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http://mathematica.stackexchange.com/questions/17506/how-to-expand-a-function-into-a-power-series-with-negative-powers?answertab=active | # How to expand a function into a power series with negative powers?
Is there any way to expand this expression
a+b(1-Exp[-T/(b c)]/(z-Exp[-T/ (b c)])
(where a, b, c, and T are constants) as a series in negative powers of $z$? The result should be in the form
a0 + a1 z^(-1) + a2 z^(-2) + a3 z^(-3) + ... + an z^(-n)
I tried solutions like Series[a + b (1 - k/(z - k)), {z, 0, -5}] but this did not work.
Thank you.
-
An expansion around zero Series[a + b (1 - k/(z - k)), {z, 0, 5}] gives $(a+2 b)+\frac{b z}{k}+\frac{b z^2}{k^2}+\frac{b z^3}{k^3}+\frac{b z^4}{k^4}+\frac{b z^5}{k^5}+O\left(z^6\right)$ – image_doctor Jan 9 '13 at 12:45
Welcome to Mathematica.SE! In the current form, your question does not fit the standards of this site and will most probably be downvoted or closes. Would you please consider to read the faq! and improve your answer by giving some background information? Furthermore when you see good answers, vote them up by clicking the gray triangles, because the credibility of the system is based on the reputation gained by users sharing their knowledge. Also, please remember to accept the answer by clicking the checkmark sign! – halirutan Jan 9 '13 at 13:02
@image_doctor thank you for the answer. but i want a serie with negative power. Series[a + b (1 - k/(z - k)), {z, 0, -5}] don't work – Mag Num Jan 9 '13 at 14:03
I was going to suggest expanding at infinity, then noticed @whuber said it better. – Daniel Lichtblau Jan 9 '13 at 15:18
## 1 Answer
You want first to fix any typographical errors (such as the unbalanced parentheses) and it's also wise to avoid symbol names beginning with capital letters. Then, to obtain a series expansion in powers of $1/z$, expand the expression around infinity, not zero:
Series[a + b (1 - Exp[-t/(b c)]/(z - Exp[-t/(b c)])) , {z, Infinity, 5}]
$(a+b)-\frac{b e^{-\frac{t}{b c}}}{z}-\frac{b e^{-\frac{2 t}{b c}}}{z^2}-\frac{b e^{-\frac{3 t}{b c}}}{z^3}-\frac{b e^{-\frac{4 t}{b c}}}{z^4}-\frac{b e^{-\frac{5 t}{b c}}}{z^5}+O\left[\frac{1}{z}\right]^6$
To confirm this, we could also replace $z$ by $1/z$, expand the series in non-negative powers of $z$, and then substitute $1/z$ back for $z$ once more:
Series[a + b (1 - Exp[-t/(b c)]/(z - Exp[-t/(b c)])) /. z -> (1/z), {z, 0, 5}] /. z -> 1/z
$(a+b)-\frac{b e^{-\frac{t}{b c}}}{z}-b e^{-\frac{2 t}{b c}} \left(\frac{1}{z}\right)^2-b e^{-\frac{3 t}{b c}} \left(\frac{1}{z}\right)^3-b e^{-\frac{4 t}{b c}} \left(\frac{1}{z}\right)^4-b e^{-\frac{5 t}{b c}} \left(\frac{1}{z}\right)^5+O\left[\frac{1}{z}\right]^6$
The two results are clearly equivalent expressions of the same series. If the terminal O[] term is undesirable, remove it by applying Normal to the output.
-
Thank you whuber, thats what am looking for :-) – Mag Num Jan 10 '13 at 8:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.5450334548950195, "perplexity": 1979.1618703288402}, "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-2015-32/segments/1438042988312.76/warc/CC-MAIN-20150728002308-00243-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://www.sciencebits.com/webpageworth | How much is a web page worth?
How much is a webpage worth?
First, some data
There are several estimates for the current size of the internet. These estimates range from about 12 billion indexable pages in early 2005 (that is, pages which can be indexed and not accessed through queries, logins, etc., not all of which are actually indexed by Google and their less efficient competitors) to as much as 25 billion pages indexed just by google (in late 2005). For the sake of simplicity, let us take 20 billion pages as a typical number.
The estimated amount of money spend on advertisement on the web also varies from estimate to estimate. Moreover, many estimates pertain to ads in american internet only (americans often mistake the national for global and vice versa, e.g., the World series in baseball). This could explain the factor two difference from low estimates of order $8 billion per year to$18 billion per year. A quick browse at the main advertisers responsible for the first figure reveals only american companies, \$8 billion is indeed the american value. Thus, \$18 Billion is a good estimate for the amount of money spend on online ads. This may sound like a very high number, but relative to the amount of content, it isn't!
This number, however, includes all types of online advertisements. For example, payed searches obviously don't increase the average worth of web pages, nor do classified ads. A quick search reveals that at least in the US, banners make up 20% of the revenue (40% goes to search engines, no wonder google is doing so great!).
Average value of a single indexable webpage
The above numbers imply that about a webpage should earn its owner a staggering 20% * \$18 billion / 20 billion pages = 18cents per year per page. This is of course an average number. Many websites don't have any advertisements, while other, like cnn.com, could be making a fortune in advertisements. Average value of a single domain O.k., so one web page is not worth a lot, at least on average, and since we don't know how many pages are not worth at all, lets look at something else, at the average value of a domain. According to this statistic, there are somewhat more than 50,000,000 .com domains. Given that the number of USA IP's is more than half of the number in the world, there are probably around 100,000,000 domains globally. Certainly, the number cannot be much higher than that. This means that every domain earns on average, 20% * \$18 billion / 100 million domains = \\$36 per year per domain, still not much, not even enough to cover the average cost of hosting a domain.
Blogroll
Sensible Climate
Physics and more Other | {"extraction_info": {"found_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": 0.342229425907135, "perplexity": 1179.8131455363164}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818687906.71/warc/CC-MAIN-20170921205832-20170921225832-00677.warc.gz"} |
https://jp.maplesoft.com/support/help/view.aspx?path=examples%2FCalculus1Tangents | Calculus1 Tangents - Maple Help
Calculus 1: Tangents, Inverses, and Sampling
The Student[Calculus1] package contains three routines that can be used to both work with and visualize the concepts of tangents, the inverses of functions, and the errors of plotting a function by sampling. This worksheet demonstrates this functionality.
For further information about any command in the Calculus1 package, see the corresponding help page. For a general overview, see Calculus1.
Getting Started
While any command in the package can be referred to using the long form, for example, Student[Calculus1][Tangent], it is easier, and often clearer, to load the package, and then use the short form command names.
> $\mathrm{restart}$
> $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{Calculus1}\right]\right):$
The following sections show how the routines work.
Main: Visualization
Next: Derivatives | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 21, "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": 0.9922307133674622, "perplexity": 2589.489243439026}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1669446710968.29/warc/CC-MAIN-20221204072040-20221204102040-00872.warc.gz"} |
http://www.ipodphysics.com/math-right-hand-rule.php | Cross Product / The Right Hand Rule
Important Equation
a x b = c
Cross Product
Right Hand Rule
In mathematics and physics, the right-hand rule is a common mnemonic for understanding notation conventions for vectors in 3 dimensions. It was invented for use in electromagnetism by British physicist John Ambrose Fleming in the late 19th century.
When choosing three vectors that must be at right angles to each other, there are two distinct solutions, so when expressing this idea in mathematics, one must remove the ambiguity of which solution is meant.
There are variations on the mnemonic depending on context, but all variations are related to the one idea of choosing a convention.
Definition by Wikipedia
One form of the right-hand rule is used in situations in which an ordered operation must be performed on two vectors a and b that has a result which is a vector c perpendicular to both a and b. The most common example is the vector cross product. The right-hand rule imposes the following procedure for choosing one of the two directions.
$\vec{a} \times \vec{b} = \vec{c}$
• With the thumb, index, and middle fingers at right angles to each other (with the index finger pointed straight), the middle finger points in the direction of c when the thumb representsa and the index finger represents b.
Other (equivalent) finger assignments are possible. For example, the first (index) finger can represent a, the first vector in the product; the second (middle) finger, b, the second vector; and the thumb, c, the product.
Definition by Wikipedia | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 1, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8739866018295288, "perplexity": 620.7653626252533}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891807660.32/warc/CC-MAIN-20180217185905-20180217205905-00263.warc.gz"} |
https://homework.cpm.org/category/CON_FOUND/textbook/mc2/chapter/5/lesson/5.1.3/problem/5-36 | Home > MC2 > Chapter 5 > Lesson 5.1.3 > Problem5-36
5-36.
Alden found a partially completed 5-D chart:
Define
Do
Decide
Target $74$
Trial 1:
$15$
$2(15) = 30$
$15 + 2 = 17$
$15 + 30 + 17 =$
$62$
too small
Trial 2:
$18$
$2(18) = 36$
$18 + 2 = 20$
$18 + 36 + 20 =$
$74$
just right
1. Create a word problem that could have been solved using this chart.
Examine the chart in the Define and Do sections to figure out what the expression from the 5-D chart is, and the relationship between each of the units.
If the first unit is the variable, $x$, and the three units are added together for the target value, $74$, think of any possible word problems that could illustrate this 5-D chart.
The farthest house is twice as far as the closest house. The middle house is two more miles farther than the closest house. If the total distance is $74$ miles, how far is the closest house?
2. What words would you put above the numbers in the three empty sections in the "Trial" and "Define" parts of the chart?
Based on the expression and the word problem you created, which sections are the shortest, medium, or longest distance?
3. What word(s) would you put above the "Do" column?
Based on the word problem you created, what would this total represent?
From the example answer, the ''Do'' column could be the total distance for all three houses. | {"extraction_info": {"found_math": true, "script_math_tex": 14, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8042080402374268, "perplexity": 1424.866409001209}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882570692.22/warc/CC-MAIN-20220807181008-20220807211008-00704.warc.gz"} |
https://math.stackexchange.com/questions/3061781/number-of-ways-of-coloring-n-objects-which-are-laid-in-a-row-with-k-colors-such | # Number of ways of coloring n objects which are laid in a row with k colors such that the adjacent objects are of different colors
Given n objects, which are lying in a straight line next to each other, in how many ways we can color them with k colors (all must be painted) such that the adjacent boxes not of same colors.
I can feel that inclusion exclusion principle will apply here but I am not able to figure out where to start. Its been a while since I read about them.
• What you want is the chromatic polynomial for the path graph. – Gerry Myerson Jan 4 at 15:58
• I guess so. This is the chromatic polynomial as the adjacent colors have to be different – Brij Raj Kishore Jan 4 at 16:03
You can color the first box in any of $$k$$ colors availble to you. The second box can be colored with one of the remaining $$k-1$$ colors. The same is true for the third, fourth... So the total number of colorings is $$k\times(k-1)^{n-1}$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 3, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.40596553683280945, "perplexity": 126.6653531511846}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1563195527000.10/warc/CC-MAIN-20190721123414-20190721145414-00533.warc.gz"} |
https://mersenneforum.org/showthread.php?s=63fe71e53668534ba386c9b3a3219dd0&p=477354 | mersenneforum.org being able to use a latex package
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2018-01-12, 18:03 #1 wildrabbitt Jul 2014 3×149 Posts being able to use a latex package Hi, I'm trying to write in an image file into a pdf file made with pdflatex from a dvi file. I need to be able to use the graphicx package but I don't think it's installed on my ubuntu computer. Can someone tell me how to achieve this (getting the package installed). Will
2018-01-12, 18:22 #2 Nick Dec 2012 The Netherlands 2·809 Posts On my Linux system (not Ubuntu but another distro), it is contained in the texlive-graphics package, installed in the same way as all packages here. Have you checked for the file graphicx.sty in your packages directory? Or tried dvipdfm on the dvi file and then take the image from pdf?
2018-01-12, 18:56 #3 wildrabbitt Jul 2014 3×149 Posts Thanks. I've just tried : william@william-GA-78LMT-USB3:~/Desktop/graphics$sudo apt-get install texlive-graphics [sudo] password for william: Reading package lists... Done Building dependency tree Reading state information... Done E: Unable to locate package texlive-graphics william@william-GA-78LMT-USB3:~/Desktop/graphics$ with self-evident results. Can you tell me where the sty file for graphicx would be if it exists on my filesystem? Please do if you can. Last fiddled with by wildrabbitt on 2018-01-12 at 18:57
2018-01-12, 19:07 #4
Nick
Dec 2012
The Netherlands
65216 Posts
Quote:
Originally Posted by wildrabbitt Can you tell me where the sty file for graphicx would be if it exists on my filesystem? Please do if you can.
On my system, it's location is:
Code:
/usr/share/texmf/tex/latex/graphics/graphicx.sty
Do you get an error if you simply try to insert a jpg image in latex, for example?
2018-01-12, 19:11 #5 wildrabbitt Jul 2014 44710 Posts william@william-GA-78LMT-USB3:~$latex quadratic.tex This is pdfTeX, Version 3.14159265-2.6-1.40.17 (TeX Live 2016/Debian) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./quadratic.tex LaTeX2e <2017/01/01> patch level 3 Babel <3.9r> and hyphenation patterns for 3 language(s) loaded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size10.clo)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the ?' option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) ! LaTeX Error: File graphicsx.sty' not found. Type X to quit or to proceed, or enter new name. (Default extension: sty) Enter file name: X 2018-01-12, 19:18 #6 paulunderwood Sep 2002 Database er0rr 3,593 Posts Quote: Originally Posted by Nick On my system, it's location is: Code: /usr/share/texmf/tex/latex/graphics/graphicx.sty Do you get an error if you simply try to insert a jpg image in latex, for example? Code: locate graphicx.sty /usr/share/texlive/texmf-dist/tex/latex/graphics/graphicx.sty on my Debian 9 system... Code: apt-cache search texlive | grep Graphics texlive-pictures - TeX Live: Graphics, pictures, diagrams texlive-font-utils - TeX Live: Graphics and font utilities It is graphicx.sty -- there is no extra "s" Last fiddled with by paulunderwood on 2018-01-12 at 19:19 2018-01-12, 19:39 #7 wildrabbitt Jul 2014 1101111112 Posts william@william-GA-78LMT-USB3:~/Desktop/graphics$ apt-cache search texlive | grep Graphics texlive-font-utils - TeX Live: Graphics and font utilities texlive-pictures - TeX Live: Graphics, pictures, diagrams The only directories I've got in /usr/share/texmf/tex/latex are gnuplot, lm, prosper, tex-gyre and tipa so I'm wondering why theres no graphics folder. I have downloaded latex-graphics.tds.zip but I extracted it to home. thanks for your patience
2018-01-12, 19:46 #8 paulunderwood Sep 2002 Database er0rr 3,593 Posts Code: ls /usr/share/texlive/texmf-dist/tex/latex/ ae babelbib etex-pkg graphics latexconfig natbib pspicture amscls base fancyhdr graphics-cfg ltxmisc oberdiek tools amsfonts carlisle fix2col graphics-def mflogo pslatex url amsmath colortbl geometry hyperref mfnfss psnfss Note: see my note about the extraneous "s" you put in what should be graphicx.sty (and not graphicsx.sty). Last fiddled with by paulunderwood on 2018-01-12 at 19:50
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Tue Mar 2 09:01:05 UTC 2021 up 89 days, 5:12, 0 users, load averages: 1.79, 1.33, 1.17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.6624798774719238, "perplexity": 15475.28279371107}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1614178363782.40/warc/CC-MAIN-20210302065019-20210302095019-00132.warc.gz"} |
http://math.stackexchange.com/questions/144267/wkb-approximation-question/144277 | # WKB approximation question
I was reading some stuff on asymptotic analysis, but how do you get from the 1st line to the 2nd line?
$y \sim \frac{1+x}{2\lambda}\exp\left(\frac{\lambda x}{1+x}\right) - \frac{1+x}{2\lambda}\exp\left(\frac{-\lambda x}{1+x}\right)\\y(1) \sim\frac{1}{\lambda}\exp\left(\frac{\lambda}{2}\right) \text{as } \lambda \rightarrow \infty$
I see they substituted $x=1$, but where does the $- \frac{1+x}{2\lambda}\exp\left(\frac{-\lambda x}{1+x}\right)$ term go?
