field, is exposed in a generally covariant form. Under a natural restriction on the parameters, the redefined theory possesses in the linearized approximation a residual transverse-diffeomorphism invariance, and consistently comprises the massless tensor graviton and a massive scalar one as a DM particle. The number of adjustable parameters in the full nonlinear theory and a partial decoupling of the latter from its weak-field limit noticeably extend the perspectives for the unified description of the gravity DM and DE in the various phenomena at the different scales. (orig.) 1. Generalized Analysis of a Distributed Energy Efficient Algorithm for Change Detection CERN Document Server Banerjee, Taposh 2009-01-01 An energy efficient distributed Change Detection scheme based on Page's CUSUM algorithm was presented in \\cite{icassp}. In this paper we consider a nonparametric version of this algorithm. In the algorithm in \\cite{icassp}, each sensor runs CUSUM and transmits only when the CUSUM is above some threshold. The transmissions from the sensors are fused at the physical layer. The channel is modeled as a Multiple Access Channel (MAC) corrupted with noise. The fusion center performs another CUSUM to detect the change. In this paper, we generalize the algorithm to also include nonparametric CUSUM and provide a unified analysis. 2. On the stability of the dark energy based on generalized uncertainty principle CERN Document Server Pasqua, Antonio; Khomenko, Iuliia 2013-01-01 The new agegraphic Dark Energy (NADE) model (based on generalized uncertainty principle) interacting with Dark Matter (DM) is considered in this study via power-law form of the scale factor$a(t)$. The equation of state (EoS) parameter$\\omega_{G}$is observed to have a phantom-like behaviour. The stability of this model is investigated through the squared speed of sound$v_{s}^{2}$: it is found that$v_{s}^{2}$always stays at negative level, which indicates instability of the considered model. 3. Generalized Least Energy of Separation for Desalination and Other Chemical Separation Processes Directory of Open Access Journals (Sweden) Karan H. Mistry 2013-05-01 Full Text Available Increasing global demand for fresh water is driving the development and implementation of a wide variety of seawater desalination technologies driven by different combinations of heat, work, and chemical energy. This paper develops a consistent basis for comparing the energy consumption of such technologies using Second Law efficiency. The Second Law efficiency for a chemical separation process is defined in terms of the useful exergy output, which is the minimum least work of separation required to extract a unit of product from a feed stream of a given composition. For a desalination process, this is the minimum least work of separation for producing one kilogram of product water from feed of a given salinity. While definitions in terms of work and heat input have been proposed before, this work generalizes the Second Law efficiency to allow for systems that operate on a combination of energy inputs, including fuel. The generalized equation is then evaluated through a parametric study considering work input, heat inputs at various temperatures, and various chemical fuel inputs. Further, since most modern, large-scale desalination plants operate in cogeneration schemes, a methodology for correctly evaluating Second Law efficiency for the desalination plant based on primary energy inputs is demonstrated. It is shown that, from a strictly energetic point of view and based on currently available technology, cogeneration using electricity to power a reverse osmosis system is energetically superior to thermal systems such as multiple effect distillation and multistage flash distillation, despite the very low grade heat input normally applied in those systems. 4. The sensitivity of BAO dark energy constraints to general isocurvature perturbations Energy Technology Data Exchange (ETDEWEB) Kasanda, S. Muya; Zunckel, C.; Moodley, K. [School of Mathematical Sciences, University of KwaZulu-Natal, University Road, Durban, 4041 (South Africa); Bassett, B.A.; Okouma, P., E-mail: [email protected], E-mail: [email protected], E-mail: [email protected], E-mail: [email protected], E-mail: [email protected] [Dept. of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, Cape Town (South Africa) 2012-07-01 Baryon Acoustic Oscillation (BAO) surveys will be a leading method for addressing the dark energy challenge in the next decade. We explore in detail the effect of allowing for small amplitude admixtures of general isocurvature perturbations in addition to the dominant adiabatic mode. We find that non-adiabatic initial conditions leave the sound speed unchanged but instead excite different harmonics. These harmonics couple differently to Silk damping, altering the form and evolution of acoustic waves in the baryon-photon fluid prior to decoupling. This modifies not only the scale on which the sound waves imprint onto the baryon distribution, which is used as the standard ruler in BAO surveys, but also the shape, width and height of the BAO peak. We discuss these effects in detail and show how more general initial conditions impact our interpretation of cosmological data in dark energy studies. We find that the inclusion of these additional isocurvature modes leads to a decrease in the Dark Energy Task Force figure of merit (FoM) by 46% i.e., FoM{sub ISO} = 0.54 × FoM{sub AD} and 53% for the BOSS and ADEPT experiments respectively when considered in conjunction with PLANK data. We also show that the incorrect assumption of adiabaticity has the potential to bias our estimates of the dark energy parameters by 2.7σ (2.2σ) for a single correlated isocurvature mode (CDM isocurvature), and up to 4.9σ (5.7σ) for three correlated isocurvature modes in the case of the BOSS (ADEPT) experiment. We find that the use of the large scale structure data in conjunction with CMB data improves our ability to measure the contributions of different modes to the initial conditions by as much as 95% for certain modes in the fully correlated case. 5. Logarithmic conformal field theory Science.gov (United States) Gainutdinov, Azat; Ridout, David; Runkel, Ingo 2013-12-01 Conformal field theory (CFT) has proven to be one of the richest and deepest subjects of modern theoretical and mathematical physics research, especially as regards statistical mechanics and string theory. It has also stimulated an enormous amount of activity in mathematics, shaping and building bridges between seemingly disparate fields through the study of vertex operator algebras, a (partial) axiomatisation of a chiral CFT. One can add to this that the successes of CFT, particularly when applied to statistical lattice models, have also served as an inspiration for mathematicians to develop entirely new fields: the Schramm-Loewner evolution and Smirnov's discrete complex analysis being notable examples. When the energy operator fails to be diagonalisable on the quantum state space, the CFT is said to be logarithmic. Consequently, a logarithmic CFT is one whose quantum space of states is constructed from a collection of representations which includes reducible but indecomposable ones. This qualifier arises because of the consequence that certain correlation functions will possess logarithmic singularities, something that contrasts with the familiar case of power law singularities. While such logarithmic singularities and reducible representations were noted by Rozansky and Saleur in their study of the U (1\|1) Wess-Zumino-Witten model in 1992, the link between the non-diagonalisability of the energy operator and logarithmic singularities in correlators is usually ascribed to Gurarie's 1993 article (his paper also contains the first usage of the term 'logarithmic conformal field theory'). The class of CFTs that were under control at this time was quite small. In particular, an enormous amount of work from the statistical mechanics and string theory communities had produced a fairly detailed understanding of the (so-called) rational CFTs. However, physicists from both camps were well aware that applications from many diverse fields required significantly more 6. Non-conformable, partial and conformable transposition DEFF Research Database (Denmark) König, Thomas; Mäder, Lars Kai 2013-01-01 Although member states are obliged to transpose directives into domestic law in a conformable manner and receive considerable time for their transposition activities, we identify three levels of transposition outcomes for EU directives: conformable, partially conformable and non-conformable....... Compared with existing transposition models, which do not distinguish between different transposition outcomes, we examine the factors influencing each transposition process by means of a competing risk analysis. We find that preference-related factors, in particular the disagreement of a member state...... and the Commission regarding a directive’s outcome, play a much more strategic role than has to date acknowledged in the transposition literature. Whereas disagreement of a member state delays conformable transposition, it speeds up non-conformable transposition. Disagreement of the Commission only prolongs... 7. Conformational landscape of an amyloid intra-cellular domain and Landau-Ginzburg-Wilson paradigm in protein dynamics Science.gov (United States) Dai, Jin; Niemi, Antti J.; He, Jianfeng 2016-07-01 The Landau-Ginzburg-Wilson paradigm is proposed as a framework, to investigate the conformational landscape of intrinsically unstructured proteins. A universal Cα-trace Landau free energy is deduced from general symmetry considerations, with the ensuing all-atom structure modeled using publicly available reconstruction programs Pulchra and Scwrl. As an example, the conformational stability of an amyloid precursor protein intra-cellular domain (AICD) is inspected; the reference conformation is the crystallographic structure with code 3DXC in Protein Data Bank (PDB) that describes a heterodimer of AICD and a nuclear multi-domain adaptor protein Fe65. Those conformations of AICD that correspond to local or near-local minima of the Landau free energy are identified. For this, the response of the original 3DXC conformation to variations in the ambient temperature is investigated, using the Glauber algorithm. The conclusion is that in isolation the AICD conformation in 3DXC must be unstable. A family of degenerate conformations that minimise the Landau free energy is identified, and it is proposed that the native state of an isolated AICD is a superposition of these conformations. The results are fully in line with the presumed intrinsically unstructured character of isolated AICD and should provide a basis for a systematic analysis of AICD structure in future NMR experiments. 8. Conformational landscape of an amyloid intra-cellular domain and Landau-Ginzburg-Wilson paradigm in protein dynamics. Science.gov (United States) Dai, Jin; Niemi, Antti J; He, Jianfeng 2016-07-28 The Landau-Ginzburg-Wilson paradigm is proposed as a framework, to investigate the conformational landscape of intrinsically unstructured proteins. A universal Cα-trace Landau free energy is deduced from general symmetry considerations, with the ensuing all-atom structure modeled using publicly available reconstruction programs Pulchra and Scwrl. As an example, the conformational stability of an amyloid precursor protein intra-cellular domain (AICD) is inspected; the reference conformation is the crystallographic structure with code 3DXC in Protein Data Bank (PDB) that describes a heterodimer of AICD and a nuclear multi-domain adaptor protein Fe65. Those conformations of AICD that correspond to local or near-local minima of the Landau free energy are identified. For this, the response of the original 3DXC conformation to variations in the ambient temperature is investigated, using the Glauber algorithm. The conclusion is that in isolation the AICD conformation in 3DXC must be unstable. A family of degenerate conformations that minimise the Landau free energy is identified, and it is proposed that the native state of an isolated AICD is a superposition of these conformations. The results are fully in line with the presumed intrinsically unstructured character of isolated AICD and should provide a basis for a systematic analysis of AICD structure in future NMR experiments. 9. Cavity as a source of conformational fluctuation and high-energy state: High-pressure NMR study of a cavity-enlarged mutant of T4 lysozyme CERN Document Server Maeno, Akihiro; Hirata, Fumio; Otten, Renee; Dahlquist, Frederick W; Yokoyama, Shigeyuki; Akasaka, Kazuyuki; Mulder, Frans A A; Kitahara, Ryo 2014-01-01 Although the structure, function, conformational dynamics, and controlled thermodynamics of proteins are manifested by their corresponding amino acid sequences, the natural rules for molecular design and their corresponding interplay remain obscure. In this study, we focused on the role of internal cavities of proteins in conformational dynamics. We investigated the pressure-induced responses from the cavity-enlarged L99A mutant of T4 lysozyme, using high-pressure NMR spectroscopy. The signal intensities of the methyl groups in the 1H/13C HSQC spectra, particularly those around the enlarged cavity, decreased with the increasing pressure, and disappeared at 200 MPa, without the appearance of new resonances, thus indicating the presence of heterogeneous conformations around the cavity within the ground state ensemble. Above 200 MPa, the signal intensities of more than 20 methyl groups gradually decreased with the increasing pressure, without the appearance of new resonances. Interestingly, these residues closel... 10. Dissipation, generalized free energy, and a self-consistent nonequilibrium thermodynamics of chemically driven open subsystems. Science.gov (United States) Ge, Hao; Qian, Hong 2013-06-01 Nonequilibrium thermodynamics of a system situated in a sustained environment with influx and efflux is usually treated as a subsystem in a larger, closed "universe." A question remains with regard to what the minimally required description for the surrounding of such an open driven system is so that its nonequilibrium thermodynamics can be established solely based on the internal stochastic kinetics. We provide a solution to this problem using insights from studies of molecular motors in a chemical nonequilibrium steady state (NESS) with sustained external drive through a regenerating system or in a quasisteady state (QSS) with an excess amount of adenosine triphosphate (ATP), adenosine diphosphate (ADP), and inorganic phosphate (Pi). We introduce the key notion of minimal work that is needed, W(min), for the external regenerating system to sustain a NESS (e.g., maintaining constant concentrations of ATP, ADP and Pi for a molecular motor). Using a Markov (master-equation) description of a motor protein, we illustrate that the NESS and QSS have identical kinetics as well as the second law in terms of the same positive entropy production rate. The heat dissipation of a NESS without mechanical output is exactly the W(min). This provides a justification for introducing an ideal external regenerating system and yields a free-energy balance equation between the net free-energy input F(in) and total dissipation F(dis) in an NESS: F(in) consists of chemical input minus mechanical output; F(dis) consists of dissipative heat, i.e. the amount of useful energy becoming heat, which also equals the NESS entropy production. Furthermore, we show that for nonstationary systems, the F(dis) and F(in) correspond to the entropy production rate and housekeeping heat in stochastic thermodynamics and identify a relative entropy H as a generalized free energy. We reach a new formulation of Markovian nonequilibrium thermodynamics based on only the internal kinetic equation without further 11. Impact of Dust on Mars Surface Albedo and Energy Flux with LMD General Circulation Model Science.gov (United States) Singh, D.; Flanner, M.; Millour, E.; Martinez, G. 2015-12-01 Mars, just like Earth experience different seasons because of its axial tilt (about 25°). This causes growth and retreat of snow cover (primarily CO2) in Martian Polar regions. The perennial caps are the only place on the planet where condensed H2O is available at surface. On Mars, as much as 30% atmospheric CO2 deposits in each hemisphere depending upon the season. This leads to a significant variation on planet's surface albedo and hence effecting the amount of solar flux absorbed or reflected at the surface. General Circulation Model (GCM) of Laboratoire de Météorologie Dynamique (LMD) currently uses observationally derived surface albedo from Thermal Emission Spectrometer (TES) instrument for the polar caps. These TES albedo values do not have any inter-annual variability, and are independent of presence of any dust/impurity on surface. Presence of dust or other surface impurities can significantly reduce the surface albedo especially during and right after a dust storm. This change will also be evident in the surface energy flux interactions. Our work focuses on combining earth based Snow, Ice, and Aerosol Radiation (SNICAR) model with current state of GCM to incorporate the impact of dust on Martian surface albedo, and hence the energy flux. Inter-annual variability of surface albedo and planet's top of atmosphere (TOA) energy budget along with their correlation with currently available mission data will be presented. 12. Generalized gradient approximation exchange energy functional with correct asymptotic behavior of the corresponding potential Energy Technology Data Exchange (ETDEWEB) Carmona-Espíndola, Javier, E-mail: [email protected] [Departamento de Química, Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186, México D. F. 09340, México (Mexico); Gázquez, José L., E-mail: [email protected] [Departamento de Química, Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186, México D. F. 09340, México (Mexico); Departamento de Química, Centro de Investigación y de Estudios Avanzados, Av. Instituto Politécnico Nacional 2508, México D. F. 07360, México (Mexico); Vela, Alberto [Departamento de Química, Centro de Investigación y de Estudios Avanzados, Av. Instituto Politécnico Nacional 2508, México D. F. 07360, México (Mexico); Trickey, S. B. [Quantum Theory Project, Department of Physics and Department of Chemistry, University of Florida, P.O. Box 118435, Gainesville, Florida 32611-8435 (United States) 2015-02-07 A new non-empirical exchange energy functional of the generalized gradient approximation (GGA) type, which gives an exchange potential with the correct asymptotic behavior, is developed and explored. In combination with the Perdew-Burke-Ernzerhof (PBE) correlation energy functional, the new CAP-PBE (CAP stands for correct asymptotic potential) exchange-correlation functional gives heats of formation, ionization potentials, electron affinities, proton affinities, binding energies of weakly interacting systems, barrier heights for hydrogen and non-hydrogen transfer reactions, bond distances, and harmonic frequencies on standard test sets that are fully competitive with those obtained from other GGA-type functionals that do not have the correct asymptotic exchange potential behavior. Distinct from them, the new functional provides important improvements in quantities dependent upon response functions, e.g., static and dynamic polarizabilities and hyperpolarizabilities. CAP combined with the Lee-Yang-Parr correlation functional gives roughly equivalent results. Consideration of the computed dynamical polarizabilities in the context of the broad spectrum of other properties considered tips the balance to the non-empirical CAP-PBE combination. Intriguingly, these improvements arise primarily from improvements in the highest occupied and lowest unoccupied molecular orbitals, and not from shifts in the associated eigenvalues. Those eigenvalues do not change dramatically with respect to eigenvalues from other GGA-type functionals that do not provide the correct asymptotic behavior of the potential. Unexpected behavior of the potential at intermediate distances from the nucleus explains this unexpected result and indicates a clear route for improvement. 13. The Energy-Momentum Tensor for a Dissipative Fluid in General Relativity CERN Document Server Pimentel, Oscar M; Lora-Clavijo, F D 2016-01-01 Considering the growing interest of the astrophysicist community in the study of dissipative fluids with the aim of getting a more realistic description of the universe, we present in this paper a physical analysis of the energy-momentum tensor of a viscous fluid with heat flux. We introduce the general form of this tensor and, using the approximation of small velocity gradients, we relate the stresses of the fluid with the viscosity coefficients, the shear tensor and the expansion factor. Exploiting these relations, we can write the stresses in terms of the extrinsic curvature of the normal surface to the 4-velocity vector of the fluid, and we can also establish a connection between the perfect fluid and the symmetries of the spacetime. On the other hand, we calculate the energy conditions for a dissipative fluid through contractions of the energy-momentum tensor with the 4-velocity vector of an arbitrary observer. This method is interesting because it allows us to compute the conditions in a reasonable easy... 14. Charged Axially Symmetric Solution and Energy in Teleparallel Theory Equivalent to General Relativity CERN Document Server Nashed, Gamal Gergess Lamee 2007-01-01 An exact charged solution with axial symmetry is obtained in the teleparallel equivalent of general relativity (TEGR). The associated metric has the structure function$G(\\xi)=1-{\\xi}^2-2mA{\\xi}^3-q^2A^2{\\xi}^4$. The fourth order nature of the structure function can make calculations cumbersome. Using a coordinate transformation we get a tetrad whose metric has the structure function in a factorisable form$(1-{\\xi}^2)(1+r_{+}A\\xi)(1+r_{-}A\\xi)$with$r_{\\pm}$as the horizons of Reissner-Nordstr$\\ddot{o}$m space-time. This new form has the advantage that its roots are now trivial to write down. Then, we study the singularities of this space-time. Using another coordinate transformation, we obtain a tetrad field. Its associated metric yields the Reissner-Nordstr$\\ddot{o}$m black hole. In Calculating the energy content of this tetrad field using the gravitational energy-momentum, we find that the resulting form depends on the radial coordinate! Using the regularized expression of the gravitational energy-moment... 15. Generalized Extreme Value Distribution Models for the Assessment of Seasonal Wind Energy Potential of Debuncha, Cameroon Directory of Open Access Journals (Sweden) Nkongho Ayuketang Arreyndip 2016-01-01 Full Text Available The method of generalized extreme value family of distributions (Weibull, Gumbel, and Frechet is employed for the first time to assess the wind energy potential of Debuncha, South-West Cameroon, and to study the variation of energy over the seasons on this site. The 29-year (1983–2013 average daily wind speed data over Debuncha due to missing values in the years 1992 and 1994 is gotten from NASA satellite data through the RETScreen software tool provided by CANMET Canada. The data is partitioned into min-monthly, mean-monthly, and max-monthly data and fitted using maximum likelihood method to the two-parameter Weibull, Gumbel, and Frechet distributions for the purpose of determining the best fit to be used for assessing the wind energy potential on this site. The respective shape and scale parameters are estimated. By making use of the P values of the Kolmogorov-Smirnov statistic (K-S and the standard error (s.e analysis, the results show that the Frechet distribution best fits the min-monthly, mean-monthly, and max-monthly data compared to the Weibull and Gumbel distributions. Wind speed distributions and wind power densities of both the wet and dry seasons are compared. The results show that the wind power density of the wet season was higher than in the dry season. The wind speeds at this site seem quite low; maximum wind speeds are listed as between 3.1 and 4.2 m/s, which is below the cut-in wind speed of many modern turbines (6–10 m/s. However, we recommend the installation of low cut-in wind turbines like the Savonius or Aircon (10 KW for stand-alone low energy need. 16. The conformal approach to asymptotic analysis CERN Document Server Nicolas, Jean-Philippe 2015-01-01 This essay was written as an extended version of a talk given at a conference in Strasbourg on "Riemann, Einstein and geometry", organized by Athanase Papadopoulos in September 2014. Its aim is to present Roger Penrose's approach to asymptotic analysis in general relativity, which is based on conformal geometric techniques, focusing on historical and recent aspects of two specialized topics~: conformal scattering and peeling. 17. Asymptotic symmetry algebra of conformal gravity CERN Document Server Irakleidou, M 2016-01-01 We compute asymptotic symmetry algebras of conformal gravity. Due to more general boundary conditions allowed in conformal gravity in comparison to those in Einstein gravity, we can classify the corresponding algebras. The highest algebra for non-trivial boundary conditions is five dimensional and it leads to global geon solution with non-vanishing charges. 18. An extension theorem for conformal gauge singularities CERN Document Server Tod, Paul 2007-01-01 We analyse conformal gauge, or isotropic, singularities in cosmological models in general relativity. Using the calculus of tractors, we find conditions in terms of tractor curvature for a local extension of the conformal structure through a cosmological singularity and prove a local extension theorem. 19. Using Graphs of Gibbs Energy versus Temperature in General Chemistry Discussions of Phase Changes and Colligative Properties Science.gov (United States) Hanson, Robert M.; Riley, Patrick; Schwinefus, Jeff; Fischer, Paul J. 2008-01-01 The use of qualitative graphs of Gibbs energy versus temperature is described in the context of chemical demonstrations involving phase changes and colligative properties at the general chemistry level. (Contains 5 figures and 1 note.) 20. Measuring the mechanical properties of molecular conformers Science.gov (United States) Jarvis, S. P.; Taylor, S.; Baran, J. D.; Champness, N. R.; Larsson, J. A.; Moriarty, P. 2015-09-01 Scanning probe-actuated single molecule manipulation has proven to be an exceptionally powerful tool for the systematic atomic-scale interrogation of molecular adsorbates. To date, however, the extent to which molecular conformation affects the force required to push or pull a single molecule has not been explored. Here we probe the mechanochemical response of two tetra(4-bromophenyl)porphyrin conformers using non-contact atomic force microscopy where we find a large difference between the lateral forces required for manipulation. Remarkably, despite sharing very similar adsorption characteristics, variations in the potential energy surface are capable of prohibiting probe-induced positioning of one conformer, while simultaneously permitting manipulation of the alternative conformational form. Our results are interpreted in the context of dispersion-corrected density functional theory calculations which reveal significant differences in the diffusion barriers for each conformer. These results demonstrate that conformational variation significantly modifies the mechanical response of even simple porpyhrins, potentially affecting many other flexible molecules. 1. Fast adaptive principal component extraction based on a generalized energy function Institute of Scientific and Technical Information of China (English) 欧阳缮; 保铮; 廖桂生 2003-01-01 By introducing an arbitrary diagonal matrix, a generalized energy function (GEF) is proposed for searching for the optimum weights of a two layer linear neural network. From the GEF, we derive a recur- sive least squares (RLS) algorithm to extract in parallel multiple principal components of the input covari-ance matrix without designing an asymmetrical circuit. The local stability of the GEF algorithm at the equilibrium is analytically verified. Simulation resultsshow that the GEF algorithm for parallel multiple principal components extraction exhibits the fast convergence and has the improved robustness resis- tance tothe eigenvalue spread of the input covariance matrix as compared to the well-known lateral inhi- bition model (APEX) and least mean square error reconstruction(LMSER) algorithms. 2. Effect of vacancy defects on generalized stacking fault energy of fcc metals. Science.gov (United States) Asadi, Ebrahim; Zaeem, Mohsen Asle; Moitra, Amitava; Tschopp, Mark A 2014-03-19 Molecular dynamics (MD) and density functional theory (DFT) studies were performed to investigate the influence of vacancy defects on generalized stacking fault (GSF) energy of fcc metals. MEAM and EAM potentials were used for MD simulations, and DFT calculations were performed to test the accuracy of different common parameter sets for MEAM and EAM potentials in predicting GSF with different fractions of vacancy defects. Vacancy defects were placed at the stacking fault plane or at nearby atomic layers. The effect of vacancy defects at the stacking fault plane and the plane directly underneath of it was dominant compared to the effect of vacancies at other adjacent planes. The effects of vacancy fraction, the distance between vacancies, and lateral relaxation of atoms on the GSF curves with vacancy defects were investigated. A very similar variation of normalized SFEs with respect to vacancy fractions were observed for Ni and Cu. MEAM potentials qualitatively captured the effect of vacancies on GSF. 3. Derrida's Generalized Random Energy models; 4, Continuous state branching and coalescents CERN Document Server Bovier, A 2003-01-01 In this paper we conclude our analysis of Derrida's Generalized Random Energy Models (GREM) by identifying the thermodynamic limit with a one-parameter family of probability measures related to a continuous state branching process introduced by Neveu. Using a construction introduced by Bertoin and Le Gall in terms of a coherent family of subordinators related to Neveu's branching process, we show how the Gibbs geometry of the limiting Gibbs measure is given in terms of the genealogy of this process via a deterministic time-change. This construction is fully universal in that all different models (characterized by the covariance of the underlying Gaussian process) differ only through that time change, which in turn is expressed in terms of Parisi's overlap distribution. The proof uses strongly the Ghirlanda-Guerra identities that impose the structure of Neveu's process as the only possible asymptotic random mechanism. 4. Generalized ghost pilgrim dark energy in F(T,TG) cosmology Science.gov (United States) Sharif, M.; Nazir, Kanwal 2016-07-01 This paper is devoted to study the generalized ghost pilgrim dark energy (PDE) model in F(T,TG) gravity with flat Friedmann-Robertson-Walker (FRW) universe. In this scenario, we reconstruct F(T,TG) models and evaluate the corresponding equation of state (EoS) parameter for different choices of the scale factors. We assume power-law scale factor, scale factor for unification of two phases, intermediate and bouncing scale factor. We study the behavior of reconstructed models and EoS parameters graphically. It is found that all the reconstructed models show decreasing behavior for PDE parameter u = -2. On the other hand, the EoS parameter indicates transition from dust-like matter to phantom era for all choices of the scale factor except intermediate for which this is less than - 1. We conclude that all the results are in agreement with PDE phenomenon. 5. Axial symmetry and conformal Killing vectors CERN Document Server Mars, M; Mars, Marc; Senovilla, Jose M.M. 1993-01-01 Axisymmetric spacetimes with a conformal symmetry are studied and it is shown that, if there is no further conformal symmetry, the axial Killing vector and the conformal Killing vector must commute. As a direct consequence, in conformally stationary and axisymmetric spacetimes, no restriction is made by assuming that the axial symmetry and the conformal timelike symmetry commute. Furthermore, we prove that in axisymmetric spacetimes with another symmetry (such as stationary and axisymmetric or cylindrically symmetric spacetimes) and a conformal symmetry, the commutator of the axial Killing vector with the two others mush vanish or else the symmetry is larger than that originally considered. The results are completely general and do not depend on Einstein's equations or any particular matter content. 6. Einstein and the conservation of energy-momentum in general relativity CERN Document Server Weinstein, Galina 2013-01-01 The main purpose of the present paper is to show that a correction of one mistake was crucial for Einstein's pathway to the first version of the 1915 general theory of relativity, but also might have played a role in obtaining the final version of Einstein's 1915 field equations. In 1914 Einstein wrote the equations for conservation of energy-momentum for matter, and established a connection between these equations and the components of the gravitational field. He showed that a material point in gravitational fields moves on a geodesic line in space-time, the equation of which is written in terms of the Christoffel symbols. By November 4, 1915, Einstein found it advantageous to use for the components of the gravitational field, not the previous equation, but the Christoffel symbols. He corrected the 1914 equations of conservation of energy-momentum for matter. Einstein had already basically possessed the field equations in 1912 together with his mathematician friend Marcel Grossman, but because he had not rec... 7. On the optimization of the generalized coplanar Hohmann impulsive transfer adopting energy change concept Energy Technology Data Exchange (ETDEWEB) Kamel, Osman M. [Cairo Univ., Astronomy and Space Sciences Dept., Giza (Egypt); Soliman, Adel S. [National Research Center, Theoretical Physics Dept., Dokki, Giza (Egypt) 2005-02-15 We considered the problem of transferring the rocket's orbit to higher energy orbit, using minimum fuel cost, as a problem in change of energy, since this is most convenient. For the generalized Hohmann case (the departure; the transferring and the destination orbits are ellipses), we adopt the first configuration only, when the apogee of transfer orbit, and the apogee of destination orbit are coincident. Firstly, we assign the {delta}v{sub A}, {delta}v{sub B} increments in velocity at points A,B (the position of peri-apse and apo-apse impulses respectively), as functions of the eccentricity of the transfer orbit, e{sub T}. Subsequently, we apply the optimum condition leading to the derivation of the quartic equation in e{sub T}, and showed how to deduce ({delta}v{sub A}+{delta}v{sub B}){sub Min}. A numerical example is presented, in which we determined the four roots of the quartic equation, by a numerical Mathematica Version 2.2. We selected the adequate consistent root, only one in this case, and evaluated ({delta}v{sub A}+{delta}v{sub B}){sub Min} for the two orbits of the couple Earth and Mars. This article is a new approach and leads to new discoveries involved in the problem, consequently adds new insight and avoids complexities of previous procedures. (Author) 8. Lectures on Conformal Field Theory CERN Document Server Qualls, Joshua D 2015-01-01 These lectures notes are based on courses given at National Taiwan University, National Chiao-Tung University, and National Tsing Hua University in the spring term of 2015. Although the course was offered primarily for graduate students, these lecture notes have been prepared for a more general audience. They are intended as an introduction to conformal field theories in various dimensions, with applications related to topics of particular interest: topics include the conformal bootstrap program, boundary conformal field theory, and applications related to the AdS/CFT correspondence. We assume the reader to be familiar with quantum mechanics at the graduate level and to have some basic knowledge of quantum field theory. Familiarity with string theory is not a prerequisite for this lectures, although it can only help. 9. Renyi entropy and conformal defects Energy Technology Data Exchange (ETDEWEB) Bianchi, Lorenzo [Humboldt-Univ. Berlin (Germany). Inst. fuer Physik; Hamburg Univ. (Germany). II. Inst. fuer Theoretische Physik; Meineri, Marco [Scuola Normale Superiore, Pisa (Italy); Perimeter Institute for Theoretical Physics, Waterloo, ON (Canada); Istituto Nazionale di Fisica Nucleare, Pisa (Italy); Myers, Robert C. [Perimeter Institute for Theoretical Physics, Waterloo, ON (Canada); Smolkin, Michael [California Univ., Berkely, CA (United States). Center for Theoretical Physics and Department of Physics 2016-04-18 We propose a field theoretic framework for calculating the dependence of Renyi entropies on the shape of the entangling surface in a conformal field theory. Our approach rests on regarding the corresponding twist operator as a conformal defect and in particular, we define the displacement operator which implements small local deformations of the entangling surface. We identify a simple constraint between the coefficient defining the two-point function of the displacement operator and the conformal weight of the twist operator, which consolidates a number of distinct conjectures on the shape dependence of the Renyi entropy. As an example, using this approach, we examine a conjecture regarding the universal coefficient associated with a conical singularity in the entangling surface for CFTs in any number of spacetime dimensions. We also provide a general formula for the second order variation of the Renyi entropy arising from small deformations of a spherical entangling surface, extending Mezei's results for the entanglement entropy. 10. Calculation of positron binding energies using the generalized any particle propagator theory Energy Technology Data Exchange (ETDEWEB) Romero, Jonathan; Charry, Jorge A. [Department of Chemistry, Universidad Nacional de Colombia, Av. Cra. 30 #45-03, Bogotá (Colombia); Flores-Moreno, Roberto [Departamento de Química, Universidad de Guadalajara, Blvd. Marcelino García Barragán 1421, Guadalajara Jal., C. P. 44430 (Mexico); Varella, Márcio T. do N. [Instituto de Física, Universidade de São Paulo, CP 66318, 05315-970 São Paulo, SP (Brazil); Reyes, Andrés, E-mail: [email protected] [Department of Chemistry, Universidad Nacional de Colombia, Av. Cra. 30 #45-03, Bogotá (Colombia); Instituto de Física, Universidade de São Paulo, CP 66318, 05315-970 São Paulo, SP (Brazil) 2014-09-21 We recently extended the electron propagator theory to any type of quantum species based in the framework of the Any-Particle Molecular Orbital (APMO) approach [J. Romero, E. Posada, R. Flores-Moreno, and A. Reyes, J. Chem. Phys. 137, 074105 (2012)]. The generalized any particle molecular orbital propagator theory (APMO/PT) was implemented in its quasiparticle second order version in the LOWDIN code and was applied to calculate nuclear quantum effects in electron binding energies and proton binding energies in molecular systems [M. Díaz-Tinoco, J. Romero, J. V. Ortiz, A. Reyes, and R. Flores-Moreno, J. Chem. Phys. 138, 194108 (2013)]. In this work, we present the derivation of third order quasiparticle APMO/PT methods and we apply them to calculate positron binding energies (PBEs) of atoms and molecules. We calculated the PBEs of anions and some diatomic molecules using the second order, third order, and renormalized third order quasiparticle APMO/PT approaches and compared our results with those previously calculated employing configuration interaction (CI), explicitly correlated and quantum Montecarlo methodologies. We found that renormalized APMO/PT methods can achieve accuracies of ∼0.35 eV for anionic systems, compared to Full-CI results, and provide a quantitative description of positron binding to anionic and highly polar species. Third order APMO/PT approaches display considerable potential to study positron binding to large molecules because of the fifth power scaling with respect to the number of basis sets. In this regard, we present additional PBE calculations of some small polar organic molecules, amino acids and DNA nucleobases. We complement our numerical assessment with formal and numerical analyses of the treatment of electron-positron correlation within the quasiparticle propagator approach. 11. Calculation of positron binding energies using the generalized any particle propagator theory Science.gov (United States) Romero, Jonathan; Charry, Jorge A.; Flores-Moreno, Roberto; Varella, Márcio T. do N.; Reyes, Andrés 2014-09-01 We recently extended the electron propagator theory to any type of quantum species based in the framework of the Any-Particle Molecular Orbital (APMO) approach [J. Romero, E. Posada, R. Flores-Moreno, and A. Reyes, J. Chem. Phys. 137, 074105 (2012)]. The generalized any particle molecular orbital propagator theory (APMO/PT) was implemented in its quasiparticle second order version in the LOWDIN code and was applied to calculate nuclear quantum effects in electron binding energies and proton binding energies in molecular systems [M. Díaz-Tinoco, J. Romero, J. V. Ortiz, A. Reyes, and R. Flores-Moreno, J. Chem. Phys. 138, 194108 (2013)]. In this work, we present the derivation of third order quasiparticle APMO/PT methods and we apply them to calculate positron binding energies (PBEs) of atoms and molecules. We calculated the PBEs of anions and some diatomic molecules using the second order, third order, and renormalized third order quasiparticle APMO/PT approaches and compared our results with those previously calculated employing configuration interaction (CI), explicitly correlated and quantum Montecarlo methodologies. We found that renormalized APMO/PT methods can achieve accuracies of ˜0.35 eV for anionic systems, compared to Full-CI results, and provide a quantitative description of positron binding to anionic and highly polar species. Third order APMO/PT approaches display considerable potential to study positron binding to large molecules because of the fifth power scaling with respect to the number of basis sets. In this regard, we present additional PBE calculations of some small polar organic molecules, amino acids and DNA nucleobases. We complement our numerical assessment with formal and numerical analyses of the treatment of electron-positron correlation within the quasiparticle propagator approach. 12. Energy Conversions in the Atmosphere on the Scale of the General Circulation OpenAIRE Mieghem, Jacques Van 2011-01-01 From the equations of balance established for the different forms of energy (potential, kinetic and internal energies), the energy fluxes, the rates of energy production and conversion are deduced. Provided weighted mean values are considered the zonally averaged value of the kinetic energy may be decomposed into kinetic energy of the mean motion and the mean kinetic energy of the large-scale eddies. The corresponding equations of balance are established. The axial symmetry with respect to th... 13. 76 FR 50204 - Decision and Order Granting a Waiver to Fujitsu General Limited From the Department of Energy... Science.gov (United States) 2011-08-12 ... No. CAC-033, which grants Fujitsu General Limited (Fujitsu) a waiver from the existing DOE test... Energy. Decision and Order In the Matter of: Fujitsu General Limited (Fujitsu) (Case No. CAC- 033... petition for waiver filed by Fujitsu (Case No. CAC-033) is hereby granted as set forth in the... 14. Energy, momentum and angular momentum in the dyadosphere of a charged spacetime in teleparallel equivalent of general relativity Institute of Scientific and Technical Information of China (English) Gamal G.L.Nashed 2012-01-01 We apply the energy momentum and angular momentum tensor to a tetrad field,with two unknown functions of radial coordinate,in the framework of a teleparallel equivalent of general relativity (TEGR).The definition of the gravitational energy is used to investigate the energy within the external event horizon of the dyadosphere region for the Reissner-Nordstr(o)m black hole.We also calculate the spatial momentum and angular momentum. 15. Viscous conformal gauge theories DEFF Research Database (Denmark) Toniato, Arianna; Sannino, Francesco; Rischke, Dirk H. 2017-01-01 We present the conformal behavior of the shear viscosity-to-entropy density ratio and the fermion-number diffusion coefficient within the perturbative regime of the conformal window for gauge-fermion theories.......We present the conformal behavior of the shear viscosity-to-entropy density ratio and the fermion-number diffusion coefficient within the perturbative regime of the conformal window for gauge-fermion theories.... 16. Conformal gravity and "gravitational bubbles" CERN Document Server Berezin, V A; Eroshenko, Yu N 2015-01-01 We describe the general structure of the spherically symmetric solutions in the Weyl conformal gravity. The corresponding Bach equations are derived for the special type of metrics, which can be considered as the representative of the general class. The complete set of the pure vacuum solutions, consisting of two classes, is found. The first one contains the solutions with constant two-dimensional curvature scalar, and the representatives are the famous Robertson--Walker metrics. We called one of them the "gravitational bubbles", which is compact and with zero Weyl tensor. These "gravitational bubbles" are the pure vacuum curved space-times (without any material sources, including the cosmological constant), which are absolutely impossible in General Relativity. This phenomenon makes it easier to create the universe from "nothing". The second class consists of the solutions with varying curvature scalar. We found its representative as the one-parameter family, which can be conformally covered by the thee-para... 17. Implications of conformal symmetry in quantum mechanics Science.gov (United States) Okazaki, Tadashi 2017-09-01 In conformal quantum mechanics with the vacuum of a real scaling dimension and with a complete orthonormal set of energy eigenstates, which is preferable under the unitary evolution, the dilatation expectation value between energy eigenstates monotonically decreases along the flow from the UV to the IR. In such conformal quantum mechanics, there exist bounds on scaling dimensions of the physical states and the gauge operators. 18. Representation of target-bound drugs by computed conformers: implications for conformational libraries Directory of Open Access Journals (Sweden) Goede Andrean 2006-06-01 Full Text Available Abstract Background The increasing number of known protein structures provides valuable information about pharmaceutical targets. Drug binding sites are identifiable and suitable lead compounds can be proposed. The flexibility of ligands is a critical point for the selection of potential drugs. Since computed 3D structures of millions of compounds are available, the knowledge of their binding conformations would be a great benefit for the development of efficient screening methods. Results Integration of two public databases allowed superposition of conformers for 193 approved drugs with 5507 crystallised target-bound counterparts. The generation of 9600 drug conformers using an atomic force field was carried out to obtain an optimal coverage of the conformational space. Bioactive conformations are best described by a conformational ensemble: half of all drugs exhibit multiple active states, distributed over the entire range of the reachable energy and conformational space. A number of up to 100 conformers per drug enabled us to reproduce the bound states within a similarity threshold of 1.0 Å in 70% of all cases. This fraction rises to about 90% for smaller or average sized drugs. Conclusion Single drugs adopt multiple bioactive conformations if they interact with different target proteins. Due to the structural diversity of binding sites they adopt conformations that are distributed over a broad conformational space and wide energy range. Since the majority of drugs is well represented by a predefined low number of conformers (up to 100 this procedure is a valuable method to compare compounds by three-dimensional features or for fast similarity searches starting with pharmacophores. The underlying 9600 generated drug conformers are downloadable from the Super Drug Web site 1. All superpositions are visualised at the same source. Additional conformers (110,000 of 2400 classified WHO-drugs are also available. 19. Five-dimensional teleparallel theory equivalent to general relativity, the axially symmetric solution,energy and spatial momentum Institute of Scientific and Technical Information of China (English) Gamal G.L. Nashed 2011-01-01 A theory of (4+1)-dimensional gravity is developed on the basis of the teleparallel theory equivalent to general relativity.The fundamental gravitational field variables are the five-dimensional vector fields (pentad),defined globally on a manifold M,and gravity is attributed to the torsion.The Lagrangian density is quadratic in the torsion tensor.We then give the exact five-dimensional solution.The solution is a generalization of the familiar Schwarzschild and Kerr solutions of the four-dimensional teleparallel equivalent of general relativity.We also use the definition of the gravitational energy to calculate the energy and the spatial momentum. 20. Conformational and Vibrational Studies of Triclosan Science.gov (United States) Özişik, Haci; Bayari, S. Haman; Saǧlam, Semran 2010-01-01 The conformational equilibrium of triclosan (5-chloro-2-(2, 4-dichlorophenoxy) phenol) have been calculated using density functional theory (DFTe/B3LYP/6-311++G(d, p)) method. Four different geometries were found to correspond to energy minimum conformations. The IR spectrum of triclosan was measured in the 4000-400 cm-1 region. We calculated the harmonic frequencies and intensities of the most stable conformers in order to assist in the assignment of the vibrational bands in the experimental spectrum. The fundamental vibrational modes were characterized depending on their total energy distribution (TED%) using scaled quantum mechanical (SQM) force field method. 1. Hot Conformal Gauge Theories DEFF Research Database (Denmark) Mojaza, Matin; Pica, Claudio; Sannino, Francesco 2010-01-01 We compute the nonzero temperature free energy up to the order g^6 \\ln(1/g) in the coupling constant for vector like SU(N) gauge theories featuring matter transforming according to different representations of the underlying gauge group. The number of matter fields, i.e. flavors, is arranged in s.......e. they are independent on the specific matter representation.......We compute the nonzero temperature free energy up to the order g^6 \\ln(1/g) in the coupling constant for vector like SU(N) gauge theories featuring matter transforming according to different representations of the underlying gauge group. The number of matter fields, i.e. flavors, is arranged...... in such a way that the theory develops a perturbative stable infrared fixed point at zero temperature. Due to large distance conformality we trade the coupling constant with its fixed point value and define a reduced free energy which depends only on the number of flavors, colors and matter representation. We... 2. Conformal mapping for multiple terminals Science.gov (United States) Wang, Weimin; Ma, Wenying; Wang, Qiang; Ren, Hao 2016-11-01 Conformal mapping is an important mathematical tool that can be used to solve various physical and engineering problems in many fields, including electrostatics, fluid mechanics, classical mechanics, and transformation optics. It is an accurate and convenient way to solve problems involving two terminals. However, when faced with problems involving three or more terminals, which are more common in practical applications, existing conformal mapping methods apply assumptions or approximations. A general exact method does not exist for a structure with an arbitrary number of terminals. This study presents a conformal mapping method for multiple terminals. Through an accurate analysis of boundary conditions, additional terminals or boundaries are folded into the inner part of a mapped region. The method is applied to several typical situations, and the calculation process is described for two examples of an electrostatic actuator with three electrodes and of a light beam splitter with three ports. Compared with previously reported results, the solutions for the two examples based on our method are more precise and general. The proposed method is helpful in promoting the application of conformal mapping in analysis of practical problems. 3. A general method to derive tissue parameters for Monte Carlo dose calculation with multi-energy CT Science.gov (United States) Lalonde, Arthur; Bouchard, Hugo 2016-11-01 To develop a general method for human tissue characterization with dual- and multi-energy CT and evaluate its performance in determining elemental compositions and quantities relevant to radiotherapy Monte Carlo dose calculation. Ideal materials to describe human tissue are obtained applying principal component analysis on elemental weight and density data available in literature. The theory is adapted to elemental composition for solving tissue information from CT data. A novel stoichiometric calibration method is integrated to the technique to make it suitable for a clinical environment. The performance of the method is compared with two techniques known in literature using theoretical CT data. In determining elemental weights with dual-energy CT, the method is shown to be systematically superior to the water-lipid-protein material decomposition and comparable to the parameterization technique. In determining proton stopping powers and energy absorption coefficients with dual-energy CT, the method generally shows better accuracy and unbiased results. The generality of the method is demonstrated simulating multi-energy CT data to show the potential to extract more information with multiple energies. The method proposed in this paper shows good performance to determine elemental compositions from dual-energy CT data and physical quantities relevant to radiotherapy dose calculation. The method is particularly suitable for Monte Carlo calculations and shows promise in using more than two energies to characterize human tissue with CT. 4. A general method to derive tissue parameters for Monte Carlo dose calculation with multi-energy CT. Science.gov (United States) Lalonde, Arthur; Bouchard, Hugo 2016-11-21 To develop a general method for human tissue characterization with dual- and multi-energy CT and evaluate its performance in determining elemental compositions and quantities relevant to radiotherapy Monte Carlo dose calculation. Ideal materials to describe human tissue are obtained applying principal component analysis on elemental weight and density data available in literature. The theory is adapted to elemental composition for solving tissue information from CT data. A novel stoichiometric calibration method is integrated to the technique to make it suitable for a clinical environment. The performance of the method is compared with two techniques known in literature using theoretical CT data. In determining elemental weights with dual-energy CT, the method is shown to be systematically superior to the water-lipid-protein material decomposition and comparable to the parameterization technique. In determining proton stopping powers and energy absorption coefficients with dual-energy CT, the method generally shows better accuracy and unbiased results. The generality of the method is demonstrated simulating multi-energy CT data to show the potential to extract more information with multiple energies. The method proposed in this paper shows good performance to determine elemental compositions from dual-energy CT data and physical quantities relevant to radiotherapy dose calculation. The method is particularly suitable for Monte Carlo calculations and shows promise in using more than two energies to characterize human tissue with CT. 5. Econometrically calibrated computable general equilibrium models: Applications to the analysis of energy and climate politics Science.gov (United States) Schu, Kathryn L. Economy-energy-environment models are the mainstay of economic assessments of policies to reduce carbon dioxide (CO2) emissions, yet their empirical basis is often criticized as being weak. This thesis addresses these limitations by constructing econometrically calibrated models in two policy areas. The first is a 35-sector computable general equilibrium (CGE) model of the U.S. economy which analyzes the uncertain impacts of CO2 emission abatement. Econometric modeling of sectors' nested constant elasticity of substitution (CES) cost functions based on a 45-year price-quantity dataset yields estimates of capital-labor-energy-material input substitution elasticities and biases of technical change that are incorporated into the CGE model. I use the estimated standard errors and variance-covariance matrices to construct the joint distribution of the parameters of the economy's supply side, which I sample to perform Monte Carlo baseline and counterfactual runs of the model. The resulting probabilistic abatement cost estimates highlight the importance of the uncertainty in baseline emissions growth. The second model is an equilibrium simulation of the market for new vehicles which I use to assess the response of vehicle prices, sales and mileage to CO2 taxes and increased corporate average fuel economy (CAFE) standards. I specify an econometric model of a representative consumer's vehicle preferences using a nested CES expenditure function which incorporates mileage and other characteristics in addition to prices, and develop a novel calibration algorithm to link this structure to vehicle model supplies by manufacturers engaged in Bertrand competition. CO2 taxes' effects on gasoline prices reduce vehicle sales and manufacturers' profits if vehicles' mileage is fixed, but these losses shrink once mileage can be adjusted. Accelerated CAFE standards induce manufacturers to pay fines for noncompliance rather than incur the higher costs of radical mileage improvements 6. Generalized hard-core dimer model approach to low-energy Heisenberg frustrated antiferromagnets: General properties and application to the kagome antiferromagnet Science.gov (United States) Schwandt, David; Mambrini, Matthieu; Poilblanc, Didier 2010-06-01 We propose a general nonperturbative scheme that quantitatively maps the low-energy sector of spin-1/2 frustrated Heisenberg antiferromagnets to effective generalized quantum dimer models. We develop the formal lattice-independent frame and establish some important results on (i) the locality of the generated Hamiltonians, (ii) how full resummations can be performed in this renormalization scheme. The method is then applied to the much debated kagome antiferromagnet for which a fully resummed effective Hamiltonian—shown to capture the essential properties and provide deep insights on the microscopic model [D. Poilblanc, M. Mambrini, and D. Schwandt, Phys. Rev. B 81, 180402(R) (2010)]—is derived. 7. Phenomenology of dark energy: general features of large-scale perturbations Science.gov (United States) Pèrenon, Louis; Piazza, Federico; Marinoni, Christian; Hui, Lam 2015-11-01 We present a systematic exploration of dark energy and modified gravity models containing a single scalar field non-minimally coupled to the metric. Even though the parameter space is large, by exploiting an effective field theory (EFT) formulation and by imposing simple physical constraints such as stability conditions and (sub-)luminal propagation of perturbations, we arrive at a number of generic predictions. (1) The linear growth rate of matter density fluctuations is generally suppressed compared to ΛCDM at intermediate redshifts (0.5 lesssim z lesssim 1), despite the introduction of an attractive long-range scalar force. This is due to the fact that, in self-accelerating models, the background gravitational coupling weakens at intermediate redshifts, over-compensating the effect of the attractive scalar force. (2) At higher redshifts, the opposite happens; we identify a period of super-growth when the linear growth rate is larger than that predicted by ΛCDM. (3) The gravitational slip parameter η—the ratio of the space part of the metric perturbation to the time part—is bounded from above. For Brans-Dicke-type theories η is at most unity. For more general theories, η can exceed unity at intermediate redshifts, but not more than about 1.5 if, at the same time, the linear growth rate is to be compatible with current observational constraints. We caution against phenomenological parametrization of data that do not correspond to predictions from viable physical theories. We advocate the EFT approach as a way to constrain new physics from future large-scale-structure data. 8. BayeSED: A General Approach to Fitting the Spectral Energy Distribution of Galaxies Science.gov (United States) Han, Yunkun; Han, Zhanwen 2014-11-01 We present a newly developed version of BayeSED, a general Bayesian approach to the spectral energy distribution (SED) fitting of galaxies. The new BayeSED code has been systematically tested on a mock sample of galaxies. The comparison between the estimated and input values of the parameters shows that BayeSED can recover the physical parameters of galaxies reasonably well. We then applied BayeSED to interpret the SEDs of a large Ks -selected sample of galaxies in the COSMOS/UltraVISTA field with stellar population synthesis models. Using the new BayeSED code, a Bayesian model comparison of stellar population synthesis models has been performed for the first time. We found that the 2003 model by Bruzual & Charlot, statistically speaking, has greater Bayesian evidence than the 2005 model by Maraston for the Ks -selected sample. In addition, while setting the stellar metallicity as a free parameter obviously increases the Bayesian evidence of both models, varying the initial mass function has a notable effect only on the Maraston model. Meanwhile, the physical parameters estimated with BayeSED are found to be generally consistent with those obtained using the popular grid-based FAST code, while the former parameters exhibit more natural distributions. Based on the estimated physical parameters of the galaxies in the sample, we qualitatively classified the galaxies in the sample into five populations that may represent galaxies at different evolution stages or in different environments. We conclude that BayeSED could be a reliable and powerful tool for investigating the formation and evolution of galaxies from the rich multi-wavelength observations currently available. A binary version of the BayeSED code parallelized with Message Passing Interface is publicly available at https://bitbucket.org/hanyk/bayesed. 9. Simple, yet powerful methodologies for conformational sampling of proteins. Science.gov (United States) Harada, Ryuhei; Takano, Yu; Baba, Takeshi; Shigeta, Yasuteru 2015-03-07 Several biological functions, such as molecular recognition, enzyme catalysis, signal transduction, allosteric regulation, and protein folding, are strongly related to conformational transitions of proteins. These conformational transitions are generally induced as slow dynamics upon collective motions, including biologically relevant large-amplitude fluctuations of proteins. Although molecular dynamics (MD) simulation has become a powerful tool for extracting conformational transitions of proteins, it might still be difficult to reach time scales of the biological functions because the accessible time scales of MD simulations are far from biological time scales, even if straightforward conventional MD (CMD) simulations using massively parallel computers are employed. Thus, it is desirable to develop efficient methods to achieve canonical ensembles with low computational costs. From this perspective, we review several enhanced conformational sampling techniques of biomolecules developed by us. In our methods, multiple independent short-time MD simulations are employed instead of single straightforward long-time CMD simulations. Our basic strategy is as follows: (i) selection of initial seeds (initial structures) for the conformational sampling in restarting MD simulations. Here, the seeds should be selected as candidates with high potential to transit. (ii) Resampling from the selected seeds by initializing velocities in restarting short-time MD simulations. A cycle of these simple protocols might drastically promote the conformational transitions of biomolecules. (iii) Once reactive trajectories extracted from the cycles of short-time MD simulations are obtained, a free energy profile is evaluated by means of umbrella sampling (US) techniques with the weighted histogram analysis method (WHAM) as a post-processing technique. For the selection of the initial seeds, we proposed four different choices: (1) Parallel CaScade molecular dynamics (PaCS-MD), (2) Fluctuation 10. CO2, energy and economy interactions: A multisectoral, dynamic, computable general equilibrium model for Korea Science.gov (United States) Kang, Yoonyoung While vast resources have been invested in the development of computational models for cost-benefit analysis for the "whole world" or for the largest economies (e.g. United States, Japan, Germany), the remainder have been thrown together into one model for the "rest of the world." This study presents a multi-sectoral, dynamic, computable general equilibrium (CGE) model for Korea. This research evaluates the impacts of controlling COsb2 emissions using a multisectoral CGE model. This CGE economy-energy-environment model analyzes and quantifies the interactions between COsb2, energy and economy. This study examines interactions and influences of key environmental policy components: applied economic instruments, emission targets, and environmental tax revenue recycling methods. The most cost-effective economic instrument is the carbon tax. The economic effects discussed include impacts on main macroeconomic variables (in particular, economic growth), sectoral production, and the energy market. This study considers several aspects of various COsb2 control policies, such as the basic variables in the economy: capital stock and net foreign debt. The results indicate emissions might be stabilized in Korea at the expense of economic growth and with dramatic sectoral allocation effects. Carbon dioxide emissions stabilization could be achieved to the tune of a 600 trillion won loss over a 20 year period (1990-2010). The average annual real GDP would decrease by 2.10% over the simulation period compared to the 5.87% increase in the Business-as-Usual. This model satisfies an immediate need for a policy simulation model for Korea and provides the basic framework for similar economies. It is critical to keep the central economic question at the forefront of any discussion regarding environmental protection. How much will reform cost, and what does the economy stand to gain and lose? Without this model, the policy makers might resort to hesitation or even blind speculation. With 11. 40 CFR 51.857 - Frequency of conformity determinations. Science.gov (United States) 2010-07-01 ... 40 Protection of Environment 2 2010-07-01 2010-07-01 false Frequency of conformity determinations... Conformity of General Federal Actions to State or Federal Implementation Plans § 51.857 Frequency of conformity determinations. Link to an amendment published at 75 FR 17272, April 5, 2010. (a) The... 12. 40 CFR 93.157 - Frequency of conformity determinations. Science.gov (United States) 2010-07-01 ... 40 Protection of Environment 20 2010-07-01 2010-07-01 false Frequency of conformity determinations... PROGRAMS (CONTINUED) DETERMINING CONFORMITY OF FEDERAL ACTIONS TO STATE OR FEDERAL IMPLEMENTATION PLANS Determining Conformity of General Federal Actions to State or Federal Implementation Plans § 93.157... 13. 40 CFR 86.407-78 - Certificate of conformity required. Science.gov (United States) 2010-07-01 ... 40 Protection of Environment 18 2010-07-01 2010-07-01 false Certificate of conformity required. 86... Regulations for 1978 and Later New Motorcycles, General Provisions § 86.407-78 Certificate of conformity... conformity issued pursuant to this subpart, except as specified in paragraph (b) of this section,... 14. Conserved Quantities and Conformal Mechanico-Electrical Systems Institute of Scientific and Technical Information of China (English) FU Jing-Li; WANG Xian-Jun; XIE Feng-Ping 2008-01-01 The conformal mechanico-electrical systems are presented by infinitesimal point transformations of time and generalized coordinates. The necessary and sufficient conditions that the conformal mechanico-electrical systems possess Lie symmetry are given. The Noether conserved quantities of the conformal mechanico-electrical systems are obtained from Lie symmetries. 15. Entanglement Temperature in Non-conformal Cases CERN Document Server He, Song; Wu, Jun-Bao 2013-01-01 Potential reconstruction can be used to find various analytical asymptotical AdS solutions in Einstein dilation system generally. We have generated two simple solutions without physical singularity called zero temperature solutions. We also proposed a numerical way to obtain black hole solution in Einstein dilaton system with special dilaton potential. By using this method, we obtain the corresponding black hole solutions numerically and investigate the thermal stability of the black hole by comparing the free energy of thermal gas and the corresponding black hole. In two groups of non-conformal gravity solutions obtained in this paper, we find that the two thermal gas solutions are more unstable than black hole solutions respectively. Finally, we consider black hole solutions as a thermal state of zero temperature solutions to check that the first thermal dynamical law exists in entanglement system from holographic point of view. 16. Phenomenology of dark energy: general features of large-scale perturbations CERN Document Server Perenon, Louis; Marinoni, Christian; Hui, Lam 2015-01-01 We present a systematic exploration of dark energy and modified gravity models containing a single scalar field non-minimally coupled to the metric. Even though the parameter space is large, by exploiting an effective field theory (EFT) formulation and by imposing simple physical constraints such as stability conditions and (sub-)luminal propagation of perturbations, we arrive at a number of generic predictions. (1) The linear growth rate of matter density fluctuations is generally suppressed compared to$\\Lambda$CDM at intermediate redshifts ($0.5 \\lesssim z \\lesssim 1$), despite the introduction of an attractive long-range scalar force. This is due to the fact that, in self-accelerating models, the background gravitational coupling weakens at intermediate redshifts, over-compensating the effect of the attractive scalar force. (2) At higher redshifts, the opposite happens; we identify a period of super-growth when the linear growth rate is larger than that predicted by$\\Lambda$CDM. (3) The gravitational sli... 17. BayeSED: A General Approach to Fitting the Spectral Energy Distribution of Galaxies CERN Document Server Han, Yunkun 2014-01-01 We present a newly developed version of BayeSED, a general Bayesian approach to the spectral energy distribution (SED) fitting of galaxies. The new BayeSED code has been systematically tested on a mock sample of galaxies. The comparison between estimated and inputted value of the parameters show that BayeSED can recover the physical parameters of galaxies reasonably well. We then applied BayeSED to interpret the SEDs of a large Ks-selected sample of galaxies in the COSMOS/UltraVISTA field with stellar population synthesis models. With the new BayeSED code, a Bayesian model comparison of stellar population synthesis models has been done for the first time. We found that the model by Bruzual & Charlot (2003), statistically speaking, has larger Bayesian evidence than the model by Maraston (2005) for the Ks-selected sample. Besides, while setting the stellar metallicity as a free parameter obviously increases the Bayesian evidence of both models, varying the IMF has a notable effect only on the Maraston (2005... 18. Consumer protection issues in energy: a guide for attorneys general. Insulation, solar, automobile device, home devices Energy Technology Data Exchange (ETDEWEB) Cohen, Harry I.; Hulse, William S.; Jones, Robert R.; Langer, Robert M.; Petrucelli, Paul J.; Schroeder, Robert J. 1979-11-01 The guide attempts to bring together two important and current issues: energy and consumer protection. Perhaps the most basic consumer-protection issue in the energy area is assuring adequate supplies at adequate prices. It is anticipated, though, that consumers will want to consider new ways to lower enegy consumption and cost, and will thus be susceptible to fraudulent energy claims. Information is prepared on insulation, solar, energy-saving devices for the home, and energy-saving devices for the automobile. 19. Office of Inspector General audit report on the U.S. Department of Energys consolidated financial statements for fiscal year 1998 Energy Technology Data Exchange (ETDEWEB) NONE 1999-02-01 The Department prepared the Fiscal Year 1998 Accountability Report to combine critical financial and program performance information in a single report. The Departments consolidated financial statements and the related audit reports are included as major components of the Accountability Report. The Office of Inspector General audited the Departments consolidated financial statements as of and for the years ended September 30, 1998 and 1997. In the opinion of the Office of Inspector General, except for the environmental liabilities lines items in Fiscal year 1998, these financial statements present fairly, in all material respects, the financial position of the Department as of September 30, 1998 and 1997, and its consolidated net cost, changes in net position, budgetary resources, financing activities, and custodial activities for the years then ended in conformity with Federal accounting standards. In accordance with Government Auditing Standards, the Office of Inspector General issued a separate report on the Department internal controls. This report discusses needed improvements to the environmental liabilities estimating process and the reporting of performance measure information. 20. Expansion Formulation of General Relativity: the Gauge Functions for Energy-Momentum Tensor Science.gov (United States) Beloushko, Konstantin; Karbanovski, Valeri At present the one of the GR (General Relativity) basic problem remains a definition of the gravitation field (GF) energy. We shall analyze this content. As well known, the energy-momentum tensor'' (EMT) of GF was introduced by Einstein [1] with purpose of the SRT (Special Relativity Theory) generalization. It supposed also, that EMT of matter satisfy to the condition begin{equation} ⪉bel{GrindEQ__1_1_} T^{ik} _{;i} =0 (a semicolon denotes a covariant differentiation with respect to coordinates). In absence of GF the equation (ref{GrindEQ__1_1_}) reduces to a corresponding SRT expression begin{equation} ⪉bel{GrindEQ__1_2_} T^{ik} _{,i} =0 (a comma denotes a differentiation with respect to coordinates of space-time). Obviously, the conservation law'' (ref{GrindEQ__1_2_}) is not broken by transformation begin{equation} ⪉bel{GrindEQ__1_3_} T^{ik} to tilde{T}^{ik} =T^{ik} +h^{ikl} _{,l} , where for h(ikl) takes place a constrain begin{equation} ⪉bel{GrindEQ__1_4_} h^{ikl} =-h^{ilk} Later the given property has been used for a construction pseudo-tensor'' tau (ik) of pure'' GF [2, S 96] begin{equation} ⪉bel{GrindEQ__1_5_} -gleft(frac{c^{4} }{8pi G} left(R^{ik} -frac{1}{2} g^{ik} Rright)+tau ^{ik} right)=h^{ikl} _{,l} However such definition was a consequence of non-covariant transition from a reference system with condition g(ik) _{,l} =0 to an arbitrary frame. Therefore the Landau-Lifshitz pseudo-tensor has no physical contents and considered problem remains actual. The non-covariant character'' of GF energy was the reason for criticism of GR as Einstein's contemporaries [3, 4], as and during the subsequent period (see, for example, [5]). In [6] were analyzed the grounds of given problem, which are connected with a formulation indefiniteness of the conservation law'' in curved space-time. In [7] contends, that the gravitational energy in EMT can be separated only artificially'' by a choice of the certain coordinate system. In [8] is concluded 1. On functional representations of the conformal algebra Science.gov (United States) Rosten, Oliver J. 2017-07-01 Starting with conformally covariant correlation functions, a sequence of functional representations of the conformal algebra is constructed. A key step is the introduction of representations which involve an auxiliary functional. It is observed that these functionals are not arbitrary but rather must satisfy a pair of consistency equations corresponding to dilatation and special conformal invariance. In a particular representation, the former corresponds to the canonical form of the exact renormalization group equation specialized to a fixed point whereas the latter is new. This provides a concrete understanding of how conformal invariance is realized as a property of the Wilsonian effective action and the relationship to action-free formulations of conformal field theory. Subsequently, it is argued that the conformal Ward Identities serve to define a particular representation of the energy-momentum tensor. Consistency of this construction implies Polchinski's conditions for improving the energy-momentum tensor of a conformal field theory such that it is traceless. In the Wilsonian approach, the exactly marginal, redundant field which generates lines of physically equivalent fixed points is identified as the trace of the energy-momentum tensor. 2. Conformational elasticity theory of chain molecules Institute of Scientific and Technical Information of China (English) YANG; Xiaozhen 2001-01-01 This paper develops a conformational elasticity theory of chain molecules, which is based on three key points: (ⅰ) the molecular model is the rotational isomeric state (RIS) model; (ⅱ) the conformational distribution function of a chain molecule is described by a function of two variables, the end-to-end distance of a chain conformation and the energy of the conformation; (ⅲ) the rule of changes in the chain conformational states during deformation is that a number of chain conformations would vanish. The ideal deformation behavior calculated by the theory shows that the change in chain conformations is physically able to make the upward curvature of the stress-strain curve at the large-scale deformation of natural rubber. With the theory, different deformation behaviors between polymers with different chemical structures can be described, the energy term of the stress in the deformations can be predicted, and for natural rubber the fraction of the energy term is around 13%, coinciding with the experimental results. 3. Conformal theory of galactic halos CERN Document Server Nesbet, R K 2011-01-01 Current cosmological theory describes an isolated galaxy as an observable central galaxy, surrounded by a large spherical halo attributed to dark matter. Galaxy formation by condensation of mass-energy out of a primordial uniform background is shown here to leave a scar, observed as a centripetal gravitational field halo in anomalous galactic rotation and in gravitational lensing. Conformal theory accounts for the otherwise counterintuitive centripetal effect. 4. Energy Science.gov (United States) 2003-01-01 Canada, Britain, and Spain. We found that the energy industry is not in crisis ; however, U.S. government policies, laws, dollars, and even public...CEIMAT (Centro de Investagaciones Energeticas , Medioambeintales y Tecnologicas) Research and development Page 3 of 28ENERGY 8/10/04http://www.ndu.edu...meet an emerging national crisis (war), emergency (natural disaster), or major impact event (Y2K). Certain resources are generally critical to the 5. Evolution of holographic dark energy with interaction term$Q \\propto H\\rho_{\\rm de}$and generalized second law Indian Academy of Sciences (India) Praseetha P; Mathew Titus K 2016-03-01 A flat FLRW Universe with dark matter and dark energy, which are interacting witheach other, is considered. The dark energy is represented by the holographic dark energy model and the interaction term is taken as proportional to the dark energy density. We have studied the cosmological evolution and analysed the validity of the generalized second law of thermodynamics (GSL) under thermal equilibrium conditions and non-equilibrium conditions. We have found thatthe GSL is completely valid at the apparent horizon but violated at the event horizon under thermal equilibrium condition. Under thermal non-equilibrium condition, for the GSL to be valid, we found out that the temperature of the dark energy must be greater than the temperature of the apparent horizon if the dark energy behaves as a quintessence fluid. 6. Temperature effects on the generalized planar fault energies and twinnabilities of Al, Ni and Cu: First principles calculations KAUST Repository Liu, Lili 2014-06-01 Based on the quasiharmonic approach from first-principles phonon calculations, the volume versus temperature relations for Al, Ni and Cu are obtained. Using the equilibrium volumes at temperature T, the temperature dependences of generalized planar fault energies have also been calculated by first-principles calculations. It is found that the generalized planar fault energies reduce slightly with increasing temperature. Based on the calculated generalized planar fault energies, the twinnabilities of Al, Ni and Cu are discussed with the three typical criteria for crack tip twinning, grain boundary twinning and inherent twinning at different temperatures. The twinnabilities of Al, Ni and Cu also decrease slightly with increasing temperature. Ni and Cu have the inherent twinnabilities. But, Al does not exhibit inherent twinnability. These results are in agreement with the previous theoretical studies at 0 K and experimental observations at ambient temperature. © 2014 Elsevier B.V. All rights reserved. 7. Radiation Analysis and Characteristics of Conformal Reflectarray Antennas Directory of Open Access Journals (Sweden) Payam Nayeri 2012-01-01 Full Text Available This paper investigates the feasibility of designing reflectarray antennas on conformal surfaces. A generalized analysis approach is presented that can be applied to compute the radiation performance of conformal reflectarray antennas. Using this approach, radiation characteristics of conformal reflectarray antennas on singly curved platforms are studied and the performances of these designs are compared with planar designs. It is demonstrated that a conformal reflectarray antenna can be a suitable choice for applications requiring high-gain antennas on curved platforms. 8. A study of the kinetic energy generation with general circulation models Science.gov (United States) Chen, T.-C.; Lee, Y.-H. 1983-01-01 The history data of winter simulation by the GLAS climate model and the NCAR community climate model are used to examine the generation of atmospheric kinetic energy. The contrast between the geographic distributions of the generation of kinetic energy and divergence of kinetic energy flux shows that kinetic energy is generated in the upstream side of jets, transported to the downstream side and destroyed there. The contributions from the time-mean and transient modes to the counterbalance between generation of kinetic energy and divergence of kinetic energy flux are also investigated. It is observed that the kinetic energy generated by the time-mean mode is essentially redistributed by the time-mean flow, while that generated by the transient flow is mainly responsible for the maintenance of the kinetic energy of the entire atmospheric flow. 9. Conformant Planning via Symbolic Model Checking CERN Document Server Cimatti, A; 10.1613/jair.774 2011-01-01 We tackle the problem of planning in nondeterministic domains, by presenting a new approach to conformant planning. Conformant planning is the problem of finding a sequence of actions that is guaranteed to achieve the goal despite the nondeterminism of the domain. Our approach is based on the representation of the planning domain as a finite state automaton. We use Symbolic Model Checking techniques, in particular Binary Decision Diagrams, to compactly represent and efficiently search the automaton. In this paper we make the following contributions. First, we present a general planning algorithm for conformant planning, which applies to fully nondeterministic domains, with uncertainty in the initial condition and in action effects. The algorithm is based on a breadth-first, backward search, and returns conformant plans of minimal length, if a solution to the planning problem exists, otherwise it terminates concluding that the problem admits no conformant solution. Second, we provide a symbolic representation ... 