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http://mathhelpforum.com/advanced-statistics/50331-plz-help-required-expectation-print.html	# Plz Help is required in expectation • September 23rd 2008, 05:09 PM UMD Plz Help is required in expectation I shall be very grateful to anyone who can guide me regarding the solution of the following problems. Show that if Xn converges to X in rth mean ( r>=1), then E ( \|Xn^r\| )converges to E ( \| X ^r \|) Show that if Xn converges to X in rth mean with r=1, then E(Xn) Converges to E(X). Give an example to show that the converse is false • September 24th 2008, 02:13 PM Laurent For the first question: rewrite the conclusion and hypothesis in terms of $L^r$ norm, and it should become straightforward. --------------------- Don't read the following if you want to find it out by yourself --------------------- By the triangle inequality, $\|\\|X_n\\|_r-\\|X\\|_r\|\leq \\|X_n-X\\|_r\to_n 0$, so that $\\|X_n\\|_r\to_n\\|X\\|_r$, which directly implies $E[\|X_n\|^r]\to_n E[\|X\|^r]$. For the second one, I think you can prove the first part by yourself. And for the reverse, almost every example works: $X_n=\pm1$ with probability 1/2, and $X=0$ may be the easiest. • September 27th 2008, 11:50 AM UMD @ Laurant Thanks alot Brother. U r providing great help for which i am grateful • September 27th 2008, 12:17 PM UMD @ Laurant one last question Show that if Xn converges to X in probability, then Xn converges mutually in probability . • September 27th 2008, 02:03 PM Laurent Quote: Originally Posted by UMD Show that if Xn converges to X in probability, then Xn converges mutually in probability . I don't know what it means to "converge mutually in probability". It seems to be an unusual expression. Could you define that? • September 29th 2008, 10:57 AM Player1 I think "mutual convergence in probability" means $P(\|X_m - X_n\| > \varepsilon) \rightarrow_{m,n \rightarrow \infty} 0$ They use the triangle inequality to argue something related here. However, 1) I don't know what they mean by norm of a random variable, and 2) I don't know if or how you can use that here... • September 29th 2008, 01:20 PM Laurent Quote: Originally Posted by Player1 I think "mutual convergence in probability" means $P(\|X_m - X_n\| > \varepsilon) \rightarrow_{m,n \rightarrow \infty} 0$ Thank you, I was suspecting something like this as well but I was pretty unsure. So "mutually" must refer to a "Cauchy sequence"-like behaviour. Suppose $(X_n)_n$ converges to $X$ in probability. Let $\varepsilon>0$. For any $m,n$, $P(\|X_n-X_m\|>\varepsilon)=P(\|(X_n-X)-(X_m-X)\|>\varepsilon)$ $\leq P(\|X_n-X\|+\|X_m-X\|>\varepsilon)\leq P(\|X_n-X\|>\varepsilon/2\mbox{ or }\|X_m-X\|>\varepsilon/2)$ $\leq P(\|X_n-X\|>\varepsilon/2)+P(\|X_m-X\|>\varepsilon/2)$. And you can easily conclude from here. • October 4th 2008, 04:23 PM UMD @ Laurant Thanks for ur reply. I am sorry for not replying on time. Yes u were right that was the thing that i exactly needed. But i submitted my assignment without it. Anyways many many thanks for ur reply. i hav a few more problems to solve if u have time to think over it, I shall be grateful. I have to submit this assignment the day after tomorrow. Lastly how to use mathematical symbols in this forum ? 1) If Xn converges in quadratic mean to X and Xn is a guassian random variable with mean Un and variance ( the symbol of variance), and if variance = lim ( n approaches to infinity) variance is non zero, show that X is Guassian 2) the second question i cannot write bcz of lack of mathematical symbols • October 4th 2008, 04:46 PM Player1 Hi! $X_n$ converges in the second mean is telling you 1) The variance is finite 2) The mean is finite 3) $X_n$ converges in probability and in distribution to $X$ Then write the PDF of $X_n$. As you have convergence in distribution, and 1) and 2), then $\mu_n \rightarrow \mu$ and $\sigma_n^2 \rightarrow \sigma^2$ as $n \rightarrow \infty$. And that's gaussian too. Please somebody correct my post if I am wrong. Thanks. To write math symbols, type "latex math symbols" in google. Then, enclose your symbols between opensquarebracket math closesquarebracket and opensquarebracket /math closesquarebracket. You get me, just replace the brackets with real brackets ;) • October 4th 2008, 05:10 PM UMD @player1 (Sleepy)Thanks. U said take its pdf. can u elaborate this . • October 4th 2008, 05:57 PM Player1 Each of $X_n$ has probability density function (or PDF) $f_{X_n}(x) = \frac{1}{\sqrt{2 \pi \sigma_n ^2}} exp\{-\frac{(x-\mu_n)^2}{2 \sigma_n^2}\}$ and by 1), 2) and 3), the PDF of $X$ is $f_{X_n}(x) = \frac{1}{\sqrt{2 \pi \sigma ^2}} exp\{-\frac{(x-\mu)^2}{2 \sigma^2}\}$ which is gaussian too. • October 4th 2008, 06:02 PM UMD 1 Attachment(s) @laurant,Player1 Please see the attached word document for question. I shall be very grateful to anyone who can guide me regarding its solution • October 4th 2008, 06:05 PM UMD @player1 Plz if u can see the second one , the attached word file • October 5th 2008, 05:18 AM Laurent Quote: Originally Posted by Player1 Each of $X_n$ has probability density function (or PDF) $f_{X_n}(x) = \frac{1}{\sqrt{2 \pi \sigma_n ^2}} exp\{-\frac{(x-\mu_n)^2}{2 \sigma_n^2}\}$ and by 1), 2) and 3), the PDF of $X$ is $f_{X_n}(x) = \frac{1}{\sqrt{2 \pi \sigma ^2}} exp\{-\frac{(x-\mu)^2}{2 \sigma^2}\}$ which is gaussian too. Beware! It is common misbelief that the probability density function converges if there is convergence in distribution. What is for sure is that for instance the cumulative distribution function converges, as well as the characteristic function. Because $\\|X_n-X\\|_2\to_n 0$, you know (by a previous question) that $E[X_n^2]\to_n E[X^2]$. Because $\\| X_n-X\\|_1\leq \\|X_n-X\\|_2$ (as Cauchy-Schwarz inequality shows), and because of a previous question of yours, you know that the mean $\mu_n$ of $X_n$ converges to $\mu=E[X]$. And hence the same with variances using the previous paragraph: $\sigma_n^2\to_n \sigma^2={\rm Var}(X)$. As for the distribution of the limit, it results from considering the characteristic function: because of the convergence in distribution, for all $t\in\mathbb{R}$, $E[e^{it X_n}]\to_n E[e^{itX}]$. However, you know that $E[e^{itX_n}]=e^{it\mu_n-t^2\sigma_n^2/2}\to_n e^{it\mu-t^2\sigma^2/2}$ because $X_n$ is Gaussian and because of what we said about the convergence of the mean and variance. So the characteristic function of $X$ is $e^{i t\mu-t^2\sigma^2/2}$, which is the characteristic function of a Gaussian random variable of mean $\mu$ and variance $\sigma^2$. The fact that the characteristic function characterizes the distribution allows to conclude.	{"extraction_info": {"found_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": 45, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9817889928817749, "perplexity": 794.6367154620411}, "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-2016-07/segments/1454701158601.61/warc/CC-MAIN-20160205193918-00099-ip-10-236-182-209.ec2.internal.warc.gz"}
https://conan777.wordpress.com/2011/06/07/a-remark-on-a-mini-course-by-kleiner-in-sullivans-70th-birthday/	## A remark on a mini-course by Kleiner in Sullivan’s 70th birthday June 7, 2011 I spent the last week on Long Island for Dennis Sullivan’s birthday conference. The conference is hosted in the brand new Simons center where great food is served everyday in the cafe (I think life-wise it’s a wonderful choice for doing a post-doc). Anyways, aside from getting to know this super-cool person named Dennis, the talks there were interesting~ There are many things I found so exciting and can’t help to not say a few words about, however due to my laziness, I can only select one item to give a little stupid remark on: So Bruce Kleiner gave a 3-lecture mini-course on boundaries of Gromov hyperbolic spaces (see this related post on a piece of his pervious work in the subject) Cannon’s conjecture: Any Gromov hyperbolic group with $\partial_\infty G \approx \mathbb{S}^2$ acts discretely and cocompactly by isometries on $\mathbb{H}^3$. As we all know, in the theory of Gromov hyperbolic spaces, we have the basic theorem that says if a groups acts on a space discretely and cocompactly by isometries, then the group (equipped with any word metric on its Cayley graph) is quasi-isometric to the space it acts on. Since I borrowed professor Sullivan as an excuse for writing this post, let’s also state a partial converse of this theorem (which is more in the line of Cannon’s conjecture): Theorem: (Sullivan, Gromov, Cannon-Swenson) For $G$ finitely generated, if $G$ is quasi-isometric to $\mathbb{H}^n$ for some $n \geq 3$, then $G$ acts on $\mathbb{H}^n$ discretely cocompactly by isometries. This essentially says that due to the strong symmetries and hyperbolicity of $\mathbb{H}^n$, in this case quasi-isometry is enough to guarantee an action. (Such thing is of course not true in general, for example any finite group is quasi-isometric to any compact metric space, there’s no way such action exists.) In some sense being quasi-isometric is a much stronger condition once the spaces has large growth at infinity. In light of the above two theorems we know that Cannon’s conjecture is equivalent to saying that any hyperbolic group with boundary $\mathbb{S}^2$ is quasi-isometric to $\mathbb{H}^3$. At first glance this seems striking since knowing only the topology of the boundary and the fact that it’s hyperbolic, we need to conclude what the whole group looks like geometrically. However, the pervious post on one dimensional boundaries perhaps gives us some hint on the boundary can’t be anything we want. In fact it’s rather rigid due to the large symmetries of our hyperbolic group structure. Having Cannon’s conjecture as a Holy Grail, they developed tools that give raise to some very elegant and inspring proofs of the conjecture in various special cases. For example: Definition: A metric space $M$, is said to be Alfors $\alpha$-regular where $\alpha$ is its Hausdorff dimension, if there exists constant $C$ s.t. for any ball $B(p, R)$ with $R \leq \mbox{Diam}(M)$, we have: $C^{-1}R^\alpha \leq \mu(B(p,R)) \leq C R^\alpha$ This is saying it’s of Hausdorff dimension $\alpha$ in a very strong sense. (i.e. the Hausdorff $\alpha$ measure behaves exactly like the regular Eculidean measure everywhere and in all scales). For two disjoint continua $C_1, C_2$ in $M$, let $\Gamma(C_1, C_2)$ denote the set of rectifiable curves connecting $C_1$ to $C_2$. For any density function $\rho: M \rightarrow \mathbb{R}^+$, we define the $\rho$-distance between $C_1, C_2$ to be $\displaystyle \mbox{dist}_\rho(C_1, C_2) = \inf_{\gamma \in \Gamma(C_1, C_2)} \int_\gamma \rho$. Definition: The $\alpha$-modulus between $C_1, C_2$ is $\mbox{Mod}_\alpha(C_1, C_2) = \inf \{ \int_M \rho^\alpha \ \| \ \mbox{dist}_\rho(C_1, C_2) \geq 1 \}$, OK…I know this is a lot of seemingly random definitions to digest, let’s pause a little bit: Given two continua in our favorite $\mathbb{R}^n$, new we are of course Hausdorff dimension $n$, what’s the $n$-modulus between them? This is equivalent to asking for a density function for scaling the metric so that the total n-dimensional volume of $\mathbb{R}^n$ is as small as possible but yet the length of any curve connecting $C_1, \ C_2$ is larger than $1$. So intuitively we want to put large density between the sets whenever they are close together. Since we are integrating the $n$-th power for volume (suppose $n>1$, since our set is path connected it’s dimension is at least 1), we would want the density as ‘spread out’ as possible while keeping the arc-length property. Hence one observation is this modulus depends on the pair of closest points and the diameter of the sets. The relative distance between $C_1, C_2$ is $\displaystyle \Delta (C_1, C_2) = \frac{\inf \{ d(p_1, p_2) \ \| \ p_1 \in C_1, \ p_2 \in C_2 \} }{ \min \{ \mbox{Diam}(C_1), \mbox{Diam}(C_2) \} }$ We say $M$ is $\alpha$-Loewner if the $\alpha$ modulus between any two continua is controlled above and below by their relative distance, i.e. there exists increasing functions $\phi, \psi: [0, \infty) \rightarrow [0, \infty)$ s.t. for all $C_1, C_2$, $\phi(\Delta(C_1, C_2)) \leq \mbox{Mod}_\alpha(C_1, C_2) \leq \psi(\Delta(C_1, C_2))$ Those spaces are, in some sense, regular with respect to it’s metric and measure. Theorem: If $\partial_\infty G$ is Alfors 2-regular and 2-Loewner, homeomorphic to $\mathbb{S}^2$, then $G$ acts discrete cocompactly on $\mathbb{H}^3$ by isometries. Most of the material appeared in the talk can be found in their paper. There are many other talks I found very interesting, especially that of Kenneth Bromberg, Mario Bonk and Peter Jones. Unfortunately I had to miss Curt McMullen, Yair Minski and Shishikura… ### 2 Responses to “A remark on a mini-course by Kleiner in Sullivan’s 70th birthday” 1. tushar Says: hi conan, they’ve put up the videos of the lectures and talks at: http://www.math.sunysb.edu/Videos/dennisfest/ best, tushar 2. 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http://tex.stackexchange.com/users/9524/fluffy?tab=activity&sort=comments	fluffy Reputation 238 Top tag Next privilege 250 Rep. Sep 9 comment typing with latex It would be helpful if you provided the error. Also, you seem to be missing a number of \s and \$s, and you've misspelled documentclass and a4paper. Aug 20 comment Using a custom TrueType font in spans of text @LeoLiu I'm just using pdfTeX because of familiarity and not bothering to learn about the newer things. :) I'll look into them now, thanks. Aug 20 comment Using a custom TrueType font in spans of text @LeoLiu I'm not a TeX beginner - this just isn't something I've had to do before (and it seems like nobody ever does this). Does that linked-to article also apply to TeXLive on OSX? I'm not a fan of mucking about with packaged installation files. There's a reason I don't run Slackware anymore. :) Aug 20 comment Using a custom TrueType font in spans of text @karlhoeller Unfortunately, no, the stuff there didn't help. It's not actually selecting the font I provided - it's just complaining about a missing T5xxx.fd file, and that t.se page doesn't explain anything about generating them. Aug 20 comment Using a custom TrueType font in spans of text @karlhoeller That looks like it might be helpful, thanks. Why didn't that come up in my extensive searching? Jul 25 comment Separating plural possessive apostrophe from closing double-quote Thanks. For some reason the linked-to one didn't come up in my search - I figured someone would have asked it before but I had a hell of a time finding it. Jan 24 comment How can I explain the meaning of LaTeX to my grandma? Just because LaTeX was designed for technical documents doesn't mean that's all its good for. I laid out a comic book in LaTeX. (Because scripting makes things easier.) Nov 13 comment Change background colour for entire document Somewhat off topic, but the best way to change the background color of an entire document is to print it on colored paper. :) Feb 13 comment How can I introduce a non-technical person to LaTeX? +1 here as well. I used LyX to write my various academic papers back when I was still an academic and it certainly saved me a lot of grief. Dec 17 comment The effect of the anonymous letter Man, that kerning is TERRIBLE. Nov 28 comment How to replace chapter boilerplate with full-page image? Okay, good to know. I guess I should have just tried it straight away. Sorry to be such a bother. Nov 28 comment How to replace chapter boilerplate with full-page image? So, I finally tried switching over to this method (since it seems a bit cleaner), but I couldn't figure out how to use \AddToShipoutPicture or \includegraphics or whatever instead of \includepdf (since my images are in .png format, not .pdf). Nov 26 comment How to replace chapter boilerplate with full-page image? Oh, so it is, thanks. Sorry about that. I think I just got overwhelmed by the verbosity of the first two approaches that I must have overlooked the third. Nov 25 comment How to replace chapter boilerplate with full-page image? @egreg Yeah, the last \picturechapter macro is pretty decent. +1 for that. When I commented on this answer only the first example code was here. I definitely like \picturechapter a lot better than the other ones. Nov 23 comment How to replace chapter boilerplate with full-page image? That certainly seems to be the proper TeX-y way to do it: overwrought, overly complicated, and requires a Makefile to update things correctly. ;) Nov 23 comment How to replace chapter boilerplate with full-page image? For now just doing \def\chapterimage{chapter-1.png} \chapter{Chapter Title} seems to work. Thanks! Nov 23 comment How to replace chapter boilerplate with full-page image? Thanks! How do I define \chapterimage? Just doing \chapter[chapterimage=foo.png]{ChapterTitle} didn't work (I got a macro error, "Undefined control sequence \chapterimage"). Also I'm including my individual chapters in separate .tex files, if that makes a difference.	{"extraction_info": {"found_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": 0.8325352668762207, "perplexity": 2253.712121083854}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443736677342.4/warc/CC-MAIN-20151001215757-00164-ip-10-137-6-227.ec2.internal.warc.gz"}
http://www.freemathhelp.com/algebra-formulas.html	# Common Algebra Formulas Here are some of the most commonly used formulas in algebra. If you have one you'd like to be added, or find an error, please contact me. ## Laws of Exponents • $$a^ma^n=a^{m+n}$$ • $$(ab)^m=a^mb^m$$ • $$(a^m)^n=a^{mn}$$ • $$a^{\frac{m}{n}}=\sqrt[n]{a^m}$$ • $$a^0=1$$ • $$\frac{a^m}{a^n}=a^{m-n}$$ • $$a^{-m}=\frac{1}{a^m}$$ For an equation of the form $$ax^2+bx+c=0$$, you can solve for x using the Quadratic Formula: $$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$$ ## Binomial Theorem • $$(a+b)^1= a + b$$ • $$(a+b)^2=a^2+2ab+b^2$$ • $$(a+b)^3=a^3+3a^2b+3ab^2+b^3$$ • $$(a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4$$ ## Difference of Squares • $$a^2-b^2=(a-b)(a+b)$$ ## Rules of Zero • $$\frac{0}{x} = 0\text{, where }x \neq 0.$$ • $$a^0=1$$ • $$0^a=0$$ • $$a0 = 0$$ • $$\frac{a}{0}\text{ is undefined (you can't do it)}$$	{"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": 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": 0.8689721822738647, "perplexity": 731.0217036034485}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988719877.27/warc/CC-MAIN-20161020183839-00345-ip-10-171-6-4.ec2.internal.warc.gz"}
https://mathoverflow.net/questions/93140/associative-yang-baxter-on-ug	# associative Yang-Baxter on U(g) Consider $\mathfrak{g}$ a finite-dimensional Lie algebra over the field $\textbf{k}$. If A is an associative algebra, we are searching for functions from $\textbf{C}\times \textbf{C}$ to $A\otimes A$ such that: $$r^{12}(-u',v)r^{13}(u+u',v+v')-r^{23}(u+u',v')r^{23}(u,v)+r^{13}(u,v+v')r^{23}(u',v')=0$$ For $u,u',v,v'\in\mathbb{C}$. This is known as the associative Yang-Baxter equation with spectral parameters. Has the set of the solutions been unravelled when $A=U(\mathfrak{g})$ is the universal envelopping algebra of $\mathfrak{g}$? In fact, I am searching for solutions which have the following unitarity condition: $$r^{12}(x,y)=-r^{21}(-x,-y)$$ • What is the relation between $A$ and $g$? – Bruce Westbury Apr 4 '12 at 16:21 • I reformulated a bit the question accordingly to your comment. I hope bob would agree with the changes I made. – DamienC Apr 5 '12 at 7:43 • bob: If you register an account, it will be easier to edit your own question. – S. Carnahan Apr 6 '12 at 2:28 I don't know about full classification results for $U(\mathfrak{g})$ in general (this is probably out of reach), but there are a bunch of very interesting constructions (together with partial classification resultas) when $A=M_n(\textbf{k})$ and/or when $\mathfrak{g}$ is a semi-simple Lie algebra:	{"extraction_info": {"found_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": 0.8248997926712036, "perplexity": 279.0270765532111}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655924908.55/warc/CC-MAIN-20200711064158-20200711094158-00255.warc.gz"}
https://www.aimsciences.org/journal/1930-8337/2020/14/1	# American Institute of Mathematical Sciences ISSN: 1930-8337 eISSN: 1930-8345 All Issues ## Inverse Problems & Imaging February 2020 , Volume 14 , Issue 1 Select all articles Export/Reference: 2020, 14(1): 1-26 doi: 10.3934/ipi.2019061 +[Abstract](865) +[HTML](159) +[PDF](1127.94KB) Abstract: We study the integral transform over a general family of broken rays in \begin{document}$\mathbb{R}^2$\end{document}. One example of the broken rays is the family of rays reflected from a curved boundary once. There is a natural notion of conjugate points for broken rays. If there are conjugate points, we show that the singularities conormal to the broken rays cannot be recovered from local data and therefore artifacts arise in the reconstruction. As for global data, more singularities might be recoverable. We apply these conclusions to two examples, the V-line transform and the parallel ray transform. In each example, a detailed discussion of the local and global recovery of singularities is given and we perform numerical experiments to illustrate the results. 2020, 14(1): 27-52 doi: 10.3934/ipi.2019062 +[Abstract](843) +[HTML](150) +[PDF](822.45KB) Abstract: Existing reconstruction methods for single photon emission computed tomography (SPECT) are most based on discrete models, leading to low accuracy in reconstruction. Reconstruction methods based on integral equation models (IEMs) with a higher order piecewise polynomial discretization on the pixel grid for SEPCT imaging were recently proposed to overcome the accuracy deficiency of the discrete models. Discretization of IEMs based on the pixel grid leads to a system of a large dimension, which may require higher computational costs to solve. We develop a SPECT reconstruction method which employs an IEM of the SPECT data acquisition process and discretizes it on a content-adaptive unstructured grid (CAUG) with the total variation (TV) regularization aiming at reducing computational costs of the integral equation method. Specifically, we design a CAUG of the image domain for the discretization of the IEM, and propose a TV regularization defined on the CAUG for the resulting ill-posed problem. We then apply a preconditioned fixed-point proximity algorithm to solve the resulting non-smooth optimization problem, and provide convergence analysis of the algorithm. Numerical experiments are presented to demonstrate the superiority of the proposed method over the competing methods in terms of suppressing noise, preserving edges and reducing computational costs. 2020, 14(1): 53-75 doi: 10.3934/ipi.2019063 +[Abstract](1246) +[HTML](143) +[PDF](409.27KB) Abstract: In this article we study the inverse problem of determining the convection term and the time-dependent density coefficient appearing in the convection-diffusion equation. We prove the unique determination of these coefficients from the knowledge of solution measured on a subset of the boundary. 2020, 14(1): 77-96 doi: 10.3934/ipi.2019064 +[Abstract](1776) +[HTML](398) +[PDF](1475.63KB) Abstract: Poisson noise is an important type of electronic noise that is present in a variety of photon-limited imaging systems. Different from the Gaussian noise, Poisson noise depends on the image intensity, which makes image restoration very challenging. Moreover, complex geometry of images desires a regularization that is capable of preserving piecewise smoothness. In this paper, we propose a Poisson denoising model based on the fractional-order total variation (FOTV). The existence and uniqueness of a solution to the model are established. To solve the problem efficiently, we propose three numerical algorithms based on the Chambolle-Pock primal-dual method, a forward-backward splitting scheme, and the alternating direction method of multipliers (ADMM), each with guaranteed convergence. Various experimental results are provided to demonstrate the effectiveness and efficiency of our proposed methods over the state-of-the-art in Poisson denoising. 2020, 14(1): 97-115 doi: 10.3934/ipi.2019065 +[Abstract](926) +[HTML](173) +[PDF](944.67KB) Abstract: This paper addresses the problem of the electro-communication for weakly electric fish. In particular we aim at sheding light on how the fish circumvent the jamming issue for both electro-communication and active electro-sensing. Our main result is a real-time tracking algorithm, which provides a new approach to the communication problem. It finds a natural application in robotics, where efficient communication strategies are needed to be implemented by bio-inspired underwater robots. 2020, 14(1): 117-132 doi: 10.3934/ipi.2019066 +[Abstract](1098) +[HTML](379) +[PDF](2355.85KB) Abstract: Bayesian inference methods have been widely applied in inverse problems due to the ability of uncertainty characterization of the estimation. The prior distribution of the unknown plays an essential role in the Bayesian inference, and a good prior distribution can significantly improve the inference results. In this paper, we propose a hybrid prior distribution on combining the nonlocal total variation regularization (NLTV) and the Gaussian distribution, namely NLTG prior. The advantage of this hybrid prior is two-fold. The proposed prior models both texture and geometric structures present in images through the NLTV. The Gaussian reference measure also provides a flexibility of incorporating structure information from a reference image. Some theoretical properties are established for the hybrid prior. We apply the proposed prior to limited tomography reconstruction problem that is difficult due to severe data missing. Both maximum a posteriori and conditional mean estimates are computed through two efficient methods and the numerical experiments validate the advantages and feasibility of the proposed NLTG prior. 2020, 14(1): 133-152 doi: 10.3934/ipi.2019067 +[Abstract](733) +[HTML](146) +[PDF](905.3KB) Abstract: We analyze the Factorization method to reconstruct the geometry of a local defect in a periodic absorbing layer using almost only incident plane waves at a fixed frequency. A crucial part of our analysis relies on the consideration of the range of a carefully designed far field operator, which characterizes the geometry of the defect. We further provide some validating numerical results in a two dimensional setting. 2020, 14(1): 153-169 doi: 10.3934/ipi.2019068 +[Abstract](1678) +[HTML](167) +[PDF](340.84KB) Abstract: The Sturm-Liouville pencil is studied with arbitrary entire functions of the spectral parameter, contained in one of the boundary conditions. We solve the inverse problem, that consists in recovering the pencil coefficients from a part of the spectrum satisfying some conditions. Our main results are 1) uniqueness, 2) constructive solution, 3) local solvability and stability of the inverse problem. Our method is based on the reduction to the Sturm-Liouville problem without the spectral parameter in the boundary conditions. We use a special vector-functional Riesz-basis for that reduction. 2018 Impact Factor: 1.469	{"extraction_info": {"found_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": 0.8211681246757507, "perplexity": 502.11251707254024}, "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-2020-24/segments/1590347389309.17/warc/CC-MAIN-20200525161346-20200525191346-00140.warc.gz"}
https://www.sawaal.com/probability-questions-and-answers/a-bag-contains-8-red-and-4-green-balls-find-the-probability-that-two-balls-are-red-and-one-ball-is-g_9582	5 Q: # A bag contains 8 red and 4 green balls. Find the probability that two balls are red and one ball is green when three balls are drawn at random. A) 56/99 B) 112/495 C) 78/495 D) None of these Explanation: $∴P(E)=112495$ Q: In a purse there are 30 coins, twenty one-rupee and remaining 50-paise coins. Eleven coins are picked simultaneously at random and are placed in a box. If a coin is now picked from the box, find the probability of it being a rupee coin? A) 4/7 B) 2/3 C) 1/2 D) 5/6 Explanation: Total coins 30 In that, 1 rupee coins 20 50 paise coins 10 Probability of total 1 rupee coins = 20C11 Probability that 11 coins are picked = 30C11 Required probability of a coin now picked from the box is 1 rupee = 20C11/30C11 = 2/3. 1 108 Q: In a box, there are 9 blue, 6 white and some black stones. A stone is randomly selected and the probability that the stone is black is ¼. Find the total number of stones in the box? A) 15 B) 18 C) 20 D) 24 Explanation: We know that, Total probability = 1 Given probability of black stones = 1/4 => Probability of blue and white stones = 1 - 1/4 = 3/4 But, given blue + white stones = 9 + 6 = 15 Hence, 3/4 ----- 15 1 ----- ? => 15 x 4/3 = 20. Hence, total number of stones in the box = 20. 5 342 Q: What is the probability of an impossible event? A) 0 B) -1 C) 0.1 D) 1 Explanation: The probability of an impossible event is 0. The event is known ahead of time to be not possible, therefore by definition in mathematics, the probability is defined to be 0 which means it can never happen. The probability of a certain event is 1. 8 775 Q: In a box, there are four marbles of white color and five marbles of black color. Two marbles are chosen randomly. What is the probability that both are of the same color? A) 2/9 B) 5/9 C) 4/9 D) 0 Explanation: Number of white marbles = 4 Number of Black marbles = 5 Total number of marbles = 9 Number of ways, two marbles picked randomly = 9C2 Now, the required probability of picked marbles are to be of same color = 4C2/9C2 + 5C2/9C2 = 1/6 + 5/18 = 4/9. 7 1085 Q: A bag contains 3 red balls, 5 yellow balls and 7 pink balls. If one ball is drawn at random from the bag, what is the probability that it is either pink or red? A) 2/3 B) 1/8 C) 3/8 D) 3/4 Explanation: Given number of balls = 3 + 5 + 7 = 15 One ball is drawn randomly = 15C1 probability that it is either pink or red = 14 997 Q: Two letters are randomly chosen from the word TIME. Find the probability that the letters are T and M? A) 1/4 B) 1/6 C) 1/8 D) 4 Explanation: Required probability is given by P(E) = 15 1553 Q: 14 persons are seated around a circular table. Find the probability that 3 particular persons always seated together. A) 11/379 B) 21/628 C) 24/625 D) 26/247 Explanation: Total no of ways = (14 – 1)! = 13! Number of favorable ways = (12 – 1)! = 11! So, required probability = $11!×3!13!$ = $39916800×66227020800$ = $24625$ 15 1549 Q: Two dice are rolled simultaneously. Find the probability of getting the sum of numbers on the on the two faces divisible by 3 or 4? A) 3/7 B) 7/11 C) 5/9 D) 6/13 Explanation: Here n(S) = 6 x 6 = 36 E={(1,2),(1,5),(2,1),(2,4),(3,3),(3,6),(4,2),(4,5),(5,1),(5,4),(6,3) ,(6,6),(1,3),(2,2),(2,6),(3,1),(3,5), (4,4),(5,3),(6,2)} => n(E)=20 Required Probability n(P) = n(E)/n(S) = 20/36 = 5/9.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 4, "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": 0.8884996175765991, "perplexity": 1046.7542882768219}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676591831.57/warc/CC-MAIN-20180720193850-20180720213850-00115.warc.gz"}
http://soft-matter.seas.harvard.edu/index.php?title=Percolation_Model_for_Slow_Dynamics_in_Glass-Forming_Materials&diff=next&oldid=15935	Difference between revisions of "Percolation Model for Slow Dynamics in Glass-Forming Materials" Entry: Chia Wei Hsu, AP 225, Fall 2010 G. Lois, J. Blawzdziewicz, and C. S. O'Hern, "Percolation Model for Slow Dynamics in Glass-Forming Materials", Phys. Rev. Lett. 102, 015702 (2009). Summary In this work, the authors propose an alternate approach to understand the glass transition. Instead of looking at the real space, they focus on the configuration space of the system. There are mobility regions in the configuration space, and the percolation of these regions corresponds to a glass-to-liquid transition. With a mean-field description of such percolation, they show that the stretched-exponential response functions typical of glassy systems can be explained. Background Glassy systems exhibit several unique properties. During a glass transition, the structural relaxation time increases by several orders of magnitude. Also, the structural correlations display an anomalous stretched-exponential time decay: $exp(-t/\tau_{\alpha})^{\beta}$, where $\beta$ is called the stretching exponent, and $\tau_{\alpha}$ is called the $\alpha$-relaxation time. Although the stretched-exponential relaxation is universal among glass-forming materials, $\beta$ and $\tau_{\alpha}$ are not. They depend on temperature and density, and they vary from one material to another. Basic Idea The basic idea is that, instead of focusing on the heterogeneous dynamics and percolation in real space (the traditional approach), the authors focus on the configuration space and its connection to the anomalous dynamics. For complete relaxation, the system must be able to diffuse over the whole configuration space. That is, there has to be a path that percolates the configuration space. Thus, we can think of a "percolation transition in the configuration space," which corresponds to the glass transition in real space. Hard spheres Fig 1. Schematic of allowed regions in configuration space for hard spheres. (a) $\phi = \phi_J$, only jammed states (discrete points in the config space) are allowed. (b) $\phi < \phi_J$, motion occurs in closed mobility domains surrounding jammed states. (c) Even smaller $\phi$, bridges between mobility domains occur. (d) Even smaller $\phi$, percolation occurs (shaded yellow). Hard spheres interact with infinite repulsion upon contact. At small volume fraction $\phi$, hard spheres behave like fluids. As $\phi$ increases to $\phi_J \approx 0.64$, the system becomes collectively jammed. In such state no motion can occur, because any particle displacement will lead to overlap. Therefore only discrete set of points in the configuration space are allowed (fig 1a). At slightly lower $\phi$, particles can move around a little bit. Therefore, the discrete points become mobility domains in the configuration space (fig 1b). Further decrease of $\phi$ lead to connection between these mobility domains (fig 1c), and eventually to a percolation between these domains (fig 1d) at $\phi=\phi_P$. Denote the volume fraction of mobility domains in the configuration by $\Pi$. Percolation occurs at a critical $\Pi_P$. When $\Pi > \Pi_P$, the system can explore the whole configuration space, and it is a metastable liquid. When $\Pi < \Pi_P$, the system can only diffuse in finite regions of the configuration space, and it is a glass. The distance it can explore is set by the percolation correlation length $\xi$, which diverges to infinite at $\Pi_P$. The authors then describe the percolation with mean-field theory. Following several known scaling laws, they show that the stretching exponent varies with time and satisfies $1/3 \leq \beta \leq 1$, agreeing with experimental observations. With a few more assumptions, they further predicts the scaling of $\alpha$-relaxation time to be $q^2 \tau_{\alpha} \propto \left\{ \begin{array}{ll} \exp\left(\frac{A \phi_J}{\phi_J-\phi}\right)(\phi_P-\phi)^{-2} & \textrm{for } \phi_P-\phi \ll \phi_J-\phi_P\\ \exp\left(\frac{B \phi_J}{\phi_J-\phi}\right) & \textrm{for } \phi_P-\phi \gg \phi_J-\phi_P \end{array} \right.$, where $q$ is the scattering wave number, and $A$, $B$ are positive constants. Finite energy barriers Bonds form for systems with a finite energy barrier. In such case, the configuration space can be decomposed into basins of attraction surrounding each local energy minimum. At short times the system is confined to a basin, whereas at long times it can hop from one basin to another. Again using mean-field arguments, the authors come up with analogous expression for $\beta$ and for $\tau_{\alpha}$. Soft matter discussion The glass transition is one of the largest outstanding questions in soft matter. The approach proposed by these authors are dramatically different from the traditional standpoint, yet explains the observed phenomena equally well. This approach may shed some light, and hopefully lead to new predictions and better understanding of the glass-formation materials.	{"extraction_info": {"found_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": 0.9108719825744629, "perplexity": 938.7516759610008}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439737039.58/warc/CC-MAIN-20200806210649-20200807000649-00249.warc.gz"}
http://openstudy.com/updates/4fa5b375e4b029e9dc35a2cf	## Got Homework? ### Connect with other students for help. It's a free community. • across Online now • laura* Helped 1,000 students Online now • Hero College Math Guru Online now Here's the question you clicked on: 55 members online • 0 viewing ## AccessDenied Group Title Note: This is NOT a Question! I am just joining the cool new trend of having a tutorial for something, and it gives me a good excuse to make a question. This Tutorial is for LaTeX formatting (mainly without Equation Editor). There are three sections in total, and I'll post them individually, along with the introduction for a total of four posts. Enjoy! 2 years ago 2 years ago Edit Question Delete Cancel Submit • This Question is Closed 1. AccessDenied Group Title Best Response You've already chosen the best response. 54 $$\color{gold}{\star \ \star} \quad \large{ \mathbb{\text{Doing That } \LaTeX \text{Thing}} } % Because knowing is half the battle! % % I hope you enjoy the tutorial as much as I did writing it. It's been fun! % % - Written by: AccessDenied, master of using an excessive amount of work in things %$$ LaTeX is a very powerful and versatile tool for writing Mathematics. The Equation Editor gives you only a small glimpse into the true power of the language. My goal is to give you the power to write this LaTeX on your own and tap into that potential! Prepare to delve into the wonderful world of LaTeX! $$\color{orange}{\cdot} \quad \small{ \text{Experience will best supplement this tutorial. Try things out!} }$$ • 2 years ago 2. AccessDenied Group Title Best Response You've already chosen the best response. 54 $$\color{goldenrod}{\star} \quad \mathbb{\LaTeX \text{ Syntax; Formatting, Commands, and Environments}}$$ $$\color{orange}{\cdot} \quad \text{Adding LaTeX to your Posts}$$ Creating fine posts that will coexist with the LaTeX content requires that you know how to declare it in the posts correctly and effectively. There are two ways to declare it: block content and in-line content (sometimes referred to as "displayed" and "text," respectively). $$\cdot \quad$$ Block content is inserted into the post as its own stand-alone element. The LaTeX is put on its own line. This type of LaTeX is applied by wrapping the code between the delimiters: $$\backslash [ \cdots \backslash ]$$. $$\cdot \quad$$ In-line content is inserted into the post with respect to its surroundings. This allows LaTeX to be inserted directly into paragraph explanations. This type is applied by using the delimiters: $$\backslash ( \cdots \backslash )$$. Which method you use is largely based on your purposes. Just remember that in-line is treated a tad differently from block content in that, it is made smaller and more compact than the block content by nature. This often affects placement of things like the stuff usually under the limit $$\lim$$ where the in-line method places it to the side, as so: $$\lim_{x \to 4}$$. This is fixed using a command (If you're interested, it is called $$\textbf{\limits}$$ -- I won't go over it here). Another note (it is a minor point), in-line text will not wrap around the text-space unless it is preceded by something. Block content will always wrap around, however. One of the staples for LaTeX is the backslash $$\backslash$$ symbol. It comes in on the declaration and the commands. It is also used to start a new line (double-backslash). That should help conserving the need for many individual LaTeX spaces. Okay, we got that out of the way! So, what can we do? You can type some basic equations in the LaTeX for the pretty, italicized font, but you'll quickly notice the limitations -- no spaces and no fancy font things... where is the fancy?! Well, we're getting there. • 2 years ago 3. AccessDenied Group Title Best Response You've already chosen the best response. 54 $$\color{orange}{\cdot} \quad \text{Commands -- Adding in that functionality}$$ Commands are the masters of the functionality you really want in the posts. They're very simple, too. The basic syntax of a command is as follows: $$\quad \boxed{ \quad \color{green}{\star} \quad \backslash \underline {\textit{name}} \{ \textit{argument}_1 \} \{ \textit{argument}_2 \} \ ... \qquad }$$ It is most common for the command to have from zero to two arguments. These arguments are usually either switches for how the command works or the target of the command. There are many, many commands for LaTeX, more than I can even discuss honestly. We'll touch on the important / awesome ones, though. I leave it to you to search for the rest as you please. $$% Just AccessDenied adding some random comments to make sure you can't reuse it! %$$ The first commands we should learn is for inserting text into the LaTeX correctly. You could try writing a message like "I love LaTeX!" directly, but it doesn't turn out right, as I shall present: $$\quad \boxed{ \quad \color{green}{\Rightarrow} \quad \text{I love LaTeX!} \qquad }$$ $$\quad \boxed{ \quad \color{lime}{\Leftarrow} \quad I love LaTeX! \qquad }$$ I'll use this kind of representation a lot for this tutorial. Just internalize that this is "input =>" "output <=" and you'll be good. On a more relevant note, however, we can see there is a serious lack of spacing. It's nowhere near as pretty as just typing it out... well, let's try out a command. $$\textbf{\text} \{ ... \}$$ Well, it sounds aptly named. Let's try it out. $$\quad \boxed{ \quad \color{green}{\Rightarrow} \quad \text{\text{I love LaTeX!}} \qquad }$$ $$\quad \boxed{ \quad \color{lime}{\Leftarrow} \quad \text{I love LaTeX!} \qquad }$$ We have our spaces in and everything! Just for novelty, let's also use the \LaTeX command, which adds a nice label for the LaTeX name. $$\quad \boxed{ \quad \color{green}{\Rightarrow} \quad \text{\text{I love } \LaTeX !} \qquad }$$ $$\quad \boxed{ \quad \color{lime}{\Leftarrow} \quad \text{I love } \LaTeX !} \qquad$$ Notice that some commands are written with specific capitalized letters. Command names are case-sensitive, so be wary that some functions will not work without addressing that capital letter. Also notice that we have an extra space after "love." That space carries even outside of the command. Just keep those things in mind! The text command creates an environment apart from the Math space, which omits the spaces. However, spacing is still possible even without the text command. In fact, there are a lot of commands for spacing in the Math space. I'll just list them here. $\quad \begin{array}{\|c\|c\|} \hline \textbf{Command} & \textbf{Effect} \\ \hline \backslash & \text{ Single space equivalent } \\ \backslash quad & \text{ Creates large horizontal space } \\ \backslash qquad & \text{ Two \quad } \\ \backslash , & \text{ Equals 3/18 a \quad } \\ \backslash : & \text{ Equals 4/18 a \quad } \\ \backslash ; & \text{ Equals 5/18 a \quad } \\ \backslash ! & \text{ Equals -3/18 a \quad } \\ \hline \end{array}$ What? An antispace!? Well, I guess antispaces may have uses somewhere. I never particularly found one, though. I guess if you wanted to create a 1/18 a \quad space or even -36/18... oh nevermind, not important! There are also the Mathematical functions. You could type them normally, but they're usually better oriented with the command. $\quad \begin{array}{\|c\|c\|c\|} \hline \text{\frac} & \text{_} & \text{^} \\ \hline \text{\sqrt} & & \\ \hline \text{\sin} & \text{arcsin} & \text{sinh} \\ \hline \text{\log} & \text{\ln} & \text{\exp} \\ \hline \text{\lim} & \text{\int} & \\ \hline \text{\sum} & \text{\prod} & \\ \hline \text{\mod} & \text{\inf} & \text{\sup} \\ \hline \end{array}$ That frac function is probably one of the big secrets of the Equation Editor and LaTeX here. The syntax of the fraction function is \frac{numerator}{denominator}. In addition, the root's index actually is placed into a weird place in square brackets (\sqrt[index] {content}) The majority of the others do come up on Equation Editor, however, and should be familiar if you've played with it before. $$% AccessDenied, adding in some special comments here and there %$$ Lastly, I want to go over some modifying commands. This should wrap up the (useful) commands section. These commands are for changing the size and style of your writing in LaTeX. $\quad \begin{array}{\|c\|c\|} \hline \textbf{Name} & \textbf{Explained} \\ \hline \text{rm} & \mathrm{Roman} \\ \text{it} & \mathit{Italicized} \\ \text{bf} & \mathbf{Bold} \\ \text{frak} & \mathfrak{Fraktur} \\ \text{cal} & \mathcal{Calligraphy} \\ \text{sf} & \mathsf{\text{Sans-serif}} \\ \hline \text{tiny} & \text{Tiny font} \\ \text{scriptsize} & \text{Size of a sub/super script} \\ \text{small} & \text{Small font} \\ \text{normalsize} & \text{Normal} \\ \text{large} & \text{Larger font} \\ \text{Large} & \text{Even larger font} \\ \text{LARGE} & \text{LARGE font} \\ \hline \text{color} & \color{green}{\text{Makes colors!}} \\ \hline \end{array}$ The first set of commands can come in a few forms, although some are exclusive to one form. All of the above forms may be stand-alone. These will not have arguments and apply to all the content in the space. The first three may be used appended to \text (i.e. \textrm{}, \textbf{}) for the same effect as if you used the stand-alone version and then \text{}, and only apply to a target-argument. All of these are also applicable at the end of a function "\math__" and will also apply to a target-argument. The sizes can either be left stand-alone to apply to their respective line, or be fed a target-argument for what should be made bigger. There are even larger sizes, but I find them overkill. The names are intuitive, though. Anything as "huge" as LARGE is absurd. The color command takes a color argument in either hexadecimal or nominal format (\color{#FF0000}{text} and \color{red}{text} do the same thing). As a test of our newfound skills, let's try to make a (considerably fancy) display for the slope-intercept form of lines! We would expect this to fail miserably, typing it directly into LaTeX: $$\quad \boxed{ \quad \color{green}{\Rightarrow} \quad \text{Slope-intercept form: y = mx + b} \qquad }$$ $$\quad \boxed{ \quad \color{lime}{\Leftarrow} \quad Slope-intercept form: y = mx + b \qquad }$$ And it does. Now, let's surround the "Slope-intercept form: " in a \text{} command. $$\quad \boxed{ \quad \color{green}{\Rightarrow} \quad \text{\text{Slope-intercept form: } y = mx + b} \qquad }$$ $$\quad \boxed{ \quad \color{lime}{\Leftarrow} \quad \text{Slope-intercept form: } y = mx + b \qquad }$$ There we go! Finally, we'll make it bold by replacing \text with \textbf and add a \quad between the text and formula for horizontal spacing. $$\quad \boxed{ \quad \color{green}{\Rightarrow} \quad \text{\textbf{Slope-intercept form: } \quad y = mx + b} \qquad }$$ $$\quad \boxed{ \quad \color{lime}{\Leftarrow} \quad \textbf{Slope-intercept form: } \quad y = mx + b \qquad }$$ We're done! Looks a lot nicer than just writing it out, which would take way too little time! Well, let's end this section off and move onto the final part: Environments! • 2 years ago 4. AccessDenied Group Title Best Response You've already chosen the best response. 54 $$\color{orange}{\cdot} \quad \text{Eco-friendly / Safe Environments}$$ So, there's just one more thing to cover, called "environments." The name may sound a little funny, but I assure you that these are anything but lame! Environments allow for special conditions to be applied to the LaTeX space. There are two very useful ones. The first one is called "align." The alignment environment allows for the use of a special operator "&" that forces each line to line up at these points. Such an example for this environment could be a solution to the equation "2x + 3 = 9." \quad \boxed{ \begin{align} \quad & \color{green}{\Rightarrow}\quad \text{\begin{align}} \qquad \\ & \color{green}{\Rightarrow} \quad \ \text{ 2x + 3 &= 9} \qquad \\ & \color{green}{\Rightarrow} \quad \ \text{ 2x &= 6} \qquad \\ & \color{green}{\Rightarrow} \quad \ \text{ x &= 3} \qquad \\ & \color{green}{\Rightarrow} \quad \text{\end{align}} \qquad \end{align} } \quad \boxed{ \begin{align} \quad & \color{lime}{\Leftarrow} \quad & 2x + 3 &= 9 \qquad \\ & \color{lime}{\Leftarrow} & \quad 2x &= 6 \qquad \\ & \color{lime}{\Leftarrow} & \quad x &= 3 \qquad \end{align} } Notice that format. It starts with a "begin{align}" and ends with a "end{align}." This is how the environments will always be declared. It is mostly the same way with the array environment. Instead of "align", you just have the "array." However, the Array environment uses an extra parameter after the "begin{array}" part. We have to declare the alignment of each element in the table. We may optionally specify vertical separation. Also, the environment supports a command called $$\textbf{\hline}$$ to make horizontal lines. The possibilities are: "l", "c", and "r." They stand for "left," "center," and "right alignment, respectively. So, to write the array header for a two-column array with all elements aligned to the right and no vertical bars, we'd use "begin{array}{rr} ... \end{array}." Similarly, if we wanted to add a vertical bar between the first and second item of each row, we'd use "begin{array}{r\|r}." I will not create another example of arrays for the sake of making sure the post isn't going to explode. Instead, I want to give you the very plaintext version of my tutorial that includes a few uses of arrays already, so that you may see how it was all written as well! Enjoy! This will conclude the tutorial! I hope you have learned a thing or two on making LaTeX. Just remember that your creativity will drive the effectiveness of what you do with this. Personally, I thought the diagram of a rectangle I made with arrays was fairly nifty. $$% Disclaimer: I love you all very much for checking out the guide %$$ Plaintext Source; Tutorial: http://pastebin.com/raw.php?i=v8GUrZi5 • 2 years ago 5. zepp Group Title Best Response You've already chosen the best response. 0 AccessDenied had lot of issues when writing the tutorial.. but he did it! Applauses ;D • 2 years ago 6. AccessDenied Group Title Best Response You've already chosen the best response. 54 two revisions and a weird error concerning environments going after LaTeX • 2 years ago 7. alexwee123 Group Title Best Response You've already chosen the best response. 0 wall of text didn't read :D • 2 years ago 8. Hero Group Title Best Response You've already chosen the best response. 0 I scanned through it in about a minute. Good info. • 2 years ago 9. asnaseer Group Title Best Response You've already chosen the best response. 0 Great stuff AccessDenied. There is a $$\LaTeX$$ practising group where you could post this as well: http://openstudy.com/study?login#/groups/LaTeX%20Practicing!%20%3A) • 2 years ago 10. Hero Group Title Best Response You've already chosen the best response. 0 They should feature these tutorials on the main page along with creating a special group. All of the tutorials so far have been pretty impressive. • 2 years ago 11. AccessDenied Group Title Best Response You've already chosen the best response. 54 Thanks. I don't think I made any more mistakes after revising again, but I hope if I did, people can catch them. The biggest deal making it was just writing the latex in latex. There was a lot of weird issues that would come up doing that. :P There was also a weird thing that happened when I tried to type the environment declarations after I introduced an independent LaTeX string before it, where the environments would become LaTeX delimiters outside of the regular ones. D: • 2 years ago 12. KingGeorge Group Title Best Response You've already chosen the best response. 0 Very impressive. And very helpful as well. Excellent job. • 2 years ago 13. lgbasallote Group Title Best Response You've already chosen the best response. 0 just when i thought ishy and myin were the best in latex you proved me wrong • 2 years ago 14. KingGeorge Group Title Best Response You've already chosen the best response. 0 I enjoy your comments in the plaintext source. • 2 years ago 15. Hero Group Title Best Response You've already chosen the best response. 0 I'm nitpicking, but you should have stated at the beginning of your tutorial that right arrow means input, and left arrow means output. @lgbasallote, there are several people who are really good at Latex, not just those you have specified. • 2 years ago 16. Hero Group Title Best Response You've already chosen the best response. 0 Belay my last • 2 years ago 17. lgbasallote Group Title Best Response You've already chosen the best response. 0 fine..hero is good in latex too...i know you wanted to be mentioned :P haha jkjk • 2 years ago 18. Hero Group Title Best Response You've already chosen the best response. 0 I didn't say it was me :P • 2 years ago 19. AccessDenied Group Title Best Response You've already chosen the best response. 54 Oh, its actually just if there are closed LaTeX strings anywhere in the post, \begin{} and \end{} become delimiters for their own LaTeX environments. Yeah, that might be a good idea for the future. Most ideas came as the thing progressed, those arrowed boxes being one of them. :D • 2 years ago 20. AccessDenied Group Title Best Response You've already chosen the best response. 54 What! Fine then. Anytime. • 2 years ago 21. lgbasallote Group Title Best Response You've already chosen the best response. 0 i hope with these the number of ambiguous questions would drop • 2 years ago 22. AccessDenied Group Title Best Response You've already chosen the best response. 54 Notes for improvements in possible future revisions * Fix mis-spacing in title (L 1) * Fix qquad position (L 58) (should precede closing bracket) * Add in the diagram i referred to, (L 130) * . Add introductory note for used notations * C: possibly more elaboration in the environments section * C: Could probably add an example for arrays, it wasn't too bad to post as predicted * C: possibly more explanation of the line break, wasnt emphasized very well. * Add sources or helpful sites to check out: --> http://en.wikibooks.org/wiki/LaTeX/Mathematics (Originally taught me a lot of the possible commands, also found some interesting commands recently) --> http://insti.physics.sunysb.edu/latex-help/ltx-176.html (Text size, usable fonts) • 2 years ago 23. AccessDenied Group Title Best Response You've already chosen the best response. 54 I bump in hopes that somebody does catch any mistakes, would help in making a revision that I don't miss something. D: • 2 years ago 24. AccessDenied Group Title Best Response You've already chosen the best response. 54 I'm assuming that most of the things to tweak have been plucked out now, so I can probably just adjust and post in the other section as mentioned earlier. D: • 2 years ago 25. TheViper Group Title Best Response You've already chosen the best response. 2 $\Huge{\color{red}{\mathbb{LaTex}}}$ • 2 years ago 26. TheViper Group Title Best Response You've already chosen the best response. 2 wow! • 2 years ago 27. TheViper Group Title Best Response You've already chosen the best response. 2 I didn't know that I can do like that in the Equation Editor! • 2 years ago 28. TheViper Group Title Best Response You've already chosen the best response. 2 Now, I too love $\Huge{\mathbb{\LaTeX}}$ • 2 years ago 29. TheViper Group Title Best Response You've already chosen the best response. 2 So much interesting!! • 2 years ago 30. TheViper Group Title Best Response You've already chosen the best response. 2 I can't tell in words that how interesting is LaTeX! • 2 years ago 31. KingGeorge Group Title Best Response You've already chosen the best response. 0 not to be rude, but it would be nice if you were to take your practicing to the $$\LaTeX$$ practice group at openstudy.com/study#/groups/LaTeX instead of repeatedly posting here. • 2 years ago 32. TheViper Group Title Best Response You've already chosen the best response. 2 $\Huge{\color{red}{\rightarrow \boxed{\mathbb{\text{I loveLaTeX}}}}}$ • 2 years ago 33. TheViper Group Title Best Response You've already chosen the best response. 2 $\Huge{\color{orange}{\star \star}{\text{{Very Excellent Job}}}}$ • 2 years ago 34. TheViper Group Title Best Response You've already chosen the best response. 2 So useful! Thanx so much! • 2 years ago 35. TheViper Group Title Best Response You've already chosen the best response. 2 I can't thank u in my words! I m so happy to know it! & thanx @ParthKohli for telling me this $\Huge{\color{red}{\star}{Tutorial}}$ • 2 years ago 36. sami-21 Group Title Best Response You've already chosen the best response. 0 $\Huge \color{blue}{nice}\color{red} {tutorial}$ ! • one year ago 37. AccessDenied Group Title Best Response You've already chosen the best response. 54 $$\large \frak{\text{Thanks, glad you learned from it!}}$$ :) I should start on a second version, so I can add more / fix some of those things. • one year ago 38. mathslover Group Title Best Response You've already chosen the best response. 0 $\huge{\frak{Hey, nice}\mathsf{Tutorial}\mathbb{Great!!!!}}$ • one year ago 39. ghazi Group Title Best Response You've already chosen the best response. 0 how you guys do it? • one year ago 40. mathslover Group Title Best Response You've already chosen the best response. 0 @ghazi prefer this group : this will help you http://openstudy.com/study#/groups/LaTeX%20Practicing!%20%3A) • one year ago 41. TheViper Group Title Best Response You've already chosen the best response. 2 Wait if u want to know how they do just right click on LaTeX & then click show math as then click text commands @ghazi :) • one year ago 42. TheViper Group Title Best Response You've already chosen the best response. 2 like • one year ago ##### 1 Attachment 43. TheViper Group Title Best Response You've already chosen the best response. 2 After clicking on text commands a box will appear then copy everything in the box & paste in equation editor @ghazi :) • one year ago 44. TheViper Group Title Best Response You've already chosen the best response. 2 @ghazi does that helped ?? • one year ago 45. ghazi Group Title Best Response You've already chosen the best response. 0 @TheViper that was really helpful...thanks dude....i am grateful to this ....i shall make use of it... • one year ago 46. TheViper Group Title Best Response You've already chosen the best response. 2 $$\Huge{\color{green}{\ddot{\smile}}}$$ • one year ago 47. TheViper Group Title Best Response You've already chosen the best response. 2 wait @ghazi :) • one year ago 48. ghazi Group Title Best Response You've already chosen the best response. 0 wow !! • one year ago 49. TheViper Group Title Best Response You've already chosen the best response. 2 If you want to write anything in 'Code Block' you have write between these :- \|dw:1347024632753:dw\|Like this :- http://assets.openstudy.com/updates/attachments/503dfd2ce4b0074824ff598c-theviper-1346930814196-untitled.png It will be shown as :- GHAZI • one year ago 50. ghazi Group Title Best Response You've already chosen the best response. 0 cool....let me try this • one year ago 51. TheViper Group Title Best Response You've already chosen the best response. 2 try :) • one year ago 52. ghazi Group Title Best Response You've already chosen the best response. 0 \|dw:1346936212442:dw\| • one year ago 53. ghazi Group Title Best Response You've already chosen the best response. 0 it didn't work :( • one year ago 54. TheViper Group Title Best Response You've already chosen the best response. 2 sorry u should not type in drawing :) • one year ago 55. TheViper Group Title Best Response You've already chosen the best response. 2 work as in the attachment :) • one year ago 56. TheViper Group Title Best Response You've already chosen the best response. 2 • one year ago 57. TheViper Group Title Best Response You've already chosen the best response. 2 try now :) • one year ago 58. ghazi Group Title Best Response You've already chosen the best response. 0 okay wait a sec • one year ago 59. TheViper Group Title Best Response You've already chosen the best response. 2 fast • one year ago 60. TheViper Group Title Best Response You've already chosen the best response. 2 • one year ago ##### 1 Attachment 61. ghazi Group Title Best Response You've already chosen the best response. 0 like this ,,,,,,,,,,, Type here ,,,, Viper • one year ago 62. ghazi Group Title Best Response You've already chosen the best response. 0 damn i am too stupid at this • one year ago 63. ghazi Group Title Best Response You've already chosen the best response. 0 by the way thanks for your time dude :) • one year ago 64. TheViper Group Title Best Response You've already chosen the best response. 2 hey @ghazi its not apostrophe its this • one year ago ##### 1 Attachment 65. TheViper Group Title Best Response You've already chosen the best response. 2 & u must type it only 3 times like this :- • one year ago ##### 1 Attachment 66. ghazi Group Title Best Response You've already chosen the best response. 0 @the viper thanks man • one year ago 67. TheViper Group Title Best Response You've already chosen the best response. 2 ok try here:) • one year ago 68. sauravshakya Group Title Best Response You've already chosen the best response. 0 $$\checkmark$$ • one year ago 69. MathSofiya Group Title Best Response You've already chosen the best response. 0 Thanks soo much @AccessDenied :D • one year ago 70. jiteshmeghwal9 Group Title Best Response You've already chosen the best response. 0 $\huge{\color{red}{\frak{So \space \text{great} \space tutorial \space}\mathbb{Thanx !!!!!!}}}$ @AccessDenied :) • one year ago 71. blackops2luvr Group Title Best Response You've already chosen the best response. 0 $$\bbox [ 15pt, #000033 ,border: 15px solid #aa8866 ] {\Huge\sf\color{white} {\ hi \ there \ :D }}$$ • 4 months ago • Attachments: ## See more questions >>> ##### spraguer (Moderator) 5→ View Detailed Profile 23 • Teamwork 19 Teammate • Problem Solving 19 Hero • You have blocked this person. • ✔ You're a fan Checking fan status... Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.	{"extraction_info": {"found_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": 2, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9963624477386475, "perplexity": 4407.820338997046}, "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-2014-23/segments/1405997873839.53/warc/CC-MAIN-20140722025753-00077-ip-10-33-131-23.ec2.internal.warc.gz"}
https://neurips.cc/Conferences/2019/ScheduleMultitrack?event=13911	Timezone: » Poster Energy-Inspired Models: Learning with Sampler-Induced Distributions John Lawson · George Tucker · Bo Dai · Rajesh Ranganath Wed Dec 11 10:45 AM -- 12:45 PM (PST) @ East Exhibition Hall B + C #120 Energy-based models (EBMs) are powerful probabilistic models, but suffer from intractable sampling and density evaluation due to the partition function. As a result, inference in EBMs relies on approximate sampling algorithms, leading to a mismatch between the model and inference. Motivated by this, we consider the sampler-induced distribution as the model of interest and maximize the likelihood of this model. This yields a class of energy-inspired models (EIMs) that incorporate learned energy functions while still providing exact samples and tractable log-likelihood lower bounds. We describe and evaluate three instantiations of such models based on truncated rejection sampling, self-normalized importance sampling, and Hamiltonian importance sampling. These models out-perform or perform comparably to the recently proposed Learned Accept/RejectSampling algorithm and provide new insights on ranking Noise Contrastive Estimation and Contrastive Predictive Coding. Moreover, EIMs allow us to generalize a recent connection between multi-sample variational lower bounds and auxiliary variable variational inference. We show how recent variational bounds can be unified with EIMs as the variational family.	{"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": 0.8868663311004639, "perplexity": 2858.7394704352223}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988882.94/warc/CC-MAIN-20210508151721-20210508181721-00518.warc.gz"}
https://www.physicsforums.com/threads/mathematical-induction.95577/	# Mathematical Induction 1. Oct 19, 2005 ### dglee Show taht for every natural numbers n>=2 $$n \geq 2$$ the number 2^2^n - 6 $$2^{2^n} -6$$ is a multiple of 10. Using mathematical induction. Okay i got no clue how to start this question. Ahhh Is there a series where the x^n is a series? Well this stuff really sucks. Last edited: Oct 19, 2005 2. Oct 19, 2005 ### Tide HINT: $$2^{2^{n+1}} = \left( 2^{2^n}\right)^2$$ 3. Oct 19, 2005 ### dglee Oh WOw... that helped a lot.. now i can show that its inductive hmm but how would i show its a multiple of 10? hmm maybe if i said 10^2 is 100 so thats a multiple of 10. $$10n\leq2^{2^n}-6$$ if i proved that.. hmm i wonder if that would be right... $$10(n+1)\leq2^{2^(n+1)}$$ hmm if i proved that would i have solved the question? wow if hmm well ahaha thanks a LOT!!!!!!!! YOUR AWSOME. that little hint helped a lot but i got no clue if im actually doing it right. Last edited: Oct 19, 2005 4. Oct 19, 2005 For Mathematical Induction, you assume that P_{k} is true, in this case for n greater or equal to 2. With this, you can immediately say that 2^(2^k) - 6 is equal to 10a, for some a, where a is a positive integer. Can you then use this fact, and the hint provided, to prove the desired result for 2^(2^(k+1)) - 6? 5. Oct 19, 2005 ### dglee hmm should could i say that $$10(a+x)\leq (\left2^{2^n}\right )^2 - 6$$ where x is some positive number and should that it is inductive to prove that 10 is a multiple? soo $$10(a)\leq2^{2^n} - 6$$ then $$10(a+x)\leq(\left 2^{2^n}\right)^2 - 6$$ then show that $$10(a+x)\leq2^{2^{n+1}} -6$$ ahhh im confusd now.. ahh Last edited: Oct 19, 2005 6. Oct 19, 2005 ### Tide HINT: If M - 6 is a multiple of 10 then M ends in the digit 6! :) 7. Oct 19, 2005 Hmmm... Why are there so many inequalitites in your working? From what I know, the only inequality to appear in your solution should be the fact that n is greater than or equal to 2, but this is just a specification, and should not appear in your proof. 8. Oct 19, 2005 Steps in Mathematical Induction 1) Let $$P_{n}$$ be the statement $$2^{2^n}-6$$ is divisible by 10, for n$$\geq$$ 2. 2) Check that the result you want to prove is valid for n=2, so $$P_{2}$$ is true. 3) Assume $$P_{k}$$ is true, for some n $$\geq$$ 2. So, $$2^{2^k}-6$$ = 10a, for some a, which is a positive integer. 4) Using this result, you must somehow prove that $$2^{2^{k+1}}-6$$ is a multiple of 10. How would you go around doing it? Look at the first hint provided and observe... What has been done to the term $$2^{2^n}$$ ? USE BOTH THE RESULT FROM STEP 3 AND THE FIRST HINT 5) Once you have proven step 4, give a conclusion. "Since $$P_{2}$$ is true, and for some n$$\geq$$2, $$P_{k}$$ is true $$\Longrightarrow$$ $$P_{k+1}$$ is true. By Mathematical Induction, $$P_{n}$$ is true for all n$$\geq$$2." Last edited: Oct 19, 2005 9. Oct 19, 2005 ### dglee Wow thanks a lot for your help. I will try to figure this out. You helped a lot pizzasky and Tide.	{"extraction_info": {"found_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": 0.8393678665161133, "perplexity": 714.6971180550121}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376823442.17/warc/CC-MAIN-20181210191406-20181210212906-00408.warc.gz"}
http://math.stackexchange.com/users/44220/brian?tab=activity	# Brian less info reputation 3 bio website location age 31 member for 2 years, 1 month seen Feb 16 at 6:46 profile views 10 # 8 Actions Feb16 awarded Supporter Oct10 comment Prove by induction $\sum_{i=1}^ni^3=\frac{n^2(n+1)^2}{4}$ for $n\ge1$ My algebra is rusty and the intellectual leap to $\frac{(n+1)^2}{4}[n^2+4(n+1)]$ in the first equality escapes me. Can you explain further? Oct10 revised Prove by induction $\sum_{i=1}^ni^3=\frac{n^2(n+1)^2}{4}$ for $n\ge1$ deleted 3 characters in body Oct10 awarded Editor Oct10 revised Prove by induction $\sum_{i=1}^ni^3=\frac{n^2(n+1)^2}{4}$ for $n\ge1$ Fixed formatting of equalities in second equation Oct10 comment Prove by induction $\sum_{i=1}^ni^3=\frac{n^2(n+1)^2}{4}$ for $n\ge1$ @MichaelHardy Thanks for pointing out my misused arrow. The post has been edited to fix this. Oct10 awarded Student Oct10 asked Prove by induction $\sum_{i=1}^ni^3=\frac{n^2(n+1)^2}{4}$ for $n\ge1$	{"extraction_info": {"found_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": 0.8288465142250061, "perplexity": 2299.5371043094738}, "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-2014-49/segments/1416400380233.64/warc/CC-MAIN-20141119123300-00088-ip-10-235-23-156.ec2.internal.warc.gz"}
https://planetmath.org/limitfunctionofsequence	# limit function of sequence ###### Theorem 1. Let $f_{1},\,f_{2},\,\ldots$ be a sequence of real functions all defined in the interval$[a,\,b]$. This function sequence converges uniformly to the limit function $f$ on the interval $[a,\,b]$ if and only if $\lim_{n\to\infty}\sup\{\|f_{n}(x)-f(x)\|\vdots\,\,a\leqq x\leqq b\}=0.$ If all functions $f_{n}$ are continuous in the interval $[a,\,b]$ and $\lim_{n\to\infty}f_{n}(x)=f(x)$ in all points $x$ of the interval, the limit function needs not to be continuous in this interval; example $f_{n}(x)=\sin^{n}x$ in $[0,\,\pi]$: ###### Theorem 2. If all the functions $f_{n}$ are continuous and the sequence $f_{1},\,f_{2},\,\ldots$ converges uniformly to a function $f$ in the interval $[a,\,b]$, then the limit function $f$ is continuous in this interval. Note. The notion of can be extended to the sequences of complex functions (the interval is replaced with some subset $G$ of $\mathbb{C}$). The limit function of a uniformly convergent sequence of continuous functions is continuous in $G$. Title limit function of sequence LimitFunctionOfSequence 2013-03-22 14:37:45 2013-03-22 14:37:45 pahio (2872) pahio (2872) 22 pahio (2872) Theorem msc 26A15 msc 40A30 LimitOfAUniformlyConvergentSequenceOfContinuousFunctionsIsContinuous PointPreventingUniformConvergence function sequence limit function	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 19, "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": 0.987941563129425, "perplexity": 1288.1540655199094}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202628.42/warc/CC-MAIN-20190322034516-20190322060516-00450.warc.gz"}
http://www.satishkashyap.com/2014/01/video-lectures-on-classical-field.html	## Search this site ### Video Lectures on "Classical Field Theory" by Prof. Suresh Govindarajan sir Video Lecture Series from IIT Professors : Classical Field Theory by Prof. Suresh Govindarajan sir Prof. Suresh Govindarajan Dr. Suresh Govindarajan – Suresh Govindarajan is undoubtedly one of the most brilliant String Theorists in India. He has been at the forefront of String Theory research and has a lot of publications in Superstring Theory and related fields. He had completed his bachelor's in Electrical Engineering from IIT Madras and went on to get a PhD in Physics from University of Pennsylvania. He has worked in many Institutions which are leaders in the field of research including CERN, TIFR and IIT Madras. Currently, he is an Associate Professor in the Dept. Of Physics, IIT Madras. People who have attended his classes will vouch for the fact that he is a wonderful and enthusiastic teacher • High School (1982, Atomic Energy Central School, Hyderabad) • B.Tech. in Electrical Engineering (1986, Indian Institute of Technology Madras) • Ph. D. in Theoretical Physics (1991, University of Pennsylvania) COURSE OUTLINE The course introduces the student to relativistic classical field theory. The basic object is a field (such as the electromagnetic field) which possesses infinite degrees of freedom. The use of local and global symmetries (such as rotations) forms an underlying theme in the discussion. Concepts such as conservation laws, spontaneous breakdown of symmetry, Higgs mechanism etc. are discussed in this context. Several interesting solutions to the Euler-Lagrange equations of motion such as kinks, vortices, monopoles and instantons are discussed along with their applications. The Standard Model of particle physics is used to illustrate how the various concepts discussed in this course are combined in real applications. All necessary mathematical background is provided to make the course self-contained. This course may also be considered as a prelude to Quantum Field Theory. 1. Classical Mechanics, Electromagnetism (and possibly the special theory of relativity). 1. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Pergamon (1975). 2. M. R. Spiegel, Vector Analysis, Schaum Outline Series, McGraw-Hill (1974). 3. M. Carmeli, Classical Fields, Wiley (1982). 4. A. O. Barut, Electrodynamics and Classical Theory of Fields, Chap. 1,Macmillan (1986). 5. C. Itzykson and J. B. Zuber, Quantum Field Theory, Chap. 1, McGraw-Hill (1986). 6. S. Coleman, Aspects of Symmetry, Cambridge Univ. Press. 7. R. Rajaraman, Solitons and Instantons, North-Holland. Module 1 Introduction to Classical Field Theory (1 Lecture) Lecture Number Content of the Lecture Additional Info Lecture 1: What is Classical Field Theory? Review of classical mechanics, Particle Trajectories and the Principle of least action, Feynman's description of QM, Classical Mechanics to Classical Fields. Do Problem Set 1 before viewing Lecture 2! Module 2 Symmetries and Group Theory (6 Lectures) Lecture Number Content of the Lecture Additional Info Lecture 2: Symmetries and Invariances - I Symmetries, Invariances of Newton's EOM vs Maxwell's Equations, The Galilean Group. Lecture 3: Symmetries and Invariances - II Invariances of Maxwell's Equations continued, Common Four Vectors, Covariant Formulation of Maxwell's Equations,Lorentz and Poincare Groups, Rotation Group and vectors under rotation. Attempt Problem Set 2 after viewing Lecs. 2 and 3. Lecture 4: Group Theory in Physics - I Definition of a Group, Antisymmetric Matrices and SO(d),Vectors and Tensors of SO(d), Parity: Polar and Axial Vectors. Solve Problem Set 3 while/after viewing lecs. 4 and 5. Lecture 5 Group Theory in Physics - II Generalizations of SO(d) (specifically the Lorentz Group),Simple Boost Matrices and Rapidity, SO(p,q) with general signatures in metric,The Symplectic Group. 40:30 The matrix should be symmetric and non-degenerate. Symplectic matrices have det=1 Lecture 6: Finite Groups - I Finite Groups of low order : Cyclic and Coxeter(specifically Dihedral) Groups,Definition of a Subgroup, Equivalence relation and Cosets. 49:19 Left coset wrongly called right coset. Corrected at the start of lec. 7 Lecture 7: Finite Groups - II Left and Right Cosets, Permutation Group,Normal Subgroups, Classification of Finite Simple Groups, Monstrous moonshine. Solve Problem Set 4 while/after viewing lecs. 6 and 7. Normal Subgroups Module 3 Actions for Classical Field Theory (3 Lectures) Lecture Number Content of the Lecture Additional Info Lecture 8: Basics of CFT - I Classical Mechanics of Fields, Structure of the KE term in the Lagrangian density, the ultra-local term, and Lorentz invariance of the Lagrangian. Solve Problem Set 5 after viewing lecs. 8 and 9. Lecture 9: Basics of CFT - II Action Principle for fields, Conditions on Lagrangian density for no surface contribution, Conserved Currents,Hamiltonian density, Conditions for Finite Energy. at 50:46 and 51:33 min[finite energy cond — wrong power] Lecture 10: Basics of CFT - III Definition of Vacuum and examples, Vacuum Solutions for quartic potential, Topological Currents and Charges, Noether's Theorem, Application to translational invariance. Solve Problem Set 6 after viewing lec. 10 but before lec. 15 where it is discussed. Module 4 Green Functions for the Klein-Gordon Operator (2 Lectures) Lecture Number Content of the Lecture Additional Info Lecture 11 Green Functions - I Inhomogenous Klein-Gordon Equation, Method of the Green functions, Advanced and Retarded Green Functions. Lecture 12 Green Functions - II Green Functions of the KG operator, Closing the contour,The Feynman propagator. Module 5 Symmetries and Conserved quantities (2 Lectures) Lecture Number Content of the Lecture Additional Info Lecture 13 Noether's Theorem - I Types of Symmetries, Internal Symmetries, Notion of "small", Transformations to first order (for Lorentz Group), Formulation to derive the Master formula. Lecture 14 Noether's Theorem - II Derivation of the Master formula for the Noether current, The energy-momentum tensor and the generalized angular momentum tensor as examples. Solve Problem Set 7 after viewing lec. 14 but before lec. 20 where it is discussed. Module 6 Solitons - I (Kink soliton) (1 lecture) Lecture Number Content of the Lecture Additional Info Lecture 15 Kink Soliton Studying time-independent, finite energy solutions to the Euler-Lagrange equations of motion,the kink soliton, Derrick's theorem and its proof. Module 7 Hidden Symmetry (Spontaneous Symmetry Breaking) & the abelian Higgs mechanism (3 Lectures) Lecture Number Content of the Lecture Additional Info Lecture 16: Hidden Symmetry Spontaneous symmetry breaking and statement of Goldstone's theorem. Lecture 17: Local Symmetries Symmetry breaking continued, The Mermin-Wagner-Coleman theorem,The ideas of global and local symmetries, the covariant derivative,Minimal prescription for the covariant derivative. 23:13 Should change the location of the minus sign in the SO(2) matrix or equivalently take q to −q. Lecture 18 The Abelian Higgs model Definition of field strength using the covariant derivative, Small fluctuations about the vacuum solution,The Higgs mechanism in the U(1) case 29:25: Index mismatch in cov. current μ/ν on LHS/RHS. Module 8 Lie algebras, symmetry breaking and Noether's theorem for Maxwell Equations (2 Lectures) Lecture Number Content of the Lecture Additional Info Lecture 19: Lie Algebras - I Recap of symmetries and Noether's theorem, Lie algebras and finite-dimensional representations, su(2) Lie Algebra. Solve Problem Set 8 while viewing lectures 19/20. Lecture 20 Lie Algebras - II su(3) Lie Algebra ; Symmetry breaking in terms of Lie algebras, Conserved currents for the Proca action: energy-momentum, generalized angular momentum and the symmetric energy-momentum tensors. Module 9 Solitons — II (Magnetic Vortices) (2 Lectures) Lecture Number Content of the Lecture Additional Info Lecture 21: Magnetic Vortices - I Finite energy, time-independent solutions in the Abelian Higgs model in 2+1 dimensions, Topological charge == Magnetic flux, Quantization of magnetic flux, The Bogomol'nyi-Prasad-Sommerfield(BPS) bound for energy, Saturation of the BPS bound. Solve Problem Set 9 before viewing lecture 30. Lecture 22: Magnetic Vortices - II Vortices in the Abelian Higgs model applied to superconducting materials,characteristic lengths in the problem, "size" of a vortex, Description of vortex number using the fundamental group of the gauge group U(1), or the circle. 35:38 and 36:03 min[finite energy cond — wrong power] Module 10 Towards Non-abelian gauge theories (2 Lectures) Lecture Number Content of the Lecture Additional Info Lecture 23: Non-abelian gauge theories - I Non-abelian gauge symmetry with SU(2) as an example, Covariant derivative in the non-abelian case, Construction of a locally SU(2) invariant Lagrangian, Transformation of the gauge fields under local gauge transformations. Solve Problem Set 10 while viewing lectures 23/24. Lecture 24: Non-abelian gauge theories - II Transformation of the gauge fields(continued), Derivation of the field strength for the gauge field, Symmetry breaking in the non-abelian case, Goldstone's theorem in terms of Lie Algebras. Module 11 Representation theory of Lie Algebras (2 Lectures) Lecture Number Content of the Lecture Additional Info Lecture 25: Irreps of Lie algebras - I Representation theory of su(2) and su(3), the Cartan subalgebra, the adjoint representation. 3:00 - Misleading statement: Map from G to GL(N). While GL(N) is set of all linear maps on V, the map from G to GL(N) is not linear.28:00 - Blocks "0" and "*" in the matrix have been interchanged. Lecture 26: Irreps of Lie algebras - II Representation theory continued, Ferrer's diagrams. Quiz (Test yourself) If you have gotten this far, you can test you understanding by taking this Quiz. It is meant to be an open notes (i.e., your own notes) examination and the expected duration is one and half hours. I don't intend to post the solutions online but they will be provided on request. This is to counter the natural human tendency to look at solutions if they are available! What is a good score? I would say anything over 50% is acceptable. Module 12 The Standard Model of Particle Physics (2 Lectures) Lecture Number Content of the Lecture Additional Info Lecture 27 The Standard Model - I su(3) multiplets, Motivation for the Standard Model, Colour confinement, Gell-Mann—Nishijima relation. Lecture 28 The Standard Model - II Electroweak symmetry breaking — An application of symmetry breaking in the non-abelian case. Module 13 The Lorentz and Poincare Lie Algebras (1 Lecture) Lecture Number Content of the Lecture Additional Info Lecture 29: Irreps of the Lorentz/Poincare algebras The Lorentz and Poincare algebras and their representations. Module 14 Solitons — III (Monopoles and Dyons) (3 Lectures) Lecture Number Content of the Lecture Additional Info Lecture 30: The Dirac mononpole Magnetically charged solutions: The Dirac monopole, Flux quantization. Solve Problem Set 11 while viewing lectures 30/31/32. Lecture 31: The 't Hooft-Polaykov monopole Magnetically charged solutions: The ‘t Hooft-Polyakov monopole, The Prasad-Sommerfield limit. Lecture 32: Revisiting Derrick’s Theorem Revisiting Derrick's theorem, BPS solution Lecture 33: The Julia-Zee dyon Constructing dyonic solutions, Dirac quantization for dyons; Dimensional reduction. Module 15 Instantons and their physical interpretation (4 Lectures) Lecture Number Content of the Lecture Additional Info Lecture 34: Instantons - I Quantum mechanical tunnelling and Instantons. Solve Problem Set 12 while viewing lectures 34/37. Lecture 35:\| Instantons - II Kink soliton and tunnelling, Instantons in pure Yang-Mills theories(SU(2)). Lecture 36: - Instantons - III More on instantons, The BPS bound. Lecture 37: Instantons - IV Free parameters in instanton solutions, moduli space, Complexified Yang-Mills and theta vacua. Module 16 An introduction to some advanced topics (2 Lectures) Lecture Number Content of the Lecture Additional Info Lecture 38: Dualities Dualities in Field Theory: Ising Model; Sine-Gordon / Massive Thirring; SU(2) Yang-Mills in 3+1 dimensions. Lecture 39: Geometrization of Field Theory General relativity as a gauge theory; Geometrization of Field Theory; Glimpse into String theory and branes. The Final If you have gotten this far, you can test your understanding (of the course material) by taking this Final Examination. It is meant to be an open notes (i.e., your own notes) examination and the expected duration is three hours. I don't intend to post the solutions online but they will be provided on request.	{"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": 0.8501892685890198, "perplexity": 2096.7684134004176}, "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/1498128319265.41/warc/CC-MAIN-20170622114718-20170622134718-00566.warc.gz"}
http://math.stackexchange.com/questions/37088/integration-doubt	# Integration doubt How do I integrate $\sin(x)/x$? I tried using integration by parts, but it led me to nowhere. Please help. - You have to use the (nonelementary) sine integral in the general case; however, if you're interested in the limit from 0 to $\infty$, there is an m.SE question devoted to that topic... – Ｊ. Ｍ. May 5 '11 at 3:03 The function $f(x)=\sin(x)/x$ does not admit an elementary antiderivative, i.e., there is no formula for its integral (using quotients of polynomials, trig. functions, logartithms, exponentials, i.e., the usual functions you study in calculus). Symbolic integration is the part of calculus that deals with finding antiderivatives. There is a fairly sophisticated algorithm due to Risch and implementing it shows that there is no nice formula for $\displaystyle \int\frac{\sin(x)}x dx$. The algorithm is sufficiently elaborate that apparently no software package can currently find antiderivatives for all functions for which it is possible. The Wikipedia page I linked to has references to the original (and nice) paper. A few years ago, Matthew Wiener posted a fairly readable account of the algorithm on sci.math; here is a pdf of the post. For a nice full length exposition of the mathematics involved, I highly recommend the book by Manuel Bronstein,"Symbolic Integration 1 (transcendental functions)" (2 ed.), 1997, Springer-Verlag. Now, not all is bad news here: One can integrate term by term the power series for $\sin(x)/x$ expression and obtain the power series of its antiderivative, (that converges everywhere), and there are numerical methods to approximate very decently this function. Finally, one can compute explicitly (for example, using methods of complex analysis) that $$\int_0^\infty\frac{\sin(x)}x dx=\frac{\pi}2.$$ - The pdf was typed by Apollo Hogan. I'm afraid I lost the link to the original post. – Andres Caicedo May 5 '11 at 3:36 here is the original sci.math post. – Ｊ. Ｍ. May 7 '11 at 3:22 @J.M. : Thanks! – Andres Caicedo May 12 '11 at 16:37 – Andres Caicedo Apr 12 '13 at 2:50 There is no indefinite integral that can be written in elementary functions. However, as sometimes happens, the definite integral on certain endpoints is known; see: Solving the integral $\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$? - If you ask Wolfram Alpha, it will tell you that the integral is $\text{Si}(x)+C$. If you ask it what is $\text{Si}(x)$, it will tell you, among other things, that $$\text{Si}(x)=\int_0^x \frac{\sin t}{t}dt$$ Not very helpful! But one can get some information out of all this unhelpfulness. If there were an expression for your integral in terms of elementary functions, Wolfram Alpha, which is really pretty good, would very likely produce such an expression. And indeed it can be proved that there is no such expression. But the integral that you want shows up naturally in a number of applications, for example in optics. So it is convenient to have a name for it, and $\text{Si}(x)$ is as far as I know the only one in common use. Some definite integrals involving $\sin(x)/x$ can be evaluated explicitly, but of course not by the usual technique of finding an indefinite integral and then substituting. There is nothing particularly mysterious about a function given by a simple formula not having an indefinite integral given by a combination of elementary functions. In fact "most" elementary functions do not have an elementary antiderivative. - To add: one should consider him/herself incredibly lucky if an integral s/he encounters in applications has a (simple) closed form. – Ｊ. Ｍ. May 5 '11 at 5:49	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9713327288627625, "perplexity": 455.98392175855975}, "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/1398223210034.18/warc/CC-MAIN-20140423032010-00008-ip-10-147-4-33.ec2.internal.warc.gz"}
https://arxiv.org/abs/1204.3752	cs.IT (what is this?) # Title:GPS Information and Rate Tolerance - Clarifying Relationship between Rate Distortion and Complexity Distortion Authors:Chenguang Lu Abstract: I proposed rate tolerance and discussed its relation to rate distortion in my book "A Generalized Information Theory" published in 1993. Recently, I examined the structure function and the complexity distortion based on Kolmogorov's complexity theory. It is my understanding now that complexity-distortion is only a special case of rate tolerance while constraint sets change from fuzzy sets into clear sets that look like balls with the same radius. It is not true that the complexity distortion is generally equivalent to rate distortion as claimed by the researchers of complexity theory. I conclude that a rate distortion function can only be equivalent to a rate tolerance function and both of them can be described by a generalized mutual information formula where P(Y\|X) is equal to P(Y\|Tolerance). The paper uses GPS as an example to derive generalized information formulae and proves the above conclusions using mathematical analyses and a coding example. The similarity between the formula for measuring GPS information and the formula for rate distortion function can deepen our understanding the generalized information measure. Comments: 6 pages, 4 figures Subjects: Information Theory (cs.IT); Computational Complexity (cs.CC) ACM classes: H.1.1; H.1.2; I.4.2; I.5.0 Cite as: arXiv:1204.3752 [cs.IT] (or arXiv:1204.3752v1 [cs.IT] for this version) ## Submission history From: Chenguang Lu [view email] [v1] Tue, 17 Apr 2012 10:48:14 UTC (311 KB)	{"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": 0.8961293697357178, "perplexity": 1652.342226637382}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583657510.42/warc/CC-MAIN-20190116134421-20190116160421-00440.warc.gz"}
https://www.kasal.com/long-table-trellis-la-palma	# “The Long Table” from Under the Trellis to “La Palma” 0 66 views Wedding banquets were always clarion calls for gathering kith and kin. No table was ever long enough to seat all. Hence the conjoined table boards to make one long table. The term has come to denote a nuptial feast. Where were the “long tables” spread in Grandmother’s time? Home was the traditional venue. Those cavernous elegant mga bahay na bato (stone houses) could contain the largest of social gatherings. There were armies of servants to attend the guests. To this day, in town and barrio, there’s a leafy trellis beside the bahay kubo shading the open-air long table. In the city, however, receptions at hotels and restaurants in time became fashionable. From twilight of the reign of Spain to morning of American Empire days en grande receptions were invariably at La Palma de Mallorca, a famous Spanish hostelry in Intramurous until well into the Twenties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Source: Alvina, C. & Sta. Maria, F. 1987. Essays on Philippine Culture.	{"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": 0.8083345890045166, "perplexity": 1582.9664896851104}, "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/1524125948426.82/warc/CC-MAIN-20180426164149-20180426184149-00136.warc.gz"}
https://emacs.stackexchange.com/questions/32347/how-to-have-wrapped-text-when-exporting-from-org-to-latex	# How to have wrapped text when exporting from org to Latex? I have an org file which I export to LaTeX and then to a PDF document. The problem is I have some long text that exceeds the length of the page. How to wrap it so that it falls into the next line? First example: ``````#+BEGIN_SRC c++ <code goes here> // very long comment that doesn't wrap ........ #+END_SRC `````` The comment is very long and exceeds the length of the page, how to make it wrap? Second example: ``````\|------+------+------------------------+------\| \| text \| text \| text \| text \| \|------+------+------------------------+------\| \| text \| text \| very long texttt...... \| text \| \|------+------+------------------------+------\| `````` Some cells contain long text which also doesn't wrap, how to make it wrap? From a LaTeX point of view, these are different cases. In the first case, LaTeX isn't going to wrap -- and it shouldn't! A "verbatim" environment, which is what source-code is set in, respects lines precisely and doesn't break paragraphs, because LaTeX simply can't know where to break them. If you want to wrap the lines, you have to wrap them in the source-code itself. In the second case, you can use `#+ATTR_LATEX` to help LaTeX format the table appropriately. Simplest but least elegant approach: use `:align` with a `p` column to specify a width for the problematic column. LaTeX will then wrap that column to the specified width. So this specifies three columns: one aligned left, one a `p` column with a fixed width of 4cm, and another aligned left. ``````#+ATTR_LATEX: :align lp{4cm}l `````` More elegant. Use `tabularx` and a `X` column, and LaTeX will then set the column to a length appropriate to enable a table of a fixed overall width. You need to `(add-to-list 'org-latex-packages-alist '("" "tabularx"))` I think. ``````#+ATTR_LATEX: :environment tabularx :width \textwidth :align lXl \| column \| column with very very very very very overlong text which would flow over \| last \| \| column \| shorter column \| column \| \| column \| shorter column \| column \| ``````	{"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": 0.9239920377731323, "perplexity": 3307.5081789363944}, "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/1573496671548.98/warc/CC-MAIN-20191122194802-20191122223802-00130.warc.gz"}
http://forums.xkcd.com/viewtopic.php?f=17&t=68643	## The Shortest String Containing all Permutations of n Symbols For the discussion of math. Duh. Moderators: gmalivuk, Moderators General, Prelates NathanielJ Posts: 882 Joined: Sun Jan 13, 2008 9:04 pm UTC ### The Shortest String Containing all Permutations of n Symbols Here's an interesting (to me, anyway) combinatorics problem that has a reasonably intuitive answer that I can't for the life of me prove is correct or find a proof of. What is the length of the shortest string that contains every permutation of the digits 1,2,...,n as substrings? For example, when n = 2 a minimal string is 121, which contains each of "12" and "21" as substrings. When n = 3, a minimal string is 123121321, which contains each of "123", "132", "213", "231", "312", and "321" as substrings. The length of the shortest such string has been conjectured to be equal to$\sum_{k=1}^n k!$See the OEIS, MathExchange, StackOverflow and this PDF for similar posts by people asking about this question. Strangely, these are the only sources that I can find that mention this problem, and they all are from last year. Those posts all go into how to construct a string that by all rights seems optimal, and all of those sources conjecture that the given string is indeed of minimal length, but not one of them actually proves it. And I can't seem to prove it either, despite the fact that it looks like it should fall to a simple induction argument. So my question is -- is the conjecture true? Is the conjectured "optimal" string of length [imath]\sum_{k=1}^n k![/imath] actually optimal? I believe the reason the conjecture is more difficult to prove than would be expected is because there seems to be some sort of shift in behaviour of the optimal strings around n = 5 or n = 6, at which point it becomes infeasible to brute force what the optimal strings are, and the strings are so long as to be difficult to work with. For example, via brute-force you can show that for n = 1,2,3,4 the optimal string has length 1,3,9,33, follows the construction given in the posts linked above, and furthermore is unique (up to requiring that the string starts with "123...n"). However, when n = 5 there are two optimal strings that start with "12345", and when n = 6 there are at least 96 distinct strings of the conjectured minimal length. So does anyone have any insights on this problem or any other references that I'm not aware of? Cheers. Homepage: http://www.njohnston.ca Conway's Game of Life: http://www.conwaylife.com Charlie! Posts: 2033 Joined: Sat Jan 12, 2008 8:20 pm UTC ### Re: The Shortest String Containing all Permutations of n Sym Hm. What properties could we expect this minimal string to have? Should we expect it to have no repetitions of length n? It cannot satisfy the rule that each substring of length n has every character, since then it would just be repetitions of one template. Can we find some rule governing how many "invalid" substrings the shortst string must have? Some people tell me I laugh too much. To them I say, "ha ha ha!" NathanielJ Posts: 882 Joined: Sun Jan 13, 2008 9:04 pm UTC ### Re: The Shortest String Containing all Permutations of n Sym Charlie! wrote:Hm. What properties could we expect this minimal string to have? Should we expect it to have no repetitions of length n? It cannot satisfy the rule that each substring of length n has every character, since then it would just be repetitions of one template. Can we find some rule governing how many "invalid" substrings the shortst string must have? The shortest string seems to be the one obtained in this way (I will demonstrate the construction for n = 4, but it extends naturally for higher n): Start with "1234" and notice that if we tack a "1" on the end then we have added another permutation with only one extra symbol. Then adding "2" gives another permutation at the cost of only one more symbol, and so on until we get "1234123". But now we can't add another permutation to the string by adding only one symbol -- we have to add two symbols. So we get "123412314". And now we can go back to adding just one symbol 3 times to get "123412314231", which contains three more permutations. And then again we have to add 2 more symbols to add another permutation. And so on. If we continue in this way naively using this "greedy" algorithm where we just add as few symbols at a time as possible, we end up with the string "123412314231243121342132413214321", which is in fact known to be the unique optimal string for n = 4. For n = 4, the number of symbols that you need to add to the string to get one more permutation are as follows: 1 1 1 2 1 1 1 2 1 1 1 3 1 1 1 2 1 1 1 2 1 1 1 There is a clear pattern here that is not difficult to generalize, but the difficulty is proving that this pattern is indeed optimal in general -- it could be the case (for large n) that if we increase one of the early numbers (i.e., we don't use the greedy algorithm but rather add some extra length early on), it allows us to decrease later numbers by a larger factor, thus shortening the overall string. We can simplify this a bit even. Notice that the optimal string for n = 4 can be broken into chunks of length 2n-1 = 7 as follows: Code: Select all 1234123142312431213421324132143211234123 2314231 3124312 2134213 1324132 3214321 Each chunk has the following two properties: (1) The middle digit of each chunk is n (=4 in this case). (2) The first n-1 symbols of the chunk are the same as the last n-1 symbols of the chuck (including order) and are a permutation of the symbols 1,2,...,n-1. If we can prove that the optimal string can always be broken down into chunks of length 2n-1 that satisfy those properties, then we can prove many nice things (such as the length of the optimal strings, and even a formula for how many optimal strings there are). Homepage: http://www.njohnston.ca Conway's Game of Life: http://www.conwaylife.com undecim Posts: 289 Joined: Tue Jan 19, 2010 7:09 pm UTC Contact: ### Re: The Shortest String Containing all Permutations of n Sym Blue, blue, blue skullturf Posts: 556 Joined: Thu Dec 07, 2006 8:37 pm UTC Location: Chicago Contact: ### Re: The Shortest String Containing all Permutations of n Sym undecim wrote:http://en.wikipedia.org/wiki/De_Bruijn_sequence Note that this includes all words of specified length, including those where symbols are repeated. NathanielJ Posts: 882 Joined: Sun Jan 13, 2008 9:04 pm UTC ### Re: The Shortest String Containing all Permutations of n Sym How did I know that one of the first replies would be to the Wikipedia article on de Bruijn sequences? The de Bruijn diagram approach doesn't work because there is no way to concatenate all of the permutations next to each other with overlap n-1 each. After you use n of the permutations, you run out of options for what the next symbol in the string should be. Homepage: http://www.njohnston.ca Conway's Game of Life: http://www.conwaylife.com ++\$_ Mo' Money Posts: 2370 Joined: Thu Nov 01, 2007 4:06 am UTC ### Re: The Shortest String Containing all Permutations of n Sym The more general term for this kind of object is "universal sequence". (However, universal sequences are often forbidden from having repeats -- that is, each structure would have to appear exactly once in the sequence.) In general, not much is known about them, so any solution to this problem is probably worth publishing. In a related problem, it has been proven that when n > 2, a universal cycle of length n! (that is, a universal sequence that eats its own tail) for permutations in [imath]S_n[/imath] must have at least n+1 distinct characters in its alphabet, and n+1 is achievable for all n. silverhammermba Posts: 178 Joined: Fri Oct 13, 2006 1:16 am UTC ### Re: The Shortest String Containing all Permutations of n Sym NathanielJ wrote:There is a clear pattern here that is not difficult to generalize, but the difficulty is proving that this pattern is indeed optimal in general -- it could be the case (for large n) that if we increase one of the early numbers (i.e., we don't use the greedy algorithm but rather add some extra length early on), it allows us to decrease later numbers by a larger factor, thus shortening the overall string. Perhaps if we think about the number of "uses" of each character i.e. the number of unique permutations it is part of in the sequence. The greedy algorithm tries to maximize the use of each character, whereas the naive algorithm (concatenating all permutations) minimizes the use of each character. Some usage assignments would be impossible and perhaps we could find a restriction on the assignments that would force the string to be of the desired length. For your n=3 example (123121321), the uses we get are 123222321 which also implies a pattern. If we can find restrictions on the usage patterns, it could provide a way to show that the uses cannot be more efficiently packed. Here are the simple restrictions I've come up with so far: 1. The number of a character's uses is less than or equal to the number of length n substrings it is contained in 2. The total uses of a string must sum to nn! With those restrictions alone, for n=3 we would hope to get a string with the following uses 12333321 which is one character shorter than optimal, so hopefully we could find more restrictions on uses that would force the string to meet the hypothesized optimal length. jtillots Posts: 2 Joined: Wed Apr 20, 2011 1:59 pm UTC ### Re: The Shortest String Containing all Permutations of n Sym I along with my math professor Dan Ashlock wrote a paper about this very topic back in 1993. We called the paper Construction of Small Superpermutations and Minimal Injective Superstrings. We found the length of the shortest strings to be the sum(k!) as k goes from 1 to n. Note there are also n! such strings since you can replace 1 with 2 and 2 with 1, etc. and still have a valid shortest string. We also found algorithms for deriving the shortest strings, but we couldn't prove that our strings were shortest. We used an exhaustive computer search to prove that the algorithm generated the shortest string up to 11, but that's as far as we went. I've been trying for years to get a mathematician to take up this problem. I'm a computer scientist and cognitive scientist and don't know enough math to prove this. I've been told by many mathematicians that this problem is easy and they've generated the proof before, but no one can produce the proof. I think the problem is deceptively simple which causes mathematicians to blow it off. I periodically check to see if anyone has done any research on this topic. Thanks for the post here! It made my day. I'd also be happy to send more information. I'm new to this forum, but ping me on here if you can or leave a message in this comment. Jenett NathanielJ Posts: 882 Joined: Sun Jan 13, 2008 9:04 pm UTC ### Re: The Shortest String Containing all Permutations of n Sym jtillots wrote:I along with my math professor Dan Ashlock... Oh wow that's weird -- what a small world, Dan is a professor at my university (U of Guelph). I had no idea that he had worked on this problem! jtillots wrote:We also found algorithms for deriving the shortest strings, but we couldn't prove that our strings were shortest. We used an exhaustive computer search to prove that the algorithm generated the shortest string up to 11, but that's as far as we went. I would be very interested in seeing the algorithm for searching the space, as I've only been able to prove optimality up to n = 6; by n = 11 things are quite huge. jtillots wrote:I've been told by many mathematicians that this problem is easy and they've generated the proof before, but no one can produce the proof. I think the problem is deceptively simple which causes mathematicians to blow it off. Yep, that's been my experience with the problem too. I even found one mathematics competition that contained the problem, and then the "solutions" just hand-waved it by showing that a string of that length always exists, but they didn't prove optimality. Anyway, I would love to see any information about this problem that you have (including the paper that you co-authored with Dan ). You can e-mail me at njohns01(AT)uoguelph.ca if you don't want to post them here. Thanks, and all the best. Homepage: http://www.njohnston.ca Conway's Game of Life: http://www.conwaylife.com jtillots Posts: 2 Joined: Wed Apr 20, 2011 1:59 pm UTC ### Re: The Shortest String Containing all Permutations of n Sym I think I heard about that mathematics competition. I was talking to someone at a party about this problem and they told me about a math competition where this was one of the problems. I got super excited and asked what the proof was, he said, "Oh, I don't remember. But it was simple." sigh* I'm sorry it's taken me so long to reply to you. I don't frequent these forums very often - only when I'm looking for more info about super permutations. I've sent you some email. I'd love to see this problem move forward. Jenett tomtom2357 Posts: 563 Joined: Tue Jul 27, 2010 8:48 am UTC ### Re: The Shortest String Containing all Permutations of n Sym If these results are true, then you have an upper bound on the string, and you're conjecturing that the correct value is that upper bound. An easy lower bound is n!+n-1, but this can be improved slightly to n!+n for n>2, because after the first n symbols (which without loss of generality we can define them to be 123...n) one of the next n symbols must not make a new permutation. I am going to try to extend this in some way to make the lower bound something of the form n!+O((n-1)!). Edit: I have found a better lower bound on the length of the optimal string. I can prove that at least n!+(n-1)!+n-2 symbols are needed, by using the argument that for every n permutations in the string, there is at least one symbol after them in the string which make the sub-string of all n symbols behind that one, including itself, not be a new permutation. The n-1 comes from the fact that the first n-1 symbols do not add any new permutations without the later symbols. The -1 comes from the fact the last permutation does not have any symbols after it, therefore there is no non-permutation sub-string ending in a symbol after that. This lower bound is exact only for n<=3, so I might have to change my line of reasoning from here. Edit 2: I realized how I can improve my bound on the number of symbols needed. If the number of symbols used is exactly the lower bound, then there are exactly n permutations for every symbol that does not make a new permutation (not including the first n-1 symbols). Therefore, assuming that the string starts with 123...n, then we can generate the sequence. Setting n=4, and generating the sequence the way mentioned above gives 1234123142312431234..., which is clearly repetitive, and does not give every permutation. Therefore, there must be a corrective symbol in the string that makes the string non-repetitive, so the shortest string for n=4 is 33. I am working on generalizing this to give a lower bound of the form n!+(n-1)!+O((n-2)!). I can see that if I keep thinking, I can probably come up with bounds that contain arbitrarily many factorials in it, so all I need is an inductive proof to prove that the lower bound is the upper bound. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain. PM 2Ring Posts: 3617 Joined: Mon Jan 26, 2009 3:19 pm UTC Location: Mid north coast, NSW, Australia ### Re: The Shortest String Containing all Permutations of n Sym FWIW, Martin Gardner wrote about a closely related problem that used circular lists rather than linear ones. In his article, the puzzle was described in terms of different coloured beads on a bracelet. Sorry I can't remember any details, I read the article several decades ago. EDIT: After a bit of Googling, it appears that the bracelet in Gardner's puzzle only used beads of two colours, and it contains all possible bit strings upto a certain length, so that's probably not very helpful. Oh well. dudiobugtron Posts: 1098 Joined: Mon Jul 30, 2012 9:14 am UTC Location: The Outlier ### Re: The Shortest String Containing all Permutations of n Sym NathanielJ wrote:it looks like it should fall to a simple induction argument. I imagine the induction step would be something like: Since there are (n+1)! permutations, there must be at least [size of min string for n] + (n+1)! characters in a minimal string for n+1, otherwise we could do some fancy trick and remove (n+1)! characters and have a minimal string for the digits 1,...,n, which would be a contradiction. I haven't thought about it too much though but if I had more time that is the way I'd try to go about it. dudiobugtron Posts: 1098 Joined: Mon Jul 30, 2012 9:14 am UTC Location: The Outlier ### Re: The Shortest String Containing all Permutations of n Sym Yup. I think this is the outline of a viable proof, although I have made more than a few appeals to 'obviousness' which people are free to dispute!!! I've spoiler tagged it because it's a bit long and has a few sum symbols.: Spoiler: Base case: done, by others above. Background: Consider the permutations of the digits 1,...,n+1. There are (n+1)! of these. If you remove all of the digits 'n+1' from each permutation, you will end up with n+1 copies of each of the n! permutations of the digits of 1,...,n. (One copy for each of the n+1 positions the 'n+1' digit could occupy.) Consider a string, S_(n+1), containing all (n+1)! permutations of the digits 1,...,n+1. Each digit 'n+1' in the string can be a part of at most n+1 of these permutations (since the permutations are of length n+1), so the minimum number of 'n+1' digits in the string is (n+1)! / (n+1) = n! If you remove all of the digits 'n+1', the string will then (as above) contain at least n+1 copies of each of the n! permutations of the digits 1,...,n. It seems apparent that we can safely remove n.n! digits (corresponding to the n.n! extra copies) and still have all n! of the permutations of the digits 1,...,n contained in the string. Removing these n.n! digits, along with the (at least) n! 'n+1' digits gives a total of n! + n.n! = (n+1).n! = (n+1)! Therefore: Lemma: Given a string S_(n+1) containing all permutations of the digits 1,...,n+1 as substrings, it is possible to remove (n+1)! digits and be left with a string which contains all of the permutations of the digits 1,...,n as substrings. Induction step: Assume that for every string, S_n, containing every permutation of the digits 1,...,n as substrings, the length of S_n is at least: $\sum_{k=1}^n k!$ Now, assume that there is a string, S(n+1), which contains every permutation of the digits 1,...,n+1 as substrings, and that the length of S_(n+1) is less than: $\sum_{k=1}^{n+1} k! = \sum_{k=1}^n k! + (n+1)!$ Using the above Lemma, we can remove (n+1)! of the digits and still have a string which contains all of the permutations of the digits 1,...,n as substrings. But the length of this string is less than: $\sum_{k=1}^n k!$ Which is a contradiction. Therefore any string, S(n+1), which contains every permutation of the digits 1,...,n+1 as substrings, must be of length at least: $\sum_{k=1}^{n+1} k!$ /proof CCC Posts: 46 Joined: Thu Jun 14, 2012 12:29 pm UTC ### Re: The Shortest String Containing all Permutations of n Sym dudiobugtron wrote:If you remove all of the digits 'n+1', the string will then (as above) contain at least n+1 copies of each of the n! permutations of the digits 1,...,n. I think that's not quite right. Consider the sequence of digits 41234. This contains two unique length-four sequences (4123 and 1234), but removing the 4s leaves us with only one copy of the base length-three sequence. This only applies to the case where (n+1) is before and after the same sequence of every other digit; so the string as a whole will still contain at least n copies of each of the n! permutations. tomtom2357 Posts: 563 Joined: Tue Jul 27, 2010 8:48 am UTC ### Re: The Shortest String Containing all Permutations of n Sym CCC wrote: dudiobugtron wrote:If you remove all of the digits 'n+1', the string will then (as above) contain at least n+1 copies of each of the n! permutations of the digits 1,...,n. I think that's not quite right. Consider the sequence of digits 41234. This contains two unique length-four sequences (4123 and 1234), but removing the 4s leaves us with only one copy of the base length-three sequence. This only applies to the case where (n+1) is before and after the same sequence of every other digit; so the string as a whole will still contain at least n copies of each of the n! permutations. I think that given his assumption that the string contains all (n+1)! permutations of the digits 1 to n+1, then if you remove all the n+1's there is at least n+1 of each permutation of the digits from 1 to n. I agree with everything except: dudiobugtron wrote:It seems apparent that we can safely remove n.n! digits (corresponding to the n.n! extra copies) and still have all n! of the permutations of the digits 1,...,n contained in the string. Removing these n.n! digits, along with the (at least) n! 'n+1' digits gives a total of n! + n.n! = (n+1).n! = (n+1)! If you could prove this, I would be happy with your proof. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain. Deedlit Posts: 91 Joined: Sun Mar 08, 2009 2:55 am UTC ### Re: The Shortest String Containing all Permutations of n Sym tomtom2357 wrote: CCC wrote: dudiobugtron wrote:If you remove all of the digits 'n+1', the string will then (as above) contain at least n+1 copies of each of the n! permutations of the digits 1,...,n. I think that's not quite right. Consider the sequence of digits 41234. This contains two unique length-four sequences (4123 and 1234), but removing the 4s leaves us with only one copy of the base length-three sequence. This only applies to the case where (n+1) is before and after the same sequence of every other digit; so the string as a whole will still contain at least n copies of each of the n! permutations. I think that given his assumption that the string contains all (n+1)! permutations of the digits 1 to n+1, then if you remove all the n+1's there is at least n+1 of each permutation of the digits from 1 to n. But I think, because of CCC's objection, that statement remains unproven. There are n+1 permutations of length n+1 that collapse to the same permutation of length n, but as CCC said, two such permutations could be the same permutation when you remove n+1. I agree with everything except: dudiobugtron wrote:It seems apparent that we can safely remove n.n! digits (corresponding to the n.n! extra copies) and still have all n! of the permutations of the digits 1,...,n contained in the string. Removing these n.n! digits, along with the (at least) n! 'n+1' digits gives a total of n! + n.n! = (n+1).n! = (n+1)! If you could prove this, I would be happy with your proof. I agree that this is a problem. dudiobugtron Posts: 1098 Joined: Mon Jul 30, 2012 9:14 am UTC Location: The Outlier ### Re: The Shortest String Containing all Permutations of n Sym Deedlit wrote: tomtom2357 wrote: CCC wrote: dudiobugtron wrote:If you remove all of the digits 'n+1', the string will then (as above) contain at least n+1 copies of each of the n! permutations of the digits 1,...,n. I think that's not quite right. Consider the sequence of digits 41234. This contains two unique length-four sequences (4123 and 1234), but removing the 4s leaves us with only one copy of the base length-three sequence. This only applies to the case where (n+1) is before and after the same sequence of every other digit; so the string as a whole will still contain at least n copies of each of the n! permutations. I think that given his assumption that the string contains all (n+1)! permutations of the digits 1 to n+1, then if you remove all the n+1's there is at least n+1 of each permutation of the digits from 1 to n. But I think, because of CCC's objection, that statement remains unproven. There are n+1 permutations of length n+1 that collapse to the same permutation of length n, but as CCC said, two such permutations could be the same permutation when you remove n+1. It's obvious that any two permutations which share the same n+1 digit will 'contract' to two distinct permutations like I want them to. I had been assuming that all the interesting cases were of this sort. However, I also think CCC is right, that the issue is when you have two permutations that share all their digits except n+1. I'm thinking that these situations would be the result of 'excess' n+1 digits (ie: more than the 'n!' minimum I assumed in my 'proof'), which can be removed to compensate for the lost permutations. Also, you can really remove all occurrences of any digit you like (it doesn't have to be n+1, since the digits are all interchangible). If all of the digits had 'excess' occurrences like this, then the string probably isn't minimal. I know that's not a proof though! tomtom2357 wrote:I agree with everything except: dudiobugtron wrote:It seems apparent that we can safely remove n.n! digits (corresponding to the n.n! extra copies) and still have all n! of the permutations of the digits 1,...,n contained in the string. Removing these n.n! digits, along with the (at least) n! 'n+1' digits gives a total of n! + n.n! = (n+1).n! = (n+1)! If you could prove this, I would be happy with your proof. Yeah I wasn't happy with that part either. I realised after posting the proof that you don't actually have to prove that you can remove the digits, you merely have to prove that the size of a string which contains p more permutations than a 'minimum string' is atleast p greater. That's still not that easy either (even though it seems intuitively correct), so I'll think about it some more (unless someone beats me to it!) CCC Posts: 46 Joined: Thu Jun 14, 2012 12:29 pm UTC ### Re: The Shortest String Containing all Permutations of n Sym dudiobugtron wrote:It's obvious that any two permutations which share the same n+1 digit will 'contract' to two distinct permutations like I want them to. I had been assuming that all the interesting cases were of this sort. However, I also think CCC is right, that the issue is when you have two permutations that share all their digits except n+1. I'm thinking that these situations would be the result of 'excess' n+1 digits (ie: more than the 'n!' minimum I assumed in my 'proof'), which can be removed to compensate for the lost permutations. Hmmm. Let's take a look at the earlier-provided length-4 sequence: 123412314231243121342132413214321 In this case, (n+1)=4. Now, there's no case where a pair of sequences share every number but 4 (the 4s are all spaced just a little too widely for that). And every number except 4 does indeed occur enough times to make up for the lost permutations. But let us consider briefly the consequence of removing the threes: 123412314231243121342132413214321 Note the sequence "34123" near the start. Let's look at the 412's after removing all the 3's: 1241214212412142124121421 There are only three such sequences, less than (n+1). (In fact, since there are only six fours, there's only three of each possible three-unique-digit sequence). Hmmm. That sequence is... usefully repetitive. After the first eight digits, it simply repeats itself. Since it has the minimum possible number of 4s (i.e. six of them), I don't think there's any option; it does have to include at least n of every sequence, n=3, and the sequences 412 and 421 cannot use the same 4. (Similarly for the pairs (241; 142) and (214; 124)). NathanielJ Posts: 882 Joined: Sun Jan 13, 2008 9:04 pm UTC ### Re: The Shortest String Containing all Permutations of n Sym Be careful when assuming that the optimal string must have a certain form based on the optimal strings in the n = 1, 2, 3, 4 cases. While there are indeed exactly (n-1)! instances of the digit n for those cases, and there is tons of symmetry to be found, that breaks down for n >= 5. When n = 5, I have two fundamentally different optimal strings: one that has 24 '5's and is symmetric like we would hope... and then one that has 27 '5's and isn't symmetric. Homepage: http://www.njohnston.ca Conway's Game of Life: http://www.conwaylife.com dudiobugtron Posts: 1098 Joined: Mon Jul 30, 2012 9:14 am UTC Location: The Outlier ### Re: The Shortest String Containing all Permutations of n Sym I've been thinking about this for a while (on-and-off!), and although I haven't gotten anywhere, I thought I'd at least share my approaches in case they help anyone else do something more useful. The main idea I was trying was to look at the underlying string of permutations of n symbols, namely p_1, p_2, ..., p_(n!) , ordered by their first (and only?) appearance in the main string. I was looking at the case where the underlying string could be partitioned into (n-1)! sets of n permutations, {p_(in),...,p_((i+1)n-1)}, where each p_(jn) differed from p_((j-1)n) by merely moving the starting symbol to the end (and of course similarly for each p_(in) and p_((i+1)n-1)). I was trying to prove that finding the 'best' way for ordering these sets of partitions was suitably analogous to finding the best way for ordering the permutations of (n-1) symbols. And also of course that such a process gave a best possible ordering for the partitions of n symbols. Another related idea I tried, based on this idea as well: NathanielJ wrote:For n = 4, the number of symbols that you need to add to the string to get one more permutation are as follows: 1 1 1 2 1 1 1 2 1 1 1 3 1 1 1 2 1 1 1 2 1 1 1 was to give a value to each of those permutations based on how many symbols needed to be added to put that permutation in the string. For the example above, this would be: 4 1 1 1 2 1 1 1 2 1 1 1 3 1 1 1 2 1 1 1 2 1 1 1 For n=4 and below, it is easily observable that the number of permutations where this number is 1 or greater is 4!, the number of permutations where it is 2 or greater is 3!, etc etc... ending up with 4! + 3! + 2! + 1! symbols needed. So I guess if you could prove that a solution with this property would always be as short as possible, maybe by tying it in with the ideas above, then you'd be set. (You can also see how this illustrates the above model (6 sets of 4 'nearby' permutations, each preceded by a bigger gap).) Like I said, nothing concrete or necessarily useful, but on the chance that it may help someone I thought it was worth posting. Jorel Posts: 1 Joined: Wed Aug 27, 2014 5:01 pm UTC ### Re: The Shortest String Containing all Permutations of n Sym I couldn't help noting that each of the optimal strings posted here is palindromic. That is to say, each can be read either from left to right or from right to left with the same result: 121 123121321 123412314231243121342132413214321 Many years of recreational mathematics lead me to suspect that this will be the case for all optimal strings of this type. Does anybody happen to know whether my hunch is correct? Meanwhile, I have been working (on and off for the last fifteen years or so) on a similar problem, which is that of producing every n-digit permutation in base b. For example, there are four optimal strings in base two containing every possible two-digit permutation exactly once: 00110 11001 01100 10011 The main difference between my problem and the one being discussed here is that my optimal strings contain repeating digits. As base b increases, the number of possible optimal strings increases hyper-exponentially. In base 3 there are 216 possible solutions. In base ten, one of many, many solutions is the following 101-digit string: 00110221203323130443424140554535251506656463626160776757473727170887868584838281809989796959493929190 This string contains every single two-digit permutation from 00 to 99 in base ten, without repetition. (It is not pallindromic, of course, as no palindromic solution could possibly exist, for reasons that are not too difficult to work out.) Moole Posts: 93 Joined: Sun Jan 08, 2012 9:52 pm UTC ### Re: The Shortest String Containing all Permutations of n Sym Well, I was googling about this a little, and I found out that the value conjectured in this thread (and elsewhere) is false. So, I guess that makes this problem even harder, since we don't even have a bound to try to prove. Prior to knowing this, I spent some time thinking and came up with a novel way to look at the problem. I couldn't think of what to do next, but... Spoiler: What I'd do is to look at every occurrence of a given element. We will call this element n. For every permutation P of n-1, there must be at least one position where P immediately precedes n and one where n immediately precedes n. So, it makes sense to take the nth element and consider the string containing the n-1 preceding and succeeding elements. i.e. in 123121321, one has the substrings 12312 and 21321. Removing the middle "3" from each of these yields the tableaux Code: Select all 12122121 Or, if we, in 123121321 swapped 2s and 3s Code: Select all 1212112121 What we should note here is that, any two consecutive columns of the tableaux contain every permutation of 2, representing permutations of the form xx3, x3x, and 3xx. More generally, in any tableaux, any string of n-1 columns must contain every permutation of n-1 elements. We will refer to the right side of the tableaux as the rightmost n-1 elements (which succeeded the instance of the character n) and the left side as the leftmost n-1 elements (which preceded it). Given the tableaux (in order), we can reconstruct the relevant string (less any elements not used in any permutation) just by reinserting the n into the middle of each row, and putting successive rows together (with as much overlap as possible), which means we can work with them instead of strings. Further, removal of any cell from the tableaux always decreases the length of the string - so every row must be necessary in an optimal solution. We start by looking at a bit of a less trivial tableaux than the above. In particular, take the optimal string 412341243124132414231421342143214, make the tableaux Code: Select all 123123124412312312132132142241231231213213214421321321 Since we only care about permutations of 3 appear in the tableaux, we can safely remove any references to "4" from the tableaux, along with any elements that would use the four in any substring of length 3 (i.e. the row "132142" can have the last two elements removed): Code: Select all 12312312 123123121321321 123123121321321 21321321 Notice that, if a row ends with spaces, the next row starts with the same amount. This would be true of any tableaux reduced like so. More strongly, in fact, if the right side of a row is string A followed by k spaces, the left side of the next row is k spaces followed by A (since, for instance, the pair of rows "1231/1231" represents "123414231", so the "1" on the right of the first row is the same one as on the left of the second row). Mathematical hangover (n.): The feeling one gets in the morning when they realize that that short, elementary proof of the Riemann hypothesis that they came up with at midnight the night before is, in fact, nonsense. robinhouston Posts: 3 Joined: Thu Feb 27, 2014 9:28 am UTC ### Re: The Shortest String Containing all Permutations of n Sym Hello! Funny to see a link to my paper here while idly browsing the forum. It seems that lurkers like me are not allowed to post links, so you’ll have to Google for the things I wanted to link to. The first interesting development since the main burst of activity on this thread was the discovery by Benjamin Chaffin of eight different length-153 superpermutations on five symbols, with a (machine) proof that there are no shorter ones. Nathaniel Johnston – the OP of this thread –has details on his blog. That emboldened me to try to disprove the minimal-length conjecture for n=6, which I succeeded in doing. I think in a way this makes our quest easier, since we can stop wasting effort trying to prove something that is not true; but it means the true picture is more complicated than some people had hoped. I think the most obvious low-hanging fruit now is to prove (or disprove!) that length 872 is minimal for n=6. If anyone has access to a decent compute cluster, it should be pretty easy to hit it with Concorde (the TSP solver) till it breaks. The input file you need is 6.tsp in the list of ancillary files I included with the paper. PM 2Ring Posts: 3617 Joined: Mon Jan 26, 2009 3:19 pm UTC Location: Mid north coast, NSW, Australia ### Re: The Shortest String Containing all Permutations of n Sym Here's the link to the article on Robin Houston's blog. Tackling the Minimal Superpermutation Problem Last edited by PM 2Ring on Thu Sep 18, 2014 12:22 am UTC, edited 1 time in total. Posts: 33 Joined: Wed Apr 21, 2010 8:33 pm UTC ### Re: The Shortest String Containing all Permutations of n Sym Doesn't finding a short length-5 sequence automatically imply the existence of shorter sequences of length 6 and up? robinhouston Posts: 3 Joined: Thu Feb 27, 2014 9:28 am UTC ### Re: The Shortest String Containing all Permutations of n Sym Doesn't finding a short length-5 sequence automatically imply the existence of shorter sequences of length 6 and up? Indeed it would, but Ben Chaffin showed that there is no short superpermutation on 5 symbols: all the minimal ones have length 153, which is the same length as the recursively constructed one, i.e. 1! + 2! + 3! + 4! + 5!. The existence of a shorter-than-conjectured superpermutation on 6 symbols implies that there are shorter-than-conjectured superpermutations for all n>6 too. korona Posts: 495 Joined: Sun Jul 04, 2010 8:40 pm UTC ### Re: The Shortest String Containing all Permutations of n Sym I wrote a short conversion of the Hamiltonian path problem (see Robin Houston's paper on arXiv) to (Max-)SAT. The encoding uses roughly (n!)^2 variables, so it won't scale to large n but it may be possible to tackle the n = 6 case with this approach. However in order to do that we need to encode more domain specific knowledge. One starting point would be the following question: Can we remove edges with weight n from the graph without changing the weight of the minimal Hamiltonian path? Does any solution for n = 5 use such an edge? robinhouston Posts: 3 Joined: Thu Feb 27, 2014 9:28 am UTC ### Re: The Shortest String Containing all Permutations of n Sym Interesting! It would be great if we could dispose of n=6, and going via SAT is a reasonable idea. I haven’t proved it, but it seems plausible that it’s safe to remove edges of weight n. There are no known minimal superpermutations that use these. If removing them works, it would be worth trying to prove. There is a minimal superpermutation at n=5 that uses an edge of weight 4, namely the “obvious” recursive one. The structure is like this (read down the columns): Code: Select all 12345 25341 ... 31542 51243 ... 21534 51324 13254 3521423451 53412 31425 15423 ... 21345 15342 ... 32541 5214334512 ... 14253 54231 24315 13452 53421 32415 25413 ...45123 41235 42531 ... 43152 34521 ... 24153 54132 1432551234 12354 25314 ... 31524 45213 42135 41532 ... 43251... 23541 53142 31245 15243 52134 21354 15324 ... 3251423415 35412 ... 12453 52431 ... 13542 53241 32145 2514334152 54123 14235 24531 ... 13425 35421 ... 21453 5143241523 ... 42351 45312 43125 34251 54213 24135 14532 ...15234 ... 23514 53124 31254 42513 ... 41352 45321 4321552341 23145 35142 ... 12543 25134 ... 13524 53214 32154... 31452 51423 12435 25431 51342 13245 35241 ... 2154334125 14523 ... 24351 54312 ... 32451 52413 21435 1543241253 45231 42315 43512 ... 34215 24513 ... 14352 5432112534 52314 23154 35124 ... 42153 45132 41325 43521 In the middle there’s an edge from 54312 to 21345, which has weight 4. On the other hand, the heaviest edge in the shortest-known superpermutation for n=6 has weight 3. If you could establish a bound even assuming all edges have weight ≤3, that would still be pretty interesting I think!	{"extraction_info": {"found_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": 0.8161991834640503, "perplexity": 757.5908047761066}, "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/1505818689823.92/warc/CC-MAIN-20170924010628-20170924030628-00265.warc.gz"}