What does $y(1) \sim\frac{1}{\lambda}\exp\left(\frac{\lambda}{2}\right) \text{as } \lambda \rightarrow \infty$ actually mean? The main part im not sure about is the $\sim$.
-
$$f\sim g \quad (\text{as }x\rightarrow\infty)$$
reads "$f$ is asymptotic to $g$ as $x$ goes to infinity" . This means basically that $\lim_{x\rightarrow \infty} \frac{f}{g} = 1$ .
Because $e^x$ dominates $e^{-x}$ as $x$ becomes large (i.e. $e^{-x} \in \mathcal{o}(e^{x})$) they neglected the term with the negative exponent in the second line. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9529242515563965, "perplexity": 118.4114564855042}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042987135.9/warc/CC-MAIN-20150728002307-00157-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://blog.supplysideliberal.com/post/62036391045/quartz-29-the-complete-guide-to-getting-into-an | # Quartz #29—>The Complete Guide to Getting into an Economics PhD Program
Link to the Column on Quartz
Here is the full text of my 29th Quartz column, ”The Complete Guide to Getting into an Economics PhD Program.” I am glad to now bring it home to supplysideliberal.com, and I expect Noah to post it on his blog Noahpinion, as well. It was first published on August 16, 2013. Links to all my other columns can be found here.
Up to this point, this is by far my most popular Quartz column. In addition to great interest in the topic, I attribute the popularity of this column to Noah’s magic touch. Personally, I would rather read Noah’s blog than any other blog in cyberspace. That brilliant style shows through here; I think I managed not to spoil things too much in this column.
This column generated many reactions, two of which you can see as guest posts on supplysideliberal.com:
Jeff Smith is my colleague at the University of Michigan. He amplifies many of the things we say. For a complete guide, be sure to see what Jeff has to say, too. What Bruce Bartlett had to say is worth reading simply because of his interesting career path.
If you want to mirror the content of this post on another site, that is possible for a limited time if you read the legal notice at this link and include both a link to the original Quartz column and the following copyright notice:
© August 16, 2013: Miles Kimball and Noah Smith, as first published on Quartz. Used by permission according to a temporary nonexclusive license expiring June 30, 2014. All rights reserved.
Noah has agreed to give permission on the same terms as I do.
Back in May, Noah wrote about the amazingly good deal that is the PhD in economics. Why? Because:
1. You get a job.
2. You get autonomy.
3. You get intellectual fulfillment.
4. The risk is low.
5. Unlike an MBA, law, or medical degree, you don’t have to worry about paying the sticker price for an econ PhD: After the first year, most schools will give you teaching assistant positions that will pay for the next several years of graduate study, and some schools will take care of your tuition and expenses even in the first year. (See what is written at the end of this post, after the column proper, for more about costs of graduate study and how econ PhD’s future earnings makes it worthwhile, even if you can’t get a full ride.)
Of course, such a good deal won’t last long now that the story is out, so you need to act fast! Since he wrote his post, Noah has received a large number of emails asking the obvious follow-up question: “How do I get into an econ PhD program?” And Miles has been asked the same thing many times by undergraduates and other students at the University of Michigan. So here, we present together our guide for how to break into the academic Elysium called Econ PhD Land:
(Note: This guide is mainly directed toward native English speakers, or those from countries whose graduate students are typically fluent in English, such as India and most European countries. Almost all highly-ranked graduate programs teach economics in English, and we find that students learn the subtle non-mathematical skills in economics better if English is second nature. If your nationality will make admissions committees wonder about your English skills, you can either get your bachelor’s degree at a—possibly foreign—college or university where almost all classes are taught in English, or you will have to compensate by being better on other dimensions. On the bright side, if you are a native English speaker, or from a country whose graduate students are typically fluent in English, you are already ahead in your quest to get into an economics PhD.)
Here is the not-very-surprising list of things that will help you get into a good econ PhD program:
• good grades, especially in whatever math and economics classes you take,
• a good score on the math GRE,
• some math classes and a statistics class on your transcript,
• research experience, and definitely at least one letter of recommendation from a researcher,
• a demonstrable interest in the field of economics.
Chances are, if you’re asking for advice, you probably feel unprepared in one of two ways. Either you don’t have a sterling math background, or you have quantitative skills but are new to the field of econ. Fortunately, we have advice for both types of applicant.
## If you’re weak in math…
Fortunately, if you’re weak in math, we have good news: Math is something you can learn. That may sound like a crazy claim to most Americans, who are raised to believe that math ability is in the genes. It may even sound like arrogance coming from two people who have never had to struggle with math. But we’ve both taught people math for many years, and we really believe that it’s true. Genes help a bit, but math is like a foreign language or a sport: effort will result in skill.
Here are the math classes you absolutely should take to get into a good econ program:
• Linear algebra
• Multivariable calculus
• Statistics
Here are the classes you should take, but can probably get away with studying on your own:
• Ordinary differential equations
• Real analysis
Linear algebra (matrices, vectors, and all that) is something that you’ll use all the time in econ, especially when doing work on a computer. Multivariable calculus also will be used a lot. And stats of course is absolutely key to almost everything economists do. Differential equations are something you will use once in a while. And real analysis—by far the hardest subject of the five—is something that you will probably never use in real econ research, but which the economics field has decided to use as a sort of general intelligence signaling device.
If you took some math classes but didn’t do very well, don’t worry. Retake the classes. If you are worried about how that will look on your transcript, take the class the first time “off the books” at a different college (many community colleges have calculus classes) or online. Or if you have already gotten a bad grade, take it a second time off the books and then a third time for your transcript. If you work hard, every time you take the class you’ll do better. You will learn the math and be able to prove it by the grade you get. Not only will this help you get into an econ PhD program, once you get in, you’ll breeze through parts of grad school that would otherwise be agony.
Here’s another useful tip: Get a book and study math on your own before taking the corresponding class for a grade. Reading math on your own is something you’re going to have to get used to doing in grad school anyway (especially during your dissertation!), so it’s good to get used to it now. Beyond course-related books, you can either pick up a subject-specific book (Miles learned much of his math from studying books in the Schaum’s outline series), or get a “math for economists” book; regarding the latter, Miles recommends Mathematics for Economists by Simon and Blume, while Noah swears by Mathematical Methods and Models for Economists by de la Fuente. When you study on your own, the most important thing is to work through a bunch of problems. That will give you practice for test-taking, and will be more interesting than just reading through derivations.
This will take some time, of course. That’s OK. That’s what summer is for (right?). If you’re late in your college career, you can always take a fifth year, do a gap year, etc.
When you get to grad school, you will have to take an intensive math course called “math camp” that will take up a good part of your summer. For how to get through math camp itself, see this guide by Jérémie Cohen-Setton.
One more piece of advice for the math-challenged: Be a research assistant on something non-mathy. There are lots of economists doing relatively simple empirical work that requires only some basic statistics knowledge and the ability to use software like Stata. There are more and more experimental economists around, who are always looking for research assistants. Go find a prof and get involved! (If you are still in high school or otherwise haven’t yet chosen a college, you might want to choose one where some of the professors do experiments and so need research assistants—something that is easy to figure out by studying professors’ websites carefully, or by asking about it when you visit the college.)
## If you’re new to econ…
If you’re a disillusioned physicist, a bored biostatistician, or a neuroscientist looking to escape that evilPrincipal Investigator, don’t worry: An econ background is not necessary. A lot of the best economists started out in other fields, while a lot of undergrad econ majors are headed for MBAs or jobs in banks. Econ PhD programs know this. They will probably not mind if you have never taken an econ class.
That said, you may still want to take an econ class, just to verify that you actually like the subject, to start thinking about econ, and to prepare yourself for the concepts you’ll encounter. If you feel like doing this, you can probably skip Econ 101 and 102, and head straight for an Intermediate Micro or Intermediate Macro class.
Another good thing is to read through an econ textbook. Although economics at the PhD level is mostly about the math and statistics and computer modeling (hopefully getting back to the real world somewhere along the way when you do your own research), you may also want to get the flavor of the less mathy parts of economics from one of the well-written lower-level textbooks (either one by Paul Krugman and Robin WellsGreg Mankiw, or Tyler Cowen and Alex Tabarrok) and maybe one at a bit higher level as well, such as David Weil’s excellent book on economic growth) or Varian’s Intermediate Microeconomics.
Remember to take a statistics class, if you haven’t already. Some technical fields don’t require statistics, so you may have missed this one. But to econ PhD programs, this will be a gaping hole in your resume. Go take stats!
One more thing you can do is research with an economist. Fortunately, economists are generally extremely welcoming to undergrad RAs from outside econ, who often bring extra skills. You’ll get great experience working with data if you don’t have it already. It’ll help you come up with some research ideas to put in your application essays. And of course you’ll get another all-important letter of recommendation.
And now for…
## General tips for everyone
Here is the most important tip for everyone: Don’t just apply to “top” schools. For some degrees—an MBA for example—people question whether it’s worthwhile to go to a non-top school. But for econ departments, there’s no question. Both Miles and Noah have marveled at the number of smart people working at non-top schools. That includes some well-known bloggers, by the way—Tyler Cowen teaches at George Mason University (ranked 64th), Mark Thoma teaches at the University of Oregon (ranked 56th), and Scott Sumner teaches at Bentley, for example. Additionally, a flood of new international students is expanding the supply of quality students. That means that the number of high-quality schools is increasing; tomorrow’s top 20 will be like today’s top 10, and tomorrow’s top 100 will be like today’s top 50.
Apply to schools outside of the top 20—any school in the top 100 is worth considering, especially if it is strong in areas you are interested in. If your classmates aren’t as elite as you would like, that just means that you will get more attention from the professors, who almost all came out of top programs themselves. When Noah said in his earlier post that econ PhD students are virtually guaranteed to get jobs in an econ-related field, that applied to schools far down in the ranking. Everyone participates in the legendary centrally managed econ job market. Very few people ever fall through the cracks.
Next—and this should go without saying—don’t be afraid to retake the GRE. If you want to get into a top 10 school, you probably need a perfect or near-perfect score on the math portion of the GRE. For schools lower down the rankings, a good GRE math score is still important. Fortunately, the GRE math section is relatively simple to study for—there are only a finite number of topics covered, and with a little work you can “overlearn” all of them, so you can do them even under time pressure and when you are nervous. In any case, you can keep retaking the test until you get a good score (especially if the early tries are practice tests from the GRE prep books and prep software), and then you’re OK!
Here’s one thing that may surprise you: Getting an econ master’s degree alone won’t help. Although master’s degrees in economics are common among international students who apply to econ PhD programs, American applicants do just fine without a master’s degree on their record. If you want that extra diploma, realize that once you are in a PhD program, you will get a master’s degree automatically after two years. And if you end up dropping out of the PhD program, that master’s degree will be worth more than a stand-alone master’s would. The one reason to get a master’s degree is if it can help you remedy a big deficiency in your record, say not having taken enough math or stats classes, not having taken any econ classes, or not having been able to get anyone whose name admissions committees would recognize to write you a letter of recommendation.
For getting into grad school, much more valuable than a master’s is a stint as a research assistant in the Federal Reserve System or at a think tank—though these days, such positions can often be as hard to get into as a PhD program!
Finally—and if you’re reading this, chances are you’re already doing this—read some econ blogs. (See Miles’s speculations about the future of the econ blogosphere here.) Econ blogs are no substitute for econ classes, but they’re a great complement. Blogs are good for picking up the lingo of academic economists, and learning to think like an economist. Don’t be afraid to write a blog either, even if no one ever reads it (you don’t have to be writing at the same level as Evan Soltas orYichuan Wang); you can still put it on your CV, or just practice writing down your thoughts. And when you write your dissertation, and do research later on in your career, you are going to have to think for yourself outside the context of a class. One way to practice thinking critically is by critiquing others’ blog posts, at least in your head.
Anyway, if you want to have intellectual stimulation and good work-life balance, and a near-guarantee of a well-paying job in your field of interest, an econ PhD could be just the thing for you. Don’t be scared of the math and the jargon. We’d love to have you.
In case you are curious, let me say a little about the financial costs and benefits of an economics PhD. At Michigan and other top places, PhD students are fully funded. Here, that means that the first year’s tuition and costs are covered (including a stipend for your living expenses). In years 2 through 5 (which is enough time to finish your PhD if you work hard to stay on track), as long as you are in good standing in the program, the costs of a PhD are just the work you do as a teaching assistant. So there are no out-of-pocket costs as long as you finish within five years, which is tough but doable if you work hard to stay on track. Tuition is relatively low in year 6 (and 7) if you can’t finish in 5 years. Plus, graduate students in economics who have had that much teaching experience often find they can make about as much money by tutoring struggling undergraduates as they could have by being a teaching assistant.
When a school can’t manage full funding, the first place it adds a charge is in charging the bottom-half of the applicant pool for the first year, when a student can’t realistically teach because the courses the grad students are taking are too heavy. That might add up to a one-time expense of $40,000 or so in tuition, plus living expenses. On pay, the market price for a brand-new assistant professor at a top department seems to be at least$115,000 for 9 months, with the opportunity to earn more during the summer months. If you don’t quite make it to that level, University of Michigan PhD’s I have asked seem to get at least $80,000 starting salary, and Louis Johnston tweets that below-top liberal arts colleges pay a starting salary in the$55,000 to $60,000 range. But remember that all of these numbers are for 9-month salaries that allow for the possibility (though not the regularity) of earning more in the summer. Government jobs tend to pay 12-month salaries that are about 12/9 of 9-month academic salaries at a comparable level. There is definitely the possibility of being paid very well in academic economics, though not as well as the upside potential if you go to Wall Street. For example, with summer pay included, quite a few of the full economics professors at the University of Michigan make more than$250,000 a year. (Because we are at a state university, our salaries are public.)
The bottom line is that the financial returns are good enough that you should have no hesitation begging or borrowing to finance your Economics PhD. (Please don’t steal to finance it.)
What about the costs of the extra year it might take to study math the way we recommend? If you have been developing self-discipline like a champion, but are short on money and summers aren’t enough, you could spend a gap year right after high school just studying math, living in your parents’ house at very low cost; most colleges will let you defer admission for a year after they have let you in.
## Update:
I liked this comment that Kevin C. Smith (an MD) sent to Quartz: | {"extraction_info": {"found_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": 0.22562779486179352, "perplexity": 1206.1930062139425}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987822458.91/warc/CC-MAIN-20191022155241-20191022182741-00148.warc.gz"} |
https://www.genealogy.math.ndsu.nodak.edu/id.php?id=24631 | ## Joan Bagaria
Ph.D. University of California, Berkeley 1991
Dissertation: Definable Forcing and Regularity Properties of Projective Sets of Reals
Mathematics Subject Classification: 03—Mathematical logic and foundations
Students:
NameSchoolYearDescendants | {"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": 0.8513866662979126, "perplexity": 5374.330599923261}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499891.42/warc/CC-MAIN-20230131222253-20230201012253-00588.warc.gz"} |
http://waffle.readthedocs.io/en/v0.11/types/sample.html | # Samples¶
Samples are on a given percentage of the time. They do not require a request object and can be used in other contexts, such as management commands and tasks.
Warning
Sample values are random: if you check a Sample twice, there is no guarantee you will get the same value both times. If you need to rely on the value more than once, you should store it in a variable.