10. A geodesic model in conformal superspace CERN Document Server Gomes, Henrique de A 2016-01-01 In this paper, I look for the most general geometrodynamical symmetries compatible with spatial relational principles. I argue that they lead either to a completely static Universe, or one embodying spatial conformal diffeomorphisms. Demanding locality for an action compatible with these principles severely limits its form, both for the gravitational part as well as all matter couplings. The simplest and most natural choice for pure gravity has two propagating physical degrees of freedom (and no refoliation-invariance). The system has a geometric interpretation as a geodesic model in infinite dimensional conformal superspace. Conformal superspace is a stratified manifold, with different strata corresponding to different isometry groups. Choosing space to be (homeomorphic to)$S^3$, conformal superspace has a preferred stratum with maximal stabilizer group. This stratum consists of a single point -- corresponding to the conformal geometry of the round 3-sphere. This is the most homogeneous non-degenerate geome... 11. Transitive conformal holonomy groups CERN Document Server Alt, Jesse 2011-01-01 For$(M,[g])$a conformal manifold of signature$(p,q)$and dimension at least three, the conformal holonomy group$\\mathrm{Hol}(M,[g]) \\subset O(p+1,q+1)$is an invariant induced by the canonical Cartan geometry of$(M,[g])$. We give a description of all possible connected conformal holonomy groups which act transitively on the M\\"obius sphere$S^{p,q}$, the homogeneous model space for conformal structures of signature$(p,q)$. The main part of this description is a list of all such groups which also act irreducibly on$\\R^{p+1,q+1}$. For the rest, we show that they must be compact and act decomposably on$\\R^{p+1,q+1}$, in particular, by known facts about conformal holonomy the conformal class$[g]$must contain a metric which is locally isometric to a so-called special Einstein product. 12. LINEAR GENERAL EQUILIBRIUM MODEL OF ENERGY DEMAND AND CO2 EMISSIONS GENERATED BY THE ANDALUSIAN PRODUCTIVE SYSTEM Directory of Open Access Journals (Sweden) Manuel Alejandro Cardenete 2012-01-01 Full Text Available In this study we apply a multiplier decomposition methodology of a linear general equilibrium model based on the regional social accounting matrix to the Andalusian economy. The aim of this methodology is to separate the size of the different effects in terms of energy expenditure and total emissions generated by the whole productive system to satisfy the final demand of each branch of the Andalusian economy and the direct emissions generated to produce energy for each subsystem. 13. Stress-energy distribution for a cylindrical artificial gravity field via the Darmois-Israel junction conditions of general relativity Science.gov (United States) Istrate, Nicolae; Lindner, John 2014-03-01 We design an Earth-like artificial gravity field using the Darmois-Israel junction conditions of general relativity to connect the flat spacetime outside an infinitesimally thin cylinder to the curved spacetime inside. In the calculation of extrinsic curvature, our construction exploits Earth's weak gravity, which implies similar inside and outside curvatures, to approximate the unit normal inside by the negative unit normal outside. The stress-energy distribution on the cylinder's sides includes negative energy density. 14. Conformal dynamical equivalence and applications Science.gov (United States) Spyrou, N. K. 2011-02-01 The "Conformal Dynamical Equivalence" (CDE) approach is briefly reviewed, and some of its applications, at various astrophysical levels (Sun, Solar System, Stars, Galaxies, Clusters of Galaxies, Universe as a whole), are presented. According to the CDE approach, in both the Newtonian and general-relativistic theories of gravity, the isentropic hydrodynamic flows in the interior of a bounded gravitating perfect-fluid source are dynamically equivalent to geodesic motions in a virtual, fully defined fluid source. Equivalently, the equations of hydrodynamic motion in the former source are functionally similar to those of the geodesic motions in the latter, physically, fully defined source. The CDE approach is followed for the dynamical description of the motions in the fluid source. After an observational introduction, taking into account all the internal physical characteristics of the corresponding perfect-fluid source, and based on the property of the isentropic hydrodynamic flows (quite reasonable for an isolated physical system), we examine a number of issues, namely, (i) the classical Newtonian explanation of the celebrated Pioneer-Anomaly effect in the Solar System, (ii) the possibility of both the attractive gravity and the repulsive gravity in a non-quantum Newtonian framework, (iii) the evaluation of the masses - theoretical, dynamical, and missing - and of the linear dimensions of non-magnetized and magnetized large-scale cosmological structures, (iv) the explanation of the flat-rotation curves of disc galaxies, (v) possible formation mechanisms of winds and jets, and (vi) a brief presentation of a conventional approach - toy model to the dynamics of the Universe, characterized by the dominant collisional dark matter (with its subdominant luminous baryonic "contamination"), correctly interpreting the cosmological observational data without the need of the notions dark energy, cosmological constant, and universal accelerating expansion. 15. On Functional Representations of the Conformal Algebra CERN Document Server Rosten, Oliver J 2014-01-01 Starting with conformally covariant correlation functions, a sequence of functional representations of the conformal algebra is constructed. A key step is the introduction of representations which involve an auxiliary functional. It is observed that these functionals are not arbitrary but rather must satisfy a pair of consistency equations; one such is identified, in a particular representation, as an Exact Renormalization Group equation specialized to a fixed-point. Therefore, the associated functional is identified with the Wilsonian Effective Action and this creates a concrete link between action-free formulations of Conformal Field Theory and the cutoff-regularized path integral approach. Following this, the energy-momentum tensor is investigated, from which it becomes apparent that the conformal Ward Identities serve to define a particular representation of the energy-momentum tensor. It follows, essentially trivially, that if the Schwinger functional exists and is non-vanishing then theories exhibiting ... 16. Radical covalent organic frameworks: a general strategy to immobilize open-accessible polyradicals for high-performance capacitive energy storage. Science.gov (United States) Xu, Fei; Xu, Hong; Chen, Xiong; Wu, Dingcai; Wu, Yang; Liu, Hao; Gu, Cheng; Fu, Ruowen; Jiang, Donglin 2015-06-01 Ordered π-columns and open nanochannels found in covalent organic frameworks (COFs) could render them able to store electric energy. However, the synthetic difficulty in achieving redox-active skeletons has thus far restricted their potential for energy storage. A general strategy is presented for converting a conventional COF into an outstanding platform for energy storage through post-synthetic functionalization with organic radicals. The radical frameworks with openly accessible polyradicals immobilized on the pore walls undergo rapid and reversible redox reactions, leading to capacitive energy storage with high capacitance, high-rate kinetics, and robust cycle stability. The results suggest that channel-wall functional engineering with redox-active species will be a facile and versatile strategy to explore COFs for energy storage. 17. The Optimal Price Ratio of Typical Energy Sources in Beijing Based on the Computable General Equilibrium Model Directory of Open Access Journals (Sweden) Yongxiu He 2014-04-01 Full Text Available In Beijing, China, the rational consumption of energy is affected by the insufficient linkage mechanism of the energy pricing system, the unreasonable price ratio and other issues. This paper combines the characteristics of Beijing’s energy market, putting forward the society-economy equilibrium indicator R maximization taking into consideration the mitigation cost to determine a reasonable price ratio range. Based on the computable general equilibrium (CGE model, and dividing four kinds of energy sources into three groups, the impact of price fluctuations of electricity and natural gas on the Gross Domestic Product (GDP, Consumer Price Index (CPI, energy consumption and CO2 and SO2 emissions can be simulated for various scenarios. On this basis, the integrated effects of electricity and natural gas price shocks on the Beijing economy and environment can be calculated. The results show that relative to the coal prices, the electricity and natural gas prices in Beijing are currently below reasonable levels; the solution to these unreasonable energy price ratios should begin by improving the energy pricing mechanism, through means such as the establishment of a sound dynamic adjustment mechanism between regulated prices and market prices. This provides a new idea for exploring the rationality of energy price ratios in imperfect competitive energy markets. 18. Conformal higher-order viscoelastic fluid mechanics CERN Document Server Fukuma, Masafumi 2012-01-01 We present a generally covariant formulation of conformal higher-order viscoelastic fluid mechanics with strain allowed to take arbitrarily large values. We give a general prescription to determine the dynamics of a relativistic viscoelastic fluid in a way consistent with the hypothesis of local thermodynamic equilibrium and the second law of thermodynamics. We then elaborately study the transient time scales at which the strain almost relaxes and becomes proportional to the gradients of velocity. We particularly show that a conformal second-order fluid with all possible parameters in the constitutive equations can be obtained without breaking the hypothesis of local thermodynamic equilibrium, if the conformal fluid is defined as the long time limit of a conformal second-order viscoelastic system. We also discuss how local thermodynamic equilibrium could be understood in the context of the fluid/gravity correspondence. 19. Conformal higher-order viscoelastic fluid mechanics Science.gov (United States) Fukuma, Masafumi; Sakatani, Yuho 2012-06-01 We present a generally covariant formulation of conformal higher-order viscoelastic fluid mechanics with strain allowed to take arbitrarily large values. We give a general prescription to determine the dynamics of a relativistic viscoelastic fluid in a way consistent with the hypothesis of local thermodynamic equilibrium and the second law of thermodynamics. We then elaborately study the transient time scales at which the strain almost relaxes and becomes proportional to the gradients of velocity. We particularly show that a conformal second-order fluid with all possible parameters in the constitutive equations can be obtained without breaking the hypothesis of local thermodynamic equilibrium, if the conformal fluid is defined as the long time limit of a conformal second-order viscoelastic system. We also discuss how local thermodynamic equilibrium could be understood in the context of the fluid/gravity correspondence. 20. Conformal field theory on the plane CERN Document Server Ribault, Sylvain 2014-01-01 We provide an introduction to conformal field theory on the plane in the conformal bootstrap approach. We introduce the main ideas of the bootstrap approach to quantum field theory, and how they apply to two-dimensional theories with local conformal symmetry. We describe the mathematical structures which appear in such theories, from the Virasoro algebra and its representations, to the BPZ equations and their solutions. As examples, we study a number of models: Liouville theory, (generalized) minimal models, free bosonic theories, the$H_3^+$model, and the$SU_2$and$\\widetilde{SL}_2(\\mathbb{R})$WZW models. 1. 75 FR 62141 - In the Matter of Certain Energy Drink Products; Notice of Issuance of a Corrected General... Science.gov (United States) 2010-10-07 ... From the Federal Register Online via the Government Publishing Office INTERNATIONAL TRADE COMMISSION In the Matter of Certain Energy Drink Products; Notice of Issuance of a Corrected General Exclusion Order AGENCY: U.S. International Trade Commission. ACTION: Notice. SUMMARY: Notice is hereby... 2. Generalized Chou-Yang model for p(antip)p and. lambda. (anti. lambda. )p elastic scattering at high energies Energy Technology Data Exchange (ETDEWEB) Saleem, M.; Fazal-E-Aleem; Azhar, I.A. 1988-06-01 The various characteristics of pp and antipp elastic scattering at high energies are explained by using the generalized Chou-Yang model which takes into consideration the anisotropic scattering of objects constituting colliding particles. The model is also used to extract the form factor and radius of the ..lambda.. particle. 3. Identification of general linear relationships between activation energies and enthalpy changes for dissociation reactions at surfaces. Science.gov (United States) Michaelides, Angelos; Liu, Z-P; Zhang, C J; Alavi, Ali; King, David A; Hu, P 2003-04-02 The activation energy to reaction is a key quantity that controls catalytic activity. Having used ab inito calculations to determine an extensive and broad ranging set of activation energies and enthalpy changes for surface-catalyzed reactions, we show that linear relationships exist between dissociation activation energies and enthalpy changes. Known in the literature as empirical Brønsted-Evans-Polanyi (BEP) relationships, we identify and discuss the physical origin of their presence in heterogeneous catalysis. The key implication is that merely from knowledge of adsorption energies the barriers to catalytic elementary reaction steps can be estimated. 4. [Dosimetric evaluation of conformal radiotherapy: conformity factor]. Science.gov (United States) Oozeer, R; Chauvet, B; Garcia, R; Berger, C; Felix-Faure, C; Reboul, F 2000-01-01 The aim of three-dimensional conformal therapy (3DCRT) is to treat the Planning Target Volume (PTV) to the prescribed dose while reducing doses to normal tissues and critical structures, in order to increase local control and reduce toxicity. The evaluation tools used for optimizing treatment techniques are three-dimensional visualization of dose distributions, dose-volume histograms, tumor control probabilities (TCP) and normal tissue complication probabilities (NTCP). These tools, however, do not fully quantify the conformity of dose distributions to the PTV. Specific tools were introduced to measure this conformity for a given dose level. We have extended those definitions to different dose levels, using a conformity index (CI). CI is based on the relative volumes of PTV and outside the PTV receiving more than a given dose. This parameter has been evaluated by a clinical study including 82 patients treated for lung cancer and 82 patients treated for prostate cancer. The CI was low for lung dosimetric studies (0.35 at the prescribed dose 66 Gy) due to build-up around the GTV and to spinal cord sparing. For prostate dosimetric studies, the CI was higher (0.57 at the prescribed dose 70 Gy). The CI has been used to compare treatment plans for lung 3DCRT (2 vs 3 beams) and prostate 3DCRT (4 vs 7 beams). The variation of CI with dose can be used to optimize dose prescription. 5. Conformational stability of calreticulin DEFF Research Database (Denmark) Jørgensen, C.S.; Trandum, C.; Larsen, N. 2005-01-01 The conformational stability of calreticulin was investigated. Apparent unfolding temperatures (T-m) increased from 31 degrees C at pH 5 to 51 degrees C at pH 9, but electrophoretic analysis revealed that calreticulin oligomerized instead of unfolding. Structural analyses showed that the single C......-terminal a-helix was of major importance to the conformational stability of calreticulin.... 6. The simulated features of heliospheric cosmic-ray modulation with a time-dependent drift model. III - General energy dependence Science.gov (United States) Potgieter, M. S.; Le Roux, J. A. 1992-01-01 The time-dependent cosmic-ray transport equation is solved numerically in an axially symmetric heliosphere. Gradient and curvature drifts are incorporated, together with an emulated wavy neutral sheet. This model is used to simulate heliospheric cosmic-ray modulation for the period 1985-1989 during which drifts are considered to be important. The general energy dependence of the modulation of Galactic protons is studied as predicted by the model for the energy range 1 MeV to 10 GeV. The corresponding instantaneous radial and latitudinal gradients are calculated, and it is found that, whereas the latitudinal gradients follow the trends in the waviness of the neutral sheet to a large extent for all energies, the radial gradients below about 200 MeV deviate from this general pattern. In particular, these gradients increase when the waviness decreases for the simulated period 1985-1987.3, after which they again follow the neutral sheet by increasing rapidly. 7. N = 4 l-conformal Galilei superalgebra Science.gov (United States) Galajinsky, Anton; Masterov, Ivan 2017-08-01 An N = 4 supersymmetric extension of the l-conformal Galilei algebra is constructed. This is achieved by combining generators of spatial symmetries from the l-conformal Galilei algebra and those underlying the most general superconformal group in one dimension D (2 , 1 ; α). The value of the group parameter α is fixed from the requirement that the resulting superalgebra is finite-dimensional. The analysis reveals α = -1/2 thus reducing D (2 , 1 ; α) to OSp (4 \| 2). 8. Effective Conformal Descriptions of Black Hole Entropy Directory of Open Access Journals (Sweden) Steven Carlip 2011-07-01 Full Text Available It is no longer considered surprising that black holes have temperatures and entropies. What remains surprising, though, is the universality of these thermodynamic properties: their exceptionally simple and general form, and the fact that they can be derived from many very different descriptions of the underlying microscopic degrees of freedom. I review the proposal that this universality arises from an approximate conformal symmetry, which permits an effective “conformal dual” description that is largely independent of the microscopic details. 9. Conformally Invariant Spinorial Equations in Six Dimensions CERN Document Server Batista, Carlos 2016-01-01 This work deals with the conformal transformations in six-dimensional spinorial formalism. Several conformally invariant equations are obtained and their geometrical interpretation are worked out. Finally, the integrability conditions for some of these equations are established. Moreover, in the course of the article, some useful identities involving the curvature of the spinorial connection are attained and a digression about harmonic forms and more general massless fields is made. 10. Conformal Killing Vectors Of Plane Symmetric Four Dimensional Lorentzian Manifolds CERN Document Server Khan, Suhail; Bokhari, Ashfaque H; Khan, Gulzar Ali; Mathematics, Department of; Peshawar, University of; Pakhtoonkhwa, Peshawar Khyber; Pakistan.,; Petroleum, King Fahd University of; Minerals,; 31261, Dhahran; Arabia, Saudi 2015-01-01 In this paper, we investigate conformal Killing's vectors (CKVs) admitted by some plane symmetric spacetimes. Ten conformal Killing's equations and their general forms of CKVs are derived along with their conformal factor. The existence of conformal Killing's symmetry imposes restrictions on the metric functions. The conditions imposing restrictions on these metric functions are obtained as a set of integrability conditions. Considering the cases of time-like and inheriting CKVs, we obtain spacetimes admitting plane conformal symmetry. Integrability conditions are solved completely for some known non-conformally flat and conformally flat classes of plane symmetric spacetimes. A special vacuum plane symmetric spacetime is obtained, and it is shown that for such a metric CKVs are just the homothetic vectors (HVs). Among all the examples considered, there exists only one case with a six dimensional algebra of special CKVs admitting one proper CKV. In all other examples of non-conformally flat metrics, no proper ... 11. Generative models of conformational dynamics. Science.gov (United States) Langmead, Christopher James 2014-01-01 Atomistic simulations of the conformational dynamics of proteins can be performed using either Molecular Dynamics or Monte Carlo procedures. The ensembles of three-dimensional structures produced during simulation can be analyzed in a number of ways to elucidate the thermodynamic and kinetic properties of the system. The goal of this chapter is to review both traditional and emerging methods for learning generative models from atomistic simulation data. Here, the term 'generative' refers to a model of the joint probability distribution over the behaviors of the constituent atoms. In the context of molecular modeling, generative models reveal the correlation structure between the atoms, and may be used to predict how the system will respond to structural perturbations. We begin by discussing traditional methods, which produce multivariate Gaussian models. We then discuss GAMELAN (GRAPHICAL MODELS OF ENERGY LANDSCAPES), which produces generative models of complex, non-Gaussian conformational dynamics (e.g., allostery, binding, folding, etc.) from long timescale simulation data. 12. Geothermal energy: Technology and general studies. Citations from the NTIS data base Science.gov (United States) Hundemann, A. S. 1980-09-01 This bibliography contains 311 citations of Government-sponsored research on geothermal energy conversion, power plants, heat extraction, and space heating. Studies on fluid flow, heat transfer, rock fracturing, environmental impacts, pressure, and reservoir engineering are included. Reports on economics, legislation, technology assessment, comparative evaluation with other energy sources, Government policies, and planning are also cited. 13. Eikonalization of Conformal Blocks CERN Document Server Fitzpatrick, A Liam; Walters, Matthew T; Wang, Junpu 2015-01-01 Classical field configurations such as the Coulomb potential and Schwarzschild solution are built from the$t$-channel exchange of many light degrees of freedom. We study the CFT analog of this phenomenon, which we term the eikonalization' of conformal blocks. We show that when an operator$T$appears in the OPE$\\mathcal{O}(x) \\mathcal{O}(0)$, then the large spin$\\ell$Fock space states$[TT \\cdots T]_{\\ell}$also appear in this OPE with a computable coefficient. The sum over the exchange of these Fock space states in an$\\langle \\mathcal{O} \\mathcal{O} \\mathcal{O} \\mathcal{O} \\rangle$correlator build the classical `$T$field' in the dual AdS description. In some limits the sum of all Fock space exchanges can be represented as the exponential of a single$T$exchange in the 4-pt correlator of$\\mathcal{O}$. Our results should be useful for systematizing$1/\\ell$perturbation theory in general CFTs and simplifying the computation of large spin OPE coefficients. As examples we obtain the leading$\\log \\ell\$... 14. Energy care. The tool for structured attention for energy efficiency. For profit and non-profit organizations conform ISO 14001(2004); Energiezorg. Het middel voor structurele aandacht voor energie-efficiency. Voor profit- en non-profitorganisaties conform ISO14001:2004 Energy Technology Data Exchange (ETDEWEB) NONE 2006-06-15 An energy care ('energiezorg' in Dutch) system has been developed by means of which profit and non-profit organizations can continuously control the consumption of energy and improve the energy efficiency within their organization. [Dutch] Steeds meer organisaties en bedrijven nemen maatregelen om energie te besparen of de energie-efficiency te verbeteren. Dit levert zowel financiele als milieuwinst op. Helaas zijn de effecten van de inspanningen vaak maar tijdelijk. Zodra de aandacht voor het onderwerp wegebt, neemt het energiegebruik weer toe. Daarom heeft SenterNovem een systeem voor structurele aandacht ontwikkeld. Dit zorgt ervoor dat energiegebruik binnen de onderneming of organisatie de aandacht krijgt die het verdient en dat de aandacht nooit verslapt. Dat blijkt ook uit de ervaringen van veel bedrijven en instellingen. Zij constateren dat invoering van Energiezorg niet alleen bijdraagt aan een blijvende verlaging van het energiegebruik, maar zelfs ook extra energiebesparing oplevert van gemiddeld 3 procent. Dit geldt voor zowel profit- als non-profitorganisaties. Het instrument Energiezorg is geschikt voor kleine, middelgrote en grote organisaties en bedrijven. 15. Boundary terms of conformal anomaly Directory of Open Access Journals (Sweden) Sergey N. Solodukhin 2016-01-01 Full Text Available We analyze the structure of the boundary terms in the conformal anomaly integrated over a manifold with boundaries. We suggest that the anomalies of type B, polynomial in the Weyl tensor, are accompanied with the respective boundary terms of the Gibbons–Hawking type. Their form is dictated by the requirement that they produce a variation which compensates the normal derivatives of the metric variation on the boundary in order to have a well-defined variational procedure. This suggestion agrees with recent findings in four dimensions for free fields of various spins. We generalize this consideration to six dimensions and derive explicitly the respective boundary terms. We point out that the integrated conformal anomaly in odd dimensions is non-vanishing due to the boundary terms. These terms are specified in three and five dimensions. 16. Boundary terms of conformal anomaly Energy Technology Data Exchange (ETDEWEB) Solodukhin, Sergey N., E-mail: [email protected] 2016-01-10 We analyze the structure of the boundary terms in the conformal anomaly integrated over a manifold with boundaries. We suggest that the anomalies of type B, polynomial in the Weyl tensor, are accompanied with the respective boundary terms of the Gibbons–Hawking type. Their form is dictated by the requirement that they produce a variation which compensates the normal derivatives of the metric variation on the boundary in order to have a well-defined variational procedure. This suggestion agrees with recent findings in four dimensions for free fields of various spins. We generalize this consideration to six dimensions and derive explicitly the respective boundary terms. We point out that the integrated conformal anomaly in odd dimensions is non-vanishing due to the boundary terms. These terms are specified in three and five dimensions. 17. The impact of a stimulus to energy efficiency on the economy and the environment: A regional computable general equilibrium analysis Energy Technology Data Exchange (ETDEWEB) Hanley, Nick D. [Department of Economics, University of Stirling, Stirling FK9 4LA (United Kingdom); McGregor, Peter G.; Swales, J. Kim [Fraser of Allander Institute, CPPR and Department of Economics, University of Strathclyde, Sir William Duncan Building, 130 Rottenrow, Glasgow G4 0GE (United Kingdom); Turner, Karen [Fraser of Allander Institute and Department of Economics, University of Strathclyde, Sir William Duncan Building, 130 Rottenrow, Glasgow G4 0GE (United Kingdom) 2006-02-01 Sustainable development is a key objective of UK national and regional policies. Improvements in resource productivity have been suggested as both a measure of progress towards sustainable development and as a means of achieving sustainability. Making 'more with less' intuitively seems to be good for the environment, and this is the presumption of current UK policy. However, in a system-wide context, improvements in energy efficiency lower the cost of energy in efficiency units and may even stimulate the consumption and production of energy measured in physical units, and increase pollution. Simulations of a computable general equilibrium model of Scotland suggest that an across the board stimulus to energy efficiency there would actually stimulate energy production and consumption and lead to a deterioration in environmental indicators. The implication is that policies directed at stimulating energy efficiency are not, in themselves, sufficient to secure environmental improvements: this may require the use of complementary energy policies designed to moderate incentives to increased energy consumption. (author) 18. 75 FR 22586 - Energy Conservation Program for Consumer Products: Notice of Petition for Waiver of General... Science.gov (United States) 2010-04-29 ... asterisks, or wild cards, denote color or other features that do not affect energy performance.) DOE notes...*, ZFGP21HZ**. The asterisks, or wild cards, denote color or other features that do not affect... 19. A general scheme for the estimation of oxygen binding energies on binary transition metal surface alloys DEFF Research Database (Denmark) Greeley, Jeffrey Philip; Nørskov, Jens Kehlet 2005-01-01 A simple scheme for the estimation of oxygen binding energies on transition metal surface alloys is presented. It is shown that a d-band center model of the alloy surfaces is a convenient and appropriate basis for this scheme; variations in chemical composition, strain effects, and ligand effects...... for the estimation of oxygen binding energies on a wide variety of transition metal alloys. (c) 2005 Elsevier B.V. All rights reserved.... 20. EC declaration of conformity. Science.gov (United States) Donawa, M E 1996-05-01 The CE-marking procedure requires that manufacturers draw up a written declaration of conformity before placing their products on the market. However, some companies do not realize that this is a requirement for all devices. Also, there is no detailed information concerning the contents and format of the EC declaration of conformity in the medical device Directives or in EC guidance documentation. This article will discuss some important aspects of the EC declaration of conformity and some of the guidance that is available on its contents and format.	{"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": 0.8763025403022766, "perplexity": 3118.585857077437}, "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/1518891812913.37/warc/CC-MAIN-20180220070423-20180220090423-00683.warc.gz"}
http://mathhelpforum.com/math-topics/168492-drawing-displacement-time-wave.html	# Thread: Drawing a displacement/time wave 1. ## Drawing a displacement/time wave 1. The problem statement, all variables and given/known data Hi I need help on how to draw Displacement/time waves. I am considering this question: If a wave is travelling from left to right, draw a diagram of what the wave would look like 10ms later. The amplitude is 0.02 and wavelength is 0.2 m respectively. 2. Relevant equations Speed = distance/time 3. The attempt at a solution I can clearly work out the distance which is 0.3m. But I am not sure of the amplitude, shape and wavelength. Can someone please comment? 2. I find it strange that you don't have the speed of the wave but managed to get the distance covered within 10 ms... What you basically need to draw is a sine curve starting at the distance covered on the x axis, going towards the left. The wavelength is measured on the x-axis and consists of a complete cycle of the sine graph. The amplitude indicates the maximum and minimum points of the graph. Here, the maximum is 0.02 m and the minimum is -0.02 m. 3. Sorry. the speed wasn't included in the question (Thats by me). So, what happens if the furthest point on the sine curve is 0.3m? 4. Yes, as the wavelength is 0.2 m, it means you'll have a cycle and a half drawn. The resulting graph will be quite similar to the graph of y = 0.02sin(x) between 0 and 0.3 m, with two maxima at 0.5 and 2.5 m and a minimum at 1.5. 5. Thanks. That makes more sense.	{"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": 0.9409170746803284, "perplexity": 411.2771159572267}, "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-2017-39/segments/1505818689752.21/warc/CC-MAIN-20170923160736-20170923180736-00303.warc.gz"}
https://www.physicsforums.com/threads/velocity-profile.652866/	# Velocity Profile 1. Nov 17, 2012 ### CrazyNeutrino What is a velocity profile in fluid mechanics? Why do they give you a ratio between the fluids velocity and a random reference velocity and why is it equal to a completely arbitrary function? Does the ratio equal a function because it is an experimental fact? 2. Nov 17, 2012 ### Staff: Mentor Can you give a specific example of what is troubling you? It is difficult to understand your question otherwise.	{"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": 0.9070075750350952, "perplexity": 817.6366196862674}, "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-2017-09/segments/1487501171271.48/warc/CC-MAIN-20170219104611-00637-ip-10-171-10-108.ec2.internal.warc.gz"}
https://steamsplay.com/final-fantasy-vi-how-to-replace-the-default-font-guide/	# FINAL FANTASY VI How to Replace the Default Font Guide Replace the default font with the one from the Final Fantasy Advance and Final Fantasy DS games. # INTRO: Replaces the default font with the one from the Final Fantasy Advance and Final Fantasy DS games. Specifically the final version, from FFIV DS. A previous mod exists, but lacks support for non-English languages, and even for English, there were errors. Supports all languages that use the Latin script, the Cyrillic script, and the Greek script, so long as they don’t require letters that use multiple diacritics at the same time. Height limitations prevented them. Japanese Hiragana and Katakana also supported, but not Kanji. If you see any problems, let me know. Especially in regards to the Cyrillic script and Greek script since I’m not familiar with them. I’ll update the font and the bundle files if need be. In the download, I included a TTF file of the font, so it can be used for other purposes too. # INSTRUCTIONS: Go to your Steam folder, then continue onward to: steamapps\common\FINAL FANTASY VI PR\FINAL FANTASY VI_Data\StreamingAssets , and then copy-and-paste the BUNDLE files into the folder, replacing the pre-existing ones. You can make a backup of the originals first, if you want, though I think just deleting the new ones should result in Steam re-downloading the originals. These files also work on the first five games, in the same manner.	{"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": 0.9589086174964905, "perplexity": 2985.181677748938}, "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/1652662578939.73/warc/CC-MAIN-20220525023952-20220525053952-00226.warc.gz"}
https://dsp.stackexchange.com/questions/4825/why-is-the-fft-mirrored	Why is the FFT “mirrored”? If you do an FFT plot of a simple signal, like: t = 0:0.01:1 ; N = max(size(t)); x = 1 + sin( 2pit ) ; y = abs( fft( x ) ) ; stem( Nt, y ) FFT of above I understand that the number in the first bin is "how much DC" there is in the signal. y(1) %DC > 101.0000 The number in the second bin should be "how much 1-cycle over the whole signal" there is: y(2) %1 cycle in the N samples > 50.6665 But it's not 101! It's about 50.5. There's another entry at the end of the fft signal, equal in magnitude: y(101) > 50.2971 So 50.5 again. My question is, why is the FFT mirrored like this? Why isn't it just a 101 in y(2) (which would of course mean, all 101 bins of your signal have a 1 Hz sinusoid in it?) Would it be accurate to do: mid = round( N/2 ) ; % Prepend y(1), then add y(2:middle) with the mirror FLIPPED vector % from y(middle+1:end) z = [ y(1), y( 2:mid ) + fliplr( y(mid+1:end) ) ]; stem( z ) Flip and add-in the second half of the FFT vector I thought now, the mirrored part on the right hand side is added in correctly, giving me the desired "all 101 bins of the FFT contain a 1Hz sinusoid" >> z(2) ans = 100.5943 • A similar question has been answered here: dsp.stackexchange.com/questions/3466/… – pichenettes Oct 28 '12 at 17:35 • But this is specifically about the symmetry (I believe it's called Hermetian symmetry?) of the signal. – bobobobo Oct 28 '12 at 17:44 • For a pure real signals F(k)=conj(F(N-k)), this is why the Fourier transform of a pure real signal is symmetric. – WebMonster Oct 28 '12 at 18:02 • Ask yourself: what result would you expect if your signal was 1 + cos(2pit)... And 1 + i cos(2pit)... And 1 + i sin(2pit)... – pichenettes Oct 28 '12 at 18:21 • Because a Fourier transform breaks up a signal into complex exponentials, and a sine wave is the sum of 2 complex exponentials. dsp.stackexchange.com/a/449/29 – endolith Oct 28 '12 at 19:55 Real signals are "mirrored" in the real and negative halves of the Fourier transform because of the nature of the Fourier transform. The Fourier transform is defined as the following- $H(f) = \int h(t)e^{-j2\pi ft}dt$ Basically it correlates the signal with a bunch of complex sinusoids, each with its own frequency. So what do those complex sinusoids look like? The picture below illustrates one complex sinusoid. The "corkscrew" is the rotating complex sinusoid in time, while the two sinusoids that follow it are the extracted real and imaginary components of the complex sinusoid. The astute reader will note that the real and imaginary components are the exact same, only they are out of phase with each other by 90 degrees ($\frac{\pi}{2}$). Because they are 90 degrees out of phase they are orthogonal and can "catch" any component of the signal at that frequency. The relationship between the exponential and the cosine/sine is given by Euler's formula- $e^{jx} = cos(x) + jsin(x)$ This allows us to modify the Fourier transform as follows- $$H(f) = \int h(t)e^{-j2\pi ft}dt \\ = \int h(t)(cos(2\pi ft) - jsin(2\pi ft))dt$$ At the negative frequencies the Fourier transform becomes the following- $$H(-f) = \int h(t)(cos(2\pi (-f)t) - jsin(2\pi (-f)t))dt \\ = \int h(t)(cos(2\pi ft) + jsin(2\pi ft))dt$$ Comparing the negative frequency version with the positive frequency version shows that the cosine is the same while the sine is inverted. They are still 90 degrees out of phase with each other, though, allowing them to catch any signal component at that (negative) frequency. Because both the positive and negative frequency sinusoids are 90 degrees out of phase and have the same magnitude, they will both respond to real signals in the same way. Or rather, the magnitude of their response will be the same, but the correlation phase will be different. EDIT: Specifically, the negative frequency correlation is the conjugate of the positive frequency correlation (due to the inverted imaginary sine component) for real signals. In mathematical terms, this is, as Dilip pointed out, the following- $H(-f) = [H(f)]^$ Another way to think about it: Imaginary components are just that..Imaginary! They are a tool, which allows the employ of an extra plane to view things on and makes much of digital (and analog) signal processing possible, if not much easier than using differential equations! But we can't break the logical laws of nature, we can't do anything 'real' with the imaginary content$^\dagger$ and so it must effectively cancel itself out before returning to reality. How does this look in the Fourier Transform of a time based signal(complex frequency domain)? If we add/sum the positive and negative frequency components of the signal the imaginary parts cancel, this is what we mean by saying the positive and negative elements are conjugate to each-other. Notice that when an FT is taken of a time-signal there exists these conjugate signals, with the 'real' part of each sharing the magnitude, half in the positive domain, half in the negative, so in effect adding the conjugates together removes the imaginary content and provides the real content only. $^\dagger$ Meaning we can't create a voltage that is $5i$ volts. Obviously, we can use imaginary numbers to represent real-world signals that are two-vector-valued, such as circularly polarized EM waves. • Good answer - one slight nitpick though, I am not on-board with "Because they are the same, anything that one correlates with, the other will too with the exact same magnitude and a 90 degree phase shift.". I know what you are trying to say, however (as you know), a sine correlates with a sine (score 1), but wont correlate at all with a cosine at all, (score 0). They are the same signal, but with different phases afterall. – Spacey Oct 29 '12 at 12:54 • You're right. There's another more serious problem too. I will fix it later. – Jim Clay Oct 29 '12 at 13:23 • It would be nice if you could edit your answer to be more responsive to the question which is about DFTs (though it says FFT in the title) rather than giving the general theory of Fourier transforms. – Dilip Sarwate Oct 29 '12 at 22:35 • @DilipSarwate My goal is to help the questioner understand, and I think my approach is best for that. I have upvoted your answer, though, for doing the discrete math. – Jim Clay Oct 29 '12 at 23:01 • @JimClay Your approach is greatly appreciated by the entire readership of dsp.SE, and I hope that you will find the time to make your answer a truly great answer by explicitly including in your answer what is currently left for the reader to deduce: viz. that the equations show that $H(-f) = [H(f)]^$ (and hence $\|H(-f)\| = \|H(f)\|$) when $x(t)$ is a real-valued signal and that this is the "mirroring" that the OP was asking about. In other words, I request that you edit your answer to be more responsive to the question actually asked (as I requested in my previous comment). – Dilip Sarwate Oct 30 '12 at 12:42 The FFT (or Fast Fourier Transform) is actually an algorithm for the computation of the Discrete Fourier Transform or DFT. The typical implementation achieves speed-up over the conventional computation of the DFT by exploiting the fact that $N$, the number of data points, is a composite integer which is not the case here since $101$ is a prime number. (While FFTs exist for the case when $N$ is a prime, they use a different formulation that might or might not be implemented in MATLAB). Indeed, many people deliberately choose $N$ to be of the form $2^k$ or $4^k$ so as to speed up the DFT computation via the FFT. Turning to the question as to why the mirroring occurs, hotpaw2 has essentially stated the reason, and so the following is just a filling in of the details. The DFT of a sequence $\mathbf x = \bigr(x[0], x[1], x[2], \ldots, x[N-1]\bigr)$ of $N$ data points is defined to be a sequence $\mathbf X =\bigr(X[0], X[1], X[2], \ldots, X[N-1]\bigr)$ where $$X[m] = \sum_{n=0}^{N-1} x[n]\left(\exp\left(-j2\pi \frac{m}{N}\right)\right)^n, m = 0, 1, \ldots, N-1$$ where $j = \sqrt{-1}$. It will be obvious that $\mathbf X$ is, in general a complex-valued sequence even when $\mathbf x$ is a real-valued sequence. But note that when $\mathbf x$ is a real-valued sequence, $\displaystyle X[0]=\sum_{n=0}^{N-1} x[n]$ is a real number. Furthermore, if $N$ is an even number, then, since $\exp(-j\pi) = -1$, we also have that $$X\left[\frac{N}{2}\right] = \sum_{n=0}^{N-1} x[n]\left(\exp\left(-j2\pi \frac{N/2}{N}\right)\right)^n = \sum_{n=0}^{N-1} x[n](-1)^n$$ is a real number. But, regardless of whether $N$ is odd or even, the DFT $\mathbf X$ of a real-valued sequence $\mathbf x$ has Hermitian symmetry property that you have mentioned in a comment. We have for any fixed $m$, $1 \leq m \leq N-1$, \begin{align} X[m] &= \sum_{n=0}^{N-1} x[n]\left(\exp\left(-j2\pi \frac{m}{N}\right)\right)^n\\ X[N-m] &= \sum_{n=0}^{N-1} x[n]\left(\exp\left(-j2\pi \frac{N-m}{N}\right)\right)^n\\ &= \sum_{n=0}^{N-1} x[n]\left(\exp\left(-j2\pi + j2\pi\frac{m}{N}\right)\right)^n\\ &= \sum_{n=0}^{N-1} x[n]\left(\exp\left(j2\pi\frac{m}{N}\right)\right)^n\\ &= \left(X[m]\right)^* \end{align} Thus, for $1 \leq m \leq N-1$, $X[N-m] = \left(X[m]\right)^$. As a special case of this, note that if we choose $m = N/2$ when $N$ is even, we get that $X[N/2] = \left(X[N/2]\right)^$, thus confirming our earlier conclusion that $X[N/2]$ is a real number. Note that an effect of the Hermitian symmetry property is that the $m$-th bin in the DFT of a real-valued sequence has the same magnitude as the $(N-m)$-th bin. MATLABi people will need to translate this to account for the fact that MATLAB arrays are numbered from $1$ upwards. Turning to your actual data, your $\mathbf x$ is a DC value of $1$ plus slightly more than one period of a sinusoid of frequency $1$ Hz. Indeed, what you are getting is $$x[n] = 1 + \sin(2\pi (0.01n)), ~ 0 \leq n \leq 100$$ where $x[0] = x[100] = 1$. Thus, the first and the last of $101$ samples has the same value. The DFT that you are computing is thus given by $$X[m] = \sum_{n=0}^{100} \left(1+\sin\left(2\pi \left(\frac{n}{100}\right)\right)\right)\left(\exp\left(-j2\pi \frac{m}{101}\right)\right)^n$$ The mismatch between $100$ and $101$ causes clutter in the DFT: the values of $X[m]$ for $2 \leq m \leq 99$ are nonzero, albeit small. On the other hand, suppose you were to adjust the array t in your MATLAB program to have $100$ samples taken at $t=0, 0.01, 0.02, \ldots, 0.99$ so that what you have is $$x[n] = 1 + \sin(2\pi (0.01n)), ~ 0 \leq n \leq 99.$$ Then the DFT is $$X[m] = \sum_{n=0}^{99} \left(1+\sin\left(2\pi \left(\frac{n}{100}\right)\right)\right)\left(\exp\left(-j2\pi \frac{m}{100}\right)\right)^n,$$ you will see that your DFT will be exactly $\mathbf X = (100, -50j, 0, 0, \ldots, 0, 50j)$ (or at least within round-off error), and the inverse DFT will give that for $0 \leq n \leq 99$, \begin{align} x[n] &= \frac{1}{100}\sum_{m=0}^{99}X[m]\left(\exp\left(j2\pi \frac{n}{100}\right)\right)^m\\ &= \frac{1}{100}\left[100 - 50j\exp\left(j2\pi \frac{n}{100}\right)^1 + 50j \left(\exp\left(j2\pi \frac{n}{100}\right)\right)^{99}\right]\\ &= 1 + \frac{1}{2j}\left[\exp\left(j2\pi \frac{n}{100}\right) - \exp\left(j2\pi \frac{-n}{100}\right)\right]\\ &= 1 + \sin(2\pi (0.01n)) \end{align*} which is precisely what you started from. • So, is it possible to tell from the FFT if a signal is periodic or not? – displayname Sep 18 '16 at 14:01 • @displayname That is a separate question that should be asked in its own right (and perhaps has been asked and answered already). – Dilip Sarwate Sep 20 '16 at 15:58 • When I carefully pry out the conjugate symmetrical bins [By writing a 0 + 0i into them] and reconstruct the time domain signal using ifft, the magnitude of the reconstructed time domain signal has halved. Is this natural or is it a tooling problem? I do take care of FFT output normalization and its reverse after iFFT. – Raj Aug 15 '17 at 4:51 Note that an FFT result is mirrored (as in conjugate symmetric) only if the input data is real. For strictly real input data, the two conjugate mirror images in the FFT result cancel out the imaginary parts of any complex sinusoids, and thus sum to a strictly real sinusoid (except for tiny numerical rounding noise), thus leaving you with a representation of strictly real sine waves. If the FFT result wasn't conjugate mirrored, it would represent a waveform that had complex values (non-zero imaginary components), not something strictly real valued.	{"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": 0.8852511644363403, "perplexity": 540.2994671171074}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986693979.65/warc/CC-MAIN-20191019114429-20191019141929-00009.warc.gz"}
https://www.doubtnut.com/question-answer/in-a-school-there-are-two-sections-section-a-and-section-b-of-class-x-there-are-32-students-in-secti-1409201	Home > English > Class 10 > Maths > Chapter > Real Numbers > In a school there are two s... # In a school there are two sections section A and section B of class X . There are 32 students in section A and 36 students in section B . Determine the minimum number of books required for their class library so that they can be distributed equally among students of section A or section B . Updated On: 27-06-2022	{"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": 0.9341356754302979, "perplexity": 855.686989768533}, "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/1669446710503.24/warc/CC-MAIN-20221128102824-20221128132824-00315.warc.gz"}
http://mathhelpforum.com/advanced-algebra/176787-solving-equations-factor-ring-z-x-x.html	# Thread: Solving Equations in a factor Ring Z[x] / <x³> 1. ## Solving Equations in a factor Ring Z[x] / <x³> How do I solve y² -2y + 1 = 0 in the factor ring Z[x] / <x³> ? I know how to solve this in Z or Z_n (where n is small), but I'm not too familiar with factor rings. How do I solve y² -2y + 1 = 0 in the factor ring Z[x] / <x³> ? I know how to solve this in Z or Z_n (where n is small), but I'm not too familiar with factor rings. In any unitary ring, $y^2-2y+1=(y-1)^2$ , so... Tonio 3. I know that y = 1 is one solution, but are there others? Like in Z_n, there can be other solutions than just y = 1. You know that $(y-1)^2=0$. You now need to decide whether there are any nonzero elements in this ring whose square is 0. If not, then you can conclude that $y-1=0.$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 3, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8365876078605652, "perplexity": 470.35531375113965}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187823360.1/warc/CC-MAIN-20171019175016-20171019195016-00841.warc.gz"}
https://www.groundai.com/project/importance-resampling-for-off-policy-prediction/	Importance Resampling for Off-policy Prediction # Importance Resampling for Off-policy Prediction Matthew Schlegel University of Alberta [email protected] &Wesley Chung University of Alberta [email protected] &Daniel Graves Huawei [email protected] &Jian Qian University of Alberta [email protected] &Martha White University of Alberta [email protected] ###### Abstract Importance sampling (IS) is a common reweighting strategy for off-policy prediction in reinforcement learning. While it is consistent and unbiased, it can result in high variance updates to the weights for the value function. In this work, we explore a resampling strategy as an alternative to reweighting. We propose Importance Resampling (IR) for off-policy prediction, which resamples experience from a replay buffer and applies standard on-policy updates. The approach avoids using importance sampling ratios in the update, instead correcting the distribution before the update. We characterize the bias and consistency of IR, particularly compared to Weighted IS (WIS). We demonstrate in several microworlds that IR has improved sample efficiency and lower variance updates, as compared to IS and several variance-reduced IS strategies, including variants of WIS and V-trace which clips IS ratios. We also provide a demonstration showing IR improves over IS for learning a value function from images in a racing car simulator. Importance Resampling for Off-policy Prediction Matthew Schlegel University of Alberta [email protected] Wesley Chung University of Alberta [email protected] Daniel Graves Huawei [email protected] Jian Qian University of Alberta [email protected] Martha White University of Alberta [email protected] \@float noticebox[b]Preprint. Under review.\end@float ## 1 Introduction An emerging direction for reinforcement learning systems is to learn many predictions, formalized as value function predictions contingent on many different policies. The idea is that such predictions can provide a powerful abstract model of the world. Some examples of systems that learn many value functions are the Horde architecture composed of General Value Functions (GVFs) (Sutton et al., 2011; Modayil et al., 2014), systems that use options (Sutton et al., 1999; Schaul et al., 2015a), predictive representation approaches (Sutton et al., 2005; Schaul and Ring, 2013; Silver et al., 2017) and systems with auxiliary tasks (Jaderberg et al., 2017). Off-policy learning is critical for learning many value functions with different policies, because it enables data to be generated from one behavior policy to update the values for each target policy in parallel. The typical strategy for off-policy learning is to reweight updates using importance sampling (IS). For a given state , with action selected according to behavior , the IS ratio is the ratio between the probability of the action under the target policy and the behavior: . The update is multiplied by this ratio, adjusting the action probabilities so that the expectation of the update is as if the actions were sampled according to the target policy . Though the IS estimator is unbiased and consistent (Kahn and Marshall, 1953; Rubinstein and Kroese, 2016), it can suffer from high or even infinite variance due to large magnitude IS ratios, in theory (Andradottir et al., 1995) and in practice (Precup et al., 2001; Mahmood et al., 2014, 2017). There have been some attempts to modify off-policy prediction algorithms to mitigate this variance.111There is substantial literature on variance reduction for another area called off-policy policy evaluation, but which estimates only a single number or value for a policy (e.g., see (Thomas and Brunskill, 2016)). The resulting algorithms differ substantially, and are not appropriate for learning the value function. Weighted IS (WIS) algorithms have been introduced (Precup et al., 2001; Mahmood et al., 2014; Mahmood and Sutton, 2015), which normalize each update by the sample average of the ratios. These algorithms improve learning over standard IS strategies, but are not straightforward to extend to nonlinear function approximation. In the offline setting, a reweighting scheme, called importance sampling with unequal support (Thomas and Brunskill, 2017), was introduced to account for samples where the ratio is zero, in some cases significantly reducing variance. Another strategy is to rescale or truncate the IS ratios, as used by V-trace (Espeholt et al., 2018) for learning value functions and Tree-Backup (Precup et al., 2000), Retrace (Munos et al., 2016) and ABQ (Mahmood et al., 2017) for learning action-values. Truncation of IS-ratios in V-trace can incur significant bias, and this additional truncation parameter needs to be tuned. An alternative to reweighting updates is to instead correct the distribution before updating the estimator using weighted bootstrap sampling: resampling a new set of data from the previously generated samples (Smith et al., 1992; Arulampalam et al., 2002). Consider a setting where a buffer of data is stored, generated by a behavior policy. Samples for policy can be obtained by resampling from this buffer, proportionally to for state-action pairs in the buffer. In the sampling literature, this strategy has been proposed under the name Sampling Importance Resampling (SIR) (Rubin, 1988; Smith et al., 1992; Gordon et al., 1993), and has been particularly successful for Sequential Monte Carlo sampling (Gordon et al., 1993; Skare et al., 2003). Such resampling strategies have also been popular in classification, with over-sampling or under-sampling typically being preferred to weighted (cost-sensitive) updates (Lopez et al., 2013). A resampling strategy has several potential benefits for off-policy prediction.222We explicitly use the term prediction rather than policy evaluation to make it clear that we are not learning value functions for control. Rather, our goal is to learn value functions solely for the sake of prediction. Resampling could even have larger benefits for learning approaches, as compared to averaging or numerical integration problems, because updates accumulate in the weight vector and change the optimization trajectory of the weights. For example, very large importance sampling ratios could destabilize the weights. This problem does not occur for resampling, as instead the same transition will be resampled multiple times, spreading out a large magnitude update across multiple updates. On the other extreme, with small ratios, IS will waste updates on transitions with very small IS ratios. By correcting the distribution before updating, standard on-policy updates can be applied. The magnitude of the updates vary less—because updates are not multiplied by very small or very large importance sampling ratios—potentially reducing variance of stochastic updates and simplifying learning rate selection. We hypothesize that resampling (a) learns in a fewer number of updates to the weights, because it focuses computation on samples that are likely under the target policy and (b) is less sensitive to learning parameters and target and behavior policy specification. In this work, we investigate the use of resampling for online off-policy prediction for known, unchanging target and behavior policies. We first introduce Importance Resampling (IR), which samples transitions from a buffer of (recent) transitions according to IS ratios. These sampled transitions are then used for on-policy updates. We show that IR has the same bias as WIS, and that it can be made unbiased and consistent with the inclusion of a batch correction term—even under a sliding window buffer of experience. We provide additional theoretical results characterizing when we might expect the variance to be lower for IR than IS. We then empirically investigate IR on three microworlds and a racing car simulator, learning from images, highlighting that (a) IR is less sensitive to learning rate than IS and V-trace (IS with clipping) and (b) IR converges more quickly in terms of the number of updates. ## 2 Background We consider the problem of learning General Value Functions (GVFs) (Sutton et al., 2011). The agent interacts in an environment defined by a set of states , a set of actions and Markov transition dynamics, with probability of transitions to state when taking action in state . A GVF is defined for policy , cumulant and continuation function , with and for a (random) transition . The value for a state is V(s)\makebox[0.0pt]def=Eπ[Gt\|St=s] where Gt\makebox[0.0pt]def=Ct+1+γt+1Ct+2+γt+1γt+2Ct+3+…. The operator indicates an expectation with actions selected according to policy . GVFs encompass standard value functions, where the cumulant is a reward. Otherwise, GVFs enable predictions about discounted sums of others signals into the future, when following a target policy . These values are typically estimated using parametric function approximation, with weights defining approximate values . In off-policy learning, transitions are sampled according to behavior policy, rather than the target policy. To get an unbiased sample of an update to the weights, the action probabilities need to be adjusted. Consider on-policy temporal difference (TD) learning, with update for a given , for learning rate and TD-error . If actions are instead sampled according to a behavior policy , then we can use importance sampling (IS) to modify the update, giving the off-policy TD update for IS ratio . Given state , if when , then the expected value of these two updates are equal. To see why, notice that Eμ[αtρtδt∇θVθ(s)\|St=s]=αt∇θVθ(s)Eμ[ρtδt\|St=s] which equals because Eμ[ρtδt\|St=s] =∑a∈Aμ(a\|s)π(a\|s)μ(a\|s)E[δt\|St=s,At=a]= Eπ[δt\|St=s]. Though unbiased, IS can be high-variance. A lower variance alternative is Weighted IS (WIS). For a batch consisting of transitions , batch WIS uses a normalized estimate for the update. For example, an offline batch WIS TD algorithm, denoted WIS-Optimal below, would use update . Obtaining an efficient WIS update is not straightforward, however, when learning online and has resulted in algorithms specialized to the tabular setting (Precup et al., 2001) or linear functions (Mahmood et al., 2014; Mahmood and Sutton, 2015). We nonetheless use WIS as a baseline in the experiments and theory. ## 3 Importance Resampling In this section, we introduce Importance Resampling (IR) for off-policy prediction and characterize its bias and variance. A resampling strategy requires a buffer of samples, from which we can resample. Replaying experience from a buffer was introduced as a biologically plausible way to reuse old experience (Lin, 1992, 1993), and has become common for improving sample efficiency, particularly for control (Mnih et al., 2015; Schaul et al., 2015b). In the simplest case—which we assume here—the buffer is a sliding window of the most recent samples, , at time step . We assume samples are generated by taking actions according to behavior . The transitions are generated with probability , where is the stationary distribution for policy . The goal is to obtain samples according to , as if we had taken actions according to policy from states333The assumption that states are sampled from underlies most off-policy learning algorithms. Only a few attempt to adjust probabilities to , either by multiplying IS ratios before a transition (Precup et al., 2001) or by directly estimating state distributions (Hallak and Mannor, 2017; Liu et al., 2018). In this work, we focus on using resampling to correct the action distribution—the standard setting. We expect, however, that some insights will extend to how to use resampling to correct the state distribution, particularly because wherever IS ratios are used it should be straightforward to use our resampling approach. . The IR algorithm is simple: resample a mini-batch of size on each step from the buffer of size , proportionally to in the buffer. Standard on-policy updates, such as on-policy TD or on-policy gradient TD, can then used on this resample. The key difference to IS and WIS is that the distribution itself is corrected, before the update, whereas IS and WIS correct the update itself. This small difference, however, can have larger ramifications practically, as we show in this paper. We consider two variants of IR: with and without bias correction. For point sampled from the buffer, let be the on-policy update for that transition. For example, for TD, . The first step for either variant is to sample a mini-batch of size from the buffer, proportionally to . Bias-Corrected IR (BC-IR) additionally pre-multiplies with the average ratio in the buffer , giving the following estimators for the update direction XIR \makebox[0.0pt]def=1kk∑j=1ΔijXBC\makebox[0.0pt]def=¯ρkk∑j=1Δij BC-IR negates bias introduced by the average ratio in the buffer deviating significantly from the true mean. For reasonably large buffers, will be close to 1 making IR and BC-IR have near-identical updates. Nonetheless, they do have different theoretical properties, particularly for small buffer sizes , so we characterize both. Across all results, we make the following assumption. ###### Assumption 1. Transition tuples are sampled i.i.d. according to the distribution , for . To denote expectations under and , we overload the notation from above, using operators and respectively. To reduce clutter, we write to mean , because most expectations are under the sampling distribution. All proofs can be found in Appendix B. ### 3.1 Bias of IR We first show that IR is biased, and that its bias is actually equal to WIS-Optimal, in Theorem 3.1. ###### Theorem 3.1. [Bias for a fixed buffer of size ] Assume a buffer of transitions is sampled i.i.d., according to . Let be the WIS-Optimal estimator of the update. Then, E[XIR]=E[XWIS∗] and so the bias of is proportional to Bias(XIR)=E[XIR]−Eπ[Δ]∝1n(Eπ[Δ]σ2ρ−σρ,ΔσρσΔ) (1) where is the expected update across all transitions, with actions from taken by the target policy ; ; ; and covariance . This bias of IR will be small for reasonably large , both because it is proportional to and because larger will result in lower variance of the average ratios and average update for the buffer in Equation (1). In particular, as grows, these variances decay proportionally to . Nonetheless, for smaller buffers, such bias could have an impact. We can, however, easily mitigate this bias with a bias-correction term, as shown in the next corollary and proven in Appendix B.2. ###### Corollary 3.1.1. BC-IR is unbiased: . ### 3.2 Consistency of IR Consistency of IR in terms of an increasing buffer, with , is a relatively straightforward extension of prior results for SIR, with or without the bias correction, and from the derived bias of both estimators (see Theorem B.1 in Appendix B.3). More interesting, and reflective of practice, is consistency with a fixed length buffer and increasing interactions with the environment, . IR, without bias correction, is asymptotically biased in this case; in fact, its asymptotic bias is the one characterized above for a fixed length buffer in Theorem 3.1. BC-IR, on the other hand, is consistent, even with a sliding window, as we show in the following theorem. ###### Theorem 3.2. Let be the buffer of the most recent transitions sampled by time , i.i.d. as specified in Assumption 1. Let be the bias-corrected IR estimator, with samples from buffer . Define the sliding-window estimator . Assume there exists a such that . Then, as , converges in probability to . It might seem that resampling avoids high-variance in updates, because it does not reweight with large magnitude IS ratios. The notion of effective sample size from statistics, however, provides some intuition about why large magnitude IS ratios can also negatively affect IR, not just IS. Effective sample size is between 1 and , with one estimator (Kong et al., 1994; Martino et al., 2017). When the effective sample size is low, this indicates that most of the probability is concentrated on a few samples. For high magnitude ratios, IR will repeatedly sample the same transitions, and potentially never sample some of the transitions with small IS ratios. Fortunately, we find that, despite this dependence on effective sample size, IR can significantly reduce variance over IS. In this section, we characterize the variance of the BC-IR estimator. We choose this variant of IR, because it is unbiased and so characterizing its variance is a more fair comparison to IS. We define the mini-batch IS estimator , where indices are sampled uniformly from . This contrasts the indices for that are sampled proportionally to . We begin by characterizing the variance, under a fixed dataset . For convenience, let . We characterize the sum of the variances of each component in the update estimator, which equivalently corresponds to normed deviation of the update from its mean, V(Δ \| B)\makebox[0.0pt]% def=trCov(Δ \| B)=∑dm=1Var(Δm \| B)=E[∥Δ−μB∥22 \| B] for an unbiased stochastic update . We show two theorems that BC-IR has lower variance than IS, with two different conditions on the norm of the update. We first start with more general conditions, and then provide a theorem for conditions that are likely only true in early learning. ###### Theorem 3.3. Assume that, for a given buffer , for samples where , and that for samples where , for some . Then the BC-IR estimator has lower variance than the IS estimator: . The conditions in Theorem 3.3 preclude having update norms for samples with small be quite large—larger than a number —and a small norm for samples with large . These conditions can be relaxed to a statement on average, where the cumulative weighted magnitude of the update norm for samples with below the median needs to be smaller than for samples with above the meanx (see the proof in Appendix B.5). We next consider a setting where the magnitude of the update is independent of the given state and action. We expect this condition to hold in early learning, where the weights are randomly initialized and so potentially randomly incorrect across the state-action space. As learning progresses, and value estimates become more accurate in some states, it is unlikely for this condition to hold. ###### Theorem 3.4. Assume and the magnitude of the update are independent E[ρj∥Δj∥22 \| B]=E[ρj \| B] E[∥Δj∥22 \| B] Then the BC-IR estimator will have equal or lower variance than the IS estimator. These results have focused on variance of each estimator, for a fixed buffer, which provided insight into variance of updates when executing the algorithms. We would, however, also like to characterize variability across buffers, especially for smaller buffers. Fortunately, such a characterization is a simple extension on the above results, because variability for a given buffer already demonstrates variability due to different samples. It is easy to check that . The variances can be written using the law of total variance V(XBC) V(XIS) =E[V(XIS \| B)]+V(μB) ⟹V(XBC) −V(XIS)=E[V(XBC \| B)−V(XIS \| B)] with expectation across buffers. Therefore, the analysis of directly applies. ## 4 Empirical Results We investigate the two hypothesized benefits of resampling as compared to reweighting: improved sample efficiency and reduced variance. These benefits are tested in two microworld domains—a Markov chain and the Four Rooms domain—where exhaustive experiments can be conducted. We also provide a demonstration that IR reduces sensitivity over IS and VTrace in a car simulator, TORCs, when learning from images. We compare IR and BC-IR against several reweighting strategies, including importance sampling (IS); two online approaches to weighted important sampling, WIS-Minibatch with weighting and WIS-Buffer with weighting ; and V-trace444Retrace, ABQ and TreeBackup also use clipping to reduce variance. But, they are designed for learning action-values and for mitigating variance in eligibility traces. When trace parameter —as we assume here—there are no IS ratios and these methods become equivalent to using Sarsa(0) for learning action-values. , which corresponds to clipping importance weights (Espeholt et al., 2018). Where appropriate, we also include baselines using On-policy sampling; WIS-Optimal which uses the whole buffer to get an update; and Sarsa(0) which learns action-values—which does not require IS ratios—and then produces estimate . WIS-Optimal is included as an optimal baseline, rather than as a competitor, as it estimates the update using the whole buffer on every step. In all the experiments, the data is generated off-policy. We compute the absolute value error (AVE) on every training step. The error bars represent the standard error over runs, which are featured on every plot — although not visible in some instances. For the microworlds, the true value function is found using dynamic programming with threshold , and we compute AVE over all the states. For TORCs and continuous Four Rooms, the true value function is approximated using rollouts from a random subset of states generated when running the behavior policy . A tabular representation is used in the microworld experiments, tilecoded features with 64 tilings and 8 tiles is used in continuous Four Rooms, and a convolutional neural network is used for TORCs, with the same architecture as previously defined for self-driving cars (Bojarski et al., 2016). ### 4.1 Investigating Convergence Rate We first investigate the convergence rate of IR. We report learning curves in Four Rooms, as well as sensitivity to the learning rate. The Four Rooms domain (Stolle and Precup, 2002) has four rooms in an 11x11 grid world. The four rooms are positioned in a grid pattern with each room having two adjacent rooms. Each adjacent room is separated by a wall with a single connecting hallway. The target policy takes the down action deterministically. The cumulant for the value function is 1 when the agent hits a wall and 0 otherwise. The continuation function is , with termination when the agent hits a wall. The resulting value function can be thought of as distance to the bottom wall. The behavior policy is uniform random everywhere except for 25 randomly selected states which take the action down with probability 0.05 with remaining probability split equally amongst the other actions. The choice of behavior and target policy induce high magnitude IS ratios. As shown in Figure 1, IR has noticeable improvements over the reweighting strategies tested. The fact that IR resamples more important transitions from the replay buffer seems to significantly increase the learning speed. Further, IR has a wider range of usable learning rates. The same effect is seen even as we reduce the total number of updates, where the uniform sampling methods perform significantly worse as the interactions between updates increases—suggesting improved sample efficiency. WIS-Buffer performs almost equivalently to IS, because for reasonably size buffers, its normalization factor because . WIS-Minibatch and V-trace both reduce the variance significantly, with their bias having only a limited impact on the final performance compared to IS. Even the most aggressive clipping parameter for V-trace—a clipping of 1.0— outperforms IS. The bias may have limited impact because the target policy is deterministic, and so only updates for exactly one action in a state. Sarsa—which is the same as Retrace(0)—performs similarly to the reweighting strategies. The above results highlight the convergence rate improvements from IR, in terms of number of updates, without generalization across values. Conclusions might actually be different with function approximation, when updates for one state can be informative for others. For example, even if in one state the target policy differs significantly from the behavior policy, if they are similar in a related state, generalization could overcome effective sample size issues. We therefore further investigate if the above phenomena arise under function approximation with RMSProp learning rate selection. We conduct a similar experiment above, in a continuous state Four Rooms variant. The agent is a circle with radius 0.1, and the state consists of a continuous tuple containing the x and y coordinates of the agent’s center point. The agent takes an action in one of the 4 cardinal directions moving in that directions with random drift in the orthogonal direction sampled from . The target policy, as before, deterministically takes the down action. The representation is a tile coded feature vector with 64 tilings and 8 tiles. We find that generalization can mitigate some of the differences between IR and IS above in some settings, but in others the difference remains just as stark. If we use the behavior policy from the tabular domain, which skews the behavior in a sparse set of states, the nearby states mitigate this skew. However, if we use a behavior policy that selects all actions uniformly, then again IR obtains noticeable gains over IS and V-trace, for reducing the required number of updates, as shown in in Figure 2. Expanded results can be found in Appendix C.2. ### 4.2 Investigating Variance To better investigate the update variance we use a Markov chain, where we can more easily control dissimilarity between and , and so control the magnitude of the IS ratios. The Markov chain is composed of 8 non-terminating states and 2 terminating states on the ends of the chain, with a cumulant of 1 on the transition to the right-most terminal state and 0 everywhere else. We consider policies with probabilities [left, right] equal in all states: ; further policy settings can be found in Appendix C.1. We first measure the variance of the updates for fixed buffers. We compute the variance of the update—from a given weight vector—by simulating the many possible updates that could have occurred. We are interested in the variance of updates both for early learning—when the weight vector is quite incorrect and updates are larger—and later learning. To obtain a sequence of such weight vectors, we use the sequence of weights generated by WIS-Optimal. As shown in Figure 5, the variance of IR is lower than IS, particularly in early learning, where the difference is stark. Once the weight vector has largely converged, the variance of IR and IS is comparable and near zero. We can also evaluate the variance by proxy using learning rate sensitivity curves. As seen in Figure 5 (a) and (b), IR has the lowest sensitivity to learning rates, on-par with On-Policy sampling. IS has the highest sensitivity, along with WIS-Buffer and WIS-Minibatch. Various clipping parameters with V-trace are also tested. V-trace does provide some level of variance reduction but incurs more bias as the clipping becomes more aggressive. ### 4.3 Demonstration on a Car Simulator We use the TORCs racing car simulator to perform scaling experiments with neural networks to compare IR, IS, and V-trace. The simulator produces 64x128 cropped grayscale images. We use an underlying deterministic steering controller that produces steering actions and take an action with probability defined by a Gaussian . The target policy is a Gaussian , which corresponds to steering left. Pseudo-termination (i.e., ) occurs when the car nears the center of the road, and the cumulant becomes 1. Otherwise, the cumulant is zero and . The policy is specified using continuous action distributions and results in IS ratios as high as and highly variant updates for IS. Again, we can see that IR provides benefits over IS and V-trace, in Figure 4. There is even more generalization from the neural network in this domain, than in Four Rooms where we found generalization did reduce some of the differences between IR and IS. Yet, IR still obtains the best performance, and avoids some of the variance seen in IS for two of the learning rates. Additionally, BC-IR actually performs differently here, having worse performance for the largest learning rate. This suggest IR has an effect in reducing variance. ## 5 Conclusion In this paper we introduced a new approach to off-policy learning: Importance Resampling. We showed that IR is consistent, and that the bias is the same as for Weighted Importance Sampling. We also provided an unbiased variant of IR, called Bias-Corrected IR. We empirically showed that IR (a) has lower learning rate sensitivity than IS and V-trace, which is IS with varying clipping thresholds; (b) the variance of updates for IR are much lower in early learning than IS and (c) IR converges faster than IS and other competitors, in terms of the number of updates. These results confirm the theory presented for IR, which states that variance of updates for IR are lower than IS in two settings, one being an early learning setting. Such lower variance also explains why IR can converge faster in terms of number of updates, for a given buffer of data. The algorithm and results in this paper suggest new directions for off-policy prediction, particularly for faster convergence. Resampling is promising for scaling to learning many value functions in parallel, because many fewer updates can be made for each value function. A natural next step is a demonstration of IR, in such a parallel prediction system. 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In AAAI Conference on Artificial Intelligence, 2016. • Thomas and Brunskill [2017] Philip S Thomas and Emma Brunskill. Importance Sampling with Unequal Support. In AAAI Conference on Artificial Intelligence, 2017. ## Appendix A Weighted Importance Sampling We consider three weighted importance sampling updates as competitors to IR. is the size of the experience replay buffer, is the size of a single batch. WIS-Minibatch and WIS-Buffer both follow a similar protocol as IS, in that they uniformly sample a mini-batch from the experience replay buffer and use this to update the value functions. The difference comes in the scaling of the update. The first, WIS-Minibatch, uses the sum of the importance weights in the sampled mini-batch, while WIS-Buffer uses the sum of importance weights in the experience replay buffer. WIS-Buffer is also scaled by the size of the buffer and brought to the same effective scale as the other updates with . WIS-Optimal follows a different approach and performs the best possible version of WIS where the gradient descent update is calculated from the whole experience replay buffer. We do not provide analysis on the bias or consistency of WIS-Minibatch or WIS-Buffer, but are natural versions of WIS one might try. Δθ =∑kiρiδi∇θV(si;θ)∑kjρj WIS-Minibatch Δθ =nk∑kiρiδi∇θV(si;θ)∑njρj WIS-Buffer Δθ =∑niρiδi∇θV(si;θ)∑njρj WIS-Optimal ## Appendix B Additional Theoretical Results and Proofs ### b.1 Bias of IR Theorem 3.1(Bias for a fixed buffer of size ) Assume a buffer of transitions is sampled i.i.d., according to . Let be the WIS-Optimal estimator of the update. Then, E[XIR]=E[XWIS∗] and so the bias of is proportional to Bias(XIR)=E[XIR]−Eπ[Δ]∝1n(Eπ[Δ]σ2ρ−σρ,ΔσρσΔ) where is the expected update across all transitions, with actions from taken by the target policy ; ; ; and covariance . ###### Proof. Notice first that when we weight with , this is equivalent to weighting with , and so is the correct IS ratio for the transition. E[ XIR]=E[E[XIR\|B]]=E[E[1kk∑j=1Δij\|B]] =E[1kk∑j=1E[Δij\|B]] ▹E[Δij\|B]=n∑i=1ρi∑nj=1ρjΔi =E[n∑i=1ρi∑nj=1ρjΔi] =E[XWIS∗]. This bias of is the same as , which is characterized in Owen [2013], completing the proof. ∎ ### b.2 Proof of Unbiasedness of BC-IR Corollary 3.1.1 BC-IR is unbiased: . ###### Proof. E[XBC] =E[¯ρkk∑j=1E[Δij\|B]]=E[¯ρn∑i=1ρi∑nj=1ρjΔi] =E[1nn∑i=1ρiΔi]=1nn∑i=1E[π(Ai\|Si)μ(Ai\|Si)Δi] =1nn∑i=1E[dμ(Si)π(Ai\|Si)P(Si+1\|Si,Ai)dμ(Si)μ(Ai\|Si)P(Si+1\|Si,Ai)Δi] ### b.3 Consistency of the resampling distribution with a growing buffer We show that the distribution when following a resampling strategy is consistent: as , the resampling distribution converges to the true distribution. Our approach closely follows that of [Smith et al., 1992], but we recreate it here for convenience. ###### Theorem B.1. Let be a buffer of data sampled i.i.d. according to proposal density . Let be some distribution of interest with associated random variable and assume the proposal distribution samples everywhere where is non-zero, i.e . Also, let be a discrete random variable taking values with probability . Then, converges in distribution to as . ###### Proof. Let . From the probability mass function of , we have that: P[Y≤a] =n∑i=1P[Y=xi]\mathbbm1{xi≤a} =n−1∑ni=1ρi\mathbbm1{xi≤a}n−1∑ni=1ρi n→∞−−−→Eq[ρ(x)\mathbbm1{x≤a}]Eq[ρ(x)] =1⋅∫a−∞q(x)p(x)p(x)dx+0⋅∫∞aq(x)p(x)p(x)dx∫∞−∞q(x)p(x)p(x)dx =∫a−∞q(x)dx=P[Q≤a] Y d→Q This means a resampling strategy effectively changes the distribution of random variable to that of , meaning we can use samples from to build statistics about the target distribution . This result motivates using resampling to correct the action distribution in off-policy learning. This result can also be used to show that the IR estimators are consistent, with . ### b.4 Consistency under a sliding window Theorem 3.2 Let be the buffer of the most recent transitions sampled by time , i.i.d. as specified in Assumption 1. Let be the bias-corrected IR estimator, with samples from buffer . Define the sliding-window estimator . Assume there exists a such that . Then, as , converges in probability to . ###### Proof. Notice first that is random because is random and because transitions are sampled from . Therefore, given , is independent of other random variables and for . Now, using Corollary 3.1.1, we can show that BC-IR is unbiased for a sliding window E[Xt] =E[1tt∑i=1X(i)BC]=1tt∑i=1E[E[X(i)BC\|Bi]]=Eπ[Δ]. Next, we show that . For , and are independent, because they are disjoint sets of i.i.d. random variables. Correspondingly, is independent of . Explicitly, using the law of total covariance, we get that The first term is zero because is independent of given , and the second term is zero because and are independent. Therefore, . Using the assumption on the variance, we can apply Lemma B.2 to to get the desired result. ∎ ###### Lemma B.2. Let be random variables with mean . Suppose there exists a such that and that . Then, as , converges in probability to . ###### Proof. Let . V(SN)=N∑i=1V(Zi)+2N∑i=1N∑j=i+1Cov(Zi,Zj) The first term is bounded by from our assumption on the variance. Now, to bound the second term. Fix and choose such that , (such an must exist since ). Assuming that , we can decompose the second term into N∑i=1N∑j=i+1Cov(Zi,Zj) =N∑i=1i+M∑j=i+1Cov(Zi,Zj)+ N∑i=1N∑j=i+M+1Cov(Zi,Zj) ∣∣ ∣∣N∑i=1N∑j=i+1Cov(Zi,Zj)∣∣ ∣∣ ≤N∑i=1i+M∑j=i+1\|Cov(Zi,Zj)\|+ N∑i=1N∑j=i+M+1\|Cov(Zi,Zj)\| By the Cauchy-Schwarz inequality and our variance assumption, . So, we get ∣∣ ∣∣N∑i=1N∑j=i+1Cov(Zi,Zj)∣∣ ∣∣ ≤N∑i=1i+M∑j=i+1c+N∑i=1N∑j=i+M+1δ ≤NMc+N2δ Altogether, our upper bound is V(SNN)≤cN+McN+δ Finally, we apply Chebyshev’s inequality. For a fixed , P(∣∣∣SNN−μ∣∣∣>ϵ) ≤1ϵ2V(SNN) ≤1ϵ2(cN+McN+δ) Since we can choose to be arbitrarily small (say ), the right-hand side goes to as , concluding the proof. ∎ ### b.5 Variance of BC-IR and IS This lemma characterizes the variance of the BC-IR and IS estimators for a fixed buffer. ###### Lemma B.3. V(XIS \| B) V(XBC \| B) ###### Proof. Because we have independent samples, V(XBC \| B)=1k2k∑j=1V(¯ρΔij \| B)=1kV(¯ρΔi1 \| B) and similarly We can further simplify these expressions. For the IS estimator V(ρz1Δz1\|B) =E[ρ2z1Δ⊤z1Δz1\|B]−E[ρz1Δz1\|B]⊤E[ρz1Δz1\|B] =1nn∑j=1ρ2j∥Δj∥22−μ⊤BμB and for the BC-IR estimator, where recall , V(¯ρΔi1\|B)=E[¯ρ2Δ⊤i1Δi1\|B]−E[¯ρΔi1\|B]⊤E[¯ρΔi1\|B] =n∑j=1¯ρ2ρj∑ni=1ρi∥Δj∥22−μ⊤BμB =¯ρnn∑j=1ρj∥Δj∥22−μ⊤BμB The following two theorems present certain conditions when the BC-IR estimator would have lower variance than the IS estimator. Theorem 3.3 Assume that for samples where , and that for samples where , for some . Then the BC-IR estimator has lower variance than the IS estimator. ###### Proof. We show : V(XIS\|B)−V(XBC\|B)=1nkn∑j=1∥∥Δj∥∥22(ρ2j−¯ρρj) =1nk∑s:ρs<¯ρ∥Δs∥22≤c/ρsρs(ρs−¯ρ)≤0+1nk∑l:ρl≥¯ρ∥Δl∥22>c/ρsρl(ρl−¯ρ)≥0 >1nk∑s:ρs<¯ρcρsρs(ρs−¯ρ)+1nk∑l:ρl≥¯ρcρlρl(ρl−¯ρ) =cnkn∑j=1(ρj−¯ρ)=0 Theorem 3.4 Assume and the magnitude of the update	{"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": 0.89418625831604, "perplexity": 2239.8727286645794}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251799918.97/warc/CC-MAIN-20200129133601-20200129163601-00013.warc.gz"}