http://mathoverflow.net/questions/67917/does-specification-implies-that-entropy-map-is-upper-semicontinuous	# Does specification implies that entropy map is upper semicontinuous? Let (X,d) be a compact metric space and f a continuous transformation on X. f has the specification if one can always find a single orbit to interpolate between different pieces of orbits, up to a pre-assigned error. We call the map \mu to h_{\mu}(f) (Kolmogorov-Sinai entropy)the entropy map . Does specification implies that entropy map is upper semicontinuous? - Your definition of specification sounds more like a definition of topological transitivity -- the key function of specification is to ensure that the time you spend going from one orbit segment to the next when you approximate is uniformly bounded. – Vaughn Climenhaga Jun 16 '11 at 11:36 This sounds a little bit like a homework exercise (my apologies if it's not). A natural thing to do would be to write down a couple examples of systems whose entropy map is not upper semi-continuous, and see if you can find something from that list with specification. – Vaughn Climenhaga Jun 16 '11 at 11:45 Hi, dear Climenhaga, thank you for your answers. – ljjpfx Jun 17 '11 at 3:08	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8746165037155151, "perplexity": 587.5748565573251}, "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/1408500837094.14/warc/CC-MAIN-20140820021357-00274-ip-10-180-136-8.ec2.internal.warc.gz"}
https://proceedings.neurips.cc/paper/2020/file/92bf5e6240737e0326ea59846a83e076-MetaReview.html	NeurIPS 2020 ### Meta Review The four reviewers, all of whom are seasoned domain experts, consistently agree that this paper is conceptually solid, timely and relevant. I should thus be accepted. The reviewers also pointed out some issues with the presentation. I want to strongly encourage the authors to take these into account to increase the potential audience of this paper.	{"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": 0.8809148073196411, "perplexity": 1425.6457155060918}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039544239.84/warc/CC-MAIN-20210421130234-20210421160234-00526.warc.gz"}
http://math.stackexchange.com/questions/89563/are-there-infinitely-many-mersenne-primes	# Are there infinitely many Mersenne primes? known facts : $1.$ There are infinitely many Mersenne numbers : $M_p=2^p-1$ $2.$ Every Mersenne number greater than $7$ is of the form : $6k\cdot p +1$ , where $k$ is an odd number $3.$ There are infinitely many prime numbers of the form $6n+1$ , where $n$ is an odd number $4.$ If $p$ is prime number of the form $4k+3$ and if $2p+1$ is prime number then $M_p$ is composite What else one can include in this list above in order to prove (or disprove) that there are infinitely many Mersenne primes ? - Do you know about the LPW conjecture? – Guess who it is. Dec 8 '11 at 8:18 @J.M.,Interesting,but it isn't fact,it is conjecture... – pedja Dec 8 '11 at 8:21 As far as I know, this is still an open problem. – InterestedGuest Dec 8 '11 at 8:23 Clearly, you missed the point. There's a reason why the infinitude of Mersenne primes remains a conjecture. – Guess who it is. Dec 8 '11 at 8:30 No, he's saying we don't know. – JSchlather Dec 8 '11 at 8:41	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9530572891235352, "perplexity": 323.6951157251308}, "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/1432207930256.3/warc/CC-MAIN-20150521113210-00218-ip-10-180-206-219.ec2.internal.warc.gz"}
http://aeciosantos.com/2012/09/04/custom-latex-beamer-theme/	# Aécio Santos Just another personal web site. September 4, 2012 # A simple & clean Latex Beamer theme Beamer is a great Latex class used to create presentations. Unfortunately, most default Beamer themes are very bloated and full of stuff that waste the usefull space of the slides. Looking for simplicity, I created a custom beamer theme that is simple and clean. If you already know latex and beamer, it’s pretty easy to use it. Just download the file style.tex and put it on the same folder of you latex document. Then, you just need to include the file using the command \input{style.tex} at the begining of your latex document. The structure of your document should look like this: \documentclass[t,14pt,mathserif]{beamer} \input{style.tex} \title{Presentation Title} \author{Author Name} \begin{document}	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9094642996788025, "perplexity": 2746.127537245702}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578681624.79/warc/CC-MAIN-20190425034241-20190425060035-00021.warc.gz"}
https://par.nsf.gov/biblio/10347071	Experimental verification of the Landau–Lifshitz equation Abstract The Landau–Lifshitz (LL) equation has been proposed as the classical equation to describe the dynamics of a charged particle in a strong electromagnetic field when influenced by radiation reaction. Until recently, there has been no clear experimental verification. However, aligned crystals have remedied the situation: here, as in Nielsen et al CERN NA63 Collaboration (2020 Phys. Rev. D 102 052004), we report on a quantitative experimental test of the LL equation by measuring the emission spectra of electrons and positrons penetrating aligned single crystals. The recorded spectra are in remarkable agreement with simulations based on the LL equation of motion with moderate quantum corrections for recoil and, in the case of electrons in axially aligned crystals, spin and reduced radiation intensity. Authors: ; ; ; ; Award ID(s): Publication Date: NSF-PAR ID: 10347071 Journal Name: New Journal of Physics Volume: 23 Issue: 8 Page Range or eLocation-ID: 085001 ISSN: 1367-2630	{"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": 0.9500597715377808, "perplexity": 1568.7368600946427}, "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/1669446711114.3/warc/CC-MAIN-20221206192947-20221206222947-00272.warc.gz"}
https://www.onemathematicalcat.org/Math/Precalculus_obj/writeFctComp.htm	# WRITING A FUNCTION AS A COMPOSITION • PRACTICE (online exercises and printable worksheets) When you're given a multi-step task to perform, you may want to break it into pieces, and assign different pieces to different people. When you're given a function that does several things, you may want to break it into ‘smaller’ functions that accomplish the same job! ## EXAMPLE (breaking a composite function into pieces) Consider the function $\,h(x) = 5(x-4)^3 - 7\,$. This function $\,h\,$ does the following: 1. subtracts $\,4\,$ 2. cubes the result 3. multiplies by $\,5\,$ 4. subtracts $\,7\,$ Break these four tasks into two pieces, as shown in the mapping diagram below: • assign the first two tasks (subtract $\,4\,$, then cube) to $\,f\$: $f(x) = (x-4)^3$ • assign the last two tasks (multiply by $\,5\,$, then subtract $\,7\,$) to $\,g\$: $g(x) = 5x - 7$ With these assignments, the composite function $\,g\circ f\$ (where $\,f\,$ acts first, followed by $\,g\,$) accomplishes the same thing as $\,h\,$: $$(g\circ f\,)(x) = g(f(x)) = g\bigl((x-4)^3\bigr) = 5(x-4)^3 - 7 = h(x)$$ ## You can break a task into pieces in different ways! Of course, you can delegate the responsibilities in different ways: • $f\,$ takes the first step (subtract $4$): $f(x) = x-4$ $g\,$ takes the other three steps (cube, multiply by $5$, subtract $7$): $g(x) = 5x^3 - 7$ • $f\,$ takes the first three steps (subtract $4$, cube, multiply by $5$): $f(x) = 5(x-4)^3$ $g\,$ takes the last (subtract $7$): $g(x) = x - 7$ In both cases, be sure to check that $\,(g\circ f\,)(x) = h(x)\,$. ## You can use more than two helpers! Or, you can use more ‘helper’ functions. For example: • let $\ a\$ subtract $4$: $a(x) = x-4$ • let $\ b\$ cube: $b(x) = x^3$ • let $\ c\$ multiply by $5$ and subtract $7$: $c(x) = 5x - 7$ Then, \displaystyle \begin{align} (c\circ b\circ a\,)(x) &= c(b(a(x)))\cr &= c(b(x-4))\cr &= c((x-4)^3)\cr &= 5(x-4)^3 - 7\cr &= h(x) \end{align} The exercises in this lesson give you practice with this process of writing a function as a composition. Master the ideas from this section	{"extraction_info": {"found_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": 3, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9837517738342285, "perplexity": 3383.790488910981}, "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/1600400198213.25/warc/CC-MAIN-20200920125718-20200920155718-00481.warc.gz"}
http://mathhelpforum.com/algebra/16553-inequality.html	1. ## inequality (x-2)(x+5) < 0 do you just make the < an = and solve there? I tried expanding doing: x^2 + 3x - 10 I know (x + 2)(x + 1) = x^2 + 3x + 2 which is similar, so then (x + 2)(x + 1) - 8 < 0 .. but that doesn't really help me hmm 2. Originally Posted by flash101 (x-2)(x+5) < 0 do you just make the < an = and solve there? I tried expanding doing: x^2 + 3x - 10 I know (x + 2)(x + 1) = x^2 + 3x + 2 which is similar, so then (x + 2)(x + 1) - 8 < 0 .. but that doesn't really help me hmm you treat the < as an equal sign, then test your results at the end (x - 2)(x + 5) < 0 => x - 2 < 0 or x + 5 < 0 => x < 2 or x < -5 so now we basically think of, or draw a number line, placing the numbers -5 and 2 in there relative positions, and test what regions the original equation holds true for 3. Hello, flash! $\displaystyle (x-2)(x+5) \:< \:0$ Leave it in factored form . . . We have: .$\displaystyle (x - 2)(x + 5) \:<\:0$ It says: the product of two numbers is negative. So one factor must be positive and the other must be negative. . . There are two possible cases. [1] .$\displaystyle (x - 2)$ is positive and $\displaystyle (x + 5)$ is negative. So we have: .$\displaystyle \begin{array}{ccc}x - 2 \:>\:0 & \Rightarrow & x \:> \:2 \\ x+5 \:<\:0 & \Rightarrow & x \:<\:\text{-}5\end{array}$ Hence, $\displaystyle x$ is a number which greater than 2 and less than -5. . . This is clearly impossible. [2] .$\displaystyle (x-2)$ is negative and $\displaystyle (x+5)$ is positive. So we have: .$\displaystyle \begin{array}{ccc}x - 2 \:<\:0 & \Rightarrow & x \:<\:2 \\ x + 5 \:>\:0 & \Rightarrow & x \:>\:\text{-}5\end{array}$ Hence, $\displaystyle x$ is a number which is less than 2 and greater than -5. This is possible for $\displaystyle x$ between -5 and +2: .$\displaystyle \boxed{-5 \:<\:x\:<\:2}$ 4. Flash, remember when you multiply with a negative number, the $\displaystyle >$ will become a $\displaystyle <$ and vice versa.	{"extraction_info": {"found_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": 0.9161691665649414, "perplexity": 519.3223980066832}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647327.52/warc/CC-MAIN-20180320091830-20180320111830-00159.warc.gz"}
https://math.stackexchange.com/questions/1738696/probability-of-picking-frac13-between-the-interval-0-1/1738707	# probability of picking $\frac{1}{3}$ between the interval $[0,1]$? My professor told us a situation in class which i do not understand very well. He said "what is the probability of picking $\frac{1}{3}$ between the interval $[0,1]$?" and then he proved it to be $0$. He said something like if you continue to pick subsets of $\frac{1}{3}$ it will eventually branch off to $0$. I wanted to ask my professor to clarify but he had a meeting afterwards. Therefore, I come to you all. Does anyone think they can explain this concept to me? • There are infinite possible numbers (between 0 and 1) which can be picked. Thus the probability of picking a specific number is zero. – callculus Apr 12 '16 at 5:28 • If your professor means for the probability distribution over $[0,1]$ to be uniform, then he is correct that the probability of getting exactly $\frac{1}{3}$ is zero. One way to explain why this is the case is that measure of the set $\{ \frac{1}{3}\}$ is zero in this case. – Justin Benfield Apr 12 '16 at 5:28 • Roughly speaking, there are infinitely many points in $[0,1]$ and $\frac 13$ is just one of them, so the probability of picking it is $\frac 1{\infty}$. – BigbearZzz Apr 12 '16 at 5:28 • @JustinBenfield, i think my professor was using your idea – yaniz rakiov Apr 12 '16 at 5:30 $1/3$ lies in the interval $[0,1/2]$, and there is probability $1/2$ of any point we choose in $[0,1]$ lying in that interval. Further, $1/3$ lies in $[1/4,1/2]$, and there is probability $1/4$ of any point we choose in $[0,1]$ lying in this interval. Likewise, $1/3$ lies in $[1/4,3/8]$, which we would land in with probability $1/8$. Continue in this manner inductively constructing a sequence of nested intervals of length $1/2^n$, each containing the point $1/3$. This means that the probability of choosing $1/3$ out of the interval $[0,1]$ is smaller than the length of any of these nested intervals that contain the point $1/3$, and hence the probability is smaller than $1/2^n$ for any positive integer $n$, so the probability must be $0$. I will answer this question by asking you How many number are there in the interval $$[0,1]$$ and remember it is a continuous realnumber line	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.953511655330658, "perplexity": 115.28888180624458}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195525974.74/warc/CC-MAIN-20190719032721-20190719054721-00132.warc.gz"}
https://stacks.math.columbia.edu/tag/02KO	Definition 35.19.1. Let $\mathcal{P}$ be a property of morphisms of schemes over a base. Let $\tau \in \{ fpqc, fppf, syntomic, smooth, {\acute{e}tale}, Zariski\}$. We say $\mathcal{P}$ is $\tau$ local on the base, or $\tau$ local on the target, or local on the base for the $\tau$-topology if for any $\tau$-covering $\{ Y_ i \to Y\} _{i \in I}$ (see Topologies, Section 34.2) and any morphism of schemes $f : X \to Y$ over $S$ we have $f \text{ has }\mathcal{P} \Leftrightarrow \text{each }Y_ i \times _ Y X \to Y_ i\text{ has }\mathcal{P}.$ In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.995086133480072, "perplexity": 291.13839147173564}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780053657.29/warc/CC-MAIN-20210916145123-20210916175123-00077.warc.gz"}
https://www.lmfdb.org/GaloisGroup/?n=30	Results (displaying matches 1-50 of at least 5000) Next Label Name Order Parity Solvable Subfields Low Degree Siblings 30T1 $C_{30}$ 30 -1 Yes $C_2$, $C_3$, $C_5$, $C_6$, $C_{10}$, $C_{15}$ 30T2 $C_5\times S_3$ 30 -1 Yes $C_2$, $S_3$, $C_5$, $S_3$, $C_{10}$, $S_3 \times C_5$ 15T4 30T3 $D_{15}$ 30 -1 Yes $C_2$, $S_3$, $D_{5}$, $S_3$, $D_5$, $D_{15}$ 15T2 30T4 $C_3\times D_5$ 30 -1 Yes $C_2$, $C_3$, $D_{5}$, $C_6$, $D_5$, $D_5\times C_3$ 15T3 30T5 $C_6\times D_5$ 60 -1 Yes $C_2$, $C_3$, $D_{5}$, $C_6$, $D_{10}$, $D_5\times C_3$ 30T5 30T6 $C_3:F_5$ 60 -1 Yes $C_2$, $S_3$, $F_5$, $S_3$, $F_5$, $C_{15} : C_4$ 15T6 30T7 $C_3\times F_5$ 60 -1 Yes $C_2$, $C_3$, $F_5$, $C_6$, $F_5$, $F_5\times C_3$ 15T8 30T8 $S_3\times D_5$ 60 -1 Yes $C_2$, $S_3$, $D_{5}$, $S_3$, $D_{10}$, $D_5\times S_3$ 15T7, 30T10, 30T13 30T9 $A_5$ 60 1 No $A_5$, $\PSL(2,5)$, $A_{5}$, $A_5$ 5T4, 6T12, 10T7, 12T33, 15T5, 20T15 30T10 $S_3\times D_5$ 60 -1 Yes $C_2$, $S_3$, $D_{5}$, $D_{6}$, $D_5$, $D_5\times S_3$ 15T7, 30T8, 30T13 30T11 $C_5\times A_4$ 60 1 Yes $C_3$, $C_5$, $A_4$, $C_{15}$ 20T14 30T12 $C_{10}\times S_3$ 60 -1 Yes $C_2$, $S_3$, $C_5$, $D_{6}$, $C_{10}$, $S_3 \times C_5$ 30T12 30T13 $S_3\times D_5$ 60 -1 Yes $C_2$, $S_3$, $D_{5}$, $D_{6}$, $D_{10}$, $D_5\times S_3$ 15T7, 30T8, 30T10 30T14 $D_{30}$ 60 -1 Yes $C_2$, $S_3$, $D_{5}$, $D_{6}$, $D_{10}$, $D_{15}$ 30T14 30T15 $C_{15}\times S_3$ 90 -1 Yes $C_2$, $C_5$, $S_3\times C_3$, $C_{10}$ 45T3 30T16 $C_3\times D_{15}$ 90 -1 Yes $C_2$, $D_{5}$, $S_3\times C_3$, $D_5$ 45T5 30T17 $C_2\times C_3:F_5$ 120 -1 Yes $C_2$, $S_3$, $F_5$, $D_{6}$, $F_{5}\times C_2$, $C_{15} : C_4$ 30T17 30T18 $C_{10}\times A_4$ 120 -1 Yes $C_3$, $C_5$, $A_4\times C_2$, $C_{15}$ 40T59 30T19 $C_5:S_4$ 120 1 Yes $S_3$, $D_{5}$, $S_4$, $D_{15}$ 20T33, 30T31, 40T63 30T20 $D_5\times A_4$ 120 -1 Yes $C_3$, $D_{5}$, $A_4\times C_2$, $D_5\times C_3$ 20T37, 30T28, 40T65 30T21 $C_2\times S_3\times D_5$ 120 -1 Yes $C_2$, $S_3$, $D_{5}$, $D_{6}$, $D_{10}$, $D_5\times S_3$ 30T21 x 3 30T22 $S_5$ 120 -1 No $S_5$, $\PGL(2,5)$, $S_5$ 5T5, 6T14, 10T12, 10T13, 12T74, 15T10, 20T30, 20T32, 20T35, 24T202, 30T25, 30T27, 40T62 30T23 $S_3\times F_5$ 120 -1 Yes $C_2$, $S_3$, $F_5$, $D_{6}$, $F_{5}\times C_2$, $F_5 \times S_3$ 15T11, 30T24, 30T32 30T24 $S_3\times F_5$ 120 -1 Yes $C_2$, $S_3$, $F_5$, $D_{6}$, $F_5$, $F_5 \times S_3$ 15T11, 30T23, 30T32 30T25 $S_5$ 120 -1 No $C_2$, $S_5$, $S_5$, $S_5$ 5T5, 6T14, 10T12, 10T13, 12T74, 15T10, 20T30, 20T32, 20T35, 24T202, 30T22, 30T27, 40T62 30T26 $C_6\times F_5$ 120 -1 Yes $C_2$, $C_3$, $F_5$, $C_6$, $F_{5}\times C_2$, $F_5\times C_3$ 30T26 30T27 $S_5$ 120 1 No $S_5$, $S_5$, $S_5$ 5T5, 6T14, 10T12, 10T13, 12T74, 15T10, 20T30, 20T32, 20T35, 24T202, 30T22, 30T25, 40T62 30T28 $D_5\times A_4$ 120 1 Yes $C_3$, $D_{5}$, $A_4$, $D_5\times C_3$ 20T37, 30T20, 40T65 30T29 $C_2\times A_5$ 120 -1 No $A_5$, $A_5$ 10T11, 12T75, 12T76, 20T31, 20T36, 24T203, 30T30, 40T61 30T30 $C_2\times A_5$ 120 -1 No $C_2$, $A_5$, $A_5\times C_2$, $A_5$ 10T11, 12T75, 12T76, 20T31, 20T36, 24T203, 30T29, 40T61 30T31 $C_5:S_4$ 120 -1 Yes $S_3$, $D_{5}$, $S_4$, $D_{15}$ 20T33, 30T19, 40T63 30T32 $S_3\times F_5$ 120 -1 Yes $C_2$, $S_3$, $F_5$, $S_3$, $F_{5}\times C_2$, $F_5 \times S_3$ 15T11, 30T23, 30T24 30T33 $C_5\times S_4$ 120 1 Yes $S_3$, $C_5$, $S_4$, $S_3 \times C_5$ 20T34, 30T34, 40T64 30T34 $C_5\times S_4$ 120 -1 Yes $S_3$, $C_5$, $S_4$, $S_3 \times C_5$ 20T34, 30T33, 40T64 30T35 $C_5^2:C_6$ 150 -1 Yes $C_2$, $C_3$, $C_6$, $(C_5^2 : C_3):C_2$ 15T12 x 2, 25T15, 30T35 30T36 $C_5\times D_{15}$ 150 -1 Yes $C_2$, $S_3$, $S_3$, $D_5\times C_5$ 30T36 30T37 $C_5^2:S_3$ 150 -1 Yes $C_2$, $S_3$, $S_3$, $(C_5^2 : C_3):C_2$ 15T13, 15T14, 25T16, 30T38 30T38 $C_5^2:S_3$ 150 -1 Yes $C_2$, $S_3$, $S_3$, $(C_5^2 : C_3):C_2$ 15T13, 15T14, 25T16, 30T37 30T39 $C_{15}\times D_5$ 150 -1 Yes $C_2$, $C_3$, $C_6$, $D_5\times C_5$ 30T39 30T40 $C_2\times C_5^2:C_3$ 150 -1 Yes $C_2$, $C_3$, $C_6$, $C_5^2 : C_3$ 30T40 30T41 $C_5\times S_3^2$ 180 -1 Yes $C_2$, $C_5$, $S_3^2$, $C_{10}$ 45T15 30T42 $S_3\times D_{15}$ 180 -1 Yes $C_2$, $D_{5}$, $S_3^2$, $D_{10}$ 45T13 30T43 $D_{15}:S_3$ 180 -1 Yes $C_2$, $D_{5}$, $S_3^2$, $D_5$ 45T21 30T44 $C_3\times S_3\times D_5$ 180 -1 Yes $C_2$, $D_{5}$, $S_3\times C_3$, $D_{10}$ 45T14 30T45 $C_3\times A_5$ 180 1 No $C_3$, $A_{5}$ 15T15 x 2, 15T16, 18T90, 36T176, 45T16 30T46 $(C_3\times C_{15}):C_4$ 180 1 Yes $C_2$, $F_5$, $C_3^2:C_4$, $F_5$ 30T46, 45T27 30T47 $C_3\times C_3:F_5$ 180 -1 Yes $C_2$, $F_5$, $S_3\times C_3$, $F_5$ 45T18 30T48 $C_3^2:(C_5:C_4)$ 180 1 Yes $C_2$, $D_{5}$, $C_3^2:C_4$, $D_5$ 30T48, 45T26 30T49 $C_5\times C_3:S_3.C_2$ 180 1 Yes $C_2$, $C_5$, $C_3^2:C_4$, $C_{10}$ 30T49, 45T25 30T50 $F_{16}$ 240 1 Yes $C_3$, $C_5$, $C_{15}$ 16T447, 20T67 Next Results are complete for degrees $\leq 23$.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9998562335968018, "perplexity": 2389.0778029660323}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250625097.75/warc/CC-MAIN-20200124191133-20200124220133-00263.warc.gz"}
https://operativeneurosurgery.com/doku.php?id=atmospheric_pressure	# Operative Neurosurgery ### Site Tools atmospheric_pressure ## Atmospheric pressure Atmospheric pressure, sometimes also called barometric pressure, is the pressure exerted by the weight of air in the atmosphere of Earth (or that of another planet). In most circumstances atmospheric pressure is closely approximated by the hydrostatic pressure caused by the weight of air above the measurement point. Low-pressure areas have less atmospheric mass above their location, whereas high-pressure areas have more atmospheric mass above their location. Likewise, as elevation increases, there is less overlying atmospheric mass, so that atmospheric pressure decreases with increasing elevation. On average, a column of air one square centimeter in cross-section, measured from sea level to the top of the atmosphere, has a mass of about 1.03 kg and weight of about 10.1 N (2.27 lbf) That force over one square centimeter is a pressure of 10.1 N/cm2 or 101000 N/m2. (A column one square inch in cross-section would have a weight of about 14.7 lb, or about 65.4 N.). The cranial cavity is a closed compartment and any breach to this confined space secondary to neurosurgery or trauma cause an imbalance between atmospheric pressure and intracranial pressure. As the altitude increases, the atmospheric pressure decreases and hypoxia with hypercarbia is a well-known fact. In children, there is an argument to suggest that hypoxia can contribute to mild increase in intracranial pressure during commercial flights 1). Fodstad et al. also commented on the effect of atmospheric pressure acting directly on cerebral tissue during craniectomy. According to them, during an upright position intracranial pressure would equalize with the atmospheric pressure 2). 1) Lo Presti A, Weil AG, Ragheb J. Flying with a shunt. J Neurosurg Pediatr. 2015;15(2):223-224. 2) Fodstad H, Love JA, Ekstedt J, Fridén H, Liliequist B. Effect of cranioplasty on cerebrospinal fluid hydrodynamics in patients with the syndrome of the trephined. Acta Neurochir (Wien). 1984;70(1-2):21-30. PubMed PMID: 6741628.	{"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": 0.874378502368927, "perplexity": 2921.1542369777508}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540518627.72/warc/CC-MAIN-20191209093227-20191209121227-00116.warc.gz"}
http://mathhelpforum.com/pre-calculus/186555-writing-sum-using-sigma-notation.html	# Thread: Writing the Sum Using Sigma Notation 1. ## Writing the Sum Using Sigma Notation Alright, it goes hand-in-hand with a previous topic titled "Summation Formulas" but yet...it's different at the same time. This time they give me two series: 1. $1+1/2+2+5/2+...+6$ 2. $[1-(1/2)^2]+[1-(1/3)^2]+[1-(1/4)^2]+...$ My first reaction was that I would have to find the formula myself, where it starts, and where it ends. Simple...except not so much. The second one is possibly $\sum_{n=1}^{INF}([1-(1/(1+N)^2])$. Although that would be way to simple, hehe. The first one though, I can't seem to find a proper equation that fits the numbers given. 2. ## Re: Writing the Sum Using Sigma Notation The answer to the second one is correct. You could even say $\displaystyle \sum_{n=2}^{\infty}1-\frac{1}{n^2}$ Can't see a pattern in the first one yet. 3. ## Re: Writing the Sum Using Sigma Notation Yeah, I dunno what's going on with the first one. I spent at least an hour or two going over it but it's harder then it looks...and it looked pretty hard to start with. Oh, I also made a mistake on number one, accidently left out a number. Fixed it. Should be: $1+\frac{1}{2}+2+\frac{5}{2}+...+6$ 4. ## Re: Writing the Sum Using Sigma Notation Originally Posted by UnstoppableBeast Alright, it goes hand-in-hand with a previous topic titled "Summation Formulas" but yet...it's different at the same time. 1. $1+{\color{red}1}/2+2+5/2+...+6$ I think it ought to be $1+\frac{{\color{blue}3}}{2}+2+\frac{5}{2}+\cdots+6$. In which case it would be $\sum\limits_{k =1}^{11} {\frac{{k + 1}}{2}}$ 5. ## Re: Writing the Sum Using Sigma Notation Is it possible the second term has to be $\frac{3}{2}$ in stead of $\frac{1}{2}$? 6. ## Re: Writing the Sum Using Sigma Notation Sadly no. It would've been easier that way but I'm sure of what the question is asking, I have the paper right in front of me. Maybe the person who made the packet made a mistake, seems as likely as anything.	{"extraction_info": {"found_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": 10, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9171527028083801, "perplexity": 974.2481660252311}, "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-2017-17/segments/1492917123318.85/warc/CC-MAIN-20170423031203-00619-ip-10-145-167-34.ec2.internal.warc.gz"}
http://mathoverflow.net/questions/48488/how-does-the-mixing-time-of-a-geodesic-flow-on-a-surface-vary-with-the-genus	# How does the mixing time of a geodesic flow on a surface vary with the genus? I have been looking at the numerical behavior of a particular quantity (of no direct importance here, though if you must know the gory details start with figure 17 here) associated to the geodesic flow on a surface of constant negative curvature and genus $g$. The behavior is quantitatively similar for $g = 2,3,4$ and physical intuition based on this quantity suggests that some "intrinsic" timescale--prime candidates are the mixing or relaxation time--should therefore depend on $g$ only weakly or even not at all. So: what is known about the behavior of the mixing and relaxation (or similar) times associated to these flows as $g$ varies? - There are many different metrics of curvature -1 on a surface of genus g. Do you have particular metrics in mind? For surfaces of genus 2, the mixing time can go to infinity as you vary the metric: if there are two genus-one subsurfaces separated by a long thin tube, the portion of the unit tangent bundle on one side takes a long time to mix with the portion on the other side. – Bill Thurston Dec 8 '10 at 23:58 Let's say for concreteness that the surface of constant negative curvature is given by a identifying appropriate edges of a regular $8g-4$-gon in the disk model with metric $ds^2 = dz d\bar{z}/(1-\|z\|^2)^2$. – Steve Huntsman Dec 9 '10 at 0:59 @Steve -- do you mean the regular $4g$-gon? The rate of mixing has to depend on $g$, because the surface that the $4g$-gon gives has injectivity radius an increasing function of $g$. (It grows like $\log(g)$.) – Sam Nead Dec 9 '10 at 10:30 @Sam--thanks. I mean $8g-4$: see imgur.com/XDADm.png for the edge identification. – Steve Huntsman Dec 9 '10 at 13:45 @Steve - Thank you for the figure. So you are using the regular, right angled, $8g−4$-gon. Let's call it $P$. If you draw $P$ in the most symmetric fashion in the disc model of $H^2$ then the center $O \in P$ will be distance $\log(g)$ (more or less) from the boundary of $P$. So for subsets of the unit tangent "close to" the point $O$ you'll have to flow for at least that long before any mixing at all happens. – Sam Nead Dec 9 '10 at 18:13 Here is another thought that struck me on the way home, that I should have realized earlier. Suppose that $S$ is a closed hyperbolic surface, of genus $g$. Then the area of $S$ is $-2\pi\chi(S) = 2\pi(2g - 2)$. Since the area of a disk in the hyperbolic plane is exponential in its radius, it follows that the diameter of $S$ is at least logarithmic in $g$. The mixing time of a space has to be at least the diameter, right? So this gives a uniform lower bound on the mixing time. Thurston's comment is pointing out that there is no uniform upper bound. To see this: The injectivity radius is one-half the systole (the length of the shortest closed geodesic). For a hyperbolic surface, the collar lemma implies that as the injectivity radius goes to zero the diameter goes to infinity (this is the previously mentioned "long thin tube"). Thus the mixing time also has to grow, by the previous paragraph. I roughly expect the mixing time can be estimated from the logarithm of the genus and the inverse of the injectivity radius. One reference for the geometric facts above is Peter Buser's book "Geometry and spectra of compact Riemann surfaces". - OK, so I wanted to elaborate here on Sam's helpful comments. Set $N = 8g-4$. An explicit description of the $N$-gon $F$ that I have in mind is $$F = D \ \backslash \ \bigcup_{j=1}^{N} \left(\sqrt{a-1} \cdot D + \sqrt{a} e^{2\pi i(j-2g)/N}\right)$$ with $a = \sec \frac{2\pi}{N}$. In particular, the nearest point to the origin is at a Euclidean distance $u := \sqrt{a} - \sqrt{a-1}$, so the hyperbolic distance is $d = \int_0^u \frac{dr}{1-r^2} = \frac{1}{2}\log\frac{1+u}{1-u}$, which evidently grows as $\log g$. A bit more context also: I expect that $t_g f(g) \approx const$, where $t_g$ is whatever timescale and $f(g)$ is the quantity mentioned in the question. If $t_g \sim \log g$ then I'd expect that $f(g) \sim 1/\log g$, which is actually a weak enough dependence on $g$ to not be surprising based on the numerics alluded to in the question. -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.9502773284912109, "perplexity": 272.8810647160655}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042988308.23/warc/CC-MAIN-20150728002308-00124-ip-10-236-191-2.ec2.internal.warc.gz"}
http://mathonline.wikidot.com/dense-sets-in-finite-topological-products	Dense Sets in Finite Topological Products # Dense Sets in Finite Topological Products Consider a finite collection of topological spaces $\{ X_1, X_2, ..., X_n \}$. If $A_i \subseteq X_i$ are dense in $X_i$ for all $i \in \{1, 2, ..., n \}$, then what can we say about the product $\displaystyle{\prod_{i=1}^{n} A_i}$? Conversely, if $\displaystyle{\prod_{i=1}^{n} A_i}$ is dense in $\displaystyle{\prod_{i=1}^{n} X_i}$, what can we say about each individual set $A_i$ in $X_i$? The following theorem gives us the desired answer. $A_i \subseteq X_i$ is dense in $X_i$ for all $i \in \{1, 2, ..., n \}$ if and only if $\displaystyle{\prod_{i=1}^{n} A_i}$ is dense in $\displaystyle{\prod_{i=1}^{n} X_i}$. Theorem 1: Let $\{ X_1, X_2, ..., X_n \}$ be a finite collection of topological spaces and let $A_i \subseteq X_i$ for all $i \in \{ 1, 2, ..., n \}$. Then $A_i$ is dense in $X_i$ for all $i \in \{ 1, 2, ..., n \}$ if and only if $\displaystyle{\prod_{i=1}^{n} A_i}$ is dense in $\displaystyle{\prod_{i=1}^{n} X_i}$. • Proof: $\Rightarrow$ Suppose that $A_i$ is dense in $X_i$ for all $i \in \{1, 2, ..., n \}$. Let $\displaystyle{U = \prod_{i=1}^{n} U_i}$ be any open set in $\displaystyle{\prod_{i=1}^{n} X_i}$. Then $U_i$ is open in $X_i$ for all $i \in \{ 1, 2, ..., n \}$. So since $A_i$ is dense in $X_i$ we have that for all $i$ that: (1) \begin{align} \quad A_i \cap U_i \neq \emptyset \end{align} • So take $x_i \in A_i \cap U_i$. Then $\mathbf{x} = (x_1, x_2, ..., x_n)$ is such that: (2) \begin{align} \quad \mathbf{x} \in \left ( \prod_{i=1}^{n} A_i \right ) \cap \left ( \prod_{i=1}^{n} U_i \right ) \end{align} • Thus $\displaystyle{\left ( \prod_{i=1}^{n} A_i \right ) \cap \left ( \prod_{i=1}^{n} U_i \right ) \neq \emptyset}$ for all open sets $\displaystyle{U = \prod_{i=1}^{n} U_i}$ in $\displaystyle{\prod_{i=1}^{n} X_i}$ which shows that $\displaystyle{\prod_{i=1}^{n} A_i}$ is dense in $\displaystyle{\prod_{i=1}^{n} X_i}$. • $\Leftarrow$ Suppose that $\displaystyle{\prod_{i=1}^{n} A_i}$ is dense in $\displaystyle{\prod_{i=1}^{n} X_i}$. • Consider the set $A_i$ and let $U_i$ be any open set in $X_i$. Let: (3) \begin{align} \quad U = X_1 \times X_2 \times ... \times X_{i-1} \times U_i \times X_{i+1}, \times ... \times X_n \end{align} • Then $U$ is an open set in $\displaystyle{\prod_{i=1}^{n} X_i}$. Since $\displaystyle{\prod_{i=1}^{n} A_i}$ is dense in $\displaystyle{\prod_{i=1}^{n} X_i}$ we have that: (4) \begin{align} \quad \left ( \prod_{i=1}^{n} A_i \right ) \cap U & \neq \emptyset \\ \quad \left ( A_1 \times A_2 \times ... \times A_n \right ) \cap \left (X_1 \times X_2 \times ... \times X_{i-1} \times U_i \times X_{i+1} \times ... \times X_n \right ) & \neq \emptyset \\ \quad ( A_1 \cap X_1) \times (A_2 \cap X_2) \times ... \times (A_{i-1} \cap X_{i-1}) \times (A_i \cap U_i) \times (A_{i+1} \cap X_{i+1}) \times ... \times (A_n \cap X_n) & \neq \emptyset \\ \quad A_1 \times A_2 \times ... \times A_{i-1} \times (A_i \cap U_i) \times A_{i+1} \times ... \times A_n & \neq \emptyset \end{align} • This shows that $A_i \cap U_i \neq \emptyset$. • So for all $i \in \{ 1, 2, ..., n \}$ we have that $A_i$ is dense in $X_i$. $\blacksquare$	{"extraction_info": {"found_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": 4, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 1.0000100135803223, "perplexity": 159.92665269311414}, "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-2017-17/segments/1492917125849.25/warc/CC-MAIN-20170423031205-00451-ip-10-145-167-34.ec2.internal.warc.gz"}
https://en.m.wikipedia.org/wiki/Multivariate_analysis_of_variance	# Multivariate analysis of variance In statistics, multivariate analysis of variance (MANOVA) is a procedure for comparing multivariate sample means. As a multivariate procedure, it is used when there are two or more dependent variables,[1] and is typically followed by significance tests involving individual dependent variables separately. It helps to answer:[2] 1. Do changes in the independent variable(s) have significant effects on the dependent variables? 2. What are the relationships among the dependent variables? 3. What are the relationships among the independent variables? ## Relationship with ANOVA MANOVA is a generalized form of univariate analysis of variance (ANOVA),[1] although, unlike univariate ANOVA, it uses the covariance between outcome variables in testing the statistical significance of the mean differences. Where sums of squares appear in univariate analysis of variance, in multivariate analysis of variance certain positive-definite matrices appear. The diagonal entries are the same kinds of sums of squares that appear in univariate ANOVA. The off-diagonal entries are corresponding sums of products. Under normality assumptions about error distributions, the counterpart of the sum of squares due to error has a Wishart distribution. MANOVA is based on the product of model variance matrix, ${\displaystyle \Sigma _{model}}$ and inverse of the error variance matrix, ${\displaystyle \Sigma _{res}^{-1}}$ , or ${\displaystyle A=\Sigma _{model}\times \Sigma _{res}^{-1}}$ . The hypothesis that ${\displaystyle \Sigma _{model}=\Sigma _{residual}}$ implies that the product ${\displaystyle A\sim I}$ .[3] Invariance considerations imply the MANOVA statistic should be a measure of magnitude of the singular value decomposition of this matrix product, but there is no unique choice owing to the multi-dimensional nature of the alternative hypothesis. The most common[4][5] statistics are summaries based on the roots (or eigenvalues) ${\displaystyle \lambda _{p}}$ of the ${\displaystyle A}$ matrix: • Samuel Stanley Wilks' ${\displaystyle \Lambda _{Wilks}=\prod _{1...p}(1/(1+\lambda _{p}))=\det(I+A)^{-1}=\det(\Sigma _{res})/\det(\Sigma _{res}+\Sigma _{model})}$ distributed as lambda (Λ) • the Pillai-M. S. Bartlett trace, ${\displaystyle \Lambda _{Pillai}=\sum _{1...p}(\lambda _{p}/(1+\lambda _{p}))=\mathrm {tr} ((I+A)^{-1})}$ [6] • the Lawley-Hotelling trace, ${\displaystyle \Lambda _{LH}=\sum _{1...p}(\lambda _{p})=\mathrm {tr} (A)}$ • Roy's greatest root (also called Roy's largest root), ${\displaystyle \Lambda _{Roy}=max_{p}(\lambda _{p})=\\|A\\|_{\infty }}$ Discussion continues over the merits of each,[1] although the greatest root leads only to a bound on significance which is not generally of practical interest. A further complication is that, except for the Roy's greatest root, the distribution of these statistics under the null hypothesis is not straightforward and can only be approximated except in a few low-dimensional cases.[7] An algorithm for the distribution of the Roy's largest root under the null hypothesis was derived in [8] while the distribution under the alternative is studied in.[9] The best-known approximation for Wilks' lambda was derived by C. R. Rao. In the case of two groups, all the statistics are equivalent and the test reduces to Hotelling's T-square. ## Correlation of dependent variables MANOVA's power is affected by the correlations of the dependent variables and by the effect sizes associated with those variables. For example, when there are two groups and two dependent variables, MANOVA's power is lowest when the correlation equals the ratio of the smaller to the larger standardized effect size.[10]	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 11, "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": 0.9859305024147034, "perplexity": 925.2213975549955}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376823348.23/warc/CC-MAIN-20181210144632-20181210170132-00439.warc.gz"}
https://elytraflight.com/blog/	Velocity Vectors: Secondary Solutions! Hello, again! Encouraged by my sudden lack in homework and with interest in the problem at an all-time high, I decided to write a Part 2 about the idea of velocity vectors immediately! 😀 If you haven’t read the original post, I would highly suggest doing so first: https://elytraflight.com/2021/05/17/velocity-vectors-circles-and-calculus/ The main objective of this post is to go over two different approaches to the same problem, and to discuss their benefits and drawbacks. For now, however, let’s do this! First, let’s reiterate the problem: “A particle moves along a curve so that its position vector and velocity vector are perpendicular at all times. If the particle passes through the point (4, 3), what is the equation of the curve?” Solution 2 In the previous post, we used a solution involving linear algebra, but who wants linear algebra? After all, linear algebra is just a bunch of funny number magic, right? So, we decide to attack the problem an alternative way! Consider what it means for two lines to be perpendicular. With two lines in the form of y = mx + b, the slopes have to be negative reciprocals of each other (e.g. 1/2 and -2 in the diagram below). What does this mean for our problem? Let’s recall our position vector s(t) = (x(t), y(t)). The “slope” of our position vector, in this case, can be found with the general “rise over run” technique: m = (y-0)/(x-0) = y/x. However, as we mentioned before, our velocity vector, v(t) = (x'(t), y'(t)), is perpendicular to the position vector! Thus, we know that the “slope” of our velocity vector, y'(t)/x'(t), is the negative reciprocal of y/x, which happens to be -x/y. So, this is what we end up with: However, the second equality is more than just an equation; it’s a differential equation! A differential equation that we have the tools to solve, as a matter of fact! Our first step is to separate the variables. We can move the dx to the right side and the y to the left side, to arrive at an equation as follows: Time to integrate both sides! Voila! We have arrived at the exact same answer as we did with the other method. Just like last time, we can plug in a specific case for x and y to find a specific case. Beautiful! 😀 Solution 3 And, last but not least, we have our final solution! Suppose we are still slightly suspicious of dot products, but we’re willing to give parametric functions (and some inefficient but cool techniques) a chance. Edit: After researching this solution a bit more, I realized that this solution gets crazy quite quickly. Buckle up! Jumping right in, we reintroduce our vectors s(t) = (x(t), y(t)), and v(t) = (x'(t), y'(t)). However, this time we have a twist. This time, we don’t end up removing t to get a general form of the circle equation in terms of x and y. No; this time, we dig even deeper. What we intend to do today is find both x(t) and y(t) in general form. In the above example, we see that we are taking advantage of a very important characteristic of perpendicular vectors. We take the x position, making it negative and multiplying by some constant a (as we do not want to assume that the velocity vector is the same magnitude as the position vector). Note that we are multiplying by a constant value a. This will be important at the end. So here’s the two equations we have so far: x'(t) = a y(t) y'(t) = -a x(t) Let’s start by manipulating the first equation, moving a to the other side. y(t) = x'(t)/a Now, we take the derivative of both sides: y'(t) = x”(t)/a We now have two different formulas for y'(t). Let’s set them equal to each other! x”(t)/a = -a x(t) x”(t) = -a2x(t) We now have a seemingly simple differential equation, with the second derivative in terms of the original function. However, we can’t just integrate, as that would leave us with a first derivative and some unruly antiderivative of x(t). Our options are running out! With no options left, we have no choice but to put on our LaTeX gloves (hahaha I’m hilarious) and set out on a journey into Laplace Land. Ahh, sunlight fills our eyes, and fresh air enters our lungs. At last, after all that pain and effort, here we are with our two final, completely general equations for x(t) and y(t): And, you know what this means! If we find x2+y2, it will equal some constant not in terms of t, which is the radius of our new circle squared. Granted, this newly-found radius of the circle will be in terms of the given values for x(0), x'(0), and a, but that is to be expected. Some of the incredibly astute may recall previously experimenting with piecewise functions, and how these trig functions are by no means the only two functions that can produce circles. For instance, something as simple as can follow the path of a (semi)circle within its domain. Furthermore, parametric functions like these also satisfy the property of having position vectors and velocity vectors being perpendicular, since x2+y2=R; in the previous solutions, we demonstrated that this is enough to conclude that the velocity vector is always tangent to the position vector in such a scenario. However, our very complex-looking trigonometric formula up above is incredibly special, in that it is the only general case parametric function out there whose value for “a” stays constant everywhere. In other cases, such as x(t) = t, the two vectors are perpendicular, but the ratio in magnitudes varies wildly depending on t. In our above case, however, a stays constant. Everywhere. If one is skeptical, we can test these formulas, and demonstrate that for any value of x(0) and x'(0) and a, the difference in magnitudes of the position and velocity vectors will indeed still be a, and by taking the derivatives of the formulas for x(t) and y(t) show that the formulas do indeed work out with what they should be. A true mathematical miracle. This concludes the third and final solution to this amazing problem. What a crazy experience! This took me about 10 hours of work to finish, between the three different solutions. I now intend to rest for the time being. It’s been an honor to have all of you readers along on this journey. As always, until we meet again, best wishes, and happy exploring! – Ely Velocity Vectors, Circles, and Calculus Hello, everyone! While I have been drifting through my sea of homework, I came across a rather interesting problem involving circles and velocity vectors. The problem goes as follows: “A particle moves along a curve so that its position vector and velocity vector are perpendicular at all times. If the particle passes through the point (4, 3), what is the equation of the curve?” Building Intuition Part of what makes the problem so interesting is our ability to come up with a general case, usable for any provided point of contact, and not just (4, 3)! First, let’s imagine what such a situation with perpendicular velocity and position vectors might look like. One such possibility that fits the description would be a circle. As seen in the diagram, the velocity vectors v1, v2, and v3 (which happen to be tangent to the circle) are all perpendicular to their respective position vectors, r1, r2, and r3. However, as we can soon see, there are multiple different (and rather beautiful) ways that we can prove that the only possible general curve is a circle! Solution Let’s do this! First things first, let’s describe our particle’s motion by describing it as a vector, where the x and y positions and x(t) and y(t) respectively, where t is time. We can write our position vector as s(t) = (x(t), y(t)). What’s more, we can take the derivative of our position vector with respect to time to get a velocity vector! 😀 To accomplish that, all we need to do is take the derivative of the x and y positions separately. (Taking the derivative of a single-variable function f(t) results in f'(t), pronounced f prime of t. One might visualize the derivative as being the “rate of change” of a function, or how fast the function is changing at time t.) v(t) = (x'(t), y'(t)). From here, we can employ a neat trick from linear algebra known as the dot product. Essentially, the dot product helps us find the angle between two vectors. In this case, we know that the angle between our position vector (x(t), y(t)) and our velocity vector (x'(t), y'(t)) must be 90 degrees. This is where the fun begins! One important aspect of the dot product, as we see on the bottom left, is that the dot product of perpendicular vectors is always 0. Thus, we can say that in all cases, s(t) · v(t) = 0. The dot product of two vectors is calculated by summing the products of the same components of both vectors. For example, (3, 4) · (5, 6) = 3 * 5 + 4 * 6 = 15+24 = 39. However, (0, 2) · (3, 0) = 0 * 3 + 2 * 0 = 0. This makes sense, because (0, 2) and (3, 0) are perpendicular. If we expand out s(t) · v(t) = 0, we get (x(t), y(t)) · (x'(t), y'(t)) = 0. If we expand this out using the above definition, we get that x(t) * x'(t) + y(t) * y'(t) = 0. This is cool, but how can we proceed from here? ANTIDERIVATIVES! Indeed, if you noticed that the original functions are multiplied by their derivatives and were reminded of the chain rule, you would be onto something important. Because we have x(t) * x'(t), we are provided with a very clean antiderivative. Now we have some math steps: And we have it! Does the final equation look familiar? x2 + y2 = D, where D is the radius of the circle squared! In the above problem, we can plug in x(t) = 4 and y(t) = 3, and we get D = 9 + 16 = 25, meaning that the curve our particle is following is a circle with a radius of 5! 🙂 This post is running longer than I was originally intended, so I’ll make sure to write a Part 2 regarding the two other interesting ways of solving this problem if you’re interested. Until then, best wishes and happy exploring! – Ely Many thanks to my Calculus instructor, Ms. Nguyen, for inspiring me to write this post, as well as providing students with many different opportunities to succeed this year! For further explanation in regards to calculus and linear algebra, I would highly suggest checking out the amazing YouTube channel 3Blue1Brown for detailed and immersive visuals to help understand those sorts of topics on a deeper level! Here is his channel: https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw You can also subscribe; it’s completely free, WordPress handles all of privacy concerns, and you get notified whenever something new comes out! The Hardy-Weinberg Principle, and Mathematics within Biology When we consider low-level biology courses, we may tend to think of more memorization-based concepts, such as the names of dozens of different organelles within a plant cell, as well as their various functions. In many cases, biology courses remain completely descriptive, removing any form of mathematics to avoid confusing the students any further than they already are. There are a few cases in biology, however, where instructors are required to introduce some mathematics to the curriculum; a welcome break from all the memorization required! Consider a population of herbivorous moths, enjoying a sequestered life on a far-away island, with no predators to worry about and enough food to maintain a stable population of 1,000 moths. Inside the DNA of these moths resides a particular gene known as gene X. Gene X can have two different possible “variants”, capital A and lowercase a. Each moth cell has two copies of this gene, one from its mother and one from its father. As biologists, we know that the capital A variant corresponds to being a red-colored moth, and the lowercase a variant to being a blue-colored moth. The capital A variant is dominant over the other, meaning that if your father or mother gave you the red variant, you would also be a red moth. However, having two a variants results in a blue moth. These kinds of genes can be represented in a Punnett Square, as seen in the diagram below. When two moths reproduce, one copy of the gene from each parent is contributed (at random) to create the child moth, leading to four possible equally-likely results. This concept of how genes are transmitted will help us understand the math behind the Hardy-Weinberg Principle. Now that we understand the nature of how two individual moths reproduce, let’s zoom out to the bigger picture, considering the case of a population with 1,000 moths. Using our god-like omnipotence, we know that at this point in time there are a total of 1000 lowercase a gene variants in the population, and 1000 uppercase A variants. exactly 250 moths with two lowercase a variants, 250 moths with two uppercase A variants, and 500 with one uppercase A and one lowercase a. Following the Punnett Square logic seen above, it follows that both the 250 homozygous red moths (AA) and the 500 heterozygous moths (Aa) are red, resulting in a total of 250 + 500 = 750 red moths. Let’s unpack this information: Awesome! We see that the totals gene variants add up correctly — There should be twice as many copies of the gene as there are moths, as every moth has 2 copies. This also highlights an important aspect of observing such populations. Although there are an equal amount of A and a variants in the population, 75% of the population is still red. Why is that? Recall that the A variant is dominant. Any moths that have the Aa genotype are still completely red. One could say that some of the blue a variants are “wasted”, as they do not contribute to the color when the A variant is present. Now that we have some of the background necessary, we can begin to explore the question of what the Hardy-Weinberg Principle actually tells us. This may sound complicated, but we can break down this concept into more understandable terms. An allele is the scientific name for the different “variants” of a particular gene that we mentioned earlier. For instance, in our above example, the a and A variants are both examples of different alleles. The term “allele frequency” refers to how many of each variant of the gene are present. For instance, the a allele’s frequency would be 50%, since 1000 of the 2000 alleles in the population are type a. What does it mean for the allele frequencies to remain constant? Let’s consider the moth example again. Right now, we have a total of 1,000 moths, with 1,000 a alleles and 1,000 A alleles. Now, let’s say the moths have been reproducing a lot, and now there are 2,000 moths. Using the Hardy-Weinberg principle, we can also say there are now 2,000 a alleles and 2,000 A alleles. Furthermore, if there were 750 red moths and 250 blue moths, now there are 750 × 2 = 1,500 red moths, and 250 × 2 = 500 blue moths! This may seem fairly straightforward. Twice as many moths, twice as many of each kind of moth. Simple algebra, right? What makes Hardy-Weinberg so interesting, however, is that this neat pattern of multiplication only occurs under a specific set of circumstances — when the population is in Hardy-Weinberg Equilibrium. There are a few requirements for a population to be in Hardy-Weinberg Equilibrium, but here are the main ones: • Mate choice must be completely random. If Mr. Moth #295 has a particular affinity for Ms. Moth #296, tough luck. Everything must be random, otherwise we run the risk of mate selection rendering certain alleles “better” than others, which would disrupt the balance between the two alleles. • As we mentioned above, having one allele must provide no advantage over having the other. If being blue makes it easier to survive or reproduce than being red, then over time there will be more blue moths, disrupting the balance. • There must be no movement in or out of the population, or any other abnormal activity. That’s why our moths are on an island in the middle of nowhere; escapees or new visitors could also disrupt the balance. • The population must be fairly large. In all cases, the Hardy-Weinberg principle deals with averages, not exact numbers. Perhaps a red moth eats a poisonous plant and dies, leaving behind 749 red moths and 250 blue moths. Although it isn’t ever exact, the ratio can still be treated as approximately 3:1. In an extreme example, if our population has a total of 4 moths, the chance of a random occurrence throwing off the balance gets to be extremely likely. This concept is known as genetic drift, an interesting topic in its own right. Now that we understand what the Hardy-Weinberg Principle describes and under what criteria it occurs, how can we prove that this phenomenon would occur in this way? Now the math starts to get really interesting. Let’s return again to our example of a population of 1,000 moths which we assume are in Hardy-Weinberg equilibrium. Let’s say this time, however, there are only 500 A alleles, and 1,500 a alleles. Punnett Square time! At this point, our next step is to calculate the “expected” amount of alleles generated with a case of 16 baby moths. In math, to calculate the expected value we multiply the probability by the number of “trials”, in this case the 16 moths. Since the Punnett Square has been reduced to one row and column, the math becomes a bit less complex. The probability of the child being AA is the probability of getting two A‘s in a row (because the population is large we can treat it as an independent event). In math, the probability of one event AND another event can be calculated by multiplying the probabilities. Since we know that 1/4 of the alleles are A in this case, the probability of getting two As is 1/4 × 1/4, or 1/16. Likewise, the probability of being Aa is the same as getting either an A from the father and an a from the mother, OR getting an a from the father and an A from the mother. Since 1/4 of the alleles are A, and 3/4 of the alleles are a, and there are two ways of doing it, the probability is 2 × 1/4 × 3/4 = 3/8. Finally, the probability of being aa is the probability of getting two a‘s in a row, or 3/4 × 3/4 = 9/16. From here, we can convert these probabilities to expected values, totaling the genotypes to find the expected values for each allele. The expected value of AA genotypes will be 16 × 1/16 = 1. However, since the AA genotype contains two As, the expected value of A alleles from being AA will be 2. Next, since there is only 1 A allele in the Aa genotype, the expected value of A alleles from being Aa will be 16 × 3/8 = 6. Adding these together, we get that the total expected number of A alleles is 8! Following similar logic (that is a bit too long for the article), we get that the total expected number of a alleles is 24. Voila! Just how the original ratio of A alleles to the total was 1:4, the expected ratio of A alleles in the 16 baby moths is also 1:4! When new moths were created, the allele ratio stayed the same. This, in essence, demonstrates the Hardy-Weinberg Principle. Complicated, but beautiful! The light at the end of the tunnel! In summary, the Hardy-Weinberg principle states that whenever a population is in genetic equilibrium, any new organisms will always follow the same patterns as their predecessors. Like always, there are many interesting questions left unanswered. What practical use does this concept have? What happens to a population that isn’t in genetic equilibrium? Rest assured there will be plenty of time to go over this at a future date. Until we meet again, best wishes and happy exploring! – Ely Many thanks to my biology instructor, Mr. Lewis, for introducing me to Hardy-Weinberg, and inspiring me to create this post! For further exploration, I would recommend checking out Khan Academy’s excellent article: https://www.khanacademy.org/science/ap-biology/natural-selection/hardy-weinberg-equilibrium/a/hardy-weinberg-mechanisms-of-evolution You can also subscribe; it’s completely free, WordPress handles all of privacy concerns, and you get notified whenever something new comes out! 🙂 Test Hi! This is technically the first blog post on my website. I’m fairly new to WordPress, the service with which I am creating my blog, so this is only a trial run to see what happens when I create a blog post using their designated button. Have a good night, everyone! 🙂	{"extraction_info": {"found_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": 0.8325800895690918, "perplexity": 592.3102233437916}, "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-2022-05/segments/1642320304287.0/warc/CC-MAIN-20220123141754-20220123171754-00194.warc.gz"}
https://www.lessonplanet.com/teachers/points-of-intersection	Points of Intersection Eighth and ninth graders solve five different problems that include solving two equations. They determine the point of intersection in each of the five pairs of equations given. Pupils define point of intersection as the point where two lines cross with x and y coordinates on a graph.	{"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": 0.8478983640670776, "perplexity": 343.3576794330933}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794870470.67/warc/CC-MAIN-20180527205925-20180527225925-00494.warc.gz"}
http://mathoverflow.net/questions/60957/how-to-determine-whether-an-ideal-is-prime-or-not-by-an-algorithm/60961	# how to determine whether an ideal is prime or not by an algorithm Given polynomials $f_{1},\cdots,f_{n}\in \mathbb{C}[x_{1},\cdots,x_{m}]$, do we have an algorithm to determine whether the ideal $I=(f_{1},\cdots,f_{n})$ is prime ideal or not? Of course, we assume the polynomials are irreducible. - Macaulay2 has a function isPrime which tells you if an ideal is or not prime. – Mariano Suárez-Alvarez Apr 7 '11 at 15:59 Why do you assume that the polynomials are irreducible? – Qiaochu Yuan Apr 7 '11 at 16:36 to rule out the trivial case that its factors might not be in the ideal. – Jiang Apr 7 '11 at 17:28 It just doesn't seem to me that that assumption really helps. – Qiaochu Yuan Apr 8 '11 at 4:36 Let $R$ be a Noetherian ring and let $I$ be an ideal in $R[x]$. Then the following facts hold: • $I$ is prime in $R[x]$ $\Longleftrightarrow$ $I\cap R$ is prime in $R$ and $\overline{I}$ is prime in $R/(R\cap I)$. • If $R$ is an integral domain and $I \cap R=0$, then $I$ is prime in $R[x]$ $\Longleftrightarrow$ $I K[x]$ is prime in $K[x]$ and $I=IK[x] \cap K[x]$. Here $K$ denotes the fraction field of $R$. Using the above to successively eliminate variables, this shows that one can reduce the problem of checking primiality to the one-variable case, where many efficient methods are known. I think this is also how the Grobner basis works, since these can algorithmically compute the elimination ideals above. - Eliminate, eliminate, eliminate the eliminators of elimination theory... =) – Harry Gindi Apr 8 '11 at 6:05 There is such a test. Some explanation can be found in: "An introduction to Gröbner bases, By William Wells Adams, Philippe Loustaunau", or the original article (http://portal.acm.org/citation.cfm?id=65034) the above text is based on. See also the singular manual. - Did you look into "A Singular Introduction to Commutative Algebra"? - Buchberger's algorithm should do, Faugere's F4 is also one. However, this is generally for any ideal and not necessarily for irreducible. Is it something specific to irreducible polynomials that you are looking for? - Irreducibility of the polynomials are not necessarily in my problem, what I need is just an algorithm to test the primality of a finite generated polynomial ideal. – Jiang Apr 8 '11 at 3:44	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9200435280799866, "perplexity": 369.838656004594}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860121423.81/warc/CC-MAIN-20160428161521-00101-ip-10-239-7-51.ec2.internal.warc.gz"}
https://www.lessonplanet.com/teachers/multiplying-by-zero-1st-2nd	# Multiplying By Zero In this multiplication by zero worksheet, students problem solve ten equations associated with multiplying by zero to equal the answer of zero.	{"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": 0.9934379458427429, "perplexity": 2028.4514232795177}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794864405.39/warc/CC-MAIN-20180521142238-20180521162238-00502.warc.gz"}
https://pos.sissa.it/316/013	Volume 316 - XXVI International Workshop on Deep-Inelastic Scattering and Related Subjects (DIS2018) - WG1: Structure Functions and Parton Densities Determination and application of TMD parton densities using the Parton Branching method A. Bermudez Martinez*, P.L.S. Connor, F. Hautmann, H. Jung, A.A. Lelek, V. Radescu and R. Zlebcik Full text: pdf Pre-published on: September 20, 2018 Published on: November 23, 2018 Abstract We present a determination of parton densities at NLO obtained with the Parton Branching method using precision measurements of deep inelastic scattering cross sections at HERA. The two sets of parton densities shown in this work are obtained with the same angular angular ordering condition for the evolution scale and they differ in the chosen scale for the strong coupling evaluation, for which we consider two scenarios: the evolution scale, and the transverse momentum qT from the angular ordering prescription. The transverse momentum dependent densities obtained with the Parton Branching method are applied to two LHC processes: the Drell-Yan pT spectrum and the azimuthal correlation in high pT dijet events. For the Drell-Yan pT spectrum a significant effect from the strong coupling scale choice is observed. DOI: https://doi.org/10.22323/1.316.0013 How to cite Metadata are provided both in "article" format (very similar to INSPIRE) as this helps creating very compact bibliographies which can be beneficial to authors and readers, and in "proceeding" format which is more detailed and complete. Open Access	{"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": 0.9498782157897949, "perplexity": 2871.6062442518983}, "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/1652662520817.27/warc/CC-MAIN-20220517194243-20220517224243-00356.warc.gz"}
http://aas.org/archives/BAAS/v26n2/aas184/abs/S2901.html	Evidence for Homogeneous Ionization of Helium in the Local Interstellar Medium from EUVE Spectroscopy of Hot DA Stars Session 29 -- General Interstellar Medium Display presentation, Tuesday, 31, 1994, 9:20-6:30 ## [29.01] Evidence for Homogeneous Ionization of Helium in the Local Interstellar Medium from EUVE Spectroscopy of Hot DA Stars J.Dupuis, S.Vennes, S.Bowyer (CEA/UCB), A.K.Pradhan (OSU), P.Thejll (NBI) We present an analysis of the Extreme Ultraviolet Explorer (EUVE) spectra of the DA stars Feige~24, MCT0455--281, HZ~43, GD~153, GD~71, and Sirius~B. This sample is selected on the basis of the stars' low interstellar hydrogen column densities and because of their widely separated Galactic coordinates. With the exception of Sirius~B, these stars all have a prominent He {\sc i} photoionization edge and a well defined Lyman continuum from which we derive reliable interstellar column densities of neutral hydrogen and neutral helium. For all the stars in the sample the measured abundance ratios of He\,{\sc i} \,:\,H\,{\sc i} differ significantly from the cosmic abundance ratio. We interpret these measurements in terms of ionization of helium in the local interstellar medium. The most striking result is that for all the Galactic lines of sight investigated, the relative ionization of helium is nearly constant with He~{\sc i}/H~{\sc i}$= 0.07 \pm 0.02$\,. This is similar to the result found for the hot DA GD~246 (Vennes et al. 1993, ApJ, 410, L119) for which a direct measurement of the He~{\sc ii} column was possible. The current sample probes various Galactic latitudes and longitudes and is indicative that helium is homogeneously ionized in the local interstellar medium. In optical spectra all these stars exhibit a pure hydrogen atmosphere; in the EUV both Feige~24 and MCT0455--281 show a substantial number of metals. We do not find metals in the remainder of the objects reported here, and we provide stringent upper limits for He, C, N, and O in the photospheres of these stars. This work has been supported by NASA contract NAS5-30180.	{"extraction_info": {"found_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": 0.9296587109565735, "perplexity": 3491.7082504806363}, "config": {"markdown_headings": true, "markdown_code": false, "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/1432207927592.52/warc/CC-MAIN-20150521113207-00161-ip-10-180-206-219.ec2.internal.warc.gz"}
http://mathhelpforum.com/number-theory/96226-help-preparing-final-exam-print.html	# Help Preparing for Final Exam Show 40 post(s) from this thread on one page Page 1 of 2 12 Last • Jul 27th 2009, 11:06 AM diddledabble Help Preparing for Final Exam I am trying to prepare for my final exam in NT. Any help would be appreciated. These questions are just for study purposes, I messed them up on my homework, etc. Reduced Residue System Proof Show that if a=b(mod n) then (a,n)= (b,n) • Jul 27th 2009, 03:54 PM Gamma Quote: Originally Posted by diddledabble I am trying to prepare for my final exam in NT. Any help would be appreciated. These questions are just for study purposes, I messed them up on my homework, etc. Reduced Residue System Proof Show that if a=b(mod n) then (a,n)= (b,n) $a \equiv b$ (mod n) $\Leftrightarrow n\|(b-a) \Rightarrow nk=b-a$ for some integer k Now we show the set of divisors of a and n are the same as the set of divisors of b and n. Let d divide a and n. $nk=b-a \Rightarrow b=nk+a$ then d divides the RHS so it divides b as well. Let d divide b and n. $nk=b-a \Rightarrow b-nk=a$ then d divides the LHS so it divides a as well. Thus the set of divisors is the same, so their greatest element is the same. so (a,n)=(b,n) • Aug 3rd 2009, 10:13 PM streethot First of all, assuming that $a\equiv b \ (mod \ m)$ we have that if $d\|m$ and $d\|a$ then $d\|b$. Proof: $a-b=m.k$ $m=d.u$ and $a=d.s$ then $b=d(s-u)$ Now, let $d=(a,m)$ and $e=(b,m)$. Then $d\|m$, $d\|a$ so $d\|b$. Hence $d\|e$. Similarly, $e\|m$, $e\|b$ so $e\|a$. Hence $e\|d$. Therefore $d=e$ • Aug 4th 2009, 11:55 AM diddledabble Another Proof Show that if 2n+1 is prime, then n must be a power of 2. • Aug 4th 2009, 12:01 PM Gamma Not true Quote: Originally Posted by diddledabble Show that if 2n+1 is prime, then n must be a power of 2. $2\cdot 3 + 1=7$ is prime and 3 is not a power of 2. • Aug 4th 2009, 12:08 PM Gamma Maybe what you meant was $2^n+1$ being prime implies $n=2^i$ for $i\in \{0,1,2,...\}$. If so, I would check out this:Fermat number - Wikipedia, the free encyclopedia For completeness, it should state the n could also be 0. • Aug 4th 2009, 12:09 PM diddledabble Quote: Originally Posted by Gamma $2\cdot 3 + 1=7$ is prime and 3 is not a power of 2. Maybe I messed it up. It is supposed to be proveable. I thought of the counterwxample issue too. Maybe it should be $2^n+1$ or $2^{n+1}$ • Aug 4th 2009, 12:12 PM Gamma Quote: Originally Posted by diddledabble Maybe I messed it up. It is supposed to be proveable. I thought of the counterwxample issue too. Maybe it should be $2^n+1$ or $2^{n+1}$ You can't prove something is true if there is a counter-example... that is the beauty of math. $2\|2^{n+1}$ so it can never be prime if n is a natural number (not including 0), so that would be a pretty dull question. Probably it is what I suggested above. • Aug 4th 2009, 12:13 PM streethot Gamma, Show that IF 2n+1 is prime, then n must be a power of 2. • Aug 4th 2009, 12:16 PM diddledabble Streethot, Everytime I read it I think of a different way to go with it. I think you have it though. 2^n+1 must be a prime to start with. • Aug 4th 2009, 12:17 PM Gamma Quote: Originally Posted by streethot Gamma, Show that IF 2n+1 is prime, then n must be a power of 2. Yes, if $7=2\cdot 3 + 1$ is prime... which it is for n=3, then 3 should be a power of 2. It is not, therefore this statement is not even close to being true. need another let n=5 or n=11? need any more? • Aug 4th 2009, 12:23 PM diddledabble But if it should have been $2^n+1$ let n=2 that is not a square. I just don;t get this one and fermat seems way to tough to be on the exam. • Aug 4th 2009, 12:30 PM streethot Sorry, i thought the question was about $2^{m}+1$ with $m=2^n \ \ n\in\mathbb{N}$ • Aug 4th 2009, 12:33 PM Gamma The statement should be IF $2^n+1$ is prime, then n is a power of 2 or n=0. the proof is given here Fermat number - Wikipedia, the free encyclopedia There is nothing complicated about the proof supplied in the link. It is a proven fact, this has to be what the teacher was going for, the other possibilities are either not true or would never be asked on an exam by someone with a PhD in math. I am not following what your problem is with this, it is a very straightforward if then statement. The proof goes by contradiction in supposing that n is not a power of two, then it has an odd prime factor. Then you show how you factor $2^n+1$ which makes it NOT prime, a contradiction. • Aug 4th 2009, 12:35 PM Gamma Quote: Originally Posted by streethot Sorry, i thought the question was about $2^{m}+1$ with $m=2^n \ \ n\in\mathbb{N}$ It is, that is what it means to be a power of 2. Show 40 post(s) from this thread on one page Page 1 of 2 12 Last	{"extraction_info": {"found_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": 41, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.909927248954773, "perplexity": 754.4983851388765}, "config": {"markdown_headings": true, "markdown_code": false, "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-2016-44/segments/1476988719192.24/warc/CC-MAIN-20161020183839-00347-ip-10-171-6-4.ec2.internal.warc.gz"}
http://crypto.stackexchange.com/questions/2249/how-does-one-attack-a-two-time-pad-i-e-one-time-pad-with-key-reuse	# How does one attack a two-time pad (i.e. one time pad with key reuse)? My question might appear the same as the question Taking advantage of one-time pad key reuse?, but actually I did read all the answers and none of them helped me with the details I need. I am new to cryptography and my problem is with two time pad attacks on OTP. The problem I had in my course was that I have 10 ciphertexts encrypted with the same key $K$. I am then given another ciphertext that I should decrypt. I know that XOR-ing two ciphers gives me the XOR of their original messages. My question is what is the correct thing to do after that? I tried to take 3 ciphertexts $C_1, C_2$ and $C_3$. Then get $S_1 = C_1 \oplus C_2 \oplus$' ', also get $S_2 = C_1 \oplus C_3 \oplus$ ' '. After that I compared all corresponding characters in $S_1$ and $S_2$, and if $S_1[i] = S_2[i]$ then I calculate $S_1[i] \oplus C_2[i]$ to get $K[i]$. I tried this on paper before coding and it worked, but I might be missing something. Is this the right approach? Why does it work? - If string1[i] == string2[i], that only means that C2[i] == C3[i] (and thus M2[i] == M3[i]). You can check that fact without doing any of those XOR operations. And XORing string1 by C2 gives you back C1 XOR "space", which, at a glance, doesn't seem to be related to key[i]. I haven't looked at that EC assignment for the class in any detail yet, but I think you're supposed to use the fact that XORing by "space" only changes the case of the original character. You might use that fact to analyze the XOR or two plaintext messages, which you obtain by XORing their two respective ciphertexts. – B-Con Apr 1 '12 at 1:35 Yes, your totally right. I was mistaken with that part, but it is my very first trial in cryptography. I got it now – Samer Meggaly Apr 1 '12 at 20:21 Does "the class" include a disclaimer that the method needs to have $m_1 \neq m_2$ in order for the method to work? – Dilip Sarwate Jun 16 '12 at 11:35 @DilipSarwate: If $m_1=m_2$ then the key has still only been used once. Therefore it is implicitly clear that $m_1\neq m_2$ – Maeher Jun 16 '12 at 11:56 Well, the classical answer to "what is the correct thing to do after you have the XOR of the two original messages" is crib-dragging. That is, you take a guess of a common phrase that may appear in one of the plaintexts (the classical example against ASCII english is the 5 letter " the "), and exclusive-or that against the XOR of the two original messages in various locations. If one of the plaintexts had the text of the crib (" the " in our example), then the result of the exclusive-or is what the other plaintext had in that position; if neither plaintext had that, it's likely that the result of the exclusive-or is just gibberish. And, once you have a plausible short section, you can extend it (for example, if you know that one of the plaintexts is " na**", you can go through the dictionary of all words that start with "na", use those as cribs, and see which makes the other plaintext make sense). In addition, you can often deduce things directly from the bit pattern. For example, if the messages are in ASCII, then one thing to note that bit 6 of letters is set, but bit 6 of spaces, numbers and (most) punctuation is clear; because spaces are far more common than numbers and punctuation, then that will give you a good guess of where spaces occur in the texts (albeit without telling you which message a specific space appears in). Now, if you have 11 messages all encrypted with the same pad (an "11-time pad"), things get even easier. Obviously, you can grab a crib across one message, and check it against the other 10; if it makes all 10 make sense, then it is almost certainly accurate. Even better, by using the observation that you can distinguish spaces from letters (again, by comparing bit 6), you can find where all the spaces appear in the messages; these all act like 1 character cribs, probably revealing virtually all the text of all the messages. - Thanks, a lot i solved the problem. I started with ' the ' as you said and got 'robab' which i guessed would be 'probably'. I kept doing this with some guesses until i solved the whole message. Thanks again. – Samer Meggaly Apr 1 '12 at 20:23 Furthermore, consider contents of more common files - sometimes a bunch of binary files are XORd with the same OTP. Some header patterns make determining the key super easy if you know the type, for example some image type has a long string of null bytes. – deed02392 Mar 13 at 8:22 In general, knowledge of $m_1 \oplus m_2$ is not enough to uniquely determine $m_1$ and $m_2$, even if both are known to be, say, English text. For a simple example, $$\text{"one one"} \oplus \text{"two two"} = \text{"one two"} \oplus \text{"two one"}.$$ However, in practice it may be possible to obtain fairly good guesses for $m_1$ and $m_2$; the typical methods are similar to those used for breaking classical ciphers, and rely on the fact that there's a lot of redundancy in English text (and in many other types of data). For example, one might start by guessing that at least one of the messages is likely to contain the word "the", probably surrounded by spaces. So one can take the five-character string " the ", XOR it with every five-character substring of $m_1 \oplus m_2$ and look for results that look like English (either by eye or by computer using statistical analysis). Now, let's say that one of the five-character substrings thus obtained is, say, "messa". Now we (or a computer) could guess that the next two characters are likely to be "ge" (or perhaps "gi"). We can now XOR that with the next two characters of $m_1 \oplus m_2$ and see if the result fits naturally after " the "; if the result is, say, "la", we might tentatively assume our guess to have been right; if it's "q%", we probably guessed wrong. We can proceed in this manner to confirm and extend our guess further, and perhaps eventually to connect separate guessed fragments together until we have a reasonable guess of all, or at least most, of the content of the two messages. -	{"extraction_info": {"found_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": 0.8041526675224304, "perplexity": 737.110599646978}, "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/1405997894260.71/warc/CC-MAIN-20140722025814-00191-ip-10-33-131-23.ec2.internal.warc.gz"}
https://testbook.com/question-answer/in-a-row-of-students-a-is-8th-from-the-left-and-b--5ecd2fbdf60d5d024b9cd1c5	# In a row of students, A is 8th from the left and B is 15th form the right. If they interchange their positions. A becomes 42th form the left. How many students are in the row? Free Practice With Testbook Mock Tests ## Options: 1. 57 2. 55 3. 58 4. 56 ### Correct Answer: Option 4 (Solution Below) This question was previously asked in LMRC Assistant Manager Electrical : 2019 Paper LMRC Assistant Manager Electrical : 2018 Paper ## Solution: Position of A from the left end = 8th Position of B from the right end = 15th After interchanging positions, the new position of A = 42nd Total number of students in the row = (42 + 15) – 1 = 57 – 1 = 56 Hence, ‘56’ is the correct answer.	{"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": 0.8813962936401367, "perplexity": 3090.5730295681883}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046152236.64/warc/CC-MAIN-20210727041254-20210727071254-00395.warc.gz"}
http://farside.ph.utexas.edu/teaching/315/Waveshtml/node92.html	Next: Wavefunction Collapse Up: Wave Mechanics Previous: Wave Packets # Heisenberg's Uncertainty Principle According to the analysis contained in the previous section, a particle wave packet that is initially localized in -space, with characteristic width , is also localized in -space, with characteristic width . However, as time progresses, the width of the wave packet in -space increases [see Equation (1133)], while that of the packet in -space stays the same [because is given by Equation (1123) at all times]. Hence, in general, we can say that (1134) Furthermore, we can interpret and as characterizing our uncertainty regarding the values of the particle's position and wavenumber, respectively. A measurement of a particle's wavenumber, , is equivalent to a measurement of its momentum, , because . Hence, an uncertainty in of order translates to an uncertainty in of order . It follows, from the previous inequality, that (1135) This is the famous Heisenberg uncertainty principle, first proposed by Werner Heisenberg in 1927 (Dirac 1982). According to this principle, it is impossible to simultaneously measure the position and momentum of a particle (exactly). Indeed, a good knowledge of the particle's position implies a poor knowledge of its momentum, and vice versa. The uncertainty principle is a direct consequence of representing particles as waves. It is apparent, from Equation (1133), that a particle wave packet of initial spatial extent spreads out in such a manner that its spatial extent becomes (1136) at large . It is readily demonstrated that this spreading of the wave packet is a consequence of the uncertainty principle. Indeed, because the initial uncertainty in the particle's position is , it follows that the uncertainty in its momentum is of order . This translates to an uncertainty in velocity of . Thus, if we imagine that part of the wave packet propagates at , and another part at , where is the mean propagation velocity, then it follows that the wave packet will spread out as time progresses. Indeed, at large , we expect the width of the wave packet to be (1137) which is identical to Equation (1136). Evidently, the spreading of a particle wave packet, as time progresses, should be interpreted as representing an increase in our uncertainty regarding the particle's position, rather than an increase in the spatial extent of the particle itself. Next: Wavefunction Collapse Up: Wave Mechanics Previous: Wave Packets Richard Fitzpatrick 2013-04-08	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9802438616752625, "perplexity": 412.53688494477103}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370496669.0/warc/CC-MAIN-20200330054217-20200330084217-00008.warc.gz"}
https://rd.springer.com/chapter/10.1007/978-0-8176-8152-4_6	# The Tent Method in Finite-Dimensional Spaces Part of the Systems & Control: Foundations & Applications book series (SCFA) ## Abstract The Tent Method is shown to be a general tool for solving a wide spectrum of extremal problems. First, we show its workability in finite-dimensional spaces. Then topology is applied for the justification of some results in variational calculus. A short historical remark on the Tent Method is made and the idea of the proof of the Maximum Principle is explained in detail, paying special attention to the necessary topological tools. The finite-dimensional version of the Tent Method allows one to establish the Maximum Principle and to obtain a generalization of the Kuhn–Tucker Theorem in Euclidean spaces. ## Keywords Maximum Principle Extremal Problem Variational Calculus Conditional Extremum Tucker Theorem These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves. ## References 1. Boltyanski, V. (1958), ‘The maximum principle in the theory of optimal processes’, Dokl. Akad. Nauk SSSR 119(6), 1070–1073 (in Russian). 2. Boltyanski, V. (1975), ‘The tent method in the theory of extremal problems’, Usp. Mat. Nauk 30, 3–65 (in Russian). Google Scholar 3. Boltyanski, V. (1985), ‘Separation of a system of convex cones in a topological vector space’, Dokl. Akad. Nauk SSSR 283(5), 1044–1047 (in Russian). 4. Boltyanski, V. (1986), ‘The method of tents in topological vector spaces’, Dokl. Akad. Nauk SSSR 289(5), 1036–1039 (in Russian). 5. Boltyanski, V., & Poznyak, A. (1999b), ‘Robust maximum principle in minimax control’, Int. J. Control 72(4), 305–314. 6. Boltyanski, V., Martini, H., & Soltan, V. (1999), Geometric Methods and Optimization Problems, Kluwer Academic, Dordrecht. 7. Bressan, A., & Piccoli, B. (2007), Introduction to the Mathematical Theory of Control, Vol. 2 of Applied Mathematics, American Institute of Mathematical Sciences (AIMS), Springfield. 8. Dubovitski, A., & Milyutin, A. (1963), ‘Extremum problems with constrains’, Dokl. Akad. Nauk SSSR 149(4), 759–762 (in Russian). Google Scholar 9. Dubovitski, A., & Milyutin, A. (1965), ‘Extremum problems with constrains’, Zh. Vychisl. Mat. Mat. Fiz. 5(3), 395–453. Google Scholar 10. Fattorini, H.O. (1999), Infinite-Dimensional Optimization and Control Theory, Vol. 62 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge. 11. Feldbaum, A. (1953), ‘Optimal processes in systems of automatic control’, Avtom. Telemeh. 14(6), 712–728 (in Russian). Google Scholar 12. Hilton, P. (1988), ‘A brief, subjective history of homology and homotopy theory in this century’, Math. Mag. 60(5), 282–291. 13. Kuhn, H., & Tucker, A. (1951), ‘Nonlinear programming’, in Proceedings of the Second Berkeley Symposium on Math. Statistics and Probability, University of California Press, Berkeley, pp. 481–492. Google Scholar 14. McShane, E. (1978), ‘The calculus of variations from the beginning through optimal control theory’, in Optimal Control and Differential Equations (A.B. Schwarzkopf, W.G. Kelley & S.B. Eliason, eds.), University of Oklahoma Press, Norman. Google Scholar 15. Neustadt, L. (1969), ‘A general theory of extremals’, J. Comput. Syst. Sci. 3(1), 57–92. 16. Polyak, B.T. (1987), Introduction to Optimization, Optimization Software Publication Division, New York. Google Scholar 17. Pontryagin, L.S., Boltyansky, V.G., Gamkrelidze, R.V., & Mishenko, E.F. (1969), Mathematical Theory of Optimal Processes, Nauka, Moscow (in Russian). Google Scholar 18. Poznyak, A.S. (2008), Advanced Mathematical Tools for Automatic Control Engineers, Vol. 1: Deterministic Technique, Elsevier, Amsterdam. Google Scholar	{"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": 0.8269747495651245, "perplexity": 4706.967825746146}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257649095.35/warc/CC-MAIN-20180323220107-20180324000107-00487.warc.gz"}
https://www.lessonplanet.com/teachers/fill-in-the-missing-numbers-5th-6th	# Fill in the Missing Numbers In this math learning exercise, students complete a 36 square math puzzle. Students examine the 6 numbers on a diagonal and figure out the pattern before filling out the grid. Concepts Resource Details	{"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": 0.8754026889801025, "perplexity": 1595.061522401275}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257644877.27/warc/CC-MAIN-20180317100705-20180317120705-00617.warc.gz"}
http://perfectpuddle.blogspot.com/2015/11/kias-cfhep-workshop-liveblog-day-five.html	## Friday, 13 November 2015 ### KIAS-CFHEP Workshop Liveblog: Day Five Session Two The last day of the conference, and the day after the banquet. As seems to be traditional, I overslept and missed the first session. Oh well. I missed two ILC talks that would probably have been interesting, but so be it. 11:20 am: Physics at a 100-TeV Collider, Tao Han I'm guessing this talk will be similar to the seminar I saw Tao give recently. HEP is at an extremely exciting time. The completion of the SM means that for the first time we have a complete and consistent relativistic quantum theory. Indeed, our theory seems valid up to the Planck scale (possibly). But we've thought this kind of thing before (19th century). Indeed, the central questions today are not about details, but fundamental: origin of spacetime, UV/IR connection, real underlying theory. In this line, let us consider three questions. 1. The Nature of EWSB? All we really know is the Higgs potential in a tiny region about the vacuum. What does the full potential really look like? The first and most important question here is the nature of the Higgs triple coupling. This can consistently vary by order-1 factors relative to the SM! This would have huge effects on the cosmology, by allowing a 1st-order phase transition. 2. Naturalness. Possible theoretically nice explanations: true naturalness and NP at the TeV scale; a fundamental and deep connection between UV and IR physics; some kind of multiverse/anthropic selection effect. 3. Dark Sector. In particular, Higgs portal will be probed in the next generation of detectors. Additionally, these can be probed using measurements of Higgs physics at colliders. The immediate collider future is, of course, the LHC. Looking further, and ignoring linear colliders, two rough ideas. First is possible US/European electron-positron collider of few hundred GeV. But real interest here is, of course, a next generation hadron collider running at 100 TeV. The Chinese proposal recently confirmed is the one. Possible Europe timeline: LHC till 2020, VH-LHC till 2030. Hope to have maybe TLEP ready then for another decade, followed by ...? Chinese machine recent announcement a surprise to scientists! 100 TeV leads to qualitatively new physics. Note that top quark at these energies is as massless as the bottom quark was at the Tevatron! So all SM particles are partons, and really working in unbroken EW phase. Even pure QCD events will be studied in completely new regimes; e.g. dijet invariant masses of 50 TeV, one event per iab. Guage boson FSR will be like radiation in QCD jets. Probe of LHC self-coupling can be resolved to order 10%. Also tests compositeness of Higgs (as a function of scale). SUSY mass reach improved typically be a factor of 7. More generally, fine-tuning can be pushed to 10-4, which more or less answers that question. Dark matter searches good in regions where DD fails: very light particles, or very pure winos/Higgsinos. Finally, there's the potential to find something unexpected. Standard resonance searches give us some idea of reach. Tens of TeV in some channels. Question No interesting question, but Tao got provoked into the most animated speech I've ever seen from him. The crux is that we don't need to find something to justify this kind of experiment; simply understanding nature at a better scale than anyone else ever before is enough, at least it should be for scientists. 12:00 pm: Higgs Physics and Detector Design at CEPC, Manqi Ruan Plans to run China first as electron-positron tunnel, then upgrade to hadron collider. Detector goals: high precision vertex tracker, close to IP, for bottom, charm and tau tagging. High precision tracking to measure transverse momentum for TeV tracks at 10% level. Calorimeter suitable for jet substructure etc. Calorimeter benefit from development of microelectronics allowing ultra high granularity. Improve by ~ 1000 over LHC, gain 3D granularity. Goal: identify every detector hit. Substantial pile up. Measurement of Higgs width using production cross section (shape as function of energy) to resolve the coupling ambiguity. Also direct measurement of line width (3%) and tag of recoil to measure invisible width (better than 10%?). Leads to model independent measurements of Higgs couplings. Hadron machine run benefits from luminosity, lots of tops, and energy to measure top Yukawa. And we finish 20 minutes late.	{"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": 0.8223904967308044, "perplexity": 4875.610794579794}, "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/1495463607963.70/warc/CC-MAIN-20170525025250-20170525045250-00070.warc.gz"}
https://export.arxiv.org/abs/1912.01815	math.AP (what is this?) # Title: Extremal case of parabolic differential equations having discontinuous unbounded coefficients. Existence of fundamental solution for an initial Cauchy problem. Parametrix method Abstract: We prove in this short report the existence of a fundamental solution (F.S.) for the Cauchy initial boundary problem on the whole space for the parabolic differential equation having at origin the point of non-integrable unbounded discontinuity for coefficient before a first order derivative. We give also the non-asymptotic rapidly decreasing at infinity estimate for these function. We extend the classical parametrix method offered by E.E.Levi. Subjects: Analysis of PDEs (math.AP) Cite as: arXiv:1912.01815 [math.AP] (or arXiv:1912.01815v1 [math.AP] for this version) ## Submission history From: Leonid Sirota [view email] [v1] Wed, 4 Dec 2019 06:18:05 GMT (13kb) Link back to: arXiv, form interface, contact.	{"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": 0.8210899829864502, "perplexity": 4928.553761803948}, "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/1623487610841.7/warc/CC-MAIN-20210613192529-20210613222529-00162.warc.gz"}
https://pennstate.pure.elsevier.com/en/publications/data-driven-estimation-of-the-invisible-energy-of-cosmic-ray-show	# Data-driven estimation of the invisible energy of cosmic ray showers with the Pierre Auger Observatory The Pierre Auger Collaboration Research output: Contribution to journalArticlepeer-review 11 Scopus citations ## Abstract The determination of the primary energy of extensive air showers using the fluorescence detection technique requires an estimation of the energy carried away by particles that do not deposit all their energy in the atmosphere. This estimation is typically made using Monte Carlo simulations and thus depends on the assumed primary particle mass and on model predictions for neutrino and muon production. In this work we present a new method to obtain the invisible energy from events detected by the Pierre Auger Observatory. The method uses measurements of the muon number at ground level, and it allows us to significantly reduce the systematic uncertainties related to the mass composition and the high energy hadronic interaction models, and consequently to improve the estimation of the energy scale of the Pierre Auger Observatory. Original language English (US) 082003 Physical Review D 100 8 https://doi.org/10.1103/PhysRevD.100.082003 Published - Oct 25 2019 ## All Science Journal Classification (ASJC) codes • Physics and Astronomy (miscellaneous) ## Fingerprint Dive into the research topics of 'Data-driven estimation of the invisible energy of cosmic ray showers with the Pierre Auger Observatory'. Together they form a unique fingerprint.	{"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": 0.8319315910339355, "perplexity": 1073.6113874750704}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320300533.72/warc/CC-MAIN-20220117091246-20220117121246-00229.warc.gz"}
http://proofsfromthebook.com/2013/01/19/proof-that-vertical-angles-are-congruent/	Proof that Vertical Angles Are Congruent A pair of angles whose sides form two lines is called vertical angles. In the figure below, angles 1 and 3 are vertical angles since their sides form lines l and m. Similarly, angles 2 and 4 are vertical angles for the same reason. Vertical angles are congruent and it is easy to prove. We just use the fact that a linear pair of angles are supplementary; that is their measures add up to $180^\circ$. In the figure above, to prove that vertical angles are congruent, we have to show that $\angle 1$ and $\angle 3$ are congruent or $\angle 2$ and $\angle 4$ are congruent. Theorem Vertical angles are congruent. Proof We show that $\angle 1 \cong \angle 3$. $m \angle 1 + m \angle 2 = 180^\circ$ Linear pair of angles are supplementary. $m \angle 2 + m\angle 3 = 180^\circ$ Linear pair of angles are supplementary. $m \angle 1 + m \angle 2 = m \angle 2 + m \angle 3$ ** Substitution property of equality; that is $180^\circ = 180^\circ$. Substracting $\angle 2$ from both sides, we have $m \angle 1 = m \angle 3$. Therefore, vertical angles are congruent. $\blacksquare$ As an exercise, show that $m \angle 2 = m \angle 4$.	{"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": 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": 0.878794252872467, "perplexity": 571.277339561234}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400198287.23/warc/CC-MAIN-20200920161009-20200920191009-00540.warc.gz"}
https://math.stackexchange.com/questions/2208995/notation-for-delta-distribution	Notation For Delta Distribution I've encountered two different notations for the delta distribution: $$\delta(x,y)$$ and $$\delta(x-y)$$ What is the difference between these two notations. Do they depend on context or should I prefer one over the other? • The first one is not sufficiently standard for me to even know the intended meaning. The second one is quite standard (if the variable of integration is $x$, then this can be understood as a point mass of $1$ at $y$). – Ian Mar 29 '17 at 17:33 • If the domain / space isn't additive, then $x-y$ makes no sense. But usually one writes $\delta_y$ if the mass is centered at $y$. – user251257 Mar 29 '17 at 17:38 The second describes a delta function acting along the line $x=y$: The first notation is more convenient when working in high dimensions, since it naturally generalizes to $n$ dimensions: e.g., $\delta (x,y,z,w, ...)$. You can do this with the second notation $\delta(x) \delta(y), ...$	{"extraction_info": {"found_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": 0.9744724035263062, "perplexity": 214.40356809507486}, "config": {"markdown_headings": false, "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-2019-51/segments/1575540488620.24/warc/CC-MAIN-20191206122529-20191206150529-00301.warc.gz"}
http://math.stackexchange.com/questions/216391/how-to-show-that-frac-sinnn-is-1-as-n-rightarrow-0/216407	# How to show that $\frac{\sin(n)}{n}$ is $1$ as $n \rightarrow 0$? [duplicate] Possible Duplicate: How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$? How to show that $\frac{\sin(n)}{n}$ is $1$ as $n \rightarrow 0$? just hint. - ## marked as duplicate by Martin Sleziak, AD., rschwieb, Norbert, Jason DeVitoOct 19 '12 at 1:27 Show that $\cos{x}<\frac{\sin{x}}{x}<1,x\in(-\frac{\pi}{2},\frac{\pi}{2})$ – Golbez Oct 18 '12 at 15:52 As an alternative, you can consider that limit as a derivative. – T. Verron Oct 18 '12 at 15:56 T.Verron, that may be a bit circular. You can consider the limit as a derivative, but if you can't prove this limit you can't prove the derivative. – Isaac Solomon Oct 18 '12 at 15:57 By the way how to formulate arbitrary complex trigonometric polynom? I know that in real form it is $\sum_{n=1}^{k}cos(nx)+isin(nx)$ – alvoutila Oct 18 '12 at 16:02 It is a little confusing to use the notation '$n \rightarrow 0$'. We usually reserve $n$ for natural numbers and $x$ for real numbers. Hence, for reasons of pedagogy, it is better to write '$x \rightarrow 0$'. :) – Haskell Curry Oct 18 '12 at 16:09 Maclaurin series expansion of $\sin(n)$ is, $$\sin(n) = n - \frac{n^3}{3!} +\frac{n^5}{5!}+...$$ Hence, $$\frac{\sin(n)}{n} = 1-\frac{n^2}{3!} + \frac{n^4}{5!}+...$$ $$\lim_{n\to 0}\frac{\sin(n)}{n} = 1$$ - This reasoning is a bit circular (unless we take the series expansion as the definition of the sine function). The Maclaurin expansion depends on the derivative of the sine function, which depends on the limit we're trying to compute. – Hans Lundmark Oct 18 '12 at 16:17 First, Prove that $\sin{x}<x<\tan{x}$, when $x\in (0,\frac{\pi}{2})$ By means of drawing a circle, take an arbitary point on the circle with coordinate $A:(\cos{x},\sin{x})$, take $B:(0,1),O:(0,0),C:(\cos{x},0),D:(\sec{x},0)$ Obviously We have $\sin{x}=S_{\Delta OAC }$, $x=S_{ OAB}$ where $S_{OAB}$ denotes the area of the circular sector, $\tan{x}=S_{\Delta OAD}$ Also, it's obvious(By drawing this circle) that $S_{\Delta OAC }<S_{ OAB}<S_{\Delta OAD}$, thus\begin{align}\sin{x}<x<\tan{x},\quad(x\in(0,\frac{\pi}{2}))\end{align} By multiplying $-1$ on each side \begin{align}\sin{x}>x>\tan{x},\quad(x\in(-\frac{\pi}{2},0))\end{align}	{"extraction_info": {"found_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": 3, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9710438847541809, "perplexity": 474.10471842380156}, "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-2016-07/segments/1454701147841.50/warc/CC-MAIN-20160205193907-00247-ip-10-236-182-209.ec2.internal.warc.gz"}
https://gitlab.mpcdf.mpg.de/ift/nifty/-/commit/4e67193280166255e63e1b6b9dd774b9ecb1dd5e	Commit 4e671932 by Martin Reinecke ### Merge remote-tracking branch 'upstream/NIFTy_5' into simplify_for_const parents 0fd92060 eba1fd40 image: $CONTAINER_TEST_IMAGE variables: CONTAINER_TEST_IMAGE: gitlab-registry.mpcdf.mpg.de/ift/nifty-dev:$CI_BUILD_REF_NAME CONTAINER_TEST_IMAGE: gitlab-registry.mpcdf.mpg.de/$CI_PROJECT_PATH:$CI_BUILD_REF_NAME OMP_NUM_THREADS: 1 stages: ... ... @@ -39,9 +39,9 @@ test_serial: script: - pytest-3 -q --cov=nifty5 test - > python3 -m coverage report --omit "plot,distributed_do" python3 -m coverage report --omit "plot,distributed_do" \| tee coverage.txt - > python3 -m coverage report --omit "plot,distributed_do" \| grep TOTAL \| awk '{ print "TOTAL: "$4; }' grep TOTAL coverage.txt \| awk '{ print "TOTAL: "$4; }' test_mpi: stage: test ... ... @@ -52,17 +52,15 @@ test_mpi: pages: stage: release before_script: - ls script: - python3 setup.py install --user -f - sh docs/generate.sh - mv docs/build/ public/ artifacts: paths: - public only: - NIFTy_4 - NIFTy_5 before_script: - python3 setup.py install --user -f ... ... ... ... @@ -6,7 +6,7 @@ RUN apt-get update && apt-get install -y \ # Packages needed for NIFTy python3-scipy \ # Documentation build dependencies python3-sphinx-rtd-theme \ python3-sphinx-rtd-theme dvipng texlive-latex-base texlive-latex-extra \ # Testing dependencies python3-pytest-cov jupyter \ # Optional NIFTy dependencies ... ... ... ... @@ -41,7 +41,6 @@ Abstract base class One of the fundamental building blocks of the NIFTy5 framework is the domain. Its required capabilities are expressed by the abstract :py:class:Domain class. A domain must be able to answer the following queries: m - its total number of data entries (pixels), which is accessible via the :attr:~Domain.size property ... ... @@ -129,7 +128,7 @@ specify full field domains. In principle, a :class:~domain_tuple.DomainTuple can even be empty, which implies that the field living on it is a scalar. A :class:~domain_tuple.DomainTuple supports iteration and indexing, and also provides the properties :attr:~domain_tuple.DomainTuple.shape, provides the properties :attr:~domain_tuple.DomainTuple.shape and :attr:~domain_tuple.DomainTuple.size in analogy to the elementary :class:~domains.domain.Domain. ... ... @@ -159,10 +158,11 @@ Contractions (like summation, integration, minimum/maximum, computation of statistical moments) can be carried out either over an entire field (producing a scalar result) or over sub-domains (resulting in a field defined on a smaller domain). Scalar products of two fields can also be computed easily. See the documentation of :class:~field.Field for details. There is also a set of convenience functions to generate fields with constant values or fields filled with random numbers according to a user-specified distribution. distribution: :attr:~sugar.full, :attr:~sugar.from_random. Like almost all NIFTy objects, fields are immutable: their value or any other attribute cannot be modified after construction. To manipulate a field in ways ... ... @@ -311,11 +311,15 @@ and f1 and f2 are of type :class:~field.Field, writing:: will perform the operation suggested intuitively by the notation, checking domain compatibility while building the composed operator. The combined operator infers its domain and target from its constituents, as well as the set of operations it can support. The properties :attr:~LinearOperator.adjoint and :attr:~LinearOperator.inverse return a new operator which behaves as if it were the original operator's adjoint or inverse, respectively. The combined operator infers its domain and target from its constituents, as well as the set of operations it can support. Instantiating operator adjoints or inverses by :attr:~LinearOperator.adjoint and similar methods is to be distinguished from the instant application of operators performed by :attr:~LinearOperator.adjoint_times and similar methods. .. _minimization: ... ... @@ -368,8 +372,8 @@ failure. Sensible stopping criteria can vary significantly with the problem being solved; NIFTy provides one concrete sub-class of :class:IterationController called :class:GradientNormController, which should be appropriate in many circumstances, but users have complete freedom to implement custom sub-classes for their specific applications. circumstances, but users have complete freedom to implement custom :class:IterationController sub-classes for their specific applications. Minimization algorithms ... ... @@ -398,7 +402,7 @@ Many minimizers for nonlinear problems can be characterized as This family of algorithms is encapsulated in NIFTy's :class:~descent_minimizers.DescentMinimizer class, which currently has three concrete implementations: :class:~descent_minimizers.SteepestDescent, generally usable concrete implementations: :class:~descent_minimizers.NewtonCG, :class:~descent_minimizers.L_BFGS and :class:~descent_minimizers.VL_BFGS. Of these algorithms, only :class:~descent_minimizers.NewtonCG requires the energy object to provide ... ... @@ -424,11 +428,13 @@ the information propagator whose inverse is defined as: :math:D^{-1} = \left(R^\dagger N^{-1} R + S^{-1}\right). It needs to be applied in forward direction in order to calculate the Wiener filter solution. Only its inverse application is straightforward; to use it in forward direction, we make use of NIFTy's filter solution, but only its inverse application is straightforward. To use it in forward direction, we make use of NIFTy's :class:~operators.inversion_enabler.InversionEnabler class, which internally performs a minimization of a :class:~minimization.quadratic_energy.QuadraticEnergy by means of the :class:~minimization.conjugate_gradient.ConjugateGradient algorithm. An example is provided in applies the (approximate) inverse of the given operator :math:x = Op^{-1} (y) by solving the equation :math:y = Op (x) for :math:x. This is accomplished by minimizing a suitable :class:~minimization.quadratic_energy.QuadraticEnergy with the :class:~minimization.conjugate_gradient.ConjugateGradient algorithm. An example is provided in :func:~library.wiener_filter_curvature.WienerFilterCurvature. ... ... @@ -4,8 +4,7 @@ IFT -- Information Field Theory Theoretical Background ---------------------- Information Field Theory _ [1]_ (IFT) is information theory, the logic of reasoning under uncertainty, applied to fields. Information Field Theory _ [1]_ (IFT) is information theory, the logic of reasoning under uncertainty, applied to fields. A field can be any quantity defined over some space, e.g. the air temperature over Europe, the magnetic field strength in the Milky Way, or the matter density in the Universe. IFT describes how data and knowledge can be used to infer field properties. Mathematically it is a statistical field theory and exploits many of the tools developed for such. ... ... @@ -22,89 +21,18 @@ NIFTy comes with reimplemented MAP and VI estimators. .. tip:: In-a-nutshell introductions to information field theory can be found in [2]_, [3]_, [4]_, and [5]_, with the latter probably being the most didactical. .. [1] T.A. Enßlin et al. (2009), "Information field theory for cosmological perturbation reconstruction and nonlinear signal analysis", PhysRevD.80.105005, 09/2009; [arXiv:0806.3474] _ .. [1] T.A. Enßlin et al. (2009), "Information field theory for cosmological perturbation reconstruction and nonlinear signal analysis", PhysRevD.80.105005, 09/2009; [arXiv:0806.3474] _ .. [2] T.A. Enßlin (2013), "Information field theory", proceedings of MaxEnt 2012 -- the 32nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering; AIP Conference Proceedings, Volume 1553, Issue 1, p.184; [arXiv:1301.2556] _ .. [2] T.A. Enßlin (2013), "Information field theory", proceedings of MaxEnt 2012 -- the 32nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering; AIP Conference Proceedings, Volume 1553, Issue 1, p.184; [arXiv:1301.2556] _ .. [3] T.A. Enßlin (2014), "Astrophysical data analysis with information field theory", AIP Conference Proceedings, Volume 1636, Issue 1, p.49; [arXiv:1405.7701] _ .. [3] T.A. Enßlin (2014), "Astrophysical data analysis with information field theory", AIP Conference Proceedings, Volume 1636, Issue 1, p.49; [arXiv:1405.7701] _ .. [4] Wikipedia contributors (2018), "Information field theory" _, Wikipedia, The Free Encyclopedia. .. [5] T.A. Enßlin (2019), "Information theory for fields", accepted by Annalen der Physik; [DOI] _, [arXiv:1804.03350] _ Discretized continuum --------------------- The representation of fields that are mathematically defined on a continuous space in a finite computer environment is a common necessity. The goal hereby is to preserve the continuum limit in the calculus in order to ensure a resolution independent discretization. +-----------------------------+-----------------------------+ \| .. image:: images/42vs6.png \| .. image:: images/42vs9.png \| \| :width: 100 % \| :width: 100 % \| +-----------------------------+-----------------------------+ Any partition of the continuous position space :math:\Omega (with volume :math:V) into a set of :math:Q disjoint, proper subsets :math:\Omega_q (with volumes :math:V_q) defines a pixelization, .. math:: \Omega &\quad=\quad \dot{\bigcup_q} \; \Omega_q \qquad \mathrm{with} \qquad q \in \{1,\dots,Q\} \subset \mathbb{N} , \\ V &\quad=\quad \int_\Omega \mathrm{d}x \quad=\quad \sum_{q=1}^Q \int_{\Omega_q} \mathrm{d}x \quad=\quad \sum_{q=1}^Q V_q . Here the number :math:Q characterizes the resolution of the pixelization and the continuum limit is described by :math:Q \rightarrow \infty and :math:V_q \rightarrow 0 for all :math:q \in \{1,\dots,Q\} simultaneously. Moreover, the above equation defines a discretization of continuous integrals, :math:\int_\Omega \mathrm{d}x \mapsto \sum_q V_q. Any valid discretization scheme for a field :math:{s} can be described by a mapping, .. math:: s(x \in \Omega_q) \quad\mapsto\quad s_q \quad=\quad \int_{\Omega_q} \mathrm{d}x \; w_q(x) \; s(x) , if the weighting function :math:w_q(x) is chosen appropriately. In order for the discretized version of the field to converge to the actual field in the continuum limit, the weighting functions need to be normalized in each subset; i.e., :math:\forall q: \int_{\Omega_q} \mathrm{d}x \; w_q(x) = 1. Choosing such a weighting function that is constant with respect to :math:x yields .. [5] T.A. Enßlin (2019), "Information theory for fields", accepted by Annalen der Physik; [DOI] _, [arXiv:1804.03350] _ .. math:: s_q = \frac{\int_{\Omega_q} \mathrm{d}x \; s(x)}{\int_{\Omega_q} \mathrm{d}x} = \left< s(x) \right>_{\Omega_q} , which corresponds to a discretization of the field by spatial averaging. Another common and equally valid choice is :math:w_q(x) = \delta(x-x_q), which distinguishes some position :math:x_q \in \Omega_q, and evaluates the continuous field at this position, .. math:: s_q \quad=\quad \int_{\Omega_q} \mathrm{d}x \; \delta(x-x_q) \; s(x) \quad=\quad s(x_q) . In practice, one often makes use of the spatially averaged pixel position, :math:x_q = \left< x \right>_{\Omega_q}. If the resolution is high enough to resolve all features of the signal field :math:{s}, both of these discretization schemes approximate each other, :math:\left< s(x) \right>_{\Omega_q} \approx s(\left< x \right>_{\Omega_q}), since they approximate the continuum limit by construction. (The approximation of :math:\left< s(x) \right>_{\Omega_q} \approx s(x_q \in \Omega_q) marks a resolution threshold beyond which further refinement of the discretization reveals no new features; i.e., no new information content of the field :math:{s}.) All operations involving position integrals can be normalized in accordance with the above definitions. For example, the scalar product between two fields :math:{s} and :math:{u} is defined as .. math:: {s}^\dagger {u} \quad=\quad \int_\Omega \mathrm{d}x \; s^(x) \; u(x) \quad\approx\quad \sum_{q=1}^Q V_q^{\phantom{}} \; s_q^* \; u_q^{\phantom{}} , where :math:\dagger denotes adjunction and :math: complex conjugation. Since the above approximation becomes an equality in the continuum limit, the scalar product is independent of the pixelization scheme and resolution, if the latter is sufficiently high. The above line of argumentation analogously applies to the discretization of operators. For a linear operator :math:{A} acting on some field :math:{s} as :math:{A} {s} = \int_\Omega \mathrm{d}y \; A(x,y) \; s(y), a matrix representation discretized with constant weighting functions is given by .. math:: A(x \in \Omega_p, y \in \Omega_q) \quad\mapsto\quad A_{pq} \quad=\quad \frac{\iint_{\Omega_p \Omega_q} \mathrm{d}x \, \mathrm{d}y \; A(x,y)}{\iint_{\Omega_p \Omega_q} \mathrm{d}x \, \mathrm{d}y} \quad=\quad \big< \big< A(x,y) \big>_{\Omega_p} \big>_{\Omega_q} . The proper discretization of spaces, fields, and operators, as well as the normalization of position integrals, is essential for the conservation of the continuum limit. Their consistent implementation in NIFTy allows a pixelization independent coding of algorithms. Free Theory & Implicit Operators -------------------------------- ... ... @@ -205,11 +133,11 @@ NIFTy takes advantage of this formulation in several ways: 3) The response can be non-linear, e.g. :math:{R'(s)=R \exp(A\,\xi)}, see demos/getting_started_2.py. 4) The amplitude operator may dependent on further parameters, e.g. :math:A=A(\tau)= F\, \widehat{e^\tau} represents an amplitude operator with a positive definite, unknown spectrum defined in the Fourier domain. 4) The amplitude operator may depend on further parameters, e.g. :math:A=A(\tau)= F\, \widehat{e^\tau} represents an amplitude operator with a positive definite, unknown spectrum defined in the Fourier domain. The amplitude field :math:{\tau} would get its own amplitude operator, with a cepstrum (spectrum of a log spectrum) defined in quefrency space (harmonic space of a logarithmically binned harmonic space) to regularize its degrees of freedom by imposing some (user-defined degree of) spectral smoothness. 5) NIFTy calculates the gradient of the information Hamiltonian and the Fisher information metric with respect to all unknown parameters, here :math:{\xi} and :math:{\tau}, by automatic differentiation. The gradients are used for MAP and HMCF estimates, and the Fisher matrix is required in addition to the gradient by Metric Gaussian Variational Inference (MGVI), which is available in NIFTy as well. The gradients are used for MAP estimates, and the Fisher matrix is required in addition to the gradient by Metric Gaussian Variational Inference (MGVI), which is available in NIFTy as well. MGVI is an implicit operator extension of Automatic Differentiation Variational Inference (ADVI). The reconstruction of a non-Gaussian signal with unknown covariance from a non-trivial (tomographic) response is demonstrated in demos/getting_started_3.py. ... ... @@ -296,7 +224,7 @@ Thus, only the gradient of the KL is needed with respect to this, which can be e We stochastically estimate the KL-divergence and gradients with a set of samples drawn from the approximate posterior distribution. The particular structure of the covariance allows us to draw independent samples solving a certain system of equations. This KL-divergence for MGVI is implemented in the class MetricGaussianKL within NIFTy5. This KL-divergence for MGVI is implemented in the class :class:~minimization.metric_gaussian_kl.MetricGaussianKL within NIFTy5. The demo getting_started_3.py for example not only infers a field this way, but also the power spectrum of the process that has generated the field. ... ...	{"extraction_info": {"found_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": 0.9783794283866882, "perplexity": 4946.468495886932}, "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/1637964363791.16/warc/CC-MAIN-20211209091917-20211209121917-00363.warc.gz"}
https://lessonplanet.com/teachers/evaluating-informaton-quality	# Evaluating Informaton Quality Sixth graders investigate the concept of the quality of information that is used to conduct research. They begin to conceive the differences between information that is fact or fiction. Students write a critique of an information source and identify its point of view. Resource Details	{"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": 0.8787626028060913, "perplexity": 1508.6061748402785}, "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/1570986702077.71/warc/CC-MAIN-20191020024805-20191020052305-00322.warc.gz"}
https://www.physicsforums.com/threads/superposition-waves.198866/	# Superposition waves 1. Nov 17, 2007 ### Nghi 1. The problem statement, all variables and given/known data Two wave pulses on a string approach one another at the time t = 0, as shown in the figure below, except that pulse 2 is inverted so that it is a downward deflection of the string rather than an upward deflection. Each pulse moves with a speed of 1.0 m/s. Assume that the superposition principle holds for these waves, and that the absolute value of the height of each pulse is 1 mm in the figure below. Determine the value of the resultant wave at x = 4.1 m at t = 1.0 s, 2.0 s, 2.5 s, 3.0 s, and 4.0 s. 2. Relevant equations None? o_o 3. The attempt at a solution Sorry to bother everyone on the same day again, but everyone is just so helpful on this forum! :) This one makes my heart sad because I don't know how to find the slope of pulse 2, which is essentially a straight line. :( The only solutions I didn't get were t = 2.0 s and 2.5 s. But I think if I understood how to do 2.0 s, then 2.5 would be manageable. I understand the idea of superposition, but I don't know how to apply it, I guess. Ha ha. :'( My friend mentioned something about finding the slope of pulse 2 first, but I don't know how to do that. I think it's because I'm underthinking. 2. Nov 18, 2007 ### chaoseverlasting The slope of pulse two may be found using the graph. You know the height and the width of the pulse. Similar Discussions: Superposition waves	{"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": 0.9079399704933167, "perplexity": 370.9216329268778}, "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/1500549448198.69/warc/CC-MAIN-20170728103510-20170728123510-00058.warc.gz"}
http://ito.wias-berlin.de/publications/wias-publ/run.jsp?template=abstract&type=Preprint&year=1999&number=502	WIAS Preprint No. 502, (1999) # Simulation of Rare Events by the Stochastic Weighted Particle Method for the Boltzmann Equation Authors • Rjasanow, Sergej • Wagner, Wolfgang 2010 Mathematics Subject Classification • 65C05 76P05 82C80 Keywords • Boltzmann equation, stochastic particle method, rare events, variance reduction, numerical experiments DOI 10.20347/WIAS.PREPRINT.502 Abstract An extension of the stochastic weighted particle method for the numerical treatment of the Boltzmann equation is presented. A new procedure for modelling the inflow boundary condition is introduced and its performance is tested in a two-dimensional example with strong density gradients. A gain factor in computing time of several orders of magnitude is achieved in specific situations. Appeared in • Math.Comput. Modelling 33 (2001), no. 8-9, pp. 907-926	{"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": 0.8384100198745728, "perplexity": 2012.6445348679497}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347401260.16/warc/CC-MAIN-20200529023731-20200529053731-00093.warc.gz"}
https://tex.stackexchange.com/questions/501780/longtable-vertical-line-not-complete	# Longtable: Vertical Line not complete I am using the longtable LaTeX package to create a multi-page table with three columns, one of which contains only images. However, the vertical line between the middle and right column does not complete up to the top \hline as seen in the screenshot. This is the code I'm using to create the table. I removed most of the lines to make it viable easier because it's basically the same line repeated over. \begin{longtable}{m{0.5\textwidth}\|m{0.25\textwidth}\|m{0.15\textwidth}} Thumbnail & Description & Number of Images \\ \hline \\ \includegraphics[width=0.45\textwidth]{img/data/some_img1.png} & Description 1 & 2 \\ \includegraphics[width=0.45\textwidth]{img/data/some_img2.png} & Description 2 & 10 \\ \caption{Gathered Data} \label{tbl:some_data_table}\\ \end{longtable} Thanks! • Welcome to TeX.SE! Remove \\ after first \hline. – Zarko Jul 28 at 1:13 As Zarko already mentioned in the comments, you can get rid of the undesired gap in the vertical line by removing the \\ after \hline: \documentclass{article} \usepackage{longtable} \usepackage{graphicx} \usepackage{array} \begin{document} \begin{longtable}{m{0.5\textwidth}\|m{0.25\textwidth}\|m{0.15\textwidth}} Thumbnail & Description & Number of Images \\ \hline \includegraphics[width=0.45\textwidth]{example-image} & Description 1 & 2 \\ \includegraphics[width=0.45\textwidth]{example-image} & Description 2 & 10 \\ \caption{Gathered Data} \label{tbl:some_data_table}\\ \end{longtable} \end{document} As you can see, removing the \\ also removed the small vertical whie space between the \hline and the first image. If you want to keep such a space around the images, I'd sugggest the usage of the cellspace package. In teh following MWE, I have also used the xltabular package in order to make the table as wide as the textwidth without the need to calculate the required column widths manually. I have also added the adjustbox package in order to use the valing option. Lastly, I have also moved the \caption to the top of the table as it might be a good idea to inform the reader what the table is about before starting the actual table. Especially if your table is longer than one page. \documentclass{article} \usepackage{xltabular} \usepackage{graphicx} \usepackage{array} \usepackage{cellspace} \setlength\cellspacetoplimit{\tabcolsep} \setlength\cellspacebottomlimit{\tabcolsep} \usepackage{makecell} \begin{document} \begin{xltabular}{\textwidth}{Sc\|X\|l} \caption{Gathered Data}\label{tbl:some_data_table}\\ \makecell[l]{Thumbnail} & Description & \makecell[l]{Number of\\ Images} \\ \hline \includegraphics[width=0.45\textwidth,valign=c]{example-image} & Description 1 & 2 \\ \includegraphics[width=0.45\textwidth,valign=c]{example-image} & Description 2 & 10 \\ \end{xltabular} \end{document} Personally, I'd also remove the vertical lines entirely and use horizontal lines from the booktabs package. I would also top align the contents of the cells: \documentclass{article} \usepackage{xltabular} \usepackage{graphicx} \usepackage{array} \usepackage{cellspace} \setlength\cellspacetoplimit{\tabcolsep} \setlength\cellspacebottomlimit{\tabcolsep} \usepackage{makecell} \usepackage{booktabs} \begin{document} \begin{xltabular}{\textwidth}{@{}ScXl} \caption{Gathered Data}\label{tbl:some_data_table}\\ \toprule \multicolumn{1}{l}{Thumbnail} & Description & \makecell[l]{Number of\\ Images} \\ \midrule	{"extraction_info": {"found_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": 0.9755344986915588, "perplexity": 1277.8751284204125}, "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/1573496668416.11/warc/CC-MAIN-20191114104329-20191114132329-00412.warc.gz"}
http://www.techspot.com/community/topics/dos-help-in-xp-pro.121851/	# DOS help in XP pro By gbhall Feb 8, 2009 1. MS has really got me confused ! There is no actual Dos under XP as we all well know, but the implementation of a command window gets more extensive with every issue of Windows - as witness the ROBOCOPY command in Vista !! Now my problem is - where the heck is the PATH property specified ? When I open a command window which is running cmd.exe in System32, there is a long path name there already, with things added by certain installed programs, like (would you believe) quicktime. In addition to which I have found there is an AUTOEXEC.NT file in SYSTEM32 and an AUTOEXEC.DOS in C:\ Neither of those has a PATH line in them. If I open Control panel / system advanced tab and click the 'environment variables' button, there is a Path variable there, but by experiment, all it does is it ADDS to the path already present in any command windows EXCEPT a window which runs command.com instead of cmd.exe More and more confusing - two command language processors, and at least three places where a Path can appear, one of which - the most important - I cannot even find ! I would at this point say I need both of the command processors, because there are certain 16-bit programms I like to run occasionally which will work fine under command.com, but do nothing under cmd.exe.... I would love it if someone could point me to a tutorial on 'dos in windows XP' 2. ### rickohTS Rookie gbhall hold down win key...(between ctrl and alt) and hit r type cmd in the box oops...saw that you did that....sorry. 3. ### LookinAroundTechSpot ChancellorPosts: 8,373 +167 The Windows PATH system variable (as defined via Windows system variables) should appear exactly as you define it in all command windows you open AFTER you've changed the PATH system variable. Not anything currently open. New command windows inherit CURRENT value of the Windows PATH. You can append, change, delete whatever you please to the system PATH variable If you open a command window, remember that path is just a shell variable => From command prompt just type path to see your currrent Path variable => From command prompt, type path=yyyy to set the path variable in THAT current shell to yyyy /* EDIT / And don't forget the importance of enclosing things in quotes as otherwise you can get DOS command prompt very confused when it sees spaces or or other special characters. e.g. PATH="C:\Windows\Program Files" is right, PATH=C:\Windows\Program Files will never work 4. ### gbhallTechSpot ChancellorTopic StarterPosts: 2,348 +50 What I am asking is where is the extensive PATH defined ? The one that is there already when a command shell is opened. try it, what do you get ? I get this.... PATH=C:\WINDOWS\system32;C:\WINDOWS;C:\WINDOWS\System32\Wbem;C:\Program Files\QuickTime\QTSystem\;C:\Program Files\Pinnacle\Shared Files\InstantCDDVD\;C:\ where the last C:\ is the only thing in my windows path system variable. 5. ### rickohTS Rookie gbhall have you tried 'doskey'? as a command? 6. ### LookinAroundTechSpot ChancellorPosts: 8,373 +167 AHH.... Now i think i know what you are asking. Check out this: When you look at your Environment Variables in Windows, note there are two sets of variables • User variables for your logon (in upper window pane) • System variables (in lower window pane) Check to see if you have a PATH variable defined as each! both as a user and a system variable. The command window shell PATH is then the concatenation of both! 7. ### Bernie157TS Rookie The path is set in the System Environment variables. You get to them by right-clicking on My Computer, then Properties (at the bottom), then the Advanced tab, then the Environment Variables button near the bottom. To answer what I thought you were asking originally, you change directories from within the command prompt window with the CD command, e.g. CD \directory1\direcotry2 etc. You can use * as a wildcard, as in CD \dir* or CD \*ory1. And I think you also asked: "Where is the PATH command?" Back in the days of real DOS, many commands were imbedded in the file COMMAND.COM and maybe that's the case with Windows DOS. Bernie 8. ### gbhallTechSpot ChancellorTopic StarterPosts: 2,348 +50 Bernie, sorry to confuse, I meant the Path variable or parameter specification, not the command, which is indeed in command.com or cmd.exe LookinAround has the correct answer to my question - there are two PATH parameters in Systen environment properties - one for user and one for 'all users' or system if you will. The latter is the one installs write to if they insist on having a path extension to work properly. Those packages were written for Win98 or thereabouts Topic Status: Not open for further replies.	{"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": 0.9511334300041199, "perplexity": 3506.6192489238065}, "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/1413507444339.31/warc/CC-MAIN-20141017005724-00254-ip-10-16-133-185.ec2.internal.warc.gz"}
https://mapleprimes.com/users/Parham2016/replies	## 60 Reputation 5 years, 156 days ## @Preben Alsholm There is ... There is a physical meaning for the relation between C1 nad B in Mechanics. I do know that 'dsolve' works for differential equations, however, the desire result could be derived by it! ## @tomleslie I meant why it was ok ... I meant why it was ok and done in your computer but not in my laptop. How did you edit the file? For example: alpha1 was changed to alpha[1]... Did you write the whole file again? ## @tomleslie Hello, yes it&... Hello, yes it's ok now. It's ambiguous yet.. What have happened? ## @Kitonum Thanks. But it d... Thanks. But it doesn't work for me.... It drove me crazy!!! It workes correctly last week. I had to get some results by the file today... ## @tomleslie Great, God ble... Great, God bless you and happy new year. ## @tomleslie What a simple ... What a simple solution! Thanks. ## @_Maxim_ The integral term in F2(r)... @_Maxim_ The integral term in F2(r) must be a value necessarily as long as those two Nu are close toghether. Indeed in the loop I want this expression " Error= (Nu_final- Nu_guess)<1e-3" to be satisfied and then the loop is going to be interrupted. The program is correct in the first iteration for \phi= 0.06 and "initial guess Nu=3.6941" and the "final calculated Nu= 3.6942". I already know the Final Nu for \phi= 0.06!!!! But for other values of \phi I could not get any solution precisely. ## @_Maxim_ By the way, it does no... @_Maxim_ By the way, it does not matter whether or not k is zero @ initial guess "For The First Time", the first iteration in a potential loop. As you know the initial guess for Nu is just for calculation of the integral and after that I removed it by this command: Nu:= 'Nu'. The only important thing is this fact that the final calculated for Nu, in the end of the loop, must be equal to the initial guess. ## @_Maxim_ Hello, thanks for yo... Hello, thanks for your consideration dear. Yes, I want to find the Nu if the determinat is equal zero. And the problem is exactly here that the at the initial guess the determinant, k, might be any number, not necessarily zero. That is why I want to correct the Nu by replacing the initial guess with the final calculated Nu. It assumed to be difficult for me. ## @vv Thanks. But r is not equa... Thanks. I must change the domain then! That's 3.195986267 if 0.2 <r< 1 Thanks again. ## @Markiyan Hirnyk Thanks dear,... Thanks dear, Yes you're right. I solve that system in Matlab (written below) and those solutions are the same. [ 1.6759649064951345004378236436225, 0.7729562969870007822477027324705, 1.9940322524732747423780039106827] By the way, I wrote some indices wrongly in my question!!! Now I corrected it and we can see this is true as well. (1) (2) Best. ## How solve this PDE... @tomleslie Unfortunately I didn't write the fourth boundary conditions for you. The four boundary conditions are in here: V(r1,z)= V(r2,z)=0 V(r,0)= V(r,b)=0 and this point that the coordinates are r1 < r <r2 & 0< z < b And I want to tell you I don't have the Matlab Code I just olnly the final solution attached in the Maple primes websie. ## @Carl Love Thanks a lot Mr.... Thanks a lot Mr. Carl. God bless you ## @tomleslie Ok, thanks. But ... Ok, thanks. But I thought Maple could give the right answer instead of F_1(x+t)+ F_2(t-x)+ cos(x).	{"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": 0.9415252804756165, "perplexity": 2661.629195369924}, "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/1596439738819.78/warc/CC-MAIN-20200811180239-20200811210239-00201.warc.gz"}
https://mathstoshare.com/2020/03/23/why-is-the-fundamental-theorem-of-arithmetic-a-theorem/	# Why is the fundamental theorem of arithmetic a theorem? The fundamental theorem of arithmetic states that every natural number can be factorized uniquely as a product of prime numbers. The word “uniquely” here means unique up to rearranging. The theorem means that if you and I take the same number $n$ and I write $n = p_1p_2\ldots p_k$ and you write $n = q_1q_2\ldots q_l$ where each $p_i$ and $q_i$ is a prime number, then in fact $k=l$ and we wrote the same prime numbers (but maybe in a different order). Most people happily accept this theorem as self evident and believe it without proof. Indeed some people take it to be so self evident they feel it doesn’t really deserve the name “theorem” – hence the title of this blog post. In this post I want to highlight two situations where an analogous theorem fails. ## Situation One: The Even Numbers Imagine a world where everything comes in twos. In this world nobody knows of the number one or indeed any odd number. Their counting numbers are the even numbers $\mathbb{E} = \{2,4,6,8,\ldots\}$. People in this world can add numbers and multiply numbers just like we can. They can even talk about divisibility, for example $2$ divides $8$ since $8 = 4\cdot 2$. Note that things are already getting a bit strange in this world. Since there is no number one, numbers in this world do not divide themselves. Once people can talk about divisibility, they can talk about prime numbers. A number is prime in this world if it is not divisible by any other number. For example $2$ is prime but as we saw $8$ is not prime. Surprisingly the number $6$ is also prime in this world. This is because there are no two even numbers that multiply together to make $6$. If a number is not prime in this world, we can reduce it to a product of primes. This is because if $n$ is not prime, then there are two number $a$ and $b$ such that $n = ab$. Since $a$ and $b$ are both smaller than $n$, we can apply the same argument and recursively write $n$ as a product of primes. Now we can ask whether or not the fundamental theorem of arthimetic holds in this world. Namely we want to know if their is a unique way to factorize each number in this world. To get an idea we can start with some small even numbers. • $2$ is prime. • $4 = 2 \cdot 2$ can be factorized uniquely. • $6$ is prime. • $8 = 2\cdot 2 \cdot 2$ can be factorized uniquely. • $10$ is prime. • $12 = 2 \cdot 6$ can be factorized uniquely. • $14$ is prime. • $16 = 2\cdot 2 \cdot 2 \cdot 2$ can be factorized uniquely. • $18$ is prime. • $20 = 2 \cdot 10$ can be factorized uniquely. Thus it seems as though there might be some hope for this theorem. It at least holds for the first handful of numbers. Unfortunately we eventually get to $36$ and we have: $36 = 2 \cdot 18$ and $36 = 6 \cdot 6$. Thus there are two distinct ways of writing $36$ as a product of primes in this world and thus the fundamental theorem of arithmetic does not hold. ## Situtation Two: A Number Ring While the first example is fun and interesting, it is somewhat artificial. We are unlikely to encounter a situation where we only have the even numbers. It is however common and natural for mathematicians to be lead into certain worlds called number rings. We will see one example here and see what an effect the fundamental theorem of arithmetic can have. Consider wanting to solve the equation $x^2+19=y^3$ where $x$ and $y$ are both integers. One way to try to solve this is by rewriting the equation as $(x+\sqrt{-19})(x-\sqrt{-19}) = y^3$. With this rewriting we have left the familiar world of the whole numbers and entered the number ring $\mathbb{Z}[\sqrt{-19}]$. In $\mathbb{Z}[\sqrt{-19}]$ all numbers have the form $a + b \sqrt{-19}$, where $a$ and $b$ are integers. Addition of two such numbers is defined like so $(a+b\sqrt{-19}) + (c + d \sqrt{-19}) = (a+c) + (b+d)\sqrt{-19}$. Multiplication is define by using the distributive law and the fact that $\sqrt{-19}^2 = -19$. Thus $(a+b\sqrt{-19})(c+d\sqrt{-19}) = (ac-19bd) + (ad+bc)\sqrt{-19}$. Since we have multiplication we can talk about when a number in $\mathbb{Z}[\sqrt{-19}]$ divides another and hence define primes in $\mathbb{Z}[\sqrt{-19}]$. One can show that if $x^2 + 19 = y^3$, then $x+\sqrt{-19}$ and $x-\sqrt{-19}$ are coprime in $\mathbb{Z}[\sqrt{-19}]$ (see the references at the end of this post). This means that there are no primes in $\mathbb{Z}[\sqrt{-19}]$ that divides both $x+\sqrt{-19}$ and $x-\sqrt{-19}$. If we assume that the fundamental theorem of arthimetic holds in $\mathbb{Z}[\sqrt{-19}]$, then this implies that $x+\sqrt{-19}$ must itself be a cube. This is because $(x+\sqrt{-19})(x-\sqrt{-19})=y^3$ is a cube and if two coprime numbers multiply to be a cube, then both of those coprime numbers must be cubes. Thus we can conclude that there are integers $a$ and $b$ such that $x+\sqrt{-19} = (a+b\sqrt{-19})^3$. If we expand out this cube we can conclude that $x+\sqrt{-19} = (a^3-57ab^2)+(3a^2b-19b^3)\sqrt{-19}$. Thus in particular we have $1=3a^2b-19b^3=(3a^2-19b^2)b$. This implies that $b = \pm 1$ and $3a^2-19b^2=\pm 1$. Hence $b^2=1$ and $3a^2-19 = \pm 1$. Now if $3a^2 -19 =-1$, then $a^2=6$ – a contradiction. Similarly if $3a^2-19=1$, then $3a^2=20$ – another contradiction. Thus we can conclude there are no integer solutions to the equation $x^2+19=y^3$! Unfortunately however, a bit of searching reveals that $18^2+19=343=7^3$. Thus simply assuming that that the ring $\mathbb{Z}[\sqrt{-19}]$ has unique factorization led us to incorrectly conclude that an equation had no solutions. The question of unique factorization in number rings such as $\mathbb{Z}[\sqrt{-19}]$ is a subtle and important one. Some of the flawed proofs of Fermat’s Last Theorem incorrectly assume that certain number rings have unique factorization – like we did above. ## References The lecturer David Smyth showed us that the even integers do not have unique factorization during a lecture of the great course MATH2222. The example of $\mathbb{Z}[\sqrt{-19}]$ failing to have unique factorization and the consequences of this was shown in a lecture for a course on algebraic number theory by James Borger. In this class we followed the (freely available) textbook “Number Rings” by P. Stevenhagen. Problem 1.4 on page 8 is the example I used in this post. By viewing the textbook you can see a complete solution to the problem.	{"extraction_info": {"found_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": 78, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9363698363304138, "perplexity": 148.4490659309981}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948900.50/warc/CC-MAIN-20230328232645-20230329022645-00733.warc.gz"}