# YES
foo_on = sample_is_active('foo')
if foo_on:
pass
# ...later...
if foo_on:
pass
# NO!
if sample_is_active('foo'):
pass
# ...later...
if sample_is_active('foo'): # INDEPENDENT of the previous check
pass
## Sample Attributes¶
Samples can be administered through the Django admin site or the command line. They have the following attributes:
Name: The name of the Sample. A number from 0.0 to 100.0 that determines how often the Sample will be active. Describe where the Sample is used. | {"extraction_info": {"found_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": 0.5282379388809204, "perplexity": 1110.6003714189237}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1487501171629.92/warc/CC-MAIN-20170219104611-00599-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://gamedev.stackexchange.com/questions/62917/uncharted-2-tone-mapping-and-an-eye-adaptation | # Uncharted 2 tone mapping and an eye adaptation
I found an example of uncharted 2 tone mapping on this site. Here is the code:
float A = 0.15;
float B = 0.50;
float C = 0.10;
float D = 0.20;
float E = 0.02;
float F = 0.30;
float W = 11.2;
float3 Uncharted2Tonemap(float3 x)
{
return ((x*(A*x+C*B)+D*E)/(x*(A*x+B)+D*F))-E/F;
}
float4 ps_main( float2 texCoord : TEXCOORD0 ) : COLOR
{
float3 texColor = tex2D(Texture0, texCoord );
texColor *= 16; // Hardcoded Exposure Adjustment
float ExposureBias = 2.0f;
float3 curr = Uncharted2Tonemap(ExposureBias*texColor);
float3 whiteScale = 1.0f/Uncharted2Tonemap(W);
float3 color = curr*whiteScale;
float3 retColor = pow(color,1/2.2);
return float4(retColor,1);
}
I use Reinhard tone mapping. I calculate an average luminance and max luminance in screen space. Based on those parameters I get an eye adaptation, for example when a camera is moved from a bright outdoor area to the dark room. How an eye adaptation process is achieved in Uncharted 2 tone mapping ? In my engine I render results of the shading to the floating point texture. I assume that I don’t need hardcoded exposure adjustment (texColor *= 16) and exposure bias = 2.0f. Or maybe I should interpret those parameters based on the average luminance ? What about “w” parameter ? Can I assign to “w” a maximum luminance from the screen space ?
• @Irbis The Reinhard tone mapping operator is just L / (1 + L) (equation 3 in the paper). That's the part you replace with the Uncharted tone mapping operator. The Reinhard paper also shows how to prescale the luminances based on an adaptation value (equation 2). That part you would leave alone. – Nathan Reed Oct 1 '13 at 0:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4800362288951874, "perplexity": 6613.307552766136}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1596439739073.12/warc/CC-MAIN-20200813191256-20200813221256-00365.warc.gz"} |
https://open.library.ubc.ca/cIRcle/collections/48630/items/1.0366264 | # Open Collections
## BIRS Workshop Lecture Videos
### Stabilizing Weighted Graphs Koh, Cedric
#### Description
An edge-weighted graph G is called stable if the value of a maximum-weight matching equals the value of a maximum-weight fractional matching. Stable graphs play an important role in some interesting game theory problems, such as network bargaining games and cooperative matching games, because they characterize instances which admit stable outcomes. Motivated by this, in the last few years many researchers have investigated the algorithmic problem of turning a given graph into a stable one, via edge- and vertex-removal operations. However, all the algorithmic results developed in the literature so far only hold for unweighted instances, i.e., assuming unit weights on the edges of G. We give the first polynomial-time algorithm to find a minimum cardinality subset of vertices whose removal from G yields a stable graph, for any weighted graph G. The algorithm is combinatorial and exploits new structural properties of basic fractional matchings, which may be of independent interest. In contrast, we show that the problem of finding a minimum cardinality subset of edges whose removal from a weighted graph G yields a stable graph, does not admit any constant-factor approximation algorithm, unless P=NP. In this setting, we develop an O(Delta)-approximation algorithm for the problem, where Delta is the maximum degree of a node in G. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8932477235794067, "perplexity": 389.13780678454816}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103943339.53/warc/CC-MAIN-20220701155803-20220701185803-00442.warc.gz"} |
http://mail-archives.apache.org/mod_mbox/maven-dev/201802.mbox/%3CCAPoyBqTMyOukhMQgzFe6BgtMDzVAaGL+eyCQHrR4Gs1xnue=-Q@mail.gmail.com%3E | # maven-dev mailing list archives
##### Site index · List index
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From Olivier Lamy <[email protected]>
Subject Re: Surefire 2.21 release date ?
Date Thu, 08 Feb 2018 05:45:24 GMT
```Hi,
I'm looking now at having surefire/failsafe working with jdk10 (yup first
RC is expected really soon :-)
I can't get failsafe working with jdk10 because of a commons-lang3 bug (see
https://issues.apache.org/jira/browse/SUREFIRE-1473 due to
https://issues.apache.org/jira/browse/LANG-1365)
So I simply upgraded to commons-lang3 but now it looks enforcer is
configured to enforce 1.6.
Well as I was thinking: 1.6 really in 2018?
So I created SUREFIRE-1474 and push it.
Let me know if you have any issues with that?
As jdk10 is very soon. I'm happy to release surefire ASAP now.
WDYT?
Cheers
Olivier
On 4 February 2018 at 19:11, Enrico Olivelli <[email protected]> wrote:
> Il dom 4 feb 2018, 19:38 Tibor Digana <[email protected]> ha scritto:
>
> > Hi Enrico,
> >
> > I have got working build script and got some issues on integration tests
> in
> > TestNG & Java 10.
> > See this CLI. I think the JaCoCo Agent property is the issue because it
> is
> > not resolved. I will try to put an empty string in there. Maybe the build
> > would pass.
> >
>
>
> Thank you Tibor for the update. Unfortunately I don't have a Windows box to
> reproduce the issue.
>
> From the error I guess you are right
>
> Enrico
>
> >
> > [windows-jdk10-maven3.5.x] [ERROR] Command was cmd.exe /X /C
> > "F:\jenkins\tools\java\jdk10-ea+37\bin\java \${jacoco.agent} -jar
> >
> > C:\Users\jenkins\AppData\Local\Temp\1\surefire3375403847264435698\
> surefirebooter6458194328264691762.jar
> > C:\Users\jenkins\AppData\Local\Temp\1\surefire3375403847264435698
> > 2018-02-04T17-23-23_801-jvmRun1 surefire15708374661287747855tmp
> > surefire_015474024382399849038tmp"
> >
> >
> >
> > On Fri, Feb 2, 2018 at 12:23 AM, Tibor Digana <[email protected]>
> > wrote:
> >
> > > It's ok, not annoying.
> > > The issue SUREFIRE-1374 is quite easy to workaround and therefore I am
> > > working on more difficult issue - the build process.
> > > So regarding build process I contacted our INFRA team and found the
> > > problem. Now the build is running in [1] and the build script is in
> [2].
> > I
> > > should get this working first. We are testing JDK 7, 8, 9, 10 on Linux
> > and
> > > Windows and Maven 3.2 - 3.5.
> > >
> > > [1]: https://builds.apache.org/view/M-R/view/Maven/job/maven-
> > > surefire-pipeline/job/SUREFIRE-1463/
> > > [2]: https://github.com/apache/maven-surefire/blob/SUREFIRE-
> > > 1463/Jenkinsfile
> > >
> > >
> > > On Thu, Feb 1, 2018 at 1:55 PM, Enrico Olivelli <[email protected]>
> > > wrote:
> > >
> > >> Sorry, I don't want to annoy.
> > >> I see that SUREFIRE-1374 needs feedback from users.
> > >> I took a look but I don't know how to move forward that issue
> > >>
> > >> Is it a blocker for 2.21 release ? o can it be post-poned to 2.2 ?
> > >>
> > >> 2.21 is a great milestone to move people to Java 9
> > >>
> > >> -- Enrico
> > >>
> > >>
> > >> 2018-01-23 17:49 GMT+01:00 Enrico Olivelli <[email protected]>:
> > >>
> > >> >
> > >> >
> > >> > 2018-01-16 21:26 GMT+01:00 Enrico Olivelli <[email protected]>:
> > >> >
> > >> >>
> > >> >>
> > >> >> Il gio 11 gen 2018, 22:26 Tibor Digana <[email protected]>
ha
> > >> >> scritto:
> > >> >>
> > >> >>> I was busy at work but now I am free.
> > >> >>> I need to have a help with investigating a bug
> > >> >>> https://issues.apache.org/jira/browse/SUREFIRE-1450
> > >> >>> I should fix branches which are already in progress:
> > >> UnicodeTestNamesIT
> > >> >>> and
> > >> >>> JaCoCO (SUREFIRE-1455).
> > >> >>>
> > >> >>
> > >> > Tibor
> > >> >
> > >> > I see that now SUREFIRE-1455 (JaCoCo) is closed
> > >> > https://issues.apache.org/jira/browse/SUREFIRE-1455
> > >> >
> > >> > And I see that SUREFIRE-1461 about UnicodeTestNamesIT is closed too
> > >> > https://issues.apache.org/jira/browse/SUREFIRE-1461
> > >> >
> > >> >
> > >> > It seems that for 2.21 there is only one task open
> > >> >
> > >> > https://issues.apache.org/jira/browse/SUREFIRE-1374?jql=
> > >> > project%20%3D%20SUREFIRE%20AND%20status%20in%20(Open%
> > >> > 2C%20%22In%20Progress%22%2C%20Reopened)%20AND%20fixVersion%
> > >> 20%3D%202.21.0.
> > >> > Jigsaw
> > >> >
> > >> > Do you need some help ?
> > >> > I am looking forward a release of 2.21
> > >> >
> > >> > Thank you
> > >> > Enrico
> > >> >
> > >> >
> > >> >>
> > >> >> I can pick one on these, just tell me which is the best
> > >> >>
> > >> >> Then all ITs setup with JDK 9.
> > >> >>>
> > >> >>> On Wed, Jan 10, 2018 at 11:17 AM, Enrico Olivelli <
> > >> [email protected]>
> > >> >>> wrote:
> > >> >>>
> > >> >>> > (resending)
> > >> >>> >
> > >> >>> > Hi Tibor,
> > >> >>> > Do you have any plan about the release date of surefire
2.21.0 ?
> > >> >>> >
> > >> >>> > If you have some tasks to do I will be happy to help.
> > >> >>> >
> > >> >>> > Cheers
> > >> >>> >
> > >> >>> > Enrico
> > >> >>> >
> > >> >>>
> > >> >> --
> > >> >>
> > >> >>
> > >> >> -- Enrico Olivelli
> > >> >>
> > >> >
> > >> >
> > >>
> > >
> > >
> >
>
>
> --
>
>
> -- Enrico Olivelli
>
--
Olivier Lamy | {"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": 0.9686837792396545, "perplexity": 19664.704352430803}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1552912202433.77/warc/CC-MAIN-20190320150106-20190320172106-00287.warc.gz"} |
https://fr.maplesoft.com/support/help/maple/view.aspx?path=DEtools%2Ffirtest | DEtools - Maple Programming Help
DEtools
firtest
test a given first integral
Calling Sequence firtest(first_int, ODE, y(x))
Parameters
first_int - first integral ODE - ordinary differential equation y(x) - (optional) indeterminate function of the ODE
Description
• The firtest command checks whether a given expression is a first integral of a given ODE. Similar to odetest, firtest returns $0$ when the result is valid, or an algebraic expression obtained after simplifying the PDE for the first integral associated with the given ODE (see odepde). Among other things, firtest can be used to test the results obtained using the command firint.
• If the result returned by firtest is not zero, the expression might nevertheless be a first integral. Sometimes, with further simplification, you can obtain the desired $0$ using commands such as expand, combine, and so on.
• This function is part of the DEtools package, and so it can be used in the form firtest(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[firtest](..).
Examples
A first order ODE
> $\mathrm{with}\left(\mathrm{DEtools}\right):$
> $\mathrm{ODE}≔\mathrm{diff}\left(y\left(x\right),x\right)=y\left(x\right)a\left(x\right)$
${\mathrm{ODE}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{y}{}\left({x}\right){}{a}{}\left({x}\right)$ (1)
An integrating factor for ODE above
> $\mathrm{Μ}≔\mathrm{intfactor}\left(\mathrm{ODE}\right)$
${\mathrm{Μ}}{≔}{{ⅇ}}^{{\int }{-}{a}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}$ (2)
A related (to Mu) first integral for ODE above
> $\mathrm{ans}≔\mathrm{firint}\left(\mathrm{Μ}\mathrm{ODE}\right)$
${\mathrm{ans}}{≔}{{ⅇ}}^{{\int }{-}{a}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}{}{y}{}\left({x}\right){+}{\mathrm{_C1}}{=}{0}$ (3)
Testing the first integral
> $\mathrm{firtest}\left(\mathrm{ans},\mathrm{ODE}\right)$
${0}$ (4)
A second order ODE example
> $\mathrm{ODE}≔\mathrm{diff}\left(y\left(x\right),x,x\right)=-\frac{2\left(2\mathrm{diff}\left(y\left(x\right),x\right)+5x{y\left(x\right)}^{2}+2{x}^{2}y\left(x\right)\mathrm{diff}\left(y\left(x\right),x\right)\right)}{x}$
${\mathrm{ODE}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{-}\frac{{2}{}\left({2}{}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{5}{}{x}{}{{y}{}\left({x}\right)}^{{2}}{+}{2}{}{{x}}^{{2}}{}{y}{}\left({x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)\right)}{{x}}$ (5)
> $\mathrm{first_int}≔2{x}^{5}{y\left(x\right)}^{2}+{x}^{4}\mathrm{diff}\left(y\left(x\right),x\right)+\mathrm{_C1}=0$
${\mathrm{first_int}}{≔}{2}{}{{x}}^{{5}}{}{{y}{}\left({x}\right)}^{{2}}{+}{{x}}^{{4}}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{\mathrm{_C1}}{=}{0}$ (6)
> $\mathrm{firtest}\left(\mathrm{first_int},\mathrm{ODE}\right)$
${0}$ (7) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 17, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9986535906791687, "perplexity": 1361.2215027652294}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370521876.48/warc/CC-MAIN-20200404103932-20200404133932-00559.warc.gz"} |
https://www.spp2026.de/members-guests/15-member-pages/prof-dr-anand-dessai/ | # Members & Guests
## Prof. Dr. Anand Dessai
Université de Fribourg
E-mail: anand.dessai(at)unifr.ch
Telephone: +41 26 300-9184
Homepage: http://homeweb1.unifr.ch/dessaia/pub/...
## Publications within SPP2026
We show that the moduli space of metrics of nonnegative sectional curvature on every homotopy RP^5 has infinitely many path components. We also show that in each dimension 4k+1 there are at least 2^{2k} homotopy RP^{4k+1}s of pairwise distinct oriented diffeomorphism type for which the moduli space of metrics of positive Ricci curvature has infinitely many path components. Examples of closed manifolds with finite fundamental group with these properties were known before only in dimensions 4k+3≥7.
Journal to appear in Transactions of the AMS Link to preprint version
We show that in each dimension 4n+3, n>1, there exist infinite sequences of closed smooth simply connected manifolds M of pairwise distinct homotopy type for which the moduli space of Riemannian metrics with nonnegative sectional curvature has infinitely many path components. Closed manifolds with these properties were known before only in dimension 7, and our result also holds for moduli spaces of Riemannian metrics with positive Ricci curvature. Moreover, inconjunction with work of Belegradek, Kwasik and Schultz, we obtain that for each such M the moduli space of complete nonnegative sectional curvature metrics on the open simply connected manifold M × R also has infinitely many path components.
Journal Bulletin of the London Math. Society Volume 50 Pages 96-107 Link to preprint version Link to published version
Let M be a Milnor sphere or, more generally, the total space of a linear S^3-bundle over S^4 with H^4(M;Q) = 0. We show that the moduli space of metrics of nonnegative sectional curvature on M has infinitely many path components. The same holds true for the moduli space of m etrics of positive Ricci curvature on M.
Journal preprint arXiv Pages 11 pages Link to preprint version
• 1 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9002240896224976, "perplexity": 327.7527992568781}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875146342.41/warc/CC-MAIN-20200226115522-20200226145522-00113.warc.gz"} |
https://rd.springer.com/article/10.1007%2Fs00026-019-00414-1 | Advertisement
# Distribution of Descents in Matchings
• Gene B. Kim
Article
• 9 Downloads
## Abstract
The distribution of descents in certain conjugacy classes of $$S_n$$ has been previously studied, and it is shown that its moments have interesting properties. This paper provides a bijective proof of the symmetry of the descents and major indices of matchings (also known as fixed point free involutions) and uses a generating function approach to prove an asymptotic normality theorem for the number of descents in matchings.