http://link.springer.com/article/10.1134/S1063772911100039	, Volume 55, Issue 10, pp 867-877 Date: 29 Sep 2011 # Estimating the rate and luminosity function of all classes of GRBs Rent the article at a discount Rent now * Final gross prices may vary according to local VAT. ## Abstract The aim of the present work is to estimate the rate and luminosity functions of short, intermediate and long gamma-ray bursts (GRBs) by fitting their intensity distributions with parameterized explosion rates and luminosity functions. The results show that the parameters of the rate and luminosity function for long GRBs can be calculated with an accuracy of 10–30%. However, some parameters of intermediate and short GRBs have large uncertainties. An important conclusion is that there was initially a large outburst in the frequency of long GRBs, and consequently a large outburst in the star-formation rate, if they come from collapsars. Finally, a simulated intensity distribution has been constructed to test the ability of the method to recover the simulated parameters.	{"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": 0.8786410689353943, "perplexity": 1293.5743680808134}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042988650.6/warc/CC-MAIN-20150728002308-00269-ip-10-236-191-2.ec2.internal.warc.gz"}
https://www.intechopen.com/chapters/61684	Open access peer-reviewed chapter # Separability and Nonseparability of Elastic States in Arrays of One-Dimensional Elastic Waveguides Written By Pierre Alix Deymier, Jerome Olivier Vasseur, Keith Runge and Pierre Lucas Submitted: February 20th, 2018 Reviewed: April 13th, 2018 Published: November 5th, 2018 DOI: 10.5772/intechopen.77237 From the Edited Volume ## Phonons in Low Dimensional Structures Edited by Vasilios N. Stavrou Chapter metrics overview View Full Metrics ## Abstract We show that the directional projection of longitudinal waves propagating in a parallel array of N elastically coupled waveguides can be described by a nonlinear Dirac-like equation in a 2N dimensional exponential space. This space spans the tensor product Hilbert space of the two-dimensional subspaces of N uncoupled waveguides grounded elastically to a rigid substrate (called φ -bits). The superposition of directional states of a φ -bit is analogous to that of a quantum spin. We can construct tensor product states of the elastically coupled system that are nonseparable on the basis of tensor product states of N φ -bits. We propose a system of coupled waveguides in a ring configuration that supports these nonseparable states. ### Keywords • one-dimensional elastic waveguides • nonseparability • elastic waves • elastic pseudospin • coupled waveguides ## 1. Introduction Quantum bit-based computing platforms can capitalize on exponentially complex entangled states which allow a quantum computer to simultaneously process calculations well beyond what is achievable with serially interconnected transistor-based processors. Ironically, a pair of classical transistors can emulate some of the functions of a qubit. While current manufacturing can fabricate billions of transistors on a chip, it is inconceivable to connect them in the exponentially complex way that would be required to achieve nonseparable quantum superposition analogues. In contrast, quantum systems possess such complexity through the nature of the quantum world. Outside the quantum world, the notion of classical nonseparability [1, 2, 3] has been receiving a lot of attention from the theoretical and experimental point of views in the field of optics. Degrees of freedom of photon states that span different Hilbert spaces can be made to interact in a way that leads to local correlations. Correlation has been achieved between degrees of freedom that include spin angular momentum and orbital angular momentum (OAM) [4, 5, 6, 7, 8, 9], OAM, polarization and radial degrees of freedom of a beam of light [10] as well as propagation direction [11, 12]. Recently, we have extended this notion to correlation between directional and OAM degrees of freedom in elastic systems composed of arrays of elastic waveguides [13]. This classical nonseparability lies only in the tensor product Hilbert space of the subspaces associated with these degrees of freedom. This Hilbert space does not possess the exponential complexity of a multiqubit Hilbert space, for instance. It has been suggested theoretically and experimentally that classical systems coupled via nonlinear interactions may have computational capabilities approaching that of quantum computers [14, 15, 16]. We demonstrated in Ref. [17] that nonlinear elastic media can be used to produce phonons that can be correlated simultaneously in time and frequency. We have also shown an analogy between the propagation of elastic waves on elastically coupled one-dimensional (1D) wave guides and quantum phenomena [18, 19, 20, 21]. More specifically, the projection on the direction of propagation of elastic waves in an elastic system composed of a 1D waveguide grounded to a rigid substrate (denoted φ -bit) is isomorphic to the spin of a quantum particle. The pseudospin states of elastic waves in these systems can be described via a Dirac-like equation and possess 2 × 1 spinor amplitudes. Unlike the quantum systems, these amplitudes are, however, measurable through the measurement of transmission coefficients. The notion of measurement is an important one as it has been realized that separability is relative to the choice of the partitioning of a multipartite system. Indeed, it is known that given a multipartite physical system, whether quantum or classical, the way to subdivide it into subsystems is not unique [22, 23]. For instance, the states of a quantum system may not appear entangled relative to some decomposition but may appear entangled relative to another partitioning. The criterion for that choice may be the ability to perform observations and measurements of some degrees of freedom of the subsystems [23]. The objective of this paper is to investigate the notion of separability and nonseparability of multipartite classical mechanical systems supporting elastic waves. These systems are composed of 1D elastic waveguides that are elastically coupled along their length to each other and/or to some rigid substrate. The 1D waveguides support spinor-like amplitudes in the two-dimensional (2D) subspace of directional degrees of freedom. The amplitudes of N coupled waveguides span an N -dimensional subspace. Subsequently, the Hilbert space spanned by the elastic modes is a 2 N -dimensional space, comprised of the tensor product of the directional and waveguides subspaces. This representation is isomorphic to the degrees of freedom of photon states in a beam of light. While beams of light cannot be decomposed into subsystems, an elastic system composed of coupled 1D waveguides can. Indeed, the elastic system considered here forms a multipartite system composed of N 1D waveguide subsystems. We show that, since each waveguide possesses two directional degrees of freedom, one can represent the elastic states of the N -waveguide system in the 2 N dimensional tensor product Hilbert space of N 2D spinor subspaces associated with individual waveguides. The elastic modes in this representation obey a 2 N dimensional nonlinear Dirac-like equation. These modes span the same space as that of uncoupled waveguides grounded to a rigid substrate, i.e., N φ -bits. However, the modes’ solutions of the nonlinear Dirac equation cannot be expressed as tensor products of the states of N uncoupled grounded waveguides, i.e., φ -bit states. In Section 2 of this chapter, we introduce the mathematical formalism that is needed to demonstrate the nonseparability of elastic states of coupled elastic waveguides in an exponentially complex space. Throughout this section, we use illustrations of the concepts in the case of systems composed of small numbers of waveguides. However, the approach is fully scalable and can be generalized to any large number of coupled waveguides. In Section 3, we draw conclusions concerning the applicability of this approach to solve complex problems. ## 2. Models and methods We have previously considered systems constituted of N one-dimensional (1D) waveguides coupled elastically along their length [13]. In this section, we summarize the results of these previous investigations to develop a formalism to address our current considerations. The parallelly coupled waveguides can be arranged in any desired way. The propagation of elastic modes is limited to longitudinal modes along the waveguides in the long wavelength limit, i.e., the continuum limit. We consider the representations of the modes of the coupled waveguide systems in two spaces. The first space scales linearly with N. The second space scales as 2 N and leads to a description of the elastic system with exponential complexity. The linear representation enables us to operate easily on the states in the exponential space. ### 2.1. Representation of elastic states in a space scaling linearly with N A compact form for the equations of motion of the N coupled waveguides is: H . I N × N + α 2 M N × N u N × 1 = 0 E1 Here, the propagation of elastic waves in the direction x along the waveguides is modeled by the dynamical differential operator, H = 2 t 2 β 2 2 x 2 . The parameter β is proportional to the speed of sound in the medium constituting the waveguides and the parameter α 2 characterizes the strength of the elastic coupling between them (here, we consider that the strength is the same for all coupled waveguides). u N × 1 is a vector with components, u i , i = 1 N , representing the displacement of the ith waveguide. The coupling matrix operator M N × N describes the elastic coupling between waveguides which, in the case of N = 3 parallel waveguides in a closed ring arrangement with first neighbor coupling, takes the form: M N = 3 × N = 3 = 2 1 1 1 2 1 1 1 2 E2 Eq. (1) takes the form of a generalized Klein-Gordon (KG) equation and its Dirac factorization introduces the notion of the square root of the operator H . I N × N + α 2 M N × N . In this factorization, the dynamics of the system are represented in terms of first derivatives with respect to time, t , and position along the waveguides, x . There are two possible Dirac equations: U N × N σ x t + β U N × N i σ y x ± U 2 N × 2 N M N × N σ x Ψ 2 N × 1 = 0 E3 In Eq. (3), U N × N and U 2 N × 2 N are antidiagonal matrices with unit elements. σ x = 0 1 1 0 and σ y = 0 i i 0 are two of the Pauli matrices. Ψ 2 N × 1 is a 2 N dimensional vector which represents the modes of vibration of the N waveguides projected in the two possible directions of propagation (forward and backward) and M N × N is the square root of the coupling matrix. The square root of a matrix is not unique but we will show later that we can pick any form without loss of generality. We choose components of the Ψ 2 N × 1 vector in the form of plane waves ψ I = a I e ikx e iωt with I = 1 , , 2 N and k and ω being the wave number and angular frequency, respectively, Eq. (3) becomes: ωA 2 N × 2 N + βk B 2 N × 2 N ± α C 2 N × 2 N a 2 N × 1 = 0 E4 where A 2 N × 2 N = I N × N I 2 × 2 E5a B 2 N × 2 N = I N × N σ z E5b C 2 N × 2 N = M N × N σ x E5c In Eqs. (4) and (5), σ z = 1 0 0 1 is the third Pauli matrix, I N × N is the identity matrix of order N and a2Nx1 is a 2 N dimensional vector whose components are the amplitudes aI. In obtaining Eq. (4), we have multiplied all terms in Eq. (3) on the left by U 2 N × 2 N . Writing Eq. (4) as a linear combination of tensor products of N × N and 2 × 2 matrix operators: I N × N ωI 2 × 2 βk σ z ± α M N × N σ x a 2 N × 1 = 0 E6 we seek solutions in the form of tensor products: a 2 N × 1 = E N × 1 s 2 × 1 .E7 While the degrees of freedom associated with E N × 1 span an N dimensional Hilbert subspace, the degrees of freedom associated with s 2 × 1 span a 2D space. Replacing a 2 N × 1 from Eq. (7) in Eq. (6) yields: I N × N E N × 1 ωI 2 × 2 βk σ z s 2 × 1 ± α M N × N E N × 1 σ x s 2 × 1 = 0 . E8 Choosing E N × 1 to be an eigenvector, e n , of the matrix M N × N with eigen value λ n Eq. (8) reduces to: e n ωI 2 × 2 βk σ z ± α λ n σ x s 2 × 1 = 0 E9 For nontrivial eigenvectors e n , the problem in the space of the directions of propagation reduces to finding solutions of ωI 2 × 2 βk σ z ± α λ n σ x s 2 × 1 = 0 E10 In obtaining Eq. (9), we have also used the fact that e n is an eigen vector of I N × N with eigen value 1 and we note that Eq. (9) is the 1D Dirac equation for an elastic system which solutions, s 2 × 1 , have the properties of Dirac spinors [18, 19, 20, 21]. The components of the spinor represent the amplitude of the elastic waves in the positive and negative directions along the waveguides, respectively. Eq. (10), now written in the matrix form, can now be solved for a given λ n ; ω n βk ± α λ n ± α λ n ω n + βk s 1 s 2 = 0 E11 This eigen equation gives the dispersion relation ω n 2 = βk 2 + α λ n 2 (vide infra) and the following eigen vectors projected into the space of directions of propagation: s 2 × 1 = s 0 ω n + βk ± ω n βk E12 To determine the eigen vectors of M N × N , we note that they are identical to the eigen vectors of the coupling matrix M N × N and the eigen values of M N × N are also λ n 2 . These properties indicate that we do not have to determine the square root of the coupling matrix to find the solutions a 2 N × 1 . All that is required is to calculate the eigen vectors and the eigen values of the coupling matrix. Hence, the nonuniqueness of M N × N does not introduce difficulties in determining the elastic modes of the coupled system in the Dirac representation. In the case of the coupling matrix, M 3 × 3 , presented in Eq. (2), the eigen values and real eigen vectors are obtained as λ 0 2 = 0 , λ 1 2 = λ 2 2 = 3 , and e 0 = 1 3 1 1 1 , e 1 = 2 3 1 1 2 1 2 and e 2 = 2 3 1 2 1 1 2 E13 Eq. (3) being linear, its solutions can be written as linear combinations of elastic wave functions in the form: Ψ 2 N × 1 n k = e n N s 2 × 1 k e ikx e i ω n k t E14 In Eq. (14), we have expressed the dependencies on the wave number k and the number of waveguides N. The eigen vectors e n N depend on the connectivity of the N waveguides. The space spanned by these solutions scales linearly with the number of waveguides, i.e., as 2N. ### 2.2. Representation of elastic states in a space scaling as 2 N We first illustrate the notion of exponential space in the case of three waveguides. Each guide is connected to a rigid substrate and therefore constitutes a φ -bit. The waveguides are not coupled to each other. The dynamics of the system can be described by a single equation which is constructed as follows: σ x σ x σ x t + σ y σ x σ x x 1 + σ x σ y σ x x 2 + σ x σ x σ y x 3 ± I 2 × 2 σ x σ x ± σ x I 2 × 2 σ x ± σ x σ x I 2 × 2 Ψ 8 × 1 = 0 E15 In Eq. (15), we are now defining a positional variable for each waveguide, namely, x 1 , x 2 , x 3 . The quantity α is a measure of the strength of the elastic coupling to the rigid substrate. The solutions are the 8 × 1 vectors Ψ 8 × 1 = Ψ 1 Ψ 2 Ψ 3 Ψ 4 Ψ 5 Ψ 6 Ψ 7 Ψ 8 . When seeking solutions in the form of tensor products of spinor solutions for the three waveguides (as indicated by the upper scripts) Ψ 8 × 1 = ψ 1 ψ 2 ψ 3 = ψ 1 1 ψ 2 1 ψ 1 2 ψ 2 2 ψ 1 3 ψ 2 3 = ψ 1 1 ψ 1 2 ψ 1 3 ψ 1 1 ψ 1 2 ψ 2 3 ψ 1 1 ψ 2 2 ψ 1 3 ψ 1 1 ψ 2 2 ψ 2 3 ψ 2 1 ψ 1 2 ψ 1 3 ψ 2 1 ψ 1 2 ψ 2 3 ψ 2 1 ψ 2 2 ψ 1 3 ψ 2 1 ψ 2 2 ψ 2 3 E16 it is straightforward to show that one recovers from Eq. (15), the six Dirac equations of Eq. (3) with M N = 3 × N = 3 = I 3 × 3 . The solutions of Eq. (16) are obtained from the spinor solution for individual waveguides (j): ψ j = s 0 ω + βk ± ω βk e ikx e iωt E17 The Hilbert space spanned by the solutions of Eq. (15) is the product space of the three 2D subspaces associated with each waveguide. The states of a system composed of N φ -bits span a space when dimension is 2 N . The question that arises then concerns the possibility of writing an equation in the exponential Hilbert space for N waveguides coupled to each other. For instance, we wish to obtain the states of the system composed of three waveguides coupled in a ring arrangement from an equation of the form: σ x σ x σ x t + σ y σ x σ x x 1 + σ x σ y σ x x 2 + σ x σ x σ y x 3 ± ε 8 × 8 Ψ 8 × 1 = 0 E18 The matrix αε 8 × 8 represents the coupling between the waveguides in the 2 N = 3 space. We are still seeking solutions in the form of tensor products (Eq. (16)). After a lengthy algebraic manipulation, we find that we can reproduce Eq. (3) with the coupling matrix of Eq. (2) if one chooses ε ij = 0 excepting ε 14 = ε 41 = ε 23 = ε 31 = 2 ψ 1 1 ψ 1 2 ψ 1 3 ψ 1 1 ; ε 16 = ε 61 = ε 25 = ε 52 = 2 ψ 1 2 ψ 1 1 ψ 1 3 ψ 1 2 ; ε 17 = ε 71 = ε 35 = ε 53 = 2 ψ 1 3 ψ 1 1 ψ 1 2 ψ 1 3 ; ε 28 = ε 82 = ε 46 = ε 64 = 2 ψ 2 3 ψ 2 1 ψ 2 2 ψ 2 3 ; ε 38 = ε 83 = ε 47 = ε 74 = 2 ψ 2 2 ψ 2 1 ψ 2 3 ψ 2 2 ; ε 58 = ε 85 = ε 67 = ε 76 = 2 ψ 2 1 ψ 2 2 ψ 2 3 ψ 2 1 E19 The Dirac equation of the three coupled waveguides in the exponential space is therefore nonlinear. Generalization to N coupled chains will result in the following nonlinear equation: σ x N t + σ y σ x N 1 x 1 + σ x σ y σ x N 2 x 2 + + σ x N 1 σ y x N ± i αε 2 N × 2 N Ψ 2 N × 1 = 0 E20 where nonzero components of ε 2 N × 2 N depend on the ψ i j , i = 1 , 2 ; j = 1 , N that appear in the solution Ψ 2 N × 1 = ψ 1 ψ 2 ψ N . The solutions of the nonlinear Dirac equation for the coupled waveguides span the same space as that of the system of φ -bits, i.e., uncoupled waveguides connected to rigid substrates. The next subsection addresses the question of separability of the coupled waveguide system into a system of uncoupled φ -bits. ### 2.3. Elastic states in the exponential space For a system of waveguides that are not coupled, the elastic states, solutions of linear equations of the form of Eq. (15), are tensor products but also linear combinations of tensor products of spinor solution for individual waveguides (see Eq. (17)). It is therefore possible to construct nonseparable states in the exponential space for systems of uncoupled waveguides. For example, if we consider a system of two uncoupled waveguides, a possible state of the system in the 2 2 space can be constructed in the form of the following linear combination of tensor products: Ψ 4 × 1 = s 0 2 ω + βk ± ω βk ω + βk ± ω βk e i 2 kx e i 2 ωt s 0 2 ω βk ± ω + βk ω βk ± ω + βk e i 2 kx e i 2 ωt E21 Choosing s 0 = s 0 and writing Eq. (21) at the location x = 0 , one gets: Ψ 4 × 1 x = 0 = ω + βk ω + βk ± ω + βk ω βk ± ω βk ω + βk ω βk ω βk ω βk ω βk ± ω βk ω + βk ± ω + βk ω βk ω + βk ω + βk e i 2 ωt E22 The bracket takes the form: ω + βk ω + βk ω βk ω βk 1 0 0 1 E23 The vector 1 0 0 1 is not separable into a tensor product of two 2 × 1 vectors. Considering on the basis 0 = 1 0 and 1 = 0 1 , one can write the state given in Eq. (22) in the form of the nonseparable Bell state: Ψ 4 × 1 x = 0 = ω + βk ω + βk ω βk ω βk 0 0 1 1 e i 2 ωt E24 Since the waveguides are not coupled, it is, however, not possible to manipulate the state of one of the waveguides by manipulating the state of the other one. Simultaneous manipulation of the state of waveguides in the exponential space requires coupling. We now address elastic states in the coupled waveguides system. For a system of N coupled waveguides, we construct a solution of Eq. (3) that takes the form of a linear combination of solutions given in Eq. (14): Ψ 2 N × 1 n n k k = χ n e n N s 2 × 1 k e ikx e i ω n k t + χ n e n N s 2 × 1 k e ik x e i ω n k t E25 The n and n correspond to two different nonzero eigen values, λ n and λ n , i.e., they correspond to two different dispersion relations ω n k and ω n k . We also choose the wave number k such that ω n k = ω n k = ω 0 . These modes are illustrated in Figure 1 in the case of an N = 9 waveguide system. χ n and χ n are the coefficients of the linear combination. With e n N = A 1 A 2 A N and e n N = A 1 A 2 A N where the specific values of the components A I and A I are determined by the connectivity and coupling of the waveguides, the state of Eq. (25) can be rewritten as: Ψ 2 N × 1 n n k k = χ n A 1 ω 0 + βk e ikx + χ n A 1 ω 0 + βk e ik x χ n A 1 ω 0 βk e ikx + χ n A 1 ω 0 βk e ik x χ n A N ω 0 + βk e ikx + χ n A N ω 0 + βk e ik x χ n A N ω 0 βk e ikx + χ n A N ω 0 βk e ik x e i ω 0 t = φ 1 1 φ 2 1 φ 1 N φ 2 N E26 Here, we have chosen, for the sake of simplicity, the + of the ± in the s 2 × 1 terms. The first two terms in Eq. (26) form a 2 × 1 spinor, φ 1 = φ 1 1 φ 2 1 , which corresponds to the first waveguide, the next two terms form a spinor φ 2 for the second waveguide, etc. We can then construct a solution of the nonlinear Dirac Eq. (20) in the exponential space as the tensor product: Φ 2 N × 1 = φ 1 φ 2 φ N E27 Since Eq. (20) is nonlinear, linear combinations of tensor product solutions of the form above are not solutions. Solutions of the nonlinear Dirac equation always take the form of a tensor product when the spinor wave functions φ j are expressed on the basis of 2 × 1 vectors. 0 = 1 0 and 1 = 0 1 . If one desires to express Φ 2 N × 1 as a nonseparable state, one has to define a new basis in which this wave function cannot be expressed as a tensor product. This is done in Section 2.5. However, prior to demonstrating this, we illustrate in the next subsection how one can manipulate states of the form Φ 2 N × 1 in the exponential space. ### 2.4. Operating on exponentially-complex tensor product elastic states In this subsection, we expand tensor product states of the form given in Eq. (27) in linear combinations of tensor products of pure states in the exponential space. We illustrate this expansion in the case of three parallel waveguides elastically coupled to each other. Each waveguide is also coupled elastically to a rigid substrate. We treat the case where the strength of all the couplings is the same. In that case, the coupling matrix is: M N = 3 × N = 3 = 3 1 1 1 3 1 1 1 3 This matrix has three nonzero eigen values λ 0 2 = 1 , and λ 1 2 = λ 2 2 = 4 corresponding to two dispersion relations ω n 2 = βk 2 + α λ n 2 with cutoff frequencies. The second band is doubly degenerate. The eigen vectors are also given in Eq. (13). We now consider an elastic mode in the linear space that is a linear combination of these eigen modes (see Eq. (26)): Ψ 6 × 1 n n k k = φ 1 1 φ 2 1 φ 1 2 φ 2 2 φ 1 3 φ 2 3 = χ n A 1 ω 0 + βk e ikx + χ n A 1 ω 0 + βk e ik x χ n A 1 ω 0 βk e ikx + χ n A 1 ω 0 βk e ik x χ n A 2 ω 0 + βk e ikx + χ n A 2 ω 0 + βk e ik x χ n A 2 ω 0 βk e ikx + χ n A 2 ω 0 βk e ik x χ n A 3 ω 0 + βk e ikx + χ n A 3 ω 0 + βk e ik x χ n A 3 ω 0 βk e ikx + χ n A 3 ω 0 βk e ik x e i ω 0 t E28 In Eq. (28), the A I ’s can be the components of the eigen vector e 0 and the A I ’s can be linear combinations of the components of the eigen vectors e 1 and e 2 . We can calculate the tensor product of the spinor components in the form of Eq. (27) Φ 2 3 × 1 = φ 1 φ 2 φ 3 E29 Eq. (29) can be rewritten after some algebraic manipulations in the form of the linear combination: Φ 2 3 × 1 = { ζ 1 ξ 1 ζ 2 ξ 2 ζ 3 ξ 3 + ζ 1 ξ 1 ζ 2 ξ 2 ζ 3 ξ 3 + ζ 1 ξ 1 ζ 2 ξ 2 ζ 3 ξ 3 + ζ 1 ξ 1 ζ 2 ξ 2 ζ 3 ξ 3 + ζ 1 ξ 1 ζ 2 ξ 2 ζ 3 ξ 3 + ζ 1 ξ 1 ζ 2 ξ 2 ζ 3 ξ 3 + ζ 1 ξ 1 ζ 2 ξ 2 ζ 3 ξ 3 + ζ 1 ξ 1 ζ 2 ξ 2 ζ 3 ξ 3 } e i 3 ω 0 t E30 In Eq. (30), we have defined ζ I ξ I = χ n A I e ikx ω 0 + βk ω 0 βk = χ n A I e ikx s 2 × 1 E31a ζ I ξ I = χ n A 1 e i k x ω 0 + βk ω 0 βk = χ n A 1 e i k x s 2 × 1 E31b The tensor product of Eq. (30) then reduces to Φ 2 3 × 1 = { χ n 3 A 1 A 2 A 3 e i 3 kx s 2 × 1 s 2 × 1 s 2 × 1 + χ n 2 χ n A 1 A 2 A 3 e i 2 kx e i k x s 2 × 1 s 2 × 1 s 2 × 1 + χ n 2 χ n A 1 A 2 A 3 e i 2 kx e i k x s 2 × 1 s 2 × 1 s 2 × 1 + χ n 2 χ n A 1 A 2 A 3 e i 2 kx e i k x s 2 × 1 s 2 × 1 s 2 × 1 + χ n 2 χ n A 1 A 2 A 3 e ikx e i 2 k x s 2 × 1 s 2 × 1 s 2 × 1 + χ n 2 χ n A 1 A 2 A 3 e ikx e i 2 k x s 2 × 1 s 2 × 1 s 2 × 1 + χ n 2 χ n A 1 A 2 A 3 e ikx e i 2 k x s 2 × 1 s 2 × 1 s 2 × 1 + χ n 3 A 1 A 2 A 3 e i 3 k x s 2 × 1 s 2 × 1 s 2 × 1 } e i 3 ω 0 t E32 The spinors s 2 × 1 and s 2 × 1 can be expressed on the basis 0 = 1 0 and 1 = 0 1 : s 2 × 1 = s 1 0 + s 2 1 E33a s 2 × 1 = s 1 0 + s 2 1 E33b With s 1 = ω 0 + βk , s 2 = ω 0 βk , s 1 = ω 0 + βk and s 2 = ω 0 βk . Inserting Eqs. (33a) and (33b) into Eq. (32), we can express the tensor product Φ 2 3 × 1 on the basis 0 0 0 0 0 1 0 1 0 1 0 0 0 1 1 1 0 1 1 1 0 1 1 1 . In defining the basis vectors for the exponential space, we have omitted the symbols . It is also implicit that the left, middle, and right elements in the tensor product a b c correspond to the first, second, and third waveguides, respectively. We find Φ 2 3 × 1 = T 1 0 0 0 + T 2 0 0 1 + + T 8 1 1 1 e i 3 ω 0 t E34 with T 1 = Q 1 s 1 s 1 s 1 + Q 2 s 1 s 1 s 1 + Q 3 s 1 s 1 s 1 + Q 4 s 1 s 1 s 1 + Q 5 s 1 s 1 s 1 + Q 6 s 1 s 1 s 1 + Q 7 s 1 s 1 s 1 + Q 8 s 1 s 1 s 1 E35a T 2 = Q 1 s 1 s 1 s 2 + Q 2 s 1 s 1 s 2 + Q 3 s 1 s 1 s 2 + Q 4 s 1 s 1 s 2 + Q 5 s 1 s 1 s 2 + Q 6 s 1 s 1 s 2 + Q 7 s 1 s 1 s 2 + Q 8 s 1 s 1 s 2 E35b T 3 = Q 1 s 1 s 2 s 1 + Q 2 s 1 s 2 s 1 + Q 3 s 1 s 2 s 1 + Q 4 s 1 s 2 s 1 + Q 5 s 1 s 2 s 1 + Q 6 s 1 s 2 s 1 + Q 7 s 1 s 2 s 1 + Q 8 s 1 s 2 s 1 E35c T 4 = Q 1 s 2 s 1 s 1 + Q 2 s 2 s 1 s 1 + Q 3 s 2 s 1 s 1 + Q 4 s 2 s 1 s 1 + Q 5 s 2 s 1 s 1 + Q 6 s 2 s 1 s 1 + Q 7 s 2 s 1 s 1 + Q 8 s 2 s 1 s 1 E35d T 5 = Q 1 s 1 s 2 s 2 + Q 2 s 1 s 2 s 2 + Q 3 s 1 s 2 s 2 + Q 4 s 1 s 2 s 2 + Q 5 s 1 s 2 s 2 + Q 6 s 1 s 2 s 2 + Q 7 s 1 s 2 s 2 + Q 8 s 1 s 2 s 2 E35e T 6 = Q 1 s 2 s 1 s 2 + Q 2 s 2 s 1 s 2 + Q 3 s 2 s 1 s 2 + Q 4 s 2 s 1 s 2 + Q 5 s 2 s 1 s 2 + Q 6 s 2 s 1 s 2 + Q 7 s 2 s 1 s 2 + Q 8 s 2 s 1 s 2 E35f T 7 = Q 1 s 2 s 2 s 1 + Q 2 s 2 s 2 s 1 + Q 3 s 2 s 2 s 1 + Q 4 s 2 s 2 s 1 + Q 5 s 2 s 2 s 1 + Q 6 s 2 s 2 s 1 + Q 7 s 2 s 2 s 1 + Q 8 s 2 s 2 s 1 E35g T 8 = Q 1 s 2 s 2 s 2 + Q 2 s 2 s 2 s 2 + Q 3 s 2 s 2 s 2 + Q 4 s 2 s 2 s 2 + Q 5 s 2 s 2 s 2 + Q 6 s 2 s 2 s 2 + Q 7 s 2 s 2 s 2 + Q 8 s 2 s 2 s 2 E35h with Q 1 = χ n 3 A 1 A 2 A 3 e i 3 kx ; Q 2 = χ n 2 χ n e i 2 kx e i k x A 1 A 2 A 3 ; Q 3 = χ n 2 χ n e i 2 kx e i k x A 1 A 2 A 3 ; Q 4 = χ n 2 χ n e i 2 kx e i k x A 1 A 2 A 3 ; Q 5 = χ n 2 χ n e ikx e i 2 k x A 1 A 2 A 3 ; Q 6 = χ n 2 χ n e ikx e i 2 k x A 1 A 2 A 3 ; Q 7 = χ n 2 χ n e ikx e i 2 k x A 1 A 2 A 3 ; Q 8 = χ n 3 A 1 A 2 A 3 e i 3 k x . In a true quantum system composed of three spins for instance, states can be created in the form of linear combinations like m 1 0 0 0 + m 2 0 0 1 + + m 8 1 1 1 . For the quantum system, the linear coefficients m 1 , m 2 ,…, m 8 are independent. The classical elastic analogue, introduced here, states which are given in Eq. (34) possesses linear coefficients T 1 ,…, T 8 are interdependent. While somewhat restrictive compared to true quantum systems, the coefficients T I depend on an extraordinary number of degrees of freedom which allows exploration of a large volume of the exponential tensor product space. In the case of the three waveguides, these degrees of freedom include (a) the components (or linear combinations) of the eigen vectors of the coupling matrix through the choice of the eigen modes or the application of a rotational operation that creates cyclic permutations of the eigen vector components, (b) the linear coefficients χ n and χ n used to form the multiband linear superposition of states in the linear space, (c) the frequency and therefore wave number which affect the spinor states and the phase factors e ikx and e i k x , and (d) a phase added to the terms e ikx and e i k x . In the case of N > 3 , Eq. (21) can be extended to linear combinations of more than two modes with the same frequency, leading to additional freedom in the control of the T I . Furthermore, the elastic coefficients β of the waveguides and the coupling elastic coefficient α could also be modified by using constitutive materials with tunable elastic properties via, for instance, the piezoelectric, magneto-elastic or photoelastic effects [24, 25, 26]. Also note that in all the examples we considered, the coupling of the waveguides had the same strength. Tunability of the coupling elastic medium would lead to the ability to modify the connectivity of the waveguides and therefore the coupling matrix. Exploration of the elastic modes given in Eq. (34) can be realized by varying any number of these variables. We illustrate in Figure 2 an example of operation in a very simple case. Figure 2b shows that by varying a single parameter one may achieve a wide variety of states. For instance, one can obtain states with T 1 > 0 and T 2 > 0 or T 1 > 0 and T 2 = 0 or T 1 > 0 and T 2 < 0 or T 1 = 0 and T 2 = 0 . Another interesting example occurs at χ n ˜ 0.6 , there only T 5 and T 8 are different from zero. Then, Φ 2 3 × 1 = T 5 0.6 0 1 1 + T 8 0.6 1 1 1 e i 3 ω 0 t can be written as the tensor product state T 5 0.6 0 + T 8 0.6 1 1 1 e i 3 ω 0 t . A similar state is also obtained for χ n 0.25 . This is the state T 1 0.25 0 + T 4 0.25 1 0 0 e i 3 ω 0 t . Varying χ n can be visualized as a matrix operator. For example, in this latter case, one can define the operation: q 11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 q 44 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 T 1 χ n T 2 χ n T 3 χ n T 4 χ n T 5 χ n T 6 χ n T 7 χ n T 8 χ n = T 1 0.25 T 2 0.25 T 3 0.25 T 4 0.25 T 5 0.25 T 6 0.25 T 7 0.25 T 8 0.25 = T 1 0.25 0 0 T 4 0.25 0 0 0 0 E36 with q 11 = T 1 0.25 T 1 χ n and q 44 = T 4 0.25 T 4 χ n . Another interesting state occurs at χ n = 0.5 . Here, we have T 1 = T 5 , T 2 = T 3 , T 4 = T 8 and T 6 = T 7 . We also have T 3 = T 5 and T 4 = T 6 This state can be written as the tensor product T 1 0.5 0 T 6 0.5 1 0 1 0 1 . This simple example indicates the large variability in T I ’s (i.e., of states) that we can achieve with a single variable. The large number of available variables will lead to even more flexibility in defining states and operators in the exponential tensor product space. ### 2.5. Nonseparability of states in exponentially complex space States given in Eq. (27) are tensor products on the basis 0 0 0 0 0 1 0 1 0 1 1 1 . They are therefore always separable in that basis. Consequently these states cannot be written as nonseparable Bell states. However, we might be able to identify a basis in which Eq. (27) is not separable. Since the Dirac equation (Eq. (15)) for the uncoupled waveguides is linear, its solutions can be a tensor product of a linear combination of N different individual spinors. For instance, we can write: Ψ 2 N × 1 = ρ 1 ω 1 + β k 1 e i k 1 x + μ 1 ω 1 β k 1 e i k 1 x ρ 1 ω 1 β k 1 e i k 1 x + μ 1 ω 1 + β k 1 e i k 1 x e i ω 1 t ρ N ω N + β k N e i k N x + μ N ω N β k N e i k N x ρ N ω N β k N e i k N x + μ N ω N + β k N e i k N x e i ω N t E37 ρ I and μ I , I = 1 , , N are linear coefficients. The state of the coupled waveguide system will be separable in the exponential space into the state of φ -bits if Φ 2 N × 1 = Ψ 2 N × 1 E38 A necessary condition for satisfying Eq. (38) is that N ω 0 = ω 1 + + ω N . Furthermore, the first two terms in the column vector Φ 2 N × 1 of the coupled system are φ 1 1 φ 1 2 φ 1 N and φ 1 1 φ 1 2 φ 2 N . Their ratio is simply equal to r N c = φ 1 N φ 2 N . Similarly, the ratio of the first two terms in the vector Ψ 2 N × 1 of the φ -bit system is given by r N u = ρ N ω N + β k N e i k N x + μ N ω N β k N e i k N x ρ N ω N β k N e i k N x + μ N ω N + β k N e i k N x . Anecessary condition for Eq. (38) to be satisfied is that r N C = r N u E39 χ n A N ω 0 + βk e ikx + χ n A N ω 0 + βk e ik x χ n A N ω 0 βk e ikx + χ n A N ω 0 βk e ik x = ρ N ω N + β k N e i k N x + μ N ω N β k N e i k N x ρ N ω N β k N e i k N x + μ N ω N + β k N e i k N x which can be rewritten as: u e ikx + u e ik x v e ikx + v e ik x = γ e i k N x + γ e i k N x δ e i k N x + δ e i k N x This condition takes the more compact form: P e i k + k N x + Q e i k k N x + R e i k + k N x + S e i k k N x = 0 E40 P , Q , R , S are real. Eq. (40) is true for all values of position x . At x = 0 , we obtain the relation P + Q + R + S = 0 . Inserting that relation into Eq. (27) and eliminating Q yields: {(R+S)coskNx[cos(kk)x1]+(RS)sin(kk)sinkNx}+i{(R+S)sin(kk)x+sinkNx[(RS)cos(kk)x+(R+S)]} For this condition to be satisfied, one needs the real part of the right-hand side of the equation to be equal to zero. This can be achieved for all x ’s by setting k = k . In this case, equating the imaginary parts leads to R = P . However, when k k , Eq. (39) and therefore Eq. (38) are not satisfied. When k k which corresponds to considering a linear combination of multiband states, Φ 2 N × 1 is not separable into the tensor product of individually uncoupled φ -bit waveguides. Therefore, we conclude that there are a large number of solutions of the nonlinear Dirac equation (Eq. (20)) representing states of arrangements of elastically coupled 1-D waveguides that are not separable in the 2 N dimensional tensor product Hilbert space of individual φ -bits. We illustrate the notion of nonseparability of exponentially complex states of a coupled system composed of N = 2 waveguides on a basis in the exponential Hilbert space of two individual φ -bits. The waveguides are coupled to each other but also to a rigid substrate such that the coupling matrix, M N × N , takes the form: M 2 × 2 = 2 1 1 2 The eigen values and real eigen vectors of this coupling matrix are λ 0 2 = 1 , and λ 1 2 = 3 and e 0 = A 1 A 2 = 1 2 1 1 , e 1 = A 1 A 2 = 1 2 1 1 E41 Following the procedure of Section 2.4, we construct a tensor product state in the 2 2 exponential space: Φ 2 2 × 1 = { χ n 2 A 1 A 2 e i 2 kx s 2 × 1 s 2 × 1 + χ n χ n A 1 A 2 e ikx e ik x s 2 × 1 s 2 × 1 + χ n χ n A 1 A 2 e i k x e ikx s 2 × 1 s 2 × 1 + χ n 2 A 1 A 2 e i 2 k x s 2 × 1 s 2 × 1 } e i 2 ω 0 t E42 Eq. (42) is equivalent to Eq. (32) but for two coupled waveguides. On the basis, η 1 = e i 2 ω 0 t e i 2 kx s 2 × 1 s 2 × 1 , η 2 = e i 2 ω 0 t e ikx e ik x s 2 × 1 s 2 × 1 , η 3 = e i 2 ω 0 t e i k x e ikx s 2 × 1 s 2 × 1 , and η 4 = e i 2 ω 0 t e i 2 k x s 2 × 1 s 2 × 1 , Eq. (42) can be rewritten as: Φ 2 2 × 1 = a 11 η 1 + a 12 η 2 + a 21 η 3 + a 22 η 4 E43 with a 11 = χ n 2 A 1 A 2 = 1 2 χ n 2 , a 12 = χ n χ n A 1 A 2 = 1 2 χ n χ n , a 21 = χ n χ n A 1 A 2 = 1 2 χ n χ n , and a 22 = χ n 2 A 1 A 2 = 1 2 χ n 2 . It is then easy to demonstrate that det a 11 a 12 a 21 a 22 = 1 2 χ n 2 1 2 χ n χ n 1 2 χ n χ n 1 2 χ n 2 = 0 , which indicates that the state Φ 2 2 × 1 is separable on the basis η 1 η 2 η 3 η 4 . At this stage, there is nothing surprising as the state Φ 2 2 × 1 was constructed as a tensor product. We now try to express the state given in Eq. (42) on a basis of two individually uncoupled φ -bits. Considering the Hilbert space of the first φ -bit, H 1 , we use the spinor solutions for uncoupled waveguides given in Eq. (17) to construct the orthonormal basis ψ 1 1 = 1 2 ω 1 ω 1 + β 1 k 1 ω 1 β 1 k 1 e i k 1 x e i ω 1 t = s 1 1 k 1 e i k 1 x e i ω 1 t E44a ψ 2 1 = 1 2 ω 1 ω 1 β 1 k 1 ω 1 + β 1 k 1 e i k 1 x e i ω 1 t = s 2 1 k 1 e i k 1 x e i ω 1 t E44b Similarly, we define the orthonormal basis in the Hilbert space, H 2 , of the second φ -bit, ψ 1 2 = 1 2 ω 2 ω 2 + β 2 k 2 ω 2 β 2 k 2 e i k 2 x e i ω 2 t = s 1 2 k 2 e i k 2 x e i ω 2 t E45a ψ 2 2 = 1 2 ω 2 ω 2 β 2 k 2 ω 2 + β 2 k 2 e i k 2 x e i ω 2 t = s 2 2 k 2 e i k 2 x e i ω 2 t E45b In these equations, we have used s 1 1 k 1 , s 2 1 k 1 , s 1 2 k 2 , and s 2 2 k 2 as short-hands for the spinor parts of the basis functions. The basis in the tensor product space H 1 H 2 is given by the four functions: τ 1 = ψ 1 1 ψ 1 2 , τ 2 = ψ 1 1 ψ 2 2 , τ 3 = ψ 2 1 ψ 1 2 , τ 4 = ψ 2 1 ψ 2 2 E46 We have τ 1 = s 1 1 k 1 s 1 2 k 2 e i k 1 + k 2 x e i ω 1 + ω 2 t E47a τ 2 = s 1 1 k 1 s 2 2 k 2 e i k 1 k 2 x e i ω 1 + ω 2 t E47b τ 3 = s 2 1 k 1 s 1 2 k 2 e i k 1 + k 2 x e i ω 1 + ω 2 t E47c τ 4 = s 2 1 k 1 s 2 2 k 2 e i k 1 k 2 x e i ω 1 + ω 2 t E47d It is straightforward to show that τ 1 τ 2 τ 3 τ 4 form an orthogonal basis. That is, τ i τ j = 0 if i j , where τ i is the Hermitian conjugate of τ i . We want now to express the state Φ 2 2 × 1 in the τ basis: Φ 2 2 × 1 = b 11 τ 1 + b 12 τ 2 + b 21 τ 3 + b 22 τ 4 E48 For this, we now need to expand the basis vectors η 1 η 2 η 3 η 4 on the basis τ 1 τ 2 τ 3 τ 4 We define the expansions: η 1 = c 11 τ 1 + c 12 τ 2 + c 13 τ 3 + c 14 τ 4 E49a η 2 = c 21 τ 1 + c 22 τ 2 + c 23 τ 3 + c 24 τ 4 E49b η 3 = c 31 τ 1 + c 32 τ 2 + c 33 τ 3 + c 34 τ 4 E49c η 4 = c 41 τ 1 + c 42 τ 2 + c 43 τ 3 + c 44 τ 4 E49d Note that the c ij ’s are functions of k 1 , k 2 , x, and t. We can find the coefficients c ij by exploiting the orthogonality of the τ i s . For instance, we can multiply Eq. (49a) to the left by the Hermitian conjugate τ 1 , leading to τ 1 η 1 = c 11 τ 1 τ 1 + c 12 τ 1 τ 2 + c 13 τ 1 τ 3 + c 14 τ 1 τ 4 = c 11 τ 1 τ 1 E50 or c 11 k 1 k 2 x t = s 1 1 k 1 s 1 2 k 2 s 2 × 1 s 2 × 1 e i k 1 + k 2 x e i 2 kx e i ω 1 + ω 2 t e i 2 ω 0 t / s 1 1 k 1 s 1 2 k 2 s 1 1 k 1 s 1 2 k 2 E51 We can obtain all other c ij ’s in a similar fashion. Eqs. (49a)–(49d) can be rewritten in the form: η 1 η 2 η 3 η 1 = η 4 × 1 = c 11 c 12 c 13 c 14 c 21 c 22 c 23 c 24 c 31 c 32 c 33 c 34 c 41 c 42 c 43 c 44 τ 1 τ 2 τ 3 τ 4 = C 4 × 4 τ 4 × 1 E52 The matrix C 4 × 4 can be diagonalized. Let d 1 , d 2 , d 3 , and d 4 be the four eigen values of C 4 × 4 with their associated eigen vectors v 11 v 12 v 13 v 14 , v 21 v 22 v 23 v 24 , v 31 v 32 v 33 v 34 and v 41 v 42 v 43 v 44 . We can construct the following matrix out of the four eigen vectors: V 4 × 4 = v 11 v 21 v 31 v 41 v 12 v 22 v 32 v 42 v 13 v 23 v 33 v 43 v 14 v 24 v 34 v 44 On the new basis τ ˜ 1 τ ˜ 2 τ ˜ 3 τ ˜ 4 constructed by using the relation τ ˜ 4 × 1 = V 4 × 4 1 τ 4 × 1 V 4 × 4 , the matrix that couples the η basis and the τ basis takes the form: C ˜ 4 × 4 = d 1 0 0 0 0 d 2 0 0 0 0 d 3 0 0 0 0 d 4 so η 1 = d 1 τ ˜ 1 , η 2 = d 2 τ ˜ 2 , η 3 = d 3 τ ˜ 3 and η 4 = d 4 τ ˜ 4 . On the τ ˜ basis, Eq. (43) can be rewritten as Φ 2 2 × 1 = a 11 d 1 τ ˜ 1 + a 12 d 2 τ ˜ 2 + a 21 d 3 τ ˜ 3 + a 22 d 4 τ ˜ 4 E53 Then on the basis τ ˜ , we can investigate the separability or nonseparability of Φ 2 2 × 1 by calculating the determinant of the linear coefficients in Eq. (53), that is det a 11 d 1 a 12 d 2 a 21 d 3 a 22 d 4 = 1 2 χ n 2 d 1 1 2 χ n χ n d 2 1 2 χ n χ n d 3 1 2 χ n 2 d 4 = 1 4 χ n 2 χ n 2 d 1 d 4 d 2 d 3 E54 Only in the unlikely event of degenerate eigen values, d 1 , d 2 , d 3 , and d 4 , would this determinant be equal to zero. A nonzero determinant given in Eq. (54) indicates that the state Φ 2 2 × 1 is nonseparable on the basis τ ˜ 1 τ ˜ 2 τ ˜ 3 τ ˜ 4 . The existence of nonseparable solutions to the nonlinear Dirac equation raises the possibility of exploiting these solutions for storing and manipulating data within the 2N dimensional tensor product Hilbert space. The exploration of algorithms for exploiting these solutions is beyond the scope of this chapter; however, we note that these solutions may well be observed in physical systems including elastic waveguides which are embedded in a coupling matrix. The manipulation of the system could be achieved either by externally altering the parameters of the system, i.e., the elastic properties of the waveguides, or by changing the frequency and wavenumber of input waves. These possibilities are illustrated for a five-waveguide system driven by transducers in Section 2.6. ### 2.6. Physical realization and actuation Figure 3 illustrates a possible realization of a five waveguide system. The parallel elastic waveguides are embedded in an elastic medium which couples them elastically. The waveguides are arranged in a ring pattern. Modes of the form given in Eq. (21) can be excited with N transducers attached to the input ends of the N waveguides and connected to N phase-locked signal generators to excite the appropriate eigen vectors e n and e n . These modes can be excited by applying a superposition of signals on the transducers with the appropriate phase, amplitude and frequency relations. The frequencies ω n k and ω n k are used to control the spinor parts of the wave functions s 2 × 1 k and s 2 × 1 k . The spinor components which represent a quasistanding wave can be quantified by measuring the transmission coefficient (normalized transmitted amplitude) along any one of the waveguides. It is then possible to operate on the eigen vectors e n and e n without affecting the spinor states or vice versa. For instance, one could apply a rotation that permutes cyclically the components of e n by changing the phase of the signal generators. Such an operation could be quantified by measuring the phase of the transmission amplitude at the output end of the waveguides. ## 3. Conclusions We have shown that the directional projection of elastic waves supported by a parallel array of N elastically coupled waveguides can be described by a nonlinear Dirac-like equation in a 2 N dimensional exponential space. This space spans the tensor product Hilbert space of the two-dimensional subspaces of N uncoupled waveguides grounded elastically to a rigid substrate (which we called φ-bits). We demonstrate that we can construct tensor product states of the elastically coupled system that are nonseparable on the basis of tensor product states of N uncoupled φ-bits. A φ-bit exhibits superpositions of directional states that are analogous to those of a quantum spin, hence it acts as a pseudospin. Since parallel arrays of coupled waveguides span the same exponentially complex space as that of uncoupled pseudospins, the type of elastic systems described here may serve as a simulator of interacting spin networks. The possibility of tuning the elastic coefficients and the elastic coupling constants of the waveguides would allow us to explore the properties of spin networks with variable connectivity and coupling strength. The mapping between the 2 N dimensional and the 2 N dimensional representations of the elastic system leads to the capacity for exploring an exponentially scaling space by handling a linearly growing number of waveguides (i.e., preparation, manipulation, and measurement of these states). 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Crystals. 2016;6:44 21. 21. Deymier PA, Runge K. Sound topology, duality, coherence and wave-mixing: An introduction to the emerging new science of sound. In: Springer Series in Solid State Sciences. Vol. 188. Berlin: Springer; 2017 22. 22. Zanardi P. Virtual quantum systems. Physical Review Letters. 2001;87:077901 23. 23. Zanardi P, Lidar DA, Lloyd S. Quantum tensor product structures are observable induced. Physical Review Letters. 2004;92:060402 24. 24. Croënne C, Vasseur JO, Bou Matar O, Ponge M-F, Deymier PA, Hladky-Hennion A-C, Dubus B. Brillouin scattering-like effect and non-reciprocal propagation of elastic waves due to spatio-temporal modulation of electrical boundary conditions in piezoelectric media. Applied Physics Letters. 2017;110:061901 25. 25. Bou Matar O, Robillard JF, Vasseur JO, Hladky-Hennion A-C, Deymier PA, Pernod P, Preobrazhensky V. Band gap tunability of magneto-elastic phononic crystal. Journal of Applied Physics. 2012;111:054901 26. 26. Swinteck N, Matsuo S, Runge K, Vasseur JO, Lucas P, Deymier PA. Bulk elastic waves with unidirectional backscattering-immune topological states in a time-dependent superlattice. Journal of Applied Physics. 2015;118:063103 Written By Pierre Alix Deymier, Jerome Olivier Vasseur, Keith Runge and Pierre Lucas Submitted: February 20th, 2018 Reviewed: April 13th, 2018 Published: November 5th, 2018	{"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": 0.9409633278846741, "perplexity": 1173.6800078636593}, "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/1664030337731.82/warc/CC-MAIN-20221006061224-20221006091224-00030.warc.gz"}
http://www.koreascience.or.kr/article/ArticleFullRecord.jsp?cn=DBSHCJ_2013_v28n4_799	AN ELEMENTARY PROOF OF SFORZA-SANTALÓ RELATION FOR SPHERICAL AND HYPERBOLIC POLYHEDRA Title & Authors AN ELEMENTARY PROOF OF SFORZA-SANTALÓ RELATION FOR SPHERICAL AND HYPERBOLIC POLYHEDRA Cho, Yunhi; Abstract We defined and studied a naturally extended hyperbolic space (see [1] and [2]). In this study, we describe Sforza`s formula [7] and Santal$\small{\acute{o}}$`s formula [6], which were rediscovered and later discussed by many mathematicians (Milnor [4], Su$\small{\acute{a}}$rez-Peir$\small{\acute{o}}$ [8], J. Murakami and Ushijima [5], and Mednykh [3]) in the spherical space in an elementary way. Thereafter, using the extended hyperbolic space, we apply the same method to prove their results in the hyperbolic space. Keywords hyperbolic space;spherical space;polyhedron;volume; Language English Cited by References 1. Y. Cho, Trigonometry in extended hyperbolic space and extended de Sitter space, Bull. Korean Math. Soc. 46 (2009), no. 6, 1099-1133. 2. Y. Cho and H. Kim, The analytic continuation of hyperbolic space, Geom. Dedicata 161 (2012), no. 1, 129-155. 3. A. D. Mednykh, Hyperbolic and spherical volume for knots, links and polyhedra, Summer school and conference on Geometry and Topology of 3-manifolds, Trieste-Italy, 6-24 June 2005. 4. J. Milnor, The Schlafli Differential Equality, Collected papers, Vol. 1, Publish or Perish, Houston, Texas, 1994. 5. J. Murakami and A. Ushijima, A volume formula for hyperbolic tetrahedra in terms of edge lengths, J. Geom. 83 (2005), no. 1-2, 153-163. 6. L. Santalo, Integral Geometry and Geometric Probability, Encyclopedia of Mathematics and its Applications, Vol. 1, Addison-Wesley, 1976. 7. G. Sforza, Spazi metrico-proiettivi, Ricerche di Estensionimetria differenziale, Serie III, VIII (1906), 3-66. 8. E. Suarez-Peiro, A Schlafli differential formula for simplices in semi-Riemannian hyperquadrics, Gauss-Bonnet formulas for simplices in the de Sitter sphere and the dual volume of a hyperbolic simplex, Pacific J. Math. 194 (2000), no. 1, 229-255.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 3, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9803687334060669, "perplexity": 2117.223547705715}, "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-2018-43/segments/1539583512161.28/warc/CC-MAIN-20181018235424-20181019020924-00160.warc.gz"}
https://math.stackexchange.com/questions/3828479/definition-of-scalar-coextension?noredirect=1	# Definition of "Scalar coextension" Let $$\phi: A \rightarrow B$$ be a ring homomorphism. Here $$A$$ and $$B$$ are commutative rings. Suppose we have an $$A$$-module $$M$$, and we hope to construct a $$B$$-module. A traditional way is via base change, namely construct the module $$M_B:= B \otimes_A M$$. Then this is a $$B$$-module. Yet in the posts and MSE 2780723: Induced representation: which adjoint is it? People also constructed a $$B$$-module $$M^B := \text{Hom}_A(B, M)$$ and call this a scalar coextension (as in the math.SE post quoted above answered by @Qiaochu Yuan). My question is: What is the $$B$$-module structure on $$M^B$$? It is indeed an $$A$$-module, but I looked up many posts and books but have not found any definition of it. So could someone provide me with some references or give me the definition of it? My guess: Let $$f \in M^B := \text{Hom}_A(B, M)$$ and $$b \in B$$, then the $$A$$-module homomorphism $$b \cdot f$$ is defined by sending any element $$r \in B$$ to $$f(br)$$. Is this definition correct? It seems that there should be a correct definition, since people usually regard (or define) this as the right adjoint to the functor of the restriction of scalars, i.e. view a $$B$$-module $$N$$ as an $$A$$-module via $$\phi$$, by defining $$a \cdot x := \phi(a) \cdot x$$ for any $$a \in A$$ and $$x \in N$$. • Your guess is correct. Sep 16, 2020 at 13:00 • Coextension of scalars makes sense with no commutativity assumptions and then your guess is not correct; you should multiply on the right if you want a left module. Sep 16, 2020 at 16:30 • @QiaochuYuan Thank you so much! Sep 18, 2020 at 2:40 • @Hanno Thank you for your comments! Sep 18, 2020 at 2:40	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 26, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9654693603515625, "perplexity": 250.73957323121653}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573630.12/warc/CC-MAIN-20220819070211-20220819100211-00533.warc.gz"}
https://www.physicsforums.com/threads/complex-representation-of-fourier-series.180556/	# Complex representation of fourier series 1. Aug 15, 2007 ### tronxo 1. The problem statement, all variables and given/known data Using the complex representation of fourier series, find the Fourier coefficients of the periodic function shown below. Hence, sketch the magnitude and phase spectra for the first five terms of the series, indicating clearly the spectral lines and their magnitudes 2. Relevant equations Firstable, I dont know how indicate the spectral lines. The other problem that i have is when i try to calculate the Cn coefficient and, therefore, the final serie. I dont know if it is right or not, and in case of it is right, im not able to rewrite "my final function" into the correct answer, which i have it in one of my books. 3. The attempt at a solution what i have done is: Cn= 1/T∫(from 0 to T) f(t)e^(-jnomegat) dt Cn=1/T ( ∫(from 0 to d) Vme^(-jnomegat) dt + ∫(from d to T) 0 e^(-jnomegat) dt ) The second part of the integral is equal to 0, therefore: Cn=1/T ∫(from 0 to d) Vme^(-jnomegat) dt where omega= (2pi/T) Cn=1/T ∫(from 0 to d) Vme^(-jn(2pi/T)t) dt Cn= Vm/T ∫(from 0 to d) e^(-jn(2pi/T)t) dt Cn= Vm/(-jn(2pi/T)T) (limits of the resulting integral from 0 to d)[e^(-jn(2pi/T)t)] Cn= Vm/ (-jn2pi) [e^(-jn(2pi/T)d) - 1] what is next? #### Attached Files: • ###### funcion.JPG File size: 2.7 KB Views: 75 Last edited: Aug 15, 2007 2. Apr 1, 2009 ### haydez98 bump... i am having a similar problem... i am given a signal which can be written as: s(t) = -1 {-1 < t < 0}, 1 {0 < t < 1}, 0 {1 < t < 2} [it's a pulse train] the period, T, is 3. i have calculated the trig. fourier series representation, which in matlab turns out to be correct, yet when i calculate the exponentical fsr, i get a version of the trig. fsr which has its amplitude halved. for the trig fsr: s(t) = 2/(pi n) * (1 - cos((2 * pi * n)/3)) * sin((2 * pi * n * t)/3); for the exp fsr: s(t) = -1/(i * pi * n) * (cos((2 * pi * n)/3) - 1) * exp((i * 2 * pi * n * t)/3) i also tried c_n = 0.5 (a_n - i * b_n) = -0.5 * i * ( 2/(pi * n) * (1 - cos((2 * pi * n)/3)) either case, my complex fsr was a scaled amplitude version of my trig fsr any guidance would be much appreciated Similar Discussions: Complex representation of fourier series	{"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": 0.9583724737167358, "perplexity": 2402.790313563508}, "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/1502886118195.43/warc/CC-MAIN-20170823094122-20170823114122-00235.warc.gz"}
http://math.stackexchange.com/questions/720310/can-i-use-strong-induction-to-prove-graph-theory-hamilton-path-tournament-gra	# Can I use Strong Induction to prove graph theory? - Hamilton Path/Tournament Graphs I am working on the below proof. I am still new to proves so I am wondering if you guys can answer the following questions: 1) Did I use the inductive hypothesis correctly here? (By the inductive hypothesis, we know that both From and To are smaller versions of the larger tournament so there is a Hamiltonian path in both From and To.) 1b) is there a better way to show that these subsets have a Hamiltonian path? 2) Does my proof flow logically and is it clear? 3) Is this proof correct? Graph to represent the outcome of a round-robin tournament, in which every player plays every other player. An edge goes from the victor of each match to the loser. Definition 1. A tournament graph is a directed graph G = (V, E) without loops, in which for every two vertices u and v, exactly one of (u, v) or (v, u) are in E. Definition 3. A Hamiltonian path through a graph is a path that visits each vertex exactly once. Note that a Hamiltonian path is not a cycle. Prove using strong induction that every tournament graph has a Hamiltonian path. For a graph with the number of vertices n = 1 then there is a Hamiltonian path of that single vertex. When n > 1, we can consider a graph with n + 1 vertices where the vertices 1…..n contain a Hamiltonian path. An arbitrary vertex v can be chosen from this graph in which, by the definition of a tournament graph, every other vertex has an edge from v or to v. These vertices are be split up into two sets, From and To, where From contains all the vertices with an edge from v and To contains all the vertices with an edge to v. By the inductive hypothesis, we know that both From and To are smaller versions of the larger tournament so there is a Hamiltonian path in both From and To. These two Hamiltonian paths can be join by the edge To v and the edge From v to form a Hamiltonian path throughout the entire graph. Thus there is a Hamiltonian path in the tournament graph n + 1 so, by strong induction, there is a Hamiltonian path in every tournament graph. - Definition 1 looks weird, what is $e$? – fgp Mar 20 at 20:29 I'm guessing that was edges but I am not sure myself. – DrJonesYu Mar 20 at 20:31 According to en.wikipedia.org/wiki/Tournament_%28graph_theory%29, it should read "...exactly one of $(u,v)$ or $(v,u)$ are in E.", not "... or $(v,e)$". Thus, a turnament graph is a directed graph that becomes the complete graph if you ignore the directions... – fgp Mar 20 at 20:33 Thanks for the catch. – DrJonesYu Mar 20 at 20:35 And you state the induction hypothesis incorrectly. It's true that vertices $1,\ldots,n$ contain a hamiltonian path, but that not sufficient for the rest of your proof - you need that all subgraphs contain one. You actually do state that later, so just drop the confusing statement about the first $n$ vertices. Just say "Pick a vertex, partition the graph into sets TO and FROM (disjoint!), both contain a hamiltonian path per the induction hypothesis, and thus $P_{FROM} \to v \to P_{TO}$ is a hamiltonian path." – fgp Mar 20 at 20:37	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9077067375183105, "perplexity": 329.1492032132353}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802766295.3/warc/CC-MAIN-20141217075246-00048-ip-10-231-17-201.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/145120/why-are-the-gegenbauer-polynomials-called-ultraspherical	# Why are the Gegenbauer polynomials called “ultraspherical”? There has to be a good reason why the Gegenbauer polynomials were also named "ultraspherical" polynomials. I am aware that when $\alpha=\frac{1}{2}$, the Gegenbauer polynomials reduce to the Legendre polynomials, and the Legendre polynomials are used in defining Spherical harmonics. But that is as far as I know how to take that reasoning. Is there a visualization of these polynomials that fits on a sphere? What is an ultrasphere anyway? - I am aware that when $\alpha=\frac12$, the Gegenbauer polynomials reduce to the Legendre polynomials, and the Legendre polynomials are used in defining spherical harmonics. You pretty much nailed it. From here: In the theory of hyperspherical harmonics, Gegenbauer polynomials play a role which is analogous to the role played by Legendre polynomials in the theory of the familiar three-dimensional spherical harmonics. As a compressed version of the discussion in the book (look there for more details), the ultraspherical/hyperspherical harmonics involve Gegenbauer polynomials of the form $C_n^{\frac{d}{2}-1}(x)$, where $d$ is the dimension of the hyperspherical harmonics being considered. For the usual case of $d=3$, we have $C_n^{\frac{3}{2}-1}(x)=P_n(x)$. - Legendre polynomials arise when solving 3D Laplace's equation in spherical coordinates by separation of variables, i.e. $f(x,y,z) = f_1(r) Y(\theta, \phi)$. Legendre polynomials appear in spherical harmonics, specifically for $\ell \in \mathbb{Z}_{\geqslant 0}$ and $m\in \mathbb{Z}$, such that $-\ell \leqslant m \leqslant \ell$: $$Y_{\ell,m}(\theta, \phi) = \sqrt{\frac{2 \ell+1}{4 \pi}} \sqrt{\frac{(\ell-m)!}{(\ell+m)!}} P_\ell^m(\cos \theta) \mathrm{e}^{i m \phi}$$ Solving Laplace's equation in $\mathbb{R}^d$ in hyper-spherical coordinates (in older literature a.k.a ultra-spherical coordinates) by separation of variables, gives rise to ultraspherical polynomials. -	{"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": 0.9575398564338684, "perplexity": 323.57344977108846}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500829754.11/warc/CC-MAIN-20140820021349-00309-ip-10-180-136-8.ec2.internal.warc.gz"}
http://mathoverflow.net/questions/107753/geodesics-and-paths-for-non-unit-quaternions	# Geodesics and paths for non-unit quaternions Given two unit quaternions $\mathbf{q}_1$ and $\mathbf{q}_2$ $\in \mathbb{H}_{1}$, it is well known that the unique geodesic parametrized by the length, $t \mapsto \gamma(t)$, joining the two quaternions is $\gamma(t) = \mathbf{q}_{1}\left(\mathbf{q}_{1}^{-1}\mathbf{q}_{2}\right)^{t}$ It corresponds to the geodesic in $\mathbb{S}^3$. The corresponding geodesic distance can be written using the logarithm of quaternions $d(\mathbf{q}_1 , \mathbf{q}_2 )=\|\|\log\left( \mathbf{q}_{1}^{-1} \mathbf{q}_{2} \right) \|\|$ Let now us consider that we are dealing with general quaternions in $\mathbb{H}$, of type $\mathbf{q}_i = \|\mathbf{q}_i \|U\mathbf{q}_i$, where $\mathbf{q}_i$ is the norm and $U\mathbf{q}_i$ the corresponding versor. Ìf we consider now the same expression than above for a parametrized path between $\|\mathbf{q}_1 \|U\mathbf{q}_1$ and $\|\mathbf{q}_2 \|U\mathbf{q}_2$, i.e., $\gamma(t) = \|\mathbf{q}_1 \|^{1-t} \|\mathbf{q}_2 \|^t U\mathbf{q}_{1}\left(U\mathbf{q}_{1}^{}U\mathbf{q}_{2}\right)^{t}$ Thus, a path defined by the product of a geodesic in $\mathbb{R}^+$ (weighted geometric mean of the norms) and a geodesic in $\mathbb{S}^3$. The length of the path involves also a decoupling between both manifolds, i.e., ${\cal l}(\mathbf{q}_1,\mathbf{q}_1 )^2 =$ $\|\log(\|\mathbf{q}_2\|) - \log(\|\mathbf{q}_1\|)\|^2 + \|\|\log\left( U\mathbf{q}_{1}^{} U\mathbf{q}_{2} \right) \|\|^2$ Curved paths $\gamma(t)$ are of course longer than the straight lines in $\mathbb{R}^4$ and therefore this is not the minimal geodesic of $\mathbb{H}$ Questions: • Is the path $\gamma(t)$ a geodesic of the embedding of $\mathbb{H}$ in $\mathbb{R}^+ \times \mathbb{S}^3$? • What happens in particular if $\|\mathbf{q}_i\| \leq 1$ (i.e., quaternions lying inside the sphere $\mathbb{S}^3$) • Is the corresponding line element given by $ds^2 = (d\log(\|\mathbf{q}_{i}\|))^2 + (dU\mathbf{q}_{i})^2$ ? • More generally, what are the strucutre of the manifold $\mathbb{R}^+ \times \mathbb{S}^3$ with this kind of metric?... compact, completeness, bounds of curvature, etc. - The metric you have written down is the standard bi-invariant metric on the Lie group of nonzero quaternions. I.e., it is the metric such that the left-invariant 1-form $\omega = \mathbf{q}^{-1}\ d\mathbf{q}$ with values in $T_{\mathbf{1}}\mathbb{H}\simeq\mathbb{H}$ is an isometry at every point. I.e., for every $\mathbf{p}$ a nonzero quaternion, $\omega_{\mathbf{p}}:T_{\mathbf{p}}\mathbb{H}\to \mathbb{H}$ induces an isometry between $T_{\mathbf{p}}\mathbb{H}$ under this metric with $\mathbb{H}$ given the standard quaternion norm. This metric (which is complete and homogeneous) is a product metric and the Riemannian manifold is isometric to $\mathbb{R}\times\mathbb{S}^3$. Thus, it is not compact, it is complete, the sectional curvature everywhere is bounded by $1$, etc.	{"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": 0.9898565411567688, "perplexity": 237.8640857118988}, "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/1438042991951.97/warc/CC-MAIN-20150728002311-00145-ip-10-236-191-2.ec2.internal.warc.gz"}
https://socratic.org/questions/how-do-you-solve-2x-2-7x-9-0-algebraically#513610	Algebra Topics # How do you solve 2x^2+7x+9=0 algebraically? Nov 30, 2017 In fraction form: $\setminus \rightarrow x = \setminus \frac{i \setminus \sqrt{23}}{4} - \setminus \frac{7}{4}$ Simplified form: $\setminus \rightarrow x \setminus \approx - 1.75 \setminus \pm 1.19895788 i$ #### Explanation: We’ll solve this two ways: by completing the square and using the quadratic formula. Here’s the first way: $2 {x}^{2} + 7 x + 9 = 0$ First divide everything by $2$, since $a$ cannot have a coefficient: $\setminus \rightarrow {x}^{2} + \setminus \frac{7}{2} x + \setminus \frac{9}{2} = 0$ Now, move $c$ to the RHS: $\setminus \rightarrow {x}^{2} + \setminus \frac{7}{2} x = - \setminus \frac{9}{2}$ Now add ${\left(\setminus \frac{b}{2}\right)}^{2}$ to both sides. Here, $b = 7$. $\setminus \rightarrow {x}^{2} + \setminus \frac{7}{2} x + \setminus \frac{49}{16} = \setminus \frac{49}{16} - \setminus \frac{9}{2}$ Simplify the RHS: $\setminus \rightarrow {x}^{2} + \setminus \frac{7}{2} x + \setminus \frac{49}{16} = - \setminus \frac{23}{16}$ Factor the LHS into ${\left(x + \setminus \frac{b}{2}\right)}^{2}$: $\setminus \rightarrow {\left(x + \setminus \frac{\setminus \frac{7}{2}}{2}\right)}^{2} = - \setminus \frac{23}{16}$ $\setminus \rightarrow {\left(x + \setminus \frac{7}{4}\right)}^{2} = - \setminus \frac{23}{16}$ Take the square root of both sides: $\setminus \rightarrow x + \setminus \frac{7}{4} = \setminus \sqrt{- \setminus \frac{23}{16}}$ Isolate $x$: $\setminus \rightarrow x = \setminus \sqrt{- \setminus \frac{23}{16}} - \setminus \frac{7}{4}$ $\setminus \rightarrow x = i \setminus \sqrt{\setminus \frac{23}{16}} - \setminus \frac{7}{4}$ $\setminus \rightarrow x = i \setminus \frac{\setminus \sqrt{23}}{4} - \setminus \frac{7}{4}$ $\setminus \rightarrow x \setminus \approx - 1.75 \setminus \pm 1.19895788 i$ Here’s the second way, using the quadratic formula: $2 {x}^{2} + 7 x + 9 = 0$ First, we need to identify $a$, $b$, and $c$: $a = 2$ $b = 7$ $c = 9$ Now, plug them into the formula: $x = \setminus \frac{- b \setminus \pm \setminus \sqrt{{b}^{2} - 4 a c}}{2 a}$ $\setminus \rightarrow x = \setminus \frac{- 7 \setminus \pm \setminus \sqrt{{7}^{2} - 4 \left(2\right) \left(9\right)}}{2 \left(2\right)}$ $\setminus \rightarrow x = \setminus \frac{- 7 \setminus \pm \setminus \sqrt{49 - 72}}{4}$ $\setminus \rightarrow x = \setminus \frac{- 7 \setminus \pm \setminus \sqrt{- 23}}{4}$ $\setminus \rightarrow x = \setminus \frac{- 7 \setminus \pm i \setminus \sqrt{23}}{4}$ Rearranging yields: $\setminus \rightarrow x = \setminus \frac{i \setminus \sqrt{23}}{4} - \setminus \frac{7}{4}$ $\setminus \rightarrow x \setminus \approx - 1.75 \setminus \pm 1.19895788 i$ It’s the same answer we got by completing the square, so we know the answer is correct. ##### Impact of this question 9497 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": 35, "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": 0.8596160411834717, "perplexity": 1210.5626839521415}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585441.99/warc/CC-MAIN-20211021195527-20211021225527-00320.warc.gz"}
http://quickmath.com/webMathematica3/quickmath/numbers/percentages/basic.jsp	Algebra Equations Inequalities Calculus Matrices Graphs Numbers #### Numbers : Percentages Enter the value(s) for the required question and click the adjacent Go button. What is % of ? Go is what percent of ? Go is % of what number? Go What is the fraction / as a percentage? Go What is the decimal or integer as a percentage? Go What is % as a fraction? Go What is % as a decimal? Go What is the percentage change when becomes ? Go Tutors	{"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": 0.8923810124397278, "perplexity": 2481.697758940738}, "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-2014-23/segments/1406510267876.49/warc/CC-MAIN-20140728011747-00201-ip-10-146-231-18.ec2.internal.warc.gz"}
http://clay6.com/qa/45063/three-charges-q-q-and-q-are-placed-as-shown-at-equal-distances-on-a-straigh	# Three charges – q. Q and – q are placed as shown at equal distances on a straight line. If the total potential energy of the system of three charges is zero. What is the ratio Q : q? $(B) \frac{1}{4}$	{"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": 0.8876146674156189, "perplexity": 266.7724491701536}, "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/1502886104631.25/warc/CC-MAIN-20170818082911-20170818102911-00465.warc.gz"}
https://www.deepdyve.com/lp/ou_press/high-dimensional-finite-elements-for-multiscale-maxwell-type-equations-labSJOLTE4	# High-dimensional finite elements for multiscale Maxwell-type equations High-dimensional finite elements for multiscale Maxwell-type equations Abstract We consider multiscale Maxwell-type equations in a domain $$D\subset\mathbb{R}^d$$ ($$d=2,3$$), which depend on $$n$$ microscopic scales. Using multiscale convergence, we derive the multiscale homogenized problem, which is posed in $$\mathbb{R}^{(n+1)d}$$. Solving it, we get all the necessary macroscopic and microscopic information. Sparse tensor product finite elements (FEs) are employed, using edge FEs. The method achieves a required level of accuracy with essentially an optimal number of degrees of freedom, which, apart from a multiplying logarithmic term, is equal to that for solving a problem in $$\mathbb{R}^d$$. Numerical correctors are constructed from the FE solutions. In the two-scale case, an explicit homogenization error is deduced. To get this error, the standard procedure in the homogenization literature requires the solution $$u_0$$ of the homogenized problem to belong to $$H^1({\rm curl\,},D)$$. However, in polygonal domains, $$u_0$$ belongs only to a weaker regularity space $$H^s({\rm curl\,},D)$$ for $$0<s<1$$. We derive a homogenization error estimate for this case. Though we prove the result for two-scale Maxwell-type equations, the approach works verbatim for elliptic and elasticity problems when the solution to the homogenized equation belongs to $$H^{1+s}(D)$$ (standard procedure requires $$H^2(D)$$ regularity). This homogenization error estimate is new in the literature. Thus, for two-scale problems, an explicit error for the numerical corrector is obtained; it is of the order of the sum of the homogenization error and the FE error. For the case of more than two scales, we construct a numerical corrector, albeit without a rate of convergence, as such a homogenization error is not available. Numerical experiments confirm the theoretical results. 1. Introduction We consider Maxwell-type equations that depend on $$n$$ separable microscopic scales in a domain $$D\in \mathbb{R}^d,$$ where $$d=2,3$$. The coefficients are assumed to be locally periodic with respect to each microscopic scale. We use the multiscale convergence of a bounded sequence in $$H({\rm curl\,},D)$$ to derive the multiscale homogenized equation, which contains all the necessary information. Solving it, we get the solution of the homogenized equation that describes the multiscale solution macroscopically and the scale interacting terms (the corrector terms) that encode the multiscale information. However, this equation is posed in high-dimensional product domains. It depends on $$n+1$$ variables in $$\mathbb{R}^d$$, one for each scale that the original multiscale problem depends on. The full tensor product finite element (FE) method requires a large number of degrees of freedom, and thus is prohibitively expensive. We develop the sparse tensor FE product approach, using edge FEs, for this multiscale homogenized Maxwell-type equation. The approach achieves accuracy essentially equal to that obtained by the full tensor product FEs but requires an essentially optimal level of complexity that is essentially equal to that for solving a problem in $$\mathbb{R}^d$$ only. Analytic homogenization for two-scale Maxwell-type equations is well developed. We mention the standard references Bensoussan et al. (1978), Sanchez-Palencia (1980) and Jikov et al. (1994). However, there has been little effort on numerical analysis of multiscale Maxwell-type equations. As for other multiscale problems, a direct numerical treatment needs a fine mesh which is at most of the order of the smallest scale, leading to a prohibitive level of complexity. The multiscale FE method (Hou & Wu, 1997; Efendiev & Hou, 2009) and the heterogeneous multiscale method (E & Engquist, 2003; Abdulle et al., 2012) are designed to overcome this difficulty but their applications to multiscale Maxwell-type equations have not been adequately studied. Solving cell problems to establish the homogenized equation and using the cell problems’ solutions to compute the correctors for two-scale Maxwell-type equations are performed in Zhang et al. (2010). However, as for other multiscale problems, this approach is rather expensive, especially when the coefficients are only locally periodic, as for each macroscopic point, several cell problems need to be solved. We contribute in this article a feasible general numerical method for locally periodic multiscale Maxwell-type problems. We employ the sparse tensor product FE approach developed by Hoang & Schwab (2004/05) for multiscale elliptic equations (see also Hoang, 2008; Harbrecht & Schwab, 2011; Xia & Hoang, 2014, 2015a,b). It achieves the required level of accuracy with an essentially optimal number of degrees of freedom. We note that sparse tensor edge FEs are considered in Hiptmair et al. (2013) in the context of computing the moments of the solutions to stochastic Maxwell-type problems. However, our setting is quite different, and does not require constructing the detail spaces for edge FEs. We only need the detail spaces for the nodal FEs that approximate functions in the Lebesgue spaces $$L^2$$. We then construct a numerical corrector for the solution of the original multiscale problem, using the FE solutions of the multiscale homogenized problem. In the case of two scales, we derive an explicit error estimate in terms of the homogenization error and the FE error. It is well known that for two-scale elliptic problems in a domain $$D$$, if the solution of the homogenized problem belongs to $$H^2(D)$$, and the solutions to the cell problems are sufficiently smooth, the homogenization error in the $$H^1(D)$$ norm is $${\mathcal O}(\varepsilon^{1/2}),$$ where $$\varepsilon$$ is the microscopic scale (Bensoussan et al., 1978; Jikov et al., 1994). For two-scale Maxwell-type equations, the $${\mathcal O}(\varepsilon^{1/2})$$ homogenization error in the $$H({\rm curl},D)$$ norm is obtained when the solution $$u_0$$ of the homogenized problem (4.7) belongs to $$H^1({\rm curl},D)$$. However, for polygonal domains that are of interest in FE discretization, $$u_0$$ generally belongs only to a weaker regularity space $$H^s({\rm curl\,},D)$$ for $$0<s<1$$ (see e.g., Hiptmair, 2002). For this case, we develop an approach to deriving a new homogenization error estimate. Though we present the result for Maxwell-type equations, the approach works verbatim for two-scale elliptic and elasticity problems when the solution to the homogenized problem is in $$H^{1+s}(D)$$. As far as we are aware, this is a new result in the homogenization theory and forms another main contribution of the article. For the case of more than two scales, an analytic homogenization error is not available. However, we can still derive a corrector from the FE solution of the multiscale homogenized problem, albeit without an explicit rate of convergence. This article is organized as follows. In the next section, we formulate the multiscale Maxwell-type equation. Homogenization of the multiscale Maxwell-type equation (2.4) is studied in Bensoussan et al. (1978) in the two-scale case, using two-scale asymptotic expansion. Here, we use the multiscale convergence method to study (2.4) in the general multiscale setting. We thus develop multiscale convergence for a bounded sequence in $$H({\rm curl\,},D)$$. Two-scale convergence for a bounded sequence in $$H({\rm curl\,},D)$$ is developed in Wellander & Kristensson (2003). Since we consider the general $$(n+1)$$-scale convergence and the limiting result that we use is in a slightly different form from that of Wellander & Kristensson (2003) in the two-scale case, so we present the proofs in full. FE approximations of the multiscale homogenized Maxwell-type problem are studied in Section 3. We prove the FE error estimates in cases of both full and sparse tensor product FE approximations. The errors are essentially equal (apart from a logarithmic multiplying factor), but the dimension of the sparse tensor product FE space is much lower than that of the full tensor product FE space and is essentially equal to that for solving a problem in $$\mathbb{R}^d$$ only. In Section 4, we construct numerical correctors for the solution to the original multiscale problem. For two-scale problems, we prove the general homogenization error estimate for the case where $$u_0$$ belongs to the weaker regularity space $$H^s({\rm curl\,},D),$$ where $$0<s<1$$. From that we deduce the error estimate for the numerical corrector, which is of the order of the sum of the homogenization error estimate and the FE error. For the case of more than two scales, we derive a numerical corrector but without a rate of convergence. In Section 5, we prove that the regularity required to get the FE error estimate for the sparse tensor product FEs and to get the homogenization error in the two-scale case is achievable. Section 6 contains numerical experiments that confirm our analysis. Finally, the two Appendices A and B contain the long proofs of some previous results: the proof of the homogenization error when $$u_0$$ belongs to a weaker regularity space is presented in Appendix A. Throughout the article, by $$\#$$ we denote the spaces of functions that are periodic with the period being the unit cube $$Y\subset \mathbb{R}^d$$. Repeated indices indicate summation. The notations $$\nabla$$ and $${\rm curl\,}$$ without indicating the variable explicitly denote the gradient and the $${\rm curl\,}$$ operator with respect to $$x$$ of a function of $$x$$ only, where $$\nabla_x$$ and $${\rm curl\,}_{\!x}$$ denote the partial gradient and partial $${\rm curl\,}$$ of a function depending on $$x$$ and also on other variables. We generally present the theoretical results for the three-dimensional case and mention the two-dimensional case only when it is necessary, as the two cases are largely similar. 2. Problem setting 2.1 Multiscale Maxwell-type problems Let $$D$$ be a domain in $$\mathbb{R}^d$$ ($$d=2,3$$). Let $$Y$$ be the unit cube in $$\mathbb{R}^d$$. By $$Y_1,\ldots,Y_n$$ we denote $$n$$ copies1 of $$Y$$. We denote by $${\bf Y}$$ the product set $$Y_1\times Y_2\times\cdots\times Y_n$$ and by $$\boldsymbol{y}\in{\bf Y}$$ the vector $$\boldsymbol{y}=(y_1,y_2,\ldots,y_n)$$. For each $$i=1,\ldots,n$$, we denote by $${\bf Y}_i$$ the set of vectors $$\boldsymbol{y}_i=(y_1,\ldots,y_i),$$ where $$y_j\in Y_j$$ for $$j=1,\ldots,i$$. For $$d=3$$, let $$a$$ and $$b$$ be functions with symmetric matrix values from $$D\times {\bf Y}$$ to $$\mathbb{R}^{d\times d}_{\rm sym}$$; $$a$$ and $$b$$ are continuous in $$D\times {\bf Y}$$ and are periodic with respect to each variable $$y_i$$ with the period being $$Y_i$$. We assume that for all $$x\in D$$ and $$\boldsymbol{y}\in{\bf Y}$$, and all $$\xi,\zeta\in \mathbb{R}^d$$, c∗\|ξ\|2≤aij(x,y)ξiξj, aij(x,y)ξiζj≤c∗\|ξ\|\|ζ\|,c∗\|ξ\|2≤bij(x,y)ξiξj, bij(x,y)ξiζj≤c∗\|ξ\|\|ζ\|, (2.1) where $$c_$$ and $$c^$$ are positive numbers; $$\|\cdot\|$$ denotes the Euclidean norm in $$\mathbb{R}^3$$. Let $$\varepsilon$$ be a small positive value, and $$\varepsilon_1,\ldots,\varepsilon_n$$ be $$n$$ functions of $$\varepsilon$$ that denote the $$n$$ microscopic scales that the problem depends on. We assume the following scale separation properties: for all $$i=1,\ldots,n-1$$, limε→0εi+1(ε)εi(ε)=0. (2.2) Without loss of generality, we assume that $$\varepsilon_1=\varepsilon$$. We define $$a^\varepsilon, b^\varepsilon: D\to\mathbb{R}^{d\times d}_{\rm sym}$$ as aε(x)=a(x,xε1,…,xεn), bε(x)=b(x,xε1,…,xεn). (2.3) Let W=H0(curl,D)={u∈L2(D)3, curlu∈L2(D)3, u×ν=0}, where $$\nu$$ denotes the outward normal vector on the boundary $$\partial D$$. Let $$f\in W'$$. We consider the problem curl(aε(x)curluε(x))+bε(x)uε(x)=f(x), (2.4) with the boundary condition $$u^{\varepsilon}\times \nu=0$$ on $$\partial D$$. We formulate this problem in the variational form as follows: find $$u^{\varepsilon}\in W$$ so that ∫D[aε(x)curluε(x)⋅curlϕ(x)+bε(x)uε(x)⋅ϕ(x)]dx=∫Df(x)⋅ϕ(x)dx (2.5) for all $$\phi\in W$$ (by $$\int_D\,f\cdot\phi\, {\rm d}x$$ we denote the duality pairing between $$W'$$ and $$W$$). The Lax–Milgram lemma guarantees the existence of a unique solution $$u^{\varepsilon}$$ that satisfies ‖uε‖W≤c‖f‖W′, (2.6) where the constant $$c$$ depends only on $$c_$$ and $$c^$$ in (2.1). For $$d=2$$, the matrix function $$b^\varepsilon:D\times{\bf Y}\to \mathbb{R}^{2\times 2}$$ is defined as above. As $${\rm curl\,}u^{\varepsilon}$$ is now a scalar function, $$a(x,\boldsymbol{y})$$ is a continuous function from $$D\times{\bf Y}$$ to $$\mathbb{R},$$ which is periodic with respect to each variable $$y_i$$ with the period being $$Y_i$$. In the place of (2.1), we have c∗≤a(x,y)≤c∗ ∀x∈D and y∈Y. The variational formulation in two dimensions becomes ∫D[aε(x)curluε(x)curlϕ(x)+bε(x)uε(x)⋅ϕ(x)]dx=∫Df(x)⋅ϕ(x)dx ∀ϕ∈W. (2.7) In the rest of the article, we present the results for the three-dimensional case and only mention the two-dimensional case when necessary; the results for two dimensions are similar. 2.2 Multiscale convergence We use multiscale convergence to derive the homogenized equation. We first recall the definition of multiscale convergence (see Nguetseng, 1989; Allaire, 1992; Allaire & Briane, 1996). Definition 2.1 A sequence of functions $$\{w^\varepsilon\}_\varepsilon\subset L^2(D)$$$$(n+1)$$-scale converges to a function $$w^0\in L^2(D\times {\bf Y})$$ if for all smooth functions $$\phi\in C^\infty(D\times{\bf Y}),$$ which are periodic with respect to $$y_i$$ with the period being $$Y_i$$ for $$i=1,\ldots,n$$, limε→0∫Dwε(x)ϕ(x,xε1,…,xεn)dx=∫D∫Yw0(x,y)ϕ(x,y)dydx. We have the following result. Proposition 2.2 From a bounded sequence in $$L^2(D),$$ we can extract an $$(n+1)$$-scale convergent subsequence. For a bounded sequence in $$H({\rm curl\,},D)$$, we have the following results on $$(n+1)$$-scale convergence. These results were first established in Wellander & Kristensson (2003) for the two-scale case. We present below the multiscale convergence of a bounded sequence in $$H({\rm curl\,},D),$$ which will be used to study the multiscale equations (2.5) and (2.7). By $$\tilde H_\#({\rm curl\,},Y_i)$$ we denote the equivalent classes of functions in $$H_\#({\rm curl\,},Y_i)$$ such that if $${\rm curl\,} v={\rm curl\,} w$$ we regard $$v=w$$ in $$\tilde H_\#({\rm curl\,},Y_i)$$. Proposition 2.3 Let $$\{w^\varepsilon\}_\varepsilon$$ be a bounded sequence in $$H({\rm curl\,},D)$$. There is a subsequence (not renumbered), a function $$w_0\in H({\rm curl\,},D)$$, $$n$$ functions $${\frak w_i}\in L^2(D\times Y_1\times\cdots\times Y_{i-1},H^1_\#(Y_i)/\mathbb{R})$$ such that wε⟶(n+1)−scalew0+∑i=1n∇yiwi. Further, there are $$n$$ functions $$w_i\in L^2(D\times Y_1\times\cdots\times Y_{i-1},\tilde H_\#({\rm curl\,},Y_i))$$ such that curlwε⟶(n+1)−scalecurlw0+∑i=1ncurlyiwi. Proof. Let $$\xi\in L^2(D\times{\bf Y})^3$$ be the $$(n+1)$$-scale limit of $$\{w^\varepsilon\}_\varepsilon$$. Consider the function $$\phi=\varepsilon_n{\it{\Phi}}(x,y_1,\ldots,y_n),$$ where $${\it{\Phi}}$$ is a function in $$C^\infty_0(D,C^\infty_\#(Y_1,\ldots,C^\infty_\#(Y_n),\ldots))^3$$ and is periodic with respect to $$y_1, \ldots,y_n$$ with the period being $$Y_1,\ldots,Y_n,$$ respectively. We then have limε→0∫Dcurlwε⋅εnΦ(x,xε1,…,xεn)dx=0. On the other hand, limε→0∫Dcurlwε⋅εnΦ(x,xε1,…,xεn)dx = limε→0∫Dwε⋅εncurlΦ(x,xε1,…,xεn)dx = limε→0∫Dwε⋅curlynΦ(x,xε1,…,xεn)dx = ∫D∫Yξ(x,y)⋅curlynΦ(x,y)dydx. Thus, there is a function $$\xi_{n-1}(x,\boldsymbol{y}_{n-1})\in L^2(D\times{\bf Y}_{n-1})$$ and a function $${\frak w_n}(x,\boldsymbol{y}_n)\in L^2(D\times{\bf Y}_{n-1},H^1_\#(Y_n)/\mathbb{R})$$ such that ξ(x,y)=ξn−1(x,yn−1)+∇ynwn(x,y). Next we choose $$\phi=\varepsilon_{n-1}{\it{\Phi}}(x,y_1,\ldots,y_{n-1})$$ for a function $${\it{\Phi}}\in C^\infty_0(D,C^\infty_\#(Y_1,\ldots,C^\infty_\#(Y_{n-1}),\ldots),$$ which is periodic with respect to $$y_1,\ldots,y_{n-1}$$. We then have 0 = limε→0∫Dcurlwε⋅εn−1Φ(x,xε1,…,xεn−1)=limε→0∫Dwε⋅curlyn−1Φ(x,xε1,…,xεn−1)dx = ∫D∫Y(ξn−1(x,yn−1)+∇ynwn(x,y))⋅curlyn−1Φ(x,y1,…,yn−1)dyn−1dx = ∫D∫Yn−1ξn−1(x,yn−1)⋅curlyn−1Φ(x,y1,…,yn−1)dyn−1dx. From this, there is a function $$\xi_{n-2}(x,\boldsymbol{y}_{n-2})\in L^2(D\times {\bf Y}_{n-2})$$ and a function $${\frak w_{n-1}}(x,\boldsymbol{y}_{n-1})\in L^2(D\times {\bf Y}_{n-2},H^1_\#(Y_{n-1})/\mathbb{R})$$ so that ξn−1(x,yn−1)=ξn−2(x,yn−2)+∇yn−1wn−1(x,yn−1), so ξ(x,y)=ξn−2(x,yn−2)+∇yn−1wn−1(x,yn−1)+∇ynwn(x,y). Continuing this process, we have ξ(x,y)=w0(x)+∑i=1n∇yiwi(x,yi), where $$w_0\in L^2(D)^3$$ and $${\frak w_i}(x,\boldsymbol{y}_i)\in L^2(D\times {\bf Y}_{i-1},H^1_\#(Y_i))$$. As $$\int_Y\xi(x,\boldsymbol{y})\,{\rm d}\boldsymbol{y}=w_0(x)$$, $$w_0$$ is the weak limit of $$w^\varepsilon$$ in $$L^2(D)^3$$. Let $$\eta(x,\boldsymbol{y})$$ be the $$(n+1)$$-scale convergence limit of $${\rm curl\,} w^\varepsilon$$ in $$L^2(D\times{\bf Y})$$. Let $${\it{\Phi}}(x,y_1,\ldots,y_n)\in C^\infty_0(D,C^\infty_\#(Y_1,\ldots,C^\infty_\#(Y_n),\ldots))$$. We have ∫Dcurlwε⋅∇Φ(x,xε1,…,xεn)dx =∫Dwε⋅curl∇Φ(x,xε1,…,xεn)dx−∫∂D(wε×ν)⋅∇Φ(x,xε1,…,xεn)ds=0. Thus, 0 = limε→ 0∫Dcurlwε⋅εn∇Φ(x,xε1,…,xεn)dx=limε→0∫Dcurlwε⋅∇ynΦ(x,xε1,…,xεn)dx = ∫D∫Yη(x,y)⋅∇ynΦ(x,y1,…,yn)dydx. Therefore, there is a function $$w_n(x,\boldsymbol{y}_n)\in L^2(D\times{\bf Y}_{n-1},\tilde H_\#({\rm curl\,},Y_n))$$ and a function $$\eta_{n-1}(x,\boldsymbol{y}_{n-1})\in L^2(D\times{\bf Y}_{n-1})$$ such that η(x,y)=ηn−1(x,yn−1)+curlynwn(x,y). Let $${\it{\Phi}}(x,y_1,\ldots,y_{n-1})\in C^\infty_0(D,C^\infty_\#(Y_1,\ldots,C^\infty_\#(Y_{n-1}),\ldots))$$. We have 0 = limε→ 0∫Dcurlwε⋅εn−1∇Φ(x,xε1,…,xεn−1)dx=limε→0∫Dcurlwε⋅∇yn−1Φ(x,xε1,…,xεn−1)dx = ∫D∫Y(ηn−1(x,yn−1)+curlynwn(x,y))⋅∇yn−1Φ(x,y1,…,yn−1)dydx = ∫D∫Yn−1ηn−1(x,yn−1)⋅∇yn−1ϕ(x,y1,…,yn−1)dyn−1dx. Therefore, there is a function $$w_{n-1}(x,\boldsymbol{y}_{n-1})\in L^2(D\times{\bf Y}_{n-2},\tilde H_\#({\rm curl\,},Y_{n-1}))$$ and a function $$\eta_{n-2}(x,\boldsymbol{y}_{n-2})\in L^2(D\times {\bf Y}_{n-2})^3$$ so that ηn−1(x,yn−1)=ηn−2(x,yn−2)+curlyn−1wn−1(x,yn−1) so η(x,y)=ηn−2(x,yn−2)+curlyn−1wn−1(x,yn−1)+curlynwn(x,y). Continuing, we find that there is a function $$\eta_0(x)\in L^2(D)^3$$ and functions $$w_i(x,\boldsymbol{y}_{i})\in L^2(D\times{\bf Y}_{i-1},\tilde H_\#({\rm curl\,},Y_i))$$ so that η(x,y)=η0(x)+∑i=1ncurlyiwi(x,yi). As for all $$\phi(x)\in C^\infty_0(D)^3$$ limε→0∫Dcurlwε(x)⋅ϕ(x)dx=∫Dη0(x)⋅ϕ(x)dx,$$\eta_0$$ is the weak limit of $${\rm curl\,} w^\varepsilon$$ in $$L^2(D)^3$$. Thus, $$\eta_0={\rm curl\,} w_0$$. We then get the conclusion. □ 2.3 Multiscale homogenized Maxwell-type problem From (2.6) and Proposition 2.3, we can extract a subsequence (not renumbered), a function $$u_0\in H_0({\rm curl\,}, D)$$, $$n$$ functions $${\frak u_i}\in L^2(D\times Y_1\times\cdots\times Y_{i-1},H^1_\#(Y_i)/\mathbb{R})$$ and $$n$$ functions $$u_i\in L^2(D\times Y_1\times\cdots\times Y_{i-1},\tilde H_\#({\rm curl\,},Y_i))$$ such that uε⟶(n+1)−scaleu0+∑i=1n∇yiui (2.8) and curluε⟶(n+1)−scalecurlu0+∑i=1ncurlyiui. (2.9) For $$i=1,\ldots,n$$, let $$W_i=L^2(D\times Y_1\times\cdots\times Y_{i-1},\tilde H_\#({\rm curl\,},Y_i))$$ and $$V_i=L^2(D\times Y_1\times\cdots\times Y_{i-1},H^1_\#(Y_i)/\mathbb{R})$$. We define the space $${\bf V}$$ as V=W×W1×⋯×Wn×V1×⋯×Vn. For $$\boldsymbol{v}=(v_0,\{v_i\},\{\frak v_i\})\in {\bf V}$$, we define the norm \|\|\|v\|\|\|=‖v0‖H(curl,D)+∑i=1n‖vi‖L2(D×Yi−1,H~#(curl,Yi))+∑i=1n‖vi‖L2(D×Yi−1,H#1(Yi)/R). We then have the following result. Proposition 2.4 We define $$\boldsymbol{u}=(u_0,\{u_i\}, \{\frak u_i\})\in{\bf V}$$. Then $$\boldsymbol{u}$$ satisfies B(u,v):=∫D∫Y[a(x,y)(curlu0+∑i=1ncurlyiui)⋅(curlv0+∑i=1ncurlyivi) +b(x,y) (u0+∑i=1n∇yiui)⋅(v0+∑i=1n∇yivi)]dydx=∫Df(x)⋅v0(x)dx (2.10) for all $$\boldsymbol{v}=(v_0,\{v_i\},\{\frak v_i\})\in {\bf V}$$. Proof. Let $$v_0\in C^\infty_0(D)^3$$, $$v_i\in C^\infty_0(D, C^\infty_\#(Y_1,\ldots,C^\infty_\#(Y_i),\ldots))^3$$ and $${\frak v_i}\in C^\infty_0(D,C^\infty_\#(Y_1,\ldots,C^\infty_\# (Y_i),\ldots))$$ for $$i=1,\ldots,n$$. Let the test function $$v$$ in (2.5) be v(x)=v0(x)+∑i=1nεi(vi(x,xε1,…,xεi)+∇vi(x,xε1,…,xεi)). We have ∫D[aε(x)curluε(x)⋅(curlv0(x)+∑i=1nεicurlxvi(x,xε1,…,xεi)+∑i=1n∑j=1iεiεjcurlyjvi(x,xε1,…,xεi) + ∑i=1nεicurl∇vi(x,xε1,…,xεi)) +bε(x)uε(x)⋅(v0(x)+∑i=1nεivi(x,xε1,…,xεi) +∑i=1nεi∇xvi(x,xε1,…,xεi)+∑i=1n∑j=1iεiεj∇yjvi(x,xε1,…,xεi))]dx =∫Df(x)⋅(v0(x)+∑i=1nεivi(x,xε1,…,xεi) +∑i=1nεi∇xvi(x,xε1,…,xεn)+∑i=1n∑j=1iεiεj∇yjvi(x,xε1,…,xεi)). Using multiscale convergence and the scale separation (2.2), letting $$\varepsilon$$ go to 0, we have ∫D∫Y[a(x,y)(curlu0+∑i=1ncurlyiui)⋅(curlv0+∑i=1ncurlyivi) +b(x,y)(u0+∑i=1n∇yiui)⋅(v0+∑i=1n∇yivi)]dydx =∫Df(x)⋅v0(x)dx+∫D∫Yf(x)⋅∑i=1n∇yivi(x,y1,…,yi)dydx =∫Df(x)⋅v0(x)dx. Using a density argument, we have (2.10). □ Proposition 2.5 The bilinear form $$B:{\bf V}\times{\bf V}\to \mathbb{R}$$ is coercive and bounded, i.e., there are positive constants $$C^$$ and $$C_$$ so that B(u,v)≤C∗\|\|\|u\|\|\|\|\|\|v\|\|\|andC∗\|\|\|u\|\|\|\|\|\|u\|\|\|≤B(u,u) (2.11) for all $$\boldsymbol{u},\boldsymbol{v}\in {\bf V}$$. Problem (2.10) thus has a unique solution. The convergence relations (2.8) and (2.9) hold for the whole sequence $$\{u^{\varepsilon}\}_\varepsilon$$. Proof. It is easy to see that there is a positive constant $$C^$$ such that B(u,v)≤C∗\|\|\|u\|\|\|\|\|\|v\|\|\|. Now we show that $$B$$ is coercive. We have from (2.1), B(u,u) ≥c∗∫D∫Y(\|curlu0+∑i=1ncurlyiui\|2+\|u0+∑i=1n∇yiui\|2)dydx ≥c∫D∫Y(\|curlu0\|2+∑i=1n\|curlyiui\|2+\|u0\|2+\|∇yiui\|2)dydx≥c\|\|\|u\|\|\|2. We then get the conclusion from Lax–Milgram lemma. □ 3. FE discretization Let $$D$$ be a polygonal domain in $$\mathbb{R}^3$$. We consider a hierarchy of simplices $$\mathcal{T}^l$$ ($$l=0,1,\ldots$$), where $$\mathcal{T}^{l+1}$$ is obtained from $$\mathcal{T}^l$$ by dividing each simplex in $$\mathcal{T}^l$$ into eight tedrahedra. The mesh size of $$\mathcal{T}^l$$ is $$h_l={\mathcal O}(2^{-l})$$. For each tedrahedron $$T$$, we consider the edge FE space R(T)={v: v=α+β×x, α,β∈R3}. When $$d=2$$, $$\mathcal{T}^{l+1}$$ is obtained from $$\mathcal{T}^l$$ by dividing each simplex in $$\mathcal{T}^l$$ into four congruent triangles. For each triangle $$T$$, we consider the edge FE space R(T)={v: v=(α1α2)+β(x2−x1)}, where $$\alpha_1,\alpha_2$$ and $$\beta$$ are constants. Alternatively, if the domain can be partitioned into a set of cubes, we can use edge FE on a cubic mesh instead (see Monk, 2003). We denote by $$\mathcal{P}_1(T)$$ the set of linear polynomials in each simplex $$T$$. In the following, we present the analysis for the three-dimensional case only; the two-dimensional case is similar. For the cube $$Y$$, we partition it into a hierarchy of simplices $$\mathcal{T}^l_\#,$$ which are distributed periodically. We consider the FE spaces Wl ={v∈H0(curl,D), v\|T∈R(T) ∀T∈Tl},Vl ={v∈H1(D), v\|T∈P1(T) ∀T∈Tl},W#l ={v∈H#(curl,Y), v\|T∈R(T) ∀T∈T#l} and V#l={v∈H#1(Y), v\|T∈P1(T) ∀T∈T#l}. For $$d=2,3$$, we have the following estimates (see Ciarlet, 1978; Monk, 2003): infvl∈Wl‖v−vl‖H(curl,D)≤chls(‖v‖Hs(D)d+‖curlv‖Hs(D)d) for all $$v\in H_0({\rm curl\,}, D)\bigcap H^s({\rm curl\,},D)$$; infvl∈W#l‖v−vl‖H#(curl,Y)≤chls(‖v‖Hs(Y)d+‖curlv‖Hs(Y)d) for all $$v\in H_\#({\rm curl\,}, Y)\bigcap H^s({\rm curl\,}, Y)$$; infvl∈Vl‖v−vl‖L2(D)≤chls‖v‖Hs(D) for all $$v\in H^s(D)$$; infvl∈V#l‖v−vl‖L2(Y)≤chls‖v‖Hs(Y) for all $$v\in H^s_\#(Y)$$ and infvl∈V#l‖v−vl‖H#1(Y)≤chls‖v‖H1+s(Y) for all $$v\in H^1_\#(Y)\bigcap H^{1+s}(Y)$$. 