http://umj.imath.kiev.ua/article/?lang=en&article=9988	2018 Том 70 № 8 # Investigation of the solutions of a system of $n + m$ nonlineai differential equations in the vicinity of an integral manifold Lykova O. B. Abstract For a system of $n + m$ equations $$\frac{dx}{dt} = X(y)x + \varepsilon X(t, x, y),$$ $$\frac{dy}{dt} = \varepsilon Y(t, x, y),$$ where $x, X, y, Y$ are respectively $n$ and $m$ vectors, $X — n \times n$ is the matrix, $\varepsilon$ is a small parameter, the author proves the theorem of the existence and properties of a two-dimensional local integral manifold in the neighbourhood of family of periodic solutions $$x = 0,\; y = y^0(\psi, a)$$ oi the lollowing auxiliary system $$\frac{dx}{dt} = X(y)x,$$ $$\frac{dy}{dt} = \varepsilon Y_0(x, y),$$ where $$Y_0(x, y) = \lim_{T\rightarrow 0}\int_0^T Y(t, x,y)dt.$$ Citation Example: Lykova O. B. Investigation of the solutions of a system of $n + m$ nonlineai differential equations in the vicinity of an integral manifold // Ukr. Mat. Zh. - 1964. - 16, № 1. - pp. 13-30. Full text	{"extraction_info": {"found_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": 0.9883484840393066, "perplexity": 452.3384100009818}, "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-2018-39/segments/1537267158450.41/warc/CC-MAIN-20180922142831-20180922163231-00397.warc.gz"}
https://www.physicsforums.com/threads/physics-question-regarding-impluse-momentum-and-velocity.278222/	# Physics Question Regarding Impluse, Momentum, and Velocity. 1. Dec 8, 2008 ### Jim4592 I'm not sure exactly sure how to set up this problem... 1. The problem statement, all variables and given/known data A hockey puck with a mass of 0.160 kg is at rest at the origin (x=0) on a horizontal, frictionless surface at time t=0, a force of 0.250 N is applied to the puck parallel to the X axis, the force is applied until time = 2.00s. 2. Relevant equations What is the acceleration, position, and speed of the puck at t=2.00 s 3. The attempt at a solution Impulse (J) = (F) * (∆T) J = (0.250N)(2.00s) J = .5 NS I know J = ∆ Momentum (P) ; and P = Mass Velocity, but i'm not exactly sure how to calculate what it wants me to find. Any help will be appreciated. Thanks, Jim 2. Dec 8, 2008 ### Staff: Mentor You have the force. What does that tell you about the acceleration? You found the impulse. What does that tell you about the speed at t=2.00s? Use a little kinematics to find the distance traveled. 3. Dec 8, 2008 ### Jim4592 I think i get it but aren't completly sure so using P=mv .5 NS = 0.160 kg * v v= 3.125 m/s ? A = (3.125 m/s) / 2.00s A = 1.563 m/s^2 and position: X(final) = X(initial) + V(i)t + 1/2 at^2 X = 1/2 * (1.563 m/s) * (2.00s)^2 X = 3.125m Could someone confirm that is correct please? 4. Dec 8, 2008 All good! 5. Dec 8, 2008 ### Jim4592 wow that was alot easier than i thought. Thanks for your help. Similar Discussions: Physics Question Regarding Impluse, Momentum, and Velocity.	{"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": 0.9280861020088196, "perplexity": 2395.2550174836238}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698540928.63/warc/CC-MAIN-20161202170900-00009-ip-10-31-129-80.ec2.internal.warc.gz"}
https://math.stackexchange.com/questions/963299/true-or-false-a-relative-maximum-or-minimum-must-occur-at-a-critical-point	# True or false? A relative maximum or minimum must occur at a critical point. I'm taking calculus one and I have to determine if this statement is true or false. A relative maximum or minimum must occur at a critical point. I believe it is false. The answer key says it is true so I am curious if I am right (low probability) or if I can get this clarified (the answer key has always been right when I thought it was wrong). For example for y = $\frac{1}{x}$ there is a critical point at x=0 (because the derivative at this point does not exist) even though there is no relative maximum or minimum (or is there?). I think if the statement was "A relative maximum or minimum must occur at a critical point when the derivative at that point exists." • A critical point is when the derivative is 0, not when it doesn't exist. There is no critical point at x=0. – Loocid Oct 8 '14 at 5:25 • Hmm then there is a lot of wrong information out on the internet if what you say is true.mathwords.com/c/critical_point.htm or tutorial.math.lamar.edu/Classes/CalcI/CriticalPoints.aspx – Alex Oct 8 '14 at 5:27 • Apologies, you are correct. But your source also says: "But may be neither". – Loocid Oct 8 '14 at 5:30 • Thanks for pointing that out. Didn't notice that. – Alex Oct 8 '14 at 5:30 • Its interesting that Wikipedia states critical points are only at 0's and doesn't mention when it doesn't exist, and they reference "Calculus: A Complete Course". Would be interesting to see if that's actually what they say in the book. – Loocid Oct 8 '14 at 5:34 It depends on what definition of "critical point" you are using. If you mean the same thing as "stationary point" (i.e., a point where the derivative exists and equals zero), then you are right, but not for the reason that you give (see below). However, some people use the phrase "critical point" also for a point where the function $f$ is defined, but not its derivative $f'$ (including endpoints of an interval in the domain $D_f$), and if this is the definition used in your course, then you are wrong. In your example, $f(x)=1/x$ is undefined at $x=0$, so it doesn't even make sense to talk about maximum or minimum there. Consider instead the absolute value function $f(x)=\|x\|$, which has a minimum at $x=0$ despite the fact that $x=0$ is not a stationary point ($f'(0)$ doesn't exist). [On the other hand, if $f'(a)$ exists and $f$ has a local max/min at $a$, then $a$ must be a stationary point (i.e., $f'(a)=0$).]	{"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": 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": 0.8851754069328308, "perplexity": 199.39340884156005}, "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/1600400198942.13/warc/CC-MAIN-20200921050331-20200921080331-00073.warc.gz"}
https://worldwidescience.org/topicpages/e/excitation+systems.html	#### Sample records for excitation systems 1. Excitation system testing in HPP 'Uvac' Directory of Open Access Journals (Sweden) Milojčić Nemanja 2011-01-01 Full Text Available The excitation system of hydro unit in HPP 'Uvac' and results of testings of excitation system performed for achieving of unit's mathematical model are presented in this paper. Description of excitation system equipment, parameters of regulators and results obtained after testings are presented. The presented results showed that the regulators are properly adjusted and that the excitation system is completely functional and reliable. 2. Excited states in biological systems International Nuclear Information System (INIS) Cilento, G.; Zinner, K.; Bechara, E.J.H.; Duran, N.; Baptista, R.C. de; Shimizu, Y.; Augusto, O.; Faljoni-Alario, A.; Vidigal, C.C.C.; Oliveira, O.M.M.F.; Haun, M. 1979-01-01 Some aspects of bioluminescence related to bioenergetics are discussed: 1. chemical generation of excited species, by means of two general processes: electron transference and cyclic - and linear peroxide cleavage; 2. biological systems capable of generating excited states and 3. biological functions of these states, specially the non-emissive ones (tripletes). The production and the role of non-emissive excited states in biological systems are analysed, the main purpose of the study being the search for non-emissive states. Experiences carried out in biological systems are described; results and conclusions are given. (M.A.) [pt 3. Fission fragment excited laser system Science.gov (United States) McArthur, David A.; Tollefsrud, Philip B. 1976-01-01 A laser system and method for exciting lasing action in a molecular gas lasing medium which includes cooling the lasing medium to a temperature below about 150 K and injecting fission fragments through the lasing medium so as to preferentially excite low lying vibrational levels of the medium and to cause population inversions therein. The cooled gas lasing medium should have a mass areal density of about 5 .times. 10.sup.-.sup.3 grams/square centimeter, relaxation times of greater than 50 microseconds, and a broad range of excitable vibrational levels which are excitable by molecular collisions. 4. Effects of noise in excitable systems International Nuclear Information System (INIS) Lindner, B.; Garcia-Ojalvo, J.; Neiman, A.; Schimansky-Geier, L. 2004-01-01 We review the behavior of theoretical models of excitable systems driven by Gaussian white noise. We focus mainly on those general properties of such systems that are due to noise, and present several applications of our findings in biophysics and lasers. As prototypes of excitable stochastic dynamics we consider the FitzHugh-Nagumo and the leaky integrate-and-fire model, as well as cellular automata and phase models. In these systems, taken as individual units or as networks of globally or locally coupled elements, we study various phenomena due to noise, such as noise-induced oscillations, stochastic resonance, stochastic synchronization, noise-induced phase transitions and noise-induced pulse and spiral dynamics. Our approach is based on stochastic differential equations and their corresponding Fokker-Planck equations, treated by both analytical calculations and/or numerical simulations. We calculate and/or measure the rate and diffusion coefficient of the excitation process, as well as spectral quantities like power spectra and degree of coherence. Combined with a multiparametric bifurcation analysis of the corresponding cumulant equations, these approaches provide a comprehensive picture of the multifaceted dynamical behaviour of noisy excitable systems 5. Equivalent non-Gaussian excitation method for response moment calculation of systems under non-Gaussian random excitation International Nuclear Information System (INIS) Tsuchida, Takahiro; Kimura, Koji 2015-01-01 Equivalent non-Gaussian excitation method is proposed to obtain the moments up to the fourth order of the response of systems under non-Gaussian random excitation. The excitation is prescribed by the probability density and power spectrum. Moment equations for the response can be derived from the stochastic differential equations for the excitation and the system. However, the moment equations are not closed due to the nonlinearity of the diffusion coefficient in the equation for the excitation. In the proposed method, the diffusion coefficient is replaced with the equivalent diffusion coefficient approximately to obtain a closed set of the moment equations. The square of the equivalent diffusion coefficient is expressed by the second-order polynomial. In order to demonstrate the validity of the method, a linear system to non-Gaussian excitation with generalized Gaussian distribution is analyzed. The results show the method is applicable to non-Gaussian excitation with the widely different kurtosis and bandwidth. (author) 6. Response moments of dynamic systems under non-Gaussian random excitation by the equivalent non-Gaussian excitation method International Nuclear Information System (INIS) Tsuchida, Takahiro; Kimura, Koji 2016-01-01 Equivalent non-Gaussian excitation method is proposed to obtain the response moments up to the 4th order of dynamic systems under non-Gaussian random excitation. The non-Gaussian excitation is prescribed by the probability density and the power spectrum, and is described by an Ito stochastic differential equation. Generally, moment equations for the response, which are derived from the governing equations for the excitation and the system, are not closed due to the nonlinearity of the diffusion coefficient in the equation for the excitation even though the system is linear. In the equivalent non-Gaussian excitation method, the diffusion coefficient is replaced with the equivalent diffusion coefficient approximately to obtain a closed set of the moment equations. The square of the equivalent diffusion coefficient is expressed by a quadratic polynomial. In numerical examples, a linear system subjected to nonGaussian excitations with bimodal and Rayleigh distributions is analyzed by using the present method. The results show that the method yields the variance, skewness and kurtosis of the response with high accuracy for non-Gaussian excitation with the widely different probability densities and bandwidth. The statistical moments of the equivalent non-Gaussian excitation are also investigated to describe the feature of the method. (paper) 7. Semiconductor-machine system for controlling excitation of synchronous medium power generators Energy Technology Data Exchange (ETDEWEB) Vrtikapa, G 1982-01-01 A system for controlling excitation (ARP-29/1) is described which was developed at the ''Nikola Tesla'' institute (Czechoslavakia) for rebuilding the Zvornik hydroelectric plant with 30 MV X A units. The system corresponds to the modern level of automation and considers positive characteristics of existing equipment, it is easily included in a technological process, has small dimensions and is easily installed during overhaul of a electric generating plant, and it allows one to obtain good economic results. Two years of use have confirmed the high reliability and quality of the excitation. The excitation control system consists of synchronous motor, excitation system, automatic control of voltage, manual control of excitation unit, unit for automatic following and switching, relay automatic device with protection and warning. The excitation system of the generator has: thyristor rectifier, thyristor converter, a bridge with thyristor control unit, machine excitation generator, switch for demagnetization. The excitation system is supplied from an electric power network or from a three phase generator with permanent magnets. 8. Analysis about modeling MEC7000 excitation system of nuclear power unit Science.gov (United States) Liu, Guangshi; Sun, Zhiyuan; Dou, Qian; Liu, Mosi; Zhang, Yihui; Wang, Xiaoming 2018-02-01 Aiming at the importance of accurate modeling excitation system in stability calculation of nuclear power plant inland and lack of research in modeling MEC7000 excitation system,this paper summarize a general method to modeling and simulate MEC7000 excitation system. Among this method also solve the key issues of computing method of IO interface parameter and the conversion process of excitation system measured model to BPA simulation model. At last complete the simulation modeling of MEC7000 excitation system first time in domestic. By used No-load small disturbance check, demonstrates that the proposed model and algorithm is corrective and efficient. 9. Power system stabilization by superconducting magnetic energystorage connected to rotating exciter OpenAIRE Mitani, Yasunori; Tsuji, K 1993-01-01 The authors describe a combination of a rotating exciter and a superconducting magnetic energy storage (SMES) system for efficient power system stabilization. A SMES system connected to an exciter rotating with a turbine-rotor shaft is proposed. The exciter is installed exclusively to supply current for the SMES. Since electrical power output from the SMES is converted into a mechanical torque of the generator directly by the exciter, it is expected that power swings of the generator will be ... 10. Programmable logic controller based synchronous motor excitation system Directory of Open Access Journals (Sweden) Janda Žarko 2011-01-01 Full Text Available This paper presents a 3.5 MW synchronous motor excitation system reconstruction. In the proposed solution programmable logic controller is used to control motor, which drives the turbo compressor. Comparing to some other solutions that are used in similar situations, the proposed solution is superior due to its flexibility and usage of mass-production hardware. Moreover, the implementation of PLC enables easy integration of the excitation system with the other technological processes in the plant as well as in the voltage regulation of 'smart grid' system. Also, implementation of various optimization algorithms can be done comfortably and it does not require additional investment in hardware. Some experimental results that depict excitation current during motor start-up, as well as, measured static characteristics of the motor, were presented. 11. Excited, bound and resonant positron-atom systems International Nuclear Information System (INIS) Bromley, M W J; Mitroy, J 2010-01-01 Calculations have demonstrated that eleven neutral atoms can bind positrons, while many more can bind positronium. This is a short review of recent progress made in understanding some of the underlying mechanisms. The emphasis here being on configuration interaction calculations with excited state configurations. These have demonstrated the existence of a 2 P o excited state of e + Ca, which consists predominantly of a positronium cluster orbiting the Ca + ion in the L = 1 partial wave. Preliminary results are presented of excited state positron binding to a model alkali atom, where the excited 1 P o states are stable over a limited region. Implications for the unnatural parity, 2,4 S o , states of PsH, LiPs, NaPs and KPs are also discussed. The e + Mg, e + Cu, e + Zn and e + Cd systems show a lack of a 2 P o excited state, each instead possessing a low-energy p-wave shape resonance of varying strength. 12. Rated power factor and excitation system of large turbine generator International Nuclear Information System (INIS) Tokumitsu, Iwao; Watanabe, Takashi; Banjou, Minoru. 1979-01-01 As for the rated power factor of turbine generators for thermal power stations, 90% has been adopted since around 1960. On the other hand, power transmission system has entered 500 kV age, and 1,000 kV transmission is expected in the near future. As for the supply of reactive power from thermal and nuclear turbine generators, the necessity of supplying leading reactive power has rather increased. Now, the operating power factor of thermal and nuclear generators becomes 96 to 100% actually. As for the excess stability of turbine generators owing to the strengthening of transmission system and the adoption of super-high voltage, the demand of strict conditions can be dealt with by the adoption of super-fast response excitation system of thyristor shunt winding self exciting type. The adoption of the turbine generators with 90 to 95% power factor and the adoption of the thyristor shunt winding self exciting system were examined and evaluated. The rated power factor of generators, excitation system and economy of adopting these systems are explained. When the power factor of generators is increased from 0.9 to 0.95, about 6% of saving can be obtained in the installation cost. When the thyristor shunt winding self excitation is adopted, it is about 10% more economical than AC excitation. (Kako, I.) 13. Methodology to estimate parameters of an excitation system based on experimental conditions Energy Technology Data Exchange (ETDEWEB) Saavedra-Montes, A.J. [Carrera 80 No 65-223, Bloque M8 oficina 113, Escuela de Mecatronica, Universidad Nacional de Colombia, Medellin (Colombia); Calle 13 No 100-00, Escuela de Ingenieria Electrica y Electronica, Universidad del Valle, Cali, Valle (Colombia); Ramirez-Scarpetta, J.M. [Calle 13 No 100-00, Escuela de Ingenieria Electrica y Electronica, Universidad del Valle, Cali, Valle (Colombia); Malik, O.P. [2500 University Drive N.W., Electrical and Computer Engineering Department, University of Calgary, Calgary, Alberta (Canada) 2011-01-15 A methodology to estimate the parameters of a potential-source controlled rectifier excitation system model is presented in this paper. The proposed parameter estimation methodology is based on the characteristics of the excitation system. A comparison of two pseudo random binary signals, two sampling periods for each one, and three estimation algorithms is also presented. Simulation results from an excitation control system model and experimental results from an excitation system of a power laboratory setup are obtained. To apply the proposed methodology, the excitation system parameters are identified at two different levels of the generator saturation curve. The results show that it is possible to estimate the parameters of the standard model of an excitation system, recording two signals and the system operating in closed loop with the generator. The normalized sum of squared error obtained with experimental data is below 10%, and with simulation data is below 5%. (author) 14. Excitation power quantities in phase resonance testing of nonlinear systems with phase-locked-loop excitation Science.gov (United States) Peter, Simon; Leine, Remco I. 2017-11-01 Phase resonance testing is one method for the experimental extraction of nonlinear normal modes. This paper proposes a novel method for nonlinear phase resonance testing. Firstly, the issue of appropriate excitation is approached on the basis of excitation power considerations. Therefore, power quantities known from nonlinear systems theory in electrical engineering are transferred to nonlinear structural dynamics applications. A new power-based nonlinear mode indicator function is derived, which is generally applicable, reliable and easy to implement in experiments. Secondly, the tuning of the excitation phase is automated by the use of a Phase-Locked-Loop controller. This method provides a very user-friendly and fast way for obtaining the backbone curve. Furthermore, the method allows to exploit specific advantages of phase control such as the robustness for lightly damped systems and the stabilization of unstable branches of the frequency response. The reduced tuning time for the excitation makes the commonly used free-decay measurements for the extraction of backbone curves unnecessary. Instead, steady-state measurements for every point of the curve are obtained. In conjunction with the new mode indicator function, the correlation of every measured point with the associated nonlinear normal mode of the underlying conservative system can be evaluated. Moreover, it is shown that the analysis of the excitation power helps to locate sources of inaccuracies in the force appropriation process. The method is illustrated by a numerical example and its functionality in experiments is demonstrated on a benchmark beam structure. 15. Comparison of multiple support excitation solution techniques for piping systems International Nuclear Information System (INIS) Sterkel, H.P.; Leimbach, K.R. 1980-01-01 Design and analysis of nuclear power plant piping systems exposed to a variety of dynamic loads often require multiple support excitation analysis by modal or direct time integration methods. Both methods have recently been implemented in the computer program KWUROHR for static and dynamic analysis of piping systems, following the previous implementation of the multiple support excitation response spectrum method (see papers K 6/15 and K 6/15a of the SMiRT-4 Conference). The results of multiple support excitation response spectrum analyses can be examined by carrying out the equivalent time history analyses which do not distort the time phase relationship between the excitations at different support points. A frequent point of discussion is multiple versus single support excitation. A single support excitation analysis is computationally straightforward and tends to be on the conservative side, as the numerical results show. A multiple support excitation analysis, however, does not incur much more additional computer cost than the expenditure for an initial static solution involving three times the number, L, of excitation levels, i.e. 3L static load cases. The results are more realistic than those from a single support excitation analysis. A number of typical nuclear plant piping systems have been analyzed using single and multiple support excitation algorithms for: (1) the response spectrum method, (2) the modal time history method via the Wilson, Newmark and Goldberg integration operators and (3) the direct time history method via the Wilson integration operator. Characteristic results are presented to compare the computational quality of all three methods. (orig.) 16. Thermal Excitation System for Shearography (TESS) Science.gov (United States) Lansing, Matthew D.; Bullock, Michael W. 1996-01-01 One of the most convenient and effective methods of stressing a part or structure for shearographic evaluation is thermal excitation. This technique involves heating the part, often convectively with a heat gun, and then monitoring with a shearography device the deformation during cooling. For a composite specimen, unbonds, delaminations, inclusions, or matrix cracking will deform during cooling differently than other more structurally sound regions and thus will appear as anomalies in the deformation field. However, one of the difficulties that cause this inspection to be dependent on the operator experience is the conventional heating process. Fanning the part with a heat gun by hand introduces a wide range of variability from person to person and from one inspection to the next. The goal of this research effort was to conduct research in the methods of thermal excitation for shearography inspection. A computerized heating system was developed for inspection of 0.61 m (24 in.) square panels. The Thermal Excitation System for Shearography (TESS) provides radiant heating with continuous digital measurement of the surface temperature profile to ensure repeatability. The TESS device functions as an accessory to any electronic shearography device. 17. Excited, bound and resonant positron-atom systems Energy Technology Data Exchange (ETDEWEB) Bromley, M W J [Department of Physics and Computational Science Research Center, San Diego State University, San Diego CA 92182 (United States); Mitroy, J, E-mail: [email protected] [ARC Centre for Antimatter-Matter Studies and Faculty of Education, Health and Science, Charles Darwin University, Darwin NT 0909 (Australia) 2010-01-01 Calculations have demonstrated that eleven neutral atoms can bind positrons, while many more can bind positronium. This is a short review of recent progress made in understanding some of the underlying mechanisms. The emphasis here being on configuration interaction calculations with excited state configurations. These have demonstrated the existence of a {sup 2}P{sup o} excited state of e{sup +}Ca, which consists predominantly of a positronium cluster orbiting the Ca{sup +} ion in the L = 1 partial wave. Preliminary results are presented of excited state positron binding to a model alkali atom, where the excited {sup 1}P{sup o} states are stable over a limited region. Implications for the unnatural parity, {sup 2,4}S{sup o}, states of PsH, LiPs, NaPs and KPs are also discussed. The e{sup +}Mg, e{sup +}Cu, e{sup +}Zn and e{sup +}Cd systems show a lack of a {sup 2}P{sup o} excited state, each instead possessing a low-energy p-wave shape resonance of varying strength. 18. Modernization, reconstruction and development of excitation systems for synchronous generators Directory of Open Access Journals (Sweden) Arnautović Dušan 2011-01-01 Full Text Available This paper presents the previous results of work and development of excitation systems with digital automatic voltage regulators regarding their design, development, manufacturing and commissioning. A special attention was paid to the characteristics of excitation system voltage regulator. 19. Synchronization of chaos in non-identical parametrically excited systems International Nuclear Information System (INIS) Idowu, B.A.; Vincent, U.E.; Njah, A.N. 2009-01-01 In this paper, we investigate the synchronization of chaotic systems consisting of non-identical parametrically excited oscillators. The active control technique is employed to design control functions based on Lyapunov stability theory and Routh-Hurwitz criteria so as to achieve global chaos synchronization between a parametrically excited gyroscope and each of the parametrically excited pendulum and Duffing oscillator. Numerical simulations are implemented to verify the results. 20. Excitation of graphene plasmons as an analogy with the two-level system International Nuclear Information System (INIS) Fu, Jiahui; Lv, Bo; Li, Rujiang; Ma, Ruyu; Chen, Wan; Meng, Fanyi 2016-01-01 The excitation of graphene plasmons (GPs) is presented as an interaction between the GPs and the incident electromagnetic field. In this Letter, the excitation of GPs in a plasmonic system is interpreted as an analogy with the two-level system by taking the two-coupled graphene-covered gratings as an example. Based on the equivalent circuit theory, the excitation of GPs in the graphene-covered grating is equivalent to the resonance of an oscillator. Thus, according to the governing equation, the electric currents at the resonant frequencies for two-coupled graphene-covered gratings correspond to the energy states in a two-level system. In addition, the excitation of GPs in different two-coupled graphene-covered gratings is numerically studied to validate our theoretical model. Our work provides an intuitive understanding of the excitation of GPs using an analogy with the two-level system. - Highlights: • The excitation of graphene plasmons (GPs) in graphene-covered grating is equivalent to the resonance of an oscillator. • We establish the equivalent circuit of two-level system to analyze the resonant character. • The excitation of GPs in different two-coupled graphene-covered gratings are numerically studied to validate our theoretical model. 1. Excitation of graphene plasmons as an analogy with the two-level system Energy Technology Data Exchange (ETDEWEB) Fu, Jiahui [Microwave and Electromagnetic Laboratory, Harbin Institute of Technology, No. 92, Xidazhi Street, Nangang District, Harbin City, Heilongjiang Province (China); Lv, Bo, E-mail: [email protected] [Microwave and Electromagnetic Laboratory, Harbin Institute of Technology, No. 92, Xidazhi Street, Nangang District, Harbin City, Heilongjiang Province (China); Li, Rujiang [College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310027 (China); Ma, Ruyu; Chen, Wan; Meng, Fanyi [Microwave and Electromagnetic Laboratory, Harbin Institute of Technology, No. 92, Xidazhi Street, Nangang District, Harbin City, Heilongjiang Province (China) 2016-02-15 The excitation of graphene plasmons (GPs) is presented as an interaction between the GPs and the incident electromagnetic field. In this Letter, the excitation of GPs in a plasmonic system is interpreted as an analogy with the two-level system by taking the two-coupled graphene-covered gratings as an example. Based on the equivalent circuit theory, the excitation of GPs in the graphene-covered grating is equivalent to the resonance of an oscillator. Thus, according to the governing equation, the electric currents at the resonant frequencies for two-coupled graphene-covered gratings correspond to the energy states in a two-level system. In addition, the excitation of GPs in different two-coupled graphene-covered gratings is numerically studied to validate our theoretical model. Our work provides an intuitive understanding of the excitation of GPs using an analogy with the two-level system. - Highlights: • The excitation of graphene plasmons (GPs) in graphene-covered grating is equivalent to the resonance of an oscillator. • We establish the equivalent circuit of two-level system to analyze the resonant character. • The excitation of GPs in different two-coupled graphene-covered gratings are numerically studied to validate our theoretical model. 2. Nontrivial effects of high-frequency excitation for strongly damped mechanical systems DEFF Research Database (Denmark) Fidlin, Alexander; Thomsen, Jon Juel Some nontrivial effects are investigated, which can occur if strongly damped mechanical systems are subjected to strong high-frequency (HF) excitation. The main result is a theoretical prediction, supported by numerical simulation, that for such systems the (quasi-)equilibrium states can change...... that can be substantial (depending on the strength of the HF excitation) for finite values of the damping. The analysis is focused on the differences between the classic results for weakly damped systems, and new effects for which the strong damping terms are responsible. The analysis is based...... on a slightly modified averaging technique, and includes an elementary example of an elliptically excited pendulum for illustration, alongside with a generalization to a broader class of strongly damped dynamical systems with HF excitation. As an application example, the nontrivial behavior of a classical... 3. Vibratory synchronization transmission of a cylindrical roller in a vibrating mechanical system excited by two exciters Science.gov (United States) Zhang, Xueliang; Wen, Bangchun; Zhao, Chunyu 2017-11-01 In present work vibratory synchronization transmission (VST) of a cylindrical roller with dry friction in a vibrating mechanical system excited by two exciters, is studied. Using the average method, the criterion of implementing synchronization of two exciters and that of ensuring VST of a roller, are achieved. The criterion of stability of the synchronous states satisfies the Routh-Hurwitz principle. The influences of the structural parameters of the system to synchronization and stability, are discussed numerically, which can be served as the theoretical foundation for engineering designs. An experiment is carried out, which approximately verify the validity of the theoretical and numerical results, as well as the feasibility of the method used. Utilizing the VST theory of a roller, some types of vibrating crushing or grinding equipments, etc., can be designed. 4. Nontrivial effects of high-frequency excitation for strongly damped mechanical systems DEFF Research Database (Denmark) Fidlin, Alexander; Thomsen, Jon Juel 2008-01-01 Some non-trivial effects are investigated, which can occur if strongly damped mechanical systems are subjected to strong high-frequency (HF) excitation. The main result is a theoretical prediction, supported by numerical simulation, that for such systems the (quasi-)equilibrium states can change...... that can be substantial depending on the strength of the HF excitation) for finite values of the damping. The analysis is focused on the differences between the classic results for weakly damped systems, and new effects for which the strong damping terms are responsible. The analysis is based on a slightly...... modified averaging technique, and includes an elementary example of an elliptically excited pendulum for illustration, alongside with a generalization to a broader class of strongly damped dynamical systems with HF excitation. As an application example, the nontrivial behavior of a classical optimally... 5. Encryption in Chaotic Systems with Sinusoidal Excitations Directory of Open Access Journals (Sweden) G. Obregón-Pulido 2014-01-01 Full Text Available In this contribution an encryption method using a chaotic oscillator, excited by “n” sinusoidal signals, is presented. The chaotic oscillator is excited by a sum of “n” sinusoidal signals and a message. The objective is to encrypt such a message using the chaotic behavior and transmit it, and, as the chaotic system is perturbed by the sinusoidal signal, the transmission security could be increased due to the effect of such a perturbation. The procedure is based on the regulation theory and consider that the receiver knows the frequencies of the perturbing signal, with this considerations the algorithm estimates the excitation in such a way that the receiver can cancel out the perturbation and all the undesirable dynamics in order to produce only the message. In this way we consider that the security level is increased. 6. Coherence resonance in an excitable system with time delay International Nuclear Information System (INIS) Sethia, Gautam C.; Kurths, Juergen; Sen, Abhijit 2007-01-01 We study the noise activated dynamics of a model excitable system that consists of a subcritical Hopf oscillator with a time delayed nonlinear feedback. The coherence of the noise driven pulses of the system exhibits a novel double peaked structure as a function of the noise amplitude. The two peaks correspond to separate optimal noise levels for excitation of single spikes and multiple spikes (bursts) respectively. The relative magnitudes of these peaks are found to be a sensitive function of time delay. The physical significance of our results and its practical implications in various real life systems are discussed 7. New formalism for determining excitation spectra of many-body systems International Nuclear Information System (INIS) Saito, S.; Zhang, S.B.; Louie, S.G.; Cohen, M.L. 1990-01-01 We present a new general formalism for determining the excitation spectrum of interacting many-body systems. The basic assumption is that the number of the excitations is equal to the number of sites. Within this approximation, it is shown that the density-density response functions with two different pure-imaginary energies determine the excitation spectrum. The method is applied to the valence electrons of sodium clusters of differing sizes in the time-dependent local-density approximation (TDLDA). A jellium-sphere background model is used for the ion cores. The excitation spectra obtained in this way represent well the excitation spectra given by the full TDLDA calculation along the real energy axis. Important collective modes are reproduced very well 8. Nuclear excited power generation system International Nuclear Information System (INIS) Parker, R.Z.; Cox, J.D. 1989-01-01 A power generation system is described, comprising: a gaseous core nuclear reactor; means for passing helium through the reactor, the helium being excited and forming alpha particles by high frequency radiation from the core of the gaseous core nuclear reactor; a reaction chamber; means for coupling chlorine and hydrogen to the reaction chamber, the helium and alpha particles energizing the chlorine and hydrogen to form a high temperature, high pressure hydrogen chloride plasma; means for converting the plasma to electromechanical energy; means for coupling the helium back to the gaseous core nuclear reactor; and means for disassociating the hydrogen chloride to form molecular hydrogen and chlorine, to be coupled back to the reaction chamber in a closed loop. The patent also describes a power generation system comprising: a gaseous core nuclear reactor; means for passing hydrogen through the reactor, the hydrogen being excited by high frequency radiation from the core; means for coupling chlorine to a reaction chamber, the hydrogen energizing the chlorine in the chamber to form a high temperature, high pressure hydrogen chloride plasma; means for converting the plasma to electromechanical energy; means for disassociating the hydrogen chloride to form molecular hydrogen and chlorine, and means for coupling the hydrogen back to the gaseous core nuclear reactor in a closed loop 9. Picosecond excitation transport in disordered systems International Nuclear Information System (INIS) Hart, D.E. 1987-11-01 Time-resolved fluorescence decay profiles are used to study excitation transport in 2- and 3-dimensional disordered systems. Time-correlated single photon counting detection is used to collect the fluorescence depolarization data. The high signal-to-noise ratios afforded by this technique makes it possible to critically examine current theories of excitation transport. Care has been taken to eliminate or account for the experimental artifacts common to this type of study. Solutions of 3,3'-diethyloxadicarbocyanine iodide (DODCI) in glycerol serve as a radomly distributed array of energy donors in 3-dimensions. A very thin sample cell (/approximately/ 2 μm) is used to minimize the effects of fluorescence self-absorption on the decay kinetics. Evidence of a dynamic shift of the fluorescence spectrum of DODCI in glycerol due to solvent reorganization is presented. The effects of excitation trapping on the decay profiles is minimized in the data analysis procedure. The 3-body theory of Gochanour, Andersen, and Fayer (GAF) and the far less complex 2-particle analytic theory of Huber, Hamilton, and Barnett yield indistinguishable fits to the data over the wide dynamic range of concentrations and decay times studied 10. A portable tube exciting X-ray fluorescence analysis system International Nuclear Information System (INIS) Yang Qiang; Lai Wanchang; Ge Liangquan 2009-01-01 Article introduced a portable tube exciting X-ray fluorescence analysis system which is based on arm architecture. Also, we designed Tube control circuit and finished preliminary application. The energy and the intensity of the photon can be adjusted continuously by using the tube. Experiments show that high excitation efficiency obtained by setting the appropriate parameters of the tube for the various elements. (authors) 11. Excitation processes in organic systems under irradiation with vacuum ultraviolet radiation International Nuclear Information System (INIS) Shefer, Y. 1983-08-01 The subject of this work is the fluorescence of organic systems in the excitation range where phenomena of photon multiplication begin. It was hoped to reach the excitation energy above which the distribution of the various phenomena was constant and as a result, a linear function between the variation of the fluorescence intensity with variations of the excitation, would be obtained. The experimental set-up consisted mainly of suitable light sources, monochromators and detectors. The gated measuring system consisted of an oscilloscope, integrator and recorder. The material predominantly used in the experiments was anthracene whose absorption spectrum was investigated and calculated. The absorption spectra of various polycrystalline layers were also calculated. The absorption spectrum of a randomly ordered polycrystalline layer was compared with that of a hexane solution and a good correlation between the two spectra was obtained. For the study of the relationship between the excitation spectrum of anthracene and the order of crystal, the excitation spectrum of single crystals of anthracene was measured from 4 eV to 107 eV. For the excitation region from 10 eV to 23 eV the efficiency of exciting a singlet level by a photoelectron was calculated as a function of the kinetic energy of the photoelectron, assuming the efficiency of the recombination to be constant. The excitation spectra of single crystals of p-terphenyl, pyrene and phenanthrene were also examined. In all four crystals the excitation spectrum rises monotonically with an increase in the energy of the exciting photon. (author) 12. Design for the excitation systems of three generator sets of HL-2A International Nuclear Information System (INIS) Li Huajun; Xu Lirong; Liu Xuemei; You Tianxue 2001-01-01 The design for the excitation systems have been made on the basis of respective features and load demands of three motor-generator sets of HL-2A. The excitation systems of No.1 and No.2 generator sets which supply for toroidal field are discussed in detail and three feasible excitation plans have been proposed according to the investment 13. Dynamic performance estimation of stator voltage regulator in rotary exciter system with DC exciter Directory of Open Access Journals (Sweden) Stojić Đorđe 2011-01-01 Full Text Available In this paper, procedure for AVR parameter estimation is proposed, based on step responses when synchronous generator in idle run. The exciter system includes AVR, thyristor rectifier and DC exciter. AVR is realized in the form of cascade control structure with two control loops. PID controller in the outer loop represents the primary controller. P controller in the inner loop represents secondary controller which enables the faster field current response time. The aim of procedure is to determine equivalent gain of PID controller and thyristor rectifier. The measurements used in the parameter estimation procedure are taken from fossil power plant 'Kolubara A', aggregate A5. 14. Multiple excitation of supports - Part 2 : Implementation in TUBO system International Nuclear Information System (INIS) Galeao, A.C.N.R.; Barbosa, H.J.C. 1980-12-01 From the formulation of multiple excitation support problem, discussed in the first part of this work, and with the use of numerical techniques presented there, we discuss in this second part, the implementation in TUBO system, the follow procedure: Direct integration, Modal overlap, Spectral response emphasizing the aspects related to supports excitation. Finally, we present two numerical examples of TUBO system utilization in the solution of support movement problem. The several implemented computational procedures are compared. (E.G.) [pt 15. Bifurcation and chaos in neural excitable system International Nuclear Information System (INIS) Jing Zhujun; Yang Jianping; Feng Wei 2006-01-01 In this paper, we investigate the dynamical behaviors of neural excitable system without periodic external current (proposed by Chialvo [Generic excitable dynamics on a two-dimensional map. Chaos, Solitons and Fractals 1995;5(3-4):461-79] and with periodic external current as system's parameters vary. The existence and stability of three fixed points, bifurcation of fixed points, the conditions of existences of fold bifurcation, flip bifurcation and Hopf bifurcation are derived by using bifurcation theory and center manifold theorem. The chaotic existence in the sense of Marotto's definition of chaos is proved. We then give the numerical simulated results (using bifurcation diagrams, computations of Maximum Lyapunov exponent and phase portraits), which not only show the consistence with the analytic results but also display new and interesting dynamical behaviors, including the complete period-doubling and inverse period-doubling bifurcation, symmetry period-doubling bifurcations of period-3 orbit, simultaneous occurrence of two different routes (invariant cycle and period-doubling bifurcations) to chaos for a given bifurcation parameter, sudden disappearance of chaos at one critical point, a great abundance of period windows (period 2 to 10, 12, 19, 20 orbits, and so on) in transient chaotic regions with interior crises, strange chaotic attractors and strange non-chaotic attractor. In particular, the parameter k plays a important role in the system, which can leave the chaotic behavior or the quasi-periodic behavior to period-1 orbit as k varies, and it can be considered as an control strategy of chaos by adjusting the parameter k. Combining the existing results in [Generic excitable dynamics on a two-dimensional map. Chaos, Solitons and Fractals 1995;5(3-4):461-79] with the new results reported in this paper, a more complete description of the system is now obtained 16. Structural System Identification with Extended Kalman Filter and Orthogonal Decomposition of Excitation Directory of Open Access Journals (Sweden) Y. Ding 2014-01-01 Full Text Available Both the structural parameter and external excitation have coupling influence on structural response. A new system identification method in time domain is proposed to simultaneously evaluate structural parameter and external excitation. The method can be used for linear and hysteresis nonlinear structural condition assessment based on incomplete structural responses. In this method, the structural excitation is decomposed by orthogonal approximation. With this approximation, the strongly time-variant excitation identification is transformed to gentle time-variant, even constant parameters identification. Then the extended Kalman filter is applied to simultaneously identify state vector including the structural parameters and excitation orthogonal parameters in state space based on incomplete measurements. The proposed method is validated numerically with the simulation of three-story linear and nonlinear structures subject to external force. The external force on the top floor and the structural parameters are simultaneously identified with the proposed system identification method. Results from both simulations indicate that the proposed method is capable of identifing the dynamic load and structural parameters fairly accurately with contaminated incomplete measurement for both of the linear and nonlinear structural systems. 17. Identification of the low-energy excitations in a quantum critical system Directory of Open Access Journals (Sweden) Tom Heitmann 2017-05-01 Full Text Available We have identified low-energy magnetic excitations in a doped quantum critical system by means of polarized neutron scattering experiments. The presence of these excitations could explain why Ce(Fe0.76Ru0.242Ge2 displays dynamical scaling in the absence of local critical behavior or long-range spin-density wave criticality. The low-energy excitations are associated with the reorientations of the superspins of fully ordered, isolated magnetic clusters that form spontaneously upon lowering the temperature. The system houses both frozen clusters and dynamic clusters, as predicted by Hoyos and Vojta [Phys. Rev. B 74, 140401(R (2006]. 18. Excitation model of pacemaker cardiomyocytes of cardiac conduction system Science.gov (United States) Grigoriev, M.; Babich, L. 2015-11-01 Myocardium includes typical and atypical cardiomyocytes - pacemakers, which form the cardiac conduction system. Excitation from the atrioventricular node in normal conditions is possible only in one direction. Retrograde direction of pulses is impossible. The most important prerequisite for the work of cardiomyocytes is the anatomical integrity of the conduction system. Changes in contractile force of the cardiomyocytes, which appear periodically, are due to two mechanisms of self-regulation - heterometric and homeometric. Graphic course of the excitation pulse propagation along the heart muscle more accurately reveals the understanding of the arrhythmia mechanism. These models have the ability to visualize the essence of excitation dynamics. However, they do not have the proper forecasting function for result estimation. Integrative mathematical model enables further investigation of general laws of the myocardium active behavior, allows for determination of the violation mechanism of electrical and contractile function of cardiomyocytes. Currently, there is no full understanding of the topography of pacemakers and ionic mechanisms. There is a need for the development of direction of mathematical modeling and comparative studies of the electrophysiological arrangement of cells of atrioventricular connection and ventricular conduction system. 19. Synchronization of Two Asymmetric Exciters in a Vibrating System Directory of Open Access Journals (Sweden) Zhaohui Ren 2011-01-01 Full Text Available We investigate synchronization of two asymmetric exciters in a vibrating system. Using the modified average method of small parameters, we deduce the non-dimensional coupling differential equations of the two exciters (NDDETE. By using the condition of existence for the zero solutions of the NDDETE, the condition of implementing synchronization is deduced: the torque of frequency capture is equal to or greater than the difference in the output electromagnetic torque between the two motors. Using the Routh-Hurwitz criterion, we deduce the condition of stability of synchronization that the inertia coupling matrix of the two exciters is positive definite. A numeric result shows that the structural parameters can meet the need of synchronization stability. 20. Development of the Fragment Molecular Orbital Method for Calculating Nonlocal Excitations in Large Molecular Systems. Science.gov (United States) Fujita, Takatoshi; Mochizuki, Yuji 2018-04-19 We developed the fragment-based method for calculating nonlocal excitations in large molecular systems. This method is based on the multilayer fragment molecular orbital method and the configuration interaction single (CIS) wave function using localized molecular orbitals. The excited-state wave function for the whole system is described as a superposition of configuration state functions (CSFs) for intrafragment excitations and for interfragment charge-transfer excitations. The formulation and calculations of singlet excited-state Hamiltonian matrix elements in the fragment CSFs are presented in detail. The efficient approximation schemes for calculating the matrix elements are also presented. The computational efficiency and the accuracy were evaluated using the molecular dimers and molecular aggregates. We confirmed that absolute errors of 50 meV (relative to the conventional calculations) are achievable for the molecular systems in their equilibrium geometries. The perturbative electron correlation correction to the CIS excitation energies is also demonstrated. The present theory can compute a large number of excited states in large molecular systems; in addition, it allows for the systematic derivation of a model exciton Hamiltonian. These features are useful for studying excited-state dynamics in condensed molecular systems based on the ab initio electronic structure theory. 1. Dynamic response of piping system subject to flow acoustic excitation International Nuclear Information System (INIS) Wang, T.; Sun, Y.S. 1988-01-01 Through the use of a theoretically derived and test data-calibrated forcing function, the dynamic response of a piping system subject to flow-acoustic induced vibration is analyzed. It is shown that the piping behavior can be predicted when consideration is given to both the wall flexural vibration and the piping system vibration. Piping responded as a system to the transversal excitation due to the swirling motion of the fluid flow, as well as flexurally to the high-frequency acoustic excitations. The transverse piping system response was calculated using a lumped mass piping model. The piping model has more stringent requirements than its counterpart for waterhammer and seismic modeling due to the shorter spiral wavelength and higher frequency of the forcing function. Proper modeling ensured that both the moment stress caused by system excitation and the local stress induced by the support reaction load were properly accounted for. Flexural vibration not only poses a threat to nipples and branch connections, but also contributes substantially to the resultant total stress experienced by the pipe. The forcing function approach has the advantage that the critical locations on the piping system can be identified by means of analysis, facilitating surveillance and inspection, as well as fatigue evaluation 2. Heat capacity for systems with excited-state quantum phase transitions Energy Technology Data Exchange (ETDEWEB) Cejnar, Pavel; Stránský, Pavel, E-mail: [email protected] 2017-03-18 Heat capacities of model systems with finite numbers of effective degrees of freedom are evaluated using canonical and microcanonical thermodynamics. Discrepancies between both approaches, which are observed even in the infinite-size limit, are particularly large in systems that exhibit an excited-state quantum phase transition. The corresponding irregularity of the spectrum generates a singularity in the microcanonical heat capacity and affects smoothly the canonical heat capacity. - Highlights: • Thermodynamics of systems with excited-state quantum phase transitions • ESQPT-generated singularities of the microcanonical heat capacity • Non-monotonous dependences of the canonical heat capacity • Discord between canonical and microcanonical pictures in the infinite-size limit. 3. The design of a sensor with flexible circuit excitation in electromagnetic tomography system International Nuclear Information System (INIS) Liu Ze; He Min; Xiong Hanliang 2005-01-01 A novel sensor structure of electromagnetic tomography system is presented in this paper. Flexible circuit straps are used in the excitation layer of the sensor and current of each strip can be controlled independently according to the excitation protocol matrix. In the sensor three kinds of excitation protocols: parallel, quasi-parallel and coil pair can be generated. Furthermore excitation field simulation and image reconstruction experiments have been done for analyzing the performance of the different excitation protocols 4. Stochastic stability of mechanical systems under renewal jump process parametric excitation DEFF Research Database (Denmark) Iwankiewicz, R.; Nielsen, Søren R.K.; Larsen, Jesper Winther 2005-01-01 A dynamic system under parametric excitation in the form of a non-Erlang renewal jump process is considered. The excitation is a random train of nonoverlapping rectangular pulses with equal, deterministic heights. The time intervals between two consecutive jumps up (or down), are the sum of two... 5. Excitation transfer in two two-level systems coupled to an oscillator International Nuclear Information System (INIS) Hagelstein, P L; Chaudhary, I U 2008-01-01 We consider a generalization of the spin-boson model in which two different two-level systems are coupled to an oscillator, under conditions where the oscillator energy is much less than the two-level system energies, and where the oscillator is highly excited. We find that the two-level system transition energy is shifted, producing a Bloch-Siegert shift in each two-level system similar to what would be obtained if the other were absent. At resonances associated with energy exchange between a two-level system and the oscillator, the level splitting is about the same as would be obtained in the spin-boson model at a Bloch-Siegert resonance. However, there occur resonances associated with the transfer of excitation between one two-level system and the other, an effect not present in the spin-boson model. We use a unitary transformation leading to a rotated system in which terms responsible for the shift and splittings can be identified. The level splittings at the anticrossings associated with both energy exchange and excitation transfer resonances are accounted for with simple two-state models and degenerate perturbation theory using operators that appear in the rotated Hamiltonian 6. Research on the reliability of friction system under combined additive and multiplicative random excitations Science.gov (United States) Sun, Jiaojiao; Xu, Wei; Lin, Zifei 2018-01-01 In this paper, the reliability of a non-linearly damped friction oscillator under combined additive and multiplicative Gaussian white noise excitations is investigated. The stochastic averaging method, which is usually applied to the research of smooth system, has been extended to the study of the reliability of non-smooth friction system. The results indicate that the reliability of friction system can be improved by Coulomb friction and reduced by random excitations. In particular, the effect of the external random excitation on the reliability is larger than the effect of the parametric random excitation. The validity of the analytical results is verified by the numerical results. 7. Distribution function of excitations in systems with fractional statistics International Nuclear Information System (INIS) Protogenov, A.P. 1992-08-01 The distribution function of low-energy excitations in 2+1D systems has been considered. It is shown that in these systems the quantum distribution function differs from the usual one by having a finite value of the entropy of linked braids. (author). 47 refs 8. Study on the adjustment capability of the excitation system located inside superconducting machine electromagnetic shield Science.gov (United States) Xia, D.; Xia, Z. 2017-12-01 The ability for the excitation system to adjust quickly plays a very important role in maintaining the normal operation of superconducting machines and power systems. However, the eddy currents in the electromagnetic shield of superconducting machines hinder the exciting magnetic field change and weaken the adjustment capability of the excitation system. To analyze this problem, a finite element calculation model for the transient electromagnetic field with moving parts is established. The effects of three different electromagnetic shields on the exciting magnetic field are analyzed using finite element method. The results show that the electromagnetic shield hinders the field changes significantly, the better its conductivity, the greater the effect on the superconducting machine excitation. 9. Relationships between resting conductances, excitability, and t-system ionic homeostasis in skeletal muscle. Science.gov (United States) Fraser, James A; Huang, Christopher L-H; Pedersen, Thomas H 2011-07-01 Activation of skeletal muscle fibers requires rapid sarcolemmal action potential (AP) conduction to ensure uniform excitation along the fiber length, as well as successful tubular excitation to initiate excitation-contraction coupling. In our companion paper in this issue, Pedersen et al. (2011. J. Gen. Physiol. doi:10.1085/jgp.201010510) quantify, for subthreshold stimuli, the influence upon both surface conduction velocity and tubular (t)-system excitation of the large changes in resting membrane conductance (G(M)) that occur during repetitive AP firing. The present work extends the analysis by developing a multi-compartment modification of the charge-difference model of Fraser and Huang to provide a quantitative description of the conduction velocity of actively propagated APs; the influence of voltage-gated ion channels within the t-system; the influence of t-system APs on ionic homeostasis within the t-system; the influence of t-system ion concentration changes on membrane potentials; and the influence of Phase I and Phase II G(M) changes on these relationships. Passive conduction properties of the novel model agreed with established linear circuit analysis and previous experimental results, while key simulations of AP firing were tested against focused experimental microelectrode measurements of membrane potential. This study thereby first quantified the effects of the t-system luminal resistance and voltage-gated Na(+) channel density on surface AP propagation and the resultant electrical response of the t-system. Second, it demonstrated the influence of G(M) changes during repetitive AP firing upon surface and t-system excitability. Third, it showed that significant K(+) accumulation occurs within the t-system during repetitive AP firing and produces a baseline depolarization of the surface membrane potential. Finally, it indicated that G(M) changes during repetitive AP firing significantly influence both t-system K(+) accumulation and its influence on the 10. Seismic response analysis of structural system subjected to multiple support excitation International Nuclear Information System (INIS) Wu, R.W.; Hussain, F.A.; Liu, L.K. 1978-01-01 In the seismic analysis of a multiply supported structural system subjected to nonuniform excitations at each support point, the single response spectrum, the time history, and the multiple response spectrum are the three commonly employed methods. In the present paper the three methods are developed, evaluated, and the limitations and advantages of each method assessed. A numerical example has been carried out for a typical piping system. Considerably smaller responses have been predicted by the time history method than that by the single response spectrum method. This is mainly due to the fact that the phase and amplitude relations between the support excitations are faithfully retained in the time history method. The multiple response spectrum prediction has been observed to compare favourably with the time history method prediction. Based on the present evaluation, the multiple response spectrum method is the most efficient method for seismic response analysis of structural systems subjected to multiple support excitation. (Auth.) 11. Chaos excited chaos synchronizations of integral and fractional order generalized van der Pol systems International Nuclear Information System (INIS) Ge Zhengming; Hsu Maoyuan 2008-01-01 In this paper, chaos excited chaos synchronizations of generalized van der Pol systems with integral and fractional order are studied. Synchronizations of two identified autonomous generalized van der Pol chaotic systems are obtained by replacing their corresponding exciting terms by the same function of chaotic states of a third nonautonomous or autonomous generalized van der Pol system. Numerical simulations, such as phase portraits, Poincare maps and state error plots are given. It is found that chaos excited chaos synchronizations exist for the fractional order systems with the total fractional order both less than and more than the number of the states of the integer order generalized van der Pol system 12. Adaptive Dynamic Surface Control for Generator Excitation Control System Directory of Open Access Journals (Sweden) Zhang Xiu-yu 2014-01-01 Full Text Available For the generator excitation control system which is equipped with static var compensator (SVC and unknown parameters, a novel adaptive dynamic surface control scheme is proposed based on neural network and tracking error transformed function with the following features: (1 the transformation of the excitation generator model to the linear systems is omitted; (2 the prespecified performance of the tracking error can be guaranteed by combining with the tracking error transformed function; (3 the computational burden is greatly reduced by estimating the norm of the weighted vector of neural network instead of the weighted vector itself; therefore, it is more suitable for the real time control; and (4 the explosion of complicity problem inherent in the backstepping control can be eliminated. It is proved that the new scheme can make the system semiglobally uniformly ultimately bounded. Simulation results show the effectiveness of this control scheme. 13. Analysis of piping system response to seismic excitations International Nuclear Information System (INIS) Wang, C.Y. 1987-01-01 This paper describes a numerical algorithm for analyzing piping system response to seismic excitations. The numerical model of the piping considers hoop, flexural, axial, and torsional modes of deformation. Hoop modes generated from internal hydrodynamic loading are superimposed on the bending and twisting modes by two extra degrees of freedom. A time-history analysis technique using the implicit temporal integration scheme is addressed. The time integrator uses a predictor-corrector successive iterative scheme which satisfies the equation of motion. Both geometrical and material nonlinearities are considered. Multiple support excitations, fluid effect, piping insulation, and material dampings can be included in the analysis. Two problems are presented to illustrate the method. The results are discussed in detail 14. Seismic analysis of piping systems subjected to multiple support excitations International Nuclear Information System (INIS) Sundararajan, C.; Vaish, A.K.; Slagis, G.C. 1981-01-01 The paper presents the results of a comparative study between the multiple response spectrum method and the time-history method for the seismic analysis of nuclear piping systems subjected to different excitation at different supports or support groups. First, the necessary equations for the above analysis procedures are derived. Then, three actual nuclear piping systems subjected to single and multiple excitations are analyzed by the different methods, and extensive comparisons of the results (stresses) are made. Based on the results, it is concluded that the multiple response spectrum analysis gives acceptable results as compared to the ''exact'', but much more costly, time-history analysis. 6 refs 15. Excitation functions of the systems 12C+14C and 13C+12C International Nuclear Information System (INIS) Haindl, E. 1975-01-01 The excitation functions of the systems 12 C+ 14 C and 13 C+ 12 C are investigated for different exit channels. The excitation functions measured do not show correlated structures as in the system 12 C+ 12 C. (WL/AK) [de 16. Spirals in a reaction-diffusion system: Dependence of wave dynamics on excitability Science.gov (United States) Mahanta, Dhriti; Das, Nirmali Prabha; Dutta, Sumana 2018-02-01 A detailed study of the effects of excitability of the Belousov-Zhabotinsky (BZ) reaction on spiral wave properties has been carried out. Using the Oregonator model, we explore the various regimes of wave activity, from sustained oscillations to wave damping, as the system undergoes a Hopf bifurcation, that is achieved by varying the excitability parameter, ɛ . We also discover a short range of parameter values where random oscillations are observed. With an increase in the value of ɛ , the frequency of the wave decreases exponentially, as the dimension of the spiral core expands. These numerical results are confirmed by carrying out experiments in thin layers of the BZ system, where the excitability is changed by varying the concentrations of the reactant species. Effect of reactant concentrations on wave properties like time period and wavelength are also explored in detail. Drifting and meandering spirals are found in the parameter space under investigation, with the excitability affecting the tip trajectory in a way predicted by the numerical studies. This study acts as a quantitative evidence of the relationship between the excitability parameter, ɛ , and the substrate concentrations. 17. Spirals in a reaction-diffusion system: Dependence of wave dynamics on excitability. Science.gov (United States) Mahanta, Dhriti; Das, Nirmali Prabha; Dutta, Sumana 2018-02-01 A detailed study of the effects of excitability of the Belousov-Zhabotinsky (BZ) reaction on spiral wave properties has been carried out. Using the Oregonator model, we explore the various regimes of wave activity, from sustained oscillations to wave damping, as the system undergoes a Hopf bifurcation, that is achieved by varying the excitability parameter, ε. We also discover a short range of parameter values where random oscillations are observed. With an increase in the value of ε, the frequency of the wave decreases exponentially, as the dimension of the spiral core expands. These numerical results are confirmed by carrying out experiments in thin layers of the BZ system, where the excitability is changed by varying the concentrations of the reactant species. Effect of reactant concentrations on wave properties like time period and wavelength are also explored in detail. Drifting and meandering spirals are found in the parameter space under investigation, with the excitability affecting the tip trajectory in a way predicted by the numerical studies. This study acts as a quantitative evidence of the relationship between the excitability parameter, ε, and the substrate concentrations. 18. Production of excited double hypernuclei via Fermi breakup of excited strange systems International Nuclear Information System (INIS) Sanchez Lorente, Alicia; Botvina, Alexander S.; Pochodzalla, Josef 2011-01-01 Precise spectroscopy of multi-strange hypernuclei provides a unique chance to explore the hyperon-hyperon interaction. In the present work we explore the production of excited states in double hypernuclei following the micro-canonical break-up of an initially excited double hypernucleus which is created by the absorption and conversion of a stopped Ξ - hyperon. Rather independent on the spectrum of possible excited states in the produced double hypernuclei the formation of excited states dominates in our model. For different initial target nuclei which absorb the Ξ - , different double hypernuclei nuclei dominate. Thus the ability to assign the various observable γ-transitions in a unique way to a specific double hypernuclei by exploring various light targets as proposed by the PANDA Collaboration seems possible. We also confront our predictions with the correlated pion spectra measured by the E906 Collaboration. 19. Experimental and theoretical evidence for fluctuation driven activations in an excitable chemical system Science.gov (United States) Hastings, Harold; Sobel, Sabrina; Field, Richard; Minchenberg, Scott; Spinelli, Nicole; Zauderer, Keith 2011-03-01 An excitable medium is a system in which small perturbations die out, but sufficiently large perturbations generate large ``excitations.'' Biological examples include neurons and the heart; the latter supports waves of excitation normally generated by the sinus node, but occasionally generated by other mechanisms. The ferroin-catalyzed Belousov-Zhabotinsky reaction is the prototype chemical excitable medium. We present experimental and theoretical evidence for that random fluctuations can generate excitations in the Belousov-Zhabothinsky reaction. Although the heart is significantly different, there are some scaling analogies. This material is based upon work supported by the Department of Energy under Award Number DE-FG02-08ER64623. 20. Excitation of electrostatic ion cyclotron wave in electron beam plasma system International Nuclear Information System (INIS) Fukumura, Takashi; Takamoto, Teruo 1984-01-01 The electrostatic ion cyclotron waves excited in an electron beam plasma system was investigated. The excitation condition of the waves was calculated by using Harris type dispersion relation under some assumption, and its comparison with the experimental result was made. Beam plasma discharge is a kind of RF discharge, and it is caused by the waves generated by the interaction of electron beam with plasma. It was shown that electrostatic ion cyclotron waves seemed to be the most probable as excited waves. But the excitation mechanism of these waves has not been concretely investigated. In this study, the excitation condition of electrostatic ion cyclotron waves was calculated as described above. The experimental apparatus and the results of potential, electric field and ion saturation current in beam plasma, electron drift motion in azimuthal direction and the waves excited in beam plasma are reported. The frequency of oscillation observed in beam plasma corresponds to the harmonics or subharmonics of ion cyclotron frequency. The calculation of Harris type dispersion relation, the numerical calculation and the comparison of the experimental result with the calculated result are described. (Kako, I.) 1. Modernization of the Control Systems of High-Frequency, Brush-Free, and Collector Exciters of Turbogenerators Energy Technology Data Exchange (ETDEWEB) Popov, E. N., E-mail: [email protected]; Komkov, A. L.; Ivanov, S. L.; Timoshchenko, K. P. [JSC “Scientific and Industrial Enterprise “Rusélprom-Élektromash” (Russian Federation) 2016-11-15 Methods of modernizing the regulation systems of electric machinery exciters with high-frequency, brush-free, and collector exciters by means of microprocessor technology are examined. The main problems of modernization are to increase the response speed of a system and to use a system stabilizer to increase the stability of the power system. 2. Spike Bursts from an Excitable Optical System Science.gov (United States) Rios Leite, Jose R.; Rosero, Edison J.; Barbosa, Wendson A. S.; Tredicce, Jorge R. Diode Lasers with double optical feedback are shown to present power drop spikes with statistical distribution controllable by the ratio of the two feedback times. The average time between spikes and the variance within long time series are studied. The system is shown to be excitable and present bursting of spikes created with specific feedback time ratios and strength. A rate equation model, extending the Lang-Kobayashi single feedback for semiconductor lasers proves to match the experimental observations. Potential applications to construct network to mimic neural systems having controlled bursting properties in each unit will be discussed. Brazilian Agency CNPQ. 3. Base excitation testing system using spring elements to pivotally mount wind turbine blades Science.gov (United States) Cotrell, Jason; Hughes, Scott; Butterfield, Sandy; Lambert, Scott 2013-12-10 A system (1100) for fatigue testing wind turbine blades (1102) through forced or resonant excitation of the base (1104) of a blade (1102). The system (1100) includes a test stand (1112) and a restoring spring assembly (1120) mounted on the test stand (1112). The restoring spring assembly (1120) includes a primary spring element (1124) that extends outward from the test stand (1112) to a blade mounting plate (1130) configured to receive a base (1104) of blade (1102). During fatigue testing, a supported base (1104) of a blad (1102) may be pivotally mounted to the test stand (1112) via the restoring spring assembly (1120). The system (1100) may include an excitation input assembly (1140) that is interconnected with the blade mouting plate (1130) to selectively apply flapwise, edgewise, and/or pitch excitation forces. The restoring spring assemply (1120) may include at least one tuning spring member (1127) positioned adjacent to the primary spring element (1124) used to tune the spring constant or stiffness of the primary spring element (1124) in one of the excitation directions. 4. Exploring excited eigenstates of many-body systems using the functional renormalization group Science.gov (United States) Klöckner, Christian; Kennes, Dante Marvin; Karrasch, Christoph 2018-05-01 We introduce approximate, functional renormalization group based schemes to obtain correlation functions in pure excited eigenstates of large fermionic many-body systems at arbitrary energies. The algorithms are thoroughly benchmarked and their strengths and shortcomings are documented using a one-dimensional interacting tight-binding chain as a prototypical testbed. We study two "toy applications" from the world of Luttinger liquid physics: the survival of power laws in lowly excited states as well as the spectral function of high-energy "block" excitations, which feature several single-particle Fermi edges. 5. A study on an object transport system using ultrasonic wave excitation International Nuclear Information System (INIS) Jeong, Sang Hwa; Kim, Gwang Ho; Choi, Suk Bong; Park, Jun Ho; Cha, Kyoung Rae 2007-01-01 The development of information and telecommunication industries leads to the development of semiconductor and optical industries. In recent years, the demand of optical components is growing due to the demand of faster network. On the other hand, conventional transport systems are not adequate for transporting precision optical components and semiconductors. Because the conveyor belt can damage precision optical components with contact force and a magnetic system would destroy the inner structure of semiconductor with magnetic field, a new system for transporting optical components and semiconductors is required. One of the alternatives to the existing systems is a transport system using ultrasonic wave excitation since it can transport precision components such as semiconductors and optical components without damage. In this paper, a transport system using 2-mode ultrasonic wave excitation was developed for transporting optical components and semiconductor, and its performance was evaluated. The relationship between transporting characteristics and flexural beam shapes were evaluated 6. FSI analysis of piping systems under seismic excitation International Nuclear Information System (INIS) Uras, R.A.; Ma, D.C.; Chang, Yao W.; Liu, Wing Kam 1991-01-01 A formulation which accounts for fluid-structure interaction of piping system under seismic excitation is presented. The governing equations of the fluid and the structure to model the pipe are stated. Using the finite element method the discretized equations are obtained. A transformation procedure for proper assembly of matrices is introduced. A solution algorithm is described. 9 refs., 2 figs 7. The conditions for attaining the greatest degree of system stability with strict generator excitation control Energy Technology Data Exchange (ETDEWEB) Gruzdev, I.A.; Ekimova, M.M.; Truspekova, G.A. 1982-01-01 Expressions are derived for an idealized model of a complex electric power system; these expressions define the greatest level of stability of an electric power system and the optimum combination of stabilization factors with automatic excitation control in a single power system. The possibility of increasing the level of stability of an electric power system with simultaneous strict automatic excitation control of the synychronous generators in several power systems is analyzed. 8. Non-linear operation of nanomechnical systems combining photothermal excitation and magneto-motive detection International Nuclear Information System (INIS) Koenig, Daniel R; Metzger, Constanze; Camerer, Stephan; Kotthaus, Joerg P 2006-01-01 We present a non-linear operation of a nanomechanical beam resonator by photothermal excitation at 4 K. The resonators dimensions are 10 μm in length, 200 nm in width, and 200 nm in height. The actuation mechanism is based on a pulsed diode laser focused onto the centre of the beam resonator. Thermally induced stress caused by the different thermal expansion coefficients of the bi-layer system periodically deflects the resonator. Magnetomotively detected amplitudes up to 150 nm are reached at the fundamental resonance mode at a frequency of 8.9 MHz. Furthermore, the third eigenmode of the resonator at a frequency 36 MHz is also excited. We conclude that the photothermal excitation at 4 K should be applicable up to the GHz regime, the operation in the non-linear regime can be used for performance enhancement of nanomechanical systems, and the combination of photothermal excitation and magneto-motive detection avoids undesired cross talk 9. The excitation system of 727.5 MVA synchronous generator of the unit B1 in TPP 'Nikola Tesla B' Directory of Open Access Journals (Sweden) Ćirić Zoran 2013-01-01 Full Text Available This paper presents a technical solution for the replacement of the excitation system of the unit B1 in TPP 'Nikola Tesla B' as a part of the maintenance service in 2012. Since the generators of TPP 'Nikola Tesla B' have the greatest power in the power system of Serbia, it was necessary to achieve high reliability of the excitation system so that the process of producing electricity is not endangered Considering this, the implemented excitation system uses modern technology with redundancy both in the power and control blocks, which resulted in an increase in the hot reserve by 100%. In addition, it was necessary to adjust the excitation system to increased generator power and performance from 618MW to 667.5MW. In this paper, the main parameters of the excitation system are given: the power, the excitation system control, the thyristor ignition system, the event recorder system, the digital relay protection, as well as the measuring and signaling functions. 10. Control of base-excited dynamical systems through piezoelectric energy harvesting absorber Science.gov (United States) Abdelmoula, H.; Dai, H. L.; Abdelkefi, A.; Wang, L. 2017-09-01 The spring-mass absorber usually offers a good control to dynamical systems under direct base excitations for a specific value of the excitation frequency. As the vibrational energy of a primary dynamical system is transferred to the absorber, it gets dissipated. In this study, this energy is no longer dissipated but converted to available electrical power by designing efficient energy harvesters. A novel design of a piezoelectric beam installed inside an elastically-mounted dynamical system undergoing base excitations is considered. A design is carried out in order to determine the properties and dimensions of the energy harvester with the constraint of simultaneously decreasing the oscillating amplitudes of the primary dynamical system and increasing the harvested power of the energy harvesting absorber. An analytical model for the coupled system is constructed using Euler-Lagrange principle and Galerkin discretization. Different strategies for controlling the primary structure displacement and enhancing the harvested power as functions of the electrical load resistance and thickness of the beam substrate are performed. The linear polynomial approximation of the system’s key parameters as a function of the beam’s substrate thickness is first carried out. Then, the gradient method is applied to determine the adequate values of the electrical load resistance and thickness of the substrate under the constraints of minimizing the amplitudes of the primary structure or maximizing the levels of the harvested power. After that, an iterative strategy is considered in order to simultaneously minimize the amplitudes of the primary structure and maximize the levels of the harvested power as functions of the thickness of the substrate and electrical load resistance. In addition to harmonic excitations, the coupled system subjected to a white noise is explored. Through this analysis, the load resistance and thickness of the substrate of the piezoelectric energy harvester 11. Vibrational-rotational excitation: chemical reactions of vibrationally excited molecules International Nuclear Information System (INIS) Moore, C.B.; Smith, I.W.M. 1979-03-01 This review considers a limited number of systems, particularly gas-phase processes. Excited states and their preparation, direct bimolecular reactions, reactions of highly excited molecules, and reactions in condensed phases are discussed. Laser-induced isotope separation applications are mentioned briefly. 109 references 12. Optimization of tube parameters in a tube excited X-ray fluorescence (TEXRF) system using secondary fluorescers International Nuclear Information System (INIS) Islam, A.; Biswas, S.K. 1995-12-01 A study of the optimization of excitation parameters in a tube excited X-ray fluorescence system (TEXRF) having Mo as the primary target has been carried out for biological matrix. Fe, Zn and Mo were used as the secondary fluorecers. For the present investigation a cellulose based synthetic standard containing K, Cr, Ni, Zn, Se and Y was excited with the TEXRF system. All experiments were carried out under the same experimental conditions except the tube potential. For each fluorescer the minimum detection limits (MDL) of excited elements were calculated for the corresponding tube voltage. The MDLs were found to be increasing with decreasing atomic number and it was also observed that the maximum sensitivity with Fe and Zn secondary fluorescers for elements analyzed occurred around 35 kV of the excitation potential. For Mo secondary fluorescer maximum sensitivity was found at higher excitation potential. In most cases MDLs were minimum at 40-45 kV of the excitation potential. 5 refs., 12 figs 13. Using axicons for depth discrimination in excitation-emission laser scanning imaging systems Science.gov (United States) Iglesias, Ignacio 2017-10-01 Besides generating good approximations to zero-order Bessel beams, an axicon lens coupled to a spatial filter can be used to collect light while preserving information on the depth coordinate of the source location. To demonstrate the principle, we describe an experimental excitation-emission fluorescence imaging system that uses an axicon twice: to generate an excitation Bessel beam and to collect the emitted light. 14. Applicability of annular-source excited systems in quantitative XRF analysis International Nuclear Information System (INIS) Mahmoud, A.; Bernasconi, G.; Bamford, S.A.; Dosan, B.; Haselberger, N.; Markowicz, A. 1996-01-01 Radioisotope-excited XRF systems, using annular sources, are widely used in view of their simplicity, wide availability, relatively low price for the complete system and good overall performance with respect to accuracy and detection limits. However some problems arise when the use of fundamental parameter techniques for quantitative analysis is attempted. These problems are due to the fact that the systems operate with large solid angles for incoming and emerging radiation and both the incident and take-off angles are not trivial. In this paper an improved way to calculate effective values for the incident and take-off angles, using monte Carlo (M C) integration techniques is shown. In addition, a study of the applicability of the effective angles for analysing different samples, or standards was carried out. The M C method allows also calculation of the excitation-detection efficiency for different parts of the sample and estimation of the overall efficiency of a source-excited XRF setup. The former information is useful in the design of optimized XRF set-ups and prediction of the response of inhomogeneous samples. A study of the sensitivity of the results due to sample characteristics and a comparison of the results with experimentally determined values for incident and take-off angles is also presented. A flexible and user-friendly computer program was developed in order to perform efficiently the lengthy calculation involved. (author). 14 refs. 5 figs 15. Multi-frequency excitation KAUST Repository 2016-03-10 Embodiments of multi-frequency excitation are described. In various embodiments, a natural frequency of a device may be determined. In turn, a first voltage amplitude and first fixed frequency of a first source of excitation can be selected for the device based on the natural frequency. Additionally, a second voltage amplitude of a second source of excitation can be selected for the device, and the first and second sources of excitation can be applied to the device. After applying the first and second sources of excitation, a frequency of the second source of excitation can be swept. Using the methods of multi- frequency excitation described herein, new operating frequencies, operating frequency ranges, resonance frequencies, resonance frequency ranges, and/or resonance responses can be achieved for devices and systems. 16. The Modeling and Analysis for the Self-Excited Vibration of the Maglev Vehicle-Bridge Interaction System Directory of Open Access Journals (Sweden) Jinhui Li 2015-01-01 Full Text Available This paper addresses the self-excited vibration problems of maglev vehicle-bridge interaction system which greatly degrades the stability of the levitation control, decreases the ride comfort, and restricts the cost of the whole system. Firstly, two levitation models with different complexity are developed, and the comparison of the energy curves associated with the two models is carried out. We conclude that the interaction model with a single levitation control unit is sufficient for the study of the self-excited vibration. Then, the principle underlying the self-excited vibration is explored from the standpoint of work acting on the bridge done by the levitation system. Furthermore, the influences of the parameters, including the modal frequency and modal damping of bridge, the gain of the controller, the sprung mass, and the unsprung mass, on the stability of the interaction system are carried out. The study provides a theoretical guidance for solving the self-excited vibration problems of the vehicle-bridge interaction systems. 17. H∞ Excitation Control Design for Stochastic Power Systems with Input Delay Based on Nonlinear Hamiltonian System Theory Directory of Open Access Journals (Sweden) Weiwei Sun 2015-01-01 Full Text Available This paper presents H∞ excitation control design problem for power systems with input time delay and disturbances by using nonlinear Hamiltonian system theory. The impact of time delays introduced by remote signal transmission and processing in wide-area measurement system (WAMS is well considered. Meanwhile, the systems under investigation are disturbed by random fluctuation. First, under prefeedback technique, the power systems are described as a nonlinear Hamiltonian system. Then the H∞ excitation controller of generators connected to distant power systems with time delay and stochasticity is designed. Based on Lyapunov functional method, some sufficient conditions are proposed to guarantee the rationality and validity of the proposed control law. The closed-loop systems under the control law are asymptotically stable in mean square independent of the time delay. And we through a simulation of a two-machine power system prove the effectiveness of the results proposed in this paper. 18. Design, fabrication and testing of a 5-Hz acoustic exciter system Science.gov (United States) Lundy, D. H.; Robinson, G. D. 1973-01-01 A 5-Hz acoustic excitation system was designed, fabricated and checked out for use in the modulation of a stagnant gas volume contained in an absorption cell. A detailed system description of the test equipment, both mechanical and electronic, and an operating procedure are included. Conclusions are also presented. 19. Fragment emission from modestly excited nuclear systems Energy Technology Data Exchange (ETDEWEB) Lou, Y. [Indiana Univ., Bloomington, IN (United States). Dept. of Chemistry]\|[Indiana Univ., Bloomington, IN (United States). Cyclotron Facility; Souza, R.T. de [Indiana Univ., Bloomington, IN (United States). Dept. of Chemistry]\|[Indiana Univ., Bloomington, IN (United States). Cyclotron Facility; Chen, S.L. [Indiana Univ., Bloomington, IN (United States). Dept. of Chemistry]\|[Indiana Univ., Bloomington, IN (United States). Cyclotron Facility; Cornell, E.W. [Indiana Univ., Bloomington, IN (United States). Dept. of Chemistry]\|[Indiana Univ., Bloomington, IN (United States). Cyclotron Facility; Davin, B. [Indiana Univ., Bloomington, IN (United States). Dept. of Chemistry]\|[Indiana Univ., Bloomington, IN (United States). Cyclotron Facility; Fox, D. [Indiana Univ., Bloomington, IN (United States). Dept. of Chemistry]\|[Indiana Univ., Bloomington, IN (United States). Cyclotron Facility; Hamilton, T.M. [Indiana Univ., Bloomington, IN (United States). Dept. of Chemistry]\|[Indiana Univ., Bloomington, IN (United States). Cyclotron Facility; Mcdonald, K. [Indiana Univ., Bloomington, IN (United States). Dept. of Chemistry]\|[Indiana Univ., Bloomington, IN (United States). Cyclotron Facility; Tsang, M.B. [Michigan State Univ., East Lansing, MI (United States). National Superconducting Cyclotron Lab.; Glasmacher, T. [Michigan State Univ., East Lansing, MI (United States). National Superconducting Cyclotron Lab.; Dinius, J. [Michigan State Univ., East Lansing, MI (United States). National Superconducting Cyclotron Lab.; Gelbke, C.K. [Michigan State Univ., East Lansing, MI (United States). National Superconducting Cyclotron Lab.; Handzy, D.O. [Indiana Univ., Bloomington, IN (United States). Dept. of Chemistry]\|[Indiana Univ., Bloomington, IN (United States). Cyclotron Facility]\|[Michigan State Univ., East Lansing, MI (United States). National Superconducting Cyclotron Lab.; Hsi, W.C. 1996-07-08 Fragment emission patterns occurring in nuclear systems of modest excitation are studied. Exclusive measurement of fragment emission in {sup 14}N+{sup 197}Au reactions at E/A=100, 130 and 156 MeV allows selection of central collisions where a single source dominates the decay. Low threshold measurement of IMF emission for these events allows investigation of the influence of detector threshold effects. The time scale of fragment emission is deduced using fragment-fragment velocity correlations. Comparisons are made to the predictions of a statistical decay model. (orig.). 20. Beam excitation and damping with the transverse feedback system International Nuclear Information System (INIS) Pellegrin, J.L.; Rees, J.R. 1979-08-01 The questions often come up, ''What is the strength if the beam excitation system? How much damping can the transverse feedback provide?'' The design is now advanced enough to answer these questions; also, laboratory tests of some components have been conducted and we know what can be expected of the hardware. This paper discusses these questions 1. Indirect control of quantum systems via an accessor: pure coherent control without system excitation International Nuclear Information System (INIS) Fu, H C; Dong Hui; Sun, C P; Liu, X F 2009-01-01 A pure indirect control of quantum systems via a quantum accessor is investigated. In this control scheme, we do not apply any external classical excitation fields on the controlled system and we control a quantum system via a quantum accessor and classical control fields control the accessor only. Complete controllability is investigated for arbitrary finite-dimensional quantum systems and exemplified by two- and three-dimensional systems. The scheme exhibits some advantages; it uses less qubits in the accessor and does not depend on the energy-level structure of the controlled system 2. Pure-Phase Selective Excitation in Fast-Relaxing Systems Science.gov (United States) Zangger, Klaus; Oberer, Monika; Sterk, Heinz 2001-09-01 Selective pulses have been used frequently for small molecules. However, their application to proteins and other macromolecules has been limited. The long duration of shaped-selective pulses and the short T2 relaxation times in proteins often prohibited the use of highly selective pulses especially on larger biomolecules. A very selective excitation can be obtained within a short time by using the selective excitation sequence presented in this paper. Instead of using a shaped low-intensity radiofrequency pulse, a cluster of hard 90° pulses, delays of free precession, and pulsed field gradients can be used to selectively excite a narrow chemical shift range within a relatively short time. Thereby, off-resonance magnetization, which is allowed to evolve freely during the free precession intervals, is destroyed by the gradient pulses. Off-resonance excitation artifacts can be removed by random variation of the interpulse delays. This leads to an excitation profile with selectivity as well as phase and relaxation behavior superior to that of commonly used shaped-selective pulses. Since the evolution of scalar coupling is inherently suppressed during the double-selective excitation of two different scalar-coupled nuclei, the presented pulse cluster is especially suited for simultaneous highly selective excitation of N-H and C-H fragments. Experimental examples are demonstrated on hen egg white lysozyme (14 kD) and the bacterial antidote ParD (19 kD). 3. Application of Excitation from Multiple Locations on a Simplified High-Lift System Science.gov (United States) Melton, LaTunia Pack; Yao, Chung-Sheng; Seifert, Avi 2004-01-01 A series of active flow control experiments were recently conducted on a simplified high-lift system. The purpose of the experiments was to explore the prospects of eliminating all but simply hinged leading and trailing edge flaps, while controlling separation on the supercritical airfoil using multiple periodic excitation slots. Excitation was provided by three. independently controlled, self-contained, piezoelectric actuators. Low frequency excitation was generated through amplitude modulation of the high frequency carrier wave, the actuators' resonant frequencies. It was demonstrated, for the first time, that pulsed modulated signal from two neighboring slots interact favorably to increase lift. Phase sensitivity at the low frequency was measured, even though the excitation was synthesized from the high-frequency carrier wave. The measurements were performed at low Reynolds numbers and included mean and unsteady surface pressures, surface hot-films, wake pressures and particle image velocimetry. A modest (6%) increase in maximum lift (compared to the optimal baseline) was obtained due t o the activation of two of the three actuators. 4. Line shape of magnetic excitations in singlet-ground-state systems International Nuclear Information System (INIS) Bak, P. 1976-08-01 The excitation spectrum in a paramagnetic singlet doublet system is calculated using a diagrammatic expansion technique, and the theoretical predictions are compared with experiments on praseodymium. The theory gives an accurate description of the dramatic temperature dependence of the energies and lineshapes for the exciton modes 5. Excited State Structural Dynamics of Carotenoids and Charge Transfer Systems International Nuclear Information System (INIS) Van Tassle, Aaron Justin 2006-01-01 This dissertation describes the development and implementation of a visible/near infrared pump/mid-infrared probe apparatus. Chapter 1 describes the background and motivation of investigating optically induced structural dynamics, paying specific attention to solvation and the excitation selection rules of highly symmetric molecules such as carotenoids. Chapter 2 describes the development and construction of the experimental apparatus used throughout the remainder of this dissertation. Chapter 3 will discuss the investigation of DCM, a laser dye with a fluorescence signal resulting from a charge transfer state. By studying the dynamics of DCM and of its methyl deuterated isotopomer (an otherwise identical molecule), we are able to investigate the origins of the charge transfer state and provide evidence that it is of the controversial twisted intramolecular (TICT) type. Chapter 4 introduces the use of two-photon excitation to the S1 state, combined with one-photon excitation to the S2 state of the carotenoid beta-apo-8'-carotenal. These 2 investigations show evidence for the formation of solitons, previously unobserved in molecular systems and found only in conducting polymers Chapter 5 presents an investigation of the excited state dynamics of peridinin, the carotenoid responsible for the light harvesting of dinoflagellates. This investigation allows for a more detailed understanding of the importance of structural dynamics of carotenoids in light harvesting 6. Response analysis of a class of quasi-linear systems with fractional derivative excited by Poisson white noise Science.gov (United States) Yang, Yongge; Xu, Wei; Yang, Guidong; Jia, Wantao 2016-08-01 The Poisson white noise, as a typical non-Gaussian excitation, has attracted much attention recently. However, little work was referred to the study of stochastic systems with fractional derivative under Poisson white noise excitation. This paper investigates the stationary response of a class of quasi-linear systems with fractional derivative excited by Poisson white noise. The equivalent stochastic system of the original stochastic system is obtained. Then, approximate stationary solutions are obtained with the help of the perturbation method. Finally, two typical examples are discussed in detail to demonstrate the effectiveness of the proposed method. The analysis also shows that the fractional order and the fractional coefficient significantly affect the responses of the stochastic systems with fractional derivative. 7. Response analysis of a class of quasi-linear systems with fractional derivative excited by Poisson white noise International Nuclear Information System (INIS) Yang, Yongge; Xu, Wei; Yang, Guidong; Jia, Wantao 2016-01-01 The Poisson white noise, as a typical non-Gaussian excitation, has attracted much attention recently. However, little work was referred to the study of stochastic systems with fractional derivative under Poisson white noise excitation. This paper investigates the stationary response of a class of quasi-linear systems with fractional derivative excited by Poisson white noise. The equivalent stochastic system of the original stochastic system is obtained. Then, approximate stationary solutions are obtained with the help of the perturbation method. Finally, two typical examples are discussed in detail to demonstrate the effectiveness of the proposed method. The analysis also shows that the fractional order and the fractional coefficient significantly affect the responses of the stochastic systems with fractional derivative. 8. Response analysis of a class of quasi-linear systems with fractional derivative excited by Poisson white noise Energy Technology Data Exchange (ETDEWEB) Yang, Yongge; Xu, Wei, E-mail: [email protected]; Yang, Guidong; Jia, Wantao [Department of Applied Mathematics, Northwestern Polytechnical University, Xi' an 710072 (China) 2016-08-15 The Poisson white noise, as a typical non-Gaussian excitation, has attracted much attention recently. However, little work was referred to the study of stochastic systems with fractional derivative under Poisson white noise excitation. This paper investigates the stationary response of a class of quasi-linear systems with fractional derivative excited by Poisson white noise. The equivalent stochastic system of the original stochastic system is obtained. Then, approximate stationary solutions are obtained with the help of the perturbation method. Finally, two typical examples are discussed in detail to demonstrate the effectiveness of the proposed method. The analysis also shows that the fractional order and the fractional coefficient significantly affect the responses of the stochastic systems with fractional derivative. 9. Localized excitations in nonlinear complex systems current state of the art and future perspectives CERN Document Server Cuevas-Maraver, Jesús; Frantzeskakis, Dimitri; Karachalios, Nikos; Kevrekidis, Panayotis; Palmero-Acebedo, Faustino 2014-01-01 The study of nonlinear localized excitations is a long-standing challenge for research in basic and applied science, as well as engineering, due to their importance in understanding and predicting phenomena arising in nonlinear and complex systems, but also due to their potential for the development and design of novel applications. This volume is a compilation of chapters representing the current state-of-the-art on the field of localized excitations and their role in the dynamics of complex physical systems. 10. Study of resonant magnet exciting system for the 3 GeV proton synchrotron Energy Technology Data Exchange (ETDEWEB) Koseki, Shoichiro; Zhang, Fengqing; Watanabe, Yasuhiro; Tani, Norio [Japan Atomic Energy Research Inst., Tokai, Ibaraki (Japan). Tokai Research Establishment; Adachi, Toshikazu; Someya, Hirohiko [High Energy Accelerator Research Organization, Tsukuba, Ibaraki (Japan) 2001-07-01 Exciting system for magnets of the 3 GeV Proton synchrotron is under consideration. A resonant exciting system is studied, and two type of power supply are compared. One is a parallel supply that is used generally. Another is a modified series supply. Either of them uses IGBT sinusoidal converters. Capacity of the power converter of the series supply for bending magnets becomes 28.8 MVAp. This is lager more than twice compared with the parallel supply. In the other hand, the series supply has good control performance and flexibility. More study is necessary to decide finally. (author) 11. Assessment of time-dependent density functional theory with the restricted excitation space approximation for excited state calculations of large systems Science.gov (United States) Hanson-Heine, Magnus W. D.; George, Michael W.; Besley, Nicholas A. 2018-06-01 The restricted excitation subspace approximation is explored as a basis to reduce the memory storage required in linear response time-dependent density functional theory (TDDFT) calculations within the Tamm-Dancoff approximation. It is shown that excluding the core orbitals and up to 70% of the virtual orbitals in the construction of the excitation subspace does not result in significant changes in computed UV/vis spectra for large molecules. The reduced size of the excitation subspace greatly reduces the size of the subspace vectors that need to be stored when using the Davidson procedure to determine the eigenvalues of the TDDFT equations. Furthermore, additional screening of the two-electron integrals in combination with a reduction in the size of the numerical integration grid used in the TDDFT calculation leads to significant computational savings. The use of these approximations represents a simple approach to extend TDDFT to the study of large systems and make the calculations increasingly tractable using modest computing resources. 12. Stand-alone front-end system for high- frequency, high-frame-rate coded excitation ultrasonic imaging. Science.gov (United States) Park, Jinhyoung; Hu, Changhong; Shung, K Kirk 2011-12-01 A stand-alone front-end system for high-frequency coded excitation imaging was implemented to achieve a wider dynamic range. The system included an arbitrary waveform amplifier, an arbitrary waveform generator, an analog receiver, a motor position interpreter, a motor controller and power supplies. The digitized arbitrary waveforms at a sampling rate of 150 MHz could be programmed and converted to an analog signal. The pulse was subsequently amplified to excite an ultrasound transducer, and the maximum output voltage level achieved was 120 V(pp). The bandwidth of the arbitrary waveform amplifier was from 1 to 70 MHz. The noise figure of the preamplifier was less than 7.7 dB and the bandwidth was 95 MHz. Phantoms and biological tissues were imaged at a frame rate as high as 68 frames per second (fps) to evaluate the performance of the system. During the measurement, 40-MHz lithium niobate (LiNbO(3)) single-element lightweight (<;0.28 g) transducers were utilized. The wire target measure- ment showed that the -6-dB axial resolution of a chirp-coded excitation was 50 μm and lateral resolution was 120 μm. The echo signal-to-noise ratios were found to be 54 and 65 dB for the short burst and coded excitation, respectively. The contrast resolution in a sphere phantom study was estimated to be 24 dB for the chirp-coded excitation and 15 dB for the short burst modes. In an in vivo study, zebrafish and mouse hearts were imaged. Boundaries of the zebrafish heart in the image could be differentiated because of the low-noise operation of the implemented system. In mouse heart images, valves and chambers could be readily visualized with the coded excitation. 13. Formation of excited states in high-Z helium-like systems International Nuclear Information System (INIS) Fritzsche, S.; Fricke, B.; Brinzanescu, O. 1999-12-01 High-Z helium-like ions represent the simplest multi-electron systems for studying the interplay between electron-electron correlations, relativistic as well as quantum electrodynamical effects in strong fields. In contrast to the adjacent lithium-like ions, however, almost no experimental information is available about the excited states in the high-Z domain of the helium sequence. Here, we present a theoretical analysis of the X-ray production and decay dynamics of the excited states in helium-like uranium. Emphasize has been paid particularly to the formation of the 3 P 0 and 3 P 2 levels by using electron capture into hydrogen-like U 91+ . Both states are of interest for precise measurements on high-Z helium-like ions in the future. (orig.) 14. Excitations in superfluid systems: contributions of the nuclear structure; Excitations dans les systemes superfluides: contributions de la structure nucleaire Energy Technology Data Exchange (ETDEWEB) Khan, E 2005-12-15 The author presents successively the theoretical aspect, the experimental aspect and the applied aspect of excitations in nuclear structures. The quasi-particle random phase approximation (QRPA) tool is first described. Recent approaches on QRPA are based on the theory of the density function where the ground state and excited states are described from the same nucleon-nucleon interaction. 2 methods for measuring the collective excitations are then presented: the proton scattering that has the potentiality to investigate the evolution of magicity, the second method is in fact a new method for measuring the giant mono-polar resonance (GMP) in exotic nuclei. Nuclear reactions are considered as a compulsory step on the way from observables like cross-sections to nuclear structure. The author highlights the assets of the convolution model that can generate the optical potential from the effective nucleon-nucleon interaction and from proton and neutron densities of the nuclei involved. R-processes in nucleosynthesis and neutron stars are reviewed as applications of collective excitations in the field of nuclear astrophysics. (A.C.) 15. Studies of isotopic effects in the excited electronic states of molecular systems International Nuclear Information System (INIS) 1982-01-01 Rare gas halogen (RGH) lasers serve as convenient tools for a range of photophysical processes which exhibit isotope effects. This document summarizes progress in the production of molecular systems in their electronic excited states with the aid of RGH lasers, and the various isotopic effects one can study under these conditions. We conclude that the basic physical mechanisms involved in the isotopically sensitive characteristics of excited molecular electronic states are sufficiently selective to be useful in both the detection and separation of many atomic materials 16. A High-Voltage SOI CMOS Exciter Chip for a Programmable Fluidic Processor System. Science.gov (United States) Current, K W; Yuk, K; McConaghy, C; Gascoyne, P R C; Schwartz, J A; Vykoukal, J V; Andrews, C 2007-06-01 A high-voltage (HV) integrated circuit has been demonstrated to transport fluidic droplet samples on programmable paths across the array of driving electrodes on its hydrophobically coated surface. This exciter chip is the engine for dielectrophoresis (DEP)-based micro-fluidic lab-on-a-chip systems, creating field excitations that inject and move fluidic droplets onto and about the manipulation surface. The architecture of this chip is expandable to arrays of N X N identical HV electrode driver circuits and electrodes. The exciter chip is programmable in several senses. The routes of multiple droplets may be set arbitrarily within the bounds of the electrode array. The electrode excitation waveform voltage amplitude, phase, and frequency may be adjusted based on the system configuration and the signal required to manipulate a particular fluid droplet composition. The voltage amplitude of the electrode excitation waveform can be set from the minimum logic level up to the maximum limit of the breakdown voltage of the fabrication technology. The frequency of the electrode excitation waveform can also be set independently of its voltage, up to a maximum depending upon the type of droplets that must be driven. The exciter chip can be coated and its oxide surface used as the droplet manipulation surface or it can be used with a top-mounted, enclosed fluidic chamber consisting of a variety of materials. The HV capability of the exciter chip allows the generated DEP forces to penetrate into the enclosed chamber region and an adjustable voltage amplitude can accommodate a variety of chamber floor thicknesses. This demonstration exciter chip has a 32 x 32 array of nominally 100 V electrode drivers that are individually programmable at each time point in the procedure to either of two phases: 0deg and 180deg with respect to the reference clock. For this demonstration chip, while operating the electrodes with a 100-V peak-to-peak periodic waveform, the maximum HV electrode 17. An optical authentication system based on imaging of excitation-selected lanthanide luminescence. Science.gov (United States) Carro-Temboury, Miguel R; Arppe, Riikka; Vosch, Tom; Sørensen, Thomas Just 2018-01-01 Secure data encryption relies heavily on one-way functions, and copy protection relies on features that are difficult to reproduce. We present an optical authentication system based on lanthanide luminescence from physical one-way functions or physical unclonable functions (PUFs). They cannot be reproduced and thus enable unbreakable encryption. Further, PUFs will prevent counterfeiting if tags with unique PUFs are grafted onto products. We have developed an authentication system that comprises a hardware reader, image analysis, and authentication software and physical keys that we demonstrate as an anticounterfeiting system. The physical keys are PUFs made from random patterns of taggants in polymer films on glass that can be imaged following selected excitation of particular lanthanide(III) ions doped into the individual taggants. This form of excitation-selected imaging ensures that by using at least two lanthanide(III) ion dopants, the random patterns cannot be copied, because the excitation selection will fail when using any other emitter. With the developed reader and software, the random patterns are read and digitized, which allows a digital pattern to be stored. This digital pattern or digital key can be used to authenticate the physical key in anticounterfeiting or to encrypt any message. The PUF key was produced with a staggering nominal encoding capacity of 7 3600 . Although the encoding capacity of the realized authentication system reduces to 6 × 10 104 , it is more than sufficient to completely preclude counterfeiting of products. 18. Excited species in the FBX dosimeter system International Nuclear Information System (INIS) Gupta, B.L. 2003-01-01 In the FBX dosimeter solution, the excitation of xylenol orange (XO) produces maximum emission at 550-575 nm both at room and liquid nitrogen temperatures (about 85%) having a lifetime of 0.20-0.36 ns. In addition, at room temperature there is an emission at 350 nm for the excitation at 260 nm (about 15%) having a longer lifetime of 3.71-4.01 ns. Benzoic acid (BA) has excitation at 284-295 nm and emission at 320-365 nm having a lifetime of 1.38 ns. In an aqueous solution containing 5x10 -3 mol dm -3 BA, 2x10 -4 mol dm -3 XO and 0.04 mol dm -3 H 2 SO 4 there is no XO emission at 550 nm due to UV absorption at 260 nm by BA. In this solution, 2 emissions are observed near 350-360 nm, having lifetimes of 1.25 ns (89%) and 2.86 ns (11%). The wavelengths for the emission of XO and absorption of ferric-XO complex are nearly the same. Excited XO produces oxidation of ferrous ions and BA increases the chain length 19. A new autogenous mobile system driven by vibration without impacts, excited by an impulse periodic force Directory of Open Access Journals (Sweden) Duong The-Hung 2018-01-01 Full Text Available This report describes a new proposed design for autogenous mobile systems which can move without any external mechanisms such as legs or wheels. A Duffing oscillator with a cubic spring, which is excited by an impulse periodic force, is utilized to drive the whole system. The rectilinear motion of the system is performed employing the periodically oscillation of the internal mass interacting without collisions with the main body. Utilizing the nonlinear restoring force of the cubic spring, the system can move in desired directions. When the ratio between the excitation force and the friction force is smaller than 2.5, backward or forward motion can be easily achieved by applying an excitation force in the same desired direction. Different from other vibro-impact drifting devices, no impact needed to drive the new proposed system. This novel structure allows to miniaturize the device as well as to simplify the control algorithm thus can significantly expand applicability of the proposed system. 20. Electronic-excitation energy transfer in heterogeneous dye solutions under laser excitation International Nuclear Information System (INIS) Levshin, L.V.; Mukushev, B.T.; Saletskii, A.M. 1995-01-01 An experimental study has been made of electronic-excitation energy transfer (EEET) among dye molecules of different types for different exciting-fight wavelengths and temperatures. Upon selective laser excitation of the donor, the inhomogeneous broadening of molecular levels increases the probability of EEET from the donor to acceptor molecules. The efficiency of this process is directly proportional to the acceptor molecule concentration and is temperature dependent. The EEET is accompanied by the spectral migration of energy among donor molecules, which reduces the fluorescence quantum efficiency of the donor. Increasing the frequency of the exciting light decreases in the donor fluorescence quantum efficiency. An increase in the acceptor molecule concentration results in a decrease of the spectral migration of excitation in the donor molecule system. 5 refs., 5 figs 1. Magnetic excitations and exchange interactions in the spin-gap system TlCuCl sub 3 CERN Document Server Oosawa, A; Kato, T; Kakurai, K; Müller, M; Mikeska, H J 2002-01-01 The magnetic excitations from the gapped ground state in TlCuCl sub 3 have been investigated by means of inelastic neutron scattering experiments. The excitation data were collected along four different directions in the a sup * -c sup * plane. A well-defined single magnetic excitation mode was observed. The lowest excitation occurs at Q=(h,0,l) with integer h and odd l, as observed in KCuCl sub 3. The dispersion relations were analyzed by the cluster-series expansion up to the sixth order, so that the individual exchange interactions were evaluated. It was demonstrated that TlCuCl sub 3 is a strongly coupled spin-dimer system. (orig.) 2. Some features of excited states density matrix calculation and their pairing relations in conjugated systems International Nuclear Information System (INIS) Giambiagi, M.S. de; Giambiagi, M. 1982-01-01 Direct PPP-type calculations of self-consistent (SC) density matrices for excited states are described and the corresponding 'thawn' molecular orbitals (MO) are discussed. Special attention is addressed to particular solutions arising in conjugated systems of a certain symmetry, and to their chemical implications. The U(2) and U(3) algebras are applied respectively to the 4-electron and 6-electron cases: a natural separation of excited states in different cases follows. A simple approach to the convergence problem for excited states is given. The complementarity relations, an alternative formulation of the pairing theorem valid for heteromolecules and non-alternant systems, allow some fruitful experimental applications. Together with the extended pairing relations shown here, they may help to rationalize general trends. (Author) [pt 3. Synchronisation and general dynamic symmetry of a vibrating system with two exciters rotating in opposite directions International Nuclear Information System (INIS) Chun-Yu, Zhao; Yi-Min, Zhang; Bang-Chun, Wen 2010-01-01 We derive the non-dimensional coupling equation of two exciters, including inertia coupling, stiffness coupling and load coupling. The concept of general dynamic symmetry is proposed to physically explain the synchronisation of the two exciters, which stems from the load coupling that produces the torque of general dynamic symmetry to force the phase difference between the two exciters close to the angle of general dynamic symmetry. The condition of implementing synchronisation is that the torque of general dynamic symmetry is greater than the asymmetric torque of the two motors. A general Lyapunov function is constructed to derive the stability condition of synchronisation that the non-dimensional inertia coupling matrix is positive definite and all its elements are positive. Numeric results show that the structure of the vibrating system can guarantee the stability of synchronisation of the two exciters, and that the greater the distances between the installation positions of the two exciters and the mass centre of the vibrating system are, the stronger the ability of general dynamic symmetry is 4. Synchronization of Two Non-Identical Coupled Exciters in a Non-Resonant Vibrating System of Linear Motion. Part II: Numeric Analysis Directory of Open Access Journals (Sweden) Chunyu Zhao 2009-01-01 Full Text Available The paper focuses on the quantitative analysis of the coupling dynamic characteristics of two non-identical exciters in a non-resonant vibrating system. The load torque of each motor consists of three items, including the torque of sine effect of phase angles, that of coupling sine effect and that of coupling cosine effect. The torque of frequency capture results from the torque of coupling cosine effect, which is equal to the product of the coupling kinetic energy, the coefficient of coupling cosine effect, and the sine of phase difference of two exciters. The motions of the system excited by two exciters in the same direction make phase difference close to π and that in opposite directions makes phase difference close to 0. Numerical results show that synchronous operation is stable when the dimensionless relative moments of inertia of two exciters are greater than zero and four times of their product is greater than the square of their coefficient of coupling cosine effect. The stability of the synchronous operation is only dependent on the structural parameters of the system, such as the mass ratios of two exciters to the vibrating system, and the ratio of the distance between an exciter and the centroid of the system to the equivalent radius of the system about its centroid. 5. Theoretical and Experimental Study on Synchronization of the Two Homodromy Exciters in a Non-Resonant Vibrating System Directory of Open Access Journals (Sweden) Xue-Liang Zhang 2013-01-01 Full Text Available In this paper we give some theoretical analyses and experimental results on synchronization of the two non-identical exciters. Using the average method of modified small parameters, the dimensionless coupling equation of the two exciters is deduced. The synchronization criterion for the two exciters is derived as the torque of frequency capture being equal to or greater than the absolute value of difference between the residual electromagnetic torques of the two motors. The stability criterion of synchronous state is verified to satisfy the Routh-Hurwitz criterion. The regions of implementing synchronization and that of stability of phase difference for the two exciters are manifested by numeric method. Synchronization of the two exciters stems from the coupling dynamic characteristic of the vibrating system having selecting motion, especially, under the condition that the parameters of system are complete symmetry, the torque of frequency capture stemming from the circular motion of the rigid frame drives the phase difference to approach PI and carry out the swing of the rigid frame; that from the swing of the rigid frame forces the phase difference to near zero and achieve the circular motion of the rigid frame. In the steady state, the motion of rigid frame will be one of three types: pure swing, pure circular motion, swing and circular motion coexistence. The numeric and experiment results derived thereof show that the two exciters can operate synchronously as long as the structural parameters of system satisfy the criterion of stability in the regions of frequency capture. In engineering, the distance between the centroid of the rigid frame and the rotational centre of exciter should be as far as possible. Only in this way, can the elliptical motion of system required in engineering be realized. 6. Four-nucleon system with Δ-isobar excitation International Nuclear Information System (INIS) Deltuva, A.; Fonseca, A.C.; Sauer, P.U. 2008-01-01 The four-nucleon bound state and scattering below three-body breakup threshold are described based on the realistic coupled-channel potential CD Bonn+Δ which allows the excitation of a single nucleon to a Δ isobar. The Coulomb repulsion between protons is included. In the four-nucleon system the two-baryon coupled-channel potential yields effective two-, three- and four-nucleon forces, mediated by the Δ isobar and consistent with each other and with the underlying two-nucleon force. The effect of the four-nucleon force on the studied observables is much smaller than the effect of the three-nucleon force. The inclusion of the Δ isobar is unable to resolve the existing discrepancies with the experimental data 7. Controllable excitation of higher-order rogue waves in nonautonomous systems with both varying linear and harmonic external potentials Science.gov (United States) Jia, Heping; Yang, Rongcao; Tian, Jinping; Zhang, Wenmei 2018-05-01 The nonautonomous nonlinear Schrödinger (NLS) equation with both varying linear and harmonic external potentials is investigated and the semirational rogue wave (RW) solution is presented by similarity transformation. Based on the solution, the interactions between Peregrine soliton and breathers, and the controllability of the semirational RWs in periodic distribution and exponential decreasing nonautonomous systems with both linear and harmonic potentials are studied. It is found that the harmonic potential only influences the constraint condition of the semirational solution, the linear potential is related to the trajectory of the semirational RWs, while dispersion and nonlinearity determine the excitation position of the higher-order RWs. The higher-order RWs can be partly, completely and biperiodically excited in periodic distribution system and the diverse excited patterns can be generated for different parameter relations in exponential decreasing system. The results reveal that the excitation of the higher-order RWs can be controlled in the nonautonomous system by choosing dispersion, nonlinearity and external potentials. 8. Simultaneous excitation system for efficient guided wave structural health monitoring Science.gov (United States) Hua, Jiadong; Michaels, Jennifer E.; Chen, Xin; Lin, Jing 2017-10-01 Many structural health monitoring systems utilize guided wave transducer arrays for defect detection and localization. Signals are usually acquired using the ;pitch-catch; method whereby each transducer is excited in turn and the response is received by the remaining transducers. When extensive signal averaging is performed, the data acquisition process can be quite time-consuming, especially for metallic components that require a low repetition rate to allow signals to die out. Such a long data acquisition time is particularly problematic if environmental and operational conditions are changing while data are being acquired. To reduce the total data acquisition time, proposed here is a methodology whereby multiple transmitters are simultaneously triggered, and each transmitter is driven with a unique excitation. The simultaneously transmitted waves are captured by one or more receivers, and their responses are processed by dispersion-compensated filtering to extract the response from each individual transmitter. The excitation sequences are constructed by concatenating a series of chirps whose start and stop frequencies are randomly selected from a specified range. The process is optimized using a Monte-Carlo approach to select sequences with impulse-like autocorrelations and relatively flat cross-correlations. The efficacy of the proposed methodology is evaluated by several metrics and is experimentally demonstrated with sparse array imaging of simulated damage. 