## Keywords
Enumerative combinatorics Probabilistic combinatorics Central limit theorem
## Mathematics Subject Classification
Primary 05A19 60F05 Secondary 60C05 62E20
## Notes
### Acknowledgements
The author would like to thank Brendon Rhoades for introducing the author to the world of Young tableaux and helping come up with the bijection in Sect. 3. The author would also like to thank Sangchul Lee for the suggestion of considering moment generating functions instead of characteristic functions and helping with the calculations of bounding the integrals in Sect. 4. Finally, the author would like to thank Jason Fulman for the suggestion of the problem and his guidance.
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Chen, W.Y.C., Deng, E.Y.P., Du, R.R.X., Stanley, R.P., Yan, C.H.: Crossings and nestings of matchings and partitions. Trans. Amer. Math. Soc. 359(4), 1555–1575 (2007)
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Curtiss, J.H.: A note on the theory of moment generating functions. Ann. Math. Statist. 13, 430–433 (1942)
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Diaconis, P., McGrath, M., Pitman, J.: Riffle shuffles, cycles, and descents. Combinatorica 15(1), 11–29 (1995)
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Fulman, J.: Stein’s method and non-reversible Markov chains. In: Diaconis, P., Holmes, S. (eds.) Stein’s Method: Expository Lectures and Applications, IMS Lecture Notes Monogr. Ser., Vol. 46, pp. 69–77. Inst. Math. Statist., Beachwood, OH (2004)Google Scholar
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Garsia, A.M., Gessel, I.: Permutation statistics and partitions. Adv. Math. 31(3), 288–305 (1979)
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Gessel, I.M., Reutenauer, C.: Counting permutations with given cycle structure and descent set. J. Combin. Theory Ser. A 64(2), 189–215 (1993)
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Goldstein, L., Rinott, Y.: A permutation test for matching and its asymptotic distribution. Metron 61(3), 375–388 (2004)
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Harper, L.H.: Stirling behavior is asymptotically normal. Ann. Math. Statist. 38, 410–414 (1966)
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Knuth, D.E.: The Art of Computer Programming. Vol. 3. Sorting and Searching. Addison-Wesley Series in Computer Science and Information Processing. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. (1973)Google Scholar
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Vatutin, V.A.: The numbers of ascending segments in a random permutation and in one inverse to it are asymptotically independent. Discrete Math. Appl. 6(1), 41–52 (1996)
## Copyright information
© Springer Nature Switzerland AG 2019
## Authors and Affiliations
1. 1.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA | {"extraction_info": {"found_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": 0.7470290064811707, "perplexity": 3776.3709551847523}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247481992.39/warc/CC-MAIN-20190217111746-20190217133746-00365.warc.gz"} |
https://eprint.iacr.org/2008/268 | ## Cryptology ePrint Archive: Report 2008/268
Craig Gentry and Brent Waters
Abstract: We present new techniques for achieving adaptive security in broadcast encryption systems. Previous work on fully-collusion resistant broadcast encryption with short ciphertexts was limited to only considering static security.
First, we present a new definition of security that we call semi-static security and show a generic two-key" transformation from semi-statically secure systems to adaptively secure ones that have comparable-sized ciphertexts. Using bilinear maps, we then construct broadcast encryption systems that are semi-statically secure in the standard model and have constant size ciphertexts. Our semi-static constructions work when the number of indices or identifiers in the system is polynomial in the security parameter.
For identity-based broadcast encryption, where the number of potential indices or identifiers may be exponential, we present the first adaptively secure system with sublinear ciphertexts. We prove security in the standard model.
Category / Keywords:
Date: received 11 Jun 2008, last revised 23 Jun 2008
Contact author: bwaters at csl sri com
Available format(s): PDF | BibTeX Citation
Short URL: ia.cr/2008/268
[ Cryptology ePrint archive ] | {"extraction_info": {"found_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": 0.24691298604011536, "perplexity": 7896.416746736301}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084890795.64/warc/CC-MAIN-20180121155718-20180121175718-00647.warc.gz"} |
http://mathhelpforum.com/advanced-statistics/193391-probability-distribution.html | ## Probability distribution
1. You go to the supermarket with your friend. After picking up something, each of you arrives at a different queue at the same time. Assume that the time you need to wait(measured in minutes) has a geometric distribution with mean 2 and the waiting time of your friend is also geometrically distributed but with mean 4.
a) Find the pmf of the difference between the waiting times of you and your friend.
b) Find the probability that you wait longer than your friend.
2. Suppose that 40% of voters are in favor of certain legislation. A large number of voters are polled and a relative frequency estimate for the above proportion is obtained. Use the Chebyshev inequality to determine how many voters should be polled in order to make sure that the probability that the relative frequency estimate differs from the actual probability by
less than 0.01 is at least 0.95.
3. Let the number of widgets tested in an assembly line in 1 hour be a binomial random variable with parameter n=600 and p . Suppose that the probability that a widget is faulty is q. Denote S as the number of widgets that are found faulty in a 1-hour period. Find the mean and variance of S. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9260799288749695, "perplexity": 326.1033046354893}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657139669.58/warc/CC-MAIN-20140914011219-00177-ip-10-234-18-248.ec2.internal.warc.gz"} |
http://nrich.maths.org/6949/solution | ### Number Detective
Follow the clues to find the mystery number.
### Eight Dominoes
Using the 8 dominoes make a square where each of the columns and rows adds up to 8
### Online
A game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter.
# Which Numbers? (2)
##### Stage: 2 Challenge Level:
In a similar way to Which Numbers? (1) the solutions we had tended to identify correctly two of the sets but struggled with the third.
Joshua of Crookhill Primary School said:
For the red group it is all the multiples of $6$.
For the blue group it is $+ 13$ every time.
and for the black we have no idea what so ever!
Sophie and Jo of Huish Primary continued:
The blue set's give away numbers are $26, 39, 65$ and $91$. We first looked at the end digits and saw they were going up by $3$ each time. We then knew it was a multiple of somthing with a $3$ on the end. We then knew they were going up by $10$ each time. We added the $10$ and the $3$ together to get $13$. So the blue set is going up by $13$ each time: $\{13,26,39,52,65,78,91\}$. There are $7$ numbers in the blue which is the same as on the sheet.
The red set's give away numbers are $12, 18, 30, 42, 66, 78, 84$. We knew they were even, so it would be in either the $2$s, $4$s, $6$s or $8$s. We narrowed it down to the $6$s and the $2$s. The $2$s has $50$ numbers less than $101$, so we knew it was the $6$s. There were $16$ numbers in the red set like it said on the sheet: $\{6,12,18,24,30,36,42,48,54,60,66,72,78,84,90,96\}$.
The black set's give away numbers are $14, 17, 33, 38, 51, 57, 74, 79, 94, 99$. We thought a long time about what it could be. As we looked closer we realised that the $10$s digit was always odd. We also realised that there are $50$ numbers with an odd $10$s digit before $101$. So the black set is all the $10$s, $30$s, $50$s, $70$s and $90$s.
Do you agree with Sophie and Jo? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8343479633331299, "perplexity": 572.2404594345279}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320915.38/warc/CC-MAIN-20170627032130-20170627052130-00077.warc.gz"} |
http://www.computer.org/csdl/trans/ts/2012/03/tts2012030629-abs.html | Subscribe
Issue No.03 - May-June (2012 vol.38)
pp: 629-641
Robert Mark Hierons , Brunel University, Middlesex
ABSTRACT
The problem of deciding whether an observed behavior is acceptable is the oracle problem. When testing from a finite state machine (FSM), it is easy to solve the oracle problem and so it has received relatively little attention for FSMs. However, if the system under test has physically distributed interfaces, called ports, then in distributed testing, we observe a local trace at each port and we compare the set of local traces with the set of allowed behaviors (global traces). This paper investigates the oracle problem for deterministic and nondeterministic FSMs and for two alternative definitions of conformance for distributed testing. We show that the oracle problem can be solved in polynomial time for the weaker notion of conformance ({\sqsubseteq_w}) but is NP-hard for the stronger notion of conformance ({\sqsubseteq_s}), even if the FSM is deterministic. However, when testing from a deterministic FSM with controllable input sequences, the oracle problem can be solved in polynomial time and similar results hold for nondeterministic FSMs. Thus, in some cases, the oracle problem can be efficiently solved when using \sqsubseteq_s and where this is not the case, we can use the decision procedure for \sqsubseteq_w as a sound approximation.
INDEX TERMS
Software engineering/software/program verification, software engineering/testing and debugging, systems and software, distributed systems, finite state machine, nondeterminism, test oracle, controllability, local observability.
CITATION
Robert Mark Hierons, "Oracles for Distributed Testing", IEEE Transactions on Software Engineering, vol.38, no. 3, pp. 629-641, May-June 2012, doi:10.1109/TSE.2011.45
REFERENCES
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Ural, "The Synchronization Problem in Protocol Testing and Its Complexity," Information Processing Letters, vol. 40, no. 3, pp. 131-136, 1991. [19] D. Lee and M. Yannakakis, "Principles and Methods of Testing Finite-State Machines—A Survey," Proc. IEEE, vol. 84, no. 8, pp. 1089-1123, Aug. 1996. [20] R.M. Hierons, M.G. Merayo, and M. Núñez, "Implementation Relations for the Distributed Test Architecture," Proc. 20th IFIP TC 6/WG 6.1 Int'l Conf. Testing of Software and Comm. Systems, pp. 200-215, 2008. [21] R.M. Hierons, M.G. Merayo, and M. Núñez, "Implementation Relations and Test Generation for Systems with Distributed Interfaces," submitted for publication. [22] R.M. Hierons, "Controllable Testing from Nondeterministic Finite State Machines with Multiple Ports," IEEE Trans. Computers, vol. 60, no. 12, pp. 1818-1822, Dec. 2011. [23] J.L. Jacob, "Refinement of Shared Systems," Theory and Practice of Refinement: Approaches to the Formal Development of Large-Scale Software Systems, J. McDermid, ed., pp. 27-36, Butterworths, 1989. [24] C. Kloukinas, G. Spanoudakis, and K. Mahbub, "Estimating Event Lifetimes for Distributed Runtime Verification," Proc. 20th Int'l Conf. Software Eng. and Knowledge Eng., pp. 117-122, 2008. [25] L. Cacciari and O. Rafiq, "Controllability and Observability in Distributed Testing," Information and Software Technology, vol. 41, nos. 11/12, pp. 767-780, 1999. [26] O. Rafiq and L. Cacciari, "Coordination Algorithm for Distributed Testing," The J. Supercomputing, vol. 24, no. 2, pp. 203-211, 2003. [27] A. Bauer, M. Leucker, and C. Schallhart, "Model-Based Runtime Analysis of Distributed Reactive Systems," Proc. 17th Australian Software Eng. Conf., pp. 243-252, 2006. [28] P.S. Dodd and C.V. Ravishankar, "Monitoring and Debugging Distributed Real-Time Programs," Software—Practice and Experience, vol. 22, no. 10, pp. 863-877, 1992. [29] D. Gunter, B. Tierney, B. Crowley, M. Holding, and J. Lee, "Netlogger: A Toolkit for Distributed System Performance Analysis," Proc. Eighth Int'l Symp. Modeling, Analysis and Simulation of Computer and Telecomm. Systems, pp. 267-273, 2000. [30] M. Mansorui-Samani and M. Sloman, "Monitoring Distributed Systems," IEEE Network, vol. 7, no. 6, pp. 20-30, Nov. 1993. [31] M. Zulkernine and R.E. Seviora, "A Compositional Approach to Monitoring Distributed Systems," Proc. Int'l Conf. Dependable Systems and Networks, pp. 763-772, 2002. [32] E. Brinksma, L. Heerink, and J. Tretmans, "Factorized Test Generation for Multi-Input/Output Transition Systems," Proc. IFIP TC6 11th Int'l Workshop Testing Communicating Systems, pp. 67-82, 1998. [33] L. Heerink and J. Tretmans, "Refusal Testing for Classes of Transition Systems with Inputs and Outputs," Proc. Int'l Conf. Formal Description Techniques for Distributed Systems and Comm. Protocols and Protocol Specification, Testing and Verification, pp. 23-38, 1997. [34] Z. Li, J. Wu, and X. Yin, "Testing Multi Input/Output Transition System with All-Observer," Proc. 16th IFIP Int'l Conf. Testing of Communicating Systems, pp. 95-111, 2004. [35] T.J. Schaefer, "The Complexity of Satisfiability Problems," Proc. 10th Ann. ACM Symp. Theory of Computing, pp. 216-226, 1978. [36] S. Haar, C. Jard, and G.-V. Jourdan, "Testing Input/Output Partial Order Automata," Proc. Int'l Conf. Testing of Software and Communicating Systems, pp. 171-185, 2007. [37] G. von Bochmann, S. Haar, C. Jard, and G.-V. Jourdan, "Testing Systems Specified as Partial Order Input/Output Automata," Proc. 20th IFIP TC 6/WG 6.1 Int'l Conf. Testing of Software and Communicating Systems, pp. 169-183, 2008. | {"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": 0.9323804974555969, "perplexity": 9481.062423885374}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1422115855845.27/warc/CC-MAIN-20150124161055-00081-ip-10-180-212-252.ec2.internal.warc.gz"} |
https://export.arxiv.org/list/math-ph/new | # Mathematical Physics
## New submissions
[ total of 23 entries: 1-23 ]
[ showing up to 2000 entries per page: fewer | more ]
### New submissions for Fri, 4 Dec 20
[1]
Title: On the Near-Critical Behavior of Continuous Polymers
Comments: arXiv admin note: text overlap with arXiv:2008.04493
Subjects: Mathematical Physics (math-ph)
The aim of this paper is to investigate the distribution of a continuous polymer in the presence of an attractive finitely supported potential. The most intricate behavior can be observed if we simultaneously and independently vary two parameters: the temperature, which approaches the critical value, and the length of the polymer chain, which tends to infinity. We describe how the typical size of the polymer depends on the two parameters.
[2]
Title: On the Lyapunov-Perron reducible Markovian Master Equation
Comments: 22 pages, no figures
Subjects: Mathematical Physics (math-ph); Quantum Physics (quant-ph)
We consider an open quantum system in $M_{d}(\mathbb{C})$ governed by quasiperiodic Hamiltonian with rationally independent frequencies and under assumption of Lyapunov-Perron reducibility of associated Schroedinger equation. We construct the Markovian Master Equation and resulting CP-divisible evolution in weak coupling limit regime, generalizing our previous results from periodic case. The analysis is conducted with application of projection operator techniques and concluded with some results regarding stability of solutions and existence of quasiperiodic global steady state.
[3]
Title: A geometric approach to Wigner-type theorems
Subjects: Mathematical Physics (math-ph)
Let $H$ be a complex Hilbert space and let ${\mathcal P}(H)$ be the associated projective space (the set of rank-one projections). Suppose that $\dim H\ge 3$. We prove the following Wigner-type theorem: if $H$ is finite-dimensional, then every orthogonality preserving transformation of ${\mathcal P}(H)$ is induced by a unitary or anti-unitary operator. This statement will be obtained as a consequence of the following result: every orthogonality preserving lineation of ${\mathcal P}(H)$ to itself is induced by a linear or conjugate-linear isometry ($H$ is not assumed to be finite-dimensional). As an application, we describe (not necessarily injective) transformations of Grassmannians preserving some types of principal angles.