3.1 Full tensor product FEs As $$L^2(D\times{\bf Y}_{i-1},\tilde H_\#({\rm curl\,},Y_i))\cong L^2(D)\otimes L^2(Y_1)\otimes\cdots\otimes L^2(Y_{i-1})\otimes \tilde H_\#({\rm curl\,},Y_i),$$ we use the tensor product FE space Wil=Vl⊗V#l⊗⋯⊗V#l⏟i−1 times ⊗W#l to approximate $$u_i$$. Similarly, as $${\frak u_i}\in L^2(D\times{\bf Y}_{i-1},H^1_\#(Y))$$, we use the FE space Vil=Vl⊗V#l⊗⋯⊗V#l⏟i times to approximate $${\frak u_i}$$. We define the space Vl=Wl×W1l×⋯×Wnl×V1l×⋯×Vnl. The full tensor product FE approximating problem is, find $$\boldsymbol{u}^L\in{\bf V}^L$$ so that B(uL,vL)=∫Df(x)⋅v0L(x)dx ∀vL=(v0L,{viL},viL)∈VL. (3.1) To get an error estimate for this FE approximating problem, we define the following regularity spaces for $${\frak u_i}$$ and $$u_i$$. For the functions $$u_i$$, we define the regularity space $$\mathcal{H}_i$$ of functions $$w$$ in $$L^2(D\times Y_1\times\cdots\times Y_{i-1},H^1_\#({\rm curl\,},Y_i))$$ such that for all $$k=1,2,3$$, ∂w∂xk∈L2(D×Y1×⋯×Yi−1,H~#(curl,Yi)) and for all $$j=1,\ldots,i-1$$ and $$k=1,2,3$$, ∂w∂(yj)k∈L2(D×Y1×⋯×Yi−1,H~#(curl,Yi)). In other words, for all $$w\in \mathcal{H}_i$$, $$w$$ belongs to $$L^2(D\times Y_1\times\cdots\times Y_{i-1},H^1_\#({\rm curl\,},Y_i))$$, $$L^2(Y_1\times\cdots\times Y_{i-1},H^1(D,\tilde H_\#({\rm curl\,},Y_i)))$$ and $$L^2(D\times\prod_{k<i,k\ne j}Y_k,H^1_\#(Y_j,\tilde H_\#({\rm curl\,},Y_i)))$$ for $$j=1,\ldots,i-1$$. For $$0<s<1$$, we define the space $$\mathcal{H}^s_i$$ by interpolation. It consists of functions $$w$$ such that $$w$$ belongs to $$L^2(D\times Y_1\times\cdots\times Y_{i-1},H^s_\#({\rm curl\,},Y_i))$$, $$L^2(Y_1\times\cdots\times Y_{i-1},H^s(D,\tilde H_\#({\rm curl\,},Y_i)))$$ and $$L^2(D\times\prod_{k<i,k\ne j}Y_k,H^s_\#(Y_j,\tilde H_\#({\rm curl\,},Y_i)))$$. We equip $$\mathcal{H}_i^s$$ with the norm ‖w‖His =‖w‖L2(D×Y1×⋯×Yi−1,H#s(curl,Yi))+‖w‖L2(Y1×⋯×Yi−1,Hs(D,H~#(curl,Yi))) +∑j=1i−1‖w‖L2(D×∏k<i,k≠j,H#s(Yj,H~#(curl,Yi))). We then have the following lemma. Lemma 3.1 For $$w\in \mathcal{H}_i^s$$, infwl∈Wil‖w−wl‖L2(D×Y1×⋯×Yi−1,H~#(curl,Yi))≤chls‖w‖His. The proof of this lemma is similar to that for full tensor product FEs in Hoang & Schwab (2004/05) and Bungartz & Griebel (2004), using orthogonal projection. We refer to Hoang & Schwab (2004/05) and Bungartz & Griebel (2004) for details. We define $${\frak H_i}^s$$ as the space of functions $$w\in L^2(D\times Y_1\times\cdots\times Y_{i-1},H^{1+s}_\#(Y_i))$$ such that $$w\in L^2(Y_1\times\cdots\times Y_{i-1},H^s(D,H^1_\#(Y_i)))$$ and for all $$j=1,\ldots,i-1$$, $$w\in L^2(D\times\prod_{k<i,k\ne j}Y_k,H^s_\#(Y_{j},H^1_\#(Y_i)))$$. We then define the norm ‖w‖His =‖w‖L2(D×Y1×⋯×Yi−1,H#1+s(Yi))+‖w‖L2(Y1×⋯×Yi−1,Hs(D,H1(Yi))) +∑j=1i−1‖w‖L2(D×∏k<i,k≠jYk,Hs(Yj,H1(Yi))). We have the following result. Lemma 3.2 For $$w\in {\frak H_i}^s$$, infwl∈Vil‖w−wl‖L2(D×Y1×⋯×Yi−1,H#1(Yi))≤chls‖w‖His. We then define the regularity space Hs=Hs(curl,D)×H1s×⋯×Hns×H1s×⋯×Hns with the norm ‖w‖Hs=‖w0‖Hs(curl,D)+∑i=1n‖wi‖His+∑i=1n‖wi‖His for $$\boldsymbol{w}=(w_0,\{w_i\},\{\frak w_i\})\in \boldsymbol{\mathcal{H}}^s$$. We have the following approximation result. Lemma 3.3 For $$\boldsymbol{w}\in \boldsymbol{\mathcal{H}}^s$$ infwl∈Vl‖w−wl‖V≤chls‖w‖Hs. From the boundedness and coerciveness conditions (2.11), using Cea’s lemma, we deduce the following result. Proposition 3.4 If $$\boldsymbol{u}\in \boldsymbol{\mathcal{H}}^s$$, for the full tensor product FE approximating problem (3.1) we have the error estimate ‖u−uL‖V≤chLs‖u‖Hs. (3.2) 3.2 Sparse tensor product FEs We define the following orthogonal projection: Pl0:L2(D)→Vl,P#l0:L2(Y)→V#l with the convention $$P^{-10}=0$$, $$P^{-10}_\#=0$$. We define the following detail spaces: Vl=(Pl0−P(l−1)0)Vl, V#l=(P#l0−P#(l−1)0)Vl. Since Vl=⨁0≤i≤lViandV#l=⨁0≤i≤lV#i, the full tensor product spaces $$W_i^L$$ and $$V_i^L$$ are defined as WiL=(⨁0≤l0,…,li−1≤LVl0⊗V#l1⊗⋯⊗V#li−1)⊗W#L and ViL=(⨁0≤l0,…,li−1≤LVl0⊗V#l1⊗⋯⊗V#li−1)⊗V#L. We then define the sparse tensor product FE spaces as W^iL=⨁l0+⋯+li−1≤LVl0⊗V#l1⊗⋯⊗V#li−1⊗W#L−(l0+⋯+li−1) and V^iL=⨁l0+⋯+li−1≤LVl0⊗V#l1⊗⋯⊗V#li−1⊗V#L−(l0+⋯+li−1). The function $$\boldsymbol{u}$$ is approximated by the space V^L=WL⊗W^1L⊗⋯⊗W^nL⊗V^1L⊗⋯⊗V^nL. The sparse tensor product FE approximating problem is, find $$\widehat {\bf{u}}^L\in \hat{\bf V}^L$$ such that B(u^L,v^L)=∫Df(x)⋅v^0L(x)dx ∀v^L=(v^0L,{v^iL},{v^iL})∈V^L. (3.3) From the coerciveness and boundedness conditions in (2.11), using Cea’s lemma we deduce the error estimate for the sparse tensor product approximating problem ‖u−u^L‖V≤cinfv^L∈VL‖u−v^L‖V. To quantify the error estimate, we use the following regularity spaces. We define $$\hat{\mathcal{H}}_i$$ as the space of functions $$w\in L^2(D\times Y_1\times\cdots\times Y_{i-1},H^1_\#({\rm curl\,},Y_i)),$$ which are periodic with respect to $$y_j$$ with the period being $$Y_j$$ ($$j=1,\ldots,i-1$$) such that for any $$\alpha_0,\alpha_1,\ldots,\alpha_{i-1}\in \mathbb{N}_0^d$$ with $$\|\alpha_k\|\le 1$$ for $$k=0,\ldots,i-1$$, ∂\|α0\|+\|α1\|+⋯+\|αi−1\|∂xα0∂y1α1⋯∂yi−1αi−1w∈L2(D×Y1×⋯×Yi−1,H#1(curl,Yi)). We equip $$\hat{\mathcal{H}}_i$$ with the norm ‖w‖H^i=∑αj∈Rd,\|αj\|≤10≤j≤i−1‖∂\|α0\|+\|α1\|+⋯+\|αi−1\|∂xα0∂y1α1⋯∂yi−1αi−1w‖L2(D×Y1×⋯×Yi−1,H#1(curl,Yi)). We can write $$\hat{\mathcal{H}}_i$$ as $$H^1(D,H^1_\#(Y_1,\ldots,H^1_\#(Y_{i-1},H^1_\#({\rm curl\,},Y_i)),\ldots))$$. By interpolation, we define $$\hat{\mathcal{H}}_i^s=H^s(D,H^s_\#(Y_1,\ldots,H^s_\#(Y_{i-1},H^s_\#({\rm curl\,},Y_i)),\ldots))$$ for $$0<s<1$$. We define $$\hat{\frak H}_i$$ as the space of functions $$w\in L^2(D\times Y_1\times\cdots\times Y_{i-1},H^2_\#(Y_i))$$ that are periodic with respect to $$y_j$$ with the period being $$Y_j$$ for $$j=1,\ldots,i-1$$ such that $$\alpha_0,\alpha_1,\ldots,\alpha_{i-1}\in \mathbb{N}_0^d$$ with $$\|\alpha_k\|\le 1$$ for $$k=0,\ldots,i-1$$, ∂\|α0\|+\|α1\|+⋯+\|αi−1\|∂xα0∂y1α1⋯∂yi−1αi−1w∈L2(D×Y1×⋯×Yi−1,H#2(Yi)). The space $$\hat{\frak H}_i$$ is equipped with the norm ‖w‖H^i=∑αj∈Rd,\|αj\|≤10≤j≤i−1‖∂\|α0\|+\|α1\|+⋯+\|αi−1\|∂xα0∂y1α1⋯∂yi−1αi−1w‖L2(D×Y1×⋯×Yi−1,H#2(Yi)). We can write $$\hat{\frak H}_i$$ as $$H^1(D,H^1_\#(Y_1,\ldots,H^1_\#(Y_{i-1},H^2_\#(Y_i))))$$. By interpolation, we define the space $$\hat{\frak H}_i^s:=H^s(D,H^s(Y_1,\ldots,H^s(Y_{i-1},H^{1+s}_\#(Y_i))))$$. The regularity space $$\hat{\boldsymbol{\mathcal{H}}}^s$$ is defined as H^s=Hs(curl,D)×H^1s×⋯H^ns×H^1s×⋯×H^ns. Lemmas 3.5 and 3.6 present the approximating properties of functions in $$\hat{\mathcal{H}}_i^s$$ and $${\hat{\frak H}_i^s}$$. The proofs follow from those for sparse tensor products in Hoang & Schwab (2004/05) and Bungartz & Griebel (2004). Lemma 3.5 For $$w\in \hat{\mathcal{H}}_i^s$$, infwL∈W^iL‖w−wL‖L2(D×Y1×⋯×Yi−1,H#(curl,Yi))≤cLi/2hLs‖w‖H^is. Similarly we have the following lemma. Lemma 3.6 For $$w\in \hat{\frak H}_i^s$$, infwL∈V^iL‖w−wL‖L2(D×Y1×⋯×Yi−1,H#1(Yi))≤cLi/2hLs‖w‖H^is. From these lemmas we deduce the following result. Lemma 3.7 For $$\boldsymbol{w}\in \hat{\boldsymbol{\mathcal{H}}}^s$$, infwL∈V^L‖w−wL‖V≤cLn/2hLs‖w‖H^s. From this we deduce the following error estimate for the sparse tensor product FE problem (3.3). Proposition 3.8 If the solution $$\boldsymbol{u}$$ of problem (2.10) belongs to $$\hat{\boldsymbol{\mathcal{H}}}^s$$ then ‖u−u^‖V≤cLn/2hLs‖u‖H^s. Remark 3.9 The dimension of the full tensor product FE space $${\bf V}^L$$ is $${\mathcal O}(2^{dnL}),$$ which is very large when $$L$$ is large. The dimension of the sparse tensor product FE space $$\hat{\bf V}^L$$ is $${\mathcal O}(L^n2^{dL}),$$ which is essentially equal to the number of degrees of freedom for solving a problem in $$\mathbb{R}^d$$ obtaining the same level of accuracy. 4. Convergence in physical variables We employ the FE solutions for the multiscale homogenized Maxwell-type equation (2.10) in the previous section to derive numerical correctors for the solution $$u^{\varepsilon}$$ of the multiscale problem (2.4). In the two-scale case, we derive the homogenization error explicitly in terms of $$\varepsilon$$ so that an error in terms of the microscopic scale $$\varepsilon$$ and the mesh size is obtained for the numerical corrector. We consider the general case, where the solution $$u^0$$ of the homogenized problem belongs to the space $$H^s({\rm curl\,},D)$$ for $$0<s\le 1$$, thus generalizing the standard homogenization rate of convergence $$\varepsilon^{1/2}$$ for elliptic problems (see e.g, Bensoussan et al., 1978; Jikov et al., 1994). This is a new result in homogenization theory. We present it for two-scale Maxwell-type equations, but the procedure works verbatim for two-scale elliptic and elasticity problems, where the solutions of the homogenized problems belong to $$H^{1+s}(D)$$. We present this section for the case $$d=3$$; the case $$d=2$$ is similar. 4.1 Two-scale problems For the two-scale case, we denote the function $$a(x,\boldsymbol{y})$$ by $$a(x,y)$$. The two-scale homogenized equation becomes ∫D∫Y[a(x,y)(curlu0+curlyu1)⋅(curlv0+curlyv1)+b(x,y)(u0+∇yu1)⋅(v0+∇yv1)]dydx =∫Df(x)⋅v0(x)dx. We first let $$v_0=0$$, $$v_1=0$$ and deduce that ∫D∫Yb(x,y)(u0+∇yu1)⋅∇yv1dydx=0. For each $$r=1,2,3$$, let $$w^r(x,\cdot)\in L^2(D, H^1_\#(Y)/\mathbb{R})$$ be the solution of the problem ∫D∫Yb(x,y)(er+∇ywr)⋅∇yψdydx=0 ∀ψ∈L2(D,H#1(Y)/R), (4.1) where $$e_r$$ is the vector in $$\mathbb{R}^3$$ with all the components being 0, except the $$r$$th component, which equals 1. This is the standard cell problem in elliptic homogenization. From this we have u1(x,y)=wr(x,y)u0r(x). (4.2) Therefore, ∫D∫Yb(x,y)(u0+∇yu1)⋅v0dxdy=∫Db0(x)u0(x)⋅v0(x)dx, where the positive-definite matrix $$b^0(x)$$ is defined as bij0(x)=∫Yb(x,y)(ej+∇wj(x,y))⋅(ei+∇ywi(x,y))dy, (4.3) which is the usual homogenized coefficient for elliptic problems with the two-scale coefficient matrix $$b^\varepsilon$$. Let $$v_0=0$$ and $${\frak v_1}=0$$. We have ∫D∫Ya(x,y)(curlu0+curlyu1)⋅curlyv1dydx=0 for all $$v_1\in L^2(D, \tilde H_\#({\rm curl\,},Y))$$. For each $$r=1,2,3$$, let $$N^r\in L^2(D,\tilde H_\#({\rm curl\,},Y))$$ be the solution of ∫D∫Ya(x,y)(er+curlyNr)⋅curlyvdydx=0 (4.4) for all $$v\in L^2(D, \tilde H_\#({\rm curl\,},Y))$$. We have u1=(curlu0(x))rNr(x,y). (4.5) The homogenized coefficient $$a^0$$ is determined by aij0(x)=∫Ya(x,y)ip(ejp+(curlyNj)p)dy=∫Ya(x,y)(ej+curlyNj)⋅(ei+curlyNi)dy. (4.6) We have ∫D∫Ya(x,y)(curlu0+curlyu1)⋅curlv0dxdy=∫Da0(x)curlu0(x)⋅curlv0(x)dx. The homogenized problem is ∫D[a0(x)curlu0(x)⋅curlv0(x)+b0(x)u0(x)⋅v0(x)]dx=∫Df(x)⋅v0(x)dx ∀v0∈H0(curl,D). (4.7) Following the procedure for deriving the homogenization error (Bensoussan et al., 1978; Jikov et al., 1994), we have the following homogenization error estimate. Theorem 4.1 Assume that $$a\in C(\bar D, C(\bar Y))^{3\times 3}$$, $$u_0\in H^1({\rm curl\,};D)$$, $$N^r\in C^1(\bar D,C(\bar Y))^3$$, $${\rm curl}_yN^r\in C^1(\bar D,C(\bar Y))^3$$, $$w^r\in C^1(\bar D,C^1(\bar Y))$$ for all $$r=1,2,3$$, then2 ‖uε−[u0+∇yu1(⋅,⋅ε)]‖L2(D)3≤cε1/2 and ‖curluε−[curlu0+curlyu1(⋅,⋅ε)]‖L2(D)3≤cε1/2. The proof of this theorem uses the functions $$G_r$$ and $$g_r$$ defined in (A.2) and (A.3) below. For $$u_0\in H^s({\rm curl\,},D)$$ when $$0<s<1$$, we have the following homogenization error estimate. Theorem 4.2 Assume that $$a\in C(\bar D,C(\bar Y))^{3\times 3}$$, $$u_0\in H^s({\rm curl\,},D)$$, $$N^r\in C^1(\bar D, C(\bar Y))^3$$, $${\rm curl}_yN^r\in C^1(\bar D,C(\bar Y))^3$$ and $$w^r\in C^1(\bar D,C^1(\bar Y))$$ for all $$r=1,2,3$$, then ‖uε−[u0+∇yu1(⋅,⋅ε)‖L2(D)3≤cεs/(1+s) and ‖curluε−[curlu0+curlyu1(⋅,⋅ε)]‖L2(D)3≤cεs/(1+s). We present the proof of this theorem in Appendix A. To employ the FE solutions to construct numerical correctors for $$u^{\varepsilon}$$, we define the following operator: Uε(Φ)(x)=∫YΦ(ε[xε]+εz,{xε})dz. (4.8) Let $$D^\varepsilon$$ be a $$2\varepsilon$$ neighbourhood of $$D$$. Regarding $${\it{\Phi}}$$ as zero when $$x$$ is outside $$D$$, we have ∫DεUε(Φ)(x)dx=∫D∫YΦ(x,y)dxdy. (4.9) The proof of (4.9) may be found in Cioranescu et al. (2008). We have the following result. Lemma 4.3 Assume that for $$r=1,2,3$$, $${\rm curl}_yN^r(x,y)\in C^1(\bar D, C(\bar Y))^3$$ and $$u_0\in H^s({\rm curl\,},D)$$, then ‖curlyu1(⋅,⋅ε)−Uε(curlyu1)‖L2(D)3≤cεs. We prove this lemma in Appendix B. We then have the following result. Theorem 4.4 Assume that $$a\in C(\bar D,C(\bar Y))^{3\times 3}$$, $$u_0\in H^s({\rm curl\,},D)$$, $$N^r\in C^1(\bar D,C(\bar Y))^3$$, $${\rm curl}_yN^r\in C^1(\bar D,C(\bar Y))^3$$ and $$w^r\in C^1(\bar D,C^1(\bar Y))$$ for all $$r=1,2,3$$. Then for the full tensor product FE solution $$(u_0^L,u_1^L,{\frak u_1}^L)$$ we have ‖uε−u0L−Uε(∇uu1L)‖L2(D)3≤c(εs/(1+s)+hLs) and ‖curluε−curlu0L−Uε(curlyu1L)‖L2(D)3≤c(εs/(1+s)+hLs). Proof. From Lemma 4.3, we have ‖curluε−curlu0L−Uε(curlyu1L)‖L2(D)3 ≤‖curluε−curlu0−curlyu1(⋅,⋅ε)‖L2(D)3 +‖curlu0−curlu0L‖L2(D)3+‖curlyu1(⋅,⋅ε)−Uε(curlyu1)‖L2(D)3 +‖Uε(curlyu1)−Uε(curlyu1L)‖L2(D)3. Using the fact that $$(\mathcal{U}^\varepsilon({\it{\Phi}}))^2\le \mathcal{U}({\it{\Phi}}^2)$$ and (4.9), we have ‖Uε(curlyu1)−Uε(curlyu1L)‖L2(D)3≤‖curlyu1−curlyu1L‖L2(D×Y)3≤chLs. This together with (3.2), Theorem 4.2 and Lemma 4.3 gives ‖curluε−curlu0L−Uε(curlyu1L)‖L2(D)3≤c(εs/(1+s)+hLs). Similarly, we have ‖uε−u0L−Uε(∇uu1L)‖L2(D)3≤c(εs/(1+s)+hLs). □ For the sparse tensor product FE approximation, we have the following result. Theorem 4.5 Assume that $$a\in C(\bar D,C(\bar Y))^{3\times 3}$$, $$u_0\in H^s({\rm curl\,},D)$$, $$N^r\in C^1(\bar D,C(\bar Y))^3$$, $${\rm curl}_yN^r\in C^1(\bar D,C(\bar Y))^3$$ and $$w^r\in C^1(\bar D,C^1(\bar Y))$$ for all $$r=1,2,3$$. Then for the sparse tensor product FE solution $$(\hat u_0^L,\hat u_1^L,\hat{\frak u}_1^L)$$ we have ‖uε−u^0L−Uε(∇yu^1L)‖L2(D)3≤c(εs/(1+s)+Ln/2hLs) and ‖curluε−curlu^0L−Uε(curlyu^1L)‖L2(D)3≤c(εs/(1+s)+Ln/2hLs). 4.2 Multiscale problems For multiscale problems, we do not have an explicit homogenization rate of convergence. However, for the case where $$\varepsilon_i/\varepsilon_{i+1}$$ is an integer for all $$i=1,\ldots,n-1$$ we can derive a corrector for the solution $$u^{\varepsilon}$$ of the multiscale problem from the FE solutions of the multiscale homogenized problem. For each function $$\phi\in L^1(D),$$ which is understood as 0 outside $$D$$, we define a function in $$L^1(D\times{\bf Y})$$: Tnε(ϕ)(x,y)=ϕ(ε1[xε1]+ε2[y1ε2/ε1]+⋯+εn[yn−1εn/εn−1]+εnyn). Letting $$D^{\varepsilon_1}$$ be the $$2\varepsilon_1$$ neighbourhood of $$D$$, we have ∫Dϕdx=∫Dε1∫Y1⋯∫YnTnε(ϕ)dyn⋯dy1dx (4.10) for all $$\phi \in L^1(D)$$. If a sequence $$\{\phi^\varepsilon\}_\varepsilon$$$$(n+1)$$-scale converges to $$\phi(x,y_1,\ldots,y_n)$$ then Tnε(ϕ)⇀ϕ(x,y1,…,yn) in $$L^2(D\times Y_1\times\ldots\times Y_n)$$. Thus, when $$\varepsilon\to 0$$, Tnε(curluε)⇀curlu0+curly1u1+⋯+curlynun (4.11) and Tnε(uε)⇀u0+∇y1u1+⋯+∇ynun (4.12) in $$L^2(D\times{\bf Y})^3$$. To deduce an approximation of $$u^{\varepsilon}$$ in $$H({\rm curl\,},D)$$ in terms of the FE solution, we use the operator $$\mathcal{U}_n^\varepsilon$$ , which is defined as Unε(Φ)(x) =∫Y1⋯∫YnΦ(ε1[xε1]+ε1t1,ε2ε1[ε1ε2{xε1}] +ε2ε1t2,⋯, εnεn−1[εn−1εn{xεn−1}]+εnεn−1tn,{xεn})dtn⋯dt1 for all functions $${\it{\Phi}}\in L^1(D\times{\bf Y})$$. For each function $${\it{\Phi}}\in L^1(D\times{\bf Y})$$ we have ∫Dε1Unε(Φ)dx=∫D∫YΦ(x,y)dydx. (4.13) The proofs for these facts may be found in Cioranescu et al. (2008). We then have the following corrector result. Proposition 4.6 The solution $$u^{\varepsilon}$$ of problem (2.5) and the solution $$(u_0,\{u_i\}\,\{{\frak u_i}\})$$ of problem (2.10) satisfies limε→0‖uε−[u0+Unε(∇y1u1)+⋯+Unε(∇ynun)‖L2(D)3=0 (4.14) and limε→0‖curluε−[curlu0+Unε(curly1u1)+⋯+Unε(curlynun)]‖L2(D)3=0. (4.15) Proof. We consider the expression ∫D∫Y[Tnε(aε)(Tnε(curluε)−(curlu0+curly1u1+⋯+curlynun)) ⋅(Tnε(curluε)−(curlu0+curly1u1+⋯+curlynun)) +Tnε(bε)(Tnε(uε)−(u0+∇y1u1+⋯+∇ynun))⋅(Tnε(uε)−(u0+∇y1u1+⋯+∇ynun))dydx. Using (2.5), (2.10), (4.10), (4.11) and (4.12), we deduce that this expression converges to 0. From (2.1) we have limε→0‖Tnε(curluε)−(curlu0+curly1u1+⋯+curlynun)‖L2(D×Y1×…×Yn)3=0 and limε→0‖Tnε(uε)−(u0+∇y1u1+⋯+∇ynun)‖L2(D×Y1×⋯×Yn)3=0. From (4.13) and the fact that $$\mathcal{U}^\varepsilon_n({\it{\Phi}})^2\le \mathcal{U}^\varepsilon_n({\it{\Phi}}^2)$$, we have ∫D\|Unε(Tnε(curluε)−(curlu0+curly1u1+⋯+curlynun)(x)\|2dx ≤∫DUnε(\|Tnε(curluε)−(curlu0+curly1u1+⋯+curlynun\|2)(x)\|dx ≤∫D∫Y\|Tnε(curluε)−(curlu0+curly1u1+⋯+curlynun\|2)dydx, which converges to 0 when $$\varepsilon\to 0$$. Using $$\mathcal{U}^\varepsilon_n(\mathcal{T}^\varepsilon_n({\it{\Phi}}))={\it{\Phi}}$$, we get (4.15). We derive (4.14) similarly. □ We then deduce the numerical corrector result. Theorem 4.7 For the full tensor product FE approximation solution $$\boldsymbol{u}^L=(u_0^L,\{u_i^L\},\{{\frak u_i}^L\})$$ in (3.1), we have limε→0L→∞‖uε−[u0L+Unε(∇y1u1L)+⋯+Unε(∇ynunL)]‖L2(D)3=0 (4.16) and limε→0L→∞‖curluε−[curlu0L+Unε(curly1u1L)+⋯+Unε(curlynunL)]‖L2(D)3=0. (4.17) Proof. We note that ‖Unε(curly1u1+⋯+curlynun)−Unε(curly1u1L+⋯+curlynunL)‖L2(D)3 ≤∫DUnε(\|(curly1u1+⋯+curlynun)−(curly1u1L+⋯+curlynunL)\|2)(x)dx ≤∫D∫Y\|(curly1u1+⋯+curlynun)−(curly1u1L+⋯+curlynunL)\|2dydx, which converges to 0 when $$L\to \infty$$. From this and (4.15), we get (4.17). We obtain (4.16) in the same way. □ Remark 4.8 As $${\bf V}^{\lceil L/n\rceil}\subset\hat{\bf V}^L$$, the result in Theorem 4.7 also holds for the sparse tensor product FE solution $$\widehat {\bf{u}}^L$$. Since we do not have an explicit homogenization error for problems with more than two scales, we do not distinguish the two cases of full and sparse tensor FE approximations. 5. Regularity of $$\boldsymbol{N^r}$$, $$\boldsymbol{w^r}$$ and $$\boldsymbol{u_0}$$ We show in this section that the regularity requirements for obtaining the sparse tensor product FE error estimate and the homogenization error estimate in the previous sections are achievable. We present the results for the two-scale case in detail. The multiscale case is similar; we summarize it in Remark 5.7. We first prove the following lemma. Lemma 5.1 Let $$\psi\in H_\#({\rm curl\,},Y)\bigcap H_\#({\rm div},Y)$$. Assume further that $$\int_Y\psi(y)\,{\rm d}y=0$$. Then $$\psi\in H^1_\#(Y)^3$$ and ‖ψ‖H1(Y)3≤c(‖curlyψ‖L2(Y)3+‖divyψ‖L2(Y)). Proof. Let $$\omega\subset\mathbb{R}^3$$ be a smooth domain such that $$\omega\supset Y$$. Let $$\eta\in \mathcal{D}(\omega)$$ be such that $$\eta(y)=1$$ when $$y\in Y$$. We have curly(ηψ)=ηcurlyψ+∇yη×ψ∈L2(ω)3 and divy(ηψ)=∇yη⋅ψ+ηdivyψ∈L2(ω)3. Together with the zero boundary condition, we conclude that $$\eta\psi\in H^1(\omega)^3$$ so $$\psi\in H^1(Y)^3$$. We note that ∫Y(divyψ(y)2+\|curlyψ(y)\|2)dy=∑i,j=13∫Y(∂ψi∂yj)2+∑i≠j∫Y∂ψi∂yi∂ψj∂yjdy−∑i≠j∫Y∂ψj∂yi∂ψi∂yjdy. Assume that $$\psi$$ is a smooth periodic function. We have ∫Y∂ψi∂yi∂ψj∂yjdy=∫Y[∂∂yi(ψi∂ψj∂yj)−ψi∂2ψj∂yi∂yj]dy=−∫Yψi∂2ψj∂yi∂yjdy as $$\psi$$ is periodic. Similarly, we have ∫Y∂ψi∂yj∂ψj∂yidy=∫Y[∂∂yj(ψi∂ψj∂yi)−ψi∂2ψj∂yj∂yi]dy=−∫Yψi∂2ψj∂yj∂yidy. Thus, ∫Y∂ψi∂yi∂ψj∂yjdy=∫Y∂ψi∂yj∂ψj∂yidy. Therefore, ‖∇yψ‖L2(Y)32=‖divyψ‖L2(Y)2+‖curlyψ‖L2(Y)32. Using a density argument, this holds for all $$\psi\in H^1_\#(Y)^3$$. As $$\int_Y\psi(y)\,{\rm d}y=0$$, from the Poincaré inequality we deduce ‖ψ‖H1(Y)3≤c(‖divyψ‖L2(Y)+‖curlyψ‖L2(Y)3). □ Lemma 5.2 Let $$\alpha\in C^1_\#(\bar Y)^{3\times 3}$$ be uniformly bounded, positive definite and symmetric for all $$y\in \bar Y$$. Let $$F\in L^2(Y)$$, extending periodically to $$\mathbb{R}^3$$. Let $$\psi\in H^1_\#(Y)^3$$ satisfy the equation curly(α(y)curlyψ(y))=F(y). Then $${\rm curl}_y\psi\in H^1_\#(Y)^3$$ and ‖curlyψ‖H1(Y)3≤c(‖F‖L2(Y)3+‖ψ‖H1(Y)3). Proof. Let $$\omega\supset Y$$ be a smooth domain. Let $$\eta\in \mathcal{D}(\omega)$$ be such that $$\eta(y)=1$$ for $$y\in Y$$. We have curly(αcurly(ηψ)) =curly(αηcurlyψ)+curly(α∇yη×ψ) =ηcurly(αcurlyψ)+∇yη×(αcurlyψ)+curly(α∇yη×ψ). Let $$U=\alpha{\rm curl}_y(\eta\psi)$$. We have ‖curlyU‖L2(ω)3≤c(‖F‖L2(ω)+‖ψ‖H1(ω)3)≤c(‖F‖L2(Y)3+‖ψ‖H1(Y)3). Further, ‖U‖L2(ω)3=‖α(∇yη×ψ+ηcurlyψ)‖\|L2(ω)3≤c‖ψ‖H1(ω)3≤c‖ψ‖H1(Y)3. As $$\eta\in\mathcal{D}(\omega)$$, $$U$$ has compact support in $$\omega$$ so $$U$$ belongs to $$H_0({\rm curl\,},\omega)$$. Thus, we can write U=z+∇Φ, where $$z\in H^1_0(\omega)^3$$ and $${\it{\Phi}}\in H^1_0(\omega)$$ satisfy ‖z‖H1(ω)3≤c‖U‖H(curl,ω) and ‖Φ‖H1(ω)≤c‖U‖H(curl,ω). From $${\rm div}_y(\alpha^{-1}U)=0$$ we deduce that divy(α−1∇Φ)=−divy(α−1z)∈L2(ω). Since $$\alpha\in C^1(\bar\omega)^{3\times 3}$$ and is uniformly bounded and positive definite, $$\alpha^{-1}\in C^1(\bar\omega)^{3\times 3}$$ and is uniformly positive definite. Therefore, $${\it{\Phi}}\in H^2(\omega)$$ and satisfies ‖Φ‖H2(ω)≤c‖z‖H1(ω)3≤c‖U‖H(curl,ω). Thus, $$U\in H^1(\omega)^3$$ and $$\\|U\\|_{H^1(\omega)^3}\le c\\|U\\|_{H({\rm curl\,},\omega)}\le c(\\|F\\|_{L^2(Y)^3}+\\|\psi\\|_{H^1(Y)^3})$$. From $${\rm curl}_y(\eta\psi)=\alpha^{-1}U$$, we deduce that $${\rm curl}_y(\eta\psi)\in H^1(\omega)^3$$ so $${\rm curl}_y\psi\in H^1(Y)^3$$ and ‖curlyψ‖H1(Y)3≤c(‖F‖L2(Y)3+‖ψ‖H1(Y)3). □ We then prove the following result on the regularity of $$N^r$$. Proposition 5.3 Assume that $$a(x,y)\in C^1(\bar D,C^2(\bar Y))^{3\times 3}$$; then $${\rm curl}_yN^r(x,y)\in C^1(\bar D,C(\bar Y))^3$$ and we can choose a version of $$N^r$$ in $$L^2(D,\tilde H_\#({\rm curl\,},Y))$$ so that $$N^r(x,y)\in C^1(\bar D, C(\bar Y))^3$$. Proof. We can choose a version of $$N^r$$ so that $${\rm div}_yN^r=0$$. Indeed, let $${\it{\Phi}}(x,\cdot)\in L^2(D, H^1_\#(Y))$$ be such that $$\Delta_y{\it{\Phi}}=-{\rm div}_yN^r$$; then $${\rm curl}_y(N^r+\nabla_y{\it{\Phi}})={\rm curl}_y N^r$$ and $${\rm div}_y(N^r+\nabla_y{\it{\Phi}})=0$$. Further we can choose $$N^r$$ so that $$\int_YN^r(x,y)\,{\rm d}y=0$$. From Lemma 5.1, we have ‖Nr(x,⋅)‖H1(Y)3≤c‖curlyNr(x,⋅)‖L2(Y)3, which is uniformly bounded with respect to $$x$$. From (4.4) and Lemma 5.2, we deduce that ‖curlyNr(x,⋅)‖H1(Y)3≤c‖curly(a(x,⋅)er‖L2(Y)3+‖Nr(x,⋅)‖H1(Y)3, which is uniformly bounded with respect to $$x$$. For each index $$q=1,2,3$$, we have that $${\rm curl}_y{\partial\over\partial y_q}N^r(x,\cdot)$$ is uniformly bounded in $$L^2(Y)$$ and $${\rm div}_y{\partial\over\partial y_q}N^r(x,\cdot)=0$$. Therefore, from Lemma 5.1, $${\partial\over\partial y_q}N^r(x,\cdot)$$ is uniformly bounded in $$H^1(Y)$$. We note that ∂∂yq(curly(a(x,⋅)curlyNr))=−∂∂yqcurly(a(x,⋅)er)∈L2(Y). Thus, curly(acurly∂Nr∂yq)=∂∂yq(curly(a(x,y)curlyNr))−curly(∂a∂yqcurlyNr)∈L2(Y). From Lemma 5.2 we deduce that $${\rm curl}_y{\partial N^r\over\partial y_q}(x,\cdot)$$ is uniformly bounded in $$H^1(Y)$$ so that $${\rm curl}_y N^r(x,\cdot)$$ is uniformly bounded in $$H^2(Y)\subset C(\bar Y)$$. We now show that $${\rm curl}_yN^r\in C^1(\bar D,H^2(Y))^3\subset C^1(\bar D,C(\bar Y))^3$$. Fix $$h\in\mathbb{R}^3$$. From (4.4) we have curly(a(x,y)curly(Nr(x+h,y)−Nr(x,y))) =−curly((a(x+h,y)−a(x,y))er) −curly((a(x+h,y)−a(x,y))curlyNr(x+h,y)). The smoothness of $$a$$ and the uniform boundedness of $${\rm curl}_yN^r(x,\cdot)$$ in $$L^2(Y)^3$$ gives limh→0‖curly(Nr(x+h,⋅)−Nr(x,⋅))‖L2(Y)3=0. (5.1) From Lemma 5.1 we have that $$N^r(x+h,\cdot)-N^r(x,\cdot)\in H^1(Y)^3$$ and ‖Nr(x+h,⋅)−Nr(x,⋅)‖H1(Y)3≤c‖curly(Nr(x+h,⋅)−Nr(x,⋅))‖L2(Y)3, which converges to 0 when $$\|h\|\to 0$$. From Lemma 5.2, we have ‖curly(Nr(x+h,⋅)−Nr(x,⋅))‖H1(Y)3 ≤‖−curly((a(x+h,⋅)−a(x,⋅))er)−curly((a(x+h,⋅)−a(x,⋅))curlyNr(x+h,⋅))‖L2(Y)3 +‖Nr(x+h,⋅)−Nr(x,⋅)‖H1(Y)3→0 when \|h\|→0. (5.2) We have further that curly(a(x,y)curly∂∂yq(Nr(x+h,y)−Nr(x,y)))=−curly(∂a∂yq(x,y)curly(Nr(x+h,y)−Nr(x,y))) −∂∂yqcurly((a(x+h,y)−a(x,y))er)−∂∂yqcurly((a(x+h,y)−a(x,y))curlyNr(x+h,y)). (5.3) From this we have ‖curly∂∂yq(Nr(x+h,y)−Nr(x,y))‖L2(Y)3 ≤c‖curly(Nr(x+h,⋅)−Nr(x,⋅))‖L2(Y)3 +c‖a(x+h,⋅)−a(x,⋅)‖W1,∞(Y)3→0 when \|h\|→0, so from Lemma 5.1 we have ‖∂∂yq(Nr(x+h,y)−Nr(x,y))‖H1(Y)3→0 when \|h\|→0. As the right-hand side of (5.3) converges to 0 in the $$L^2(Y)^3$$ norm when $$\|h\|\to 0$$, we deduce from Lemma 5.2 that ‖curly∂∂yq(Nr(x+h,y)−Nr(x,y))‖H1(Y)3→0 when \|h\|→0. (5.4) We have curly[a(x,y)curly(Nr(x+h,y)−Nr(x,y)h)] =−curly((a(x+h,y)−a(x,y)h)er) −curly(a(x+h,y)−a(x,y)hcurlyNr(x+h,y)). Let $$\chi^r(x,\cdot)\in \tilde H_\#({\rm curl\,},Y)$$ with $${\rm div}_y\chi^r(x,)=0$$ be the solution of the problem curly(a(x,y)curlyχr(x,⋅))=−curly(∂a∂xqer)−curly(∂a∂xqcurlyNr(x,y)). We deduce that curly(a(x,y)curly(Nr(x+h,y)−Nr(x,y)h−χr(x,y))) =−curly((a(x+h,y)−a(x,y)h−∂a∂xq(x,y))er) −curly((a(x+h,y)−a(x,y)h−∂a∂xq(x,y))curlyNr(x+h,y)) −curly(∂a∂xq(x,y)curly(Nr(x+h,y)−Nr(x,y))):=I1. (5.5) Let $$h\in \mathbb{R}^3$$ be a vector with all components 0 except for the $$q$$th component. We have ‖curly(Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅))‖L2(Y)3 ≤c‖a(x+h,⋅)−a(x,⋅)h−∂a∂xq(x,⋅)‖L∞(Y) +c‖curly(Nr(x+h,⋅)−Nr(x,⋅)‖L2(Y)3, (5.6) which converges to 0 when $$\|h\|\to 0$$ due to (5.1). Thus, we deduce from Lemma 5.1 that lim\|h\|→0‖Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅)‖H1(Y)3=0. (5.7) From Lemma 5.2, we have lim\|h\|→0‖curly(Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅))‖H1(Y)3 ≤lim\|h\|→0‖I1(x,⋅)‖L2(Y)3+‖Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅)‖H1(Y)3=0 (5.8) due to (5.2) and (5.7). Let $$p=1,2,3$$. We then have curly(a(x,y)curly∂∂yp(Nr(x+h,y)−Nr(x,y)h−χr(x,y))) =−curly(∂a∂yp(x,y)curly(Nr(x+h,y)−Nr(x,y)h−χr(x,y))) −∂∂ypcurly((a(x+h,y)−a(x,y)h−∂a(x,y)∂xq)er) −∂∂ypcurly((a(x+h,y)−a(x,y)h−∂a∂xq(x,y))curlyNr(x+h,y)) −∂∂ypcurly(∂a∂xq(x,y)curly(Nr(x+h,y)−Nr(x,y))), which converges to 0 in $$L^2(Y)$$ for each $$x$$ due to (5.4), (5.8) and the uniform boundedness of $$\\|{\rm curl\,} N^r(x,\cdot)\\|_{H^2(Y)^3}$$. We have ‖curly∂∂yp(Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅))‖L2(Y)3 ≤c‖curly(Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅))‖L2(Y)3 +c‖a(x+h,⋅)−a(x,⋅)h−∂a(x,⋅)∂xq‖W1,∞(Y)3+c‖curly(Nr(x+h,⋅)−Nr(x,⋅)‖H1(Y)3, which converges to 0 when $$\|h\|\to 0$$, so from Lemma 5.1, lim\|h\|→0‖∂∂yp(Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅))‖H1(Y)3=0. (5.9) Therefore, $$N^r\in C^1(\bar D,H^2(Y))^3\subset C^1(\bar D,C(\bar Y))^3$$. We then get from Lemma 5.2 that lim\|h\|→0‖curly∂∂yp(Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅))‖H1(Y)3=0. Thus, $${\rm curl}_yN^r\in C^1(\bar D,H^2(Y))^3\subset C^1(\bar D,C(\bar Y))^3$$. □ Proposition 5.4 Assume that $$b(x,y)\in C^1(\bar D,C^2(\bar Y))^{3\times 3}$$. The solution $$w^r$$ of cell problem (4.1) belongs to $$C^1(\bar D, C^1(\bar Y))$$. Proof. The cell problem (4.1) can be written as −∇y⋅(b(x,y)∇ywr(x,y))=∇y(b(x,y)er). Fixing $$x\in \bar D$$, the right-hand side is bounded uniformly in $$H^1(Y)$$ so $$w^r(x,\cdot)$$ is uniformly bounded in $$H^3(Y)$$ from elliptic regularity (see McLean, 2000, Theorem 4.16). For $$h\in \mathbb{R}^3$$, we note that −∇y⋅[b(x,y)∇y(wr(x+h,y)−wr(x,y))] =∇y⋅[(b(x+h,y)−b(x,y))er] +∇y⋅[(b(x+h,y)−b(x,y))∇ywr(x+h,y)]:=i1. As $$\int_Yw^r(x,y)\,{\rm d}y=0$$, we have ‖wr(x+h,⋅)−wr(x,⋅)‖H1(Y) ≤c‖∇y(wr(x+h,⋅)−wr(x,⋅))‖L2(Y) ≤c‖(b(x+h,⋅)−b(x,⋅))er‖L2(Y) +c‖(b(x+h,⋅)−b(x,⋅))∇ywr(x+h,⋅)‖L2(Y), which converges to 0 when $$\|h\|\to 0$$. Fixing $$x\in\bar D$$, we then have from McLean (2000, Theorem 4.16) that ‖wr(x+h,⋅)−wr(x,⋅)‖H3(Y)≤‖wr(x+h,⋅)−wr(x,⋅)‖H1(Y)+‖i1(x,⋅)‖H1(Y), (5.10) which converges to 0 when $$\|h\|\to 0$$. Fixing an index $$q=1,2,3$$, let $$h\in \mathbb{R}^3$$ be a vector whose components are all zero except the $$q$$th component. Let $$\eta(x,\cdot)\in H^1_\#(Y)/\mathbb{R}$$ be the solution of the problem −∇y⋅[b(x,y)∇yη(x,y)]=∇y⋅[∂b∂xqer]+∇y⋅[∂b∂xq∇ywr(x,y)]. We have −∇y⋅[b(x,y)∇y(wr(x+h,y)−wr(x,y)h−η(x,y))] =∇y⋅[(b(x+h,y)−b(x,y)h−∂b∂xq(x,y))er] +∇y⋅[(b(x+h,y)−b(x,y)h−∂b∂xq(x,y))∇ywr(x+h,y)] +∇y⋅[∂b(x,y)∂xq(∇ywr(x+h,y)−∇ywr(x,y))]:=i2. From (5.10) and the regularity of $$b$$, $$\lim_{\|h\|\to 0}\\|i_2(x,\cdot)\\|_{H^1(Y)}=0$$. As $$\int_Yw^r(x,y)\,{\rm d}y=0$$ and $$\int_Y\eta(x,y)\,{\rm d}y=0$$, we have lim\|h\|→0‖wr(x+h,⋅)−wr(x,⋅)h−η(x,⋅)‖H1(Y)=0. Therefore from McLean (2000, Theorem 4.16), we have ‖wr(x+h,⋅)−wr(x,⋅)h−η(x,⋅)‖H3(Y)≤‖wr(x+h,⋅)−wr(x,⋅)h−η(x,⋅)‖H1(Y)+‖i2(x,⋅)‖H1(Y), which converges to 0 when $$\|h\|\to 0$$. Thus, $$w^r\in C^1(\bar D,H^3(Y))\subset C^1(\bar D,C^1(\bar Y))$$. □ For the regularity of the solution $$u_0$$ of the homogenized problem (4.7) we have the following result. Proposition 5.5 Assume that $$D$$ is a Lipschitz polygonal domain, and the coefficient $$a(\cdot,y)$$, as a function of $$x$$, is Lipschitz, uniformly with respect to $$y$$; then there is a constant $$0<s<1$$ so that $${\rm curl\,} u_0\in H^s(D)$$. Proof. When $$a(x,y)$$ is Lipschitz with respect to $$x$$, from (4.4), $$\\|{\rm curl}_y N^r(x,\cdot)\\|_{L^2(Y)}$$ is a Lipschitz function of $$x$$, so from (4.6) we have that $$a^0$$ is Lipschitz with respect to $$x$$. As $$a^0$$ is positive definite, $$(a^0)^{-1}$$ is Lipschitz. Let $$U=a^0{\rm curl\,} u_0$$. We have from (4.7) that $$U\in H({\rm curl\,},D)$$, $${\rm div}((a^0)^{-1}U)=0$$ and $$(a^0)^{-1}U\cdot \nu=0$$ on $$\partial D,$$ where $$\nu$$ is the outward normal vector on $$\partial D$$. The conclusion follows from Hiptmair (2002, Lemma 4.2). □ Remark 5.6 If $$a^0$$ is isotropic, we have from (4.7) that curlcurlu0=−(a0)−1∇a0×curlu0−(a0)−1b0u0+(a0)−1f∈L2(D)3 so $$u_0\in H^1({\rm curl},D)$$. However, even if $$a$$ is isotropic, $$a^0$$ may not be isotropic. Remark 5.7 The homogenized equation for the multiscale case is determined as follows. We denote by $$a^n(x,\boldsymbol{y})=a(x,\boldsymbol{y})$$. Recursively, for $$i=1,\ldots,n-1$$, the $$i$$-th level homogenized coefficient is determined as follows. For $$r=1,2,3$$, let $$N_{i+1}^r\in L^2(D\times{\bf Y}_i,\tilde H({\rm curl}_{y_{i+1}},Y_{i+1}))$$ be the solution of the cell problem ∫D∫Y1…∫Yi+1ai+1(x,yi,yi+1)(er+curlyi+1Ni+1r)⋅curlyi+1ψdyi+1dyidx=0 for all $$\psi\in L^2(D\times{\bf Y}_i,\tilde H({\rm curl}_{y_{i+1}},Y_{i+1}))$$. The $$i$$-th level homogenized coefficient $$a^{i}$$ is determined by arsi(x,yi)=∫Yi+1ai+1(x,yi,yi+1)(es+curlyi+1Ni+1s)⋅(er+curlyi+1Ni+1r)dyi+1. Let $$b^n(x,\boldsymbol{y})=b(x,\boldsymbol{y})$$. Similarly, let $$w_{i+1}^{r}\in L^2(D\times{\bf Y}_i,H^1_\#(Y_{i+1})/\mathbb{R})$$ be the solution of the problem ∫D∫Y1…∫Yi+1bi+1(x,yi,yi+1)(er+∇yi+1wi+1r)⋅∇yi+1ψdyi+1dyidx=0 for all $$\psi\in L^2(D\times{\bf Y}_i,H^1_\#(Y_{i+1})/\mathbb{R})$$. The $$i$$-th level homogenized coefficient $$b^i$$ is determined by brsi(x,yi)=∫Yi+1bi+1(x,yi,yi+1)(es+∇yi+1wi+1s)⋅(er+∇yi+1wi+1r)dyi+1. We then have the equation ∫D∫Yi[ai(x,yi)(curlu0+curly1u1+⋯+curlyiui)⋅(curlv0+curly1v1+⋯+curlyivi)dyidx +bi(x,yi)(u0+∇y1u1+⋯+∇yiui)⋅(v0+∇y1v1+⋯+∇yivi)]dyidx=∫Df(x)⋅v0(x)dx. The coefficients $$a^0(x)$$ and $$b^0(x)$$ are the homogenized coefficients. We have ui(x,yi) = [curlu0(x)r+curly1u1(x,y1)r+⋯+curlyi−1ui−1(x,yi−1)r]Nir(x,yi) =curlu0(x)r0(δr0r1+curly1N1r0(x,y1)r1)(δr1r2+curly2N2r1(x,y2)r2)⋯ (δri−2ri−1+curlyi−1Ni−1ri−2(x,yi−1)ri−1)Niri−1(x,yi). If $$a(x,\boldsymbol{y})\in C^1(\bar D,C^2(\bar Y_1,\ldots,C^2(\bar Y_n),\ldots))^{3\times 3}$$, by following the same procedure as above, we can show inductively that $${\rm curl}_{y_i}N_i^r(x,\boldsymbol{y}_i)\in C^1(\bar D,C^2(\bar Y_1,\ldots,C^2(\bar Y_{i-1},H^2(Y_i)),\ldots))$$ and $$a^i(x,\boldsymbol{y}_i)\in C^1(\bar D,C^2(\bar Y_1,\ldots,C^2(\bar Y_i),\ldots))$$. Thus, if $$u_0\in H^s({\rm curl\,},D)$$ for $$0<s\le 1$$, $$u_i\in \hat{\mathcal{H}}_i^s$$. Similarly, we can show that if $$b\in C^1(\bar D, C^2(\bar Y_1,\ldots,C^2(\bar Y_n)\ldots))$$ then $$w^{ir}\in C^1(\bar D,C^2(\bar Y_1,\ldots,C^2(\bar Y_{i-1},H^3(Y_i))\ldots))$$. As ui=u0r0(x)(δr0r1+∂w1r0∂y1r1(x,y1))…(δri−2ri−1+∂wi−1ri−2∂y(i−1)ri−1(x,yi−1))wiri−1(x,yi), if $$u_0\in H^s(D)$$, $${\frak u_i}\in \hat{\frak H}_i^s$$. 6. Numerical results The detail spaces $$\mathcal{V}^l$$ and $$\mathcal{V}^l_\#$$, which are difficult to construct in numerical implementations, are defined via orthogonal projection in Section 3.2. We employ Riesz basis functions and define equivalent norms, which facilitate the construction of these spaces. We make the following assumption. Assumption 6.1 (i) For each multidimensional vector $$j \in \mathbb{N}_0^d$$, there exists a set of indices $$I^j \subset \mathbb{N}^d_0$$ and a set of basis functions $$\phi^{jk}\in L^2(D)$$ for $$k\in I^j$$, such that $$V^l = \text{span}\left\{\phi^{jk} : \|\,j\|_{\infty}\le l\right\}$$. There are constants $$c_2>c_1>0$$ such that if $$\phi = \sum_{\|\,j\|_{\infty}\leq l,k\in I^j}\phi^{jk}c_{jk}\in V^l$$, then the following norm equivalences hold: c1∑\|j\|∞≤lk∈Ij\|cjk\|2≤‖ϕ‖L2(D)2≤c2∑\|j\|∞≤lk∈Ij\|cjk\|2, where $$c_1$$ and $$c_2$$ are independent of $$\phi$$ and $$l$$. (ii) For the space $$L^2(Y)$$, for each $$j \in \mathbb{N}_0^d$$, there exists a set of indices $$I^j_0 \subset \mathbb{N}_0^d$$ and a set of basis functions $$\phi^{jk}_0\in L^2(Y)$$, $$k\in I^j_0$$, such that $$V^l_\# = \text{span}\{\phi^{jk}_0 : \|\,j\|_{\infty}\le l\}$$. There are constants $$c_4>c_3>0$$ such that if $$\phi = \sum_{\|\,j\|_{\infty}\leq l,k\in I^{\,j}_{\,0}}\phi^{jk}_0c_{jk}\in V^l$$ then c3∑\|j\|∞≤lk∈I0j\|cjk\|2≤‖ϕ‖L2(Y)2≤c4∑\|j\|∞≤lk∈I0j\|cjk\|2, where $$c_3$$ and $$c_4$$ are independent of $$\phi$$ and $$l$$. Because of the norm equivalence, we can use $$\mathcal{V}^l=\text{span}\{\phi^{jk} : \|\,j\|_{\infty}= l\}$$ and $$\mathcal{V}^l_\#=\text{span}\{\phi^{jk}_0 : \|\,j\|_{\infty}= l\}$$ to construct the sparse tensor product FE spaces. Example 6.2 (i) We can construct a hierarchical basis for $$L^2(0,1)$$ as follows. We first take three piecewise linear functions as the basis for level $$j=0$$: $$\psi^{01}$$ obtains values $$(1,0)$$ at $$(0,1/2)$$ and is 0 in $$(1/2,1)$$, $$\psi^{02}$$ is piecewise linear and obtains values $$(0, 1, 0)$$ at $$(0, 1/2, 1)$$ and $$\psi^{03}$$ obtains values $$(0,1)$$ at $$(1/2, 1)$$ and is 0 in $$(0,1/2)$$. The basis functions for other levels are constructed from the wavelet function $$\psi$$ that takes values $$(0,-1,2,-1,0)$$ at $$(0,1/2,1,3/2,2)$$, the left boundary function $$\psi^{\rm left}$$ taking values $$(-2,2,-1,0)$$ at $$(0,1/2,1,3/2)$$ and the right boundary function $$\psi^{\rm right}$$ taking values $$(0, -1,2,-2)$$ at $$(1/2,1,3/2,2)$$. For levels $$j\geq 1$$, $$I^j=\{1,2,\ldots,2^j\}$$. The wavelet basis functions are defined as $$\psi^{j1}(x) = 2^{j/2}\psi^{\rm left}(2^j x)$$, $$\psi^{jk}(x)=2^{j/2}\psi(2^j x - k + 3/2)$$ for $$k = 2, \ldots, 2^j-1$$ and $$\psi^{j2^j} = 2^{j/2}\psi^{\rm right}(2^j x - 2^j+2)$$. This base satisfies Assumption 6.1 (i). (ii) For $$Y = (0,1)$$, we can construct a hierarchy of periodic basis functions for $$L^2(Y)$$ that satisfies Assumption 6.1 (ii) from those in (i). For level 0, we exclude $$\psi^{01}$$, $$\psi^{03}$$ and include the periodic piecewise linear function that takes values $$(1,0,1)$$ at $$(0,1/2,1),$$ respectively. At other levels, the functions $$\psi^{\rm left}$$ and $$\psi^{\rm right}$$ are replaced by the piecewise linear functions that take values $$(0,2, -1, 0)$$ at $$(0,1/2,1,3/2)$$ and values $$(0, -1,2,0)$$ at $$(1/2,1, 3/2 ,2),$$ respectively. When $$D=(0,1)^d$$, the basis functions can be constructed by taking the tensor products of the basis functions in $$(0,1)$$. They satisfy Assumption 6.1 after appropriate scaling (see Griebel & Oswald, 1995). Remark 6.3 When the norm equivalence for the basis functions in $$L^2(D)$$ and in $$L^2(Y)$$ does not hold, in many cases, we can still prove a rate of convergence similar to those in Lemmas 3.5 and 3.6 for the sparse tensor product FE approximations. For example, with the division of the domain $$D$$ into sets of triangles $$\mathcal{T}^l$$, the set of continuous piecewise linear functions with value 1 at one vertex and 0 at all the others forms a basis of $$V^l$$. Let $$S^l$$ be the set of vertices of the set of simplices $$\mathcal{T}^l$$. We can define $$\mathcal{V}^l$$ as the linear span of functions that are 1 at a vertex in $$S^l\setminus S^{l-1}$$ and 0 at all the other vertices. We can then construct the sparse tensor product FE approximations with these spaces but the norm equivalence does not hold. A rate of convergence for sparse tensor product FEs similar to those in Lemmas 3.5 and 3.6 can be deduced (see e.g, Hoang, 2008). In the first example, we consider a two-scale Maxwell-type equation in the two-dimensional domain $$D=(0,1)^2$$. The coefficients a(x,y)=(1+x1)(1+x2)(1+cos2⁡2πy1)(1+cos2⁡2πy2) and b(x,y)=1(1+x1)(1+x2)(1+cos2⁡2πy1)(1+cos2⁡2πy2). We can compute the homogenized coefficients exactly. In this case, a0=4(1+x1)(1+x2)9 and b0=23(1+x1)(1+x2). We choose f=(49(1+x1)(1+2x2−x1)+23(1+x1)(1+x2)x1x2(1−x2)49(1+x2)(1+2x1−x2)+23(1+x1)(1+x2)x1x2(1−x1)) so that the solution to the homogenized equation is u0=(x1x2(1−x2)x1x2(1−x1)). In Fig. 1, we plot the energy error versus the mesh size for the sparse tensor product FE approximations of the two-scale homogenized Maxwell-type problem. The figure agrees with the error estimate in Proposition 3.8. Fig. 1. View largeDownload slide The sparse tensor energy error $$B(\boldsymbol{u}-\widehat {\bf{u}}^L,\boldsymbol{u}-\widehat {\bf{u}}^L).$$ Fig. 1. View largeDownload slide The sparse tensor energy error $$B(\boldsymbol{u}-\widehat {\bf{u}}^L,\boldsymbol{u}-\widehat {\bf{u}}^L).$$ In the second example, we consider the case where $$b$$ is the identity matrix, i.e., it does not depend on $$y$$. In this case, from (2.10) we note that the function $${\frak u_1}=0$$. We choose a(x,y)=(1+x1)(1+x2)(1+cos2⁡2πy1)(1+cos2⁡2πy2) and f=(4(2π(1+x1)(1+x2)sin⁡2πx2+(1+x1)(cos⁡2πx1−cos⁡2πx2))9+12πsin⁡2πx24(2π(1+x1)(1+x2)sin⁡2πx1−(1+x2)(cos⁡2πx1−cos⁡2πx2))9+12πsin⁡2πx1) so that the solution to the homogenized problem is u0=(12πsin⁡2πx212πsin⁡2πx1). Figure 2 plots the energy error versus the mesh size for the sparse tensor product FE approximations for the two-scale homogenized Maxwell-type problem. The plot confirms the analysis. Fig. 2. View largeDownload slide The sparse tensor energy error $$B(\boldsymbol{u}-\widehat {\bf{u}}^L,\boldsymbol{u}-\widehat {\bf{u}}^L).$$ Fig. 2. View largeDownload slide The sparse tensor energy error $$B(\boldsymbol{u}-\widehat {\bf{u}}^L,\boldsymbol{u}-\widehat {\bf{u}}^L).$$ Acknowledgements The authors gratefully acknowledge a postgraduate scholarship of Nanyang Technological University, the AcRF Tier 1 grant RG69/10, the Singapore AStar SERC grant 122-PSF-0007 and the AcRF Tier 2 grant MOE 2013-T2-1-095 ARC 44/13. Footnotes 1 The notations $$Y_1,\ldots,Y_n$$, which denote the same unit cube $$Y$$, are introduced for convenience only, especially in the case where the Cartesian product of several of them is used, to avoid the necessity of indicating how many times the unit cube appears in the product. The functions $$a$$ and $$b$$ depend on the macroscopic scale only and are periodic with respect to $$y_i$$ with the period being the unit cube $$Y$$. 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( 2010) Multiscale computations for 3D time-dependent Maxwell’s equations in composite materials. SIAM J. Sci. Comput. , 32, 2560– 2583. Google Scholar CrossRef Search ADS Appendix A. We present the proof of Theorem 4.2 in this appendix. We consider a set of $$M$$ open cubes $$Q_i$$ ($$i=1,\ldots,M$$) of size $$\varepsilon^t$$ for $$t>0$$ to be chosen later such that $$D\subset\bigcup_{i=1}^MQ_i$$ and $$Q_i\bigcap D\ne\emptyset$$. Each cube $$Q_i$$ intersects with only a finite number, which does not depend on $$\varepsilon$$, of other cubes. We consider a partition of unity that consists of $$M$$ functions $$\rho_i$$ such that $$\rho_i$$ has support in $$Q_i$$, $$\sum_{i=1}^M\rho_i(x)=1$$ for all $$x\in D$$ and $$\|\nabla\rho_i(x)\|\le c\varepsilon^{-t}$$ for all $$x$$ (indeed such a set of cubes $$Q_i$$ and a partition of unity can be constructed from a fixed set of cubes of size $${\mathcal O}(1)$$ by rescaling). For $$r=1,2,3$$ and $$i=1,\ldots,M$$, we define Uir=1\|Qi\|∫Qicurlu0(x)rdx and Vir=1\|Qi\|∫Qiu0(x)rdx (as $$u_0\in H^s(D)^3$$ and $${\rm curl\,} u_0\in H^s(D)^3$$, for the Lipschitz domain $$D$$, we can extend each of them, separately, continuously outside $$D$$ and understand $$u_0$$ and $${\rm curl\,} u_0$$ as these extensions; see Wloka, 1987, Theorem 5.6). Let $$U_i$$ and $$V_i$$ denote the vectors $$(U_i^1, U_i^2,U_i^3)$$ and $$(V_i^1,V_i^2,V_i^3),$$ respectively. Let $$B$$ be the unit cube in $$\mathbb{R}^3$$. From the Poincaré inequality, we have ∫B\|ϕ−∫Bϕ(x)dx\|2dx≤c∫B\|∇ϕ(x)\|2dx ∀ϕ∈H1(B). By translation and scaling, we deduce that ∫Qi\|ϕ−1\|Qi\|∫Qiϕ(x)dx\|2dx≤cε2t∫Qi\|∇ϕ(x)\|2dx ∀ϕ∈H1(Qi), i.e., ‖ϕ−1\|Qi\|∫Qiϕ(x)dx‖L2(Qi)≤cεt‖ϕ‖H1(Qi). Together with ‖ϕ−1\|Qi\|∫Qiϕ(x)dx‖L2(Qi)≤c‖ϕ‖L2(Qi), we deduce from interpolation that ‖ϕ−1\|Qi\|∫Qiϕ(x)dx‖L2(Qi)≤cεts‖ϕ‖Hs(Qi) ∀ϕ∈Hs(Qi). Thus, ∫Qi\|curlu0(x)r−Uir\|2dx≤cε2ts‖(curlu0)r‖Hs(Qi)2. (A.1) Let u1ε(x)=u0(x)+εNr(x,xε)Ujrρj(x)+ε∇[wr(x,xε)Vjrρj(x)]. We have curl(aε(x)curlu1ε(x))+bε(x)u1ε(x) =curla(x,xε)[curlu0(x)+εcurlxNr(x,xε)Ujrρj(x)+curlyNr(x,xε)Ujrρj+ε(Ujr∇ρj)×Nr(x,xε)] +b(x,xε)[u0(x)+εNr(x,xε)Ujrρj(x)+ε∇xwr(x,xε)Vjrρj(x)+∇ywr(x,xε)Vjrρj(x) + εwr(x,xε)Vjr∇ρj(x)] =curl(a0(x)curlu0(x))+b0(x)u0(x)+curl[Gr(x,xε)Ujrρj(x)]+gr(x,xε)Vjrρj(x)+εcurlI(x) +εJ(x)+curl[(aε(x)−a0(x))(curlu0(x)−Ujρj(x))]+(bε(x)−b0(x))(u0(x)−Vjρj(x)), where the vector functions $$G_r(x,y)$$ and $$g_r(x,y)$$ are defined by (Gr)i(x,y) =air(x,y)+aij(x,y)curlyNr(x,y)j−air0(x), (A.2) (gr)i(x,y) =bir(x,y)+bij(x,y)∂wr∂yj(x,y)−bir0(x) (A.3) and I(x) =a(x,xε)[curlxNr(x,xε)Ujrρj(x)+(Ujr∇ρj(x))×Nr(x,xε)],J(x) =b(x,xε)[Nr(x,xε)Ujrρj(x)+∇xwr(x,xε)Vjrρj(x)+wr(x,xε)Vjr∇ρj(x)]. Therefore, for $$\phi\in W$$, ⟨curl(aεcurlu1ε)+bεu1ε−curl(a0curlu0)−b0u0,ϕ⟩ =∫DUjrρj(x)Gr(x,xε)⋅curlϕdx+∫DVjrρj(x)gr(x,xε)⋅ϕ(x)dx +ε∫DI(x)⋅curlϕ(x)dx+ε∫DJ(x)⋅ϕ(x)dx+∫D(aε−a0)(curlu0(x)−Ujρj)⋅curlϕ(x)dx +∫D(bε−b0)(u0−Vjρj)⋅ϕdx (here $$\langle\cdot\rangle$$ denotes the duality pairing between $$W'$$ and $$W$$). From (4.4), we have that $${\rm curl}_yG_r(x,y)=0$$. Further, from (4.6) $$\int_YG_r(x,y)\,{\rm d}y=0$$. Therefore, there is a function $$\tilde G_r(x,y)$$ such that $$G_r(x,y)=\nabla_y \tilde G_r(x,y)$$. From (4.1), we have $${\rm div}_yg_r(x,y)=0$$ and from (4.6) $$\int_Yg_r(x,y)\,{\rm d}y=0$$. Hence, there is a function $$\tilde g_r$$ such that $$g_r(x,y)={\rm curl}_y\tilde g_r(x,y)$$. As $$\nabla_y\tilde G_r(x,\cdot)=G_r(x,\cdot)\in H^1(Y)^3$$ so $$\Delta_y\tilde G_r(x,\cdot)\in L^2(Y)$$. Thus, $$\tilde G_r(x,\cdot)\in H^2(Y),$$ which implies $$\tilde G_r(x,\cdot)\in C(\bar Y)$$. As $$G_r(x,\cdot)\in C^1(\bar D,H^1_\#(Y)^3)$$, we deduce that $$\tilde G_r(x,y)\in C^1(\bar D, H^2(Y))\subset C^1(\bar D,C(\bar Y))$$. The construction of $$\tilde g_r$$ in Jikov et al. (1994) implies that $$\tilde g_r\in C^1(\bar D, C(\bar Y))$$ (see Hoang & Schwab, 2013). We have ∫DUjrρjGr(x,xε)⋅curlϕdx=∫DUjrρj(x)[ε∇G~r(x,xε)−ε∇xG~r(x,xε)]⋅curlϕdx =−ε∫DG~r(x,xε)div[(Ujrρj)curlϕ]dx−ε∫DUjrρj∇xG~r(x,xε)⋅curlϕdx. We note that \|∫DUjrρj∇xG~r(x,xε)⋅curlϕdx\|≤c‖(Ujrρj)‖L2(D)‖curlϕ‖L2(D)3. From ‖Ujrρj‖L2(D)2=∫D(Ujr)2ρj(x)2dx+∑i≠j∫DUirUjrρi(x)ρj(x)dx, and the fact that the support of each function $$\rho_i$$ intersects only with the support of a finite number (which does not depend on $$\varepsilon$$) of other functions $$\rho_j$$ in the partition of unity, we deduce ‖Ujrρj‖L2(D)2 ≤c∑j=1M(Ujr)2\|Qj\| =c∑j=1M1\|Qj\|(∫Qjcurlu0(x)rdx)2≤c∑j=1M∫Qjcurlu0(x)r2dx≤c∫Dcurlu0(x)r2dx. Thus, \|ε∫DUjrρj∇xG~r(x,xε)⋅curlϕdx\|≤cε‖curlϕ‖L2(D)3. We have further that ε∫DG~r(x,xε)div[(Ujrρj)curlϕ]dx =ε∫DG~r(x,xε)[(Ujr∇ρj(x))]⋅curlϕdx ≤cε‖Ujr∇ρj‖L2(D)3‖curlϕ‖L2(D)3. As the support of each function $$\rho_i$$ intersects with the support of a finite number of other functions $$\rho_j$$ and $$\\|\nabla\rho_j\\|_{L^\infty(D)}\le c\varepsilon^{-t}$$, we have ‖Ujr∇ρj‖L2(D)32≤c∑j=1M(Ujr)2\|Qj\|‖∇ρj‖L∞(D)2≤cε−2t∑j=1M(Ujr)2\|Qj\|≤cε−2t, so ε∫DG~r(x,xε)div[(Ujrρj)curlϕ]dx≤cε‖Ujr∇ρj‖L2(D)3‖curlϕ‖L2(D)3≤cε1−t‖curlϕ‖L2(D)3. We therefore deduce that \|∫DUjrρjGr(x,xε)⋅curlϕdx\|≤cε1−t‖curlϕ‖L2(D)3. We have ∫DVjrρjgr(x,xε)⋅ϕ(x)dx=∫DVjrρj[εcurlg~r(x,xε)−εcurlxg~r(x,xε)]⋅ϕdx. Arguing similarly to above, we have \|ε∫DVjrρjcurlxg~r(x,xε)⋅ϕdx\|≤cε‖Vjrρj‖L2(D)3‖ϕ‖L2(D)3≤cε‖ϕ‖L2(D)3 and \|ε∫DVjrρjcurlg~r(x,xε)⋅ϕdx\| = \|ε∫Dg~r(x,xε)⋅curl[(Vjrρj)ϕ]dx\| ≤ \|ε∫Dg~r(x,xε)⋅[(Vjrρj)curlϕ+ϕ×(Vjr∇ρj)]dx ≤c(ε‖curlϕ‖L2(D)3+cε1−t‖ϕ‖L2(D)3)(∑j=1M(Vjr)2\|Qj\|)1/2 ≤c(ε‖curlϕ‖L2(D)3+cε1−t‖ϕ‖L2(D)3). We note that ‖I‖L2(D)3≤csupr[‖Ujrρj‖L2(D)+‖Ujr∇ρj‖L2(D)]≤cε−t and ‖J‖L2(D)3≤csupr[‖Ujrρj‖L2(D)+c‖Vjrρj‖L2(D)+c‖Vjr∇ρj‖L2(D)]≤cε−t. We have further that ⟨curl((aε−a0)(curlu0−Ujρj)),ϕ⟩≤c‖curlu0−(Ujρj))‖L2(D)3‖curlϕ‖L2(D)3. From ∫D\|(curlu0)r−(Ujrρj)\|2dx=∫D\|∑j=1M((curlu0)r−Ujr)ρj\|2dx, using the support property of $$\rho_j$$, we have from (A.1), ∫D\|(curlu0)r−(Ujrρj)\|2dx ≤c∑j=1M∫Qj\|(curlu0)r−Ujr\|2dx≤cε2st∑j=1M‖(curlu0)r‖Hs(Qj)2 =cε2st∑j=1M[∫Qj(curlu0)r2dx+∫Qj×Qj(curlu0(x)r−curlu0(x′)r)2\|x−x′\|3+2sdxdx′] ≤cε2st[‖(curlu0)r‖L2(D)2+∫D×D(curlu0(x)r−curlu0(x′)r)2\|x−x′\|3+2sdxdx′] =cε2st‖(curlu0)r‖Hs(D)2. (A.4) Thus, ⟨curl((aε−a0)(curlu0−Ujρj)),ϕ⟩≤cεst‖curlϕ‖L2(D)3. Similarly, we have \|∫D(bε−b0)(u0−∑j=1MVjρj)⋅ϕdx\|≤c‖∑j=1M(u0−Vj)ρj‖L2(D)3‖ϕ‖L2(D)3≤cεst‖ϕ‖L2(D)3. Therefore, \|⟨curl(aεcurlu1ε)+bεu1ε−curl(a0curlu0)−b0u0,ϕ⟩\|≤c(ε1−t+εst)‖ϕ‖V i.e., ‖curl(aεcurlu1ε)+bεu1ε−curl(a0curlu0)−b0u0‖W′≤c(ε1−t+εst). Thus, ‖curl(aεcurlu1ε)+bεu1ε−curl(aεcurluε)−bεuε‖W′≤c(ε1−t+εst). (A.5) Let $$\tau^\varepsilon(x)$$ be a function in $$\mathcal{D}(D)$$ such that $$\tau^\varepsilon(x)=1$$ outside an $$\varepsilon$$ neighbourhood of $$\partial D$$ and $${\rm sup}_{x\in D}\varepsilon\|\nabla\tau^\varepsilon(x)\|<c,$$ where $$c$$ is independent of $$\varepsilon$$. We consider the function w1ε(x)=u0(x)+ετε(x)Ujrρj(x)Nr(x,xε)+ε∇[Vjrρjτε(x)wr(x,xε)]. We then have u1ε−w1ε=ε(1−τε(x))Ujrρj(x)Nr(x,xε)+ε∇[(1−τε(x))Vjrρjwr(x,xε)] and curl(u1ε−w1ε) =εcurlxNr(x,xε)Ujrρj(x)(1−τε(x))+curlyNr(x,xε)Ujrρj(x)(1−τε(x)) −εUjrρj(x)∇τε(x)×Nr(x,xε)+ε(1−τε(x))Ujr∇ρj(x)×Nr(x,xε). As shown above, $$\\|U_j^r\rho_j\\|_{L^2(D)}$$ is uniformly bounded, so ‖εcurlxNr(x,xε)(Ujrρj)(1−τε(x))‖L2(D)3≤cε. Let $$\tilde D^\varepsilon$$ be the $$3\varepsilon^{t}$$ neighbourhood of $$\partial D$$. We note that $${\rm curl\,} u_0$$ is extended continuously into a function in $$H^s(\mathbb{R}^3)$$ outside $$D$$. As shown in Hoang & Schwab (2013), for $$\phi\in H^1(\tilde D^\varepsilon)$$, ‖ϕ‖L2(D~ε)≤cεt/2‖ϕ‖H1(D~ε). From this and ‖ϕ‖L2(D~ε)≤‖ϕ‖L2(D~ε), using interpolation, we get ‖ϕ‖L2(D~ε)≤cεst/2‖ϕ‖Hs(D~ε)≤cεst/2‖ϕ‖Hs(D) for all $$\phi\in H^s(D)$$ extended continuously outside $$D$$. We then have ‖Ujrρj‖L2(Dε)2 ≤c∑j=1M∫Qj⋂Dε(Ujr)2ρj2dx ≤c∑j=1M\|Qj⋂Dε\|1\|Qj\|2(∫Qj(curlu0)rdx)2 ≤c∑Qj⋂Dε≠∅\|Qj⋂Dε\|\|Qj\|∫Qj(curlu0)r2dx. As $$D^\varepsilon$$ is the $$\varepsilon$$ neighbourhood of $$\partial D$$, $$\partial D$$ is Lipschitz and $$Q_j$$ has size $$\varepsilon^t$$, $$\|Q_j\bigcap D^\varepsilon\|\le c\varepsilon^{1+(d-1)t}$$ so $$\|Q_j\bigcap D^\varepsilon\|/\|Q_j\|\le c\varepsilon^{1-t}$$. When $$Q_j\bigcap D^\varepsilon\ne\emptyset$$, $$Q_j\subset\tilde D^\varepsilon$$. Thus, ‖Ujrρj‖L2(Dε)2≤cε1−t‖(curlu0)r‖L2(D~ε)2≤cε1−t+st‖curlu0‖Hs(D)32. Therefore, ‖curlyNr(x,xε)(Ujrρj)(1−τε(x))‖L2(D)3≤cε(1−t+st)/2 and ‖ε(Ujrρj)∇τε(x)×Nr(x,xε)‖L2(D)3≤cε(1−t+st)/2. Similarly, we have ‖Ujr∇ρj‖L2(Dε)32 ≤cε−2t∑Qj⋂Dε≠∅\|Qj⋂Dε\|\|Qj\|∫Qj(curlu0)r2dx ≤cε−2t+1−t‖curlu0‖L2(D~ε)32≤cε1−3t+st‖curlu0‖Hs(D)32. Thus, ‖ε(1−τε(x))(Ujr∇ρj)×Nr(x,xε)‖L2(D)≤cε(1−t)+(1−t+st)/2. Therefore, ‖curl(u1ε−w1ε)‖L2(D)3≤c(ε(1−t+st)/2+ε(1−t)+(1−t+st)/2). We further have ε∇[(1−τε(x))wr(x,xε)(Vjrρj)] = −ε∇τε(x)wr(x,xε)(Vjrρj)+ε(1−τε(x))∇xwr(x,xε)(Vjrρj) +(1−τε(x))∇ywr(x,xε)(Vjrρj)+ε(1−τε(x))wr(x,xε)(Vjr∇ρj). Arguing as above, we deduce that ‖Vjrρj‖L2(Dε)≤cε(1−t+st)/2, ‖Vjr∇ρj‖L2(Dε)≤cε(1−t+st)/2−t. Therefore, ‖ε∇[(1−τε(x))wr(x,xε)(Vjrρj)]‖L2(D)3≤c(ε(1−t+st)/2+ε1−t+(1−t+st)/2). Thus, ‖u1ε−w1ε‖L2(D)3≤c(ε(1−t+st)/2+ε(1−t)+(1−t+st)/2). Choosing $$t=1/(s+1)$$ we have ‖curl(aεcurl(u1ε−w1ε))+bε(u1ε−w1ε)‖W′≤cεs/(s+1). This together with (A.5) gives ‖curl(aεcurl(uε−w1ε))+bε(uε−w1ε)‖W′≤cεs/(s+1). Thus, ‖uε−w1ε‖W≤cεs/(s+1), which implies ‖uε−u1ε‖W≤cεs/(s+1). (A.6) We note that curlu1ε=curlu0(x)+curlyNr(x,xε)(Ujrρj)+εcurlxNr(x,xε)(Ujrρj)+ε(Ujr∇ρj)×Nr(x,xε). From ‖εcurlxNr(x,xε)(Ujrρj)‖L2(D)3≤cε and ‖ε(Ujr∇ρj)×Nr(x,xε)‖L2(D)3≤cεε−t=cεs/(1+s), we deduce that ‖curlu1ε−curlu0−curlyNr(x,xε)(Ujrρj)‖L2(D)3≤cεs/(s+1). From (A.4), ‖curlu0−(Ujrρj)‖L2(D)3≤cεts=cεs/(s+1), we get ‖curlu1ε−[curlu0+curlyNr(x,xε)(curlu0)r‖L2(D)3≤cεs/(s+1). This together with (A.6) implies ‖curluε−[curlu0+curlyNr(x,xε)(curlu0)r‖L2(D)3≤cεs/(s+1). □ Appendix B. We prove Lemma 4.3 in this appendix. We adapt the proof of Hoang & Schwab (2013, Lemma 5.5). As u1(x,y)=∑r=13curlu0(x)rNr(x,y), it is sufficient to show that for each $$r=1,2,3$$, ∫D\|curlu0(x)rcurlyNr(x,xε)−∫Ycurlu0(ε[xε]+εt)rcurlyNr(ε[xε]+εt,xε)dt\|2dx≤cε2s. The expression on the left-hand side is bounded by ∫D∫Y\|curlu0(x)rcurlyNr(x,xε)−curlu0(ε[xε]+εt)rcurlyNr(ε[xε]+εt,xε)\|2dtdx ≤2∫D∫Y\|(curlu0(x)r−curlu0(ε[xε]+εt)r)curlyNr(ε[xε]+εt,xε)\|2dtdx +2∫D∫Y\|curlu0(x)r\|2\|curlyNr(x,xε)−curlyNr(ε[xε]+εt,xε)\|2dtdx. As $${\rm curl}_yN^r\in C^1(\bar D,C(\bar Y))^3$$, there exists a constant $$c$$ such that supx∈Dsupt∈Y\|curlyNr(x,xε)−curlyNr(ε[xε]+εt,xε)\|≤cε. From this we have ∫D\|curlu0(x)rcurlyNr(x,xε)−Uε(curlu0(⋅)rcurlyNr(⋅,⋅))(x)\|2dx≤c∫D∫Y\|curlu0(x)r−curlu0(ε[xε]+εt)r\|2dtdx+cε2. We now show that for $${\rm curl\,} u_0\in H^s(D)$$, ∫D∫Y\|curlu0(x)r−curlu0(ε[xε]+εt)r\|2dtdx≤cε2s. (B.1) Letting $$\phi(x)$$ be a smooth function, we have ∫D∫Y\|ϕ(x)−ϕ(ε[xε]+εt)\|2dtdx ≤∑i=1d∫D∫Y\|ϕ(ε[x1ε]+εt1,…,ε[xi−1ε]+εti−1,xi,…,xd) −ϕ(ε[x1ε]+εt1,…,ε[xiε]+εti,xi+1,…,xd)\|2dtdx ≤∑i=1d∫D∫Y\|ε∫ti{xi/ε}∂ϕ∂xi(ε[x1ε]+εt1,…,ε[xiε]+εζi,xi+1,…,xd)dζi\|2dtdx ≤ε2∑i=1d∫D∫Y∫01\|∂ϕ∂xi(ε[x1ε]+εt1,…,ε[xiε]+εζi,xi+1,…,xd)\|2dζidtdx ≤ε2∑i=1d∫D\|∂ϕ∂xi\|2dx, which follows from (4.13); here we freeze the variables $$x_{i+1},\ldots,x_d$$. Let $$\psi\in H^1(D)$$. We consider a sequence $$\{\phi_n\}_n\subset C^\infty(\bar D),$$ which converges to $$\psi$$ in $$H^1(D)$$. As $$n\rightarrow \infty$$, ∫D∫Y(ϕn(ε[xε]+εt)−ψ(ε[xε]+εt))2dtdx = ∫DUε((ϕn−ψ)2)(x)dx ≤ ∫D(ϕn(x)−ψ(x))2dx→0. Therefore, ∫D∫Y(ψ(x)−ψ(ε[xε]+εt))2dtdx ≤3∫D(ψ−ϕn)2dx+3∫D∫Y(ϕn−ϕn(ε[xε]+εt))2dtdx +3∫D∫Y(ϕn(ε[xε]+εt)−ψ(ε[xε]+εt))2dtdx ≤6∫D(ψ−ϕn)2dx+3ε2∑i=1d∫D\|∂ϕn∂xi\|2dx. Letting $$n\to\infty$$, we have ∫D∫Y(ψ(x)−ψ(ε[xε]+εt))2dtdx≤3ε2∑i=1d∫D\|∂ψ∂xi\|2dx. Let $$T$$ be the linear map from $$L^2(D)$$ to $$L^2(D\times Y)$$ so that T(ϕ)(x,y)=ϕ(x)−ϕ(ε[xε]+εt). We thus have ‖T‖H1(D)→L2(D×Y)≤cε. On the other hand, ‖T‖L2(D)→L2(D×Y)≤c. From interpolation theory, we deduce that ‖T‖Hs(D)→L2(D×Y)≤cεs. We then get (B.1). The conclusion follows. □ Notes added after the proof stage: After the article is accepted, we learnt about the related recent article: P. Henning, M. Ohlberger and B. Verfürth (2016), A new heterogeneous multiscale method for time-harmonic Maxwell’s equations, SIAM J. Numer. Anal., 54, 3493–3522. This article considers a locally periodic two-scale time harmonic Maxwell equation, but the variational form is still assumed to be strictly coercive, uniformly with respect to the microscopic scale, similar to the equation considered in our present article. These authors formulate the two-scale homogenized equation in a slightly different manner. The Heterogeneous Multiscale Method (HMM) is used to solve the two-scale problem, and is shown to be equivalent to solving the two-scale homogenized equation by using the full tensor finite element spaces. © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Numerical Analysis Oxford University Press # High-dimensional finite elements for multiscale Maxwell-type equations , Volume 38 (1) – Jan 1, 2018 44 pages /lp/ou_press/high-dimensional-finite-elements-for-multiscale-maxwell-type-equations-labSJOLTE4 Publisher Oxford University Press ISSN 0272-4979 eISSN 1464-3642 D.O.I. 10.1093/imanum/drx001 Publisher site See Article on Publisher Site ### Abstract Abstract We consider multiscale Maxwell-type equations in a domain $$D\subset\mathbb{R}^d$$ ($$d=2,3$$), which depend on $$n$$ microscopic scales. Using multiscale convergence, we derive the multiscale homogenized problem, which is posed in $$\mathbb{R}^{(n+1)d}$$. Solving it, we get all the necessary macroscopic and microscopic information. Sparse tensor product finite elements (FEs) are employed, using edge FEs. The method achieves a required level of accuracy with essentially an optimal number of degrees of freedom, which, apart from a multiplying logarithmic term, is equal to that for solving a problem in $$\mathbb{R}^d$$. Numerical correctors are constructed from the FE solutions. In the two-scale case, an explicit homogenization error is deduced. To get this error, the standard procedure in the homogenization literature requires the solution $$u_0$$ of the homogenized problem to belong to $$H^1({\rm curl\,},D)$$. However, in polygonal domains, $$u_0$$ belongs only to a weaker regularity space $$H^s({\rm curl\,},D)$$ for $$0<s<1$$. We derive a homogenization error estimate for this case. Though we prove the result for two-scale Maxwell-type equations, the approach works verbatim for elliptic and elasticity problems when the solution to the homogenized equation belongs to $$H^{1+s}(D)$$ (standard procedure requires $$H^2(D)$$ regularity). This homogenization error estimate is new in the literature. Thus, for two-scale problems, an explicit error for the numerical corrector is obtained; it is of the order of the sum of the homogenization error and the FE error. For the case of more than two scales, we construct a numerical corrector, albeit without a rate of convergence, as such a homogenization error is not available. Numerical experiments confirm the theoretical results. 1. Introduction We consider Maxwell-type equations that depend on $$n$$ separable microscopic scales in a domain $$D\in \mathbb{R}^d,$$ where $$d=2,3$$. The coefficients are assumed to be locally periodic with respect to each microscopic scale. We use the multiscale convergence of a bounded sequence in $$H({\rm curl\,},D)$$ to derive the multiscale homogenized equation, which contains all the necessary information. Solving it, we get the solution of the homogenized equation that describes the multiscale solution macroscopically and the scale interacting terms (the corrector terms) that encode the multiscale information. However, this equation is posed in high-dimensional product domains. It depends on $$n+1$$ variables in $$\mathbb{R}^d$$, one for each scale that the original multiscale problem depends on. The full tensor product finite element (FE) method requires a large number of degrees of freedom, and thus is prohibitively expensive. We develop the sparse tensor FE product approach, using edge FEs, for this multiscale homogenized Maxwell-type equation. The approach achieves accuracy essentially equal to that obtained by the full tensor product FEs but requires an essentially optimal level of complexity that is essentially equal to that for solving a problem in $$\mathbb{R}^d$$ only. Analytic homogenization for two-scale Maxwell-type equations is well developed. We mention the standard references Bensoussan et al. (1978), Sanchez-Palencia (1980) and Jikov et al. (1994). However, there has been little effort on numerical analysis of multiscale Maxwell-type equations. As for other multiscale problems, a direct numerical treatment needs a fine mesh which is at most of the order of the smallest scale, leading to a prohibitive level of complexity. The multiscale FE method (Hou & Wu, 1997; Efendiev & Hou, 2009) and the heterogeneous multiscale method (E & Engquist, 2003; Abdulle et al., 2012) are designed to overcome this difficulty but their applications to multiscale Maxwell-type equations have not been adequately studied. Solving cell problems to establish the homogenized equation and using the cell problems’ solutions to compute the correctors for two-scale Maxwell-type equations are performed in Zhang et al. (2010). However, as for other multiscale problems, this approach is rather expensive, especially when the coefficients are only locally periodic, as for each macroscopic point, several cell problems need to be solved. We contribute in this article a feasible general numerical method for locally periodic multiscale Maxwell-type problems. We employ the sparse tensor product FE approach developed by Hoang & Schwab (2004/05) for multiscale elliptic equations (see also Hoang, 2008; Harbrecht & Schwab, 2011; Xia & Hoang, 2014, 2015a,b). It achieves the required level of accuracy with an essentially optimal number of degrees of freedom. We note that sparse tensor edge FEs are considered in Hiptmair et al. (2013) in the context of computing the moments of the solutions to stochastic Maxwell-type problems. However, our setting is quite different, and does not require constructing the detail spaces for edge FEs. We only need the detail spaces for the nodal FEs that approximate functions in the Lebesgue spaces $$L^2$$. We then construct a numerical corrector for the solution of the original multiscale problem, using the FE solutions of the multiscale homogenized problem. In the case of two scales, we derive an explicit error estimate in terms of the homogenization error and the FE error. It is well known that for two-scale elliptic problems in a domain $$D$$, if the solution of the homogenized problem belongs to $$H^2(D)$$, and the solutions to the cell problems are sufficiently smooth, the homogenization error in the $$H^1(D)$$ norm is $${\mathcal O}(\varepsilon^{1/2}),$$ where $$\varepsilon$$ is the microscopic scale (Bensoussan et al., 1978; Jikov et al., 1994). For two-scale Maxwell-type equations, the $${\mathcal O}(\varepsilon^{1/2})$$ homogenization error in the $$H({\rm curl},D)$$ norm is obtained when the solution $$u_0$$ of the homogenized problem (4.7) belongs to $$H^1({\rm curl},D)$$. However, for polygonal domains that are of interest in FE discretization, $$u_0$$ generally belongs only to a weaker regularity space $$H^s({\rm curl\,},D)$$ for $$0<s<1$$ (see e.g., Hiptmair, 2002). For this case, we develop an approach to deriving a new homogenization error estimate. Though we present the result for Maxwell-type equations, the approach works verbatim for two-scale elliptic and elasticity problems when the solution to the homogenized problem is in $$H^{1+s}(D)$$. As far as we are aware, this is a new result in the homogenization theory and forms another main contribution of the article. For the case of more than two scales, an analytic homogenization error is not available. However, we can still derive a corrector from the FE solution of the multiscale homogenized problem, albeit without an explicit rate of convergence. This article is organized as follows. In the next section, we formulate the multiscale Maxwell-type equation. Homogenization of the multiscale Maxwell-type equation (2.4) is studied in Bensoussan et al. (1978) in the two-scale case, using two-scale asymptotic expansion. Here, we use the multiscale convergence method to study (2.4) in the general multiscale setting. We thus develop multiscale convergence for a bounded sequence in $$H({\rm curl\,},D)$$. Two-scale convergence for a bounded sequence in $$H({\rm curl\,},D)$$ is developed in Wellander & Kristensson (2003). Since we consider the general $$(n+1)$$-scale convergence and the limiting result that we use is in a slightly different form from that of Wellander & Kristensson (2003) in the two-scale case, so we present the proofs in full. FE approximations of the multiscale homogenized Maxwell-type problem are studied in Section 3. We prove the FE error estimates in cases of both full and sparse tensor product FE approximations. The errors are essentially equal (apart from a logarithmic multiplying factor), but the dimension of the sparse tensor product FE space is much lower than that of the full tensor product FE space and is essentially equal to that for solving a problem in $$\mathbb{R}^d$$ only. In Section 4, we construct numerical correctors for the solution to the original multiscale problem. For two-scale problems, we prove the general homogenization error estimate for the case where $$u_0$$ belongs to the weaker regularity space $$H^s({\rm curl\,},D),$$ where $$0<s<1$$. From that we deduce the error estimate for the numerical corrector, which is of the order of the sum of the homogenization error estimate and the FE error. For the case of more than two scales, we derive a numerical corrector but without a rate of convergence. In Section 5, we prove that the regularity required to get the FE error estimate for the sparse tensor product FEs and to get the homogenization error in the two-scale case is achievable. Section 6 contains numerical experiments that confirm our analysis. Finally, the two Appendices A and B contain the long proofs of some previous results: the proof of the homogenization error when $$u_0$$ belongs to a weaker regularity space is presented in Appendix A. Throughout the article, by $$\#$$ we denote the spaces of functions that are periodic with the period being the unit cube $$Y\subset \mathbb{R}^d$$. Repeated indices indicate summation. The notations $$\nabla$$ and $${\rm curl\,}$$ without indicating the variable explicitly denote the gradient and the $${\rm curl\,}$$ operator with respect to $$x$$ of a function of $$x$$ only, where $$\nabla_x$$ and $${\rm curl\,}_{\!x}$$ denote the partial gradient and partial $${\rm curl\,}$$ of a function depending on $$x$$ and also on other variables. We generally present the theoretical results for the three-dimensional case and mention the two-dimensional case only when it is necessary, as the two cases are largely similar. 2. Problem setting 2.1 Multiscale Maxwell-type problems Let $$D$$ be a domain in $$\mathbb{R}^d$$ ($$d=2,3$$). Let $$Y$$ be the unit cube in $$\mathbb{R}^d$$. By $$Y_1,\ldots,Y_n$$ we denote $$n$$ copies1 of $$Y$$. We denote by $${\bf Y}$$ the product set $$Y_1\times Y_2\times\cdots\times Y_n$$ and by $$\boldsymbol{y}\in{\bf Y}$$ the vector $$\boldsymbol{y}=(y_1,y_2,\ldots,y_n)$$. For each $$i=1,\ldots,n$$, we denote by $${\bf Y}_i$$ the set of vectors $$\boldsymbol{y}_i=(y_1,\ldots,y_i),$$ where $$y_j\in Y_j$$ for $$j=1,\ldots,i$$. For $$d=3$$, let $$a$$ and $$b$$ be functions with symmetric matrix values from $$D\times {\bf Y}$$ to $$\mathbb{R}^{d\times d}_{\rm sym}$$; $$a$$ and $$b$$ are continuous in $$D\times {\bf Y}$$ and are periodic with respect to each variable $$y_i$$ with the period being $$Y_i$$. We assume that for all $$x\in D$$ and $$\boldsymbol{y}\in{\bf Y}$$, and all $$\xi,\zeta\in \mathbb{R}^d$$, c∗\|ξ\|2≤aij(x,y)ξiξj, aij(x,y)ξiζj≤c∗\|ξ\|\|ζ\|,c∗\|ξ\|2≤bij(x,y)ξiξj, bij(x,y)ξiζj≤c∗\|ξ\|\|ζ\|, (2.1) where $$c_$$ and $$c^$$ are positive numbers; $$\|\cdot\|$$ denotes the Euclidean norm in $$\mathbb{R}^3$$. Let $$\varepsilon$$ be a small positive value, and $$\varepsilon_1,\ldots,\varepsilon_n$$ be $$n$$ functions of $$\varepsilon$$ that denote the $$n$$ microscopic scales that the problem depends on. We assume the following scale separation properties: for all $$i=1,\ldots,n-1$$, limε→0εi+1(ε)εi(ε)=0. (2.2) Without loss of generality, we assume that $$\varepsilon_1=\varepsilon$$. We define $$a^\varepsilon, b^\varepsilon: D\to\mathbb{R}^{d\times d}_{\rm sym}$$ as aε(x)=a(x,xε1,…,xεn), bε(x)=b(x,xε1,…,xεn). (2.3) Let W=H0(curl,D)={u∈L2(D)3, curlu∈L2(D)3, u×ν=0}, where $$\nu$$ denotes the outward normal vector on the boundary $$\partial D$$. Let $$f\in W'$$. We consider the problem curl(aε(x)curluε(x))+bε(x)uε(x)=f(x), (2.4) with the boundary condition $$u^{\varepsilon}\times \nu=0$$ on $$\partial D$$. We formulate this problem in the variational form as follows: find $$u^{\varepsilon}\in W$$ so that ∫D[aε(x)curluε(x)⋅curlϕ(x)+bε(x)uε(x)⋅ϕ(x)]dx=∫Df(x)⋅ϕ(x)dx (2.5) for all $$\phi\in W$$ (by $$\int_D\,f\cdot\phi\, {\rm d}x$$ we denote the duality pairing between $$W'$$ and $$W$$). The Lax–Milgram lemma guarantees the existence of a unique solution $$u^{\varepsilon}$$ that satisfies ‖uε‖W≤c‖f‖W′, (2.6) where the constant $$c$$ depends only on $$c_$$ and $$c^$$ in (2.1). For $$d=2$$, the matrix function $$b^\varepsilon:D\times{\bf Y}\to \mathbb{R}^{2\times 2}$$ is defined as above. As $${\rm curl\,}u^{\varepsilon}$$ is now a scalar function, $$a(x,\boldsymbol{y})$$ is a continuous function from $$D\times{\bf Y}$$ to $$\mathbb{R},$$ which is periodic with respect to each variable $$y_i$$ with the period being $$Y_i$$. In the place of (2.1), we have c∗≤a(x,y)≤c∗ ∀x∈D and y∈Y. The variational formulation in two dimensions becomes ∫D[aε(x)curluε(x)curlϕ(x)+bε(x)uε(x)⋅ϕ(x)]dx=∫Df(x)⋅ϕ(x)dx ∀ϕ∈W. (2.7) In the rest of the article, we present the results for the three-dimensional case and only mention the two-dimensional case when necessary; the results for two dimensions are similar. 2.2 Multiscale convergence We use multiscale convergence to derive the homogenized equation. We first recall the definition of multiscale convergence (see Nguetseng, 1989; Allaire, 1992; Allaire & Briane, 1996). Definition 2.1 A sequence of functions $$\{w^\varepsilon\}_\varepsilon\subset L^2(D)$$$$(n+1)$$-scale converges to a function $$w^0\in L^2(D\times {\bf Y})$$ if for all smooth functions $$\phi\in C^\infty(D\times{\bf Y}),$$ which are periodic with respect to $$y_i$$ with the period being $$Y_i$$ for $$i=1,\ldots,n$$, limε→0∫Dwε(x)ϕ(x,xε1,…,xεn)dx=∫D∫Yw0(x,y)ϕ(x,y)dydx. We have the following result. Proposition 2.2 From a bounded sequence in $$L^2(D),$$ we can extract an $$(n+1)$$-scale convergent subsequence. For a bounded sequence in $$H({\rm curl\,},D)$$, we have the following results on $$(n+1)$$-scale convergence. These results were first established in Wellander & Kristensson (2003) for the two-scale case. We present below the multiscale convergence of a bounded sequence in $$H({\rm curl\,},D),$$ which will be used to study the multiscale equations (2.5) and (2.7). By $$\tilde H_\#({\rm curl\,},Y_i)$$ we denote the equivalent classes of functions in $$H_\#({\rm curl\,},Y_i)$$ such that if $${\rm curl\,} v={\rm curl\,} w$$ we regard $$v=w$$ in $$\tilde H_\#({\rm curl\,},Y_i)$$. Proposition 2.3 Let $$\{w^\varepsilon\}_\varepsilon$$ be a bounded sequence in $$H({\rm curl\,},D)$$. There is a subsequence (not renumbered), a function $$w_0\in H({\rm curl\,},D)$$, $$n$$ functions $${\frak w_i}\in L^2(D\times Y_1\times\cdots\times Y_{i-1},H^1_\#(Y_i)/\mathbb{R})$$ such that wε⟶(n+1)−scalew0+∑i=1n∇yiwi. Further, there are $$n$$ functions $$w_i\in L^2(D\times Y_1\times\cdots\times Y_{i-1},\tilde H_\#({\rm curl\,},Y_i))$$ such that curlwε⟶(n+1)−scalecurlw0+∑i=1ncurlyiwi. Proof. Let $$\xi\in L^2(D\times{\bf Y})^3$$ be the $$(n+1)$$-scale limit of $$\{w^\varepsilon\}_\varepsilon$$. Consider the function $$\phi=\varepsilon_n{\it{\Phi}}(x,y_1,\ldots,y_n),$$ where $${\it{\Phi}}$$ is a function in $$C^\infty_0(D,C^\infty_\#(Y_1,\ldots,C^\infty_\#(Y_n),\ldots))^3$$ and is periodic with respect to $$y_1, \ldots,y_n$$ with the period being $$Y_1,\ldots,Y_n,$$ respectively. We then have limε→0∫Dcurlwε⋅εnΦ(x,xε1,…,xεn)dx=0. On the other hand, limε→0∫Dcurlwε⋅εnΦ(x,xε1,…,xεn)dx = limε→0∫Dwε⋅εncurlΦ(x,xε1,…,xεn)dx = limε→0∫Dwε⋅curlynΦ(x,xε1,…,xεn)dx = ∫D∫Yξ(x,y)⋅curlynΦ(x,y)dydx. Thus, there is a function $$\xi_{n-1}(x,\boldsymbol{y}_{n-1})\in L^2(D\times{\bf Y}_{n-1})$$ and a function $${\frak w_n}(x,\boldsymbol{y}_n)\in L^2(D\times{\bf Y}_{n-1},H^1_\#(Y_n)/\mathbb{R})$$ such that ξ(x,y)=ξn−1(x,yn−1)+∇ynwn(x,y). Next we choose $$\phi=\varepsilon_{n-1}{\it{\Phi}}(x,y_1,\ldots,y_{n-1})$$ for a function $${\it{\Phi}}\in C^\infty_0(D,C^\infty_\#(Y_1,\ldots,C^\infty_\#(Y_{n-1}),\ldots),$$ which is periodic with respect to $$y_1,\ldots,y_{n-1}$$. We then have 0 = limε→0∫Dcurlwε⋅εn−1Φ(x,xε1,…,xεn−1)=limε→0∫Dwε⋅curlyn−1Φ(x,xε1,…,xεn−1)dx = ∫D∫Y(ξn−1(x,yn−1)+∇ynwn(x,y))⋅curlyn−1Φ(x,y1,…,yn−1)dyn−1dx = ∫D∫Yn−1ξn−1(x,yn−1)⋅curlyn−1Φ(x,y1,…,yn−1)dyn−1dx. From this, there is a function $$\xi_{n-2}(x,\boldsymbol{y}_{n-2})\in L^2(D\times {\bf Y}_{n-2})$$ and a function $${\frak w_{n-1}}(x,\boldsymbol{y}_{n-1})\in L^2(D\times {\bf Y}_{n-2},H^1_\#(Y_{n-1})/\mathbb{R})$$ so that ξn−1(x,yn−1)=ξn−2(x,yn−2)+∇yn−1wn−1(x,yn−1), so ξ(x,y)=ξn−2(x,yn−2)+∇yn−1wn−1(x,yn−1)+∇ynwn(x,y). Continuing this process, we have ξ(x,y)=w0(x)+∑i=1n∇yiwi(x,yi), where $$w_0\in L^2(D)^3$$ and $${\frak w_i}(x,\boldsymbol{y}_i)\in L^2(D\times {\bf Y}_{i-1},H^1_\#(Y_i))$$. As $$\int_Y\xi(x,\boldsymbol{y})\,{\rm d}\boldsymbol{y}=w_0(x)$$, $$w_0$$ is the weak limit of $$w^\varepsilon$$ in $$L^2(D)^3$$. Let $$\eta(x,\boldsymbol{y})$$ be the $$(n+1)$$-scale convergence limit of $${\rm curl\,} w^\varepsilon$$ in $$L^2(D\times{\bf Y})$$. Let $${\it{\Phi}}(x,y_1,\ldots,y_n)\in C^\infty_0(D,C^\infty_\#(Y_1,\ldots,C^\infty_\#(Y_n),\ldots))$$. We have ∫Dcurlwε⋅∇Φ(x,xε1,…,xεn)dx =∫Dwε⋅curl∇Φ(x,xε1,…,xεn)dx−∫∂D(wε×ν)⋅∇Φ(x,xε1,…,xεn)ds=0. Thus, 0 = limε→ 0∫Dcurlwε⋅εn∇Φ(x,xε1,…,xεn)dx=limε→0∫Dcurlwε⋅∇ynΦ(x,xε1,…,xεn)dx = ∫D∫Yη(x,y)⋅∇ynΦ(x,y1,…,yn)dydx. Therefore, there is a function $$w_n(x,\boldsymbol{y}_n)\in L^2(D\times{\bf Y}_{n-1},\tilde H_\#({\rm curl\,},Y_n))$$ and a function $$\eta_{n-1}(x,\boldsymbol{y}_{n-1})\in L^2(D\times{\bf Y}_{n-1})$$ such that η(x,y)=ηn−1(x,yn−1)+curlynwn(x,y). Let $${\it{\Phi}}(x,y_1,\ldots,y_{n-1})\in C^\infty_0(D,C^\infty_\#(Y_1,\ldots,C^\infty_\#(Y_{n-1}),\ldots))$$. We have 0 = limε→ 0∫Dcurlwε⋅εn−1∇Φ(x,xε1,…,xεn−1)dx=limε→0∫Dcurlwε⋅∇yn−1Φ(x,xε1,…,xεn−1)dx = ∫D∫Y(ηn−1(x,yn−1)+curlynwn(x,y))⋅∇yn−1Φ(x,y1,…,yn−1)dydx = ∫D∫Yn−1ηn−1(x,yn−1)⋅∇yn−1ϕ(x,y1,…,yn−1)dyn−1dx. Therefore, there is a function $$w_{n-1}(x,\boldsymbol{y}_{n-1})\in L^2(D\times{\bf Y}_{n-2},\tilde H_\#({\rm curl\,},Y_{n-1}))$$ and a function $$\eta_{n-2}(x,\boldsymbol{y}_{n-2})\in L^2(D\times {\bf Y}_{n-2})^3$$ so that ηn−1(x,yn−1)=ηn−2(x,yn−2)+curlyn−1wn−1(x,yn−1) so η(x,y)=ηn−2(x,yn−2)+curlyn−1wn−1(x,yn−1)+curlynwn(x,y). Continuing, we find that there is a function $$\eta_0(x)\in L^2(D)^3$$ and functions $$w_i(x,\boldsymbol{y}_{i})\in L^2(D\times{\bf Y}_{i-1},\tilde H_\#({\rm curl\,},Y_i))$$ so that η(x,y)=η0(x)+∑i=1ncurlyiwi(x,yi). As for all $$\phi(x)\in C^\infty_0(D)^3$$ limε→0∫Dcurlwε(x)⋅ϕ(x)dx=∫Dη0(x)⋅ϕ(x)dx,$$\eta_0$$ is the weak limit of $${\rm curl\,} w^\varepsilon$$ in $$L^2(D)^3$$. Thus, $$\eta_0={\rm curl\,} w_0$$. We then get the conclusion. □ 2.3 Multiscale homogenized Maxwell-type problem From (2.6) and Proposition 2.3, we can extract a subsequence (not renumbered), a function $$u_0\in H_0({\rm curl\,}, D)$$, $$n$$ functions $${\frak u_i}\in L^2(D\times Y_1\times\cdots\times Y_{i-1},H^1_\#(Y_i)/\mathbb{R})$$ and $$n$$ functions $$u_i\in L^2(D\times Y_1\times\cdots\times Y_{i-1},\tilde H_\#({\rm curl\,},Y_i))$$ such that uε⟶(n+1)−scaleu0+∑i=1n∇yiui (2.8) and curluε⟶(n+1)−scalecurlu0+∑i=1ncurlyiui. (2.9) For $$i=1,\ldots,n$$, let $$W_i=L^2(D\times Y_1\times\cdots\times Y_{i-1},\tilde H_\#({\rm curl\,},Y_i))$$ and $$V_i=L^2(D\times Y_1\times\cdots\times Y_{i-1},H^1_\#(Y_i)/\mathbb{R})$$. We define the space $${\bf V}$$ as V=W×W1×⋯×Wn×V1×⋯×Vn. For $$\boldsymbol{v}=(v_0,\{v_i\},\{\frak v_i\})\in {\bf V}$$, we define the norm \|\|\|v\|\|\|=‖v0‖H(curl,D)+∑i=1n‖vi‖L2(D×Yi−1,H~#(curl,Yi))+∑i=1n‖vi‖L2(D×Yi−1,H#1(Yi)/R). We then have the following result. Proposition 2.4 We define $$\boldsymbol{u}=(u_0,\{u_i\}, \{\frak u_i\})\in{\bf V}$$. Then $$\boldsymbol{u}$$ satisfies B(u,v):=∫D∫Y[a(x,y)(curlu0+∑i=1ncurlyiui)⋅(curlv0+∑i=1ncurlyivi) +b(x,y) (u0+∑i=1n∇yiui)⋅(v0+∑i=1n∇yivi)]dydx=∫Df(x)⋅v0(x)dx (2.10) for all $$\boldsymbol{v}=(v_0,\{v_i\},\{\frak v_i\})\in {\bf V}$$. Proof. Let $$v_0\in C^\infty_0(D)^3$$, $$v_i\in C^\infty_0(D, C^\infty_\#(Y_1,\ldots,C^\infty_\#(Y_i),\ldots))^3$$ and $${\frak v_i}\in C^\infty_0(D,C^\infty_\#(Y_1,\ldots,C^\infty_\# (Y_i),\ldots))$$ for $$i=1,\ldots,n$$. Let the test function $$v$$ in (2.5) be v(x)=v0(x)+∑i=1nεi(vi(x,xε1,…,xεi)+∇vi(x,xε1,…,xεi)). We have ∫D[aε(x)curluε(x)⋅(curlv0(x)+∑i=1nεicurlxvi(x,xε1,…,xεi)+∑i=1n∑j=1iεiεjcurlyjvi(x,xε1,…,xεi) + ∑i=1nεicurl∇vi(x,xε1,…,xεi)) +bε(x)uε(x)⋅(v0(x)+∑i=1nεivi(x,xε1,…,xεi) +∑i=1nεi∇xvi(x,xε1,…,xεi)+∑i=1n∑j=1iεiεj∇yjvi(x,xε1,…,xεi))]dx =∫Df(x)⋅(v0(x)+∑i=1nεivi(x,xε1,…,xεi) +∑i=1nεi∇xvi(x,xε1,…,xεn)+∑i=1n∑j=1iεiεj∇yjvi(x,xε1,…,xεi)). Using multiscale convergence and the scale separation (2.2), letting $$\varepsilon$$ go to 0, we have ∫D∫Y[a(x,y)(curlu0+∑i=1ncurlyiui)⋅(curlv0+∑i=1ncurlyivi) +b(x,y)(u0+∑i=1n∇yiui)⋅(v0+∑i=1n∇yivi)]dydx =∫Df(x)⋅v0(x)dx+∫D∫Yf(x)⋅∑i=1n∇yivi(x,y1,…,yi)dydx =∫Df(x)⋅v0(x)dx. Using a density argument, we have (2.10). □ Proposition 2.5 The bilinear form $$B:{\bf V}\times{\bf V}\to \mathbb{R}$$ is coercive and bounded, i.e., there are positive constants $$C^$$ and $$C_$$ so that B(u,v)≤C∗\|\|\|u\|\|\|\|\|\|v\|\|\|andC∗\|\|\|u\|\|\|\|\|\|u\|\|\|≤B(u,u) (2.11) for all $$\boldsymbol{u},\boldsymbol{v}\in {\bf V}$$. Problem (2.10) thus has a unique solution. The convergence relations (2.8) and (2.9) hold for the whole sequence $$\{u^{\varepsilon}\}_\varepsilon$$. Proof. It is easy to see that there is a positive constant $$C^$$ such that B(u,v)≤C∗\|\|\|u\|\|\|\|\|\|v\|\|\|. Now we show that $$B$$ is coercive. We have from (2.1), B(u,u) ≥c∗∫D∫Y(\|curlu0+∑i=1ncurlyiui\|2+\|u0+∑i=1n∇yiui\|2)dydx ≥c∫D∫Y(\|curlu0\|2+∑i=1n\|curlyiui\|2+\|u0\|2+\|∇yiui\|2)dydx≥c\|\|\|u\|\|\|2. We then get the conclusion from Lax–Milgram lemma. □ 3. FE discretization Let $$D$$ be a polygonal domain in $$\mathbb{R}^3$$. We consider a hierarchy of simplices $$\mathcal{T}^l$$ ($$l=0,1,\ldots$$), where $$\mathcal{T}^{l+1}$$ is obtained from $$\mathcal{T}^l$$ by dividing each simplex in $$\mathcal{T}^l$$ into eight tedrahedra. The mesh size of $$\mathcal{T}^l$$ is $$h_l={\mathcal O}(2^{-l})$$. For each tedrahedron $$T$$, we consider the edge FE space R(T)={v: v=α+β×x, α,β∈R3}. When $$d=2$$, $$\mathcal{T}^{l+1}$$ is obtained from $$\mathcal{T}^l$$ by dividing each simplex in $$\mathcal{T}^l$$ into four congruent triangles. For each triangle $$T$$, we consider the edge FE space R(T)={v: v=(α1α2)+β(x2−x1)}, where $$\alpha_1,\alpha_2$$ and $$\beta$$ are constants. Alternatively, if the domain can be partitioned into a set of cubes, we can use edge FE on a cubic mesh instead (see Monk, 2003). We denote by $$\mathcal{P}_1(T)$$ the set of linear polynomials in each simplex $$T$$. In the following, we present the analysis for the three-dimensional case only; the two-dimensional case is similar. For the cube $$Y$$, we partition it into a hierarchy of simplices $$\mathcal{T}^l_\#,$$ which are distributed periodically. We consider the FE spaces Wl ={v∈H0(curl,D), v\|T∈R(T) ∀T∈Tl},Vl ={v∈H1(D), v\|T∈P1(T) ∀T∈Tl},W#l ={v∈H#(curl,Y), v\|T∈R(T) ∀T∈T#l} and V#l={v∈H#1(Y), v\|T∈P1(T) ∀T∈T#l}. For $$d=2,3$$, we have the following estimates (see Ciarlet, 1978; Monk, 2003): infvl∈Wl‖v−vl‖H(curl,D)≤chls(‖v‖Hs(D)d+‖curlv‖Hs(D)d) for all $$v\in H_0({\rm curl\,}, D)\bigcap H^s({\rm curl\,},D)$$; infvl∈W#l‖v−vl‖H#(curl,Y)≤chls(‖v‖Hs(Y)d+‖curlv‖Hs(Y)d) for all $$v\in H_\#({\rm curl\,}, Y)\bigcap H^s({\rm curl\,}, Y)$$; infvl∈Vl‖v−vl‖L2(D)≤chls‖v‖Hs(D) for all $$v\in H^s(D)$$; infvl∈V#l‖v−vl‖L2(Y)≤chls‖v‖Hs(Y) for all $$v\in H^s_\#(Y)$$ and infvl∈V#l‖v−vl‖H#1(Y)≤chls‖v‖H1+s(Y) for all $$v\in H^1_\#(Y)\bigcap H^{1+s}(Y)$$. 3.1 Full tensor product FEs As $$L^2(D\times{\bf Y}_{i-1},\tilde H_\#({\rm curl\,},Y_i))\cong L^2(D)\otimes L^2(Y_1)\otimes\cdots\otimes L^2(Y_{i-1})\otimes \tilde H_\#({\rm curl\,},Y_i),$$ we use the tensor product FE space Wil=Vl⊗V#l⊗⋯⊗V#l⏟i−1 times ⊗W#l to approximate $$u_i$$. Similarly, as $${\frak u_i}\in L^2(D\times{\bf Y}_{i-1},H^1_\#(Y))$$, we use the FE space Vil=Vl⊗V#l⊗⋯⊗V#l⏟i times to approximate $${\frak u_i}$$. We define the space Vl=Wl×W1l×⋯×Wnl×V1l×⋯×Vnl. The full tensor product FE approximating problem is, find $$\boldsymbol{u}^L\in{\bf V}^L$$ so that B(uL,vL)=∫Df(x)⋅v0L(x)dx ∀vL=(v0L,{viL},viL)∈VL. (3.1) To get an error estimate for this FE approximating problem, we define the following regularity spaces for $${\frak u_i}$$ and $$u_i$$. For the functions $$u_i$$, we define the regularity space $$\mathcal{H}_i$$ of functions $$w$$ in $$L^2(D\times Y_1\times\cdots\times Y_{i-1},H^1_\#({\rm curl\,},Y_i))$$ such that for all $$k=1,2,3$$, ∂w∂xk∈L2(D×Y1×⋯×Yi−1,H~#(curl,Yi)) and for all $$j=1,\ldots,i-1$$ and $$k=1,2,3$$, ∂w∂(yj)k∈L2(D×Y1×⋯×Yi−1,H~#(curl,Yi)). In other words, for all $$w\in \mathcal{H}_i$$, $$w$$ belongs to $$L^2(D\times Y_1\times\cdots\times Y_{i-1},H^1_\#({\rm curl\,},Y_i))$$, $$L^2(Y_1\times\cdots\times Y_{i-1},H^1(D,\tilde H_\#({\rm curl\,},Y_i)))$$ and $$L^2(D\times\prod_{k<i,k\ne j}Y_k,H^1_\#(Y_j,\tilde H_\#({\rm curl\,},Y_i)))$$ for $$j=1,\ldots,i-1$$. For $$0<s<1$$, we define the space $$\mathcal{H}^s_i$$ by interpolation. It consists of functions $$w$$ such that $$w$$ belongs to $$L^2(D\times Y_1\times\cdots\times Y_{i-1},H^s_\#({\rm curl\,},Y_i))$$, $$L^2(Y_1\times\cdots\times Y_{i-1},H^s(D,\tilde H_\#({\rm curl\,},Y_i)))$$ and $$L^2(D\times\prod_{k<i,k\ne j}Y_k,H^s_\#(Y_j,\tilde H_\#({\rm curl\,},Y_i)))$$. We equip $$\mathcal{H}_i^s$$ with the norm ‖w‖His =‖w‖L2(D×Y1×⋯×Yi−1,H#s(curl,Yi))+‖w‖L2(Y1×⋯×Yi−1,Hs(D,H~#(curl,Yi))) +∑j=1i−1‖w‖L2(D×∏k<i,k≠j,H#s(Yj,H~#(curl,Yi))). We then have the following lemma. Lemma 3.1 For $$w\in \mathcal{H}_i^s$$, infwl∈Wil‖w−wl‖L2(D×Y1×⋯×Yi−1,H~#(curl,Yi))≤chls‖w‖His. The proof of this lemma is similar to that for full tensor product FEs in Hoang & Schwab (2004/05) and Bungartz & Griebel (2004), using orthogonal projection. We refer to Hoang & Schwab (2004/05) and Bungartz & Griebel (2004) for details. We define $${\frak H_i}^s$$ as the space of functions $$w\in L^2(D\times Y_1\times\cdots\times Y_{i-1},H^{1+s}_\#(Y_i))$$ such that $$w\in L^2(Y_1\times\cdots\times Y_{i-1},H^s(D,H^1_\#(Y_i)))$$ and for all $$j=1,\ldots,i-1$$, $$w\in L^2(D\times\prod_{k<i,k\ne j}Y_k,H^s_\#(Y_{j},H^1_\#(Y_i)))$$. We then define the norm ‖w‖His =‖w‖L2(D×Y1×⋯×Yi−1,H#1+s(Yi))+‖w‖L2(Y1×⋯×Yi−1,Hs(D,H1(Yi))) +∑j=1i−1‖w‖L2(D×∏k<i,k≠jYk,Hs(Yj,H1(Yi))). We have the following result. Lemma 3.2 For $$w\in {\frak H_i}^s$$, infwl∈Vil‖w−wl‖L2(D×Y1×⋯×Yi−1,H#1(Yi))≤chls‖w‖His. We then define the regularity space Hs=Hs(curl,D)×H1s×⋯×Hns×H1s×⋯×Hns with the norm ‖w‖Hs=‖w0‖Hs(curl,D)+∑i=1n‖wi‖His+∑i=1n‖wi‖His for $$\boldsymbol{w}=(w_0,\{w_i\},\{\frak w_i\})\in \boldsymbol{\mathcal{H}}^s$$. We have the following approximation result. Lemma 3.3 For $$\boldsymbol{w}\in \boldsymbol{\mathcal{H}}^s$$ infwl∈Vl‖w−wl‖V≤chls‖w‖Hs. From the boundedness and coerciveness conditions (2.11), using Cea’s lemma, we deduce the following result. Proposition 3.4 If $$\boldsymbol{u}\in \boldsymbol{\mathcal{H}}^s$$, for the full tensor product FE approximating problem (3.1) we have the error estimate ‖u−uL‖V≤chLs‖u‖Hs. (3.2) 3.2 Sparse tensor product FEs We define the following orthogonal projection: Pl0:L2(D)→Vl,P#l0:L2(Y)→V#l with the convention $$P^{-10}=0$$, $$P^{-10}_\#=0$$. We define the following detail spaces: Vl=(Pl0−P(l−1)0)Vl, V#l=(P#l0−P#(l−1)0)Vl. Since Vl=⨁0≤i≤lViandV#l=⨁0≤i≤lV#i, the full tensor product spaces $$W_i^L$$ and $$V_i^L$$ are defined as WiL=(⨁0≤l0,…,li−1≤LVl0⊗V#l1⊗⋯⊗V#li−1)⊗W#L and ViL=(⨁0≤l0,…,li−1≤LVl0⊗V#l1⊗⋯⊗V#li−1)⊗V#L. We then define the sparse tensor product FE spaces as W^iL=⨁l0+⋯+li−1≤LVl0⊗V#l1⊗⋯⊗V#li−1⊗W#L−(l0+⋯+li−1) and V^iL=⨁l0+⋯+li−1≤LVl0⊗V#l1⊗⋯⊗V#li−1⊗V#L−(l0+⋯+li−1). The function $$\boldsymbol{u}$$ is approximated by the space V^L=WL⊗W^1L⊗⋯⊗W^nL⊗V^1L⊗⋯⊗V^nL. The sparse tensor product FE approximating problem is, find $$\widehat {\bf{u}}^L\in \hat{\bf V}^L$$ such that B(u^L,v^L)=∫Df(x)⋅v^0L(x)dx ∀v^L=(v^0L,{v^iL},{v^iL})∈V^L. (3.3) From the coerciveness and boundedness conditions in (2.11), using Cea’s lemma we deduce the error estimate for the sparse tensor product approximating problem ‖u−u^L‖V≤cinfv^L∈VL‖u−v^L‖V. To quantify the error estimate, we use the following regularity spaces. We define $$\hat{\mathcal{H}}_i$$ as the space of functions $$w\in L^2(D\times Y_1\times\cdots\times Y_{i-1},H^1_\#({\rm curl\,},Y_i)),$$ which are periodic with respect to $$y_j$$ with the period being $$Y_j$$ ($$j=1,\ldots,i-1$$) such that for any $$\alpha_0,\alpha_1,\ldots,\alpha_{i-1}\in \mathbb{N}_0^d$$ with $$\|\alpha_k\|\le 1$$ for $$k=0,\ldots,i-1$$, ∂\|α0\|+\|α1\|+⋯+\|αi−1\|∂xα0∂y1α1⋯∂yi−1αi−1w∈L2(D×Y1×⋯×Yi−1,H#1(curl,Yi)). We equip $$\hat{\mathcal{H}}_i$$ with the norm ‖w‖H^i=∑αj∈Rd,\|αj\|≤10≤j≤i−1‖∂\|α0\|+\|α1\|+⋯+\|αi−1\|∂xα0∂y1α1⋯∂yi−1αi−1w‖L2(D×Y1×⋯×Yi−1,H#1(curl,Yi)). We can write $$\hat{\mathcal{H}}_i$$ as $$H^1(D,H^1_\#(Y_1,\ldots,H^1_\#(Y_{i-1},H^1_\#({\rm curl\,},Y_i)),\ldots))$$. By interpolation, we define $$\hat{\mathcal{H}}_i^s=H^s(D,H^s_\#(Y_1,\ldots,H^s_\#(Y_{i-1},H^s_\#({\rm curl\,},Y_i)),\ldots))$$ for $$0<s<1$$. We define $$\hat{\frak H}_i$$ as the space of functions $$w\in L^2(D\times Y_1\times\cdots\times Y_{i-1},H^2_\#(Y_i))$$ that are periodic with respect to $$y_j$$ with the period being $$Y_j$$ for $$j=1,\ldots,i-1$$ such that $$\alpha_0,\alpha_1,\ldots,\alpha_{i-1}\in \mathbb{N}_0^d$$ with $$\|\alpha_k\|\le 1$$ for $$k=0,\ldots,i-1$$, ∂\|α0\|+\|α1\|+⋯+\|αi−1\|∂xα0∂y1α1⋯∂yi−1αi−1w∈L2(D×Y1×⋯×Yi−1,H#2(Yi)). The space $$\hat{\frak H}_i$$ is equipped with the norm ‖w‖H^i=∑αj∈Rd,\|αj\|≤10≤j≤i−1‖∂\|α0\|+\|α1\|+⋯+\|αi−1\|∂xα0∂y1α1⋯∂yi−1αi−1w‖L2(D×Y1×⋯×Yi−1,H#2(Yi)). We can write $$\hat{\frak H}_i$$ as $$H^1(D,H^1_\#(Y_1,\ldots,H^1_\#(Y_{i-1},H^2_\#(Y_i))))$$. By interpolation, we define the space $$\hat{\frak H}_i^s:=H^s(D,H^s(Y_1,\ldots,H^s(Y_{i-1},H^{1+s}_\#(Y_i))))$$. The regularity space $$\hat{\boldsymbol{\mathcal{H}}}^s$$ is defined as H^s=Hs(curl,D)×H^1s×⋯H^ns×H^1s×⋯×H^ns. Lemmas 3.5 and 3.6 present the approximating properties of functions in $$\hat{\mathcal{H}}_i^s$$ and $${\hat{\frak H}_i^s}$$. The proofs follow from those for sparse tensor products in Hoang & Schwab (2004/05) and Bungartz & Griebel (2004). Lemma 3.5 For $$w\in \hat{\mathcal{H}}_i^s$$, infwL∈W^iL‖w−wL‖L2(D×Y1×⋯×Yi−1,H#(curl,Yi))≤cLi/2hLs‖w‖H^is. Similarly we have the following lemma. Lemma 3.6 For $$w\in \hat{\frak H}_i^s$$, infwL∈V^iL‖w−wL‖L2(D×Y1×⋯×Yi−1,H#1(Yi))≤cLi/2hLs‖w‖H^is. From these lemmas we deduce the following result. Lemma 3.7 For $$\boldsymbol{w}\in \hat{\boldsymbol{\mathcal{H}}}^s$$, infwL∈V^L‖w−wL‖V≤cLn/2hLs‖w‖H^s. From this we deduce the following error estimate for the sparse tensor product FE problem (3.3). Proposition 3.8 If the solution $$\boldsymbol{u}$$ of problem (2.10) belongs to $$\hat{\boldsymbol{\mathcal{H}}}^s$$ then ‖u−u^‖V≤cLn/2hLs‖u‖H^s. Remark 3.9 The dimension of the full tensor product FE space $${\bf V}^L$$ is $${\mathcal O}(2^{dnL}),$$ which is very large when $$L$$ is large. The dimension of the sparse tensor product FE space $$\hat{\bf V}^L$$ is $${\mathcal O}(L^n2^{dL}),$$ which is essentially equal to the number of degrees of freedom for solving a problem in $$\mathbb{R}^d$$ obtaining the same level of accuracy. 