9. Non-contact test set-up for aeroelasticity in a rotating turbomachine combining a novel acoustic excitation system with tip-timing International Nuclear Information System (INIS) Freund, O; Seume, J R; Montgomery, M; Mittelbach, M 2014-01-01 10. Electronic excited states and relaxation dynamics in polymer heterojunction systems Science.gov (United States) Ramon, John Glenn Santos The potential for using conducting polymers as the active material in optoelectronic devices has come to fruition in the past few years. Understanding the fundamental photophysics behind their operations points to the significant role played by the polymer interface in their performance. Current device architectures involve the use of bulk heterojunctions which intimately blend the donor and acceptor polymers to significantly increase not only their interfacial surface area but also the probability of exciton formation within the vicinity of the interface. In this dissertation, we detail the role played by the interface on the behavior and performance of bulk heterojunction systems. First, we explore the relation between the exciton binding energy to the band offset in determining device characteristics. As a general rule, when the exciton binding energy is greater than the band offset, the exciton remains the lowest energy excited state leading to efficient light-emitting properties. On the other hand, if the offset is greater than the binding energy, charge separation becomes favorable leading to better photovoltaic behavior. Here, we use a Wannier function, configuration interaction based approach to examine the essential excited states and predict the vibronic absorption and emission spectra of the PPV/BBL, TFB/F8BT and PFB/F8BT heterojunctions. Our results underscore the role of vibrational relaxation in the formation of charge-transfer states following photoexcitation. In addition, we look at the relaxation dynamics that occur upon photoexcitation. For this, we adopt the Marcus-Hush semiclassical method to account for lattice reorganization in the calculation of the interconversion rates in TFB/F8BT and PFB/F8BT. We find that, while a tightly bound charge-transfer state (exciplex) remains the lowest excited state, a regeneration pathway to the optically active lowest excitonic state in TFB/F8BT is possible via thermal repopulation from the exciplex. Finally 11. Effects of Isospin on Pre-scission Particle Multiplicity of Heavy Systems and Its Excitation Energy Dependence Institute of Scientific and Technical Information of China (English) YE Wei; CHEN Na 2004-01-01 Isospin effects on particle emission of fissioning isobaric sources 202Fr, 202po, 202Tl and isotopic sources 189,202,212Po, and its dependence on the excitation energy are studied via Smoluchowski equations. It is shown that with increasing the isospin of fissioning systems, charged-particle emission is not sensitive to the strength of nuclear dissipation. In addition, we have found that increasing the excitation energy not only increases the influence of nuclear dissipation on particle emission but also greatly enhances the sensitivity of the emission of pre-scission neutrons or charged particles to the isospin of the system. Therefore, in order to extract dissipation strength more accurately by taking light particle multiplicities it is important to choose both a highly excited compound nucleus and a proper kind of particles for systems with different isospins. 12. Excitation-scanning hyperspectral imaging system for microscopic and endoscopic applications Science.gov (United States) Mayes, Sam A.; Leavesley, Silas J.; Rich, Thomas C. 2016-04-01 Current microscopic and endoscopic technologies for cancer screening utilize white-light illumination sources. Hyper-spectral imaging has been shown to improve sensitivity while retaining specificity when compared to white-light imaging in both microscopy and in vivo imaging. However, hyperspectral imaging methods have historically suffered from slow acquisition times due to the narrow bandwidth of spectral filters. Often minutes are required to gather a full image stack. We have developed a novel approach called excitation-scanning hyperspectral imaging that provides 2-3 orders of magnitude increased signal strength. This reduces acquisition times significantly, allowing for live video acquisition. Here, we describe a preliminary prototype excitation-scanning hyperspectral imaging system that can be coupled with endoscopes or microscopes for hyperspectral imaging of tissues and cells. Our system is comprised of three subsystems: illumination, transmission, and imaging. The illumination subsystem employs light-emitting diode arrays to illuminate at different wavelengths. The transmission subsystem utilizes a unique geometry of optics and a liquid light guide. Software controls allow us to interface with and control the subsystems and components. Digital and analog signals are used to coordinate wavelength intensity, cycling and camera triggering. Testing of the system shows it can cycle 16 wavelengths at as fast as 1 ms per cycle. Additionally, more than 18% of the light transmits through the system. Our setup should allow for hyperspectral imaging of tissue and cells in real time. 13. Faraday waves under time-reversed excitation. Science.gov (United States) Pietschmann, Dirk; Stannarius, Ralf; Wagner, Christian; John, Thomas 2013-03-01 Do parametrically driven systems distinguish periodic excitations that are time mirrors of each other? Faraday waves in a Newtonian fluid are studied under excitation with superimposed harmonic wave forms. We demonstrate that the threshold parameters for the stability of the ground state are insensitive to a time inversion of the driving function. This is a peculiarity of some dynamic systems. The Faraday system shares this property with standard electroconvection in nematic liquid crystals [J. Heuer et al., Phys. Rev. E 78, 036218 (2008)]. In general, time inversion of the excitation affects the asymptotic stability of a parametrically driven system, even when it is described by linear ordinary differential equations. Obviously, the observed symmetry has to be attributed to the particular structure of the underlying differential equation system. The pattern selection of the Faraday waves above threshold, on the other hand, discriminates between time-mirrored excitation functions. 14. Generalization of the variational principle and the Hohenberg and Kohn theorems for excited states of Fermion systems Energy Technology Data Exchange (ETDEWEB) Gonis, A., E-mail: [email protected] 2017-01-05 Through the entanglement of a collection of K non-interacting replicas of a system of N interacting Fermions, and making use of the properties of reduced density matrices the variational principle and the theorems of Hohenberg and Kohn are generalized to excited states. The generalization of the variational principle makes use of the natural orbitals of an N-particle density matrix describing the state of lowest energy of the entangled state. The extension of the theorems of Hohenberg and Kohn is based on the ground-state formulation of density functional theory but with a new interpretation of the concept of a ground state: It is the state of lowest energy of a system of KN Fermions that is described in terms of the excited states of the N-particle interacting system. This straightforward implementation of the line of reasoning of ground-state density functional theory to a new domain leads to a unique and logically valid extension of the theory to excited states that allows the systematic treatment of all states in the spectrum of the Hamiltonian of an interacting system. - Highlights: • Use of entanglement in connection with the properties of density matrices. • An anti-symmetric entangled state of order KN expressed in terms of excited states of an interacting N-particle system. 15. Measurement of fusion excitation functions in the system {sup 78}Kr + {sup 100}Mo Energy Technology Data Exchange (ETDEWEB) Rehm, K.E.; Jiang, C.L.; Esbensen, H. [and others 1995-08-01 Earlier measurements of fusion reactions involving {sup 78}Kr and {sup 100}Mo projectiles and Ni-targets showed surprisingly large fusion yields at low energies which could not be explained by coupled-channels calculations. The main difference to similar measurements involving the neighboring {sup 86}Kr and {sup 92}Mo isotopes was the different slope of the excitation functions at sub-barrier energies. An analysis of a variety of experiments showed a correlation between the nuclear structure and the slope of the excitation functions, with the {open_quotes}soft{close_quotes} transitional nuclei ({sup 78}Kr, {sup 100}Mo) exhibiting shallower slopes than the {open_quotes}stiff{close_quotes} nuclei ({sup 86}Kr, {sup 92}Mo) measured at the same energies with respect to the barrier. In this experiment we studied the fusion excitation function involving two transitional nuclei {sup 78}Kr + {sup 100}Mo. The measurements were performed with {sup 78}Kr beams from the ECR source at energies between 285-370 MeV. Separation of the evaporation nucleus from the elastically scattered particles was achieved by measuring time-of-flight and magnetic rigidity in the gas-filled spectrograph. The data were completely analyzed. A comparison of the cross sections with measurements for the system {sup 86}Kr + {sup 92}Mo populating the same compound nucleus {sup 178}Pt. It shows good agreement at the highest energies, but quite different falloffs of the excitation functions toward lower energies. Coupled-channels calculations, including multi-phonon excitation for the two systems, are being performed. 16. Development of the system for excitation function automatic measurement of nuclear reactions International Nuclear Information System (INIS) Sapozhnikov, A.B. 2004-01-01 Full text: The resonance nuclear reaction method is applied at the tandem accelerator UKP-2-1 to determinate films thickness and obtain light element depth distribution. The system for automatic measurement of the nuclear reaction excitation curve has been developed. It allowed to obtain an excitation function of nuclear reaction using continuous changing potential of the target with energy step of 6 eV. Saw-tooth voltage with amplitude up to 6 kV from the block of scanning beam is fed to a target. The amplitude is determined by constant voltage from the scanning beam block control. Nal(Tl) detector detects gamma quanta - the products of a nuclear reaction and transforms they in voltage impulses. The impulses through the amplifier income in the single-channel analyzer which forms impulses to start the analog-to-digital converter. The value of saw-tooth voltage corresponding to the moment of gamma quantum detection is read by the analog-to-digital converter, where it is transformed to digital code and transmitted to the computer. The computer program has been developed to control the process of accumulation of excitation function. The dependence a detected γ-quanta yield from a target potential is automatically plotted by the program. This dependence corresponds to the nuclear reaction excitation function. If scanning amplitude is not enough in order to scan need depth of a sample, an operator increases energy of the proton beam changing high voltage potential of the terminal up 3 keV and measures the nuclear reaction excitation function with the new energy. This procedure can be repeated some times. After that 'sewing' of excitation functions is carried out by the program under the hypothesis that nuclear reaction yield in last points be identical 17. Role of vortex structures in excitation of self-oscillating combustion of condensed systems International Nuclear Information System (INIS) Samsonov, V.P.; Murunov, E.Yu.; Alekseev, M.V. 2008-01-01 One studied experimentally the effect of the free convection and the eddy structures occurring near the gasoline burner singing flame on the excitation conditions of thermal self-oscillations in a tube-resonator. One introduces a procedure to measure the gas column oscillation amplitude. The singing flame height and the flame mass speed at the excitation of the acoustic oscillations are revealed to reduce, while the gasoline burning efficiency is found to increase. By means of the digital photometry one studied the mechanisms of the singing flame temperature field changes within one oscillation period. One derived the hysteresis dependences of the amplitude of the acoustic oscillations on the gasoline diffusion flame thermal power. One brings to the notice a mechanism of the effect of the eddy structures of the excitation of the burning self-oscillation mode of the condensed systems [ru 18. Bifurcations, chaos and adaptive backstepping sliding mode control of a power system with excitation limitation Energy Technology Data Exchange (ETDEWEB) Min, Fuhong, E-mail: [email protected]; Wang, Yaoda; Peng, Guangya; Wang, Enrong [School of Electrical and Automation Engineering, Nanjing Normal University, Jiangsu, 210042 (China) 2016-08-15 The bifurcation and Lyapunov exponent for a single-machine-infinite bus system with excitation model are carried out by varying the mechanical power, generator damping factor and the exciter gain, from which periodic motions, chaos and the divergence of system are observed respectively. From given parameters and different initial conditions, the coexisting motions are developed in power system. The dynamic behaviors in power system may switch freely between the coexisting motions, which will bring huge security menace to protection operation. Especially, the angle divergences due to the break of stable chaotic oscillation are found which causes the instability of power system. Finally, a new adaptive backstepping sliding mode controller is designed which aims to eliminate the angle divergences and make the power system run in stable orbits. Numerical simulations are illustrated to verify the effectivity of the proposed method. 19. Bifurcations, chaos and adaptive backstepping sliding mode control of a power system with excitation limitation Directory of Open Access Journals (Sweden) Fuhong Min 2016-08-01 Full Text Available The bifurcation and Lyapunov exponent for a single-machine-infinite bus system with excitation model are carried out by varying the mechanical power, generator damping factor and the exciter gain, from which periodic motions, chaos and the divergence of system are observed respectively. From given parameters and different initial conditions, the coexisting motions are developed in power system. The dynamic behaviors in power system may switch freely between the coexisting motions, which will bring huge security menace to protection operation. Especially, the angle divergences due to the break of stable chaotic oscillation are found which causes the instability of power system. Finally, a new adaptive backstepping sliding mode controller is designed which aims to eliminate the angle divergences and make the power system run in stable orbits. Numerical simulations are illustrated to verify the effectivity of the proposed method. 20. Quantification of entanglement entropies for doubly excited resonance states in two-electron atomic systems International Nuclear Information System (INIS) Ho, Yew Kam; Lin, Chien-Hao 2015-01-01 In this work, we study the quantum entanglement for doubly excited resonance states in two-electron atomic systems such as the H - and Ps - ions and the He atom by using highly correlated Hylleraas type functions The resonance states are determined by calculation of density of resonance states with the stabilization method. The spatial (electron-electron orbital) entanglement entropies (linear and von Neumann) for the low-lying doubly excited states are quantified using the Schmidt-Slater decomposition method. (paper) 1. X-ray excited optical luminescence studies on the system BaXY (X ... Home; Journals; Pramana – Journal of Physics; Volume 65; Issue 2. X-ray excited optical luminescence studies on the system Ba (, =F, Cl, Br, I) ... India; Department of Chemical Engineering, National Taiwan University, Republic of China ... Proceedings of the International Workshop/Conference on Computational ... 2. Hysteresis-induced bifurcation and chaos in a magneto-rheological suspension system under external excitation International Nuclear Information System (INIS) Zhang Hailong; Zhang Ning; Wang Enrong; Min Fuhong 2016-01-01 The magneto-rheological damper (MRD) is a promising device used in vehicle semi-active suspension systems, for its continuous adjustable damping output. However, the innate nonlinear hysteresis characteristic of MRD may cause the nonlinear behaviors. In this work, a two-degree-of-freedom (2-DOF) MR suspension system was established first, by employing the modified Bouc–Wen force–velocity (F–v) hysteretic model. The nonlinear dynamic response of the system was investigated under the external excitation of single-frequency harmonic and bandwidth-limited stochastic road surface. The largest Lyapunov exponent (LLE) was used to detect the chaotic area of the frequency and amplitude of harmonic excitation, and the bifurcation diagrams, time histories, phase portraits, and power spectrum density (PSD) diagrams were used to reveal the dynamic evolution process in detail. Moreover, the LLE and Kolmogorov entropy (K entropy) were used to identify whether the system response was random or chaotic under stochastic road surface. The results demonstrated that the complex dynamical behaviors occur under different external excitation conditions. The oscillating mechanism of alternating periodic oscillations, quasi-periodic oscillations, and chaotic oscillations was observed in detail. The chaotic regions revealed that chaotic motions may appear in conditions of mid-low frequency and large amplitude, as well as small amplitude and all frequency. The obtained parameter regions where the chaotic motions may appear are useful for design of structural parameters of the vibration isolation, and the optimization of control strategy for MR suspension system. (paper) 3. Isovector excitations in charge independent systems International Nuclear Information System (INIS) Menezes, D.P. 1986-01-01 A method for building states with good isospin, from states given by the action of an isovector excitation operator on states of the parent multiplet is developed. This new method is a generalization of Toki's method and is applicable to cases involving any isovector excitation operator and a parent state, which is not a double magic nucleus. Once obtained these states with well defined isospin, it is shown how to do a Tamm-Dancoff calculation for determining the energy levels. The transition matrix elements of an isotensor operator are also calculated. An application of this formalism to the Gamow-Teller transition strength in 90 Zr is studied. In this case, besides the double magic configuration, the 2 particles - 2 holes (Π1g 9/2 ) 2 (υ 2p 1/2 -1 ) 2 configuration is also considered. (author) [pt 4. Excitation dynamics and relaxation in a molecular heterodimer International Nuclear Information System (INIS) Balevičius, V.; Gelzinis, A.; Abramavicius, D.; Mančal, T.; Valkunas, L. 2012-01-01 Highlights: ► Dynamics of excitation within a heterogenous molecular dimer. ► Excited states can be swapped due to different reorganization energies of monomers. ► Conventional excitonic basis becomes renormalized due to interaction with the bath. ► Relaxation is independent of mutual positioning of monomeric excited states. -- Abstract: The exciton dynamics in a molecular heterodimer is studied as a function of differences in excitation and reorganization energies, asymmetry in transition dipole moments and excited state lifetimes. The heterodimer is composed of two molecules modeled as two-level systems coupled by the resonance interaction. The system-bath coupling is taken into account as a modulating factor of the molecular excitation energy gap, while the relaxation to the ground state is treated phenomenologically. Comparison of the description of the excitation dynamics modeled using either the Redfield equations (secular and full forms) or the Hierarchical quantum master equation (HQME) is demonstrated and discussed. Possible role of the dimer as an excitation quenching center in photosynthesis self-regulation is discussed. It is concluded that the system-bath interaction rather than the excitonic effect determines the excitation quenching ability of such a dimer. 5. New excitation equipment for 220 MW generators in Kozloduy NPP International Nuclear Information System (INIS) Tomerlin, D. 2001-01-01 Rehabilitation on the excitation equipment for Generator 5, Reactor Unit 3, in Kozloduy NPP was completed in November 2000. ABB's Static Excitation System based on UNITROL 5000 technology has been chosen by the Bulgarian National Utility and Kozloduy NPP to substitute the original Russian excitation system equipment with electro-magnetic voltage regulators. The substitution is in a rehabilitation package of four excitation system equipment for Generator 5 and 6 of Reactor Unit 3 and Generator 7 and 8 of Reactor Unit 4 after a short overview of the original excitation system this paper describes the new Static Excitation System UNITROL 5000 including configuration with block diagram, its main features and merits such as modes of operation, limiter, special control functions and diagnostic facilities. Furthermore, new facilities, which are implemented in UNITROL 5000, such as dynamic current distribution among the thyristors working in parallel as well as the start-up from the residual magnetism are mentioned. Special functions including a so-called free-running mode of operation and automatic change over sequence from new excitation system to the stand-by excitation system, which is DC exciter machine, are described. Some records of the transient responses performed during the commissioning and a photograph of a manufactured system are provided. (author) 6. Optical studies of multiply excited states International Nuclear Information System (INIS) Mannervik, S. 1989-01-01 Optical studies of multiply-excited states are reviewed with emphasis on emission spectroscopy. From optical measurements, properties such as excitation energies, lifetimes and autoionization widths can be determined with high accuracy, which constitutes a challenge for modern computational methods. This article mainly covers work on two-, three- and four-electron systems, but also sodium-like quartet systems. Furthermore, some comments are given on bound multiply-excited states in negative ions. Fine structure effects on transition wavelengths and lifetimes (autoionization) are discussed. In particular, the most recent experimental and theoretical studies of multiply-excited states are covered. Some remaining problems, which require further attention, are discussed in more detail. (orig.) With 228 refs 7. Oxygen auroral transition laser system excited by collisional and photolytic energy transfer International Nuclear Information System (INIS) Murray, J.R.; Powell, H.T.; Rhodes, C.K. 1975-06-01 The properties of laser media involving the auroral transition of atomic oxygen and analogous systems are examined. A discussion of the atomic properties, collisional mechanisms, excitation processes, and collisionally induced radiative phenomena is given. Crossing phenomena play a particularly important role in governing the dynamics of the medium 8. Early warning signal for interior crises in excitable systems. Science.gov (United States) Karnatak, Rajat; Kantz, Holger; Bialonski, Stephan 2017-10-01 The ability to reliably predict critical transitions in dynamical systems is a long-standing goal of diverse scientific communities. Previous work focused on early warning signals related to local bifurcations (critical slowing down) and nonbifurcation-type transitions. We extend this toolbox and report on a characteristic scaling behavior (critical attractor growth) which is indicative of an impending global bifurcation, an interior crisis in excitable systems. We demonstrate our early warning signal in a conceptual climate model as well as in a model of coupled neurons known to exhibit extreme events. We observed critical attractor growth prior to interior crises of chaotic as well as strange-nonchaotic attractors. These observations promise to extend the classes of transitions that can be predicted via early warning signals. 9. Effect of power frequency excitation character on ferroresonance in neutral-grounded system International Nuclear Information System (INIS) Hui Meng; Liu Chong-Xin 2010-01-01 In most earlier ferroresonance studies the traditional excitation characteristic of iron core, in which the traditional excitation characteristic contains harmonic voltages or currents, has been used as if it were made up of pure fundamental voltage or current. However, this is not always true. In comparison with traditional excitation characteristics, this paper introduces the power frequency excitation characteristic of the iron core, which contains no harmonics. The power frequency excitation characteristic of iron core has been obtained by Elector Magnetic Transient Program, resulting in discrete voltage and current pairs. Extensive simulations are carried out to analyse the effect of power frequency excitation characteristic on potential transformer ferroresonance. A detailed analysis of simulation results demonstrates that with power frequency excitation characteristic of iron core inclusion at certain excitation voltage the ferroresonance may happen, conversely it may not happen with traditional excitation characteristic inclusion. (general) 10. Photo-redox activated drug delivery systems operating under two photon excitation in the near-IR. Science.gov (United States) Guardado-Alvarez, Tania M; Devi, Lekshmi Sudha; Vabre, Jean-Marie; Pecorelli, Travis A; Schwartz, Benjamin J; Durand, Jean-Olivier; Mongin, Olivier; Blanchard-Desce, Mireille; Zink, Jeffrey I 2014-05-07 We report the design and synthesis of a nano-container consisting of mesoporous silica nanoparticles with the pore openings covered by "snap-top" caps that are opened by near-IR light. A photo transducer molecule that is a reducing agent in an excited electronic state is covalently attached to the system. Near IR two-photon excitation causes inter-molecular electron transfer that reduces a disulfide bond holding the cap in place, thus allowing the cargo molecules to escape. We describe the operation of the "snap-top" release mechanism by both one- and two-photon activation. This system presents a proof of concept of a near-IR photoredox-induced nanoparticle delivery system that may lead to a new type of photodynamic drug release therapy. 11. Effect of Various Excitation Conditions on Vibrational Energy in a Multi-Degree-of-Freedom Torsional System with Piecewise-Type Nonlinearities Directory of Open Access Journals (Sweden) Jong-Yun Yoon 2015-09-01 Full Text Available Dynamic behaviors in practical driveline systems for wind turbines or vehicles are inherently affected by multiple nonlinearities such as piecewise-type torsional springs. However, various excitation conditions with different levels of magnitudes also show strong relationships to the dynamic behaviors when system responses are examined in both frequency and time domains. This study investigated the nonlinear responses of torsional systems under various excitations by using the harmonic balance method and numerical analysis. In order to understand the effect of piecewise-type nonlinearities on vibrational energy with different excitations, the nonlinear responses were investigated with various comparisons. First, two different jumping phenomena with frequency up- and down-sweeping conditions were determined under severe excitation levels. Second, practical system analysis using the phase plane and Poincaré map was conducted in various ways. When the system responses were composed of quasi-periodic components, Poincaré map analysis clearly revealed the nonlinear dynamic characteristics and thus it is suggested to investigate complicated nonlinear dynamic responses in practical driveline systems. 12. Remodelling of cellular excitation (reaction) and intercellular coupling (diffusion) by chronic atrial fibrillation represented by a reaction-diffusion system Science.gov (United States) Zhang, Henggui; Garratt, Clifford J.; Kharche, Sanjay; Holden, Arun V. 2009-06-01 Human atrial tissue is an excitable system, in which myocytes are excitable elements, and cell-to-cell electrotonic interactions are via diffusive interactions of cell membrane potentials. We developed a family of excitable system models for human atrium at cellular, tissue and anatomical levels for both normal and chronic atrial fibrillation (AF) conditions. The effects of AF-induced remodelling of cell membrane ionic channels (reaction kinetics) and intercellular gap junctional coupling (diffusion) on atrial excitability, conduction of excitation waves and dynamics of re-entrant excitation waves are quantified. Both ionic channel and gap junctional coupling remodelling have rate dependent effects on atrial propagation. Membrane channel conductance remodelling allows the propagation of activity at higher rates than those sustained in normal tissue or in tissue with gap junctional remodelling alone. Membrane channel conductance remodelling is essential for the propagation of activity at rates higher than 300/min as seen in AF. Spatially heterogeneous gap junction coupling remodelling increased the risk of conduction block, an essential factor for the genesis of re-entry. In 2D and 3D anatomical models, the dynamical behaviours of re-entrant excitation waves are also altered by membrane channel modelling. This study provides insights to understand the pro-arrhythmic effects of AF-induced reaction and diffusion remodelling in atrial tissue. 13. Synchronization of uncoupled excitable systems induced by white and coloured noise International Nuclear Information System (INIS) Zambrano, Samuel; Marino, Ines P; Seoane, Jesus M; Sanjuan, Miguel A F; Euzzor, Stefano; Geltrude, Andrea; Meucci, Riccardo; Arecchi, Fortunato T 2010-01-01 We study, both numerically and experimentally, the synchronization of uncoupled excitable systems due to a common noise. We consider two identical FitzHugh-Nagumo systems, which display both spiking and non-spiking behaviours in chaotic or periodic regimes. An electronic circuit provides a laboratory implementation of these dynamics. Synchronization is tested with both white and coloured noise, showing that coloured noise is more effective in inducing synchronization of the systems. We also study the effects on the synchronization of parameter mismatch and of the presence of intrinsic (not common) noise, and we conclude that the best performance of coloured noise is robust under these distortions. 14. Comparison of sensitivities and detection limits between direct excitation and secondary excitation modes in energy dispersive x-ray fluorescence analysis International Nuclear Information System (INIS) Artz, B.E.; Short, M.A. 1976-01-01 A comparison was made between the direct tube excitation mode and the secondary target excitation mode using a Kevex 0810 energy dispersive x-ray fluorescence system. Relative sensitivities and detection limits were determined with two system configurations. The first configuration used a standard, high power, x-ray fluorescence tube to directly excite the specimen. Several x-ray tubes, including chromium, molybdenum, and tungsten, both filtered and not filtered, were employed. The second configuration consisted of using the x-ray tube to excite a secondary target which in turn excited the specimen. Appropriate targets were compared to the direct excitation results. Relative sensitivities and detection limits were determined for K-series lines for elements from magnesium to barium contained in a low atomic number matrix and in a high atomic number matrix 15. Electron excitation of alkali atoms International Nuclear Information System (INIS) Ormonde, S. 1979-02-01 The development and testing of a synthesized close-coupling effective model potential ten-channel electron-atom scattering code and some preliminary calculations of resonances in cross sections for the excitation of excited states of potassium by low energy electrons are described. The main results obtained are: identification of 1 S and 1 D structures in excitation cross sections below the 5 2 S threshold of neutral potassium; indications of additional structures - 1 P and 1 D between the 5 2 S and 5 2 D thresholds; and a suggested explanation of anomalously high interstate-electron impact excitation cross sections inferred from experiments on potassium-seeded plasmas. The effective potential model imbedded in the code can be used to simulate any atomic system that can be approximated by a single bound electron outside an ionic core. All that is needed is a set of effective potential parameters--experimental or theoretical. With minor modifications the code could be adapted to calculations of electron scattering by two-electron systems 16. Critical Assessment of TD-DFT for Excited States of Open-Shell Systems: I. Doublet-Doublet Transitions. Science.gov (United States) Li, Zhendong; Liu, Wenjian 2016-01-12 A benchmark set of 11 small radicals is set up to assess the performance of time-dependent density functional theory (TD-DFT) for the excited states of open-shell systems. Both the unrestricted (U-TD-DFT) and spin-adapted (X-TD-DFT) formulations of TD-DFT are considered. For comparison, the well-established EOM-CCSD (equation-of-motion coupled-cluster with singles and doubles) is also used. In total, 111 low-lying singly excited doublet states are accessed by all the three approaches. Taking the MRCISD+Q (multireference configuration interaction with singles and doubles plus the Davidson correction) results as the benchmark, it is found that both U-TD-DFT and EOM-CCSD perform well for those states dominated by singlet-coupled single excitations (SCSE) from closed-shell to open-shell, open-shell to vacant-shell, or closed-shell to vacant-shell orbitals. However, for those states dominated by triplet-coupled single excitations (TCSE) from closed-shell to vacant-shell orbitals, both U-TD-DFT and EOM-CCSD fail miserably due to severe spin contaminations. In contrast, X-TD-DFT provides balanced descriptions of both SCSE and TCSE. As far as the functional dependence is concerned, it is found that, when the Hartree-Fock ground state does not suffer from the instability problem, both global hybrid (GH) and range-separated hybrid (RSH) functionals perform grossly better than pure density functionals, especially for Rydberg and charge-transfer excitations. However, if the Hartree-Fock ground state is instable or nearly instable, GH and RSH tend to underestimate severely the excitation energies. The SAOP (statistically averaging of model orbital potentials) performs more uniformly than any other density functionals, although it generally overestimates the excitation energies of valence excitations. Not surprisingly, both EOM-CCSD and adiabatic TD-DFT are incapable of describing excited states with substantial double excitation characters. 17. Nonlinear Dynamical Analysis for the Cable Excited with Parametric and Forced Excitation Directory of Open Access Journals (Sweden) C. Z. Qian 2014-01-01 Full Text Available Considering the deck vibration effect on the cable in cable-stayed bridge, using nonlinear structure dynamics theory, the nonlinear dynamical equation for the stayed cable excited with deck vibration is proposed. Research shows that the vertical vibration of the deck has a combined parametric and forced excitation effect on the cable when the angle of the cable is taken into consideration. Using multiscale method, the 1/2 principle parametric resonance is studied and the bifurcation equation is obtained. Despite the parameters analysis, the bifurcation characters of the dynamical system are studied. At last, by means of numerical method and software MATHMATIC, the effect rules of system parameters to the dynamical behavior of the system are studied, and some useful conclusions are obtained. 18. Characterizing Plasmonic Excitations of Quasi-2D Chains Science.gov (United States) Townsend, Emily; Bryant, Garnett A quantum description of the optical response of nanostructures and other atomic-scale systems is desirable for modeling systems that use plasmons for quantum information transfer, or coherent transport and interference of quantum states, as well as systems small enough for electron tunneling or quantum confinement to affect the electronic states of the system. Such a quantum description is complicated by the fact that collective and single-particle excitations can have similar energies and thus will mix. We seek to better understand the excitations of nanosystems to identify which characteristics of the excitations are most relevant to modeling their behavior. In this work we use a quasi 2-dimensional linear atomic chain as a model system, and exact diagonalization of the many-body Hamiltonian to obtain its excitations. We compare this to previous work in 1-d chains which used a combination of criteria involving a many-body state's transfer dipole moment, balance, transfer charge, dynamical response, and induced-charge distribution to identify which excitations are plasmonic in character. 19. Synaptic control of motoneuronal excitability DEFF Research Database (Denmark) Rekling, J C; Funk, G D; Bayliss, D A 2000-01-01 important in understanding the transformation of neural activity to motor behavior. Here, we review recent studies on the control of motoneuronal excitability, focusing on synaptic and cellular properties. We first present a background description of motoneurons: their development, anatomical organization......, and membrane properties, both passive and active. We then describe the general anatomical organization of synaptic input to motoneurons, followed by a description of the major transmitter systems that affect motoneuronal excitability, including ligands, receptor distribution, pre- and postsynaptic actions...... and norepinephrine, and neuropeptides, as well as the glutamate and GABA acting at metabotropic receptors, modulate motoneuronal excitability through pre- and postsynaptic actions. Acting principally via second messenger systems, their actions converge on common effectors, e.g., leak K(+) current, cationic inward... Science.gov (United States) Johns, C. E. 1987-01-01 The development of an X-band exciter, for use in the X-Band Uplink Subsystem, was completed. The exciter generates the drive signal for the X-band transmitter and also generates coherent test signals for the S- and X-band Block 3 translator and a Doppler reference signal for the Doppler extractor system. In addition to the above, the exciter generates other reference signals that are described. Also presented is an overview of the exciter design and some test data taken on the prototype. A brief discussion of the Block 3 Doppler extractor is presented. 1. ANALYSIS OF THE PROCESSES IN AN INDUCTOR SYSTEM WITH AN ATTRACTING SCREEN EXCITED BY THE EXTERNAL CIRCULAR SOLENOID Directory of Open Access Journals (Sweden) E.A. Chaplygin 2015-12-01 Full Text Available Introduction. Developments in the field of magnetic-pulse treatment of metals (MPTM are increasingly used in the modern technologies of production and repair of the aviation, automotive and other machinery, as they are environmentally friendly and energy-efficient in comparison with classical approaches. One of the main components of the device MPTM is a tool – inductor or the inductor system with an attractive screen (ISAS. The calculated dependences to calculate the inductor system with an attractive screen were taken from previous works. The ratios were obtained for the low-frequency mode of the excited fields, when is place their significant penetration through a thin-walled metal screen and a deformed workpiece. As it was shown earlier this mode is the most efficient from point of view of a force action on the object of a processing. Purpose. The theoretical analysis of the spatial-temporal distributions of the induced currents and forces of an attraction in the inductor system with an attractive screen excited by a flat circular solenoid located on the outside of the auxiliary screen. Methodology. The calculations are shown that the induced currents both in the screen and the workpiece are unidirectional and their interaction, in accordance with the law of Ampere determines the amplitudes of excited forces of attraction. Let’s note the effective validity of the considered inductor system excited by an external circular solenoid. With sufficient simplicity of the design take place rather high values of the developed forces of attraction and their averages. Results. Physically, a higher power efficiency of the system with an «external» coil in comparison with a system where coil is located in the internal cavity, can be accounted for lade of «failure» in the radial distribution of the excited forces. This «failure» in the design with a coil between the sheet metal is caused by its screening action against the forces of attraction 2. The dynamic behaviour of a non-stationary elevator compensating rope system under harmonic and stochastic excitations Energy Technology Data Exchange (ETDEWEB) Kaczmarczyk, S [School of Applied Sciences, University of Northampton, St. George' s Avenue, Northampton NN2 6JD (United Kingdom); Iwankiewicz, R [Institute of Mechanics and Ocean Engineering, Hamburg University of Technology, Eissendorfer Strasse 42 D-21073, Hamburg (Germany); Terumichi, Y, E-mail: [email protected] [Faculty of Science and Technology, Sophia University, 7-1 KIOI-CHO, CHIYODAKU, Tokyo, 102-8554 Japan (Japan) 2009-08-01 Moving slender elastic elements such as ropes, cables and belts are pivotal components of vertical transportation systems such as traction elevators. Their lengths vary within the host building structure during the elevator operation which results in the change of the mass and stiffness characteristics of the system. The structure of modern high-rise buildings is flexible and when subjected to loads due to strong winds and earthquakes it vibrates at low frequencies. The inertial load induced by the building motion excites the flexible components of the elevator system. The compensating ropes due to their lower tension are particularly affected and undergo large dynamic deformations. The paper focuses on the presentation of the non-stationary model of a building-compensating rope system and on the analysis to predict its dynamic response. The excitation mechanism is represented by a harmonic process and the results of computer simulations to predict transient resonance response are presented. The analysis of the simulation results leads to recommendations concerning the selection of the weight of the compensation assembly to minimize the effects of an adverse dynamic response of the system. The scenario when the excitation is represented as a narrow-band stochastic process with the state vector governed by stochastic equations is then discussed and the stochastic differential equations governing the second-order statistical moments of the state vector are developed. 3. Lattice relaxation theory of localized excitations in quasi-one-dimensional systems International Nuclear Information System (INIS) Wang Chuilin; Su Zhaobin; Yu Lu. 1993-04-01 The lattice relaxation theory developed earlier by Su and Yu for solitons and polarons in conducting polymers is applied to systems with both electron-phonon and electron-electron interactions, described by a single band Peierls-Hubbard model. The localized excitations in the competing bond-order-wave (BOW), charge-density-wave (CDW) and spin-density-wave (SDW) systems show interesting new features in their dynamics. In particular, a non-monotonic dependence of the relaxation rate on the coupling strength is predicted from the theory. The possible connection of this effect with photo-luminescence experiments is discussed. Similar phenomena may occur in other quasi-one-dimensional systems as well. (author). 21 refs, 4 figs 4. Charge transfer excitations from excited state Hartree-Fock subsequent minimization scheme International Nuclear Information System (INIS) Theophilou, Iris; Tassi, M.; Thanos, S. 2014-01-01 Photoinduced charge-transfer processes play a key role for novel photovoltaic phenomena and devices. Thus, the development of ab initio methods that allow for an accurate and computationally inexpensive treatment of charge-transfer excitations is a topic that nowadays attracts a lot of scientific attention. In this paper we extend an approach recently introduced for the description of single and double excitations [M. Tassi, I. Theophilou, and S. Thanos, Int. J. Quantum Chem. 113, 690 (2013); M. Tassi, I. Theophilou, and S. Thanos, J. Chem. Phys. 138, 124107 (2013)] to allow for the description of intermolecular charge-transfer excitations. We describe an excitation where an electron is transferred from a donor system to an acceptor one, keeping the excited state orthogonal to the ground state and avoiding variational collapse. These conditions are achieved by decomposing the space spanned by the Hartree-Fock (HF) ground state orbitals into four subspaces: The subspace spanned by the occupied orbitals that are localized in the region of the donor molecule, the corresponding for the acceptor ones and two more subspaces containing the virtual orbitals that are localized in the neighborhood of the donor and the acceptor, respectively. Next, we create a Slater determinant with a hole in the subspace of occupied orbitals of the donor and a particle in the virtual subspace of the acceptor. Subsequently we optimize both the hole and the particle by minimizing the HF energy functional in the corresponding subspaces. Finally, we test our approach by calculating the lowest charge-transfer excitation energies for a set of tetracyanoethylene-hydrocarbon complexes that have been used earlier as a test set for such kind of excitations 5. Time-dependent theory of Raman scattering for systems with several excited electronic states: Application to a H+3 model system International Nuclear Information System (INIS) Heather, R.; Metiu, H. 1989-01-01 The time-dependent formulation of Raman scattering theory is used to study how nonadiabatic interactions affect the Raman spectrum of a model H + 3 system, which has two excited electronic states. We start with a formula derived by Heller which gives the Raman scattering cross section as the Fourier transform (over time) of a time-dependent overlap integral. The latter is calculated with a method proposed by Fleck, Morris, and Feit, and extended to curve crossing by Alvarellos and Metiu. In performing these calculations we are especially interested in displaying effects typical of systems having more than one upper state. If the incident laser populates two electronic states there are several ways (i.e., excite to state one and emit from state two, excite to state one, and emit from state one, etc.) by which the Raman process can reach a given final state, and this leads to quantum interference. This interference is manifested in the Raman cross section as approximate selection rules controlling which final states can be reached through the Raman process. These selection rules depend on the relative orientation of the transition dipoles that radiatively couple the ground electronic state with the excited electronic states. The magnitude of the nonadiabatic contribution to the Raman emission, e.g., the contribution from absorbing to state one and emitting from state two, can be determined from the polarization dependence of the Raman emission if the transition dipoles have neither parallel nor antiparallel relative orientation 6. Quinary excitation method for pulse compression ultrasound measurements. Science.gov (United States) Cowell, D M J; Freear, S 2008-04-01 A novel switched excitation method for linear frequency modulated excitation of ultrasonic transducers in pulse compression systems is presented that is simple to realise, yet provides reduced signal sidelobes at the output of the matched filter compared to bipolar pseudo-chirp excitation. Pulse compression signal sidelobes are reduced through the use of simple amplitude tapering at the beginning and end of the excitation duration. Amplitude tapering using switched excitation is realised through the use of intermediate voltage switching levels, half that of the main excitation voltages. In total five excitation voltages are used creating a quinary excitation system. The absence of analogue signal generation and power amplifiers renders the excitation method attractive for applications with requirements such as a high channel count or low cost per channel. A systematic study of switched linear frequency modulated excitation methods with simulated and laboratory based experimental verification is presented for 2.25 MHz non-destructive testing immersion transducers. The signal to sidelobe noise level of compressed waveforms generated using quinary and bipolar pseudo-chirp excitation are investigated for transmission through a 0.5m water and kaolin slurry channel. Quinary linear frequency modulated excitation consistently reduces signal sidelobe power compared to bipolar excitation methods. Experimental results for transmission between two 2.25 MHz transducers separated by a 0.5m channel of water and 5% kaolin suspension shows improvements in signal to sidelobe noise power in the order of 7-8 dB. The reported quinary switched method for linear frequency modulated excitation provides improved performance compared to pseudo-chirp excitation without the need for high performance excitation amplifiers. 7. On One Means of Hard Excitation of Oscillations in Nonlinear Flutter Systems Directory of Open Access Journals (Sweden) S. D. Glyzin 2014-01-01 Full Text Available Considered are so-called finite-dimensional flutter systems, i.e. systems of ordinary differential equations, arising from Galerkin approximations of certain boundary value problems of aeroelasticity theory as well as from a number of radiophysics applications. We study small oscillations of these equations in case of 1 : 3 resonance. By combining analytical and numerical methods, it is concluded that the mentioned resonance can cause a hard excitation of oscillations. Namely, for flutter systems shown is the possibility of coexistence, along with the stable zero state, of stable invariant tori of arbitrary finite dimension as well as chaotic attractors. 8. Excitation methods for energy dispersive analysis International Nuclear Information System (INIS) Jaklevic, J.M. 1976-01-01 The rapid development in recent years of energy dispersive x-ray fluorescence analysis has been based primarily on improvements in semiconductor detector x-ray spectrometers. However, the whole analysis system performance is critically dependent on the availability of optimum methods of excitation for the characteristic x rays in specimens. A number of analysis facilities based on various methods of excitation have been developed over the past few years. A discussion is given of the features of various excitation methods including charged particles, monochromatic photons, and broad-energy band photons. The effects of the excitation method on background and sensitivity are discussed from both theoretical and experimental viewpoints. Recent developments such as pulsed excitation and polarized photons are also discussed 9. RESONANT POST-NEWTONIAN ECCENTRICITY EXCITATION IN HIERARCHICAL THREE-BODY SYSTEMS Energy Technology Data Exchange (ETDEWEB) Naoz, Smadar; Kocsis, Bence; Loeb, Abraham [Institute for Theory and Computation, Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 (United States); Yunes, Nicolas, E-mail: [email protected] [Department of Physics, Montana State University, Bozeman, MT 59718 (United States) 2013-08-20 We study the secular, hierarchical three-body problem to first-order in a post-Newtonian expansion of general relativity (GR). We expand the first-order post-Newtonian Hamiltonian to leading-order in the ratio of the semi-major axis of the two orbits. In addition to the well-known terms that correspond to the GR precession of the inner and outer orbits, we find a new secular post-Newtonian interaction term that can affect the long-term evolution of the triple. We explore the parameter space for highly inclined and eccentric systems, where the Kozai-Lidov mechanism can produce large-amplitude oscillations in the eccentricities. The standard lore, i.e., that GR effects suppress eccentricity, is only consistent with the parts of phase space where the GR timescales are several orders of magnitude shorter than the secular Newtonian one. In other parts of phase space, however, post-Newtonian corrections combined with the three-body ones can excite eccentricities. In particular, for systems where the GR timescale is comparable to the secular Newtonian timescales, the three-body interactions give rise to a resonant-like eccentricity excitation. Furthermore, for triples with a comparable-mass inner binary, where the eccentric Kozai-Lidov mechanism is suppressed, post-Newtonian corrections can further increase the eccentricity and lead to orbital flips even when the timescale of the former is much longer than the timescale of the secular Kozai-Lidov quadrupole perturbations. 10. Symmetry characterization of electrons and lattice excitations Directory of Open Access Journals (Sweden) Schober H. 2012-03-01 Full Text Available Symmetry concerns all aspects of a physical system from the electronic orbitals to structural and magnetic excitations. In this article we will try to elaborate the fundamental connection between symmetry and excitations. As excitations are manyfold in physical systems it is impossible to treat them exhaustively. We thus concentrate on the two topics of Bloch electrons and phonons. These two examples are complementary in the sense that Bloch electrons describe single particles in an external periodic potential while phonons exemplify a decoupled system of interacting particles. The way we develop the argument gives as by-product a short account of molecular orbitals and molecular vibrations. 11. Topological excitations in magnetic materials Energy Technology Data Exchange (ETDEWEB) Bazeia, D., E-mail: [email protected] [Departamento de Física, Universidade Federal da Paraíba, 58051-970 João Pessoa, PB (Brazil); Doria, M.M. [Instituto de Física, Universidade Federal do Rio de Janeiro, Rio de Janeiro (Brazil); Dipartimento di Fisica, Università di Camerino, I-62032 Camerino (Italy); Rodrigues, E.I.B. [Departamento de Física, Universidade Federal da Paraíba, 58051-970 João Pessoa, PB (Brazil) 2016-05-20 In this work we propose a new route to describe topological excitations in magnetic systems through a single real scalar field. We show here that spherically symmetric structures in two spatial dimensions, which map helical excitations in magnetic materials, admit this formulation and can be used to model skyrmion-like structures in magnetic materials. 12. Controllability of multi-partite quantum systems and selective excitation of quantum dots International Nuclear Information System (INIS) Schirmer, S G; Pullen, I C H; Solomon, A I 2005-01-01 We consider the degrees of controllability of multi-partite quantum systems, as well as necessary and sufficient criteria for each case. The results are applied to the problem of simultaneous control of an ensemble of quantum dots with a single laser pulse. Finally, we apply optimal control techniques to demonstrate selective excitation of individual dots for a simultaneously controllable ensemble of quantum dots 13. Bound state and localization of excitation in many-body open systems Science.gov (United States) Cui, H. T.; Shen, H. Z.; Hou, S. C.; Yi, X. X. 2018-04-01 We study the exact bound state and time evolution for single excitations in one-dimensional X X Z spin chains within a non-Markovian reservoir. For the bound state, a common feature is the localization of single excitations, which means the spontaneous emission of excitations into the reservoir is prohibited. Exceptionally, the pseudo-bound state can be found, for which the single excitation has a finite probability of emission into the reservoir. In addition, a critical energy scale for bound states is also identified, below which only one bound state exists, and it is also the pseudo-bound state. The effect of quasirandom disorder in the spin chain is also discussed; such disorder induces the single excitation to locate at some spin sites. Furthermore, to display the effect of bound state and disorder on the preservation of quantum information, the time evolution of single excitations in spin chains is studied exactly. An interesting observation is that the excitation can stay at its initial location with high probability only when the bound state and disorder coexist. In contrast, when either one of them is absent, the information of the initial state can be erased completely or becomes mixed. This finding shows that the combination of bound state and disorder can provide an ideal mechanism for quantum memory. 14. Hysteresis-induced bifurcation and chaos in a magneto-rheological suspension system under external excitation Science.gov (United States) Hailong, Zhang; Enrong, Wang; Fuhong, Min; Ning, Zhang 2016-03-01 The magneto-rheological damper (MRD) is a promising device used in vehicle semi-active suspension systems, for its continuous adjustable damping output. However, the innate nonlinear hysteresis characteristic of MRD may cause the nonlinear behaviors. In this work, a two-degree-of-freedom (2-DOF) MR suspension system was established first, by employing the modified Bouc-Wen force-velocity (F-v) hysteretic model. The nonlinear dynamic response of the system was investigated under the external excitation of single-frequency harmonic and bandwidth-limited stochastic road surface. The largest Lyapunov exponent (LLE) was used to detect the chaotic area of the frequency and amplitude of harmonic excitation, and the bifurcation diagrams, time histories, phase portraits, and power spectrum density (PSD) diagrams were used to reveal the dynamic evolution process in detail. Moreover, the LLE and Kolmogorov entropy (K entropy) were used to identify whether the system response was random or chaotic under stochastic road surface. The results demonstrated that the complex dynamical behaviors occur under different external excitation conditions. The oscillating mechanism of alternating periodic oscillations, quasi-periodic oscillations, and chaotic oscillations was observed in detail. The chaotic regions revealed that chaotic motions may appear in conditions of mid-low frequency and large amplitude, as well as small amplitude and all frequency. The obtained parameter regions where the chaotic motions may appear are useful for design of structural parameters of the vibration isolation, and the optimization of control strategy for MR suspension system. Projects supported by the National Natural Science Foundation of China (Grant Nos. 51475246, 51277098, and 51075215), the Research Innovation Program for College Graduates of Jiangsu Province China (Grant No. KYLX15 0725), and the Natural Science Foundation of Jiangsu Province of China (Grant No. BK20131402). 15. Excitability of the T-tubular system in rat skeletal muscle DEFF Research Database (Denmark) Nielsen, O B; Ørtenblad, Niels; Lamb, G D 2004-01-01 Strenuous exercise causes an increase in extracellular [K(+)] and intracellular Na(+) ([Na(+)](i)) of working muscles, which may reduce sarcolemma excitability. The excitability of the sarcolemma is, however, to some extent protected by a concomitant increase in the activity of muscle Na(+)-K(+) ... 16. Lattice Boltzmann simulation for the energy and entropy of excitable systems Institute of Scientific and Technical Information of China (English) Deng Min-Yi; Tang Guo-Ning; Kong Ling-Jiang; Liu Mu-Ren 2011-01-01 The internal energy and the spatiotemporal entropy of excitable systems are investigated with the lattice Boltzmann method. The numerical results show that the breakup of spiral wave is attributed to the inadequate supply of energy, i.e., the internal energy of system is smaller than the energy of self-sustained spiral wave. It is observed that the average internal energy of a regular wave state reduces with its spatiotemporal entropy decreasing. Interestingly, although the energy difference between two regular wave states is very small, the different states can be distinguished obviously due to the large difference between their spatiotemporal entropies. In addition, when the unstable spiral wave converts into the spatiotemporal chaos, the internal energy of system decreases, while the spatiotemporal entropy increases, which behaves as the thermodynamic entropy in an isolated system. 17. Multireference Density Functional Theory with Generalized Auxiliary Systems for Ground and Excited States. Science.gov (United States) Chen, Zehua; Zhang, Du; Jin, Ye; Yang, Yang; Su, Neil Qiang; Yang, Weitao 2017-09-21 To describe static correlation, we develop a new approach to density functional theory (DFT), which uses a generalized auxiliary system that is of a different symmetry, such as particle number or spin, from that of the physical system. The total energy of the physical system consists of two parts: the energy of the auxiliary system, which is determined with a chosen density functional approximation (DFA), and the excitation energy from an approximate linear response theory that restores the symmetry to that of the physical system, thus rigorously leading to a multideterminant description of the physical system. The electron density of the physical system is different from that of the auxiliary system and is uniquely determined from the functional derivative of the total energy with respect to the external potential. Our energy functional is thus an implicit functional of the physical system density, but an explicit functional of the auxiliary system density. We show that the total energy minimum and stationary states, describing the ground and excited states of the physical system, can be obtained by a self-consistent optimization with respect to the explicit variable, the generalized Kohn-Sham noninteracting density matrix. We have developed the generalized optimized effective potential method for the self-consistent optimization. Among options of the auxiliary system and the associated linear response theory, reformulated versions of the particle-particle random phase approximation (pp-RPA) and the spin-flip time-dependent density functional theory (SF-TDDFT) are selected for illustration of principle. Numerical results show that our multireference DFT successfully describes static correlation in bond dissociation and double bond rotation. 18. Scalable implementations of accurate excited-state coupled cluster theories: application of high-level methods to porphyrin based systems Energy Technology Data Exchange (ETDEWEB) Kowalski, Karol; Krishnamoorthy, Sriram; Olson, Ryan M.; Tipparaju, Vinod; Apra, Edoardo 2011-11-30 The development of reliable tools for excited-state simulations is emerging as an extremely powerful computational chemistry tool for understanding complex processes in the broad class of light harvesting systems and optoelectronic devices. Over the last years we have been developing equation of motion coupled cluster (EOMCC) methods capable of tackling these problems. In this paper we discuss the parallel performance of EOMCC codes which provide accurate description of the excited-state correlation effects. Two aspects are discuss in details: (1) a new algorithm for the iterative EOMCC methods based on the novel task scheduling algorithms, and (2) parallel algorithms for the non-iterative methods describing the effect of triply excited configurations. We demonstrate that the most computationally intensive non-iterative part can take advantage of 210,000 cores of the Cray XT5 system at OLCF. In particular, we demonstrate the importance of non-iterative many-body methods for achieving experimental level of accuracy for several porphyrin-based system. 19. Critical Assessment of Time-Dependent Density Functional Theory for Excited States of Open-Shell Systems: II. Doublet-Quartet Transitions. Science.gov (United States) Li, Zhendong; Liu, Wenjian 2016-06-14 Compared with closed-shell systems, open-shell systems place three additional challenges to time-dependent density functional theory (TD-DFT) for electronically excited states: (a) the spin-contamination problem is a serious issue; (b) the exchange-correlation (XC) kernel may be numerically instable; and (c) the single-determinant description of open-shell ground states readily becomes energetically instable. Confined to flip-up single excitations, the spin-contamination problem can largely be avoided by using the spin-flip TD-DFT (SF-TD-DFT) formalism, provided that a noncollinear XC kernel is employed. As for the numerical instabilities associated with such a kernel, only an ad hoc scheme has been proposed so far, viz., the ALDA0 kernel, which amounts to setting the divergent components (arising from density gradients and kinetic energy density) simply to zero. The ground-state instability problem can effectively be avoided by introducing the Tamm-Dancoff approximation (TDA) to TD-DFT. Therefore, on a general basis, the SF-TDA/ALDA0 Ansatz is so far the only promising means within the TD-DFT framework for flip-up single excitations of open-shell systems. To assess systematically the performance of SF-TDA/ALDA0, in total 61 low-lying quartet excited states of the benchmark set of 11 small radicals [J. Chem. Theory Comput. 2016, 12, 238] are investigated with various XC functionals. Taking the MRCISD+Q (multireference configuration interaction with singles and doubles plus the Davidson correction) results as benchmark, it is found that the mean absolute errors of SF-TDA/ALDA0 with the SAOP (statistical averaging of model orbital potentials), global hybrid, and range-separated hybrid functionals are in the range of 0.2-0.4 eV. This is in line not only with the typical accuracy of TD-DFT for singlet and triplet excited states of closed-shell systems but also with the gross accuracy of spin-adapted TD-DFT for spin-conserving excited states of open-shell systems. 20. Broad-Band Analysis of Polar Motion Excitations Science.gov (United States) Chen, J. 2016-12-01 Earth rotational changes, i.e. polar motion and length-of-day (LOD), are driven by two types of geophysical excitations: 1) mass redistribution within the Earth system, and 2) angular momentum exchange between the solid Earth (more precisely the crust) and other components of the Earth system. Accurate quantification of Earth rotational excitations has been difficult, due to the lack of global-scale observations of mass redistribution and angular momentum exchange. The over 14-years time-variable gravity measurements from the Gravity Recovery and Climate Experiment (GRACE) have provided a unique means for quantifying Earth rotational excitations from mass redistribution in different components of the climate system. Comparisons between observed Earth rotational changes and geophysical excitations estimated from GRACE, satellite laser ranging (SLR) and climate models show that GRACE-derived excitations agree remarkably well with polar motion observations over a broad-band of frequencies. GRACE estimates also suggest that accelerated polar region ice melting in recent years and corresponding sea level rise have played an important role in driving long-term polar motion as well. With several estimates of polar motion excitations, it is possible to estimate broad-band noise variance and noise power spectra in each, given reasonable assumptions about noise independence. Results based on GRACE CSR RL05 solutions clearly outperform other estimates with the lowest noise levels over a broad band of frequencies. 1. Can Measured Synergy Excitations Accurately Construct Unmeasured Muscle Excitations? Science.gov (United States) Bianco, Nicholas A; Patten, Carolynn; Fregly, Benjamin J 2018-01-01 Accurate prediction of muscle and joint contact forces during human movement could improve treatment planning for disorders such as osteoarthritis, stroke, Parkinson's disease, and cerebral palsy. Recent studies suggest that muscle synergies, a low-dimensional representation of a large set of muscle electromyographic (EMG) signals (henceforth called "muscle excitations"), may reduce the redundancy of muscle excitation solutions predicted by optimization methods. This study explores the feasibility of using muscle synergy information extracted from eight muscle EMG signals (henceforth called "included" muscle excitations) to accurately construct muscle excitations from up to 16 additional EMG signals (henceforth called "excluded" muscle excitations). Using treadmill walking data collected at multiple speeds from two subjects (one healthy, one poststroke), we performed muscle synergy analysis on all possible subsets of eight included muscle excitations and evaluated how well the calculated time-varying synergy excitations could construct the remaining excluded muscle excitations (henceforth called "synergy extrapolation"). We found that some, but not all, eight-muscle subsets yielded synergy excitations that achieved >90% extrapolation variance accounted for (VAF). Using the top 10% of subsets, we developed muscle selection heuristics to identify included muscle combinations whose synergy excitations achieved high extrapolation accuracy. For 3, 4, and 5 synergies, these heuristics yielded extrapolation VAF values approximately 5% lower than corresponding reconstruction VAF values for each associated eight-muscle subset. These results suggest that synergy excitations obtained from experimentally measured muscle excitations can accurately construct unmeasured muscle excitations, which could help limit muscle excitations predicted by muscle force optimizations. 2. Evolutional Properties of Localized Excitations for Generalized Broer-Kaup System in (2+1) Dimensions International Nuclear Information System (INIS) Zheng Chunlong; Ye Jianfeng; Xu Yuan 2006-01-01 Using a special Painleve-Baecklund transformation as well as the extended mapping approach and the linear superposition theorem, we obtain new families of variable separation solutions to the (2+1)-dimensional generalized Broer-Kaup (GBK) system. Based on the derived exact solution, we reveal some novel evolutional behaviors of localized excitations, i.e. fission and fusion phenomena in the (2+1)-dimensional GBK system. 3. Influence of Voltage Dips on the Operation of Brushless Exciter System of Synchronous Machines Directory of Open Access Journals (Sweden) Fedotov A. 2016-06-01 Full Text Available This paper presents a mathematical model with continuous variables for brushless exciter system of synchronous machines, containing the thyristor elements. Discrete Laplace transform is used for transition from a mathematical model of a system with variable structure in continuous variables to equation finite difference with permanent structure. Then inverse transition is made to a mathematical model in continuous variables with permanent structure. 4. [Study on relationship between emotional stability in flight and nerve system excitability]. Science.gov (United States) Liu, Fang; Huang, Wei-fen; Jing, Xiao-lu; Zhang, Ping 2003-06-01 To study the related factors of emotional stability in flight. Based on the operable definition of emotional stability in flight and the related literature review, 63 experienced pilots and flight coaches were investigated and the other-rating questionnaire of emotional stability in flight was established. To test the senior nerve system, Uchida Kraeplin (UK) test was administrated on 153 19-21 years old male student pilots of the second grade in the department of flight technique in China Civil Aviation College, who were selected through 13 h flight, 35 h solo flight, and acted as the standardization group. In the end, the correlation was explored between the testing results and their emotional behavioral characteristics in flight. Significant positive correlation was found between emotional feature indexes of emotional stability in flight and excitability in UK test. The excitability in UK test are good predictors for emotional stability in flight. 5. A Dynamic Branch-Switching Method for Parametrically Excited Systems Directory of Open Access Journals (Sweden) A.Y.T. Leung 1999-01-01 Full Text Available The branch-switching algorithm in static is applied to steady state dynamic problems. The governing ordinary differential equations are transformed to nonlinear algebraic equations by means of harmonic balance method using multiple frequency components. The frequency components of the (irrational nonlinearity of oscillator are obtained by Fast Fourier Transform and Toeplitz Jacobian method (FFT/TJM. All singularities, folds, flips, period doubling and period bubbling, are computed accurately in an analytical manner. Coexisting solutions can be predicted without using initial condition search. The consistence of both stability criteria in time and frequency domains is discussed. A highly nonlinear parametrically excited system is given as example. All connected solution paths are predicted. 6. Effects of intermediate load on performance limitations in excitation control Directory of Open Access Journals (Sweden) Pichai Aree 2008-05-01 Full Text Available The stability of excitation control systems is of great concern in power system operations. In this paper, the effects of intermediate load on performance limitation in excitation control are studied. The results reveal that the open-loop characteristic of synchronous machines flux linkage can be changed from minimum to non-minimum phase at a high level of intermediate load. This change leads to instability of synchronous machines under manual excitation control. A particular emphasis is also given to investigate the fundamental limitations in excitation control, imposed by non-minimum phases with regard to the open-loop right-half-plane (ORHP pole. The study demonstrates the difficulties of excitation control tuning to achieve the desired performance and robustness under the ORHP pole occurrence. Moreover, this paper shows the conditional stability in excitation control loop, where either an increase or decrease of the exciter gain causes a destabilization of the systems stability. Frequency response techniques are used for these investigations. 7. The DSS-14 C-band exciter Science.gov (United States) Rowan, D. R. 1989-01-01 The development and implementation of a C-band exciter for use with the Block IV Receiver-Exciter Subsystem at Deep Space Station 14 (DSS-14) has been completed. The exciter supplements the standard capabilities of the Block IV system by providing a drive signal for the C-band transmitter while generating coherent translation frequencies for C-band (5-GHz) to S-band (2.2- to 2.3-GHz) Doppler extraction, C-band to L-band (1.6-GHz) zero delay measurements, and a level calibrated L-band test signal. Exciter functions are described, and a general explanation and description of the C-band uplink controller is presented. 8. Stabilization of nonlinear excitations by disorder DEFF Research Database (Denmark) Christiansen, Peter Leth; Gaididei, Yuri Borisovich; Johansson, M. 1998-01-01 Using analytical and numerical techniques we analyze the static and dynamical properties of solitonlike excitations in the presence of parametric disorder in the one-dimensional nonlinear Schrodinger equation with a homogeneous power nonlinearity. Both the continuum and the discrete problem...... are investigated. We find that otherwise unstable excitations can be stabilized by the presence of disorder in the continuum problem. For the very narrow excitations of the discrete problem we find that the disorder has no effect on the averaged behavior. Finally, we show that the disorder can be applied to induce...... a high degree of controllability of the spatial extent of the stable excitations in the continuum system.... 9. On Emulation of Flueric Devices in Excitable Chemical Medium. Science.gov (United States) 2016-01-01 Flueric devices are fluidic devices without moving parts. Fluidic devices use fluid as a medium for information transfer and computation. A Belousov-Zhabotinsky (BZ) medium is a thin-layer spatially extended excitable chemical medium which exhibits travelling excitation wave-fronts. The excitation wave-fronts transfer information. Flueric devices compute via jets interaction. BZ devices compute via excitation wave-fronts interaction. In numerical model of BZ medium we show that functions of key flueric devices are implemented in the excitable chemical system: signal generator, and, xor, not and nor Boolean gates, delay elements, diodes and sensors. Flueric devices have been widely used in industry since late 1960s and are still employed in automotive and aircraft technologies. Implementation of analog of the flueric devices in the excitable chemical systems opens doors to further applications of excitation wave-based unconventional computing in soft robotics, embedded organic electronics and living technologies. 10. On Emulation of Flueric Devices in Excitable Chemical Medium. Directory of Open Access Journals (Sweden) Full Text Available Flueric devices are fluidic devices without moving parts. Fluidic devices use fluid as a medium for information transfer and computation. A Belousov-Zhabotinsky (BZ medium is a thin-layer spatially extended excitable chemical medium which exhibits travelling excitation wave-fronts. The excitation wave-fronts transfer information. Flueric devices compute via jets interaction. BZ devices compute via excitation wave-fronts interaction. In numerical model of BZ medium we show that functions of key flueric devices are implemented in the excitable chemical system: signal generator, and, xor, not and nor Boolean gates, delay elements, diodes and sensors. Flueric devices have been widely used in industry since late 1960s and are still employed in automotive and aircraft technologies. Implementation of analog of the flueric devices in the excitable chemical systems opens doors to further applications of excitation wave-based unconventional computing in soft robotics, embedded organic electronics and living technologies. 11. Cryogenic exciter Science.gov (United States) Bray, James William [Niskayuna, NY; Garces, Luis Jose [Niskayuna, NY 2012-03-13 The disclosed technology is a cryogenic static exciter. The cryogenic static exciter is connected to a synchronous electric machine that has a field winding. The synchronous electric machine is cooled via a refrigerator or cryogen like liquid nitrogen. The static exciter is in communication with the field winding and is operating at ambient temperature. The static exciter receives cooling from a refrigerator or cryogen source, which may also service the synchronous machine, to selected areas of the static exciter and the cooling selectively reduces the operating temperature of the selected areas of the static exciter. 12. Process and system for isotope separation using the selective vibrational excitation of molecules International Nuclear Information System (INIS) Woodroffe, J.A.; Keck, J.C. 1976-01-01 This invention concerns the separation of isotopes by using the isotopically selective vibrational excitation and the vibration-translation reactions of the excited particles. UF 6 molecular mixed with a carrier gas, such as argon, are directed through a refrigerated chamber lighted by a laser radiation tuned to excite vibrationally the uranium hexafluoride molecules of a particular uranium isotope. The density of the carrier gas is preferably maintained above the density of the uranium hexafluoride to allow a greater collision probability of the vibrationally excited molecules with a carried molecule. In such a case, the vibrationally excited uranium hexafluoride will collide with a carrier gas molecule provoking the conversion of the excitation energy into a translation of the excited molecule, resulting in thermal energy or greater diffusibility than that of the other uranium hexafluoride molecules [fr 13. Amplitude control of the track-induced self-excited vibration for a maglev system. Science.gov (United States) Zhou, Danfeng; Li, Jie; Zhang, Kun 2014-09-01 The Electromagnet Suspension (EMS) maglev train uses controlled electromagnetic forces to achieve suspension, and self-excited vibration may occur due to the flexibility of the track. In this article, the harmonic balance method is applied to investigate the amplitude of the self-excited vibration, and it is found that the amplitude of the vibration depends on the voltage of the power supplier. Based on this observation, a vibration amplitude control method, which controls the amplitude of the vibration by adjusting the voltage of the power supplier, is proposed to attenuate the vibration. A PI controller is designed to control the amplitude of the vibration at a given level. The effectiveness of this method shows a good prospect for its application to commercial maglev systems. Copyright © 2013 ISA. Published by Elsevier Ltd. All rights reserved. International Nuclear Information System (INIS) Burden, M.St.J.; Cross, K.B. 1979-01-01 An investigation into the use of rf sputtering for ion cleaning of insulating substrates before ion plating is reported. Initial experiments consisted of sputtering metals with rf power followed by the deposition of copper onto glass slides using rf plasma excitation and biasing supply. It was found that good quality films were obtained by rf ion plating onto plastics with excellent adhesion over a wide operating pressure range. A block schematic of the rf plasma excitation system is shown. (UK) 15. Subharmonic response of a single-degree-of-freedom nonlinear vibro-impact system to a narrow-band random excitation. Science.gov (United States) Haiwu, Rong; Wang, Xiangdong; Xu, Wei; Fang, Tong 2009-08-01 The subharmonic response of single-degree-of-freedom nonlinear vibro-impact oscillator with a one-sided barrier to narrow-band random excitation is investigated. The narrow-band random excitation used here is a filtered Gaussian white noise. The analysis is based on a special Zhuravlev transformation, which reduces the system to one without impacts, or velocity jumps, thereby permitting the applications of asymptotic averaging over the "fast" variables. The averaged stochastic equations are solved exactly by the method of moments for the mean-square response amplitude for the case of linear system with zero offset. A perturbation-based moment closure scheme is proposed and the formula of the mean-square amplitude is obtained approximately for the case of linear system with nonzero offset. The perturbation-based moment closure scheme is used once again to obtain the algebra equation of the mean-square amplitude of the response for the case of nonlinear system. The effects of damping, detuning, nonlinear intensity, bandwidth, and magnitudes of random excitations are analyzed. The theoretical analyses are verified by numerical results. Theoretical analyses and numerical simulations show that the peak amplitudes may be strongly reduced at large detunings or large nonlinear intensity. 16. Excitation-energy influence at the scission configuration Directory of Open Access Journals (Sweden) Ramos D. 2017-01-01 Full Text Available Transfer- and fusion-induced fission in inverse kinematics was proven to be a powerful tool to investigate nuclear fission, widening the information of the fission fragments and the access to unstable fissioning systems with respect to other experimental approaches. An experimental campaign for fission investigation has being carried out at GANIL with this technique since 2008. In these experiments, a beam of 238U, accelerated to 6.1 MeV/u, impinges on a 12C target. Fissioning systems from U to Cf are populated through transfer and fusion reactions, with excitation energies that range from few MeV up to 46 MeV. The use of inverse kinematics, the SPIDER telescope, and the VAMOS spectrometer permitted the characterization of the fissioning system in terms of mass, nuclear charge, and excitation energy, and the isotopic identification of the full fragment distribution. The neutron excess, the total neutron multiplicity, and the even-odd staggering in the nuclear charge of fission fragments are presented as a function of the excitation energy of the fissioning system. Structure effects are observed at Z∼50 and Z∼55, where their impact evolves with the excitation energy. 17. Few Issues Related to an Electrodynamic Exciter Control OpenAIRE Čala, M. 2015-01-01 There are multiple problems to solve when controlling an electromagnetic exciter for vibrations generation. Main challenge is to straighten a frequency response of an exciter which is normally not uniform due to resonances resulting from the mechanical construction of an exciter, specimen to test, or mounting fixture. This paper describes number of aspects to consider, which arose during implementation of the control system for small electrodynamic exciter on the Department of Control and Ins... 18. Excited State Spectra and Dynamics of Phenyl-Substituted Butadienes DEFF Research Database (Denmark) Wallace-Williams, Stacie E.; Schwartz, Benjamin J.; Møller, Søren 1994-01-01 indicate that phenyl torsional motion is not important to the excited-state dynamics and reveal alternative excited-state reaction pathways. The results demonstrate how molecular systems that are structually similar can exhibit different electronic properties and excited-state dynamics.... 19. Nonlinear response to the multiple sine wave excitation of a softening--hardening system International Nuclear Information System (INIS) Koplik, B.; Subudhi, M.; Curreri, J. 1979-01-01 In studying the earthquake response of the HTGR core, it was observed that the system can display softening--hardening characteristics. This is of great consequence in evaluating the structural safety aspects of the core. In order to obtain a better understanding of the governing parameters, an investigation was undertaken with a single-degree-of-freedom system having a softening--hardening spring characteristic and excited by multiple sine waves. A parametric study varying the input amplitudes and the spring characteristic was performed. Transients were introduced into the system, and the jump phenomena between the lower softening characteristics to the higher hardening curve was studied 20. Increasing Mud Pump Motor Reliability against Malfunctions of DC Motor Excitation System Science.gov (United States) Nikulin, O. V.; Shabanov, V. A. 2017-10-01 The most widely used drilling machinery, such as mud pumps, draw-works, and rotors, use direct-current (DC) motors with independent excitation as the electric drive. Drilling machinery drives operate in harsh ambient conditions, including those with the presence of moisture, dust and vibration, which increases the malfunction rate of both drilling equipment and their electric drives. One of the frequently encountered malfunctions are DC motor excitation coil faults, which disrupt the normal functioning of electric drives, often leading to shutdown of the drilling process. In a four-pole DC motor, the malfunction of one coil leads to lack of excitation current in just one coil pair, while the other pair remains functional. In this case, DC motors and drilling equipment can remain operational, which would allow for continuing the drilling process. This paper considers the possibility of operation of a DC motor on a drilling rig in those cases when one pair of excitation coils is non-functional, and describes the device for switching between the excitation coils and the auxiliary winding in a DC motor with independent excitation. 1. A truncated spherical shell model for nuclear collective excitations: Applications to the odd-mass systems, neutron-proton systems, and other topics International Nuclear Information System (INIS) Wu, Hua. 1989-01-01 One of the most elusive quantum system in nature is the nucleus, which is a strongly interacting many body system. In the hadronic (a la neutrons and protons) phase, the primary concern of this thesis, the nucleus' single particle excitations are intertwined with their various collective excitations. Although the underpinning of the nucleus is the spherical shell model, it is rendered powerless without a severe, but intelligent truncation of the infinite Hilbert space. The recently proposed Fermion Dynamical Symmetry Model (FDSM) is precisely such a truncation scheme and in which a symmetry-dictated truncation scheme is introduced in nuclear physics for the first time. In this thesis, extensions and explorations of the FDSM are made to specifically study the odd mass (where the most intricate mixing of the single particle and the collective excitations are observed) and the neutron-proton systems. In particular, the author finds that the previously successful phenomenological particle-rotor-model of the Copenhagen school can now be well understood microscopically via the FDSM. Furthermore, the well known Coriolis attenuation and variable moment of inertia effects are naturally understood from the model as well. A computer code FDUO was written by one of us to study, for the first time, the numerical implications of the FDSM. Several collective modes were found even when the system does not admit a group chain description. In addition, the code is most suitable to study the connection between level statistical behavior (a at Gaussian Orthogonal Ensemble) and dynamical symmetry. It is found that there exist critical region of the interaction parameter space were the system behaves chaotically. This information is certainly crucial to understanding quantum chaotic behavior 2. Vector boson excitations near deconfined quantum critical points. Science.gov (United States) Huh, Yejin; Strack, Philipp; Sachdev, Subir 2013-10-18 We show that the Néel states of two-dimensional antiferromagnets have low energy vector boson excitations in the vicinity of deconfined quantum critical points. We compute the universal damping of these excitations arising from spin-wave emission. Detection of such a vector boson will demonstrate the existence of emergent topological gauge excitations in a quantum spin system. 3. Ion-Beam-Excited Electrostatic Ion Cyclotron Waves DEFF Research Database (Denmark) Michelsen, Poul; Pécseli, Hans; Juul Rasmussen, Jens 1976-01-01 Self-excited electrostatic ion cyclotron waves were observed in an ion-beam-plasma system produced in a DP-operated Q-machine. The frequency of the waves showed the theoretically predicted variation with the magnetic field.......Self-excited electrostatic ion cyclotron waves were observed in an ion-beam-plasma system produced in a DP-operated Q-machine. The frequency of the waves showed the theoretically predicted variation with the magnetic field.... 4. Charge and energy dynamics in photo-excited poly(para-phenylenevinylene) systems International Nuclear Information System (INIS) Gisslen, L.; Johansson, A.; Stafstroem, S. 2004-01-01 We report results from simulations of charge and energy dynamics in poly(para-phenylenevinylene) (PPV) and PPV interacting with C 60 . The simulations were performed by solving the time-dependent Schroedinger equation and the lattice equation of motion simultaneously and nonadiabatically. The electronic system and the coupling of the electrons to the lattice were described by an extended three-dimensional version of the Su-Schrieffer-Heeger model, which also included an external electric field. Electron and lattice dynamics following electronic excitations at different energies have been simulated. The effect of additional lattice energy was also included in the simulations. Our results show that both exciton diffusion and transitions from high to lower lying excitations are stimulated by increasing the lattice energy. Also field induced charge separation occurs faster if the lattice energy is increased. This separation process is highly nonadiabatic and involves a significant rearrangement of the electron distribution. In the case of PPV coupled to C 60 , we observe a spontaneous charge separation. The separation time is in this case limited by the local concentration of C 60 molecules close to the PPV chain 5. Dynamic response of tertiary systems in structures subjected to base excitation International Nuclear Information System (INIS) Hernried, A.G.; Kai-sing Lau 1988-01-01 The dynamic response of very lightweight equipment (tertiary subsystem) attached to light equipment (secondary subsystem) which in turn is attached to a heavier structure (primary subsystem) that is subjected to ground shock or earthquake excitation is investigated. Both the single-degree-of-freedom and multi-degree-of-freedom subsystem models are considered. The systems are damped as well as undamped, completely detuned (all natural frequencies of the subsystems well spaced), singly tuned (one natural frequency of each subsystem equal or close to one another), or multiply tuned (more than one natural frequency of the subsystems close to each other). Efficient techniques for the determination of the tertiary subsystem response that avoid a computationally intensive numerical integration of the combined system equations are presented. (author) 6. Crossed-coil detection of two-photon excited nuclear quadrupole resonance Science.gov (United States) Eles, Philip T.; Michal, Carl A. 2005-08-01 Applying a recently developed theoretical framework for determining two-photon excitation Hamiltonians using average Hamiltonian theory, we calculate the excitation produced by half-resonant irradiation of the pure quadrupole resonance of a spin-3/2 system. This formalism provides expressions for the single-quantum and double-quantum nutation frequencies as well as the Bloch-Siegert shift. The dependence of the excitation strength on RF field orientation and the appearance of the free-induction signal along an axis perpendicular to the excitation field provide an unmistakable signature of two-photon excitation. We demonstrate single- and double-quantum excitation in an axially symmetric system using 35Cl in a single crystal of potassium chlorate ( ωQ = 28 MHz) with crossed-coil detection. A rotation plot verifies the orientation dependence of the two-photon excitation, and double-quantum coherences are observed directly with the application of a static external magnetic field. 7. Performance Evaluation on Transmission Tower-Line System with Passive Friction Dampers Subjected to Wind Excitations Directory of Open Access Journals (Sweden) Bo Chen 2015-01-01 Full Text Available The vibration control and performance evaluation on a transmission-tower line system by using friction dampers subjected to wind excitations are carried out in this study. The three-dimensional finite element (FE model of a transmission tower is firstly constructed. A two-dimensional lumped mass model of a transmission tower is developed for dynamic analysis. The analytical model of transmission tower-line system is proposed by taking the dynamic interaction between the tower and the transmission lines into consideration. The mechanical model of passive friction damper is presented by involving the effects of damper axial stiffness. The equation of motion of the transmission tower-line system incorporated with the friction dampers disturbed by wind excitations is established. A real transmission tower-line system is taken as an example to examine the feasibility and reliability of the proposed control approach. An extensive parameter study is carried out to find the optimal parameters of friction damper and to assess the effects of slipping force axial stiffness and hysteresis loop on control performance. The work on an example structure indicates that the application of friction dampers with optimal parameters could significantly reduce wind-induced responses of the transmission tower-line system. 8. Orientation of nuclei excited by polarized neutrons International Nuclear Information System (INIS) Lifshits, E.P. 1986-01-01 Polarization and radiation angular distribution of oriented nuclei in inelastic scattering of polarized neutrons were investigated. Nucleus orientation in the final state was described by polarization density matrix (PDM). If PDM is known, angular distributions, linear and circular polarization of γ-quanta emitted by a nucleus can be determined. Analytical expression for PDM, conditions of its diagonalization in the case of direct nucleus excitation and excitation by the stage of compound nucleus were obtained. Orientation of 12 C nuclei in the excited state 4.439 MeV, 2 + at energy of incident neutrons in the laboratory system from 4.8 MeV (excitation threshold) upt to 9 MeV was calculated as an example. Neutrons in initial state are completely polarized along Z axis. Calculations showed that excitation proceeds mainly by the stage of compound nucleus formation and 12 C nucleus is highly polarized in excited state 9. Adapting a compact confocal microscope system to a two-photon excitation fluorescence imaging architecture. Science.gov (United States) Diaspro, A; Corosu, M; Ramoino, P; Robello, M 1999-11-01 Within the framework of a national National Institute of Physics of Matter (INFM) project, we have realised a two-photon excitation (TPE) fluorescence microscope based on a new generation commercial confocal scanning head. The core of the architecture is a mode-locked Ti:Sapphire laser (Tsunami 3960, Spectra Physics Inc., Mountain View, CA) pumped by a high-power (5 W, 532 nm) laser (Millennia V, Spectra Physics Inc.) and an ultracompact confocal scanning head, Nikon PCM2000 (Nikon Instruments, Florence, Italy) using a single-pinhole design. Three-dimensional point-spread function has been measured to define spatial resolution performances. The TPE microscope has been used with a wide range of excitable fluorescent molecules (DAPI, Fura-2, Indo-1, DiOC(6)(3), fluoresceine, Texas red) covering a single photon spectral range from UV to green. An example is reported on 3D imaging of the helical structure of the sperm head of the Octopus Eledone cirrhosa labelled with an UV excitable dye, i.e., DAPI. The system can be easily switched for operating both in conventional and two-photon mode. Copyright 1999 Wiley-Liss, Inc. 10. Excited State Structural Dynamics of Carotenoids and ChargeTransfer Systems Energy Technology Data Exchange (ETDEWEB) Van Tassle, Aaron Justin [Univ. of California, Berkeley, CA (United States) 2006-01-01 This dissertation describes the development andimplementation of a visible/near infrared pump/mid-infrared probeapparatus. Chapter 1 describes the background and motivation ofinvestigating optically induced structural dynamics, paying specificattention to solvation and the excitation selection rules of highlysymmetric molecules such as carotenoids. Chapter 2 describes thedevelopment and construction of the experimental apparatus usedthroughout the remainder of this dissertation. Chapter 3 will discuss theinvestigation of DCM, a laser dye with a fluorescence signal resultingfrom a charge transfer state. By studying the dynamics of DCM and of itsmethyl deuterated isotopomer (an otherwise identical molecule), we areable to investigate the origins of the charge transfer state and provideevidence that it is of the controversial twisted intramolecular (TICT)type. Chapter 4 introduces the use of two-photon excitation to the S1state, combined with one-photon excitation to the S2 state of thecarotenoid beta-apo-8'-carotenal. These 2 investigations show evidencefor the formation of solitons, previously unobserved in molecular systemsand found only in conducting polymers Chapter 5 presents an investigationof the excited state dynamics of peridinin, the carotenoid responsiblefor the light harvesting of dinoflagellates. This investigation allowsfor a more detailed understanding of the importance of structuraldynamics of carotenoids in light harvesting. 11. The Design and Implementation of Test System Based on Programmable Excitation Power Supply for Mining Comprehensive Protector Directory of Open Access Journals (Sweden) Zhi-jie Zhang 2013-11-01 Full Text Available As comprehensive protectors for coal mining (referred to comprehensive protectors in use are prone to fail, it can timely screen out the invalid comprehensive protector by periodic functional test when it is used (it is called test in use to ensure the production safety. The test in use needs the specialized test equipment, which is not used in delivery inspection by the manufacturers of comprehensive protectors. Thus, testing excitation power becomes a constraint for the improvement of the accuracy of test in use and the degree of automation. To solve the problem, this paper developed a power frequency programmed input-output testing excitation power supply, and on that basis it also realized the mining comprehensive protector test system in use with the excitation circuit and voltage program-controlled output. 12. Low-lying excited states by constrained DFT Science.gov (United States) Ramos, Pablo; Pavanello, Michele 2018-04-01 Exploiting the machinery of Constrained Density Functional Theory (CDFT), we propose a variational method for calculating low-lying excited states of molecular systems. We dub this method eXcited CDFT (XCDFT). Excited states are obtained by self-consistently constraining a user-defined population of electrons, Nc, in the virtual space of a reference set of occupied orbitals. By imposing this population to be Nc = 1.0, we computed the first excited state of 15 molecules from a test set. Our results show that XCDFT achieves an accuracy in the predicted excitation energy only slightly worse than linear-response time-dependent DFT (TDDFT), but without incurring into problems of variational collapse typical of the more commonly adopted ΔSCF method. In addition, we selected a few challenging processes to test the limits of applicability of XCDFT. We find that in contrast to TDDFT, XCDFT is capable of reproducing energy surfaces featuring conical intersections (azobenzene and H3) with correct topology and correct overall energetics also away from the intersection. Venturing to condensed-phase systems, XCDFT reproduces the TDDFT solvatochromic shift of benzaldehyde when it is embedded by a cluster of water molecules. Thus, we find XCDFT to be a competitive method among single-reference methods for computations of excited states in terms of time to solution, rate of convergence, and accuracy of the result. 13. Coulomb excitation International Nuclear Information System (INIS) McGowan, F.K.; Stelson, P.H. 1974-01-01 The theory of Coulomb excitation and a brief review of pertinent treatments of the Coulomb excitation process that are useful for the analysis of experiments are given. Examples demonstrating the scope of nuclear structure information obtainable from gamma spectroscopy are presented. Direct Elambda excitation of 232 Th is discussed in terms of the one phonon octupole vibrational spectrum. B(MI) reduced transition probabilities resulting from Coulomb excitation of odd-A deformed nuclei with heavy ions are presented as a test of the rotational model. The use of gamma ray coincidence and particle-gamma coincidence as tools for investigating Coulomb excitation is discussed. (U.S.) 14. Nested variant of the method of moments of coupled cluster equations for vertical excitation energies and excited-state potential energy surfaces. Science.gov (United States) Kowalski, Karol 2009-05-21 In this article we discuss the problem of proper balancing of the noniterative corrections to the ground- and excited-state energies obtained with approximate coupled cluster (CC) and equation-of-motion CC (EOMCC) approaches. It is demonstrated that for a class of excited states dominated by single excitations and for states with medium doubly excited component, the newly introduced nested variant of the method of moments of CC equations provides mathematically rigorous way of balancing the ground- and excited-state correlation effects. The resulting noniterative methodology accounting for the effect of triples is tested using its parallel implementation on the systems, for which iterative CC/EOMCC calculations with full inclusion of triply excited configurations or their most important subset are numerically feasible. 15. Synchronization of Two Non-Identical Coupled Exciters in a Non-Resonant Vibrating System of Linear Motion. Part I: Theoretical Analysis Directory of Open Access Journals (Sweden) Chunyu Zhao 2009-01-01 Full Text Available In this paper an analytical approach is proposed to study the feature of frequency capture of two non-identical coupled exciters in a non-resonant vibrating system. The electromagnetic torque of an induction motor in the quasi-steady-state operation is derived. With the introduction of two perturbation small parameters to average angular velocity of two exciters and their phase difference, we deduce the Equation of Frequency Capture by averaging two motion equations of two exciters over their average period. It converts the synchronization problem of two exciters into that of existence and stability of zero solution for the Equation of Frequency Capture. The conditions of implementing frequency capture and that of stabilizing synchronous operation of two motors have been derived. The concept of torque of frequency capture is proposed to physically explain the peculiarity of self-synchronization of the two exciters. An interesting conclusion is reached that the moments of inertia of the two exciters in the Equation of Frequency Capture reduce and there is a coupling moment of inertia between the two exciters. The reduction of moments of inertia and the coupling moment of inertia have an effect on the stability of synchronous operation. 16. Self-excited vibration control for axially fast excited beam by a time delay state feedback International Nuclear Information System (INIS) Hamdi, Mustapha; Belhaq, Mohamed 2009-01-01 This work examines the control of self-excited vibration of a simply-supported beam subjected to an axially high-frequency excitation. The investigation of the resonant cases are not considered in this paper. The control is implemented via a corrective position feedback with time delay. The objective of this control is to eliminate the undesirable self-excited vibrations with an appropriate choice of parameters. The issue of stability is also addressed in this paper. Using the technique of direct partition of motion, the dynamic of discretized equations is separated into slow and fast components. The multiple scales method is then performed on the slow dynamic to obtain a slow flow for the amplitude and phase. Analysis of this slow flow provides analytical approximations locating regions in parameters space where undesirable self-excited vibration can be eliminated. A numerical study of these regions is performed on the original discretized system and compared to the analytical prediction showing a good agreement. 17. Multireference excitation energies for bacteriochlorophylls A within light harvesting system 2 DEFF Research Database (Denmark) Anda, Andre; Hansen, Thorsten; De Vico, Luca 2016-01-01 Light-harvesting system 2 (LH2) of purple bacteria is one of the most popular antenna complexes used to study Nature's way of collecting and channeling solar energy. The dynamics of the absorbed energy is probed by ultrafast spectroscopy. Simulation of these experiments relies on fitting a range...... bacteriochlorophylls in LH2. We find that the excitation energies vary among the bacteriochlorophyll monomers and that they are regulated by the curvature of the macrocycle ring and the dihedral angle of an acetyl moiety. Increasing the curvature lifts the ground state energy, which causes a red shift... 18. Slow cortical potential and theta/beta neurofeedback training in adults: effects on attentional processes, and motor system excitability Directory of Open Access Journals (Sweden) Petra eStuder 2014-07-01 Full Text Available Neurofeedback (NF is being successfully applied, among others, in children with ADHD and as a peak performance training in healthy subjects. However, the neuronal mechanisms mediating a successful NF training have not yet been sufficiently uncovered for both theta/beta (T/B, and slow cortical potential (SCP training, two protocols established in NF in ADHD. In the present randomized controlled investigation in adults without a clinical diagnosis (n = 59, the specificity of the effects of these two NF protocols on attentional processes, and motor system excitability were to be examined, focusing on the underlying neuronal mechanisms. NF training consisted of 10 double sessions, and self-regulation skills were analyzed. Pre- and post-training assessments encompassed performance and event-related potential measures during an attention task, and motor system excitability assessed by transcranial magnetic stimulation. Some NF protocol specific effects have been obtained. However, due to the limited sample size medium effects didn’t reach the level of significance. Self-regulation abilities during negativity trials of the SCP training were associated with increased contingent negative variation amplitudes, indicating improved resource allocation during cognitive preparation. Theta/beta training was associated with increased response speed and decreased target-P3 amplitudes after successful theta/beta regulation suggested reduced attentional resources necessary for stimulus evaluation. Motor system excitability effects after theta/beta training paralleled the effects of methylphenidate. Overall, our results are limited by the non-sufficiently acquired self-regulation skills, but some specific effects between good and poor learners could be described. Future studies with larger sample sizes and sufficient acquisition of self-regulation skills are needed to further evaluate the protocol specific effects on attention and motor system excitability 19. Response of a Duffing—Rayleigh system with a fractional derivative under Gaussian white noise excitation International Nuclear Information System (INIS) Zhang Ran-Ran; Xu Wei; Yang Gui-Dong; Han Qun 2015-01-01 In this paper, we consider the response analysis of a Duffing–Rayleigh system with fractional derivative under Gaussian white noise excitation. A stochastic averaging procedure for this system is developed by using the generalized harmonic functions. First, the system state is approximated by a diffusive Markov process. Then, the stationary probability densities are derived from the averaged Itô stochastic differential equation of the system. The accuracy of the analytical results is validated by the results from the Monte Carlo simulation of the original system. Moreover, the effects of different system parameters and noise intensity on the response of the system are also discussed. (paper) 20. Exciter For X-Band Transmitter And Receiver Science.gov (United States) Johns, Carl E. 1989-01-01 Report describes developmental X-band exciter for X-band uplink subsystem of Deep Space Network. X-band transmitter-exciting signal expected to have fractional frequency stability of 5.2 X 10 to negative 15th power during 1,000-second integration period. Generates coherent test signals for S- and X-band Block III translator of Deep Space Network, Doppler-reference signal for associated Doppler-extractor system, first-local-oscillator signal for associated receiver, and reference signal for associated ranging subsystem. Tests of prototype exciter show controlling and monitoring and internal phase-correcting loops perform according to applicable design criteria. Measurements of stability of frequency and of single-sideband noise spectral density of transmitter-exciting signal made subsequently. 1. Excitations and phase transitions in random anti-ferromagnets International Nuclear Information System (INIS) Cowley, R.A.; Birgeneau, R.J.; Shirane, G. 1979-01-01 Neutron scattering techniques can be used to study the magnetic excitations and phase transitions in the randomly mixed transition metal fluorides. The results for the excitations of samples with two different types of magnetic ions show two bands of excitations; each associated with excitations propagating largely on one type of ion. In the diluted salts the spectra show a complex line shape and greater widths. These results are in good accord with computer simulations showing that linear spin wave theory can be used, but have not been described satisfactorily using the coherent potential approximation. The phase transitions in these materials are always smeared, but it is difficult to ascertain if this smearing is due to macroscopic fluctuations in the concentration or of an intrinsic origin. Studies of these systems close to the percolation point have shown that the thermal disorder is associated with the one-dimensional weak links of the large clusters. Currently theory and experiment are in accord for the two-dimensional Ising system but features are still not understood in Heisenberg systems in both two and three dimensions 2. Wind turbine blade testing system using base excitation Science.gov (United States) Cotrell, Jason; Thresher, Robert; Lambert, Scott; Hughes, Scott; Johnson, Jay 2014-03-25 An apparatus (500) for fatigue testing elongate test articles (404) including wind turbine blades through forced or resonant excitation of the base (406) of the test articles (404). The apparatus (500) includes a testing platform or foundation (402). A blade support (410) is provided for retaining or supporting a base (406) of an elongate test article (404), and the blade support (410) is pivotally mounted on the testing platform (402) with at least two degrees of freedom of motion relative to the testing platform (402). An excitation input assembly (540) is interconnected with the blade support (410) and includes first and second actuators (444, 446, 541) that act to concurrently apply forces or loads to the blade support (410). The actuator forces are cyclically applied in first and second transverse directions. The test article (404) responds to shaking of its base (406) by oscillating in two, transverse directions (505, 507). 3. Wideband MEMS Resonator Using Multifrequency Excitation KAUST Repository Jaber, Nizar; Ramini, Abdallah; Al Hennawi, Qais M.; Younis, Mohammad I. 2016-01-01 We demonstrate the excitation of combination resonances of additive and subtractive types and their exploitations to realize a large bandwidth micro-machined resonator of large amplitude even at higher harmonic modes of vibrations. The investigation is conducted on a Microelectromechanical systems (MEMS) clamped-clamped microbeam fabricated using polyimide as a structural layer coated with nickel from top and chromium and gold layers from bottom. The microbeam is excited by a two-source harmonic excitation, where the first frequency source is swept around the targeted resonance (first or third mode of vibration) while the second source frequency is kept fixed. We report for the first time a large bandwidth and large amplitude response near the higher order modes of vibration. Also, we show that by properly tuning the frequency and amplitude of the excitation force, the frequency bandwidth of the resonator is controlled. 4. Wideband MEMS Resonator Using Multifrequency Excitation KAUST Repository Jaber, Nizar 2016-03-09 We demonstrate the excitation of combination resonances of additive and subtractive types and their exploitations to realize a large bandwidth micro-machined resonator of large amplitude even at higher harmonic modes of vibrations. The investigation is conducted on a Microelectromechanical systems (MEMS) clamped-clamped microbeam fabricated using polyimide as a structural layer coated with nickel from top and chromium and gold layers from bottom. The microbeam is excited by a two-source harmonic excitation, where the first frequency source is swept around the targeted resonance (first or third mode of vibration) while the second source frequency is kept fixed. We report for the first time a large bandwidth and large amplitude response near the higher order modes of vibration. Also, we show that by properly tuning the frequency and amplitude of the excitation force, the frequency bandwidth of the resonator is controlled. 5. Language identification using excitation source features CERN Document Server Rao, K Sreenivasa 2015-01-01 This book discusses the contribution of excitation source information in discriminating language. The authors focus on the excitation source component of speech for enhancement of language identification (LID) performance. Language specific features are extracted using two different modes: (i) Implicit processing of linear prediction (LP) residual and (ii) Explicit parameterization of linear prediction residual. The book discusses how in implicit processing approach, excitation source features are derived from LP residual, Hilbert envelope (magnitude) of LP residual and Phase of LP residual; and in explicit parameterization approach, LP residual signal is processed in spectral domain to extract the relevant language specific features. The authors further extract source features from these modes, which are combined for enhancing the performance of LID systems. The proposed excitation source features are also investigated for LID in background noisy environments. Each chapter of this book provides the motivatio... 6. Optimal placement of excitations and sensors for verification of large dynamical systems Science.gov (United States) Salama, M.; Rose, T.; Garba, J. 1987-01-01 The computationally difficult problem of the optimal placement of excitations and sensors to maximize the observed measurements is studied within the framework of combinatorial optimization, and is solved numerically using a variation of the simulated annealing heuristic algorithm. Results of numerical experiments including a square plate and a 960 degrees-of-freedom Control of Flexible Structure (COFS) truss structure, are presented. Though the algorithm produces suboptimal solutions, its generality and simplicity allow the treatment of complex dynamical systems which would otherwise be difficult to handle. 7. Energy-optimal electrical excitation of nerve fibers. Science.gov (United States) Jezernik, Saso; Morari, Manfred 2005-04-01 We derive, based on an analytical nerve membrane model and optimal control theory of dynamical systems, an energy-optimal stimulation current waveform for electrical excitation of nerve fibers. Optimal stimulation waveforms for nonleaky and leaky membranes are calculated. The case with a leaky membrane is a realistic case. Finally, we compare the waveforms and energies necessary for excitation of a leaky membrane in the case where the stimulation waveform is a square-wave current pulse, and in the case of energy-optimal stimulation. The optimal stimulation waveform is an exponentially rising waveform and necessitates considerably less energy to excite the nerve than a square-wave pulse (especially true for larger pulse durations). The described theoretical results can lead to drastically increased battery lifetime and/or decreased energy transmission requirements for implanted biomedical systems. 8. Coulomb excitation of {sup 8}Li Energy Technology Data Exchange (ETDEWEB) Assuncao, Marlete; Britos, Tatiane Nassar [Universidade Federal de Sao Paulo (UNIFESP), SP (Brazil). Dept. de Ciencias Exatas e da Terra; Descouvemont, Pierre [Universite Libre de Bruxelles (ULB), Brussels (Belgium). Physique Nucleaire Theorique et Physique Mathematique; Lepine-Szily, Alinka; Lichtenthaler Filho, Rubens; Barioni, Adriana; Silva, Diego Medeiros da; Pereira, Dirceu; Mendes Junior, Djalma Rosa; Pires, Kelly Cristina Cezaretto; Gasques, Leandro Romero; Morais, Maria Carmen; Added, Nemitala; Neto Faria, Pedro; Rec, Rafael [Universidade de Sao Paulo (IF/USP), SP (Brazil). Inst. de Fisica. Dept. de Fisica Nuclear 2012-07-01 Full text: This work shows the Coulomb Excitation of {sup 8}Li on targets that have effectively behavior of Rutherford in angles and energies of interest for determining the value of the B(E2) electromagnetic transition. Theoretical aspects involved in this type of measure, known as COULEX [1], and some results in the literature [2-3] will be presented. Some problems with the targets and measurement system while performing an experiment on Coulomb Excitation of {sup 8}Li will be discussed: the energy resolution, background, possible contributions of the primary beam and also the excited states of the target near the region of elastic and inelastic peaks. They will be illustrated by measurements of the Coulomb Excitation of {sup 8}Li on targets of {sup 197}Au and {sup 208}Pb using the system RIBRAS(Brazilian Radioactive Ion Beam). In this case, the {sup 8}Li beam(T{sub 1/2} = 838 ms)is produced by {sup 9}Be({sup 7}Li;{sup 8} Li){sup 8}Be reaction from RIBRAS system which is installed at Instituto de Fisica of the Universidade de Sao Paulo. The primary {sup 7L}i beam is provided by Pelletron Accelerator. [1] K. Alder and A. Winther, Electromagnetic Excitation, North-Holland, New York, 1975; [2] P. Descouvemont and D. Baye, Phys. Letts. B 292, 235-238, 1992; [3] J. A. Brown, F. D. Becchetti, J. W. Jaenecke, K, Ashktorab, and D. A. Roberts, J. J. Kolata, R. J. Smith, and K. Lamkin, R. E. Warner, Phys. Rev. Letts., 66, 19, 1991; [4] R. J. Smith, J. J Kolata, K. Lamkin and A. Morsard, F. D. Becchetti, J. A. Brown, W. Z. Liu, J. W. Jaenecke, and D. A. Roberts, R. E. Warner, Phys. Rev. C, 43, 5, 1991. (author) 9. Nonlinear stability of spin-flip excitations International Nuclear Information System (INIS) Arunasalam, V. 1975-01-01 A rather complete discussion of the nonlinear electrodynamic behavior of a negative-temperature spin system is presented. The method presented here is based on a coupled set of master equations, one describing the time evolution of the photon (i.e., the spin-flip excitation) distribution function and the other describing the time evolution of the particle distribution function. It is found that the initially unstable (i.e., growing) spin-flip excitations grow to such a large amplitude that their nonlinear reaction on the particle distribution function becomes important. It is then shown that the initially totally inverted two-level spin system evolves rapidly (through this nonlinear photon-particle coupling) towards a quasilinear steady state where the populations of the spin-up and the spin-down states are equal to each other. Explicit expressions for the time taken to reach this quasilinear steady state and the energy in the spin-flip excitations at this state are also presented 10. A benchmark study of electronic excitation energies, transition moments, and excited-state energy gradients on the nicotine molecule Energy Technology Data Exchange (ETDEWEB) Egidi, Franco, E-mail: [email protected]; Segado, Mireia; Barone, Vincenzo, E-mail: [email protected] [Scuola Normale Superiore, Piazza dei Cavalieri, 7 I-56126 Pisa (Italy); Koch, Henrik [Department of Chemistry, Norwegian University of Science and Technology, 7491 Trondheim (Norway); Cappelli, Chiara [Dipartimento di Chimica e Chimica Industriale, Università di Pisa, via G. Moruzzi, 3 I-56124 Pisa (Italy) 2014-12-14 In this work, we report a comparative study of computed excitation energies, oscillator strengths, and excited-state energy gradients of (S)-nicotine, chosen as a test case, using multireference methods, coupled cluster singles and doubles, and methods based on time-dependent density functional theory. This system was chosen because its apparent simplicity hides a complex electronic structure, as several different types of valence excitations are possible, including n-π{sup }, π-π{sup }, and charge-transfer states, and in order to simulate its spectrum it is necessary to describe all of them consistently well by the chosen method. 11. Light emitting diode excitation emission matrix fluorescence spectroscopy. Science.gov (United States) Hart, Sean J; JiJi, Renée D 2002-12-01 An excitation emission matrix (EEM) fluorescence instrument has been developed using a linear array of light emitting diodes (LED). The wavelengths covered extend from the upper UV through the visible spectrum: 370-640 nm. Using an LED array to excite fluorescence emission at multiple excitation wavelengths is a low-cost alternative to an expensive high power lamp and imaging spectrograph. The LED-EEM system is a departure from other EEM spectroscopy systems in that LEDs often have broad excitation ranges which may overlap with neighboring channels. The LED array can be considered a hybrid between a spectroscopic and sensor system, as the broad LED excitation range produces a partially selective optical measurement. The instrument has been tested and characterized using fluorescent dyes: limits of detection (LOD) for 9,10-bis(phenylethynyl)-anthracene and rhodamine B were in the mid parts-per-trillion range; detection limits for the other compounds were in the low parts-per-billion range (LED-EEMs were analyzed using parallel factor analysis (PARAFAC), which allowed the mathematical resolution of the individual contributions of the mono- and dianion fluorescein tautomers a priori. Correct identification and quantitation of six fluorescent dyes in two to six component mixtures (concentrations between 12.5 and 500 ppb) has been achieved with root mean squared errors of prediction (RMSEP) of less than 4.0 ppb for all components. 12. Comparison among nonlinear excitation control strategies used for damping power system oscillations International Nuclear Information System (INIS) Leon, A.E.; Solsona, J.A.; Valla, M.I. 2012-01-01 Highlights: ► A description and comparison of nonlinear control strategies for synchronous generators are presented. ► Advantages of using nonlinear controllers are emphasized against the use of classical PSSs. ► We find that a particular selection of IDA gains achieve the same performance that FL controllers. - Abstract: This work is focused on the problem of power system stability. A thorough description of nonlinear control strategies for synchronous generator excitation, which are designed for damping oscillations and improving transient stability on power systems, is presented along with a detailed comparison among these modern strategies and current solutions based on power system stabilizers. The performance related to damping injection in each controller, critical time enhancement, robustness against parametric uncertainties, and control signal energy consumption is analyzed. Several tests are presented to validate discussions on various advantages and disadvantages of each control strategy. 13. Exciter switch Science.gov (United States) Mcpeak, W. L. 1975-01-01 A new exciter switch assembly has been installed at the three DSN 64-m deep space stations. This assembly provides for switching Block III and Block IV exciters to either the high-power or 20-kW transmitters in either dual-carrier or single-carrier mode. In the dual-carrier mode, it provides for balancing the two drive signals from a single control panel located in the transmitter local control and remote control consoles. In addition to the improved switching capabilities, extensive monitoring of both the exciter switch assembly and Transmitter Subsystem is provided by the exciter switch monitor and display assemblies. 14. 340nm UV LED excitation in time-resolved fluorescence system for europium-based immunoassays detection OpenAIRE Rodenko, Olga; Fodgaard, Henrik; Tidemand-Lichtenberg, Peter; Pedersen, Christian 2017-01-01 In immunoassay analyzers for in-vitro diagnostics, Xenon flash lamps have been widely used as excitation light sources. Recent advancements in UV LED technology and its advantages over the flash lamps such as smaller footprint, better wall-plug efficiency, narrow emission spectrum, and no significant afterglow, have made them attractive light sources for gated detection systems. In this paper, we report on the implementation of a 340 nm UV LED based time-resolved fluorescence system based on ... 15. Terahertz Solitons in Biomolecular Systems and their Excitation by External Electromagnetic Field Directory of Open Access Journals (Sweden) Bugay А.N. 2015-01-01 Full Text Available Nonlinear dynamics of charge and acoustic excitations in cellular microtubules is considered. Different types of nonlinear solitary waves were studied taking account for dissipation. The mechanism of electro-acoustic pulse excitation by external electromagnetic field of terahertz frequency is recognized. 16. A high excitation magnetic quadrupole lens quadruplet incorporating a single octupole lens for a low spherical aberration probe forming lens system Science.gov (United States) Dou, Yanxin; Jamieson, David N.; Liu, Jianli; Li, Liyi 2018-03-01 This paper describes the design of a new probe forming lens system consisting of a high excitation magnetic quadrupole lens quadruplet that incorporates a single magnetic octupole lens. This system achieves both a high demagnification and a low spherical aberration compared to conventional high excitation systems and is intended for deployment for the Harbin 300 MeV proton microprobe for applications in space science and ion beam therapy. This relative simplicity of the ion optical design to include a single octupole lens minimizes the risks associated with the constructional and operational precision usually needed for the probe forming lens system and this system could also be deployed in microprobe systems that operate with less magnetically rigid ions. The design of the new system is validated with reference to two independent ion optical computer codes. 17. Autoresonant Excitation of Antiproton Plasmas CERN Document Server Andresen, Gorm B; Baquero-Ruiz, Marcelo; Bertsche, William; Bowe, Paul D; Butler, Eoin; Carpenter, P T; Cesar, Claudio L; Chapman, Steven; Charlton, Michael; Fajans, Joel; Friesen, Tim; Fujiwara, Makoto C; Gill, David R; Hangst, Jeffrey S; Hardy, Walter N; Hayden, Michael E; Humphries, Andrew J; Hurt, J L; Hydomako, Richard; Jonsell, Svante; Madsen, Niels; Menary, Scott; Nolan, Paul; Olchanski, Konstantin; Olin, Art; Povilus, Alexander; Pusa, Petteri; Robicheaux, Francis; Sarid, Eli; Silveira, Daniel M; So, Chukman; Storey, James W; Thompson, Robert I; van der Werf, Dirk P; Wurtele, Jonathan S; Yamazaki, Yasunori 2011-01-01 We demonstrate controllable excitation of the center-of-mass longitudinal motion of a thermal antiproton plasma using a swept-frequency autoresonant drive. When the plasma is cold, dense, and highly collective in nature, we observe that the entire system behaves as a single-particle nonlinear oscillator, as predicted by a recent theory. In contrast, only a fraction of the antiprotons in a warm plasma can be similarly excited. Antihydrogen was produced and trapped by using this technique to drive antiprotons into a positron plasma, thereby initiating atomic recombination. 