### Cross-lists for Fri, 4 Dec 20
[4] arXiv:2012.01453 (cross-list from quant-ph) [pdf, other]
Title: Constructing quantum codes from any classical code and their embedding in ground space of local Hamiltonians
Comments: 29 pages + references ; 7 figures
Subjects: Quantum Physics (quant-ph); Strongly Correlated Electrons (cond-mat.str-el); Mathematical Physics (math-ph)
We introduce a framework for constructing a quantum error correcting code from {\it any} classical error correcting code. This includes CSS codes and goes beyond the stabilizer formalism to allow quantum codes to be constructed from classical codes that are not necessarily linear or self-orthogonal (Fig. 1). We give an algorithm that explicitly constructs quantum codes with linear distance and constant rate from classical codes with a linear distance and rate. As illustrations for small size codes, we obtain Steane's $7-$qubit code uniquely from Hamming's [7,4,3] code, and obtain other error detecting quantum codes from other explicit classical codes of length 4 and 6. Motivated by quantum LDPC codes and the use of physics to protect quantum information, we introduce a new 2-local frustration free quantum spin chain Hamiltonian whose ground space we analytically characterize completely. By mapping classical codewords to basis states of the ground space, we utilize our framework to demonstrate that the ground space contains explicit quantum codes with linear distance. This side-steps the Bravyi-Terhal no-go theorem because our work allows for more general quantum codes beyond the stabilizer and/or linear codes. We hesitate to call this an example of {\it subspace} quantum LDPC code with linear distance.
[5] arXiv:2012.01550 (cross-list from math.DG) [pdf, ps, other]
Title: Bochner-Kodaira Formulas and the Type IIA Flow
Comments: 36 pages, comments welcome!
Subjects: Differential Geometry (math.DG); Mathematical Physics (math-ph); Analysis of PDEs (math.AP); Symplectic Geometry (math.SG)
A new derivation of the flow of metrics in the Type IIA flow is given. It is adapted to the formulation of the flow as a variant of a Laplacian flow, and it uses the projected Levi-Civita connection of the metrics themselves instead of their conformal rescalings.
[6] arXiv:2012.01593 (cross-list from math.DS) [pdf, ps, other]
Title: Logarithmic capacity of random $G_δ$-sets
Subjects: Dynamical Systems (math.DS); Mathematical Physics (math-ph); Probability (math.PR)
We study the logarithmic capacity of $G_\delta$ subsets of the interval $[0,1].$ Let $S$ be of the form \begin{align*} S=\bigcap_m \bigcup_{k\ge m} I_k, \end{align*} where each $I_k$ is an interval in $[0,1]$ with length $l_k$ that decrease to $0$. We provide sufficient conditions for $S$ to have full capacity, i.e. $\mathop{\mathrm{Cap}}(S)=\mathop{\mathrm{Cap}}([0,1])$. We consider the case when the intervals decay exponentially and are placed in $[0,1]$ randomly with respect to some given distribution. The random $G_\delta$ sets generated by such distribution satisfy our sufficient conditions almost surely and hence, have full capacity almost surely. This study is motivated by the $G_\delta$ set of exceptional energies in the parametric version of the Furstenberg theorem on random matrix products. We also study the family of $G_\delta$ sets $\{S(\alpha)\}_{\alpha>0}$ that are generated by setting the decreasing speed of the intervals to $l_k=e^{-k^\alpha}.$ We observe a sharp transition from full capacity to zero capacity by varying $\alpha>0$.
[7] arXiv:2012.01818 (cross-list from math.DG) [pdf, ps, other]
Title: Port-Hamiltonian Modeling of Ideal Fluid Flow: Part I. Foundations and Kinetic Energy
Comments: This is a preprint submitted to the journal of Geometry and Physics. Please do not CITE this version, but only the published manuscript
Subjects: Differential Geometry (math.DG); Mathematical Physics (math-ph); Fluid Dynamics (physics.flu-dyn)
In this two-parts paper, we present a systematic procedure to extend the known Hamiltonian model of ideal inviscid fluid flow on Riemannian manifolds in terms of Lie-Poisson structures to a port-Hamiltonian model in terms of Stokes-Dirac structures. The first novelty of the presented model is the inclusion of non-zero energy exchange through, and within, the spatial boundaries of the domain containing the fluid. The second novelty is that the port-Hamiltonian model is constructed as the interconnection of a small set of building blocks of open energetic subsystems. Depending only on the choice of subsystems one composes and their energy-aware interconnection, the geometric description of a wide range of fluid dynamical systems can be achieved. The constructed port-Hamiltonian models include a number of inviscid fluid dynamical systems with variable boundary conditions. Namely, compressible isentropic flow, compressible adiabatic flow, and incompressible flow. Furthermore, all the derived fluid flow models are valid covariantly and globally on n-dimensional Riemannian manifolds using differential geometric tools of exterior calculus.
[8] arXiv:2012.01827 (cross-list from physics.flu-dyn) [pdf, ps, other]
Title: Port-Hamiltonian Modeling of Ideal Fluid Flow: Part II. Compressible and Incompressible Flow
Comments: This is a prevprint submitted to the journal of Geometry and Physics. Please DO NOT CITE this version, but only the published manuscript
Subjects: Fluid Dynamics (physics.flu-dyn); Mathematical Physics (math-ph); Differential Geometry (math.DG)
Part I of this paper presented a systematic derivation of the Stokes Dirac structure underlying the port-Hamiltonian model of ideal fluid flow on Riemannian manifolds. Starting from the group of diffeomorphisms as a configuration space for the fluid, the Stokes Dirac structure is derived by Poisson reduction and then augmented by boundary ports and distributed ports. The additional boundary ports have been shown to appear naturally as surface terms in the pairings of dual maps, always neglected in standard Hamiltonian theory. The port-Hamiltonian model presented in Part I corresponded only to the kinetic energy of the fluid and how its energy variables evolve such that the energy is conserved.
In Part II, we utilize the distributed port of the kinetic energy port-Hamiltonian system for representing a number of fluid-dynamical systems. By adding internal energy we model compressible flow, both adiabatic and isentropic, and by adding constraint forces we model incompressible flow. The key tools used are the interconnection maps relating the dynamics of fluid motion to the dynamics of advected quantities.
[9] arXiv:2012.01858 (cross-list from math.RT) [pdf, ps, other]
Title: Local opers with two singularities: the case of $\mathfrak{sl}(2)$
Comments: 56 pages. Comments are very welcome!
Subjects: Representation Theory (math.RT); Mathematical Physics (math-ph); Algebraic Geometry (math.AG)
We study local opers with two singularities for the case of the Lie algebra sl(2), and discuss their connection with a two-variables extension of the affine Lie algebra. We prove an analogue of the Feigin-Frenkel theorem describing the centre at the critical level, and an analogue of a result by Frenkel and Gaitsgory that characterises the endomorphism rings of Weyl modules in terms of functions on the space of opers.
[10] arXiv:2012.01889 (cross-list from gr-qc) [pdf, ps, other]
Title: Null infinity as an open Hamiltonian system
Authors: Wolfgang Wieland
Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
When a system emits gravitational radiation, the Bondi mass decreases. If the Bondi energy is Hamiltonian, it can thus only be a time dependent Hamiltonian. In this paper, we show that the Bondi energy can be understood as a time-dependent Hamiltonian on the covariant phase space. Our derivation starts from the Hamiltonian formulation in domains with boundaries that are null. We introduce the most general boundary conditions on a generic such null boundary, and compute quasi-local charges for boosts, energy and angular momentum. Initially, these domains are at finite distance, such that there is a natural IR regulator. To remove the IR regulator, we introduce a double null foliation together with an adapted Newman--Penrose null tetrad. Both null directions are surface orthogonal. We study the falloff conditions for such specific null foliations and take the limit to null infinity. At null infinity, we recover the Bondi mass and the usual covariant phase space for the two radiative modes at the full non-perturbative level. Apart from technical results, the framework gives two important physical insights. First of all, it explains the physical significance of the corner term that is added in the Wald--Zoupas framework to render the quasi-conserved charges integrable. The term to be added is simply the derivative of the Hamiltonian with respect to the background fields that drive the time-dependence of the Hamiltonian. Secondly, we propose a new interpretation of the Bondi mass as the thermodynamical free energy of gravitational edge modes at future null infinity. The Bondi mass law is then simply the statement that the free energy always decreases on its way towards thermal equilibrium.
[11] arXiv:2012.01943 (cross-list from math.CA) [pdf, other]
Title: Finite-Part Integration in the Presence of Competing Singularities: Transformation Equations for the hypergeometric functions arising from Finite-Part Integration
Comments: 44 pages, 3 figures
Subjects: Classical Analysis and ODEs (math.CA); Mathematical Physics (math-ph); Complex Variables (math.CV)
Finite-part integration is a recently introduced method of evaluating convergent integrals by means of the finite part of divergent integrals [E.A. Galapon, {\it Proc. R. Soc. A 473, 20160567} (2017)]. Current application of the method involves exact and asymptotic evaluation of the generalized Stieltjes transform $\int_0^a f(x)/(\omega + x)^{\rho} \, \mathrm{d}x$ under the assumption that the extension of $f(x)$ in the complex plane is entire. In this paper, the method is elaborated further and extended to accommodate the presence of competing singularities of the complex extension of $f(x)$. Finite part integration is then applied to derive consequences of known Stieltjes integral representations of the Gauss function and the generalized hypergeometric function which involve Stieltjes transforms of functions with complex extensions having singularities in the complex plane. Transformation equations for the Gauss function are obtained from which known transformation equations are shown to follow. Also, building on the results for the Gauss function, transformation equations involving the generalized hypergeometric function $\,_3F_2$ are derived.
[12] arXiv:2012.01995 (cross-list from math.CO) [pdf, other]
Title: Multicritical random partitions
Comments: 12 pages, 3 figures
Subjects: Combinatorics (math.CO); Mathematical Physics (math-ph); Probability (math.PR)
We study two families of probability measures on integer partitions, which are Schur measures with parameters tuned in such a way that the edge fluctuations are characterized by a critical exponent different from the generic $1/3$. We find that the first part asymptotically follows a "higher-order analogue" of the Tracy-Widom GUE distribution, previously encountered by Le Doussal, Majumdar and Schehr in quantum statistical physics. We also compute limit shapes, and discuss an exact mapping between one of our families and the multicritical unitary matrix models introduced by Periwal and Shevitz.
[13] arXiv:2012.02079 (cross-list from cond-mat.stat-mech) [pdf, other]
Title: Effective free-fermionic form factors and the XY spin chain
Subjects: Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)
We introduce effective form factors for one-dimensional lattice fermions with arbitrary phase shifts. We study tau functions defined as series of these form factors. On the one hand we perform the exact summation and present tau functions as Fredholm determinants in the thermodynamic limit. On the other hand simple expressions of form factors allow us to present the corresponding series as integrals of elementary functions. Using this approach we re-derive the asymptotics of static correlation functions of the XY quantum chain at finite temperature.
### Replacements for Fri, 4 Dec 20
[14] arXiv:1801.05183 (replaced) [pdf, ps, other]
Title: Riemannian exponential and quantization
Comments: Important changes have been made with respect to the previous version, including 1) An improved demonstration of the equivalence between the two proposed quantizations and 2) A major extension of one of them to a much broader set of functions. The title, the abstract and the introduction have been modified, making them more in line with the new content
Subjects: Mathematical Physics (math-ph)
[15] arXiv:1811.02551 (replaced) [pdf, ps, other]
Title: Topological defects in lattice models and affine Temperley-Lieb algebra
Comments: 44 pages, v2: much improved version with few sections rewritten, new result in Theorem 2.1, many typos fixed
Subjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Quantum Algebra (math.QA); Representation Theory (math.RT)
[16] arXiv:2003.10526 (replaced) [pdf, ps, other]
Title: Hessian metric via transport information geometry
Authors: Wuchen Li
Subjects: Differential Geometry (math.DG); Information Theory (cs.IT); Mathematical Physics (math-ph)
[17] arXiv:2004.08934 (replaced) [pdf, ps, other]
Title: Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications
Comments: 70 pages
Subjects: Metric Geometry (math.MG); Mathematical Physics (math-ph); Differential Geometry (math.DG); Optimization and Control (math.OC)
[18] arXiv:2008.11884 (replaced) [pdf, ps, other]
Title: Orthogonal rational functions with real poles, root asymptotics, and GMP matrices
Subjects: Spectral Theory (math.SP); Mathematical Physics (math-ph); Classical Analysis and ODEs (math.CA)
[19] arXiv:2009.04783 (replaced) [pdf, ps, other]
Title: Bounds on amplitude damping channel discrimination
Comments: 15 pages. 7 figures
Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
[20] arXiv:2011.08830 (replaced) [pdf, other]
Title: Stable maps to Looijenga pairs
Comments: 114 pages (80pp+appendices), 40 figures. v2: minor changes, references added
Subjects: Algebraic Geometry (math.AG); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
[21] arXiv:2011.10554 (replaced) [pdf, other]
Title: Multi-orbital Flat Band Ferromagnetism with a Provable Percolation Representation
Subjects: Strongly Correlated Electrons (cond-mat.str-el); Mathematical Physics (math-ph)
[22] arXiv:2011.11402 (replaced) [pdf, other]
Title: The linear and nonlinear instability of the Akhmediev breather
Authors: P. G. Grinevich (1 and 2), P. M. Santini (3 and 4) ((1) Steklov Mathematical Institute of Russian Academy of Sciences, (2) L.D. Landau Institute for Theoretical Physics of Russian Academy of Sciences, (3) Dipartimento di Fisica, Università di Roma "La Sapienza", (4) Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Roma)
Comments: 31 pages, 4 figures, Simplification of final formulas was made in this version
Subjects: Pattern Formation and Solitons (nlin.PS); Mathematical Physics (math-ph); Fluid Dynamics (physics.flu-dyn); Optics (physics.optics)
[23] arXiv:2011.13499 (replaced) [pdf, ps, other]
Title: Contact Geometry in Superconductors and New Massive Gravity
Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
[ total of 23 entries: 1-23 ]
[ showing up to 2000 entries per page: fewer | more ]
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Links to: arXiv, form interface, find, math-ph, recent, 2012, contact, help (Access key information) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8644097447395325, "perplexity": 1253.8547626797408}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141743438.76/warc/CC-MAIN-20201204193220-20201204223220-00220.warc.gz"} |
http://crypto.stackexchange.com/questions?page=35&sort=newest | All Questions
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El-Gamal signature with two messages
Alice uses an ElGamal signature with base the group $Z^*_{107}$ and parameter $g=3$ of order $q=53$.The private key of Alice is some $x \in \{0,1,.....,52\}$ and the public key of her is $y=10$. To ...
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Can a 1 byte difference in AES 128 bit keys make huge difference in output?
If we take some randomly generated key of AES-128 and we change any random 1 byte of that 16 byte key, will this make huge difference in the AES cipher text generated over same input string? Does ...
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Terminology: differences between the terms “pre-master secret”, “master secret”, “private key”, and “shared secret”?
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Transforming Gaussian random $[0,1]$ numbers to uniform $[0,255]$
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Proving existence of an encryption scheme that has indistinghuishable multiple encryptions in the presence of an eavesdropper, but is not CPA-secure [duplicate]
I got stuck in trying to find a solution to the 3.7 exercise of the Katz-Lindell book. The exercise also assumes the existence of a pseudorandom function. The problem is that a multiple messages ...
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How can I generate a good password from a SHA512 hash?
I have to change local administrator passwords on machines. I don't want to store password in a database. I have to generate a password that I can find later to connect to the machine again. So I ...
77 views | {"extraction_info": {"found_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": 0.34079885482788086, "perplexity": 1961.195428766196}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701153998.27/warc/CC-MAIN-20160205193913-00141-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://lavelle.chem.ucla.edu/forum/viewtopic.php?p=212683 | ## H20 in the ICE table
Alexis Robles 2k
Posts: 100
Joined: Wed Sep 18, 2019 12:18 am
### H20 in the ICE table
is H20 the only thing we leave out when doing an ICE table?
205007651
Posts: 53
Joined: Tue Nov 13, 2018 12:17 am
### Re: H20 in the ICE table
Yes, you will only use the other three compounds in an ICE table.
Ashley Nguyen 2L
Posts: 103
Joined: Sat Aug 17, 2019 12:18 am
Been upvoted: 1 time
### Re: H20 in the ICE table
When doing an ICE table, do not include water, liquids, or solids, as these do not have concentrations.