4. Convergence in physical variables We employ the FE solutions for the multiscale homogenized Maxwell-type equation (2.10) in the previous section to derive numerical correctors for the solution $$u^{\varepsilon}$$ of the multiscale problem (2.4). In the two-scale case, we derive the homogenization error explicitly in terms of $$\varepsilon$$ so that an error in terms of the microscopic scale $$\varepsilon$$ and the mesh size is obtained for the numerical corrector. We consider the general case, where the solution $$u^0$$ of the homogenized problem belongs to the space $$H^s({\rm curl\,},D)$$ for $$0<s\le 1$$, thus generalizing the standard homogenization rate of convergence $$\varepsilon^{1/2}$$ for elliptic problems (see e.g, Bensoussan et al., 1978; Jikov et al., 1994). This is a new result in homogenization theory. We present it for two-scale Maxwell-type equations, but the procedure works verbatim for two-scale elliptic and elasticity problems, where the solutions of the homogenized problems belong to $$H^{1+s}(D)$$. We present this section for the case $$d=3$$; the case $$d=2$$ is similar. 4.1 Two-scale problems For the two-scale case, we denote the function $$a(x,\boldsymbol{y})$$ by $$a(x,y)$$. The two-scale homogenized equation becomes ∫D∫Y[a(x,y)(curlu0+curlyu1)⋅(curlv0+curlyv1)+b(x,y)(u0+∇yu1)⋅(v0+∇yv1)]dydx =∫Df(x)⋅v0(x)dx. We first let $$v_0=0$$, $$v_1=0$$ and deduce that ∫D∫Yb(x,y)(u0+∇yu1)⋅∇yv1dydx=0. For each $$r=1,2,3$$, let $$w^r(x,\cdot)\in L^2(D, H^1_\#(Y)/\mathbb{R})$$ be the solution of the problem ∫D∫Yb(x,y)(er+∇ywr)⋅∇yψdydx=0 ∀ψ∈L2(D,H#1(Y)/R), (4.1) where $$e_r$$ is the vector in $$\mathbb{R}^3$$ with all the components being 0, except the $$r$$th component, which equals 1. This is the standard cell problem in elliptic homogenization. From this we have u1(x,y)=wr(x,y)u0r(x). (4.2) Therefore, ∫D∫Yb(x,y)(u0+∇yu1)⋅v0dxdy=∫Db0(x)u0(x)⋅v0(x)dx, where the positive-definite matrix $$b^0(x)$$ is defined as bij0(x)=∫Yb(x,y)(ej+∇wj(x,y))⋅(ei+∇ywi(x,y))dy, (4.3) which is the usual homogenized coefficient for elliptic problems with the two-scale coefficient matrix $$b^\varepsilon$$. Let $$v_0=0$$ and $${\frak v_1}=0$$. We have ∫D∫Ya(x,y)(curlu0+curlyu1)⋅curlyv1dydx=0 for all $$v_1\in L^2(D, \tilde H_\#({\rm curl\,},Y))$$. For each $$r=1,2,3$$, let $$N^r\in L^2(D,\tilde H_\#({\rm curl\,},Y))$$ be the solution of ∫D∫Ya(x,y)(er+curlyNr)⋅curlyvdydx=0 (4.4) for all $$v\in L^2(D, \tilde H_\#({\rm curl\,},Y))$$. We have u1=(curlu0(x))rNr(x,y). (4.5) The homogenized coefficient $$a^0$$ is determined by aij0(x)=∫Ya(x,y)ip(ejp+(curlyNj)p)dy=∫Ya(x,y)(ej+curlyNj)⋅(ei+curlyNi)dy. (4.6) We have ∫D∫Ya(x,y)(curlu0+curlyu1)⋅curlv0dxdy=∫Da0(x)curlu0(x)⋅curlv0(x)dx. The homogenized problem is ∫D[a0(x)curlu0(x)⋅curlv0(x)+b0(x)u0(x)⋅v0(x)]dx=∫Df(x)⋅v0(x)dx ∀v0∈H0(curl,D). (4.7) Following the procedure for deriving the homogenization error (Bensoussan et al., 1978; Jikov et al., 1994), we have the following homogenization error estimate. Theorem 4.1 Assume that $$a\in C(\bar D, C(\bar Y))^{3\times 3}$$, $$u_0\in H^1({\rm curl\,};D)$$, $$N^r\in C^1(\bar D,C(\bar Y))^3$$, $${\rm curl}_yN^r\in C^1(\bar D,C(\bar Y))^3$$, $$w^r\in C^1(\bar D,C^1(\bar Y))$$ for all $$r=1,2,3$$, then2 ‖uε−[u0+∇yu1(⋅,⋅ε)]‖L2(D)3≤cε1/2 and ‖curluε−[curlu0+curlyu1(⋅,⋅ε)]‖L2(D)3≤cε1/2. The proof of this theorem uses the functions $$G_r$$ and $$g_r$$ defined in (A.2) and (A.3) below. For $$u_0\in H^s({\rm curl\,},D)$$ when $$0<s<1$$, we have the following homogenization error estimate. Theorem 4.2 Assume that $$a\in C(\bar D,C(\bar Y))^{3\times 3}$$, $$u_0\in H^s({\rm curl\,},D)$$, $$N^r\in C^1(\bar D, C(\bar Y))^3$$, $${\rm curl}_yN^r\in C^1(\bar D,C(\bar Y))^3$$ and $$w^r\in C^1(\bar D,C^1(\bar Y))$$ for all $$r=1,2,3$$, then ‖uε−[u0+∇yu1(⋅,⋅ε)‖L2(D)3≤cεs/(1+s) and ‖curluε−[curlu0+curlyu1(⋅,⋅ε)]‖L2(D)3≤cεs/(1+s). We present the proof of this theorem in Appendix A. To employ the FE solutions to construct numerical correctors for $$u^{\varepsilon}$$, we define the following operator: Uε(Φ)(x)=∫YΦ(ε[xε]+εz,{xε})dz. (4.8) Let $$D^\varepsilon$$ be a $$2\varepsilon$$ neighbourhood of $$D$$. Regarding $${\it{\Phi}}$$ as zero when $$x$$ is outside $$D$$, we have ∫DεUε(Φ)(x)dx=∫D∫YΦ(x,y)dxdy. (4.9) The proof of (4.9) may be found in Cioranescu et al. (2008). We have the following result. Lemma 4.3 Assume that for $$r=1,2,3$$, $${\rm curl}_yN^r(x,y)\in C^1(\bar D, C(\bar Y))^3$$ and $$u_0\in H^s({\rm curl\,},D)$$, then ‖curlyu1(⋅,⋅ε)−Uε(curlyu1)‖L2(D)3≤cεs. We prove this lemma in Appendix B. We then have the following result. Theorem 4.4 Assume that $$a\in C(\bar D,C(\bar Y))^{3\times 3}$$, $$u_0\in H^s({\rm curl\,},D)$$, $$N^r\in C^1(\bar D,C(\bar Y))^3$$, $${\rm curl}_yN^r\in C^1(\bar D,C(\bar Y))^3$$ and $$w^r\in C^1(\bar D,C^1(\bar Y))$$ for all $$r=1,2,3$$. Then for the full tensor product FE solution $$(u_0^L,u_1^L,{\frak u_1}^L)$$ we have ‖uε−u0L−Uε(∇uu1L)‖L2(D)3≤c(εs/(1+s)+hLs) and ‖curluε−curlu0L−Uε(curlyu1L)‖L2(D)3≤c(εs/(1+s)+hLs). Proof. From Lemma 4.3, we have ‖curluε−curlu0L−Uε(curlyu1L)‖L2(D)3 ≤‖curluε−curlu0−curlyu1(⋅,⋅ε)‖L2(D)3 +‖curlu0−curlu0L‖L2(D)3+‖curlyu1(⋅,⋅ε)−Uε(curlyu1)‖L2(D)3 +‖Uε(curlyu1)−Uε(curlyu1L)‖L2(D)3. Using the fact that $$(\mathcal{U}^\varepsilon({\it{\Phi}}))^2\le \mathcal{U}({\it{\Phi}}^2)$$ and (4.9), we have ‖Uε(curlyu1)−Uε(curlyu1L)‖L2(D)3≤‖curlyu1−curlyu1L‖L2(D×Y)3≤chLs. This together with (3.2), Theorem 4.2 and Lemma 4.3 gives ‖curluε−curlu0L−Uε(curlyu1L)‖L2(D)3≤c(εs/(1+s)+hLs). Similarly, we have ‖uε−u0L−Uε(∇uu1L)‖L2(D)3≤c(εs/(1+s)+hLs). □ For the sparse tensor product FE approximation, we have the following result. Theorem 4.5 Assume that $$a\in C(\bar D,C(\bar Y))^{3\times 3}$$, $$u_0\in H^s({\rm curl\,},D)$$, $$N^r\in C^1(\bar D,C(\bar Y))^3$$, $${\rm curl}_yN^r\in C^1(\bar D,C(\bar Y))^3$$ and $$w^r\in C^1(\bar D,C^1(\bar Y))$$ for all $$r=1,2,3$$. Then for the sparse tensor product FE solution $$(\hat u_0^L,\hat u_1^L,\hat{\frak u}_1^L)$$ we have ‖uε−u^0L−Uε(∇yu^1L)‖L2(D)3≤c(εs/(1+s)+Ln/2hLs) and ‖curluε−curlu^0L−Uε(curlyu^1L)‖L2(D)3≤c(εs/(1+s)+Ln/2hLs). 4.2 Multiscale problems For multiscale problems, we do not have an explicit homogenization rate of convergence. However, for the case where $$\varepsilon_i/\varepsilon_{i+1}$$ is an integer for all $$i=1,\ldots,n-1$$ we can derive a corrector for the solution $$u^{\varepsilon}$$ of the multiscale problem from the FE solutions of the multiscale homogenized problem. For each function $$\phi\in L^1(D),$$ which is understood as 0 outside $$D$$, we define a function in $$L^1(D\times{\bf Y})$$: Tnε(ϕ)(x,y)=ϕ(ε1[xε1]+ε2[y1ε2/ε1]+⋯+εn[yn−1εn/εn−1]+εnyn). Letting $$D^{\varepsilon_1}$$ be the $$2\varepsilon_1$$ neighbourhood of $$D$$, we have ∫Dϕdx=∫Dε1∫Y1⋯∫YnTnε(ϕ)dyn⋯dy1dx (4.10) for all $$\phi \in L^1(D)$$. If a sequence $$\{\phi^\varepsilon\}_\varepsilon$$$$(n+1)$$-scale converges to $$\phi(x,y_1,\ldots,y_n)$$ then Tnε(ϕ)⇀ϕ(x,y1,…,yn) in $$L^2(D\times Y_1\times\ldots\times Y_n)$$. Thus, when $$\varepsilon\to 0$$, Tnε(curluε)⇀curlu0+curly1u1+⋯+curlynun (4.11) and Tnε(uε)⇀u0+∇y1u1+⋯+∇ynun (4.12) in $$L^2(D\times{\bf Y})^3$$. To deduce an approximation of $$u^{\varepsilon}$$ in $$H({\rm curl\,},D)$$ in terms of the FE solution, we use the operator $$\mathcal{U}_n^\varepsilon$$ , which is defined as Unε(Φ)(x) =∫Y1⋯∫YnΦ(ε1[xε1]+ε1t1,ε2ε1[ε1ε2{xε1}] +ε2ε1t2,⋯, εnεn−1[εn−1εn{xεn−1}]+εnεn−1tn,{xεn})dtn⋯dt1 for all functions $${\it{\Phi}}\in L^1(D\times{\bf Y})$$. For each function $${\it{\Phi}}\in L^1(D\times{\bf Y})$$ we have ∫Dε1Unε(Φ)dx=∫D∫YΦ(x,y)dydx. (4.13) The proofs for these facts may be found in Cioranescu et al. (2008). We then have the following corrector result. Proposition 4.6 The solution $$u^{\varepsilon}$$ of problem (2.5) and the solution $$(u_0,\{u_i\}\,\{{\frak u_i}\})$$ of problem (2.10) satisfies limε→0‖uε−[u0+Unε(∇y1u1)+⋯+Unε(∇ynun)‖L2(D)3=0 (4.14) and limε→0‖curluε−[curlu0+Unε(curly1u1)+⋯+Unε(curlynun)]‖L2(D)3=0. (4.15) Proof. We consider the expression ∫D∫Y[Tnε(aε)(Tnε(curluε)−(curlu0+curly1u1+⋯+curlynun)) ⋅(Tnε(curluε)−(curlu0+curly1u1+⋯+curlynun)) +Tnε(bε)(Tnε(uε)−(u0+∇y1u1+⋯+∇ynun))⋅(Tnε(uε)−(u0+∇y1u1+⋯+∇ynun))dydx. Using (2.5), (2.10), (4.10), (4.11) and (4.12), we deduce that this expression converges to 0. From (2.1) we have limε→0‖Tnε(curluε)−(curlu0+curly1u1+⋯+curlynun)‖L2(D×Y1×…×Yn)3=0 and limε→0‖Tnε(uε)−(u0+∇y1u1+⋯+∇ynun)‖L2(D×Y1×⋯×Yn)3=0. From (4.13) and the fact that $$\mathcal{U}^\varepsilon_n({\it{\Phi}})^2\le \mathcal{U}^\varepsilon_n({\it{\Phi}}^2)$$, we have ∫D\|Unε(Tnε(curluε)−(curlu0+curly1u1+⋯+curlynun)(x)\|2dx ≤∫DUnε(\|Tnε(curluε)−(curlu0+curly1u1+⋯+curlynun\|2)(x)\|dx ≤∫D∫Y\|Tnε(curluε)−(curlu0+curly1u1+⋯+curlynun\|2)dydx, which converges to 0 when $$\varepsilon\to 0$$. Using $$\mathcal{U}^\varepsilon_n(\mathcal{T}^\varepsilon_n({\it{\Phi}}))={\it{\Phi}}$$, we get (4.15). We derive (4.14) similarly. □ We then deduce the numerical corrector result. Theorem 4.7 For the full tensor product FE approximation solution $$\boldsymbol{u}^L=(u_0^L,\{u_i^L\},\{{\frak u_i}^L\})$$ in (3.1), we have limε→0L→∞‖uε−[u0L+Unε(∇y1u1L)+⋯+Unε(∇ynunL)]‖L2(D)3=0 (4.16) and limε→0L→∞‖curluε−[curlu0L+Unε(curly1u1L)+⋯+Unε(curlynunL)]‖L2(D)3=0. (4.17) Proof. We note that ‖Unε(curly1u1+⋯+curlynun)−Unε(curly1u1L+⋯+curlynunL)‖L2(D)3 ≤∫DUnε(\|(curly1u1+⋯+curlynun)−(curly1u1L+⋯+curlynunL)\|2)(x)dx ≤∫D∫Y\|(curly1u1+⋯+curlynun)−(curly1u1L+⋯+curlynunL)\|2dydx, which converges to 0 when $$L\to \infty$$. From this and (4.15), we get (4.17). We obtain (4.16) in the same way. □ Remark 4.8 As $${\bf V}^{\lceil L/n\rceil}\subset\hat{\bf V}^L$$, the result in Theorem 4.7 also holds for the sparse tensor product FE solution $$\widehat {\bf{u}}^L$$. Since we do not have an explicit homogenization error for problems with more than two scales, we do not distinguish the two cases of full and sparse tensor FE approximations. 5. Regularity of $$\boldsymbol{N^r}$$, $$\boldsymbol{w^r}$$ and $$\boldsymbol{u_0}$$ We show in this section that the regularity requirements for obtaining the sparse tensor product FE error estimate and the homogenization error estimate in the previous sections are achievable. We present the results for the two-scale case in detail. The multiscale case is similar; we summarize it in Remark 5.7. We first prove the following lemma. Lemma 5.1 Let $$\psi\in H_\#({\rm curl\,},Y)\bigcap H_\#({\rm div},Y)$$. Assume further that $$\int_Y\psi(y)\,{\rm d}y=0$$. Then $$\psi\in H^1_\#(Y)^3$$ and ‖ψ‖H1(Y)3≤c(‖curlyψ‖L2(Y)3+‖divyψ‖L2(Y)). Proof. Let $$\omega\subset\mathbb{R}^3$$ be a smooth domain such that $$\omega\supset Y$$. Let $$\eta\in \mathcal{D}(\omega)$$ be such that $$\eta(y)=1$$ when $$y\in Y$$. We have curly(ηψ)=ηcurlyψ+∇yη×ψ∈L2(ω)3 and divy(ηψ)=∇yη⋅ψ+ηdivyψ∈L2(ω)3. Together with the zero boundary condition, we conclude that $$\eta\psi\in H^1(\omega)^3$$ so $$\psi\in H^1(Y)^3$$. We note that ∫Y(divyψ(y)2+\|curlyψ(y)\|2)dy=∑i,j=13∫Y(∂ψi∂yj)2+∑i≠j∫Y∂ψi∂yi∂ψj∂yjdy−∑i≠j∫Y∂ψj∂yi∂ψi∂yjdy. Assume that $$\psi$$ is a smooth periodic function. We have ∫Y∂ψi∂yi∂ψj∂yjdy=∫Y[∂∂yi(ψi∂ψj∂yj)−ψi∂2ψj∂yi∂yj]dy=−∫Yψi∂2ψj∂yi∂yjdy as $$\psi$$ is periodic. Similarly, we have ∫Y∂ψi∂yj∂ψj∂yidy=∫Y[∂∂yj(ψi∂ψj∂yi)−ψi∂2ψj∂yj∂yi]dy=−∫Yψi∂2ψj∂yj∂yidy. Thus, ∫Y∂ψi∂yi∂ψj∂yjdy=∫Y∂ψi∂yj∂ψj∂yidy. Therefore, ‖∇yψ‖L2(Y)32=‖divyψ‖L2(Y)2+‖curlyψ‖L2(Y)32. Using a density argument, this holds for all $$\psi\in H^1_\#(Y)^3$$. As $$\int_Y\psi(y)\,{\rm d}y=0$$, from the Poincaré inequality we deduce ‖ψ‖H1(Y)3≤c(‖divyψ‖L2(Y)+‖curlyψ‖L2(Y)3). □ Lemma 5.2 Let $$\alpha\in C^1_\#(\bar Y)^{3\times 3}$$ be uniformly bounded, positive definite and symmetric for all $$y\in \bar Y$$. Let $$F\in L^2(Y)$$, extending periodically to $$\mathbb{R}^3$$. Let $$\psi\in H^1_\#(Y)^3$$ satisfy the equation curly(α(y)curlyψ(y))=F(y). Then $${\rm curl}_y\psi\in H^1_\#(Y)^3$$ and ‖curlyψ‖H1(Y)3≤c(‖F‖L2(Y)3+‖ψ‖H1(Y)3). Proof. Let $$\omega\supset Y$$ be a smooth domain. Let $$\eta\in \mathcal{D}(\omega)$$ be such that $$\eta(y)=1$$ for $$y\in Y$$. We have curly(αcurly(ηψ)) =curly(αηcurlyψ)+curly(α∇yη×ψ) =ηcurly(αcurlyψ)+∇yη×(αcurlyψ)+curly(α∇yη×ψ). Let $$U=\alpha{\rm curl}_y(\eta\psi)$$. We have ‖curlyU‖L2(ω)3≤c(‖F‖L2(ω)+‖ψ‖H1(ω)3)≤c(‖F‖L2(Y)3+‖ψ‖H1(Y)3). Further, ‖U‖L2(ω)3=‖α(∇yη×ψ+ηcurlyψ)‖\|L2(ω)3≤c‖ψ‖H1(ω)3≤c‖ψ‖H1(Y)3. As $$\eta\in\mathcal{D}(\omega)$$, $$U$$ has compact support in $$\omega$$ so $$U$$ belongs to $$H_0({\rm curl\,},\omega)$$. Thus, we can write U=z+∇Φ, where $$z\in H^1_0(\omega)^3$$ and $${\it{\Phi}}\in H^1_0(\omega)$$ satisfy ‖z‖H1(ω)3≤c‖U‖H(curl,ω) and ‖Φ‖H1(ω)≤c‖U‖H(curl,ω). From $${\rm div}_y(\alpha^{-1}U)=0$$ we deduce that divy(α−1∇Φ)=−divy(α−1z)∈L2(ω). Since $$\alpha\in C^1(\bar\omega)^{3\times 3}$$ and is uniformly bounded and positive definite, $$\alpha^{-1}\in C^1(\bar\omega)^{3\times 3}$$ and is uniformly positive definite. Therefore, $${\it{\Phi}}\in H^2(\omega)$$ and satisfies ‖Φ‖H2(ω)≤c‖z‖H1(ω)3≤c‖U‖H(curl,ω). Thus, $$U\in H^1(\omega)^3$$ and $$\\|U\\|_{H^1(\omega)^3}\le c\\|U\\|_{H({\rm curl\,},\omega)}\le c(\\|F\\|_{L^2(Y)^3}+\\|\psi\\|_{H^1(Y)^3})$$. From $${\rm curl}_y(\eta\psi)=\alpha^{-1}U$$, we deduce that $${\rm curl}_y(\eta\psi)\in H^1(\omega)^3$$ so $${\rm curl}_y\psi\in H^1(Y)^3$$ and ‖curlyψ‖H1(Y)3≤c(‖F‖L2(Y)3+‖ψ‖H1(Y)3). □ We then prove the following result on the regularity of $$N^r$$. Proposition 5.3 Assume that $$a(x,y)\in C^1(\bar D,C^2(\bar Y))^{3\times 3}$$; then $${\rm curl}_yN^r(x,y)\in C^1(\bar D,C(\bar Y))^3$$ and we can choose a version of $$N^r$$ in $$L^2(D,\tilde H_\#({\rm curl\,},Y))$$ so that $$N^r(x,y)\in C^1(\bar D, C(\bar Y))^3$$. Proof. We can choose a version of $$N^r$$ so that $${\rm div}_yN^r=0$$. Indeed, let $${\it{\Phi}}(x,\cdot)\in L^2(D, H^1_\#(Y))$$ be such that $$\Delta_y{\it{\Phi}}=-{\rm div}_yN^r$$; then $${\rm curl}_y(N^r+\nabla_y{\it{\Phi}})={\rm curl}_y N^r$$ and $${\rm div}_y(N^r+\nabla_y{\it{\Phi}})=0$$. Further we can choose $$N^r$$ so that $$\int_YN^r(x,y)\,{\rm d}y=0$$. From Lemma 5.1, we have ‖Nr(x,⋅)‖H1(Y)3≤c‖curlyNr(x,⋅)‖L2(Y)3, which is uniformly bounded with respect to $$x$$. From (4.4) and Lemma 5.2, we deduce that ‖curlyNr(x,⋅)‖H1(Y)3≤c‖curly(a(x,⋅)er‖L2(Y)3+‖Nr(x,⋅)‖H1(Y)3, which is uniformly bounded with respect to $$x$$. For each index $$q=1,2,3$$, we have that $${\rm curl}_y{\partial\over\partial y_q}N^r(x,\cdot)$$ is uniformly bounded in $$L^2(Y)$$ and $${\rm div}_y{\partial\over\partial y_q}N^r(x,\cdot)=0$$. Therefore, from Lemma 5.1, $${\partial\over\partial y_q}N^r(x,\cdot)$$ is uniformly bounded in $$H^1(Y)$$. We note that ∂∂yq(curly(a(x,⋅)curlyNr))=−∂∂yqcurly(a(x,⋅)er)∈L2(Y). Thus, curly(acurly∂Nr∂yq)=∂∂yq(curly(a(x,y)curlyNr))−curly(∂a∂yqcurlyNr)∈L2(Y). From Lemma 5.2 we deduce that $${\rm curl}_y{\partial N^r\over\partial y_q}(x,\cdot)$$ is uniformly bounded in $$H^1(Y)$$ so that $${\rm curl}_y N^r(x,\cdot)$$ is uniformly bounded in $$H^2(Y)\subset C(\bar Y)$$. We now show that $${\rm curl}_yN^r\in C^1(\bar D,H^2(Y))^3\subset C^1(\bar D,C(\bar Y))^3$$. Fix $$h\in\mathbb{R}^3$$. From (4.4) we have curly(a(x,y)curly(Nr(x+h,y)−Nr(x,y))) =−curly((a(x+h,y)−a(x,y))er) −curly((a(x+h,y)−a(x,y))curlyNr(x+h,y)). The smoothness of $$a$$ and the uniform boundedness of $${\rm curl}_yN^r(x,\cdot)$$ in $$L^2(Y)^3$$ gives limh→0‖curly(Nr(x+h,⋅)−Nr(x,⋅))‖L2(Y)3=0. (5.1) From Lemma 5.1 we have that $$N^r(x+h,\cdot)-N^r(x,\cdot)\in H^1(Y)^3$$ and ‖Nr(x+h,⋅)−Nr(x,⋅)‖H1(Y)3≤c‖curly(Nr(x+h,⋅)−Nr(x,⋅))‖L2(Y)3, which converges to 0 when $$\|h\|\to 0$$. From Lemma 5.2, we have ‖curly(Nr(x+h,⋅)−Nr(x,⋅))‖H1(Y)3 ≤‖−curly((a(x+h,⋅)−a(x,⋅))er)−curly((a(x+h,⋅)−a(x,⋅))curlyNr(x+h,⋅))‖L2(Y)3 +‖Nr(x+h,⋅)−Nr(x,⋅)‖H1(Y)3→0 when \|h\|→0. (5.2) We have further that curly(a(x,y)curly∂∂yq(Nr(x+h,y)−Nr(x,y)))=−curly(∂a∂yq(x,y)curly(Nr(x+h,y)−Nr(x,y))) −∂∂yqcurly((a(x+h,y)−a(x,y))er)−∂∂yqcurly((a(x+h,y)−a(x,y))curlyNr(x+h,y)). (5.3) From this we have ‖curly∂∂yq(Nr(x+h,y)−Nr(x,y))‖L2(Y)3 ≤c‖curly(Nr(x+h,⋅)−Nr(x,⋅))‖L2(Y)3 +c‖a(x+h,⋅)−a(x,⋅)‖W1,∞(Y)3→0 when \|h\|→0, so from Lemma 5.1 we have ‖∂∂yq(Nr(x+h,y)−Nr(x,y))‖H1(Y)3→0 when \|h\|→0. As the right-hand side of (5.3) converges to 0 in the $$L^2(Y)^3$$ norm when $$\|h\|\to 0$$, we deduce from Lemma 5.2 that ‖curly∂∂yq(Nr(x+h,y)−Nr(x,y))‖H1(Y)3→0 when \|h\|→0. (5.4) We have curly[a(x,y)curly(Nr(x+h,y)−Nr(x,y)h)] =−curly((a(x+h,y)−a(x,y)h)er) −curly(a(x+h,y)−a(x,y)hcurlyNr(x+h,y)). Let $$\chi^r(x,\cdot)\in \tilde H_\#({\rm curl\,},Y)$$ with $${\rm div}_y\chi^r(x,)=0$$ be the solution of the problem curly(a(x,y)curlyχr(x,⋅))=−curly(∂a∂xqer)−curly(∂a∂xqcurlyNr(x,y)). We deduce that curly(a(x,y)curly(Nr(x+h,y)−Nr(x,y)h−χr(x,y))) =−curly((a(x+h,y)−a(x,y)h−∂a∂xq(x,y))er) −curly((a(x+h,y)−a(x,y)h−∂a∂xq(x,y))curlyNr(x+h,y)) −curly(∂a∂xq(x,y)curly(Nr(x+h,y)−Nr(x,y))):=I1. (5.5) Let $$h\in \mathbb{R}^3$$ be a vector with all components 0 except for the $$q$$th component. We have ‖curly(Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅))‖L2(Y)3 ≤c‖a(x+h,⋅)−a(x,⋅)h−∂a∂xq(x,⋅)‖L∞(Y) +c‖curly(Nr(x+h,⋅)−Nr(x,⋅)‖L2(Y)3, (5.6) which converges to 0 when $$\|h\|\to 0$$ due to (5.1). Thus, we deduce from Lemma 5.1 that lim\|h\|→0‖Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅)‖H1(Y)3=0. (5.7) From Lemma 5.2, we have lim\|h\|→0‖curly(Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅))‖H1(Y)3 ≤lim\|h\|→0‖I1(x,⋅)‖L2(Y)3+‖Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅)‖H1(Y)3=0 (5.8) due to (5.2) and (5.7). Let $$p=1,2,3$$. We then have curly(a(x,y)curly∂∂yp(Nr(x+h,y)−Nr(x,y)h−χr(x,y))) =−curly(∂a∂yp(x,y)curly(Nr(x+h,y)−Nr(x,y)h−χr(x,y))) −∂∂ypcurly((a(x+h,y)−a(x,y)h−∂a(x,y)∂xq)er) −∂∂ypcurly((a(x+h,y)−a(x,y)h−∂a∂xq(x,y))curlyNr(x+h,y)) −∂∂ypcurly(∂a∂xq(x,y)curly(Nr(x+h,y)−Nr(x,y))), which converges to 0 in $$L^2(Y)$$ for each $$x$$ due to (5.4), (5.8) and the uniform boundedness of $$\\|{\rm curl\,} N^r(x,\cdot)\\|_{H^2(Y)^3}$$. We have ‖curly∂∂yp(Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅))‖L2(Y)3 ≤c‖curly(Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅))‖L2(Y)3 +c‖a(x+h,⋅)−a(x,⋅)h−∂a(x,⋅)∂xq‖W1,∞(Y)3+c‖curly(Nr(x+h,⋅)−Nr(x,⋅)‖H1(Y)3, which converges to 0 when $$\|h\|\to 0$$, so from Lemma 5.1, lim\|h\|→0‖∂∂yp(Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅))‖H1(Y)3=0. (5.9) Therefore, $$N^r\in C^1(\bar D,H^2(Y))^3\subset C^1(\bar D,C(\bar Y))^3$$. We then get from Lemma 5.2 that lim\|h\|→0‖curly∂∂yp(Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅))‖H1(Y)3=0. Thus, $${\rm curl}_yN^r\in C^1(\bar D,H^2(Y))^3\subset C^1(\bar D,C(\bar Y))^3$$. □ Proposition 5.4 Assume that $$b(x,y)\in C^1(\bar D,C^2(\bar Y))^{3\times 3}$$. The solution $$w^r$$ of cell problem (4.1) belongs to $$C^1(\bar D, C^1(\bar Y))$$. Proof. The cell problem (4.1) can be written as −∇y⋅(b(x,y)∇ywr(x,y))=∇y(b(x,y)er). Fixing $$x\in \bar D$$, the right-hand side is bounded uniformly in $$H^1(Y)$$ so $$w^r(x,\cdot)$$ is uniformly bounded in $$H^3(Y)$$ from elliptic regularity (see McLean, 2000, Theorem 4.16). For $$h\in \mathbb{R}^3$$, we note that −∇y⋅[b(x,y)∇y(wr(x+h,y)−wr(x,y))] =∇y⋅[(b(x+h,y)−b(x,y))er] +∇y⋅[(b(x+h,y)−b(x,y))∇ywr(x+h,y)]:=i1. As $$\int_Yw^r(x,y)\,{\rm d}y=0$$, we have ‖wr(x+h,⋅)−wr(x,⋅)‖H1(Y) ≤c‖∇y(wr(x+h,⋅)−wr(x,⋅))‖L2(Y) ≤c‖(b(x+h,⋅)−b(x,⋅))er‖L2(Y) +c‖(b(x+h,⋅)−b(x,⋅))∇ywr(x+h,⋅)‖L2(Y), which converges to 0 when $$\|h\|\to 0$$. Fixing $$x\in\bar D$$, we then have from McLean (2000, Theorem 4.16) that ‖wr(x+h,⋅)−wr(x,⋅)‖H3(Y)≤‖wr(x+h,⋅)−wr(x,⋅)‖H1(Y)+‖i1(x,⋅)‖H1(Y), (5.10) which converges to 0 when $$\|h\|\to 0$$. Fixing an index $$q=1,2,3$$, let $$h\in \mathbb{R}^3$$ be a vector whose components are all zero except the $$q$$th component. Let $$\eta(x,\cdot)\in H^1_\#(Y)/\mathbb{R}$$ be the solution of the problem −∇y⋅[b(x,y)∇yη(x,y)]=∇y⋅[∂b∂xqer]+∇y⋅[∂b∂xq∇ywr(x,y)]. We have −∇y⋅[b(x,y)∇y(wr(x+h,y)−wr(x,y)h−η(x,y))] =∇y⋅[(b(x+h,y)−b(x,y)h−∂b∂xq(x,y))er] +∇y⋅[(b(x+h,y)−b(x,y)h−∂b∂xq(x,y))∇ywr(x+h,y)] +∇y⋅[∂b(x,y)∂xq(∇ywr(x+h,y)−∇ywr(x,y))]:=i2. From (5.10) and the regularity of $$b$$, $$\lim_{\|h\|\to 0}\\|i_2(x,\cdot)\\|_{H^1(Y)}=0$$. As $$\int_Yw^r(x,y)\,{\rm d}y=0$$ and $$\int_Y\eta(x,y)\,{\rm d}y=0$$, we have lim\|h\|→0‖wr(x+h,⋅)−wr(x,⋅)h−η(x,⋅)‖H1(Y)=0. Therefore from McLean (2000, Theorem 4.16), we have ‖wr(x+h,⋅)−wr(x,⋅)h−η(x,⋅)‖H3(Y)≤‖wr(x+h,⋅)−wr(x,⋅)h−η(x,⋅)‖H1(Y)+‖i2(x,⋅)‖H1(Y), which converges to 0 when $$\|h\|\to 0$$. Thus, $$w^r\in C^1(\bar D,H^3(Y))\subset C^1(\bar D,C^1(\bar Y))$$. □ For the regularity of the solution $$u_0$$ of the homogenized problem (4.7) we have the following result. Proposition 5.5 Assume that $$D$$ is a Lipschitz polygonal domain, and the coefficient $$a(\cdot,y)$$, as a function of $$x$$, is Lipschitz, uniformly with respect to $$y$$; then there is a constant $$0<s<1$$ so that $${\rm curl\,} u_0\in H^s(D)$$. Proof. When $$a(x,y)$$ is Lipschitz with respect to $$x$$, from (4.4), $$\\|{\rm curl}_y N^r(x,\cdot)\\|_{L^2(Y)}$$ is a Lipschitz function of $$x$$, so from (4.6) we have that $$a^0$$ is Lipschitz with respect to $$x$$. As $$a^0$$ is positive definite, $$(a^0)^{-1}$$ is Lipschitz. Let $$U=a^0{\rm curl\,} u_0$$. We have from (4.7) that $$U\in H({\rm curl\,},D)$$, $${\rm div}((a^0)^{-1}U)=0$$ and $$(a^0)^{-1}U\cdot \nu=0$$ on $$\partial D,$$ where $$\nu$$ is the outward normal vector on $$\partial D$$. The conclusion follows from Hiptmair (2002, Lemma 4.2). □ Remark 5.6 If $$a^0$$ is isotropic, we have from (4.7) that curlcurlu0=−(a0)−1∇a0×curlu0−(a0)−1b0u0+(a0)−1f∈L2(D)3 so $$u_0\in H^1({\rm curl},D)$$. However, even if $$a$$ is isotropic, $$a^0$$ may not be isotropic. Remark 5.7 The homogenized equation for the multiscale case is determined as follows. We denote by $$a^n(x,\boldsymbol{y})=a(x,\boldsymbol{y})$$. Recursively, for $$i=1,\ldots,n-1$$, the $$i$$-th level homogenized coefficient is determined as follows. For $$r=1,2,3$$, let $$N_{i+1}^r\in L^2(D\times{\bf Y}_i,\tilde H({\rm curl}_{y_{i+1}},Y_{i+1}))$$ be the solution of the cell problem ∫D∫Y1…∫Yi+1ai+1(x,yi,yi+1)(er+curlyi+1Ni+1r)⋅curlyi+1ψdyi+1dyidx=0 for all $$\psi\in L^2(D\times{\bf Y}_i,\tilde H({\rm curl}_{y_{i+1}},Y_{i+1}))$$. The $$i$$-th level homogenized coefficient $$a^{i}$$ is determined by arsi(x,yi)=∫Yi+1ai+1(x,yi,yi+1)(es+curlyi+1Ni+1s)⋅(er+curlyi+1Ni+1r)dyi+1. Let $$b^n(x,\boldsymbol{y})=b(x,\boldsymbol{y})$$. Similarly, let $$w_{i+1}^{r}\in L^2(D\times{\bf Y}_i,H^1_\#(Y_{i+1})/\mathbb{R})$$ be the solution of the problem ∫D∫Y1…∫Yi+1bi+1(x,yi,yi+1)(er+∇yi+1wi+1r)⋅∇yi+1ψdyi+1dyidx=0 for all $$\psi\in L^2(D\times{\bf Y}_i,H^1_\#(Y_{i+1})/\mathbb{R})$$. The $$i$$-th level homogenized coefficient $$b^i$$ is determined by brsi(x,yi)=∫Yi+1bi+1(x,yi,yi+1)(es+∇yi+1wi+1s)⋅(er+∇yi+1wi+1r)dyi+1. We then have the equation ∫D∫Yi[ai(x,yi)(curlu0+curly1u1+⋯+curlyiui)⋅(curlv0+curly1v1+⋯+curlyivi)dyidx +bi(x,yi)(u0+∇y1u1+⋯+∇yiui)⋅(v0+∇y1v1+⋯+∇yivi)]dyidx=∫Df(x)⋅v0(x)dx. The coefficients $$a^0(x)$$ and $$b^0(x)$$ are the homogenized coefficients. We have ui(x,yi) = [curlu0(x)r+curly1u1(x,y1)r+⋯+curlyi−1ui−1(x,yi−1)r]Nir(x,yi) =curlu0(x)r0(δr0r1+curly1N1r0(x,y1)r1)(δr1r2+curly2N2r1(x,y2)r2)⋯ (δri−2ri−1+curlyi−1Ni−1ri−2(x,yi−1)ri−1)Niri−1(x,yi). If $$a(x,\boldsymbol{y})\in C^1(\bar D,C^2(\bar Y_1,\ldots,C^2(\bar Y_n),\ldots))^{3\times 3}$$, by following the same procedure as above, we can show inductively that $${\rm curl}_{y_i}N_i^r(x,\boldsymbol{y}_i)\in C^1(\bar D,C^2(\bar Y_1,\ldots,C^2(\bar Y_{i-1},H^2(Y_i)),\ldots))$$ and $$a^i(x,\boldsymbol{y}_i)\in C^1(\bar D,C^2(\bar Y_1,\ldots,C^2(\bar Y_i),\ldots))$$. Thus, if $$u_0\in H^s({\rm curl\,},D)$$ for $$0<s\le 1$$, $$u_i\in \hat{\mathcal{H}}_i^s$$. Similarly, we can show that if $$b\in C^1(\bar D, C^2(\bar Y_1,\ldots,C^2(\bar Y_n)\ldots))$$ then $$w^{ir}\in C^1(\bar D,C^2(\bar Y_1,\ldots,C^2(\bar Y_{i-1},H^3(Y_i))\ldots))$$. As ui=u0r0(x)(δr0r1+∂w1r0∂y1r1(x,y1))…(δri−2ri−1+∂wi−1ri−2∂y(i−1)ri−1(x,yi−1))wiri−1(x,yi), if $$u_0\in H^s(D)$$, $${\frak u_i}\in \hat{\frak H}_i^s$$. 6. Numerical results The detail spaces $$\mathcal{V}^l$$ and $$\mathcal{V}^l_\#$$, which are difficult to construct in numerical implementations, are defined via orthogonal projection in Section 3.2. We employ Riesz basis functions and define equivalent norms, which facilitate the construction of these spaces. We make the following assumption. Assumption 6.1 (i) For each multidimensional vector $$j \in \mathbb{N}_0^d$$, there exists a set of indices $$I^j \subset \mathbb{N}^d_0$$ and a set of basis functions $$\phi^{jk}\in L^2(D)$$ for $$k\in I^j$$, such that $$V^l = \text{span}\left\{\phi^{jk} : \|\,j\|_{\infty}\le l\right\}$$. There are constants $$c_2>c_1>0$$ such that if $$\phi = \sum_{\|\,j\|_{\infty}\leq l,k\in I^j}\phi^{jk}c_{jk}\in V^l$$, then the following norm equivalences hold: c1∑\|j\|∞≤lk∈Ij\|cjk\|2≤‖ϕ‖L2(D)2≤c2∑\|j\|∞≤lk∈Ij\|cjk\|2, where $$c_1$$ and $$c_2$$ are independent of $$\phi$$ and $$l$$. (ii) For the space $$L^2(Y)$$, for each $$j \in \mathbb{N}_0^d$$, there exists a set of indices $$I^j_0 \subset \mathbb{N}_0^d$$ and a set of basis functions $$\phi^{jk}_0\in L^2(Y)$$, $$k\in I^j_0$$, such that $$V^l_\# = \text{span}\{\phi^{jk}_0 : \|\,j\|_{\infty}\le l\}$$. There are constants $$c_4>c_3>0$$ such that if $$\phi = \sum_{\|\,j\|_{\infty}\leq l,k\in I^{\,j}_{\,0}}\phi^{jk}_0c_{jk}\in V^l$$ then c3∑\|j\|∞≤lk∈I0j\|cjk\|2≤‖ϕ‖L2(Y)2≤c4∑\|j\|∞≤lk∈I0j\|cjk\|2, where $$c_3$$ and $$c_4$$ are independent of $$\phi$$ and $$l$$. Because of the norm equivalence, we can use $$\mathcal{V}^l=\text{span}\{\phi^{jk} : \|\,j\|_{\infty}= l\}$$ and $$\mathcal{V}^l_\#=\text{span}\{\phi^{jk}_0 : \|\,j\|_{\infty}= l\}$$ to construct the sparse tensor product FE spaces. Example 6.2 (i) We can construct a hierarchical basis for $$L^2(0,1)$$ as follows. We first take three piecewise linear functions as the basis for level $$j=0$$: $$\psi^{01}$$ obtains values $$(1,0)$$ at $$(0,1/2)$$ and is 0 in $$(1/2,1)$$, $$\psi^{02}$$ is piecewise linear and obtains values $$(0, 1, 0)$$ at $$(0, 1/2, 1)$$ and $$\psi^{03}$$ obtains values $$(0,1)$$ at $$(1/2, 1)$$ and is 0 in $$(0,1/2)$$. The basis functions for other levels are constructed from the wavelet function $$\psi$$ that takes values $$(0,-1,2,-1,0)$$ at $$(0,1/2,1,3/2,2)$$, the left boundary function $$\psi^{\rm left}$$ taking values $$(-2,2,-1,0)$$ at $$(0,1/2,1,3/2)$$ and the right boundary function $$\psi^{\rm right}$$ taking values $$(0, -1,2,-2)$$ at $$(1/2,1,3/2,2)$$. For levels $$j\geq 1$$, $$I^j=\{1,2,\ldots,2^j\}$$. The wavelet basis functions are defined as $$\psi^{j1}(x) = 2^{j/2}\psi^{\rm left}(2^j x)$$, $$\psi^{jk}(x)=2^{j/2}\psi(2^j x - k + 3/2)$$ for $$k = 2, \ldots, 2^j-1$$ and $$\psi^{j2^j} = 2^{j/2}\psi^{\rm right}(2^j x - 2^j+2)$$. This base satisfies Assumption 6.1 (i). (ii) For $$Y = (0,1)$$, we can construct a hierarchy of periodic basis functions for $$L^2(Y)$$ that satisfies Assumption 6.1 (ii) from those in (i). For level 0, we exclude $$\psi^{01}$$, $$\psi^{03}$$ and include the periodic piecewise linear function that takes values $$(1,0,1)$$ at $$(0,1/2,1),$$ respectively. At other levels, the functions $$\psi^{\rm left}$$ and $$\psi^{\rm right}$$ are replaced by the piecewise linear functions that take values $$(0,2, -1, 0)$$ at $$(0,1/2,1,3/2)$$ and values $$(0, -1,2,0)$$ at $$(1/2,1, 3/2 ,2),$$ respectively. When $$D=(0,1)^d$$, the basis functions can be constructed by taking the tensor products of the basis functions in $$(0,1)$$. They satisfy Assumption 6.1 after appropriate scaling (see Griebel & Oswald, 1995). Remark 6.3 When the norm equivalence for the basis functions in $$L^2(D)$$ and in $$L^2(Y)$$ does not hold, in many cases, we can still prove a rate of convergence similar to those in Lemmas 3.5 and 3.6 for the sparse tensor product FE approximations. For example, with the division of the domain $$D$$ into sets of triangles $$\mathcal{T}^l$$, the set of continuous piecewise linear functions with value 1 at one vertex and 0 at all the others forms a basis of $$V^l$$. Let $$S^l$$ be the set of vertices of the set of simplices $$\mathcal{T}^l$$. We can define $$\mathcal{V}^l$$ as the linear span of functions that are 1 at a vertex in $$S^l\setminus S^{l-1}$$ and 0 at all the other vertices. We can then construct the sparse tensor product FE approximations with these spaces but the norm equivalence does not hold. A rate of convergence for sparse tensor product FEs similar to those in Lemmas 3.5 and 3.6 can be deduced (see e.g, Hoang, 2008). In the first example, we consider a two-scale Maxwell-type equation in the two-dimensional domain $$D=(0,1)^2$$. The coefficients a(x,y)=(1+x1)(1+x2)(1+cos2⁡2πy1)(1+cos2⁡2πy2) and b(x,y)=1(1+x1)(1+x2)(1+cos2⁡2πy1)(1+cos2⁡2πy2). We can compute the homogenized coefficients exactly. In this case, a0=4(1+x1)(1+x2)9 and b0=23(1+x1)(1+x2). We choose f=(49(1+x1)(1+2x2−x1)+23(1+x1)(1+x2)x1x2(1−x2)49(1+x2)(1+2x1−x2)+23(1+x1)(1+x2)x1x2(1−x1)) so that the solution to the homogenized equation is u0=(x1x2(1−x2)x1x2(1−x1)). In Fig. 1, we plot the energy error versus the mesh size for the sparse tensor product FE approximations of the two-scale homogenized Maxwell-type problem. The figure agrees with the error estimate in Proposition 3.8. Fig. 1. View largeDownload slide The sparse tensor energy error $$B(\boldsymbol{u}-\widehat {\bf{u}}^L,\boldsymbol{u}-\widehat {\bf{u}}^L).$$ Fig. 1. View largeDownload slide The sparse tensor energy error $$B(\boldsymbol{u}-\widehat {\bf{u}}^L,\boldsymbol{u}-\widehat {\bf{u}}^L).$$ In the second example, we consider the case where $$b$$ is the identity matrix, i.e., it does not depend on $$y$$. In this case, from (2.10) we note that the function $${\frak u_1}=0$$. We choose a(x,y)=(1+x1)(1+x2)(1+cos2⁡2πy1)(1+cos2⁡2πy2) and f=(4(2π(1+x1)(1+x2)sin⁡2πx2+(1+x1)(cos⁡2πx1−cos⁡2πx2))9+12πsin⁡2πx24(2π(1+x1)(1+x2)sin⁡2πx1−(1+x2)(cos⁡2πx1−cos⁡2πx2))9+12πsin⁡2πx1) so that the solution to the homogenized problem is u0=(12πsin⁡2πx212πsin⁡2πx1). Figure 2 plots the energy error versus the mesh size for the sparse tensor product FE approximations for the two-scale homogenized Maxwell-type problem. The plot confirms the analysis. Fig. 2. View largeDownload slide The sparse tensor energy error $$B(\boldsymbol{u}-\widehat {\bf{u}}^L,\boldsymbol{u}-\widehat {\bf{u}}^L).$$ Fig. 2. View largeDownload slide The sparse tensor energy error $$B(\boldsymbol{u}-\widehat {\bf{u}}^L,\boldsymbol{u}-\widehat {\bf{u}}^L).$$ Acknowledgements The authors gratefully acknowledge a postgraduate scholarship of Nanyang Technological University, the AcRF Tier 1 grant RG69/10, the Singapore AStar SERC grant 122-PSF-0007 and the AcRF Tier 2 grant MOE 2013-T2-1-095 ARC 44/13. Footnotes 1 The notations $$Y_1,\ldots,Y_n$$, which denote the same unit cube $$Y$$, are introduced for convenience only, especially in the case where the Cartesian product of several of them is used, to avoid the necessity of indicating how many times the unit cube appears in the product. The functions $$a$$ and $$b$$ depend on the macroscopic scale only and are periodic with respect to $$y_i$$ with the period being the unit cube $$Y$$. 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( 2010) Multiscale computations for 3D time-dependent Maxwell’s equations in composite materials. SIAM J. Sci. Comput. , 32, 2560– 2583. Google Scholar CrossRef Search ADS Appendix A. We present the proof of Theorem 4.2 in this appendix. We consider a set of $$M$$ open cubes $$Q_i$$ ($$i=1,\ldots,M$$) of size $$\varepsilon^t$$ for $$t>0$$ to be chosen later such that $$D\subset\bigcup_{i=1}^MQ_i$$ and $$Q_i\bigcap D\ne\emptyset$$. Each cube $$Q_i$$ intersects with only a finite number, which does not depend on $$\varepsilon$$, of other cubes. We consider a partition of unity that consists of $$M$$ functions $$\rho_i$$ such that $$\rho_i$$ has support in $$Q_i$$, $$\sum_{i=1}^M\rho_i(x)=1$$ for all $$x\in D$$ and $$\|\nabla\rho_i(x)\|\le c\varepsilon^{-t}$$ for all $$x$$ (indeed such a set of cubes $$Q_i$$ and a partition of unity can be constructed from a fixed set of cubes of size $${\mathcal O}(1)$$ by rescaling). For $$r=1,2,3$$ and $$i=1,\ldots,M$$, we define Uir=1\|Qi\|∫Qicurlu0(x)rdx and Vir=1\|Qi\|∫Qiu0(x)rdx (as $$u_0\in H^s(D)^3$$ and $${\rm curl\,} u_0\in H^s(D)^3$$, for the Lipschitz domain $$D$$, we can extend each of them, separately, continuously outside $$D$$ and understand $$u_0$$ and $${\rm curl\,} u_0$$ as these extensions; see Wloka, 1987, Theorem 5.6). Let $$U_i$$ and $$V_i$$ denote the vectors $$(U_i^1, U_i^2,U_i^3)$$ and $$(V_i^1,V_i^2,V_i^3),$$ respectively. Let $$B$$ be the unit cube in $$\mathbb{R}^3$$. From the Poincaré inequality, we have ∫B\|ϕ−∫Bϕ(x)dx\|2dx≤c∫B\|∇ϕ(x)\|2dx ∀ϕ∈H1(B). By translation and scaling, we deduce that ∫Qi\|ϕ−1\|Qi\|∫Qiϕ(x)dx\|2dx≤cε2t∫Qi\|∇ϕ(x)\|2dx ∀ϕ∈H1(Qi), i.e., ‖ϕ−1\|Qi\|∫Qiϕ(x)dx‖L2(Qi)≤cεt‖ϕ‖H1(Qi). Together with ‖ϕ−1\|Qi\|∫Qiϕ(x)dx‖L2(Qi)≤c‖ϕ‖L2(Qi), we deduce from interpolation that ‖ϕ−1\|Qi\|∫Qiϕ(x)dx‖L2(Qi)≤cεts‖ϕ‖Hs(Qi) ∀ϕ∈Hs(Qi). Thus, ∫Qi\|curlu0(x)r−Uir\|2dx≤cε2ts‖(curlu0)r‖Hs(Qi)2. (A.1) Let u1ε(x)=u0(x)+εNr(x,xε)Ujrρj(x)+ε∇[wr(x,xε)Vjrρj(x)]. We have curl(aε(x)curlu1ε(x))+bε(x)u1ε(x) =curla(x,xε)[curlu0(x)+εcurlxNr(x,xε)Ujrρj(x)+curlyNr(x,xε)Ujrρj+ε(Ujr∇ρj)×Nr(x,xε)] +b(x,xε)[u0(x)+εNr(x,xε)Ujrρj(x)+ε∇xwr(x,xε)Vjrρj(x)+∇ywr(x,xε)Vjrρj(x) + εwr(x,xε)Vjr∇ρj(x)] =curl(a0(x)curlu0(x))+b0(x)u0(x)+curl[Gr(x,xε)Ujrρj(x)]+gr(x,xε)Vjrρj(x)+εcurlI(x) +εJ(x)+curl[(aε(x)−a0(x))(curlu0(x)−Ujρj(x))]+(bε(x)−b0(x))(u0(x)−Vjρj(x)), where the vector functions $$G_r(x,y)$$ and $$g_r(x,y)$$ are defined by (Gr)i(x,y) =air(x,y)+aij(x,y)curlyNr(x,y)j−air0(x), (A.2) (gr)i(x,y) =bir(x,y)+bij(x,y)∂wr∂yj(x,y)−bir0(x) (A.3) and I(x) =a(x,xε)[curlxNr(x,xε)Ujrρj(x)+(Ujr∇ρj(x))×Nr(x,xε)],J(x) =b(x,xε)[Nr(x,xε)Ujrρj(x)+∇xwr(x,xε)Vjrρj(x)+wr(x,xε)Vjr∇ρj(x)]. Therefore, for $$\phi\in W$$, ⟨curl(aεcurlu1ε)+bεu1ε−curl(a0curlu0)−b0u0,ϕ⟩ =∫DUjrρj(x)Gr(x,xε)⋅curlϕdx+∫DVjrρj(x)gr(x,xε)⋅ϕ(x)dx +ε∫DI(x)⋅curlϕ(x)dx+ε∫DJ(x)⋅ϕ(x)dx+∫D(aε−a0)(curlu0(x)−Ujρj)⋅curlϕ(x)dx +∫D(bε−b0)(u0−Vjρj)⋅ϕdx (here $$\langle\cdot\rangle$$ denotes the duality pairing between $$W'$$ and $$W$$). From (4.4), we have that $${\rm curl}_yG_r(x,y)=0$$. Further, from (4.6) $$\int_YG_r(x,y)\,{\rm d}y=0$$. Therefore, there is a function $$\tilde G_r(x,y)$$ such that $$G_r(x,y)=\nabla_y \tilde G_r(x,y)$$. From (4.1), we have $${\rm div}_yg_r(x,y)=0$$ and from (4.6) $$\int_Yg_r(x,y)\,{\rm d}y=0$$. Hence, there is a function $$\tilde g_r$$ such that $$g_r(x,y)={\rm curl}_y\tilde g_r(x,y)$$. As $$\nabla_y\tilde G_r(x,\cdot)=G_r(x,\cdot)\in H^1(Y)^3$$ so $$\Delta_y\tilde G_r(x,\cdot)\in L^2(Y)$$. Thus, $$\tilde G_r(x,\cdot)\in H^2(Y),$$ which implies $$\tilde G_r(x,\cdot)\in C(\bar Y)$$. As $$G_r(x,\cdot)\in C^1(\bar D,H^1_\#(Y)^3)$$, we deduce that $$\tilde G_r(x,y)\in C^1(\bar D, H^2(Y))\subset C^1(\bar D,C(\bar Y))$$. The construction of $$\tilde g_r$$ in Jikov et al. (1994) implies that $$\tilde g_r\in C^1(\bar D, C(\bar Y))$$ (see Hoang & Schwab, 2013). We have ∫DUjrρjGr(x,xε)⋅curlϕdx=∫DUjrρj(x)[ε∇G~r(x,xε)−ε∇xG~r(x,xε)]⋅curlϕdx =−ε∫DG~r(x,xε)div[(Ujrρj)curlϕ]dx−ε∫DUjrρj∇xG~r(x,xε)⋅curlϕdx. We note that \|∫DUjrρj∇xG~r(x,xε)⋅curlϕdx\|≤c‖(Ujrρj)‖L2(D)‖curlϕ‖L2(D)3. From ‖Ujrρj‖L2(D)2=∫D(Ujr)2ρj(x)2dx+∑i≠j∫DUirUjrρi(x)ρj(x)dx, and the fact that the support of each function $$\rho_i$$ intersects only with the support of a finite number (which does not depend on $$\varepsilon$$) of other functions $$\rho_j$$ in the partition of unity, we deduce ‖Ujrρj‖L2(D)2 ≤c∑j=1M(Ujr)2\|Qj\| =c∑j=1M1\|Qj\|(∫Qjcurlu0(x)rdx)2≤c∑j=1M∫Qjcurlu0(x)r2dx≤c∫Dcurlu0(x)r2dx. Thus, \|ε∫DUjrρj∇xG~r(x,xε)⋅curlϕdx\|≤cε‖curlϕ‖L2(D)3. We have further that ε∫DG~r(x,xε)div[(Ujrρj)curlϕ]dx =ε∫DG~r(x,xε)[(Ujr∇ρj(x))]⋅curlϕdx ≤cε‖Ujr∇ρj‖L2(D)3‖curlϕ‖L2(D)3. As the support of each function $$\rho_i$$ intersects with the support of a finite number of other functions $$\rho_j$$ and $$\\|\nabla\rho_j\\|_{L^\infty(D)}\le c\varepsilon^{-t}$$, we have ‖Ujr∇ρj‖L2(D)32≤c∑j=1M(Ujr)2\|Qj\|‖∇ρj‖L∞(D)2≤cε−2t∑j=1M(Ujr)2\|Qj\|≤cε−2t, so ε∫DG~r(x,xε)div[(Ujrρj)curlϕ]dx≤cε‖Ujr∇ρj‖L2(D)3‖curlϕ‖L2(D)3≤cε1−t‖curlϕ‖L2(D)3. We therefore deduce that \|∫DUjrρjGr(x,xε)⋅curlϕdx\|≤cε1−t‖curlϕ‖L2(D)3. We have ∫DVjrρjgr(x,xε)⋅ϕ(x)dx=∫DVjrρj[εcurlg~r(x,xε)−εcurlxg~r(x,xε)]⋅ϕdx. Arguing similarly to above, we have \|ε∫DVjrρjcurlxg~r(x,xε)⋅ϕdx\|≤cε‖Vjrρj‖L2(D)3‖ϕ‖L2(D)3≤cε‖ϕ‖L2(D)3 and \|ε∫DVjrρjcurlg~r(x,xε)⋅ϕdx\| = \|ε∫Dg~r(x,xε)⋅curl[(Vjrρj)ϕ]dx\| ≤ \|ε∫Dg~r(x,xε)⋅[(Vjrρj)curlϕ+ϕ×(Vjr∇ρj)]dx ≤c(ε‖curlϕ‖L2(D)3+cε1−t‖ϕ‖L2(D)3)(∑j=1M(Vjr)2\|Qj\|)1/2 ≤c(ε‖curlϕ‖L2(D)3+cε1−t‖ϕ‖L2(D)3). We note that ‖I‖L2(D)3≤csupr[‖Ujrρj‖L2(D)+‖Ujr∇ρj‖L2(D)]≤cε−t and ‖J‖L2(D)3≤csupr[‖Ujrρj‖L2(D)+c‖Vjrρj‖L2(D)+c‖Vjr∇ρj‖L2(D)]≤cε−t. We have further that ⟨curl((aε−a0)(curlu0−Ujρj)),ϕ⟩≤c‖curlu0−(Ujρj))‖L2(D)3‖curlϕ‖L2(D)3. From ∫D\|(curlu0)r−(Ujrρj)\|2dx=∫D\|∑j=1M((curlu0)r−Ujr)ρj\|2dx, using the support property of $$\rho_j$$, we have from (A.1), ∫D\|(curlu0)r−(Ujrρj)\|2dx ≤c∑j=1M∫Qj\|(curlu0)r−Ujr\|2dx≤cε2st∑j=1M‖(curlu0)r‖Hs(Qj)2 =cε2st∑j=1M[∫Qj(curlu0)r2dx+∫Qj×Qj(curlu0(x)r−curlu0(x′)r)2\|x−x′\|3+2sdxdx′] ≤cε2st[‖(curlu0)r‖L2(D)2+∫D×D(curlu0(x)r−curlu0(x′)r)2\|x−x′\|3+2sdxdx′] =cε2st‖(curlu0)r‖Hs(D)2. (A.4) Thus, ⟨curl((aε−a0)(curlu0−Ujρj)),ϕ⟩≤cεst‖curlϕ‖L2(D)3. Similarly, we have \|∫D(bε−b0)(u0−∑j=1MVjρj)⋅ϕdx\|≤c‖∑j=1M(u0−Vj)ρj‖L2(D)3‖ϕ‖L2(D)3≤cεst‖ϕ‖L2(D)3. Therefore, \|⟨curl(aεcurlu1ε)+bεu1ε−curl(a0curlu0)−b0u0,ϕ⟩\|≤c(ε1−t+εst)‖ϕ‖V i.e., ‖curl(aεcurlu1ε)+bεu1ε−curl(a0curlu0)−b0u0‖W′≤c(ε1−t+εst). Thus, ‖curl(aεcurlu1ε)+bεu1ε−curl(aεcurluε)−bεuε‖W′≤c(ε1−t+εst). (A.5) Let $$\tau^\varepsilon(x)$$ be a function in $$\mathcal{D}(D)$$ such that $$\tau^\varepsilon(x)=1$$ outside an $$\varepsilon$$ neighbourhood of $$\partial D$$ and $${\rm sup}_{x\in D}\varepsilon\|\nabla\tau^\varepsilon(x)\|<c,$$ where $$c$$ is independent of $$\varepsilon$$. We consider the function w1ε(x)=u0(x)+ετε(x)Ujrρj(x)Nr(x,xε)+ε∇[Vjrρjτε(x)wr(x,xε)]. We then have u1ε−w1ε=ε(1−τε(x))Ujrρj(x)Nr(x,xε)+ε∇[(1−τε(x))Vjrρjwr(x,xε)] and curl(u1ε−w1ε) =εcurlxNr(x,xε)Ujrρj(x)(1−τε(x))+curlyNr(x,xε)Ujrρj(x)(1−τε(x)) −εUjrρj(x)∇τε(x)×Nr(x,xε)+ε(1−τε(x))Ujr∇ρj(x)×Nr(x,xε). As shown above, $$\\|U_j^r\rho_j\\|_{L^2(D)}$$ is uniformly bounded, so ‖εcurlxNr(x,xε)(Ujrρj)(1−τε(x))‖L2(D)3≤cε. Let $$\tilde D^\varepsilon$$ be the $$3\varepsilon^{t}$$ neighbourhood of $$\partial D$$. We note that $${\rm curl\,} u_0$$ is extended continuously into a function in $$H^s(\mathbb{R}^3)$$ outside $$D$$. As shown in Hoang & Schwab (2013), for $$\phi\in H^1(\tilde D^\varepsilon)$$, ‖ϕ‖L2(D~ε)≤cεt/2‖ϕ‖H1(D~ε). From this and ‖ϕ‖L2(D~ε)≤‖ϕ‖L2(D~ε), using interpolation, we get ‖ϕ‖L2(D~ε)≤cεst/2‖ϕ‖Hs(D~ε)≤cεst/2‖ϕ‖Hs(D) for all $$\phi\in H^s(D)$$ extended continuously outside $$D$$. We then have ‖Ujrρj‖L2(Dε)2 ≤c∑j=1M∫Qj⋂Dε(Ujr)2ρj2dx ≤c∑j=1M\|Qj⋂Dε\|1\|Qj\|2(∫Qj(curlu0)rdx)2 ≤c∑Qj⋂Dε≠∅\|Qj⋂Dε\|\|Qj\|∫Qj(curlu0)r2dx. As $$D^\varepsilon$$ is the $$\varepsilon$$ neighbourhood of $$\partial D$$, $$\partial D$$ is Lipschitz and $$Q_j$$ has size $$\varepsilon^t$$, $$\|Q_j\bigcap D^\varepsilon\|\le c\varepsilon^{1+(d-1)t}$$ so $$\|Q_j\bigcap D^\varepsilon\|/\|Q_j\|\le c\varepsilon^{1-t}$$. When $$Q_j\bigcap D^\varepsilon\ne\emptyset$$, $$Q_j\subset\tilde D^\varepsilon$$. Thus, ‖Ujrρj‖L2(Dε)2≤cε1−t‖(curlu0)r‖L2(D~ε)2≤cε1−t+st‖curlu0‖Hs(D)32. Therefore, ‖curlyNr(x,xε)(Ujrρj)(1−τε(x))‖L2(D)3≤cε(1−t+st)/2 and ‖ε(Ujrρj)∇τε(x)×Nr(x,xε)‖L2(D)3≤cε(1−t+st)/2. Similarly, we have ‖Ujr∇ρj‖L2(Dε)32 ≤cε−2t∑Qj⋂Dε≠∅\|Qj⋂Dε\|\|Qj\|∫Qj(curlu0)r2dx ≤cε−2t+1−t‖curlu0‖L2(D~ε)32≤cε1−3t+st‖curlu0‖Hs(D)32. Thus, ‖ε(1−τε(x))(Ujr∇ρj)×Nr(x,xε)‖L2(D)≤cε(1−t)+(1−t+st)/2. Therefore, ‖curl(u1ε−w1ε)‖L2(D)3≤c(ε(1−t+st)/2+ε(1−t)+(1−t+st)/2). We further have ε∇[(1−τε(x))wr(x,xε)(Vjrρj)] = −ε∇τε(x)wr(x,xε)(Vjrρj)+ε(1−τε(x))∇xwr(x,xε)(Vjrρj) +(1−τε(x))∇ywr(x,xε)(Vjrρj)+ε(1−τε(x))wr(x,xε)(Vjr∇ρj). Arguing as above, we deduce that ‖Vjrρj‖L2(Dε)≤cε(1−t+st)/2, ‖Vjr∇ρj‖L2(Dε)≤cε(1−t+st)/2−t. Therefore, ‖ε∇[(1−τε(x))wr(x,xε)(Vjrρj)]‖L2(D)3≤c(ε(1−t+st)/2+ε1−t+(1−t+st)/2). Thus, ‖u1ε−w1ε‖L2(D)3≤c(ε(1−t+st)/2+ε(1−t)+(1−t+st)/2). Choosing $$t=1/(s+1)$$ we have ‖curl(aεcurl(u1ε−w1ε))+bε(u1ε−w1ε)‖W′≤cεs/(s+1). This together with (A.5) gives ‖curl(aεcurl(uε−w1ε))+bε(uε−w1ε)‖W′≤cεs/(s+1). Thus, ‖uε−w1ε‖W≤cεs/(s+1), which implies ‖uε−u1ε‖W≤cεs/(s+1). (A.6) We note that curlu1ε=curlu0(x)+curlyNr(x,xε)(Ujrρj)+εcurlxNr(x,xε)(Ujrρj)+ε(Ujr∇ρj)×Nr(x,xε). From ‖εcurlxNr(x,xε)(Ujrρj)‖L2(D)3≤cε and ‖ε(Ujr∇ρj)×Nr(x,xε)‖L2(D)3≤cεε−t=cεs/(1+s), we deduce that ‖curlu1ε−curlu0−curlyNr(x,xε)(Ujrρj)‖L2(D)3≤cεs/(s+1). From (A.4), ‖curlu0−(Ujrρj)‖L2(D)3≤cεts=cεs/(s+1), we get ‖curlu1ε−[curlu0+curlyNr(x,xε)(curlu0)r‖L2(D)3≤cεs/(s+1). This together with (A.6) implies ‖curluε−[curlu0+curlyNr(x,xε)(curlu0)r‖L2(D)3≤cεs/(s+1). □ Appendix B. We prove Lemma 4.3 in this appendix. We adapt the proof of Hoang & Schwab (2013, Lemma 5.5). As u1(x,y)=∑r=13curlu0(x)rNr(x,y), it is sufficient to show that for each $$r=1,2,3$$, ∫D\|curlu0(x)rcurlyNr(x,xε)−∫Ycurlu0(ε[xε]+εt)rcurlyNr(ε[xε]+εt,xε)dt\|2dx≤cε2s. The expression on the left-hand side is bounded by ∫D∫Y\|curlu0(x)rcurlyNr(x,xε)−curlu0(ε[xε]+εt)rcurlyNr(ε[xε]+εt,xε)\|2dtdx ≤2∫D∫Y\|(curlu0(x)r−curlu0(ε[xε]+εt)r)curlyNr(ε[xε]+εt,xε)\|2dtdx +2∫D∫Y\|curlu0(x)r\|2\|curlyNr(x,xε)−curlyNr(ε[xε]+εt,xε)\|2dtdx. As $${\rm curl}_yN^r\in C^1(\bar D,C(\bar Y))^3$$, there exists a constant $$c$$ such that supx∈Dsupt∈Y\|curlyNr(x,xε)−curlyNr(ε[xε]+εt,xε)\|≤cε. From this we have ∫D\|curlu0(x)rcurlyNr(x,xε)−Uε(curlu0(⋅)rcurlyNr(⋅,⋅))(x)\|2dx≤c∫D∫Y\|curlu0(x)r−curlu0(ε[xε]+εt)r\|2dtdx+cε2. We now show that for $${\rm curl\,} u_0\in H^s(D)$$, ∫D∫Y\|curlu0(x)r−curlu0(ε[xε]+εt)r\|2dtdx≤cε2s. (B.1) Letting $$\phi(x)$$ be a smooth function, we have ∫D∫Y\|ϕ(x)−ϕ(ε[xε]+εt)\|2dtdx ≤∑i=1d∫D∫Y\|ϕ(ε[x1ε]+εt1,…,ε[xi−1ε]+εti−1,xi,…,xd) −ϕ(ε[x1ε]+εt1,…,ε[xiε]+εti,xi+1,…,xd)\|2dtdx ≤∑i=1d∫D∫Y\|ε∫ti{xi/ε}∂ϕ∂xi(ε[x1ε]+εt1,…,ε[xiε]+εζi,xi+1,…,xd)dζi\|2dtdx ≤ε2∑i=1d∫D∫Y∫01\|∂ϕ∂xi(ε[x1ε]+εt1,…,ε[xiε]+εζi,xi+1,…,xd)\|2dζidtdx ≤ε2∑i=1d∫D\|∂ϕ∂xi\|2dx, which follows from (4.13); here we freeze the variables $$x_{i+1},\ldots,x_d$$. Let $$\psi\in H^1(D)$$. We consider a sequence $$\{\phi_n\}_n\subset C^\infty(\bar D),$$ which converges to $$\psi$$ in $$H^1(D)$$. As $$n\rightarrow \infty$$, ∫D∫Y(ϕn(ε[xε]+εt)−ψ(ε[xε]+εt))2dtdx = ∫DUε((ϕn−ψ)2)(x)dx ≤ ∫D(ϕn(x)−ψ(x))2dx→0. Therefore, ∫D∫Y(ψ(x)−ψ(ε[xε]+εt))2dtdx ≤3∫D(ψ−ϕn)2dx+3∫D∫Y(ϕn−ϕn(ε[xε]+εt))2dtdx +3∫D∫Y(ϕn(ε[xε]+εt)−ψ(ε[xε]+εt))2dtdx ≤6∫D(ψ−ϕn)2dx+3ε2∑i=1d∫D\|∂ϕn∂xi\|2dx. Letting $$n\to\infty$$, we have ∫D∫Y(ψ(x)−ψ(ε[xε]+εt))2dtdx≤3ε2∑i=1d∫D\|∂ψ∂xi\|2dx. Let $$T$$ be the linear map from $$L^2(D)$$ to $$L^2(D\times Y)$$ so that T(ϕ)(x,y)=ϕ(x)−ϕ(ε[xε]+εt). We thus have ‖T‖H1(D)→L2(D×Y)≤cε. On the other hand, ‖T‖L2(D)→L2(D×Y)≤c. From interpolation theory, we deduce that ‖T‖Hs(D)→L2(D×Y)≤cεs. We then get (B.1). The conclusion follows. □ Notes added after the proof stage: After the article is accepted, we learnt about the related recent article: P. Henning, M. Ohlberger and B. Verfürth (2016), A new heterogeneous multiscale method for time-harmonic Maxwell’s equations, SIAM J. Numer. Anal., 54, 3493–3522. This article considers a locally periodic two-scale time harmonic Maxwell equation, but the variational form is still assumed to be strictly coercive, uniformly with respect to the microscopic scale, similar to the equation considered in our present article. These authors formulate the two-scale homogenized equation in a slightly different manner. The Heterogeneous Multiscale Method (HMM) is used to solve the two-scale problem, and is shown to be equivalent to solving the two-scale homogenized equation by using the full tensor finite element spaces. © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. ### Journal IMA Journal of Numerical AnalysisOxford University Press Published: Jan 1, 2018 ## You’re reading a free preview. 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DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month \$360/year Save searches from PubMed Create lists to Export lists, citations	{"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": 0.9901425838470459, "perplexity": 434.4491168926726}, "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-2018-39/segments/1537267158011.18/warc/CC-MAIN-20180922024918-20180922045318-00040.warc.gz"}
https://socratic.org/questions/how-do-you-find-the-equation-x-intercept-and-the-y-intercept-for-the-line-that-p	Algebra Topics # How do you find the equation, x-intercept, and the y-intercept for the line that passes through (-3,1) with a slope of zero? Jul 24, 2017 See a solution process below: #### Explanation: A line with slope $0$ by definition is a horizontal line. Horizontal lines have the same value for $y$ for each and every value of $x$. Because the $y$ value of the point given in the problem is $1$ then the equation is: $y = 1$ y-intercept Because the value of the equation is the same for each and every value of $x$, when we set $x$ equal to $0$ to find the $y$ intercept, $y = 1$ Therefore, the $y$-intercept is: $1$ or $\left(0 , 1\right)$ x-intercept Because this is a horizontal line it is by definition parallel to the $x$ axis. Because it is parallel to the $x$ axis it never crosses the $x$ axis. Therefore the is no $x$-intercept. ##### Impact of this question 829 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": 18, "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": 0.9160181879997253, "perplexity": 289.23082950785874}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487582767.0/warc/CC-MAIN-20210612103920-20210612133920-00513.warc.gz"}
http://math.stackexchange.com/questions/48508/how-are-the-integral-parts-of-9-4-sqrt5n-and-9-4-sqrt5n-relate?answertab=active	# How are the integral parts of $(9 + 4\sqrt{5})^n$ and $(9 − 4\sqrt{5})^n$ related to the parity of $n$? I am stuck on this question, The integral parts of $(9 + 4\sqrt{5})^n$ and $(9 − 4\sqrt{5})^n$ are: 1. even and zero if $n$ is even; 2. odd and zero if $n$ is even; 3. even and one if $n$ is even; 4. odd and one if $n$ is even. I think either the problem or the options are wrong. To me it seems that answer should be odd irrespective of $n$. Consider the following: \begin{align} (9 \pm 4 \sqrt{5})^4 &= 51841 \pm 23184\sqrt{5} \\ (9 \pm 4 \sqrt{5})^5 &= 930249 \pm 416020\sqrt{5} \end{align} Am I missing something? - I think you are confusing "integral part". You are interpreting it as the value of $a$ if you write $(9\pm 4\sqrt{5})^n = a+b\sqrt{n}$ with $a,b\in\mathbb{Z}$. But what they mean is the floor of $(9+4\sqrt{5})^n$, i.e., $\lfloor (9\pm 4\sqrt{5})^n\rfloor$. With that in mind, see Beni Bogosel's answer. – Arturo Magidin Jun 29 '11 at 20:26 @Arturo Magidin:Yes,I confused on "integral part".Thanks! – Quixotic Jun 30 '11 at 11:17 add comment ## 3 Answers The idea is to see that $(9+4\sqrt{5})^n+(9-4\sqrt{5})^n=2d_n$ is an even number for every $n$. This can be seen using the binomial expansion formula, and seeing that odd terms appear once with $+$ once with $-$, and even terms are always with $+$ and integers. Moreover, $0<(9-4\sqrt{5})=9-\sqrt{80}=\frac{1}{\sqrt{81}+\sqrt{80}}<1$. This means that the integer part of the second term is zero. And for the other one, think as this $$2d_n-1<(9+4\sqrt{5})^n<2d_n$$ so the integer part of the first term is always odd. The correct answer would be the second one (although this happens for every $n$). As Arturo Magidin wrote in his comment, the integer part of a real number $x$, often denoted $\lfloor x \rfloor$ is the unique integer $\lfloor x \rfloor=k$ such that $k \leq x < k+1$, and it does not equal $a$ from the expansion $(9\pm 4\sqrt{5})^n=a\pm b\sqrt{5},\ a,b \in \Bbb{Z}$. - add comment Alternative to binomial expansion, one can deduce that $\rm\:w = (9+4\sqrt{5})^n + (9-4\sqrt{5})^n\:$ is even by noting that the notion of parity uniquely extends from $\;\mathbb Z\:$ to $\:\mathbb Z[\sqrt{5}]\:$ by defining $\rm\:\sqrt{5}\:$ to be odd. Hence, since $\rm\:w'= w\:,\:$ we infer that $\rm\:w\in \mathbb Z\:$ with parity $\rm\ odd^n +\: odd^n\: =\ odd+odd\ =\ even\:.\$ Then the desired result follows easily as Beni explained. Notice, in particular, how this viewpoint is a very natural higher-degree extension of ubiquitous parity-based proofs in $\mathbb Z\:.$ Alternatively, it follows immediately from the fact that, mod $2$, the sequence $\rm\:w_n\:$ satisfies a monic integer-coefficient recurrence, and the first $2$ (= degree) terms are $\equiv 0\:.\:$ Hence by induction so are all subsequent terms, viz. $\rm\ f_{n+2} \equiv\ a\ f_{n+1} + b\ f_{n} \equiv\ 0\$ since by induction $\rm\ f_{n+1},\ f_{n}\ \equiv\ 0\:.\:$ Said equivalently $\rm\:f_n \equiv 0\$ by the uniqueness theorem for solutions of difference equations (recurrences). As I frequently emphasize uniqueness theorems provide very powerful tools for proving equalities. Here the uniqueness theorem is rather trivial, amounting to the trivial induction that if two solutions of a degree $\rm\:d\:$ monic integer coefficient recurrence agree for $\rm\:d\:$ initial values then they agree at all subsequent values; equivalently, taking differences, if a solution is $\:0\:$ for $\rm\:d\:$ initial values then it is identically $\:0\:$. More generally the same holds true for integer-linear combinations of roots of any monic integer coefficient equation (i.e. algebraic integer roots) since they too will satisfy a monic integer coefficient recurrence, viz. the characteristic equation associated to the polynomial having said roots (the quadratic case is the widely studied Lucas sequence). Thus every term of the sequence will be divisible by $\rm\:m\:$ iff it is true for the first $\rm\:d\:$ (= degree) terms. More generally one easily checks that the gcd of all terms is simply the gcd of the initial values. Note that, as above, one requires only the knowledge of the existence of such a recurrence. There is no need to explicitly calculate the coefficients of the recurrence; rather, only its degree is employed. - Nice point of view :) – Beni Bogosel Jun 30 '11 at 11:44 add comment To prove that $$a_n:=\left(9+4\sqrt5\right)^n+\left(9-4\sqrt5\right)^n$$ is an even integer, it suffices to observe that $a_0=2,a_1=18$ and $$a_{n+2}=18\ a_{n+1}-a_n$$ for $n\ge0$. Variation. Note that $a_n$ is an integer. Indeed $a_n$ is of the form $b_n+c_n\sqrt5$ with $b_n,c_n$ integers, and $a_n$ doesn't depend on the choice of the square root of $5$, so $c_n=0$. Once we know that $a_n$ is an integer, we can compute it in the field $\mathbb F_2$ with two elements (in which $\sqrt5$ exists). EDIT. The above arguments show this: If $a,b,d,n$ are integers, and if $n\ge0$, then $$\left(a+b\sqrt d\right)^n+\left(a-b\sqrt d\right)^n$$ is an even integer. - The recursion comes from the quadratic equation that defines $9 \pm 4 \sqrt{5}$. – lhf Dec 23 '11 at 11:32 Dear @lhf: Yes, the minimal polynomial is $X^2-\tau\ X+\nu$, where $\tau$ is the trace and $\nu$ the norm. – Pierre-Yves Gaillard Dec 23 '11 at 11:38 add 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": 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": 0.9878459572792053, "perplexity": 245.87995148797575}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1398223201753.19/warc/CC-MAIN-20140423032001-00128-ip-10-147-4-33.ec2.internal.warc.gz"}
https://www.ias.ac.in/listing/bibliography/pram/S_V_Moharil	• S V Moharil Articles written in Pramana – Journal of Physics • Reflectance spectra and thermoluminescence of NaF coloured in an electrodeless discharge A comparative study ofγ-ray colouration and electrodeless discharge excitation is reported for NaF. New absorption bands and glow peaks were found. These are shown to be characteristic of electrodeless discharge method of colouration. These are attributed to the multiple types of defects. Further, it is shown that the process by which such defects are formed is strongly temperature dependant. A tentative explanation for the peculiar characteristic of the electrodeless discharge excitation is put forth. The possibility of exploiting these peculiarities for the study of certain properties of colour centres is pointed out. • Colour centres in powdered KI Fine powder of KI was coloured in an electrodeless discharge. Due to quick bleaching of the F centres produced in this method, it was possible to prepare the samples that were almost free from the F centres. High concentration of the electron deficient centres could be produced, which were studied by measuring the diffused reflectance. A band at 354 nm is shown to be composed of two overlapping bands. Further, growth of a band appearing at 265 nm is studied. Bleaching characteristics of the samples are studied and it is shown that, similar to F centre bleaching, bleaching of these samples also proceeds in at least two steps. The difference between two components of the bleaching curves is quite marked. Further it is showed that the components are related to the presence of different absorption bands and appear at different stages of colouration. • Reflectance spectra and thermoluminescence of NaBr coloured in an electrodeless discharge A comparative study of the optical and thermoluminescent properties ofγ irradiated NaBr and NaBr coloured in an electrodeless discharge is reported. In discharge coloured NaBrF centre absorption maxima shifted with the colouration time. This is tentatively attributed to the formation ofE centres. Correspondingly, an additional peak was observed in thermoluminescence glow curve. It is suggested that the results are not characteristic of the method of colouration but rather of the imperfections inherent to the powders, and in a perfect (undeformed) single crystal such a phenomenon should not be observed. • Reflectance spectra and thermoluminescence of alkali halides coloured in an electrodeless discharge Microcrystalline powders of NaCl, KCl and KBr are coloured in electrodeless discharge. Reflectance and TL studies of these coloured powders are reported. It is concluded that colouration of powders can be understood by considering them as an admixture of perfect and imperfect lattices, and differs from that of single crystals. It is suggested that some of the descrepancies reported on TL data may be due to such a difference. Further, it is shown that a better correlation can be had if TL data are presented along with the corresponding optical measurements. Adoption of such a procedure may help to remove the descrepancy in TL data. • Mechanoluminescence excitation in alkali halide crystals and colouration decay in microcrystalline powders The mechanoluminescence (ML) of NaCl, NaBr, NaF, LiCl and LiF crystals ceases at 105, 58, 170, 151 and 175°C respectively. Both the temperatureTc at whichML disappears and the temperatureTs required to induce a particular percentage of colouration decay in a given time, decreases with increasing nearest neighbour distance in alkali halide crystals. This perhaps suggests that similar processes cause the disappearance ofml in alkali halide crystals and the colouration decay in their microcrystalline powders. It is shown that mobile dislocations may cause the leakage of surface charge and the decay of colouration in microcrystalline powders. • Pramana – Journal of Physics Volume 96, 2022 All articles Continuous Article Publishing mode • Editorial Note on Continuous Article Publication Posted on July 25, 2019	{"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": 0.8185836672782898, "perplexity": 2842.9088070916587}, "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/1659882571472.69/warc/CC-MAIN-20220811133823-20220811163823-00622.warc.gz"}
http://mathhelpforum.com/pre-calculus/93158-how-find-changed-base.html	# Thread: How to find the Changed Base 1. ## How to find the Changed Base The formula A = P(1.09)^t is an example of exponential growth with base 1.09. Determine an equivalent continuous growth formula using base e. Also, what if it asks me to use a different base? (eg. 2) 2. Originally Posted by AlphaRock The formula A = P(1.09)^t is an example of exponential growth with base 1.09. Determine an equivalent continuous growth formula using base e. Also, what if it asks me to use a different base? (eg. 2) 3. Hello, AlphaRock! The formula $A = P(1.09)^t$ is an example of exponential growth with base 1.09. Determine an equivalent continuous growth formula using base $e$. Also, what if it asks me to use a different base? Suppose we wish to change to base $b.$ Instead of $(1.09)^t$, we want $b^{kt}$, where $k$ is a constant to be determined. We have: . $b^{kt} \:=\:(1.09)^t$ Take logs (base $b$): . $\log_b\left(b^{kt}\right) \:=\:\log_b(1.09)^t$ . . $\text{We have: }\;kt\underbrace{\log_b(b)}_{\text{This is 1}} \:=\:t\log_b(1.09) \quad\Rightarrow\quad kt \:=\:t\log_b(1.09)$ . . Hence: . $k \:=\:\log_b(1.09)$ Therefore: . $A \;=\;P\,b^{\log_b(1.09)\cdot t}$ 4. Originally Posted by Soroban Hello, AlphaRock! Suppose we wish to change to base $b.$ Instead of $(1.09)^t$, we want $b^{kt}$, where $k$ is a constant to be determined. We have: . $b^{kt} \:=\1.09)^t" alt="b^{kt} \:=\1.09)^t" /> Take logs (base $b$): . $\log_b\left(b^{kt}\right) \:=\:\log_b(1.09)^t$ . . $\text{We have: }\;kt\underbrace{\log_b(b)}_{\text{This is 1}} \:=\:t\log_b(1.09) \quad\Rightarrow\quad kt \:=\:t\log_b(1.09)$ . . Hence: . $k \:=\:\log_b(1.09)$ Therefore: . $A \;=\;P\,b^{\log_b(1.09)\cdot t}$ Un, Soroban, since $b^{\log_b(a)}= a$, that reduces to $A= 1.09^t$, the formula we started with. If we want to write $1.09^t$ as an exponential with base b, we have, as Soroban said, $1.09^t= b^{kt}$. Now take the natural logarithm of both sides: $ln(1.09^t)= t ln(1.09)$ and $ln(b^{kt}= kt ln(b)$ so the equation becomes $t ln(1.09)= kt ln(b)$ and $k= \frac{ln(1.09)}{ln(b)}$. The advantage is that your calculator has a "ln" key but not a " $log_b$" key! In particular, if b= e, then ln(b)= 1 so $k= ln(1.09)= 0.0862$ and $1.09^t= e^{t ln(1.09)}= e^{0.0862 t}$. If b= 2, then ln(b)= ln(2)= 0.6931 so $k= \frac{ln(1.09)}{ln(2)}= \frac{0.0862}{0.6931}= 0.1243$ and $1.09^t= 2^{0.1243 t}$.	{"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": 35, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9758851528167725, "perplexity": 4160.826623760525}, "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/1495463608058.57/warc/CC-MAIN-20170525102240-20170525122240-00057.warc.gz"}
http://www.maa.org/publications/periodicals/convergence/benjamin-bannekers-inscribed-equilateral-triangle-transcription-of-bannekers-problem	# Benjamin Banneker's Inscribed Equilateral Triangle - Transcription of Banneker's Problem Author(s): John F. Mahoney (Benjamin Banneker Academic High School) Required the Lengths of the Sides of an Equilateral Triangle inscribed in a Circle whose Diameter is 200 perches, with a general Theorem for all such Questions Solution of the above problem 10.00 ….. 3.142 …… 200 200 1000)628400 Lenght of the periphery 628.400 1/3 of the length of the periphery 209.466 1/3 of 1/3 of the periphery 69.822 2 2/3 of 1/3 of the periphery 139.644 Length of the Sides required 349.110 Note: In Banneker's journal there is a big X crossing out the last five numbers in his calculations and adjacent to each of the 175's in his figure is 349.110 This example deals with the problem of finding the lengths of the sides of an inscribed equilateral triangle in a circle of diameter 200 perches. A perch, a synonym of a rod, is a unit of measurement equal to 16½ feet. There are 320 perches in a mile. A surveyor could find the number of acres in a rectangular piece of land by multiplying the length in perches by the width in perches and dividing the product by 160. What did Banneker do to solve this problem? First of all he calculated the circumference, which he calls periphery, by multiplying an approximation of $$\pi$$ (3.142) by 200. He did this by multiplying 3.142 by 200 to get 628.400 and then dividing that by 1000 to get 628.400 (How often have do we, as teachers, pull our hair out when we see our students multiplying or dividing by powers of 10?) He then finds 1/3 the length (misspelled in his journal) of the circumference and later 2/9 of the circumference. He adds those numbers together to get 349.110, but clearly he knew that the side of the equilateral triangle had to be less than the diameter. He crossed out his work and then labeled the sides of triangle 175 which is approximately half of 349.110. Using the properties of a 30º-60º-90º triangle and a circle of radius 100, one can see that the length of the side of the equilateral triangle is 173.205 which shows that Banneker's solution is within 1% of the actual one. The side of the equilateral triangle is the square root of 3 times the radius of the circumscribed circle. How did Banneker figure this out without using the properties of a 30º-60º-90º triangle or trigonometry? Banneker calculated 5/9 of the circumference to get 349.110 and by taking half of that he essentially computed 5/18 of the circumference. Since (5/18)(2$$\pi$$)100 is approximately equal to 174.533, Banneker's method is quite good. It would have been even better if he had taken exactly half of 349.110 to get 174.555. Banneker's method essentially uses (5/9)$$\pi$$ to approximate the square root of 3. Perhaps this approximation was a rule of thumb that surveyors used in the 18th century. John F. Mahoney (Benjamin Banneker Academic High School), "Benjamin Banneker's Inscribed Equilateral Triangle - Transcription of Banneker's Problem," Loci (July 2010)	{"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": 0.8654367327690125, "perplexity": 824.2854854953267}, "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/1438042981921.1/warc/CC-MAIN-20150728002301-00005-ip-10-236-191-2.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/530747/logical-proof-of-the-statement-xy-0-implies-x-0-text-or-y-0/530749	# Logical proof of the statement $xy = 0 \implies x=0\text{ or } y=0$ Claim: If $xy=0$, then $x=0$ or $y=0$. My proof is as follows: • case 1: $x=0$, so $0y=0$ • case 2: $y=0$, so $x0=0$ Either way, $xy=0$. I'm very confused by this myself. So if I let $xy=0$ be $P$, and $x=0$ or $y=0$ be $Q$, then the claim "if $xy=0$, then either $x=0$ or $y=0$" is asking me to prove that $P \implies Q$. But I feel as if the proof I gave is $Q \implies P$, which is very different from $P \implies Q$. Can anyone enlighten me on the subject of proving if-then statements? - You have to specify the domain you are working in. An example: In ${\mathbb Z}_6$ we have $2\ne0$, $3\ne0$, but $2\cdot 3=0$. – Christian Blatter Oct 18 '13 at 9:17 Your feeling is right: What you have done is the opposite direction. Your argument proves $$(x = 0 \text{ or } y = 0) \implies xy = 0.$$ How can you prove a statement of the form $$A \implies B$$ in general? The direct method is to assume that $A$ is true, and then to conclude that also $B$ is true under this assumption. Let's apply this to prove the statement $$xy = 0 \implies (x = 0 \text{ or } y = 0).$$ In this case \begin{align} A & = \text{''}xy = 0\text{''} \\ B & = \text{''}x = 0\text{ or }y = 0\text{''} \end{align} So we assume that $xy = 0$ is true. Now we have to show that $(x = 0 \text{ or } y = 0)$ is true. The nature of an "or"-statement often involves a case by case study: 1. If $x = 0$, then of course $(x = 0 \text{ or } y = 0)$ is true. 2. Otherwise, we have $x \neq 0$. Now we may divide our assumption (the equation $xy = 0$) by $x$ to get $y = 0$, so $(x = 0 \text{ or } y = 0)$ is true also in this case. We have just proven $$xy = 0 \implies (x = 0 \text{ or } y = 0),$$ and the argument in your question proves $$xy = 0 \Longleftarrow (x = 0 \text{ or } y = 0).$$ So in fact, we have an equivalence, which we can write down as $$xy = 0 \iff (x = 0 \text{ or } y = 0).$$ - well, it only happens in an integral domain, or fields as we generally works in which are integral domains automatically. if you are in, say, Z6 , its not true, as take 32=6mod6=0 where none of 3 and 2 are zero. but i guess u have already assumed it over reals which is a field, in that case, T.P xy=0⟹x=0 or y=0, you can assume y≠0, and then multiply on both sides by y^-1 which will give you x.1=0y^-1=0 implies x=0. for integral domains, its basically the definition if an integral domain. - You should consider using MathJax. – Student Jul 8 '14 at 18:47 Not quite right, as you have it written you merely show that if either $x$ or $y$ are zero then their product is zero. Not that if the product is zero then either $x$ or $y$ must necessarily be zero. You can show this by contradiction; assume $xy=0$ but $x\neq0$ and $y\neq 0$ then $xy\neq 0$ which contradicts our original assumption that $xy=0$ so we cannot have that both $x$ and $y$ are non zero. In other words at if $xy=0$ at least one of $x$ and $y$ must be zero. - ‘Not quite right’ is an understatement: it’s simply wrong, I’m afraid. – Brian M. Scott Oct 18 '13 at 8:16 If $x\ne0$, what happens if you multiply both sides by $\frac1x$? - Just another way. It might not be a formal method. The solution satisfying the following equation $$A \times B =0$$ is $A=0$ (for any $B$) or $B=0$ (for any $A$). You cannot apply the same pattern for the case in which the right hand side is not zero. Why? For example, $$A\times B = 2$$ If you choose $A=2$ then $B$ must be $1$ (rather than for any $B$). If you choose $B=2$ then $A$ must be $1$ (rather than for any $A$). -	{"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": 0.9911600351333618, "perplexity": 172.81201045882614}, "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-2016-26/segments/1466783402479.21/warc/CC-MAIN-20160624155002-00194-ip-10-164-35-72.ec2.internal.warc.gz"}
http://www.ams.org/joursearch/servlet/PubSearch?f1=msc&pubname=all&v1=13A15&startRec=31	# American Mathematical Society My Account · My Cart · Customer Services · FAQ Publications Meetings The Profession Membership Programs Math Samplings Washington Office In the News About the AMS You are here: Home > Publications AMS eContent Search Results Matches for: msc=(13A15) AND publication=(all) Sort order: Date Format: Standard display Results: 31 to 60 of 77 found Go to page: 1 2 3 [31] Henrik Bresinsky and Lê Tuân Hoa. On the reduction number of some graded algebras. Proc. Amer. Math. Soc. 127 (1999) 1257-1263. MR 1473657. Abstract, references, and article information View Article: PDF This article is available free of charge [32] Alberto Corso, William Heinzer and Craig Huneke. A generalized Dedekind-Mertens lemma and its converse . Trans. Amer. Math. Soc. 350 (1998) 5095-5109. MR 1473435. Abstract, references, and article information View Article: PDF This article is available free of charge [33] William Heinzer and Craig Huneke. The Dedekind-Mertens Lemma and the contents of polynomials. Proc. Amer. Math. Soc. 126 (1998) 1305-1309. MR 1425124. Abstract, references, and article information View Article: PDF This article is available free of charge [34] D. D. Anderson and Moshe Roitman. A characterization of cancellation ideals. Proc. Amer. Math. Soc. 125 (1997) 2853-2854. MR 1415571. Abstract, references, and article information View Article: PDF This article is available free of charge [35] William Heinzer and Craig Huneke. Gaussian polynomials and content ideals. Proc. Amer. Math. Soc. 125 (1997) 739-745. MR 1401742. Abstract, references, and article information View Article: PDF This article is available free of charge [36] Paul-Jean Cahen, Evan G. Houston and Thomas G. Lucas. Discrete valuation overrings of Noetherian domains. Proc. Amer. Math. Soc. 124 (1996) 1719-1721. MR 1317033. Abstract, references, and article information View Article: PDF This article is available free of charge [37] D. D. Anderson and Bernadette Mullins. Finite factorization domains . Proc. Amer. Math. Soc. 124 (1996) 389-396. MR 1322910. Abstract, references, and article information View Article: PDF This article is available free of charge [38] William Heinzer and Sylvia Wiegand. Prime ideals in polynomial rings over one-dimensional domains . Trans. Amer. Math. Soc. 347 (1995) 639-650. MR 1242087. Abstract, references, and article information View Article: PDF This article is available free of charge [39] D. D. Anderson. A note on minimal prime ideals . Proc. Amer. Math. Soc. 122 (1994) 13-14. MR 1191864. Abstract, references, and article information View Article: PDF This article is available free of charge [40] Songqing Ding. Auslander's $\delta$-invariants of Gorenstein local rings . Proc. Amer. Math. Soc. 122 (1994) 649-656. MR 1203983. Abstract, references, and article information View Article: PDF This article is available free of charge [41] Henri Dichi and Daouda Sangare. Filtrations, asymptotic and Pr\"uferian closures, cancellation laws . Proc. Amer. Math. Soc. 113 (1991) 617-624. MR 1064901. Abstract, references, and article information View Article: PDF This article is available free of charge [42] Raymond C. Heitmann and Stephen McAdam. Comaximizable primes . Proc. Amer. Math. Soc. 112 (1991) 661-669. MR 1049136. Abstract, references, and article information View Article: PDF This article is available free of charge [43] Mutsumi Amasaki. Application of the generalized Weierstrass preparation theorem to the study of homogeneous ideals . Trans. Amer. Math. Soc. 317 (1990) 1-43. MR 992603. Abstract, references, and article information View Article: PDF This article is available free of charge [44] Melvin Hochster and Craig Huneke. Tight closure, invariant theory, and the Brian\c con-Skoda theorem . J. Amer. Math. Soc. 3 (1990) 31-116. MR 1017784. Abstract, references, and article information View Article: PDF This article is available free of charge [45] William J. Heinzer and Ira J. Papick. Remarks on a remark of Kaplansky . Proc. Amer. Math. Soc. 105 (1989) 1-9. MR 973834. Abstract, references, and article information View Article: PDF This article is available free of charge [46] Louis J. Ratliff. $\Delta$-closures of ideals and rings . Trans. Amer. Math. Soc. 313 (1989) 221-247. MR 961595. Abstract, references, and article information View Article: PDF This article is available free of charge [47] Budh Nashier. On choosing generating sets for ideals . Proc. Amer. Math. Soc. 100 (1987) 233-234. MR 884457. Abstract, references, and article information View Article: PDF This article is available free of charge [48] Stephen McAdam and L. J. Ratliff. Sporadic and irrelevant prime divisors . Trans. Amer. Math. Soc. 303 (1987) 311-324. MR 896024. 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Abstract, references, and article information View Article: PDF This article is available free of charge Results: 31 to 60 of 77 found Go to page: 1 2 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": 0.9279045462608337, "perplexity": 1996.069679790251}, "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/1498128321025.86/warc/CC-MAIN-20170627064714-20170627084714-00558.warc.gz"}
https://www.rieti.go.jp/jp/publications/summary/13110014.html	# Does Competition Improve Industrial Productivity? An analysis of Japanese industries on the basis of the industry-level panel data 執筆者安橋正人 (コンサルティングフェロー) 2013年11月 13-E-098 ## 概要（英語） This study mainly investigates the causal relation between the degree of competition, which is measured by the Lerner index, and the total factor productivity (TFP) growth rate on the basis of the Japanese industry-level panel data (Japan Industrial Productivity (JIP) Database) from 1980 to 2008. The central finding indicates that, although a positive effect of competition on the TFP growth rate is clearly observable in the manufacturing industries throughout the sample period, such effect in the non-manufacturing industries may be slightly negative in the latter half of the sample period (1995-2008). This finding of a negative competition effect may lend support to the claim that the Schumpeterian hypothesis can be applied in the case of the non-manufacturing industries. Furthermore, a weak inverted-U shape relation between the competition measure and TFP growth proposed by Aghion et al. (2005) can be seen limitedly almost exclusively in all industries. Published: Ambashi, Masahito, 2017. "Competition effects and industrial productivity: Lessons from Japanese industry," Asian Economic Paper Vol. 16(3), pp. 214-249 http://www.mitpressjournals.org/doi/abs/10.1162/asep_a_00568	{"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": 0.9197221398353577, "perplexity": 4503.3668479593125}, "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/1652662562410.53/warc/CC-MAIN-20220524014636-20220524044636-00119.warc.gz"}
http://blog.plover.com/2012/08/24/	# The Universe of Discourse Fri, 24 Aug 2012 More about ZF's asymmetry between union and intersection In an article earlier this week, I explored some oddities of defining a toplogy in terms of closed sets rather than open sets, mostly as a result of analogous asymmetry in the ZF set theory axioms. Let's review those briefly. The relevant axioms concern the operations by which sets can be constructed. There are two that are important. First is the axiom of union, which says that if !!{\mathcal F}!! is a family of sets, then we can form !!\bigcup {\mathcal F}!!, which is the union of all the sets in the family. The other is actually a family of axioms, the specification axiom schema. It says that for any one-place predicate !!\phi(x)!! and any set !!X!! we can construct the subset of !!X!! for which !!\phi!! holds: $$\{ x\in X \;\|\; \phi(x) \}$$ Both of these are required. The axiom of union is for making bigger sets out of smaller ones, and the specification schema is for extracting smaller sets from bigger ones. (Also important is the axiom of pairing, which says that if !!x!! and !!y!! are sets, then so is the two-element set !!\{x, y\}!!; with pairing and union we can construct all the finite sets. But we won't need it in this article.) Conspicuously absent is an axiom of intersection. If you have a family !!{\mathcal F}!! of sets, and you want a set of every element that is in some member of !!{\mathcal F}!!, that is easy; it is what the axiom of union gets you. But if you want a set of every element that is in every member of !!{\mathcal F}!!, you have to use specification. Let's begin by defining this compact notation: $$\bigcap_{(X)} {\mathcal F}$$ for this longer formula: $$\{ x\in X \;\|\; \forall f\in {\mathcal F} . x\in f \}$$ This is our intersection of the members of !!{\mathcal F}!!, taken "relative to !!X!!", as we say in the biz. It gives us all the elements of !!X!! that are in every member of !!{\mathcal F}!!. The !!X!! is mandatory in !!\bigcap_{(X)}!!, because ZF makes it mandatory when you construct a set by specification. If you leave it out, you get the Russell paradox. Most of the time, though, the !!X!! is not very important. When !!{\mathcal F}!! is nonempty, we can choose some element !!f\in {\mathcal F}!!, and consider !!\bigcap_{(f)} {\mathcal F}!!, which is the "normal" intersection of !!{\mathcal F}!!. We can easily show that $$\bigcap_{(X)} {\mathcal F}\subseteq \bigcap_{(f)} {\mathcal F}$$ for any !!X!! whatever, and this immediately implies that $$\bigcap_{(f)} {\mathcal F} = \bigcap_{(f')}{\mathcal F}$$ for any two elements of !!{\mathcal F}!!, so when !!{\mathcal F}!! contains an element !!f!!, we can omit the subscript and just write $$\bigcap {\mathcal F}$$ for the usual intersection of members of !!{\mathcal F}!!. Even the usually troublesome case of an empty family !!{\mathcal F}!! is no problem. In this case we have no !!f!! to use for !!\bigcap_{(f)} {\mathcal F}!!, but we can still take some other set !!X!! and talk about !!\bigcap_{(X)} \emptyset!!, which is just !!X!!. Now, let's return to topology. I suggested that we should consider the following definition of a topology, in terms of closed sets, but without an a priori notion of the underlying space: A co-topology is a family !!{\mathcal F}!! of sets, called "closed" sets, such that: 1. The union of any two elements of !!{\mathcal F}!! is again in !!{\mathcal F}!!, and 2. The intersection of any subfamily of !!{\mathcal F}!! is again in !!{\mathcal F}!!. Item 2 begs the question of which intersection we are talking about here. But now that we have nailed down the concept of intersections, we can say briefly and clearly what we want: It is the intersection relative to !!\bigcup {\mathcal F}!!. This set !!\bigcup {\mathcal F}!! contains anything that is in any of the closed sets, and so !!\bigcup {\mathcal F}!!, which I will henceforth call !!U!!, is effectively a universe of discourse. It is certainly big enough that intersections relative to it will contain everything we want them to; remember that intersections of subfamilies of !!{\mathcal F}!! have a maximum size, so there is no way to make !!U!! too big. It now immediately follows that !!U!! itself is a closed set, since it is the intersection !!\bigcap_{(U)} \emptyset!! of the empty subfamily of !!{\mathcal F}!!. If !!{\mathcal F}!! itself is empty, then so is !!U!!, and !!\bigcap_{(U)} {\mathcal F} = \emptyset!!, so that is all right. From here on we will assume that !!{\mathcal F}!! is nonempty, and therefore that !!\bigcap {\mathcal F}!!, with no relativization, is well-defined. We still cannot prove that the empty set is closed; indeed, it might not be, because even !!M = \bigcap {\mathcal F}!! might not be empty. But as David Turner pointed out to me in email, the elements of !!M!! play a role dual to the extratoplogical points of a topological space that has been defined in terms of open sets. There might be points that are not in any open set anywhere, but we may as well ignore them, because they are topologically featureless, and just consider the space to be the union of the open sets. Analogously and dually, we can ignore the points of !!M!!, which are topologically featureless in the same way. Rather than considering !!{\mathcal F}!!, we should consider !!{\mathcal F}HAT!!, whose members are the members of !!{\mathcal F}!!, but with !!M!! subtracted from each one: $${\mathcal F}HAT = \{\hat{f}\in 2^U \;\|\; \exists f\in {\mathcal F} . \hat{f} = f\setminus M \}$$ So we may as well assume that this has been done behind the scenes and so that !!\bigcap {\mathcal F}!! is empty. If we have done this, then the empty set is closed. Now we move on to open sets. An open set is defined to be the complement of a closed set, but we have to be a bit careful, because ZF does not have a global notion of the complement !!S^C!! of a set. Instead, it has only relative complements, or differences. !!X\setminus Y!! is defined as: $$X\setminus Y = \{ x\in X \;\|\; x\notin Y\}$$ Here we say that the complement of !!Y!! is taken relative to !!X!!. For the definition of open sets, we will say that the complement is taken relative to the universe of discourse !!U!!, and a set !!G!! is open if it has the form !!U\setminus f!! for some closed set !!f!!. Anatoly Karp pointed out on Twitter that we know that the empty set is open, because it is the relative complement of !!U!!, which we already know is closed. And if we ensure that !!\bigcap {\mathcal F}!! is empty, as in the previous paragraph, then since the empty set is closed, !!U!! is open, and we have recovered all the original properties of a topology. Order General Topology with kickback no kickback But gosh, what a pain it was; in contrast recovering the missing axioms from the corresponding open-set definition of a topology was painless. (John Armstrong said it was bizarre, and probably several other people were thinking that too. But I did not invent this bizarre idea; I got it from the opening paragraph of John L. Kelley's famous book General Topology, which has been in print since 1955. Here Kelley deals with the empty set and the universe in two sentences, and never worries about them again. In contrast, doing the same thing for closed sets was fraught with technical difficulties, mostly arising from ZF. (The exception was the need to repair the nonemptiness of the minimal closed set !!M!!, which was not ZF's fault.) Order On Numbers and Games with kickback no kickback I don't think I have much of a conclusion here, except that whatever the advantages of ZF as a millieu for doing set theory, it is overrated as an underlying formalism for actually doing mathematics. (Another view on this is laid out by J.H. Conway in the Appendix to Part Zero of On Numbers and Games (Academic Press, 1976).) None of the problems we encountered were technically illuminating, and nothing was clarified by examining them in detail. On the other hand, perhaps this conclusion is knocking down a straw man. I think working mathematicians probably don't concern themselves much with whether their stuff works in ZF, much less with what silly contortions are required to make it work in ZF. I think day-to-day mathematical work, to the extent that it needs to deal with set theory at all, handles it in a fairly naïve way, depending on a sort of folk theory in which there is some reasonably but not absurdly big universe of discourse in which one can take complements and intersections, and without worrying about this sort of technical detail. [ MathJax doesn't work in Atom or RSS syndication feeds, and can't be made to work, so if you are reading a syndicated version of this article, such as you would in Google Reader, or on Planet Haskell or PhillyLinux, you are seeing inlined images provided by the Google Charts API. The MathJax looks much better, and if you would like to compare, please visit my blog's home site. ]	{"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": 0.8271920084953308, "perplexity": 851.4039261507569}, "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-2017-09/segments/1487501170569.99/warc/CC-MAIN-20170219104610-00097-ip-10-171-10-108.ec2.internal.warc.gz"}
http://clay6.com/qa/3240/find-the-equation-of-the-curve-through-the-point-1-0-if-the-slope-of-the-ta	# Find the equation of the curve through the point (1,0) if the slope of the tangent to the curve at any point (x,y) is $\Large \frac{y-1}{x^2+x}.$ Toolbox: • The slope of the tangent to the curve is $\large\frac{dy}{dx}$ • If a proper rational expression is $\large\frac{1}{(x+a)(x+b)},$ then it can be resolved into its partial fraction as $\large\frac{A}{x+a}+\frac{B}{x+b}$ • A linear differential equation of the form $\large\frac{dy}{dx}$$=f(x) can be solved by seperating the variables and then integrating it . Given :The slope of the tangent to the curve at any point is \large\frac{y-1}{x^2+x} The slope of the tangent to the curve is \large\frac{dy}{dx} Therefore \large\frac{dy}{dx}=\frac{y-1}{x^2+x} Seperating the variables we get \large\frac{dy}{y-1}=\frac{dx}{x(x+1)} Now \large\frac{1}{x(x+1)} can be resolved into paritial fractions as \large\frac{A}{x}+\frac{B}{x+1} Hence 1=A(x+1)+B(x) Now let us equate the coefficient of like terms First let us equate the 'x' term 0=A+B-----(1) Equating the constant term 1=+A\qquad B=-1 Hence \large\frac{1}{x(x+1)}=\frac{1}{x}-\frac{1}{x+1} Therefore \large \frac{dy}{y-1}$$=+\large\frac{dx}{x}-\frac{dx}{x+1}$ Integrating on both sides we get, $\int \large \frac{dy}{y-1}$$=+\int \large\frac{dx}{x}-\int \frac{dx}{x+1}$ =>$\log (y-1)=+\log x-\log (x+1)+\log c$ =>$\log (y-1)=\log \large\frac{c(x)}{(x+1)}$ =>$(y-1)=\Large\frac{cx}{x+1}$ =>$(y-1)(x+1)=cx$ To evaluate the value of c,let us now substitute the value of x and y with the given point (1,0) $(0-1)(1+1)=c(1)$ =>$c=-2$ $(y-1)(x+1)=-2x$ $(y-1)(x+1)+2x=0$ is the required solution.	{"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": 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": 0.9846416711807251, "perplexity": 730.2353580159837}, "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-2017-51/segments/1512948545526.42/warc/CC-MAIN-20171214163759-20171214183759-00349.warc.gz"}