18. Modeling pulsed excitation for gas-phase laser diagnostics International Nuclear Information System (INIS) Settersten, Thomas B.; Linne, Mark A. 2002-01-01 Excitation dynamics for pulsed optical excitation are described with the density-matrix equations and the rate equations for a two-level system. A critical comparison of the two descriptions is made with complete and consistent formalisms that are amenable to the modeling of applied laser-diagnostic techniques. General solutions, resulting from numerical integration of the differential equations describing the excitation process, are compared for collisional conditions that range from the completely coherent limit to the steady-state limit, for which the two formalisms are identical. This analysis demonstrates the failure of the rate equations to correctly describe the transient details of the excitation process outside the steady-state limit. However, reasonable estimates of the resultant population are obtained for nonsaturating (linear) excitation. This comparison provides the laser diagnostician with the means to evaluate the appropriate model for excitation through a simple picture of the breakdown of the rate-equation validity 19. Clustered chimera states in systems of type-I excitability International Nuclear Information System (INIS) Vüllings, Andrea; Omelchenko, Iryna; Hövel, Philipp; Hizanidis, Johanne 2014-01-01 The chimera state is a fascinating phenomenon of coexisting synchronized and desynchronized behaviour that was discovered in networks of nonlocally coupled identical phase oscillators over ten years ago. Since then, chimeras have been found in numerous theoretical and experimental studies and more recently in models of neuronal dynamics as well. In this work, we consider a generic model for a saddle-node bifurcation on a limit cycle representative of neural excitability type I. We obtain chimera states with multiple coherent regions (clustered chimeras/multi-chimeras) depending on the distance from the excitability threshold, the range of nonlocal coupling and the coupling strength. A detailed stability diagram for these chimera states and other interesting coexisting patterns (like traveling waves) is presented. (paper) 20. Spectroscopy of collective excitations in interacting low-dimensional many-body systems using quench dynamics. Science.gov (United States) Gritsev, Vladimir; Demler, Eugene; Lukin, Mikhail; Polkovnikov, Anatoli 2007-11-16 We study the problem of rapid change of the interaction parameter (quench) in a many-body low-dimensional system. It is shown that, measuring the correlation functions after the quench, the information about a spectrum of collective excitations in a system can be obtained. This observation is supported by analysis of several integrable models and we argue that it is valid for nonintegrable models as well. Our conclusions are supplemented by performing exact numerical simulations on finite systems. We propose that measuring the power spectrum in a dynamically split 1D Bose-Einsten condensate into two coupled condensates can be used as an experimental test of our predictions. 1. Magnetic excitations in thulium metal International Nuclear Information System (INIS) Fernandez-Baca, J.A.; Nicklow, R.M.; Rhyne, J.J. 1989-01-01 We have performed inelastic neutron scattering measurements on a single crystal specimen of Tm at wavevectors rvec κ = (1,1, ζ) and (0,0,2 + ζ) (ζ = 0, hor-ellipsis, 1). Most of the measurements have been made at T = 5K, where Tm exhibits a seven layer ferrimagnetic-antiphase-domain structure (four moments up, parallel to the c-axis, followed by three moments down). At this temperature the excitation spectra consist of three peaks. The two lower energy excitations have been identified as originating from magneto-vibrational scattering from the TA phonon, while the higher energy excitation is magnetic and exhibits only a weak dispersion (between 8.3 and 9.6 meV). At T = 50K, a temperature at which the system exhibits a c-axis sinusoidally modulated structure, the magnetic mode shows significant softening and broadening. The magneto-vibrational scattering vanishes above the Neel temperature (T N = 58.5K) while the magnetic mode persists at least up to T = 70K. These results suggest that the Hamiltonian in this system is dominated by the crystal-field-anistropy energy, and that the exchange interaction is relatively weak. 9 refs., 2 figs 2. Relay protection coordination with generator capability curve, excitation system limiters and power system relay protections settings Directory of Open Access Journals (Sweden) Buha Danilo 2016-01-01 Full Text Available The relay protection settings performed in the largest thermal powerplant (TE "Nikola Tesla B" are reffered and explained in this paper. The first calculation step is related to the coordination of the maximum stator current limiter settings, the overcurrent protection with inverse characteristics settings and the permitted overload of the generator stator B1. In the second calculation step the settings of impedance generator protection are determined, and the methods and criteria according to which the calculations are done are described. Criteria used to provide the protection to fulfill the backup protection role in the event of malfunction of the main protection of the transmission system. are clarified. The calculation of all protection functions (32 functions of generator B1 were performed in the project "Coordination of relay protection blocks B1 and B2 with the system of excitation and power system protections -TENT B". 3. Seismic structural response analysis for multiple support excitation International Nuclear Information System (INIS) Shaw, D.E. 1975-01-01 In the seismic analysis of nuclear power plant equipment such as piping systems situations often arise in which piping systems span between adjacent structures or between different elevations in the same structure. Owing to the differences in the seismic time history response of different structures or different elevations of the same structure, the input support motion will differ for different supports. The concept of a frequency dependent participation factor and rotational response spectra accounting for phase differences between support excitations is developed by using classical equations of motion to formulate the seismic response of a structure subjected to multiple support excitation. The essence of the method lies in describing the seismic excitation of a multiply excited structure in terms of translational and rotational spectra used at every support and a frequency dependent spatial distribution function derived from the phase relationships of the different support time histories. In this manner it is shown that frequency dependent participation factors can be derived from the frequency dependent distribution functions. Examples are shown and discussed relative to closed form solutions and the state-of-the-art techniques presently being used for the solution of problems of multiply excited structures 4. Core excitation and de-excitation spectroscopies of free atoms and molecules International Nuclear Information System (INIS) Ueda, Kiyoshi 2006-01-01 This article provides a review of the current status of core excitation and de-excitation spectroscopy studies of free atoms molecules using a high-resolution soft X-ray monochromator and a high-resolution electron energy analyzer, installed in the soft X-ray photochemistry beam line at SPring-8. Experimental results are discussed for 1s excitation of Ne, O 1s excitation of CO and H 2 O, and F 1s excitation of CF 4 . (author) 5. Robust structural design against self-excited vibrations CERN Document Server Spelsberg-Korspeter, Gottfried 2013-01-01 This book studies methods for a robust design of rotors against self-excited vibrations. The occurrence of self-excited vibrations in engineering applications if often unwanted and in many cases difficult to model. Thinking of complex systems such as machines with many components and mechanical contacts, it is important to have guidelines for design so that the functionality is robust against small imperfections. This book discusses the question on how to design a structure such that unwanted self-excited vibrations do not occur. It shows theoretically and practically that the old design rule to avoid multiple eigenvalues points toward the right direction and have optimized structures accordingly. This extends results for the well-known flutter problem in which equations of motion with constant coefficients occur to the case of a linear conservative system with arbitrary time periodic perturbations. 6. Excited-state quantum phase transitions in systems with two degrees of freedom: II. Finite-size effects Energy Technology Data Exchange (ETDEWEB) Stránský, Pavel [Institute of Particle and Nuclear Physics, Faculty of Mathematics and Physics, Charles University, V Holešovičkách 2, 18000 Prague (Czech Republic); Macek, Michal [Institute of Particle and Nuclear Physics, Faculty of Mathematics and Physics, Charles University, V Holešovičkách 2, 18000 Prague (Czech Republic); Center for Theoretical Physics, Sloane Physics Laboratory, Yale University, New Haven, CT 06520-8120 (United States); Leviatan, Amiram [Racah Institute of Physics, The Hebrew University, 91904 Jerusalem (Israel); Cejnar, Pavel, E-mail: [email protected] [Institute of Particle and Nuclear Physics, Faculty of Mathematics and Physics, Charles University, V Holešovičkách 2, 18000 Prague (Czech Republic) 2015-05-15 This article extends our previous analysis Stránský et al. (2014) of Excited-State Quantum Phase Transitions (ESQPTs) in systems of dimension two. We focus on the oscillatory component of the quantum state density in connection with ESQPT structures accompanying a first-order ground-state transition. It is shown that a separable (integrable) system can develop rather strong finite-size precursors of ESQPT expressed as singularities in the oscillatory component of the state density. The singularities originate in effectively 1-dimensional dynamics and in some cases appear in multiple replicas with increasing excitation energy. Using a specific model example, we demonstrate that these precursors are rather resistant to proliferation of chaotic dynamics. - Highlights: • Oscillatory components of state density and spectral flow studied near ESQPTs. • Enhanced finite-size precursors of ESQPT caused by fully/partly separable dynamics. • These precursors appear due to criticality of a subsystem with lower dimension. • Separability-induced finite-size effects disappear in case of fully chaotic dynamics. 7. From fusion hierarchy to excited state TBA International Nuclear Information System (INIS) Juettner, G.; Kluemper, A. 1998-01-01 Functional relations among the fusion hierarchy of quantum transfer matrices give a novel derivation of the TBA equations, namely without string hypothesis. This is demonstrated for two important models of 1D highly correlated electron systems, the supersymmetric t-J model and the supersymmetric extended Hubbard model. As a consequence, ''the excited state TBA'' equations, which characterize correlation lengths, are explicitly derived for the t-J model. To the authors' knowledge, this is the first explicit derivation of excited state TBA equations for 1D lattice electron systems. (orig.) 8. Doubly and triply excited states for different plasma sources International Nuclear Information System (INIS) More, R.M.; Safronova, U.I. 2000-01-01 Autoionizing rates of doubly excited states as nln'l' configurations with n=2-9 and n'=2-9 are calculated. Analytical expressions of decay amplitude for two-electron system are derived. Expressions for autoionizing rates with averaging over LS are obtained for many-electron systems. The n and l dependence of doubly excited states as nln'l' configurations are investigated. (author) 9. An Improved Multidimensional MPA Procedure for Bidirectional Earthquake Excitations OpenAIRE Wang, Feng; Sun, Jian-Gang; Zhang, Ning 2014-01-01 Presently, the modal pushover analysis procedure is extended to multidimensional analysis of structures subjected to multidimensional earthquake excitations. an improved multidimensional modal pushover analysis (IMMPA) method is presented in the paper in order to estimate the response demands of structures subjected to bidirectional earthquake excitations, in which the unidirectional earthquake excitation applied on equivalent SDOF system is replaced by the direct superposition of two compone... 10. Automatic vibration mode selection and excitation; combining modal filtering with autoresonance Science.gov (United States) Davis, Solomon; Bucher, Izhak 2018-02-01 Autoresonance is a well-known nonlinear feedback method used for automatically exciting a system at its natural frequency. Though highly effective in exciting single degree of freedom systems, in its simplest form it lacks a mechanism for choosing the mode of excitation when more than one is present. In this case a single mode will be automatically excited, but this mode cannot be chosen or changed. In this paper a new method for automatically exciting a general second-order system at any desired natural frequency using Autoresonance is proposed. The article begins by deriving a concise expression for the frequency of the limit cycle induced by an Autoresonance feedback loop enclosed on the system. The expression is based on modal decomposition, and provides valuable insight into the behavior of a system controlled in this way. With this expression, a method for selecting and exciting a desired mode naturally follows by combining Autoresonance with Modal Filtering. By taking various linear combinations of the sensor signals, by orthogonality one can "filter out" all the unwanted modes effectively. The desired mode's natural frequency is then automatically reflected in the limit cycle. In experiment the technique has proven extremely robust, even if the amplitude of the desired mode is significantly smaller than the others and the modal filters are greatly inaccurate. 11. Stochastic responses of Van der Pol vibro-impact system with fractional derivative damping excited by Gaussian white noise Energy Technology Data Exchange (ETDEWEB) Xiao, Yanwen; Xu, Wei, E-mail: [email protected]; Wang, Liang [Department of Applied Mathematics, Northwestern Polytechnical University, Xi' an 710072 (China) 2016-03-15 This paper focuses on the study of the stochastic Van der Pol vibro-impact system with fractional derivative damping under Gaussian white noise excitation. The equations of the original system are simplified by non-smooth transformation. For the simplified equation, the stochastic averaging approach is applied to solve it. Then, the fractional derivative damping term is facilitated by a numerical scheme, therewith the fourth-order Runge-Kutta method is used to obtain the numerical results. And the numerical simulation results fit the analytical solutions. Therefore, the proposed analytical means to study this system are proved to be feasible. In this context, the effects on the response stationary probability density functions (PDFs) caused by noise excitation, restitution condition, and fractional derivative damping are considered, in addition the stochastic P-bifurcation is also explored in this paper through varying the value of the coefficient of fractional derivative damping and the restitution coefficient. These system parameters not only influence the response PDFs of this system but also can cause the stochastic P-bifurcation. 12. Stochastic responses of Van der Pol vibro-impact system with fractional derivative damping excited by Gaussian white noise. Science.gov (United States) Xiao, Yanwen; Xu, Wei; Wang, Liang 2016-03-01 This paper focuses on the study of the stochastic Van der Pol vibro-impact system with fractional derivative damping under Gaussian white noise excitation. The equations of the original system are simplified by non-smooth transformation. For the simplified equation, the stochastic averaging approach is applied to solve it. Then, the fractional derivative damping term is facilitated by a numerical scheme, therewith the fourth-order Runge-Kutta method is used to obtain the numerical results. And the numerical simulation results fit the analytical solutions. Therefore, the proposed analytical means to study this system are proved to be feasible. In this context, the effects on the response stationary probability density functions (PDFs) caused by noise excitation, restitution condition, and fractional derivative damping are considered, in addition the stochastic P-bifurcation is also explored in this paper through varying the value of the coefficient of fractional derivative damping and the restitution coefficient. These system parameters not only influence the response PDFs of this system but also can cause the stochastic P-bifurcation. 13. Controlling flexible rotor vibrations using parametric excitation Energy Technology Data Exchange (ETDEWEB) Atepor, L, E-mail: [email protected] [Department of Mechanical Engineering, University of Glasgow, G12 8QQ (United Kingdom) 2009-08-01 This paper presents both theoretical and experimental studies of an active vibration controller for vibration in a flexible rotor system. The paper shows that the vibration amplitude can be modified by introducing an axial parametric excitation. The perturbation method of multiple scales is used to solve the equations of motion. The steady-state responses, with and without the parametric excitation terms, is investigated. An experimental test machine uses a piezoelectric exciter mounted on the end of the shaft. The results show a reduction in the rotor response amplitude under principal parametric resonance, and some good correlation between theory and experiment. 14. Relative excitation functions for singly-excited and core-excited levels of S V--S IX populated by the beam-foil interaction International Nuclear Information System (INIS) Moenke, D.; Bengtsson, P.; Engstroem, L.; Hutton, R.; Jupen, C.; Kirm, M.; Westerlind, M. 1994-01-01 We have investigated the relative excitation functions for low-lying singly excited and low-lying core-excited levels in S V (S 4+ ) to S IX (S 8+ ) after beam-foil excitation using ions in the energy range 2--10 MeV. The spectral line intensities have been normalized to the same number of particles at each ion energy and corrections for the level lifetimes have been made. The overall accuracy of the measured relative excitation function at each energy and charge state is estimated to be better than 2%. A comparison of the relative excitation functions for singly excited and core-excited lines shows a difference in S VII, but not in S VI 15. Measurement and analysis of excitation functions in 16O + 103Rh system in the excitation energy range ≅ 2-4 MeV/A International Nuclear Information System (INIS) Singh, Devendra P.; Unnati; Sharma, Manoj Kumar; Singh, Pushpendra P.; Singh, B.P.; Prasad, R.; Gupta, Sunita; Rakesh Kumar; Bhardwaj, H.D. 2006-01-01 In the present work, excitation functions for seven evaporation residues (ERs) produced via complete fusion and incomplete fusion processes in 16 O + 103 Rh system have been measured in the energy range ≅ 47-85 MeV, using recoil catcher technique followed by off-line gamma-ray spectrometry. Comparison of the experimental data with statistical model based computer code PACE 2 revealed dominance of incomplete fusion in reactions involving alpha-emission channels. To the best of our knowledge these reactions are being reported for the first time Science.gov (United States) Jansma, P. A. 1982-01-01 A description of the general design of both the block 3 and block 4 receiver-exciter controllers for the Deep Space Network (DSN) Mark IV-A System is presented along with the design approach. The controllers are designed to enable the receiver-exciter subsystem (RCV) to be configured, calibrated, initialized and operated from a central location via high level instructions. The RECs are designed to be operated under the control of the DMC subsystem. The instructions are in the form of standard subsystem blocks (SSBs) received via the local area network (LAN). The centralized control provided by RECs and other DSCC controllers in Mark IV-A is intended to reduce DSN operations costs from the Mark III era. 17. Low-frequency excitations in zirconium hydrides International Nuclear Information System (INIS) Radulescu, A.; Padureanu, I.; Rapeanu, S.N.; Beldiman, A.; Kozlov, Zh.A.; Semenov, V.A. 1999-01-01 The slow inelastic neutron scattering (INS) on ZrH x systems (x = 0.38, 0.52) revealed new excitations located within the energy range 2-10 MeV. Besides the acoustic vibrations specific to α-HCP Zr and γ-FCO Zr hydride the fine structure of these excitations is clearly observed. The origin of the new observed peaks is not very clear but a proton tunneling or a resonance effect in α-Zr lattice could be taken into account 18. Test and Control System for Chlorophyll Fluorescence Parameters Using LED as Excitation Source Directory of Open Access Journals (Sweden) Zou Qiuying 2014-05-01 Full Text Available A new scheme on test and control system for chlorophyll fluorescence is presented in this work, which uses light-emitting diode (LED excitation by means of measuring the fluorescence parameter fpsII. The system takes programmable power supply as LEDs illumination drive power with high sensitivity and signal-to-noise ratio. MINIPAM is used to measure fluorescence parameter fpsII and keeps communication with upper PC by serial port. The upper PC can control the power supply and process the data received from MINIPAM by software which is programmed in VB6. The results show that the system has a lot of advantages such as high accuracy and convenience. The effect of environmental factors on fluorescence parameters is analyzed comprehensively. It will be a practical measurement and control system for photosynthetic ability and have wide application foreground. 19. Magnetic measurement of soft magnetic composites material under 3D SVPWM excitation Science.gov (United States) Zhang, Changgeng; Jiang, Baolin; Li, Yongjian; Yang, Qingxin 2018-05-01 The magnetic properties measurement and analysis of soft magnetic material under the rotational space-vector pulse width modulation (SVPWM) excitation are key factors in design and optimization of the adjustable speed motor. In this paper, a three-dimensional (3D) magnetic properties testing system fit for SVPWM excitation is built, which includes symmetrical orthogonal excitation magnetic circuit and cubic field-metric sensor. Base on the testing system, the vector B and H loci of soft magnetic composite (SMC) material under SVPWM excitation are measured and analyzed by proposed 3D SVPWM control method. Alternating and rotating core losses under various complex excitation with different magnitude modulation ratio are calculated and compared. 20. Stochastic stability of mechanical systems under renewal jump process parametric excitation DEFF Research Database (Denmark) Iwankiewicz, R.; Nielsen, Søren R.K.; Larsen, Jesper Winther 2005-01-01 independent, negative exponential distributed variables; hence, the arrival process may be termed as a generalized Erlang renewal process. The excitation process is governed by the stochastic equation driven by two independent Poisson processes, with different parameters. If the response in a single mode...... is investigated, the problem is governed in the state space by two stochastic equations, because the stochastic equation for the excitation process is autonomic. However due to the parametric nature of the excitation, the nonlinear term appears at the right-hand sides of the equations. The equations become linear...... of the stochastic equation governing the natural logarithm of the hyperspherical amplitude process and using the modification of the method wherein the time averaging of the pertinent expressions is replaced by ensemble averaging. It is found that the direct simulation is more suitable and that the asymptotic mean... 1. Mechanism of spiral formation in heterogeneous discretized excitable media. Science.gov (United States) Kinoshita, Shu-ichi; Iwamoto, Mayuko; Tateishi, Keita; Suematsu, Nobuhiko J; Ueyama, Daishin 2013-06-01 Spiral waves on excitable media strongly influence the functions of living systems in both a positive and negative way. The spiral formation mechanism has thus been one of the major themes in the field of reaction-diffusion systems. Although the widely believed origin of spiral waves is the interaction of traveling waves, the heterogeneity of an excitable medium has recently been suggested as a probable cause. We suggest one possible origin of spiral waves using a Belousov-Zhabotinsky reaction and a discretized FitzHugh-Nagumo model. The heterogeneity of the reaction field is shown to stochastically generate unidirectional sites, which can induce spiral waves. Furthermore, we found that the spiral wave vanished with only a small reduction in the excitability of the reaction field. These results reveal a gentle approach for controlling the appearance of a spiral wave on an excitable medium. 2. Examination of excited state populations in sputtering using Multiphoton Resonance Ionization International Nuclear Information System (INIS) Kimock, F.M.; Baxter, J.P.; Pappas, D.L.; Kobrin, P.H.; Winograd, N. 1984-01-01 Multiphoton Resonance Ionization has been employed to study the populations of excited state atoms ejected from ion bombarded metal surfaces. Preliminary investigations have focused on three model systems: aluminum, indium and cobalt. In this paper the authors examine the effect of primary ion energy (2 to 12 keV Ar + ) on excited state yields for these three systems. The influence of the sample matrix on excited state populations of sputtered atoms is also discussed 3. Photoinduced Ultrafast Intramolecular Excited-State Energy Transfer in the Silylene-Bridged Biphenyl and Stilbene (SBS) System: A Nonadiabatic Dynamics Point of View. Science.gov (United States) Wang, Jun; Huang, Jing; Du, Likai; Lan, Zhenggang 2015-07-09 The photoinduced intramolecular excited-state energy-transfer (EET) process in conjugated polymers has received a great deal of research interest because of its important role in the light harvesting and energy transport of organic photovoltaic materials in photoelectric devices. In this work, the silylene-bridged biphenyl and stilbene (SBS) system was chosen as a simplified model system to obtain physical insight into the photoinduced intramolecular energy transfer between the different building units of the SBS copolymer. In the SBS system, the vinylbiphenyl and vinylstilbene moieties serve as the donor (D) unit and the acceptor (A) unit, respectively. The ultrafast excited-state dynamics of the SBS system was investigated from the point of view of nonadiabatic dynamics with the surface-hopping method at the TDDFT level. The first two excited states (S1 and S2) are characterized by local excitations at the acceptor (vinylstilbene) and donor (vinylbiphenyl) units, respectively. Ultrafast S2-S1 decay is responsible for the intramolecular D-A excitonic energy transfer. The geometric distortion of the D moiety play an essential role in this EET process, whereas the A moiety remains unchanged during the nonadiabatic dynamics simulation. The present work provides a direct dynamical approach to understand the ultrafast intramolecular energy-transfer dynamics in SBS copolymers and other similar organic photovoltaic copolymers. 4. Piping damping tests evaluating influence of types of support and excitation International Nuclear Information System (INIS) Arendts, J.G.; Ware, A.G.; Gorman, V.W. 1985-01-01 The United States Nuclear Regulatory Commission and the Electric Power Research Institute have jointly sponsored construction of two laboratory piping systems at the ANCO Engineers facility in California. EG and G Idaho used the second of these systems to obtain piping system damping data using different supports and methods of excitation. The 6-in. carbon steel piping system was approximately 50 ft in length with two 3-in. branch lines. It was supported at five locations and excited using a single electrohydraulic shaker. Both random and swept sine methods of excitations were used. A variable support attached near the shaker location allowed four different configurations to be tested: a rigid strut, a mechanical snubber, a hydraulic snubber, and a rigid strut with a gap. Data were recorded for the lowest nine significant modes. Damping for the first three modes ranged for 1 to 3% of critical damping and decreased as frequency increased. The random excitation produced a slightly higher average overall damping of the system 5. Design considerations for highly effective fluorescence excitation and detection optical systems for molecular diagnostics Science.gov (United States) Kasper, Axel; Van Hille, Herbert; Kuk, Sola 2018-02-01 Modern instruments for molecular diagnostics are continuously optimized for diagnostic accuracy, versatility and throughput. The latest progress in LED technology together with tailored optics solutions allows developing highly efficient photonics engines perfectly adapted to the sample under test. Super-bright chip-on-board LED light sources are a key component for such instruments providing maximum luminous intensities in a multitude of narrow spectral bands. In particular the combination of white LEDs with other narrow band LEDs allows achieving optimum efficiency outperforming traditional Xenon light sources in terms of energy consumption, heat dissipation in the system, and switching time between spectral channels. Maximum sensitivity of the diagnostic system can only be achieved with an optimized optics system for the illumination and imaging of the sample. The illumination beam path must be designed for optimum homogeneity across the field while precisely limiting the angular distribution of the excitation light. This is a necessity for avoiding spill-over to the detection beam path and guaranteeing the efficiency of the spectral filtering. The imaging optics must combine high spatial resolution, high light collection efficiency and optimized suppression of excitation light for good signal-to-noise ratio. In order to achieve minimum cross-talk between individual wells in the sample, the optics design must also consider the generation of stray light and the formation of ghost images. We discuss what parameters and limitations have to be considered in an integrated system design approach covering the full path from the light source to the detector. 6. Kinetics of excited levels in copper-vapor laser International Nuclear Information System (INIS) Smilanski, I. 1981-10-01 A full and representative description of the excited copper level kinetics in a copper-vapor laser is presented. The research was carried out in three stages. The first stage was the development of a representative and reliable measurement cell. A laser tube constructed of refractory materials and an excitation circuit which provides short pulses at a high repetition rate to heat the tube and excite the copper atoms were developed. This stage was also dedicated to characterizing the laser and studying its scaling laws. In the second stage a rapid neasuring system which avoids the problem of spectral line shape was developed. The system is based on the 'hook' method, which utilizes the anomalous dispersion in the vicinity of an atomic line. The light source, a wide band nitrogen-laser-pumped dye laser, ensures a short sampling time, and the recording system, with a television camera face as the recording medium, allows precise data reduction. In the third stage the excited copper level kinetics in a copper vapor laser is measured. The principal conclusions, that only a small part of the energy in the discharge is utilized to populate the upper laser levels and that the lower laser level population is very large at the end of the excitation pulse and cannot be attributed to relaxation of the upper levels, necessitate a new kinetic description of the copper-vapor laser. The laser is not self-terminating; it is activated and terminated by the electrical discharge 7. Validations of CNDOL approximate Hamiltonian as a fast and reliable method to obtain vertical excitation energies in polyatomic systems International Nuclear Information System (INIS) Montero-Alejo, Ana L.; Gonzalez-Santana, Susana; Montero-Cabrera, Luis A.; Hernandez-Rodriguez, Erix Wiliam; Fuentes-Montero, Maria Elena; Bunge-Molina, Carlos F.; Gonzalez, Augusto 2008-01-01 Theoretical prediction of vertical excitation energies and an estimation of charge distributions of polyatomic systems can be calculated, through the configuration interaction of single (CIS) excited determinants procedure, with the CNDOL (Complete Neglect of Differential Overlap considering the l azimuthal quantum number) Hamiltonians. This method does not use adjusted parameters to fit experimental data and only employ a priori data on atomic orbitals and simple formulas to substitute large computations of electronic integrals. In this sense, different functions for bi-electron integrals have been evaluated in order to improve the approximate Hamiltonian. The reliability of predictions and theoretical consistence has been tested with a benchmark set of organic molecules that covers important classes of chromophores including polyenes and other unsaturated aliphatic compounds, aromatic, hydrocarbons, heterocycles, carbonyl compounds, and nucleobases. The calculations are done at identical geometries (MP2) with the same basis set (6-31G) for these medium-sized molecules and the obtained results were statistically compared with other analogous methods and experimental data. The accuracy of prediction of each CNDOL vertical transitions energy increases while the active space is more complete allowing the best variational optimization of CIS matrices i.e. molecular excited states. Moreover and due to the feasible computation procedure for large polyatomic systems, the studies have been extended, as a preliminary work, in the field of optoelectronic materials for photovoltaic applications. Hence, the excitation energies of different conjugated Phenyl-cored Thiophene Dendrimers optimized by DFT (Density Functional Theory) were calculated and show good agreement with the experiment data. The predicted charge distribution during the excitation contributes to understand the photophysics process on these kind materials. (Full text) 8. Design and Implementation of Wideband Exciter for an Ultra-high Resolution Airborne SAR System Directory of Open Access Journals (Sweden) Jia Ying-xin 2013-03-01 Full Text Available According to an ultra-high resolution airborne SAR system with better than 0.1 m resolution, a wideband Linear Frequency Modulated (LFM pulse compression exciter with 14.8 GHz carrier and 3.2 GHz bandwidth is designed and implemented. The selection of signal generation scheme and some key technique points for wideband LFM waveform is presented in detail. Then, an acute test and analysis of the LFM signal is performed. The final airborne experiments demonstrate the validity of the LFM source which is one of the subsystems in an ultra-high resolution airborne SAR system. 9. Study on antilock brake system with elastic membrane vibration generated by controlled solenoid excitation International Nuclear Information System (INIS) Wibowo,; Zakaria,; Lambang, Lullus; Triyono,; Muhayat, Nurul 2016-01-01 The most effective chassis control system for improving vehicle safety during severe braking is anti-lock braking system (ABS). Antilock effect can be gained by vibrate the pad brake at 7 to 20 cycle per second. The aim of this study is to design a new method of antilock braking system with membrane elastic vibrated by solenoid. The influence of the pressure fluctuations of brake fluid is investigated. Vibration data is collected using a small portable accelerometer-slam stick. The experiment results that the vibration of brake pad caused by controlled solenoid excitation at 10 Hz is obtained by our new method. The result of measurements can be altered by varying brake fluid pressure. 10. Study on antilock brake system with elastic membrane vibration generated by controlled solenoid excitation Energy Technology Data Exchange (ETDEWEB) Wibowo,, E-mail: [email protected]; Zakaria,, E-mail: [email protected]; Lambang, Lullus, E-mail: [email protected]; Triyono,, E-mail: [email protected]; Muhayat, Nurul, E-mail: [email protected] [Mechanical Engineering Department, Sebelas Maret University, Surakarta 57128 (Indonesia) 2016-03-29 The most effective chassis control system for improving vehicle safety during severe braking is anti-lock braking system (ABS). Antilock effect can be gained by vibrate the pad brake at 7 to 20 cycle per second. The aim of this study is to design a new method of antilock braking system with membrane elastic vibrated by solenoid. The influence of the pressure fluctuations of brake fluid is investigated. Vibration data is collected using a small portable accelerometer-slam stick. The experiment results that the vibration of brake pad caused by controlled solenoid excitation at 10 Hz is obtained by our new method. The result of measurements can be altered by varying brake fluid pressure. 11. Approximate Forward Difference Equations for the Lower Order Non-Stationary Statistics of Geometrically Non-Linear Systems subject to Random Excitation DEFF Research Database (Denmark) Köylüoglu, H. U.; Nielsen, Søren R. K.; Cakmak, A. S. Geometrically non-linear multi-degree-of-freedom (MDOF) systems subject to random excitation are considered. New semi-analytical approximate forward difference equations for the lower order non-stationary statistical moments of the response are derived from the stochastic differential equations...... of motion, and, the accuracy of these equations is numerically investigated. For stationary excitations, the proposed method computes the stationary statistical moments of the response from the solution of non-linear algebraic equations.... 12. Examination of excited state populations in sputtering using multiphoton resonance ionization International Nuclear Information System (INIS) Kimock, F.M.; Baxter, J.P.; Pappas, D.L.; Kobrin, P.H.; Winograd, N. 1984-01-01 Multiphoton Resonance Ionization has been employed to study the populations of excited state atoms ejected from ion bombarded metal surfaces. Preliminary investigations have focused on three model systems: aluminum, indium and cobalt. In this paper we examine the effect of primary ion energy (2 to 12 keV Ar + ) on excited state yields for these three systems. The influence of the sample matrix on excited state populations of sputtered atoms is also discussed. 8 refs., 4 figs 13. Magnetic excitations in low-dimensional spin systems: neutron scattering study on AV2O5 International Nuclear Information System (INIS) Nakajima, Kenji 1997-01-01 Recent experiments on vanadium oxide bronzes AV 2 O 5 (A=Na, Mg, Li) are reviewed. Experiments are carried out combining two triple-axis spectrometers installed at a thermal beam port and a cold neutron guide at JRR-3M. Spin-wave excitations in single crystals NaV 2 O 5 in the spin-Peierls state shows a steep intra-chain dispersion, which is consistent with estimated exchange interaction from magnetization measurement, and a weak inter-chain dispersion. In the low energy excitation measurement on powder sample of MgV 2 O 5 , we have observed energy gap of 2 meV, which indicates that this material is a ladder system with strong 1D character. Preliminary result on LiV 2 O 5 , which is expected to be a simple 1D antiferromagnet or a zig-zag chain, is also mentioned 14. Surface and bulk excitations in condensed matter International Nuclear Information System (INIS) Ritchie, R.H. 1988-01-01 In this lecture collective and single-particle electron excitations of solids will be discussed with emphasis on the properties of metallic and semiconducting materials. However, some of the general properties of long-wavelength collective modes to be discussed are valid for insulators as well, and some considerations apply to nuclear excitations such as optical or acoustical phonons, dipolar plasmons, etc. The concept of elementary excitations in solids, pioneered by Bohm and Pines almost 4 decades ago, has proved to be extremely useful in understanding the properties of systems of many particles, especially in respect to the response to the action of external probes. 32 refs., 12 figs 15. Excitations of Bose-Einstein condensates at finite temperatures International Nuclear Information System (INIS) Rusch, M. 2000-01-01 Recent experimental observations of collective excitations of Bose condensed atomic vapours have stimulated interest in the microscopic description of the dynamics of a Bose-Einstein condensate confined in an external potential. We present a finite temperature field theory for collective excitations of trapped Bose-Einstein condensates and use a finite-temperature linear response formalism, which goes beyond the simple mean-field approximation of the Gross-Pitaevskii equation. The effect of the non-condensed thermal atoms we include using perturbation theory in a quasiparticle basis. This presents a simple scheme to understand the interaction between condensate and non-condensed atoms and enables us to include the effect the condensate has on collision dynamics. At first we limit our treatment to the case of a spatially homogeneous Bose gas. We include the effect of pair and triplet anomalous averages and thus obtain a gapless theory for the excitations of a weakly interacting system, which we can link to well known results for Landau and Beliaev damping rates. A gapless theory for trapped systems with a static thermal component follows straightforwardly. We then investigate finite temperature excitations of a condensate in a spherically symmetric harmonic trap. We avoid approximations to the density of states and thus emphasise finite size aspects of the problem. We show that excitations couple strongly to a restricted number of modes, giving rise to resonance structure in their frequency spectra. Where possible we derive energy shifts and lifetimes of excitations. For one particular mode, the breathing mode, the effects of the discreteness of the system are sufficiently pronounced that the simple picture of an energy shift and width fails. Experiments in spherical traps have recently become feasible and should be able to test our detailed quantitative predictions. (author) 16. Resonantly enhanced production of excited fragments of gaseous molecules following core-level excitation International Nuclear Information System (INIS) Chen, J.M.; Lu, K.T.; Lee, J.M.; Ho, S.C.; Chang, H.W.; Lee, Y.Y. 2005-01-01 State-selective dissociation dynamics for the excited fragments of gaseous Si(CH 3 ) 2 Cl 2 following Cl 2p and Si 2p core-level excitations have been investigated by resonant photoemission spectroscopy and dispersed UV/optical fluorescence spectroscopy. The main features in the gaseous Si(CH 3 ) 2 Cl 2 fluorescence spectrum are identified as the emission from excited Si, Si + , CH* and H. The core-to-Rydberg excitations at both Si 2p and Cl 2p edges lead to a noteworthy production of not only the excited atomic fragments, neutral and ionic (Si, Si + ) but also the excited diatomic fragments (CH). In particular, the excited neutral atomic fragments Si* are significantly reinforced. The experimental results provide deeper insight into the state-selective dissociation dynamics for the excited fragments of molecules via core-level excitation 17. Development of tunable flashlamp excited dye laser system International Nuclear Information System (INIS) Bhanthumnavin, V.; Apikitmata, S.; Kochareon, P. 1991-01-01 A tunable flashlamp excited dye laser (FEDL) was successfully developed for the first time in Thailand by Thai scientists at KMIT Thonburi (Bangmod). The Rhodamine 6G dissolved in ethyl alcohol was utilized as a laser medium and circulated by a pump through a laser head. The dye cuvette had an inner diameter of 4.0 mm and was 90 mm long. The cavity mirrors M 1 , and M 2 were concave mirrors with reflectivities of 100% and 73% respectively. A power supply of 0-20 kV and current of 0-50 mA charged a capacitor of 0.3 μ f at 10-15 kV which was then discharged via a spark gap through the flashlamp. The output laser wavelengths was tunable from λ = 550-640 nm. It is the first FEDL system, locally developed, which has a tunable wavelength for the laser output. The laser pulse width is about 1.0 μs with energy of 20 mJ and peak power pf 20 KW. The repetition rate of the laser is 1/15 Hz. (author). 14 refs, 7 figs 18. Generation of spiral waves pinned to obstacles in a simulated excitable system Science.gov (United States) Phantu, Metinee; Kumchaiseemak, Nakorn; Porjai, Porramain; Sutthiopad, Malee; Müller, Stefan C.; Luengviriya, Chaiya; Luengviriya, Jiraporn 2017-09-01 Pinning phenomena emerge in many dynamical systems. They are found to stabilize extreme conditions such as superconductivity and super fluidity. The dynamics of pinned spiral waves, whose tips trace the boundary of obstacles, also play an important role in the human health. In heart, such pinned waves cause longer tachycardia. In this article, we present two methods for generating pinned spiral waves in a simulated excitable system. In method A, an obstacle is set in the system prior to an ignition of a spiral wave. This method may be suitable only for the case of large obstacles since it often fails when used for small obstacles. In method B, a spiral wave is generated before an obstacle is placed at the spiral tip. With this method, a pinned spiral wave is always obtained, regardless the obstacle size. We demonstrate that after a transient interval the dynamics of the pinned spiral waves generated by the methods A and B are identical. The initiation of pinned spiral waves in both two- and three-dimensional systems is illustrated. 19. Observation and quantification of the quantum dynamics of a strong-field excited multi-level system. Science.gov (United States) Liu, Zuoye; Wang, Quanjun; Ding, Jingjie; Cavaletto, Stefano M; Pfeifer, Thomas; Hu, Bitao 2017-01-04 The quantum dynamics of a V-type three-level system, whose two resonances are first excited by a weak probe pulse and subsequently modified by another strong one, is studied. The quantum dynamics of the multi-level system is closely related to the absorption spectrum of the transmitted probe pulse and its modification manifests itself as a modulation of the absorption line shape. Applying the dipole-control model, the modulation induced by the second strong pulse to the system's dynamics is quantified by eight intensity-dependent parameters, describing the self and inter-state contributions. The present study opens the route to control the quantum dynamics of multi-level systems and to quantify the quantum-control process. 20. Nonlinear excitations in biomolecules International Nuclear Information System (INIS) Peyrard, M. 1995-01-01 The aim of the workshop entitled ''Nonlinear Excitations in Biomolecules'' is to attempt to bridge the gap between the physicists and biologists communities which is mainly due to language and cultural barriers. The progress of nonlinear science in the last few decades which have shown that the combination of nonlinearity, which characterize most biological phenomena, and cooperative effects in a system having a large number of degrees of freedom, can give rise to coherent excitations with remarkable properties. New concepts, such as solitons nd nonlinear energy localisation have become familiar to physicists and applied mathematicians. It is thus tempting to make an analogy between these coherent excitations and the exceptional stability of some biological processes, such as for instance DNA transcription, which require the coordination of many events in the ever changing environment of a cell. Physicists are now invoking nonlinear excitations to describe and explain many bio-molecular processes while biologists often doubt that the seemingly infinite variety of phenomena that they are attempting to classify can be reduced to such simple concepts. A large part of the meeting is devoted to tutorial lectures rather than to latest research results. The book provides a pedagogical introduction to the two topics forming the backbone of the meeting: the theory of nonlinear excitations and solitons, and their application in biology; and the structure and function of biomolecules, as well as energy and charge transport in biophysics. In order to emphasize the link between physics and biology, the volume is not divided along these two topics but according to biological subjects. Each chapter starts with a short introduction attempting to help the reader to find his way among the contributions and point out the connection between them. 23 lectures over the 32 presented have been selected and refers to quantum properties of macro-molecules. (J.S.) 1. Dissociative Excitation of Thymine by Electron Impact Science.gov (United States) McConkey, William; Tiessen, Collin; Hein, Jeffrey; Trocchi, Joshuah; Kedzierski, Wladek 2014-05-01 A crossed electron-gas beam system coupled to a VUV spectrometer has been used to investigate the dissociation of thymine (C5H6N2O2) into excited atomic fragments in the electron-impact energy range from threshold to 375 eV. A special stainless steel oven is used to vaporize the thymine and form it into a beam where it is intersected by a magnetically collimated electron beam, typical current 50 μA. The main features in the spectrum are the H Lyman series lines. The probability of extracting excited C or N atoms from the ring is shown to be very small. In addition to spectral data, excitation probability curves as a function of electron energy will be presented for the main emission features. Possible dissociation channels and excitation mechanisms in the parent molecule will be discussed. The authors thank NSERC (Canada) for financial support. 2. Excitation function of elastic scattering on 12C + 4He system, at low energies International Nuclear Information System (INIS) Perez-Torres, R.; Aguilera, E. F.; Martinez-Quiroz, E.; Murillo, G.; Belyaeva, T. L.; Maldonado-Velazquez, M. 2011-01-01 Interactions in the 12 C + 4 He system are of great interest in astrophysics and to help determine the relative abundances of elements in stars, at the end of helium burning [1, 2]. The Instituto Nacional de Investigaciones Nucleares (ININ) in Mexico, have made measurements of elastic scattering for this system, using the inverse kinematics method with thick white gas [3, 4], for E CM (0.5 - 4 MeV) θ CM = 180 o . In this work we obtain excitation functions of elastic scattering of 12 C + 4 He system with angular and energy dependence; E CM = 0.5 - 4 MeV and θ CM 100 o -170 o .Using inverse kinematics method with thick white gas and energy loss tables. (Author) 3. Self-excitation of single nanomechanical pillars Science.gov (United States) Kim, Hyun S.; Qin, Hua; Blick, Robert H. 2010-03-01 Self-excitation is a mechanism that is ubiquitous for electromechanical power devices such as electrical generators. This is conventionally achieved by making use of the magnetic field component in electrical generators (Nedic and Lipo 2000 IEEE/IAS Conf. Records (Rome, Italy) vol 1 pp 51-6), a good and widely visible example of which is the wind turbine farm (Muljadi et al 2005 J. Sol. Energy Eng. 127 581-7). In other words, a static force, such as the wind acting on rotor blades, can generate a resonant excitation at a certain mechanical frequency. For nanomechanical systems (Craighead 2000 Science 290 1532-5 Roukes 2001 Phys. World 14 25-31 Cleland 2003 Foundations of Nanomechanics (Berlin: Springer); Ayari et al 2007 Nano Lett. 7 2252-7 Koenig et al 2008 Nat. Nanotechnol. 3 482-4) such a self-excitation (SE) mechanism is also highly desirable, because it can generate mechanical oscillations at radio frequencies by simply applying a dc bias voltage. This is of great importance for low-power signal communication devices and detectors, as well as for mechanical computing elements. For a particular nanomechanical system—the single electron shuttle—this effect was predicted some time ago by Gorelik et al (Phys. Rev. Lett. 80 4526-9). Here, we use a nanoelectromechanical single electron transistor (NEMSET) to demonstrate self-excitation for both the soft and hard regimes, respectively. The ability to use self-excitation in nanomechanical systems may enable the detection of quantum mechanical backaction effects (Naik et al 2006 Nature 443 193-6) in direct tunneling, macroscopic quantum tunneling (Savelev et al 2006 New J. Phys. 8 105-15) and rectification (Pistolesi and Fazio 2005 Phys. Rev. Lett. 94 036806-4). All these effects have so far been overshadowed by the large driving voltages that had to be applied. 4. Microstructure ion Nuclear Spectra at High Excitation International Nuclear Information System (INIS) Ericson, T.E.O. 1969-01-01 The statistical microstructure of highly excited systems is illustrated by the distribution and fluctuations of levels, widths and cross-sections of nuclei both for the case of sharp resonances and the continuum case. The coexistence of simple modes of excitation with statistical effects in terms of strength functions is illustrated by isobaric analogue states. The analogy is made with similar phenomena for coherent light, is solid-state physics and high-energy physics. (author) 5. Stationary responses of a Rayleigh viscoelastic system with zero barrier impacts under external random excitation. Science.gov (United States) Wang, Deli; Xu, Wei; Zhao, Xiangrong 2016-03-01 This paper aims to deal with the stationary responses of a Rayleigh viscoelastic system with zero barrier impacts under external random excitation. First, the original stochastic viscoelastic system is converted to an equivalent stochastic system without viscoelastic terms by approximately adding the equivalent stiffness and damping. Relying on the means of non-smooth transformation of state variables, the above system is replaced by a new system without an impact term. Then, the stationary probability density functions of the system are observed analytically through stochastic averaging method. By considering the effects of the biquadratic nonlinear damping coefficient and the noise intensity on the system responses, the effectiveness of the theoretical method is tested by comparing the analytical results with those generated from Monte Carlo simulations. Additionally, it does deserve attention that some system parameters can induce the occurrence of stochastic P-bifurcation. 6. Multiflavour excited mesons from the fifth dimension International Nuclear Information System (INIS) Paredes, Angel; Talavera, Pere 2005-01-01 We study the Regge trajectories and the quark-antiquark energy in excited hadrons composed by different dynamical mass constituents via the gauge/string correspondence. First we exemplify the procedure in a supersymmetric system, D3-D7, in the extremal case. Afterwards we discuss the model dual to large-N c QCD, D4-D6 system. In the latter case we find the field theory expected gross features of vector like theories: the spectrum resembles that of heavy quarkonia and the Chew-Frautschi plot of the singlet and first excited states is in qualitative agreement with those of lattice QCD. We stress the salient points of including different constituents masses 7. Immersion and Invariance-Based Coordinated Generator Excitation and SVC Control for Power Systems Directory of Open Access Journals (Sweden) 2014-01-01 Full Text Available A nonlinear coordinated control of excitation and SVC of an electrical power system is proposed for transient stability, and voltage regulation enhancement after the occurrence of a large disturbance and a small perturbation. Using the concept of Immersion and Invariance (I&I design methodology, the proposed nonlinear controller is used to not only achieve power angle stability, frequency and voltage regulation but also ensure that the closed-loop system is transiently and asymptotically stable. In order to show the effectiveness of the proposed controller design, the simulation results illustrate that, in spite of the case where a large perturbation occurs on the transmission line or there is a small perturbation to mechanical power inputs, the proposed controller can not only keep the system transiently stable but also simultaneously accomplish better dynamic properties of the system as compared to operation with the existing controllers designed through a coordinated passivation technique controller and a feedback linearization scheme, respectively. 8. Design and development of a parametrically excited nonlinear energy harvester International Nuclear Information System (INIS) Yildirim, Tanju; Ghayesh, Mergen H.; Li, Weihua; Alici, Gursel 2016-01-01 Highlights: • A parametrically broadband energy harvester was fabricated. • Strong softening-type nonlinear behaviour was observed. • Experiments were conducted showing the large bandwidth of the device. - Abstract: An energy harvester has been designed, fabricated and tested based on the nonlinear dynamical response of a parametrically excited clamped-clamped beam with a central point-mass; magnets have been used as the central point-mass which pass through a coil when parametrically excited. Experiments have been conducted for the energy harvester when the system is excited (i) harmonically near the primary resonance; (ii) harmonically near the principal parametric resonance; (iii) by means of a non-smooth periodic excitation. An electrodynamic shaker was used to parametrically excite the system and the corresponding displacement of the magnet and output voltages of the coil were measured. It has been shown that the system displays linear behaviour at the primary resonance; however, at the principal parametric resonance, the motion characteristic of the magnet substantially changed displaying a strong softening-type nonlinearity. Theoretical simulations have also been conducted in order to verify the experimental results; the comparison between theory and experiment were within very good agreement of each other. The energy harvester developed in this paper is capable of harvesting energy close to the primary resonance as well as the principal parametric resonance; the frequency-band has been broadened significantly mainly due to the nonlinear effects as well as the parametric excitation. 9. Excited states 2 CERN Document Server Lim, Edward C 2013-01-01 Excited States, Volume 2 is a collection of papers that deals with molecules in the excited states. The book describes the geometries of molecules in the excited electronic states. One paper describes the geometries of a diatomic molecule and of polyatomic molecules; it also discusses the determination of the many excited state geometries of molecules with two, three, or four atoms by techniques similar to diatomic spectroscopy. Another paper introduces an ordered theory related to excitons in pure and mixed molecular crystals. This paper also presents some experimental data such as those invo 10. Flow-excited acoustic resonance excitation mechanism, design guidelines, and counter measures International Nuclear Information System (INIS) 2014-01-01 The excitation mechanism of acoustic resonances has long been recognized, but the industry continues to be plagued by its undesirable consequences, manifested in severe vibration and noise problems in a wide range of industrial applications. This paper focuses on the nature of the excitation mechanism of acoustic resonances in piping systems containing impinging shear flows, such as flow over shallow and deep cavities. Since this feedback mechanism is caused by the coupling between acoustic resonators and shear flow instabilities, attention is focused first on the nature of various types of acoustic resonance modes and then on the aero-acoustic sound sources, which result from the interaction of the inherently unstable shear flow with the sound field generated by the resonant acoustic modes. Various flow-sound interaction patterns are discussed, in which the resonant sound field can be predominantly parallel or normal to the mean flow direction and the acoustic wavelength can be an order of magnitude longer than the length scale of the separated shear flow or as short as the cavity length scale. Since the state of knowledge in this field has been recently reviewed by Tonon et al. (2011, 'Aero-acoustics of Pipe Systems With Closed Branches', Int. J. Aeroacoust., 10(2), pp. 201-276), this article focuses on the more practical aspects of the phenomenon, including various flow sound interaction patterns and the resulting aero-acoustic sources, which are relevant to industrial applications. A general design guide proposal and practical means to alleviate the excitation mechanism are also presented. These are demonstrated by two examples of recent industrial case histories dealing with acoustic fatigue failure of the steam dryer in a boiling water reactor (BWR) due to acoustic resonance in the main steam piping and acoustic resonances in the roll posts of the Short Take-Off and Vertical Lift Joint Strike Fighter (JSF). (authors) 11. Nerve conduction and excitability studies in peripheral nerve disorders DEFF Research Database (Denmark) Krarup, Christian; Moldovan, Mihai 2009-01-01 counterparts in the peripheral nervous system, in some instances without peripheral nervous system symptoms. Both hereditary and acquired demyelinating neuropathies have been studied and the effects on nerve pathophysiology have been compared with degeneration and regeneration of axons. SUMMARY: Excitability......PURPOSE OF REVIEW: The review is aimed at providing information about the role of nerve excitability studies in peripheral nerve disorders. It has been known for many years that the insight into peripheral nerve pathophysiology provided by conventional nerve conduction studies is limited. Nerve...... excitability studies are relatively novel but are acquiring an increasingly important role in the study of peripheral nerves. RECENT FINDINGS: By measuring responses in nerve that are related to nodal function (strength-duration time constant, rheobase and recovery cycle) and internodal function (threshold... 12. Sampling system for pulsed signals. Study of the radioactive lifetimes of excited 32P1/2 and 32P3/2 states of Na, excited by a tunable dye laser International Nuclear Information System (INIS) Thomas, P.; Campos, J. 1979-01-01 A system for sampling and averaging repetitive signals in the order of nanoseconds is discussed. The system uses as storage memory a multichannel analyzer operating in multi scaling mode. This instrument is employed for the measurement of atomic level lifetimes using a dye laser to excite the atoms and is applied to the study of lifetimes of the 3 2 P1/2 and 3 2 P3/2 states of sodium. (Author) 32 refs 13. Sweep excitation with order tracking: A new tactic for beam crack analysis Science.gov (United States) Wei, Dongdong; Wang, KeSheng; Zhang, Mian; Zuo, Ming J. 2018-04-01 Crack detection in beams and beam-like structures is an important issue in industry and has attracted numerous investigations. A local crack leads to global system dynamics changes and produce non-linear vibration responses. Many researchers have studied these non-linearities for beam crack diagnosis. However, most reported methods are based on impact excitation and constant frequency excitation. Few studies have focused on crack detection through external sweep excitation which unleashes abundant dynamic characteristics of the system. Together with a signal resampling technique inspired by Computed Order Tracking, this paper utilize vibration responses under sweep excitations to diagnose crack status of beams. A data driven method for crack depth evaluation is proposed and window based harmonics extracting approaches are studied. The effectiveness of sweep excitation and the proposed method is experimentally validated. 14. New features of nuclear excitation by {alpha} particles scattering; Nouveaux aspects de l'excitation nucleaire par diffusion de particules {alpha} Energy Technology Data Exchange (ETDEWEB) Saudinos, J [Commissariat a l' Energie Atomique, Saclay (France). Centre d' Etudes Nucleaires 1962-07-01 Inelastic scattering of medium energy a particles by nuclei is known to excite preferentially levels of collective character. We have studied the scattering of isotopically enriched targets of Ca, Fe, Ni, Cu, Zn. In part I, we discuss the theoretical features of the interaction. In part II, we describe the experimental procedure. Results are presented and analysed in part III. {alpha} particles scattering by Ca{sup 40} is showed to excite preferentially odd parity levels. In odd nuclei we have observed multiplets due to the coupling of the odd nucleon with the even-even core vibrations. For even-even nuclei, a few levels are excited with lower cross-sections between the well-known first 2{sup +} and 3{sup -} states. Some could be members of the two phonon quadrupole excitation and involve a double nuclear excitation process. (author) [French] On sait que la diffusion inelastique des particules alpha de moyenne energie excite preferentiellement des niveaux de caractere collectif. Nous avons etudie la diffusion des particules alpha de 44 MeV du cyclotron de Saclay par des isotopes separes de Ca, Fe, Ni, Cu, Zn. Dans la premiere partie nous exposons les theories de cette interaction. Dans la seconde nous decrivons le systeme experimental. Les resultats sont donnes dans la troisieme partie. Nous montrons que les niveaux excites preferentiellement pour {sup 40}Ca par diffusion ({alpha},{alpha}') sont de parite negative. Dans les noyaux pair-impair nous avons observe des multiplets dus au couplage du nucleon celibataire avec les vibrations du coeur pair-pair. Pour les noyaux pair-pair nous avons pu etudier entre le premier niveau 2{sup +} et le niveau 3{sup -} deja bien connus certains etats plus faiblement excites. Il semble qu'ils sont dus a une excitation quadrupolaire a deux phonons et impliquent un processus de double excitation nucleaire. (auteur) 15. Suppression of radiation excitation in focusing environment International Nuclear Information System (INIS) Huang, Z.; Ruth, R.D. 1996-12-01 Radiation damping and quantum excitation in an electron damping ring and a straight focusing channel are reviewed. They are found to be the two limiting cases in the study of a general bending and focusing combined system. In the intermediate regime where the radiation formation length is comparable to the betatron wavelength, quantum excitation can be exponentially suppressed by focusing field. This new regime may have interesting applications in the generation of ultra-low emittance beams 16. Method of producing excited states of atomic nuclei International Nuclear Information System (INIS) Morita, M.; Morita, R. 1976-01-01 A method is claimed of producing excited states of atomic nuclei which comprises bombarding atoms with x rays or electrons, characterized in that (1) in the atoms selected to be produced in the excited state of their nuclei, (a) the difference between the nuclear excitation energy and the difference between the binding energies of adequately selected two electron orbits is small enough to introduce the nuclear excitation by electron transition, and (b) the system of the nucleus and the electrons in the case of ionizing an orbital electron in said atoms should satisfy the spin and parity conservation laws; and (2) the energy of the bombarding x rays or electrons should be larger than the binding energy of one of the said two electron orbits which is located at shorter distance from the atomic nucleus. According to the present invention, atomic nuclei can be excited in a relatively simple manner without requiring the use of large scale apparatus, equipment and production facilities, e.g., factories. It is also possible to produce radioactive substances or separate a particular isotope with an extremely high purity from a mixture of isotopes by utilizing nuclear excitation 17. Study of the excitation mechanisms of the second positive system in the negative glow of a N2-Ar discharge International Nuclear Information System (INIS) Isola, L; Lopez, M; Gomez, B J 2011-01-01 In an Ar-N 2 discharge, the high excitation transfer from Ar( 3 P 2,0 ) to N 2 produces an overpopulation of the high rotational levels of the bands of the second positive system (SPS), and so the spectra interpretation is not straightforward. This paper presents a fit function for the SPS bands measured in Ar-N 2 , which allows us to study the excitation process contributions to the N 2 (C) level. The procedure was tested in the negative glow of a pulsed Ar-N 2 discharge at a pressure of 2.5 Torr, for different mixture concentrations. In this discharge, through the fitting, it was possible to calculate the variation of the N 2 (C) densities produced by different excitation processes as well as the variation of Ar metastable density. 18. Anisotropy of electronic states excited in ion-atom collisions International Nuclear Information System (INIS) Boskamp, E.B. 1983-01-01 The author reports coincidence measurements made on the He + + Ne and He + + He systems. The complex population amplitudes for the magnetic sublevels of the investigated excited states, Ne(2p 4 3s 2 ) 1 D and He(2p 2 ) 1 D, were completely determined and possible excitation mechanisms are described. (Auth.) 19. On isospin excitation energy International Nuclear Information System (INIS) Li Wenfei; Zhang Fengshou; Chen Liewen 2001-01-01 Within the framework of Hartree-Fock theory using the extended Skyrme effective interaction, the isospin excitation energy as a function of relative neutron excess δ was investigated at different temperatures and densities. It was found that the isospin excitation energy decreased with the increment of temperature and/or the decrement of density. The authors pointed out that the decrement of isospin excitation energy was resulted from the weakening of quantum effect with increment of temperature and/or decrement of density. Meanwhile, the relationship between the isospin excitation energy and the symmetry energy was discussed and found that the symmetry energy was just a part of the isospin excitation energy. With increasing temperature and decreasing density, the contribution of the symmetry energy to the isospin excitation energy becomes more and more important. The isospin excitation energy as a function of relative neutron excess was also investigated using different potential parameters. The results shows that the isospin excitation energy is almost independent of the incompressibility and the effective mass, but strongly depends on the symmetry energy strength coefficient, which indicates that it is possible to extract the symmetry energy of the nuclear equation of state by investigating the isospin excitation energy in experiments 20. Analysis and control of the effects of over excitation limiters on the stability of the Itaipu HVAC transmission system Energy Technology Data Exchange (ETDEWEB) Jardim, J L; Macedo, N J; Santo, S E; Praca, A S [FURNAS Centrais Eletricas S.A., Rio de Janeiro, RJ (Brazil) 1994-12-31 The effect of over excitation limiters on power system voltage stability is presented in this paper. A linear analysis based on system eigenvalues for various operating conditions shows that voltage collapse is essentially a dynamic phenomenon. Time simulations using digital tools and real-time simulator were performed to verify lin ear results and study large disturbances. A control system designed to keep system in secure region is proposed. (author) 3 refs., 9 figs. 1. Statistical properties of highly excited quantum eigenstates of a strongly chaotic system International Nuclear Information System (INIS) Aurich, R.; Steiner, F. 1992-06-01 Statistical properties of highly excited quantal eigenstates are studied for the free motion (geodesic flow) on a compact surface of constant negative curvature (hyperbolic octagon) which represents a strongly chaotic system (K-system). The eigenstates are expanded in a circular-wave basis, and it turns out that the expansion coefficients behave as Gaussian pseudo-random numbers. It is shown that this property leads to a Gaussian amplitude distribution P(ψ) in the semiclassical limit, i.e. the wavefunctions behave as Gaussian random functions. This behaviour, which should hold for chaotic systems in general, is nicely confirmed for eigenstates lying 10000 states above the ground state thus probing the semiclassical limit. In addition, the autocorrelation function and the path-correlation function are calculated and compared with a crude semiclassical Bessel-function approximation. Agreement with the semiclassical prediction is only found, if a local averaging is performed over roughly 1000 de Broglie wavelengths. On smaller scales, the eigenstates show much more structure than predicted by the first semiclassical approximation. (orig.) 2. Multiflavour excited mesons from the fifth dimension Energy Technology Data Exchange (ETDEWEB) Paredes, Angel [Centre de Physique Theorique, Ecole Polytechnique, 91128 Palaiseau (France)]. E-mail: [email protected]; Talavera, Pere [Departament de Fisica i Enginyeria Nuclear, Universitat Politecnica de Catalunya, Jordi Girona 1-3, E-08034 Barcelona (Spain)]. E-mail: [email protected] 2005-05-02 We study the Regge trajectories and the quark-antiquark energy in excited hadrons composed by different dynamical mass constituents via the gauge/string correspondence. First we exemplify the procedure in a supersymmetric system, D3-D7, in the extremal case. Afterwards we discuss the model dual to large-N{sub c} QCD, D4-D6 system. In the latter case we find the field theory expected gross features of vector like theories: the spectrum resembles that of heavy quarkonia and the Chew-Frautschi plot of the singlet and first excited states is in qualitative agreement with those of lattice QCD. We stress the salient points of including different constituents masses. 3. Anisotropy in the simultaneous excitation of two colliding atoms to various substate combinations International Nuclear Information System (INIS) Moorman, L. 1987-01-01 In this thesis double-atom excitation (DAE) processes in atomic collision experiments are studied by measuring the angular correlation of two coincident photons emitted by both excited collision particles. The analytical expression for the angular correlation function is derived which contains as adjustable parameters the various (complex) excitation amplitudes integrated over all scattering angles. The He+He system is investigated, for projectile energies between 0.5 and 3.5 keV, in which both particles are excited simultaneously to the 2 1 P state. The relation between photon correlations and atomic state correlations is investigated and the density matrix elements are calculated for a statistical distribution of the excited atomic substates into which a certain symmetry is incorporated. Collisions between metastable and groundstate He atoms are considered. Single-photon spectra are presented and compared with spectra from the He+He collision system. Coincidence measurements were performed on these collision systems to study possible double-atom excitations. Coincidences between two ultraviolet as well as an ultraviolet and a visible photon were measu0515 Also a measurement is reported of the relative population of the magnetic substates of the 3 1 D state of helium. Coincidence measurements on two ultraviolet photons emitted upon Ne-Ne and He-Ne collisions are described and the double-atom excitations for these systems are studied. For Ne+Ne no coincidence peaks were found. For He+Ne double-atom excitation was observed and from the measured angular correlations the corresponding density matrix elements for some kinetic energies of the projectile. (Auth.) 4. Asymmetric excitation of surface plasmons by dark mode coupling KAUST Repository Zhang, X. 2016-02-19 Control over surface plasmons (SPs) is essential in a variety of cutting-edge applications, such as highly integrated photonic signal processing systems, deep-subwavelength lasing, high-resolution imaging, and ultrasensitive biomedical detection. Recently, asymmetric excitation of SPs has attracted enormous interest. In free space, the analog of electromagnetically induced transparency (EIT) in metamaterials has been widely investigated to uniquely manipulate the electromagnetic waves. In the near field, we show that the dark mode coupling mechanism of the classical EIT effect enables an exotic and straightforward excitation of SPs in a metasurface system. This leads to not only resonant excitation of asymmetric SPs but also controllable exotic SP focusing by the use of the Huygens-Fresnel principle. Our experimental findings manifest the potential of developing plasmonic metadevices with unique functionalities. 5. Asymmetric excitation of surface plasmons by dark mode coupling KAUST Repository Zhang, X.; Xu, Q.; Li, Q.; Xu, Y.; Gu, J.; Tian, Z.; Ouyang, C.; Liu, Y.; Zhang, S.; Zhang, Xixiang; Han, J.; Zhang, W. 2016-01-01 Control over surface plasmons (SPs) is essential in a variety of cutting-edge applications, such as highly integrated photonic signal processing systems, deep-subwavelength lasing, high-resolution imaging, and ultrasensitive biomedical detection. Recently, asymmetric excitation of SPs has attracted enormous interest. In free space, the analog of electromagnetically induced transparency (EIT) in metamaterials has been widely investigated to uniquely manipulate the electromagnetic waves. In the near field, we show that the dark mode coupling mechanism of the classical EIT effect enables an exotic and straightforward excitation of SPs in a metasurface system. This leads to not only resonant excitation of asymmetric SPs but also controllable exotic SP focusing by the use of the Huygens-Fresnel principle. Our experimental findings manifest the potential of developing plasmonic metadevices with unique functionalities. 6. Study of a Quantum Dot in an Excited State Science.gov (United States) Slamet, Marlina; Sahni, Viraht We have studied the first excited singlet state of a quantum dot via quantal density functional theory (QDFT). The quantum dot is represented by a 2D Hooke's atom in an external magnetostatic field. The QDFT mapping is from an excited singlet state of this interacting system to one of noninteracting fermions in a singlet ground state. The results of the study will be compared to (a) the corresponding mapping from a ground state of the quantum dot and (b) to the similar mapping from an excited singlet state of the 3D Hooke's atom. 7. Non-orthogonal configuration interaction for the calculation of multielectron excited states Energy Technology Data Exchange (ETDEWEB) Sundstrom, Eric J., E-mail: [email protected]; Head-Gordon, Martin [Department of Chemistry, University of California Berkeley, Berkeley, California 94720, USA and Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720 (United States) 2014-03-21 We apply Non-orthogonal Configuration Interaction (NOCI) to molecular systems where multielectron excitations, in this case double excitations, play a substantial role: the linear polyenes and β-carotene. We demonstrate that NOCI when applied to systems with extended conjugation, provides a qualitatively correct wavefunction at a fraction of the cost of many other multireference treatments. We also present a new extension to this method allowing for purification of higher-order spin states by utilizing Generalized Hartree-Fock Slater determinants and the details for computing 〈S{sup 2}〉 for the ground and excited states. 8. Effects of Energy Dissipation on the Parametric Excitation of a Coupled Qubit-Cavity System Science.gov (United States) Remizov, S. V.; Zhukov, A. A.; Shapiro, D. S.; Pogosov, W. V.; Lozovik, Yu. E. 2018-02-01 We consider a parametrically driven system of a qubit coupled to a cavity taking into account different channels of energy dissipation. We focus on the periodic modulation of a single parameter of this hybrid system, which is the coupling constant between the two subsystems. Such a modulation is possible within the superconducting realization of qubit-cavity coupled systems, characterized by an outstanding degree of tunability and flexibility. Our major result is that energy dissipation in the cavity can enhance population of the excited state of the qubit in the steady state, while energy dissipation in the qubit subsystem can enhance the number of photons generated from vacuum. We find optimal parameters for the realization of such dissipation-induced amplification of quantum effects. Our results might be of importance for the full control of quantum states of coupled systems as well as for the storage and engineering of quantum states. 9. Consideration of Gyroscopic Effect in Fault Detection and Isolation for Unbalance Excited Rotor Systems Directory of Open Access Journals (Sweden) Zhentao Wang 2012-01-01 Full Text Available Fault detection and isolation (FDI in rotor systems often faces the problem that the system dynamics is dependent on the rotor rotary frequency because of the gyroscopic effect. In unbalance excited rotor systems, the continuously distributed unbalances are hard to be determined or estimated accurately. The unbalance forces as disturbances make fault detection more complicated. The aim of this paper is to develop linear time invariant (LTI FDI methods (i.e., with constant parameters for rotor systems under consideration of gyroscopic effect and disturbances. Two approaches to describe the gyroscopic effect, that is, as unknown inputs and as model uncertainties, are investigated. Based on these two approaches, FDI methods are developed and the results are compared regarding the resulting FDI performances. Results are obtained by the application in a rotor test rig. Restrictions for the application of these methods are discussed. 10. Elementary excitations in nuclei International Nuclear Information System (INIS) Lemmer, R.H. 1987-01-01 The role of elementary quasi-particle and quasi-hole excitations is reviewed in connection with the analysis of data involving high-lying nuclear states. This article includes discussions on: (i) single quasi-hole excitations in pick-up reactions, (ii) the formation of single quasi-hole and quasi-particle excitations (in different nuclei) during transfer reactions, followed by (iii) quasi-particle quasi-hole excitations in the same nucleus that are produced by photon absorption. Finally, the question of photon absorption in the vicinity of the elementary Δ resonance is discussed, where nucleonic as well as nuclear degrees of freedom can be excited 11. Soil-structure interaction effects for laterally excited liquid-tank system International Nuclear Information System (INIS) Tang, Yu; Veletsos, A.S. 1992-01-01 Following a brief review of the mechanical model for liquid-storage tanks which permits consideration of the effects of tank and ground flexibility, and lateral and rocking base excitations, the effects of both kinematic and inertia interaction effects on the response of the tank-liquid system are examined and elucidated. The free-field motion is defined by a power spectral density function and an incoherence function, which characterizes the spatial variability of the ground motion due to the vertically incident incoherence waves. The quantities examined are the ensemble means of the peak values of the response. The results are compared with those obtained for no soil-structure interaction and for kinematic interaction to elucidate the nature and relative importance of the two interactions. Only the impulsive actions are examined, the convective actions are for all practical purposes unaffected by both kinematic and inertia interactions. It is shown that the major reduction of the response is attributed to inertia interaction. 20 refs 12. Electronic excitations in metallic systems: from defect annihilation to track formation International Nuclear Information System (INIS) Dunlop, A.; Lesueur, D. 1991-01-01 This paper presents an overview of the effects of high electronic energy deposition in metallic targets irradiated with GeV heavy ions. The main result of these investigations is that high electronic excitations lead to various and sometimes conflicting effects according to the nature of the target: - partial annealing of the defects induced by elastic collisions, - creation of additional disorder, - phase transformation (tracks formation and amorphization), - anisotropic growth. These different effects of high electronic energy deposition in metallic targets are probably manifestations at various degrees of the same basic energy transfer process between the excited electrons and the target atoms. Up to now no theoretical model explains these effects. 24 refs 13. Pure odd-order oscillators with constant excitation Science.gov (United States) Cveticanin, L. 2011-02-01 In this paper the excited vibrations of a truly nonlinear oscillator are analyzed. The excitation is assumed to be constant and the nonlinearity is pure (without a linear term). The mathematical model is a second-order nonhomogeneous differential equation with strong nonlinear term. Using the first integral, the exact value of period of vibration i.e., angular frequency of oscillator described with a pure nonlinear differential equation with constant excitation is analytically obtained. The closed form solution has the form of gamma function. The period of vibration depends on the value of excitation and of the order and coefficient of the nonlinear term. For the case of pure odd-order-oscillators the approximate solution of differential equation is obtained in the form of trigonometric function. The solution is based on the exact value of period of vibration. For the case when additional small perturbation of the pure oscillator acts, the so called 'Cveticanin's averaging method' for a truly nonlinear oscillator is applied. Two special cases are considered: one, when the additional term is a function of distance, and the second, when damping acts. To prove the correctness of the method the obtained results are compared with those for the linear oscillator. Example of pure cubic oscillator with constant excitation and linear damping is widely discussed. Comparing the analytically obtained results with exact numerical ones it is concluded that they are in a good agreement. The investigations reported in the paper are of special interest for those who are dealing with the problem of vibration reduction in the oscillator with constant excitation and pure nonlinear restoring force the examples of which can be found in various scientific and engineering systems. For example, such mechanical systems are seats in vehicles, supports for machines, cutting machines with periodical motion of the cutting tools, presses, etc. The examples can be find in electronics 14. Excitation equilibria in plasmas: a classification International Nuclear Information System (INIS) Mullen, J.-J.A.M. van der. 1986-01-01 In this thesis the author presents a classification of plasmas based on the atomic state distribution function. The study is based on the relation between the distribution function and the underlying processes and starts with the proper understanding of thermodynamic equilibrium (TE). Four types of proper balances are relevant: The 'Maxwell balance' of kinetic energy transfer, the 'Boltzmann balance' of excitation/deexcitation, the 'Saha balance' of ionization/recombination and the 'Planck balance' for interaction of atoms with radiation. Special attention is paid to the distribution function of the ionizing excitation saturation balance. The classification theory of the distribution functions in relation with underlying balances is supported by experimental evidence in an ionizing argon plasma. The AR I system provides a pertinent support of the theory. Experimental facts found in the AR II system can be interpreted in global terms. (Auth.) 15. Development and performance test of picosecond pulse x-ray excited streak camera system for scintillator characterization International Nuclear Information System (INIS) Yanagida, Takayuki; Fujimoto, Yutaka; Yoshikawa, Akira 2010-01-01 To observe time and wavelength-resolved scintillation events, picosecond pulse X-ray excited streak camera system is developed. The wavelength range spreads from vacuum ultraviolet (VUV) to near infrared region (110-900 nm) and the instrumental response function is around 80 ps. This work describes the principle of the newly developed instrument and the first performance test using BaF 2 single crystal scintillator. Core valence luminescence of BaF 2 peaking around 190 and 220 nm is clearly detected by our system, and the decay time turned out to be of 0.7 ns. These results are consistent with literature and confirm that our system properly works. (author) 16. 340nm UV LED excitation in time-resolved fluorescence system for europium-based immunoassays detection DEFF Research Database (Denmark) Rodenko, Olga; Fodgaard, Henrik; Tidemand-Lichtenberg, Peter 2017-01-01 In immunoassay analyzers for in-vitro diagnostics, Xenon flash lamps have been widely used as excitation light sources. Recent advancements in UV LED technology and its advantages over the flash lamps such as smaller footprint, better wall-plug efficiency, narrow emission spectrum......, and no significant afterglow, have made them attractive light sources for gated detection systems. In this paper, we report on the implementation of a 340 nm UV LED based time-resolved fluorescence system based on europium chelate as a fluorescent marker. The system performance was tested with the immunoassay based...... on the cardiac marker, TnI. The same signal-to-noise ratio as for the flash lamp based system was obtained, operating the LED below specified maximum current. The background counts of the system and its main contributors were measured and analyzed. The background of the system of the LED based unit was improved... 17. Clinical Comparison of Pulse and Chirp Excitation DEFF Research Database (Denmark) Pedersen, Morten Høgholm; Misaridis, T.; Jensen, Jørgen Arendt 2002-01-01 Coded excitation (CE) using frequency modulated signals (chirps) combined with modified matched filtering has earlier been presented showing promising results in simulations and in-vitro. In this study an experimental ultrasound system is evaluated in a clinical setting, where image sequences...... and short pulse excitation to simultaneously produce identical image sequences using both techniques. Nine healthy male volunteers were scanned in abdominal locations. All sequences were evaluated by 3 skilled medical doctors, blinded to each other and to the technique used. They assessed the depth (1... 18. Pilot testing of a hydraulic bridge exciter Directory of Open Access Journals (Sweden) 2015-01-01 Full Text Available This paper describes the development of a hydraulic bridge exciter and its first pilot testing on a full scale railway bridge in service. The exciter is based on a hydraulic load cylinder with a capacity of 50 kN and is intended for controlled dynamic loading up to at least 50 Hz. The load is applied from underneath the bridge, enabling testing while the railway line is in service. The system is shown to produce constant load amplitude even at resonance. The exciter is used to experimentally determine frequency response functions at all sensor locations, which serve as valuable input for model updating and verification. An FE-model of the case study bridge has been developed that is in good agreement with the experimental results. 19. High Tc Superconducting Magnet Excited by a Semiconductor Thermoelectric Element Science.gov (United States) Kuriyama, T.; Ono, M.; Tabe, S.; Oguchi, A.; Okamura, T. 2006-04-01 A high Tc superconducting (HTS) magnet excited by a thermal electromotive force of a thermoelectric element is studied. This HTS magnet has the advantages of compactness, lightweight and continuous excitation in comparison with conventional HTS magnets, because this HTS magnet does not need a large external power source. In this system, a heat input into the cryogenic environment is necessary to excite the thermoelectric element for constant operation. This heat generation, however, causes a rise in temperature of an HTS coil and reduces the system performance. In this paper, a newly designed magnet system which adopted a two-stage GM cryocooler was investigated. It enabled us to control the temperature of a thermoelectric element and that of an HTS coil independently. The temperature of the HTS coil could be kept at 10-20 K at the second stage of the GM cryocooler, while the thermoelectric element could be excited at higher temperature in the range of 50-70 K at the first stage, where the performance of the thermoelectric element was higher. The experimental results on this HTS magnet are shown and the possibility of the thermoelectric element as a main power source of the HTS magnets is discussed. 20. Energy harvesting from coherent resonance of horizontal vibration of beam excited by vertical base motion Energy Technology Data Exchange (ETDEWEB) Lan, C. B.; Qin, W. Y. [Department of Engineering Mechanics, Northwestern Polytechnical University, Xi' an 710072 (China) 2014-09-15 This letter investigates the energy harvesting from the horizontal coherent resonance of a vertical cantilever beam subjected to the vertical base excitation. The potential energy of the system has two symmetric potential wells. So, under vertical excitation, the system can jump between two potential wells, which will lead to the large vibration in horizontal direction. Two piezoelectric patches are pasted to harvest the energy. From experiment, it is found that the vertical excitation can make the beam turn to be bistable. The system can transform vertical vibration into horizontal vibration of low frequency when excited by harmonic motion. The horizontal coherence resonance can be observed when excited by a vertical white noise. The corresponding output voltages of piezoelectric films reach high values. 1. Comparison of exciplex generation under optical and X-ray excitation Science.gov (United States) Kipriyanov, A. A.; Melnikov, A. R.; Stass, D. V.; Doktorov, A. B. 2017-09-01 Exciplex generation under optical and X-ray excitation in identical conditions is experimentally compared using a specially chosen model donor-acceptor system, anthracene (electron acceptor) and N,N-dimethylaniline (electron donor) in non-polar solution, and the results are analyzed and interpreted based on analytically calculated luminescence quantum yields. Calculations are performed on the basis of kinetic equations for multistage schemes of bulk exciplex production reaction under optical excitation and combination of bulk and geminate reactions of radical ion pairs under X-ray excitation. These results explain the earlier experimentally found difference in the ratio of the quantum yields of exciplexes and excited electron acceptors (exciplex generation efficiency) and the corresponding change in the exciplex generation efficiency under X-irradiation as compared to the reaction under optical excitation. 2. The CLAS Excited Baryon Program at Jefferson Laboratory International Nuclear Information System (INIS) Crede, Volker 2009-01-01 Nucleons are complex systems of confined quarks and exhibit characteristic spectra of excited states. Highly excited nucleon states are sensitive to details of quark confinement which is poorly understood within Quantum Chromodynamics (QCD), the fundamental theory of strong interactions. Thus, measurements of excited states and the corresponding determination of their properties are needed to come to a better understanding of how confinement works in nucleons. However, the excited states of the nucleon cannot simply be inferred from cleanly separated spectral lines. Quite the contrary, a spectral analysis in nucleon resonance physics is challenging because of the fact that the resonances are broadly overlapping states which decay into a multitude of final states involving mesons and baryons. To provide a consistent and complete picture of an individual nucleon resonance, the various possible production and decay channels must be treated in a multi-channel framework that permits separat 3. Sub-50 fs excited state dynamics of 6-chloroguanine upon deep ultraviolet excitation. Science.gov (United States) Mondal, Sayan; Puranik, Mrinalini 2016-05-18 The photophysical properties of natural nucleobases and their respective nucleotides are ascribed to the sub-picosecond lifetime of their first singlet states in the UV-B region (260-350 nm). Electronic transitions of the ππ* type, which are stronger than those in the UV-B region, lie at the red edge of the UV-C range (100-260 nm) in all isolated nucleobases. The lowest energetic excited states in the UV-B region of nucleobases have been investigated using a plethora of experimental and theoretical methods in gas and solution phases. The sub-picosecond lifetime of these molecules is not a general attribute of all nucleobases but specific to the five primary nucleobases and a few xanthine and methylated derivatives. To determine the overall UV photostability, we aim to understand the effect of more energetic photons lying in the UV-C region on nucleobases. To determine the UV-C initiated photophysics of a nucleobase system, we chose a halogen substituted purine, 6-chloroguanine (6-ClG), that we had investigated previously using resonance Raman spectroscopy. We have performed quantitative measurements of the resonance Raman cross-section across the Bb absorption band (210-230 nm) and constructed the Raman excitation profiles. We modeled the excitation profiles using Lee and Heller's time-dependent theory of resonance Raman intensities to extract the initial excited state dynamics of 6-ClG within 30-50 fs after photoexcitation. We found that imidazole and pyrimidine rings of 6-ClG undergo expansion and contraction, respectively, following photoexcitation to the Bb state. The amount of distortions of the excited state structure from that of the ground state structure is reflected by the total internal reorganization energy that is determined at 112 cm(-1). The contribution of the inertial component of the solvent response towards the total reorganization energy was obtained at 1220 cm(-1). In addition, our simulation also yields an instantaneous response of the first 4. A density matrix renormalization group study of low-lying excitations ... Symmetrized density-matrix-renormalization-group calculations have been carried out, within Pariser-Parr-Pople Hamiltonian, to explore the nature of the ground and low-lying excited states of long polythiophene oligomers. We have exploited 2 symmetry and spin parity of the system to obtain excited states of ... 5. Integrated modeling and analysis of ball screw feed system and milling process with consideration of multi-excitation effect Science.gov (United States) Zhang, Xing; Zhang, Jun; Zhang, Wei; Liang, Tao; Liu, Hui; Zhao, Wanhua 2018-01-01 The present researches about feed drive system and milling process are almost independent with each other, and ignore the interaction between the two parts, especially the influence of nonideal motion of feed drive system on milling process. An integrated modeling method of ball screw feed system and milling process with multi-excitation effect is proposed in this paper. In the integrated model, firstly an analytical model of motor harmonic torque with consideration of asymmetrical drive circuit and asymmetrical permanent magnet is given. Then, the numerical simulation procedure of cutter/workpiece engagement during milling process with displacement fluctuation induced by harmonic torque is put forward, which is followed by the solving flow for the proposed integrated model. Based on the integrated model, a new kind of quality defect shown as contour low frequency oscillation on machined surface is studied by experiments and simulations. The results demonstrate that the forming mechanism of the contour oscillation can be ascribed to the multi-excitation effect with motor harmonic torque and milling force. Moreover, the influence of different milling conditions on the contour oscillation characteristics, particularly on surface roughness, are further discussed. The results indicate that it is necessary to explain the cause of the new kind of quality defect with a view of system integration. 6. Adaptive transition rates in excitable membranes Directory of Open Access Journals (Sweden) Shimon Marom 2009-02-01 Full Text Available Adaptation of activity in excitable membranes occurs over a wide range of timescales. Standard computational approaches handle this wide temporal range in terms of multiple states and related reaction rates emanating from the complexity of ionic channels. The study described here takes a different (perhaps complementary approach, by interpreting ion channel kinetics in terms of population dynamics. I show that adaptation in excitable membranes is reducible to a simple Logistic-like equation in which the essential non-linearity is replaced by a feedback loop between the history of activation and an adaptive transition rate that is sensitive to a single dimension of the space of inactive states. This physiologically measurable dimension contributes to the stability of the system and serves as a powerful modulator of input-output relations that depends on the patterns of prior activity; an intrinsic scale free mechanism for cellular adaptation that emerges from the microscopic biophysical properties of ion channels of excitable membranes. 7. Dynamics of the edge excitations in the FQH effects International Nuclear Information System (INIS) Wen, X.G. 1994-01-01 Fractional quantum Hall effects (FQHE) discovered by Tsui, Stormer and Gossard open a new era in theory of strongly correlated system. In the first time the authors have to completely abandon the theories based on the single-body picture and use an intrinsic many-body theory proposed by Laughlin and others to describe the FQHE. Due to the repulsive interaction, the strongly correlated FQH liquid is an incompressible state despite the first Landau level is only partially filled. All the bulk excitations in the FQH states have finite energy gaps. The FQH states and insulators are similar in the sense that both states have finite energy gap and short ranged electron propagators. Because of this similarity, it is puzzling that the FQH systems apparently have very different transport properties than ordinary insulators. Halperin first point out that the integral quantum Hall (IQH) states contain gapless edge excitations. Although the electronic states in the bulk are localized, the electronic states at the edge of the sample are extended. Therefore the nontrivial transport properties of the IQH states come from the gapless edge excitations. Such an edge transport picture has been supported by many experiments. One also found that the edge excitations in the IQH states are described by a chiral 1D Fermi liquid theory. Here, the authors review the dynamical theory of the edge excitations in the FQH effects 8. Electron spectroscopy of collisional excited atoms International Nuclear Information System (INIS) Straten, P. van der. 1987-01-01 In this thesis measurements are described in which coincidences are detected between scattered projectiles and emitted electrons. This yields information on two-electron excitation processes. In order to show what can be learnt from coincidence experiments a detailed theoretical analysis is given. The transition amplitudes, which contain all the information, are introduced (ch.2). In ch.3 the experimental set-up is shown. The results for the Li + -He system are shown in ch. 7 and are compared with predictions based on the Molecular-Orbitalmodel which however does not account for two-excitation mechanisms. With the transition amplitudes also the wave function of the excited atom has been completely determined. In ch.8 the shape of the electron cloud, induced by the collision, is derived from the amplitudes. The relation between the oscillatory motion of this cloud after the collision and the correlation between the two electrons of the excited atom is discussed. In ch. 6 it is shown that the broad structures in the non-coincident energy spectra of the Li + -He system are erroneously interpretated as a result of electron emission from the (Li-He) + -quasimolecule. A model is presented which explains, based on the results obtained from the coincidence measurements, these broad structures. In ch. 4 the Post-Collision Interaction process is treated. It is shown that for high-energy collisions, in contrast with general assumptions, PCI is important. In ch. 5 the importance of PCI-processes in photoionization of atoms, followed by Auger decay, are studied. From the formulas derived in ch. 4 simple analytical results are obtained. These are applied to recent experiments and good agreement is achieved. 140 refs.; 55 figs.; 9 tabs 9. Experimental study on the kinetically induced electronic excitation in atomic collisional cascades International Nuclear Information System (INIS) Meyer, S. 2006-01-01 the present thesis deals with the ion-collision-induced electronic excitation of metallic solids. For this for the first time metal-insulator-metal layer systems are used for the detection of this electronic excitation. The here applied aluminium/aluminium oxide/silver layer sytems have barrier heights of 2.4 eV on the aluminium respectively 3.3 eV on the silver side. With the results it could uniquely be shown that the electronic excitation is generated by kinetic processes, this excitation dependenc on the kinetic energy of the colliding particles, and the excitation dependes on the charge state of the projectile 10. A scalable piezoelectric impulse-excited energy harvester for human body excitation International Nuclear Information System (INIS) Pillatsch, P; Yeatman, E M; Holmes, A S 2012-01-01 Harvesting energy from low-frequency and non-harmonic excitations typical of human motion presents specific challenges. While resonant devices do have an advantage in environments where the excitation frequency is constant, and while they can make use of the entire proof mass travel range in the case of excitation amplitudes that are smaller than the internal displacement limit, they are not suitable for body applications since the frequencies are random and the amplitudes tend to be larger than the device size. In this paper a piezoelectric, impulse-excited approach is presented. A cylindrical proof mass actuates an array of piezoelectric bi-morph beams through magnetic attraction. After the initial excitation these transducers are left to vibrate at their natural frequency. This increases the operational frequency range as well as the electromechanical coupling. The principle of impulse excitation is discussed and a centimetre-scale functional model is introduced as a proof of concept. The obtained data show the influence of varying the frequency, acceleration and proof mass. Finally, a commercially available integrated circuit for voltage regulation is tested. At a frequency of 2 Hz and an acceleration of 2.7 m s −2 a maximal power output of 2.1 mW was achieved. (paper) 11. Magnetic field effects on exciplex-forming systems: the effect on the locally excited fluorophore and its dependence on free energy. Science.gov (United States) Kattnig, Daniel R; Rosspeintner, Arnulf; Grampp, Günter 2011-02-28 This study addresses magnetic field effects in exciplex forming donor-acceptor systems. For moderately exergonic systems, the exciplex and the locally excited fluorophore emission are found to be magneto-sensitive. A previously introduced model attributing this finding to excited state reversibility is confirmed. Systems characterised by a free energy of charge separation up to approximately -0.35 eV are found to exhibit a magnetic field effect on the fluorophore. A simple three-state model of the exciplex is introduced, which uses the reaction distance and the asymmetric electron transfer reaction coordinate as pertinent variables. Comparing the experimental emission band shapes with those predicted by the model, a semi-quantitative picture of the formation of the magnetic field effect is developed based on energy hypersurfaces. The model can also be applied to estimate the indirect contribution of the exchange interaction, even if the perturbative approach fails. The energetic parameters that are essential for the formation of large magnetic field effects on the exciplex are discussed. 12. Electron impact excitation and ionization of laser-excited sodium atoms Na(7d) International Nuclear Information System (INIS) Nienhaus, J.; Dorn, A.; Mehlhorn, W.; Zatsarinny, O.I. 1997-01-01 We have investigated the ejected-electron spectrum following impact excitation and ionization of laser-excited Na (nl) atoms by 1.5 keV electrons. By means of two-laser excitation 3s → 3p 3/2 → 7d and subsequent cascading transitions about 8% (4%) of the target atoms were in excited states with n > 3 (7d). The experimental ejected-electron spectrum due to the decay of Auger and autoionization states of laser-excited atoms Na * (nl) with n = 4-7 has been fully interpreted by comprehensive calculations of the energies, cross sections and decay probabilities of the corresponding states. The various processes contributing to the ejected-electron spectrum are with decreasing magnitude: 2s ionization leading to 2s2p 6 nl Auger states, 2p → 3s excitation leading to 2p 5 3s( 1 P)nl autoionization states and 2s → 3l' excitation leading to 2s2p 6 3l'( 1 L)nl autoionization states. (Author) 13. High-Resolution Spectroscopy of Jet-Cooled 1,1 '-Diphenylethylene: Electronically Excited and Ionic States of a Prototypical Cross-Conjugated System NARCIS (Netherlands) Smolarek, S.; Vdovin, A.; Rijs, A.; van Walree, C. A.; Zgierski, M. Z.; Buma, W. J. 2011-01-01 The photophysics of a prototypical cross-conjugated pi-system, 1,1'-diphenylethylene, have been studied using high-resolution resonance enhanced multiphoton ionization excitation spectroscopy and zero kinetic energy photoelectron spectroscopy, in combination with advanced ab initio 14. Perturbation expansion theory corrected from basis set superposition error. I. Locally projected excited orbitals and single excitations. Science.gov (United States) Nagata, Takeshi; Iwata, Suehiro 2004-02-22 The locally projected self-consistent field molecular orbital method for molecular interaction (LP SCF MI) is reformulated for multifragment systems. For the perturbation expansion, two types of the local excited orbitals are defined; one is fully local in the basis set on a fragment, and the other has to be partially delocalized to the basis sets on the other fragments. The perturbation expansion calculations only within single excitations (LP SE MP2) are tested for water dimer, hydrogen fluoride dimer, and colinear symmetric ArM+ Ar (M = Na and K). The calculated binding energies of LP SE MP2 are all close to the corresponding counterpoise corrected SCF binding energy. By adding the single excitations, the deficiency in LP SCF MI is thus removed. The results suggest that the exclusion of the charge-transfer effects in LP SCF MI might indeed be the cause of the underestimation for the binding energy. (c) 2004 American Institute of Physics. 15. The blue light indicator in rubidium 5S-5P-5D cascade excitation Science.gov (United States) Raja, Waseem; Ali, Md. Sabir; Chakrabarti, Alok; Ray, Ayan 2017-07-01 The cascade system has played an important role in contemporary research areas related to fields like Rydberg excitation, four wave mixing and non-classical light generation, etc. Depending on the specific objective, co or counter propagating pump-probe laser experimental geometry is followed. However, the stepwise excitation of atoms to states higher than the first excited state deals with increasingly much fewer number of atoms even compared to the population at first excited level. Hence, one needs a practical indicator to study the complex photon-atom interaction of the cascade system. Here, we experimentally analyze the case of rubidium 5S → 5P → 5D as a specimen of two-step excitation and highlight the efficacy of monitoring one branch, which emits 420 nm, of associated cascade decay route 5D → 6P → 5S, as an effective monitor of the coherence in the system. 16. Multipole giant resonances in highly excited nuclei International Nuclear Information System (INIS) Xia Keding; Cai Yanhuang 1989-01-01 The isoscalar giant surface resonance and giant dipole resonance in highly excited nuclei are discussed. Excitation energies of the giant modes in 208 Pb are calculated in a simplified model, using the concept of energy wieghted sum rule (EWSR), and the extended Thomas-Fermi approximation at the finite temperature is employed to describe the finite temperature is employed to describe the finite temperature equilibrium state. It is shown that EWSR and the energy of the resonance depend only weakly on temperature in the system. This weak dependence is analysed 17. A Solution Method for Linear and Geometrically Nonlinear MDOF Systems with Random Properties subject to Random Excitation DEFF Research Database (Denmark) Micaletti, R. C.; Cakmak, A. S.; Nielsen, Søren R. K. structural properties. The resulting state-space formulation is a system of ordinary stochastic differential equations with random coefficient and deterministic initial conditions which are subsequently transformed into ordinary stochastic differential equations with deterministic coefficients and random......A method for computing the lower-order moments of randomly-excited multi-degree-of-freedom (MDOF) systems with random structural properties is proposed. The method is grounded in the techniques of stochastic calculus, utilizing a Markov diffusion process to model the structural system with random...... initial conditions. This transformation facilitates the derivation of differential equations which govern the evolution of the unconditional statistical moments of response. Primary consideration is given to linear systems and systems with odd polynomial nonlinearities, for in these cases... 18. Non-linear vibrating systems excited by a nonideal energy source with a large slope characteristic Science.gov (United States) González-Carbajal, Javier; Domínguez, Jaime 2017-11-01 This paper revisits the problem of an unbalanced motor attached to a fixed frame by means of a nonlinear spring and a linear damper. The excitation provided by the motor is, in general, nonideal, which means it is affected by the vibratory response. Since the system behaviour is highly dependent on the order of magnitude of the motor characteristic slope, the case of large slope is considered herein. Some Perturbation Methods are applied to the system of equations, which allows transforming the original 4D system into a much simpler 2D system. The fixed points of this reduced system and their stability are carefully studied. We find the existence of a Hopf bifurcation which, to the authors' knowledge, has not been addressed before in the literature. These analytical results are supported by numerical simulations. We also compare our approach and results with those published by other authors. 19. Stick-Slip Analysis of a Drill String Subjected to Deterministic Excitation and Stochastic Excitation Directory of Open Access Journals (Sweden) Hongyuan Qiu 2016-01-01 Full Text Available Using a finite element model, this paper investigates the torsional vibration of a drill string under combined deterministic excitation and random excitation. The random excitation is caused by the random friction coefficients between the drill bit and the bottom of the hole and assumed as white noise. Simulation shows that the responses under random excitation become random too, and the probabilistic distribution of the responses at each discretized time instant is obtained. The two points, entering and leaving the stick stage, are examined with special attention. The results indicate that the two points become random under random excitation, and the distributions are not normal even when the excitation is assumed as Gaussian white noise. 20. Cerebellum tunes the excitability of the motor system: evidence from peripheral motor axons. Science.gov (United States) Nodera, Hiroyuki; Manto, Mario 2014-12-01 Cerebellum is highly connected with the contralateral cerebral cortex. So far, the motor deficits observed in acute focal cerebellar lesions in human have been mainly explained on the basis of a disruption of the cerebello-thalamo-cortical projections. Cerebellar circuits have also numerous anatomical and functional interactions with brainstem nuclei and projects also directly to the spinal cord. Cerebellar lesions alter the excitability of peripheral motor axons as demonstrated by peripheral motor threshold-tracking techniques in cerebellar stroke. The biophysical changes are correlated with the functional scores. Nerve excitability measurements represent an attractive tool to extract the rules underlying the tuning of excitability of the motor pathways by the cerebellum and to discover the contributions of each cerebellar nucleus in this key function, contributing to early plasticity and sensorimotor learning. 1. Improvement of low speed induction generator performances and reducing the power of excitation and voltage control system Energy Technology Data Exchange (ETDEWEB) Budisan, N. [Politechnica Univ. of Timisoara (Romania); Hentea, T.; Mahil, S. [Purdue Univ. Calumet, Hammond, IN (United States); Madescu, G. [Romanian Academy, Timisoara (Romania) 1996-12-31 In this paper we present the results of our investigations concerning the utilization of induction generators at very low speed. It is shown that, by proper design, it is possible to obtain high efficiency and high power factor values. The optimized induction generators require lower reactive power resulting in lower size and price of the excitation control system. 4 refs., 2 figs. 2. Isotope separation using vibrationally excited molecules International Nuclear Information System (INIS) Woodroffe, J.A.; Keck, J.C. 1977-01-01 A system for isotope separation or enrichment wherein molecules of a selected isotope type in a flow of molecules of plural isotope types are vibrationally excited and collided with a background gas to provide enhanced diffusivity for the molecules of the selected isotope type permitting their separate collection. The system typically is for the enrichment of uranium using a uranium hexafluoride gas in combination with a noble gas such as argon. The uranium hexafluoride molecules having a specific isotope of uranium are vibrationally excited by laser radiation. The vibrational energy is converted to a translation energy upon collision with a particle of the background gas and the added translation energy enhances the diffusivity of the selected hexafluoride molecules facilitating its condensation on collection surfaces provided for that purpose. This process is periodically interrupted and the cryogenic flow halted to permit evaporation of the collected molecules to provide a distinct, enriched flow 3. Electron-excited molecule interactions International Nuclear Information System (INIS) Christophorou, L.G.; Tennessee Univ., Knoxville, TN 1991-01-01 In this paper the limited but significant knowledge to date on electron scattering from vibrationally/rotationally excited molecules and electron scattering from and electron impact ionization of electronically excited molecules is briefly summarized and discussed. The profound effects of the internal energy content of a molecule on its electron attachment properties are highlighted focusing in particular on electron attachment to vibrationally/rotationally and to electronically excited molecules. The limited knowledge to date on electron-excited molecule interactions clearly shows that the cross sections for certain electron-molecule collision processes can be very different from those involving ground state molecules. For example, optically enhanced electron attachment studies have shown that electron attachment to electronically excited molecules can occur with cross sections 10 6 to 10 7 times larger compared to ground state molecules. The study of electron-excited molecule interactions offers many experimental and theoretical challenges and opportunities and is both of fundamental and technological significance. 54 refs., 15 figs 4. Linear-scaling quantum mechanical methods for excited states. Science.gov (United States) Yam, ChiYung; Zhang, Qing; Wang, Fan; Chen, GuanHua 2012-05-21 The poor scaling of many existing quantum mechanical methods with respect to the system size hinders their applications to large systems. In this tutorial review, we focus on latest research on linear-scaling or O(N) quantum mechanical methods for excited states. Based on the locality of quantum mechanical systems, O(N) quantum mechanical methods for excited states are comprised of two categories, the time-domain and frequency-domain methods. The former solves the dynamics of the electronic systems in real time while the latter involves direct evaluation of electronic response in the frequency-domain. The localized density matrix (LDM) method is the first and most mature linear-scaling quantum mechanical method for excited states. It has been implemented in time- and frequency-domains. The O(N) time-domain methods also include the approach that solves the time-dependent Kohn-Sham (TDKS) equation using the non-orthogonal localized molecular orbitals (NOLMOs). Besides the frequency-domain LDM method, other O(N) frequency-domain methods have been proposed and implemented at the first-principles level. Except one-dimensional or quasi-one-dimensional systems, the O(N) frequency-domain methods are often not applicable to resonant responses because of the convergence problem. For linear response, the most efficient O(N) first-principles method is found to be the LDM method with Chebyshev expansion for time integration. For off-resonant response (including nonlinear properties) at a specific frequency, the frequency-domain methods with iterative solvers are quite efficient and thus practical. For nonlinear response, both on-resonance and off-resonance, the time-domain methods can be used, however, as the time-domain first-principles methods are quite expensive, time-domain O(N) semi-empirical methods are often the practical choice. Compared to the O(N) frequency-domain methods, the O(N) time-domain methods for excited states are much more mature and numerically stable, and 5. a simple a simple excitation control excitation control excitation African Journals Online (AJOL) eobe field voltages determined follow a simple quadratic relationship that offer a very simple control scheme, dependent on only the stator current. Keywords: saturated reactances, no-load field voltage, excitation control, synchronous generators. 1. Introduction. Introduction. Introduction. The commonest generator in use today is ... 6. Excitation transfer pathways in excitonic aggregates revealed by the stochastic Schrödinger equation Energy Technology Data Exchange (ETDEWEB) Abramavicius, Vytautas, E-mail: [email protected]; Abramavicius, Darius, E-mail: [email protected] [Faculty of Physics, Department of Theoretical Physics, Vilnius University, Saulėtekio 9, LT-10222 Vilnius (Lithuania) 2014-02-14 We derive the stochastic Schrödinger equation for the system wave vector and use it to describe the excitation energy transfer dynamics in molecular aggregates. We suggest a quantum-measurement based method of estimating the excitation transfer time. Adequacy of the proposed approach is demonstrated by performing calculations on a model system. The theory is then applied to study the excitation transfer dynamics in a photosynthetic pigment-protein Fenna-Matthews-Olson (FMO) aggregate using both the Debye spectral density and the spectral density obtained from earlier molecular dynamics simulations containing strong vibrational high-frequency modes. The obtained results show that the excitation transfer times in the FMO system are affected by the presence of the vibrational modes; however, the transfer pathways remain the same. 7. Excited states v.6 CERN Document Server Lim, Edward C 1982-01-01 Excited States, Volume 6 is a collection of papers that discusses the excited states of molecules. The first paper discusses the linear polyene electronic structure and potential surfaces, considering both the theoretical and experimental approaches in such electronic states. This paper also reviews the theory of electronic structure and cites some experimental techniques on polyene excitations, polyene spectroscopic phenomenology, and those involving higher states of polyenes and their triplet states. Examples of these experimental studies of excited states involve the high-resolution one-pho 8. Portable vibration exciter Science.gov (United States) Beecher, L. C.; Williams, F. T. 1970-01-01 Gas-driven vibration exciter produces a sinusoidal excitation function controllable in frequency and in amplitude. It allows direct vibration testing of components under normal loads, removing the possibility of component damage due to high static pressure. 9. Impulses and pressure waves cause excitement and conduction in the nervous system. Science.gov (United States) Barz, Helmut; Schreiber, Almut; Barz, Ulrich 2013-11-01 It is general accepted, that nerval excitement and conduction is caused by voltage changes. However, the influx of fluid into an elastical tube releases impulses or pressure waves. Therefore an influx of ion currents, respectively fluid motions into the elastic neuronal cells and fibres also induce impulses. This motion of charge carriers are measured by voltage devices as oscillations or action potentials, but the voltage changes may be an epiphenomenon of the (mechanical) impulses. Impulse waves can have a high speed. As stiffer or inelastic a tube wall, the greater is the speed of the impulse. Myelin sheaths cause a significant stiffening of the nerve fibre wall and myelinated fibres have a conduction velocity up to 120 m/s. The influx of fluid at the nodes of Ranvier intensifies periodically the impulse wave in the nerve fibres. The authors suggest that also the muscle end-plate acts as a conductor of axonal impulses to the inner of the muscle fibres and that the exocytosis of acetylcholine into the synaptic cleft may be an amplifier of the axonal impulse. It is discussed that intracellular actin filaments may also influence motions at the neuronal membrane. Many sensory nerve cells are excited due to exogenous or endogenous mechanical impulses. It may plausible that such impulses are conducted directly to the sensory nerve cell bodies in the dorsal root ganglia without the transformation in electric energy. Excitation conduction happens without noteworthy energy consumption because the flow of ion currents through the membranes takes place equivalent to the concentration gradient. Impulse waves cause short extensions of the lipid membranes of the cell- and fibres walls and therefore they can induce opening and closing of the included ion channels. This mechanism acts to "voltage gated" and "ligand-gated" channels likewise. The concept of neuronal impulses can be helpful to the understanding of other points of neurophysiology or neuronal diseases. This includes 10. High-resolution spectroscopy of jet-cooled 1,1 '-diphenylethylene: electronically excited and ionic states of a prototypical cross-conjugated system NARCIS (Netherlands) Smolarek, S.; Vdovin, A.; Rijs, A.; van Walree, C.A.; Zgierski, M.Z.; Buma, W.J. 2011-01-01 The photophysics of a prototypical cross-conjugated π-system, 1,1′-diphenylethylene, have been studied using high-resolution resonance enhanced multiphoton ionization excitation spectroscopy and zero kinetic energy photoelectron spectroscopy, in combination with advanced ab initio calculations. We 11. Dynamics of Solid Body in Magnetic Suspension under Periodic Excitation Directory of Open Access Journals (Sweden) A. M. Gouskov 2017-01-01 Full Text Available The article studies dynamics of ferromagnetic body in hybrid magnetic suspension (HMS. The body is supposed to have one degree of freedom and a nonlinear magnetic force dependence on the current and displacement. The magnetic force induced in the HMS is divided into a passive component and an active one. Specifying the law of current variation in the coil allows us to generate nonlinear oscillations under electromagnet action. To provide periodic excitation the appropriate law of the current variation in the electromagnet coil is proposed. The mathematical model includes external periodic step-excitation. The equation of motion is formed. The scales of similarity are highlighted in the system, and the equation of motion is reduced to dimensionless form.The motion dynamics is studied numerically. The relaxation method was used to determine the periodic motions at different values of dimensionless frequency of the electromagnet excitation as well as to estimate the influence of other dimensionless parameters on the system dynamics. The amplitude-frequency curve analysis allows us to come to conclusion that the nature of system nonlinearity is rigid. Adding the external periodic step-excitation leads to the qualitative change in the nature of movement. This points to the occurrence of bifurcation. 12. Classification of a Supersolid: Trial Wavefunctions, Symmetry Breakings and Excitation Spectra Science.gov (United States) Chen, Yu; Ye, Jinwu; Tian, Guangshan 2012-11-01 A state of matter is characterized by its symmetry breaking and elementary excitations. A supersolid is a state which breaks both translational symmetry and internal U(1) symmetry. Here, we review some past and recent works in phenomenological Ginsburg-Landau theories, ground state trial wavefunctions and microscopic numerical calculations. We also write down a new effective supersolid Hamiltonian on a lattice. The eigenstates of the Hamiltonian contains both the ground state wavefunction and all the excited states (supersolidon) wavefunctions. We contrast various kinds of supersolids in both continuous systems and on lattices, both condensed matter and cold atom systems. We provide additional new insights in studying their order parameters, symmetry breaking patterns, the excitation spectra and detection methods. 13. Amplitudes and state parameters from ion- and atom-atom excitation processes International Nuclear Information System (INIS) Andersen, T.; Horsdal-Pedersen, E. 1984-01-01 This chapter examines single collisions between two atomic species, one of which is initially in a 1 S state (there is only one initial spin channel). The collisions are characterized by a definite scattering plane and a definite orientation. Topics considered include an angular correlation between scattered particles and autoionization electrons or polarized photons emitted from states excited in atomic collisions (photon emission, electron emission, selectivity excited target atoms), experimental methods for obtaining information on the alignment and orientation parameters of atoms or ions excited in specific collisions, results of experiments and numerical calculations (quasi-oneelectron systems, He + -He collisions, other collision systems), and future aspects and possible applications of the polarizedphoton, scattered-particle coincidence techniques to atomic spectroscopy 14. Theory of nuclear excitation by electron capture for heavy ions International Nuclear Information System (INIS) Palffy, Adriana; Scheid, Werner; Harman, Zoltan 2006-01-01 We investigate the resonant process of nuclear excitation by electron capture (NEEC), in which a continuum electron is captured into a bound state of an ion with the simultaneous excitation of the nucleus. In order to derive the cross section a Feshbach projection operator formalism is introduced. Nuclear states and transitions are described by a nuclear collective model and making use of experimental data. Transition rates and total cross sections for NEEC followed by the radiative decay of the excited nucleus are calculated for various heavy-ion collision systems 15. Study of the excitation mechanisms of the second positive system in the negative glow of a N{sub 2}-Ar discharge Energy Technology Data Exchange (ETDEWEB) Isola, L; Lopez, M; Gomez, B J, E-mail: [email protected] [Instituto de Fisica Rosario (CONICET-UNR) 27 Febrero 210 Bis. (S2000EZP) Rosario (Argentina) 2011-09-21 In an Ar-N{sub 2} discharge, the high excitation transfer from Ar({sup 3}P{sub 2,0}) to N{sub 2} produces an overpopulation of the high rotational levels of the bands of the second positive system (SPS), and so the spectra interpretation is not straightforward. This paper presents a fit function for the SPS bands measured in Ar-N{sub 2}, which allows us to study the excitation process contributions to the N{sub 2}(C) level. The procedure was tested in the negative glow of a pulsed Ar-N{sub 2} discharge at a pressure of 2.5 Torr, for different mixture concentrations. In this discharge, through the fitting, it was possible to calculate the variation of the N{sub 2}(C) densities produced by different excitation processes as well as the variation of Ar metastable density. 