Anna Chen 1K
Posts: 104
Joined: Thu Jul 25, 2019 12:18 am
### Re: H20 in the ICE table
Yes H2O is not included in the ICE tables along with solids and liquids.
Alexaaguilera
Posts: 7
Joined: Fri Sep 28, 2018 12:20 am
### Re: H20 in the ICE table
I think you only include gases when doing an ICE
SimranSangha4I
Posts: 99
Joined: Sat Sep 14, 2019 12:17 am
### Re: H20 in the ICE table
Yep! Leave out H2O, solids, and other liquids.
RasikaObla_4I
Posts: 100
Joined: Thu Jul 25, 2019 12:15 am
### Re: H20 in the ICE table
Yes, you should leave out H2O because we assume that it is in excess so the concentration doesn't affect the reaction.
Mitchell Koss 4G
Posts: 128
Joined: Sat Jul 20, 2019 12:17 am
### Re: H20 in the ICE table
Leave H2O out only if it is a liquid. It must be included if it is a gas.
Julia Holsinger_1A
Posts: 50
Joined: Tue Feb 26, 2019 12:16 am
### Re: H20 in the ICE table
I believe ICE table only pertains to gases
stephaniekim2K
Posts: 79
Joined: Fri Aug 09, 2019 12:16 am
### Re: H20 in the ICE table
You include any gases, but since solids and liquids have negligible change they are not included in ice box or the equilibrium constant.
nicole-2B
Posts: 103
Joined: Fri Aug 30, 2019 12:18 am
### Re: H20 in the ICE table
when performing a ice table you should only worry about the gasses and aqueous molecules
anjali41
Posts: 109
Joined: Fri Aug 09, 2019 12:15 am
### Re: H20 in the ICE table
Typically in the textbook problems, water is seen in liquid form. Since liquid and solids are not included in ice tables, do not include water. However if water was in gas form and depending on the problem, I think you might have to include it.
205291012
Posts: 50
Joined: Mon Nov 11, 2019 12:17 am
### Re: H20 in the ICE table
Only if H20 is in the liquid form then do not include it in the ICE table but if it is in gaseous form then include it. Also, don't include any solids or liquids in the ICE table.
Sally Qiu 2E
Posts: 105
Joined: Fri Aug 30, 2019 12:18 am
### Re: H20 in the ICE table
leave out pure liquids and solids
alex_4l
Posts: 109
Joined: Thu Sep 26, 2019 12:18 am
### Re: H20 in the ICE table
Leave out water, solids and other liquids but aqueous and gases can be used in the table
Jainam Shah 4I
Posts: 130
Joined: Fri Aug 30, 2019 12:16 am
### Re: H20 in the ICE table
Leave out pure liquids and solids. Water as a liquid isn't included, because it remains in excess. Its acting as a solute and the changes in concentration of water are so minute we don't have to account for them.
Joanne Lee 1J
Posts: 100
Joined: Thu Jul 25, 2019 12:15 am
### Re: H20 in the ICE table
In addition to leaving out water, you should also leave out solids and liquids in the ICE table.
Jasmine Fendi 1D
Posts: 108
Joined: Sat Aug 24, 2019 12:15 am
### Re: H20 in the ICE table
When filling out and ICE table, only include gases and aqueous molecules. In addition to the comments above, you want to leave our solids because you cannot really condense or expand solids since they are in a fixed state
Matthew Tsai 2H
Posts: 101
Joined: Wed Sep 18, 2019 12:20 am
### Re: H20 in the ICE table
H2O could be included if it is in the gas phase and you are calculating Kp. Otherwise it should be excluded if it is liquid (which most of the time it will be).
Junwei Sun 4I
Posts: 125
Joined: Wed Oct 02, 2019 12:16 am
### Re: H20 in the ICE table
In an ICE table we leave about liquids and solids. If the H2O is in gas phase then it will be included in the ICE table as well.
JohannaPerezH2F
Posts: 94
Joined: Wed Sep 18, 2019 12:18 am
### Re: H20 in the ICE table
yes you don't include water but also don't include other liquids and solids, just gases.
kristi le 2F
Posts: 102
Joined: Thu Jul 11, 2019 12:15 am
### Re: H20 in the ICE table
whether we leave out water in the ICE table is dependent on the question, so make sure to read the chemical equation carefully.
Mariana Fuentes 1L
Posts: 43
Joined: Wed Nov 15, 2017 3:00 am
### Re: H20 in the ICE table
Do not include H2O.
Posts: 125
Joined: Sat Aug 17, 2019 12:17 am
### Re: H20 in the ICE table
Alexis Robles 2k wrote:is H20 the only thing we leave out when doing an ICE table?
In ICE tables, you leave out liquids and solids, so if that water is in liquid form, you leave it out. If that water is in gaseous form, you do have to include it in the ICE table.
805097738
Posts: 180
Joined: Wed Sep 18, 2019 12:20 am
### Re: H20 in the ICE table
you leave out any molecule that is a pure liquid or solid. If H20 is in a gas phase, then it is included
vibha gurunathan 1h
Posts: 51
Joined: Mon Oct 07, 2019 12:15 am
### Re: H20 in the ICE table
Alexaaguilera wrote:I think you only include gases when doing an ICE
you include gases as well as aqueous reactants/products
Pablo 1K
Posts: 118
Joined: Sat Feb 02, 2019 12:15 am
### Re: H20 in the ICE table
You do not include liquids, solids, and water even in expressions. Aqueous is included though. Water in gas phase is included though. | {"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": 0.8261533379554749, "perplexity": 6428.948380282329}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703517966.39/warc/CC-MAIN-20210119042046-20210119072046-00644.warc.gz"} |
https://fake.build/dotnet-nuget.html | # NuGet package restore
Note: This documentation is for FAKE.exe before version 5 (or the non-netcore version). The documentation needs te be updated, please help!
If you are using a source control system like git you probably don't want to store all binary dependencies in it. With FAKE you can use NuGet to download all dependent packages during the build.
## Setting the stage for NuGet
In order to download the packages during the build we need to add NuGet.exe to our repository. You can download the "NuGet.exe Command Line Tool" from the release page.
## Restore packages from the build script
Modify your build script and add RestorePackages() near the beginning of the script. This will use the default parameters to retrieve all NuGet packages specified in "./**/packages.config" files.
If you need to use different parameters please use the RestorePackage task directly.
If you don't want to store FAKE.exe and its components in your repository, you can use a batch file which downloads it before the build:
1: 2: 3: 4: 5: @echo off cls "tools\nuget\nuget.exe" "install" "FAKE" "-OutputDirectory" "tools" "-ExcludeVersion" "tools\FAKE\tools\Fake.exe" build.fsx pause
# Creating NuGet packages
Note: This documentation is for FAKE.exe before version 5 (or the non-netcore version). The documentation needs te be updated, please help!
## Creating a .nuspec template
The basic idea to create nuget packages is to create a .nuspec template and let FAKE fill out the missing parts. The following code shows such .nuspec file from the OctoKit project.
1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: @[email protected] @[email protected] @[email protected] @[email protected] @[email protected] https://github.com/octokit/octokit.net/blob/master/LICENSE.txt https://github.com/octokit/octokit.net https://github.com/octokit/octokit.net/icon.png false @[email protected] @[email protected] Copyright GitHub 2013 GitHub API Octokit @[email protected] @[email protected] @[email protected]
The .nuspec template contains some placeholders like @[email protected] which can be replaced later by the build script. It also contains some specific information like the copyright which is not handled by FAKE.
The following table gives the correspondence between the placeholders and the fields of the record type used by the NuGet task.
Placeholder
replaced by (NuGetParams record field)
@[email protected]
Version
@[email protected]
Authors
@[email protected]
Project
@[email protected]
Summary
@[email protected]
Description
@[email protected]
Tags
@[email protected]
ReleaseNotes
@[email protected]
Copyright
@[email protected]
a combination of Dependencies and DependenciesByFramework
@[email protected]
a combination of References and ReferencesByFramework
@[email protected]
a list of source, target, and exclude strings for files to be included in the nuget package
## Setting up the build script
In the build script you need to create a target which executes the NuGet task:
1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: Target "CreatePackage" (fun _ -> // Copy all the package files into a package folder CopyFiles packagingDir allPackageFiles NuGet (fun p -> {p with Authors = authors Project = projectName Description = projectDescription OutputPath = packagingRoot Summary = projectSummary WorkingDir = packagingDir Version = buildVersion AccessKey = myAccesskey Publish = true }) "myProject.nuspec" )
There are a couple of interesting things happening here. In this sample FAKE created:
• a copy of the .nuspec file
• filled in all the specified parameters
• created the NuGet package
• pushed it to nuget.org using the given myAccessKey.
## Handling package dependencies
If your project dependends on other projects it is possible to specify these dependencies in the .nuspec definition (see also Nuget docs). Here is a small sample which sets up dependencies for different framework versions:
1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: NuGet (fun p -> {p with Authors = authors // ... Dependencies = // fallback - for all unspecified frameworks ["Octokit", "0.1" "Rx-Main", GetPackageVersion "./packages/" "Rx-Main"] DependenciesByFramework = [{ FrameworkVersion = "net40" Dependencies = ["Octokit", "0.1" "Rx-Main", GetPackageVersion "./packages/" "Rx-Main" "SignalR", GetPackageVersion "./packages/" "SignalR"]} { FrameworkVersion = "net45" Dependencies = ["Octokit", "0.1" "SignalR", GetPackageVersion "./packages/" "SignalR"]}] // ... Publish = true }) "myProject.nuspec"
## Explicit assembly references
If you want to have auxiliary assemblies next to the ones that get referenced by the target project, you can place all the needed files in the lib directory and explicitly specify which of them should be referenced (see Nuget docs) via the References and ReferencesByFramework fields. Here is a code snippet showing how to use these:
1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: NuGet (fun p -> {p with Authors = authors // ... References = ["a.dll"] ReferencesByFramework = [{ FrameworkVersion = "net40"; References = ["b.dll"]} { FrameworkVersion = "net45"; References = ["c.dll"]}] // ... Publish = false }) "template.nuspec"
## Explicit file specifications
If you want to specify exactly what files are packaged and where they are placed in the resulting NuGet package you can specify the Files property directly. This is exactly like having the Files element of a nuspec filled out ahead of time. Here is a code snippet showing how to use this:
1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: // Here we are specifically only taking the js and css folders from our project and placing them in matching target folder in the resulting nuspec. // Note that the include paths are relative to the location of the .nuspec file // See [Nuget docs](http://docs.nuget.org/docs/reference/nuspec-reference#Specifying_Files_to_Include_in_the_Package) for more detailed examples of how to specify file includes, as this follows the same syntax. NuGet (fun p -> {p with // ... Files = [ (@"tools\**\*.*", None, None) (@"bin\Debug\*.dll", Some "lib", Some "badfile.css;otherbadfile.css") ] // ... }) "template.nuspec"
union case Option.None: Option<'T>
union case Option.Some: Value: 'T -> Option<'T> | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.44908177852630615, "perplexity": 5360.2784485628845}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948512208.1/warc/CC-MAIN-20171211052406-20171211072406-00281.warc.gz"} |
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## jordin Group Title triangle A,B,C is similar triangle S,T,R . Gary builds the proportion below to solve for the missing side AB. Explain what mistake Gary made and how you would correct it. AB OVER BC = ST OVER SR one year ago one year ago Edit Question Delete Cancel Submit
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1. hartnn Group Title
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when triangles are similar , "corresponding " sides are proportional AB corresponds to ST BC corresponds to TR so can u now find the error?
• one year ago
2. hartnn Group Title
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3
so it actually should be $$\frac{AB}{BC}=\frac{ST}{TR}$$
• one year ago
3. Denebel Group Title
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|dw:1347082841678:dw|
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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": 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": 0.9884203672409058, "perplexity": 19147.338723891287}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1406510273513.48/warc/CC-MAIN-20140728011753-00200-ip-10-146-231-18.ec2.internal.warc.gz"} |
http://annals.math.princeton.edu/2006/164-3 | Volume 164 Issue 3 – November 2006
## Global hyperbolicity of renormalization for $C^r$ unimodal mappings
Pages 731-824 by Edson de Faria, Welington de Melo, Alberto Pinto | From volume 164-3
## Orbit equivalence rigidity and bounded cohomology
Pages 825-878 by Nicolas Monod, Yehuda Shalom | From volume 164-3
## A Paley–Wiener theorem for reductive symmetric spaces
Pages 879-909 by Erik P. van den Ban, Henrik Schlichtkrull | From volume 164-3
## Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles
Pages 911-940 by Artur Ávila, Raphaël Krikorian | From volume 164-3
## Isometric actions of simple Lie groups on pseudoRiemannian manifolds
Pages 941-969 by Raul Quiroga-Barranco | From volume 164-3
## A Mass Transference Principle and the Duffin–Schaeffer conjecture for Hausdorff measures
Pages 971-992 by Victor Beresnevich, Sanju Velani | From volume 164-3
## Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing
Pages 993-1032 by Martin Hairer, Jonathan C. Mattingly | From volume 164-3
## Analytic representation of functions and a new quasi-analyticity threshold
Pages 1033-1064 by Gady Kozma, Alexander Olevskiĭ | From volume 164-3
## Proofs without syntax
Pages 1065-1076 by Dominic J. D. Hughes | From volume 164-3
## Non-quasi-projective moduli spaces
Pages 1077-1096 by János Kollár | From volume 164-3
## Erratum to “Generalizations of the Poincaré–Birkhoff theorem”
Pages 1097-1098 by John Franks | From volume 164-3 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3536631464958191, "perplexity": 19987.60707115316}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1454701153736.68/warc/CC-MAIN-20160205193913-00117-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://matheducators.stackexchange.com/questions/14869/how-to-deal-with-why-cant-i-just-do-in-real-analysis | # How to deal with “Why can't I just do …” in real analysis?
I'm teaching introductory real analysis at a large public university in the US. A common question from students is of the form
"Why can't I just do it like this?".
i.e. Often a student has come up with a not fully rigorous but quick or easy version of a correct analysis proof and then demand to know exactly "where" the mistake is.
A concrete example would be e.g. showing that the series $$\sum_{n=1}^{\infty}(a_n + b_n)$$ diverges to $$+\infty$$ say by using $$\sum_{n=1}^{\infty}(a_n + b_n) = \sum_{n=1}^{\infty}a_n + \sum_{n=1}^{\infty}b_n$$ and then saying that $$\sum_{n=1}^{\infty}a_n$$ converges and $$\sum_{n=1}^{\infty}b_n$$ diverges to $$+\infty$$.
One of the difficulties of the teacher is that the student happens not to have made a fatal mistake... they have "got the right answer" and now you are telling them that something is still wrong. The issue of course this isn't a real equation; it's a nonsense equation like "$$+\infty = 1+ \infty"$$ and really they need to argue with the partial sums. i.e. The real mistake is usually just... "you've used this like its a general fact but it isn't a general fact"... and that usually seems like a weak answer to the student (in my experience). I think this is partly because until you are more experienced and have done analysis in an unknown setting, you are not used to deliberately considering all of the things that could go wrong, so they are focussed on the fact that in this example nothing went wrong so what's the big deal?
Clearly I can't be expected to come up with the exact explicit example that highlights where their thinking would have let them down, but sometimes it feels like I'm trapped into doing that (and coming up with an example live at the blackboard is tricky!).
Are there are any good tricks or tips to deal with this sort of issue when teaching analysis?