https://www.physicsforums.com/threads/elementary-doubts-i-i-got-confronted-with-while-reading-qft.769395/	# Elementary doubts i i got confronted with while reading qft 1. Sep 6, 2014 ### kau why should Proca eqn be like ∂γ Fγμ + m2 Aμ = 0 but not ∂γ Fμγ + m2 Aμ = 0 ???? another doubt is (λ-1 ω λ)μγ = λ-1 ρμ ωρσ λσγ ??? why in λ-1 transformation got upper index in the second place but not in the first place???? if someone clear my doubts...I would be thanful... regards.. Kau 2. Sep 6, 2014 ### Freddieknets Your first problem is easily solved: First note that the gauge field tensor is antisymmetric, i.e. $F_{\gamma\mu}=-F_{\mu\gamma}$. If $\partial^\gamma F_{\gamma\mu} + m^2 A_\mu =0$, then $\partial^\gamma F_{\gamma\mu} = -m^2 A_\mu$. Filling this in the second equation, we have $\partial^\gamma F_{\mu\gamma} +m^2 A_\mu=-\partial^\gamma F_{\gamma\mu} +m^2 A_\mu =m^2 A_\mu+m^2 A_\mu\neq 0$. The second equation needs a minus sign, i.e. $$\partial^\gamma F_{\gamma\mu} + m^2 A_\mu =0$$ $$\partial^\gamma F_{\mu\gamma} - m^2 A_\mu =0$$ are both correct. For your second question, i.e. $(\lambda^{-1}\omega\lambda)_{\mu\gamma}$ I don't understand what the problem is. Last edited: Sep 6, 2014 3. Sep 6, 2014 Just use $$at the beginning and [ /tex] (without the space) at the end. Or $for inline equations. The is not rendered by MathJax. 4. Sep 6, 2014 ### Freddieknets Jup, actually I knew that, switched too fast from thesis writing to physicsforums.. :-) 5. Sep 6, 2014 ### kau I said in the second one suppose λ is lorentz transformation , λ-1 is inverse lorentz transformation and δωμγ is some parameter boost or rotation. now if you write its components explicitly. then it would be λ-1 μρ δωμγ λγ σ ... ok.. look the difference in λ-1 μρ μ is written upstair in 2nd position and ρ is written downstair ad 1st position which says that λ-1 μ ρ = λμ ρ ... but in λγ σ here y is in upstair 1st postion and σ is in downstair second position .. but this I understand it represents direct transformation. so if you write γ in the second position and σ in the 1st position that would imply and inverse lorentz transformation. my ques when we wrote that inverse transformation part why we wrote λ-1 μρ but not λ-1μ ρ ? I am sorry if you still didn't get it.. I am using this first time.. so don't know proper way to write eqns. so my eqns may look misleading... but if you get my point please answer me. thanks. Last edited: Sep 6, 2014 6. Sep 6, 2014 ### ChrisVer Another way to see the Proca equation, is by taking the Lagrangian of a spin-1 massive vector field without sources: [itex] L= -aF_{\mu \nu} F^{\mu \nu} + \frac{1}{2} m^{2} A_{\mu} A^{\mu}$ The first term contains the gauge invariant kinetic terms and the second is the mass term. $F_{\mu \nu} \equiv \partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ Now if you try to find the EOM: $\partial_{\rho}\frac{\partial L}{\partial (\partial_{\rho} A_{\sigma})}- \frac{ \partial L}{\partial A_{\sigma}}=0$ You have for the 2nd term: $\frac{ \partial L}{\partial A_{\sigma}}= m^{2} A^{\sigma}$ and for the 1st term: $-a \partial_{\rho}\frac{\partial (F_{\mu \nu} F^{\mu \nu})}{\partial (\partial_{\rho} A_{\sigma})}=-2a \partial_{\rho} F^{\mu \nu} \frac{\partial (\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})}{\partial (\partial_{\rho} A_{\sigma})} = -2a \partial_{\rho} F^{\mu \nu} (\delta^{\rho}_{\mu} \delta^{\sigma}_{\nu}-\delta^{\rho}_{\nu} \delta^{\sigma}_{\mu})=-2a \partial_{\rho} (F^{\rho \sigma} - F^{\sigma \rho})= -4 a \partial_{\rho} F^{\rho \sigma}$ So that's a reason why someone can choose $a= \frac{1}{4}$ and have: $-\partial_{\rho} F^{\rho \sigma} - m^{2} A^{\sigma} =0$ And you get what you ask for. Now what would your "2nd" choice mean? it would have to mean that $a= - \frac{1}{4} <0$ and so the 1st term in the Lagrangian would have a positive sign. That's bad I guess because then the vacuum would have to have negative energy ... Also in general you can reach this lagrangian (without the mass) from the "classical" electromagnetic lagrangian with the correct signs which leads to the term: $-\frac{1}{4}F^{2}$ As for your second question... The positioning of the indices can be very tough to follow, and sometimes people in notes or books can mix them up without a problem, because afterall what they want to say becomes clear... The only time someone has to be careful with the indices is when they denote the representation and not the rows-columns. It's difficult for you to try explaining in words what you are asking, so it's better to show us what you mean. To me, as you write them, there is no 1st or 2nd position. In general what you wrote as $(\Lambda ^{-1} \omega \Lambda)_{\mu \nu}$ means the mu nu component of the matrix $(\Lambda ^{-1} \omega \Lambda)$...So take the $\Lambda$ its inverse and mix them with the $\omega_{\rho \sigma}$ as: $[\Lambda^{-1}] [\omega] [\Lambda]$ where by brackets I mean matrices. Last edited: Sep 6, 2014 7. Sep 7, 2014 ### samalkhaiat The second equation leads to wrong sign for the mass term. Remember, you always need to satisfy the dispersion relation $E^{2} = p^{2} + m^{2}$. The “first place” is already occupied by the index $\mu$. [tex]\left( \lambda^{ - 1 } \omega \lambda \right)_{ \mu \nu } = ( \lambda^{ - 1 } )_{ \mu }{}^{ \rho } \omega_{ \rho \sigma } \lambda^{ \sigma }{}_{ \nu } .$$ For Lorentz transformation, we have $$( \lambda^{ - 1 } )_{ \mu }{}^{ \rho } = \lambda^{ \rho }{}_{ \mu } .$$ So, the $\mu \nu$ matrix element is $$\left( \lambda^{ - 1 } \omega \lambda \right)_{ \mu \nu } = \omega_{ \rho \sigma } \lambda^{ \rho }{}_{ \mu } \lambda^{ \sigma }{}_{ \nu } .$$ In the Euclidean space, i.e. when we don’t distinguish between upstairs from down stairs indices, the $\mu \nu$ matrix element of $(ABC)$ is found by usual multiplication of matrices $$( A B C )_{ \mu \nu } = A_{ \mu \rho } B_{ \rho \sigma } C_{ \sigma \nu } .$$ In Minkowski space, this matrix element has the following equivalent forms $$A_{ \mu }{}^{ \rho } B_{ \rho \sigma } C^{ \sigma }{}_{ \nu } = A_{ \mu \rho } B^{ \rho \sigma } C_{ \sigma \nu } = A_{ \mu \rho } B^{ \rho }{}_{ \sigma } C^{ \sigma }{}_{ \nu } = A_{ \mu }{}^{ \rho } B_{ \rho }{}^{ \sigma } C_{ \sigma \nu } .$$ Sam 8. Sep 7, 2014 ### kau tensor notation contraction and ordering of elements Last edited: Sep 7, 2014 9. Sep 7, 2014 ### ChrisVer You can write it as such, you only have to be careful about what you actually did... In your case you renamed mu into nu and vice versa, and also exchanged the indices of the omega [which would have to give you a minus which I don't see]. As for the other question: $\lambda^{a}_{b} C_{ac} = C_{bc}$. Do you want more explanation on that? As for the position of the elements of the tensor, yes in most cases you can do that. Except for if your elements are not commuting. No such case comes in my mind now. Last edited: Sep 7, 2014 10. Sep 7, 2014 ### kau here you have assume the metric to be of form (-,+,+,+). if you work with (+,-,-,-) probably you could have written it like $-\partial_{\rho} F^{\sigma \rho} - m^{2} A^{\sigma} =0$ ~ to get the energy relation correct. so in any case checking energy relation would give the right choice. isn't it?? 11. Sep 7, 2014 ### ChrisVer I don't see where you used the metric in what you did [flipping the indices of the F without changing the sign is not a part of metric choice, but the fact that F is an antisymmetric tensor]. In fact the metric choice doesn't really play any difference - the ones you said are opposite to each other, and in the Lagrangian the metrics appear twice in the kinetic term: $F_{\mu \nu} F^{\mu \nu} = \eta^{\mu \rho} \eta^{\nu \sigma} F_{\mu \nu} F_{\rho \sigma}$ Also energy results don't depend on the choice of the metric. Last edited: Sep 7, 2014 12. Sep 7, 2014 ### kau yeah if you change ${ \mu}$ and ${ \nu}$ in w and x and ∂ in all places then you will get $-L^{ \nu \mu}$ which is equivalent to $L^{ \mu \nu}$ .. but my question is in the first place why I can't write $δω_{ \mu \nu} x^{ \nu} ∂^{ \mu}$??? why writing in this way leads to an expression which is not quite right??? [/QUOTE] if you want to explain it,please do. I understand this result. but really i do not have clue why that quantity should not be$C_{cb}$?? Last edited: Sep 7, 2014 13. Sep 7, 2014 ### kau so, I have following eqn of motion $-∂_{\rho}F^{\rho \sigma }-m^{2}A^{\sigma}=0$ therefore, $-∂^{2} A^{/sigma}-m^{2}A^{\sigma}=0$ using Lorentz gauge. now if $∂^{2}=-E^{2}+p^{2}$ then I will have correct form $E^{2}-p^{2}=m^{2}$ for other choice of metric I will get an extra negative sign. 14. Sep 7, 2014 ### ChrisVer Because $\lambda$ tensor will act on the 1st components of the $C$... Or in other words you will have the tensor product which will give you [in components] $S^{a}_{dac} = S_{dc}$. At least that's how I understand it. As for your last, then my metric choice is (+ - - - ), because the final result was $(\partial F)^{\sigma} + m^{2} A^{\sigma} =0$ Now if you suddenly change the sign of the metric, there should appear an extra change in the sign of the mass term... because in the EOM you had to take the derivative of $A^{2} = \eta_{\mu \nu} A^{\mu} A^{\nu}$ wrt $A^{\sigma}$. The result always remains the wanted one, that's why I said the energy result doesn't depend on the metric choice... Otherwise you wouldn't be able to compare energetically results of different metric choices. Ah and I forgot the other question... Because then you are not getting a well defined matrix multiplication... you have to take the transpose of omega ... or else you are writting that the rows will multiply rows or collumns will multiply collumns... In matrices when you have $A^a C_{ba} B^{b}= A^{T} C^{T} B \ne A^{T}CB$ Last edited: Sep 7, 2014 15. Sep 7, 2014 ### kau ok .. I got you. but for the last question I still find any reason which stops me from writing $∂ω_{\mu \nu} x^{\nu} ∂^{\mu}$ .... say$∂ω = C_{ba}$ and $x =A^a$ and $∂{\mu}=B^{b}$ then it's all fine. I can understand that what you are saying is true but I am not getting what is wrong in my choice?? would you little elaborate this part?? 16. Sep 7, 2014 ### ChrisVer Nevertheless, in Schrednicki's you have 34.1, where you insert 34.5: $(I - \frac{i}{2} \delta \omega_{\mu \nu} M^{\mu \nu}) \psi_{a}(x) (I + \frac{i}{2} \delta \omega_{\mu \nu} M^{\mu \nu}) = (\delta^{~b}_{a} + \frac{i}{2} \delta \omega_{\mu \nu} (S^{\mu \nu})^{~b}_{a}) \psi_{b}$ $\psi_{a}\frac{i}{2} \delta \omega_{\mu \nu} M^{\mu \nu}-\frac{i}{2} \delta \omega_{\mu \nu} M^{\mu \nu} \psi_{a} = -\delta \omega_{\nu \mu} x^{\nu} \partial^{\mu} \psi(x)_{a}+ \frac{i}{2} \delta \omega_{\mu \nu} (S^{\mu \nu})^{~b}_{a}\psi_{b}$ So you are asking why you get it like that? In general what he does is a Taylor expansion of $\psi ( (I+\delta \omega^{T}) x)$ from that you get the indices... Last edited: Sep 7, 2014 17. Sep 7, 2014 ### samalkhaiat No, you can’t. $\omega_{ \mu \nu } x^{ \nu } \partial^{ \mu } = - \omega_{ \mu \nu } x^{ \mu } \partial^{ \nu }$ Use the fact that contracted (dummy) indices can be relabelled freely. So, you can write $$\omega_{ \mu \nu } x^{ \mu } \ \partial^{ \nu } \phi = \omega_{ \rho \sigma } \ x^{ \rho } \ \partial^{ \sigma } \phi . \ \ \ (1)$$ Using the identity, $A = ( A + A ) / 2$, we can rewrite (1) as $$\omega_{ \mu \nu } \ x^{ \mu } \ \partial^{ \nu } \phi = \frac{ 1 }{ 2 } \left( \omega_{ \mu \nu } \ x^{ \mu } \ \partial^{ \nu } \phi + \omega_{ \rho \sigma } \ x^{ \rho } \ \partial^{ \sigma } \phi \right) .$$ Now, letting $\rho = \nu$ and $\sigma = \mu$, we get $$\omega_{ \mu \nu } \ x^{ \mu } \ \partial^{ \nu } \phi = \frac{ 1 }{ 2 } \left( \omega_{ \mu \nu } \ x^{ \mu } \ \partial^{ \nu } \phi + \omega_{ \nu \mu } \ x^{ \nu } \ \partial^{ \mu } \phi \right) .$$ But $\omega_{ \nu \mu } = - \omega_{ \mu \nu }$. Thus $$\omega_{ \mu \nu } \ x^{ \mu } \ \partial^{ \nu } \phi = \frac{ 1 }{ 2 } \omega_{ \mu \nu } \left( x^{ \mu } \partial^{ \nu } - x^{ \nu } \partial^{ \mu } \right) \phi . \ \ \ \ \ \ (2)$$ You don’t need to mention any textbook for me. Instead, I think it is better to show you a step by step derivation of this equation. Under the Lorentz group, finite-component fields on space-time transform by finite-dimensional (matrix) representations $$\bar{ \phi }_{ a } ( \bar{ x } ) = D_{ a }{}^{ b } \phi_{ b } ( x ) , \ \ \ \ \ \ \ (3)$$ where $D$ is a representation matrix $$D_{ a }{}^{ c }( \Lambda_{ 1 } ) \ D_{ c }{}^{ b }( \Lambda_{ 2 } ) = D_{ a }{}^{ b }( \Lambda_{ 1 } \Lambda_{ 2 } ) .$$ Infinitesimally, we may write $$D_{ a }{}^{ b } = \delta_{ a }^{ b } - \frac{ i }{ 2 } \omega_{ \mu \nu } ( S^{ \mu \nu } )_{ a }{}^{ b } , \ \ \ \ \ \ \ (4)$$ where $S^{ \mu \nu }$ are the appropriate spin matrices for the field $\phi$. They satisfy the Lorentz algebra. However, since $\phi_{ a }( x )$ (for all a’s) is an operator-valued field, it also transforms by (infinite-dimensional) unitary representation,$U( \Lambda )$, of the Lorentz group. $$\bar{ \phi }_{ a } ( \bar{ x } ) = U^{ - 1 } \ \phi_{ a } ( \bar{ x } ) \ U . \ \ \ \ \ (5)$$ In terms of the abstract Lorentz generators, $M^{ \mu \nu }$, and the infinitesimal parameters $\omega_{ \mu \nu }$, we may write the unitary operator as $$U( \Lambda ) = 1 - \frac{ i }{ 2 } \omega_{ \mu \nu } \ M^{ \mu \nu } . \ \ \ \ \ (6)$$ From (3) and (5), we find the finite transformation rule for the field operator $$\bar{ \phi }_{ a } ( \bar{ x } ) = U^{ - 1 } \ \phi_{ a } ( \bar{ x } ) \ U = D_{ a }{}^{ b } \ \phi_{ b } ( x ) . \ \ \ (7)$$ Using $$x = \Lambda^{ - 1 } \ \bar{ x } ,$$ we rewrite (7) as $$\bar{ \phi }_{ a } ( \bar{ x } ) = U^{ - 1 } \ \phi_{ a } ( \bar{ x } ) \ U = D_{ a }{}^{ b } \phi_{ b } ( \Lambda^{ - 1 } \ \bar{ x } ) .$$ Dropping the bars from the coordinates, we get $$\bar{ \phi }_{ a } ( x ) = U^{ - 1 } \ \phi_{ a } ( x ) \ U = D_{ a }{}^{ b } \ \phi_{ b } ( \Lambda^{ - 1 } x ) . \ \ \ \ (8)$$ Now, we will try to find the infinitesimal version of (8). Using $$( \Lambda^{ - 1 } )^{ \mu \nu } x_{ \nu } = x^{ \mu } - \omega^{ \mu \nu } \ x_{ \nu } ,$$ we expand $\phi ( \Lambda^{ - 1 } x )$ to first order as $$\phi_{ b } ( \Lambda^{ - 1 } x ) = \phi_{ b } ( x ) - \omega^{ \mu \nu } \ x_{ \nu } \ \partial_{ \mu } \phi_{ b } ( x ) . \ \ \ \ \ (9)$$ Substituting the equations (4), (6) and (9) in equation (8) and keeping only first order terms, we find $$\bar{ \phi }_{ a } ( x ) - \phi_{ a } ( x ) = \frac{ i }{ 2 } \omega_{ \mu \nu } \ [ M^{ \mu \nu } , \phi_{ a } ( x ) ] = - \omega^{ \mu \nu } \ x_{ \nu } \ \partial_{ \mu } \phi_{ a } - \frac{ i }{ 2 } \omega_{ \mu \nu } ( S^{ \mu \nu } )_{ a }{}^{ b } \phi_{ b } ( x ) . \ (10)$$ Using $\omega^{ \mu \nu } = - \omega^{ \nu \mu }$ and $$\omega^{ \nu \mu } \ x_{ \nu } \ \partial_{ \mu } = \omega_{ \nu \mu } \ x^{ \nu } \ \partial^{ \mu } = \omega_{ \mu \nu } \ x^{ \mu } \ \partial^{ \nu } ,$$ equation (10) becomes $$\delta \phi_{ a } ( x ) = \frac{ i }{ 2 } \omega_{ \mu \nu } \ [ M^{ \mu \nu } , \phi_{ a } ( x ) ] = \omega_{ \mu \nu } \ x^{ \mu } \ \partial^{ \nu } \phi_{ a } - \frac{ i }{ 2 } \omega_{ \mu \nu } ( S^{ \mu \nu } )_{ a }{}^{ b } \phi_{ b } ( x ) .$$ Now, if we use (2) in the first term on the right-hand-side, we find $$\delta \phi_{ a } ( x ) = \frac{ i }{ 2 } \omega_{ \mu \nu } \ [ M^{ \mu \nu } , \phi_{ a } ( x ) ] = \frac{ 1 }{ 2 } \omega_{ \mu \nu } \ ( x^{ \mu } \partial^{ \nu } - x^{ \nu } \partial^{ \mu } ) \phi_{ a } ( x ) - \frac{ i }{ 2 } \omega_{ \mu \nu } \ ( S^{ \mu \nu } )_{ a }{}^{ b } \phi_{ b } ( x ) .$$ So, the equation you are after follows from $$[ i M^{ \mu \nu } , \phi_{ a } ( x ) ] = ( x^{ \mu } \partial^{ \nu } - x^{ \nu } \partial^{ \mu } ) \phi_{ a } ( x ) - i ( S^{ \mu \nu } )_{ a }{}^{ b } \phi_{ b } ( x ) .$$ Neither! The question does not make sense. $\lambda^{ n }{}_{ a } C_{ n u } = C_{ a u }$ if and only if $\lambda^{ n }{}_{ a }$ is equal to the Kronecker delta $\delta^{ n }_{ a }$. Yes, for numerical tensors and matrix elements the order does not matter. Sam 18. Sep 7, 2014 ### ChrisVer Samal I think the problem is that you derived: $[\phi_{a}, M^{\mu \nu}] = L^{\nu \mu} \phi_{a} + i (S^{\mu \nu})^{~~b}_{a} \phi_{b}$ while in Schrednicki [the one kau mentions to have problem with] is: $[\phi_{a}, M^{\mu \nu}] = L^{\mu \nu} \phi_{a} + i (S^{\mu \nu})^{~~b}_{a} \phi_{b}$ However I also reached your result. 19. Sep 7, 2014 ### samalkhaiat With the D matrix and the U operator are given by $$D = 1 - \frac{ i }{ 2 } \omega_{ \mu \nu } \ S^{ \mu \nu } \ , \ \ U = 1 - \frac{ i }{ 2 } \omega_{ \mu \nu } \ M^{ \mu \nu } ,$$ and in any metric convention, the correct transformation is given by $$[i M^{ \mu \nu } , \phi_{ a } ] = ( x^{ \mu } \ \partial^{ \nu } - x^{ \nu } \ \partial^{ \mu } ) \phi_{ a } - i ( S^{ \mu \nu } )_{ a }{}^{ b } \phi_{ b } .$$ Notice that I did not write $L^{ \mu \nu }$ because its definition depends on the metric one uses. Any way, who is Steven ? Sam 20. Sep 12, 2014 ### kau let me tell you what I actually mean in this part. $\lambda^{\mu \nu} X^{\alpha}{}_{mu}= Y^{\nu \alpha} or y^{\alpha \nu}$ now I guess my question makes sense.. and you have written that numerical and matrix component tensor cases order does not matter. ok fine.. but for just to get one clarification i am asking you this.. so the point is we should only worry about tensor or matrix component ordering when they are non commutating.??? thanks. Know someone interested in this topic? 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https://www.physicsforums.com/threads/entropy-is-a-measure-of-energy-availiable-for-work.659598/	# Entropy is a measure of energy availiable for work ? 1. ### Rap 789 Entropy is a measure of energy availiable for work ???? "Entropy is a measure of energy availiable for work". Can someone explain this to me? Give some examples that show in what sense it is true. It has to come with a lot of caveats, proviso's etc. because its simply not true on its face. I mean, if I have a container of gas at some temperature above 0 K, then I can extract all of its internal energy as work, just let it quasistatically expand to infinity. 2. ### Studiot Re: Entropy is a measure of energy availiable for work ???? Such a container has internal pressure P. Expanding from P into a vacuum does no work. Expanding from P against an external pressure P' < P does work, but as this happens P diminishes until P = P' when the system is in equilibrium. How would you proceed from this equilibrium to infinity, where P = 0 ? 3. ### Rap 789 Re: Entropy is a measure of energy availiable for work ???? Yes, you would need initial pressure (P) greater than zero, and ambient pressure (P') equal to zero, i.e. in outer space. But the point remains, the statement that "entropy is a measure of energy unavailable for work" is contradicted by this example. 4. ### Studiot Re: Entropy is a measure of energy availiable for work ???? This is fundamental, pushing against something that offers no resistance does no work. ### Staff: Mentor Re: Entropy is a measure of energy availiable for work ???? Where did you get that quote? It is wrong: it is missing the word "not"! http://en.wikipedia.org/wiki/Entropy 6. ### Studiot Re: Entropy is a measure of energy availiable for work ???? Agreed - I think that was a typo -, but we also need to dispel the misconception in the proposed counterexample that follows. Once that is done it is easy to explain the correct reasoning. 7. ### Rap 789 Re: Entropy is a measure of energy availiable for work ???? Yes, sorry, I misquoted it. It should be something like "Entropy is a measure of the energy NOT available for useful work". I also see the point that the example I gave is not good. In order for the expansion to be slow, there has to be an opposing force almost equal to the pressure force, and that force has to diminish as the volume increases (and pressure decreases). This opposing force would be the mechanism by which work was done on the environment. Something like a mass in a gravitational field, in which the mass is slowly being reduced by removal. I found a web site http://web.mit.edu/16.unified/www/SPRING/propulsion/notes/node48.html which seems to give an explanation, I will have to look at it closely. Also the Wikipedia quote says it is "energy per unit temperature" not available for work, which I cannot immediately decipher. 8. ### Rap 789 Re: Entropy is a measure of energy availiable for work ???? Looking at the above site, it seems to me what it is saying is that the amount of energy unavailable for useful work in a Carnot cycle is equal to the entropy extracted from the hot reservoir (which is equal to the entropy deposited in the cold reservoir) times the temperature of the cold reservoir. How you get from this to the idea that "Entropy is a measure of the energy unavailable for work" still eludes me. Entropy of what? I tried assuming that the hot reservoir was actually a hot system with a finite amount of energy and entropy. Again, you can prove that if you extract entropy ∆S from the hot body, the amount of unavailable energy is Tc ∆S where Tc is the temperature of the cold reservoir. But you can only extract so much entropy from the hot body using a working body at the cold reservoir temperature. If their temperatures are very close, you can extract very little entropy and energy. The great majority of the internal energy of the hot body is unavailable for work in this case. If you set the cold reservoir to zero degrees, then the amount of energy unavailable for work is zero. You could extract all of the internal energy of the hot body as work. (right?). I still don't get it. 9. ### Andrew Mason 6,846 Re: Entropy is a measure of energy availiable for work ???? We have hashed this over before. The problem with "entropy is a measure of the energy unavailable for work" is that standing alone it is not clear and open to a number of interpretations. It requires a special definition of "energy unavailable for work" as the explanation shows. Without that explanation it can easily lead to misinterpretation. For example, between two temperatures, Th and Tc, the heat flow Q is capable of producing an amount of work, W = Q(1-Tc/Th) ie. with a Carnot engine. So there is energy E = QTc/Th energy that is, in that sense, "unavailable" for doing work. Yet, as we know, ΔS = 0 for a Carnot engine. So one could well ask, how is 0 a measure of QTc/Th? The answer is: "well, that is not what we mean by 'energy unavailable to do work'. We really mean 'lost work' which is the potential work that could be extracted minus the work that was actually extracted, or the amount of additional work required to restore the system and surroundings to their original state if you saved the output work and used it to drive the process in reverse." Hence the confusion. So, as I have said before, this particular statement should not be used to introduce the concept of entropy. By itself it explains nothing and leads to great confusion. AM Last edited: Dec 19, 2012 10. ### Rap 789 Re: Entropy is a measure of energy availiable for work ???? Well, I kind of thought that was case, but there was also the possibility I was missing something. Thanks for the clarification. 11. ### Studiot Re: Entropy is a measure of energy availiable for work ???? The time for hand waving is over, here is some mathematics. Consider a universe consisting of a system contained in a heat bath or reservoir at uniform constant temperature T. Consider changes in the function Z = entropy of bath plus entropy of the system = entropy of this universe. dZ = dSb + dSs.................1 Where b refers to the bath and s refers to the system. If the system absorbs heat dq the same amount of heat is lost by the bath so the entropy change dSb =-dq/T...................2 substituting this into equation 1 dZ = dSs - dq/T..............3 Now consider a change of state of the system from state A to state B By the first law dUs = dq - dw ..................4 Combining equations 3 and 4 and rearranging dSs = (dUs+dw)/T + dZ re-arranging dw = TdSs - dUs -TdZ Since TdZ is always a positive quantity or zero dw ≤ TdSs - dUs Where dw is the work done by the system This is the principle of maximum work and calculates the maximum work that can be obtained from the system. As you can see it has two components vis from the entropy created and from the change in internal energy and these act in opposite directions so in this sense the TdSs term reduces the amount of work obtainable from the internal energy of the system and accounts for unavailable energy since TdSs has the dimensions of energy. Note the usual caveat The inequality refers to irreversible processes, the equality to reversible ones. Last edited: Dec 19, 2012 12. ### Darwin123 741 Re: Entropy is a measure of energy availiable for work ???? What you said is only possible if the gas expands both adiabatically and reversibly. In an adiabatic and reversible expansion, the change in entropy of the gas is zero. Under that condition, one could turn all the internal energy into work. Any deviation from the conditions of adiabatic and reversible would result in some internal energy not being turned to work. First, I prove that one can extract all the internal energy from a monotonic ideal gas using an expansion that is BOTH adiabatic and reversible. Suppose one were to take an ideal gas in a closed chamber and expand it both adiabatically and slowly, so that it is in a state near thermal equilibrium at all times. No entropy goes in or out of the chamber. At the end of the expansion, even in the limit of infinite volume, you would end up with a gas of finite temperature. The ideal gas law is: 1) PV=nRT where P is the pressure of the gas, V is the volume of the chamber, n is the molarity of the gas, R is the gas constant and T is the temperature. The internal energy of the ideal gas is: 2) U=(3/2)nRT where U is the internal energy of the gas and everything else is the same. Substituting equation 2 into equation 1: 3)U=(3/2)PV Before the gas starts expanding, let P=P0, V=V0, T=T0, and U=U0. The chamber is closed, so "n" is constant the entire time. There are three degrees of freedom for each atom in a mono atomic gas. Therefore, for a mono atomic gas: 3) U0=1.5 P0 V0 The adiabatic expansion of an mono-atomic gas is: 4) P0 V0 ^(5/3)= P V^(1.5) Therefore, 5) P = P0 (V0/V)^(5/3) The work, W, done by the gas is 6) W = ∫[V0→∞] P dV Substituting equation 5 into equation 6: 7) W = (P0 V0^(5/3)))∫[V0→∞] V^(-5/3) dV Evaluating the integral in equation 6: 8) W = (3/2)(P0 V0^5/3)V0^(-2/3) 9) W=(1.5) P0 V0 The expression for W in equation 9 is the same as the expression for U0 in equation 3. Therefore, the internal energy has been taken out completely. I’ll solve the problem later (a week or so) for an expansion with sliding friction. There, the increase in entropy characterizes the amount of internal energy not turned into work. However, I will set up the problem. I will show the two equations that makes the case with sliding friction different from the reversible condition. One remove the reversibility condition by adding sliding friction, 10) Wf = ∫[V0→Vf] (P-Pf) dV where Pf is the pressure due to sliding friction and Vf is the volume when the P=Pf. The chamber stops expanding when P=Pf. However, another expression is necessary to describe how much entropy is created. 11) dQ = Pf dV Equation merely says that the energy used up by the sliding friction causes entropy to be created. The heat energy, dQ, is the energy used up by friction. I don’t have time now, so I leave it as an exercise. Honest, moderator, I promise to get back to it. However, he wants an example where the creation of entropy limits the work that can be extracted. This is a good one. Spoiler Wf<W. Not all the internal energy is turned into work with sliding friction included. Let Q be the work done by the sliding force alone. The increase in entropy is enough to explain why the internal energy is not being turned into work. 13. ### Studiot Re: Entropy is a measure of energy availiable for work ???? Do you not agree that the maximum possible work is extracted in a reversible isothermal expansion? Are you sure you mean this: you have different values of gamma on each side? 14. ### Darwin123 741 Re: Entropy is a measure of energy availiable for work ???? In an isothermal expansion, energy is entering the gas from a hot reservoir. Therefore, one can't say that one is extracting the energy from the ideal gas. Most of the energy is being extracted from the hot reservoir, not from the ideal gas. In the corresponding isothermal expansion, the ideal gas is acting like a conduit for energy and entropy. The ideal gas is not acting as a storage matrix for the energy. The OP was saying that the work was being extracted from the internal energy that was initially embedded in the ideal gas. All work energy comes from the container of gas, not outside reservoirs. So the walls of the container have to be thermal insulators. If you allow heat energy to conduct through the walls of the container, then the work may exceed the initial value of the internal energy. The hot reservoir can keep supplying energy long after the internal energy is used up. So I stick to my guns with regards to the specificity of the OPs hypothesis. He was unconsciously assuming that the expansion is both adiabatic and reversible. I diagree with those people who said that the gas would remain the same temperature during the expansion, and that not all the energy could be extracted from the internal energy. I showed that the internal energy could be entirely extracted for an ideal monotonic gas under adiabatic and reversible conditions. The big problem that I have with the OP's question is with the word "quasistatic". I conjecture that the OP thought that "quasistatic" meant "both adiabatic and reversible". A quasistatic process can be both nonadiabatic and irreversible. However, "quasistatic" is a useful hypothesis. The word "quasistatic" implies spatial uniformity. In this case, the word quasistatic implies that the temperature and the pressure of the ideal gas is uniform in the chamber. Quasistatic implies that enough time has passed between steps that both temperature and pressure are effectively constant in space. Thus, temperature is not a function of position. Pressure is not a function of position. I think this is correct. I didn't spend much time checking my work. If you see an arithmetic blunder, feel free to correct me. Also, I specified a specific case to simplify the problem. So even if I did it correctly, the gamma value that I used was atypical. Next time, I will let gamma be a parameter of arbitrary value. I specified a monotonic gas. The gas is comprise of individual atoms. There are no internal degrees of freedom in these atoms. Any correlation between coefficients may be due to my choice. There are many ways an expansion can be irreversible. I think the one most people think of is where the expansion is not quasistatic. Suppose the gas is allowed to expand freely, so that temperature and pressure are not uniform. This is irreversible. However, the mathematics is way beyond my level of expertise. The problem is tractable if the process is quasistatic. So, I think the best thing would be to show how it works with an quasistatic but irreversible. For instance, what happens if one turns on the sliding friction and the static friction in this expansion. That would result in a process where entropy is created. In other words, friction would result in an irreversible expansion even under quasistatic conditions. I don't have time now. I will post a solution to that later. For now, just remember that not all quasistatic processes are reversible. 15. ### Studiot Re: Entropy is a measure of energy availiable for work ???? I wanted to be more subtle and polite but $${P_1}{V_1}^\gamma = {P_2}{V_2}^\gamma$$ Whereas you have $${P_0}{V_0}^{\frac{5}{3}} = {P_1}{V_1}^{\frac{3}{2}}$$ Is this a rejection of Joule's experiment and the definition of an ideal gas? If so you should make it clear that your view is not mainstream physics. It should be noted that Joules experiment was both adiabatic and isothermal and has been repeated successfully many times. You are correct in observing that during an isothermal expansion neither q nor w are zero. However what makes you think the internal energy is the same at the beginning and end, in the light of your above statement? One definition (or property derivable from an equivalent definition) of an ideal gas is that its internal energy is a function of temperature alone so if the temperature changes the internal energy changes. To remove all the internal energy you would have to remove all the kinetic energy of all its molecules. 16. ### Andrew Mason 6,846 Re: Entropy is a measure of energy availiable for work ???? I think it was a typo. Darwin did write: P = P0 (V0/V)^(5/3) a little farther down. I am not sure that you are both talking about the same process. Darwin was referring to a quasi-static adiabatic expansion of an ideal gas. Temperature is given by the adiabatic condition: $$TV^{(\gamma - 1)} = \text{constant}$$ So, if volume changes this cannot be isothermal. Temperature has to change. I think you (Studiot) may be talking about free expansion, not quasi-static expansion, in which case T is constant for an ideal gas. AM 17. ### Rap 789 Re: Entropy is a measure of energy availiable for work ???? Studiot - I agree with your derivation but with regard to the OP, the correct statement would then be: "an infinitesimal change in entropy is a measure of the minimum infinitesimal amount of energy unavailable for work given a particular ambient temperature." Darwin123 - Thank you for clarifying the muddled OP. Depending on conditions, all, some, or none of a system's internal energy can be converted to work, and so the statement "Entropy is a measure of the energy unavailable for work" is ambiguous at best, wrong at worst. You are right, I was assuming adiabatic and reversible. Adiabatic or else you are potentially using energy from somewhere else to do the work. Reversible, because it will give the miniumum amount of work unavailable. I should have said that instead of quasistatic. I take quasistatic to mean, by definition, a process can be described as a continuum of equilibrium states. 18. ### Darwin123 741 Re: Entropy is a measure of energy availiable for work ???? Oops. My typo. I meant, No, it was a typo. Certainly not mainstream physics. However, I did not insert that mistake into my later equations. Good for Joule! More power to him! The internal energy of an ideal gas varies only with its temperature. The internal energy is not explicitly determined by either its pressure or its temperature. If you know how many atoms are in a molecule of the ideal, the number of molecules and the temperature, then you can uniquely determine the internal energy. The quantity of the ideal gas in the closed container is constant. The number of atoms per molecule is constant. For an isothermal expansion, the temperature of the ideal gas is constant. Therefore, the internal energy of the ideal gas is constant for an isothermal expansion. The internal energy never changes during the entire expansion, even in the limit of infinite volume. However, work is being done for the entire time. Therefore, the work can't come from the internal energy. An isothermal expansion is actually the most inefficient way to use the internal energy of the ideal gas. None of the internal energy of the ideal gas becomes work in an isothermal expansion. All heat energy absorbed by the ideal gas instantly turns into work on the surroundings. In an isothermal expansion, not a single picojoule of work comes from the internal energy of the gas. It all comes from the heat reservoir connected to the ideal gas. One should note that the total amount of work done by the gas in the isothermal expansion is infinite. The work done by the gas increases with the logarithm of volume. So an infinite volume means that an infinite amount of work is performed. Do the calculations for yourself. Obviously, the infinite energy that is going to become work is not in the internal energy of the gas. In fact, the internal energy of the gas remains the same even in the extreme limit of infinite work. The energy for work is being supplied by the heat reservoir, not the ideal gas. In order to maintain a constant temperature, the container has to be in thermal contact with Therefore, the kinetic energy of the molecules in the gas can not change during an isothermal expansion. If you want to extract all the internal energy of the ideal gas to work on the surroundings, then you have to decrease the temperature to absolute zero. This can be done in an adiabatic expansion. It can't be done in an isothermal expansion. Entropy can change only two ways. It can move or it can be created. In an adiabatic process, entropy can't move. The temperature changes instead. In an isothermal process, the entropy moves in such a way as to keep the temperature constant. 19. ### Rap 789 Re: Entropy is a measure of energy availiable for work ???? Note, that is only for a simple system (homogeneous). For complex systems contained inside a thermally insulating boundary, entropy may move around inside the system, driven by temperature differences inside the system, equalizing them when possible, but can never be transferred across the boundary. During these internal sub-processes, entropy may also be created. In an isothermal process, the boundary is thermally open, and entropy may move across the boundary, again driven by temperature differences between the system and the environment, in such a way as to equalize internal temperatures at the constant temperature of the environment, when possible. Entropy transfer goes hand in hand with energy transfer via dU=T dS. If a process is converting energy to work, and you want to know how much of that energy is converted to work, in order to keep the bookwork straight, you cannot bring in energy or entropy from somewhere else to accomplish that work. To ensure this, the process has to be adiabatic, i.e. inside a thermally insulating boundary which prevents entropy and energy coming in from somewhere else. For finite temperature differences, transfer of entropy across a thermally open boundary causes creation of entropy at the boundary which is transferred to the lower temperature system. The transfer of entropy is of order ∆T, the creation of entropy at the boundary is of order ∆T^2, so, in the limit of small ∆T, entropy may be transferred without creation at the boundary. If ∆T is identically zero, there will be no transfer of entropy, since only temperature differences will drive entropy transfer. Last edited: Dec 20, 2012 20. ### Andrew Mason 6,846 Re: Entropy is a measure of energy availiable for work ???? I think you have to qualify this statement. You have to speaking about a reversible process. Entropy can and certainly does change in an adiabatic irreversible expansion. And I wouldn't say it moves because entropy is not a conserved quantity such as energy or momentum. We can speak of energy or momentum transfer because the loss of energy/momentum must result in the gain of energy/momentum of some other body so it behaves as if it moves. Entropy does not behave like. So I would suggest that the concept of entropy moving is not a particularly helpful one. In a reversible isothermal process total entropy change is 0. In a real isothermal process, the entropy of the system + surroundings inevitably increases. It is not entropy that moves. It is energy. And the faster the energy moves, the greater the increase in entropy. So I might suggest that entropy increase is related more to the speed of energy transfer (heat flow) than to the fact that a body remains at the same temperature. AM	{"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": 0.918388307094574, "perplexity": 468.7400551344971}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246639414.6/warc/CC-MAIN-20150417045719-00203-ip-10-235-10-82.ec2.internal.warc.gz"}
http://www.reference.com/browse/electric%20currents	Definitions # Electric current Electric current is the flow (movement) of electric charge. The SI unit of electric current is the ampere. Electric current is measured using an ammeter. The electric charge may be either electrons or ions. The nature of the electric current is basically the same for either type. ## Current in a metal wire A solid conductive metal contains a large population of mobile, or free, electrons. These electrons are bound to the metal lattice but not to any individual atom. Even with no external electric field applied, these electrons move about randomly due to thermal energy but, on average, there is zero net current within the metal. Given a plane through which the wire passes, the number of electrons moving from one side to the other in any period of time is on average equal to the number passing in the opposite direction. When a metal wire is connected across the two terminals of a DC voltage source such as a battery, the source places an electric field across the conductor. The moment contact is made, the free electrons of the conductor are forced to drift toward the positive terminal under the influence of this field. The free electrons are therefore the current carrier in a typical solid conductor. For an electric current of 1 ampere, 1 coulomb of electric charge (which consists of about 6.242 × 1018 electrons) drifts every second through any plane through which the conductor passes. The current I in amperes can be calculated with the following equation: $I = \left\{Q over t\right\}$ where $Q !$ is the electric charge in coulombs (ampere seconds) $t !$ is the time in seconds It follows that: $Q=It !$ and $t = \left\{Q over I\right\}$ More generally, electric current can be represented as the time rate of change of charge, or $I = frac\left\{dQ\right\}\left\{dt\right\}$. ## Current density Current density is a measure of the density of electric current. It is defined as a vector whose magnitude is the electric current per cross-sectional area. In SI units, the current density is measured in amperes per square meter. ## The drift speed of electric charges The mobile charged particles within a conductor move constantly in random directions, like the particles of a gas. In order for there to be a net flow of charge, the particles must also move together with an average drift rate. Electrons are the charge carriers in metals and they follow an erratic path, bouncing from atom to atom, but generally drifting in the direction of the electric field. The speed at which they drift can be calculated from the equation: $I=nAvQ !$ where $I !$ is the electric current $n !$ is number of charged particles per unit volume $A !$ is the cross-sectional area of the conductor $v !$ is the drift velocity, and $Q !$ is the charge on each particle. Electric currents in solids typically flow very slowly. For example, in a copper wire of cross-section 0.5 mm², carrying a current of 5 A, the drift velocity of the electrons is of the order of a millimetre per second. To take a different example, in the near-vacuum inside a cathode ray tube, the electrons travel in near-straight lines ("ballistically") at about a tenth of the speed of light. Any accelerating electric charge, and therefore any changing electric current, gives rise to an electromagnetic wave that propagates at very high speed outside the surface of the conductor. This speed is usually a significant fraction of the speed of light, as can be deduced from Maxwell's Equations, and is therefore many times faster than the drift velocity of the electrons. For example, in AC power lines, the waves of electromagnetic energy propagate through the space between the wires, moving from a source to a distant load, even though the electrons in the wires only move back and forth over a tiny distance. The ratio of the speed of the electromagnetic wave to the speed of light in free space is called the velocity factor, and depends on the electromagnetic properties of the conductor and the insulating materials surrounding it, and on their shape and size. The nature of these three velocities can be illustrated by an analogy with the three similar velocities associated with gases. The low drift velocity of charge carriers is analogous to air motion; in other words, winds. The high speed of electromagnetic waves is roughly analogous to the speed of sound in a gas; while the random motion of charges is analogous to heat - the thermal velocity of randomly vibrating gas particles. ## Ohm's law Ohm's law predicts the current in an (ideal) resistor (or other ohmic device) to be the applied voltage divided by resistance: I = frac {V}{R} where I is the current, measured in amperes V is the potential difference measured in volts R is the resistance measured in ohms ## Conventional current A flow of positive charge gives the same electric current as an opposite flow of negative charge. Thus, opposite flows of opposite charges contribute to a single electric current. For this reason, the polarity of the flowing charges can usually be ignored during measurements. All the flowing charges are assumed to have positive polarity, and this flow is called Conventional current. In solid metals such as wires, the positive charge carriers are immobile, and only the negatively charged electrons flow. Because the electron carries negative charge, the electron motion in a metal is in the direction opposite to that of conventional (or electric) current. In many other conductive materials, the electric current is due to the flow of both positively and negatively charged particles at the same time. In still others, the current is entirely due to positive charge flow. For example, the electric currents in electrolytes are flows of electrically charged atoms (ions), which exist in both positive and negative varieties. In a common lead-acid electrochemical cell, electric currents are composed of positive hydrogen ions (protons) flowing in one direction, and negative sulfate ions flowing in the other. Electric currents in sparks or plasma are flows of electrons as well as positive and negative ions. In ice and in certain solid electrolytes, the electric current is entirely composed of flowing protons. For conceptual simplicity, Conventional current is used to conceal these issues by summing the various currents together into a single value. There are also materials where the electric current is due to the flow of electrons, and yet it is conceptually easier to think of the current as due to the flow of positive "holes" (the spots that should have an electron to make the conductor neutral). This is the case in a p-type semiconductor. ## Examples Natural examples include lightning and the solar wind, the source of the polar auroras (the aurora borealis and aurora australis). The artificial form of electric current is the flow of conduction electrons in metal wires, such as the overhead power lines that deliver electrical energy across long distances and the smaller wires within electrical and electronic equipment. In electronics, other forms of electric current include the flow of electrons through resistors or through the vacuum in a vacuum tube, the flow of ions inside a battery or a neuron, and the flow of holes within a semiconductor. ## Electromagnetism Electric current produces a magnetic field. The magnetic field can be visualized as a pattern of circular field lines surrounding the wire. Electric current can be directly measured with a galvanometer, but this method involves breaking the circuit, which is sometimes inconvenient. Current can also be measured without breaking the circuit by detecting the magnetic field associated with the current. Devices used for this include Hall effect sensors, current clamps, current transformers, and Rogowski coils. ## Reference direction When solving electrical circuits, the actual direction of current through a specific circuit element is usually unknown. Consequently, each circuit element is assigned a current variable with an arbitrarily chosen reference direction. When the circuit is solved, the circuit element currents may have positive or negative values. A negative value means that the actual direction of current through that circuit element is opposite that of the chosen reference direction. ## Electrical safety The most obvious hazard is electrical shock, where a current passes through part of the body. It is the amount of current passing through the body that determines the effect, and this depends on the nature of the contact, the condition of the body part, the current path through the body and the voltage of the source. While a very small amount can cause a slight tingle, too much can cause severe burns if it passes through the skin or even cardiac arrest if enough passes through the heart. The effect also varies considerably from individual to individual. (For approximate figures see Shock Effects under electric shock.) Due to this and the fact that passing current cannot be easily predicted in most practical circumstances, any supply of over 50 volts should be considered a possible source of dangerous electric shock. In particular, note that 110 volts (a minimum voltage at which AC mains power is distributed in much of the Americas, and 4 other countries, mostly in Asia) can certainly cause a lethal amount of current to pass through the body. Electric arcs, which can occur with supplies of any voltage (for example, a typical arc welding machine has a voltage between the electrodes of just a few tens of volts), are very hot and emit ultra-violet (UV) and infra-red radiation (IR). Proximity to an electric arc can therefore cause severe thermal burns, and UV is damaging to unprotected eyes and skin. Accidental electric heating can also be dangerous. An overloaded power cable is a frequent cause of fire. A battery as small as an AA cell placed in a pocket with metal coins can lead to a short circuit heating the battery and the coins which may inflict burns. NiCad, NiMh cells, and lithium batteries are particularly risky because they can deliver a very high current due to their low internal resistance.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 13, "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": 0.9034416079521179, "perplexity": 429.70059725326547}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207929956.54/warc/CC-MAIN-20150521113209-00110-ip-10-180-206-219.ec2.internal.warc.gz"}
https://www.mathswithmum.com/subtracting-unlike-fractions/	# How to Subtract Fractions with Unlike Denominators How to Subtract Fractions with Unlike Denominators • To subtract fractions, they must first have the same denominator on the bottom. • Find the first number that is in the times tables of each denominator. We call this the least common denominator. • The first number in both the 2 and 5 times table is 10. 10 is the least common denominator. • Write each fraction out of this least common denominator using equivalent fractions. • 1/2 can be written as 5/10 by multiplying the numbers by 5. • 1/5 can be written as 2/10 by multiplying the numbers by 2. • Once the denominators are the same, the fractions can be subtracted. • Keep the denominator on the bottom the same and subtract the numerators on top. • 5/10 - 2/10 = 3/10 and therefore 1/2 - 1/5 = 3/10 . Convert the fractions to like fractions with the same denominator before subtracting them. • To subtract fractions with different denominators, first write them as equivalent fractions with the same denominator. • The denominators of 6 and 8 have a least common denominator of 24. This means that 24 is the first number in both the 6 and 8 times table. • 4/6 can be written as 16/24 by multiplying the values by 4. • 3/8 can be written as 9/24 by multiplying the values by 3. • 16/24 - 9/24 = 7/24 . • Therefore 4/6 - 3/8 = 7/24 . Supporting Lessons # Subtracting Fractions with Unlike Denominators ## How to Subtract Fractions with Unlike Denominators To subtract fractions with unlike denominators, follow these steps: 1. Find the least common multiple of the denominators of the fractions. 2. Write equivalent fractions that have the least common multiple as the denominator. 3. Keep this least common multiple as the denominator of the answer. 4. Subtract the numerators in the question to get the numerator of the answer. The most important rule when subtracting fractions is to first make a common denominator. The denominators of the fractions must be the same before subtracting them. For example, here is 4/6 - 3/8 . Denominators are the numbers on the bottom of the fraction. The denominators are 6 and 8, which are different. To subtract fractions, we need the denominators to be the same. A common denominator is a number that is a multiple of all of the other denominators. It is usually chosen as the first number that appears in all of the times tables of the denominators. For example, with 4/6 - 3/8 , the common denominator is 24 because 24 is the first number in the 6 and 8 times tables. To find the least common denominator, list the multiples of each denominator and write down the first number to appear in every list. Alternatively, a common denominator can be found by multiplying the denominators together. The multiples of 6 are 6, 12, 18, 24 and the multiples of 8 are 8, 16, 24. 24 is the first number to appear in each list and so, it is the least common denominator of 6 and 8. The least common denominator is also commonly known as the lowest common denominator. It is the smallest common denominator that can be found. The next step is to find equivalent fractions that have the least common denominator. The first fraction denominator has been multiplied by 4. Therefore we need to multiply the numerator by 4 as well. The denominator calculation is 6 × 4 = 24. The numerator calculation is 4 × 4 = 16. Both the numerator and denominator are multiplied by 4. The second fraction denominator has been multiplied by 3. Therefore we need to multiply the numerator by 3 as well. The denominator calculation is 8 × 3 = 24. The numerator calculation is 3 × 3 = 9. Both the numerator and denominator are multiplied by 3. Now that the fractions have common denominators, the subtraction can be done. The denominator of the answer is the same as the common denominator. The lowest common denominator is 24 and so, the denominator of the answer is also 24. To find the numerator of the answer, simply subtract the numerators in the question. 16 - 9 = 7 and so, the numerator of the answer is 7. 4/6 - 3/8 = 7/24 . Here is another example of subtracting unlike fractions step by step. We have 1/2 - 1/5 . The denominators are different and so, we need to find equivalent fractions with a common denominator. The multiples of 2 are 2, 4, 6, 8, 10 and the multiples of 5 are 5 and 10. We can see that 10 is the first number in both lists and so, 10 is the least common denominator of 2 and 5. We write 1/2 as 5/10 . We write 1/5 as 2/10 . Now that we have a common denominator, we can subtract the fractions 5/10 - 2/10 = 3/10 . If both denominators are prime, the least common denominator is found by multiplying the denominators together. For example in 1/2 - 1/5 , the least common denominator is 10 because 2 × 5 = 10. ## Subtracting Fractions with the Butterfly Method The butterfly method is a short method that can be used for adding or subtracting 2 fractions. It involves multiplying the numerator of one fraction by the denominator of the other with the bubbles around each multiplication drawn to make an image of a butterfly. To subtract fractions using the butterfly method, follow these steps: 1. Multiply the two denominators together to find the denominator of the answer. 2. Multiply the first numerator by the second denominator. 3. Multiply the second numerator by the first denominator. 4. Write both of these two answers on the numerator, separated by a subtraction sign. 5. Work out the subtraction to get one number as the numerator. 6. Simplify the fraction if possible. For example, we have 4/5 - 2/3 . The diagram below shows how the butterfly method works. We first multiply the denominators of 5 and 3. 5 × 3 = 15 and so the denominator of the answer is 15. Next we multiply the numerator of the first fraction by the denominator of the second fraction. 4 × 3 = 12, so we write 12 on the numerator of the fraction. Next we multiply the numerator of the second fraction by the denominator of the first fraction. 2 × 5 = 10 and so we write a 10 alongside the 12 with a subtraction sign in between. For each multiplication in the butterfly method, draw a bubble around the numbers. This makes the overall calculation look like a butterfly and can help make the method easier to remember and learn. Finally, we work out the subtraction on the numerator. 12 - 10 = 2 and so, 2 is the numerator. The result of the butterfly method calculation is 2/15 The butterfly method is an easy way to routinely solve the addition and subtraction of 2 fractions. The benefits of the butterfly method are that it reduces working out and the method is easier to remember due to the symmetrical butterfly pattern. It is a useful method to teach when the initial understanding of how to add and subtract fractions has been learnt. The main problem with the butterfly method is that it only allows for the addition and subtraction of two fractions. It is not recommended to introduce adding and subtracting fractions with the butterfly method because it does not allow for a strong understanding of why the method works and it is limited to use on specific types of question. ## Subtracing Unlike Fractions Easy Examples Here are some easier examples to practise with. When teaching subtracting fractions with unlike denominations, it is helpful to start with examples where only one fraction needs changing. These examples only require one fraction to change in order to find a common denominator. If one fraction denominator is a multiple of the other, only one fraction needs to be changed. The first easy example is 1/2 - 1/4 . We can see that the denominator of 4 is a multiple of the denominator of 2. This means that we can simply double the values in the fraction 1/2 to get a common denominator of 4. 1/2 = 2/4 . We rewrite 1/2 - 1/4 as 2/4 - 1/4 . 2 - 1 = 1 and so, the numerator of the answer is 1. The denominator remains as 4. Here is another easy example of subtracting fractions with different denominators. We have 11/12 - 3/4 . We can see that 12 is a multiple of 4 and so, only one fraction needs changing. The 4 needs to be multiplied by 3 to make 12. We rewrite the fraction 3/4 as 9/12 . 11/12 - 9/12 = 2/12 . It is possible to simplify this answer by halving both the numerator and denominator. 2/12 simplifies to 1/6 . ## How to Subtract Mixed Numbers with Unlike Denominators To subtract mixed numbers with unlike denominators, follow these steps: 1. Write the mixed numbers as improper fractions. 2. Find the least common denominator. 3. Write the improper fractions as equivalent fractions that have the least common denominator. 4. The denominator of the answer is the same as this least common denominator. 5. Subtract the numerators to find the numerator of the answer. For example, here is 5 1/4 - 2 2/3 . The first step is to convert the mixed numbers into improper fractions. To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator. This result is the new numerator and the denominator is the same as the denominator of the mixed number. 5 × 4 = 20 and then 20 + 1 = 21. The mixed number of 5 1/4 can be rewritten as an improper fraction as 21/4 . 2 × 3 = 6 and then 6 + 2 = 8. The mixed number of 2 2/3 can be rewritten as an improper fraction as 8/3 . Now that the mixed numbers have been written as improper fractions, the next step in the subtraction is to find the lowest common denominator. The first number in both the 4 and 3 times table is 12. The least common denominator is 12. We multiply the denominator and numerator of the first fraction by 3 and the second fraction by 4. 21/4 is rewritten as 63/12 . 8/3 is rewritten as 32/12 . Now that the mixed numbers have been converted into improper fractions and the improper fractions now have common denominators, we can finally perform the subtraction. The denominator remains the same and we subtract the numerators. 63/12 - 32/12 = 31/12 . The final step is to write the improper fraction as a mixed number if necessary. 31 ÷ 12 = 2 remainder 7. We write the whole number down and the remainder as the numerator of the fraction. The improper fraction of 31/12 is written as the mixed number of 2 7/12 .	{"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": 0.977627694606781, "perplexity": 883.5216562474812}, "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/1664030334515.14/warc/CC-MAIN-20220925070216-20220925100216-00027.warc.gz"}
http://mathhelpforum.com/advanced-algebra/7064-elementary-functions-solvable-group.html	# Math Help - Elementary Functions as a Solvable Group 1. ## Elementary Functions as a Solvable Group I am not so happy with the defintion for a elementary function viewed as a finite number of combinations of addition, subtraction, multiplication, division and roots. It sounds familar to the meaning of a "solutions by radicals" of a polynomial. Though it does has an alternate defintion, it is defined as a solvable group. Is there a way to define these functions also as some solvable group? The only thing I can think of is that the number of solutions to a polynomial is finite. Where the number of elementary functions is not. Thus, the compositions series does not exist. But if you can find one will it not make the proves based on showing there is no such elementary function simpler? This is my 31th Post!!! 2. Originally Posted by ThePerfectHacker I am not so happy with the defintion for a elementary function viewed as a finite number of combinations of addition, subtraction, multiplication, division and roots. It sounds familar to the meaning of a "solutions by radicals" of a polynomial. Though it does has an alternate defintion, it is defined as a solvable group. Is there a way to define these functions also as some solvable group? The only thing I can think of is that the number of solutions to a polynomial is finite. Where the number of elementary functions is not. Thus, the compositions series does not exist. But if you can find one will it not make the proves based on showing there is no such elementary function simpler? This is my 31th Post!!! Just to get this straight, are you suggesting that we no longer consider the sine function elementary? -Dan 3. Originally Posted by topsquark Just to get this straight, are you suggesting that we no longer consider the sine function elementary? -Dan No, it is. To be elementary we define a function as a finite sum difference multiplication division and composition of: polynomials,trigonometric,inverse trigonometric, exponential, logarithmic. 4. I guess I was right there is such an area in mathematics. Look Here.	{"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": 0.8750512599945068, "perplexity": 269.83995285564635}, "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-2015-27/segments/1435375096944.75/warc/CC-MAIN-20150627031816-00079-ip-10-179-60-89.ec2.internal.warc.gz"}
http://cmup.fc.up.pt/main/content/quadric-bundles-and-their-moduli-spaces	Quadric bundles and their moduli spaces Title Quadric bundles and their moduli spaces Publication Type Preprint Year of Preprint 2016 Authors Oliveira A. Abstract We consider quadric bundles on a compact Riemann surface X. These generalise orthogonal bundles and arise naturally in the study of the moduli space of representations of \pi_1(X) in Sp(2n,R). We prove some basic results on the moduli spaces of quadric bundles over X of arbitrary rank and survey deeper results about these moduli spaces, for rank 2. [2016-10] Geometry	{"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": 0.995516300201416, "perplexity": 746.637097826414}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818686983.8/warc/CC-MAIN-20170920085844-20170920105844-00450.warc.gz"}
http://iloctech.com/nabil-elouahabi-mupsg/6922e5-perceptron-convergence-theorem-ques10	m[��]�sv��,�L�Ӥ!s�'�F�{�>��֨��1�>�� 0N1Š�� 1415–1442, (1990). Algorithms: Discrete and Continuous Perceptron Networks, Perceptron Convergence theorem, Limitations of the Perceptron Model, Applications. 0000009440 00000 n , zp ... Q NA RMA recurrent perceptron, convergence towards a point in the FPI sense does not depend on the number of external input signals (i.e. Fig. Convergence Theorem: if the training data is linearly separable, the algorithm is guaranteed to converge to a solution. /10 be such that-1 "/, Then Perceptron makes at most 243658795:3; 3 mistakes on this example sequence. << /BBox [ 0 0 612 792 ] /Filter /FlateDecode /FormType 1 /Matrix [ 1 0 0 1 0 0 ] /Resources << /Font << /F34 311 0 R /F35 283 0 R >> /ProcSet [ /PDF /Text ] >> /Subtype /Form /Type /XObject /Length 866 >> 0000063075 00000 n Unit- IV: Multilayer Feed forward Neural Networks Credit Assignment Problem, Generalized Delta Rule, Derivation of Backpropagation (BP) Training, Summary of Backpropagation Algorithm, Kolmogorov Theorem, Learning Difficulties and … Assume D is linearly separable, and let be w be a separator with \margin 1". Logical functions are a great starting point since they will bring us to a natural development of the theory behind the perceptron and, as a consequence, neural networks. 0000004302 00000 n When the set of training patterns is linearly non-separable, then for any set of weights, W. there will exist some training example. << /Ascent 668 /CapHeight 668 /CharSet (/A/L/M/P/one/quoteright/seven) /Descent -193 /Flags 4 /FontBBox [ -169 -270 1010 924 ] /FontFile 286 0 R /FontName /TVDNNQ+NimbusRomNo9L-ReguItal /ItalicAngle -15 /StemV 78 /Type /FontDescriptor /XHeight 441 >> %PDF-1.4 << /Annots [ 289 0 R 290 0 R 291 0 R 292 0 R 293 0 R 294 0 R 295 0 R 296 0 R 297 0 R 298 0 R 299 0 R 300 0 R 301 0 R 302 0 R 303 0 R 304 0 R ] /Contents [ 287 0 R 307 0 R 288 0 R ] /MediaBox [ 0 0 612 792 ] /Parent 257 0 R /Resources << /ExtGState 306 0 R /Font 305 0 R /ProcSet [ /PDF /Text /ImageB /ImageC /ImageI ] /XObject << /Xi0 282 0 R >> >> /Type /Page >> Symposium on the Mathematical Theory of Automata, 12, 615–622. Perceptron Convergence Due to Rosenblatt (1958). The corresponding test must be introduced in the above pseudocode to make it stop and to transform it into a fully-fledged algorithm. Proof. 0000062734 00000 n I will not develop such proof, because involves some advance mathematics beyond what I want to touch in an introductory text. 0000038487 00000 n << /BaseFont /TVDNNQ+NimbusRomNo9L-ReguItal /Encoding 312 0 R /FirstChar 39 /FontDescriptor 285 0 R /LastChar 80 /Subtype /Type1 /Type /Font /Widths 284 0 R >> 0000011051 00000 n xref NOT logical function. 0000017806 00000 n 282 0 obj Introduction: The Perceptron Haim Sompolinsky, MIT October 4, 2013 1 Perceptron Architecture The simplest type of perceptron has a single layer of weights connecting the inputs and output. 6.c Delta Learning Rule (5 marks) 00. Step size = 1 can be used. 0000037666 00000 n 6.d McCulloh Pitts neuron model (5 marks) 00. question paper mumbai university (mu) • 2.3k views. 0000008943 00000 n Pages 43–50. << /Filter /FlateDecode /Length1 1647 /Length2 2602 /Length3 0 /Length 3406 >> 0000010275 00000 n Polytechnic Institute of Brooklyn. The routine can be stopped when all vectors are classified correctly. ��7�[s�8M�p� �� ~��{�6m7 �� E�J��̸H�u��s��0�?he7��:@l:3>�Ǆ��r�y�>�¯�Â�Z�(x�< 0000009939 00000 n stream Collins, M. 2002. visualization in open space. 0000047049 00000 n 0000010605 00000 n Theorem: Suppose data are scaled so that kx ik 2 1. Let’s start with a very simple problem: Can a perceptron implement the NOT logical function? Perceptron convergence. 0000073290 00000 n 0000047745 00000 n 0000021688 00000 n Perceptron Convergence Theorem [ 41. Másképpen fogalmazva: 2.1.2 Tétel: perceptron konvergencia tétel: Legyen The perceptron convergence theorem guarantees that if the two sets P and N are linearly separable the vector w is updated only a finite number of times. If PCT holds, then: jj1 T P T t=1 v tjj˘O(1=T). And explains the convergence theorem of perceptron and its proof. endobj endobj Download our mobile app and study on-the-go. Convergence Proof for the Perceptron Algorithm Michael Collins Figure 1 shows the perceptron learning algorithm, as described in lecture. Widrow, B., Lehr, M.A., "30 years of Adaptive Neural Networks: Perceptron, Madaline, and Backpropagation," Proc. 0000010440 00000 n ADD COMMENT Continue reading. 0000018412 00000 n 0000010772 00000 n Let-. << /Metadata 276 0 R /Outlines 258 0 R /PageLabels << /Nums [ 0 << /P () >> ] >> /Pages 257 0 R /Type /Catalog >> Theory and Examples 4-2 Learning Rules 4-2 Perceptron Architecture 4-3 Single-Neuron Perceptron 4-5 Multiple-Neuron Perceptron 4-8 Perceptron Learning Rule 4-8 Test Problem 4-9 Constructing Learning Rules 4-10 Unified Learning Rule 4-12 Training Multiple-Neuron Perceptrons 4-13 Proof of Convergence 4-15 Notation 4-15 Proof 4-16 Limitations 4-18 Summary of Results 4-20 Solved … IEEE, vol 78, no 9, pp. 0000004570 00000 n 0000065821 00000 n That is, there exist a finite such that : = 0: Statistical Machine Learning (S2 2017) Deck 6: Perceptron convergence theorem • Assumptions ∗Linear separability: There exists ∗ so that : : ∗′ p-the AR part of the NARMA (p,q) process (411, nor on their values, QS long QS they are finite. You'll get subjects, question papers, their solution, syllabus - All in one app. ��z��p�B[�� M��]�-p�ϐ�Su��./ْ��-KL�b�0��\|g}�[(n��E��Z��_��X�f��,zt:�^[ 4�ۊZ�Hxh)mNI ��q"k��?�?��2��Q�D��RW�;e;}��1ʟge��BE0�� B]��lr�W��u�dAkB�oLJ��7��\��E��'�ͨ�0V��M#� �ֲ9�ߢ�Zpl,(R2�P ��˘w��endstream 0000020876 00000 n 0000010107 00000 n x�mUK��6��W�P��HJ�� Alߒh��X��n��;�P^o�0�y�y��)��_;�e@��Q��l �u"j�r�t�.�y]�DF+�4���Y6��Nx�0AIU�d�'_�m㜙�,/�:��A}�M5J�9�.(L�Y��n��v�zD�.?��.�lb�S8k��P:^C�u�xs��PZ. It's the best way to discover useful content. Sengupta, Department of Electronics and Electrical Communication Engineering, IIT Kharagpur. Frank Rosenblatt invented the perceptron algorithm in 1957 as part of an early attempt to build brain models'', artificial neural networks. 0000000015 00000 n Verified perceptron convergence theorem. ABSTRACT. Previous Chapter Next Chapter. The PCT immediately leads to the following result: Convergence Theorem. 0000010937 00000 n stream endobj . Definition of perceptron. Find answer to specific questions by searching them here. 0000008171 00000 n 279 0 obj Mumbai University > Computer Engineering > Sem 7 > Soft Computing. << /Linearized 1 /L 287407 /H [ 1812 637 ] /O 281 /E 73886 /N 8 /T 281727 >> 0000040138 00000 n Obviously, the author was looking at the materials from multiple different sources but did not generalize it very well to match his proceeding writings in the book. 0000009108 00000 n NOT(x) is a 1-variable function, that means that we will have one input at a time: N=1. The Winnow algorithm [4] has a very similar structure. On the other hand, it is possible to construct an additive algorithm that never makes more than N + 0( klog N) mistakes. 0000020703 00000 n (large margin = very The famous Perceptron Convergence Theorem [6] bounds the number of mistakes which the Perceptron algorithm can make: Theorem 1 Let be a sequence of labeled examples with! It's the best way to discover useful content. 278 0 obj The Perceptron learning algorithm has been proved for pattern sets that are known to be linearly separable. Perceptron algorithm is used for supervised learning of binary classification. PERCEPTRON CONVERGENCE THEOREM: Says that there if there is a weight vector wsuch that f(wp(q)) = t(q) for all q, then for any starting vector w, the perceptron learning rule will converge to a weight vector (not necessarily unique and not necessarily w) that gives the correct response for all training patterns, and it will do so in a finite number of steps. D lineárisan szeparálható X 0 és X 1 halmazokra, hogyha: ahol ’’ a skaláris szorzás felett. Lecture Notes: http://www.cs.cornell.edu/courses/cs4780/2018fa/lectures/lecturenote03.html input x = $( I_1, I_2, I_3) = ( 5, 3.2, 0.1 ).$, Summed input $$= \sum_i w_iI_i = 5 w_1 + 3.2 w_2 + 0.1 w_3$$. endobj 0000009274 00000 n "# % & and (') +* for all,. ��@4��* ��"��2"�JA�!��:�"��IŢ�[�)D?�CDӶZ�� Aԭ\� ��($��Hdh�"��@�Qd�P�{�v~� �K�( Gߎ&n{�UD��8?E.U8'� Xk, such that Wk misclassifies Xk. Like all structured prediction learning frameworks, the structured perceptron can be costly to train as training complexity is proportional to inference, which is frequently non-linear in example sequence length. 8t 0: If wT tv 0, then there exists a constant M>0 such that kw t w 0k�m�8,��ǚ��@�a&��4)��&&E��#�[�AY�'=��ٮ��cs�� 2 Perceptron konvergencia tétel 2.1 A tétel kimondása 2.1.1 Definíció: lineáris szeparálhatóság (5) Legyen . << /Filter /FlateDecode /S 383 /O 610 /Length 549 >> 0000073517 00000 n 6.b Binary Hopfield Network (5 marks) 00. , y(k - q + l), l,q,. [ 333 333 333 500 675 250 333 250 278 500 500 500 500 500 500 500 500 500 500 333 333 675 675 675 500 920 611 611 667 722 611 611 722 722 333 444 667 556 833 667 722 611 ] When the set of training patterns is linearly non-separable, then for any set of weights, W. there will exist some training example. 0000066348 00000 n Convergence. endobj endobj 0000041214 00000 n 0000008279 00000 n ��r�� Yֈ_\|��f��a?� S�&C+��X�l�\� ��w�LNf0_�h��8Er�A� ��s�a�q�� d2��a^��\|H� 021�X� 2�8T 3�� 0000008609 00000 n 0000009773 00000 n 285 0 obj Legyen D két diszjunkt részhalmaza X 0 és X 1 (azaz ). By formalizing and proving perceptron convergence, we demon-strate a proof-of-concept architecture, using classic programming languages techniques like proof by reﬁnement, by which further machine-learning algorithms with sufﬁciently developed metatheory can be implemented and veriﬁed. %�� 0000018127 00000 n 0000065914 00000 n It is immediate from the code that should the algorithm terminate and return a weight vector, then the weight vector must separate the points from the points. No such guarantees exist for the linearly non-separable case because in weight space, no solution cone exists. Chapters 1–10 present the authors' perceptron theory through proofs, Chapter 11 involves learning, Chapter 12 treats linear separation problems, and Chapter 13 discusses some of the authors' thoughts on simple and multilayer perceptrons and pattern recognition. 0000063410 00000 n ��D����P�Ӹ�Ï��m�B����ʖ� γ • The perceptron algorithm is trying to ﬁnd a weight vector w that points roughly in the same direction as w. Perceptron algorithm in a fresh light: the language of dependent type theory as implemented in Coq (The Coq Development Team 2016). 0000039694 00000 n the data is linearly separable), the perceptron algorithm will converge. 0000008089 00000 n Subject: Electrical Courses: Neural Network and Applications. The proof that the perceptron will find a set of weights to solve any linearly separable classification problem is known as the perceptron convergence theorem. 0000040698 00000 n x�c�gacP�d�0��dٙɨQ��aKM��I��a'��tȦ�I�?p��\��d��&jg�Yo�U٧��_X�5�k��޾��n9��]z�B^��g��\|b�ʨ��oH:9�m�\�J��_.�[u�M�ּg��_��"��F�\��\2�� Formally, the perceptron is deﬁned by y = sign(PN i=1 wixi ) or y = sign(wT x ) (1) where w is the weight vector and is the threshold. %%EOF Consequently, the Perceptron learning algorithm will continue to make weight changes indefinitely. The theorem still holds when V is a ﬁnite set in a Hilbert space. 278 64 0000009606 00000 n trailer << /Info 277 0 R /Root 279 0 R /Size 342 /Prev 281717 /ID [<58ec75fda24c432cc812dba252618c1f><1aefbf0404691781113e5401cf827802>] >> 8��:�{��5�>k 6ں��V�O��;�K��r�w�{��r K2��i��qs�a o��h�)�]@��8c֝ ��"��G"�� The Perceptron learning algorithm has been proved for pattern sets that are known to be linearly separable. We also show that the Perceptron algorithm in its basic form can make 2k( N - k + 1) + 1 mistakes, so the bound is essentially tight. 0000003936 00000 n I found the authors made some errors in the mathematical derivation by introducing some unstated assumptions. 6.a Explain perceptron convergence theorem (5 marks) 00. 0000011087 00000 n ��\J[�bI�#��O,$o_��E�0D�@?.%;"N ��w+�}"� �-�-��o��ѿ. 0000002449 00000 n I was reading the perceptron convergence theorem, which is a proof for the convergence of perceptron learning algorithm, in the book “Machine Learning - An Algorithmic Perspective” 2nd Ed. Rosenblatt’s Perceptron Convergence Theorem γ−2 γ > 0 x ∈ D The idea of the proof: • If the data is linearly separable with margin , then there exists some weight vector w that achieves this margin. By formalizing and proving perceptron convergence, we demon-strate a proof-of-concept architecture, using classic programming languages techniques like proof by reﬁnement, by which further machine-learning algorithms with sufﬁciently developed metatheory can be implemented and veriﬁed. 0000047161 00000 n Lecture Series on Neural Networks and Applications by Prof.S. ۘ��Ħ��ɜ��ԫU��d��T2��-�~a��h��l�uq��r��=��)�� 0000002830 00000 n This post is the summary of “Mathematical principles in Machine Learning” The perceptron convergence theorem was proved for single-layer neural nets. . endstream Perceptron convergence theorem COMP 652 - Lecture 12 9 / 37 The perceptron convergence theorem states that if the perceptron learning rule is applied to a linearly separable data set, a solution will be found after some finite number of updates. 0000039169 00000 n 0000001812 00000 n 0000021215 00000 n endobj 0000073192 00000 n startxref Perceptron Cycling Theorem (PCT). . Find more. 0 Explain the perceptron learning with example. 0000056654 00000 n xڭTgXTY�DAT��Cɱ�Cjr�i�/��N_�%�� J�"%6(iz�I�QA��^pg��~꭪��)�_��0D_I$PT�u ;�K�8�vD��#�O��p �ipIK��A"LQTPp1�)�TU�% �It2䏥�.�nr��~X�\ _��I�� # �Ix�@�)��@'�X��p b��aigȚ۹ �$�M8�\|q�� ~D2��~ �D�j��sQ @!�h�� i:�@2�P�o � �d� 0000008444 00000 n 3�#0��o�9L�5��whƢ��a�F=n�� According to the perceptron convergence theorem, the perceptron learning rule guarantees to find a solution within a finite number of steps if the provided data set is linearly separable. I then tried to look up the right derivation on the i… 283 0 obj 0000008776 00000 n 0000040791 00000 n The Perceptron Convergence Theorem is, from what I understand, a lot of math that proves that a perceptron, given enough time, will always be able to find a … Winnow maintains … Then the perceptron algorithm will converge in at most kw k2 epochs. 0000001681 00000 n You must be logged in to read the answer. . �C�� lJ� 3 Theorem 3 (Perceptron convergence). No such guarantees exist for the linearly non-separable case because in weight space, no solution cone exists. endobj 284 0 obj [We’re not going to prove this, because perceptrons are obsolete.] 0000056131 00000 n 281 0 obj The Perceptron Learning Algorithm makes at most R2 2 updates (after which it returns a separating hyperplane). 0000040630 00000 n In this post, it will cover the basic concept of hyperplane and the principle of perceptron based on the hyperplane. 0000063827 00000 n 286 0 obj Convergence Convergence theorem –If there exist a set of weights that are consistent with the data (i.e. Perceptron Convergence Theorem: If data is linearly separable, perceptron algorithm will ﬁnd a linear classiﬁer that classiﬁes all data correctly in at most O(R2/2) iterations, where R = max\|X i\| is “radius of data” and is the “maximum margin.” [I’ll deﬁne “maximum margin” shortly.] 0000056022 00000 n stream 0000004113 00000 n For the Perceptron learning algorithm, as described in lecture it stop and to transform it into a algorithm! When all vectors are classified correctly //www.cs.cornell.edu/courses/cs4780/2018fa/lectures/lecturenote03.html Perceptron algorithm is trying to ﬁnd a weight vector w that roughly... Function, that means that we will have one input at a time: N=1 correct categories a! On Neural Networks of an early attempt to build brain models '' artificial. Tron given by ( 9 ) where XE = [ y ( k ), of Binary.... Their solution, syllabus - all in one app a convergence proof for the (! The data is linearly separable ), l, q, beyond what i to!: can a Perceptron implement the not logical function result: convergence theorem was proved for pattern that. Mathematics beyond what i want to touch in an introductory text there exists a M! Department of Electronics and Electrical Communication Engineering, IIT Kharagpur Collins Figure shows! 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D két diszjunkt részhalmaza X 0 és X 1 halmazokra, hogyha perceptron convergence theorem ques10 ’., l, q, be w be a separator with \margin 1 '' because...	{"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": 0.850111722946167, "perplexity": 4103.766674336016}, "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-2021-17/segments/1618038061820.19/warc/CC-MAIN-20210411085610-20210411115610-00040.warc.gz"}