16. Multi-frequency excitation KAUST Repository 2016-01-01 Embodiments of multi-frequency excitation are described. In various embodiments, a natural frequency of a device may be determined. In turn, a first voltage amplitude and first fixed frequency of a first source of excitation can be selected 17. Excited-state quantum phase transitions in systems with two degrees of freedom: Level density, level dynamics, thermal properties International Nuclear Information System (INIS) Stránský, Pavel; Macek, Michal; Cejnar, Pavel 2014-01-01 Quantum systems with a finite number of freedom degrees f develop robust singularities in the energy spectrum of excited states as the system’s size increases to infinity. We analyze the general form of these singularities for low f, particularly f=2, clarifying the relation to classical stationary points of the corresponding potential. Signatures in the smoothed energy dependence of the quantum state density and in the flow of energy levels with an arbitrary control parameter are described along with the relevant thermodynamical consequences. The general analysis is illustrated with specific examples of excited-state singularities accompanying the first-order quantum phase transition. -- Highlights: •ESQPTs found in infinite-size limit of systems with low numbers of freedom degrees f. •ESQPTs related to non-analytical evolutions of classical phase–space properties. •ESQPT signatures analyzed for general f, particularly f=2, extending known case f=1. •ESQPT signatures identified in smoothened density and flow of energy spectrum. •ESQPTs shown to induce a new type of thermodynamic anomalies 18. Seismic enhancement of multi-span continuous bridges subjected to three-directional excitations Science.gov (United States) Aryan, H.; Ghassemieh, M. 2015-04-01 Considering the seismic ground motions as the excitations in only two principal horizontal directions of the bridges and ignoring the third vertical direction is a disregard for the seismic conditions of the region and the bridge distance from epicenter. Numerous cases of substantial damages have been reported among the bridges tremendously suffered from being exposed to the simultaneous three-directional seismic ground motions. Besides the significant compression and tension damages in the columns due to the presence of vertical excitation, it could lead to unexpected shear and flexural failures in the columns and other components as well. Because the axial force variation in the columns due to three-directional excitations, could affect the demands and capacities of the bridge’s components. With respect to this issue, several studies on the bridge damages during the earthquakes have urged researchers to offer efficient methods for bridges handling of the three-directional seismic excitations. Thus, this paper presents and evaluates a superelastic based system for designing as well as retrofitting the multi-span continuous (MSC) bridges that can cope with two- and three-directional seismic excitations. Efficiency evaluation of the proposed system is conducted through various nonlinear time history analyses on a three-dimensional model of a detailed MSC bridge using a suite of developed ground motions for the bridge region. Also, all the analyses are fulfilled based on variation of one influential design characteristic of the proposed system in order to achieve the optimal design. Several pertinent assessment parameters are used during the evaluation of the proposed system. Finally, the efficiency of the new system subjected to the vertical and horizontal seismic excitations is confirmed according to reduction of the bridge responses and improvement in nonlinear performance of the columns in comparison with the as-built bridge results. 19. Excited charmed mesons International Nuclear Information System (INIS) Butler, J.N.; Shukla, S. 1995-05-01 The experimental status of excited charmed mesons is reviewed and is compared to theoretical expectations. Six states have been observed and their properties are consistent with those predicted for excited charmed states with orbital angular momentum equal to one 20. Nonlinear characteristics of the rotating exciter system of power plant generators in case of electricity accidents; Transientes Verhalten des rotierenden Erregersystems von Kraftwerksgeneratoren bei elektrischen Stoerfaellen Energy Technology Data Exchange (ETDEWEB) 2006-05-09 Different types of exciter are used for voltage supply to the synchronous generators of power stations depending on the required power and design. The exciter system of the generator, which as a rule consists syncronous motors and commutators, is commonly modeled in conventional models by control units with nonlinear characteristics which do not give an accurate picture of the dynamic processes inside the exciter motor. It was not possible to assess the component loads of the exciter components and the physical characteristics within the exciter system. In this study, a brushless exciter for the grid-connected synchronous generator was investigated which consists of two synchronous motors as primary and secondary exciter and two commutator bridges. A dynamic simulation model was developed for calculating the interactions between the grid, generator and exciter unit in consideration of electromagnetic and galvanic coupling. For this, the normal control units were replaced by physical components of the exciter system, i.e. electric exciter motors and commutators. The study was carried out using an enhanced version of the Siemens NETOMAC software, which provided information on the loads on the exciter components in case of internal and external failures. In particular, loads in coils and commutators were calculated that could not be measured before. The findings enable more accurate dimensioning of the exciter unit making it more fail-safe, and the protective systems can be adjusted more accurately. One important result of the investigation was the identification of all dynamic processes going on between the exciter motors, commutators, generator and grid induced by external and internal failures. (orig.) [German] Zur Spannungsversorgung der Synchrongeneratoren in Kraftwerken werden je nach Leistungsanforderung und Baukonzept unterschiedliche Erregereinrichtungen verwendet. Das Erregersystem des Generators, das in der Regel aus Erregersynchronmaschinen und 1. Electric quadrupole excitation of the first excited state of 11B International Nuclear Information System (INIS) Fewell, M.P.; Spear, R.H.; Zabel, T.H.; Baxter, A.M. 1980-02-01 The Coulomb excitation of backscattered 11 B projectiles has been used to measure the reduced E2 transition probability B(E2; 3/2 - →1/2 - ) between the 3/2 - ground state and the 1/2 - first excited state of 11 B. It is found that B(E2; 3/2 - →1/2 - ) = 2.1 +- 0.4 e 2 fm 4 , which agrees with shell-model predictions but is a factor of 10 larger than the prediction of the core-excitation model 2. Proposition for sensorless self-excitation by a piezoelectric device Science.gov (United States) Tanaka, Y.; Kokubun, Y.; Yabuno, H. 2018-04-01 In this paper, we propose a method to realize self-excitation in an oscillator actuated by a piezoelectric device without a sensor. In general, the positive feedback associated with the oscillator velocity causes the self-excitation. Instead of measuring the velocity with a sensor, we utilize the electro-mechanical coupling effect in the oscillator and piezoelectric device. We drive the piezoelectric device with a current proportional to the linear combination of the voltage across the terminals of the piezoelectric device and its differential voltage signal. Then, the oscillator with the piezoelectric device behaves like a third-order system, which has three eigenvalues. The self-excitation can be realized because appropriate feedback gains can set two of the eigenvalues to be conjugate complex roots with a positive real part and the other eigenvalue to be a negative real root. To confirm the validity of the proposed method, we experimentally demonstrated the sensorless self-excitation and, as an application example, carried out mass sensing in a sensorless self-excited macrocantilever. 3. MEMS Logic Using Mixed-Frequency Excitation KAUST Repository 2017-06-22 We present multi-function microelectromechanical systems (MEMS) logic device that can perform the fundamental logic gate AND, OR, universal logic gates NAND, NOR, and a tristate logic gate using mixed-frequency excitation. The concept is based on exciting combination resonances due to the mixing of two or more input signals. The device vibrates at two steady states: a high state when the combination resonance is activated and a low state when no resonance is activated. These vibration states are assigned to logical value 1 or 0 to realize the logic gates. Using ac signals to drive the resonator and to execute the logic inputs unifies the input and output wave forms of the logic device, thereby opening the possibility for cascading among logic devices. We found that the energy consumption per cycle of the proposed logic resonator is higher than those of existing technologies. Hence, integration of such logic devices to build complex computational system needs to take into consideration lowering the total energy consumption. [2017-0041 4. Charge transfer excitations from exact and approximate ensemble Kohn-Sham theory Science.gov (United States) Gould, Tim; Kronik, Leeor; Pittalis, Stefano 2018-05-01 By studying the lowest excitations of an exactly solvable one-dimensional soft-Coulomb molecular model, we show that components of Kohn-Sham ensembles can be used to describe charge transfer processes. Furthermore, we compute the approximate excitation energies obtained by using the exact ensemble densities in the recently formulated ensemble Hartree-exchange theory [T. Gould and S. Pittalis, Phys. Rev. Lett. 119, 243001 (2017)]. Remarkably, our results show that triplet excitations are accurately reproduced across a dissociation curve in all cases tested, even in systems where ground state energies are poor due to strong static correlations. Singlet excitations exhibit larger deviations from exact results but are still reproduced semi-quantitatively. 5. Ultrafast electronic relaxation of excited state vitamin B12 in the gas phase International Nuclear Information System (INIS) Shafizadeh, Niloufar; Poisson, Lionel; Soep, Benoit 2008-01-01 The time evolution of electronically excited vitamin B 12 (cyanocobalamin) has been observed for the first time in the gas phase. It reveals an ultrafast decay to a state corresponding to metal excitation. This decay is interpreted as resulting from a ring to metal electron transfer. This opens the observation of the excited state of other complex biomimetic systems in the gas phase, the key to the characterisation of their complex evolution through excited electronic states 6. Springing response due to bidirectional wave excitation DEFF Research Database (Denmark) Vidic-Perunovic, Jelena 2005-01-01 theories deal with the unidirectional wave excitation. This is quite standard. The problem is how to include more than one directional wave systems described by a wave spectrum with arbitrary heading. The main objective of the present work has been to account for the additional second-order springing......-linear (second order) high frequency springing analyses with unidirectional wave excitation are much more scattered. Some of the reasons are different level of wave excitation accounted in the different Executive Summary ivtheories, inclusion of additional hydrodynamic phenomena e.g. slamming in the time...... because, to the author's knowledge, this is the first time that the wave data were collected simultaneously with stress records on the deck of the ship. This is highly appreciated because one can use the precise input and not only the most probable sea state statistics. The actual picture of the sea waves... 7. DESIGN METHODOLOGY OF SELF-EXCITED ASYNCHRONOUS GENERATOR Directory of Open Access Journals (Sweden) Berzan V. 2012-04-01 Full Text Available The paper sets out the methodology of designing an asynchronous generator with capacitive self-excitation. It is known that its design is possible on the basis of serial synchronous motor with squirrel cage rotor. With this approach, the design reworked only the stator winding of electrical machines, making it cost-effectively implement the creation of the generator. Therefore, the methodology for the design, optimization calculations, the development scheme and the stator winding excitation system gain, not only of practical interest, and may also be useful for specialists in the field of electrical machines in the design of asynchronous generators. 8. Dissociative Excitation of Acetylene Induced by Electron Impact: Excitation-emission Cross-sections Energy Technology Data Exchange (ETDEWEB) Országh, Juraj; Danko, Marián; Čechvala, Peter; Matejčík, Štefan, E-mail: [email protected] [Department of Experimental Physics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynská dolina F-2, 842 48 Bratislava (Slovakia) 2017-05-20 The optical emission spectrum of acetylene excited by monoenergetic electrons was studied in the range of 190–660 nm. The dissociative excitation and dissociative ionization associated with excitation of the ions initiated by electron impact were dominant processes contributing to the spectrum. The spectrum was dominated by the atomic lines (hydrogen Balmer series, carbon) and molecular bands (CH(A–X), CH(B–X), CH{sup +}(B–A), and C{sub 2}). Besides the discrete transitions, we have detected the continuum emission radiation of ethynyl radical C{sub 2}H(A–X). For most important lines and bands of the spectrum we have measured absolute excitation-emission cross sections and determined the energy thresholds of the particular dissociative channels. 9. Dynamics of excited instantons in the system of forced Gursey nonlinear differential equations Energy Technology Data Exchange (ETDEWEB) Aydogmus, F., E-mail: [email protected] [Istanbul University, Department of Physics, Faculty of Science (Turkey) 2015-02-15 The Gursey model is a 4D conformally invariant pure fermionic model with a nonlinear spinor self-coupled term. Gursey proposed his model as a possible basis for a unitary description of elementary particles following the “Heisenberg dream.” In this paper, we consider the system of Gursey nonlinear differential equations (GNDEs) formed by using the Heisenberg ansatz. We use it to understand how the behavior of spinor-type Gursey instantons can be affected by excitations. For this, the regular and chaotic numerical solutions of forced GNDEs are investigated by constructing their Poincaré sections in phase space. A hierarchical cluster analysis method for investigating the forced GNDEs is also presented. 10. Modeling And Control Of Excitation And Governor Based On PSO For MHPP Directory of Open Access Journals (Sweden) 2013-07-01 Full Text Available This paper presents the modeling and control of the excitation system via the automatic voltage regulator (AVR and governor system through the automatic generation control (AGC or frequency load control (FLC to improve stability on a micro hydro power plant (MHPP. Three main parts of the generation system are synchronous generator, AVR/excitation, AGC modelled linearly. Generator is modelled by a single machine connected to infinite bus (SMIB which is equipped by AVR and excitation linear model. Excitation control system made ??by optimizing the gain of the AVR (KA and the governor with the gain of the AGC (Ki. Optimization is done using the method improved particle swam optimization (IPSO. The main purpose of setting the gain of the AVR-AGC is to stabilize the oscillation frequency of the MHPP is connected to an infinite bus. Simulations are conducted by inputting step function with 5% load fluctuations as a representation of dynamic load. The simulation results show that the proposed method effectively raises the level of electromechanical damping oscillations the SMIB by generating the comprehensive damping index (CDI is minimum. 11. Excitation spectrum of Heisenberg spin ladders International Nuclear Information System (INIS) Barnes, T.; Dagotto, E.; Riera, J.; Swanson, E.S. 1993-01-01 Heisenberg antiferromagnetic spin ''ladders'' (two coupled spin chains) are low-dimensional magnetic systems which for S=1/2 interpolate between half-integer-spin chains, when the chains are decoupled, and effective integer-spin one-dimensional chains in the strong-coupling limit. The spin-1/2 ladder may be realized in nature by vanadyl pyrophosphate, (VO) 2 P 2 O 7 . In this paper we apply strong-coupling perturbation theory, spin-wave theory, Lanczos techniques, and a Monte Carlo method to determine the ground-state energy and the low-lying excitation spectrum of the ladder. We find evidence of a nonzero spin gap for all interchain couplings J perpendicular >0. A band of spin-triplet excitations above the gap is also analyzed. These excitations are unusual for an antiferromagnet, since their long-wavelength dispersion relation behaves as (k-k 0 ) 2 (in the strong-coupling limit J perpendicular much-gt J, where J is the in-chain antiferromagnetic coupling). Their band is folded, with a minimum energy at k 0 =π, and a maximum between k 1 =π/2 (for J perpendicular =0) and 0 (for J perpendicular =∞). We also give numerical results for the dynamical structure factor S(q,ω), which can be determined in neutron scattering experiments. Finally, possible experimental techniques for studying the excitation spectrum are discussed 12. Negative ion formation in collisions involving excited alkali atoms International Nuclear Information System (INIS) Cheret, M. 1988-01-01 Ion-pair production is considered as the prototype of the crossing problem between potential energy curves. In general an alkali atom is one of the reactants the other being an halogen, hydrogen atom or molecule. Experimental results are generally analyzed in the framework of the Landau-Zener-Stuekelberg theory, ionization potential and electron affinity, being the most important parameters. In order to vary these parameters over a wide range two experimental works have been devoted to systems of excited alkali atoms colliding with ground state alkali atoms. In the first study Rb atoms are excited to various ns or nd states from Rb(5d) to Rb(9s) in a cell. The second study is devoted to the Na(3p)-Na(3s) system, in this study also the possibility of creating excited negative ions (Na - (3s3p)) has been investigated. These results are presented and analyzed. Finally further developments of the subject are suggested. 17 refs.; 8 figs.; 1 table 13. Nuclear fission fragment excitation of electronic transition laser media International Nuclear Information System (INIS) Lorents, D.C.; McCusker, M.V.; Rhodes, C.K. 1976-01-01 The properties of high energy electronic transition lasers excited by fission fragments are expanded. Specific characteristics of the media including density, excitation rates, wavelength, kinetics, fissile material, scale size, and medium uniformity are assessed. The use of epithermal neutrons, homogeneously mixed fissile material, and special high cross section nuclear isotopes to optimize coupling of the energy to the medium are shown to be important considerations maximizing the scale size, energy deposition, and medium uniformity. A performance limit point of approximately 1000 J/l in approximately 100 μs pulses is established for a large class of systems operating in the near ultraviolet and visible spectral regions. It is demonstrated that e-beam excitation can be used to simulate nuclear pumping conditions to facilitate the search for candidate media. Experimental data for the kinetics of a XeF* laser operating in Ar/Xe/F 2 /UF 6 mixtures are given. These reactor-pumped systems are suitable for scaling to volumes on the order of (meters) 3 14. Photo-excitation of carotenoids causes cytotoxicity via singlet oxygen production International Nuclear Information System (INIS) Yoshii, Hiroshi; Yoshii, Yukie; Asai, Tatsuya; Furukawa, Takako; Takaichi, Shinichi; Fujibayashi, Yasuhisa 2012-01-01 Highlights: ► Some photo-excited carotenoids have photosensitizing ability. ► They are able to produce ROS. ► Photo-excited fucoxanthin can produce singlet oxygen through energy transfer. -- Abstract: Carotenoids, natural pigments widely distributed in algae and plants, have a conjugated double bond system. Their excitation energies are correlated with conjugation length. We hypothesized that carotenoids whose energy states are above the singlet excited state of oxygen (singlet oxygen) would possess photosensitizing properties. Here, we demonstrated that human skin melanoma (A375) cells are damaged through the photo-excitation of several carotenoids (neoxanthin, fucoxanthin and siphonaxanthin). In contrast, photo-excitation of carotenoids that possess energy states below that of singlet oxygen, such as β-carotene, lutein, loroxanthin and violaxanthin, did not enhance cell death. Production of reactive oxygen species (ROS) by photo-excited fucoxanthin or neoxanthin was confirmed using a reporter assay for ROS production with HeLa Hyper cells, which express a fluorescent indicator protein for intracellular ROS. Fucoxanthin and neoxanthin also showed high cellular penetration and retention. Electron spin resonance spectra using 2,2,6,6-tetramethil-4-piperidone as a singlet oxygen trapping agent demonstrated that singlet oxygen was produced via energy transfer from photo-excited fucoxanthin to oxygen molecules. These results suggest that carotenoids such as fucoxanthin, which are capable of singlet oxygen production through photo-excitation and show good penetration and retention in target cells, are useful as photosensitizers in photodynamic therapy for skin disease. 15. Low-energy d-d excitations in MnO studied by resonant x-ray fluorescence spectroscopy Energy Technology Data Exchange (ETDEWEB) Butorin, S.M.; Guo, J.; Magnuson, M. [Uppsala Univ. (Sweden)] [and others 1997-04-01 Resonant soft X-ray emission spectroscopy has been demonstrated to possess interesting abilities for studies of electronic structure in various systems, such as symmetry probing, alignment and polarization dependence, sensitivity to channel interference, etc. In the present abstract the authors focus on the feasibility of resonant soft X-ray emission to probe low energy excitations by means of resonant electronic X-ray Raman scattering. Resonant X-ray emission can be regarded as an inelastic scattering process where a system in the ground state is transferred to a low excited state via a virtual core excitation. The energy closeness to a core excitation of the exciting radiation enhances the (generally) low probability for inelastic scattering at these wavelengths. Therefore soft X-ray emission spectroscopy (in resonant electronic Raman mode) can be used to study low energy d-d excitations in transition metal systems. The involvement of the intermediate core state allows one to use the selection rules of X-ray emission, and the appearance of the elastically scattered line in the spectra provides the reference to the ground state. 16. Low-energy d-d excitations in MnO studied by resonant x-ray fluorescence spectroscopy International Nuclear Information System (INIS) Butorin, S.M.; Guo, J.; Magnuson, M. 1997-01-01 Resonant soft X-ray emission spectroscopy has been demonstrated to possess interesting abilities for studies of electronic structure in various systems, such as symmetry probing, alignment and polarization dependence, sensitivity to channel interference, etc. In the present abstract the authors focus on the feasibility of resonant soft X-ray emission to probe low energy excitations by means of resonant electronic X-ray Raman scattering. Resonant X-ray emission can be regarded as an inelastic scattering process where a system in the ground state is transferred to a low excited state via a virtual core excitation. The energy closeness to a core excitation of the exciting radiation enhances the (generally) low probability for inelastic scattering at these wavelengths. Therefore soft X-ray emission spectroscopy (in resonant electronic Raman mode) can be used to study low energy d-d excitations in transition metal systems. The involvement of the intermediate core state allows one to use the selection rules of X-ray emission, and the appearance of the elastically scattered line in the spectra provides the reference to the ground state 17. Electron-helium S-wave model benchmark calculations. II. Double ionization, single ionization with excitation, and double excitation Science.gov (United States) Bartlett, Philip L.; Stelbovics, Andris T. 2010-02-01 The propagating exterior complex scaling (PECS) method is extended to all four-body processes in electron impact on helium in an S-wave model. Total and energy-differential cross sections are presented with benchmark accuracy for double ionization, single ionization with excitation, and double excitation (to autoionizing states) for incident-electron energies from threshold to 500 eV. While the PECS three-body cross sections for this model given in the preceding article [Phys. Rev. A 81, 022715 (2010)] are in good agreement with other methods, there are considerable discrepancies for these four-body processes. With this model we demonstrate the suitability of the PECS method for the complete solution of the electron-helium system. 18. Strategies to enhance the excitation energy-transfer efficiency in a light-harvesting system using the intra-molecular charge transfer character of carotenoids Energy Technology Data Exchange (ETDEWEB) Yukihira, Nao [Department of Applied Chemistry for Environment; School of Science and Technology; Kwansei Gakuin University; Sanda; Japan; Sugai, Yuko [Department of Applied Chemistry for Environment; School of Science and Technology; Kwansei Gakuin University; Sanda; Japan; Fujiwara, Masazumi [Department of Applied Chemistry for Environment; School of Science and Technology; Kwansei Gakuin University; Sanda; Japan; Kosumi, Daisuke [Institute of Pulsed Power Science; Kumamoto University; Kumamoto; Japan; Iha, Masahiko [South Product Co. Ltd.; Uruma-shi; Japan; Sakaguchi, Kazuhiko [Department of Chemistry; Graduate School of Science; Osaka City University; Osaka 558-8585; Japan; Katsumura, Shigeo [Department of Chemistry; Graduate School of Science; Osaka City University; Osaka 558-8585; Japan; Gardiner, Alastair T. [Glasgow Biomedical Research Centre; University of Glasgow; 126 University Place; Glasgow, G12 8QQ; UK; Cogdell, Richard J. [Glasgow Biomedical Research Centre; University of Glasgow; 126 University Place; Glasgow, G12 8QQ; UK; Hashimoto, Hideki [Department of Applied Chemistry for Environment; School of Science and Technology; Kwansei Gakuin University; Sanda; Japan 2017-01-01 Fucoxanthin is a carotenoid that is mainly found in light-harvesting complexes from brown algae and diatoms. Due to the presence of a carbonyl group attached to polyene chains in polar environments, excitation produces an excited intra-molecular charge transfer. This intra-molecular charge transfer state plays a key role in the highly efficient (~95%) energy-transfer from fucoxanthin to chlorophyllain the light-harvesting complexes from brown algae. In purple bacterial light-harvesting systems the efficiency of excitation energy-transfer from carotenoids to bacteriochlorophylls depends on the extent of conjugation of the carotenoids. In this study we were successful, for the first time, in incorporating fucoxanthin into a light-harvesting complex 1 from the purple photosynthetic bacterium,Rhodospirillum rubrumG9+ (a carotenoidless strain). Femtosecond pump-probe spectroscopy was applied to this reconstituted light-harvesting complex in order to determine the efficiency of excitation energy-transfer from fucoxanthin to bacteriochlorophyllawhen they are bound to the light-harvesting 1 apo-proteins. 19. An Improved Multidimensional MPA Procedure for Bidirectional Earthquake Excitations Directory of Open Access Journals (Sweden) Feng Wang 2014-01-01 Full Text Available Presently, the modal pushover analysis procedure is extended to multidimensional analysis of structures subjected to multidimensional earthquake excitations. an improved multidimensional modal pushover analysis (IMMPA method is presented in the paper in order to estimate the response demands of structures subjected to bidirectional earthquake excitations, in which the unidirectional earthquake excitation applied on equivalent SDOF system is replaced by the direct superposition of two components earthquake excitations, and independent analysis in each direction is not required and the application of simplified superposition formulas is avoided. The strength reduction factor spectra based on superposition of earthquake excitations are discussed and compared with the traditional strength reduction factor spectra. The step-by-step procedure is proposed to estimate seismic demands of structures. Two examples are implemented to verify the accuracy of the method, and the results of the examples show that (1 the IMMPA method can be used to estimate the responses of structure subjected to bidirectional earthquake excitations. (2 Along with increase of peak of earthquake acceleration, structural response deviation estimated with the IMMPA method may also increase. (3 Along with increase of the number of total floors of structures, structural response deviation estimated with the IMMPA method may also increase. 20. Range-separated density-functional theory for molecular excitation energies International Nuclear Information System (INIS) Rebolini, E. 2014-01-01 Linear-response time-dependent density-functional theory (TDDFT) is nowadays a method of choice to compute molecular excitation energies. However, within the usual adiabatic semi-local approximations, it is not able to describe properly Rydberg, charge-transfer or multiple excitations. Range separation of the electronic interaction allows one to mix rigorously density-functional methods at short range and wave function or Green's function methods at long range. When applied to the exchange functional, it already corrects most of these deficiencies but multiple excitations remain absent as they need a frequency-dependent kernel. In this thesis, the effects of range separation are first assessed on the excitation energies of a partially-interacting system in an analytic and numerical study in order to provide guidelines for future developments of range-separated methods for excitation energy calculations. It is then applied on the exchange and correlation TDDFT kernels in a single-determinant approximation in which the long-range part of the correlation kernel vanishes. A long-range frequency-dependent second-order correlation kernel is then derived from the Bethe-Salpeter equation and added perturbatively to the range-separated TDDFT kernel in order to take into account the effects of double excitations. (author) 1. Multipurpose exciter with low phase noise Science.gov (United States) Conroy, B.; Le, D. 1989-01-01 Results of an effort to develop a lower-cost exciter with high stability, low phase noise, and controllable phase and frequency for use in Deep Space Network and Goldstone Solar System Radar applications are discussed. Included is a discussion of the basic concept, test results, plans, and concerns. 2. Melnikov's criteria, parametric control of chaos, and stationary chaos occurrence in systems with asymmetric potential subjected to multiscale type excitation. Science.gov (United States) Kwuimy, C A Kitio; Nataraj, C; Litak, G 2011-12-01 We consider the problems of chaos and parametric control in nonlinear systems under an asymmetric potential subjected to a multiscale type excitation. The lower bound line for horseshoes chaos is analyzed using the Melnikov's criterion for a transition to permanent or transient nonperiodic motions, complement by the fractal or regular shape of the basin of attraction. Numerical simulations based on the basins of attraction, bifurcation diagrams, Poincaré sections, Lyapunov exponents, and phase portraits are used to show how stationary dissipative chaos occurs in the system. Our attention is focussed on the effects of the asymmetric potential term and the driven frequency. It is shown that the threshold amplitude ∣γ(c)∣ of the excitation decreases for small values of the driven frequency ω and increases for large values of ω. This threshold value decreases with the asymmetric parameter α and becomes constant for sufficiently large values of α. γ(c) has its maximum value for asymmetric load in comparison with the symmetric load. Finally, we apply the Melnikov theorem to the controlled system to explore the gain control parameter dependencies. 3. Semi-active control of a cable-stayed bridge under multiple-support excitations. Science.gov (United States) Dai, Ze-Bing; Huang, Jin-Zhi; Wang, Hong-Xia 2004-03-01 This paper presents a semi-active strategy for seismic protection of a benchmark cable-stayed bridge with consideration of multiple-support excitations. In this control strategy, Magnetorheological (MR) dampers are proposed as control devices, a LQG-clipped-optimal control algorithm is employed. An active control strategy, shown in previous researches to perform well at controlling the benchmark bridge when uniform earthquake motion was assumed, is also used in this study to control this benchmark bridge with consideration of multiple-support excitations. The performance of active control system is compared to that of the presented semi-active control strategy. Because the MR fluid damper is a controllable energy- dissipation device that cannot add mechanical energy to the structural system, the proposed control strategy is fail-safe in that bounded-input, bounded-output stability of the controlled structure is guaranteed. The numerical results demonstrated that the performance of the presented control design is nearly the same as that of the active control system; and that the MR dampers can effectively be used to control seismically excited cable-stayed bridges with multiple-support excitations. 4. Excited-state dynamics of acetylene excited to individual rotational level of the V04K01 subband Science.gov (United States) Makarov, Vladimir I.; Kochubei, Sergei A.; Khmelinskii, Igor V. 2006-01-01 Dynamics of the IR emission induced by excitation of the acetylene molecule using the (32Ka0,1,2,ÃAu1←41la1,X˜Σg+1) transition was investigated. The observed IR emission was assigned to transitions between the ground-state vibrational levels. Acetylene fluorescence quenching induced by external electric and magnetic fields acting upon the system prepared using the (34Ka1,ÃAu1←00la0,X˜Σg+1) excitation was also studied. External electric field creates an additional radiationless pathway to the ground-state levels, coupling levels of the ÃAu1 excited state to the quasiresonant levels of the X˜Σg+1 ground state. The level density of the ground state in the vicinity of the excited state is very high, thus the electric-field-induced transition is irreversible, with the rate constant described by the Fermi rule. Magnetic field alters the decay profile without changing the fluorescence quantum yield in collisionless conditions. IR emission from the CCH transient was detected, and was also affected by the external electric and magnetic fields. Acetylene predissociation was demonstrated to proceed by the direct S1→S0 mechanism. The results were explained using the previously developed theoretical approach, yielding values of the relevant model parameters. 5. Localized excitations in superconducting point contacts: probing the Andreev doublet International Nuclear Information System (INIS) Bretheau, L. 2013-01-01 The Josephson effect describes the coherent coupling between superconductors and the resulting supercurrent. Microscopically, it arises from the existence of discrete quasiparticle states, localized at the weak link, the Andreev bound states. They come in doublets in each conduction channel of the weak link, with energies symmetric about the Fermi energy and opposite supercurrents. Each Andreev doublet gives rise to four states: the ground state \|-> and the excited state \|+>, with even parity, and the excited odd states \|↑> and \|↓>. Is it possible to address and control Andreev doublets? This thesis describes two sets of experiments designed to answer this question using the most basic Josephson element, a one-atom contact between two superconducting electrodes. In a first experiment, we have observed and characterized the excited odd states \|↑> and \|↓>. As expected for a spin-degenerate system, they do not carry supercurrent. In this experiment the excitation was uncontrolled and resulted from trapping of spurious quasiparticles. We have measured the lifetime of the odd states: under some condition, it is found to exceed 100 μs. The second experiment is a photon-absorption spectroscopy of the Andreev doublet. It was performed by using a Josephson junction as an integrated on-chip microwave emitter and detector. The observed Andreev transitions correspond to excitation from the ground state \|->to the excited even state \|+>, and are well accounted for by our quantum model. This result opens the way to coherent manipulation of this two level system. The direct observation of the excited Andreev state, either by quasiparticle-injection or photon-absorption, strongly supports the mesoscopic theory of the Josephson effect. It shows that in addition to the phase difference, each channel of a Josephson weak link possesses an internal fermionic degree of freedom. It could be used to code information in a novel type of superconducting qubit. (author) [fr 6. Excitation of solar and stellar oscillations International Nuclear Information System (INIS) Baudin, Frederic 2009-01-01 In this report for an Accreditation to Supervise Research (HDR), and after an introduction which outlines the potential of helio-seismology, the author addresses the problem of excitation and amplitude of stellar oscillations with respect to their most important aspects, i.e. the theoretical framework of the present understanding of excitation mechanisms, and instrumental influences on measurements which are used to assess excitation rates, the difficulty to perform these measurements, and their analysis in some various cases. Thus, the author addresses excitation mechanisms of stellar oscillation (stochastic excitation, opacity- related excitation, and other excitation mechanisms), the excitation of solar modes (observation and theoretical predictions, influence of magnetic phenomena, solar g modes), and the excitation of modes in other stars (solar-type pulsators, red giants, and not so conventional pulsators such as HD180642 and Be stars like HD49330) 7. Mixed frequency excitation of an electrostatically actuated resonator KAUST Repository Ramini, Abdallah 2015-04-24 We investigate experimentally and theoretically the dynamics of a capacitive resonator under mixed frequency excitation of two AC harmonic signals. The resonator is composed of a proof mass suspended by two cantilever beams. Experimental measurements are conducted using a laser Doppler vibrometer to reveal the interesting dynamics of the system when subjected to two-source excitation. A nonlinear single-degree-of-freedom model is used for the theoretical investigation. The results reveal combination resonances of additive and subtractive type, which are shown to be promising to increase the bandwidth of the resonator near primary resonance frequency. Our results also demonstrate the ability to shift the combination resonances to much lower or much higher frequency ranges. We also demonstrate the dynamic pull-in instability under mixed frequency excitation. © 2015 Springer-Verlag Berlin Heidelberg 8. Excitation in the radiation chemistry of inorganic gases International Nuclear Information System (INIS) Willis, C.; Boyd, A.W. 1976-01-01 Gas phase radiation chemistry yield data and electron impact cross-section data are used to derive excitation mechanisms and to discuss the role of excited states in the radiation chemistry of O 2 , N 2 , N 2 O, CO, CO 2 , H 2 S, H 2 O and NH 3 . For each of these systems available cross-sections for ionization and neutral excitation are listed, together with relevant reaction rate data and a summary of the radiation chemistry studies at both high and low dose rates. In general, fairly complete mechanisms are derived and further tested by energy balance calculations. In order to present as complete a picture as possible, a summary of rates and products of ion-neutralization reactions is given at the end of the paper. (author) 9. Integrated parallel reception, excitation, and shimming (iPRES). Science.gov (United States) Han, Hui; Song, Allen W; Truong, Trong-Kha 2013-07-01 To develop a new concept for a hardware platform that enables integrated parallel reception, excitation, and shimming. This concept uses a single coil array rather than separate arrays for parallel excitation/reception and B0 shimming. It relies on a novel design that allows a radiofrequency current (for excitation/reception) and a direct current (for B0 shimming) to coexist independently in the same coil. Proof-of-concept B0 shimming experiments were performed with a two-coil array in a phantom, whereas B0 shimming simulations were performed with a 48-coil array in the human brain. Our experiments show that individually optimized direct currents applied in each coil can reduce the B0 root-mean-square error by 62-81% and minimize distortions in echo-planar images. The simulations show that dynamic shimming with the 48-coil integrated parallel reception, excitation, and shimming array can reduce the B0 root-mean-square error in the prefrontal and temporal regions by 66-79% as compared with static second-order spherical harmonic shimming and by 12-23% as compared with dynamic shimming with a 48-coil conventional shim array. Our results demonstrate the feasibility of the integrated parallel reception, excitation, and shimming concept to perform parallel excitation/reception and B0 shimming with a unified coil system as well as its promise for in vivo applications. Copyright © 2013 Wiley Periodicals, Inc. 10. Performance Analysis of a Hybrid One-Sided Magnetic Exciter Mounted on a Piezoelectric Stack Directory of Open Access Journals (Sweden) A. Nandi 2010-01-01 Full Text Available The present work proposes a non-contact hybrid exciter especially useful for harmonic excitation of lightly damped structures/rotors. In the proposed exciter an electromagnet is placed on a piezoelectric stack and the extension of the piezoelectric stack is made almost equal to the displacement of the structure using a simple tracking control. This largely eliminates stiffness coupling between the structure/rotor and the exciter and non-linearity in the excitation force due to the vibration of the structure/rotor. The stiffness and inertia of the piezoelectric stack is considered in the analysis. A SIMULINK model of the combined structure and the exciter is developed for a full time-domain simulation of the excitation system. 11. Characterization of weakly excited final states by shakedown spectroscopy of laser-excited potassium International Nuclear Information System (INIS) Schulz, J.; Heinaesmaeki, S.; Aksela, S.; Aksela, H.; Sankari, R.; Rander, T.; Lindblad, A.; Bergersen, H.; Oehrwall, G.; Svensson, S.; Kukk, E. 2006-01-01 3p shakedown spectra of laser excited potassium atoms as well as direct 3p photoemission of ground state potassium have been studied. These two excitation schemes lead to the same final states and thereby provide a good basis for a detailed study of the 3p 5 (4s3d) 1 configurations of singly ionized potassium and the photoemission processes leading to these configurations. The comparison of direct photoemission from the ground state and conjugate shakedown spectra from 4p 1/2 laser excited potassium made it possible to experimentally determine the character of final states that are only weakly excited in the direct photoemission but have a much higher relative intensity in the shakedown spectrum. Based on considerations of angular momentum and parity conservation the excitation scheme of the final states can be understood 12. Design of excitation signals for active system monitoring in a performance assessment setup DEFF Research Database (Denmark) 2011-01-01 This paper investigates how the excitation signal should be chosen for a active performance setup. The signal is used in a setup where the main purpose is to detect whether a parameter change of the controller has changed the global performance significantly. The signal has to be able to excite...... the dynamics of the subsystem under investigation both before and after the parameter change. The controller is well know, but there exists no detailed knowledge about the dynamics of the subsystem.... 13. Quenching reactions of electronically excited atoms International Nuclear Information System (INIS) Setser, D.W. 2001-01-01 The two-body, thermal quenching reactions of electronically excited atoms are reviewed using excited states of Ar, Kr, and Xe atoms as examples. State-specific interstate relaxation and excitation-transfer reactions with atomic colliders are discussed first. These results then are used to discuss quenching reactions of excited-state atoms with diatomic and polyatomic molecules, the latter have large cross sections, and the reactions can proceed by excitation transfer and by reactive quenching. Excited states of molecules are not considered; however, a table of quenching rate constants is given for six excited-state molecules in an appendix 14. Giant resonances on excited states International Nuclear Information System (INIS) Besold, W.; Reinhard, P.G.; Toepffer, C. 1984-01-01 We derive modified RPA equations for small vibrations about excited states. The temperature dependence of collective excitations is examined. The formalism is applied to the ground state and the first excited state of 90 Zr in order to confirm a hypothesis which states that not only the ground state but every excited state of a nucleus has a giant resonance built upon it. (orig.) 15. Nano-optical conveyor belt with waveguide-coupled excitation. Science.gov (United States) Wang, Guanghui; Ying, Zhoufeng; Ho, Ho-pui; Huang, Ying; Zou, Ningmu; Zhang, Xuping 2016-02-01 We propose a plasmonic nano-optical conveyor belt for peristaltic transport of nano-particles. Instead of illumination from the top, waveguide-coupled excitation is used for trapping particles with a higher degree of precision and flexibility. Graded nano-rods with individual dimensions coded to have resonance at specific wavelengths are incorporated along the waveguide in order to produce spatially addressable hot spots. Consequently, by switching the excitation wavelength sequentially, particles can be transported to adjacent optical traps along the waveguide. The feasibility of this design is analyzed using three-dimensional finite-difference time-domain and Maxwell stress tensor methods. Simulation results show that this system is capable of exciting addressable traps and moving particles in a peristaltic fashion with tens of nanometers resolution. It is the first, to the best of our knowledge, report about a nano-optical conveyor belt with waveguide-coupled excitation, which is very important for scalability and on-chip integration. The proposed approach offers a new design direction for integrated waveguide-based optical manipulation devices and its application in large scale lab-on-a-chip integration. 16. Laser-excited atomic-fluorescence spectrometry with electrothermal tube atomization. Science.gov (United States) Vera, J A; Leong, M B; Stevenson, C L; Petrucci, G; Winefordner, J D 1989-12-01 The performance of graphite-tube electrothermal atomizers is evaluated for laser-excited atomic-fluorescence spectrometry for several elements. Three pulsed laser systems are used to pump tunable dye lasers which subsequently are used to excite Pb, Ga, In, Fe, Ir, and Tl atoms in the hot graphite tube. The dye laser systems used are pumped by nitrogen, copper vapour and Nd:YAG lasers. Detection limits in the femtogram and subfemtogram range are typically obtained for all elements. A commercial graphite-tube furnace is important for the successful utilization of the laser-based method when the determination of trace elements is intended, especially when complicated matrices may be present. 17. Excited states CERN Document Server Lim, Edward C 1974-01-01 Excited States, Volume I reviews radiationless transitions, phosphorescence microwave double resonance through optical spectra in molecular solids, dipole moments in excited states, luminescence of polar molecules, and the problem of interstate interaction in aromatic carbonyl compounds. The book discusses the molecular electronic radiationless transitions; the double resonance techniques and the relaxation mechanisms involving the lowest triplet state of aromatic compounds; as well as the optical spectra and relaxation in molecular solids. The text also describes dipole moments and polarizab 18. Real-space visualization of remnant Mott gap and magnon excitations. Science.gov (United States) Wang, Y; Jia, C J; Moritz, B; Devereaux, T P 2014-04-18 We demonstrate the ability to visualize real-space dynamics of charge gap and magnon excitations in the Mott phase of the single-band Hubbard model and the remnants of these excitations with hole or electron doping. At short times, the character of magnetic and charge excitations is maintained even for large doping away from the Mott and antiferromagnetic phases. Doping influences both the real-space patterns and long timescales of these excitations with a clear carrier asymmetry attributable to particle-hole symmetry breaking in the underlying model. Further, a rapidly oscillating charge-density-wave-like pattern weakens, but persists as a visible demonstration of a subleading instability at half-filling which remains upon doping. The results offer an approach to analyzing the behavior of systems where momentum space is either inaccessible or poorly defined. 19. Saturated excitation of Fluorescence to quantify excitation enhancement in aperture antennas KAUST Repository Aouani, Heykel 2012-07-23 Fluorescence spectroscopy is widely used to probe the electromagnetic intensity amplification on optical antennas, yet measuring the excitation intensity amplification is a challenge, as the detected fluorescence signal is an intricate combination of excitation and emission. Here, we describe a novel approach to quantify the electromagnetic amplification in aperture antennas by taking advantage of the intrinsic non linear properties of the fluorescence process. Experimental measurements of the fundamental f and second harmonic 2f amplitudes of the fluorescence signal upon excitation modulation are used to quantify the electromagnetic intensity amplification with plasmonic aperture antennas. © 2012 Optical Society of America. 20. Saturated excitation of Fluorescence to quantify excitation enhancement in aperture antennas KAUST Repository Aouani, Heykel; Hostein, Richard; Mahboub, Oussama; Devaux, Eloï se; Rigneault, Hervé ; Ebbesen, Thomas W.; Wenger, Jé rô me 2012-01-01 Fluorescence spectroscopy is widely used to probe the electromagnetic intensity amplification on optical antennas, yet measuring the excitation intensity amplification is a challenge, as the detected fluorescence signal is an intricate combination of excitation and emission. Here, we describe a novel approach to quantify the electromagnetic amplification in aperture antennas by taking advantage of the intrinsic non linear properties of the fluorescence process. Experimental measurements of the fundamental f and second harmonic 2f amplitudes of the fluorescence signal upon excitation modulation are used to quantify the electromagnetic intensity amplification with plasmonic aperture antennas. © 2012 Optical Society of America. 1. Spike latency and response properties of an excitable micropillar laser Science.gov (United States) Selmi, F.; Braive, R.; Beaudoin, G.; Sagnes, I.; Kuszelewicz, R.; Erneux, T.; Barbay, S. 2016-10-01 We present experimental measurements concerning the response of an excitable micropillar laser with saturable absorber to incoherent as well as coherent perturbations. The excitable response is similar to the behavior of spiking neurons but with much faster time scales. It is accompanied by a subnanosecond nonlinear delay that is measured for different bias pump values. This mechanism provides a natural scheme for encoding the strength of an ultrafast stimulus in the response delay of excitable spikes (temporal coding). Moreover, we demonstrate coherent and incoherent perturbations techniques applied to the micropillar with perturbation thresholds in the range of a few femtojoules. Responses to coherent perturbations assess the cascadability of the system. We discuss the physical origin of the responses to single and double perturbations with the help of numerical simulations of the Yamada model and, in particular, unveil possibilities to control the relative refractory period that we recently evidenced in this system. Experimental measurements are compared to both numerical simulations of the Yamada model and analytic expressions obtained in the framework of singular perturbation techniques. This system is thus a good candidate to perform photonic spike processing tasks in the framework of novel neuroinspired computing systems. 2. Optimum design of a Lanchester damper for a viscously damped single degree of freedom system subjected to inertial excitation Science.gov (United States) Bapat, V. A.; Prabhu, P. 1980-11-01 The problem of designing an optimum Lanchester damper for a viscously damped single degree of freedom system subjected to inertial harmonic excitation is investigated. Two criteria are used for optimizing the performance of the damper: (i) minimum motion transmissibility; (ii) minimum force transmissibility. Explicit expressions are developed for determining the absorber parameters. 3. Global Analysis of Response in the Piezomagnetoelastic Energy Harvester System under Harmonic and Poisson White Noise Excitations International Nuclear Information System (INIS) Yue Xiao-Le; Xu Wei; Zhang Ying; Wang Liang 2015-01-01 The piezomagnetoelastic energy harvester system subjected to harmonic and Poisson white noise excitations is studied by using the generalized cell mapping method. The transient and stationary probability density functions (PDFs) of response based on the global viewpoint are obtained by the matrix analysis method. Monte Carlo simulation results verify the accuracy of this method. It can be observed that evolutionary direction of transient and stationary PDFs is in accordance with the unstable manifold for this system, and a stochastic P-bifurcation occurs as the intensity of Poisson white noise increases. This study presents an efficient numerical tool to solve the stochastic response of a three-dimensional dynamical system and provides a new idea to analyze the energy harvester system. (paper) 4. Excitation and photon decay of giant resonances excited by intermediate energy heavy ions International Nuclear Information System (INIS) Bertrand, F.E.; Beene, J.R. 1987-01-01 Inelastic scattering of medium energy heavy ions provides very large cross sections and peak-to-continuum ratios for excitation of giant resonances. For energies above about 50 MeV/nucleon, giant resonances are excited primarily through Coulomb excitation, which is indifferent to isospin, thus providing a good probe for the study of isovector giant resonances. The extremely large cross sections available from heavy ion excitation permit the study of rare decay modes of the giant resonances. In particular, recent measurements have been made of the photon decay of giant resonances following excitation by 22 and 84 MeV/nucleon 17 O projectiles. The singles results at 84 MeV/nucleon yield peak cross sections for the isoscalar giant quadrupole resonance and the isovector giant dipole resonance of approximately 0.8 and 3 barns/sr, respectively. Data on the ground state decay of the isoscalar giant quadrupole and isovector giant dipole resonances are presented and compared with calculations. Decays to low-lying excited states are also discussed. Preliminary results from an experiment to isolate the 208 Pb isovector quadrupole resonance using its gamma decay are presented. 22 refs., 19 figs., 1 tab 5. Electron collisions and internal excitation in stored molecular ion beams International Nuclear Information System (INIS) Buhr, H. 2006-01-01 In storage ring experiments the role, which the initial internal excitation of a molecular ion can play in electron collisions, and the effect of these collisions on the internal excitation are investigated. Dissociative recombination (DR) and inelastic and super-elastic collisions are studied in the system of He + 2 . The DR rate coefficient at low energies depends strongly on the initial vibrational excitation in this system. Therefore changes in the DR rate coefficient are a very sensitive probe for changes in the vibrational excitation in He + 2 , which is used to investigate the effects of collisions with electrons and residual gas species. The low-energy DR of HD + is rich with resonances from the indirect DR process, when certain initial rotational levels in the molecular ion are coupled to levels in neutral Rydberg states lying below the ion state. Using new procedures for high-resolution electron-ion collision spectroscopy developed here, these resonances in the DR cross section can be measured with high energy sensitivity. This allows a detailed comparison with results of a MQDT calculation in an effort to assign some or all of the resonances to certain intermediate Rydberg levels. (orig.) 6. Electron collisions and internal excitation in stored molecular ion beams Energy Technology Data Exchange (ETDEWEB) Buhr, H. 2006-07-26 In storage ring experiments the role, which the initial internal excitation of a molecular ion can play in electron collisions, and the effect of these collisions on the internal excitation are investigated. Dissociative recombination (DR) and inelastic and super-elastic collisions are studied in the system of He{sup +}{sub 2}. The DR rate coefficient at low energies depends strongly on the initial vibrational excitation in this system. Therefore changes in the DR rate coefficient are a very sensitive probe for changes in the vibrational excitation in He{sup +}{sub 2}, which is used to investigate the effects of collisions with electrons and residual gas species. The low-energy DR of HD{sup +} is rich with resonances from the indirect DR process, when certain initial rotational levels in the molecular ion are coupled to levels in neutral Rydberg states lying below the ion state. Using new procedures for high-resolution electron-ion collision spectroscopy developed here, these resonances in the DR cross section can be measured with high energy sensitivity. This allows a detailed comparison with results of a MQDT calculation in an effort to assign some or all of the resonances to certain intermediate Rydberg levels. (orig.) 7. Elementary spin excitations in ultrathin itinerant magnets Energy Technology Data Exchange (ETDEWEB) Zakeri, Khalil, E-mail: [email protected] 2014-12-10 Elementary spin excitations (magnons) play a fundamental role in condensed matter physics, since many phenomena e.g. magnetic ordering, electrical (as well as heat) transport properties, ultrafast magnetization processes, and most importantly electron/spin dynamics can only be understood when these quasi-particles are taken into consideration. In addition to their fundamental importance, magnons may also be used for information processing in modern spintronics. Here the concept of spin excitations in ultrathin itinerant magnets is discussed and reviewed. Starting with a historical introduction, different classes of magnons are introduced. Different theoretical treatments of spin excitations in solids are outlined. Interaction of spin-polarized electrons with a magnetic surface is discussed. It is shown that, based on the quantum mechanical conservation rules, a magnon can only be excited when a minority electron is injected into the system. While the magnon creation process is forbidden by majority electrons, the magnon annihilation process is allowed instead. These fundamental quantum mechanical selection rules, together with the strong interaction of electrons with matter, make the spin-polarized electron spectroscopies as appropriate tools to excite and probe the elementary spin excitations in low-dimensional magnets e.g ultrathin films and nanostructures. The focus is put on the experimental results obtained by spin-polarized electron energy loss spectroscopy and spin-polarized inelastic tunneling spectroscopy. The magnon dispersion relation, lifetime, group and phase velocity measured using these approaches in various ultrathin magnets are discussed in detail. The differences and similarities with respect to the bulk excitations are addressed. The role of the temperature, atomic structure, number of atomic layers, lattice strain, electronic complexes and hybridization at the interfaces are outlined. A possibility of simultaneous probing of magnons and phonons 8. Charmonium non-potential excitations International Nuclear Information System (INIS) Borue, V.Y.; Khokhlachev, S.B. 1990-01-01 Within the framework of an effective theory of quantum gluodynamics formulated earlier in terms of the glueball degrees of freedom, the excitations of gluon bunch formed by heavy quark and antiquark are considered. It is shown that these excitations correspond to the vibration of the gluon bunch shape and lie nearly 800 MeV higher than the charmonium ground state. The consequences of the existence of these excitations are discussed 9. Response of HDR-VKL piping system to seismic test excitations: Comparison of analytical predictions and test measurements International Nuclear Information System (INIS) Srinivasan, M.G.; Kot, C.A.; Hsieh, B.J. 1989-01-01 As part of the earthquake investigations at the HDR (Heissdampfreaktor) Test Facility in Kahl/Main, FRG, simulated seismic tests (SHAM) were performed during April--May 1988 on the VKL (Versuchskreislauf) piping system. The purpose of these experiments was to study the behavior of piping subjected to a range of seismic excitation levels including those that exceed design levels manifold and that might induce failure of pipe supports or plasticity in the pipe runs, and to establish seismic margins for piping and pipe supports. Data obtained in the tests are also used to validate analysis methods. Detailed reports on the SHAM experiments are given elsewhere. The objective of this document is to evaluate a subsystem analysis module of the SMACS code. This module is a linear finite-element based program capable of calculating the response of nuclear power plant subsystems subjected to independent multiple-acceleration input excitation. The evaluation is based on a comparison of computational results of simulation of SHAM tests with corresponding test measurements 10. Antiferromagnetic resonance excited by oscillating electric currents Science.gov (United States) Sluka, Volker 2017-12-01 In antiferromagnetic materials the order parameter exhibits resonant modes at frequencies that can be in the terahertz range, making them interesting components for spintronic devices. Here, it is shown that antiferromagnetic resonance can be excited using the inverse spin-Hall effect in a system consisting of an antiferromagnetic insulator coupled to a normal-metal waveguide. The time-dependent interplay between spin torque, ac spin accumulation, and magnetic degrees of freedom is studied. It is found that the dynamics of the antiferromagnet affects the frequency-dependent conductivity of the normal metal. Further, a comparison is made between spin-current-induced and Oersted-field-induced excitation under the condition of constant power injection. 11. Dissociative Excitation of Adenine by Electron Impact Science.gov (United States) McConkey, J. William; Trocchi, Joshuah; Dech, Jeffery; Kedzierski, Wladek 2017-04-01 Dissociative excitation of adenine (C6H5NH2) into excited atomic fragments has been studied in the electron impact energy range from threshold to 300 eV. A crossed beam system coupled to a vacuum ultraviolet (VUV) monochromator is used to study emissions in the wavelength range from 110 to 200 nm. The beam of adenine vapor from a stainless steel oven is crossed at right angles by the electron beam and the resultant UV radiation is detected in a mutually orthogonal direction. The strongest feature in the spectrum is H Lyman- α. Financial support from NSERC and CFI, Canada, is gratefully acknowledged. 12. Phased Array Excitations For Efficient Near Field Wireless Power Transmission Science.gov (United States) 2016-09-01 channeled to the battery or power plant. Figure 2. WPT System Block Diagram for Battery Charging. Source : [2]. Wireless power transfer has gained...EXCITATIONS FOR EFFICIENT NEAR-FIELD WIRELESS POWER TRANSMISSION by Sean X. Hong September 2016 Thesis Advisor: David Jenn Second Reader...TYPE AND DATES COVERED Master’s thesis 4. TITLE AND SUBTITLE PHASED ARRAY EXCITATIONS FOR EFFICIENT NEAR-FIELD WIRELESS POWER TRANSMISSION 5 13. Electromagnetic excitation of 136Xe in relativistic heavy ion collisions International Nuclear Information System (INIS) Schmidt, R.D. 1991-11-01 In the framework of the experimental program at the accelerator facilities SIS/ESR at the Society for Heavy Ion Research in Darmstadt a detector system for relativistic neutrons was developed, constructed, and applied in first experiments. An essential research aim is the study of collective states after electromagnetic excitation in relativistic heavy ion collisions. In peripheral collisions high-energy virtual photons are exchanged. This leads to the excitation of giant resonances, especially of the giant dipole and quadrupole resonance. An essential decay channel of giant resonances in heavy nuclei is the emission of neutrons, followed by the emission of γ radiation below the particle threshold. These decay channels were studied with the detector system developed by the LAND collaboration. A first experiment on the electromagnetic excitation was performed with a 136 Xe beam at an energy of 700 MeV/u and Pb respectively C targets. (orig./HSI) [de 14. Multi-quantum excitation in optically pumped alkali atom: rare gas mixtures Science.gov (United States) Galbally-Kinney, K. L.; Rawlins, W. T.; Davis, S. J. 2014-03-01 Diode-pumped alkali laser (DPAL) technology offers a means of achieving high-energy gas laser output through optical pumping of the D-lines of Cs, Rb, and K. The exciplex effect, based on weak attractive forces between alkali atoms and polarizable rare gas atoms (Ar, Kr, Xe), provides an alternative approach via broadband excitation of exciplex precursors (XPAL). In XPAL configurations, we have observed multi-quantum excitation within the alkali manifolds which result in infrared emission lines between 1 and 4 μm. The observed excited states include the 42FJ states of both Cs and Rb, which are well above the two-photon energy of the excitation laser in each case. We have observed fluorescence from multi-quantum states for excitation wavelengths throughout the exciplex absorption bands of Cs-Ar, Cs-Kr, and Cs-Xe. The intensity scaling is roughly first-order or less in both pump power and alkali concentration, suggesting a collisional energy pooling excitation mechanism. Collisional up-pumping appears to present a parasitic loss term for optically pumped atomic systems at high intensities, however there may also be excitation of other lasing transitions at infrared wavelengths. 15. Trapping time statistics and efficiency of transport of optical excitations in dendrimers OpenAIRE Heijs, D.J.; Malyshev, V.A.; Knoester, J. 2004-01-01 We theoretically study the trapping time distribution and the efficiency of the excitation energy transport in dendritic systems. Trapping of excitations, created at the periphery of the dendrimer, on a trap located at its core, is used as a probe of the efficiency of the energy transport across the dendrimer. The transport process is treated as incoherent hopping of excitations between nearest-neighbor dendrimer units and is described using a rate equation. We account for radiative and non-r... 16. The MSINDO-sCIS and MSINDO-UCIS methods. Procedures for the calculation of properties of excited states in molecules and periodic systems by a semiempirical approach International Nuclear Information System (INIS) 2013-01-01 Theoretical background, parameterization and performance of the newly developed semiempirical configuration interaction singles (CIS) method MSINDO-sCIS (scaled configuration interaction singles) are presented. The CIS Hamiltonian is modified by scaling of the Coulomb and exchange integrals and a semiempirical correction of the diagonal elements. For a recently proposed benchmark set of 28 medium-sized organic molecules, vertical excitation energies for singlet and triplet states have been calculated and statistically evaluated. A full reparameterization of the MSINDO method for both ground and excited state properties was performed. The results of the reparameterized MSINDO-sCIS method are compared to the currently best semiempirical method for excited states, OM3-CISDTQ by Thiel et al., and to other standard methods, such as time-dependent density- functional theory. The mean absolute deviation with respect to the theoretical best estimates (TBEs) for MSINDO-sCIS is 0.44 eV, comparable to the OM3 method but significantly smaller than for Zerner's INDO/S. The computational effort is strongly reduced compared to OM3-CISDTQ and OM3-MRCISD, since only single excitations are taken into account. Higher excitations are implicitly included by parameterization and the empirical correction term. By application of the Davidson-Liu block diagonalization method high computational efficiency is achieved. Furthermore it is demonstrated, that the MSINDO-sCIS method correctly describes charge-transfer (CT) states, that represent a crucial problem for time-dependent density functional theory (TD-DFT) methods. Additionally this method is extended to open-shell systems by the UCIS (unrestricted CIS) approach. MSINDO allows the calculation of periodic systems via the cyclic cluster model (CCM) which is a direct-space approach and therefore can be in principle combined with all molecular quantum-chemical techniques. The sCIS/UCIS equations are solved for a cluster with periodic 17. Using narrowband excitation to confirm that the S∗ state in carotenoids is not a vibrationally-excited ground state species Science.gov (United States) Jailaubekov, Askat E.; Song, Sang-Hun; Vengris, Mikas; Cogdell, Richard J.; Larsen, Delmar S. 2010-02-01 The hypothesis that S∗ is a vibrationally-excited ground-state population is tested and discarded for two carotenoid samples: β-carotene in solution and rhodopin glucoside embedded in the light harvesting 2 protein from Rhodopseudomonas acidophila. By demonstrating that the transient absorption signals measured in both systems that are induced by broadband (1000 cm -1) and narrowband (50 cm -1) excitation pulses are near identical and hence bandwidth independent, the impulsive stimulated Raman scattering mechanism proposed as the primary source for S∗ generation is discarded. To support this conclusion, previously published multi-pulse pump-dump-probe signals [17] are revisited to discard secondary mechanisms for S∗ formation. 18. Connectivity, excitability and activity patterns in neuronal networks International Nuclear Information System (INIS) Le Feber, Joost; Stoyanova, Irina I; Chiappalone, Michela 2014-01-01 Extremely synchronized firing patterns such as those observed in brain diseases like epilepsy may result from excessive network excitability. Although network excitability is closely related to (excitatory) connectivity, a direct measure for network excitability remains unavailable. Several methods currently exist for estimating network connectivity, most of which are related to cross-correlation. An example is the conditional firing probability (CFP) analysis which calculates the pairwise probability (CFP i,j ) that electrode j records an action potential at time t = τ, given that electrode i recorded a spike at t = 0. However, electrode i often records multiple spikes within the analysis interval, and CFP values are biased by the on-going dynamic state of the network. Here we show that in a linear approximation this bias may be removed by deconvoluting CFP i,j with the autocorrelation of i (i.e. CFP i,i ), to obtain the single pulse response (SPR i,j )—the average response at electrode j to a single spike at electrode i. Thus, in a linear system SPRs would be independent of the dynamic network state. Nonlinear components of synaptic transmission, such as facilitation and short term depression, will however still affect SPRs. Therefore SPRs provide a clean measure of network excitability. We used carbachol and ghrelin to moderately activate cultured cortical networks to affect their dynamic state. Both neuromodulators transformed the bursting firing patterns of the isolated networks into more dispersed firing. We show that the influence of the dynamic state on SPRs is much smaller than the effect on CFPs, but not zero. The remaining difference reflects the alteration in network excitability. We conclude that SPRs are less contaminated by the dynamic network state and that mild excitation may decrease network excitability, possibly through short term synaptic depression. (papers) 19. Passive listening to preferred motor tempo modulates corticospinal excitability. Science.gov (United States) Michaelis, Kelly; Wiener, Martin; Thompson, James C 2014-01-01 Rhythms are an essential characteristic of our lives, and auditory-motor coupling affects a variety of behaviors. Previous research has shown that the neural regions associated with motor system processing are coupled to perceptual rhythmic and melodic processing such that the perception of rhythmic stimuli can entrain motor system responses. However, the degree to which individual preference modulates the motor system is unknown. Recent work has shown that passively listening to metrically strong rhythms increases corticospinal excitability, as indicated by transcranial magnetic stimulation (TMS). Furthermore, this effect is modulated by high-groove music, or music that inspires movement, while neuroimaging evidence suggests that premotor activity increases with tempos occurring within a preferred tempo (PT) category. PT refers to the rate of a hypothetical endogenous oscillator that may be indicated by spontaneous motor tempo (SMT) and preferred perceptual tempo (PPT) measurements. The present study investigated whether listening to a rhythm at an individual's PT preferentially modulates motor system excitability. SMT was obtained in human participants through a tapping task in which subjects were asked to tap a response key at their most comfortable rate. Subjects listened a 10-beat tone sequence at 11 log-spaced tempos and rated their preference for each (PPT). We found that SMT and PPT measurements were correlated, indicating that preferred and produced tempos occurred at a similar rate. Crucially, single-pulse TMS delivered to left M1 during PPT judgments revealed that corticospinal excitability, measured by motor-evoked potentials (MEPs), was modulated by tempos traveling closer to individual PT. However, the specific nature of this modulation differed across individuals, with some exhibiting an increase in excitability around PT and others exhibiting a decrease. These findings suggest that auditory-motor coupling induced by rhythms is preferentially 20. Stochastic stability of mechanical systems under renewal jump process parametric excitation DEFF Research Database (Denmark) Iwankiewicz, R.; Nielsen, Søren R.K.; Larsen, Jesper Winther 2005-01-01 independent, negative exponential distributed variables; hence, the arrival process may be termed as a generalized Erlang renewal process. The excitation process is governed by the stochastic equation driven by two independent Poisson processes, with different parameters. If the response in a single mode... 1. Photoionization study of doubly-excited helium at ultra-high resolution Energy Technology Data Exchange (ETDEWEB) Kaindl, G.; Schulz, K.; Domke, M. [Freie Universitaet Berlin (Germany)] [and others 1997-04-01 Ever since the pioneering work of Madden & Codling and Cooper, Fano & Prats on doubly-excited helium in the early sixties, this system may be considered as prototypical for the study of electron-electron correlations. More detailed insight into these states could be reached only much later, when improved theoretical calculations of the optically-excited {sup 1}P{sup 0} double-excitation states became available and sufficiently high energy resolution ({delta}E=4.0 meV) was achieved. This allowed a systematic investigation of the double-excitation resonances of He up to excitation energies close to the double-ionization threshold, I{sub infinity}=79.003 eV, which stimulated renewed theoretical interest into these correlated electron states. The authors report here on striking progress in energy resolution in this grazing-incidence photon-energy range of grating monochromators and its application to hitherto unobservable states of doubly-excited He. By monitoring an extremely narrow double-excitation resonance of He, with a theoretical lifetime width of less than or equal to 5 {mu}eV, a resolution of {delta}E=1.0 meV (FWHM) at 64.1 eV could be achieved. This ultra-high spectral resolution, combined with high photon flux, allowed the investigation of new Rydberg resonances below the N=3 ionization threshold, I{sub 3}, as well as a detailed comparison with ab-initio calculations. 2. Surface boiling - an obvious but like no other decay mode of highly excited atomic nuclei International Nuclear Information System (INIS) Toke, J. 2012-01-01 Essentials of a generalized compound nucleus model are introduced based on a concept of an open microcanonical ensemble which considers explicitly the role of the diffuse surface domain and of the thermal expansion of nuclear systems in the quest for maximum entropy. This obvious generalization offers a unique and universal thermodynamic framework for understanding the changes in the gross behavior of excited nuclear systems with increasing excitation energy and, specifically, the competition between different statistical decay modes, including classical evaporation and binary fission, but also the Coulomb fragmentation of excited systems into multiple fragments - the famed multifragmentation. Importantly, the formalism offers a natural explanation, in terms of boiling or spinodal vaporization, for the experimentally observed appearance of limiting excitation energy that can be thermalized by an exited nuclear system and the associated limiting temperature. It is shown that it is the thermal expansion that leads to volume boiling in an infinite matter and surface boiling in finite nuclei. The latter constitutes an important and universal, but hitherto unappreciated decay mode of highly excited nuclei, a mode here named surface spinodal vaporization. It is also shown that in iso-asymmetric systems, thermal expansion leads to what constitutes distillation - a decay mode here named distillative spinodal vaporization 3. 340nm UV LED excitation in time-resolved fluorescence system for europium-based immunoassays detection Science.gov (United States) Rodenko, Olga; Fodgaard, Henrik; Tidemand-Lichtenberg, Peter; Pedersen, Christian 2017-02-01 In immunoassay analyzers for in-vitro diagnostics, Xenon flash lamps have been widely used as excitation light sources. Recent advancements in UV LED technology and its advantages over the flash lamps such as smaller footprint, better wall-plug efficiency, narrow emission spectrum, and no significant afterglow, have made them attractive light sources for gated detection systems. In this paper, we report on the implementation of a 340 nm UV LED based time-resolved fluorescence system based on europium chelate as a fluorescent marker. The system performance was tested with the immunoassay based on the cardiac marker, TnI. The same signal-to-noise ratio as for the flash lamp based system was obtained, operating the LED below specified maximum current. The background counts of the system and its main contributors were measured and analyzed. The background of the system of the LED based unit was improved by 39% compared to that of the Xenon flash lamp based unit, due to the LEDs narrower emission spectrum and longer pulse width. Key parameters of the LED system are discussed to further optimize the signal-to-noise ratio and signal-to-background, and hence the sensitivity of the instrument. 4. Efficient primary and parametric resonance excitation of bistable resonators KAUST Repository Ramini, Abdallah 2016-09-12 We experimentally demonstrate an efficient approach to excite primary and parametric (up to the 4th) resonance of Microelectromechanical system MEMS arch resonators with large vibrational amplitudes. A single crystal silicon in-plane arch microbeam is fabricated such that it can be excited axially from one of its ends by a parallel-plate electrode. Its micro/nano scale vibrations are transduced using a high speed camera. Through the parallel-plate electrode, a time varying electrostatic force is applied, which is converted into a time varying axial force that modulates dynamically the stiffness of the arch resonator. Due to the initial curvature of the structure, not only parametric excitation is induced, but also primary resonance. Experimental investigation is conducted comparing the response of the arch near primary resonance using the axial excitation to that of a classical parallel-plate actuation where the arch itself forms an electrode. The results show that the axial excitation can be more efficient and requires less power for primary resonance excitation. Moreover, unlike the classical method where the structure is vulnerable to the dynamic pull-in instability, the axial excitation technique can provide large amplitude motion while protecting the structure from pull-in. In addition to primary resonance, parametrical resonances are demonstrated at twice, one-half, and two-thirds the primary resonance frequency. The ability to actuate primary and/or parametric resonances can serve various applications, such as for resonator based logic and memory devices. (C) 2016 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license 5. Mean excitation energies for molecular ions Energy Technology Data Exchange (ETDEWEB) Jensen, Phillip W.K.; Sauer, Stephan P.A. [Department of Chemistry, University of Copenhagen, Copenhagen (Denmark); Oddershede, Jens [Department of Physics, Chemistry, and Pharmacy, University of Southern Denmark, Odense (Denmark); Quantum Theory Project, Departments of Physics and Chemistry, University of Florida, Gainesville, FL (United States); Sabin, John R., E-mail: [email protected] [Department of Physics, Chemistry, and Pharmacy, University of Southern Denmark, Odense (Denmark); Quantum Theory Project, Departments of Physics and Chemistry, University of Florida, Gainesville, FL (United States) 2017-03-01 The essential material constant that determines the bulk of the stopping power of high energy projectiles, the mean excitation energy, is calculated for a range of smaller molecular ions using the RPA method. It is demonstrated that the mean excitation energy of both molecules and atoms increase with ionic charge. However, while the mean excitation energies of atoms also increase with atomic number, the opposite is the case for mean excitation energies for molecules and molecular ions. The origin of these effects is explained by considering the spectral representation of the excited state contributing to the mean excitation energy. 6. Strange effects of strong high-frequency excitation DEFF Research Database (Denmark) Thomsen, Jon Juel 2003-01-01 Three general effects of mechanical high-frequency excitation (HFE) are described: Stiffening - an apparent change in the stiffness associated with an equilibrium; Biasing - a tendency for a system to move towards a particular state which does not exist or is unstable without HFE; and Smoothening... 7. Perturbation Solutions for Random Linear Structural Systems subject to Random Excitation using Stochastic Differential Equations DEFF Research Database (Denmark) Köyluoglu, H.U.; Nielsen, Søren R.K.; Cakmak, A.S. 1994-01-01 perturbation method using stochastic differential equations. The joint statistical moments entering the perturbation solution are determined by considering an augmented dynamic system with state variables made up of the displacement and velocity vector and their first and second derivatives with respect......The paper deals with the first and second order statistical moments of the response of linear systems with random parameters subject to random excitation modelled as white-noise multiplied by an envelope function with random parameters. The method of analysis is basically a second order...... to the random parameters of the problem. Equations for partial derivatives are obtained from the partial differentiation of the equations of motion. The zero time-lag joint statistical moment equations for the augmented state vector are derived from the Itô differential formula. General formulation is given... 8. Electro-mechanical impact system excited by a source of limited power Czech Academy of Sciences Publication Activity Database 2008-01-01 Roč. 15, č. 6 (2008), s. 1-10 ISSN 1802-1484 R&D Projects: GA ČR GA101/06/0063 Institutional research plan: CEZ:AV0Z20760514 Keywords : mechanical oscillations * impacts * limited power of exciter * electro-mechanical interaction Subject RIV: BI - Acoustics 9. Excited fermions International Nuclear Information System (INIS) Boudjema, F.; Djouadi, A.; Kneur, J.L. 1992-01-01 The production of excited fermions with mass above 100 GeV is considered. f→Vf (1) decay widths are calculated where V=γ, Z or W. Excited fermion pair production in e + e - annihilation and in γγ collisions, and single production in e + e - annihilation, eγ and γγ collisions is also discussed. Cross sections are calculated for all these cases. The discovery potential of the NLC at 500 GeV is compared with that of other colliders. (K.A.) 15 refs., 5 figs., 2 tabs 10. Deviations from excitation equilibrium in optically thick mercury arc plasmas International Nuclear Information System (INIS) Karabourniotis, D.; Couris, S.; Damelincourt, J.J. 1989-01-01 Up to date mercury arcs at pressure greater than 1 atm have been investigated as plasma systems in local thermodynamic equilibrium (LTE) state. These studies have been motivated by the applications of mercury arcs, e.g., in the lighting industry. The LTE-assumption simplifies the use of spectroscopic diagnostics and the performance of species-concentration calculations. A high pressure mercury arc of about 1 atm had been considered in two possibilities: excitation and gas temperatures are the same, the electron temperature is higher and excitation and electron temperatures are the same, the gas temperature is lower. Recent measurements in mercury arcs reveal the existence of severe departures from thermal equilibrium and suggest the absence of excitation equilibrium in the axis and in the periphery in such an arc. The deviation from equilibrium leads to complicated distributions, such that the system cannot be described correctly by any single temperature. This becomes quite complicated when plasma inhomogeneity and strong reabsorption of the radiation are present 11. Development and Implementation of Biological Circuits Using Excitable and Non-Excitable Cells Energy Technology Data Exchange (ETDEWEB) Casasnovas-Orus, V.; Gomez-Cid, L.; Hernandez-Romero, I.; Fuentes, L.; Guillem, M.S.; Atienza, F.; Fernandez-Aviles, F.; Climent, A.M. 2016-07-01 Compared to conventional computation systems, living beings require reduced power and raw materials consumption, inviting to explore the concept of biological circuits. In this project, a proof-of-concept of logical biocircuits using cell patterns has been developed. These were based upon differential ionic communication between cells, being the cells types used excitable and non-excitable, modeled by cardiomyocytes and fibroblasts correspondingly. To begin, patterns for the basic logic computation blocks were designed, including the OR gate, AND gate and logic memory. The designs were evaluated with mathematical models and in vitro experiments. Results of mathematical modeling indicated that theoretical approval of the biocircuit function. Regarding in vitro biocircuit implementation, three different selective cell localization techniques proved useful for the pattern creation. Evaluation with optical mapping confirmed the operation of the OR gate and logic memory. More resolution in the cell placement strategy will be needed to observe the proper AND gate operation. Thus, fine-tuning of the implementation process will enable the construction of more complex biocircuits that will take on clinical applications relating to electric stimulation of tissues and programmed drug delivery. (Author) 12. Excitation of autoionizing states of helium by 100 keV proton impact: II. Excitation cross sections and mechanisms of excitation Energy Technology Data Exchange (ETDEWEB) Godunov, A.L. [Department of Physics, Tulane University, New Orleans, LA 70118-5698 (United States); Ivanov, P.B.; Schipakov, V.A. [Troitsk Institute of Innovation and Fusion Research Troitsk, Moscow region, 142092 (Russian Federation); Moretto-Capelle, P.; Bordenave-Montesquieu, D.; Bordenave-Montesquieu, A. [Laboratoire Collisions, Agregats, Reactivite, IRSAMC, UMR 5589, CNRS-Universite Paul Sabatier, 31062 Toulouse Cedex (France) 2000-03-14 Mechanisms of two-electron excitation of the (2s{sup 2}){sup 1} S, (2p{sup 2} ){sup 1} D and (2s2p){sup 1} P autoionizing states of helium are studied both experimentally and theoretically. It is shown that an explicit introduction of a kinematic factor, with a process-specific phase leads to a productive parametrization of experimental cross sections of ionization, allowing one to extract cross sections of excitation of autoionizing states. Using a new fitting procedure together with the proposed parametrization made it possible to obtain the excitation cross sections and magnetic sublevel population from electron spectra as well as, for the first time, to resolve the contribution of resonance and interference components to resonance profiles. Interference with direct ionization is shown to contribute significantly into resonance formation even for backward ejection angles. We demonstrate theoretically that the excitation cross sections thus extracted from experimental electron spectra hold information about the interaction of autoionizing states with an adjacent continuum. (author) 13. Depolarization of a photoelectret under depth-nonuniform excitation of the sample International Nuclear Information System (INIS) Vavrek, A.F.; Khristova, K.K. 1988-01-01 A simple theoretical model is given and explaination is made of the experimental observations of the recently carried out destroying the photoelectret state (PES) in Bi 1 2SiO 2 0 (BSO) by X-ray irradiation. It is assumed that during the irradiatoin two regions are formed divided by a sharp boundary - an excited region I with mobile non-equilibrium carriers and non-excited region II without mobile carriers. According to the experimental conditions, the isolating layers are between the sample and the electrodes, the total photoelectret charge is zero and the PE charge before the irradiation have a barrier distribution. For the determination of a PE charge the method of photodepolarization is used. When the photoelectret is irradiated in region I, mobile carriers are generated which move under the influence of the electrical field in this region and begin to accumulate on the boundary plane between the excited and non-excited regions, thus forming a 'shifted' charge layer. There is no movement of charges in region II. The distribution of the charges and the electric field in such a multilayer system is described by a system of equations. It is established that during the X-ray irradiation the PE charge gradually decreases. However, the maximum charge which can be destroyed is found to be a function of the thickness of the excited region and becomes equal to the initial charge when an excitation of the whole sample takes place. The consideration done explains the experimentally observed seeming loss of sensitivity of the BSO to the X radiation 14. A ballistic transport model for electronic excitation following particle impact Science.gov (United States) Hanke, S.; Heuser, C.; Weidtmann, B.; Wucher, A. 2018-01-01 We present a ballistic model for the transport of electronic excitation energy induced by keV particle bombardment onto a solid surface. Starting from a free electron gas model, the Boltzmann transport equation (BTE) is employed to follow the evolution of the temporal and spatial distribution function f (r → , k → , t) describing the occupation probability of an electronic state k → at position r → and time t. Three different initializations of the distribution function are considered: i) a thermal distribution function with a locally and temporally elevated electron temperature, ii) a peak excitation at a specific energy above the Fermi level with a quasi-isotropic distribution in k-space and iii) an anisotropic peak excitation with k-vectors oriented in a specific transport direction. While the first initialization resembles a distribution function which may, for instance, result from electronic friction of moving atoms within an ion induced collision cascade, the peak excitation can in principle result from an autoionization process after excitation in close binary collisions. By numerically solving the BTE, we study the electronic energy exchange along a one dimensional transport direction to obtain a time and space resolved excitation energy distribution function, which is then analyzed in view of general transport characteristics of the chosen model system. 15. Collective excitations of harmonically trapped ideal gases NARCIS (Netherlands) Van Schaeybroeck, B.; Lazarides, A. 2009-01-01 We theoretically study the collective excitations of an ideal gas confined in an isotropic harmonic trap. We give an exact solution to the Boltzmann-Vlasov equation; as expected for a single-component system, the associated mode frequencies are integer multiples of the trapping frequency. We show 16. The Mean Excitation Energy of Atomic Ions DEFF Research Database (Denmark) Sauer, Stephan; Oddershede, Jens; Sabin, John R. 2015-01-01 A method for calculation of the mean excitation energies of atomic ions is presented, making the calculation of the energy deposition of fast ions to plasmas, warm, dense matter, and complex biological systems possible. Results are reported to all ions of helium, lithium, carbon, neon, aluminum... 17. Lie algebraic approach to valence bond theory of π-electron systems: a preliminary study of excited states Science.gov (United States) Paldus, J.; Li, X. 1992-10-01 Following a brief outline of various developments and exploitations of the unitary group approach (UGA), and its extension referred to as Clifford algebra UGA (CAUGA), in molecular electronic structure calculations, we present a summary of a recently introduced implementation of CAUGA for the valence bond (VB) method based on the Pariser-Parr-Pople (PPP)-type Hamiltonian. The existing applications of this PPP-VB approach have been limited to groundstates of various π-electron systems or, at any rate, to the lowest states of a given multiplicity. In this paper the method is applied to the low-lying excited states of several archetypal models, namely cyclobutadiene and benzene, representing antiaromatic and aromatic systems, hexatriene, representing linear polyenic systems and, finally, naphthalene, representing polyacenes. 18. Subsurface excitations in a metal DEFF Research Database (Denmark) Ray, M. P.; Lake, R. E.; Sosolik, C. E. 2009-01-01 We investigate internal hot carrier excitations in a Au thin film bombarded by hyperthermal and low energy alkali and noble gas ions. Excitations within the thin film of a metal-oxide-semiconductor device are measured revealing that ions whose velocities fall below the classical threshold given...... by the free-electron model of a metal still excite hot carriers. Excellent agreement between these results and a nonadiabatic model that accounts for the time-varying ion-surface interaction indicates that the measured excitations are due to semilocalized electrons near the metal surface.... 19. Excited-state density functional theory International Nuclear Information System (INIS) Harbola, Manoj K; Hemanadhan, M; Shamim, Md; Samal, P 2012-01-01 Starting with a brief introduction to excited-state density functional theory, we present our method of constructing modified local density approximated (MLDA) energy functionals for the excited states. We show that these functionals give accurate results for kinetic energy and exchange energy compared to the ground state LDA functionals. Further, with the inclusion of GGA correction, highly accurate total energies for excited states are obtained. We conclude with a brief discussion on the further direction of research that include the construction of correlation energy functional and exchange potential for excited states. 20. Theory of singlet-doublet excitations in praseodymium International Nuclear Information System (INIS) Bak, P. 1975-10-01 The magnetic excitation spectrum in a paramagnetic singlet-doublet system is calculated using a diagrammatic high density expansion technique. The lowest order diagrams, which correspond to the random phase approximation (RPA), give a detailed description of the wave vector and temperature dependence of the four exciton modes in praseodymium in terms of a Hamiltonian including isotropic Heisenberg exchange interactions and anisotropic, dipolar-like interactions. The leading contributions to the linewidths of the excitations are obtained by extending the 1/Z expansion of the generalized susceptibility propagators one order beyond the random phase approximation. This damping corresponds to spin wave scattering on single-site fluctuations. The theoretical spectral functions are in detailed agreement with experiment 1. Excitable particles in an optical torque wrench Science.gov (United States) Pedaci, Francesco; Huang, Zhuangxiong; van Oene, Maarten; Barland, Stephane; Dekker, Nynke H. 2011-03-01 The optical torque wrench is a laser trapping technique capable of applying and directly measuring torque on microscopic birefringent particles using spin momentum transfer, and has found application in the measurement of static torsional properties of biological molecules such as single DNAs. Motivated by the potential of the optical torque wrench to access the fast rotational dynamics of biological systems, a result of its all-optical manipulation and detection, we focus on the angular dynamics of the trapped birefringent particle, demonstrating its excitability in the vicinity of a critical point. This links the optical torque wrench to nonlinear dynamical systems such as neuronal and cardiovascular tissues, nonlinear optics and chemical reactions, all of which display an excitable binary (`all-or-none') response to input perturbations. On the basis of this dynamical feature, we devise and implement a conceptually new sensing technique capable of detecting single perturbation events with high signal-to-noise ratio and continuously adjustable sensitivity. 2. Disorder-induced localization of excitability in an array of coupled lasers Science.gov (United States) Lamperti, M.; Perego, A. M. 2017-10-01 We report on the localization of excitability induced by disorder in an array of coupled semiconductor lasers with a saturable absorber. Through numerical simulations we show that the exponential localization of excitable waves occurs if a certain critical amount of randomness is present in the coupling coefficients among the lasers. The results presented in this Rapid Communication demonstrate that disorder can induce localization in lattices of excitable nonlinear oscillators, and can be of interest in the study of photonics-based random networks, neuromorphic systems, and, by analogy, in biology, in particular, in the investigation of the collective dynamics of neuronal cell populations. 3. The method of varying amplitudes for solving (non)linear problems involving strong parametric excitation DEFF Research Database (Denmark) 2015-01-01 Parametrically excited systems appear in many fields of science and technology, intrinsically or imposed purposefully; e.g. spatially periodic structures represent an important class of such systems [4]. When the parametric excitation can be considered weak, classical asymptotic methods like...... the method of averaging [2] or multiple scales [6] can be applied. However, with many practically important applications this simplification is inadequate, e.g. with spatially periodic structures it restricts the possibility to affect their effective dynamic properties by a structural parameter modulation...... of considerable magnitude. Approximate methods based on Floquet theory [4] for analyzing problems involving parametric excitation, e.g. the classical Hill’s method of infinite determinants [3,4], can be employed also in cases of strong excitation; however, with Floquet theory being applicable only for linear... 4. Optimal Excitation Controller Design for Wind Turbine Generator Directory of Open Access Journals (Sweden) A. K. Boglou 2011-01-01 Full Text Available An optimal excitation controller design based on multirate-output controllers (MROCs having a multirate sampling mechanismwith different sampling period in each measured output of the system is presented. The proposed H∞ -control techniqueis applied to the discrete linear open-loop system model which represents a wind turbine generator supplying an infinite busthrough a transmission line. 5. Multi-frequency exciting and spectrogram-based ECT method CERN Document Server 2000-01-01 The purpose of this paper is to experimentally demonstrate advantages of a multi-frequency ECT system. In this system, a precise crack imaging was achieved by using spectrograms obtained from an eddy-current probe multi-frequency response. A complex signal containing selected sinusoidal components was used as an excitation. The results of measurements for various test specimens are presented. 6. Does intrinsic motivation enhance motor cortex excitability? Science.gov (United States) Radel, Rémi; Pjevac, Dusan; Davranche, Karen; d'Arripe-Longueville, Fabienne; Colson, Serge S; Lapole, Thomas; Gruet, Mathieu 2016-11-01 Intrinsic motivation (IM) is often viewed as a spontaneous tendency for action. Recent behavioral and neuroimaging evidence indicate that IM, in comparison to extrinsic motivation (EM), solicits the motor system. Accordingly, we tested whether IM leads to greater excitability of the motor cortex than EM. To test this hypothesis, we used two different tasks to induce the motivational orientation using either words representing each motivational orientation or pictures previously linked to each motivational orientation through associative learning. Single-pulse transcranial magnetic stimulation over the motor cortex was applied when viewing the stimuli. Electromyographic activity was recorded on the contracted first dorsal interosseous muscle. Two indexes of corticospinal excitability (the amplitude of motor-evoked potential and the length of cortical silent period) were obtained through unbiased automatic detection and analyzed using a mixed model that provided both statistical power and a high level of control over all important individual, task, and stimuli characteristics. Across the two tasks and the two indices of corticospinal excitability, the exposure to IM-related stimuli did not lead to a greater corticospinal excitability than EM-related stimuli or than stimuli with no motivational valence (ps > .20). While these results tend to dismiss the advantage of IM at activating the motor cortex, we suggest alternative hypotheses to explain this lack of effect, which deserves further research. © 2016 Society for Psychophysiological Research. 7. Breakup excitation function at backward angles from α-spectra in the 6Li + 144Sm system International Nuclear Information System (INIS) Capurro, O.A.; Pacheco, A.J.; Arazi, A.; Figueira, J.M.; Martinez Heimann, D.; Negri, A.E. 2011-01-01 Breakup cross sections were obtained for the 6 Li + 144 Sm system at energies above and below the Coulomb barrier from a detailed analysis of the data recorded at backward angles. These cross sections are compared with inelastic target excitations previously reported revealing a similar behavior as a function of the bombarding energy but a large absolute difference between them. Using kinematical considerations we have analyzed possible contributions from different breakup channels and we have extracted information on magnitudes such as the relative kinetic energies of the corresponding breakup fragments. 8. Excitation and decay of correlated atomic states International Nuclear Information System (INIS) Rau, A.R.P. 1992-01-01 Doubly excited states of atoms and ions in which two electrons are excited from the ground configuration display strong radial and angular electron correlations. They are prototypical examples of quantum-mechanical systems with strong coupling. Two distinguishing characteristics of these states are: (1) their organization into successive families, with only weak coupling between families, and (2) a hierarchical nature of this coupling, with states from one family decaying primarily to those in the next lower family. A view of the pair of electrons as a single entity, with the electron-electron repulsion between them divided into a adiabatic and nonadiabatic piece, accounts for many of the dominant features. The stronger, adiabatic part determines the family structure and the weaker, nonadiabatic part the excitation and decay between successive families. Similar considerations extend to three-electron atomic states, which group into five different classes. They are suggestive of composite models for quarks in elementary particle physics, which exhibit analogous groupings into families with a hierarchical arrangement of masses and electroweak decays. 49 refs., 6 figs., 2 tabs 9. Fermionic Collective Excitations in a Lattice Gas of Rydberg Atoms International Nuclear Information System (INIS) Olmos, B.; Gonzalez-Ferez, R.; Lesanovsky, I. 2009-01-01 We investigate the many-body quantum states of a laser-driven gas of Rydberg atoms confined to a large spacing ring lattice. If the laser driving is much stronger than the van der Waals interaction among the Rydberg atoms, these many-body states are collective fermionic excitations. The first excited state is a spin wave that extends over the entire lattice. We demonstrate that our system permits us to study fermions in the presence of disorder although no external atomic motion takes place. We analyze how this disorder influences the excitation properties of the fermionic states. Our work shows a route towards the creation of complex many-particle states with atoms in lattices. 10. Dynamical analysis of highly excited molecular spectra Energy Technology Data Exchange (ETDEWEB) Kellman, M.E. [Univ. of Oregon, Eugene (United States) 1993-12-01 The goal of this program is new methods for analysis of spectra and dynamics of highly excited vibrational states of molecules. In these systems, strong mode coupling and anharmonicity give rise to complicated classical dynamics, and make the simple normal modes analysis unsatisfactory. New methods of spectral analysis, pattern recognition, and assignment are sought using techniques of nonlinear dynamics including bifurcation theory, phase space classification, and quantization of phase space structures. The emphasis is chaotic systems and systems with many degrees of freedom. 11. Deexcitation of single excited nuclei in the QMD model International Nuclear Information System (INIS) Mueller, W.; Begemann-Blaich, M.; Aichelin, J. 1992-10-01 We investigate the emission pattern of a single excited nucleus in the QMD model and compare the results with several statistical and phenomenological models. We find that the number of intermediate mass fragments as a function of the excitation energy is in very good agreement with the results of statistical models in which the emission pattern is governed by phase space only. This allows two conclusions: (a) The microscopic dynamical description of the disintegration of static excited nuclei in the QMD yields directly the emission pattern expected from phase space decay. This is the case despite of the fact that nuclear level densities are not given directly but are modeled semiclassically by the nucleon-nucleon interaction. Thus there is no need to supplement the QMD calculations by an additional evaporation model. (b) Differences between the QMD results and the data are not due to insufficiencies in the description of the disintegration of excited systems. Thus other possible reasons, like a substantial change of the free cross section in the nuclear environment have to be investigated. (orig.) 12. Field-dependent molecular ionization and excitation energies: Implications for electrically insulating liquids Directory of Open Access Journals (Sweden) N. Davari 2014-03-01 Full Text Available The molecular ionization potential has a relatively strong electric-field dependence as compared to the excitation energies which has implications for electrical insulation since the excited states work as an energy sink emitting light in the UV/VIS region. At some threshold field, all the excited states of the molecule have vanished and the molecule is a two-state system with the ground state and the ionized state, which has been hypothesized as a possible origin of different streamer propagation modes. Constrained density-functional theory is used to calculate the field-dependent ionization potential of different types of molecules relevant for electrically insulating liquids. The low singlet-singlet excitation energies of each molecule have also been calculated using time-dependent density functional theory. It is shown that low-energy singlet-singlet excitation of the type n → π* (lone pair to unoccupied π* orbital has the ability to survive at higher fields. This type of excitation can for example be found in esters, diketones and many color dyes. For alkanes (as for example n-tridecane and cyclohexane on the other hand, all the excited states, in particular the σ → σ* excitations vanish in electric fields higher than 10 MV/cm. Further implications for the design of electrically insulating dielectric liquids based on the molecular ionization potential and excitation energies are discussed. 13. Hydrological excitation of polar motion by different variables from the GLDAS models Science.gov (United States) Winska, Malgorzata; Nastula, Jolanta; Salstein, David 2017-12-01 Continental hydrological loading by land water, snow and ice is a process that is important for the full understanding of the excitation of polar motion. In this study, we compute different estimations of hydrological excitation functions of polar motion (as hydrological angular momentum, HAM) using various variables from the Global Land Data Assimilation System (GLDAS) models of the land-based hydrosphere. The main aim of this study is to show the influence of variables from different hydrological processes including evapotranspiration, runoff, snowmelt and soil moisture, on polar motion excitations at annual and short-term timescales. Hydrological excitation functions of polar motion are determined using selected variables of these GLDAS realizations. Furthermore, we use time-variable gravity field solutions from the Gravity Recovery and Climate Experiment (GRACE) to determine the hydrological mass effects on polar motion excitation. We first conduct an intercomparison of the maps of variations of regional hydrological excitation functions, timing and phase diagrams of different regional and global HAMs. Next, we estimate the hydrological signal in geodetically observed polar motion excitation as a residual by subtracting the contributions of atmospheric angular momentum and oceanic angular momentum. Finally, the hydrological excitations are compared with those hydrological signals determined from residuals of the observed polar motion excitation series. The results will help us understand the relative importance of polar motion excitation within the individual hydrological processes, based on hydrological modeling. This method will allow us to estimate how well the polar motion excitation budget in the seasonal and inter-annual spectral ranges can be closed. 14. Rydberg energies using excited state density functional theory International Nuclear Information System (INIS) Cheng, C.-L.; Wu Qin; Van Voorhis, Troy 2008-01-01 We utilize excited state density functional theory (eDFT) to study Rydberg states in atoms. We show both analytically and numerically that semilocal functionals can give quite reasonable Rydberg energies from eDFT, even in cases where time dependent density functional theory (TDDFT) fails catastrophically. We trace these findings to the fact that in eDFT the Kohn-Sham potential for each state is computed using the appropriate excited state density. Unlike the ground state potential, which typically falls off exponentially, the sequence of excited state potentials has a component that falls off polynomially with distance, leading to a Rydberg-type series. We also address the rigorous basis of eDFT for these systems. Perdew and Levy have shown using the constrained search formalism that every stationary density corresponds, in principle, to an exact stationary state of the full many-body Hamiltonian. In the present context, this means that the excited state DFT solutions are rigorous as long as they deliver the minimum noninteracting kinetic energy for the given density. We use optimized effective potential techniques to show that, in some cases, the eDFT Rydberg solutions appear to deliver the minimum kinetic energy because the associated density is not pure state v-representable. We thus find that eDFT plays a complementary role to constrained DFT: The former works only if the excited state density is not the ground state of some potential while the latter applies only when the density is a ground state density. 15. Experimental system to measure excitation cross-sections by electron impact. Measurements for ArI and ArII International Nuclear Information System (INIS) Blanco, F.; Sanchez, J.A.; Aguilera, J.A.; Campos, J. 1989-01-01 An experimental set-up to measure excitation cross-section of atomic and molecular levels by electron impact based on the optical method is reported. We also present some measurements on the excitation cross-section for ArI 5p'(1/2)0 level, and for simultaneous ionization and excitation of Ar leading to ArII levels belonging to the 3p 4 4p and 3p 4 4d configurations. (Author) 16. Nonlinear Response of Vibrational Conveyers with Nonideal Vibration Exciter: Superharmonic and Subharmonic Resonance Directory of Open Access Journals (Sweden) H. Bayıroğlu 2012-01-01 Full Text Available Vibrational conveyers with a centrifugal vibration exciter transmit their load based on the jumping method. Common unbalanced-mass driver oscillates the trough. The motion is strictly related to the vibrational parameters. The transition over resonance of a vibratory system, excited by rotating unbalances, is important in terms of the maximum vibrational amplitude produced and the power demand on the drive for the crossover. The mechanical system is driven by the DC motor. In this study, the working ranges of oscillating shaking conveyers with nonideal vibration exciter have been analyzed analytically for superharmonic and subharmonic resonances by the method of multiple scales and numerically. The analytical results obtained in this study agree well with the numerical results. 17. Excited waves in shear layers Science.gov (United States) Bechert, D. W. 1982-01-01 The generation of instability waves in free shear layers is investigated. The model assumes an infinitesimally thin shear layer shed from a semi-infinite plate which is exposed to sound excitation. The acoustical shear layer excitation by a source further away from the plate edge in the downstream direction is very weak while upstream from the plate edge the excitation is relatively efficient. A special solution is given for the source at the plate edge. The theory is then extended to two streams on both sides of the shear layer having different velocities and densities. Furthermore, the excitation of a shear layer in a channel is calculated. A reference quantity is found for the magnitude of the excited instability waves. For a comparison with measurements, numerical computations of the velocity field outside the shear layer were carried out. 18. Non-linear quenching of current fluctuations in a self-exciting homopolar dynamo, proved by feedback system theory Science.gov (United States) de Paor, A. M. Hide (Nonlinear Processes in Geophysics, 1998) has produced a new mathematical model of a self-exciting homopolar dynamo driving a series- wound motor, as a continuing contribution to the theory of the geomagnetic field. By a process of exact perturbation analysis, followed by combination and partial solution of differential equations, the complete nonlinear quenching of current fluctuations reported by Hide in the case that a parameter ɛ has the value 1 is proved via the Popov theorem from feedback system stability theory. 19. Efficient excitation of nonlinear phonons via chirped pulses: Induced structural phase transitions Science.gov (United States) Itin, A. P.; Katsnelson, M. I. 2018-05-01 Nonlinear phononics play important role in strong laser-solid interactions. We discuss a dynamical protocol for efficient phonon excitation, considering recent inspiring proposals: inducing ferroelectricity in paraelectric perovskites, and inducing structural deformations in cuprates [Subedi et al., Phys. Rev. B 89, 220301(R) (2014), 10.1103/PhysRevB.89.220301; Phys. Rev. B 95, 134113 (2017), 10.1103/PhysRevB.95.134113]. High-frequency phonon modes are driven by midinfrared pulses, and coupled to lower-frequency modes those indirect excitations cause structural deformations. We study in more detail the case of KTaO3 without strain, where it was not possible to excite the needed low-frequency phonon mode by resonant driving of the higher frequency one. Behavior of the system is explained using a reduced model of coupled driven nonlinear oscillators. We find a dynamical mechanism which prevents effective excitation at resonance driving. To induce ferroelectricity, we employ driving with sweeping frequency, realizing so-called capture into resonance. The method can be applied to many other related systems. 20. Effects of dynamic aspects on fusion excitation functions International Nuclear Information System (INIS) Hassan, G.S. 2008-01-01 As an extension of the macroscopic theory, the nucleus- nucleus fusion has been described in terms of the chaotic regime dynamics (liquid drop potential energy plus one body dissipation).Three milestone configurations are attended : the touching , the conditional saddle point and the unconditional saddle one. We would like to deduce the associated extra push and extra-extra push energy values required to carry the system between these configurations, respectively. The next step is to light on the effect of these limiting values on the fusion excitation functions and their significance for accurate fitting of the measured functions for larger values of the angular momentum. It is found that there is a limiting values of excitation energy and angular momentum for each interacting pair, over which these aspects must be considered to fit the excitation functions of different nucleus nucleus fusion .These values were found to be in relation with the limiting angular momentum for fusion in major cases 1. EPR studies of excited state exchange and crystal-field effects in rare earth compounds International Nuclear Information System (INIS) Huang, C.Y.; Sugawara, K.; Cooper, B.R. 1976-01-01 EPR in excited crystal-field states of Tm 3+ , Pr 3+ , and Tb 3+ in singlet-ground-state systems and in the excited state of Ce 3+ in CeP are reviewed. Because one is looking at a crystal-field excited state resonance, the exchange, even if isotropic, does not act as a secular perturbation. This means that one obtains different effects and has access to more information about the dynamic effects of exchange than in conventional paramagnetic resonance experiments. The Tm and Pr monopnictides studied are paramagnetic at all temperatures. The most striking feature of the behavior of the GAMMA 5 /sup (2)/ EPR in the Tm compounds is the presence of an anomalous maximum in the temperature dependence of the g-factor. The relationship of this effect to anisotropic exchange is discussed. The results of the EPR of the excited GAMMA 5 /sup (2)/ level of Tb 3 + (g-factor becomes very large at T/sub N/ in antiferromagnetic TbX (X = P, As, Sb) and that of the excited GAMMA 8 level of Ce 3+ in antiferromagnetic CeP will also be reported. For sufficient dilution of the Tb 3+ in the terbium monopnictides, the systems become paramagnetic (Van Vleck paramagnets) down to 0 0 K. The Tb 3+ excited state resonance EPR in Tb/sub 0.1/ La/sub 0.9/P was studied as an example of behavior in such systems. 10 fig 2. Power Management of Islanded Self-Excited Induction Generator Reinforced by Energy Storage Systems Directory of Open Access Journals (Sweden) Nachat N. Nasser 2018-02-01 Full Text Available Self-Excited Induction Generators (SEIGs, e.g., Small-Scale Embedded wind generation, are increasingly used in electricity distribution networks. The operational stability of stand-alone SEIG is constrained by the local load conditions: stability can be achieved by maintaining the load’s active and reactive power at optimal values. Changes in power demand are dependent on customers’ requirements, and any deviation from the pre-calculated optimum setting will affect a machine’s operating voltage and frequency. This paper presents an investigation of the operation of the SEIG in islanding mode of operation under different load conditions, with the aid of batteries as an energy storage source. In this research a current-controlled voltage-source converter is proposed to regulate the power exchange between a direct current (DC energy storage source and an alternating current (AC grid, the converter’s controller is driven by any variation between machine capability and load demand. In order to prolong the system stability when the battery reaches its operation constraints, it is recommended that an ancillary generator and a dummy local load be embedded in the system. The results show the robustness and operability of the proposed system in the islanding mode of the SEIG under different load conditions. 3. Proving test on the performance of a Multiple-Excitation Simulator International Nuclear Information System (INIS) Fujita, Katsuhisa; Ito, Tomohiro; Kojima, Nobuyuki; Sasaki, Yoichi; Abe, Hiroshi; Kuroda, Katsuhiko 1995-01-01 Seismic excitation test on large scale piping systems is scheduled to be carried out by the Nuclear power Engineering Corporation (NUPEC) using the large-scale, high-performance vibration table at the Tadotsu Engineering Laboratory, under the sponsorship of the Ministry of International Trade and Industry (MITI). In the test, the piping systems simulate the main steam piping system and the main feed water piping system in the nuclear power plants. In this study, a fundamental test was carried out to prove the performance of the Multiple Excitation Simulator which consists of the hydraulic actuator and the control system. An L-shaped piping system and a hydraulic actuator were installed on the shaking table. Acceleration and displacement generated by the actuator were measured. The performance of the actuator and the control system was discussed comparing the measured values and the target values on the time histories and the response spectrum of the acceleration. As a result, it was proved that the actuator and the control system have good performance and will be applicable to the verification test 4. Interfacial Phenomena of Magnetic Fluid with Permanent Magnet in a Longitudinally Excited Container International Nuclear Information System (INIS) Sudo, Seiichi; Wakuda, Hirofumi; Yano, Tetsuya 2008-01-01 This paper describes the magnetic fluid sloshing in a longitudinally excited container. Liquid responses of magnetic fluid with a permanent magnet in a circular cylindrical container subject to vertical vibration are investigated. Experiments are performed on a vibration- testing system which provided longitudinal excitation. A cylindrical container made with the acrylic plastic is used in the experiment. A permanent magnet is in the state of floating in a magnetic fluid. The disk-shaped and ring-shaped magnets are examined. The different interfacial phenomena from the usual longitudinal liquid sloshing are observed. It is found that the wave motion frequency of magnetic fluid with a disk-shaped magnet in the container subject to vertical vibration is exactly same that of the excitation. In the case of ring-shaped magnet, the first symmetrical mode of one-half subharmonic response is dominant at lower excitation frequencies. The magnetic fluid disintegration of the free surface was also observed by a high-speed video camera system 5. Interfacial Phenomena of Magnetic Fluid with Permanent Magnet in a Longitudinally Excited Container Science.gov (United States) Sudo, Seiichi; Wakuda, Hirofumi; Yano, Tetsuya 2008-02-01 This paper describes the magnetic fluid sloshing in a longitudinally excited container. Liquid responses of magnetic fluid with a permanent magnet in a circular cylindrical container subject to vertical vibration are investigated. Experiments are performed on a vibration- testing system which provided longitudinal excitation. A cylindrical container made with the acrylic plastic is used in the experiment. A permanent magnet is in the state of floating in a magnetic fluid. The disk-shaped and ring-shaped magnets are examined. The different interfacial phenomena from the usual longitudinal liquid sloshing are observed. It is found that the wave motion frequency of magnetic fluid with a disk-shaped magnet in the container subject to vertical vibration is exactly same that of the excitation. In the case of ring-shaped magnet, the first symmetrical mode of one-half subharmonic response is dominant at lower excitation frequencies. The magnetic fluid disintegration of the free surface was also observed by a high-speed video camera system. 6. Feasibility for detection of autofluorescent signatures in rat organs using a novel excitation-scanning hyperspectral imaging system Science.gov (United States) Favreau, Peter F.; Deal, Joshua A.; Weber, David S.; Rich, Thomas C.; Leavesley, Silas J. 2016-04-01 The natural fluorescence (autofluorescence) of tissues has been noted as a biomarker for cancer for several decades. Autofluorescence contrast between tumors and healthy tissues has been of significant interest in endoscopy, leading to development of autofluorescence endoscopes capable of visualizing 2-3 fluorescence emission wavelengths to achieve maximal contrast. However, tumor detection with autofluorescence endoscopes is hindered by low fluorescence signal and limited quantitative information, resulting in prolonged endoscopic procedures, prohibitive acquisition times, and reduced specificity of detection. Our lab has designed a novel excitation-scanning hyperspectral imaging system with high fluorescence signal detection, low acquisition time, and enhanced spectral discrimination. In this study, we surveyed a comprehensive set of excised tissues to assess the feasibility of detecting tissue-specific pathologies using excitation-scanning. Fresh, untreated tissue specimens were imaged from 360 to 550 nm on an inverted fluorescence microscope equipped with a set of thin-film tunable filters (Semrock, A Unit of IDEX). Images were subdivided into training and test sets. Automated endmember extraction (ENVI 5.1, Exelis) with PCA identified endmembers within training images of autofluorescence. A spectral library was created from 9 endmembers. The library was used for identification of endmembers in test images. Our results suggest (1) spectral differentiation of multiple tissue types is possible using excitation scanning; (2) shared spectra between tissue types; and (3) the ability to identify unique morphological features in disparate tissues from shared autofluorescent components. Future work will focus on isolating specific molecular signatures present in tissue spectra, and elucidating the contribution of these signatures in pathologies. 7. Development of 70 MW class superconducting generator with quick-response excitation Science.gov (United States) Miyaike, Kiyoshi; Kitajima, Toshio; Ito, Tetsuo 2002-03-01 The development of a superconducting generator had been carried out for 12 years under the first stage of a Super GM project. The 70 MW class model machine with quick response excitation was manufactured and evaluated in the project. This type of superconducting generator improves power system stability against rapid load fluctuations at the power system faults. This model machine achieved all development targets including high stability during rapid excitation control. It was also connected to the actual 77 kV electrical power grid as a synchronous condenser and proved advantages and high-operation reliability of the superconducting generator. 8. Nonlinear dynamic analysis of cantilevered piezoelectric energy harvesters under simultaneous parametric and external excitations Science.gov (United States) Fang, Fei; Xia, Guanghui; Wang, Jianguo 2018-02-01 The nonlinear dynamics of cantilevered piezoelectric beams is investigated under simultaneous parametric and external excitations. The beam is composed of a substrate and two piezoelectric layers and assumed as an Euler-Bernoulli model with inextensible deformation. A nonlinear distributed parameter model of cantilevered piezoelectric energy harvesters is proposed using the generalized Hamilton's principle. The proposed model includes geometric and inertia nonlinearity, but neglects the material nonlinearity. Using the Galerkin decomposition method and harmonic balance method, analytical expressions of the frequency-response curves are presented when the first bending mode of the beam plays a dominant role. Using these expressions, we investigate the effects of the damping, load resistance, electromechanical coupling, and excitation amplitude on the frequency-response curves. We also study the difference between the nonlinear lumped-parameter and distributed-parameter model for predicting the performance of the energy harvesting system. Only in the case of parametric excitation, we demonstrate that the energy harvesting system has an initiation excitation threshold below which no energy can be harvested. We also illustrate that the damping and load resistance affect the initiation excitation threshold. 9. 46 CFR 111.12-3 - Excitation. Science.gov (United States) 2010-10-01 ... 46 CFR 110.10-1). In particular, no static exciter may be used for excitation of an emergency generator unless it is provided with a permanent magnet or a residual-magnetism-type exciter that has the... 10. Plasmon assisted control of photo-induced excitation energy transfer in a molecular chain Science.gov (United States) Wang, Luxia; May, Volkhard 2017-08-01 The strong and ultrafast laser pulse excitation of a molecular chain in close vicinity to a spherical metal nano-particle (MNP) is studied theoretically. Due to local-field enhancement around the MNP, pronounced excited-state formation has to be expected for the part of the chain which is in proximity to the MNP. Here, the description of this phenomenon will be based on a uniform quantum theory of the MNP-molecule system. It accounts for local-field effects due to direct consideration of the strong excitation energy transfer coupling between the MNP and the various molecules. The molecule-MNP distances are chosen in such a way as to achieve a correct description of the MNP via dipole-plasmon excitations. Short plasmon life-times are incorporated in the framework of a density matrix approach. By extending earlier work the present description allows for multi-exciton formation and multiple dipole-plasmon excitation. The region of less intense and not-too-short optical excitation is identified as being best suited for excitation energy localization in the chain. 