• What is the problem with the concrete example? – Tommi Brander Dec 6 '18 at 19:33
• This just isn't an equation. Both sides are "$+\infty$" and I think that when learning analysis, you should not be manipulating equations like "$\infty = 1 + \infty$". It like putting $\lim$ infront of everything you are manipulating and then in the end you are proved right... but to begin with its not actually a valid argument unless all the limits exist. – T_M Dec 6 '18 at 19:49
• I wonder if asking "why" is the right answer. In some sense, what we're doing in analysis is saying "For this answer to be correct, you must show me that it satisfies the definition of ___. You have not shown me that it does, therefore its not correct." Asking why might lead them there: Why is $\sum b_n = \infty$? What does that mean? – Nate Bade Dec 6 '18 at 20:48
• You could show how this additivity isn't true in general (use $a_n = (-1)^n$ and $b_n = (-1)^{n+1} = -(-1)^{n}),$ although a problem for you in this case is that (I think) it actually is true that if one series converges and the other doesn't, then the term-by-term series doesn't converge, at least, it's a fairly general rule in math that "nice" combined nicely with "bad" produces "bad". Since this is a real analysis class, they should be seeing and constructing proofs. Ask them to give you an $\epsilon$-$N$ proof of what they claim is obvious, and maybe they'll see it's not entirely obvious. – Dave L Renfro Dec 6 '18 at 22:57
• @DaveLRenfro Yeah somehow this is the things that's unconvincing for the student. You are saying " 1. Your argument is a case of this general claim. 2. This general claim is false, here is a counterexample". But the student isn't thinking of a general claim, they are just manipulating whatever they see and then ending up with the correct answer. As a teacher we freak out because its clear to us that they subconsciously used some generally false claim, but they don't see it that way. – T_M Dec 6 '18 at 23:43
The issue here is that the student is still trying to learn a fundamental property of mathematics as a field of study - that the truth of any claim can, in principle, be reduced back to a relatively simple set of axioms and relatively simple rules of inference. Furthermore, this is something one should be capable of doing to any claim one wants to make use of.
This is the rigorous stage of learning mathematics, according to the classification of Terence Tao: https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/
At this stage, the appropriate response to an uncertain claim is to provide a proof. If something is obviously true, then it should be proven from first principles or reduced to known facts proven from them. This is something you have to tell to the student, and also to demonstrate, until they understand the idea.
As for the specific example, the reasoning is correct on a moral level: A converging series is essentially just a constant (up to an epsilon of error when the index is big enough), which will not affect the convergence of the summed series, so the diverging part dominates.
The proof could go by reducing the problem to sequences and showing that the sum of a converging and a diverging sequence diverges (maybe only in the case the divergence is towards plus infinity). You want the sum to be eventually bigger than a fixed $$M \in \mathbb{R}$$, and the converging sequence is almost $$A$$ (up to some small epsilon that you can choose to be less than one) when the index is big enough, so consider indices so large that the diverging sequence is bigger than $$M-A+1$$.
Some arguments that might help the student see this:
• What if you have a sum of two converging sequences, or two diverging ones, or one diverging to infinity and the other oscillating within some bounded interval, or one converging to plus infinity and the other to minus infinity, etc. Can they answer what happens in each of those cases?
• There is a lot to remember unless they learn to prove things. It is a compression technique - large amounts of information are compressed to a much smaller number of proof techniques. This is especially true, since there will be more and more mathematics to master in further courses and studies.
• Someone needs to know and verify which computational techniques and shortcuts are viable. You are training to become that one.
• It increases one's confidence when doing mathematics when one can simply prove things whenever doubt arises. As a working mathematician I do this often for quite simple claims, even though I have a good idea what the correct answer is, because this helps to avoid simple mistakes.
One should remember that not everyone can learn this, or at least it is terribly difficult to learn for some people (non-mathematicians I have talked to); maybe they could learn the techniques, but the philosophy (of being able to prove anything, in principle) is genuinely difficult for some.
• I'm finding the confidence point particularly true at the moment. I "learnt" engineering maths a few decades ago but really, I want to be able to do maths for its own sake and know that it's valid. So I've been trying to identify the chain of proofs all the way back to definitions and axioms for anything I use. This is giving me (i) hugely increased confidence in using the steps I now know how to prive, and (ii) greater confidence in writing proofs, as a result of practising by proving basics. – timtfj Jan 14 at 16:35
Disclaimer: I'm only really at the beginning of teaching myself analysis.
BUT I have a copy of Gelbaum & Olmsted, Counterexamples in Analysis which contains such examples as "Two uniformly continuous functions whose product is not uniformly continuous" ($$x$$ and $$\sin x$$), "A convergent series with a divergent rearrangement" (any conditionally convergent series, with a demonstration of how to get any predetermined limit as well), "A power series convergent at only one point", etc. (There is a whole chapter of power series examples.)
I believe there are other books in the series, covering different areas of mathematics.
Maybe books like this could be a useful resource? I didn't see a counterexample specific to your example, but I wouldn't be surprised if some of them might come in handy. And they're nicely collected together by subject, which might help towards being pre-armed with examples of what can go wrong.
(I know this is not an answer, but for readibility reasons, I'm putting this comment as an answer)
So, the student is saying:
$$\sum_{n=1}^{\infty}a_n$$ converges
$$\sum_{n=1}^{\infty}b_n$$ diverges to $$+\infty$$
Hence, the conclusion of the student is:
$$\sum_{n=1}^{\infty}(a_n + b_n)$$ diverges to $$+\infty$$
And your reaction is: this is wrong.
In my opinion, this is right. Can you give me an example of why this is wrong?
• I believe the issue is that for the instructor, this is obviously wrong because the student hasn't proven/disproven the claim using strict definitions, like one usually does in analysis. Instead, the student has used an argument that is true most of the time without considering these tiny corner cases where his argument is not actually true. (Similar to saying, "I have a continuous function, so let me take the derivative of it" without realizing that continuous doesn't always imply differentiable, such as absolute value not being differentiable at $0$. – ruferd Dec 7 '18 at 13:17
• Perhaps that splitting up the sum is not justified unless proven correct for the given sequences. – Jasper Dec 7 '18 at 13:21
• I think you've rephrased what I said that the student said. I deliberately wrote that the student used the displayed equation in my question. So, asked to show that $\sum_n (a_n + b_n)$ diverges and you start by saying: "First write $\sum_{n=1}^{\infty}(a_n + b_n) = \sum_{n=1}^{\infty}a_n + \sum_{n=1}^{\infty}b_n$... and then...."etc. The first step is not OK because you've re-ordered an infinite series without knowing anything about its convergence (yet). – T_M Dec 7 '18 at 17:03
• I think the student is basically saying "$\sum a_n$ converges to $A$, $\sum b_n$ diverges to $\infty$, therefore $\sum (a_n+b_n)$ diverges to $A+\infty$". – timtfj Jan 14 at 16:43
It sounds like the students may be treating sums with $$\infty$$ in them as just as easy to manipulate as the finite algebraic versions - presumably not just in this particular example, right? Which is very unsurprising, since a lot of times we do treat them this way and because they "look" like things you can treat wholly algebraically. (To really confuse them, spend a week on generating functions!)
More seriously, here is a possible thing to try if this happens in class: Can they write the (in general wrong) equation
$$\sum_{n=1}^{\infty}(a_n + b_n) = \sum_{n=1}^{\infty}a_n + \sum_{n=1}^{\infty}b_n$$
in terms of the definition of the infinite sum/series? This is a good in-class "ask not a specific student, but everyone" question - restate this step in terms of its definition.
Once you do that (and if nobody can, then it's time to review it slowly), things look messier and presumably less "obvious" to the student(s). Now we are saying "limit = limit + limit". "Okay, can anyone think of a case back in calculus where the limit of a sum doesn't equal the sum of the limits?" Hopefully someone comes up with a situation like $$\frac{1}{x}+\frac{-1}{x}=0$$ at $$x=0$$.
Now you can say "in this case, we have to be careful too", and hopefully this analogy from something they should recall from freshman calculus can get you to convince them of it. And note that we are not asking them to remember things about series, which presumably they are still gnawing on, but function limits. (Hopefully someone will think of connecting the calculus example to the series with $$\sum \frac{1}{n}+\sum \frac{-1}{n}\neq \sum 0$$, but that will depend on the class.)
As a bonus, presumably you have a nice theorem about when the equation is true (such as, if both sums converge or whatever) which, when applied, would make their proof complete. Then on an exam you can give one where they are allowed to use that theorem, and one where they are not (do it "from the definitions").
I admit this is just one possible idea, but for some students it might snap them out of the blind algebraic manipulation, which I believe lies at the heart of many challenges in teaching real analysis. Analysis is really topology and the Archimedean axiom applied to the real line, but it still looks as algebraic as $$(x^2)'=2x$$ in many texts because of the notation coming (historically) before the proofs. | {"extraction_info": {"found_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": 23, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7459612488746643, "perplexity": 420.30726507623893}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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-20190419041415-20190419063415-00529.warc.gz"} |
https://en.wikipedia.org/wiki/Catalyst_poisoning | # Catalyst poisoning
Catalyst poisoning refers to the partial or total deactivation of a catalyst caused by exposure to a range of chemical compounds. Poisoning may be desirable when it results in improved selectivity (e.g. Lindlar's catalyst) but may be undesirable when the catalyst is rendered ineffective (e.g. Lead in catalytic converters). Poisoning refers specifically to chemical deactivation, rather than other mechanism of catalyst degradation such as thermal decomposition or physical damage.[1]
## Poisoning process
Poisoning involves compounds which bond chemically to the active surface sites of a catalyst. This may have two effects: the total number of catalytic sites or the fraction of the total surface area that has the capability of promoting reaction always decreases, and the average distance that a reactant molecule must diffuse through the pore structure before undergoing reaction may increase.[2] Poisoned sites can no longer accelerate the reaction with which the catalyst was supposed to catalyze. [3] Large scale production of substances such as ammonia in the Haber–Bosch process include steps to remove potential poisons from the product stream.
The poisoning reaction should be viewed like any other chemical reaction between a gas phase reactant and the solid surface, where the poisoned sites are distributed throughout the catalyst pore structure as a function of poison diffusion into the catalyst and the rate of the poisoning reaction. At the two extremes, this gives rise to two scenarios. First, when the poisoning reaction rate is slow relative to the rate of diffusion, the poison will be evenly distributed throughout the catalyst and will result in homogeneous poisoning of the catalyst. Conversely, if the reaction rate is fast compared to the rate of diffusion, a poisoned shell will form on the exterior layers of the catalyst, a situation known as "pore-mouth" poisoning, and the rate of catalytic reaction may become limited by the rate of diffusion through the inactive shell.[2]
## Common catalyst poisons [4]
Organic functional groups and inorganic ions with lone pairs often have the ability to strongly adsorb to metal surfaces, thus prohibiting access to catalyst sites. Common catalyst poisons include the following: carbon monoxide, inorganic ions such as halide, cyanide, sulfide, sulfite, and phosphite and organic molecules such as nitriles, nitros, oximes and nitrogen-containing heterocycles. Agents vary their catalytic properties because of the nature of the transition metal.
Ruthenium, palladium, nickel, cobalt, and platinum are common catalysts used in many organic reactions. Some of the common poisoning reagents for each of the compounds are as follows:
Ruthenium
• Very resistant to poisoning.
• Can be poisoned by amines, sulfides, thiols, lead and certain metal oxides.
Nickel and Cobalt
• Can be poisoned by amines, carbon monoxide, nitrogen monoxide, and halide ions.
Platinum
• Can be poisoned by amines, sulfides, thiols, aluminum, cobalt, and bismuth.
## Selective poisoning
If the catalyst and reaction conditions are indicative of a low effectiveness factor, selective poisoning may be observed, which is a phenomenon where poisoning of only a small fraction of the catalyst surface gives a disproportionately large drop in activity. Mathematical models exist which describe the cases where the interaction of the poisoning process with the influence of the intraparticle diffusion on the rates of the primary and poisoning reactions leads to an interesting relations between observed catalytic activity and the fraction of surface poisoned.[2]
By combining a material balance over a differential element of pore length and the Thiele modulus, the equation is found:
${\displaystyle \eta ={\frac {\tanh(h_{p})}{h_{p}}}}$
where η is the effectiveness factor of the poisoned surface and hp is the Thiele modulus for the poisoned case.
When the ratio of the reaction rate for the poisoned pore to the unpoisoned pore is considered, the following equation can be found:
${\displaystyle F={\frac {\tanh(h_{t}\cdot {\sqrt {1-\alpha }})\cdot {\sqrt {1-\alpha }}}{\tanh(h_{t})}}}$
where F is the ratio of rates of poisoned and unpoisoned pores, ht is the Thiele modulus for the unpoisoned case, and α is the fraction of the surface that is poisoned.
The above equation simplifies depending on the value of ht. When ht is small, meaning that the surface is available, the equation becomes:
${\displaystyle F=1-\alpha }$
This represents the "classical case" of nonselective poisoning where the fraction of the activity remaining is equal to the fraction of the unpoisoned surface remaining.
When ht is very large, it becomes:
${\displaystyle F={\sqrt {1-\alpha }}}$
In this case, the catalyst effectiveness factors are considerably less than unity, and the effects of the portion of the poison adsorbed near the closed end of the pore are not as apparent as when ht is small.
Delving further into the mathematical relationships of selective poisoning, or "Pore-Mouth" poisoning, looking at the steady-state conditions, the rate of diffusion of the reactant through the poisoned region is equal to the rate of reaction. The rate of diffusion is given by:
${\displaystyle DiffusionRate=-\pi \cdot r_{avg}^{2}\cdot D_{c}\cdot {\frac {dC}{dx}}}$
And the rate of reaction within a pore is given by:
${\displaystyle ReactionRate=\eta \cdot \pi \cdot r_{avg}\cdot (1-\alpha )\cdot L_{avg}\cdot k_{1}''\cdot C_{c}}$
Through further manipulation and substitution, the fraction of the catalyst surface available for reaction can be obtained from the ratio of the poisoned reaction rate to the unpoisoned reaction rate:
${\displaystyle F={\frac {r_{poisoned}}{r_{unpoisoned}}}}$
or
${\displaystyle F={\frac {\tanh[(1-\alpha )\cdot h_{t}]}{\tanh(h_{t})}}\cdot {\frac {1}{1+\alpha \cdot h_{t}\cdot \tanh[(1-\alpha )\cdot h_{t}]}}}$
where, as before, ht is the Thiele modulus for the unpoisoned case, and α is the fraction of the surface that is poisoned.[2]:465
## Benefits of selective poisoning
Usually, catalyst poisoning is undesirable as it leads to a loss of usefulness of expensive noble metals or their complexes. However, poisoning of catalysts can be used to improve selectivity of reactions. Poisoning can allow for selective intermediates to be isolated and final products with desirable stereochemistry to be achieved.
## Examples
Poisoning of palladium and platinum catalysts has been extensively researched. As a rule of thumb, platinum (as Adams' catalyst, platinum oxide finely divided on carbon) is less susceptible. Common poisons for these two metals are sulfur and nitrogen-heterocycles like pyridine and quinoline.
In some cases, a highly active catalyst can lead to undesirable secondary reactions with desired product. In some of these cases, the addition of a small amount of a catalyst poison increase the yield of the desired product by lower the catalyst activity. For example, in the classical "Rosenmund reduction" of an acyl chloride to the corresponding aldehyde, the palladium catalyst (over barium sulfate or calcium carbonate) is intentionally poisoned by the addition of sulfur or quinoline in order to lower the catalyst activity and thereby prevent further reduction of the aldehyde product to yield a primary alcohol. In the case of Lindlar's catalyst, palladium is poisoned with a lead salt to allow reduction of an alkyne to the corresponding alkene while preventing reduction of the alkene product to the corresponding alkane.
### Hydrodesulfurization catalysts
In the purification of crude petroleum products the process of hydrodesulfurization is utilized.[5] Thiol containing hydrocarbons, such as thiophene, are reduced using H2 in order to produce H2S and different length chains of hydrocarbons. Common catalyst used are tungsten and molybdenum sulfide particles. By adding cobalt and nickel [6] nuclei to either edge s or partially incorporating them into the crystal lattice structure can create more efficient catalyst. The synthesis of the catalyst creates a supported hybrid that prevents poisoning of the cobalt nuclei that may be unstable in the mono-nuclear form.
### Catalytic converter
A catalytic converter for an automobile can be poisoned if the vehicle is operated on gasoline containing lead additives. Fuel cells running on hydrogen must use very pure reactants, free of sulfur and carbon compounds.