https://pressbooks.online.ucf.edu/algphysics/chapter/applications-of-electrostatics/	Chapter 18 Electric Charge and Electric Field # 18.8 Applications of Electrostatics ### Summary • Name several real-world applications of the study of electrostatics. The study of electrostatics has proven useful in many areas. This module covers just a few of the many applications of electrostatics. # The Van de Graaff Generator Van de Graaff generators (or Van de Graaffs) are not only spectacular devices used to demonstrate high voltage due to static electricity—they are also used for serious research. The first was built by Robert Van de Graaff in 1931 (based on original suggestions by Lord Kelvin) for use in nuclear physics research. Figure 1 shows a schematic of a large research version. Van de Graaffs utilize both smooth and pointed surfaces, and conductors and insulators to generate large static charges and, hence, large voltages. A very large excess charge can be deposited on the sphere, because it moves quickly to the outer surface. Practical limits arise because the large electric fields polarize and eventually ionize surrounding materials, creating free charges that neutralize excess charge or allow it to escape. Nevertheless, voltages of 15 million volts are well within practical limits. ### Take-Home Experiment: Electrostatics and Humidity Rub a comb through your hair and use it to lift pieces of paper. It may help to tear the pieces of paper rather than cut them neatly. Repeat the exercise in your bathroom after you have had a long shower and the air in the bathroom is moist. Is it easier to get electrostatic effects in dry or moist air? Why would torn paper be more attractive to the comb than cut paper? Explain your observations. # Xerography Most copy machines use an electrostatic process called xerography—a word coined from the Greek words xeros for dry and graphos for writing. The heart of the process is shown in simplified form in Figure 2. A selenium-coated aluminum drum is sprayed with positive charge from points on a device called a corotron. Selenium is a substance with an interesting property—it is a photoconductor. That is, selenium is an insulator when in the dark and a conductor when exposed to light. In the first stage of the xerography process, the conducting aluminum drum is grounded so that a negative charge is induced under the thin layer of uniformly positively charged selenium. In the second stage, the surface of the drum is exposed to the image of whatever is to be copied. Where the image is light, the selenium becomes conducting, and the positive charge is neutralized. In dark areas, the positive charge remains, and so the image has been transferred to the drum. The third stage takes a dry black powder, called toner, and sprays it with a negative charge so that it will be attracted to the positive regions of the drum. Next, a blank piece of paper is given a greater positive charge than on the drum so that it will pull the toner from the drum. Finally, the paper and electrostatically held toner are passed through heated pressure rollers, which melt and permanently adhere the toner within the fibers of the paper. # Laser Printers Laser printers use the xerographic process to make high-quality images on paper, employing a laser to produce an image on the photoconducting drum as shown in Figure 3. In its most common application, the laser printer receives output from a computer, and it can achieve high-quality output because of the precision with which laser light can be controlled. Many laser printers do significant information processing, such as making sophisticated letters or fonts, and may contain a computer more powerful than the one giving them the raw data to be printed. # Ink Jet Printers and Electrostatic Painting The ink jet printer, commonly used to print computer-generated text and graphics, also employs electrostatics. A nozzle makes a fine spray of tiny ink droplets, which are then given an electrostatic charge. (See Figure 4.) Once charged, the droplets can be directed, using pairs of charged plates, with great precision to form letters and images on paper. Ink jet printers can produce color images by using a black jet and three other jets with primary colors, usually cyan, magenta, and yellow, much as a color television produces color. (This is more difficult with xerography, requiring multiple drums and toners.) Electrostatic painting employs electrostatic charge to spray paint onto odd-shaped surfaces. Mutual repulsion of like charges causes the paint to fly away from its source. Surface tension forms drops, which are then attracted by unlike charges to the surface to be painted. Electrostatic painting can reach those hard-to-get at places, applying an even coat in a controlled manner. If the object is a conductor, the electric field is perpendicular to the surface, tending to bring the drops in perpendicularly. Corners and points on conductors will receive extra paint. Felt can similarly be applied. # Smoke Precipitators and Electrostatic Air Cleaning Another important application of electrostatics is found in air cleaners, both large and small. The electrostatic part of the process places excess (usually positive) charge on smoke, dust, pollen, and other particles in the air and then passes the air through an oppositely charged grid that attracts and retains the charged particles. (See Figure 5.) Large electrostatic precipitators are used industrially to remove over 99% of the particles from stack gas emissions associated with the burning of coal and oil. Home precipitators, often in conjunction with the home heating and air conditioning system, are very effective in removing polluting particles, irritants, and allergens. ### Problem-Solving Strategies for Electrostatics 1. Examine the situation to determine if static electricity is involved. This may concern separated stationary charges, the forces among them, and the electric fields they create. 2. Identify the system of interest. This includes noting the number, locations, and types of charges involved. 3. Identify exactly what needs to be determined in the problem (identify the unknowns). A written list is useful. Determine whether the Coulomb force is to be considered directly—if so, it may be useful to draw a free-body diagram, using electric field lines. 4. Make a list of what is given or can be inferred from the problem as stated (identify the knowns). It is important to distinguish the Coulomb force ${F}$ from the electric field ${E}$, for example. 5. Solve the appropriate equation for the quantity to be determined (the unknown) or draw the field lines as requested. 6. Examine the answer to see if it is reasonable: Does it make sense? Are units correct and the numbers involved reasonable? # Integrated Concepts The Integrated Concepts exercises for this module involve concepts such as electric charges, electric fields, and several other topics. Physics is most interesting when applied to general situations involving more than a narrow set of physical principles. The electric field exerts force on charges, for example, and hence the relevance of Chapter 4 Dynamics: Force and Newton’s Laws of Motion. The following topics are involved in some or all of the problems labeled “Integrated Concepts”: The following worked example illustrates how this strategy is applied to an Integrated Concept problem: ### Example 1: Acceleration of a Charged Drop of Gasoline If steps are not taken to ground a gasoline pump, static electricity can be placed on gasoline when filling your car’s tank. Suppose a tiny drop of gasoline has a mass of ${4.00 \times 10^{-15} \;\text{kg}}$ and is given a positive charge of ${3.20 \times 10^{-19} \;\text{C}}$. (a) Find the weight of the drop. (b) Calculate the electric force on the drop if there is an upward electric field of strength ${3.00 \times 10^5 \;\text{N} / \text{C}}$ due to other static electricity in the vicinity. (c) Calculate the drop’s acceleration. Strategy To solve an integrated concept problem, we must first identify the physical principles involved and identify the chapters in which they are found. Part (a) of this example asks for weight. This is a topic of dynamics and is defined in Chapter 4 Dynamics: Force and Newton’s Laws of Motion. Part (b) deals with electric force on a charge, a topic of Chapter 18 Electric Charge and Electric Field. Part (c) asks for acceleration, knowing forces and mass. These are part of Newton’s laws, also found in Chapter 4 Dynamics: Force and Newton’s Laws of Motion. The following solutions to each part of the example illustrate how the specific problem-solving strategies are applied. These involve identifying knowns and unknowns, checking to see if the answer is reasonable, and so on. Solution for (a) Weight is mass times the acceleration due to gravity, as first expressed in ${w = mg.}$ Entering the given mass and the average acceleration due to gravity yields ${w = (4.00 \times 10^{-15} \;\text{kg})(9.80 \;\text{m} / \text{s}^2) = 3.92 \times 10^{-14} \;\text{N}. }$ Discussion for (a) This is a small weight, consistent with the small mass of the drop. Solution for (b) The force an electric field exerts on a charge is given by rearranging the following equation: ${F = qE.}$ Here we are given the charge (${3.20 \times 10^{-19} \;\text{C}}$ is twice the fundamental unit of charge) and the electric field strength, and so the electric force is found to be ${F=(3.20 \times 10^{-19} \;\text{C})(3.00 \times 10^5 \;\text{N} / \text{C}) = 9.60 \times 10^{-14} \;\text{N}.}$ Discussion for (b) While this is a small force, it is greater than the weight of the drop. Solution for (c) The acceleration can be found using Newton’s second law, provided we can identify all of the external forces acting on the drop. We assume only the drop’s weight and the electric force are significant. Since the drop has a positive charge and the electric field is given to be upward, the electric force is upward. We thus have a one-dimensional (vertical direction) problem, and we can state Newton’s second law as ${a =}$ ${\frac{F_{net}}{m}}$. where ${F_{net} = F - w}$. Entering this and the known values into the expression for Newton’s second law yields $\begin{array}{r @{{}={}}l} {a}= & {\frac{F - w}{m}} \\[1em]= & {\frac{9.60 \times 10^{-14} \;\text{N} - 3.92 \times 10^{-14} \;\text{N}}{4.00 \times 10^{-15} \;\text{kg}}} \\[1em]= & {14.2 \;\text{m} / \text{s}^2}. \end{array}$ Discussion for (c) This is an upward acceleration great enough to carry the drop to places where you might not wish to have gasoline. This worked example illustrates how to apply problem-solving strategies to situations that include topics in different chapters. The first step is to identify the physical principles involved in the problem. The second step is to solve for the unknown using familiar problem-solving strategies. These are found throughout the text, and many worked examples show how to use them for single topics. In this integrated concepts example, you can see how to apply them across several topics. You will find these techniques useful in applications of physics outside a physics course, such as in your profession, in other science disciplines, and in everyday life. The following problems will build your skills in the broad application of physical principles. ### Unreasonable Results The Unreasonable Results exercises for this module have results that are unreasonable because some premise is unreasonable or because certain of the premises are inconsistent with one another. Physical principles applied correctly then produce unreasonable results. The purpose of these problems is to give practice in assessing whether nature is being accurately described, and if it is not to trace the source of difficulty. Problem-Solving Strategy To determine if an answer is reasonable, and to determine the cause if it is not, do the following. 1. Solve the problem using strategies as outlined above. Use the format followed in the worked examples in the text to solve the problem as usual. 2. Check to see if the answer is reasonable. Is it too large or too small, or does it have the wrong sign, improper units, and so on? 3. If the answer is unreasonable, look for what specifically could cause the identified difficulty. Usually, the manner in which the answer is unreasonable is an indication of the difficulty. For example, an extremely large Coulomb force could be due to the assumption of an excessively large separated charge. # Section Summary • Electrostatics is the study of electric fields in static equilibrium. • In addition to research using equipment such as a Van de Graaff generator, many practical applications of electrostatics exist, including photocopiers, laser printers, ink-jet printers and electrostatic air filters. ### Problems & Exercises 1: (a) What is the electric field 5.00 m from the center of the terminal of a Van de Graaff with a 3.00 mC charge, noting that the field is equivalent to that of a point charge at the center of the terminal? (b) At this distance, what force does the field exert on a ${2.00 \;\mu \text{C}}$ charge on the Van de Graaff’s belt? 2: (a) What is the direction and magnitude of an electric field that supports the weight of a free electron near the surface of Earth? (b) Discuss what the small value for this field implies regarding the relative strength of the gravitational and electrostatic forces. 3: A simple and common technique for accelerating electrons is shown in Figure 6, where there is a uniform electric field between two plates. Electrons are released, usually from a hot filament, near the negative plate, and there is a small hole in the positive plate that allows the electrons to continue moving. (a) Calculate the acceleration of the electron if the field strength is ${2.50 \times 10^4 \;\text{N} / \text{C}}$. (b) Explain why the electron will not be pulled back to the positive plate once it moves through the hole. 4: Earth has a net charge that produces an electric field of approximately 150 N/C downward at its surface. (a) What is the magnitude and sign of the excess charge, noting the electric field of a conducting sphere is equivalent to a point charge at its center? (b) What acceleration will the field produce on a free electron near Earth’s surface? (c) What mass object with a single extra electron will have its weight supported by this field? 5: Point charges of ${25.0 \;\mu \text{C}}$ and ${45.0 \;\mu \text{C}}$ are placed 0.500 m apart. (a) At what point along the line between them is the electric field zero? (b) What is the electric field halfway between them? 6: What can you say about two charges q1 and q2, if the electric field one-fourth of the way from q1 to q2 is zero? ### Problems & Exercises Integrated Concepts 1: Calculate the angular velocity $\omega$ of an electron orbiting a proton in the hydrogen atom, given the radius of the orbit is ${0.530 \times 10^{-10} \;\text{m}}$. You may assume that the proton is stationary and the centripetal force is supplied by Coulomb attraction. 2: An electron has an initial velocity of ${5.00 \times 10^{6} \;\text{m} / \text{s}}$ in a uniform ${2.00 \times 10^5 \;\text{N} / \text{C}}$ strength electric field. The field accelerates the electron in the direction opposite to its initial velocity. (a) What is the direction of the electric field? (b) How far does the electron travel before coming to rest? (c) How long does it take the electron to come to rest? (d) What is the electron’s velocity when it returns to its starting point? 3: The practical limit to an electric field in air is about ${3.00 \times 10^6 \;\text{N} / \text{C}}$. Above this strength, sparking takes place because air begins to ionize and charges flow, reducing the field. (a) Calculate the distance a free proton must travel in this field to reach 3.00% of the speed of light, starting from rest. (b) Is this practical in air, or must it occur in a vacuum? 4: A 5.00 g charged insulating ball hangs on a 30.0 cm long string in a uniform horizontal electric field as shown in Figure 7. Given the charge on the ball is ${1.00 \;\mu \text{C}}$, find the strength of the field. 5: Figure 8 shows an electron passing between two charged metal plates that create an 100 N/C vertical electric field perpendicular to the electron’s original horizontal velocity. (These can be used to change the electron’s direction, such as in an oscilloscope.) The initial speed of the electron is ${3.00 \times 10^6 \;\text{m} / \text{s}}$, and the horizontal distance it travels in the uniform field is 4.00 cm. (a) What is its vertical deflection? (b) What is the vertical component of its final velocity? (c) At what angle does it exit? Neglect any edge effects. 6: The classic Millikan oil drop experiment was the first to obtain an accurate measurement of the charge on an electron. In it, oil drops were suspended against the gravitational force by a vertical electric field. (See Figure 9.) Given the oil drop to be ${1.00 \;\mu \text{m}}$ in radius and have a density of 920 kg/m3: (a) Find the weight of the drop. (b) If the drop has a single excess electron, find the electric field strength needed to balance its weight. 7: (a) In Figure 10, four equal charges q lie on the corners of a square. A fifth charge Q is on a mass mm directly above the center of the square, at a height equal to the length d of one side of the square. Determine the magnitude of q in terms of Q, m, and d, if the Coulomb force is to equal the weight of m. (b) Is this equilibrium stable or unstable? Discuss. ### Unreasonable Results 1: (a) Calculate the electric field strength near a 10.0 cm diameter conducting sphere that has 1.00 C of excess charge on it. (b) What is unreasonable about this result? (c) Which assumptions are responsible? 2: (a) Two 0.500 g raindrops in a thunderhead are 1.00 cm apart when they each acquire 1.00 mC charges. Find their acceleration. (b) What is unreasonable about this result? (c) Which premise or assumption is responsible? 3: A wrecking yard inventor wants to pick up cars by charging a 0.400 m diameter ball and inducing an equal and opposite charge on the car. If a car has a 1000 kg mass and the ball is to be able to lift it from a distance of 1.00 m: (a) What minimum charge must be used? (b) What is the electric field near the surface of the ball? (c) Why are these results unreasonable? (d) Which premise or assumption is responsible? 1: Consider two insulating balls with evenly distributed equal and opposite charges on their surfaces, held with a certain distance between the centers of the balls. Construct a problem in which you calculate the electric field (magnitude and direction) due to the balls at various points along a line running through the centers of the balls and extending to infinity on either side. Choose interesting points and comment on the meaning of the field at those points. For example, at what points might the field be just that due to one ball and where does the field become negligibly small? Among the things to be considered are the magnitudes of the charges and the distance between the centers of the balls. Your instructor may wish for you to consider the electric field off axis or for a more complex array of charges, such as those in a water molecule. 2: Consider identical spherical conducting space ships in deep space where gravitational fields from other bodies are negligible compared to the gravitational attraction between the ships. Construct a problem in which you place identical excess charges on the space ships to exactly counter their gravitational attraction. Calculate the amount of excess charge needed. Examine whether that charge depends on the distance between the centers of the ships, the masses of the ships, or any other factors. Discuss whether this would be an easy, difficult, or even impossible thing to do in practice. ## Glossary Van de Graaff generator a machine that produces a large amount of excess charge, used for experiments with high voltage electrostatics the study of electric forces that are static or slow-moving photoconductor a substance that is an insulator until it is exposed to light, when it becomes a conductor xerography a dry copying process based on electrostatics grounded connected to the ground with a conductor, so that charge flows freely to and from the Earth to the grounded object laser printer uses a laser to create a photoconductive image on a drum, which attracts dry ink particles that are then rolled onto a sheet of paper to print a high-quality copy of the image ink-jet printer small ink droplets sprayed with an electric charge are controlled by electrostatic plates to create images on paper electrostatic precipitators filters that apply charges to particles in the air, then attract those charges to a filter, removing them from the airstream ### Solutions Problems & Exercises 2: (a) ${5.58 \times 10^{-11} \;\text{N} / \text{C}}$ (b)the coulomb force is extraordinarily stronger than gravity 4: (a) ${-6.76 \times 10^5 \;\text{C}}$ (b) ${2.63 \times 10^{13} \;\text{m} / \text{s}^2 \;\text{(upward)}}$ (c) ${2.45 \times 10^{-18} \;\text{kg}}$ 6: The charge q2 is 9 times greater than q1.	{"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": 0.8304648399353027, "perplexity": 593.7712591172957}, "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/1620243991537.32/warc/CC-MAIN-20210513045934-20210513075934-00599.warc.gz"}
https://en.wikipedia.org/wiki/Secant_method	# Secant method The first two iterations of the secant method. The red curve shows the function f and the blue lines are the secants. For this particular case, the secant method will not converge. In numerical analysis, the secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function f. The secant method can be thought of as a finite difference approximation of Newton's method. However, the method was developed independently of Newton's method, and predated the latter by over 3,000 years.[1] ## The method The secant method is defined by the recurrence relation $x_n =x_{n-1}-f(x_{n-1})\frac{x_{n-1}-x_{n-2}}{f(x_{n-1})-f(x_{n-2})} =\frac{x_{n-2}f(x_{n-1})-x_{n-1}f(x_{n-2})}{f(x_{n-1})-f(x_{n-2})}$ As can be seen from the recurrence relation, the secant method requires two initial values, x0 and x1, which should ideally be chosen to lie close to the root. ## Derivation of the method Starting with initial values x0 and x1, we construct a line through the points (x0, f(x0)) and (x1, f(x1)), as demonstrated in the picture on the right. In point-slope form, this line has the equation $y = \frac{f(x_1)-f(x_0)}{x_1-x_0}(x-x_1) + f(x_1)$ We find the root of this line – the value of x such that y = 0 – by solving the following equation for x: $0 = \frac{f(x_1)-f(x_0)}{x_1-x_0}(x-x_1) + f(x_1)$ The solution is $x = x_1 - f(x_1)\frac{x_1-x_0}{f(x_1)-f(x_0)}$ We then use this new value of x as x2 and repeat the process using x1 and x2 instead of x0 and x1. We continue this process, solving for x3, x4, etc., until we reach a sufficiently high level of precision (a sufficiently small difference between xn and xn - 1). $x_2 = x_1 - f(x_1)\frac{x_1-x_0}{f(x_1)-f(x_0)}$ $x_3 = x_2 - f(x_2)\frac{x_2-x_1}{f(x_2)-f(x_1)}$ $x_n = x_{n-1} - f(x_{n-1})\frac{x_{n-1}-x_{n-2}}{f(x_{n-1})-f(x_{n-2})}$ ## Convergence The iterates $x_n$ of the secant method converge to a root of $f$, if the initial values $x_0$ and $x_1$ are sufficiently close to the root. The order of convergence is α, where $\alpha = \frac{1+\sqrt{5}}{2} \approx 1.618$ is the golden ratio. In particular, the convergence is superlinear, but not quite quadratic. This result only holds under some technical conditions, namely that $f$ be twice continuously differentiable and the root in question be simple (i.e., with multiplicity 1). If the initial values are not close enough to the root, then there is no guarantee that the secant method converges. There is no general definition of "close enough", but the criterion has to do with how "wiggly" the function is on the interval $[~x_0,~x_1~]$. For example, if $f$ is differentiable on that interval and there is a point where $f^\prime = 0$ on the interval, then the algorithm may not converge. ## Comparison with other root-finding methods The secant method does not require that the root remain bracketed like the bisection method does, and hence it does not always converge. The false position method (or regula falsi) uses the same formula as the secant method. However, it does not apply the formula on $x_{n-1}$ and $x_{n-2}$, like the secant method, but on $x_{n-1}$ and on the last iterate $x_k$ such that $f(x_k)$ and $f(x_{n-1})$ have a different sign. This means that the false position method always converges. The recurrence formula of the secant method can be derived from the formula for Newton's method $x_{n} = x_{n-1} - \frac{f(x_{n-1})}{f^\prime(x_{n-1})}$ by using the finite difference approximation $f^\prime(x_{n-1}) \approx \frac{f(x_{n-1}) - f(x_{n-2})}{x_{n-1} - x_{n-2}}$. The secant method can be interpreted as a method in which the derivative is replaced by an approximation and is thus a Quasi-Newton method. If we compare Newton's method with the secant method, we see that Newton's method converges faster (order 2 against α ≈ 1.6). However, Newton's method requires the evaluation of both $f$ and its derivative $f^\prime$ at every step, while the secant method only requires the evaluation of $f$. Therefore, the secant method may occasionally be faster in practice. For instance, if we assume that evaluating $f$ takes as much time as evaluating its derivative and we neglect all other costs, we can do two steps of the secant method (decreasing the logarithm of the error by a factor α² ≈ 2.6) for the same cost as one step of Newton's method (decreasing the logarithm of the error by a factor 2), so the secant method is faster. If however we consider parallel processing for the evaluation of the derivative, Newton's method proves its worth, being faster in time, though still spending more steps. ## Generalizations Broyden's method is a generalization of the secant method to more than one dimension. The following graph shows the function f in red and the last secant line in bold blue. In the graph, the x-intercept of the secant line seems to be a good approximation of the root of f. ## A computational example The Secant method is applied to find a root of the function f(x) = x2 − 612. Here is an implementation in the Matlab language. (From calculation, we expect that the iteration converges at x = 24.7386) f=@(x) x^2 - 612; x(1)=10; x(2)=30; for i=3:7 x(i) = x(i-1) - (f(x(i-1)))*((x(i-1) - x(i-2))/(f(x(i-1)) - f(x(i-2)))); end root=x(7) ## Notes 1. ^ Papakonstantinou, J., The Historical Development of the Secant Method in 1-D, retrieved 2011-06-29	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 28, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.899657666683197, "perplexity": 231.37258156952706}, "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/1448398459214.39/warc/CC-MAIN-20151124205419-00206-ip-10-71-132-137.ec2.internal.warc.gz"}
https://www.gradesaver.com/textbooks/math/algebra/algebra-1/chapter-2-solving-equations-2-8-proportions-and-similar-figures-standardized-test-prep-page-136/33	## Algebra 1 The associative property of addition says that any pair of addends can be summed first without changing the total. This is shown below: $(a+b)+c=(b+c)+a$	{"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": 0.8608261942863464, "perplexity": 558.6791350667661}, "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-2019-35/segments/1566027313747.38/warc/CC-MAIN-20190818083417-20190818105417-00392.warc.gz"}
https://www.physicsforums.com/threads/calculate-the-change-in-the-boxs-kinetic-energy.152380/	# Calculate the change in the box's kinetic energy 1. Jan 21, 2007 ### lzh 1. The problem statement, all variables and given/known data An 72.4 N box of clothes is pulled 28.8 m up a 21.8degrees ramp by a force of 117 N that points along the ramp. The acceleration of gravity is 9.81 m/s2 : If the coefficient of kinetic friction between the box and ramp is 0.27, calculate the change in the box's kinetic energy. Answer in units of J. 3. The attempt at a solution I have already figured out everything, but i'd like to know if the change here is the sum of both the diss. energy from friction and gravitational potential. Since, all the kinetic gets converted to grav. potential and diss in the end. 2. Jan 21, 2007 ### radou The work of all forces equals the change of kinetic energy. That's all you need to know. 3. Jan 21, 2007 ### lzh so what i said was right? 4. Jan 21, 2007 ### radou Forget about conservative forces, potential changes, etc. The work of all forces equals the change in kinetic energy, as stated, independent of the nature of these forces. 5. Jan 21, 2007 ### lzh oh, i see. So i would just use W=Fdisplacemnt= 3369.6J? 6. Jan 21, 2007 ### lzh so essentially, the work of all forces is the sum of the work of mgh and F(diss)displacement? 7. Jan 21, 2007 ### radou There are three forces. You named two of them, and the third one is the force which pulls the crate up. 8. Jan 21, 2007 ### lzh oh i see! ty 9. Jan 21, 2007 ### lzh i tried adding it all up, but my hw service keeps saying that i'm wrong. heres what i found: mgh+Fdeltax+117: ->72.4(10.69539)=774.3465J 10.695(height) was founded with: 28.8sin21.8=10.6954 Fdeltax-energy of friction: first i founded the normal force: 72.4cos21.8=67.22 so force of friction is: 67.22.27=18.15N ->18.15*(28.8)=522.72J 774.3465J+522.72J+117=1414J but this isn't correct! I tried this same step on a friend's version(same quesition w/ different numbers), and it ended up being right. what did i do wrong? 10. Jan 21, 2007 ### lzh ok i figured it out Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook Have something to add? Similar Discussions: Calculate the change in the box's kinetic energy	{"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": 0.8266851305961609, "perplexity": 2682.6369514875146}, "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/1487501171166.18/warc/CC-MAIN-20170219104611-00543-ip-10-171-10-108.ec2.internal.warc.gz"}
http://quant.stackexchange.com/questions/9652/examples-of-non-increasing-variance-of-a-time-homogeneous-markovian-process	# Examples of non-increasing variance of a time homogeneous Markovian process This is an edit to the previous question, on stationary process, which was answered by Richard below. Let $x_t$ be a zero mean, time homogeneous Markovian process over time $t$ starting from $x_0=0$. What are the examples of $x_t$ where the variance at $t$ does not increase over $t$? 1) In discrete time and discrete state, the followig is a very simple example where the variance periodically oscillates over time. $$x_{t+1} = \eta(1-\|x_t\|),\, x_0=0;\, \eta\in\{-1,1\},\mbox{ with probability of } \frac{1}{2} \mbox{ on each value of }\eta.$$ 2) In continuous time, but discontinuous path setting, is the following jump diffusion process a correct example? $$dx_t = -\alpha x_t dt+dz_t+ y\eta dN_t,\, x_0 = 0,$$ where $\alpha\gg 0$, $z_t$ is the standard brownian motion with mean $0$ and standard deviation $t$, $N_t$ is the Poisson process with frequency $0<\lambda\ll 1$, $\eta$ takes on values $-1$ or $1$ with $0.5$ probability each, $z_{t_1}$, $N_{t_2}$ and $\eta$ are independent of each other at arbitrary $t_1$ and $t_2$, and constant $y\gg 1$. On second thought, this is not a correct example. One can solve this equation and one will find the variance of this process is the sum of the variance from $dz_t$ and that from $dN_t$ due their independence. We will have to make the jumps negatively correlated to $z_t$. A better setup is to shift $x_t$ beyond a barrier directly back to the $x=0$ line. So the process resides on the topology of two cylinders touched along a longitude. However, it seems to me, even this set up with $x_t$ being either a standard Browniam motion or mean reverting one without any jump process still has its variance increasing with time. Therefore, I am still without a valid example in this setup. 3) What are the examples for continuous path? I suspect it is not possible. Can anyone prove this if it is indeed impossible? - "variance periodically oscillates over time"...that doesn't sound time homogeneous to me – quasi Dec 9 '13 at 21:58 Check out my example. – Hans Dec 9 '13 at 22:23 gotcha, get it now – quasi Dec 9 '13 at 23:19 en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process; start it from its steady state distribution. note this has mean-reverting behavior, similar to your example. oops, doesn't start at 0 though – quasi Dec 9 '13 at 23:22 This does not work. The variance of the mean-reverting Ornstein-Uhlenbeck process strictly increases over time. – Hans Dec 10 '13 at 3:04 In the literature it is often dealt with the covariance function. For a stationary time series, the covariance between $X_t$ and $X_s$ only depends on the time span $\|t-s\|$. For the varianace of $X_t$ we have $t-s=0$. why is $t-a=0$? Also the autocovariance is to be a function of the time difference not necessarily a constant. – Hans Dec 7 '13 at 21:40 If we want to apply the formulation of covariance to the simple variance case, then $t=s$ and thus $t-s=0$. The ACF is a function of the time difference, true. But this is zero for the variance. A random walk e.g. is not stationary. The variance increases with the square-root of time. An white noise on the other hand is stationary - the variance at each point in time is the same. – Richard Dec 7 '13 at 22:16	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 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": 0.9501467347145081, "perplexity": 329.43950977593863}, "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/1424936461332.16/warc/CC-MAIN-20150226074101-00190-ip-10-28-5-156.ec2.internal.warc.gz"}
http://pseudomonad.blogspot.com/2011/09/rumour-of-century.html?showComment=1316474735326	## Tuesday, September 20, 2011 ### Rumour of the Century We don't usually report on unsubstantiated rumours here at AP, but this one is just too spectacular to ignore. Jester thinks its crazy, Phil sounds excited, and Graham says that Tommaso had a post up with a $6.1 \sigma$ result from OPERA, but the post has now disappeared. Given the number of false rumours that get circulated, we should be doubtful at this point. Here is OPERA's last arxiv paper. The CERN seminar is this coming Friday. So the rumour is that neutrinos arriving at OPERA have travelled at a speed greater than $c$. Forget fairy fields. If this is true, it demolishes establishment thinking in one falcon swoop. Could the minus sign in Brannen's Koide relation for neutrinos be responsible for such tachyonic behaviour? Why not? Or, perhaps all neutrinos are tachyons. Now the minus sign goes with the lightest neutrino mass state. The literature has an annoying tendency to confuse EW and mass states, but we should note that all mass states can occur in $\nu_{\mu}$-$\nu_{\tau}$ oscillations. 1. As we know, photons always travel at $c$ locally. Particles with mass, such as electrons and muons, travel at speeds less than $c$ precisely because they have mass. But until OPERA, as far as I know, nobody could actually measure neutrino speeds. And we already know that the Koide formula looks different for neutrinos. The tachyonic behaviour would not even violate Lorentzian geometry, because we would simply insist that tachyons always travel at speeds greater than $c$. 2. Even kneemo likes tachyons, because M theory has tachyons. 3. And it would strengthen the argument that the neutrino sector is crucial for understanding gravity. 4. So now we can rethink the strange MiniBooNE/LSND results, and in fact all the appearance disappearance anomalies. With tachyonic neutrinos, hopefully the ad hoc introduction of sterile states can finally be dismissed. As we said some time ago, the local neutrino gas should not behave in a standard fashion. 5. I predict that the rumored result will not stand. 6. Of course you do, Mitchell. You are still a string theorist. But there is no reason to doubt that tachyons exist in our form of M theory. Mind you, I bet OPERA will have a hard time convincing everyone that they know what they are doing. 7. Now according to the rumour, which contains only the sketchiest facts, the OPERA detector was looking for the onset of tau neutrinos from a well timed $700+$ km beam. This strongly suggests that ALL the arriving neutrinos are travelling at a speed $> c$. Duh. 8. So all mass states are tachyonic. 9. Some people are already talking about the implication of classical time travel, but there is no reason to jump to that conclusion here. There is nothing concretely acausal about sending the neutrinos from point A to point B and then measuring the apparent speed. The speed is whatever it is. Relativity is prefectly well respected by tachyons, even in interaction with ordinary matter, provided we consider a complexification of our geometry, which we do with twistor theory anyway. 10. No, wait! The onset time only gives an indication of the fastest speed. It is not true that all neutrinos need travel that fast. So perhaps OPERA actually sees something really amazing, like a triplet of arrival times, one $>c$. 11. And since the fastest speed corresponds to the lowest energy for a tachyon, the low energy excess of MiniBooNE comes from the most tachyonic neutrinos. 12. My understanding of Carl's philosophy of time is that time is absolute and Lorentz symmetry is emergent. I don't know how yours works. I don't know how to "interpret" just two timelike dimensions, let alone three. When you have a two-time signature for twistors, that's not something which shows up in any observable physical state; it's something that you analytically continue back to the physical signature. Similarly, when you have multiple times in F-theory, S-theory... they have to be hidden away with the spacelike compact dimensions. At some point, the formal mathematical concepts of time that are employed in physics have to reconnect with phenomenological time, and I do not at all see how that is possible if you have more than one "time". 13. It smells a bit bad, and rather inconsistent with previous data... There had been a delay of ~3h between the neutrinos and the gammas in SN1987A (it had been explained by astrophysical reasons, though). If one however interprets it as physical, takes the distance of the SN (168000 ly) and rescales it to the distance (700 km) between CERN and the Gran Sasso (GS), one would expect Dt ~ 10000 s * (700e3/(1680003e73e8)) ~ 5 ps which is 1 millimeter of delay... It is true that the SN neutrinos were in the MeV, but it looks strange that something dramatic happens between the MeV and the GeV... There was also a ~1-day-before claim in 1987, but then it was withdrawn - and in any case it is 2-3 orders of magnitude below the needed sensitivity. Alessandro De Angelis 14. If it is true, it would mean that neutrino can see more than any other particle. Wow. In a fundamental theory of quantum gravity c is no longer an invariant, the planck lenght or certain fundamental lenght \alpha L_P is the main quantity. Anyway, I hope this could be true. LHC is being something boring yet...The surprinsing discoveries are yet to come! Careful has to be paid to the tachyonic conclusion. OPERA is not a vaccum detector, so a bad definition of group speed in the dispersion relation could appear as a fake v>c. Is the rumour on OPERA a claim on neutrino speed at vaccum? Curiously, it is an experiment I am following less than others. And about a theory with v>c...I only know three theories ( at least published) that could obtain that claim (forget M theory at principle): Gonzalez-Mestres theory of superbradyons, varying speed-of light theories and extended relativities in C-spaces. Of course, they are non main stream theories, but they postulate from first principles the possible existence of particles with v>c without tachyons. I would like to emphasize this: if it is true, if neutrino has v>c at vacuum, it does not necessarily means they are tachyons. Questions: has someone else realized what would happen if there some other EM-like in the Universe very weakly coupled to the SM and whose "speed of light" is NOT the common speed-of-light? 15. Yes, the supernovae data are interesting, but it could be the case that the tachyonic species are only observable more locally. Juan, many people have thought about these kind of things, since the 1950s. However, very few people think about emergent geometry using modern mathematical methods, with which far more is possible. I have no more information on the rumour, since I did not even get to read Tommaso's post. I don't think the rock medium is an issue, since it is pretty transparent to neutrinos. My guess is that a satellite measurement of the distance, using an accurate angle for the two photon paths, could be used to determine a length for the other side of the triangle, as a reasonable estimate of the distance. 16. Graham knows more about supernovae. And someone at vixra mentioned fibre optics 'down the shaft' from the GPS point. 17. Now we just have to wait. As Graham says, the mistrust of $6.1 \sigma$ results is rather disturbing to those theorists that approve of the result. 18. Like EPR, till they show what really they have got, we only get romours...A pity!What time is the Friday talk, local time? I am getting interested since there are several prominent blogs commenting the rumour. Will it be real after all? Yeah, I know neutrino is almost transparent to medium ( I am doing my Master thesis on neutrinos) but group velocity and phase velocities can be greater than c. I have to read more about what they are measuring, if it proves to be a real 6.1 sigma claim. 19. By the way: note that something weird is also happening in the existent data on reactor neutrinos. Will the neutrino surprise us ...again? :D 20. Has anyone thought of the possibility of a tunneling-like event in the 'insides' of earth? It is a well known fact that light CAN travel faster than c in case of a tunneling event. (Steinber, PRL 1993 - http://prl.aps.org/abstract/PRL/v71/i5/p708_1 - Or just google 'tunneling time of single photon') In thet perspective, this data would not mean tachyonic behaviour of neutrinos, but simply the occurence of quantum barrier for the neutrinos. Just saying it...	{"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": 0.8147882223129272, "perplexity": 1133.8356458233025}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627997801.20/warc/CC-MAIN-20190616062650-20190616084650-00125.warc.gz"}
https://www.researchgate.net/profile/Sabrina-Roscani-2	# Sabrina RoscaniNational Scientific and Technical Research Council - Austral University (Argentina) · Mathematics Dr. 29 Publications 2,670 A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more 195 Citations Introduction Sabrina Roscani currently works at the Departamento de Matemática, Universidad Austral de Rosario and at the Rosario National university. Sabrina does research in Applied Mathematics, Analysis and Fractional Calculus. Their current project is 'Inecuaciones variacionales, control óptimo y problemas de frontera libre: teoría, análisis numérico y aplicaciones.'. August 2008 - present Position • Assistant Proffesor ## Publications Publications (29) Article We consider a family of initial boundary value problems governed by a fractional diffusion equation with Caputo derivative in time, where the parameter is the Newton heat transfer coefficient linked to the Robin condition on the boundary. For each problem we prove existence and uniqueness of solution by a Fourier approach. This will enable us to al... Article Full-text available In this paper we obtain self-similarity solutions for a one-phase one-dimensional fractional space Stefan problem in terms of the three parametric Mittag-Leffler function Eα,m,l(z). We consider Dirichlet and Neumann conditions at the fixed face, involving Caputo fractional space derivatives of order 0<α<1. We recover the solution for the classical... Preprint Full-text available Taking into account the recent works \cite{RoTaVe:2020} and \cite{Rys:2020}, we consider a phase-change problem for a one dimensional material with a non-local flux, expressed in terms of the Caputo derivative, which derives in a space-fractional Stefan problem. We prove existence of a unique solution to a phase-change problem with the fractional N... Article The purpose of this paper is twofold. We first provide the mathematical analysis of a dynamic contact problem in thermoelasticity, when the contact is governed by a normal damped response function and the constitutive thermoelastic law is given by the Duhamel-Neumann relation. Under suitable hypotheses on data and using a Faedo-Galerkin strategy, w... Preprint Full-text available We consider a family of initial boundary value problems governed by a fractional diffusion equation with Caputo derivative in time, where the parameter is the Newton heat transfer coefficient linked to the Robin condition on the boundary. For each problem we prove existence and uniqueness of solution by a Fourier approach. This will enable us to al... Article Full-text available In this paper we consider a family of three-dimensional problems in thermoelasticity for elliptic membrane shells and study the asymptotic behaviour of the solution when the thickness tends to zero. We fully characterize with strong convergence results the limit as the unique solution of a two-dimensional problem, where the reference domain is the... Preprint Full-text available In this paper we consider a family of three-dimensional problems in thermoelasticity for linear elliptic membrane shells and study the asymptotic behaviour of the solution when the thickness tends to zero.We fully characterize with strong convergence results the limit as the unique solution of a two-dimensional problem, where the reference domain i... Preprint Full-text available In this paper we obtain self-similarity solutions for a one-phase one-dimensional fractional space one-phase Stefan problem in terms of the three parametric Mittag-Leffer function $E_{\alpha,m;l}(z)$. We consider Dirichlet and Newmann conditions at the fixed face, involving Caputo fractional space derivatives of order $0 < \alpha < 1$. We recover t... Article Two fractional two-phase Stefan-like problems are considered by using Riemann-Liouville and Caputo derivatives of order α ∈ (0, 1) verifying that they coincide with the same classical Stefan problem at the limit case when α=1. For both problems, explicit solutions in terms of the Wright functions are presented. Even though the similarity of the two... Preprint Full-text available Two fractional two-phase Stefan-like problems are considered by using Riemann-Liouville and Caputo derivatives of order $\alpha \in (0, 1)$ verifying that they coincide with the same classical Stefan problem at the limit case when $\alpha=1$. For both problems, explicit solutions in terms of the Wright functions are presented. Even though the simil... Preprint In this paper we establish some convergence results for Riemann-Liouville, Caputo, and Caputo-Fabrizio fractional operators when the order of differentiation approaches one. We consider some errors given by $\left\|\left\| D^{1-\al}f -f'\right\|\right\|_p$ for p=1 and $p=\infty$ and we prove that for both Caputo and Caputo Fabrizio operators the order... Preprint Full-text available This paper deals with the fractional Caputo--Fabrizio derivative and some basic properties related. A computation of this fractional derivative to power functions is given in terms of Mittag--Lefler functions. The inverse operator named the fractional Integral of Caputo--Fabrizio is also analyzed. The main result consists in the proof of existence... Preprint Full-text available A mathematical model for a one-phase change problem (particularly a Stefan problem) with a memory flux, is obtained. The hypothesis that the weighted sum of fluxes back in time is proportional to the gradient of temperature is considered. The model obtained involves fractional derivatives with respect on time in the sense of Caputo and in the sense... Preprint Full-text available A generalized Neumann solution for the two-phase fractional Lam\'e--Clapeyron--Stefan problem for a semi--infinite material with constant initial temperature and a particular heat flux condition at the fixed face is obtained, when a restriction on data is satisfied. The fractional derivative in the Caputo sense of order $\al \in (0,1)$ respect on t... Article Full-text available A generalized Neumann solution for the two-phase fractional Lamé–Clapeyron–Stefan problem for a semi-infinite material with constant initial temperature and a particular heat flux condition at the fixed face is obtained, when a restriction on data is satisfied. The fractional derivative in the Caputo sense of order $$\alpha \in (0,1)$$ respect on t... Article Full-text available Two fractional Stefan problems are considered by using Riemann-Liouville and Caputo derivatives of order $\alpha \in (0,1)$ such that in the limit case ($\alpha =1$) both problems coincide with the same classical Stefan problem. For the one and the other problem, explicit solutions in terms of the Wright functions are presented. We prove that these... Article Full-text available We consider a one-dimensional moving-boundary problem for the time-fractional diffusion equation. The time-fractional derivative of order $\alpha\in (0,1)$ is taken in the sense of Caputo. We study the asymptotic behaivor, as t tends to infinity, of a general solution by using a fractional weak maximum principle. Also, we give some particular exact... Article Full-text available A one-dimensional fractional one-phase Stefan problem with a temperature boundary condition at the fixed face is considered by using the Riemann–Liouville derivative. This formulation is more convenient than the one given in Roscani and Santillan ( Fract. Calc. Appl. Anal. , 16 , No 4 (2013), 802–815) and Tarzia and Ceretani ( Fract. Calc. Appl. An... Article Full-text available We consider the time-fractional derivative in the Caputo sense of order α ∈ ( 0 , 1 ) . Taking into account the asymptotic behavior and the existence of bounds for the Mainardi and the Wright function in R + , two different initial-boundary-value problems for the time-fractional diffusion equation on the real positive semiaxis are solved. Mor... Article This paper deals with a theoretical mathematical analysis of a one-dimensional-moving-boundary problem for the time-fractional diffusion equation, where the time-fractional derivative of order $\al$ $\in (0,1)$ is taken in the Caputo's sense. A generalization of the Hopf's lemma is proved, and then this result is used to prove a monotonicity proper... Article We consider a one-dimensional moving-boundary problem for the time-fractional diffusion equation, where the time-fractional derivative of order α ∈ (0, 1) is taken in the Caputo sense. A generalization of the Hopf lemma is proved and then used to prove a monotonicity property for the free-boundary when a fractional free-boundary Stefan problem is i... Article Full-text available We obtain a generalized Neumann solution for the two-phase fractional Lam\'{e}-Clapeyron-Stefan problem for a semi-infinite material with constant boundary and initial conditions. In this problem, the two governing equations and a governing condition for the free boundary include a fractional time derivative in the Caputo sense of order $0<\al\leq... Article Full-text available A fractional Stefan's problem with a boundary convective condition is solved, where the fractional derivative of order α (0, 1) is taken in the Caputo sense. Then an equivalence with other two fractional Stefan's problems (the first one with a constant condition on x = 0 and the second with a flux condition) is proved and the convergence to the cla... Article Full-text available This paper deals with a theoretical mathematical analysis of an initial-boundary-value problem for the time-fractional diffusion equation in the quarter plane, where the time-fractional derivative is taken in the Caputo's sense of order$\al\in (0,1)$. For three different cases, changing the condition on the fixed face x=0 (temperature boundary... Article Full-text available Two Stefan's problems for the diffusion fractional equation are solved, where the fractional derivative of order$ \al \in (0,1) $is taken in the Caputo's sense. The first one has a constant condition on$ x = 0 $and the second presents a flux condition$ T_x (0, t) = \frac {q} {t ^ {\al/2}} \$. An equivalence between these problems is proved and... Cited By	{"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": 0.9658227562904358, "perplexity": 617.469721692401}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882570921.9/warc/CC-MAIN-20220809094531-20220809124531-00012.warc.gz"}
https://brilliant.org/practice/de-moivres-theorem-level-2-3-challenges/	Algebra # De Moivre's Theorem: Level 3 Challenges $\large { z }^{ 3 }=1$ What is the set of all $z$ that satisfy the equation above? Note: $\omega = \frac{ -1 + \sqrt 3 i}{2}$ where $i =\sqrt{-1}$. Find the value of $(2-\omega)(2-\omega^2)(2-\omega^{10})(2-\omega^{11}).$ Details and Assumptions: $\omega$ is a non-real cube root of unity. The five roots of the equation $z^{5}=4-4i$ each take the form $\Large \sqrt{2} e ^ { \frac{ k \pi i } { 20} },$ where $k$ is a positive integer less than 40. Find the sum of all values of $k$. $\large x + \frac 1 x = \sqrt 3\ , \ \ \ \ \ \ \ x^{200} + \frac {1}{x^{200}} = \ ?$ If the $6$ solutions of $x^{6}=-64$ are written in the form $a+ib$, where $a$ and $b$ are real, then what is the product of those solutions with $a>0$? ×	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 21, "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": 0.9598407745361328, "perplexity": 131.16851782331014}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250595787.7/warc/CC-MAIN-20200119234426-20200120022426-00378.warc.gz"}
https://gateoverflow.in/tag/load-back-cache	In the memory access time formula for the hierarchical cache - which is given as :$Emat = H1\times T1 + (1-H1)(H2\times (T1+T2) + (1-H2)\times (T1+T2+T3))$ (where Hi = Hit Ratio for the i-th level cache and Ti = Access time for i-th level ... been transferred into the cache. Is my intuition correct? If yes, then what should be the Emat formula for Load Back cache ? Should we add extra Ti values?	{"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": 0.8447291254997253, "perplexity": 943.9129737123516}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738735.44/warc/CC-MAIN-20200811055449-20200811085449-00245.warc.gz"}
http://blog.computationalcomplexity.org/2008/08/discounted-time.html	## Friday, August 08, 2008 ### Discounted Time A write-up of some ideas I presented at the Complexity Conference Rump Session. In computational complexity when we talk about time it usually represents a hard limit in the running time, solving the problem in time t(n). So we are happy, say, if we can solve the problem in one hour and miserable if it takes 61 minutes. But our real gradation of happiness over the running time is not so discontinuous. Let's take an idea from how economists deal with time. They discount the utility by a factor of δ in each time step for some δ<1. What if we did the same for complexity? Let δ = 1-ε for ε>0 and very small. Think ε about 10-12. We then discount the value of the solution by a factor δt for t steps of computation. Discounted time gives us a continuous loss due to time. It has the nice property that the future looks like the past: The discount for t steps now is the same as the the discount for the t steps already taken. When t is small, δt is about 1-εt, a linear decrease. For t large, δt is about e-εt, an exponential decrease. We can also recover traditional complexity classes. DTIME(O(m(n)) is the set of languages such that for some constant c>0, δt>c for δ=(1-1/m(n)). I'm not sure what to do with discounted time which is why this is a blog post instead of a FOCS paper. Some ideas: • What does average case and expected time mean in the discounted time model? • What if you take the value of the solution of some approximation problem and discount it with the time taken? Can you determine the optimal point to stop? 1. Instead of talking about the expected time to solve a problem, this transitions nicely to the expected value of that solution. For approximation problems, you can combine that with a function expressing the value of an approximation relative to the value of the optimal solution. Combining them you get the expected value of approximately solving the problem in time t, given the accuracy you expect to develop in time t, relative to having the optimal solution immediately. For time constrained problems, this favors algorithms that can produce more than an 1/δ improvement in the value of the approximate result in a given unit of time, over a period of time within the time constraint. This also suggests an optimal length of time to run such algorithms: until the expected improvement per unit time drops below 1/δ. 2. This reminds me of a poly-time algorithm for factoring integers that I heard from Ed Fredkin. You don't even need quantum computers for it; you only need Moore's law. If processing speed continues increasing exponentially, then you need only wait a number of years that is linear in the number of bits of your input. If you don't believe Moore's law will last for much longer, then you could instead rely on economic growth. If the economy grows in real terms by at least a constant rate, then you can invest money in a foundation where it will grow exponentially until there is enough to buy enough computers to solve the problem. (Hopefully it's clear that he was joking.) 3. Actually, the factoring algorithm is sub-linear. If it takes time exp(O(n^{1/3})) to factor an n-bit number, then the same can also be achieved with O(n^{1/3}) doublings of processor power. 4. One of the first papers I read in complexity was a paper by Levin (about complexity-theoretic cryptography, I think), where the first sentence of a proof was: (paraphrase) without loss of generality, we may assume that all algorithms run in O(1) time (/paraphrase) needless to say, i was stunned - until I realized that the quantity he was analyzing was the product of the running time and success probability of the algorithm, and was alluding to "normalizing" algorithms in the following way: if it was a worst-case t(n) time computation, toss a coin with Pr[heads] = 1/t(n), and perform the computation iff you see heads. This changes the (expected) running time to O(1) while leaving the product of the two quantities still analyzable. I suspect that one can similarly hack around with your notion of utility (which is certainly an interesting way to think about computations). 5. This concept has been explored in Databases, Real time systems and AI where it is known as a "soft deadline". Typically there is a desired time by which the result should be computed, but thereafter the utility drops according to some function, rather than a sharp threshold. They were introduced, as far as I know, by Garcia-Molina and Abbott in a SIGMOD Record paper in 1988. 6. Just a note to add that the fact that this idea has been proposed before should not stop anyone from doing research on it. In fact, if other people have proposed it this is an indication that there is interest in the study of such a measure.	{"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": 0.8834285736083984, "perplexity": 553.18024931015}, "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-2014-23/segments/1406510270399.7/warc/CC-MAIN-20140728011750-00255-ip-10-146-231-18.ec2.internal.warc.gz"}
https://www.ias.ac.in/listing/articles/pram/076/04	• Volume 76, Issue 4 April 2011, pages 533-690 • Travelling wave solutions to nonlinear physical models by means of the ﬁrst integral method This paper presents the ﬁrst integral method to carry out the integration of nonlinear partial differential equations in terms of travelling wave solutions. For illustration, three important equations of mathematical physics are analytically investigated. Through the established ﬁrst integrals, exact solutions are successfully constructed for the equations considered. • Bianchi type-I massive string magnetized barotropic perfect ﬂuid cosmological model in bimetric theory Bianchi type-I massive string cosmological model for perfect ﬂuid distribution in the presence of magnetic ﬁeld is investigated in Rosen’s [Gen. Relativ. Gravit. 4, 435 (1973)] bimetric theory of gravitation. To obtain the deterministic model in terms of cosmic time, we have used the condition $A = (B C)^n$, where n is a constant, between the metric potentials. The magnetic ﬁeld is due to the electric current produced along the 𝑥-axis with inﬁnite electrical conductivity. Some physical and geometrical properties of the exhibited model are discussed and studied. • Entropy of the Kerr–Sen black hole We study the entropy of Kerr–Sen black hole of heterotic string theory beyond semiclassical approximations. Applying the properties of exact differentials for three variables to the ﬁrst law of thermodynamics, we derive the corrections to the entropy of the black hole. The leading (logarithmic) and non-leading corrections to the area law are obtained. • A transformed rational function method for (3+1)-dimensional potential Yu–Toda–Sasa–Fukuyama equation A direct method, called the transformed rational function method, is used to construct more types of exact solutions of nonlinear partial differential equations by introducing new and more general rational functions. To illustrate the validity and advantages of the introduced general rational functions, the (3+1)-dimensional potential Yu–Toda–Sasa–Fukuyama (YTSF) equation is considered and new travelling wave solutions are obtained in a uniform way. Some of the obtained solutions, namely exponential function solutions, hyperbolic function solutions, trigonometric function solutions, Jacobi elliptic function solutions and rational solutions, contain an explicit linear function of the independent variables involved in the potential YTSF equation. It is shown that the transformed rational function method provides more powerful mathematical tool for solving nonlinear partial differential equations. • Investigation of $\Delta(3,3)$ resonance effects on the properties of neutron-rich double magic spherical ﬁnite nucleus, 132Sn, in the ground state and under compression Within the framework of the radially constrained spherical Hartree–Fock (CSHF) approximation, the resonance effects of delta on the properties of neutron-rich double magic spherical nucleus 132Sn were studied. It was found that most of the increase in the nuclear energy generated under compression was used to create massive 𝛥 particles. For 132Sn nucleus under compression at 3.19 times density of the normal nuclear density, the excited nucleons to 𝛥s were increased sharply up to 16% of the total number of constituents. This result is consistent with the values extracted from relativistic heavy-ion collisions. The single particle energy levels were calculated and their behaviours under compression were examined. A meaningful agreement was obtained between the results with effective Hamiltonian and that with the phenomenological shell model for the low-lying single-particle spectra. The results suggest considerable reduction in compressibility for the nucleus, and softening of the equation of state with the inclusion of 𝛥s in the nuclear dynamics. • Vibrational analysis of Fourier transform spectrum of the $B^3 \Sigma^-_u (0^+_u) - X^3 \Sigma^-_g (0^+_g)$ transition of 80Se2 molecule The emission spectra of $B^3 \Sigma^-_u (0^+_u) - X^3 \Sigma^-_g (0^+_g)$ transition of the isotopic species 80Se2, excited in an electrodeless discharge lamp by the microwave, was recorded on BOMEM DA8 Fourier transform spectrometer at an apodized resolution of 0.035 cm-1. Vibrational constants were improved by putting the wave number of band origins in Deslandre table. The vibrational analysis was supported by determining the Franck–Condon factor and 𝑟-centroid values. • Goos–Hänchen shift for higher-order Hermite–Gaussian beams We study the reﬂection of a Hermite–Gaussian beam at an interface between two dielectric media. We show that unlike Laguerre–Gaussian beams, Hermite–Gaussian beams undergo no signiﬁcant distortion upon reﬂection. We report Goos–Hänchen shift for all the spots of a higherorder Hermite–Gaussian beam near the critical angle. The shift is shown to be insigniﬁcant away from the critical angle. The calculations are carried out neglecting the longitudinal component along the direction of propagation for a spatially ﬁnite, s-polarized, full 3D vector beam. We brieﬂy discuss the difﬁculties associated with the paraxial approximation pertaining to a vector Gaussian beam. • Acoustic wave propagation in $Ni_3 R$ (𝑅 = Mo, Nb, Ta) compounds The ultrasonic properties of the hexagonal closed packed structured $Ni_3$Mo, $Ni_3$Nb and $Ni_3$Ta compounds were studied at room temperature for their characterization. For the investigations of ultrasonic properties, the second-order elastic constants using Lennard–Jones potential were computed. The velocities $V_1$ and $V_2$ have minima and maxima respectively at 45° with the unique axis of the crystal, while $V_3$ increases with respect to angle with the unique axis of the crystal. The inconsistent behaviour of angle-dependent velocities is associated with the action of second-order elastic constants. Debye average sound velocities of these compounds increase with the angle and has maximum at $55^{\circ}$ with the unique axis at room temperature. Hence, when a sound wave travels at $55^{\circ}$ with the unique axis of these materials, the average sound velocity is found to be maximum. The results achieved are discussed and compared with the available experimental and theoretical results. • Dielectric relaxation studies in 5CB nematic liquid crystal at 9 GH$_z$ under the inﬂuence of external magnetic ﬁeld using microwave cavity spectrometer Resonance width, shift in resonance frequency, relaxation time and activation energy of 5CB nematic liquid crystal are measured using microwave cavity technique under the inﬂuence of an external magnetic ﬁeld at 9 GHz and at different temperatures. The dielectric response in liquid crystal at different temperatures and the effects of applied magnetic ﬁeld on transition temperatures are studied in the present work. The technique needs a small quantity (< 0.001 cm3) of the sample and provides fruitful information about the macroscopic structure of the liquid crystal. • Meyer–Neldel DC conduction in chalcogenide glasses Meyer–Neldel (MN) formula for DC conductivity ($\sigma_{\text{DC}}$) of chalcogenide glasses is obtained using extended pair model and random free energy barriers. The integral equations for DC hopping conductivity and external conductance are solved by iterative procedure. It is found that MN energy ($\Delta E_{\text{MN}}$) originates from temperature-induced conﬁgurational and electronic disorders. Single polaron-correlated barrier hopping model is used to calculate $\sigma_{\text{DC}}$ and the experimental data of Se, As2S3, As2Se3 and As2Te3 are explained. The variation of attempt frequency $\upsilon_0$ and $\Delta E_{\text{MN}}$ with parameter $(r/a)$, where 𝑟 is the intersite separation and 𝑎 is the radius of localized states, is also studied. It is found that $\upsilon_0$ and $\Delta E_{\text{MN}}$ decrease with increase of $(r/a)$, and $\Delta E_{\text{MN}}$ may not be present for low density of defects. • Magnetic behaviour of AuFe and NiMo alloys We study the electronic structure and a mean-ﬁeld phase analysis based on the pair–pair energies derived from ﬁrst-principles electronic structure calculations of AuFe and NiMo alloys. We have used the tight-binding linear mufﬁn-tin orbitals-based augmented space recursion (TB-LMTO-ASR) method to do so. We investigate different behaviours of the two alloy systems by mapping the problems onto equivalent Ising models and then discuss the magnetic phase diagrams using the calculated pair energies. All three phases: paramagnetic, random ferromagnetic and spin glass, have been studied. • Mismatch of dielectric constants at the interface of nanometer metal-oxide-semiconductor devices with high-𝐾 gate dielectric impacts on the inversion charge density The comparison of the inversion electron density between a nanometer metal-oxidesemiconductor (MOS) device with high-𝐾 gate dielectric and a SiO2 MOS device with the same equivalent oxide thickness has been discussed. A fully self-consistent solution of the coupled Schrödinger–Poisson equations demonstrates that a larger dielectric-constant mismatch between the gate dielectric and silicon substrate can reduce electron density in the channel of a MOS device under inversion bias. Such a reduction in inversion electron density of the channel will increase with increase in gate voltage. A reduction in the charge density implies a reduction in the inversion electron density in the channel of a MOS device. It also implies that a larger dielectric constant of the gate dielectric might result in a reduction in the source–drain current and the gate leakage current. • A simpliﬁed approach for the generation of projection data for cone beam geometry To test a developed reconstruction algorithm for cone beam geometry, whether it is transmission or emission tomography, one needs projection data. Generally, mathematical phantoms are generated in three dimensions and the projection for all rotation angles is calculated. For non-symmetric objects, the process is cumbersome and computation intensive. This paper describes a simple methodology for the generation of projection data for cone beam geometry for both transmission and emission tomographies by knowing the object’s attenuation and/or source spatial distribution details as input. The object details such as internal geometrical distribution are nowhere involved in the projection data calculation. This simple approach uses the pixilated object matrix values in terms of the matrix indices and spatial geometrical coordinates. The projection data of some typical phantoms (generated using this approach) are reconstructed using standard FDK algorithm and Novikov’s inversion formula. Correlation between the original and reconstructed images has been calculated to compare the image quality. • Bianchi type-V string cosmological models in general relativity Bianchi type-V string cosmological models in general relativity are investigated. To get the exact solution of Einstein’s ﬁeld equations, we have taken some scale transformations used by Camci et al [Astrophys. Space Sci. 275, 391 (2001)]. It is shown that Einstein’s ﬁeld equations are solvable for any arbitrary cosmic scale function. Solutions for particular forms of cosmic scale functions are also obtained. Some physical and geometrical aspects of the models are discussed. • # Pramana – Journal of Physics Current Issue Volume 93 \| Issue 5 November 2019 • # Editorial Note on Continuous Article Publication Posted on July 25, 2019	{"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": 0.8841332197189331, "perplexity": 1537.5971650975016}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027319470.94/warc/CC-MAIN-20190824020840-20190824042840-00150.warc.gz"}
https://www.physicsforums.com/threads/preperation-of-bromine-gas.170099/	# Preperation of bromine gas 1. May 13, 2007 ### jamesyboy1990 1. The problem statement, all variables and given/known data Ok hi. I have this reaction for a lab due in about 2 weeks. It is on the preperation of bromine gas. The exact reaction was assigned (which is why i didn't choose a less-complex method, such as 2KBr + Cl2 --> 2KCl + Br2). My problem is that i need information (ie. internet websites, textbooks) becasue i need to have information such as: - nature of reaction (atomic/molecular) - bonding (intermolecular/intramolecular) - energy (endo/exothermic) - entropy (chaos) - number of particles before and after, state of particle, complexity of part - rate of reaction (actually, thats what my lab will be about) - acidic/basic/redox 2. Relevant equations 2KBr + MnO2 + 2H2SO4 --> K2SO4 + MnSO4 + 2H2O + Br2 (gas) 3. The attempt at a solution I have looked through numerous textbooks at the public library and websites and havent found any information on this reaction. If anyone has informaiton on this reaction, that would be very helpful. 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution 2. May 13, 2007 ### jamesyboy1990 oh yeah, and since this is my first time ever participating in this forum, please tell me if i'm missing anything 3. May 14, 2007 ### chemisttree You have a good start already. In what oxidation state are bromide and manganese oxide. How about manganese sulfate and bromine gas? Is anything being oxidized or reduced? Can you calculate the enthalpy of the reaction? Will the pH change during the reaction? You probably know these answers already from your coursework. How will you measure the rate of reaction in the lab?	{"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": 0.8221551179885864, "perplexity": 3162.8212567191176}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560282937.55/warc/CC-MAIN-20170116095122-00479-ip-10-171-10-70.ec2.internal.warc.gz"}
https://arxiv.org/abs/1609.02898	cs.RO (what is this?) # Title: A Linear-Time Variational Integrator for Multibody Systems Abstract: We present an efficient variational integrator for multibody systems. Variational integrators reformulate the equations of motion for multibody systems as discrete Euler-Lagrange (DEL) equations, transforming forward integration into a root-finding problem for the DEL equations. Variational integrators have been shown to be more robust and accurate in preserving fundamental properties of systems, such as momentum and energy, than many frequently used numerical integrators. However, state-of-the-art algorithms suffer from $O(n^3)$ complexity, which is prohibitive for articulated multibody systems with a large number of degrees of freedom, $n$, in generalized coordinates. Our key contribution is to derive a recursive algorithm that evaluates DEL equations in $O(n)$, which scales up well for complex multibody systems such as humanoid robots. Inspired by recursive Newton-Euler algorithm, our key insight is to formulate DEL equation individually for each body rather than for the entire system. Furthermore, we introduce a new quasi-Newton method that exploits the impulse-based dynamics algorithm, which is also $O(n)$, to avoid the expensive Jacobian inversion in solving DEL equations. We demonstrate scalability and efficiency, as well as extensibility to holonomic constraints through several case studies. Comments: Submitted to the International Workshop on the Algorithmic Foundations of Robotics (2016) Subjects: Robotics (cs.RO) Cite as: arXiv:1609.02898 [cs.RO] (or arXiv:1609.02898v1 [cs.RO] for this version) ## Submission history From: Jeongseok Lee [view email] [v1] Fri, 9 Sep 2016 19:23:54 GMT (517kb)	{"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": 0.8015760183334351, "perplexity": 1525.9048492573468}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218190181.34/warc/CC-MAIN-20170322212950-00023-ip-10-233-31-227.ec2.internal.warc.gz"}
https://encyclopediaofmath.org/index.php?title=Matrix_ring&diff=39111&oldid=36229	# Difference between revisions of "Matrix ring" 2010 Mathematics Subject Classification: Primary: 16S50 [MSN][ZBL] full matrix ring The ring of all square matrices of a fixed order over a ring $R$, with the operations of matrix addition and matrix multiplication. The ring of $(n \times n)$-dimensional matrices over $R$ is denoted by $R_n$ or $M_n(R)$. Throughout this article $R$ is an associative ring with identity. The ring $R_n$ is isomorphic to the ring $\mathop{End}(M)$ of all endomorphisms of the free right $R$-module $M = R^n$, possessing a basis with $n$ elements. The identity matrix $E_n = \text{diag}(1,\ldots,1)$ is the identity in $R_n$. An associative ring $A$ with identity 1 is isomorphic to $R_n$ if and only if there is in $A$ a set of $n^2$ elements $e_{ij}$, $i,j=1,\ldots,n$, subject to the following conditions: 1) $e_{ij}e_{kl} = \delta_{jk} e_{il}$, $\sum_{i=1}^n e_{ii}e_{ii} = 1$; 2) the centralizer of the set of elements $e_{ij}$ in $A$ is isomorphic to $R$. The centre of $R_n$ coincides with $\mathcal{Z}(R) E_n$, where $\mathcal{Z}(R)$ denotes the centre of $R$; for $n>1$ the ring $R_n$ is non-commutative. The multiplicative group of the ring $R_n$ (the group of all invertible elements), called the general linear group, is denoted by $\mathop{GL}_n(R)$. A matrix from $R_n$ is invertible in $R_n$ if and only if its columns form a basis of the free right module of all $(n \times 1)$-dimensional matrices over $R$. If $R$ is commutative, then the determinant is defined as a multiplicative map from $R_n$ to $R$ and invertibility of a matrix $X$ in $R_n$ is equivalent to the invertibility of its determinant, $\det X$, in $R$. The isomorphism $R_{mn} \sim (R_m)_n$ holds. The two-sided ideals in $R_n$ are of the form $J_n$, where $J$ is a two-sided ideal in $R$ and so the ring $R_n$ is simple if and only if $R$ is simple. An Artinian ring is simple if and only if it is isomorphic to a matrix ring over a skew-field (the Wedderburn–Artin theorem). If $\mathcal{J}(R)$ denotes the Jacobson radical of the ring $R$, then $\mathcal{J}(R_n) = \mathcal{J}(R)_n$. Consequently, every matrix ring over a semi-simple ring $R$ is semi-simple. If $R$ is a regular ring (in the sense of von Neumann) (i.e. if for every $a \in R$ there is a $b \in R$ such that $aba = a$), then so is $R_n$. If $R$ is a ring with an invariant basis number, i.e. the number of elements in a basis of each free $R$-module does not depend on the choice of the basis, then $R_n$ also has this property. The rings $R$ and $R_n$ are equivalent in the sense of Morita (see Morita equivalence): The category of $R$-modules is equivalent to the category of $R_n$-modules. However, the condition that projective $R$-modules are free does not necessarily entail that projective $R_n$-modules are free too. For instance, if $R$ is a field and $n>1$, then there exist finitely-generated projective $R_n$-modules which are not free. #### References [1] C. Faith, "Algebra: rings, modules, and categories" , 1 , Springer (1973) [2] J. Lambek, "Lectures on rings and modules" , Blaisdell (1966) [3] L.A. Bokut', "Associative rings" , 1 , Novosibirsk (1977) (In Russian)	{"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": 0.9647879600524902, "perplexity": 79.91553328037124}, "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/1656103573995.30/warc/CC-MAIN-20220628173131-20220628203131-00690.warc.gz"}
http://fluidsengineering.asmedigitalcollection.asme.org/article.aspx?articleid=1440306	0 Multiphase Flows # On the Volume Fraction Effects of Inertial Colliding Particles in Homogeneous Isotropic Turbulence [+] Author and Article Information Martin Ernst1 Mechanische Verfahrenstechnik, Zentrum für Ingenieurwissenschaften, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle (Saale), [email protected] Martin Sommerfeld Mechanische Verfahrenstechnik, Zentrum für Ingenieurwissenschaften, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle (Saale), [email protected] 1 Corresponding author. J. Fluids Eng 134(3), 031302 (Mar 23, 2012) (16 pages) doi:10.1115/1.4005681 History: Received April 19, 2011; Revised December 20, 2011; Published March 20, 2012; Online March 23, 2012 ## Abstract The main objective of the present study is the investigation of volume fraction effects on the collision statistics of nonsettling inertial particles in a granular medium as well as suspended in an unsteady homogeneous isotropic turbulent flow. For this purpose, different studies with mono-disperse Lagrangian point-particles having different Stokes numbers are considered in which the volume fraction of the dispersed phase is varied between 0.001 and 0.01. The fluid behavior is computed using a three-dimensional Lattice-Boltzmann method. The carrier-fluid turbulence is maintained at Taylor microscale Reynolds number 65.26 by applying a spectral forcing scheme. The Lagrangian particle tracking is based on considering the drag force only and a deterministic model is applied for collision detection. The influence of the particle phase on the fluid flow is neglected at this stage. The particle size is maintained at a constant value for all Stokes numbers so that the ratio of particle diameter to Kolmogorov length scale is fixed at 0.58. The variation of the particle Stokes number was realized by modifying the solids density. The observed particle Reynolds and Stokes numbers are in between [1.07, 2.61] and [0.34, 9.79], respectively. In the present simulations, the fluid flow and the particle motion including particle-particle collisions are based on different temporal discretization. Hence, an adaptive time stepping scheme is introduced. The particle motion as well as the occurrence of inter-particle collisions is characterized among others by Lagrangian correlation functions, the velocity angles between colliding particles and the collision frequencies. Initially, a fluid-free particle system is simulated and compared with the principles of the kinetic theory to validate the implemented deterministic collision model. Moreover, a selection of results obtained for homogeneous isotropic turbulence is compared with in literature available DNS and LES results as well. According to the performed simulations, the collision rate of particles with large Stokes numbers strongly depends on the adopted volume fraction, whereas for particles with small Stokes numbers the influence of particle volume fraction is less pronounced. <> ## Figures Figure 1 Velocity direction vectors of the D3Q19 model Figure 2 Program flow chart for the deterministic collision model (in analogy to Ref. [36]). Figure 3 Pictorial representation of two colliding particles [38] Figure 4 Fluctuation of collision rates N observed in a fluid-free particle system as a function of the nondimensional tracking time (closed symbols). The solid line indicates the theoretical reference value based on the kinetic theory, cf. Eq. 2. Figure 5 Comparison of the probability density functions of the velocity modulus between colliding particles with theoretical results from the kinetic theory. The kinetic theory corresponds to the velocity distribution of the injected particles. Figure 6 Fluid velocity field (vector plot) and particle field distribution (spheres, St = 2.57, αP = 0.01) for a single plane in the computational domain. Figure 7 Three-dimensional energy spectrum of the turbulent flow field (solid line with symbols: ReT = 65.26) and Kolmogorov spectrum (dashed line: universal Kolmogorov constant C = 1.5) as a function of the nondimensional wave number. Figure 8 Viscous dissipation spectrum computed from the present DNS (ReT = 65.26) and plotted against the nondimensional wave number. Figure 9 Probability density functions of the fluid velocity fluctuations for the three velocity components averaged over a single eddy turnover time. Figure 10 Probability density function of the free path between particle collisions depending on the particle Stokes number StP = 0.01). The indicated particle mean free path λFP¯ is also normalized by the Kolmogorov length scale λK . Figure 11 Effect of the particle response behavior (i.e., St) on the mean time between two particle-particle collisions τC : Here, the particle response time τP is normalized by the constant Kolmogorov timescale τK and plotted against the Stokes number St with the solid volume fraction αP as a parameter. Figure 12 Averaged collision frequencies fC , which are normalized by their corresponding particle response times τP , as a function of the Stokes number St and solid volume fraction αP . Figure 13 Computed collision frequencies fC as a function of the Stokes number StP = 0.01): Comparison of results obtained by direct numerical simulations (present study) with the analytical Saffman and Turner limit [2] as well as the kinetic theory limit [3]. Figure 14 Ratio of the computed collision frequency to the collision frequency obtained from kinetic theory plotted against the Stokes number StInt which is based on the fluid Lagrangian integral timescale: Comparison of results obtained by direct numerical simulations (open and partly filled symbols: present study), large eddy simulations for different volume fractions (closed symbols: Laviéville [21]) and analytical approximations (solid line: Kruis and Kusters [5]). Note: Symbols of one shape represent a comparable Stokes number, e.g., circle, square or triangle. Moreover, symbols of one filling level (present DNS) indicate the results for different volume fractions, i.e., P = 0.001, P = 0.005, P = 0.01. Figure 15 Averaged Lagrangian correlation functions of the particle velocities RP,u (τ) and their corresponding Lagrangian integral timescales τL,P as a function of the particle Stokes number StP = 0.01). Figure 16 Influence of the volume fraction αP on the Lagrangian correlation functions of the particles RP,u (τ) in presence as well as in absence of inter-particle collisions (St = 9.78). In case of particle-particle collision, the ratio of the particle Lagrangian integral timescale to the mean time between two successive inter-particle collisions τL,PC is given as well. Figure 17 Particle Lagrangian integral timescales τL,P as function of the Stokes number St with the solid volume fraction αP as a parameter: The calculated timescales are normalized by the Kolmogorov timescale τK as well as the mean time between successive inter-particle collisions τC Â . Figure 18 Comparison of the ratio of kinetic energy of the particle fluctuation motion kP to the turbulent kinetic energy of the flow field kF obtained by the present DNS (triangle: ReT = 65.26) with results from other DNS, published by Sundaram and Collins [12] (circle: ReT = 54.20) and Fede and Simonin [15] (square: ReT = 34.10), depending on the Stokes number St (Note: (1) symbols of one kind indicate the results for the different volume fractions, and (2) the results of all three DNS are based on a one-way momentum coupling of the dispersed phase with the fluid flow). Figure 19 Probability density function of the relative velocity modulus of colliding particles \|uPij \| which is normalized by Kolmogorov velocity uK with the Stokes number St as a parameter (αP = 0.01). In addition, the mean values of relative velocity modulus are printed for easy comparison. Figure 20 Influence of the volume fraction αP on (a) the mean relative velocity modulus \|uPij\|¯ and (b) the mean particle velocity angles ϕ¯ between colliding particles as a function of the Stokes number St. ## Discussions Some tools below are only available to our subscribers or users with an online account. ### Related Content Customize your page view by dragging and repositioning the boxes below. 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https://www.gnu.org/software/gnuastro/manual/html_node/Distance-on-a-2D-curved-space.html	GNU Astronomy Utilities Next: , Previous: , Up: CosmicCalculator [Contents][Index] 9.1.1 Distance on a 2D curved space The observations to date (for example the Plank 2013 results), have not measured the presence of a significant curvature in the universe. However to be generic (and allow its measurement if it does in fact exist), it is very important to create a framework that allows curvature. As 3D beings, it is impossible for us to mentally create (visualize) a picture of the curvature of a 3D volume in a 4D space. Hence, here we will assume a 2D surface and discuss distances on that 2D surface when it is flat, or when the 2D surface is curved (in a 3D space). Once the concepts have been created/visualized here, in Extending distance concepts to 3D, we will extend them to the real 3D universe we live in and hope to study. To be more understandable (actively discuss from an observer’s point of view) let’s assume we have an imaginary 2D friend living on the 2D space (which might be curved in 3D). So here we will be working with it in its efforts to analyze distances on its 2D universe. The start of the analysis might seem too mundane, but since it is impossible to imagine a 3D curved space, it is important to review all the very basic concepts thoroughly for an easy transition to a universe we cannot visualize any more (a curved 3D space in 4D). To start, let’s assume a static (not expanding or shrinking), flat 2D surface similar to Figure 9.1 and that our 2D friend is observing its universe from point $$A$$. One of the most basic ways to parametrize this space is through the Cartesian coordinates ($$x$$, $$y$$). In Figure 9.1, the basic axes of these two coordinates are plotted. An infinitesimal change in the direction of each axis is written as $$dx$$ and $$dy$$. For each point, the infinitesimal changes are parallel with the respective axes and are not shown for clarity. Another very useful way of parameterizing this space is through polar coordinates. For each point, we define a radius ($$r$$) and angle ($$\phi$$) from a fixed (but arbitrary) reference axis. In Figure 9.1 the infinitesimal changes for each polar coordinate are plotted for a random point and a dashed circle is shown for all points with the same radius. Figure 9.1: Two dimensional Cartesian and polar coordinates on a flat plane. Assuming a certain position, which can be parameterized as $$(x,y)$$, or $$(r,\phi)$$, a general infinitesimal change change in its position will place it in the coordinates $$(x+dx,y+dy)$$ and $$(r+dr,\phi+d\phi)$$. The distance (on the flat 2D surface) that is covered by this infinitesimal change in the static universe ($$ds_s$$, the subscript signifies the static nature of this universe) can be written as: $$ds_s=dx^2+dy^2=dr^2+r^2d\phi^2$$ The main question is this: how can our 2D friend incorporate the (possible) curvature in its universe when it is calculating distances? The universe it lives in might equally be a locally flat but globally curved surface like Figure 9.2. The answer to this question but for a 3D being (us) is the whole purpose to this discussion. So here we want to give our 2D friend (and later, ourselves) the tools to measure distances if the space (that hosts the objects) is curved. Figure 9.2 assumes a spherical shell with radius $$R$$ as the curved 2D plane for simplicity. The spherical shell is tangent to the 2D plane and only touches it at $$A$$. The result will be generalized afterwards. The first step in measuring the distance in a curved space is to imagine a third dimension along the $$z$$ axis as shown in Figure 9.2. For simplicity, the $$z$$ axis is assumed to pass through the center of the spherical shell. Our imaginary 2D friend cannot visualize the third dimension or a curved 2D surface within it, so the remainder of this discussion is purely abstract for it (similar to us being unable to visualize a 3D curved space in 4D). But since we are 3D creatures, we have the advantage of visualizing the following steps. Fortunately our 2D friend knows our mathematics, so it can follow along with us. With the third axis added, a generic infinitesimal change over the full 3D space corresponds to the distance: $$ds_s^2=dx^2+dy^2+dz^2=dr^2+r^2d\phi^2+dz^2.$$It is very important to recognize that this change of distance is for any point in the 3D space, not just those changes that occur on the 2D spherical shell of Figure 9.2. Recall that our 2D friend can only do measurements in the 2D spherical shell, not the full 3D space. So we have to constrain this general change to any change on the 2D spherical shell. To do that, let’s look at the arbitrary point $$P$$ on the 2D spherical shell. Its image ($$P'$$) on the flat plain is also displayed. From the dark triangle, we see that Figure 9.2: 2D spherical plane (centered on $$O$$) and flat plane (gray) tangent to it at point $$A$$. $$\sin\theta={r\over R},\quad\cos\theta={R-z\over R}.$$These relations allow our 2D friend to find the value of $$z$$ (an abstract dimension for it) as a function of r (distance on a flat 2D plane, which it can visualize) and thus eliminate $$z$$. From $$\sin^2\theta+\cos^2\theta=1$$, we get $$z^2-2Rz+r^2=0$$ and solving for $$z$$, we find: $$z=R\left(1\pm\sqrt{1-{r^2\over R^2}}\right).$$The $$\pm$$ can be understood from Figure 9.2: For each $$r$$, there are two points on the sphere, one in the upper hemisphere and one in the lower hemisphere. An infinitesimal change in $$r$$, will create the following infinitesimal change in $$z$$: $$dz={\mp r\over R}\left(1\over \sqrt{1-{r^2/R^2}}\right)dr.$$Using the positive signed equation instead of $$dz$$ in the $$ds_s^2$$ equation above, we get: $$ds_s^2={dr^2\over 1-r^2/R^2}+r^2d\phi^2.$$ The derivation above was done for a spherical shell of radius $$R$$ as a curved 2D surface. To generalize it to any surface, we can define $$K=1/R^2$$ as the curvature parameter. Then the general infinitesimal change in a static universe can be written as: $$ds_s^2={dr^2\over 1-Kr^2}+r^2d\phi^2.$$Therefore, we see that a positive $$K$$ represents a real $$R$$ which signifies a closed 2D spherical shell like Figure 9.2. When $$K=0$$, we have a flat plane (Figure 9.1) and a negative $$K$$ will correspond to an imaginary $$R$$. The latter two cases are open universes (where $$r$$ can extend to infinity). However, when $$K>0$$, we have a closed universe, where $$r$$ cannot become larger than $$R$$ as in Figure 9.2. A very important issue that can be discussed now (while we are still in 2D and can actually visualize things) is that $$\overrightarrow{r}$$ is tangent to the curved space at the observer’s position. In other words, it is on the gray flat surface of Figure 9.2, even when the universe if curved: $$\overrightarrow{r}=P'-A$$. Therefore for the point $$P$$ on a curved space, the raw coordinate $$r$$ is the distance to $$P'$$, not $$P$$. The distance to the point $$P$$ (at a specific coordinate $$r$$ on the flat plane) on the curved surface (thick line in Figure 9.2) is called the proper distance and is displayed with $$l$$. For the specific example of Figure 9.2, the proper distance can be calculated with: $$l=R\theta$$ ($$\theta$$ is in radians). using the $$\sin\theta$$ relation found above, we can find $$l$$ as a function of $$r$$: $$\theta=\sin^{-1}\left({r\over R}\right)\quad\rightarrow\quad l(r)=R\sin^{-1}\left({r\over R}\right)$$$$R$$ is just an arbitrary constant and can be directly found from $$K$$, so for cleaner equations, it is common practice to set $$R=1$$, which gives: $$l(r)=\sin^{-1}r$$. Also note that if $$R=1$$, then $$l=\theta$$. Generally, depending on the the curvature, in a static universe the proper distance can be written as a function of the coordinate $$r$$ as (from now on we are assuming $$R=1$$): $$l(r)=\sin^{-1}(r)\quad(K>0),\quad\quad l(r)=r\quad(K=0),\quad\quad l(r)=\sinh^{-1}(r)\quad(K<0).$$With $$l$$, the infinitesimal change of distance can be written in a more simpler and abstract form of $$ds_s^2=dl^2+r^2d\phi^2.$$ Until now, we had assumed a static universe (not changing with time). But our observations so far appear to indicate that the universe is expanding (isn’t static). Since there is no reason to expect the observed expansion is unique to our particular position of the universe, we expect the universe to be expanding at all points with the same rate at the same time. Therefore, to add a time dependence to our distance measurements, we can simply add a multiplicative scaling factor, which is a function of time: $$a(t)$$. The functional form of $$a(t)$$ comes from the cosmology and the physics we assume for it: general relativity. With this scaling factor, the proper distance will also depend on time. As the universe expands (moves), the distance will also move to larger values. We thus define a distance measure, or coordinate, that is independent of time and thus doesn’t ‘move’ which we call the comoving distance and display with $$\chi$$ such that: $$l(r,t)=\chi(r)a(t)$$. We thus shift the $$r$$ dependence of the proper distance we derived above for a static universe to the comoving distance: $$\chi(r)=\sin^{-1}(r)\quad(K>0),\quad\quad \chi(r)=r\quad(K=0),\quad\quad \chi(r)=\sinh^{-1}(r)\quad(K<0).$$ Therefore $$\chi(r)$$ is the proper distance of an object at a specific reference time: $$t=t_r$$ (the $$r$$ subscript signifies “reference”) when $$a(t_r)=1$$. At any arbitrary moment ($$t\neq{t_r}$$) before or after $$t_r$$, the proper distance to the object can simply be scaled with $$a(t)$$. Measuring the change of distance in a time-dependent (expanding) universe will also involve the speed of the object changing positions. Hence, let’s assume that we are only thinking about the change in distance caused by something (light) moving at the speed of light. This speed is postulated as the only constant and frame-of-reference-independent speed in the universe, making our calculations easier, light is also the major source of information we receive from the universe, so this is a reasonable assumption for most extra-galactic studies. We can thus parametrize the change in distance as $$ds^2=c^2dt^2-a^2(t)ds_s^2 = c^2dt^2-a^2(t)(d\chi^2+r^2d\phi^2).$$ Next: , Previous: , Up: CosmicCalculator [Contents][Index]	{"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": 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": 0.8928858041763306, "perplexity": 276.7153343628416}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187824357.3/warc/CC-MAIN-20171020211313-20171020231313-00276.warc.gz"}