11. Excited states rotational effects on the behavior of excited molecules CERN Document Server Lim, Edward C 2013-01-01 Excited States, Volume 7 is a collection of papers that discusses the excited states of molecules. The first paper reviews the rotational involvement in intra-molecular in vibrational redistribution. This paper analyzes the vibrational Hamiltonian as to its efficacy in detecting the manifestations of intra-molecular state-mixing in time-resolved and time-averaged spectroscopic measurements. The next paper examines the temporal behavior of intra-molecular vibration-rotation energy transfer (IVRET) and the effects of IVRET on collision, reaction, and the decomposition processes. This paper also 12. Non-linear quenching of current fluctuations in a self-exciting homopolar dynamo, proved by feedback system theory Directory of Open Access Journals (Sweden) A. M. de Paor 1998-01-01 Full Text Available Hide (Nonlinear Processes in Geophysics, 1998 has produced a new mathematical model of a self-exciting homopolar dynamo driving a series- wound motor, as a continuing contribution to the theory of the geomagnetic field. By a process of exact perturbation analysis, followed by combination and partial solution of differential equations, the complete nonlinear quenching of current fluctuations reported by Hide in the case that a parameter ε has the value 1 is proved via the Popov theorem from feedback system stability theory. 13. Spiral-wave dynamics in excitable medium with excitability modulated by rectangle wave International Nuclear Information System (INIS) Yuan Guo-Yong 2011-01-01 We numerically study the dynamics of spiral waves in the excitable system with the excitability modulated by a rectangle wave. The tip trajectories and their variations with the modulation period T are explained by the corresponding spectrum analysis. For a large T, the external modulation leads to the occurrence of more frequency peaks and these frequencies change with the modulation period according to their specific rules, respectively. Some of the frequencies and a primary frequency f 1 determine the corresponding curvature periods, which are locked into rational multiplies of the modulation period. These frequency-locking behaviours and the limited life-span of the frequencies in their variations with the modulation period constitute many resonant entrainment bands in the T axis. In the main bands, which follow the relation T/T 12 = m/n, the size variable R x of the tip trajectory is a monotonic increasing function of T. The rest of the frequencies are linear combinations of the two ones. Due to the complex dynamics, many unique tip trajectories appear at some certain T. We find also that spiral waves are eliminated when T is chosen from the end of the main resonant bands. This offers a useful method of controling the spiral wave. (general) 14. Low-energy scattering of excited helium atoms by rare gases International Nuclear Information System (INIS) Peach, G. 1978-01-01 The construction of semi-empirical model potentials for systems composed of helium in an excited state (Hestar) and a rare-gas atom (He or Ne) is described. The model of the atom-atom pair which has been adopted is one in which the excited electron is included explicitly, but the residual He + ion and the rare-gas atom are treated simply as cores which may be polarised. The results obtained are in satisfactory agreement with other calculations where they are available. (author) 15. Femtosecond laser excitation of dielectric materials DEFF Research Database (Denmark) Wædegaard, Kristian Juncher; Balling, Peter; Frislev, Martin Thomas 2012-01-01 We report an approach to modeling the interaction between ultrashort laser pulses and dielectric materials. The model includes the excitation of carriers by the laser through strongfield excitation, collisional excitation, and absorption in the plasma consisting of conduction-band electrons formed... 16. Equations for the kinetic modeling of supersonically flowing electrically excited lasers International Nuclear Information System (INIS) Lind, R.C. 1973-01-01 The equations for the kinetic modeling of a supersonically flowing electrically excited laser system are presented. The work focuses on the use of diatomic gases, in particular carbon monoxide mixtures. The equations presented include the vibrational rate equation which describes the vibrational population distribution, the electron, ion and electronic level rate equations, the gasdynamic equations for an ionized gas in the presence of an applied electric field, and the free electron Boltzmann equation including flow and gradient coupling terms. The model developed accounts for vibration--vibration collisions, vibration-translation collisions, electron-molecule inelastic excitation and superelastic de-excitation collisions, charge particle collisions, ionization and three body recombination collisions, elastic collisions, and radiative decay, all of which take place in such a system. A simplified form of the free electron Boltzmann equation is developed and discussed with emphasis placed on its coupling with the supersonic flow. A brief description of a possible solution procedure for the set of coupled equations is discussed 17. Thermodynamical description of excited nuclei International Nuclear Information System (INIS) Bonche, P. 1989-01-01 In heavy ion collisions it has been possible to obtain composite systems at rather high excitation energies corresponding to temperatures of several MeV. The theoretical studies of these systems are based on concepts borrowed from thermodynamics or statistical physics, such as the temperature. In these lectures, we present the concepts of statistical physics which are involved in the physics of heavy ion as they are produced nowadays in the laboratory and also during the final stage of a supernova collapse. We do not attempt to describe the reaction mechanisms which yield such nuclear systems nor their decay by evaporation or fragmentation. We shall only study their static properties. The content of these lectures is organized in four main sections. The first one gives the basic features of statistical physics and thermodynamics necessary to understand quantum mechanics at finite temperature. In the second one, we present a study of the liquid-gas phase transition in nuclear physics. A phenomenological approach of the stability of hot nuclei follows. The microscopic point of view is proposed in the third part. Starting from the basic concepts derived in the first part, it provides a description of excited or hot nuclei which confirms the qualitative results of the second part. Furthermore it gives a full description of most properties of these nuclei as a function of temperature. Finally in the last part, a microscopic derivation of the equation of state of nuclear matter is proposed to study the collapse of a supernova core 18. Trapping time statistics and efficiency of transport of optical excitations in dendrimers Science.gov (United States) Heijs, Dirk-Jan; Malyshev, Victor A.; Knoester, Jasper 2004-09-01 We theoretically study the trapping time distribution and the efficiency of the excitation energy transport in dendritic systems. Trapping of excitations, created at the periphery of the dendrimer, on a trap located at its core, is used as a probe of the efficiency of the energy transport across the dendrimer. The transport process is treated as incoherent hopping of excitations between nearest-neighbor dendrimer units and is described using a rate equation. We account for radiative and nonradiative decay of the excitations while diffusing across the dendrimer. We derive exact expressions for the Laplace transform of the trapping time distribution and the efficiency of trapping, and analyze those for various realizations of the energy bias, number of dendrimer generations, and relative rates for decay and hopping. We show that the essential parameter that governs the trapping efficiency is the product of the on-site excitation decay rate and the trapping time (mean first passage time) in the absence of decay. 19. Magnetic excitations in (VO)HPO4· 1/2 H2O International Nuclear Information System (INIS) Garrett, A.W.; Nagler, S.E.; Tennant, D.A. 1997-01-01 The magnetic excitations of an antiferromagnetic spin dimer system, (VO)HPO 4 · 1/2 H 2 O, are examined using inelastic neutron scattering. A dispersionless mode is found, consistent with expectations for a dimer excitation. The intensity variation of the mode reveals a V 4+ - V 4+ dimer separation of 4.43 angstrom, almost 50% larger than the originally expected length 20. Backreaction of excitations on a vortex OpenAIRE 1996-01-01 Excitations of a vortex are usually considered in a linear approximation neglecting their backreaction on the vortex. In the present paper we investigate backreaction of Proca type excitations on a straightlinear vortex in the Abelian Higgs model. We propose exact Ansatz for fields of the excited vortex. From initial set of six nonlinear field equations we obtain (in a limit of weak excitations) two linear wave equations for the backreaction corrections. Their approximate solutions are found ... 1. Nonlinear Dynamic Behavior of a Flexible Structure to Combined External Acoustic and Parametric Excitation Directory of Open Access Journals (Sweden) Paulo S. Varoto 2006-01-01 Full Text Available Flexible structures are frequently subjected to multiple inputs when in the field environment. The accurate determination of the system dynamic response to multiple inputs depends on how much information is available from the excitation sources that act on the system under study. Detailed information include, but are not restricted to appropriate characterization of the excitation sources in terms of their variation in time and in space for the case of distributed loads. Another important aspect related to the excitation sources is how inputs of different nature contribute to the measured dynamic response. A particular and important driving mechanism that can occur in practical situations is the parametric resonance. Another important input that occurs frequently in practice is related to acoustic pressure distributions that is a distributed type of loading. In this paper, detailed theoretical and experimental investigations on the dynamic response of a flexible cantilever beam carrying a tip mass to simultaneously applied external acoustic and parametric excitation signals have been performed. A mathematical model for transverse nonlinear vibration is obtained by employing Lagrange’s equations where important nonlinear effects such as the beam’s curvature and quadratic viscous damping are accounted for in the equation of motion. The beam is driven by two excitation sources, a sinusoidal motion applied to the beam’s fixed end and parallel to its longitudinal axis and a distributed sinusoidal acoustic load applied orthogonally to the beam’s longitudinal axis. The major goal here is to investigate theoretically as well as experimentally the dynamic behavior of the beam-lumped mass system under the action of these two excitation sources. Results from an extensive experimental work show how these two excitation sources interacts for various testing conditions. These experimental results are validated through numerically simulated results 2. A large electrically excited synchronous generator DEFF Research Database (Denmark) 2014-01-01 This invention relates to a large electrically excited synchronous generator (100), comprising a stator (101), and a rotor or rotor coreback (102) comprising an excitation coil (103) generating a magnetic field during use, wherein the rotor or rotor coreback (102) further comprises a plurality...... adjacent neighbouring poles. In this way, a large electrically excited synchronous generator (EESG) is provided that readily enables a relatively large number of poles, compared to a traditional EESG, since the excitation coil in this design provides MMF for all the poles, whereas in a traditional EESG...... each pole needs its own excitation coil, which limits the number of poles as each coil will take up too much space between the poles.... 3. Composite model describing the excitation and de-excitation of nitrogen by an electron beam International Nuclear Information System (INIS) Kassem, A.E.; Hickman, R.S. 1975-01-01 Based on recent studies, the effect of re-excited ions in the emission of electron beam induced fluorescence in nitrogen has been estimated. These effects are included in the formulation of a composite model describing the excitation and de-excitation of nitrogen by an electron beam. The shortcomings of previous models, namely the dependence of the measured temperature on true gas temperature as well as the gas density, are almost completely eliminated in the range of temperatures and densities covered by the available data. (auth) 4. Energy dependence of the ionization of highly excited atoms by collisions with excited atoms International Nuclear Information System (INIS) Shirai, T.; Nakai, Y.; Nakamura, H. 1979-01-01 Approximate analytical expressions are derived for the ionization cross sections in the high- and low-collision-energy limits using the improved impulse approximation based on the assumption that the electron-atom inelastic-scattering amplitude is a function only of the momentum transfer. Both cases of simultaneous excitation and de-excitation of one of the atoms are discussed. The formulas are applied to the collisions between two excited hydrogen atoms and are found very useful for estimating the cross sections in the wide range of collisions energies 5. Isotope separation using vibrationally excited molecules International Nuclear Information System (INIS) Woodroffe, J.A.; Keck, J.C. 1979-01-01 Vibrational excitation of molecules having components of a selected isotope type is used to produce a conversion from vibrational to translational excitation of the molecules by collision with the molecules of a heavy carrier gas. The resulting difference in translaton between the molecules of the selected isotope type and all other molecules of the same compound permits their separate collection. When applied to uranium enrichment, a subsonic cryogenic flow of molecules of uranium hexafluoride in combination with an argon carrier gas is directed through a cooled chamber that is illuminated by laser radiaton tuned to vibrationally excite the uranium hexafluoride molecules of a specific uranium isotope. The excited molecules collide with carrier gas molecules, causing a conversion of the excitation energy into a translation of the excited molecule, which results in a higher thermal energy or diffusivity than that of the other uranium hexafluoride molecules. The flowing molecules including the excited molecules directly enter a set of cryogenically cooled channels. The higher thermal velocity of the excited molecules increases the probability of their striking a collector surface. The molecules which strike this surface immediately condense. After a predetermined thickness of molecules is collected on the surface, the flow of uranium hexafluoride is interrupted and the chamber heated to the point of vaporization of the collected hexafluoride, permitting its removal. (LL) 6. Stochastic resonance and vibrational resonance in an excitable system: The golden mean barrier International Nuclear Information System (INIS) Stan, Cristina; Cristescu, C.P.; Alexandroaei, D.; Agop, M. 2009-01-01 We report on stochastic resonance and vibrational resonance in an electric charge double layer configuration as usually found in electrical discharges, biological cell membranes, chemical systems and nanostructures. The experiment and numerical computation show the existence of a barrier expressible in terms of the golden mean above which the two phenomena do not take place. We consider this as new evidence for the importance of the golden mean criticality in the oscillatory dynamics, in agreement with El Naschie's E-infinity theory. In our experiment, the dynamics of a charge double layer generated in the inter-anode space of a twin electrical discharge is investigated under noise-harmonic and harmonic-harmonic perturbations. In the first case, a Gaussian noise can enhance the response of the system to a weak injected periodic signal, a clear mark of stochastic resonance. In the second case, similar enhancement can appear if the noise is replaced by a harmonic perturbation with a frequency much higher than the frequency of the weak oscillation. The amplitude of the low frequency oscillation shows a maximum versus the amplitude of the high frequency perturbation demonstrating vibrational resonance. In order to model these dynamics, we derived an excitable system by modifying a biased van der Pol oscillator. The computational study considers the behaviour of this system under the same types of perturbation as in the experimental investigations and is found to give consistent results in both situations. 7. Excitation of resonances of microspheres on an optical fiber Science.gov (United States) Serpengüzel, A.; Arnold, S.; Griffel, G. 1995-04-01 Morphology-dependent resonances (MDR's) of solid microspheres are excited by using an optical fiber coupler. The narrowest measured MDR linewidths are limited by the excitation laser linewidth ( < 0.025 nm). Only MDR's, with an on-resonance to off-resonance intensity ratio of 104, contribute to scattering. The intensity of various resonance orders is understood by the localization principle and the recently developed generalized Lorentz-Mie theory. The microsphere fiber system has potential for becoming a building block in dispersive microphotonics. The basic physics underlying our approach may be considered a harbinger for the coupling of active photonic microstructures such as microdisk lasers. 8. Holonomic Quantum Control by Coherent Optical Excitation in Diamond Energy Technology Data Exchange (ETDEWEB) Zhou, Brian B.; Jerger, Paul C.; Shkolnikov, V. O.; Heremans, F. Joseph; Burkard, Guido; Awschalom, David D. 2017-10-01 Although geometric phases in quantum evolution are historically overlooked, their active control now stimulates strategies for constructing robust quantum technologies. Here, we demonstrate arbitrary singlequbit holonomic gates from a single cycle of nonadiabatic evolution, eliminating the need to concatenate two separate cycles. Our method varies the amplitude, phase, and detuning of a two-tone optical field to control the non-Abelian geometric phase acquired by a nitrogen-vacancy center in diamond over a coherent excitation cycle. We demonstrate the enhanced robustness of detuned gates to excited-state decoherence and provide insights for optimizing fast holonomic control in dissipative quantum systems. 9. Holonomic Quantum Control by Coherent Optical Excitation in Diamond. Science.gov (United States) Zhou, Brian B; Jerger, Paul C; Shkolnikov, V O; Heremans, F Joseph; Burkard, Guido; Awschalom, David D 2017-10-06 Although geometric phases in quantum evolution are historically overlooked, their active control now stimulates strategies for constructing robust quantum technologies. Here, we demonstrate arbitrary single-qubit holonomic gates from a single cycle of nonadiabatic evolution, eliminating the need to concatenate two separate cycles. Our method varies the amplitude, phase, and detuning of a two-tone optical field to control the non-Abelian geometric phase acquired by a nitrogen-vacancy center in diamond over a coherent excitation cycle. We demonstrate the enhanced robustness of detuned gates to excited-state decoherence and provide insights for optimizing fast holonomic control in dissipative quantum systems. 10. Interqubit coupling mediated by a high-excitation-energy quantum object NARCIS (Netherlands) Ashhab, S.; Niskanen, A.O.; Harrabi, K.; Nakamura, Y.; Picot, T.; De Groot, P.C.; Harmans, C.J.P.M.; Mooij, J.E.; Nori, F. 2008-01-01 We consider a system composed of two qubits and a high excitation energy quantum object used to mediate coupling between the qubits. We treat the entire system quantum mechanically and analyze the properties of the eigenvalues and eigenstates of the total Hamiltonian. After reproducing well known 11. Localized excitations and the geometry of the 1nπ* excited states of pyrazine International Nuclear Information System (INIS) Kleier, D.A.; Martin, R.L.; Wadt, W.R.; Moomaw, W.R. 1982-01-01 Previous theoretical work has shown that the lowest excited singlet state of pyrazine, the π* 1 B 3 u state, is best described in terms of interacting excitations localized on each nitrogen. The present work refines the localized excitation model and considers its implications for the geometry of the 1 B 3 u state. Hartree-Fock calculations show that the best single configuration description of the nπ* state has broken ( 1 B 1 ) symmetry with the excitation strongly localized at one end of the molcule. If the symmetry-restricted hf result is used for reference, this localization describes an important correlation effect. The excited-state geometry was probed using configuration interaction wave functions based on the symmetry-restricted orbitals, as well as properly symmetrized ''valance-bond'' wave functions based on the broken symmetry solutions. Both descriptions lead to a very flat potential for a b/sub 1u/ vibrational mode. This mode reduces the molecular geometry from D/sub 2h/ to C/sub 2v/. We present spectroscopic evidence of our own and of other workers which is consistent with such a flat potential 12. General theory of excitation energy transfer in donor-mediator-acceptor systems. Science.gov (United States) Kimura, Akihiro 2009-04-21 General theory of the excitation energy transfer (EET) in the case of donor-mediator-acceptor system was constructed by using generalized master equation (GME). In this theory, we consider the direct and indirect transitions in the EET consistently. Hence, our theory includes the quantum mechanical interference between the direct and indirect transitions automatically. Memory functions in the GME were expressed by the overlap integrals among the time-dependent emission spectrum of the donor, the absorption spectrum of the mediator, the time-dependent emission spectrum of the mediator, and the absorption spectrum of the acceptor. In the Markov limit of the memory functions, we obtained the rate of EET which consists of three terms due to the direct transition, the indirect transition, and the interference between them. We found that the interference works effectively in the limit of slow thermalization at the intermediate state. The formula of EET rate in this limit was expressed by the convolution of the EET interaction and optical spectra. The interference effect strongly depends on the width of the absorption spectrum of mediator molecule and the energy gap between the donor and the mediator molecules. 13. Inelastic scattering of 9Li and excitation mechanism of its first excited state International Nuclear Information System (INIS) Al Falou, H.; Kanungo, R.; Andreoiu, C.; Cross, D.S.; Davids, B.; Djongolov, M.; Gallant, A.T.; Galinski, N.; Howell, D.; Kshetri, R.; Niamir, D.; Orce, J.N.; Shotter, A.C.; Sjue, S.; Tanihata, I.; Thompson, I.J.; Triambak, S.; Uchida, M.; Walden, P.; Wiringa, R.B. 2013-01-01 The first measurement of inelastic scattering of 9 Li from deuterons at the ISAC facility is reported. The measured angular distribution for the first excited state confirms the nature of excitation to be an E2 transition. The quadrupole deformation parameter is extracted from an analysis of the angular distribution 14. VUV Study of Electron-Pyrimidine Dissociative Excitation Science.gov (United States) Hein, Jeff; Al-Khazraji, Hajar; Tiessen, Collin; Lukic, Dragan; Trocchi, Joshuah; McConkey, William 2013-05-01 A crossed electron-gas beam system coupled to a VUV spectrometer has been used to investigate the dissociation of pyrimidine (C4H4N2) into excited atomic fragments in the electron-impact energy range from threshold to 375 eV. Data have been made absolute using Lyman- α from H2 as a secondary standard. The main features in the spectrum are the H Lyman series lines. The emission cross section of Lyman- α is measured to be (2.44 +/- 0.25) 10-18 cm2 at 100 eV impact energy. The probability of extracting C or N atoms from the ring is shown to be very small. Possible dissociation channels and excitation mechanisms in the parent molecule will be discussed. The authors thank NSERC (Canada) for financial support. 15. Multiple excitation regenerative amplifier inertial confinement system International Nuclear Information System (INIS) George, V.E.; Haas, R.A.; Krupke, W.F.; Schlitt, L.G. 1980-01-01 The invention relates to apparatus and methods for producing high intensity laser radiation generation which is achieved through an optical amplifier-storage ring design. One or two synchronized, counterpropagating laser pulses are injected into a regenerative amplifier cavity and amplified by gain media which are pumped repetitively by electrical or optical means. The gain media excitation pulses are tailored to efficiently amplify the laser pulses during each transit. After the laser pulses have been amplified to the desired intensity level, they are either switched out of the cavity by some switch means, as for example an electro-optical device, for any well known laser end uses, or a target means may be injected into the regenerative amplifier cavity in such a way as to intercept simultaneously the counterpropagating laser pulses. One such well known end uses to which this invention is intended is for production of high density and temperature plasmas suitable for generating neutrons, ions and x-rays and for studying matter heated by high intensity laser radiation 16. Application brushless machines with combine excitation for a hybrid car and an electric car OpenAIRE GANDZHA S.A.; KIESSH I.E. 2015-01-01 This article shows advantages of application the brushless machines with combined excitation (excitation from permanent magnets and excitation winding) for the hybrid car and the electric car. This type of electric machine is compared with a typical brushless motor and an induction motor. The main advantage is the decrease of the dimensions of electric machine and the reduction of the price for an electronic control system. It is shown the design and the principle of operation of the electric... 17. Statistics of excitations in the electron glass model Science.gov (United States) Palassini, Matteo 2011-03-01 We study the statistics of elementary excitations in the classical electron glass model of localized electrons interacting via the unscreened Coulomb interaction in the presence of disorder. We reconsider the long-standing puzzle of the exponential suppression of the single-particle density of states near the Fermi level, by measuring accurately the density of states of charged and electron-hole pair excitations via finite temperature Monte Carlo simulation and zero-temperature relaxation. We also investigate the statistics of large charge rearrangements after a perturbation of the system, which may shed some light on the slow relaxation and glassy phenomena recently observed in a variety of Anderson insulators. In collaboration with Martin Goethe. 18. Excitation of Ion Cyclotron Waves by Ion and Electron Beams in Compensated-current System Science.gov (United States) Xiang, L.; Wu, D. J.; Chen, L. 2018-04-01 Ion cyclotron waves (ICWs) can play important roles in the energization of plasma particles. Charged particle beams are ubiquitous in space, and astrophysical plasmas and can effectively lead to the generation of ICWs. Based on linear kinetic theory, we consider the excitation of ICWs by ion and electron beams in a compensated-current system. We also investigate the competition between reactive and kinetic instabilities. The results show that ion and electron beams both are capable of generating ICWs. For ICWs driven by ion beams, there is a critical beam velocity, v bi c , and critical wavenumber, k z c , for a fixed beam density; the reactive instability dominates the growth of ICWs when the ion-beam velocity {v}{bi}> {v}{bi}c and the wavenumber {k}zz≃ 2{k}zc/3 for a given {v}{bi}> {v}{bi}c. For the slow ion beams with {v}{bi}< {v}{bi}c, the kinetic instability can provide important growth rates of ICWs. On the other hand, ICWs driven by electron beams are excited only by the reactive instability, but require a critical velocity, {v}{be}c\\gg {v}{{A}} (the Alfvén velocity). In addition, the comparison between the approximate analytical results based on the kinetic theory and the exact numerical calculation based on the fluid model demonstrates that the reactive instabilities can well agree quantitatively with the numerical results by the fluid model. Finally, some possible applications of the present results to ICWs observed in the solar wind are briefly discussed. 19. Exciplex ensemble modulated by excitation mode in intramolecular charge-transfer dyad: effects of temperature, solvent polarity, and wavelength on photochemistry and photophysics of tethered naphthalene-dicyanoethene system. Science.gov (United States) Aoki, Yoshiaki; Matsuki, Nobuo; Mori, Tadashi; Ikeda, Hiroshi; Inoue, Yoshihisa 2014-09-19 Solvent, temperature, and excitation wavelength significantly affected the photochemical outcomes of a naphthalene-dicyanoethene system tethered by different number (n) of methylene groups (1-3). The effect of irradiation wavelength was almost negligible for 2a but pronounced for 3a. The temperature dependence and theoretical calculations indicated the diversity of exciplex conformations, an ensemble of which can be effectively altered by changing excitation wavelength to eventually switch the regioselectivity of photoreactions. 20. Calibrated Noncontact Exciters for Optical Modal Analysis Directory of Open Access Journals (Sweden) Henrik O. Saldner 1996-01-01 Full Text Available Two types of exciters were investigated experimentally One of the exciters uses a small permanent magnet fastened on the object. The force is introduced by the change in the electromagnetic field from a coil via an air gap. The second exciter is an eddy-current electromagnet one. The amplitude of the forces from these exciters are calibrated by using dynamic reciprocity in conjunction with electronic holography. These forces strongly depend upon the distance between the exciter and the object. 1. Chemical modulation of electronic structure at the excited state Science.gov (United States) Li, F.; Song, C.; Gu, Y. D.; Saleem, M. S.; Pan, F. 2017-12-01 Spin-polarized electronic structures are the cornerstone of spintronics, and have thus attracted a significant amount of interest; in particular, researchers are looking into how to modulate the electronic structure to enable multifunctional spintronics applications, especially in half-metallic systems. However, the control of the spin polarization has only been predicted in limited two-dimensional systems with spin-polarized Dirac structures and is difficult to achieve experimentally. Here, we report the modulation of the electronic structure in the light-induced excited state in a typical half-metal, L a1 /2S r1 /2Mn O3 -δ . According to the spin-transport measurements, there appears a light-induced increase in magnetoresistance due to the enhanced spin scattering, which is closely associated with the excited spin polarization. Strikingly, the light-induced variation can be enhanced via alcohol processing and reduced by oxygen annealing. X-ray photoelectron spectroscopy measurements show that in the chemical process, a redox reaction occurs with a change in the valence of Mn. Furthermore, first-principles calculations reveal that the change in the valence of Mn alters the electronic structure and consequently modulates the spin polarization in the excited state. Our findings thus report a chemically tunable electronic structure, demonstrating interesting physics and the potential for multifunctional applications and ultrafast spintronics. 2. Use of polarization measurements in evaluating cascade contributions to optical excitation functions International Nuclear Information System (INIS) McConkey, J.W. 1981-01-01 Recent developments in theory and experimental measurements of rotational line polarization fractions of diatomic molecules following electron impact are used to show how in some instances cascade free optical excitation functions can be derived without additional measurements of the cascading contribution. The Lyman system of H 2 is presented as an example and some previously conflicting excitation cross-section measurements obtained by different techniques are reconciled 3. Linear and nonlinear excitations in two stacks of parallel arrays of long Josephson junctions DEFF Research Database (Denmark) Carapella, G.; Constabile, Giovanni; Latempa, R. 2000-01-01 We investigate a structure consisting of two parallel arrays of long Josephson junctions sharing a common electrode that allows inductive coupling between the arrays. A model for this structure is derived starting from the description of its continuous limit. The excitation of linear cavity modes...... known from continuous and discrete systems as well as the excitation of a new state exhibiting synchronization in two dimensions are inferred from the mathematical model of the system. The stable nonlinear solution of the coupled sine-Gordon equations describing the system is found to consist... 4. A current-excited triple-time-voltage oversampling method for bio-impedance model for cost-efficient circuit system. Science.gov (United States) Yan Hong; Yong Wang; Wang Ling Goh; Yuan Gao; Lei Yao 2015-08-01 This paper presents a mathematic method and a cost-efficient circuit to measure the value of each component of the bio-impedance model at electrode-electrolyte interface. The proposed current excited triple-time-voltage oversampling (TTVO) method deduces the component values by solving triple simultaneous electric equation (TSEE) at different time nodes during a current excitation, which are the voltage functions of time. The proposed triple simultaneous electric equations (TSEEs) allows random selections of the time nodes, hence numerous solutions can be obtained during a single current excitation. Following that, the oversampling approach is engaged by averaging all solutions of multiple TSEEs acquired after a single current excitation, which increases the practical measurement accuracy through the improvement of the signal-to-noise ratio (SNR). In addition, a print circuit board (PCB) that consists a switched current exciter and an analog-to-digital converter (ADC) is designed for signal acquisition. This presents a great cost reduction when compared against other instrument-based measurement data reported [1]. Through testing, the measured values of this work is proven to be in superb agreements on the true component values of the electrode-electrolyte interface model. This work is most suited and also useful for biological and biomedical applications, to perform tasks such as stimulations, recordings, impedance characterizations, etc. 5. System and method for controlling depth of imaging in tissues using fluorescence microscopy under ultraviolet excitation following staining with fluorescing agents Science.gov (United States) Levenson, Richard; Demos, Stavros 2018-05-08 A method is disclosed for analyzing a thin tissue sample and adapted to be supported on a slide. The tissue sample may be placed on a slide and exposed to one or more different exogenous fluorophores excitable in a range of about 300 nm-200 nm, and having a useful emission band from about 350 nm-900 nm, and including one or more fluorescent dyes or fluorescently labeled molecular probes that accumulate in tissue or cellular components. The fluorophores may be excited with a first wavelength of UV light between about 200 nm-290 nm. An optical system collects emissions from the fluorophores at a second wavelength, different from the first wavelength, which are generated in response to the first wavelength of UV light, to produce an image for analysis. 6. Back reaction of excitations on a vortex Science.gov (United States) 1997-01-01 Excitations of a vortex are usually considered in a linear approximation neglecting their back reaction on the vortex. In the present paper we investigate back reaction of Proca-type excitations on a straight linear vortex in the Abelian Higgs model. We propose an exact ansatz for fields of the excited vortex. From an initial set of six nonlinear field equations we obtain (in a limit of weak excitations) two linear wave equations for the back reaction corrections. Their approximate solutions are found in the cases of plane wave and wave-packet-type excitations. We find that the excited vortex radiates the vector field and that the Higgs field has a very broad oscillating component. 7. Theory of superconductivity and spin excitations in cuprates Science.gov (United States) Plakida, Nikolay M. 2018-06-01 A microscopic theory of high-temperature superconductivity in strongly correlated systems as cuprates is presented. The two-subband extended Hubbard model is considered where the intersite Coulomb repulsion and electron-phonon interaction are taken into account. The low-energy spin excitations are considered within the t-J model. 8. Continuum emission of excited sodium dimer International Nuclear Information System (INIS) Pardo, A.; Poyato, J.M.L.; Alonso, J.I.; Rico, F.R. 1980-01-01 A study has been made of the behaviour of excited molecular sodium using high-power Ar + laser radiation. A continuum emission was observed in the red wavelength region. This emission was thought to be caused by the formation of excited triatomic molecules. Energy transfer was observed from excited molecules to atoms. (orig.) 9. Voiced Excitations National Research Council Canada - National Science Library Holzricher, John 2004-01-01 To more easily obtain a voiced excitation function for speech characterization, measurements of skin motion, tracheal tube, and vocal fold, motions were made and compared to EM sensor-glottal derived... 10. Numerical prediction analysis of propeller exciting force for hull–propeller–rudder system in oblique flow Directory of Open Access Journals (Sweden) Shuai Sun 2018-01-01 Full Text Available In order to analyze the characteristics of propeller exciting force, the hybrid grid is adopted and the numerical prediction of KCS ship model is performed for hull–propeller–rudder system by Reynolds-Averaged Navier Stokes (RANS method and volume of fluid (VOF model. Firstly, the numerical simulation of hydrodynamics for bare hull at oblique state is carried out. The results show that with the increasing of the drift angle, the coefficients of resistance, side force and yaw moment are constantly increasing, and the bigger the drift angle, the worse the overall uniformity of propeller disk. Then, propeller bearing force for hull–propeller–rudder system in oblique flow is calculated. It is found that the propeller thrust and torque fluctuation coefficient peak in drift angle are greater than that in straight line navigation, and the negative drift angle is greater than the positive. The fluctuation peak variation law of coefficient of side force and bending moment are different due to various causes. 11. Complete fusion excitation function for the 16O + natS reaction International Nuclear Information System (INIS) Wang Sufang; Zheng Jiwen; Liu Guoxing 1994-01-01 The complete fusion excitation function for the 16 O + nat S reaction has been measured in the range of 50-75 MeV with a step of 1.0 MeV by using a position sensitive ΔE-E telescope system. The model parameters have been extracted from data analysis. The striking gross structure of the excitation function has been observed. The energies of peaks are at E CM 38,43 and 48 MeV respectively 12. The study of excited oxygen molecule gas species production and quenching on thermal protection system materials Science.gov (United States) Nordine, Paul C.; Fujimoto, Gordon T.; Greene, Frank T. 1987-01-01 The detection of excited oxygen and ozone molecules formed by surface catalyzed oxygen atom recombination and reaction was investigated by laser induced fluorescence (LIF), molecular beam mass spectrometric (MBMS), and field ionization (FI) techniques. The experiment used partially dissociated oxygen flows from a microwave discharge at pressures in the range from 60 to 400 Pa or from an inductively coupled RF discharge at atmospheric pressure. The catalyst materials investigated were nickel and the reaction cured glass coating used for Space Shuttle reusable surface insulation tiles. Nonradiative loss processes for the laser excited states makes LIF detection of O2 difficult such that formation of excited oxygen molecules could not be detected in the flow from the microwave discharge or in the gaseous products of atom loss on nickel. MBMS experiments showed that ozone was a product of heterogeneous O atom loss on nickel and tile surfaces at low temperatures and that ozone is lost on these materials at elevated temperatures. FI was separately investigated as a method by which excited oxygen molecules may be conveniently detected. Partial O2 dissociation decreases the current produced by FI of the gas. 13. Nonlinear excitation fluorescence microscopy: source considerations for biological applications Science.gov (United States) Wokosin, David L. 2008-02-01 Ultra-short-pulse solid-state laser sources have improved contrast within fluorescence imaging and also opened new windows of investigation in biological imaging applications. Additionally, the pulsed illumination enables harmonic scattering microscopy which yields intrinsic structure, symmetry and contrast from viable embryos, cells and tissues. Numerous human diseases are being investigated by the combination of (more) intact dynamic tissue imaging of cellular function with gene-targeted specificity and electrophysiology context. The major limitation to more widespread use of multi-photon microscopy has been the complete system cost and added complexity above and beyond commercial camera and confocal systems. The current status of all-solid-state ultrafast lasers as excitation sources will be reviewed since these lasers offer tremendous potential for affordable, reliable, "turnkey" multiphoton imaging systems. This effort highlights the single box laser systems currently commercially available, with defined suggestions for the ranges for individual laser parameters as derived from a biological and fluorophore limited perspective. The standard two-photon dose is defined by 800nm, 10mW, 200fs, and 80Mhz - at the sample plane for tissue culture cells, i.e. after the full scanning microscope system. Selected application-derived excitation wavelengths are well represented by 700nm, 780nm, ~830nm, ~960nm, 1050nm, and 1250nm. Many of the one-box lasers have fixed or very limited excitation wavelengths available, so the lasers will be lumped near 780nm, 800nm, 900nm, 1050nm, and 1250nm. The following laser parameter ranges are discussed: average power from 200mW to 2W, pulse duration from 70fs to 700fs, pulse repetition rate from 20MHz to 200MHz, with the laser output linearly polarized with an extinction ratio at least 100:1. 14. Two-photon excitation of argon International Nuclear Information System (INIS) Pindzola, P.S.; Payne, M.C. 1982-01-01 The authors calculate two photon excitation parameters for various excited states of argon assuming the absorption of near resonance broad-bandwidth laser radiation. Results are given for the case of two photons absorbed for the same laser beam as well as the case of absorbing photons of different frequency from each of two laser beams. The authors use multiconfiguration Hartree-Fock wave functions to evaluate the second-order sums over matrix elements. Various experimental laser schemes are suggested for the efficient excitation and subsequent ionization of argon 15. Excited state Intramolecular Proton Transfer in Anthralin DEFF Research Database (Denmark) Møller, Søren; Andersen, Kristine B.; Spanget-Larsen, Jens 1998-01-01 Quantum chemical calculations performed on anthralin (1,8-dihydroxy-9(10H)-anthracenone) predict the possibility of an excited-state intramolecular proton transfer process. Fluorescence excitation and emission spectra of the compound dissolved in n-hexane at ambient temperature results in an unus......Quantum chemical calculations performed on anthralin (1,8-dihydroxy-9(10H)-anthracenone) predict the possibility of an excited-state intramolecular proton transfer process. Fluorescence excitation and emission spectra of the compound dissolved in n-hexane at ambient temperature results......, associated with an excited-state intramolecular proton transfer process.... 16. 48-spot single-molecule FRET setup with periodic acceptor excitation Science.gov (United States) Ingargiola, Antonino; Segal, Maya; Gulinatti, Angelo; Rech, Ivan; Labanca, Ivan; Maccagnani, Piera; Ghioni, Massimo; Weiss, Shimon; Michalet, Xavier 2018-03-01 Single-molecule Förster resonance energy transfer (smFRET) allows measuring distances between donor and acceptor fluorophores on the 3-10 nm range. Solution-based smFRET allows measurement of binding-unbinding events or conformational changes of dye-labeled biomolecules without ensemble averaging and free from surface perturbations. When employing dual (or multi) laser excitation, smFRET allows resolving the number of fluorescent labels on each molecule, greatly enhancing the ability to study heterogeneous samples. A major drawback to solution-based smFRET is the low throughput, which renders repetitive measurements expensive and hinders the ability to study kinetic phenomena in real-time. Here we demonstrate a high-throughput smFRET system that multiplexes acquisition by using 48 excitation spots and two 48-pixel single-photon avalanche diode array detectors. The system employs two excitation lasers allowing separation of species with one or two active fluorophores. The performance of the system is demonstrated on a set of doubly labeled double-stranded DNA oligonucleotides with different distances between donor and acceptor dyes along the DNA duplex. We show that the acquisition time for accurate subpopulation identification is reduced from several minutes to seconds, opening the way to high-throughput screening applications and real-time kinetics studies of enzymatic reactions such as DNA transcription by bacterial RNA polymerase. 17. Back reaction of excitations on a vortex International Nuclear Information System (INIS) 1997-01-01 Excitations of a vortex are usually considered in a linear approximation neglecting their back reaction on the vortex. In the present paper we investigate back reaction of Proca-type excitations on a straight linear vortex in the Abelian Higgs model. We propose an exact ansatz for fields of the excited vortex. From an initial set of six nonlinear field equations we obtain (in a limit of weak excitations) two linear wave equations for the back reaction corrections. Their approximate solutions are found in the cases of plane wave and wave-packet-type excitations. We find that the excited vortex radiates the vector field and that the Higgs field has a very broad oscillating component. copyright 1997 The American Physical Society 18. Complex dynamics of an archetypal self-excited SD oscillator driven by moving belt friction International Nuclear Information System (INIS) Li Zhi-Xin; Cao Qing-Jie; Alain, Léger 2016-01-01 We propose an archetypal self-excited system driven by moving belt friction, which is constructed with the smooth and discontinuous (SD) oscillator proposed by the Cao et al. and the classical moving belt. The moving belt friction is modeled as the Coulomb friction to formulate the mathematical model of the proposed self-excited SD oscillator. The equilibrium states of the unperturbed system are obtained to show the complex equilibrium bifurcations. Phase portraits are depicted to present the hyperbolic structure transition, the multiple stick regions, and the friction-induced asymmetry phenomena. The numerical simulations are carried out to demonstrate the friction-induced vibration of multiple stick-slip phenomena and the stick-slip chaos in the perturbed self-excited system. The results presented here provide an opportunity for us to get insight into the mechanism of the complex friction-induced nonlinear dynamics in mechanical engineering and geography. (paper) 19. Probing thermomechanics at the nanoscale: impulsively excited pseudosurface acoustic waves in hypersonic phononic crystals. Science.gov (United States) Nardi, Damiano; Travagliati, Marco; Siemens, Mark E; Li, Qing; Murnane, Margaret M; Kapteyn, Henry C; Ferrini, Gabriele; Parmigiani, Fulvio; Banfi, Francesco 2011-10-12 High-frequency surface acoustic waves can be generated by ultrafast laser excitation of nanoscale patterned surfaces. Here we study this phenomenon in the hypersonic frequency limit. By modeling the thermomechanics from first-principles, we calculate the system's initial heat-driven impulsive response and follow its time evolution. A scheme is introduced to quantitatively access frequencies and lifetimes of the composite system's excited eigenmodes. A spectral decomposition of the calculated response on the eigemodes of the system reveals asymmetric resonances that result from the coupling between surface and bulk acoustic modes. This finding allows evaluation of impulsively excited pseudosurface acoustic wave frequencies and lifetimes and expands our understanding of the scattering of surface waves in mesoscale metamaterials. The model is successfully benchmarked against time-resolved optical diffraction measurements performed on one-dimensional and two-dimensional surface phononic crystals, probed using light at extreme ultraviolet and near-infrared wavelengths. 20. Electroluminescence from graphene excited by electron tunneling International Nuclear Information System (INIS) Beams, Ryan; Bharadwaj, Palash; Novotny, Lukas 2014-01-01 We use low-energy electron tunneling to excite electroluminescence in single layer graphene. Electrons are injected locally using a scanning tunneling microscope and the luminescence is analyzed using a wide-angle optical imaging system. The luminescence can be switched on and off by inverting the tip–sample bias voltage. The observed luminescence is explained in terms of a hot luminescence mechanism. (paper)	{"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": 0.8297527432441711, "perplexity": 2200.4732741997145}, "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/1652662521883.7/warc/CC-MAIN-20220518083841-20220518113841-00108.warc.gz"}
https://bookdown.org/egarpor/SSS2-UC3M/simplin-assumps.html	## 2.4 Assumptions of the model Why do we need assumptions? To make inference on the model parameters. In other words, to infer properties about the unknown population coefficients $$\beta_0$$ and $$\beta_1$$ from the sample $$(X_1,Y_1),\ldots,(X_n,Y_n)$$. The assumptions of the linear model are: 1. Linearity: $$\mathbb{E}[Y\|X=x]=\beta_0+\beta_1x$$. 2. Homoscedasticity: $$\mathbb{V}\text{ar}[\varepsilon_i]=\sigma^2$$, with $$\sigma^2$$ constant for $$i=1,\ldots,n$$. 3. Normality: $$\varepsilon_i\sim\mathcal{N}(0,\sigma^2)$$ for $$i=1,\ldots,n$$. 4. Independence of the errors: $$\varepsilon_1,\ldots,\varepsilon_n$$ are independent (or uncorrelated, $$\mathbb{E}[\varepsilon_i\varepsilon_j]=0$$, $$i\neq j$$, since they are assumed to be normal). A good one-line summary of the linear model is (independence is assumed) \begin{align} Y\|X=x\sim \mathcal{N}(\beta_0+\beta_1x,\sigma^2) \end{align} Recall: • Nothing is said about the distribution of X. Indeed, X could be deterministic (called fixed design) or random (random design). • The linear model assumes that Y is continuous due to the normality of the errors. However, X can be discrete! Figures 2.18 and 2.19 represent situations where the assumptions of the model are respected and violated, respectively. For the moment, we will focus on building the intuition for checking the assumptions visually. In Chapter 3 we will see more sophisticated methods for checking the assumptions. We will see also what are the possible fixes to the failure of assumptions. The dataset assumptions.RData (download) contains the variables x1, …, x9 and y1, …, y9. For each regression y1 ~ x1, …, y9 ~ x9: • Check whether the assumptions of the linear model are being satisfied (make a scatterplot with a regression line).	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 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": 0.9692782163619995, "perplexity": 880.0085769352734}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376827137.61/warc/CC-MAIN-20181215222234-20181216004234-00440.warc.gz"}
http://openstudy.com/updates/55c50f90e4b0c7f4a97911a4	Here's the question you clicked on: 55 members online • 0 viewing ## anonymous one year ago Can someone help me find an equation for the inverse of the function. g(x)=x-3/2 Delete Cancel Submit • This Question is Closed 1. campbell_st • one year ago Best Response You've already chosen the best response. 0 is the equation $g(x) = \frac{x -3}{2}$ 2. anonymous • one year ago Best Response You've already chosen the best response. 0 To find the inverse of an equation, flip the x and y values first. Your equation will turn from $g(x)=\frac{ x-3 }{ 2 }$ to $x =\frac{ y -3 }{ 2 }$ After you've done that, solve for y. 3. anonymous • one year ago Best Response You've already chosen the best response. 0 Yes @campbell_st 4. anonymous • one year ago Best Response You've already chosen the best response. 0 Is y 1/2? 5. anonymous • one year ago Best Response You've already chosen the best response. 0 To get the inverse, solve for y. By solving for y, you need to isolate it. $x =\frac{ y -3 }{ 2 }$ Multiply both sides of the equation by 2 to get rid of the division sign. 6. Not the answer you are looking for? Search for more explanations. • Attachments: Find more explanations on OpenStudy ##### spraguer (Moderator) 5→ View Detailed Profile 23 • Teamwork 19 Teammate • Problem Solving 19 Hero • You have blocked this person. • ✔ You're a fan Checking fan status... Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.	{"extraction_info": {"found_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": 0.9991968274116516, "perplexity": 4671.320843615637}, "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/1484560283301.73/warc/CC-MAIN-20170116095123-00050-ip-10-171-10-70.ec2.internal.warc.gz"}
http://mathhelpforum.com/differential-geometry/120258-poincare-find-distance-between-points.html	# Math Help - poincare: find the distance between the points 1. ## poincare: find the distance between the points I need some help finding the Poincare distance I have tried to do this problem, but I know my answer is wrong! Let Gamma be the interior of the unit disk centered at the origin in the xy-plane, (x^2+y^2 <1). Find the poincare distance between the following pairs of points. (0,0) and (1/2,0). 2. Originally Posted by mandy123 I need some help finding the Poincare distance I have tried to do this problem, but I know my answer is wrong! Let Gamma be the interior of the unit disk centered at the origin in the xy-plane, (x^2+y^2 <1). Find the poincare distance between the following pairs of points. (0,0) and (1/2,0). Okay, what have you done to get your "wrong" answer? In particular, what formulas are you using? I assume you are talking about the Poincare disk model for hyperbolic geometry. 3. Find A’ · TU is the line x= ½ · T= (½ , (√3)/2) · U=(½ , - (√3)/2) · Find the equation of the line tangent to gamma at T (Perpendicular to radius of T) o The slope of point T is √3 o Slope of the perpendicular to the radius is -1/√3 o Equation of the line: y = -1/√3 x + 2/√3 · A’ o 0= -1/√3 x + 2/√3 o X=2 o A’ =(2,0) Find midpoint of AA’ · ( 5/4 , 0) Equation of the perpendicular bisector · X= 5/4 Midpoint of BA · (¼ , 0) Intersection of midpoint and perpendicular bisector · (5/4, 0)	{"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": 0.976326048374176, "perplexity": 1731.9759309549536}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422122030742.53/warc/CC-MAIN-20150124175350-00211-ip-10-180-212-252.ec2.internal.warc.gz"}
https://math.stackexchange.com/questions/3152653/calculate-weighted-mean-based-on-simple-mean	# Calculate weighted mean based on simple mean I need to calculate the mobile weighted mean of a quantity, $$a_i$$, based on $$b_i$$. I have a tool that allows me to calculate the mobile mean of a given series, so we can say I have a function: $$F(a_i) = \frac{1}{N}\Sigma_i a_i$$ Is it right to say that $$\frac{\Sigma_i a_ib_i}{\Sigma_i b_i} = \frac{F(a_ib_i)}{F(b_i)} (1)$$ ? Is there any condition on which this equality holds that I'm assuming implicitly? I can only think of $$N \ne 0$$, is there anything else I'm missing? EDIT: To clarify the situation, I have a source of dynamic data from which I get $$a_i$$ and $$b_i$$ and I have a module which I can give these data to and that will calculate the mobile mean, taking care of excluding older samples. The function $$F(a_i)$$ is meant to represent the module. Since I only get the simple mean from the module, and not the sum, my idea was to calculate the simple mean of $$a_i*b_i$$ and of $$b_i$$ and divide the two means to obtain the weighted mean, but before doing that I wanted to be sure about the conditions at which equation (1) holds. • I don't really follow what you mean by the equation that you wrote. If $F$ is an average operator, the number of entries just divides out and your equation is true. But what is this $b_i$? Are those the weights? If you have the average of $a_i$'s and you want to to calculate the weighted average (average of $a_i \times b_i$), I don't think there's a way to do it if you don't have all the values of $a_i$'s ... – Matti P. Mar 18 at 11:23 • Edited the question to better specify the situation, thanks for the comment @MattiP. – bracco23 Mar 18 at 11:32	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 10, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9692927598953247, "perplexity": 179.3699052675252}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578527135.18/warc/CC-MAIN-20190419041415-20190419063415-00164.warc.gz"}
https://terrytao.wordpress.com/tag/sigma-algebras/	You are currently browsing the tag archive for the ‘sigma algebras’ tag. Asgar Jamneshan and I have just uploaded to the arXiv our paper “An uncountable Moore-Schmidt theorem“. This paper revisits a classical theorem of Moore and Schmidt in measurable cohomology of measure-preserving systems. To state the theorem, let ${X = (X,{\mathcal X},\mu)}$ be a probability space, and ${\mathrm{Aut}(X, {\mathcal X}, \mu)}$ be the group of measure-preserving automorphisms of this space, that is to say the invertible bimeasurable maps ${T: X \rightarrow X}$ that preserve the measure ${\mu}$: ${T_* \mu = \mu}$. To avoid some ambiguity later in this post when we introduce abstract analogues of measure theory, we will refer to measurable maps as concrete measurable maps, and measurable spaces as concrete measurable spaces. (One could also call ${X = (X,{\mathcal X}, \mu)}$ a concrete probability space, but we will not need to do so here as we will not be working explicitly with abstract probability spaces.) Let ${\Gamma = (\Gamma,\cdot)}$ be a discrete group. A (concrete) measure-preserving action of ${\Gamma}$ on ${X}$ is a group homomorphism ${\gamma \mapsto T^\gamma}$ from ${\Gamma}$ to ${\mathrm{Aut}(X, {\mathcal X}, \mu)}$, thus ${T^1}$ is the identity map and ${T^{\gamma_1} \circ T^{\gamma_2} = T^{\gamma_1 \gamma_2}}$ for all ${\gamma_1,\gamma_2 \in \Gamma}$. A large portion of ergodic theory is concerned with the study of such measure-preserving actions, especially in the classical case when ${\Gamma}$ is the integers (with the additive group law). Let ${K = (K,+)}$ be a compact Hausdorff abelian group, which we can endow with the Borel ${\sigma}$-algebra ${{\mathcal B}(K)}$. A (concrete measurable) ${K}$cocycle is a collection ${\rho = (\rho_\gamma)_{\gamma \in \Gamma}}$ of concrete measurable maps ${\rho_\gamma: X \rightarrow K}$ obeying the cocycle equation $\displaystyle \rho_{\gamma_1 \gamma_2}(x) = \rho_{\gamma_1} \circ T^{\gamma_2}(x) + \rho_{\gamma_2}(x) \ \ \ \ \ (1)$ for ${\mu}$-almost every ${x \in X}$. (Here we are glossing over a measure-theoretic subtlety that we will return to later in this post – see if you can spot it before then!) Cocycles arise naturally in the theory of group extensions of dynamical systems; in particular (and ignoring the aforementioned subtlety), each cocycle induces a measure-preserving action ${\gamma \mapsto S^\gamma}$ on ${X \times K}$ (which we endow with the product of ${\mu}$ with Haar probability measure on ${K}$), defined by $\displaystyle S^\gamma( x, k ) := (T^\gamma x, k + \rho_\gamma(x) ).$ This connection with group extensions was the original motivation for our study of measurable cohomology, but is not the focus of the current paper. A special case of a ${K}$-valued cocycle is a (concrete measurable) ${K}$-valued coboundary, in which ${\rho_\gamma}$ for each ${\gamma \in \Gamma}$ takes the special form $\displaystyle \rho_\gamma(x) = F \circ T^\gamma(x) - F(x)$ for ${\mu}$-almost every ${x \in X}$, where ${F: X \rightarrow K}$ is some measurable function; note that (ignoring the aforementioned subtlety), every function of this form is automatically a concrete measurable ${K}$-valued cocycle. One of the first basic questions in measurable cohomology is to try to characterize which ${K}$-valued cocycles are in fact ${K}$-valued coboundaries. This is a difficult question in general. However, there is a general result of Moore and Schmidt that at least allows one to reduce to the model case when ${K}$ is the unit circle ${\mathbf{T} = {\bf R}/{\bf Z}}$, by taking advantage of the Pontryagin dual group ${\hat K}$ of characters ${\hat k: K \rightarrow \mathbf{T}}$, that is to say the collection of continuous homomorphisms ${\hat k: k \mapsto \langle \hat k, k \rangle}$ to the unit circle. More precisely, we have Theorem 1 (Countable Moore-Schmidt theorem) Let ${\Gamma}$ be a discrete group acting in a concrete measure-preserving fashion on a probability space ${X}$. Let ${K}$ be a compact Hausdorff abelian group. Assume the following additional hypotheses: • (i) ${\Gamma}$ is at most countable. • (ii) ${X}$ is a standard Borel space. • (iii) ${K}$ is metrisable. Then a ${K}$-valued concrete measurable cocycle ${\rho = (\rho_\gamma)_{\gamma \in \Gamma}}$ is a concrete coboundary if and only if for each character ${\hat k \in \hat K}$, the ${\mathbf{T}}$-valued cocycles ${\langle \hat k, \rho \rangle = ( \langle \hat k, \rho_\gamma \rangle )_{\gamma \in \Gamma}}$ are concrete coboundaries. The hypotheses (i), (ii), (iii) are saying in some sense that the data ${\Gamma, X, K}$ are not too “large”; in all three cases they are saying in some sense that the data are only “countably complicated”. For instance, (iii) is equivalent to ${K}$ being second countable, and (ii) is equivalent to ${X}$ being modeled by a complete separable metric space. It is because of this restriction that we refer to this result as a “countable” Moore-Schmidt theorem. This theorem is a useful tool in several other applications, such as the Host-Kra structure theorem for ergodic systems; I hope to return to these subsequent applications in a future post. Let us very briefly sketch the main ideas of the proof of Theorem 1. Ignore for now issues of measurability, and pretend that something that holds almost everywhere in fact holds everywhere. The hard direction is to show that if each ${\langle \hat k, \rho \rangle}$ is a coboundary, then so is ${\rho}$. By hypothesis, we then have an equation of the form $\displaystyle \langle \hat k, \rho_\gamma(x) \rangle = \alpha_{\hat k} \circ T^\gamma(x) - \alpha_{\hat k}(x) \ \ \ \ \ (2)$ for all ${\hat k, \gamma, x}$ and some functions ${\alpha_{\hat k}: X \rightarrow {\mathbf T}}$, and our task is then to produce a function ${F: X \rightarrow K}$ for which $\displaystyle \rho_\gamma(x) = F \circ T^\gamma(x) - F(x)$ for all ${\gamma,x}$. Comparing the two equations, the task would be easy if we could find an ${F: X \rightarrow K}$ for which $\displaystyle \langle \hat k, F(x) \rangle = \alpha_{\hat k}(x) \ \ \ \ \ (3)$ for all ${\hat k, x}$. However there is an obstruction to this: the left-hand side of (3) is additive in ${\hat k}$, so the right-hand side would have to be also in order to obtain such a representation. In other words, for this strategy to work, one would have to first establish the identity $\displaystyle \alpha_{\hat k_1 + \hat k_2}(x) - \alpha_{\hat k_1}(x) - \alpha_{\hat k_2}(x) = 0 \ \ \ \ \ (4)$ for all ${\hat k_1, \hat k_2, x}$. On the other hand, the good news is that if we somehow manage to obtain the equation, then we can obtain a function ${F}$ obeying (3), thanks to Pontryagin duality, which gives a one-to-one correspondence between ${K}$ and the homomorphisms of the (discrete) group ${\hat K}$ to ${\mathbf{T}}$. Now, it turns out that one cannot derive the equation (4) directly from the given information (2). However, the left-hand side of (2) is additive in ${\hat k}$, so the right-hand side must be also. Manipulating this fact, we eventually arrive at $\displaystyle (\alpha_{\hat k_1 + \hat k_2} - \alpha_{\hat k_1} - \alpha_{\hat k_2}) \circ T^\gamma(x) = (\alpha_{\hat k_1 + \hat k_2} - \alpha_{\hat k_1} - \alpha_{\hat k_2})(x).$ In other words, we don’t get to show that the left-hand side of (4) vanishes, but we do at least get to show that it is ${\Gamma}$-invariant. Now let us assume for sake of argument that the action of ${\Gamma}$ is ergodic, which (ignoring issues about sets of measure zero) basically asserts that the only ${\Gamma}$-invariant functions are constant. So now we get a weaker version of (4), namely $\displaystyle \alpha_{\hat k_1 + \hat k_2}(x) - \alpha_{\hat k_1}(x) - \alpha_{\hat k_2}(x) = c_{\hat k_1, \hat k_2} \ \ \ \ \ (5)$ for some constants ${c_{\hat k_1, \hat k_2} \in \mathbf{T}}$. Now we need to eliminate the constants. This can be done by the following group-theoretic projection. Let ${L^0({\bf X} \rightarrow {\bf T})}$ denote the space of concrete measurable maps ${\alpha}$ from ${{\bf X}}$ to ${{\bf T}}$, up to almost everywhere equivalence; this is an abelian group where the various terms in (5) naturally live. Inside this group we have the subgroup ${{\bf T}}$ of constant functions (up to almost everywhere equivalence); this is where the right-hand side of (5) lives. Because ${{\bf T}}$ is a divisible group, there is an application of Zorn’s lemma (a good exercise for those who are not acquainted with these things) to show that there exists a retraction ${w: L^0({\bf X} \rightarrow {\bf T}) \rightarrow {\bf T}}$, that is to say a group homomorphism that is the identity on the subgroup ${{\bf T}}$. We can use this retraction, or more precisely the complement ${\alpha \mapsto \alpha - w(\alpha)}$, to eliminate the constant in (5). Indeed, if we set $\displaystyle \tilde \alpha_{\hat k}(x) := \alpha_{\hat k}(x) - w(\alpha_{\hat k})$ then from (5) we see that $\displaystyle \tilde \alpha_{\hat k_1 + \hat k_2}(x) - \tilde \alpha_{\hat k_1}(x) - \tilde \alpha_{\hat k_2}(x) = 0$ while from (2) one has $\displaystyle \langle \hat k, \rho_\gamma(x) \rangle = \tilde \alpha_{\hat k} \circ T^\gamma(x) - \tilde \alpha_{\hat k}(x)$ and now the previous strategy works with ${\alpha_{\hat k}}$ replaced by ${\tilde \alpha_{\hat k}}$. This concludes the sketch of proof of Theorem 1. In making the above argument rigorous, the hypotheses (i)-(iii) are used in several places. For instance, to reduce to the ergodic case one relies on the ergodic decomposition, which requires the hypothesis (ii). Also, most of the above equations only hold outside of a set of measure zero, and the hypothesis (i) and the hypothesis (iii) (which is equivalent to ${\hat K}$ being at most countable) to avoid the problem that an uncountable union of sets of measure zero could have positive measure (or fail to be measurable at all). My co-author Asgar Jamneshan and I are working on a long-term project to extend many results in ergodic theory (such as the aforementioned Host-Kra structure theorem) to “uncountable” settings in which hypotheses analogous to (i)-(iii) are omitted; thus we wish to consider actions on uncountable groups, on spaces that are not standard Borel, and cocycles taking values in groups that are not metrisable. Such uncountable contexts naturally arise when trying to apply ergodic theory techniques to combinatorial problems (such as the inverse conjecture for the Gowers norms), as one often relies on the ultraproduct construction (or something similar) to generate an ergodic theory translation of these problems, and these constructions usually give “uncountable” objects rather than “countable” ones. (For instance, the ultraproduct of finite groups is a hyperfinite group, which is usually uncountable.). This paper marks the first step in this project by extending the Moore-Schmidt theorem to the uncountable setting. If one simply drops the hypotheses (i)-(iii) and tries to prove the Moore-Schmidt theorem, several serious difficulties arise. We have already mentioned the loss of the ergodic decomposition and the possibility that one has to control an uncountable union of null sets. But there is in fact a more basic problem when one deletes (iii): the addition operation ${+: K \times K \rightarrow K}$, while still continuous, can fail to be measurable as a map from ${(K \times K, {\mathcal B}(K) \otimes {\mathcal B}(K))}$ to ${(K, {\mathcal B}(K))}$! Thus for instance the sum of two measurable functions ${F: X \rightarrow K}$ need not remain measurable, which makes even the very definition of a measurable cocycle or measurable coboundary problematic (or at least unnatural). This phenomenon is known as the Nedoma pathology. A standard example arises when ${K}$ is the uncountable torus ${{\mathbf T}^{{\bf R}}}$, endowed with the product topology. Crucially, the Borel ${\sigma}$-algebra ${{\mathcal B}(K)}$ generated by this uncountable product is not the product ${{\mathcal B}(\mathbf{T})^{\otimes {\bf R}}}$ of the factor Borel ${\sigma}$-algebras (the discrepancy ultimately arises from the fact that topologies permit uncountable unions, but ${\sigma}$-algebras do not); relating to this, the product ${\sigma}$-algebra ${{\mathcal B}(K) \otimes {\mathcal B}(K)}$ is not the same as the Borel ${\sigma}$-algebra ${{\mathcal B}(K \times K)}$, but is instead a strict sub-algebra. If the group operations on ${K}$ were measurable, then the diagonal set $\displaystyle K^\Delta := \{ (k,k') \in K \times K: k = k' \} = \{ (k,k') \in K \times K: k - k' = 0 \}$ would be measurable in ${{\mathcal B}(K) \otimes {\mathcal B}(K)}$. But it is an easy exercise in manipulation of ${\sigma}$-algebras to show that if ${(X, {\mathcal X}), (Y, {\mathcal Y})}$ are any two measurable spaces and ${E \subset X \times Y}$ is measurable in ${{\mathcal X} \otimes {\mathcal Y}}$, then the fibres ${E_x := \{ y \in Y: (x,y) \in E \}}$ of ${E}$ are contained in some countably generated subalgebra of ${{\mathcal Y}}$. Thus if ${K^\Delta}$ were ${{\mathcal B}(K) \otimes {\mathcal B}(K)}$-measurable, then all the points of ${K}$ would lie in a single countably generated ${\sigma}$-algebra. But the cardinality of such an algebra is at most ${2^{\alpha_0}}$ while the cardinality of ${K}$ is ${2^{2^{\alpha_0}}}$, and Cantor’s theorem then gives a contradiction. To resolve this problem, we give ${K}$ a coarser ${\sigma}$-algebra than the Borel ${\sigma}$-algebra, which we call the reduced ${\sigma}$-algebra ${{\mathcal B}^\otimes(K)}$, thus coarsening the measurable space structure on ${K = (K,{\mathcal B}(K))}$ to a new measurable space ${K_\otimes := (K, {\mathcal B}^\otimes(K))}$. In the case of compact Hausdorff abelian groups, ${{\mathcal B}^{\otimes}(K)}$ can be defined as the ${\sigma}$-algebra generated by the characters ${\hat k: K \rightarrow {\mathbf T}}$; for more general compact abelian groups, one can define ${{\mathcal B}^{\otimes}(K)}$ as the ${\sigma}$-algebra generated by all continuous maps into metric spaces. This ${\sigma}$-algebra is equal to ${{\mathcal B}(K)}$ when ${K}$ is metrisable but can be smaller for other ${K}$. With this measurable structure, ${K_\otimes}$ becomes a measurable group; it seems that once one leaves the metrisable world that ${K_\otimes}$ is a superior (or at least equally good) space to work with than ${K}$ for analysis, as it avoids the Nedoma pathology. (For instance, from Plancherel’s theorem, we see that if ${m_K}$ is the Haar probability measure on ${K}$, then ${L^2(K,m_K) = L^2(K_\otimes,m_K)}$ (thus, every ${K}$-measurable set is equivalent modulo ${m_K}$-null sets to a ${K_\otimes}$-measurable set), so there is no damage to Plancherel caused by passing to the reduced ${\sigma}$-algebra. Passing to the reduced ${\sigma}$-algebra ${K_\otimes}$ fixes the most severe problems with an uncountable Moore-Schmidt theorem, but one is still faced with an issue of having to potentially take an uncountable union of null sets. To avoid this sort of problem, we pass to the framework of abstract measure theory, in which we remove explicit mention of “points” and can easily delete all null sets at a very early stage of the formalism. In this setup, the category of concrete measurable spaces is replaced with the larger category of abstract measurable spaces, which we formally define as the opposite category of the category of ${\sigma}$-algebras (with Boolean algebra homomorphisms). Thus, we define an abstract measurable space to be an object of the form ${{\mathcal X}^{\mathrm{op}}}$, where ${{\mathcal X}}$ is an (abstract) ${\sigma}$-algebra and ${\mathrm{op}}$ is a formal placeholder symbol that signifies use of the opposite category, and an abstract measurable map ${T: {\mathcal X}^{\mathrm{op}} \rightarrow {\mathcal Y}^{\mathrm{op}}}$ is an object of the form ${(T^)^{\mathrm{op}}}$, where ${T^: {\mathcal Y} \rightarrow {\mathcal X}}$ is a Boolean algebra homomorphism and ${\mathrm{op}}$ is again used as a formal placeholder; we call ${T^}$ the pullback map associated to ${T}$. [UPDATE: It turns out that this definition of a measurable map led to technical issues. In a forthcoming revision of the paper we also impose the requirement that the abstract measurable map be $\sigma$-complete (i.e., it respects countable joins).] The composition ${S \circ T: {\mathcal X}^{\mathrm{op}} \rightarrow {\mathcal Z}^{\mathrm{op}}}$ of two abstract measurable maps ${T: {\mathcal X}^{\mathrm{op}} \rightarrow {\mathcal Y}^{\mathrm{op}}}$, ${S: {\mathcal Y}^{\mathrm{op}} \rightarrow {\mathcal Z}^{\mathrm{op}}}$ is defined by the formula ${S \circ T := (T^ \circ S^)^{\mathrm{op}}}$, or equivalently ${(S \circ T)^ = T^* \circ S^}$. Every concrete measurable space ${X = (X,{\mathcal X})}$ can be identified with an abstract counterpart ${{\mathcal X}^{op}}$, and similarly every concrete measurable map ${T: X \rightarrow Y}$ can be identified with an abstract counterpart ${(T^)^{op}}$, where ${T^: {\mathcal Y} \rightarrow {\mathcal X}}$ is the pullback map ${T^ E := T^{-1}(E)}$. Thus the category of concrete measurable spaces can be viewed as a subcategory of the category of abstract measurable spaces. The advantage of working in the abstract setting is that it gives us access to more spaces that could not be directly defined in the concrete setting. Most importantly for us, we have a new abstract space, the opposite measure algebra ${X_\mu}$ of ${X}$, defined as ${({\bf X}/{\bf N})^*}$ where ${{\bf N}}$ is the ideal of null sets in ${{\bf X}}$. Informally, ${X_\mu}$ is the space ${X}$ with all the null sets removed; there is a canonical abstract embedding map ${\iota: X_\mu \rightarrow X}$, which allows one to convert any concrete measurable map ${f: X \rightarrow Y}$ into an abstract one ${[f]: X_\mu \rightarrow Y}$. One can then define the notion of an abstract action, abstract cocycle, and abstract coboundary by replacing every occurrence of the category of concrete measurable spaces with their abstract counterparts, and replacing ${X}$ with the opposite measure algebra ${X_\mu}$; see the paper for details. Our main theorem is then Theorem 2 (Uncountable Moore-Schmidt theorem) Let ${\Gamma}$ be a discrete group acting abstractly on a ${\sigma}$-finite measure space ${X}$. Let ${K}$ be a compact Hausdorff abelian group. Then a ${K_\otimes}$-valued abstract measurable cocycle ${\rho = (\rho_\gamma)_{\gamma \in \Gamma}}$ is an abstract coboundary if and only if for each character ${\hat k \in \hat K}$, the ${\mathbf{T}}$-valued cocycles ${\langle \hat k, \rho \rangle = ( \langle \hat k, \rho_\gamma \rangle )_{\gamma \in \Gamma}}$ are abstract coboundaries. With the abstract formalism, the proof of the uncountable Moore-Schmidt theorem is almost identical to the countable one (in fact we were able to make some simplifications, such as avoiding the use of the ergodic decomposition). A key tool is what we call a “conditional Pontryagin duality” theorem, which asserts that if one has an abstract measurable map ${\alpha_{\hat k}: X_\mu \rightarrow {\bf T}}$ for each ${\hat k \in K}$ obeying the identity ${ \alpha_{\hat k_1 + \hat k_2} - \alpha_{\hat k_1} - \alpha_{\hat k_2} = 0}$ for all ${\hat k_1,\hat k_2 \in \hat K}$, then there is an abstract measurable map ${F: X_\mu \rightarrow K_\otimes}$ such that ${\alpha_{\hat k} = \langle \hat k, F \rangle}$ for all ${\hat k \in \hat K}$. This is derived from the usual Pontryagin duality and some other tools, most notably the completeness of the ${\sigma}$-algebra of ${X_\mu}$, and the Sikorski extension theorem. We feel that it is natural to stay within the abstract measure theory formalism whenever dealing with uncountable situations. However, it is still an interesting question as to when one can guarantee that the abstract objects constructed in this formalism are representable by concrete analogues. The basic questions in this regard are: • (i) Suppose one has an abstract measurable map ${f: X_\mu \rightarrow Y}$ into a concrete measurable space. Does there exist a representation of ${f}$ by a concrete measurable map ${\tilde f: X \rightarrow Y}$? Is it unique up to almost everywhere equivalence? • (ii) Suppose one has a concrete cocycle that is an abstract coboundary. When can it be represented by a concrete coboundary? For (i) the answer is somewhat interesting (as I learned after posing this MathOverflow question): • If ${Y}$ does not separate points, or is not compact metrisable or Polish, there can be counterexamples to uniqueness. If ${Y}$ is not compact or Polish, there can be counterexamples to existence. • If ${Y}$ is a compact metric space or a Polish space, then one always has existence and uniqueness. • If ${Y}$ is a compact Hausdorff abelian group, one always has existence. • If ${X}$ is a complete measure space, then one always has existence (from a theorem of Maharam). • If ${X}$ is the unit interval with the Borel ${\sigma}$-algebra and Lebesgue measure, then one has existence for all compact Hausdorff ${Y}$ assuming the continuum hypothesis (from a theorem of von Neumann) but existence can fail under other extensions of ZFC (from a theorem of Shelah, using the method of forcing). • For more general ${X}$, existence for all compact Hausdorff ${Y}$ is equivalent to the existence of a lifting from the ${\sigma}$-algebra ${\mathcal{X}/\mathcal{N}}$ to ${\mathcal{X}}$ (or, in the language of abstract measurable spaces, the existence of an abstract retraction from ${X}$ to ${X_\mu}$). • It is a long-standing open question (posed for instance by Fremlin) whether it is relatively consistent with ZFC that existence holds whenever ${Y}$ is compact Hausdorff. Our understanding of (ii) is much less complete: • If ${K}$ is metrisable, the answer is “always” (which among other things establishes the countable Moore-Schmidt theorem as a corollary of the uncountable one). • If ${\Gamma}$ is at most countable and ${X}$ is a complete measure space, then the answer is again “always”. In view of the answers to (i), I would not be surprised if the full answer to (ii) was also sensitive to axioms of set theory. However, such set theoretic issues seem to be almost completely avoided if one sticks with the abstract formalism throughout; they only arise when trying to pass back and forth between the abstract and concrete categories. In these notes we quickly review the basics of abstract measure theory and integration theory, which was covered in the previous course but will of course be relied upon in the current course. This is only a brief summary of the material; of course, one should consult a real analysis text for the full details of the theory.	{"extraction_info": {"found_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": 235, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9826717376708984, "perplexity": 168.52481060110915}, "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-2019-51/segments/1575540518337.65/warc/CC-MAIN-20191209065626-20191209093626-00080.warc.gz"}
http://windowsontheory.org/2014/02/17/discrepancy-bounds-from-convex-geometry/	In the last post we discussed some questions about discrepancy and the ‘Six Standard Deviations Suffice’ theorem stated below (without the ${6}$, which is not too important, but makes for a great title): Theorem 1 For vectors ${a^1,\ldots,a^n \in \{1,-1\}^n}$, there exists ${\epsilon \in \{1,-1\}^n}$ such that for every ${j \in [n]}$, ${\|\langle a^j,\epsilon\rangle\| = O(\sqrt{n})}$. In this post (second and the most technical of the three post series) we will see a proof of the above result. The theorem was proved by Spencer in 1985 using Beck’s partial coloring approach. Independently of Spencer, the result was proved by Gluskin in 1988 in a convex geometric context. Here we will review Gluskin’s proof which is quite beautiful. Gluskin’s proof will also give us an excuse to look at some elegant (and simple to describe) results in convex geometry which may be of use elsewhere. Finally, the geometric view here will actually be useful in the next post when we discuss an algorithmic proof. Gluskin’s paper is truly remarkable and seems to reinvent several key ideas from scratch such as Sidak’s lemma, a version of Kanter’s lemma for Gaussians in convex geometry and even has the partial coloring approach implicit in it. I recommend taking a look at the paper even if it is a bit of a tough read. Much of the content in the post is based on discussions with Shachar Lovett and Oded Regev, but all mistakes are mine. Gluskin’s proof follows the partial coloring approach with the crucial lemma proved using a volume argument. The partial coloring method was introduced by Beck in 1981 and all proofs of Theorem 1 and many other important discrepancy results in fact use this method. Here, instead of looking for a ${\epsilon \sim \{1,-1\}^n}$ solution as in the theorem, one looks for a ${\{1,0,-1\}^n}$ solution first. This is meaningless to begin with as we can just output the all zeros vector. The main idea is to instead look for a ${\{1,0,-1\}^n}$ solution which has ${\Omega(n)}$ support. We then recurse on the set of coordinates which are set to ${0}$. If everything goes according to plan, as we are geometrically decreasing the ambient dimension, we similarly get geometrically decreasing discrepancy bounds which we can tolerate. I won’t go into the details here, but let’s accept that it suffices to show the following: Lemma 2 For vectors ${a^1,\ldots,a^n \in \{1,-1\}^n}$, there exists ${\epsilon \in \{1,0,-1\}^n}$ such that for every ${j \in [n]}$, ${\langle a^j,\epsilon\rangle = O(\sqrt{n})}$ and ${\|Support(\epsilon)\| = \Omega(n)}$. To prove the above partial-coloring-lemma, let us first rephrase the problem in a geometric language. Let ${\mathcal{K} \subseteq R^n}$ be the symmetric convex set (symmetric meaning ${x \in \mathcal{K}}$ implies ${-x \in \mathcal{K}}$) defined as follows for ${\Delta = O(\sqrt{n})}$ to be chosen later: $\displaystyle \mathcal{K} = \{x: \|\langle a^j,x\rangle\| \leq \Delta, \forall j \in [n]\}.$ We want to show that ${\mathcal{K}}$ contains a ${\{1,0,-1\}^n}$ lattice point of large support. We show this indirectly by proving that ${\mathcal{K}}$ instead contains a lot of points from ${\{1,0,-1\}^n}$. Gluskin does this by a clever volume argument: first show that the volume of ${\mathcal{K} \cap [-1,1]^n}$ is large and then apply Minkowski’s theorem to show that there are many lattice points. To lower bound the first volume, Gluskin actually works in the Gaussian space. I don’t have a clear intuitive reason for why the Gaussian distribution is better than the Lebesgue measure in this context. But if one looks at the analysis, a clear advantage is that projections behave better (when considering volumes) in the Gaussian space. For example, if we take a set like ${S = \{x \in R^n\,:\,\|x_1\| \leq 1\}}$, then the Lebsgue volume of ${S}$ is infinite, but if we project along the first coordinate it becomes finite. In the Gaussian case, both volumes are the same. We next go over all the main pieces in Gluskin’s proof. Sidak’s Lemma Suppose we have a standard Gaussian vector ${g \sim \mathcal{N}(0,1)^n}$. Then, for any unit vector ${v \in R^n}$, ${\langle v,g\rangle}$ has the standard normal distribution. Now, suppose we have several unit vectors ${v_1,\ldots,v_m \in R^n}$. Then, the random variables ${X_i = \langle v_i,g\rangle}$ are individually standard normals, but are correlated with one another. Sidak’s lemma (1967) says that no matter what the correlations of ${X_i}$‘s are, to bound the probability that none of the ${X_i}$‘s is too large, the “worst-behaviour” one could expect is for them to be independent. Concretely: Lemma 3 (Sidak’s Lemma) Let ${v_1,\ldots,v_m \in R^n}$ and let ${g \sim \mathcal{N}(0,1)^n}$ be a standard Gaussian vector. Then, for all ${t_1,\ldots,t_m \in R_+}$, $\displaystyle Pr\left[\|\langle v_1,g\rangle\| \leq t_1 \;\wedge\; \|\langle v_2,g\rangle\| \leq t_2\;\wedge\; \cdots \;\wedge\; \|\langle v_m,g\rangle\| \leq t_m\right] \geq$ $\displaystyle \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; Pr\left[\|\langle v_1,g\rangle \|\leq t_1\right] \cdot Pr\left[\|\langle v_2,g\rangle\|\leq t_2\right] \cdots Pr\left[\|\langle v_m,g\rangle\|\leq t_m\right].$ The proof of the lemma is actually not too hard and an excellent exposition can be found in this paper. The lemma is actually a very special case of a longstanding open problem called the Correlation Conjecture. Let me digress a little bit to state this beautiful question. In the above setup, let slab ${C_i = \{x \in R^n: \|\langle v_i,x\rangle\| \leq t_i\}}$. Then, Sidak’s lemma says that for ${g \sim \mathcal{N}(0,1)^n}$, $\displaystyle Pr[g \in C_1 \wedge C_2 \wedge \cdots \wedge C_m] \geq Pr[g \in C_1] \cdot Pr[g \in C_2] \cdots Pr[g \in C_m].$ The correlation conjecture asserts that this inequality is in fact true for all symmetric convex sets (in fact, we only need to look at ${m=2}$). Sidak’s lemma says the conjecture is true for slabs. It is also known to be true for ellipsoids. The statement for ellipsoids also has a discrepancy implication leading to a vector generalization of Spencer’s theorem (pointed to me by Krzysztof Oleszkiewicz). But that’s for another day. Kanter’s Lemma The second inequality we need is a comparison inequality due to Kanter. The lemma essentially lets us lift certain relations between two distributions ${p,q}$ to their product distributions ${p^n, q^n}$ and I think should be useful in other contexts. For instance, I recently used it in this paper in a completely different context. To state the lemma, we need the notion of peakedness of distributions. Let ${p,q}$ be two symmetric distributions on ${R^m}$ for some ${m}$. We say ${p}$ is less peaked than ${q}$ (written ${p \preceq q}$) if for all symmetric convex sets ${\mathcal{K}}$, ${p(\mathcal{K}) \leq q(\mathcal{K})}$. Intuitively, this means that ${p}$ is putting less of its mass near the origin than ${q}$ (hence the term less peaked). For example, ${\mathcal{N}(0,2) \preceq \mathcal{N}(0,1)}$. Kanter’s lemma says that the peakedness relation tensorises provided we have unimodality. A univariate distribution is unimodal if the corresponding probability density function has a single maximum and no other local maxima. I won’t define what it means for a multivariate distribution to be unimodal here, but we only need the lemma for univariate distributions. See this survey for the formal definition. Lemma 4 (Kanter’s lemma) Let ${p,q}$ be two symmetric distributions on ${R^n}$ such that ${p \preceq q}$ and let ${\mu}$ be a unimodal distribution on ${R^m}$. Then, the product distributions ${p \times \mu}$, ${q \times \mu}$ on ${R^{n \times m}}$, satisfy ${p \times \mu \preceq q \times \mu}$. The proof of the lemma is not too hard, but is non-trivial in that it uses the Brunn-Minkowski’s inequality. Combining the above lemma with the not-too-hard fact that the standard Gaussian distribution is less peaked than the uniform distribution on ${[-1,1]}$, we get: Corollary 5 Let ${\mu}$ be the uniform distribution on ${[-1,1]}$. Then, ${\mathcal{N}(0,1)^n \preceq \mu^n}$. Minkowski’s Theorem The final piece we need is the classical Minkowski’s theorem form lattice geometry: Theorem 6 (Minkowski’s Theorem) Let ${C \subseteq R^n}$ be a symmetric convex set of Lebesgue volume more than ${2^n \cdot \ell}$ for an integer ${\ell \geq 1}$. Then, ${C}$ contains at least ${\ell}$ points from the integer lattice ${\mathbb{Z}^n}$. Putting Things Together We will prove the partial coloring lemma Lemma 2. The proof will be a sequence of simple implications using the above lemmas. Recall the definition of ${\mathcal{K}}$: $\displaystyle \mathcal{K} = \{x: \|\langle a^j,x\rangle\| \leq \Delta, \forall j \in [n]\}.$ Our goal (Lemma 2) is equivalent to showing that ${\mathcal{K}}$ contains a ${\{1,0,-1\}}$ point of large support. Note that ${\mathcal{K}}$ is the intersection of ${n}$ slabs. Therefore, by Sidak’s lemma, for ${g \sim \mathcal{N}(0,1)^n}$, $\displaystyle Pr[g \in \mathcal{K}] \geq \prod_{j=1}^n Pr[\|\langle a^j,g\rangle\| \leq \Delta] \geq \prod_{j=1}^n\left(1-2\exp(-\Delta^2/2n)\right),$ where the last inequality follows from the fact that ${\langle a^j,g\rangle}$ has the Gaussian distribution with standard deviation ${\sqrt{n}}$. Therefore, if we pick ${\Delta = O(\sqrt{n})}$, sufficiently big then $\displaystyle Pr[g \in \mathcal{K}] \geq (3/4)^n.$ Now, let ${\mu}$ be the uniform distribution on ${[-1,1]}$. Then, by Corollary 5, and the definition of peakedness, ${\mu^n(\mathcal{K}) \geq (3/4)^n}$. Hence, the Lebesgue volume of ${\mathcal{K}' = \mathcal{K} \cap [-1,1]^n}$ is at least ${2^n (3/4)^n = (3/2)^n}$. Therefore, for sufficiently small ${\delta > 0}$, Lebesgue volume of ${(2-\delta)\mathcal{K}' \geq (2-\delta)^n \cdot (3/2)^n = 2^n \cdot 2^{\Omega(n)}}$. Thus, by applying Minkowski’s theorem to the symmetric convex set ${(2-\delta) \mathcal{K}'}$ we get that ${(2-\delta)\mathcal{K}'}$ has at least ${2^{\Omega(n)}}$ lattice points. Now, note that the only lattice points in ${(2-\delta)\mathcal{K}'}$ are elements of ${\{1,0,-1\}^n}$ inside ${\mathcal{K}}$. Therefore ${\mathcal{K}}$ has at least ${2^{\Omega(n)}}$ points from ${\{1,0,-1\}^n}$. By a simple counting argument at least one of these lattice points, ${\epsilon}$, must ${\Omega(n)}$ non-zero coordinates – which is exactly what we need to prove Lemma 2! Discussion The above argument can actually be simplified by replacing the use of Kanter’s lemma with an appropriate version of Miknowski’s theorem for Gaussian volume as done here. But I like any excuse to discuss Kanter’s lemma. More importantly, the proof seems to be more amenable to generalization. The core of the proof really is to use Sidak’s lemma to lower bound the Gaussian volume of the convex set ${\mathcal{K}}$. Whenever you have such a statement you should even get a corresponding discrepancy statement. In particular, the matrix discrepancy conjecture from last post, essentially reduces to the following probability question: Question Is it true that for a universal constant ${C}$, for all symmetric matrices ${A_1,\ldots,A_n \in R^{n \times n}}$ with ${\\|A_i\\| \leq 1}$, $\displaystyle Pr_{g \sim \mathcal{N}^n}\left[\\|g_1 A_1 + g_2 A_2 + \cdots + g_n A_n \\| \leq C \sqrt{n}\right] \geq (3/4)^n\,?$ Acknowledgments Thanks to Shachar Lovett, Oded Regev and Nikhil Srivastava for helpful suggestions, comments, and corrections during the preparation of this post. 1. February 20, 2014 2:07 am Very nice post! I may be missing something, but I think Giannopoulos’s proof also gives the implication from Gaussian volume lower bounds to a partial coloring. Let $K$ be the set of “small discrepancy fractional colorings” (the same set you define), and assume $\Pr_g[g \in K] > 2^{-c_1n}$. Then let $f_K(g)$ be the number of sign sequences $\varepsilon \in \{-1, 1\}^n$ such that $g \in 0.01\varepsilon + K$. From looking at the pdf of the standard gaussian, $\Pr[g \in 0.01\varepsilon + K] >= 2^{-c_2n}$ for some $c_2 = 2^{(1 - c_2)n}$. So there must exist a point $\latex x$ which is in at least $N = 2^{(1-c_2)n}$ sets $\varepsilon^{1} + 100K, \ldots, \varepsilon^{N} + 100K$. For any two $i, j$, $\varepsilon^i - \varepsilon^j \in 200K$, and since $N$ is so large, there must exist two $i, j$ such that $\frac{\varepsilon^i - \varepsilon^j}{2} \in 100K$ and $\frac{\varepsilon^i - \varepsilon^j}{2}$ has large support. I chose 0.01 very conservatively, much better constants should do the job. But Kanter’s lemma is cool in any case! • February 20, 2014 6:00 am Indeed, Giannopoulos’s proof is simpler – one can view the argument you described (a typo: c_2 should be a constant) as a kind of Gaussian version of Minkowski’s theorem which by itself is a nice thing to know. • February 20, 2014 8:14 pm Sorry for the typo. I wish there was a way to preview comments by the way. I could’ve been more explicit: $\Pr[g \in x + K] \geq e^{0.5\\|x\\|_2^2}$ for any centrally symmetric $K$ and any $x$, so $c_2 = c_1 + 0.005\log_2 e \leq c_1 + 0.008$. I guess I was really conservative with constants.	{"extraction_info": {"found_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": 128, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9805178046226501, "perplexity": 229.6724386490935}, "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/1406510273676.54/warc/CC-MAIN-20140728011753-00104-ip-10-146-231-18.ec2.internal.warc.gz"}
http://mathhelpforum.com/calculus/98643-flux-integrals-verifying-stokes-theorem.html	# Math Help - Flux integrals - verifying Stokes Theorem 1. ## Flux integrals - verifying Stokes Theorem Ok, I've been stuck on this problem for hours now and its really irritating, so I need help! Here is the question: ------------------------------------------------------------- Verify Stokes Theorem for the vector field F = yi + 2zj + xzk and the surface S defined by x^2 + y^2 + z^2 = 25 & Z>=4 -------------------------------------------------------------- Right, I'm quite new to these so please explain clearly if possible, cheers! So far I've computed the line integral and got a result of -Pi Now I'm stuck on the flux integral. I've worked out that Curl F = -2i -zj -k but I'm not sure what the normal vector is or the limits. Any help much appreciated! 2. The surface is the portion of the hemi -sphere x^2 +y^2 +z^2 =25 above z= 4 z= sqrt(25-x^2 -y^2) N = -dz/dx i -dz/dy j + K N = x/sqrt(25-x^2) i + y /sqrt(25-y^2) j + k The region of integration is the circle x^2 + y^2 = 9 (using z=4) See the attachment for the calculation of the line integral using both Stokes Theorem and the definition of line integral Attached Files 3. Gah so my line integral result is incorrect? 4. you're computing the line integral around the circle of radius 3 , 4units above the x-y plane. How'd you compute the line integral ? What was your parameterization and Fdr/dt ? 5. I had circle radius 1, 4 units My parameterization was r(t) = cos(t)i +sin(t)j +4k for 0<t<2Pi F.dr = -sin^2(t) + 8cos(t) 6. x^2 + y^2 +z^2 = 25 if z = 4 x^2 + y^2 = 9 Which gives x=3cos(t) y = 3sin(t) z = 4 r ' = -3sin(t) i +3cos(t) j F = 3sin(t) i - 4 j + 12cos(t) k As in the attachment. you then end up with -9pi which is the - 28.274 in the attachment 7. How did you go about doing the first integral, ie "we obtain using the parameterization". In the word document i meant 8. The first integral is the integral curl FN Then using the pararmaterization for the bounding curve we obtain the second integral of F*dr/dt	{"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": 0.9907156825065613, "perplexity": 1849.413792762212}, "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/1432207930866.66/warc/CC-MAIN-20150521113210-00252-ip-10-180-206-219.ec2.internal.warc.gz"}
http://www.mathworks.com/help/optim/examples/analyzing-the-effect-of-uncertainty-using-semi-infinite-programming.html?prodcode=OP&language=en&requestedDomain=www.mathworks.com&nocookie=true	Accelerating the pace of engineering and science # Optimization Toolbox ## Analyzing the Effect of Uncertainty Using Semi-Infinite Programming This example shows how to use semi-infinite programming to investigate the effect of uncertainty in the model parameters of an optimization problem. We will formulate and solve an optimization problem using the function fseminf, a semi-infinite programming solver in Optimization Toolbox™. The problem illustrated in this example involves the control of air pollution. Specifically, a set of chimney stacks are to be built in a given geographic area. As the height of each chimney stack increases, the ground level concentration of pollutants from the stack decreases. However, the construction cost of each chimney stack increases with height. We will solve a problem to minimize the cumulative height of the chimney stacks, hence construction cost, subject to ground level pollution concentration not exceeding a legislated limit. This problem is outlined in the following reference: Air pollution control with semi-infinite programming, A.I.F. Vaz and E.C. Ferreira, XXVIII Congreso Nacional de Estadistica e Investigacion Operativa, October 2004 In this example we will first solve the problem published in the above article as the Minimal Stack Height problem. The models in this problem are dependent on several parameters, two of which are wind speed and direction. All model parameters are assumed to be known exactly in the first solution of the problem. We then extend the original problem by allowing the wind speed and direction parameters to vary within given ranges. This will allow us to analyze the effects of uncertainty in these parameters on the optimal solution to this problem. ### Minimal Stack Height Problem Consider a 20km-by-20km region, R, in which ten chimney stacks are to be placed. These chimney stacks release several pollutants into the atmosphere, one of which is sulfur dioxide. The x, y locations of the stacks are fixed, but the height of the stacks can vary. Constructors of the stacks would like to minimize the total height of the stacks, thus minimizing construction costs. However, this is balanced by the conflicting requirement that the concentration of sulfur dioxide at any point on the ground in the region R must not exceed the legislated maximum. First, let's plot the chimney stacks at their initial height. Note that we have zoomed in on a 4km-by-4km subregion of R which contains the chimney stacks. h0 = [210;210;180;180;150;150;120;120;90;90]; plotChimneyStacks(h0, 'Chimney Stack Initial Height'); There are two environment related parameters in this problem, the wind speed and direction. Later in this example we will allow these parameters to vary, but for the first problem we will set these parameters to typical values. % Wind direction in radians theta0 = 3.996; % Wind speed in m/s U0 = 5.64; Now let's plot the ground level concentration of sulfur dioxide (SO2) over the entire region R (remember that the plot of chimney stacks was over a smaller region). The SO2 concentration has been calculated with the chimney stacks set to their initial heights. We can see that the concentration of SO2 varies over the region of interest. There are two features of the Sulfur Dioxide graph of note: • SO2 concentration rises in the top left hand corner of the (x,y) plane • SO2 concentration is approximately zero throughout most of the region In very simple terms, the first feature is due to the prevailing wind, which is blowing SO2 toward the top left hand corner of the (x,y) plane in this example. The second factor is due to SO2 being transported to the ground via diffusion. This is a slower process compared to the prevailing wind and thus SO2 only reaches ground level in the top left hand corner of the region of interest. For a more detailed discussion of atmospheric dispersion from chimney stacks, consult the reference cited in the introduction. The pink plane indicates a SO2 concentration of . This is the legislated maximum for which the Sulfur Dioxide concentration must not exceed in the region R. It can be clearly seen from the graph that the SO2 concentration exceeds the maximum for the initial chimney stack height. Examine the MATLAB file concSulfurDioxide to see how the sulfur dioxide concentration is calculated. plotSulfurDioxide(h0, theta0, U0, ... 'Sulfur Dioxide Concentration at Initial Stack Height'); ### How fseminf Works Before we solve the minimal stack height problem, we will outline how fseminf solves a semi-infinite problem. A general semi-infinite programming problem can be stated as: such that (Linear inequality constraints) (Linear equality constraints) (Nonlinear Inequality Constraints) (Nonlinear Equality Constraints) (Bounds) and , where for (Nonlinear semi-infinite constraints) This algorithm allows you to specify constraints for a nonlinear optimization problem that must be satisfied over intervals of an auxiliary variable, . Note that for fseminf, this variable is restricted to be either 1 or 2 dimensional for each semi-infinite constraint. The function fseminf solves the general semi-infinite problem by starting from an initial value, , and using an iterative procedure to obtain an optimum solution, . The key component of the algorithm is the handling of the "semi-infinite" constraints, . At it is required that the must be feasible at every value of in the interval . This constraint can be simplified by considering all the local maxima of with respect to in the interval . The original constraint is equivalent to requiring that the value of at each of the above local maxima is feasible. fseminf calculates an approximation to all the local maximum values of each semi-infinite constraint, . To do this, fseminf first calculates each semi-infinite constraint over a mesh of values. A simple differencing scheme is then used to calculate all the local maximum values of from the evaluated semi-infinite constraint. As we will see later, you create this mesh in your constraint function. The spacing you should use for each coordinate of the mesh is supplied to your constraint function by fseminf. At each iteration of the algorithm, the following steps are performed: 1. Evaluate over a mesh of -values using the current mesh spacing for each -coordinate. 2. Calculate an approximation to all the local maximum values of using the evaluation of from step 1. 3. Replace each in the general semi-infinite problem with the set of local maximum values found in steps 1-2. The problem now has a finite number of nonlinear constraints. fseminf uses the SQP algorithm used by fmincon to take one iteration step of the modified problem. 4. Check if any of the SQP algorithm's stopping criteria are met at the new point . If any criteria are met the algorithm terminates; if not, fseminf continues to step 5. For example, if the first order optimality value for the problem defined in step 3 is less than the specified tolerance then fseminf will terminate. 5. Update the mesh spacing used in the evaluation of the semi-infinite constraints in step 1. ### Writing the Nonlinear Constraint Function Before we can call fseminf to solve the problem, we need to write a function to evaluate the nonlinear constraints in this problem. The constraint to be implemented is that the ground level Sulfur Dioxide concentration must not exceed at every point in region R. This is a semi-infinite constraint, and the implementation of the constraint function is explained in this section. For the minimal stack height problem we have implemented the constraint in the MATLAB file airPollutionCon. type airPollutionCon.m function [c, ceq, K, s] = airPollutionCon(h, s, theta, U) %AIRPOLLUTIONCON Constraint function for air pollution demo % % [C, CEQ, K, S] = AIRPOLLUTIONCON(H, S, THETA, U) calculates the % constraints for the air pollution Optimization Toolbox (TM) demo. This % function first creates a grid of (X, Y) points using the supplied grid % spacing, S. The following constraint is then calculated over each point % of the grid: % % Sulfur Dioxide concentration at the specified wind direction, THETA and % wind speed U <= 1.25e-4 g/m^3 % % See also AIRPOLLUTION % Copyright 2008 The MathWorks, Inc. % Initial sampling interval if nargin < 2 \|\| isnan(s(1,1)) s = [1000 4000]; end % Define the grid that the "infinite" constraints will be evaluated over w1x = -20000:s(1,1):20000; w1y = -20000:s(1,2):20000; [t1,t2] = meshgrid(w1x,w1y); % Maximum allowed sulphur dioxide maxsul = 1.25e-4; % Calculate the constraint over the grid K = concSulfurDioxide(t1, t2, h, theta, U) - maxsul; % Rescale constraint to make it 0(1) K = 1e4K; % No finite constraints c = []; ceq = []; This function illustrates the general structure of a constraint function for a semi-infinite programming problem. In particular, a constraint function for fseminf can be broken up into three parts: 1. Define the initial mesh size for the constraint evaluation Recall that fseminf evaluates the "semi-infinite" constraints over a mesh as part of the overall calculation of these constraints. When your constraint function is called by fseminf, the mesh spacing you should use is supplied to your function. Fseminf will initially call your constraint function with the mesh spacing, s, set to NaN. This allows you to initialize the mesh size for the constraint evaluation. Here, we have one "infinite" constraint in two "infinite" variables. This means we need to initialize the mesh size to a 1-by-2 matrix, in this case, s = [1000 4000]. 2. Define the mesh that will be used for the constraint evaluation A mesh that will be used for the constraint evaluation needs to be created. The three lines of code following the comment "Define the grid that the "infinite" constraints will be evaluated over" in airPollutionCon can be modified for most 2-d semi-infinite programming problems. 3. Calculate the constraints over the mesh Once the mesh has been defined, the constraints can be calculated over it. These constraints are then returned to fseminf from the above constraint function. Note that in this problem, we have also rescaled the constraints so that they vary on a scale which is closer to that of the objective function. This helps fseminf to avoid scaling issues associated with objectives and constraints which vary on disparate scales. ### Solve the Optimization Problem We can now call fseminf to solve the problem. The chimney stacks must all be at least 10m tall and we use the initial stack height specified earlier. Note that the third input argument to fseminf below (1) indicates that there is only one semi-infinite constraint. lb = 10ones(size(h0)); [hsopt, sumh, exitflag] = fseminf(@(h)sum(h), h0, 1, ... @(h,s) airPollutionCon(h,s,theta0,U0), [], [], [], [], lb); fprintf('\nMinimum computed cumulative height of chimney stacks : %7.2f m\n', sumh); Local minimum possible. Constraints satisfied. fseminf stopped because the predicted change in the objective function is less than the default value of the function tolerance and constraints are satisfied to within the default value of the constraint tolerance. Minimum computed cumulative height of chimney stacks : 3667.19 m The minimum cumulative height computed by fseminf is considerably higher than the initial total height of the chimney stacks. We will see how the minimum cumulative height changes when parameter uncertainty is added to the problem later in the example. For now, let's plot the chimney stacks at their optimal height. Examine the MATLAB file plotChimneyStacks to see how the plot was generated. plotChimneyStacks(hsopt, 'Chimney Stack Optimal Height'); ### Check the Optimization Results Recall that fseminf determines that the semi-infinite constraint is satisfied everywhere by ensuring that discretized maxima of the constraint are below the specified bound. We can verify that the semi-infinite constraint is satisfied everywhere by plotting the ground level sulfur dioxide concentration for the optimal stack height. Note that the sulfur dioxide concentration takes its maximum possible value in the upper left corner of the (x, y) plane, i.e. at x = -20000m, y = 20000m. This point is marked by the blue dot in the figure below and verified by calculating the sulfur dioxide concentration at this point. Examine the MATLAB file plotSulfurDioxide to see how the plots was generated. titleStr = 'Optimal Sulfur Dioxide Concentration and its maximum (blue)'; xMaxSD = [-20000 20000]; plotSulfurDioxide(hsopt, theta0, U0, titleStr, xMaxSD); SO2Max = concSulfurDioxide(-20000, 20000, hsopt, theta0, U0); fprintf('Sulfur Dioxide Concentration at x = -20000m, y = 20000m : %e g/m^3\n', SO2Max); Sulfur Dioxide Concentration at x = -20000m, y = 20000m : 1.250000e-04 g/m^3 ### Considering Uncertainty in the Environmental Factors The sulfur dioxide concentration depends on several environmental factors which were held at fixed values in the above problem. Two of the environmental factors are wind speed and wind direction. See the reference cited in the introduction for a more detailed discussion of all the problem parameters. We can investigate the change in behavior for the system with respect to the wind speed and direction. In this section of the example, we want to make sure that the sulfur dioxide limits are satisfied even if the wind direction changes from 3.82 rad to 4.18 rad and mean wind speed varies between 5 and 6.2 m/s. We need to implement a semi-infinite constraint to ensure that the sulfur dioxide concentration does not exceed the limit in region R. This constraint is required to be feasible for all pairs of wind speed and direction. Such a constraint will have four "infinite" variables (wind speed and direction and the x-y coordinates of the ground). However, any semi-infinite constraint supplied to fseminf can have no more than two "infinite" variables. To implement this constraint in a suitable form for fseminf, we recall the SO2 concentration at the optimum stack height in the previous problem. In particular, the SO2 concentration takes its maximum possible value at x = -20000m, y = 20000m. To reduce the number of "infinite" variables, we will assume that the SO2 concentration will also take its maximum value at this point when uncertainty is present. We then require that SO2 concentration at this point is below for all pairs of wind speed and direction. This means that the "infinite" variables for this problem are wind speed and direction. To see how this constraint has been implemented, inspect the MATLAB file uncertainAirPollutionCon. type uncertainAirPollutionCon.m function [c, ceq, K, s] = uncertainAirPollutionCon(h, s) %UNCERTAINAIRPOLLUTIONCON Constraint function for air pollution demo % % [C, CEQ, K, S] = UNCERTAINAIRPOLLUTIONCON(H, S) calculates the % constraints for the fseminf Optimization Toolbox (TM) demo. This % function first creates a grid of wind speed/direction points using the % supplied grid spacing, S. The following constraint is then calculated % over each point of the grid: % % Sulfur Dioxide concentration at x = -20000m, y = 20000m <= 1.25e-4 % g/m^3 % % See also AIRPOLLUTIONCON, AIRPOLLUTION % Copyright 2008 The MathWorks, Inc. % Maximum allowed sulphur dioxide maxsul = 1.25e-4; % Initial sampling interval if nargin < 2 \|\| isnan(s(1,1)) s = [0.02 0.04]; end % Define the grid that the "infinite" constraints will be evaluated over w1x = 3.82:s(1,1):4.18; % Wind direction w1y = 5.0:s(1,2):6.2; % Wind speed [t1,t2] = meshgrid(w1x,w1y); % We assume the maximum SO2 concentration is at [x, y] = [-20000, 20000] % for all wind speed/direction pairs. We evaluate the SO2 constraint over % the [theta, U] grid at this point. K = concSulfurDioxide(-20000, 20000, h, t1, t2) - maxsul; % Rescale constraint to make it 0(1) K = 1e4*K; % No finite constraints c = []; ceq = []; This constraint function can be divided into same three sections as before: 1. Define the initial mesh size for the constraint evaluation The code following the comment "Initial sampling interval" initializes the mesh size. 2. Define the mesh that will be used for the constraint evaluation The next section of code creates a mesh (now in wind speed and direction) using a similar construction to that used in the initial problem. 3. Calculate the constraints over the mesh The remainder of the code calculates the SO2 concentration at each point of the wind speed/direction mesh. These constraints are then returned to fseminf from the above constraint function. We can now call fseminf to solve the stack height problem considering uncertainty in the environmental factors. [hsopt2, sumh2, exitflag2] = fseminf(@(h)sum(h), h0, 1, ... @uncertainAirPollutionCon, [], [], [], [], lb); fprintf('\nMinimal computed cumulative height of chimney stacks with uncertainty: %7.2f m\n', sumh2); Local minimum possible. Constraints satisfied. fseminf stopped because the predicted change in the objective function is less than the default value of the function tolerance and constraints are satisfied to within the default value of the constraint tolerance. Minimal computed cumulative height of chimney stacks with uncertainty: 3812.15 m We can now look at the difference between the minimum computed cumulative stack height for the problem with and without parameter uncertainty. You should be able to see that the minimum cumulative height increases when uncertainty is added to the problem. This expected increase in height allows the SO2 concentration to remain below the legislated maximum for all wind speed/direction pairs in the specified range. We can check that the sulfur dioxide concentration does not exceed the limit over the region of interest via inspection of a sulfur dioxide plot. For a given (x, y) point, we plot the maximum SO2 concentration for the wind speed and direction in the stated ranges. Note that we have zoomed in on the upper left corner of the X-Y plane. titleStr = 'Optimal Sulfur Dioxide Concentration under Uncertainty'; thetaRange = 3.82:0.02:4.18; URange = 5:0.2:6.2; XRange = [-20000,-15000]; YRange = [15000,20000]; plotSulfurDioxideUncertain(hsopt2, thetaRange, URange, XRange, YRange, titleStr); We finally plot the chimney stacks at their optimal height when there is uncertainty in the problem definition. plotChimneyStacks(hsopt2, 'Chimney Stack Optimal Height under Uncertainty'); There are many options available for the semi-infinite programming algorithm, fseminf. Consult the Optimization Toolbox™ User's Guide for details, in the Using Optimization Toolbox Solvers chapter, under Constrained Nonlinear Optimization: fseminf Problem Formulation and Algorithm.	{"extraction_info": {"found_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": 0.9115074276924133, "perplexity": 1236.892918556407}, "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-2016-07/segments/1454701174607.44/warc/CC-MAIN-20160205193934-00270-ip-10-236-182-209.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/167498/is-this-function-decreasing-on-0-1	# Is this function decreasing on $(0,1)$? While doing some research I got stuck trying to prove that the following function is decreasing $$f(k):= k K(k) \sinh \left(\frac{\pi}{2} \frac{K(\sqrt{1-k^2})}{K(k)}\right)$$ for $k \in (0,1)$. Here $K$ is the Complete elliptic integral of the first kind, defined by $$K(k):= \int_{0}^{1} \frac{dt}{\sqrt{1-t^2} \sqrt{1-k^2t^2}}.$$ This seems to be true, as the graph below suggests : I really don't know much about elliptic integrals, so perhaps someone here can give some insight. Any relevant reference on elliptic integrals of the first kind is welcome. Thank you, Malik EDIT (2012-07-09) : Using J.M.'s suggestion to rewrite the function $f(k)$ as $$f(k) = kK(k) \frac{1-q(k)}{2 \sqrt{q(k)}}$$ and using the derivative formulas $$K'(k) = \frac{E(k)}{k(1-k^2)} - \frac{K(k)}{k},$$ $$q'(k)=\frac{\pi^2}{2} \frac{q(k)} { K(k)^2 (1-k^2)k}$$ where $E(k)$ is the Complete elliptic integral of the second kind, I was able to calculate $f'(k)$ and reduce the problem to showing that the following function is negative for $k \in (0,1)$ : $$g(k):= 4(1-q(k))K(k)E(k) - \pi^2 (1+q(k)).$$ Below is the graph of $g$ obtained with Maple : EDIT (19-07-2012) I asked the question on MathOverflow! - At least $$f(k) = \pi - \frac{\pi}{16} k^{2} - \frac{3 \pi}{128} k^{4} - \frac{27 \pi}{2048} k^{6} - \frac{575 \pi}{65536} k^{8} + \operatorname{O} \bigl(k^{10}\bigr),$$ as $k \to 0+$, so it is decreasing near $k=0$. – GEdgar Jul 6 '12 at 15:31 Note that your function can also be expressed in terms of the elliptic nome: $$k\,K(k)\,\frac{1-q(k)}{2\sqrt{q(k)}}$$ – J. M. Jul 8 '12 at 14:04 The following comment was posted by Henry Cohn on meta.MO: It's definitely possible to prove that your function is decreasing by an ugly and unilluminating calculation that shows that the derivative is nonpositive everywhere. Specifically, near $k=0$ you can compute the Taylor series expansion and bound the error. For larger $k$, you can check the values of the derivative at a bunch of points and verify that there are no sign changes in between by bounding the second derivative. So if you just need this result to get a rigorous proof of some theorem, then it will be doable... – Dan Petersen Jul 12 '12 at 6:15 On the other hand, much more seems to be true. Specifically, all the derivatives seem to be negative, not just the first derivative. You can see this in the Taylor series expansion, which has all negative coefficients beyond the constant term (well, it's an even function, so the odd terms vanish, but the even terms all have negative coefficients). And the terms are pretty nice: the coefficient of $k^{2i}$ seems to be pi times a rational number with denominator dividing $16^i$. – Dan Petersen Jul 12 '12 at 6:16 I don't know how to prove any of this, but it's more remarkable than just being a decreasing function, and all this suggests that there should be a nice way of understanding this function. Plenty of functions are decreasing for no especially good reason, but this sort of absolute monotonicity is much less common. – Dan Petersen Jul 12 '12 at 6:16 ## 2 Answers A few more terms for those investigating. From Maple. These coefficients are not listed in the On-line Encyclopedia of Integer Sequences. $$\frac{4}{\pi} \sqrt{m} \;K(4 \sqrt{m}) \sinh \biggl(\frac{\pi\; K(\sqrt{1 - 16 m})}{2\;K(4 \sqrt{m})}\biggr) \\ = 1 - m - 6 m^{2} - 54 m^{3} - 575 m^{4} - 6715 m^{5} - 83134 m^{6} - 1071482 m^{7} - \\ \quad{}\quad{} 14221974 m^{8} - 193050435 m^{9} - 2667157340 m^{10} - 37378279402 m^{11} - \\ \quad{}\quad{} 530024062361 m^{12} - 7590192561912 m^{13} - \\ \quad{}\quad{}109610113457650 m^{14} - 1594344146568120 m^{15} - \\ \quad{}\quad{}23336667998911128 m^{16} - 343468859344118109 m^{17} - \\ \quad{}\quad{}5079858166426507168 m^{18} - 75457168334744888190 m^{19} - \\ \quad{}\quad{}1125223725054635766392 m^{20} + \operatorname{O} \bigl(m^{21}\bigr)$$ added Who knows if this is relevant? See A002849 $$\frac{2}{\pi}K(4\sqrt{m}) = 1+4m+36m^2+400m^3+4900m^4+\dots =\sum_{n=0}^\infty \binom{2n}{n}^2m^n$$ - Thank you. I have absolutely no idea why once you divide $f$ by $\pi$ and make the change of variable $m=(k/4)^2$, all the coefficients seem to be negative integers... That's very interesting! – Kalim Jul 13 '12 at 14:42 Note that the last formula can also be found in the wikipedia link for elliptic integrals given in the question. – Kalim Jul 19 '12 at 19:13 See the developments here. It seems all that is left is (reasonable) numerical work. - thanks for the info! – Kalim Jul 23 '12 at 20:40	{"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": 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": 0.9048982262611389, "perplexity": 208.54135703427013}, "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-2016-22/segments/1464049278887.13/warc/CC-MAIN-20160524002118-00051-ip-10-185-217-139.ec2.internal.warc.gz"}
https://indico.nucleares.unam.mx/event/1488/contributions	# 10th International Workshop on Charm Physics (CHARM 2020) from 31 May 2021 to 4 June 2021 Mexico/General timezone Thanks for your contribution to the success of this workshop!! Home > Contribution List ## Contribution List Displaying 203 contributions out of 203 Session: In media Presented by Mr. Kaifeng SHEN on 3 Jun 2021 at 11:40 Heavy quarkonia are ideal probes of the Quark-Gluon Plasma (QGP). $J/\psi$ is the most abundantly produced quarkonium state accessible experimentally and its suppression due to the color screening effect in hot and dense medium has been suggested as a signature of the formation of the QGP. Besides the screening effect, there are other mechanisms, such as the cold nuclear effects and charm quark re ... More Presented by Mr. Kaifeng SHEN Understanding the nature of the hidden charm pentaquark(like) signals in the LHCb data for $\Lambda_b^0\to J/\psi p K^-$ is a central problem of hadron spectroscopy. We propose a scenario completely different from previous ones such as hadron molecules and compact pentaquarks. We identify relevant double triangle mechanisms with leading or lower-order singularities. The associated anomalous thre ... More Presented by Dr. Satoshi NAKAMURA Presented by Dr. Satoshi NAKAMURA on 3 Jun 2021 at 11:40 Presented by Mr. William PARROTT on 4 Jun 2021 at 13:10 Semileptonic $D \to{}K \ell \nu$ decays provide one angle of attack to get at the CKM matrix element $V_{cs}$, complementary to the study of leptonic $D_s$ decays. Here we present the results of an improved determination of $V_{cs}$, recently released on the arXiv (2104.09883). With a new, precise determination of $D\to K$ scalar and vector form factors from lattice QCD, combined with experime ... More Presented by Mr. William PARROTT We present in full analytic form the partial widths for the lepton flavour violating decays $L^\pm \to \ell^\pm \ell'^{+} \ell'^{-}$, with $L = \tau, \mu$ and $\ell(') = \mu, e$, mediated by neutrino oscillations in the one-loop diagrams. Compared to the first result by Petcov:1976ff, which was obtained in the nonphysical zero momentum limit $\mathcal{P} \ll m_{\nu} \ll M_W$, we retain full d ... More Presented by Patrick BLACKSTONE Session: Taus Presented by Patrick BLACKSTONE on 3 Jun 2021 at 12:00 We report on a precision measurement of the ratio $R_{\tau\mu} = BF(\Upsilon(3S)\to\tau^+\tau^-)/BF(\Upsilon(3S)\to\mu^+\mu^-)$ using data collected with the BABAR detector at the SLAC PEP-II $e^+e^-$ collider. The measurement is based on a 28 ${\mathrm{fb^-1}}$ data sample collected at a center-of-mass energy of 10.355 ${\mathrm{GeV}}/c^2$ which corresponds to a sample 122 million $\Upsilon(3S)$ ... More Presented by Prof. Swagato BANERJEE Session: Taus Presented by Dr. Swagato BANERJEE on 3 Jun 2021 at 11:40 Session: Exotics Presented by Prof. Tomasz SKWARNICKI on 31 May 2021 at 09:20 The suppressed helicity flip amplitude in baryon-antibaryon decays of $J/\psi$ is calculated within the effective field theory framework. It is shown that at the leading-order approximation this contribution is factorisable and the overlap with the hadronic final state can be described by collinear matrix elements. The obtained contribution depends on the nucleon light-cone distribution amplitu ... More Presented by Nikolay KIVEL Presented by Dr. Nikolay KIVEL on 4 Jun 2021 at 12:30 Session: Production We perform a combined study of $D^+ \to K^-\pi^+\pi^+$ and $D_s \to \pi^- K^+ K^+$ decays using the naive factorization approach. The formalism allows for a description of such decays in terms of the well-known vector- and scalar- $K\pi$ form factors as well as those appearing in semileptonic $D^+ \to K^-\pi^+ \ell^+\nu$ decays. We propose a useful—yet simple—parametrization to describe the l ... More Presented by Dr. Pablo SANCHEZ-PUERTAS Presented by Dr. Pablo SANCHEZ-PUERTAS on 1 Jun 2021 at 12:05 Presented by Oleg MESHKOV on 3 Jun 2021 at 12:40 The ATLAS experiment has performed accurate measurements of mixing and CP violation in the neutral B mesons,and also of rare neutral B-meson decays proceeding via electroweak FCNC-suppressed processes. This talk will focus on the latest results from ATLAS, such as rate measurements of B^0_s → mu mu and B^0 → mu mu decays; and CP violation in B_s^0 —> J/psi phi decays. In the latter, the Sta ... More Presented by Yuji ENARI Session: Production Presented by Zijun XU on 3 Jun 2021 at 12:40 Recent results from the ATLAS experiment on the charmonium production, B_c production and exotic heavy hadrons will be presented. The measurement of J/psi and psi(2S) differential cross sections at large transverse momentum values in proton-proton collisions at 13 TeV will be reported. The measurement of the ratios of the B_c+ and B+ production cross sections in proton-proton collisions at 8 TeV ... More Presented by Yuji ENARI Session: Exotics Presented by Dr. Alessandro PILLONI on 31 May 2021 at 10:20 Session: Taus Presented by Mr. Alejandro DE YTA HERNÁNDEZ on 3 Jun 2021 at 13:20 Presented by Prof. Bruno EL-BENNICH on 1 Jun 2021 at 12:30 I will discuss theoretical continuum approaches to bound-state calculations, strong and weak decays and effective couplings related to D mesons. The associated wave functions, light-front distribution amplitudes and hadronic matrix elements are expressions of nonperturbative QCD. After motivating their origin in QCD factorization and heavy quark effective theory, we retrace their evolution from ea ... More Presented by Prof. Bruno EL-BENNICH Presented by Dr. Ayan PAUL on 4 Jun 2021 at 08:30 Presented by Angelo CARBONE on 4 Jun 2021 at 07:00 Presented by Carlos RAMIREZ on 3 Jun 2021 at 13:20 The physics of CPV and oscillations in the charn sector is reviewed. Topics like the recent discover of direct CPV, the latest measurements, the respective SM predictions and the present absence of new physics in this sector. Presented by Carlos RAMIREZ Presented by Dr. Tatiana KHARLAMOVA on 1 Jun 2021 at 08:30 Presented by Dr. Stefan SCHACHT on 4 Jun 2021 at 09:00 Presented by Prof. Xiaoyan SHEN on 1 Jun 2021 at 07:30 Presented by Prof. Roy BRIERE on 31 May 2021 at 11:40 Presented by Jhovanny MEJIA on 31 May 2021 at 13:10 The production cross sections of open heavy-flavour hadrons can be obtained by the factorisation approach described as the convolution of the initial parton distribution functions of the incoming partons, the perturbative QCD partonic cross section, and the fragmentation functions for the hadronisation parametrised from measurements in $\mathrm{e^+e^-}$ collisions. Recent measurements of charm-ba ... More Presented by Ms. Jinjoo SEO Presented by Ms. Jinjoo SEO on 1 Jun 2021 at 12:20 In this contribution, we discuss the internal structure of heavy baryons and the different interpretations of the internal structure of hidden-charm pentaquarks. We present the hidden-charm pentaquarks as superpositions of meson-baryon channels coupled to a $uudc\bar{c}$ compact core by employing an interaction satisfying the heavy quark and chiral symmetries. Our model can reproduce the masses a ... More Presented by Dr. Hugo GARCIA TECOCOATZI Presented by Dr. Hugo GARCIA TECOCOATZI The LHCb experiment collected the world's largest sample of charmed hadrons during LHC Run 1 and Run 2. With this data set, LHCb is currently providing the world's most precise measurements of properties and production of known charmed baryons, as well as discovering many previously unobserved states. The latest results from the LHCb Collaboration on charmed baryons are presented Presented by Dr. Dana BOBULSKA Presented by Dr. Dana BOBULSKA on 3 Jun 2021 at 12:20 Presented by Dr. Mikhasenko MIKHAIL on 1 Jun 2021 at 10:50 Presented by Prof. Pol GOSSIAUX on 3 Jun 2021 at 09:20 Session: Production Presented by Mr. Antonio PALASCIANO on 3 Jun 2021 at 13:20 Experimental measurements of the charm quark showering processes are an important test for the current understanding of QCD. At the same time, in heavy-ion collisions, charm quarks represent ideal probes for studying the Quark-Gluon Plasma (QGP), being them produced in the very early stages of the collision from a hard-parton scattering. In particular, angular correlation measurements are sensib ... More Presented by Mr. Antonio PALASCIANO Presented by Dr. Christopher THOMAS on 2 Jun 2021 at 10:20 The measurement of charm production in hadronic collisions provides a powerful tool to understand QCD due to its creation in initial parton-parton interactions, with the heavy quark mass providing the hard scale for the process. At the LHC the measurement of the correlation of the produced $\rm c\bar{c}$ quark pairs shows sensitivity to the production mechanisms at hand. The large branching rati ... More Presented by Mr. Horst Sebastian SCHEID Session: Production Presented by Mr. Horst Sebastian SCHEID on 3 Jun 2021 at 12:00 Charm quarks, created during an early stage of the heavy-ion collision via hard scattering, have a large thermalization time within quark-gluon plasma (QGP) due to their large mass. They witness the entire evolution of QGP and hence can be used as an effective probe to study the strongly interacting matter. We studied the effect of collision and gluon radiation by charm quark on its transport coef ... More Session: In media Presented by Ms. Adiba SHAIKH on 3 Jun 2021 at 12:00 Presented by Dr. Melissa Maria CRUZ TORRES on 2 Jun 2021 at 07:30 Session: NP in charm Presented by Mr. Jitendra KUMAR on 4 Jun 2021 at 12:10 The Belle II experiment at the asymmetric $e^+e^-$ collider, SuperKEKB, aims to record 50 ab$^{-1}$ of data over the next decade, a factor of 50 more than Belle. During the first 1.5 years of operations, around 90 fb$^{-1}$ of data were collected. This dataset is used to measure the lifetimes of a few charm hadrons, confirming the expected performance of the Belle II detector, in particular the v ... More Presented by Dr. Jitendra KUMAR Presented by Frank NERLING on 31 May 2021 at 12:10 Presented by Prof. Xiao-Rui LIU on 3 Jun 2021 at 07:00 Presented by Dr. Marco Antonio BEDOLLA on 3 Jun 2021 at 13:00 Presented by Dr. Roberto MUSSA on 2 Jun 2021 at 09:50 Presented by Prof. Enrico SCOMPARIN on 3 Jun 2021 at 10:20 Results for the $\eta_c$- and $J/\Psi$-nucleus bound state energies for various nuclei are presented. These results are obtained using effective Lagrangians at the hadronic level. Essential input for the calculation, namely the medium-modified $D$ and $D^{}$ meson masses, as well as the density distributions in nuclei, are calculated within the quark-meson coupling model. The attractive potent ... More Presented by Dr. Javier COBOS-MARTINEZ Session: In media Presented by Dr. Javier COBOS-MARTINEZ on 3 Jun 2021 at 13:40 Session: Production Charmonium production studies in hadronic collisions are a powerful tool for improving our understanding of QCD, the theory of the strong interaction. The production of the charm-quark pair can be described within perturbative QCD, whereas the evolution of this pair into a colorless bound state involves soft scale processes. In addition, multiplicity dependent studies of charmonia in both proton- ... More Presented by Mr. Jon-Are SÆTRE Session: Production Presented by Mr. Jon-are SÆTRE on 1 Jun 2021 at 12:20 Presented by Dr. Geoffrey BODWIN on 2 Jun 2021 at 11:40 I will present results from the first lattice QCD+QED computations of the properties of ground-state charmonium mesons. These calculations include the effect of up, down, strange and charm quarks in the sea and cover a wide range of values of the lattice spacing enabling very accurate results in the continuum limit (with physical quark masses). We tune the charm quark's mass so that the mass of th ... More Presented by Prof. Christine DAVIES Presented by Prof. Christine DAVIES on 3 Jun 2021 at 12:00 A nonperturbative charm production contribution, known as intrinsic charm, has long been speculated but has never been satisfactorily proven. The SeaQuest experiment at FNAL is in an ideal kinematic region to provide evidence of $J/\psi$ production by intrinsic charm. Here, $J/\psi$ production in the SeaQuest kinematics is calculated with a combination of perturbative QCD and intrinsic charm ... More Presented by Ramona VOGT Session: In media Presented by Ramona Vogt VOGT on 3 Jun 2021 at 12:40 Presented by Dr. Clara MURGUI on 3 Jun 2021 at 08:30 Presented by Dr. Patricia MAGALHAES on 1 Jun 2021 at 10:20 Presented by Prof. Alex KAGAN on 4 Jun 2021 at 07:30 LHCb has collected the world's largest sample of charmed hadrons. This sample is used to measure direct $CP$ violation in $D$ mesons and charmed baryons. New measurements from several decay modes are presented, as well as prospects for future sensitivities Presented by Mr. Lorenzo PICA Presented by Mr. Lorenzo PICA on 3 Jun 2021 at 12:00 Presented by Mr. Daniel Alejandro PÉREZ NAVARRO on 2 Jun 2021 at 10:50 We suggest an efective field theory based coupled-channel approach to exotic charged Z-states and demonstrate its potential at a combined analysis of the existing experimental data on the open- and hidden-flavour decays of the Upsilon(10860) resonance. As an important ingredient of the method a dispersive approach to dipion transitions to lower lying Upsilon-resonances is developed with all imagin ... More Presented by Dr. Alexey NEFEDIEV Presented by Dr. Alexey NEFEDIEV on 1 Jun 2021 at 13:00 Session: Production Presented by Dr. Marco GIACALONE on 4 Jun 2021 at 12:50 Measurements of charm meson and baryon production in proton-proton collisions are an important test for perturbative QCD calculations. Measurements in p--Pb collisions provide important tools to disentagle cold nuclear matter effects (like nuclear modification of parton distribution functions). Furthermore, the study of the charm production in pp and p--Pb collisions as a function of multiplici ... More Presented by Dr. Marco GIACALONE We perform a global analysis of exclusive hadronic tau decays into one and two mesons using the low-energy limit of the Standard Model Effective Field Theory up to dimension six, assuming left-handed neutrinos. A controlled theoretical input on the Standard Model hadronic form factors, based on chiral symmetry, dispersion relations, data and asymptotic QCD properties, has allowed us to set bounds ... More Presented by Dr. Sergi GONZÀLEZ-SOLÍS Session: Taus Presented by Dr. Sergi GONZÀLEZ-SOLIS on 4 Jun 2021 at 12:50 Session: Production Presented by Dr. Martin HENTSCHINSKI on 3 Jun 2021 at 12:20 We investigate photo-production of vector mesons J/Psi and Upsilon measured both at HERA and LHC. We are interested in using this observable to distinguish between linear and non-linear QCD evolution at low x. The employed fits are based on non-linear Balitsky-Kovchegov evolution (Kutak-Sapeta gluon; KS) and next-to-leading order Balitsky-Fadin-Kuraev-Lipatov evolution (Hentschinski-Sabio Vera-Sal ... More Presented by Dr. Martin HENTSCHINSKI Presented by Prof. Eric BRAATEN on 31 May 2021 at 08:30 Presented by Dr. Xiaorong ZHOU on 1 Jun 2021 at 08:00 Session: In media Presented by Dr. Juan TORRES-RINCÓN on 3 Jun 2021 at 13:20 We study the spectroscopy and transport properties of charmed mesons in a thermal medium by applying an effective field theory based on chiral and heavy-quark symmetries in the imaginary time formalism. Relying on unitarity constraints and self-consistency we extract the in-medium properties (masses and widths) of D and Ds mesons and their interactions with light hadrons. We report our findings on ... More Presented by Juan TORRES-RINCON Presented by Dr. Jolanta BRODZICKA on 4 Jun 2021 at 10:50 Presented by YuLan FAN on 3 Jun 2021 at 11:40 BESIII has collected data samples corresponding to luminosities of 2.93 fb-1 and 3.19 fb-1 at center-of-mass energies of 3.773 and 4.178 GeV, respectively. The data set collected at 3.773 GeV contains quantum-correlated D0D0bar pairs that allow to access the phase differences between amplitudes. We report the measurements of strong phase differences in D0(-bar) decays, especially for K_S/Lpi+pi- ... More Presented by Jingzhi ZHANG Presented by Dr. Pere MASJUAN on 2 Jun 2021 at 08:00 Presented by Dr. Matteo FAEL on 2 Jun 2021 at 13:10 Presented by Prof. Ralf RAPP on 3 Jun 2021 at 09:50 Session: In media Presented by Peter VANDER GRIEND on 3 Jun 2021 at 13:00 Heavy quarks and their bound states are ideal probes of the quark gluon plasma formed in relativistic heavy ion collisions. Due to the hierarchy of scales of the system, the in medium dynamics can be modeled by a Langevin equation in which interactions between the medium and the heavy particle take the form of random "kicks" altering the particle's momentum. The hierarchy of scales makes the pro ... More Presented by Peter VANDER GRIEND Track: Charm meson and baryon spectroscopy (including exotica) We study the nature of the new signal reported by LHCb in the $J/\psi p$ spectrum. Based on the S-matrix principles, we perform a minimum-bias analysis of the underlying reaction amplitude, focusing on the analytic properties that can be related to the microscopic origin of the Pc(4312)+ peak. By exploring several amplitude parameterizations, we find evidence for the attractive effect of the $Σ_c ... More Presented by Mr. Jorge Antonio SILVA CASTRO Presented by Mr. Jorge Antonio SILVA CASTRO on 1 Jun 2021 at 12:40 In LEP times, hadronic tau decays were the most precise input for the (leading-order) hadronic vacuum polarization piece (HVP,LO) of the muon anomalous magnetic moment ($a_\mu$). With the advent of$\Phi$- and B-factories,$e^+e^-$hadronic cross-section surpassed them, giving the most accurate input for this piece. However, since both data-driven determinations are subject to theoretical uncertai ... More Presented by Mr. Jesús MIRANDA Session: Taus Presented by Mr. Jesús MIRANDA on 4 Jun 2021 at 12:30 Presented by Prof. Aida EL-KHADRA on 2 Jun 2021 at 12:40 We study the τ−→ντ π− π0 ℓ+ ℓ− (ℓ=e, μ) decays, which are O(α^2)-suppressed with respect to the dominant di-pion tau decay channel. Both the inner-bremsstrahlung and the structure- (and model-)dependent contributions are considered. In the ℓ=e case, structure-dependent effects are O(1%) in the decay rate, yielding a clean prediction of its branching ratio, 2.3×10^(−5), ... More Presented by Mr. Jorge Luis GUTIÉRREZ SANTIAGO Session: Taus Presented by Jorge GUTIÉRREZ SANTIAGO on 3 Jun 2021 at 12:40 Presented by Dr. Fernando SERNA on 4 Jun 2021 at 12:35 The ladder kernel of the Bethe-Salpeter equation is amended by introducing a different flavor dependence of the dressing functions in the heavy-quark sector. Compared with earlier work this allows for the simultaneous calculation of the mass spectrum and leptonic decay constants of light pseudoscalar mesons, the$D_u$,$D_s$,$B_u$,$B_s$and$B_c$mesons and the heavy quarkonia$\eta_c$and$\ ... More Presented by Dr. Fernando SERNA Presented by Prof. Shuangshi FANG on 2 Jun 2021 at 07:00 In the heavy-quark limit, the two heavy quarks in a doubly heavy baryon or a doubly heavy tetraquark are bound by their color-Coulomb potential into a compact diquark. The doubly heavy hadrons are related by the approximate heavy-quark--diquark symmetry of QCD to the heavy hadrons obtained by replacing the heavy diquark by a heavy antiquark. Effective field theories can be used to expand the masse ... More Presented by Mr. Abhishek MOHAPATRA Presented by Mr. Abhishek MOHAPATRA on 3 Jun 2021 at 12:40 Session: Production Presented by Dr. Renu BALA on 4 Jun 2021 at 12:10 In this contribution, the nuclear modification factor (RAA) and the elliptic flow (v2) of open heavy-flavour hadrons via their hadronic and semileptonic decays to electrons at mid-rapidity and to muons at forward rapidity in heavy-ion collisions will be presented. The measurements of the production of leptons from heavy-flavour hadron decays and the modification of their spectra in different colli ... More Presented by Dr. Renu BALA We study the transitions between the different color states of a static quark-antiquark pair, singlet and octet, in a thermal medium. This is done non-perturbatively exploiting the infinite mass limit of QCD. This study is interesting because it can be used for future developments within the framework of Effective Field Theories (EFTs) and because it can be combined with other techniques, like ... More Presented by Dr. Miguel Ángel ESCOBEDO ESPINOSA Session: In media Presented by Dr. Miguel Ángel ESCOBEDO on 3 Jun 2021 at 12:20 LHCb has collected the world's largest sample of charmed hadrons. This sample is used to measure $D^0 -\overline{D}^0$ mixing parameters and to search for $CP$ violation in mixing and interference. New measurements from several decay modes are presented, as well as prospects for future sensitivities. Presented by Dr. Guillaume PIETRZYK Presented by Dr. Guillaume PIETRZYK on 3 Jun 2021 at 13:00 The experimental information accumulated by the BABAR, Belle and LHCb experiments have shown disagreement on the measurements of the ratios $R(D)$ and $R(D^{})$, compared with the SM predictions. In addition, the $D^\ast$ longitudinal polarization $F_L(D^\ast)$ related with the channel $B \to D^\ast \tau \bar{\nu}_\tau$ observed by the Belle Collaboration and the ratio $R(J/\psi)$ measured by the ... More Presented by Dr. Nestor QUINTERO POVEDA Session: NP in charm Presented by Dr. Nestor QUINTERO POVEDA on 4 Jun 2021 at 13:10 We analyze the τ-→(Kπ)-ντ decays within an effective field theory description of heavy new physics (NP) modifying the SM left-handed weak charged current and include refined SM input for the participant meson form factors exploiting chiral symmetry, dispersion relations and (lattice) data. We include the leading dimension six operators and work at linear order in the effective couplings. Wit ... More Presented by Mr. Javier RENDON Session: Taus Presented by Mr. Javier RENDON on 4 Jun 2021 at 13:30 Rare $\vert \Delta c \vert=\vert \Delta u \vert=1$ processes complement flavor searches in the down-sector in a unique way. Semileptonic FCNC decays of charmed hadrons offer a large set of clean null test observables, such as CP-asymmetries, lepton-universality ratios, missing energy modes, lepton flavor violating modes and angular observables. In these observables any signal cleanly indicates Phy ... More Presented by Mr. Marcel GOLZ Session: NP in charm Presented by Mr. Marcel GOLZ on 4 Jun 2021 at 12:30 Presented by Mr. Hongrong QI psi(2S) provides good opportunities for the study of chi_cJ, eta_c, and h_c decays. These studies can be used to verify QCD based models, which provide predictions for the decay mechanism. With the world's largest sample of 4.4810^8, progress on the charmonium decays has been made. In the talk, we report the new results, such as the first measurement of the branching ratio of chi_c1,2 to mu+ mu- ... More Presented by Jingzhi ZHANG Presented by Barbara TRZECIAK on 3 Jun 2021 at 10:50 Presented by Dr. Marianna FONTANA on 3 Jun 2021 at 08:00 Presented by Prof. Jonas RADEMACKER on 4 Jun 2021 at 11:20 Session: Production Presented by Prof. Mikhail BARABANOV We compute masses of open and hidden charmed mesons in the framework of the extended Linear Sigma Model (eLSM) with (pseudo-)scalar and (axial-) vector mesons. Open charmed mesons masses turn out to be in quantitative agreement with experimental data. Whereas the masses of hidden charmed mesons, with the exception of $J/\psi$, are underpredicted by about $10$%. We calculate the (OZI-dominant) stro ... More Presented by Dr. Walaa ESHRAIM Session: Production Session: Production Presented by Dr. ESHRAIM WALAA on 4 Jun 2021 at 13:10 Session: Production Presented by Dr. Emmanuel ORTIZ-PACHECO on 4 Jun 2021 at 13:30 Presented by Andrzej KUPSC on 1 Jun 2021 at 09:50 The dependence of the production of the $X(3872)$ meson on the hadron multiplicity in $pp$ collisions has been used as evidence against $X$ being a charm-meson molecule. The argument is based in part on the incorrect assumption that the cross section for the breakup of $X$ by scattering with comovers can be approximated by a geometric cross section inversely proportional to the binding energy of $... More Presented by Mr. Kevin INGLES Session: Production Presented by Mr. Kevin INGLES on 1 Jun 2021 at 12:40 Presented by Prof. Khodjamirian ALEXANDER on 4 Jun 2021 at 08:00 Session: Production Presented by Krista SMITH on 1 Jun 2021 at 13:00 Suppression of the J/ψ nuclear modification factor has long been considered a signature of final state effects in large collision systems. In small systems, nuclear modification was assumed to be due to initial state cold nuclear matter effects, until the observation of strong differential suppression of the ψ(2S) state in p+A collisions suggested the presence of final state effects. Here we pre ... More Presented by Krista SMITH One of the best ways to understand hadronization in QCD is to study the production of quarkonium. However, the production mechanism of quarkonium is still uncertain. Spin-related measurements like the polarization are strong tests of production models. In this talk I will summarize recent results of quarkonium production and polarization in elementary collisions, including the results from NRQCD ... More Presented by Vincent CHEUNG Session: Production Presented by Vincent CHEUNG on 4 Jun 2021 at 12:30 Session: Production Presented by Dr. Vincent CHEUNG Presented by Prof. Antonio VAIRO on 2 Jun 2021 at 12:10 Presented by Dr. Isabella GARZIA on 4 Jun 2021 at 12:50 In this talk we present the latest results on radiative and rare/forbidden decays for D and D_s mesons from the BESIII experiment based on 2.92 fb$^{-ˆ’1}$and 3.19 fb$^{-1}$data samples taken at the center-of-mass energies 3.773 and 4.178 GeV, respectively. With the 4.178 GeV data, searches for the rare decay$D^+_S\to p\bar{p}e^+\nu_e$and rare radiative leptonic decay$D_S^+\to\gamma e^+\nu ... More Presented by Jingzhi ZHANG Presented by Mr. Nico ADOLPH on 4 Jun 2021 at 13:30 We compute radiative three-body decays of charmed mesons $D \to P P \gamma$ , $P=\pi, K$, in leading order QCDF, HH$\chi$PT and the soft photon approximation. We work out decay distributions and asymmetries in the standard model and with new physics in the electromagnetic dipole operators. The forward-backward asymmetry is suitable to probe the QCD frameworks, in particular the $s$-channel depende ... More We report the observation of the rare charm decay $D^0 \to K^-\pi^+e^+e^-$, a search for nine lepton-number-violating and three lepton-flavor-violating neutral charm decays of the type $D^0 \rightarrow h^- h^{'-} \ell^+ \ell^{'+}$, and $D^0 \rightarrow h^- h^{'+} \ell^+ \ell^{'-}$, and a search for seven lepton-number-violating decays of the type $D^{0}\rightarrow X^{0} e^{\pm} \mu^{\mp}$, where $... More Presented by Dr. David N. BROWN Session: NP in charm Presented by Dr. David BROWN on 4 Jun 2021 at 12:50 Presented by Prof. Svetlana FAJFER on 3 Jun 2021 at 07:30 The Belle II experiment has accumulated data corresponding to 89.99 fb-1 integrated luminosity in the past 2 years, and is performing very good. Waiting that the full planned data set will be recorded (50 ab-1), which will allow search for rare processes and will have a tremendous impact in the spectroscopy field, the Phase 3 data set allows to already perform analysis with high precision. We pres ... More Presented by Dr. Youngmin YOOK Session: Production Presented by Dr. Youngmin YOOK on 3 Jun 2021 at 11:40 Presented by Sean DOBBS on 31 May 2021 at 12:40 Presented by Dr. Longke LI on 1 Jun 2021 at 07:00 Session: Exotics Presented by Prof. Alexis POMPILI on 31 May 2021 at 10:50 A naive application of the heavy quark expansion (HQE) yields theory estimates for the decay rate of neutral$D$mesons that are four orders of magnitude below the experimental determination. It is well known that this huge suppression results from severe GIM cancellations. We find that this mismatch can be solved by individually choosing the renormalisation scale of the different internal ... More Presented by Mr. Christos VLAHOS Presented by Mr. Christos VLAHOS on 3 Jun 2021 at 12:20 Presented by Dr. Aleksey RUSOV on 4 Jun 2021 at 13:00 In this talk, I plan to present preliminary results of updated SM predictions for charm-meson lifetimes, their ratios and also the semileptonic decay widths, based on the Heavy Quark Expansion (HQE). In addition to the known contributions, for the first time we include the dimension-six Darwin term for non-leptonic charm decays and the first determinations of the dimension-six Bag paramet ... More Presented by Dr. Aleksey RUSOV Presented by Prof. Vorobiev VITALY on 1 Jun 2021 at 09:20 Many models of dark matter and hidden sectors predict new particles with masses below the electroweak scale. Low-energy electron-positron colliders such as BABAR are ideally suited to discover these hidden-sector particles. We present a recent search for prompt and long-lived hidden scalars produced in association with tau leptons and decaying into a lepton pair. This search is sensitive to viable ... More Presented by Dr. Steven ROBERTSON Session: Taus Presented by Dr. Robertson STEVEN on 3 Jun 2021 at 12:20 Presented by Dr. Huijing LI on 4 Jun 2021 at 12:10 BESIII has collected data samples corresponding to luminosities of 2.93 fb-1 and 3.19 fb-1 at center-of-mass energies of 3.773 and 4.178, respectively. We report recent measurements that include the decays D(s)+ -> l+v (l=mu, tau), D0(+) -> K-bar(pi)l+v (l=e,mu), D0(+) -> a0(980)e+v, D+->K1(1270)e+nu, D(s)+ -> eta(')e+nu, Ds+ -> K(0)enu, Ds+ -> phi enu,. The first searches for Ds+ -> gamma e+nu a ... More Presented by Jingzhi ZHANG We study the expected sensitivity at Belle and Belle II for four-body τ∓→X±l∓l∓ντ decays where l=e or μ and X=π, K, ρ and K∗ mesons. These decay processes violate the total lepton number (\|ΔL\|=2 ) and they can be induced by the exchange of Majorana neutrinos. In particular, we consider lifetimes in the accessible ranges of τN = 5, 100 ps and extract the limits on \|VℓN\|2 withou ... More Presented by Dr. Pedro PODESTA Session: Taus Presented by Mr. David RODRÍGUEZ PÉREZ on 3 Jun 2021 at 13:00 Session: Exotics Presented by Prof. Feng-Kun GUO on 31 May 2021 at 09:50 The BES III detector at Beijing Electron-Positron Collider has collected the world’s largest data sets. In this talk I will report recent results on the baryon pair production in Charmonium(-like) and in the e+e− annihilation at BESIII. Presented by Dr. Xiongfei WANG Session: Production Presented by Dr. Xiongfei WANG on 1 Jun 2021 at 11:40 We study the processes γγ → η ′ K + K − , η ′ π + π − , and ηπ + π − using a data sample of 519 f b −1 recorded with the BaBar detector operating at the SLAC PEP-II asymmetric-energy e + e − collider at center-of-mass energies at and near the Υ(nS) (n = 2, 3, 4) resonances. This is the first observation of the decay η c → η ′ K + K − and we measure the branchin ... More Presented by Prof. Antimo PALANO Presented by Prof. Antimo PALANO on 1 Jun 2021 at 11:40 Session: Production The data on tau neutrino is very scarce, only a few experiments have detected its interactions. At FNAL beam dump experiment DONUT, tau neutrino interaction cross-section was directly measured with a large systematical (~50%) and statistical (~30%) errors. The main source of systematical error is due to a poor knowledge of the tau neutrino flux. The effective way for tau neutrino production is the ... More Presented by Collaboration DSTAU Session: Taus Presented by Prof. Ali Murat GULER on 4 Jun 2021 at 13:10 Session: Production Session: Production Session: Taus Presented by Prof. Alberto LUSIANI on 4 Jun 2021 at 10:20 Presented by Emilie PASSEMAR on 4 Jun 2021 at 09:50 The Belle II experiment is a substantial upgrade of the Belle detector and will operate at the SuperKEKB energy-asymmetric e+e− collider. The design luminosity of the machine is 8 × 1035 cm−2s−1 and the Belle II experiment aims to record 50 ab−1 of data, a factor of 50 more than its predecessor. From February to July 2018, the machine has completed a commissioning run and main operation o ... More Presented by Dr. Thomas KRAETZSCHMAR Session: Taus Presented by Mr. Thomas KRAETZSCHMAR on 4 Jun 2021 at 12:10 Based on the Generalized Quantum Electrodynamics expression for the Podolsky propagator, which preserves gauge invariance for massive photons, we propose a model for the massive gluon propagator that reproduces well-known features of established strong-interaction models in the framework of the Dyson-Schwinger equation. By adjusting the Podolsky mass and the coupling strength we thus construct a m ... More Presented by Prof. Eduardo ROJAS PEÑA Presented by Prof. Eduardo ROJAS PEÑA on 4 Jun 2021 at 13:25 Session: NP in charm Presented by Mr. Marxil SÁNCHEZ GARCÍA on 4 Jun 2021 at 13:30 The$D_s$semileptonic decays provides an ideal scenario in the charm sector to search for Lepton number violation (LNV), Lepton flavor universality (LFU) tests and new contributions in Flavor changing neutral currents (FCNC) modes. In this work, we first describe the long distance (LD) contributions in the$D_{s} \rightarrow \pi l^{+} l^{-}$decay, a non-FCNC mode usually employed as normalizati ... More Presented by Mr. Marxil SÁNCHEZ Presented by Mrs. Raquel MOLINA PERALTA on 1 Jun 2021 at 12:55 In a recent paper [1], the BESIII collaboration reported the so-called first observation of pure$W$-annihilation decays$D^+_s \to a^+_0(980) \pi^0$and$D_s^+ \to a^0_0(980)\pi^+$. The measured absolute branching fractions are, however, puzzlingly larger than those of other measured pure$W$-annihilation decays by at least one order of magnitude. In addition, the relative phase between the two d ... More Presented by Dr. Raquel MOLINA PERALTA The SHiP collaboration proposes a general purpose fixed-target experiment to search for hidden particles at the new beam-dump facility at CERN SPS. For the interpretation of these searches a precise knowledge of the differential charm production cross-section in a thick target, including the cascade production, is essential. To obtain this parameter, several dedicated measurements at CERN SPS have ... More Presented by Mr. Nikolaus OWTSCHARENKO Session: Production Presented by Mr. Nikolaus OWTSCHARENKO on 3 Jun 2021 at 13:00 If the X(3872) is a weakly bound charm-meson molecule, it can be produced by the creation of$D^∗ \bar D^∗$at short distances followed by the rescattering of the charm mesons into X(3872) and a photon or a pion through a triangle loop. A triangle singularity produces narrow peak in the reaction rate in production of X(3872). The observation of this peak would provide strong evidence in suppor ... More Presented by Ms. Liping HE Session: Production Presented by Ms. Liping HE on 1 Jun 2021 at 12:00 Presented by Prof. Lisheng GENG The recent LHCb discovery of three pentaquark states may usher in a new ear in our understanding of the low energy strong interaction. These states might be part of a first complete multiplet composed of seven hadrons of molecular nature. At least this is what emerges from a leading order effective field theory using only data and heavy quark spin symmetry as constrains [1]. In addition, we will s ... More Presented by Prof. Lisheng GENG The$X(3872)$, whose mass coincides with the$D^0\bar{D}^{0}$threshold, is the most extended hadron object. Since its discovery in 2003, debates have never stopped regarding its internal structure. We propose a new object, the X atom, which is the$D^\pm D^{\mp}$composite system with positive charge parity and a mass of$(3879.89\pm0.07)$~MeV, formed mainly due to the Coulomb force. We show ... More Presented by Mr. Zhenhua ZHANG Presented by Mr. ZHANG ZHENHUA on 1 Jun 2021 at 12:00 From 2011, BESIII has taken about 20 fb^-1 data samples at center of mass energies from 3.8 to 4.6 GeV, containing 21 energy points with luminosity larger than 400 pb^-1. This makes the study of vector states Y, charged states Z, X states, as well as the connections between them through transition processes possible. Using these data samples, new information about X(3872) decays, Y states from ope ... More Presented by Jingzhi ZHANG Presented by Dr. Weimin SONG on 1 Jun 2021 at 11:40 Recently, BESIII reported observation of the structure at the kinematical threshold in the$D_{s}^{-}D^{0}+D_{s}^{-}D^{0}$mass distribution, which is interpreted as a tetraquark candidate, called$Z_{cs}(3985)^{-}\$. This is the first candidate for a tetraquark meson containing hidden-charm with non-zero strangeness. BESIII has been devoting on the studies on the nonstrange charmoniumlike Zc sta ... More	{"extraction_info": {"found_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": 0.9306180477142334, "perplexity": 4794.177789075919}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571950.76/warc/CC-MAIN-20220813111851-20220813141851-00617.warc.gz"}
https://www.physicsforums.com/threads/the-energy-of-a-point-charge.232921/	# The Energy of a Point Charge 1. May 3, 2008 ### dx Hi, I read in the Feynman Lectures Vol II that the assumption that the electron is a point leads to an infinite energy in its field, and that the difficulty hasn't been resolved. Has there been any progress on this since Feynman gave his lectures? 2. May 3, 2008 ### G01 The diverging self energy of a point charge is indeed still a very open problem in physics today. The problem arises when we consider the formula for the energy stored in an electromagetic field: $$W=\frac{\epsilon_o}{2}\int_{all space}E^2d\tau$$ For a point charge, such as an electron, this reduces to: $$W=\frac{q^2}{8\pi\epsilon_o}\int_0^{\infty}\frac{1}{r^2}dr$$ This integral diverges, leading to the infinite value for the energy contained in the field of a point charge. From a qualitative perspective: The formula derived above is derived by adding up all the energy needed to assemble a charge distribution. Well, to assemble a point charge, we would need to take a finite amount of charge and "cram" it into a point, a space infinitely small, having no dimension. Considering this, it makes sense that to "assemble" a point charge will require an infinite amount of energy, since we would need to create a distribution with infinite an charge density. Thus, the problem is that classical electromagnetism predicts that the energy required to "create" a point charge distribution is infinite, which makes no sense. This is a great example of a "singularity" appearing in a physical theory. At this singularity, the laws of physics don't work and predict ridiculous infinite answers. This problem is not just present in classical electrodynamics but is also present in the quantum theory as well. Fixing or explaining this problem is an open area of theoretical research. Last edited: May 3, 2008 3. May 4, 2008 ### Lojzek It seems to me that infinite energy is a problem only if electron is separable. If it is not (and this is what is currently assumed), then this is just a constant addition to energy: are there any troubles with that? 4. May 4, 2008 ### G01 Yes, if we consider that an electron is inseparable, then this diverging integral does not effect the rest of the calculations needed in electrostatics, but this infinite "self energy" is still embarrassing from the point of view of many physicists. This is not the only point particle inconsistency in Classical E&M. There are also other problems caused by point particles in classical electrodynamics. For instance, consider in radiation theory the Abraham-Lorentz formula which describes a "radiation reaction force" that will be experienced by a radiating point charge. This formula describes ridiculous accelerations for point particles in radiative situations, such as accelerations that happen a short time before the force causing them acts, or accelerations that spontaneously increase exponentially! Again, what this means is beyond my current knowledge, but it goes to show that electrodynamics has some problems with point charges that still need to be dealt with. 5. May 5, 2008 ### Nick M Isn't 1/r^2 a p series whose infinite sum converges to a single number because p>1? 6. May 5, 2008 ### nicksauce $$\sum_{n=1}^{n=\infty}\frac{1}{n^2}$$ converges $$\int_{0}^{\infty}\frac{1}{r^2}dr$$ diverges. Big difference between the two as the latter includes a singularity. 7. May 5, 2008 ### Redbelly98 Staff Emeritus The problem is not at $r\rightarrow \infty$ , it's at $r\rightarrow 0$. The summation you're thinking of starts at "1". Starting the integral at 0 is what is problematic. 8. May 5, 2008 ### Lojzek What about the possibility that electron is not a point particle? Can it have a very small positive radius? It is possible to calculate "classical electron radious" with the assumption that electron is a uniformly charged ball and that only electrostatic energy contributes to its mass. Does anybody know whether the experimental upper limit for electron radius has already crossed classical electron radious? 9. May 5, 2008 ### olgranpappy in quantum mechanics and QED the electron is treated as a point charge--a point particle. You can consider other quanties like strings instead of points if you like, but it's probably not worth the trouble in my opinion. 10. May 6, 2008 ### Nick M Ah... I see. Interesting! 11. May 6, 2008 ### pzlded That extrapolation applied to the mass of a electron results in a black hole. A black hole with the mass of an electron is unstable. Perhaps an electron is a stable coexistence of mass and charge. The closeness of Electron / positron annihilation to energy of mass, indicates a limit on an electron's charge energy.	{"extraction_info": {"found_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": 0.9442448616027832, "perplexity": 598.8510565937909}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794867173.31/warc/CC-MAIN-20180525180646-20180525200646-00500.warc.gz"}
http://mathhelpforum.com/advanced-algebra/81853-please-can-someone-check-see-if-my-answer-right.html	# Math Help - Please can someone check to see if my answer is right?? 1. ## Please can someone check to see if my answer is right?? So the give eigen value is λ=2. And the matrix A, what is the basis for each eigenspace? A= 0 1 1 1 0 1 1 1 0 ------------ My answer that I get after doing (A-2I)x=0 and row reducing it is a general solution and x3 is free. and my basis is 1 1 1 Can someone tell me if that's right?? please! 2. Hello, There is another eigenvalue : -1. In order to help you know if you're on the right way, you can remember that the sum of the eigenvalues equals the trace (the sum of the diagonal elements) of the matrix. And $2 \neq 0$ The eigenspace you found for 2 is correct though	{"extraction_info": {"found_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": 0.9076392650604248, "perplexity": 391.1879137968577}, "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-2014-49/segments/1416931009777.87/warc/CC-MAIN-20141125155649-00222-ip-10-235-23-156.ec2.internal.warc.gz"}
https://chem.libretexts.org/Bookshelves/Organic_Chemistry/Map%3A_Organic_Chemistry_(Bruice)/01%3A_Electronic_Structure_and_Bonding_(Acids_and_Bases)/1.13%3A_The_Bond_in_a_Hydrogen_Halide	# 1.13: The Bond in a Hydrogen Halide $$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\\| #1 \\|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\\| #1 \\|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ This page discusses the acidity of the hydrogen halides: hydrogen fluoride, hydrogen chloride, hydrogen bromide and hydrogen iodide. It begins by describing their physical properties and synthesis and then explains what happens when they react with water to make acids such as hydrofluoric acid and hydrochloric acid. ## Physical properties The hydrogen halides are colorless gases at room temperature, producing steamy fumes in moist air. The boiling points of these compounds are shown in the figure below: Hydrogen fluoride has an abnormally high boiling point for a molecule of its size(293 K or 20°C), and can condense under cool conditions. This is due to the fact that hydrogen fluoride can form hydrogen bonds. Because fluorine is the most electronegative of all the elements, the fluorine-hydrogen bond is highly polarized. The hydrogen atom carries a high partial positive charge (δ+); the fluorine is fairly negatively charged (δ-). In addition, each fluorine atom has 3 lone pairs of electrons. Fluorine's outer electrons are at the n=2 level, and the lone pairs represent small, highly charged regions of space. Hydrogen bonds form between the δ+ hydrogen on one HF molecule and a lone pair on the fluorine of another one.The figure below illustrates this association: The other hydrogen halides do not form hydrogen bonds because the larger halogens are not as electronegative as fluorine; therefore, the bonds are less polar. In addition, their lone pairs are at higher energy levels, so the halogen does not carry such an intensely concentrated negative charge; therefore, other hydrogen atoms are not attracted as strongly. ### Making hydrogen halides There are several ways of synthesizing hydrogen halides; the method considered here is the reaction between an ionic halide, like sodium chloride, and an acid like concentrated phosphoric(V) acid, H3PO4, or concentrated sulfuric acid. ### Making hydrogen chloride When concentrated sulfuric acid is added to sodium chloride under cold conditions, the acid donates a proton to a chloride ion, forming hydrogen chloride. In the gas phase, it immediately escapes from the system. $Cl^- + H_2SO_4 \rightarrow HCl + HSO_4^-$ The full equation for the reaction is: $NaCl + H_2SO_4 \rightarrow HCl + NaHSO_4$ Sodium bisulfate is also formed in the reaction. Concentrated phosphoric(V) acid reacts similarly, according to the following equation: $Cl^- + H_3PO_4 \rightarrow HCl + H_2PO_4^-$ The full ionic equation showing the formation of the salt, sodium biphosphate(V), is given below: $NaCl + H_3PO_4 \rightarrow HCl + NaH_2PO_4$ ### Making other hydrogen halides All hydrogen halides can be formed by the same method, using concentrated phosphoric(V) acid. Concentrated sulfuric acid, however, behaves differently. Hydrogen fluoride can be made with sulfuric acid, but hydrogen bromide and hydrogen iodide cannot. The problem is that concentrated sulfuric acid is an oxidizing agent, and as well as producing hydrogen bromide or hydrogen iodide, some of the halide ions are oxidized to bromine or iodine. Phosphoric acid does not have this ability because it is not an oxidant. ## The acidity of the hydrogen halides ### Hydrogen chloride as an acid By the Bronsted-Lowry definition of an acid as a proton donor, hydrogen chloride is an acid because it transfers protons to other species. Consider its reaction with water. Hydrogen chloride gas is soluble in water; its solvated form is hydrochloric acid. Hydrogen chloride fumes in moist air are caused by hydrogen chloride reacting with water vapor in the air to produce a cloud of concentrated hydrochloric acid. A proton is donated from the hydrogen chloride to one of the lone pairs on a water molecule. A coordinate (dative covalent) bond is formed between the oxygen and the transferred proton. The equation for the reaction is the following: $H_2O + HCl \rightarrow H_3O^+ + Cl^-$ The H3O+ ion is the hydroxonium ion (also known as the hydronium ion or the oxonium ion). This is the normal form of protons in water; sometimes it is shortened to the proton form, H+(aq), for brevity. When hydrogen chloride dissolves in water (to produce hydrochloric acid), almost all the hydrogen chloride molecules react in this way. Hydrochloric acid is therefore a strong acid. An acid is strong if it is fully ionized in solution. ### Hydrobromic acid and hydriodic acid as strong acids Hydrogen bromide and hydrogen iodide dissolve in (and react with) water in exactly the same way as hydrogen chloride does. Hydrogen bromide forms hydrobromic acid; hydrogen iodide gives hydriodic acid. Both of these are also strong acids. ### Hydrofluoric acid as an exception By contrast, although hydrogen fluoride dissolves freely in water, hydrofluoric acid is only a weak acid; it is similar in strength to organic acids like methanoic acid. The complicated reason for this is discussed below. ## The bond enthalpy of the H-F bond Because the fluorine atom is so small, the bond enthalpy (bond energy) of the hydrogen-fluorine bond is very high. The chart below gives values for all the hydrogen-halogen bond enthalpies: bond enthalpy (kJ mol-1) H-F +562 H-Cl +431 H-Br +366 H-I +299 In order for ions to form when the hydrogen fluoride reacts with water, the H-F bond must be broken. It would seem reasonable to say that the relative reluctance of hydrogen fluoride to react with water is due to the large amount of energy needed to break that bond, but this explanation does not hold. ## The energetics of the process from HX(g) to X-(aq) The energetics of this sequence are of interest: All of these terms are involved in the overall enthalpy change as you convert HX(g) into its ions in water. However, the terms involving the hydrogen are the same for every hydrogen halide. Only the values for the red terms in the diagram need be considered. The values are tabulated below: bond enthalpy of HX (kJ mol-1) electron affinity of X (kJ mol-1) hydration enthalpy of X- (kJ mol-1) sum of these (kJ mol-1) HF +562 -328 -506 -272 HCl +431 -349 -364 -282 HBr +366 -324 -335 -293 HI +299 -295 -293 -289 There is virtually no difference in the total HF and HCl values. The large bond enthalpy of the H-F bond is offset by the large hydration enthalpy of the fluoride ion. There is a very strong attraction between the very small fluoride ion and the water molecules. This releases a lot of heat (the hydration enthalpy) when the fluoride ion becomes wrapped in water molecules. ## Other attractions in the system The energy terms considered previously have concerned HX molecules in the gas phase. To reach a more correct explanation, the molecules must first be considered as unreacted aqueous HX molecules. The equation for this is given below: The equation is incorporated into an improved energy cycle as follows: Unfortunately, values for the first step in the reaction are not readily available. However, in each case, the initial separation of the HX from water molecules is endothermic. Energy is required to break the intermolecular attractions between the HX molecules and water. That energy is much greater for hydrogen fluoride because it forms hydrogen bonds with water. The other hydrogen halides experience only the weaker van der Waals dispersion forces or dipole-dipole attractions. The overall enthalpy changes (including all the stages in the energy cycle) for the reactions are given in the table below: $HX(aq) + H_2O (l) \rightarrow H_3O^+ (aq) + X^- (aq)$ enthalpy change (kJ mol-1) HF -13 HCl -59 HBr -63 H-I -57 The enthalpy change for HF is much smaller in magnitude than that for the other three hydrogen halides, but it is still negative exothermic change. Therefore, more information is needed to explain why HF is a weak acid. ## Entropy and free energy considerations The free energy change, not the enthalpy change, determines the extent and direction of a reaction. Free energy change is calculated from the enthalpy change, the temperature of the reaction and the entropy change during the reaction. For simplicity, entropy can be thought of as a measure of the amount of disorder in a system. Entropy is given the symbol S. If a system becomes more disordered, then its entropy increases. If it becomes more ordered, its entropy decreases. The key equation is given below: In simple terms, for a reaction to happen, the free energy change must be negative. But more accurately, the free energy change can be used to calculate a value for the equilibrium constant for a reaction using the following expression: The term Ka is the equilibrium constant for the reaction below: $HX(aq) + H_2O(l) \rightleftharpoons H_3O^+(aq) + X^-(aq)$ The values for TΔS (needed to calculate ΔG) for the four reactions at a temperature of 298 K are tabulated below: TS (kJ mol-1) HF -29 HCl -13 HBr -4 H-I +4 Notice that at the top of the group, the systems become more ordered when the HX reacts with the water. The entropy of the system (the amount of disorder) decreases, particularly for the hydrogen fluoride. The reason for this is that the very strong attraction between H3O+ and F-(aq) imposes a lot of order on the system, as does the attraction between the water molecules and the various ions present. These attractions are each greatest for the small fluoride ions. The total effect on the free energy change, and therefore the value of the equilibrium constant, can now be considered. These values are calculated in the following table: H (kJ mol-1) TS (kJ mol-1) G (kJ mol-1) Ka (mol dm-3) HF -13 -29 +16 1.6 x 10-3 HCl -59 -13 -46 1.2 x 108 HBr -63 -4 -59 2.2 x 1010 HI -57 +4 -61 5.0 x 1010 The values for these estimated equilibrium constants for HCl, HBr and HI are so high that the reaction can be considered "one-way". The ionization is virtually 100% complete. These are all strong acids, increasing in strength down the group. By contrast, the estimated Ka for hydrofluoric acid is small. Hydrofluoric acid only ionizes to a limited extent in water. Therefore, it is a weak acid. The estimated value for HF in the table can be compared to the experimental value: • Experimental value: 5.6 x 10-4 mol dm-3 • Estimated value: 1.6 x 10-3 mol dm-3 These values differ by an order of magnitude, but because of the logarithmic relationship between the free energy and the equilibrium constant, a very small change in ΔG has a very large effect on Ka. To have the values in close agreement, ΔG would must increase from +16 to +18.5 kJ mol-1. Given the uncertainty in the values used to calculate ΔG, the difference between the calculated value and the experimental value could easily fall within this range. ## Summary: Why is hydrofluoric acid a weak acid? The two main factors are: • Entropy decreases dramatically when the hydrogen fluoride reacts with water. This is particularly noticeable with hydrogen fluoride because the attraction of the small fluoride ions produced imposes significant order on the surrounding water molecules and nearby hydronium ions. The effect decreases with larger halide ions. • Very strong hydrogen bonding exists between the hydrogen fluoride molecules and water molecules. This costs a large amount of energy to break. This effect does not occur in the other hydrogen halides. ## Contributors and Attributions Jim Clark (Chemguide.co.uk) 1.13: The Bond in a Hydrogen Halide is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.	{"extraction_info": {"found_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": 0.8845311999320984, "perplexity": 2279.028774263582}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571483.70/warc/CC-MAIN-20220811164257-20220811194257-00206.warc.gz"}
https://www.physicsforums.com/threads/newton-universal-gravitation-formula-how-f-is-dimensionally-derived.768685/	# Newton universal gravitation formula, how f is dimensionally derived? 1. Sep 2, 2014 ### yeoG If f dimensions are ml/t^2, where does t^2 come from in the equation of F = Gm1m2/r^2 where I believe G to be a constant, m1 and m2 to be masses and r to be the distance between two masses - so length. To dimensionally analyse this then, where would the dimension time come from if I were to check if the equation is dimensionally correct and I cant see where it comes from? I do think that it may come from G but i have seen various answers on what the dimensions of G are so I cant cancel the dimensions out to check if it is the same as F? Help would be appreciated, many thanks. 2. Sep 2, 2014 3. Sep 2, 2014 ### Integral Staff Emeritus According to Newton F=ma since acceleration has units l t-2 the units of force must be m Lt-2. Using that you can figure out the units G needs to make the given equation dimensionally correct. Time is factored into the constant. 4. Sep 2, 2014 5. Sep 2, 2014 ### yeoG Hi thanks for the help however I focusing on F instead of G giving that G = N m^2 kg^-2 or what G is? 6. Sep 2, 2014 ### nrqed Isolating, $G= F$ $r^2 / (m_1 m_2 )$. You can tell right away the units of G. 7. Sep 2, 2014 ### yeoG ah I think I understand so by working out the dimensions of G you can show that F must equal to Gxm1xm2 because it will balance the equation to = F? 8. Sep 2, 2014 ### A.T. No. Observation shows that F equals Gxm1xm2/r^2. The dimension of G are defined based on this. So there is no point in checking if the equation is dimensionally correct, because it was defined to be correct. Similar Discussions: Newton universal gravitation formula, how f is dimensionally derived?	{"extraction_info": {"found_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": 0.9544796347618103, "perplexity": 1075.8967751771522}, "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/1508187825399.73/warc/CC-MAIN-20171022165927-20171022185927-00672.warc.gz"}
https://en.wikipedia.org/wiki/Parallelizable	# Parallelizable manifold (Redirected from Parallelizable) In mathematics, a differentiable manifold $\scriptstyle M$ of dimension n is called parallelizable[1] if there exist smooth vector fields $\{V_1, \dots,V_n\}$ on the manifold, such that at any point $\scriptstyle p$ of $\scriptstyle M$ the tangent vectors $\{V_1(p), \dots, V_n(p)\}$ provide a basis of the tangent space at $\scriptstyle p$. Equivalently, the tangent bundle is a trivial bundle,[2] so that the associated principal bundle of linear frames has a section on $\scriptstyle M.$ A particular choice of such a basis of vector fields on $\scriptstyle M$ is called a parallelization (or an absolute parallelism) of $\scriptstyle M$. ## Examples • An example with n = 1 is the circle: we can take V1 to be the unit tangent vector field, say pointing in the anti-clockwise direction. The torus of dimension n is also parallelizable, as can be seen by expressing it as a cartesian product of circles. For example, take n = 2, and construct a torus from a square of graph paper with opposite edges glued together, to get an idea of the two tangent directions at each point. More generally, any Lie group G is parallelizable, since a basis for the tangent space at the identity element can be moved around by the action of the translation group of G on G (any translation is a diffeomorphism and therefore these translations induce linear isomorphisms between tangent spaces of points in G). • A classical problem was to determine which of the spheres Sn are parallelizable. The zero-dimensional case S0 is trivially parallelizable. The case S1 is the circle, which is parallelizable as has already been explained. The hairy ball theorem shows that S2 is not parallelizable. However S3 is parallelizable, since it is the Lie group SU(2). The only other parallelizable sphere is S7; this was proved in 1958, by Michel Kervaire, and by Raoul Bott and John Milnor, in independent work. The parallelizable spheres correspond precisely to elements of unit norm in the normed division algebras of the real numbers, complex numbers, quaternions, and octonions, which allows one to construct a parallelism for each. Proving that other spheres are not parallelizable is more difficult, and requires algebraic topology. • The product of parallelizable manifolds is parallelizable. ## Remarks • The term framed manifold (occasionally rigged manifold) is most usually applied to an embedded manifold with a given trivialisation of the normal bundle, and also for an abstract (i.e. non-embedded) manifold with a given stable trivialisation of the tangent bundle.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 9, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9465168714523315, "perplexity": 295.49448393381243}, "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-2016-07/segments/1454701148428.26/warc/CC-MAIN-20160205193908-00172-ip-10-236-182-209.ec2.internal.warc.gz"}
http://math.stackexchange.com/users/73233/millardo-peacecraft?tab=activity	Millardo Peacecraft Reputation 647 Next privilege 1,000 Rep. Create tags Badges 1 4 16 Newest Impact ~6k people reached • 0 helpful flags • 48 votes cast # 271 Actions 4h awarded Yearling Mar20 awarded Inquisitive Jan19 asked Lagrange inversion theorem application Jan12 comment An application of the residue theorem or even @RonGordon? Jan12 comment An application of the residue theorem Perhaps @sos440 would know? Jan12 asked An application of the residue theorem Jan12 asked expansion in reciprocal terms Jan7 comment How to solve: $\frac{x}{\log_2(x) }= y$ @orion The OP already defined the Lambert W function in his question. Defining $W$ once more is unnecessary. Jan7 revised Root finding using Galois theory added 9 characters in body Jan7 answered How to solve: $\frac{x}{\log_2(x) }= y$ Jan7 asked Root finding using Galois theory Jan6 asked Residue of function with different branch points Dec30 comment Functional equation for polynomials @Bernard I'm aware that it is the taylor formula for polynomials. But he states that it is the functional form for a given polynomial. Hence my confusion about the $y$ Dec30 asked Functional equation for polynomials Dec30 revised How prove $\bigl(\frac{\sin x}{ x}\bigr)^{2} + \frac{\tan x }{ x} >2$ for $0 < x < \frac{\pi}{2}$ Tex Changes Dec30 suggested approved edit on How prove $\bigl(\frac{\sin x}{ x}\bigr)^{2} + \frac{\tan x }{ x} >2$ for $0 < x < \frac{\pi}{2}$ Dec25 revised Why is $x(\sqrt{x^2+1} - x )$ approaching $1/2$ when $x$ becomes infinite? tex changes Dec25 suggested approved edit on Why is $x(\sqrt{x^2+1} - x )$ approaching $1/2$ when $x$ becomes infinite? Dec16 awarded Caucus Oct24 revised Find the range of function $f(x) =\cos(\sin(\ln(\frac{x^2+e}{x^2+1})))+\sin(\cos(\ln(\frac{x^2+e}{x^2+1})))$… tex changes	{"extraction_info": {"found_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": 0.9832713007926941, "perplexity": 4570.940336551043}, "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-2015-18/segments/1429246636104.0/warc/CC-MAIN-20150417045716-00296-ip-10-235-10-82.ec2.internal.warc.gz"}
https://support.numxl.com/hc/en-us/articles/215786903-ARIMA-GOF-Goodness-of-fit-of-an-ARIMA-Model	# ARIMA_GOF - Goodness of fit of an ARIMA Model Computes the goodness of fit measure (e.g. log-likelihood function (LLF), AIC, etc.) of the estimated ARIMA model. ## Syntax ARIMA_GOF(X, Order, d, mean, sigma, phi, theta, Type) X is the univariate time series data (a one dimensional array of cells (e.g. rows or columns)). Order is the time order in the data series (i.e. the first data point's corresponding date (earliest date=1 (default), latest date=0)). Order Description 1 ascending (the first data point corresponds to the earliest date) (default) 0 descending (the first data point corresponds to the latest date) d is the degree of the differencing (i.e. d). mean is the ARMA model mean (i.e. mu). If missing, mean is assumed zero. sigma is the standard deviation value of the model's residuals/innovations. phi are the parameters of the AR(p) component model (starting with the lowest lag). theta are the parameters of the MA(q) component model (starting with the lowest lag). Type is an integer switch to select the goodness of fitness measure: (1=LLF (default), 2=AIC, 3=BIC, 4=HQC). Order Description 1 Log-Likelihood Function (LLF) (default) 2 Akaike Information Criterion (AIC) 3 Schwarz/Bayesian Information Criterion (SIC/BIC) 4 Hannan-Quinn information criterion (HQC) ## Remarks 1. The underlying model is described here. 2. The Log-Likelihood Function (LLF) is described here. 3. The time series is homogeneous or equally spaced. 4. The time series may include missing values (e.g. #N/A) at either end. 5. The residuals/innovations standard deviation (i.e. $\sigma$) should be greater than zero. 6. The ARMA model has independent and normally distributed residuals with constant variance. The ARMA log-likelihood function becomes: $$\ln L^* = -T\left(\ln 2\pi \hat \sigma^2+1\right)/2$$ Where: • $\hat \sigma$ is the standard deviation of the residuals. 7. The maximum likelihood estimation (MLE) is a statistical method for fitting a model to the data and providing estimates for the model's parameters. 8. The integration order argument (d) must be a positive integer. 9. The long-run mean can take any value or may be omitted, in which case a zero value is assumed. 10. The residuals/innovations standard deviation (sigma) must be greater than zero. 11. For the input argument (phi): • The input argument is optional and can be omitted, in which case no AR component is included. • The order of the parameters starts with the lowest lag. • One or more parameters can be missing or an error code (i.e. #NUM!, #VALUE!, etc.). • The order of the AR component model is solely determined by the order of the last value in the array with a numeric value (vs. missing or error). 12. For the input argument (theta): • The input argument is optional and can be omitted, in which case no MA component is included. • The order of the parameters starts with the lowest lag. • One or more values in the input argument can be missing or an error code (i.e. #NUM!, #VALUE!, etc.). • The order of the MA component model is solely determined by the order of the last value in the array with a numeric value (vs. missing or error). 13. The function was added in version 1.63 SHAMROCK.	{"extraction_info": {"found_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": 0.9001958966255188, "perplexity": 2160.9007340421254}, "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/1502886112539.18/warc/CC-MAIN-20170822181825-20170822201825-00371.warc.gz"}
https://www.physicsforums.com/threads/homology-of-torus-and-kleins-bottle.507446/	# Homology of torus and Klein's bottle 1. Jun 16, 2011 ### ddo 1. The problem statement, all variables and given/known data I'm trying to calculate singular homology groups of the torus and Klein's bottle using the Mayer-Vietoris sequence. 3. The attempt at a solution I represent both spaces as a rectangle with identified edges. Then I take the sets: U=rectangle without the boundary V=rectangle without the middlepoint so U is contractible thus H_n(U)=0 for n>0, H_0(U)=Z V=S1vS1 so H_1(V)=ZxZ, H_n(V)=0 and their intersection = S1, H_n(S1)=0, H_1(S1)=Z, H_0(S1)=Z Now from the M-V sequence for n>2 we get an exact sequence 0->0x0->H_n(T)->0, so H_n(T)=0. But I don't know what to do for smaller n... 2. Jun 16, 2011 ### micromass Staff Emeritus Hi ddo! Do you need to derive everything from the Mayer-Vietoris sequence? That looks quite hard. The zero'th homology group is very easy since it is the number of path connected components. The first homology group is also easy by calculating the fundamental group of $S^1\times S^2$... The second homology group can be derived from Mayer Vietoris then. 3. Jun 16, 2011 ### ddo I suppose the Mayer-Vietoris hint was there to make the task easier :) So H_0 is Z because there is only one connected component, H_1 is the abelianization of the fundamental group, both torus and Klein's bottle have abelian fundamental groups so for torus it's $Z \times Z$, for Klein's bottle $Z \times Z_2$. Now M-V gives: $0 \to H_2(T) \to Z \to 0 \times (Z \times Z) \to H_1(T) \to Z$ And I still don't know how to derive H_2(T)... 4. Jun 16, 2011 ### micromass Staff Emeritus Write out the full Mayer-Vietoris. I'll do it for the torus $$0\rightarrow H_2(X)\rightarrow \mathbb{Z}\rightarrow \mathbb{Z}\times\mathbb{Z}\rightarrow \mathbb{Z}\times\mathbb{Z}\rightarrow \mathbb{Z}\rightarrow \mathbb{Z}\times\mathbb{Z}\rightarrow \mathbb{Z} \rightarrow 0$$ Now, it is easy to see that $H_2(X)$ is either 0 or $\mathbb{Z}$. Now, reason from left to right as follows: The end of the sequence is $$\mathbb{Z}\times\mathbb{Z}\rightarrow \mathbb{Z} \rightarrow 0$$ So the the map is a surjection, and the kernel of the map is isomorphic to $\mathbb{Z}$. Consider the next part of the sequence: $$\mathbb{Z}\rightarrow \mathbb{Z}\times\mathbb{Z}\rightarrow \mathbb{Z}$$ The kernel of the right map equals the image of the left map. Thus the image of the left map is isomorphic to $\mathbb{Z}$. Thus the kernel of the left map is necessarily 0. Now consider the next part of the sequence, etc. Try to complete this argument. 5. Jun 16, 2011 ### ddo $$\mathbb{Z}\times\mathbb{Z} \rightarrow\mathbb{Z}\rightarrow \mathbb{Z}\times\mathbb{Z}$$ The kernel of the right map is 0, so the image of the left map is 0, so the kernel of the left map is $\mathbb{Z}\times\mathbb{Z}$ Next: $$\mathbb{Z}\times\mathbb{Z}\rightarrow \mathbb{Z}\times\mathbb{Z}\rightarrow \mathbb{Z}$$ The kernel of the right map is $\mathbb{Z}\times\mathbb{Z}$, so the image of the left map is $\mathbb{Z}\times\mathbb{Z}$, so the kernel of the left map is 0. $$H_2(X)\rightarrow \mathbb{Z}\rightarrow \mathbb{Z}\times\mathbb{Z}\rightarrow \mathbb{Z}\times\mathbb{Z}$$ The kernel of the right map is 0, so the kernel of the middle map is $\mathbb{Z}$, so the image of the left map is also $\mathbb{Z}$, so $H_2(X)$ can't be 0. Is that correct? 6. Jun 16, 2011 ### micromass Staff Emeritus Sounds good! So the second homology group is $\mathbb{Z}$! The thing with the Klein Bottle should be analogous! Similar Discussions: Homology of torus and Klein's bottle	{"extraction_info": {"found_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": 0.9772101044654846, "perplexity": 686.7354952050636}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934803848.60/warc/CC-MAIN-20171117170336-20171117190336-00491.warc.gz"}
http://mathhelpforum.com/geometry/99599-can-someone-please-check-my-work-calculating-area.html	# Math Help - Can someone please check my work in calculating this area? 1. ## Can someone please check my work in calculating this area? Okay, I needed to find the area of this figure: http://i25.tinypic.com/symtyt.jpg So I figured out that each side had a total area of 224 (828), so the two combined would be 448, if they were separate. Since they overlap, I needed to subtract the area of the triangle from this. Using heron's formula, I plugged in the numbers: 1/2(14+14+8) --> square root of 18(18-14)(18-14)(18-8) --> square root of 184410 --> square root of 2880. The square root of 2880 is 53.665. So, I did 448-53.665=394. Is this correct? 2. I disagree. The 2 parallelograms should have an area of 431.2 (7.7 * 28 * 2). But you are right about subtracting the triangle. There is an easier way then using hero's formula. Drop a height on the triangle and you have this: Now use the Pythagorean theorem to calculate the height. Just easier in my opinion. 3. Thank you so much!!	{"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": 0.8985596895217896, "perplexity": 611.0908012226532}, "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/1397609533308.11/warc/CC-MAIN-20140416005213-00175-ip-10-147-4-33.ec2.internal.warc.gz"}
http://mathoverflow.net/questions/106226/determine-if-a-matrix-is-unimodular	# Determine if a matrix is unimodular Is deciding if an integer square matrix has determinant $\pm 1$ faster that calculating the determinant of the matrix? - could you expand a bit on how this question is similar to detection of unimodularity, where every sub-determinant has to be in $\lbrace 0,\pm 1\rbrace$ – Suvrit Sep 3 '12 at 8:55 I don't know if this is 'faster', but consider the sequence of elementary operations which reduces a square integer matrix $A$ to it's Smith Normal form. We have an algorithm which determines the elementary divisors of $A$, and if any of these elementary divisors is $\neq \pm 1$, then we'll know $A$ is not unimodular. Any step in the reduction implements a euclidean algorithm to compute gcd's. That is, we are constantly reducing rows and colomns. If $A$ is unimodular, then we'll have computed all its elementary divisors, and their product is of course $detA$. So reducing $A$ to its Jordan – J. Martel Sep 3 '12 at 18:02 normal form, either $A$ is not unimodular and we shall eventually determine an elementary divisor $\neq \pm 1$ (which of course, may only occur at the last step), or we determine all of its elementary divisors to be $\pm 1$, and then we've just computed the determinant. So maybe the right question is simply whether or not computing the Smith normal form is faster than computing the determinant. – J. Martel Sep 3 '12 at 18:05 @J.Martel: I was thinking in terms of matrices with rational entries instead of integers; btw., computation of determinant is essentially an $O(n^3)$ procedure (Gaussian elimination); I guess the SNF is not much faster, if at all? – Suvrit Sep 4 '12 at 8:22 In fact, thanks to LU decomposition, computing the determinant is at least as fast as computing a matrix product. So we can compute the determinant exactly in \$O(n^{2.376}). See en.wikipedia.org/wiki/LU_decomposition#Theoretical_complexity. Similarly, in Storjohann's paper "Near Optimal Algorithms for Computing Smith Normal Forms of Integer Matrices" he shows a similar inequality. That is, for an integer square matrix, computing its SNF is at least as fast as computing a matrix product. – Mark Bell Sep 4 '12 at 9:43	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.977510929107666, "perplexity": 398.1373843979351}, "config": {"markdown_headings": true, "markdown_code": false, "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-2016-22/segments/1464051417337.23/warc/CC-MAIN-20160524005657-00158-ip-10-185-217-139.ec2.internal.warc.gz"}
http://mathhelpforum.com/advanced-algebra/202320-subring-section-inclusion.html	## subring - section of inclusion Let A and B be commutative rings with 1, A a subring of B. Is there a section of the inclusion, namely a ringhomomorphism from B to A, that is the identity on A? I hope not and I think that would be plaubsible. But I have no idea how to proof and I couldn't find something about it. Maybe it is easier to verify if A and B are concrete ring, A the real valued smooth and B the real valued continous functions on a differential manifold.	{"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": 0.906292200088501, "perplexity": 152.0748982278072}, "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/1487501172649.58/warc/CC-MAIN-20170219104612-00601-ip-10-171-10-108.ec2.internal.warc.gz"}
https://tex.stackexchange.com/questions/41043/macro-to-draw-a-parabola-with-pgf-tikz	# Macro to draw a parabola with pgf/TikZ I have been trying to build a macro \parabola{...} to draw a parabola passing through 3 coordinates with TikZ but without success. For example \parabola{A}{B}{C} would draw the parabola interpolating the (x,y) coordinates (A), (B) and (C). I would like also to specify the style of the curve, the plot domain, etc. The main problem I found is that I cannot figure out how to obtain the value of a given coordinate in dimensionless form (given a certain unit). • Did you try the parabola path option already? There are some examples on page 146 in the manual. What is the obstacle that prevents you to achieve what you want with parabola? – percusse Jan 13 '12 at 22:05 • Welcome to TeX.SE. It would be helpful to post a compilable MWE that illustrates what you done so far so those trying to help can have something to start with. – Peter Grill Jan 13 '12 at 22:10 • As far as I know, the parabola path operation allows to draw the parabola passing through 3 given points only if one of them is the bend. I want a way to draw a parabola using three arbitrary TikZ coordinates. – maeshtro Jan 13 '12 at 22:47 # Introduction This is an old question, but all previous answers have limitations: the main one is that all use plot. And plot command produce multiple cubic curves. But to draw a parabola a single quadratic (cubic) curve is enough. # Some explanations Any parabola can be drawn by a quadratic Bézier curve, and so by a cubic Bézier curve. (A cubic curve with control points A,B,C,D draws a quadratic one iff AD=3BC.) The "standard" parabola t(1-t) over [0,1] can be drawn by \draw (0,0) .. controls (1/3,1/3) and (2/3,1/3) .. (1,0);. Every parabola between two points can be obtained by an affine transform from this "standard one". Using this we can define a style parabola through that use a single Bézier curve to draw the desired parabola. This style can be used with to or edge in the following way (A) to[parabola through={(B)}] (C). # The code The definition of the parabola through is: \makeatletter \def\pt@get#1#2{ \tikz@scan@one@point\pgfutil@firstofone#2\relax% \csname pgf@x#1\endcsname=\pgf@x% \csname pgf@y#1\endcsname=\pgf@y% } \tikzset{ parabola through/.style={ to path={{[x={(\pgf@xc,\pgf@yc)}, y=\parabola@y, shift=(\tikztostart)] -- (0,0) .. controls (1/3,1/3) and (2/3,1/3) .. (1,0) \tikztonodes}--(\tikztotarget)} }, parabola through/.prefix code={ \pt@get{a}{(\tikztostart)}\pt@get{b}{#1}\pt@get{c}{(\tikztotarget)}% \advance\pgf@xb by-\pgf@xa\advance\pgf@yb by-\pgf@ya% \advance\pgf@xc by-\pgf@xa\advance\pgf@yc by-\pgf@ya% \pgfmathsetmacro\parabola@y{(\pgf@yc-\pgf@xc/\pgf@xb\pgf@yb)% /(\pgf@xb-\pgf@xc)\pgf@xc}% } } \makeatother Note: We can avoid \makeatletter/\makeatother and all @s by using let from the calc library. We can use (A) to[parabola through={(B)}] (C): • in every case where the parabola exists, so when the three x-coordinates are different, • the point B can be outside the drawn are, • this can be part of a general path with nodes positioned on it. Example 1: \tikz\draw[help lines] (0,0) grid (4,3) (0,0) edge[parabola through={(3,2)}, red,thick,fill=blue,fill opacity=.21] (4,1); Example 2 (Full MWE): \documentclass[tikz,border=7pt]{standalone} \makeatletter \def\pt@get#1#2{ \tikz@scan@one@point\pgfutil@firstofone#2\relax% \csname pgf@x#1\endcsname=\pgf@x% \csname pgf@y#1\endcsname=\pgf@y% } \tikzset{ parabola through/.style={ to path={{[x={(\pgf@xc,\pgf@yc)}, y=\parabola@y, shift=(\tikztostart)] -- (0,0) .. controls (1/3,1/3) and (2/3,1/3) .. (1,0) \tikztonodes}--(\tikztotarget)} }, parabola through/.prefix code={ \pt@get{a}{(\tikztostart)}\pt@get{b}{#1}\pt@get{c}{(\tikztotarget)}% \advance\pgf@xb by-\pgf@xa\advance\pgf@yb by-\pgf@ya% \advance\pgf@xc by-\pgf@xa\advance\pgf@yc by-\pgf@ya% \pgfmathsetmacro\parabola@y{(\pgf@yc-\pgf@xc/\pgf@xb\pgf@yb)% /(\pgf@xb-\pgf@xc)\pgf@xc}% } } \makeatother \begin{document} \begin{tikzpicture} \draw[help lines] (-1,-1) grid (3,3); % variations of the point "through" \foreach \y in {-1,-.9,...,1} \draw[green] (-1,1) node[black]{.} to[parabola through={(0,\y)}] node[black]{.} node[black,at end]{.} (1,.5); % variations of a boundary point \foreach \y in {1.5,1.7,...,3} \draw[purple] (-1,2) node[black]{.} to[parabola through={(0,2)}] node[black]{.} node[black,at end]{.} (1,\y); % variations of a point "trough" outside the drawn part \foreach \y in {-1,-0.5,...,3}{ \draw[red,thick] (.5,1) node[black]{.} to[parabola through={(3,\y)}] node[black]{.} node[black,at end]{.} (2,1); \draw[dashed,blue] (.5,1) node[black]{.} to[parabola through={(2,1)}] node[black]{.} node[black,at end]{.} (3,\y); } \end{tikzpicture} \end{document} # Compared to the built in parabola operation TikZ provide a parabola path operation. But it is not very well designed : • the (0,0) parabola (1,1) is supposed to draw the parabola t^2 between 0 and 1. It draws a cubic curve that is close to this parabola but it is not exactly the same, actually it draws (0,0) .. controls (.5,0) and (0.8875,0.775) .. (1,1), but the exact curve is (0,0) .. controls (1/3,0) and (2/3,1/3) .. (1,1) (not clear why this curve is not used), • when used with bend option, it use two cubic curves to approximate the parabola, but only one is enough to draw the exact one, • when used with bend=<point> option, if you do not choose well the point the curve is not a parabola. There is a situation where the original parabola is simpler to use (even if not exactly a parabola is drawn), when the bend (the extremal point) is at the start or at the end : (0,0) parabola (2,4) is simpler than (0,0) to[parabola through={(1,1)}] (2,4). • @PaulGaborit Thanks ! I changed the code following your advise putting the order "division, multiplication,division, multiplication" in the formula of\parabola@y. – Kpym May 5 '18 at 14:42 • Nice. Any idea why these can't be concatenated like other types of edges though? (E.g.,\tikz\draw (0,2) to[parabola through={(1,0)}] (2,4) to[parabola through={(3,3)}] (4,3); has a gap.) – Circumscribe Jul 3 '18 at 17:16 • @Circumscribe I have no idea why. But it is easy to correct : I have added --(\tikztotarget) to the end of to path and now it works as expected. Thank you for reporting this inconsistency. – Kpym Jul 3 '18 at 21:02 1) A variant with fp : \documentclass{article} \usepackage{tikz,fp} \FPmessagesfalse \FPdebugfalse \makeatletter \tikzset{% parabola through/.style={ to path={% \pgfextra{% \tikz@scan@one@point\pgfutil@firstofone(\tikztostart)\relax \FPeval\xa{\pgf@sys@tonumber{\pgf@x}/28.45274} \FPeval\ya{\pgf@sys@tonumber{\pgf@y}/28.45274} \tikz@scan@one@point\pgfutil@firstofone#1\relax \FPeval\xb{\pgf@sys@tonumber{\pgf@x}/28.45274} \FPeval\yb{\pgf@sys@tonumber{\pgf@y}/28.45274} \tikz@scan@one@point\pgfutil@firstofone(\tikztotarget)\relax \FPeval\xc{\pgf@sys@tonumber{\pgf@x}/28.45274} \FPeval\yc{\pgf@sys@tonumber{\pgf@y}/28.45274} \FPeval\pb@a{(\ya(\xb-\xc)+\yb(\xc-\xa)+\yc(\xa-\xb))/% ((\xa-\xb)(\xa-\xc)(\xb-\xc))} \FPeval\pb@b{(\ya(\xc+\xb)(\xc-\xb)+\yb(\xa+\xc)(\xa-\xc)+\yc(\xb+\xa)(\xb-\xa))/((\xa-\xb)(\xa-\xc)(\xb-\xc))} \FPeval\pb@c{(\ya\xb\xc(\xb-\xc)+\yb\xa\xc(\xc-\xa)+\yc\xa\xb(\xa-\xb))/((\xa-\xb)(\xa-\xc)(\xb-\xc))} \draw plot[domain=\xa:\xc] (\x,{\pb@a(\x\x)+\pb@b\x+\pb@c}) ; }(\tikztotarget) } } } \makeatother \begin{document} \begin{tikzpicture} \draw [help lines] (-3,-1) grid (7,4); \draw (-3,0) to[parabola through={(-2,2)}]% (0,-1) to[parabola through={(2,4)}] (4,0) to[parabola through={(5,3)}] (7,0); \end{tikzpicture} \end{document} 2) From maeshtro's answer with gnuplot \documentclass{article} \usepackage{tikz} \makeatletter \tikzset{% parabola through/.style={ to path={% \pgfextra{% \tikz@scan@one@point\pgfutil@firstofone(\tikztostart)\relax \edef\xa{\pgf@sys@tonumber{\pgf@x}} \edef\ya{\pgf@sys@tonumber{\pgf@y}} \tikz@scan@one@point\pgfutil@firstofone#1\relax \edef\xb{\pgf@sys@tonumber{\pgf@x}} \edef\yb{\pgf@sys@tonumber{\pgf@y}} \tikz@scan@one@point\pgfutil@firstofone(\tikztotarget)\relax \edef\xc{\pgf@sys@tonumber{\pgf@x}} \edef\yc{\pgf@sys@tonumber{\pgf@y}} \draw plot[domain=\xa/28.45274:\xc/28.45274] function{ \ya/28.45274((x28.45274-\xb)(x28.45274-\xc))/((\xa-\xb)(\xa-\xc))+ \yb/28.45274((x28.45274-\xa)(x28.45274-\xc))/((\xb-\xa)(\xb-\xc))+ \yc/28.45274((x28.45274-\xa)(x28.45274-\xb))/((\xc-\xa)(\xc-\xb)) }; }(\tikztotarget) } } } \makeatother \begin{document} \begin{tikzpicture} \draw [help lines] (-3,-1) grid (7,4); \draw (-3,0) to[parabola through={(-2,2)}] (0,-1) to[parabola through={(2,4)}] (4,0) to[parabola through={(5,3)}] (7,0); \end{tikzpicture} \end{document} Here is one solution. It essentially solves the linear equation obtained from quadratic interpolation. It is important to note however that it is far from perfect. In particular there are strong constraints on possible points because of tikz computations limitations: the numbers in the computations must be small enough. This is clearly not what tikz is made for. Using other ways to obtain the coefficients would be better (sagetex or asymptote or others). At least it is a nice example of the use of letin a tikz path. I hope the computations are clear enough. The coefficients A, B and C are the coefficients of the quadratic polynomial Ax^2 + Bx + C. The points must be entered in increasing order of the x coordinates for the code to work correctly. The code is \documentclass{article} \usepackage{tikz} \usetikzlibrary{calc} \begin{document} \begin{tikzpicture} \coordinate (1) at (0.1,0.2); \coordinate (2) at (0.2,0.7); \coordinate (3) at (0.4,-0.3); \draw let \p1 = (1), \p2 = (2), \p3 = (3), \n{denom} = {(\x1 - \x2)(\x1 - \x3)(\x2-\x3)}, \n{A} = {(\x3(\y2-\y1) + \x2(\y1-\y3) + \x1(\y3-\y2))/\n{denom}}, \n{B} = {(\x3\x3(\y1-\y2) + \x2\x2(\y3-\y1)+\x1\x1(\y2-\y3))/\n{denom}}, \n{C} = {(\x2\x3(\x2-\x3)\y1 + \x3\x1(\x3-\x1)\y2 + \x1\x2(\x1-\x2)\y3)/\n{denom}} in plot[domain=\x1:\x3] (\x,{\n{A}\x\x+\n{B}\x + \n{C}}); \end{tikzpicture} \end{document} • \coordinate (1) at (-3,0); \coordinate (2) at (-2,2); \coordinate (3) at (0,0); gives dimension too large – Alain Matthes Jan 14 '12 at 22:40 • @Altermundus: As I mentioned in my answer, there are (major) limitations on what my answer can do. That is why I used points that are close together. – Frédéric Jan 15 '12 at 4:24 Here is a solution inspired by this answer. \documentclass{standalone} \usepackage{tikz} \makeatletter \def\parabola@save@target#1{% \def\parabola@target{#1}} \def\parabola@save@start#1{% \def\parabola@start{#1}} \def\parabola@save@midpoint#1{% \def\parabola@midpoint{#1}} \tikzset{ parabola through/.style={ to path={% \pgfextra{% \edef\parabola@@target{(\tikztotarget)}% \tikz@scan@one@point\parabola@save@target\parabola@@target\relax \edef\parabola@@start{(\tikztostart)}% \tikz@scan@one@point\parabola@save@start\parabola@@start\relax \edef\parabola@@midpoint{(#1)}% \tikz@scan@one@point\parabola@save@midpoint\parabola@@midpoint\relax \parabola@start \pgfmathsetmacro{\parabola@xa}{\the\pgf@x/1cm} \pgfmathsetmacro{\parabola@ya}{\the\pgf@y/1cm} \parabola@midpoint \pgfmathsetmacro{\parabola@xb}{\the\pgf@x/1cm} \pgfmathsetmacro{\parabola@yb}{\the\pgf@y/1cm} \parabola@target \pgfmathsetmacro{\parabola@xc}{\the\pgf@x/1cm} \pgfmathsetmacro{\parabola@yc}{\the\pgf@y/1cm} % f(x) = ax^2 + bx + c % a=-(-x1y3+x3y1+x2y3+x1y2-x2y1-x3y2)/(x1x3^2-x2x3^2+x2x1^2-x3x1^2+x3x2^2-x1x2^2) % b=(-x1^2y3+x1^2y2+y1x3^2-y2x3^2+x2^2y3-y1x2^2)/((x1-x2)(-x1x3+x1x2+x3^2-x2x3)) % c=(x1^2x2y3-x1^2x3y2-x2^2x1y3+y2x1x3^2+x2^2x3y1-y1x2x3^2)/((x1-x2)(-x1x3+x1x2+x3^2-x2x3)) \pgfmathsetmacro{\parabola@a}{-(-\parabola@xa\parabola@yc+\parabola@xc\parabola@ya+\parabola@xb\parabola@yc+\parabola@xa\parabola@yb-\parabola@xb\parabola@ya-\parabola@xc\parabola@yb)/(\parabola@xa\parabola@xc^2-\parabola@xb\parabola@xc^2+\parabola@xb\parabola@xa^2-\parabola@xc\parabola@xa^2+\parabola@xc\parabola@xb^2-\parabola@xa\parabola@xb^2)} \pgfmathsetmacro{\parabola@b}{(-\parabola@xa^2\parabola@yc+\parabola@xa^2\parabola@yb+\parabola@ya\parabola@xc^2-\parabola@yb\parabola@xc^2+\parabola@xb^2\parabola@yc-\parabola@ya\parabola@xb^2)/((\parabola@xa-\parabola@xb)(-\parabola@xa\parabola@xc+\parabola@xa\parabola@xb+\parabola@xc^2-\parabola@xb\parabola@xc))} \pgfmathsetmacro{\parabola@c}{(\parabola@xa^2\parabola@xb\parabola@yc-\parabola@xa^2\parabola@xc\parabola@yb-\parabola@xb^2\parabola@xa\parabola@yc+\parabola@yb\parabola@xa\parabola@xc^2+\parabola@xb^2\parabola@xc\parabola@ya-\parabola@ya\parabola@xb\parabola@xc^2)/((\parabola@xa-\parabola@xb)(-\parabola@xa\parabola@xc+\parabola@xa\parabola@xb+\parabola@xc^2-\parabola@xb\parabola@xc))} \draw plot[samples=100,domain=\parabola@xa:\parabola@xc] function {\parabola@a(x*2)+\parabola@bx+\parabola@c}; } } } } \makeatother \begin{document} \begin{tikzpicture} \node[circle,fill=red] at (0,0) {}; \node[circle,fill=blue] at (2,2) {}; \node[circle,fill=green] at (4,0) {}; \draw (0,0) to[parabola through={(2,2)}] (4,0); \end{tikzpicture} \end{document} • \draw (-3,0) to[parabola through={(-2,2)}] (0,0); is incorrect – Alain Matthes Jan 14 '12 at 22:43 • @Altermundus You're right. maple gave me this solution. Maybe some operator precedence issue? The use of Lagrange polynomials is much better here. – cjorssen Jan 15 '12 at 21:38 You can rely on \pgfmathparse to always return any given length in pt. Take a look at the output of: \pgfmathparse{12cm+1pt} \pgfmathresult \pgfmathparse{1pt} \pgfmathresult which will output as: 342.43306 1.0 With this you can pretty much do all your calculation by transferring to the (now dimensionless) unit pt. However you should be weary of very large numbers. To circumvent this you could rely on the fpu unit of pgf. % Preamble \usepgflibrary{fpu} \begin{document} \pgfkeys{/pgf/fpu,/pgf/fpu/output format=fixed} \pgfmathparse{12cm+1pt} \pgfmathresult \pgfmathparse{1pt} \pgfmathresult 342.43306000000000 1.0000000000 Then you have access to numbers which are always in one specific unit! Thank you all for your answers. They helped me a lot. Finally I chose the following approach which uses gnuplot. It is not very elegant but it meets my needs. \documentclass{standalone} \usepackage{tikz} \usetikzlibrary{calc} \newcommand\parabola[3]{% \path[draw] let \p1=#1,\p2=#2,\p3=#3 in \pgfextra \pgfmathsetmacro{\xa}{\x1/1cm} \pgfmathsetmacro{\ya}{\y1/1cm} \pgfmathsetmacro{\xb}{\x2/1cm} \pgfmathsetmacro{\yb}{\y2/1cm} \pgfmathsetmacro{\xc}{\x3/1cm} \pgfmathsetmacro{\yc}{\y3/1cm} \endpgfextra plot[domain=\xa:\xc] function{ \ya((x-\xb)(x-\xc))/((\xa-\xb)(\xa-\xc))+ \yb((x-\xa)(x-\xc))/((\xb-\xa)(\xb-\xc))+ \yc((x-\xa)(x-\xb))/((\xc-\xa)*(\xc-\xb)) };} \begin{document} \begin{tikzpicture} \coordinate (A) at (1,0); \coordinate (B) at (3,8); \coordinate (C) at (4,-1); \fill[red] (A) circle (2pt) (B) circle (2pt) (C) circle (2pt); \parabola{(A)}{(B)}{(C)} \end{tikzpicture} \end{document} • Fine idea to use the Lagrange polynomial without develop it but you can develop the idea and gnuplot can divide for you. I update my answer. – Alain Matthes Jan 15 '12 at 7:05 • @Altermundus and Maeshtro I've tried this but my system spits this at me: Package pgf Warning: Plot data file <file name> no foundo n input line any idea why? – Pureferret Feb 11 '12 at 17:02 • It's difficult to answer with so few elements. First you need to test gnuplot, then you need to authorize TeX to launch gnuplot and to verify your path. – Alain Matthes Feb 11 '12 at 18:17	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9023674726486206, "perplexity": 2330.125224819324}, "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-2020-50/segments/1606141187753.32/warc/CC-MAIN-20201126084625-20201126114625-00009.warc.gz"}
https://sciencehouse.wordpress.com/category/physics/page/2/	# Talk in Göttingen I’m currently in Göttingen, Germany at the Bernstein Sparks Workshop: Beyond mean field theory in the neurosciences, a topic near and dear to my heart. The slides for my talk are here. Of course no trip to Göttingen would be complete without a visit to Gauss’s grave and Max Born’s house. Photos below. # New paper on path integrals Carson C. Chow and Michael A. Buice. Path Integral Methods for Stochastic Differential Equations. The Journal of Mathematical Neuroscience, 5:8 2015. Abstract: Stochastic differential equations (SDEs) have multiple applications in mathematical neuroscience and are notoriously difficult. Here, we give a self-contained pedagogical review of perturbative field theoretic and path integral methods to calculate moments of the probability density function of SDEs. The methods can be extended to high dimensional systems such as networks of coupled neurons and even deterministic systems with quenched disorder. This paper is a modified version of our arXiv paper of the same title. We added an example of the stochastically forced FitzHugh-Nagumo equation and fixed the typos. # Talk at Jackfest I’m currently in Banff, Alberta for a Festschrift for Jack Cowan (webpage here). Jack is one of the founders of theoretical neuroscience and has infused many important ideas into the field. The Wilson-Cowan equations that he and Hugh Wilson developed in the early seventies form a foundation for both modeling neural systems and machine learning. My talk will summarize my work on deriving “generalized Wilson-Cowan equations” that include both neural activity and correlations. The slides can be found here. References and a summary of the work can be found here. All videos of the talks can be found here. Addendum: 17:44. Some typos in the talk were fixed. Addendum: 18:25. I just realized I said something silly in my talk. The Legendre transform is an involution because the transform of the transform is the inverse. I said something completely inane instead. # Analytic continuation continued As I promised in my previous post, here is a derivation of the analytic continuation of the Riemann zeta function to negative integer values. There are several ways of doing this but a particularly simple way is given by Graham Everest, Christian Rottger, and Tom Ward at this link. It starts with the observation that you can write $\int_1^\infty x^{-s} dx = \frac{1}{s-1}$ if the real part of $s>0$. You can then break the integral into pieces with $\frac{1}{s-1}=\int_1^\infty x^{-s} dx =\sum_{n=1}^\infty\int_n^{n+1} x^{-s} dx$ $=\sum_{n=1}^\infty \int_0^1(n+x)^{-s} dx=\sum_{n=1}^\infty\int_0^1 \frac{1}{n^s}\left(1+\frac{x}{n}\right)^{-s} dx$ (1) For $x\in [0,1]$, you can expand the integrand in a binomial expansion $\left(1+\frac{x}{n}\right)^{-s} = 1 +\frac{sx}{n}+sO\left(\frac{1}{n^2}\right)$ (2) Now substitute (2) into (1) to obtain $\frac{1}{s-1}=\zeta(s) -\frac{s}{2}\zeta(s+1) - sR(s)$ (3) or $\zeta(s) =\frac{1}{s-1}+\frac{s}{2}\zeta(s+1) +sR(s)$ (3′) where the remainder $R$ is an analytic function when $Re s > -1$ because the resulting series is absolutely convergent. Since the zeta function is analytic for $Re s >1$, the right hand side is a new definition of $\zeta$ that is analytic for $s >0$ aside from a simple pole at $s=1$. Now multiply (3) by $s-1$ and take the limit as $s\rightarrow 1$ to obtain $\lim_{s\rightarrow 1} (s-1)\zeta(s)=1$ which implies that $\lim_{s\rightarrow 0} s\zeta(s+1)=1$ (4) Taking the limit of $s$ going to zero from the right of (3′) gives $\zeta(0^+)=-1+\frac{1}{2}=-\frac{1}{2}$ Hence, the analytic continuation of the zeta function to zero is -1/2. The analytic domain of $\zeta$ can be pushed further into the left hand plane by extending the binomial expansion in (2) to $\left(1+\frac{x}{n}\right)^{-s} = \sum_{r=0}^{k+1} \left(\begin{array}{c} -s\\r\end{array}\right)\left(\frac{x}{n}\right)^r + (s+k)O\left(\frac{1}{n^{k+2}}\right)$ Inserting into (1) yields $\frac{1}{s-1}=\zeta(s)+\sum_{r=1}^{k+1} \left(\begin{array}{c} -s\\r\end{array}\right)\frac{1}{r+1}\zeta(r+s) + (s+k)R_{k+1}(s)$ where $R_{k+1}(s)$ is analytic for $Re s>-(k+1)$. Now let $s\rightarrow -k^+$ and extract out the last term of the sum with (4) to obtain $\frac{1}{-k-1}=\zeta(-k)+\sum_{r=1}^{k} \left(\begin{array}{c} k\\r\end{array}\right)\frac{1}{r+1}\zeta(r-k) - \frac{1}{(k+1)(k+2)}$ (5) Rearranging (5) gives $\zeta(-k)=-\sum_{r=1}^{k} \left(\begin{array}{c} k\\r\end{array}\right)\frac{1}{r+1}\zeta(r-k) -\frac{1}{k+2}$ (6) where I have used $\left( \begin{array}{c} -s\\r\end{array}\right) = (-1)^r \left(\begin{array}{c} s+r -1\\r\end{array}\right)$ The righthand side of (6) is now defined for $Re s > -k$. Rewrite (6) as $\zeta(-k)=-\sum_{r=1}^{k} \frac{k!}{r!(k-r)!} \frac{\zeta(r-k)(k-r+1)}{(r+1)(k-r+1)}-\frac{1}{k+2}$ $=-\sum_{r=1}^{k} \left(\begin{array}{c} k+2\\ k-r+1\end{array}\right) \frac{\zeta(r-k)(k-r+1)}{(k+1)(k+2)}-\frac{1}{k+2}$ $=-\sum_{r=1}^{k-1} \left(\begin{array}{c} k+2\\ k-r+1\end{array}\right) \frac{\zeta(r-k)(k-r+1)}{(k+1)(k+2)}-\frac{1}{k+2} - \frac{\zeta(0)}{k+1}$ Collecting terms, substituting for $\zeta(0)$ and multiplying by $(k+1)(k+2)$ gives $(k+1)(k+2)\zeta(-k)=-\sum_{r=1}^{k-1} \left(\begin{array}{c} k+2\\ k-r+1\end{array}\right) \zeta(r-k)(k-r+1) - \frac{k}{2}$ Reindexing gives $(k+1)(k+2)\zeta(-k)=-\sum_{r'=2}^{k} \left(\begin{array}{c} k+2\\ r'\end{array}\right) \zeta(-r'+1)r'-\frac{k}{2}$ Now, note that the Bernoulli numbers satisfy the condition $\sum_{r=0}^{N-1} B_r = 0$. Hence, let $\zeta(-r'+1)=-\frac{B_r'}{r'}$ and obtain $(k+1)(k+2)\zeta(-k)=\sum_{r'=0}^{k+1} \left(\begin{array}{c} k+2\\ r'\end{array}\right) B_{r'}-B_0-(k+2)B_1-(k+2)B_{k+1}-\frac{k}{2}$ which using $B_0=1$ and $B_1=-1/2$ gives the self-consistent condition $\zeta(-k)=-\frac{B_{k+1}}{k+1}$, which is the analytic continuation of the zeta function for integers $k\ge 1$. # Analytic continuation I have received some skepticism that there are possibly other ways of assigning the sum of the natural numbers to a number other than -1/12 so I will try to be more precise. I thought it would be also useful to derive the analytic continuation of the zeta function, which I will do in a future post. I will first give a simpler example to motivate the notion of analytic continuation. Consider the geometric series $1+s+s^2+s^3+\dots$. If $\|s\| < 1$ then we know that this series is equal to $\frac{1}{1-s}$ (1) Now, while the geometric series is only convergent and thus analytic inside the unit circle, (1) is defined everywhere in the complex plane except at $s=1$. So even though the sum doesn’t really exist outside of the domain of convergence, we can assign a number to it based on (1). For example, if we set $s=2$ we can make the assignment of $1 + 2 + 4 + 8 + \dots = -1$. So again, the sum of the powers of two doesn’t really equal -1, only (1) is defined at s=2. It’s just that the geometric series and (1) are the same function inside the domain of convergence. Now, it is true that the analytic continuation of a function is unique. However, although the value of -1 for $s=-1$ is the only value for the analytic continuation of the geometric series, that doesn’t mean that the sum of the powers of 2 needs to be uniquely assigned to negative one because the sum of the powers of 2 is not an analytic function. So if you could find some other series that is a function of some parameter $z$ that is analytic in some domain of convergence and happens to look like the sum of the powers of two for some $z$ value, and you can analytically continue the series to that value, then you would have another assignment. Now consider my example from the previous post. Consider the series $\sum_{n=1}^\infty \frac{n-1}{n^{s+1}}$ (2) This series is absolutely convergent for $s>1$. Also note that if I set s=-1, I get $\sum_{n=1}^\infty (n-1) = 0 +\sum_{n'=1}^\infty n' = 1 + 2 + 3 + \dots$ which is the sum of then natural numbers. Now, I can write (2) as $\sum_{n=1}^\infty\left( \frac{1}{n^s}-\frac{1}{n^{s+1}}\right)$ and when the real part of s is greater than 1, I can further write this as $\sum_{n=1}^\infty\frac{1}{n^s}-\sum_{n=1}^\infty\frac{1}{n^{s+1}}=\zeta(s)-\zeta(s+1)$ (3) All of these operations are perfectly fine as long as I’m in the domain of absolute convergence. Now, as I will show in the next post, the analytic continuation of the zeta function to the negative integers is given by $\zeta (-k) = -\frac{B_{k+1}}{k+1}$ where $B_k$ are the Bernoulli numbers, which is given by the Taylor expansion of $\frac{x}{e^x-1} = \sum B_n \frac{x^n}{n!}$ (4) The first few Bernoulli numbers are $B_0=1, B_1=-1/2, B_2 = 1/6$. Thus using this in (4) gives $\zeta(-1)=-1/12$. A similar proof will give $\zeta(0)=-1/2$. Using this in (3) then gives the desired result that the sum of the natural numbers is (also) 5/12. Now this is not to say that all assignments have the same physical value. I don’t know the details of how -1/12 is used in bosonic string theory but it is likely that the zeta function is crucial to the calculation. # Nonuniqueness of -1/12 I’ve been asked to give an example of how the sum of the natural numbers could lead to another value in the comments to my previous post so I thought it may be of general interest to more people. Consider again $S=1+2+3+4\dots$ to be the sum of the natural numbers. The video in the previous slide gives a simple proof by combining divergent sums. In essence, the manipulation is doing renormalization by subtracting away infinities and the left over of this renormalization is -1/12. There is another video that gives the proof through analytic continuation of the Riemann zeta function $\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}$ The zeta function is only strictly convergent when the real part of s is greater than 1. However, you can use analytic continuation to extract values of the zeta function to values where the sum is divergent. What this means is that the zeta function is no longer the “same sum” per se, but a version of the sum taken to a domain where it was not originally defined but smoothly (analytically) connected to the sum. Hence, the sum of the natural numbers is given by $\zeta(-1)$ and $\zeta(0)=\sum_{n=1}^\infty 1$, (infinite sum over ones). By analytic continuation, we obtain the values $\zeta(-1)=-1/12$ and $\zeta(0)=-1/2$. Now notice that if I subtract the sum over ones from the sum over the natural numbers I still get the sum over the natural numbers, e.g. $1+2+3+4\dots - (1+1+1+1\dots)=0+1+2+3+4\dots$. Now, let me define a new function $\xi(s)=\zeta(s)-\zeta(s+1)$ so $\xi(-1)$ is the sum over the natural numbers and by analytic continuation $\xi(-1)=-1/12+1/2=5/12$ and thus the sum over the natural numbers is now 5/12. Again, if you try to do arithmetic with infinity, you can get almost anything. A fun exercise is to create some other examples. # The sum of the natural numbers is -1/12? This wonderfully entertaining video giving a proof for why the sum of the natural numbers is -1/12 has been viewed over 1.5 million times. It just shows that there is a hunger for interesting and well explained math and science content out there. Now, we all know that the sum of all the natural numbers is infinite but the beauty (insidiousness) of infinite numbers is that they can be assigned to virtually anything. The proof for this particular assignment considers the subtraction of the divergent oscillating sum $S_1=1-2+3-4+5 \dots$ from the divergent sum of the natural numbers $S = 1 + 2 + 3+4+5\dots$ to obtain $4S$. Then by similar trickery it assigns $S_1=1/4$. Solving for $S$ gives you the result $S = -1/12$. Hence, what you are essentially doing is dividing infinity by infinity and that as any school child should know, can be anything you want. The most astounding thing to me about the video was learning that this assignment was used in string theory, which makes me wonder if the calculations would differ if I chose a different assignment. Addendum: Terence Tao has a nice blog post on evaluating such sums. In a “smoothed” version of the sum, it can be thought of as the “constant” in front of an asymptotic divergent term. This constant is equivalent to the analytic continuation of the Riemann zeta function. Anyway, the -1/12 seems to be a natural way to assign a value to the divergent sum of the natural numbers.	{"extraction_info": {"found_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": 80, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9599145650863647, "perplexity": 248.03152360866815}, "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-2018-26/segments/1529267867095.70/warc/CC-MAIN-20180624215228-20180624235228-00169.warc.gz"}
https://en.m.wikipedia.org/wiki/Greedy_algorithm	# Greedy algorithm Greedy algorithms determine minimum number of coins to give while making change. These are the steps a human would take to emulate a greedy algorithm to represent 36 cents using only coins with values {1, 5, 10, 20}. The coin of the highest value, less than the remaining change owed, is the local optimum. (In general the change-making problem requires dynamic programming to find an optimal solution; however, most currency systems, including the Euro and US Dollar, are special cases where the greedy strategy does find an optimal solution.) A greedy algorithm is an algorithmic paradigm that follows the problem solving heuristic of making the locally optimal choice at each stage[1] with the intent of finding a global optimum. In many problems, a greedy strategy does not usually produce an optimal solution, but nonetheless a greedy heuristic may yield locally optimal solutions that approximate a globally optimal solution in a reasonable amount of time. For example, a greedy strategy for the traveling salesman problem (which is of a high computational complexity) is the following heuristic: "At each step of the journey, visit the nearest unvisited city." This heuristic does not intend to find a best solution, but it terminates in a reasonable number of steps; finding an optimal solution to such a complex problem typically requires unreasonably many steps. In mathematical optimization, greedy algorithms optimally solve combinatorial problems having the properties of matroids, and give constant-factor approximations to optimization problems with submodular structure. ## Specifics In general, greedy algorithms have five components: 1. A candidate set, from which a solution is created 2. A selection function, which chooses the best candidate to be added to the solution 3. A feasibility function, that is used to determine if a candidate can be used to contribute to a solution 4. An objective function, which assigns a value to a solution, or a partial solution, and 5. A solution function, which will indicate when we have discovered a complete solution Greedy algorithms produce good solutions on some mathematical problems, but not on others. Most problems for which they work will have two properties: Greedy choice property We can make whatever choice seems best at the moment and then solve the subproblems that arise later. The choice made by a greedy algorithm may depend on choices made so far, but not on future choices or all the solutions to the subproblem. It iteratively makes one greedy choice after another, reducing each given problem into a smaller one. In other words, a greedy algorithm never reconsiders its choices. This is the main difference from dynamic programming, which is exhaustive and is guaranteed to find the solution. After every stage, dynamic programming makes decisions based on all the decisions made in the previous stage, and may reconsider the previous stage's algorithmic path to solution. Optimal substructure "A problem exhibits optimal substructure if an optimal solution to the problem contains optimal solutions to the sub-problems."[2] ### Cases of failure Examples on how a greedy algorithm may fail to achieve the optimal solution. Starting from A, a greedy algorithm that tries to find the maximum by following the greatest slope will find the local maximum at "m", oblivious to the global maximum at "M". With a goal of reaching the largest-sum, at each step, the greedy algorithm will choose what appears to be the optimal immediate choice, so it will choose 12 instead of 3 at the second step, and will not reach the best solution, which contains 99. For many other problems, greedy algorithms fail to produce the optimal solution, and may even produce the unique worst possible solution. One example is the traveling salesman problem mentioned above: for each number of cities, there is an assignment of distances between the cities for which the nearest-neighbor heuristic produces the unique worst possible tour.[3] ## Types Greedy algorithms can be characterized as being 'short sighted', and also as 'non-recoverable'. They are ideal only for problems which have 'optimal substructure'. Despite this, for many simple problems, the best suited algorithms are greedy algorithms. It is important, however, to note that the greedy algorithm can be used as a selection algorithm to prioritize options within a search, or branch-and-bound algorithm. There are a few variations to the greedy algorithm: • Pure greedy algorithms • Orthogonal greedy algorithms • Relaxed greedy algorithms ## Theory Greedy algorithms have a long history of study in combinatorial optimization and theoretical computer science. Greedy heuristics are known to produce suboptimal results on many problems,[4] and so natural questions are: • For which problems do greedy algorithms perform optimally? • For which problems do greedy algorithms guarantee an approximately optimal solution? • For which problems is the greedy algorithm guaranteed not to produce an optimal solution? A large body of literature exists answering these questions for general classes of problems, such as matroids, as well as for specific problems, such as set cover. ### Matroids A matroid is a mathematical structure that generalizes the notion of linear independence from vector spaces to arbitrary sets. If an optimization problem has the structure of a matroid, then the appropriate greedy algorithm will solve it optimally.[5] ### Submodular functions A function ${\displaystyle f}$ defined on subsets of a set ${\displaystyle \Omega }$ is called submodular if for every ${\displaystyle S,T\subseteq \Omega }$ we have that ${\displaystyle f(S)+f(T)\geq f(S\cup T)+f(S\cap T)}$ . Suppose one wants to find a set ${\displaystyle S}$ which maximizes ${\displaystyle f}$ . The greedy algorithm, which builds up a set ${\displaystyle S}$ by incrementally adding the element which increases ${\displaystyle f}$ the most at each step, produces as output a set that is at least ${\displaystyle (1-1/e)\max _{X\subseteq \Omega }f(X)}$ .[6] That is, greedy performs within a constant factor of ${\displaystyle (1-1/e)\approx 0.63}$ as good as the optimal solution. Similar guarantees are provable when additional constraints, such as cardinality constraints,[7] are imposed on the output, though often slight variations on the greedy algorithm are required. See [8] for an overview. ### Other problems with guarantees Other problems for which the greedy algorithm gives a strong guarantee, but not an optimal solution, include Many of these problems have matching lower bounds; i.e., the greedy algorithm does not perform better, in the worst case, than the guarantee. ## Applications Greedy algorithms mostly (but not always) fail to find the globally optimal solution because they usually do not operate exhaustively on all the data. They can make commitments to certain choices too early which prevent them from finding the best overall solution later. For example, all known greedy coloring algorithms for the graph coloring problem and all other NP-complete problems do not consistently find optimum solutions. Nevertheless, they are useful because they are quick to think up and often give good approximations to the optimum. If a greedy algorithm can be proven to yield the global optimum for a given problem class, it typically becomes the method of choice because it is faster than other optimization methods like dynamic programming. Examples of such greedy algorithms are Kruskal's algorithm and Prim's algorithm for finding minimum spanning trees, and the algorithm for finding optimum Huffman trees. Greedy algorithms appear in network routing as well. Using greedy routing, a message is forwarded to the neighboring node which is "closest" to the destination. The notion of a node's location (and hence "closeness") may be determined by its physical location, as in geographic routing used by ad hoc networks. Location may also be an entirely artificial construct as in small world routing and distributed hash table. ## Notes 1. ^ Black, Paul E. (2 February 2005). "greedy algorithm". Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology (NIST). Retrieved 17 August 2012. 2. ^ Introduction to Algorithms (Cormen, Leiserson, Rivest, and Stein) 2001, Chapter 16 "Greedy Algorithms". 3. ^ Gutin, Gregory; Yeo, Anders; Zverovich, Alexey (2002). "Traveling salesman should not be greedy: Domination analysis of greedy-type heuristics for the TSP". Discrete Applied Mathematics. 117 (1–3): 81–86. doi:10.1016/S0166-218X(01)00195-0. 4. ^ U. Feige. A threshold of ln n for approximating set cover. Journal of the ACM (JACM), 45(4):634–652, 1998. 5. ^ Papadimitriou, Christos H., and Kenneth Steiglitz. Combinatorial optimization: algorithms and complexity. Courier Corporation, 1998. 6. ^ G. Nemhauser, L.A. Wolsey, and M.L. Fisher. "An analysis of approximations for maximizing submodular set functions—I." Mathematical Programming 14.1 (1978): 265-294. 7. ^ N. Buchbinder, et al. "Submodular maximization with cardinality constraints." Proceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms. Society for Industrial and Applied Mathematics, 2014. 8. ^ Krause, Andreas, and Daniel Golovin. "Submodular function maximization." (2014): 71-104.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 10, "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": 0.8453146815299988, "perplexity": 729.1521116878679}, "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/1566027316549.78/warc/CC-MAIN-20190821220456-20190822002456-00391.warc.gz"}
http://mathhelpforum.com/advanced-statistics/592-some-random-variable-confusion.html	# Math Help - Some random variable confusion! 1. ## Any help here would be stupendous!!! Hello, I was a bit confused by this problem that appeared in my prob and stat textbook: "Suppose that X had a uniform distribution on the interval [0,5] and that the random variable Y is defined by Y = 0 if X <= 1, Y = 5 if X >= 3, and Y = X otherwise. Sketch the d.f. of Y" I wanted to solve this problem by first finding the mixed probability function of Y, f_Y. To do this I first found the probability function of X, f_X: f_X(x) = { 0 if x < 0; 1/5 if 0 <= x <= 5; 0 if x > 5 } by the fact that X has uniform distribution on [0,5]. Then to find f_Y, the mixed probability function of Y: Pr(Y = 0) = Pr(X <= 1) = 1/5 Also, it's given that Y = 5 if X >= 3 which means that Pr(Y = 5) = Pr(X >= 3) = 2/5 The only solution I could think of was that after summing the places of non-zero probability mass at Y = 0 and Y = 5 to get 1/5 + 2/5 = 3/5, that then it forces Pr(0 < Y < 5) = 2/5. Then maybe would I interpret "Y = X otherwise" to mean that simply Y distrubutes the probability 2/5 uniformly over the interval (0,5)? In that case I would have: f_Y(y) = { 0 if y < 0; 1/5 if y = 0; 2/25 if 0 < y < 5; 2/5 if y = 5; 0 if y > 5 } And then the Pr(Y = 0) + Pr(0 < Y < 5) + Pr(Y = 5) = 1/5 + 10/25 + 2/5 = 25/25 = 1 as needed. But I'm not sure that that's what is meant by "Y = X otherwise" Oh well, I'd really appreciate any help on determining the correct formulas for the mixed probability function of Y. Thanks in advance for any help, 2. Originally Posted by epv Hello, I was a bit confused by this problem that appeared in my prob and stat textbook: "Suppose that X had a uniform distribution on the interval [0,5] and that the random variable Y is defined by Y = 0 if X <= 1, Y = 5 if X >= 3, and Y = X otherwise. Sketch the d.f. of Y" Try this: $ F_Y(x) = \begin{cases} 0 \quad (x < 0}) \\ \frac15 \quad(0 \le x < 1) \\ \frac{x}{5} \quad (1 \le x < 3) \\ \frac35 \quad(3 \le x < 5)\\	{"extraction_info": {"found_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": 0.846038818359375, "perplexity": 391.327974488443}, "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-2016-07/segments/1454701148402.62/warc/CC-MAIN-20160205193908-00272-ip-10-236-182-209.ec2.internal.warc.gz"}
http://softschools.com/math/algebra/topics/completing_the_square_when_a_notequal_1/	# Completing the Square when a ≠ 1 A quadratic equation is an equation that contains a squared variable as its highest power on any variable. The general form of a quadratic equation is: ax2 + bx + c = 0 Where a, b, and c are constants and a ≠ 0. In other words there must be a x2 term. Some examples are: x2 + 3x - 3 = 0 4x2 + 9 = 0 (Where b = 0) x2 + 5x = 0 (where c = 0) One way to solve a quadratic equation is by completing the square. ax2 + bx + c = 0 → (x- r)2 = S Where r and s are constants. PART I of this topic focused on completing the square when a, the x2-coefficient, is 1. This part, PART II, will focus on completing the square when a, the x2-coefficient, is not 1. Let's solve the following equation by completing the square: 2x2 + 8x - 5 = 0 Step 1: Write the equation in the general form ax2 + bx + c = 0. This equation is already in the proper form where a = 2 and c = -5. 2x2 + 8x - 5 = 0 Step 2: Move c, the constant term, to the right-hand side of the equation. c = -5 2x2 + 8x = 5 Step 3: Factor out a from the left-hand side. This changes the value of the x-coefficient. a = 2 2(x2 + 4x) = 5 Step 4: Complete the square of the expression in parentheses on the left-hand side of the equation. The expression is x2 + 4x. Divide the x-coefficient by two and square the result. x2 + 4x x-coefficient = 4 $\frac{4}{2}={2}\to {r}$ (2)2 = 4 Step 5: Add the result from Step 4 to the parenthetical expression on the left-hand side. Then add a x result to the right-hand side. To keep the equation true what is done to one side must also be done to the other. When adding the result to the parenthetical expression on the left-hand side the total value added is a x result. So this value must also be added to the right-hand side. 2(x2 + 4x + 4) = 5 + 2(4) Step 6: Rewrite the left-hand side as a perfect square and simplify the right-hand side. When rewriting in perfect square format the value in the parentheses is the x-coefficient of the parenthetical expression divided by 2 as found in Step 4. 2(x + 2)2 = 13 Now that the square has been completed, solve for x. Step 7: Divide both sides by a. ${\left(x+2\right)}^{2}=\frac{13}{{2}}$ Step 8: Take the square root of both sides of the equation. Remember that when taking the square root on the right-hand side the answer can be positive or negative. $x+2=±\sqrt{\frac{13}{2}}$ Step 9: Solve for x. $x=-2±\sqrt{\frac{13}{2}}$ Example 1: 3x2 = 6x + 7 Step 1: Write the equation in the general form ax2 + bx + c = 0. Where a = 3 and c = -7. 3x2 - 6x - 7 = 0 Step 2: Move c, the constant term, to the right-hand side of the equation. c = -7 3x2 - 6x = 7 Step 3: Factor out a from the left-hand side. This changes the value of the x -coefficient. a = 3 3(x2 - 2x) = 7 Step 4: Complete the square of the expression in parentheses on the left-hand side of the equation. The expression is x2 - 2x. Divide the x-coefficient by two and square the result. x2 - 2x x -coefficient = -2 $\frac{-2}{2}={-}{1}\to {r}$ (-1)2 = 1 Step 5: Add the result from Step 4 to the parenthetical expression on the left-hand side. Then add a x result to the right-hand side. To keep the equation true what is done to one side must also be done to the other. When adding the result to the parenthetical expression on the left-hand side the total value added is a x result. So this value must also be added to the right-hand side. 3(x2 - 2x + 1) = 7 + 3(1) Step 6: Rewrite the left-hand side as a perfect square and simplify the right-hand side. When rewriting in perfect square format the value in the parentheses is the x-coefficient of the parenthetical expression divided by 2, as found in Step 4. 3(x - 1)2 = 10 Now that the square has been completed, solve for x. Step 7: Divide both sides by a. ${\left(x-1\right)}^{2}=\frac{10}{{3}}$ Step 8: Take the square root of both sides of the equation. Remember that when taking the square root on the right-hand side the answer can be positive or negative. $x-1=±\sqrt{\frac{10}{3}}$ Step 9: Solve for x. $x=1±\sqrt{\frac{10}{3}}$ Example 2: 5x2 - 0.6 = 4x Step 1: Write the equation in the general form ax2 + bx + c = 0. Where a = 5 and c = 0.6. 5x2 - 4x - 0.6 = 0 Step 2: Move c, the constant term, to the right-hand side of the equation. c = -0.6 5x2 - 4x = 0.6 Step 3: Factor out a from the left-hand side. This changes the value of the x-coefficient. a = 5 5(x2 - 0.8x) = 0.6 Step 4: Complete the square of the expression in parentheses on the left-hand side of the equation. The expression is x2 - 0.8x. Divide the x-coefficient by two and square the result. x2 - 0.8x x-coefficient = -0.8 $\frac{-0.8}{2}={-}{0.4}\to {r}$ (-0.4)2 = 0.16 Step 5: Add the result from Step 4 to the parenthetical expression on the left-hand side. Then add a x result to the right-hand side. To keep the equation true what is done to one side must also be done to the other. When adding the result to the parenthetical expression on the left-hand side the total value added is a x result. So this value must also be added to the right-hand side. 5(x2 - 0.8x + 0.16) = 0.6 + 5(0.16) Step 6: Rewrite the left-hand side as a perfect square and simplify the right-hand side. When rewriting in perfect square format the value in the parentheses is the x-coefficient of the parenthetical expression divided by 2 as found in Step 4. 5(x - 0.4)2 = 1.4 Now that the square has been completed, solve for x. Step 7: Divide both sides by a. ${\left(x-0.4\right)}^{2}=\frac{1.4}{{5}}=0.28$ Step 8: Take the square root of both sides of the equation. Remember that when taking the square root on the right-hand side the answer can be positive or negative. $x-0.4=±\sqrt{0.28}$ Step 9: Solve for x. $x=0.4±\sqrt{0.28}$ Related Links: Math algebra Factoring Quadratic Equations when a equals 1 Factoring Quadratic Equations when a ≠ 1 Algebra Topics To link to this Completing the Square when a ≠ 1 page, copy the following code to your site:	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 12, "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": 0.8038483262062073, "perplexity": 222.87978538139475}, "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/1487501171176.3/warc/CC-MAIN-20170219104611-00061-ip-10-171-10-108.ec2.internal.warc.gz"}
http://mathcenter.oxford.emory.edu/site/math111/probSetImplicitDiff/	## Exercises - Implicit Differentiation 1. Suppose $y=f(x)$ to find $\displaystyle{\frac{d}{dx} 4y^3}$. $\displaystyle{12y^2 \cdot \frac{dy}{dx}}$ 2. Suppose $y=f(x)$ to find $\displaystyle{\frac{d}{dx} x^5 y^3}$. $\displaystyle{3x^5 y^2 \cdot \frac{dy}{dx} + 5x^4 y^3}$ 3. Suppose $x=g(t)$ to find $\displaystyle{\frac{d}{dt} (t + \cos x^3)}$. $\displaystyle{1 - 3x^2 \sin(x^3) \cdot \frac{dy}{dx}}$ 4. Find $\displaystyle{\frac{dy}{dx}}$ and $\displaystyle{\frac{d^2y}{dx^2}}$ if $x^2 + y^2 = r^2$ for some constant $r$. First we find $\displaystyle{\frac{dy}{dx} = \frac{-x}{y}}$. Then, upon differentiating a second time and substituting in for the first derivative, we obtain $\displaystyle{\frac{d^2 y}{dx^2} = -\frac{x^2+y^2}{y^3}}$. 5. Find the equation of the tangent line at $(1,1)$ to the graph of $x^2y + 3y^4 = x^3 y^3 + 3$. $\displaystyle{y-1 = \frac{1}{10}\left(x-1\right)}$ 6. Find $\displaystyle{\frac{d^2y}{dx^2}}$ if $x^2 + 3y^2 = xy + 3$. First, we find $\displaystyle{\frac{dy}{dx} = \frac{y-2x}{6y-x}}$. Then, upon differentiating again, simplifying, and substituting the first derivative, we obtain $\displaystyle{\frac{d^2 y}{dx^2} = -\frac{11(x^2 - xy + 6y)}{(x - 6y)^2}}$.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9966888427734375, "perplexity": 339.0717734265824}, "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/1614178356456.56/warc/CC-MAIN-20210226085543-20210226115543-00387.warc.gz"}
http://mathoverflow.net/questions/88045/quasi-dense-subsets-of-boolean-algebras/88459	## Quasi-dense subsets of boolean algebras Definition: Let $B$ be a boolean algebra. Say $X \subseteq B$ is quasi-dense in $B$ if for all $b \in B$, there is $x \in X \setminus$ { $0,1$ } such that either $x \leq b$ or $b \leq x$. Question: Suppose $A \subseteq B \subseteq C$ are atomless boolean algebras, $A$ is quasi-dense in $B$, and $B$ is dense in $C$. Does it follow that $A$ is quasi-dense in $C$? - Here is a way to construct an atomless version of Joel´s counterexample: Let $A_0 \subseteq A_1 \subseteq A_2$ be the algebras in Joel´s example (in his notation $A \subseteq B \subseteq C$) and let $X_0$,$X_1$ and $X_2$ be the corresponding Stone spaces. So $X_0$ and $X_1$ are both just a converging sequence and $X_2$ consists of two converging sequences (with different limit points). These spaces have isolated points, reflecting the fact that the algebras have atoms. So the idea now is to consider the (Stone) spaces $Z_i=X_i \times 2^\omega$ and their corresponding algebras of clopen subsets $B_i=Cl(Z_i)$ for $i \in 3$. These algebras are now atomless (since their Stone spaces have no isolated points). The inclusion maps $A_0 \subseteq A_1$ and $A_1 \subseteq A_2$ induce surjective continuous functions $\pi_1:X_1 \to X_0$ and $\pi_2:X_2 \to X_1$ respectively, which we can use to produce the (also continuous and surjective) functions: $\pi_1 \times id_{2^\omega}: Z_1 \to Z_0$ and $\pi_2 \times id_{2^\omega}:Z_2 \to Z_1$. These in turn induce "inclusions" $B_0 \subseteq B_1 \subseteq B_2$. Arguments analogous to those in Joel´s answer can be used to show that: 1) $B_0$ is quasi-dense in $B_1$ (here you use that any clopen set in $Z_1$ is a finite union of boxes). 2) $B_1$ is dense in $B_2$ (using that any clopen set in $Z_2$ contains a box), and 3) $B_0$ is not quasi-dense in $B_2$ (here a problematic clopen subset of $Z_2$ would be $K \times 2^\omega$ where $K$ is one of the two converging sequences). - Well, this still doesn't answer the atomless question, but I've got a violation of the desired implication among atomic Boolean algebras. Let $A$ consist of the finite or cofinite subsets of $\mathbb{N}$ that take $2k$ and $2k+1$ together, if at all. That is, $a\in A$ if $a\subset\mathbb{N}$ is finite or cofinite and for every $k$ we have $2k\in a\leftrightarrow 2k+1\in a$. Let $B$ be the Boolean algebra consisting of all finite or cofinite subsets of $\mathbb{N}$. Note that $A$ is quasi-dense in $B$, since if $b$ is finite, then $b$ is contained in an interval $[0,2k+1]$ for some large $k$, and this is in $A$, and if $b$ is cofinite, then $b$ contains some final segment interval $[2k,\infty)$, which is in $A$. Let $C$ be the algebra generated by the elements of $B$ together with the set $E$ of even numbers. Thus, every element of $C$ is the union of a finite or cofinite subset of $E$ with a finite or cofinite subset of $\mathbb{N}-E$. The algebra $B$ is dense in $C$, since the singletons are dense, and they are finite. Finally, $A$ is not quasi-dense in $C$, because the set $E$ contains no nonzero element of $A$, as it contains no odd numbers, and is contained in no non-unital element of $A$, as the only element of $A$ containing all the even numbers is the whole of $\mathbb{N}$. This is my original answer, which shows merely that quasi-density is not transitive. The answer is no. For a counterexample, let $A$ have at least two atoms; let $B$ split one of those atoms, and let $C$ split both of them. More explicitly, let $A$ be the 4-element Boolean algebra with atoms $\{0,1\}$ and $\{2,3\}$. Let $B$ be the $8$-element algebra with atoms $\{0\}$, $\{1\}$, $\{2,3\}$, and let $C$ be the full power set, with atoms $\{0\}$, $\{1\}$, $\{2\}$, $\{4\}$. You may observe that $A$ is quasi-dense in $B$ and $B$ is quasi-dense in $C$ by inspection. But $A$ is not quasi-dense in $C$, since $\{0,2\}$ is neither above nor below any nontrivial element of $A$. - The question was about "$B$ dense in $C$", not just quasi-dense. – Goldstern Feb 9 2012 at 23:33 But I said $B$ was dense in $C$, not just quasi-dense. This was not a typo. Also, I am only really interested in the case of atomless algebras. I will put that in the original question. – mbsq Feb 9 2012 at 23:36 Oh, sorry, I misread the question. – Joel David Hamkins Feb 9 2012 at 23:57 I have now posted a counterexample to the implication among atomic Boolean algebras. Perhaps it could be used to construct an atomless counterexample, but I don't see it yet. – Joel David Hamkins Feb 10 2012 at 4:46 It's a surprisingly difficult question! – Joel David Hamkins Feb 11 2012 at 0:16 show 1 more comment	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9778130054473877, "perplexity": 164.36132405160393}, "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-2013-20/segments/1368696382396/warc/CC-MAIN-20130516092622-00086-ip-10-60-113-184.ec2.internal.warc.gz"}
https://www.apexprojects.in/golden-section-application-architecture-design/	# Golden Section and its application in Architecture and Design The GOLDEN SECTION is nothing but division & further sub-division of a whole – 1, ½ ….. & so on; which induces a sense of proportion especially visually. Golden ratio is as such a concept derived from the nature where proportions make it symmetrical even in its natural form. The real beauty is when the same proportion is noticed even in the finer details as the portion reduces. First studied as a part of geometry & mathematics where scholars found a symmetry in polygons, this science has evolved manifolds & has been implied in various fields. This is the case when THE GOLDEN RATIO isn’t followed. There are a few standards conceptualised around this concept about the “human body”. This is the sketch of “The Vitruvian Man” first conceptualised by Da Vinci followed by renovation of the idea by Vitruvius himself. The famous architect Le Corbusier, developed a system named “MODULAR” on similar lines for studying & understanding human proportions for better architecture. In architecture, which we perceive as a scientifically conceptualised & executed art, we expect to follow the same reducing proportionate detailing to please the human eye!! It is more about the aspect ratio as we speak about than anything else. Since time immemorial, greatest of the builders have employed this technique knowingly or unknowingly in the most iconic structures which we see today. Here are a few structures that will actually make you think ?! 1. Eiffel Tower, Paris Despite its enormous height, Eiffel Tower doesn’t look out of proportion, all thanks to the golden ratio. The base is broader while it narrows down to the top, perfectly following the golden section & as needed for stability. 2. Taj Mahal, India Taj Mahal doesn’t look a misfit either despite its humungous size. The effect of the enormous dome has been subsided by complementing it with minarets. Also, if you notice the order & proportion of the arches on the main structure keep reducing &reduce proportionately. Horizontal proportionality is observed here. 3. Parthenon, Greece Ancient Greek architecture considered the concept of the Golden Ratio very much while the planning of its structures & hence it can be seen in many of the ancient Greek structures. The enormous pediment [roof] is complimented by the huge fluted columns placed at regular intervals. In case of interiors as well, there is a certain proportion-pattern that can observed… People have even tried to literally adapt the Golden Section idea in their design……let’s have a look	{"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": 0.8776854872703552, "perplexity": 2952.4081234385935}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107880038.27/warc/CC-MAIN-20201022195658-20201022225658-00273.warc.gz"}
https://rdrr.io/cran/emplik/man/emplikH.disc.html	# emplikH.disc: Empirical likelihood ratio for discrete hazard with right... Description Usage Arguments Details Value Author(s) References Examples ### Description Use empirical likelihood ratio and Wilks theorem to test the null hypothesis that ∑_i[f(x_i, θ) \log(1- dH(x_i))] = K where H(t) is the (unknown) discrete cumulative hazard function; f(t,θ) can be any predictable function of t. θ is the parameter of the function and K is a given constant. The data can be right censored and left truncated. When the given constants θ and/or K are too far away from the NPMLE, there will be no hazard function satisfy this constraint and the minus 2Log empirical likelihood ratio will be infinite. In this case the computation will stop. ### Usage 1 emplikH.disc(x, d, y= -Inf, K, fun, tola=.Machine\$double.eps^.25, theta) ### Arguments x a vector, the observed survival times. d a vector, the censoring indicators, 1-uncensor; 0-censor. y optional vector, the left truncation times. K a real number used in the constraint, sum to this value. fun a left continuous (weight) function used to calculate the weighted discrete hazard in H_0. fun(x, theta) must be able to take a vector input x, and a parameter theta. tola an optional positive real number specifying the tolerance of iteration error in solve the non-linear equation needed in constrained maximization. theta a given real number used as the parameter of the function f. ### Details The log likelihood been maximized is the ‘binomial’ empirical likelihood: ∑ D_i \log w_i + (R_i-D_i) \log [1-w_i] where w_i = Δ H(t_i) is the jump of the cumulative hazard function, D_i is the number of failures observed at t_i, R_i is the number of subjects at risk at time t_i. For discrete distributions, the jump size of the cumulative hazard at the last jump is always 1. We have to exclude this jump from the summation since \log( 1- dH(\cdot)) do not make sense. The constants theta and K must be inside the so called feasible region for the computation to continue. This is similar to the requirement that in testing the value of the mean, the value must be inside the convex hull of the observations. It is always true that the NPMLE values are feasible. So when the computation stops, try move the theta and K closer to the NPMLE. When the computation stops, the -2LLR should have value infinite. In case you do not need the theta in the definition of the function f, you still need to formally define your fun function with a theta input, just to match the arguments. ### Value A list with the following components: times the location of the hazard jumps. wts the jump size of hazard function at those locations. lambda the final value of the Lagrange multiplier. "-2LLR" The discrete -2Log Likelihood ratio. Pval P-value niters number of iterations used Mai Zhou ### References Fang, H. (2000). Binomial Empirical Likelihood Ratio Method in Survival Analysis. Ph.D. Thesis, Univ. of Kentucky, Dept of Statistics. Zhou and Fang (2001). “Empirical likelihood ratio for 2 sample problem for censored data”. Tech Report, Univ. of Kentucky, Dept of Statistics Zhou, M. and Fang, H. (2006). A comparison of Poisson and binomial empirical likelihood. Tech Report, Univ. of Kentucky, Dept of Statistics ### Examples 1 2 3 4 5 6 7 8 fun4 <- function(x, theta) { as.numeric(x <= theta) } x <- c(1, 2, 3, 4, 5, 6, 5, 4, 3, 4, 1, 2.4, 4.5) d <- c(1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1) # test if -H(4) = -0.7 emplikH.disc(x=x,d=d,K=-0.7,fun=fun4,theta=4) # we should get "-2LLR" 0.1446316 etc.... y <- c(-2,-2, -2, 1.5, -1) emplikH.disc(x=x,d=d,y=y,K=-0.7,fun=fun4,theta=4) Search within the emplik package Search all R packages, documentation and source code Questions? Problems? Suggestions? or email at [email protected]. Please suggest features or report bugs with the GitHub issue tracker. All documentation is copyright its authors; we didn't write any of that.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.8568127751350403, "perplexity": 1995.8427213843693}, "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/1487501174154.34/warc/CC-MAIN-20170219104614-00080-ip-10-171-10-108.ec2.internal.warc.gz"}
https://research.utwente.nl/en/publications/on-ising-models	# On Ising models Research output: Book/ReportReportProfessional ## Abstract For various Ising models two approaches are discussed, one is that of simulating lattices, also called gauging on exact equations, the other is that of calculating analytical expressions for the boundary free energy of Ising lattices. The first approach allows to conjecture a solution for some Ising models, that have sofar not been solved, once some exact partial result for the problem is known. The second approach aims at furnishing such a partial result in the form of a condition for the critical temperature. An example of such a result was recently given for the 2D Ising square lattice with nearest and next-nearest-neighbor interactions. The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase showed good agreement in the moderate and strong nearest neighbor coupling limit with several results obtained by Monte Carlo, transfer matrix and series expansion results. We extend the discussion of the critical line, finding an excellent fit, now also in other points, like the Padé point, as well as cusp behavior at the Onsager point where the lattice decouples into two 2D square lattices with only nearest-neighbor interaction. Combination of this result with a geometrical argument in the simulation approach leads to a critical exponent $2-\sqrt{2} \approx 0.5858,$ comparable to the exponent $4/7 \approx 0.5714$ found from renormalization arguments. Original language Undefined Enschede Universiteit Twente, Faculteit Toegepaste Wiskunde 24 Published - Jun 2006 ### Publication series Name Applied Mathematics Memoranda Department of Applied Mathematics, University of Twente 1804 0169-2690 • EWI-8055 • MSC-82B20 • METIS-237577 • IR-66581	{"extraction_info": {"found_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": 0.9004212021827698, "perplexity": 1303.0393155658733}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945333.53/warc/CC-MAIN-20230325130029-20230325160029-00217.warc.gz"}
https://brilliant.org/problems/greatest-integer-function/	# Greatest Integer Function Algebra Level 5 $$x$$ is a real number and satisfies the equation: $$\frac{1}{[x]}+\frac{1}{[2x]}=x-[x]+\frac{1}{3}$$. The sum of all such numbers can be expressed as $$\frac{m}{n}$$, where $$m$$ and $$n$$ are relatively prime positive integers. Find $$m+n$$ Details and Assumptions $$[x]$$ denotes the Greatest Integer Function: $$[x]=n$$, where $$n$$ is the greatest integer less than $$x$$. ×	{"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": 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": 0.992019772529602, "perplexity": 169.09609111028112}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578531994.14/warc/CC-MAIN-20190421160020-20190421182020-00141.warc.gz"}
https://slideplayer.com/slide/1507253/	# A graph of the instantaneous velocity of an object over a specified period of time Time is independent (x-axis) Velocity is dependent (y-axis) Remember, ## Presentation on theme: "A graph of the instantaneous velocity of an object over a specified period of time Time is independent (x-axis) Velocity is dependent (y-axis) Remember,"— Presentation transcript: A graph of the instantaneous velocity of an object over a specified period of time Time is independent (x-axis) Velocity is dependent (y-axis) Remember, velocity details magnitude (how fast) and direction (which way) Units for the slope of time vs. velocity graph (m/s)/s m/s 2 Slope of t vs. v graph tells instantaneous acceleration Straight lines on t vs. v graph = uniform (constant) acceleration Slope of t vs. v graph = acceleration Area Under t vs. v graph = displacement 16 m/s 4 s A = 32 m Instantaneous Velocity Velocity of an object at a specific moment Average Velocity Avg velocity of an object over time Instantaneous Acceleration How much objects velocity is changing at a specific moment Displacement Change in position of an object over time Download ppt "A graph of the instantaneous velocity of an object over a specified period of time Time is independent (x-axis) Velocity is dependent (y-axis) Remember," Similar presentations	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8730747103691101, "perplexity": 1525.563519185742}, "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/1634323585450.39/warc/CC-MAIN-20211022021705-20211022051705-00642.warc.gz"}
http://tex.stackexchange.com/users/17670/user-17670?tab=activity&sort=all&page=4	User 17670 Reputation 1,376 Next privilege 2,000 Rep. Apr 22 comment How to switch to upright greek in math mode? @Aydin They're both the same Apr 21 comment How to make the math font slightly thicker? Nice question! I'd like to know if there is a way to locally make certain math characters slightly thinner. For example, the letter $\Psi$ looks almost bold by default and so stands out on a page; it's kinda ugly. Apr 21 accepted How may I insert white vertical space between subequations? Apr 21 asked How may I insert white vertical space between subequations? Apr 21 accepted How can I box multiple aligned equations? Apr 21 comment How can I box multiple aligned equations? Thanks, how would I go about adjusting the vertical margins? Apr 21 comment How can I box multiple aligned equations? @cmhughes Yes, I checked previous related questions and didn't find what I am looking to achieve. Apr 21 asked How can I box multiple aligned equations? Apr 21 comment Attractive Boxed Equations DavidHammen Wrong; a load of physics and mathematics textbooks use boxes (regardless of the level). @TH- Who says it's for publication? Apr 20 comment Why does spell check not work if the figure has a label? @MarcoDaniel Any reason that should happen (is it bad LaTeX syntax?) or is this a bug? Apr 19 comment Move-around box in PDF display @AlexG To clarify: The command \tooltip{formula} produces the word "formula" in the PDF, in red, and as a clickable object. But, say I'm referring to an equation that was introduced 3 section ago, whose \label is "GoodEq" and whose number in the PDF is "(3.12)" I'd like to write, "using \eqref{GoodEq} we see that..." to give a PDF output of "using (3.12) we see that...", where the "(3.12)" part is clickable. This would be very useful and very cool. Apr 19 comment Move-around box in PDF display @AlexG Mother of God! img.mu.cdn.li/Ao/xqefEH.jpg Excellent work! Questions: Could it be made so that (in your MWE) "formula" could be replaced by \eqref{}, if you know what I mean? Also, does the end-user require anything other than a PDF viewer? Apr 19 comment Why does spell check not work if the figure has a label? @MarcoDaniel That actually fixes the problem, for whatever reason. Apr 19 asked Why does spell check not work if the figure has a label? Apr 18 comment LyX participates in the Google Summer of Code - which project ideas could be suggested? @Bugbusters In the case of a pspicture, sure, it would (I presume) need the whole environment recompiling. Like Daniel said though, the chance of that taking 'too long' is small. Apr 18 comment LyX participates in the Google Summer of Code - which project ideas could be suggested? @Bugbusters It's already been used in physics.SE and maths.SE with mathjax and it works really well. Compiling a total ball-ache. Also, would you really need to recompile everything? Apr 18 comment How can I nicely align a single split equation? +1 Nice. Visually, it's exactly what I was after (see my answer). Apr 18 answered How can I nicely align a single split equation? Apr 18 comment How can I nicely align a single split equation? +1 Thanks for the suggestions! The idea of a subexpression never occurred to me, I got too focused on the LaTeX. It's a bit awkward to introduce at this point in the document however, so I've opted for a modified version of Gonzalo's answer. I'll add it in a minute. Apr 18 accepted How can I nicely align a single split equation?	{"extraction_info": {"found_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": 0.8292393088340759, "perplexity": 2391.281330615546}, "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-2016-07/segments/1454701169261.41/warc/CC-MAIN-20160205193929-00034-ip-10-236-182-209.ec2.internal.warc.gz"}
http://mathhelpforum.com/math-topics/26036-electric-field.html	# Math Help - electric field. 1. ## electric field. An electric field of 281,000 N/C points due west at a certain point. What are the magnitude and direction of the force that acts on a charge of -7.4 µC at this spot? magnitude in Newtons direction? couldn't this be set up as: F = (-7.4uC) * (281,000 N/C) is uC the same as C? just uC is a micro Coulomb? also, do i need this expressed in scientific notation to make it correct. 2. Originally Posted by rcmango An electric field of 281,000 N/C points due west at a certain point. What are the magnitude and direction of the force that acts on a charge of -7.4 µC at this spot? magnitude in Newtons direction? couldn't this be set up as: F = (-7.4uC) * (281,000 N/C) is uC the same as C? just uC is a micro Coulomb? also, do i need this expressed in scientific notation to make it correct. To get a force in units of newton you need to express -7.4uC as $-7.4 x 10^{-6}$ C, then do the multiplication. Giving the answer using scientific notation is probably the simplest way of giving it. With the cancellation of powers of 10, it'll actually just be F = -7.4 x 0.281 N = ...... You should be able to the multiplication on paper - without a calculator And you do realise that you 'lose' the minus once you give the direction ....?	{"extraction_info": {"found_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": 0.9318487644195557, "perplexity": 1148.318534639911}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443736682947.6/warc/CC-MAIN-20151001215802-00212-ip-10-137-6-227.ec2.internal.warc.gz"}
http://aas.org/archives/BAAS/v34n4/aas201/827.htm	AAS 201st Meeting, January, 2003 Session 16. Stars in SDSS Poster, Monday, January 6, 2003, 9:20am-6:30pm, Exhibit Hall AB ## [16.04] Magnetic Activity in Low Mass Stars: SDSS Results S. L. Hawley, A. A. West, K. R. Covey, S. N. Raymond, L. M. Walkowicz (University of Washington) We present a study of the magnetic activity properties of low-mass stars in the Sloan Digital Sky Survey. Using the H\alpha emission line as an activity indicator, we examine the fraction of active stars as a function of spectral type and find a peak near type M7, confirming previous results. We investigate the ratio of the luminosity emitted in H\alpha compared to the bolometric luminosity for each star, and find a roughly constant ratio (with large scatter) over the range M0-M9. There does not appear to be a decrease in the ratio for types M8-M9, in contrast to previous determinations. We also explore the effect of metallicity on activity, and examine whether activity is correlated with changes in the SDSS colors. Bulletin of the American Astronomical Society, 34, #4 © 2002. The American Astronomical Soceity.	{"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": 0.95674067735672, "perplexity": 3561.5886934244268}, "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/1432207928864.16/warc/CC-MAIN-20150521113208-00111-ip-10-180-206-219.ec2.internal.warc.gz"}

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