http://tex.stackexchange.com/questions/57289/problem-with-shorthand-for-scope-environments	# problem with Shorthand for Scope Environments I try in this in this question to use some shorthands with the library scopes. I discover some difficulties and I can't explain why I get problems. (I use pgf 2.1 cvs) The main code comes from the pgfmanual \documentclass{article} \usepackage{tikz} \usetikzlibrary{scopes} \begin{tikzpicture} { [ultra thick] { [red] \draw (0mm,10mm) -- (10mm,10mm); \draw (0mm,8mm) -- (10mm,8mm); } \draw (0mm,6mm) -- (10mm,6mm); } { [green] \draw (0mm,4mm) -- (10mm,4mm); \draw (0mm,2mm) -- (10mm,2mm); \draw[blue] (0mm,0mm) -- (10mm,0mm); } \end{tikzpicture} \end{document} This is perfect. Now I want to draw three times these lines \documentclass{article} \usepackage{tikz} \usetikzlibrary{scopes} \begin{document} \begin{tikzpicture} \foreach \i in {0,5,10}{% \begin{scope}[xshift=\i cm] { [ultra thick] { [red] \draw (0mm,10mm) -- (10mm,10mm); \draw (0mm,8mm) -- (10mm,8mm); } \draw (0mm,6mm) -- (10mm,6mm); } { [green] \draw (0mm,4mm) -- (10mm,4mm); \draw (0mm,2mm) -- (10mm,2mm); \draw[blue] (0mm,0mm) -- (10mm,0mm); } \end{scope}} \end{tikzpicture} \end{document} This is always perfect but now if I replace \begin{scope} ...\end{scope} by {..}, the code compiles but the scopes disappear. \begin{tikzpicture} \foreach \i in {0,5,10}{% {[xshift=\i cm] { [ultra thick] { [red] \draw (0mm,10mm) -- (10mm,10mm); \draw (0mm,8mm) -- (10mm,8mm); } \draw (0mm,6mm) -- (10mm,6mm); } { [green] \draw (0mm,4mm) -- (10mm,4mm); \draw (0mm,2mm) -- (10mm,2mm); \draw[blue] (0mm,0mm) -- (10mm,0mm); } }} \end{tikzpicture} I know that { is the beginning of a TikZ's scope only if [ comes after { otherwise{..} is a simple TeX's group. Is-it possible to explain this problem? - To answer this, you need to know when TikZ looks for the {[...] ... } scoping shortcut. The TikZ parser has many different states and it has to be in the right state to recognise certain syntax, otherwise it either passes over it or complains vociferously. The check for the scoping shortcut is handled by a macro \tikz@lib@scope@check. Without the scopes library, this is empty. With the scopes library then it becomes equivalent to: • Look at the next (non-space) token on the stream. 1. Is it \tikz@intersect@finish? If so, do that and check again. 2. Is it \par? If so, do that and check again. 3. Is it \bgroup? If so, look for [ and if found, go into a scope. An important thing to note is that this is a test on the next token, and there is only limited support for delaying the test. Now let us see when this is invoked: 1. When a tikzpicture starts. So the first thing in a TikZ picture can be {[red] and that will get detected correctly. 2. When a scope starts. This can be an implicit or explicit one (so nesting of {[..] ..} syntax is fine, as is intermingling with ordinary \begin{scope} .. \end{scope} pairs). 3. When a scope ends. Actually, this is less useful than it at first seems. The problem is that an explicit scope ends within a group so the next token is not the next thing that the user thinks it is but probably an \endgroup (or something else buried in the \end{environment} code). To make use of this, the previous scope has to be either an implicit scope or of the form \scope ... \endscope. In the following, the first line is not red. Note that this can have knock-on effects: if the first implicit scope is not recognised and another follows it, that will also not be recognised. \begin{scope} \end{scope} {[red] \draw[ultra thick] (0,0) -- (0,1); } \scope \endscope {[red] \draw[ultra thick] (2,0) -- (2,1); } {[red] \draw[ultra thick] (1,0) -- (1,1); } 4. After a path. So an implicit scope can follow a path command. Unfortunately, none of those match your syntax because by the time the \foreach is processed, the check that is made at the start of the picture has been made and failed (since it found \foreach). So to get the initial implicit scope recognised (and, incidentally, the fact that the first doesn't get recognised has the knock-on effect that none of the others do either: the [ultra thick] one doesn't follow the opening of a scope, so then neither does the [red] one, and the [green] one doesn't follow the ending of a scope so it isn't recognised) we need to invoke the recognition code. Two ways to do this are by ensuring that one of the conditions is met. Either an explicit but groupless scope has to start the \foreach (so \scope\endscope) or an empty path (so \path;). These might be considered a little "hackish". There is an alternative. This is to ensure that the actual check is carried out at the start of the \foreach command. As the pgffor routines are meant to stand to one side of TikZ, this is not - and should not - be explicitly in the pgffor.code.tex file. But what is there are some hooks that can be invoked: \pgffor@beginhook and \pgffor@endhook (and \pgffor@afterhook). Indeed, these are used by TikZ for when \foreach is encountered inside a path. But they aren't used for external encounters. Since the internal one overwrites these hooks, we can just set them for the picture. As we want to make sure that the scope check is the very last thing that is added to these hooks, I've chosen a slightly circumspect way of doing it: \tikzset{every picture/.append style={ execute at begin picture={ \expandafter\def\expandafter\pgffor@beginhook\expandafter{\pgffor@beginhook\tikz@lib@scope@check} } } } This means that when the picture starts, the code for adding the code is added to the code that is executed at the start of the picture. So it will be added after anything that is added either globally or in the options at the start of the picture. I'm sure it could be done better! With that, then your code works: \documentclass{article} %\url{http://tex.stackexchange.com/q/57289/86} \usepackage{tikz} %\usepackage{trace-pgfkeys} \usetikzlibrary{scopes} \makeatletter \tikzset{every picture/.append style={ execute at begin picture={ \expandafter\def\expandafter\pgffor@beginhook\expandafter{\pgffor@beginhook\tikz@lib@scope@check} } } } \makeatother \begin{document} \begin{tikzpicture} \foreach \i in {0,5,10}{% {[xshift=\i cm] { [ultra thick] { [red] \draw (0mm,10mm) -- (10mm,10mm); \draw (0mm,8mm) -- (10mm,8mm); } \draw (0mm,6mm) -- (10mm,6mm); } { [green] \draw (0mm,4mm) -- (10mm,4mm); \draw (0mm,2mm) -- (10mm,2mm); \draw[blue] (0mm,0mm) -- (10mm,0mm); } } } \end{tikzpicture} \end{document} Finally, the trace-pgfkeys was very useful here as it showed straight away that the scopes weren't getting seen at all. - Very great answer as usual. I need to work with trace-pgfkeys` if If I want to progress. Thanks Sherlock ! – Alain Matthes May 25 '12 at 11:28	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8648639917373657, "perplexity": 2588.4860153590175}, "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-2014-42/segments/1414119647865.10/warc/CC-MAIN-20141024030047-00252-ip-10-16-133-185.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/106473/prove-that-x-12x-22x-32-1-yields-sum-i-13-fracx-i1x-i2-le	# Prove that $x_1^2+x_2^2+x_3^2=1$ yields $\sum_{i=1}^{3}\frac{x_i}{1+x_i^2} \le \frac{3\sqrt{3}}{4}$ Prove this inequality, if $x_1^2+x_2^2+x_3^2=1$: $$\sum_{i=1}^{3}\frac{x_i}{1+x_i^2} \le \frac{3\sqrt{3}}{4}$$ So far I got to $x_1^4+x_2^4+x_3^4\ge\frac{1}3$ by using QM-AM for $(2x_1^2+x_2^2, 2x_2^2+x_3^2, 2x_3^2+x_1^2)$, but to be honest I'm not sure if that's helpful at all. (You might want to "rename" those to $x,y,z$ to make writing easier): $$\frac{x}{1+x^2}+\frac{y}{1+y^2}+\frac{z}{1+z^2} \le \frac{3\sqrt3}4$$ - Do you know Lagrange Multipliers? – Will Jagy Feb 6 '12 at 23:25 Nope, I'm high school student, I only know basic inequalities (HM-GM-AM-QM, Cauchy-Schwarz-Bunyakowski, Jensen's, Minkowski, Schur and such). – Lazar Ljubenović Feb 6 '12 at 23:34 Well, maybe someone will answer by those methods. LM says that the extrema, and other critical points, of the functional occur when $x=y=z,$ which means $\pm(1/\sqrt 3, 1/\sqrt 3, 1/\sqrt 3).$ – Will Jagy Feb 6 '12 at 23:42 Suggestion: make substitution $x=r\cos\theta\sin\phi$, $y=r\sin\theta\sin\phi$, $z=r\cos\phi$. Let r = 1. Still not easy, but now a spherical trig problem. Perhaps trig identities would help. – daniel Feb 7 '12 at 9:47 We assume that $x_i\geq 0$ and let $\theta_i\in \left(0,\frac{\pi}2\right)$ sucht that $x_i=\tan\frac{\theta_i}2$. We have $\sin(\theta_i)=\frac{2x_i}{1+x_i^2}$ and since $\sin$ in concave on $\left(0,\frac{\pi}2\right)$, we have $$\sum_{i=1}^3\frac{x_i}{1+x_i^2}=\frac 32\sum_{i=1}^3\frac 13\sin(\theta_i)\leq \frac 32\sin\frac{\theta_1+\theta_2+\theta_3}3.$$ We have, using the convextiy of $x\mapsto \tan^2 x$: $$\frac 13=\frac 13\sum_{i=1}^3\tan^2\frac{\theta_i}2\geq \tan^2\frac{\theta_1+\theta_2+\theta_3}6,$$ so $\tan\frac{\theta_1+\theta_2+\theta_3}6\leq \frac 1{\sqrt 3}$ and $\frac{\theta_1+\theta_2+\theta_3}6\leq \frac{\pi}6$. Finally $$\sum_{i=1}^3\frac{x_i}{1+x_i^2}\leq \frac 32\sin \frac{\pi}3=\frac{3\sqrt 3}4,$$ with equality if and only if $(x_1,x_2,x_3)=\left(\frac 1{\sqrt 3},\frac 1{\sqrt 3},\frac 1{\sqrt 3}\right)$. The second derivative of the auxiliary function $$f(u)\ :=\ {\sqrt{u}\over 1+u}\qquad (u\geq0)$$ computes to $$f''(u)={3(u-1)^2 -4\over 4u^{3/2}(1+u)^3} \ <\ 0\qquad(0\leq u\leq 1)\ ;$$ whence $f$ is concave for $0\leq u\leq 1$. Putting $u_i:=x_i^2$ we therefore have $${1\over3}\sum_{i=1}^3{x_i\over 1+x_i^2}\leq\sum_{i=1}^3{1\over3}f(u_i)\ \leq\ f\Bigl({\sum_{i=1}^3 u_i\over3}\Bigr)=f\Bigl({1\over3}\Bigr)={\sqrt{3}\over4}\ ,$$ as claimed.	{"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": 0.9840844869613647, "perplexity": 429.94854908576883}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00428-ip-10-147-4-33.ec2.internal.warc.gz"}
http://mathhelpforum.com/statistics/3004-combinatorics.html	1. ## combinatorics A class has 10 boys and 12 girls. In how many ways can a committee of four be selected if the committe can have at most two girls? Thanks, 2. Originally Posted by gogo08 A class has 10 boys and 12 girls. In how many ways can a committee of four be selected if the committe can have at most two girls? Thanks, Possibilities 0 Girls 1 Girl 2 Girls In each case respectively we have, $\displaystyle _{10}C_4 \cdot _{12}C_0=210$ $\displaystyle _{10}C_3 \cdot _{12}C_1=1440$ $\displaystyle _{10}C_2 \cdot _{12}C_2=2970$ In total, 4260	{"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": 0.8673934936523438, "perplexity": 447.1528671614919}, "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-2018-26/segments/1529267864848.47/warc/CC-MAIN-20180623000334-20180623020334-00081.warc.gz"}
http://mathoverflow.net/users/943?tab=recent	# Dmitri 16,543 Reputation 7743 views ## Registered User Name Dmitri Member for 3 years Seen 14 hours ago Website Location London Age 36 May14 awarded ● Enlightened May13 awarded ● Nice Answer Apr7 awarded ● Popular Question Mar30 comment How many polynomial Morse functions on the sphere?If you consider the case $n=2$, I believe the precise answer to your question should be known for all $d$. V.I. Arnol'd was interested in such type of questions, originally in the case when you replace $S^1$ by $\mathbb R^1$ and homogeneous polynomials by inhomogeneous. Probably one should chase the references to Arnold's article : mathnet.ru/php/… You might also want to have a look on the article of Barannikov "On the space of real polynomials without multiple critical values" Mar29 awarded ● Nice Answer Mar23 comment finite dimensional real division algebrasThere is a readable proof in the book of Shafarevich "basic algebraic geometry" of the fact that these algebras have dimension $2^n$. The proof indeed uses Bezout's theorem. Mar23 comment cohomology of a normal crossing divisorThis is not true, consider for example a divisor $D$ on a surface that is a wheel of $\mathbb P^1$'s, i.e, each $\mathbb P^1$ intersects two neighbouring $\mathbb P^1$'s. Then $\pi_1(D)=\mathbb Z$, so $H^1(D)=\mathbb Z$. Mar21 awarded ● Nice Question Mar19 awarded ● Enlightened Mar19 accepted Solid angles of a tetrahedron Mar19 awarded ● Nice Answer Mar18 revised Solid angles of a tetrahedronadded 29 characters in body Mar18 answered Solid angles of a tetrahedron Mar16 awarded ● Popular Question Mar10 revised Bolza curve admits no anticonformal fixedpointfree involutionadded 117 characters in body Mar10 answered Bolza curve admits no anticonformal fixedpointfree involution Mar1 comment Darboux SurfaceNoam, sure I want lines to vary too, so this becomes YQ's question in dimension one less. Mar1 comment Darboux SurfaceThis a very nice question! I wonder if a similar statement holds in $\mathbb P^2$ - if you take five lines, $10$ intersection points of them and consider quartics that contain these $10$ points, is it true that double conic is not in the Zariski closure of the space of such quartics? Feb28 comment What can one say about (differentiable) topological structure of CY3s?Dear Kim, by Bogomolov-Beuaville theorem every CY manifold has a finite cover that is a product of Tori, hyperkahler manifolds and manifolds $M^n$ such that $H^k(M^n,O)=0$ for $k\ne 0,n$. So for some people "proper" CY manifolds are only those that satisfy the last condition: $H^k(M^n,O)=0$ for $k\ne 0,n$. Such manifolds also have the property that the holonomy group of CY metrics on them coincide with $SU(n)$ (and not smaller than this). Such manifolds do have finite fundamental groups. Maybe for Oguis-Sakurai a Kahler manifold is $CY$ iff it has a holomorphic volume form... Feb28 revised What can one say about (differentiable) topological structure of CY3s?the answer is corrected and expanded Feb28 answered What can one say about (differentiable) topological structure of CY3s? Feb23 accepted Betti numbers of Proper nonprojective varieties Feb23 comment Betti numbers of Proper nonprojective varietiesDonu, thanks for the reference, I'll have a look (and will try to see if indeed I have an alternative proof :) ). LMN, you are welcome :) Feb23 revised Betti numbers of Proper nonprojective varietiesadded 32 characters in body Feb23 answered Betti numbers of Proper nonprojective varieties Feb14 comment Properties of quotient varietyConsider the following example: $(x,y)\to (x^2,y)$. Then the preimage of the curve $x=y^2$ under this map is $x=\pm y$. It is singular at $(0,0)$. It seems to me that you need to make the question a bit more specific... Feb6 revised Birational Automorphisms and infinite divisibilityadded 4 characters in body Feb6 revised Birational Automorphisms and infinite divisibilityadded 102 characters in body Feb6 comment Birational Automorphisms and infinite divisibilityYves, thanks I was a bit sloppy :) . But for \mathbb Q this is ture :) Feb5 revised Birational Automorphisms and infinite divisibilityadded 165 characters in body; added 10 characters in body Feb5 comment Birational Automorphisms and infinite divisibilityDaniel, the claim is that any homomorphism from $\mathbb Z[1/2]$ to $GL(n,\mathbb Z)$ sends $\mathbb Z[1/2]$ to $1$, since $1$ in $GL(n,\mathbb Z)$ is the only infinitely divisible element. Feb5 revised Birational Automorphisms and infinite divisibilitydeleted 2 characters in body Feb5 revised Birational Automorphisms and infinite divisibilityadded 202 characters in body; added 36 characters in body Feb5 answered Birational Automorphisms and infinite divisibility Feb4 accepted singular divisors in a complete linear system Feb4 comment Are rational varieties simply connected?Thank you Vesselin. The property of been rationally connected is a birational invariant, I guess? Feb4 comment Are rational varieties simply connected?Sandor, thank you :), I completely agree with you, I added missing words. In fact I was meaning "rational complex projective varieties". I don't know what is the definition of rationally connected projective varieties in the case they are singular. For example, if you consider a cone over a genus $g>0$ curve, every to points can be connected by a two $\mathbb P^1$'s (through the center of the cone), but I don't think this variety should be called rationally connected... Feb4 revised Are rational varieties simply connected?deleted 24 characters in body Feb4 revised Are rational varieties simply connected?added 42 characters in body Feb4 comment Are rational varieties simply connected?Dear Laurent I decided to check the reference and it looks to me that the proof of the fact is not really there. It is proven in two ways that projective spaces over algebraically closed fields are simply connected SGA 1. XI. Prop. 1.1, ( arxiv.org/pdf/math/0206203v2.pdf) and then comes corollary 1.2 without an actual proof. It is just said there that the proof is the same as for projective space :). Could you indicate how to make this an actual proof? I am asking this because I want to see how to make a proof over C without Hironaka's resolution of singularities. Feb3 answered another diameter-perimeter-area inequality Feb3 comment Why are the holomorphic line bundle sections finite dimensional?I really like this reasoning with Montel theorem :) Feb3 answered Why are the holomorphic line bundle sections finite dimensional? Feb2 accepted a diameter-perimeter-area inequality for convex figures Feb2 revised singular divisors in a complete linear systemadded 132 characters in body Feb2 comment singular divisors in a complete linear systemJames, thank you! Of course this is what I meant :) Feb2 revised singular divisors in a complete linear systemadded 3 characters in body Feb2 answered singular divisors in a complete linear system Feb2 comment a diameter-perimeter-area inequality for convex figuresConnor, that is of course correct. In fact, I was thinking of exactly this example but for some reason (I guess to make the answer as short :) ) as possible put the vertices of the rombus in $\pm 1, \pm varepsilon$) instead of what I had in mind. Feb2 comment a diameter-perimeter-area inequality for convex figuresI also called rectangle what should be called a rombus :)	{"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": 0.8834314942359924, "perplexity": 1181.4388108208689}, "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/1368699755211/warc/CC-MAIN-20130516102235-00070-ip-10-60-113-184.ec2.internal.warc.gz"}
http://www.ck12.org/measurement/Conversion-Using-Unit-Analysis/lesson/Solve-Problems-Involving-Rates-and-Unit-Analysis/	<meta http-equiv="refresh" content="1; url=/nojavascript/"> # Conversion Using Unit Analysis ## Solve problems by converting units using unit analysis. 0% Progress Practice Conversion Using Unit Analysis Progress 0% Solve Problems Involving Rates and Unit Analysis “I would LOVE to climb Mount Everest!” Josh exclaimed at breakfast one morning. “Really?” his Dad said smiling. “Well son, you had better start saving now.” Josh looked up from his oatmeal with a puzzled look on his face. “What makes you say that?” Josh asked. “What makes me say that is that the going rate for one climb on Everest is about $60,000. That’s what makes me say that,” his Dad explained taking a sip of his coffee. “Really? Wow! I had no idea,” Josh said. “Well, I guess I’ll just have to make a lot of money!” Josh said leaving the table. He kept thinking about what his Dad had said all the way to school. Sixty thousand dollars was a lot of money to climb a mountain, but what really amazed Josh was thinking about the numbers of people who had climbed the mountain more than once. When he got to class, he looked up in his book that Apa Sherpa a man from Nepal had successfully climbed Everest 19 times. Now he was often a guide who was paid, but still, Josh couldn’t help thinking about how much money Apa Sherpa would have spent if he had paid to climb Everest nineteen times at the rate his father spoke about. How much would it have cost? We can use units, ratios and proportions to solve this problem. By thinking of one trip as a unit, we can look at the proportion and solve for the correct amount of money. ### Guidance A rate refers to speed or a rate can refer to the amount of money someone makes per hour. When we talk about a unit rate, we look at comparing a rate to 1, or how much it would take for 1 of something. It could be one apple, one mile, one gallon. We are comparing a quantity to one. A key word when working with unit rate is the word “per”. We can use ratios and proportions to solve problems involving rates and unit rates. Jeff makes$150.00 an hour as a consultant. What is his rate per minute? To figure this out, we have to think about the unit rate that Jeff is paid as a consultant. You will see that we have “an hour” written into the problem. This is the unit rate. Also notice that the would "per" is used in the problem. Let’s write the unit rate as a ratio compared to 1. Next, we need to think about what the problem is asking for. It is asking for his rate per minute. The given information is in hours, so we need to write a ratio that compares hours to minutes. \begin{align}\frac{1 \ hour}{60 \ minutes}\end{align} Now we can write an expression combining the two ratios. \begin{align}\frac{\ 150}{1 \ hour} \cdot \frac{1 \ hour}{60 \ minutes}\end{align} That’s a great question. We don’t compare hours to hours because we aren’t comparing hours. We are comparing money to hours and we need to figure out the rate of money per minute. You always have to think about what is being compared when working with proportions. Next, we can solve. Notice that because 1 hour is diagonal from 1 hour, we can cross cancel the hours. That leaves us with a ratio that compares money to minutes. \begin{align}\frac{\ 150}{60 \ minutes}\end{align} This also helped us to convert hours to minutes making it easier to figure out the answer to the problem. Now we can divide to figure out our answer. Jeff makes $2.50 per minute. What is unit analysis? Unit analysis is when we look at how to measure individual units in different measurement amounts and it is used to convert units of measurement. When we use unit analysis, we convert different measurement units by comparing the units using ratios and proportions. Unit analysis is very helpful when checking results. Take a look at this dilemma. Juanita worked for 18 hours. She made$116.00 at the end of her shift. Juanita was sure that her manager had made a mistake and that she should have made more money. Juanita makes 9.00 per hour. Did Juanita make the correct amount of money or was there a mistake? To work on this problem, we can use unit analysis. Let’s start by writing a ratio to compare how much Juanita made for the hours worked. \begin{align}\frac{18 \ hours}{\ 116.00}\end{align} Next, we can use her hourly rate to work with. She makes9.00 per hour. \begin{align}\frac{\ 9}{1 \ hour}\end{align} Solution: .39 #### Example B Fifteen gallons of gasoline costs $45.00. How much is it per gallon? Solution:$3.00 #### Example C Two tickets to a ballgame costs $111.50. What is the cost for one ticket? Solution:$55.75 Now let's go back to the dilemma from the beginning of the Concept. Now let’s look at solving this problem. We know that it costs 60,000 for 1 trip up Mount Everest. \begin{align}\frac{\ 60,000}{1} &= \frac{x}{19}\\ x &= \ 1,140,000\end{align} We can also use unit analysis to solve this problem.60,000 dollars \begin{align}\left( \frac{19}{x \ dollars}\right)\end{align} \begin{align}60,000 \times 19 = \ 1,140,000\end{align} is the cost of the nineteen trips. This is our solution. ### Guided Practice Here is one for you to try on your own. Solve and then check using unit analysis. Jesse has a car that holds 14 gallons of gasoline. During the first week of the month, gasoline cost $2.75 per gallon. During the second week of the month, gasoline cost$2.50 per gallon. How much was the total cost for the 28 gallons of gasoline? Solution Let’s start by writing a variable expression to work on this problem. We know that the number of gallons of gasoline does not change. That can be our variable. \begin{align}x = \end{align} number of gallons of gasoline The other parts of the expression include the different prices for the gasoline. \begin{align}2.75x+2.50x\end{align} This expression will help us to determine how much money Jesse spent on 28 gallons of gasoline. Each full tank is 14 gallons. We can substitute 14 for our variable \begin{align}x\end{align} . \begin{align}2.75(14)&+ 2.50(14)\\ \ 38.50 &+ \ 35.00\end{align} The total amount of money spent was 73.50. We can check our work by using unit analysis. \begin{align}\frac{2.75}{1 \ gallon} &= \frac{x}{14 \ gallons} = \ 38.50\\ \frac{2.50}{1 \ gallon} & = \frac{x}{14 \ gallons} = \ 35.00\end{align} The sum of the money spent was73.50. ### Explore More Directions: Use what you have learned to solve each problem. 1. Peter runs at a rate of 10 kilometers per hour. How many kilometers will he cover in 8 hours? 2. A cheetah can run at a speed of 60 miles per hour. What is his distance after 6 hours? 3. What is the distance formula? 4. If a car travels at a rate of 65 miles per hour for 30 minutes, how far will it travel? 5. A train travels at a rate of 50 miles per hour. If it needs to travel 320 miles, how many minutes will it take? 6. A car travels 65 mph for 12 hours. How many miles will it travel? 7. A bus traveled 300 miles at an average speed of 50 miles per hour. How long did this trip take the bus? 8. A car traveled at an average speed of 40 miles per hour through a construction zone. If the car traveled 20 miles at this rate, how many hours did it take to travel the 20 miles? 9. What is velocity? 10. What is the formula for velocity? 11. What is the velocity of an object that travels 500 miles in 2.5 hours? 12. If an object has a velocity of 125 miles per hour, how long will it take to travel 4,375 miles? 13. If an object has a velocity of 7 kilometers per minute, how far will it travel in 2 hours? 14. If an object has a velocity of 4 meters per second, how many kilometers will it travel in 2 days? 15. The formula for density is \begin{align}D = \frac{m}{v}\end{align} where \begin{align}D\end{align} represents the density of an object, \begin{align}m\end{align} represents the mass of the object, and \begin{align}v\end{align} represents the volume of the object. What is the density of a brick that weighs 9 pounds and has a volume of 36 cu. in.? ### Vocabulary Language: English Rate Rate A rate is a special kind of ratio that compares two quantities. Unit Analysis Unit Analysis Unit analysis is a method of converting units of measurement by using ratios and proportions. Unit Rate Unit Rate A unit rate is a ratio that compares a quantity to one. The word “per” is a key word with unit rates.	{"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": 0.8437894582748413, "perplexity": 1026.4338910137592}, "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/1438042981525.10/warc/CC-MAIN-20150728002301-00327-ip-10-236-191-2.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/miniboone-results-at-6-1-sigma-potential-evidence-for-sterile-neutrinos.948730/	FeaturedI MiniBooNE results at 6.1 sigma: Potential evidence for sterile neutrinos Tags: 1. Jun 2, 2018 2. Jun 2, 2018 Orodruin Staff Emeritus I assume that by "IceCube" you mean "MiniBooNE". You have to take these results with a (big) grain of salt. The best-fit they present is already ruled out by other experiments. Even more so if you consider experiments that measure different channels. In addition, the oscillation fit is still a pretty bad one. Although they have a $\chi^2/$dof of 35.2/28, many of the bins are not in the region where they have a signal and I imagine that if you would focus on that region the $\chi^2/$dof would be rather nasty. They also cover their bases on this in the last sentence of the abstract: "Although the data are fit with a standard oscillation model, other models may provide better fits to the data." Edit: Let me also point out that Sabine has misinterpreted some of the paper in her blog post. In particular with respect to what other experiments are shown in the "money plot". Those shown are other experiments trying to measure the exact same oscillation channel. There is no way to reconcile the best-fit with OPERA even assuming sterile neutrino oscillations. Something is going on, but it is probably not sterile neutrinos. 3. Jun 2, 2018 Staff Emeritus Also, MiniBooNE was against sterile neutrinos before they were for them. 4. Jun 2, 2018 Staff: Mentor Based on figure 4, they rule out $\Delta m^2 < 0.01 eV^2$ with more than 3 sigma. This means at least one neutrino mass eigenstate has to be at least 100 meV, right? 140 meV if we go by the 95% CL. And this is for the optimal case of $\theta = 45^\circ$. This could come in conflict with cosmological measurements in the not so distant future - in addition to the existing conflict with the other measurements of the same parameter. Edit: Oops, wrong sign Last edited: Jun 3, 2018 5. Jun 3, 2018 kimmm Is this interpretation correct? LSND experimet observed and excess number of electron neutrinos,so by considering a sterIle neutrino we can explain that the muon neutrinos oscillate 5o sterile neutrinos and then theses sterile neutrinos oscillate to electron neutrinos in the short baseline experiment? 6. Jun 3, 2018 Orodruin Staff Emeritus The best fit is ruled out by so many things (including the OPERA results that they show in the figure!) that I think cosmology would be one of the weaker ... It is completely incompatible with essentially everything else we know about neutrinos. Also, a state with maximal mixing is not very sterile ... I also believe the best fit would actually already be ruled out by Planck. A "sterile" neutrino with that kind of interactions would easily thermalise and send $\Delta N_{\rm eff}$ to at least 1. And then we have not even started to talk about atmospheric and reactor neutrino experiments ... Many would put it like this as a kind of a mental picture. However, it is a quantum process and you are never measuring the neutrino state in between production and detection so you can not say it was a sterile neutrino at some point. A more accurate way of putting it is that it would change the interference pattern among the neutrino mass eigenstates. 7. Jun 3, 2018 kimmm It is a very naive and basic question,but I just get confused, that all the neutrinos are like that, that they do not exist at some points, or just for sterile neutrinos? if for others also then how we can say we have a muon neutrino beams or electron neutrinos observed by their interactions? 8. Jun 3, 2018 Orodruin Staff Emeritus I did not say they did not exist. I said you do not measure their flavour state. 9. Jun 3, 2018 kimmm I understand,and the reason that long baseline experiments could not see any result about sterile neutrinos, is because in their baseline and their energy ranges the feature of the sterile neutrinos could not affect the probabilities of neutrino oscillation? 10. Jun 3, 2018 Orodruin Staff Emeritus At longer baselines the oscillations of eV range sterile neutrinos would be completely averaged out. That does not necessarily mean you would have no information, but it would be more difficult to extract it. 11. Jun 3, 2018 Ygggdrasil 12. Jun 3, 2018 Staff: Mentor @Ygggdrasil - that is primarily why I posted this in the first place. I was confused - the original title of the post reflected a popular science name 'Ice Cube', @mfb corrected me. Thanks for that. What also confused me is why I saw nothing on PF about it except Bee H's blog, and the kinds of articles you cited. 13. Jun 4, 2018 Orodruin Staff Emeritus A clear example of why one should not always trust the popular press. The NBC article does not mention that the interpretation as sterile neutrinos is in direct conflict with other experiments until pretty far down and you certainly do not get that impression from the first part of the article. The Quantamagazine article is a bit more forthcoming with this information. You are welcome. Also I am not sure IceCube is more "popular science" than MiniBooNE. Both are actual names of particle physics experiments. One just happens to be more known than the other. 14. Jun 4, 2018 Ygggdrasil Exactly why after seeing and skimming through the NBC article (mainly to see if the claims had any basis in a published article), I came here to see if you all had anything to say about the paper. 15. Jun 4, 2018 Chronos John Baez offers some interesting comments here: https://johncarlosbaez.wordpress.com/2018/06/02/miniboone/. The fact is MiniBooNE detected an excess of electron neutrinos over their 15 years of data collection. It isn't a huge number, but, given the confidence we have in the expected number of detections is very high, it is enough to establish a very high confidence [4.8 sigma] that something very curious is going on that is not explained by the standard model. That does not mean it is proof of sterile neutrinos, but, that is probably as good a guess as anyone has offered thus far. It should be interesting to see if MiniBooNE can nudge up the signal they currently have to up over the magic 5 sigma level with more data. 16. Jun 4, 2018 Staff: Mentor 17. Jun 4, 2018 Orodruin Staff Emeritus First of all, it is a 4.8 sigma difference with the no oscillation scenario for the best fit. This best fit happens to already be strongly excluded by other experiments, which puts the interpretation as sterile neutrinos in serious doubt. I would certainly agree that it is curious and worthy of scrutiny, but I would be very surprised if the signal is due to sterile neutrinos (see the comments I made on Backreaction). Edit: Also, it is as good a guess as someone has offered thus far and has therefore also been much more scrutinised. It has been scrutinised to the point that it seems unlikely to be able to explain the MiniBooNE low-energy excess as oscillations just don't give a good fit of the data. Last edited: Jun 4, 2018 18. Jun 4, 2018 Chronos My interest in sterile neutrinos dates back to this paper: https://arxiv.org/abs/1402.2301, Detection of An Unidentified Emission Line in the Stacked X-ray spectrum of Galaxy Clusters, which sparked an enduring controversy over the plausibility of sterile neutrinos as a component of the dark matter budget. Particle physics is not really my thing, although I have tried to keep a finger on the sterile neutrino pulse since then. 19. Jun 5, 2018 Staff: Mentor 20. Jun 5, 2018 Orodruin Staff Emeritus	{"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": 0.8268582224845886, "perplexity": 908.7114367268634}, "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-2019-13/segments/1552912201672.12/warc/CC-MAIN-20190318191656-20190318213656-00158.warc.gz"}
https://people.maths.bris.ac.uk/~matyd/GroupNames/320/Dic5.Q16.html	Copied to clipboard ## G = Dic5.Q16order 320 = 26·5 ### 2nd non-split extension by Dic5 of Q16 acting via Q16/C4=C22 Series: Derived Chief Lower central Upper central Derived series C1 — C2×C20 — Dic5.Q16 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C4×Dic5 — Dic5⋊C8 — Dic5.Q16 Lower central C5 — C2×C10 — C2×C20 — Dic5.Q16 Upper central C1 — C22 — C2×C4 — C2×Q8 Generators and relations for Dic5.Q16 G = < a,b,c,d \| a10=c8=1, b2=a5, d2=a5c4, bab-1=a-1, cac-1=a3, ad=da, cbc-1=dbd-1=a5b, dcd-1=a5bc-1 > Subgroups: 266 in 64 conjugacy classes, 26 normal (14 characteristic) C1, C2, C2, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C10, C42, C4⋊C4, C2×C8, C2×Q8, C2×Q8, Dic5, Dic5, C20, C2×C10, C4⋊C8, C4⋊Q8, C5⋊C8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C4.6Q16, C4×Dic5, C10.D4, C2×C5⋊C8, C2×Dic10, Q8×C10, Dic5⋊C8, Dic5⋊Q8, Dic5.Q16 Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, SD16, Q16, F5, C4.D4, Q8⋊C4, C2×F5, C4.6Q16, C22⋊F5, Q8⋊F5, C23.F5, Dic5.Q16 Character table of Dic5.Q16 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 5 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C 20A 20B 20C 20D 20E 20F size 1 1 1 1 4 8 10 10 10 10 40 4 20 20 20 20 20 20 20 20 4 4 4 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 -1 1 1 1 1 -1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -1 1 1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 -1 1 1 1 1 -1 1 -1 1 1 1 1 -1 -1 -1 1 1 1 -1 1 1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 -i -i -i i i i i -i 1 1 1 -1 1 1 -1 -1 -1 linear of order 4 ρ6 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 -i i i -i -i i i -i 1 1 1 1 1 1 1 1 1 linear of order 4 ρ7 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 i -i -i i i -i -i i 1 1 1 1 1 1 1 1 1 linear of order 4 ρ8 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 i i i -i -i -i -i i 1 1 1 -1 1 1 -1 -1 -1 linear of order 4 ρ9 2 2 2 2 -2 0 2 2 -2 -2 0 2 0 0 0 0 0 0 0 0 2 2 2 0 -2 -2 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 -2 0 -2 -2 2 2 0 2 0 0 0 0 0 0 0 0 2 2 2 0 -2 -2 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 -2 0 0 2 -2 0 0 0 2 0 √2 -√2 -√2 √2 0 0 0 -2 -2 2 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ12 2 -2 2 -2 0 0 0 0 -2 2 0 2 -√2 0 0 0 0 -√2 √2 √2 2 -2 -2 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ13 2 -2 2 -2 0 0 0 0 -2 2 0 2 √2 0 0 0 0 √2 -√2 -√2 2 -2 -2 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ14 2 2 -2 -2 0 0 2 -2 0 0 0 2 0 -√2 √2 √2 -√2 0 0 0 -2 -2 2 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ15 2 2 -2 -2 0 0 -2 2 0 0 0 2 0 -√-2 √-2 -√-2 √-2 0 0 0 -2 -2 2 0 0 0 0 0 0 complex lifted from SD16 ρ16 2 -2 2 -2 0 0 0 0 2 -2 0 2 -√-2 0 0 0 0 √-2 -√-2 √-2 2 -2 -2 0 0 0 0 0 0 complex lifted from SD16 ρ17 2 2 -2 -2 0 0 -2 2 0 0 0 2 0 √-2 -√-2 √-2 -√-2 0 0 0 -2 -2 2 0 0 0 0 0 0 complex lifted from SD16 ρ18 2 -2 2 -2 0 0 0 0 2 -2 0 2 √-2 0 0 0 0 -√-2 √-2 -√-2 2 -2 -2 0 0 0 0 0 0 complex lifted from SD16 ρ19 4 4 4 4 4 4 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ20 4 4 4 4 4 -4 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 -1 1 -1 -1 1 1 1 orthogonal lifted from C2×F5 ρ21 4 -4 -4 4 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 -4 4 -4 0 0 0 0 0 0 orthogonal lifted from C4.D4 ρ22 4 4 4 4 -4 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 -1 √5 1 1 √5 -√5 -√5 orthogonal lifted from C22⋊F5 ρ23 4 4 4 4 -4 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 -1 -√5 1 1 -√5 √5 √5 orthogonal lifted from C22⋊F5 ρ24 4 -4 -4 4 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 -1 1 2ζ52+2ζ5+1 -√5 √5 2ζ54+2ζ53+1 2ζ53+2ζ5+1 2ζ54+2ζ52+1 complex lifted from C23.F5 ρ25 4 -4 -4 4 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 -1 1 2ζ53+2ζ5+1 √5 -√5 2ζ54+2ζ52+1 2ζ54+2ζ53+1 2ζ52+2ζ5+1 complex lifted from C23.F5 ρ26 4 -4 -4 4 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 -1 1 2ζ54+2ζ52+1 √5 -√5 2ζ53+2ζ5+1 2ζ52+2ζ5+1 2ζ54+2ζ53+1 complex lifted from C23.F5 ρ27 4 -4 -4 4 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 -1 1 2ζ54+2ζ53+1 -√5 √5 2ζ52+2ζ5+1 2ζ54+2ζ52+1 2ζ53+2ζ5+1 complex lifted from C23.F5 ρ28 8 8 -8 -8 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 symplectic lifted from Q8⋊F5, Schur index 2 ρ29 8 -8 8 -8 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 symplectic lifted from Q8⋊F5, Schur index 2 Smallest permutation representation of Dic5.Q16 Regular action on 320 points Generators in S320 (1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30)(31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50)(51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110)(111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130)(131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150)(151 152 153 154 155 156 157 158 159 160)(161 162 163 164 165 166 167 168 169 170)(171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190)(191 192 193 194 195 196 197 198 199 200)(201 202 203 204 205 206 207 208 209 210)(211 212 213 214 215 216 217 218 219 220)(221 222 223 224 225 226 227 228 229 230)(231 232 233 234 235 236 237 238 239 240)(241 242 243 244 245 246 247 248 249 250)(251 252 253 254 255 256 257 258 259 260)(261 262 263 264 265 266 267 268 269 270)(271 272 273 274 275 276 277 278 279 280)(281 282 283 284 285 286 287 288 289 290)(291 292 293 294 295 296 297 298 299 300)(301 302 303 304 305 306 307 308 309 310)(311 312 313 314 315 316 317 318 319 320) (1 71 6 76)(2 80 7 75)(3 79 8 74)(4 78 9 73)(5 77 10 72)(11 254 16 259)(12 253 17 258)(13 252 18 257)(14 251 19 256)(15 260 20 255)(21 82 26 87)(22 81 27 86)(23 90 28 85)(24 89 29 84)(25 88 30 83)(31 54 36 59)(32 53 37 58)(33 52 38 57)(34 51 39 56)(35 60 40 55)(41 66 46 61)(42 65 47 70)(43 64 48 69)(44 63 49 68)(45 62 50 67)(91 167 96 162)(92 166 97 161)(93 165 98 170)(94 164 99 169)(95 163 100 168)(101 158 106 153)(102 157 107 152)(103 156 108 151)(104 155 109 160)(105 154 110 159)(111 139 116 134)(112 138 117 133)(113 137 118 132)(114 136 119 131)(115 135 120 140)(121 141 126 146)(122 150 127 145)(123 149 128 144)(124 148 129 143)(125 147 130 142)(171 247 176 242)(172 246 177 241)(173 245 178 250)(174 244 179 249)(175 243 180 248)(181 237 186 232)(182 236 187 231)(183 235 188 240)(184 234 189 239)(185 233 190 238)(191 219 196 214)(192 218 197 213)(193 217 198 212)(194 216 199 211)(195 215 200 220)(201 223 206 228)(202 222 207 227)(203 221 208 226)(204 230 209 225)(205 229 210 224)(261 312 266 317)(262 311 267 316)(263 320 268 315)(264 319 269 314)(265 318 270 313)(271 294 276 299)(272 293 277 298)(273 292 278 297)(274 291 279 296)(275 300 280 295)(281 308 286 303)(282 307 287 302)(283 306 288 301)(284 305 289 310)(285 304 290 309) (1 213 51 177 28 225 62 190)(2 220 60 180 29 222 61 183)(3 217 59 173 30 229 70 186)(4 214 58 176 21 226 69 189)(5 211 57 179 22 223 68 182)(6 218 56 172 23 230 67 185)(7 215 55 175 24 227 66 188)(8 212 54 178 25 224 65 181)(9 219 53 171 26 221 64 184)(10 216 52 174 27 228 63 187)(11 170 279 113 311 157 281 124)(12 167 278 116 312 154 290 127)(13 164 277 119 313 151 289 130)(14 161 276 112 314 158 288 123)(15 168 275 115 315 155 287 126)(16 165 274 118 316 152 286 129)(17 162 273 111 317 159 285 122)(18 169 272 114 318 156 284 125)(19 166 271 117 319 153 283 128)(20 163 280 120 320 160 282 121)(31 250 83 205 42 237 79 193)(32 247 82 208 43 234 78 196)(33 244 81 201 44 231 77 199)(34 241 90 204 45 238 76 192)(35 248 89 207 46 235 75 195)(36 245 88 210 47 232 74 198)(37 242 87 203 48 239 73 191)(38 249 86 206 49 236 72 194)(39 246 85 209 50 233 71 197)(40 243 84 202 41 240 80 200)(91 297 139 266 105 309 150 253)(92 294 138 269 106 306 149 256)(93 291 137 262 107 303 148 259)(94 298 136 265 108 310 147 252)(95 295 135 268 109 307 146 255)(96 292 134 261 110 304 145 258)(97 299 133 264 101 301 144 251)(98 296 132 267 102 308 143 254)(99 293 131 270 103 305 142 257)(100 300 140 263 104 302 141 260) (1 94 23 103)(2 95 24 104)(3 96 25 105)(4 97 26 106)(5 98 27 107)(6 99 28 108)(7 100 29 109)(8 91 30 110)(9 92 21 101)(10 93 22 102)(11 223 316 216)(12 224 317 217)(13 225 318 218)(14 226 319 219)(15 227 320 220)(16 228 311 211)(17 229 312 212)(18 230 313 213)(19 221 314 214)(20 222 315 215)(31 127 47 111)(32 128 48 112)(33 129 49 113)(34 130 50 114)(35 121 41 115)(36 122 42 116)(37 123 43 117)(38 124 44 118)(39 125 45 119)(40 126 46 120)(51 147 67 131)(52 148 68 132)(53 149 69 133)(54 150 70 134)(55 141 61 135)(56 142 62 136)(57 143 63 137)(58 144 64 138)(59 145 65 139)(60 146 66 140)(71 169 90 151)(72 170 81 152)(73 161 82 153)(74 162 83 154)(75 163 84 155)(76 164 85 156)(77 165 86 157)(78 166 87 158)(79 167 88 159)(80 168 89 160)(171 288 189 271)(172 289 190 272)(173 290 181 273)(174 281 182 274)(175 282 183 275)(176 283 184 276)(177 284 185 277)(178 285 186 278)(179 286 187 279)(180 287 188 280)(191 251 208 269)(192 252 209 270)(193 253 210 261)(194 254 201 262)(195 255 202 263)(196 256 203 264)(197 257 204 265)(198 258 205 266)(199 259 206 267)(200 260 207 268)(231 291 249 308)(232 292 250 309)(233 293 241 310)(234 294 242 301)(235 295 243 302)(236 296 244 303)(237 297 245 304)(238 298 246 305)(239 299 247 306)(240 300 248 307) G:=sub<Sym(320)\| (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170)(171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190)(191,192,193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208,209,210)(211,212,213,214,215,216,217,218,219,220)(221,222,223,224,225,226,227,228,229,230)(231,232,233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248,249,250)(251,252,253,254,255,256,257,258,259,260)(261,262,263,264,265,266,267,268,269,270)(271,272,273,274,275,276,277,278,279,280)(281,282,283,284,285,286,287,288,289,290)(291,292,293,294,295,296,297,298,299,300)(301,302,303,304,305,306,307,308,309,310)(311,312,313,314,315,316,317,318,319,320), (1,71,6,76)(2,80,7,75)(3,79,8,74)(4,78,9,73)(5,77,10,72)(11,254,16,259)(12,253,17,258)(13,252,18,257)(14,251,19,256)(15,260,20,255)(21,82,26,87)(22,81,27,86)(23,90,28,85)(24,89,29,84)(25,88,30,83)(31,54,36,59)(32,53,37,58)(33,52,38,57)(34,51,39,56)(35,60,40,55)(41,66,46,61)(42,65,47,70)(43,64,48,69)(44,63,49,68)(45,62,50,67)(91,167,96,162)(92,166,97,161)(93,165,98,170)(94,164,99,169)(95,163,100,168)(101,158,106,153)(102,157,107,152)(103,156,108,151)(104,155,109,160)(105,154,110,159)(111,139,116,134)(112,138,117,133)(113,137,118,132)(114,136,119,131)(115,135,120,140)(121,141,126,146)(122,150,127,145)(123,149,128,144)(124,148,129,143)(125,147,130,142)(171,247,176,242)(172,246,177,241)(173,245,178,250)(174,244,179,249)(175,243,180,248)(181,237,186,232)(182,236,187,231)(183,235,188,240)(184,234,189,239)(185,233,190,238)(191,219,196,214)(192,218,197,213)(193,217,198,212)(194,216,199,211)(195,215,200,220)(201,223,206,228)(202,222,207,227)(203,221,208,226)(204,230,209,225)(205,229,210,224)(261,312,266,317)(262,311,267,316)(263,320,268,315)(264,319,269,314)(265,318,270,313)(271,294,276,299)(272,293,277,298)(273,292,278,297)(274,291,279,296)(275,300,280,295)(281,308,286,303)(282,307,287,302)(283,306,288,301)(284,305,289,310)(285,304,290,309), (1,213,51,177,28,225,62,190)(2,220,60,180,29,222,61,183)(3,217,59,173,30,229,70,186)(4,214,58,176,21,226,69,189)(5,211,57,179,22,223,68,182)(6,218,56,172,23,230,67,185)(7,215,55,175,24,227,66,188)(8,212,54,178,25,224,65,181)(9,219,53,171,26,221,64,184)(10,216,52,174,27,228,63,187)(11,170,279,113,311,157,281,124)(12,167,278,116,312,154,290,127)(13,164,277,119,313,151,289,130)(14,161,276,112,314,158,288,123)(15,168,275,115,315,155,287,126)(16,165,274,118,316,152,286,129)(17,162,273,111,317,159,285,122)(18,169,272,114,318,156,284,125)(19,166,271,117,319,153,283,128)(20,163,280,120,320,160,282,121)(31,250,83,205,42,237,79,193)(32,247,82,208,43,234,78,196)(33,244,81,201,44,231,77,199)(34,241,90,204,45,238,76,192)(35,248,89,207,46,235,75,195)(36,245,88,210,47,232,74,198)(37,242,87,203,48,239,73,191)(38,249,86,206,49,236,72,194)(39,246,85,209,50,233,71,197)(40,243,84,202,41,240,80,200)(91,297,139,266,105,309,150,253)(92,294,138,269,106,306,149,256)(93,291,137,262,107,303,148,259)(94,298,136,265,108,310,147,252)(95,295,135,268,109,307,146,255)(96,292,134,261,110,304,145,258)(97,299,133,264,101,301,144,251)(98,296,132,267,102,308,143,254)(99,293,131,270,103,305,142,257)(100,300,140,263,104,302,141,260), (1,94,23,103)(2,95,24,104)(3,96,25,105)(4,97,26,106)(5,98,27,107)(6,99,28,108)(7,100,29,109)(8,91,30,110)(9,92,21,101)(10,93,22,102)(11,223,316,216)(12,224,317,217)(13,225,318,218)(14,226,319,219)(15,227,320,220)(16,228,311,211)(17,229,312,212)(18,230,313,213)(19,221,314,214)(20,222,315,215)(31,127,47,111)(32,128,48,112)(33,129,49,113)(34,130,50,114)(35,121,41,115)(36,122,42,116)(37,123,43,117)(38,124,44,118)(39,125,45,119)(40,126,46,120)(51,147,67,131)(52,148,68,132)(53,149,69,133)(54,150,70,134)(55,141,61,135)(56,142,62,136)(57,143,63,137)(58,144,64,138)(59,145,65,139)(60,146,66,140)(71,169,90,151)(72,170,81,152)(73,161,82,153)(74,162,83,154)(75,163,84,155)(76,164,85,156)(77,165,86,157)(78,166,87,158)(79,167,88,159)(80,168,89,160)(171,288,189,271)(172,289,190,272)(173,290,181,273)(174,281,182,274)(175,282,183,275)(176,283,184,276)(177,284,185,277)(178,285,186,278)(179,286,187,279)(180,287,188,280)(191,251,208,269)(192,252,209,270)(193,253,210,261)(194,254,201,262)(195,255,202,263)(196,256,203,264)(197,257,204,265)(198,258,205,266)(199,259,206,267)(200,260,207,268)(231,291,249,308)(232,292,250,309)(233,293,241,310)(234,294,242,301)(235,295,243,302)(236,296,244,303)(237,297,245,304)(238,298,246,305)(239,299,247,306)(240,300,248,307)>; G:=Group( (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170)(171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190)(191,192,193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208,209,210)(211,212,213,214,215,216,217,218,219,220)(221,222,223,224,225,226,227,228,229,230)(231,232,233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248,249,250)(251,252,253,254,255,256,257,258,259,260)(261,262,263,264,265,266,267,268,269,270)(271,272,273,274,275,276,277,278,279,280)(281,282,283,284,285,286,287,288,289,290)(291,292,293,294,295,296,297,298,299,300)(301,302,303,304,305,306,307,308,309,310)(311,312,313,314,315,316,317,318,319,320), (1,71,6,76)(2,80,7,75)(3,79,8,74)(4,78,9,73)(5,77,10,72)(11,254,16,259)(12,253,17,258)(13,252,18,257)(14,251,19,256)(15,260,20,255)(21,82,26,87)(22,81,27,86)(23,90,28,85)(24,89,29,84)(25,88,30,83)(31,54,36,59)(32,53,37,58)(33,52,38,57)(34,51,39,56)(35,60,40,55)(41,66,46,61)(42,65,47,70)(43,64,48,69)(44,63,49,68)(45,62,50,67)(91,167,96,162)(92,166,97,161)(93,165,98,170)(94,164,99,169)(95,163,100,168)(101,158,106,153)(102,157,107,152)(103,156,108,151)(104,155,109,160)(105,154,110,159)(111,139,116,134)(112,138,117,133)(113,137,118,132)(114,136,119,131)(115,135,120,140)(121,141,126,146)(122,150,127,145)(123,149,128,144)(124,148,129,143)(125,147,130,142)(171,247,176,242)(172,246,177,241)(173,245,178,250)(174,244,179,249)(175,243,180,248)(181,237,186,232)(182,236,187,231)(183,235,188,240)(184,234,189,239)(185,233,190,238)(191,219,196,214)(192,218,197,213)(193,217,198,212)(194,216,199,211)(195,215,200,220)(201,223,206,228)(202,222,207,227)(203,221,208,226)(204,230,209,225)(205,229,210,224)(261,312,266,317)(262,311,267,316)(263,320,268,315)(264,319,269,314)(265,318,270,313)(271,294,276,299)(272,293,277,298)(273,292,278,297)(274,291,279,296)(275,300,280,295)(281,308,286,303)(282,307,287,302)(283,306,288,301)(284,305,289,310)(285,304,290,309), (1,213,51,177,28,225,62,190)(2,220,60,180,29,222,61,183)(3,217,59,173,30,229,70,186)(4,214,58,176,21,226,69,189)(5,211,57,179,22,223,68,182)(6,218,56,172,23,230,67,185)(7,215,55,175,24,227,66,188)(8,212,54,178,25,224,65,181)(9,219,53,171,26,221,64,184)(10,216,52,174,27,228,63,187)(11,170,279,113,311,157,281,124)(12,167,278,116,312,154,290,127)(13,164,277,119,313,151,289,130)(14,161,276,112,314,158,288,123)(15,168,275,115,315,155,287,126)(16,165,274,118,316,152,286,129)(17,162,273,111,317,159,285,122)(18,169,272,114,318,156,284,125)(19,166,271,117,319,153,283,128)(20,163,280,120,320,160,282,121)(31,250,83,205,42,237,79,193)(32,247,82,208,43,234,78,196)(33,244,81,201,44,231,77,199)(34,241,90,204,45,238,76,192)(35,248,89,207,46,235,75,195)(36,245,88,210,47,232,74,198)(37,242,87,203,48,239,73,191)(38,249,86,206,49,236,72,194)(39,246,85,209,50,233,71,197)(40,243,84,202,41,240,80,200)(91,297,139,266,105,309,150,253)(92,294,138,269,106,306,149,256)(93,291,137,262,107,303,148,259)(94,298,136,265,108,310,147,252)(95,295,135,268,109,307,146,255)(96,292,134,261,110,304,145,258)(97,299,133,264,101,301,144,251)(98,296,132,267,102,308,143,254)(99,293,131,270,103,305,142,257)(100,300,140,263,104,302,141,260), (1,94,23,103)(2,95,24,104)(3,96,25,105)(4,97,26,106)(5,98,27,107)(6,99,28,108)(7,100,29,109)(8,91,30,110)(9,92,21,101)(10,93,22,102)(11,223,316,216)(12,224,317,217)(13,225,318,218)(14,226,319,219)(15,227,320,220)(16,228,311,211)(17,229,312,212)(18,230,313,213)(19,221,314,214)(20,222,315,215)(31,127,47,111)(32,128,48,112)(33,129,49,113)(34,130,50,114)(35,121,41,115)(36,122,42,116)(37,123,43,117)(38,124,44,118)(39,125,45,119)(40,126,46,120)(51,147,67,131)(52,148,68,132)(53,149,69,133)(54,150,70,134)(55,141,61,135)(56,142,62,136)(57,143,63,137)(58,144,64,138)(59,145,65,139)(60,146,66,140)(71,169,90,151)(72,170,81,152)(73,161,82,153)(74,162,83,154)(75,163,84,155)(76,164,85,156)(77,165,86,157)(78,166,87,158)(79,167,88,159)(80,168,89,160)(171,288,189,271)(172,289,190,272)(173,290,181,273)(174,281,182,274)(175,282,183,275)(176,283,184,276)(177,284,185,277)(178,285,186,278)(179,286,187,279)(180,287,188,280)(191,251,208,269)(192,252,209,270)(193,253,210,261)(194,254,201,262)(195,255,202,263)(196,256,203,264)(197,257,204,265)(198,258,205,266)(199,259,206,267)(200,260,207,268)(231,291,249,308)(232,292,250,309)(233,293,241,310)(234,294,242,301)(235,295,243,302)(236,296,244,303)(237,297,245,304)(238,298,246,305)(239,299,247,306)(240,300,248,307) ); G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30),(31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50),(51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110),(111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130),(131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150),(151,152,153,154,155,156,157,158,159,160),(161,162,163,164,165,166,167,168,169,170),(171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190),(191,192,193,194,195,196,197,198,199,200),(201,202,203,204,205,206,207,208,209,210),(211,212,213,214,215,216,217,218,219,220),(221,222,223,224,225,226,227,228,229,230),(231,232,233,234,235,236,237,238,239,240),(241,242,243,244,245,246,247,248,249,250),(251,252,253,254,255,256,257,258,259,260),(261,262,263,264,265,266,267,268,269,270),(271,272,273,274,275,276,277,278,279,280),(281,282,283,284,285,286,287,288,289,290),(291,292,293,294,295,296,297,298,299,300),(301,302,303,304,305,306,307,308,309,310),(311,312,313,314,315,316,317,318,319,320)], [(1,71,6,76),(2,80,7,75),(3,79,8,74),(4,78,9,73),(5,77,10,72),(11,254,16,259),(12,253,17,258),(13,252,18,257),(14,251,19,256),(15,260,20,255),(21,82,26,87),(22,81,27,86),(23,90,28,85),(24,89,29,84),(25,88,30,83),(31,54,36,59),(32,53,37,58),(33,52,38,57),(34,51,39,56),(35,60,40,55),(41,66,46,61),(42,65,47,70),(43,64,48,69),(44,63,49,68),(45,62,50,67),(91,167,96,162),(92,166,97,161),(93,165,98,170),(94,164,99,169),(95,163,100,168),(101,158,106,153),(102,157,107,152),(103,156,108,151),(104,155,109,160),(105,154,110,159),(111,139,116,134),(112,138,117,133),(113,137,118,132),(114,136,119,131),(115,135,120,140),(121,141,126,146),(122,150,127,145),(123,149,128,144),(124,148,129,143),(125,147,130,142),(171,247,176,242),(172,246,177,241),(173,245,178,250),(174,244,179,249),(175,243,180,248),(181,237,186,232),(182,236,187,231),(183,235,188,240),(184,234,189,239),(185,233,190,238),(191,219,196,214),(192,218,197,213),(193,217,198,212),(194,216,199,211),(195,215,200,220),(201,223,206,228),(202,222,207,227),(203,221,208,226),(204,230,209,225),(205,229,210,224),(261,312,266,317),(262,311,267,316),(263,320,268,315),(264,319,269,314),(265,318,270,313),(271,294,276,299),(272,293,277,298),(273,292,278,297),(274,291,279,296),(275,300,280,295),(281,308,286,303),(282,307,287,302),(283,306,288,301),(284,305,289,310),(285,304,290,309)], [(1,213,51,177,28,225,62,190),(2,220,60,180,29,222,61,183),(3,217,59,173,30,229,70,186),(4,214,58,176,21,226,69,189),(5,211,57,179,22,223,68,182),(6,218,56,172,23,230,67,185),(7,215,55,175,24,227,66,188),(8,212,54,178,25,224,65,181),(9,219,53,171,26,221,64,184),(10,216,52,174,27,228,63,187),(11,170,279,113,311,157,281,124),(12,167,278,116,312,154,290,127),(13,164,277,119,313,151,289,130),(14,161,276,112,314,158,288,123),(15,168,275,115,315,155,287,126),(16,165,274,118,316,152,286,129),(17,162,273,111,317,159,285,122),(18,169,272,114,318,156,284,125),(19,166,271,117,319,153,283,128),(20,163,280,120,320,160,282,121),(31,250,83,205,42,237,79,193),(32,247,82,208,43,234,78,196),(33,244,81,201,44,231,77,199),(34,241,90,204,45,238,76,192),(35,248,89,207,46,235,75,195),(36,245,88,210,47,232,74,198),(37,242,87,203,48,239,73,191),(38,249,86,206,49,236,72,194),(39,246,85,209,50,233,71,197),(40,243,84,202,41,240,80,200),(91,297,139,266,105,309,150,253),(92,294,138,269,106,306,149,256),(93,291,137,262,107,303,148,259),(94,298,136,265,108,310,147,252),(95,295,135,268,109,307,146,255),(96,292,134,261,110,304,145,258),(97,299,133,264,101,301,144,251),(98,296,132,267,102,308,143,254),(99,293,131,270,103,305,142,257),(100,300,140,263,104,302,141,260)], [(1,94,23,103),(2,95,24,104),(3,96,25,105),(4,97,26,106),(5,98,27,107),(6,99,28,108),(7,100,29,109),(8,91,30,110),(9,92,21,101),(10,93,22,102),(11,223,316,216),(12,224,317,217),(13,225,318,218),(14,226,319,219),(15,227,320,220),(16,228,311,211),(17,229,312,212),(18,230,313,213),(19,221,314,214),(20,222,315,215),(31,127,47,111),(32,128,48,112),(33,129,49,113),(34,130,50,114),(35,121,41,115),(36,122,42,116),(37,123,43,117),(38,124,44,118),(39,125,45,119),(40,126,46,120),(51,147,67,131),(52,148,68,132),(53,149,69,133),(54,150,70,134),(55,141,61,135),(56,142,62,136),(57,143,63,137),(58,144,64,138),(59,145,65,139),(60,146,66,140),(71,169,90,151),(72,170,81,152),(73,161,82,153),(74,162,83,154),(75,163,84,155),(76,164,85,156),(77,165,86,157),(78,166,87,158),(79,167,88,159),(80,168,89,160),(171,288,189,271),(172,289,190,272),(173,290,181,273),(174,281,182,274),(175,282,183,275),(176,283,184,276),(177,284,185,277),(178,285,186,278),(179,286,187,279),(180,287,188,280),(191,251,208,269),(192,252,209,270),(193,253,210,261),(194,254,201,262),(195,255,202,263),(196,256,203,264),(197,257,204,265),(198,258,205,266),(199,259,206,267),(200,260,207,268),(231,291,249,308),(232,292,250,309),(233,293,241,310),(234,294,242,301),(235,295,243,302),(236,296,244,303),(237,297,245,304),(238,298,246,305),(239,299,247,306),(240,300,248,307)]]) Matrix representation of Dic5.Q16 in GL6(𝔽41) 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 40 0 0 0 0 1 0 40 0 0 0 17 18 10 0 0 0 8 13 22 31 , 40 36 0 0 0 0 25 1 0 0 0 0 0 0 1 14 40 2 0 0 2 6 20 35 0 0 18 3 21 26 0 0 35 14 35 13 , 1 0 0 0 0 0 16 40 0 0 0 0 0 0 26 11 18 2 0 0 30 9 15 34 0 0 40 13 7 13 0 0 27 10 40 40 , 30 34 0 0 0 0 35 11 0 0 0 0 0 0 19 5 28 0 0 0 11 12 40 0 0 0 13 34 9 0 0 0 2 7 32 1 G:=sub<GL(6,GF(41))\| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,1,17,8,0,0,40,0,18,13,0,0,0,40,10,22,0,0,0,0,0,31],[40,25,0,0,0,0,36,1,0,0,0,0,0,0,1,2,18,35,0,0,14,6,3,14,0,0,40,20,21,35,0,0,2,35,26,13],[1,16,0,0,0,0,0,40,0,0,0,0,0,0,26,30,40,27,0,0,11,9,13,10,0,0,18,15,7,40,0,0,2,34,13,40],[30,35,0,0,0,0,34,11,0,0,0,0,0,0,19,11,13,2,0,0,5,12,34,7,0,0,28,40,9,32,0,0,0,0,0,1] >; Dic5.Q16 in GAP, Magma, Sage, TeX {\rm Dic}_5.Q_{16} % in TeX G:=Group("Dic5.Q16"); // GroupNames label G:=SmallGroup(320,269); // by ID G=gap.SmallGroup(320,269); # by ID G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,219,100,1571,570,136,6278,3156]); // Polycyclic G:=Group<a,b,c,d\|a^10=c^8=1,b^2=a^5,d^2=a^5c^4,bab^-1=a^-1,cac^-1=a^3,ad=da,cbc^-1=dbd^-1=a^5b,dcd^-1=a^5bc^-1>; // generators/relations Export ׿ × 𝔽	{"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": 0.8985499739646912, "perplexity": 498.0187421039998}, "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, 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https://www.physicsforums.com/threads/help-lesson-3-physics.64662/	# Help! lesson 3 physics! 1. Feb 23, 2005 ### mark9159 greetings...i need some help please and someone to check if i did the problems correctly. 1.) How much kinetic energy will a 5kg ball have after rolling down a 1m high incline? If Potential Energy=mgh, then P=(5kg)(9.81m/s^2)(1m) so P=49.05 J. If this is true, then Kinetic energy = 49.05 J = 1/2mv^2. 49.05=1/2(5)v^2. Then i divided 49.05 by 2.5 (49.05/2.5) which gave me v^2=19.62. I then took the square root of 19.62 which is approximately 4.43 J. 2.) Why are no collisions perfectly inelastic? My Answer: Almost all collisions involve some rebounding. 3.) A lever is used to raise a 300N rock a distance of 0.6 meters. You must use a force of 75N to accomplish this task. What work is done? If work can equal mass times gravity times height, then work equals 300N times 0.6 meters, which equals 180. I then did 180 divided by 75N (the amount of force needed), coming up with the solution: 2.45 J thank you very much for checking my work! mark 2. Feb 23, 2005 ### scholar 1) After 1m, all potential energy will have been converted to kinetic energy. So your calculation should read: Ep=Ek mgh=Ek Ek=5x9.81x1 Ek=49.05 J You went the whole way and calculated the velocity, which is measured in m.s^-1. It looks as though you got confused between kinetic energy and velocity. 3) This is quite clearly incorrect. You certainly used more than 2.45 J of energy to lift a 30kg rock from the ground! Read up on levers and you should be able to get this one. 3. Feb 23, 2005 ### Edgardo EDIT! Not correct, read scholar's post. Your solution would be correct if the question was: Calculate the velocity. That doesn't seem correct. First of all I can see that's not correct because of the units: You divided 180Nm=180J by 75N which gives you 2.4m and NOT 2.45J ALWAYS check the units, it gives you a first hint if your calculation is correct. Last edited: Feb 23, 2005 4. Feb 23, 2005 ### mark9159 so if Work= Force Times Distance, then Work= (75N)(0.6m) which equals 45 Joules...where exactly does the 300N rock come ino the equation? the problem im facing is that 45 J is not one of the choices i have to choose from oh and i really thank you guys for the tip/help..im going to look up more information about levers now. 5. Feb 23, 2005 ### Edgardo Mark, I just send you a private message. I think it's just 300N times 0.6m. The 75N is the force you need if you use a lever. Sorry, my fault. 6. Feb 23, 2005 ### mark9159 oh, ok. thank you edgardo. One more question. What power is required to accelerate a 500kg car from zero to 18 m/s in one minute? First i found the kinetic energy Ek=1/2mv^2 Ek=1/2(500kg)(18m/s)^2 Ek=250(324)=81000 J Power= Joules per second Power= 81000 J / 60 Power= 1350 W It takes 1350 W to accelerate a 500kg car from zero to 18 m/s in one minute. 7. Feb 23, 2005 ### Edgardo Hello Mark, that seems to be correct. Regards Edgardo Similar Discussions: Help! lesson 3 physics!	{"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": 0.9181420803070068, "perplexity": 2425.5647346910378}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549425766.58/warc/CC-MAIN-20170726042247-20170726062247-00389.warc.gz"}
http://physics.aps.org/synopsis-for/10.1103/PhysRevC.79.021302	# Synopsis: Finding the missing sign #### Sign of the overlap of Hartree-Fock-Bogoliubov wave functions L. M. Robledo Published February 20, 2009 In a many-body physics problem, the first approximation to account for interactions is to assume that each particle moves in an effective mean field created by all of the other particles. The mutual interactions of neutrons and protons in nuclei, for example, generate a mean field in which the nucleons move in single-particle orbits. Hartree-Fock is a simple mean-field theory in which the single-particle orbits are either fully occupied or not at all. A generalization of this method that can describe quasiparticles in the presence of pairing is Hartree-Fock-Bogoliubov (HFB) theory (called Bogoliubov–de Gennes theory in condensed matter physics). In going beyond mean-field theory, different HFB solutions are mixed and the overlap of HFB wave functions must be computed. However, standard formulas leave the sign of the overlap of the wave functions undetermined. This is not a problem for systems with discrete symmetries, but it is present when the HFB wave functions are triaxial, which occurs in the low-lying rotational bands of ${}^{24}\text{Mg}$, or break time-reversal symmetry. In a Rapid Communication appearing in Physical Review C, Luis Robledo of Universidad Autonoma de Madrid uses the technique of fermion coherent states to determine the sign of wave-function overlap. The overlap is given in terms of a quantity similar to a determinant called the Pfaffian of a skew-symmetric matrix. The goal is to simplify the implementation of challenging theory projects, such as those that calculate the projection of triaxial angular momentum. – John Millener and Ben Gibson	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8521015048027039, "perplexity": 665.0566977521494}, "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/1408500824209.82/warc/CC-MAIN-20140820021344-00177-ip-10-180-136-8.ec2.internal.warc.gz"}
http://physics.stackexchange.com/tags/terminology/new	# Tag Info 1 A constraint condition can reduce the DOF of the system if it can be used to express a coordinate in terms of the others. This can always be done in case of holonomic constraints which are basically just algebraic functions of the coordinates and time. This means that you just have to manipulate the constraint equation in such a way that one of the ... 0 2nd law of thermodynamics has many almost equivalent formulations. The traditional ones always assume closed system, isolation is not needed - heat and work transfer are assumed to be allowed. One formulation: When thermodynamic system goes from equilibrium state 1 to equilibrium state 2, the entropies of these states obey the relation $$S(2) - S(1) \geq ... 0 ANS: It is basically the same. All of them are anomalies for chiral fermion in even dimensional spacetime, like 1+1D, 3+1D, etc. Or, in odd dimensional space. ABJ anomaly is named after the discovers: Adler-Bell-Jackiw. - Chiral anomaly is named after and implies that the chiral current is not conserved. Chiral, means the left and the ... 0 In the context of Relativty, the adjective "coincident" characterizes all that which belongs together ("at once") to only one event without any temporal or spatial or light-like separation; i.e. indications (of any participant at this one event) or occurences (as far as they are understood to be contained within this one event). In contrast, the adjective ... 1 The "adjustable constant" in that statement is the total energy E, and they mean it's "adjustable" in that the behavior of the system is completely independent of E - this is known in physics as a symmetry, in that they system doesn't change if it has a different total amount of energy. In this case, the way to "adjust" the amount of energy would be ... 0 Positron is as "elementary" as the electron, in the current theory. Period. I took the word "elementary" in quote marks because if you hit a charged particle, it gets "broken apart" into the following pieces: the same charge and lots of neutral photons. It looks like the "target" is not that "elementary", but has "internal degrees of freedom", and sometimes ... 1 Positron IS an elementary particle, the anti-particle to the electron as you already know. But we do not get a "free positron" as a "free electron". They are usually generated through pair-production and get annihilated fast, or through radioactive decay (beta-decay) in weak interactions or in particle accelerators, and are present in cosmic rays too. A ... 1 The equation you are quoting gives the power of a lens in terms of its geometry and refractive index. Simply rearranging the terms (dividing by n) gives you an expression for \frac{1}{f} which is known as the power of the lens and is expressed in diopters. For the usual situation of a lens in air, we can put n=1 which leaves you with an even simpler ... 1 Dunno what book you're quoting, but you should realize that the index of refraction of air is n = 1+ \epsilon (where I'm using the mathematics standard of \epsilon being a tiny number). Thus the power in air is 1/F 2 As I recall, covariant refers to how an object transforms when you boost to another inertial frame. An example would be the relativistic 4-momentum P^{\mu}. Invariant refers to quantities which are unchanged under boosts to different frames. For example the product P^{\mu}P_{\mu}=m has the same numerical value in any frame. Sometimes a relativistic ... 1 "Coincident" is defined in the Google online dictionary as (1) "occurring together in space OR time" (emphasis mine), and (2) "in agreement or harmony". "Simultaneous" is defined in the same dictionary as "occurring, operating, or done at the same time". (This begs the question: "Whose time?") Unfortunately, this dictionary lists "coincident" as a synonym ... 1 Regarding your assertions: Events \varepsilon_{AJ} and \varepsilon_{BK} were simultaneous in the inertial frame of participants A, B, M. This is a perfectly reasonable statement and it is the sort of language used in everyday physics. Participant M was the middle between J and K, in the inertial frame of participants A, B, M. ... 1 I would like to bring the ladder paradox here to explain simultaneity of events.A ladder (an inertial frame) is moving horizontally with a relatively high constant speed with respect to a garage (another inertial frame). The garage has an open door where the ladder can not actually enter if the ladder was at rest in the garage's frame but that is not ... 0 It's the net resultant that a force would react with on application of a certain change in momentum in the body, on which the force is applied. 1 A site is just a place or location with given coordinates e.g. (x_0,y_0,z_0) 0 Kinetics are focused on the rate and mechanism of chemical processes, so you are definitely right to say that you gain a lot of insight about mechanism from kinetics. Many kinetic theory make extensive use of statistical thermodynamics methods, and that's why you perceive a resemblance. However, keep in mind that in kinetics the system is not in ... 4 The Weights and Measures Act (the origin of the Imperial Units) does not speak of temperature. It was intended to create a uniform system for trade. You don't sell temperature, in the way you sell a pint of milk or a yard of cloth. And frankly, when it was first conceived (before Magna Carta, which already stated: "There shall be but one Measure ... 2 According to the wiki page on Imperial and US customary units Fahrenheit is part of both the Imperial and US customary system. I can't think of any reason it wouldn't be included in the Imperial system. Note that in the wiki page on Imperial units it is mentioned that the weight's and measures act (which defined the Imperial system) explicitly used the ... 0 The effective potential is the potential of interaction you measure between two (or more) emergent physical objects when you forget (or "trace over" in the jargon) certain degrees of freedom of a more detailed model. If you take two pinned charges in vacuum for instance, they will interact with a "bare" Coulomb interaction in \sim 1/r. If you put these ... 1 As wiki says "The effective potential (also known as effective potential energy) is a mathematical expression combining multiple (perhaps opposing) effects into a single potential." Basically the concept of the effective potential simplifies the equations of motion and simplifies their analysis. 0 More on the perytons being caused by microwave ovens being opened while still operating. The microwave oven's magnetron is still generating the microwaves when the door is open before the timer has stopped the microwave. Article: http://phenomena.nationalgeographic.com/2015/04/10/rogue-microwave-ovens-are-the-culprits-behind-mysterious-radio-signals Study: ... 1 Four component formalism is the "right" formalism, but it has negative energy eigenstates corresponding to the antiparticles. Most chemists and solid state physics are not interested in the antiparticles, and such negative energy solution causes trouble for conventional variational methods, where you might end up falling to negative infinity energy. It is ... 0 Rolling friction result from for example small changes in the surface or in the wheel material (the rubber in a tire). The surface is not perfectly flat and rigid so there will be some small forces trying to stop the rotating motion: On the contrary, the static friction is not trying to stop the rotation of the wheel. Static and rolling friction are ... 0 Cicle is the smallest repatable segment between two points, where: 1. These point lies on one line and the line is parallel to direction of wave propagation. 2. These two points have the always same sign of slope. Thanks to Floris in help of derivation of the definition. 1 The below seems to be a candidate for the first use of the term 'Majorana fermion'. (I'm not sure if it satisfies your other criteria.) Salam, Abdus, and J. Strathdee. Super-symmetry and non-Abelian gauges. International Centre for Theoretical Physics, Trieste (Italy), 1974. 2 Using elementary graph theory identities one can show that the number of loops in a connected diagram is related to the number of external lines and the number of vertices of type i each of which has n_i lines attached to it, is related by$$ \sum \left(\frac{n_i}{2}-1\right) V_i -\tfrac{1}{2}E +1= L $$So you can see that for a fixed process (fixed ... 5 A paper came out this week pointing to them having a banal (if amusing) origin: they are from two 27 year old microwave ovens. When people get impatient and open the door before the timer runs down, a short burst from the ovens' magnetron is released, which appears as a peryton if the telescope is pointed in the right direction. Figure 7. shows the perytons ... 0 Let's check that parity is violated by the weak interaction lagrangian:$$\mathcal{L}(x) = \bar{\psi}(x) \gamma^\mu \frac{(1-\gamma^5)}{2} \psi(x) W_\mu(x)$$Saying that parity is violated means that the transformed lagrangian \mathcal{L}'(x) is not equal to the old lagrangian resulting from new coordinates \mathcal{L}(x') where x'^0 = x^0 and ... 2 The order of a quantity in general refers to the exponent of the quantity in an expression, ie$$x^3y^2$$would be 3rd order in x and 2nd order in y. According to the Feynman rules, each vertex in a Feynman diagrams contributes a factor of the coupling constant, so the order of each coupling constant is simply the number of vertices of that interaction. ... 0 Pressure-sensitive paints? Is that what you need? http://en.wikipedia.org/wiki/Pressure-sensitive_paint 0 I guess you have some typos in your question (or your lecture notes), since the two lines you give just differ by the index names (as already commented) . Starting from$$ R^n_{ikl;m} +R^n_{imk;l} +R^n_{ilm;k} =0 $$you can rewrite that with the symmetry relations R^n_{ikl}=- R^n_{ilk}=-R^i_{nkl} to$$ R^n_{ikl;m} - R^n_{ikm;l} +R^n_{ilm;k}=0. $$Now let ... 2 To contract a tensor is to set two of the indices equal and sum over them, so given a tensor A^i_j the contraction is A=A^i_i=A^1_1+A^2_2+A^3_3+A^4_4 The Bianchi identities you list have five indices. To contract them, you would set some pair equal and sum over them. Your second version is the same as the first, it just has the indices renamed. ... 2 Static comes from the same root as stasis, meaning stop, immovable, To create static electricity, you have to rub two different materials. At the moment you rub them, the electrons already moved Note the word "create", creation is not static, and yes there are transient fields and currents during creation of a static field. The static describes the ... 1 UV stands for Ultraviolet and it is referring to a special kind of divergences in quantum field theory. In NLO loop diagrams, we often encounter divergences (infinite integrals) when we investigate what happens at k \to \infty, where k is the internal momentum of a virtual particle in Feynman diagrams. These are precisely the divergences we call "UV ... 0 UV = Ultra-Violet = High energy = Small length-scale. 0 This is just a wild guess, but could it be the position vector of the element? 1 I think the magnon is a special case of the spin wave. Whereas spinon refers to the general quasiparticle that carries all spin of an electron, magnon refers to the limiting case of spin wave quantized in such a manner that it becomes part of an anti-magnetic cloud of quasiparticles. However, this may not be the complete story! Some usage of the terms: ... 0 In this context, it is a change of variables. The variable in the original Lagrangian is q, and Goldstein is asking you to use another variable s, which is related to the original q via the "transformation":$$s = \exp(\gamma t) \ q$$and later on, make sense of it (with the later questions). Point transformation in this context refers merely to this ... 1 i searched for the exact same problem recently after a debate with one of my colleagues. In my opinion, you already gave the answer to your question yourself. A source dipole is the flow field resulting from a sink and a source brought together. In a sink, all streamlines point radially inward to the singularity at the origin, in a source, all point ... 0 \hat{I}_D(k)={{g^2}\over4}\int_0^1d\alpha\int_0^\infty d\sigma\int\sigma\mathrm{e}^{-[q^2+\alpha(1-\alpha)k^2+m^2]\sigma}\vec{a}q. Just a wild guess. [Oh, you asked for two formulae. Sorry.] Either that, or it is \int_0^\infty\mathrm{e}^{-a\alpha}\sigma d\sigma=a^{-2}. 4 I got a translation of the article from the German Wikipedia. Here's an excerpt: Perytons are in radio astronomy short radio signals having a length of a few milliseconds, which probably terrestrial are origin. The Perytons are named after mythical creatures . In radio astronomy, terrestrial are noise is always a problem. A well-known noise signal ... 1 This paper describes them. http://www.ursi.org/proceedings/procGA11/ursi/GP2-41.pdf They were apparently given a new name because their origin was uncertain. 0 If the observer is not in free-fall, the metric-tensor g_{\mu,\nu}(s) at the observer's position, expressed in local coordinates around the observer, will not be \eta_{\mu,\nu}. Your first assumption about the path (\gamma) is wrong. I guess what you are aiming at is the notion of the space of coordinates around a point, which is indeed a flat space ... 2 When you say If something goes outside, then it will decrease inside! what you assume is exactly a conservation law. It may seem trivial, but it is not necessarily. Consider the population of a city, for example. At one point in time, you measure how many people are within the city borders; let's call this number N_0. Then, you observe all city ... 2 When we say something is conserved or that there is a conservation law for a given thing, we mean that the quantity of it does not change. You neither lose nor gain any of that thing. More specifically, conservation can come in two flavours. Something can be globally conserved. This means that the total amount of that something in the universe does not ... 1 It has a very simple yet important meaning.It simply means that the quantity that you are observing will always stay the same,even if that means that it gets transferred to another form or convert to another medium.You can not simply create more "stuff" of that quantity and you can not destroy it.It can not be created from nothing and it can not just be ... 3 Let M be your spacetime, a smooth manifold equipped with (pseudo) Riemannian metric (for example \mathbb{R}^{(1,3)} for special relativity). The set of reference frames is the frame bundle over M, usually denoted FM. Explicitly a frame at point p in M can be viewed as an ordered orthonormal basis (with respect to the the inner product defined ... 2 The kinetic term of the Lagrangian is proportional to$$g_{ij}v^iv^j$$where the vs are the generalised velocities. Writing them as the time derivative of the generalised coordinates, i.e. v^i\dot q^i, taking the square root, and multiplying by a small time lapse \epsilon you get$$\sqrt{g_{ij}\dot q^i\dot q^j}\epsilon, which is a first order ... 1 The name T-duality stands for Target-space duality, see e.g. this preprint. 1 It comes from S matrix theory, long before quarks were imagined, S,T and U characterize the type of exchange in the Feynman diagrams entering the S matrix calculation, and they are called Mandelstam variables. s channel-------------------------- t channel------------------------u channel duality meant that the sums could be done either in S ... 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