Example of Catalytic Converter used in the Automotive Industry
### Nickel
Raney nickel catalysts have reduced activity when it is in combination with mild steel. The loss in activity of catalyst can be overcome by having a lining of epoxy or other substances. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 8, "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": 0.7981799244880676, "perplexity": 2817.7885290448503}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084891546.92/warc/CC-MAIN-20180122232843-20180123012843-00284.warc.gz"} |
https://blender.stackexchange.com/questions/38861/how-to-close-a-pipe-with-a-perfect-hemisphere | # How to close a pipe with a perfect hemisphere?
I have a sort of pipe shape which I want to close with a perfectly proportional hemisphere. Problem is the pipe does not have a perfectly round cross-section. It is a little squashed, but still I need to close it with a nice proportional ideal hemisphere. Is it possible?
I tried to pull the middle vertex out with proportional editing, using smooth or inverse square falloff but the result is bad. I also tried to fit a hemisphere, and attach it to the end of the pipe, but the pipe is not a perfect circle.
Using the bevel tool is a simple way to do this. First remove all of the edge loops on the endcap so you just have a single n-gon like this:
Note that I scaled the cylinder to give it an elliptical profile and skewed the face so that it is not perpendicular to the axis of the cylinder. I did this because this is how it appears to me in your picture. If you make your endcap perpendicular to the profile axis you will get a cleaner (more spherical) result.
Be that as it may, I used the Bevel tool which is located in the menu: Mesh > Edges > Bevel with the settings shown on the bottom left of the following image:
Using 8 segments achieves a well rounded result, although as I previously mentioned, the dome shape is distorted because of the angle of the face in respect to axis of the profile. I found that the "Depth" setting worked the best in this situation because I was able to extend the bevel very close to the convergence point without getting overlap. You may want to experiment with the settings to get what you want.
Addendum: Just to clarify what I mean about getting a cleaner, more spherical result by making the end cap perpendicular to the profile axis, here is a comparison:
• Exactly the effect I'm looking for. Very flexible solution too. – Aardo Sep 20 '15 at 14:46
• LOL, my answer seems like using a chainsaw to cut a sheet of paper in comparison to this elegant fix, nice! – TLousky Sep 20 '15 at 14:55
• Glad this worked for you. That's hilarious TLousky :) I appreciate the compliment. I'm a machinist, and I too am sometimes guilty of attacking a problem with brute force when a simpler solution has not yet presented itself. – YoeyYutch Sep 21 '15 at 1:48
If you don't need a really perfect hemisphere, you can simply use the root or sphere proportional editing falloff types.
But if you really do, then follow these steps:
1. Delete all the faces at the tip of your pipe, and fill the tip with one ngon face.
2. Select the outer edge ring in the tip of your pipe, then use the looptools --> circle tool to make this loop a perfect circle ( W --> looptools --> Circle). If you don't have the looptools available, activate the addons from the user preferences).
3. Open a text editor window and paste the following script there. It will calculate the angle of the face from the face normal.
'''
Run This script after you selected the face that should be
replaced with a hemisphere (in Edit Mode of course).
we'll use its normal to align the hemisphere.
'''
import bpy, bmesh
from mathutils import Vector
from math import degrees
bm = bmesh.from_edit_mesh( bpy.context.object.data )
normal = bm.faces.active.normal.copy()
upVec = Vector( [0,0,1] )
rot = upVec.rotation_difference( normal ).to_euler()
print( "Type this into the rotation values of your hemisphere object to align with the face")
print( [ degrees(a) for a in rot ] )
4. Open a system console (Window --> Toggle System Console) to see the script's output.
5. Select the ngon face at the tip of the pipe and run the script (press the "Run Script" button in the text editor window).
6. Check the number of vertices your pipe has and use it to create a UV sphere object with the same number of segments.
7. Cut the sphere in half by deleting the lower verts / faces.
8. Copy the angles specified in your system console to the XYZ rotation values for your hemisphere (bottom two lines in the image below).
9. Select the pipe again and snap the 3D cursor to the center of your ngon (Shift + S --> Cursor to Selected).
10. Select the hemisphere, then snap it to the 3D cursor (Shift + S --> Selection to Cursor).
11. Join the pipe and hemisphere (Ctrl + J), delete the ngon you created earlier, then select the two edge loops at the tip of the pipe and start of the hemisphere, then use the looptools bridge tool to join them (W --> Looptools --> Bridge).
• This is a very handy method. Good to keep in one's toolbox. However for some reason the rotations are inaccurate. It didn't match. Also for some reason LoopTools won't make a perfect circle out of the ending. It shifts only very slightly, though I tried various settings. – Aardo Sep 20 '15 at 14:45
I would do like this:
1. Select the end face(s)
2. ShiftS > Cursor to selected
3. ShiftNumPad 7 (Align view to faces)
4. Add a sphere:
• Set the number of segments, the radius etc to match the tube
• Check "Align to View"
5. You might need to rotate the sphere to match the tube
6. Remove the unneeded part of the sphere and the end face(s) of the tube¤
7. Select the half sphere and the matching vertices of the tube
8. Press W > Remove doubles
(¤ Do this while the view still is aligned to the end faces. Tip: Enable vertex snapping (hold Ctrl) while rotating.) | {"extraction_info": {"found_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": 0.27682340145111084, "perplexity": 1416.7452462313508}, "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-2019-39/segments/1568514573735.34/warc/CC-MAIN-20190919204548-20190919230548-00316.warc.gz"} |
http://www.ck12.org/algebra/Comparing-Equations-of-Parallel-and-Perpendicular-Lines/lecture/Determine-Parallel-and-Perpendicular-Lines%3A-A-Sample-Application/r1/ | <img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
# Comparing Equations of Parallel and Perpendicular Lines
## Use slope to identify parallel and perpendicular lines
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Determine Parallel and Perpendicular Lines: A Sample Application
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Sign in to explore more, including practice questions and solutions for Comparing Equations of Parallel and Perpendicular Lines. | {"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": 0.8902269005775452, "perplexity": 13189.085649961009}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982954771.66/warc/CC-MAIN-20160823200914-00150-ip-10-153-172-175.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/orthogonality-of-functions.47923/ | # Orthogonality of Functions
1. Oct 15, 2004
### theFuture
We were doing examples in class today and showed that sin and cos were orthogonal functions. In general, is true that even and odd functions are orthogonal? I was unsure where a proof of this might begin, mostly how to generalize the notion of an even or odd function.
2. Oct 15, 2004
### matt grime
This depends on what your "inner product" is.
Let's assume it is
$$<f,g> = \int_{-a}^a f(x)g(x)dx$$
an odd function is one that satisfies f(x) = -f(-x) an even one satisfies f(x)=f(-x)
1. show that the product of an even and an odd function is odd
2. show that the integral of an odd function over any interval [-a,a] is zero.
3. Oct 15, 2004
### theFuture
Thanks. Now that I see it like that I can't believe I couldn't come up with that. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9329403042793274, "perplexity": 497.4610468375372}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988719273.37/warc/CC-MAIN-20161020183839-00490-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://cs.stackexchange.com/questions/52379/is-the-word-problem-for-regular-languages-in-alogtime | # Is the word problem for regular languages in ALogTime?
Given a regular language (by a sparse or dense matrix describing the graph of the NFA) (initially the description was supposed to be a regular expression) and a word, can the problem whether the word belongs to the regular language be decided on a (random access) alternating Turing machine in a logarithmic amount of computation time? (The many links above try to compensate for the fact that I didn't find a nice self-contained description of ALogTime.) The length of the input is the length of the description of the regular language plus the length of the word.
If the regular expression is not fixed, it is rather doubtful that your problem belongs to ALOGTIME. However, the phrase word problem usually refers to a fixed setting, which in this case means that the regular language is fixed.
If the regular language is fixed, this is apparently shown in Barrington's famous paper. Alternatively, you can describe a logtime game for deciding the word problem for regular languages. The game starts with the YES player stating the final (accepting) state and the state at the middle (the initial state is of course fixed). The NO player then chooses one of the halves, and the game continues until it comes down to a word of length 1, in which case the problem can be settled by examining one character of the input.
• If I would include the (sparse or dense matrix describing the graph of the) NFA instead of the regular expression, then it seems that checking the last step (in NLogTime) would be possible. Do you agree? (I only wrote regular expression instead of regular language, because it seemed more convenient and explicit. I didn't intent this to have any impact on the actual answer to the question.) – Thomas Klimpel Jan 28 '16 at 9:40
• Yes, this sounds reasonable. – Yuval Filmus Jan 28 '16 at 9:41
• If the NFA (or regular expression) is not fixed, then the size of a state is $\log n$. Even so the algorithm above will still work perfectly well, it will take more than $\log n$ time. Which is a good thing, since the NFA membership problem is NL-complete. (I added a community wiki answer below, which explains why this observation is nearly trivial.) – Thomas Klimpel Sep 4 '17 at 6:33
If (the regular expression or) NFA is not fixed, then the problem is NL-complete. So if the problem would be in ALogTime, then ALogTime=L=NL, which is an open problem.
It is clear that the problem is in NL, we only need one pointer into the word on which the NFA runs, the current (and guessed next) state of the NFA (its size is $\log |A|$, if $A$ is a description of the NFA by a list of possible transitions), and a pointer into the description of the NFA.
To see that it is NL complete, reduce reachability to the NFA membership problem over the unary language. Use the given digraph as NFA, and add a self-loop to the target node whose reachability is being checked. Let the source node be only state in the set of initial states, the target node be the only state in the set of (accepting) final states, and check whether this NFA accepts the word $1^n$, where $n$ is the number of nodes of the given digraph (which is also the number of states of the constructed NFA).
The reason why Yuval Filmus' algorithm (explained in his answer above, together with the comments below) doesn't solve this problem in ALogTime is not that the last step would not work. That checking can indeed be done in NLogTime. But the size of a state is $\log n$ if the NFA is not fixed, and hence the algorithm takes $\log^2 n$ time instead of $\log n$ time. | {"extraction_info": {"found_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": 0.7925804257392883, "perplexity": 383.43796303544946}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875146414.42/warc/CC-MAIN-20200226150200-20200226180200-00023.warc.gz"} |
https://fomcon.net/forums/reply/1252/ | # Reply To: Reference to fpid_optim
Home Forums FOMCON toolbox: Support forum Reference to fpid_optim Reply To: Reference to fpid_optim
#1252
mega460
Participant
Hi pritesh,
Thank you for helping me.
I already studied tha algorithm NELDER-MEAD and saw the optimization fucntion choince (ISE, ITAE, IASE, OR IAE). The problem is, for order to use this metod in my work, i need a reference on this exactly tool, and i couldnt find it in this website. | {"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": 0.8415036797523499, "perplexity": 8723.132751480633}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1652662543797.61/warc/CC-MAIN-20220522032543-20220522062543-00609.warc.gz"} |
http://math.stackexchange.com/questions/165671/convergence-of-the-sequence-sqrt12-sqrt1-sqrt12-sqrt13-sqrt1-sq | # Convergence of the sequence $\sqrt{1+2\sqrt{1}},\sqrt{1+2\sqrt{1+3\sqrt{1}}},\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1}}}},\cdots$
I recently came across this problem
Q1 Show that $\lim\limits_{n \rightarrow \infty} \underbrace{{\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots+n\sqrt{1}}}}}}}_{n \textrm{ times }} = 3$
After trying it I looked at the solution from that book which was very ingenious but it was incomplete because it assumed that the limit already exists.
So my question is
Q2 Prove that the sequence$$\sqrt{1+2\sqrt{1}},\sqrt{1+2\sqrt{1+3\sqrt{1}}},\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1}}}},\cdots,\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots+n\sqrt{1}}}}}$$ converges.
Though I only need solution for Q2, if you happen to know any complete solution for Q1 it would be a great help .
If the solution from that book is required I can post it but it is not complete as I mentioned.
Edit: I see that a similar question was asked before on this site but it was not proved that limit should exist.
-
possible duplicate of Limit of Nested Radical – Gerry Myerson Jul 2 '12 at 12:15
Did you look at all the links in all the answers to that earlier question? – Gerry Myerson Jul 2 '12 at 12:27
@GerryMyerson I think it's hard to search for the earlier posts. – Frank Science Jul 2 '12 at 13:18
@Frank While it is true that the SE search function leaves much to be desired (instead try googling with site:MSE), it does work well for reasonable unique terms like "nested radical". Indeed, it yields the cited duplicate question as first match. But, of course, one does need to know the English buzzwords for these objects. – Bill Dubuque Jul 2 '12 at 14:46
Let $f_n(0)=\sqrt{1+n}$ and $f_n(k)=\sqrt{1+(n-k)f_n(k-1)}$. Then $0<f_n(0)<n+1$ when $n>0$. Assume that $f_n(k)<n+1-k$ and we can show by induction that $$f_n(k+1) < \sqrt{1+(n-k-1)(n-k+1)} = \sqrt{1+(n-k)^2-1} = n+1-(k+1)$$ for all k. Your expression is $f_n(n-2)$ which is increasing in $n$ and bounded above by $3$, so converges.
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Note that $$x+1=\sqrt{1+x(x+2)}\tag{1}$$ Iterating $(1)$, we get $$x+1=\sqrt{1+x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{1+(x+3)\color{#C00000}{(x+5)}}}}}\tag{2}$$ Note that $$s_3=\sqrt{1+x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{1+(x+3)\color{#C00000}{\sqrt{1}}}}}}\tag{3}$$ Instead of $\color{#C00000}{\sqrt{1}}$ as in the last term of $(3)$, $(2)$ has $\color{#C00000}{(x+n+2)}$. Thus, the increasing sequence in $(3)$ is bounded above by $x+1$. Thus, the sequence in $(3)$ has a limit.
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I try to compliment this every time - but I love the custom-dulled colors. – mixedmath Dec 15 '12 at 19:40
$x + 1 = \sqrt {1 + x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{}.....}}}$. Put $x=2$ gives you the solution. For proof see http://zariski.files.wordpress.com/2010/05/sr_nroots.pdf
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It is the same proof that I know but discussion of convergence is not complete still +1 for dicussion of numerical convergence. – sabertooth Jul 2 '12 at 12:52
Note that $\frac{x+y}{x+z} < \frac{y}{z}$ for positive $x, y, z$; thus $\frac{\sqrt{x+y}}{\sqrt{x+z}} < \frac{\sqrt{y}}{\sqrt{z}}$ for $y > z$.
Thus we get $\frac{a_{n+1}}{a_n} = \frac{\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots+n\sqrt{1+(n+1)\sqrt{1}}}}}}}{\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots+n\sqrt{1}}}}}} < \frac{\sqrt{\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots+n\sqrt{1+(n+1)\sqrt{1}}}}}}}{\sqrt{\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots+n\sqrt{1}}}}}} < \frac{\sqrt{\sqrt{\cdots\sqrt{n+2}}}}{\sqrt{\sqrt{\cdots\sqrt{1}}}} = O(n^{(2^{-n})})$.
Next, $\ln{a_{n+1}}-\ln{a_n} = O(\frac{\ln_n}{2^n})$. Summing the equations we get $\ln{a_n} - \ln{a_1} = O(\frac{\ln_1}{2^1} + \cdots + \frac{\ln_n}{2^n})$; letting $n$ to approach $\infty$, we get $\ln{a} - \ln{a_1} = O(1)$, thus there is a finite limit of $a_i$.
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.950838565826416, "perplexity": 360.693942906651}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398459333.27/warc/CC-MAIN-20151124205419-00172-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://yutsumura.com/exponential-functions-form-a-basis-of-a-vector-space/ | # Exponential Functions Form a Basis of a Vector Space
## Problem 590
Let $C[-1, 1]$ be the vector space over $\R$ of all continuous functions defined on the interval $[-1, 1]$. Let
$V:=\{f(x)\in C[-1,1] \mid f(x)=a e^x+b e^{2x}+c e^{3x}, a, b, c\in \R\}$ be a subset in $C[-1, 1]$.
(a) Prove that $V$ is a subspace of $C[-1, 1]$.
(b) Prove that the set $B=\{e^x, e^{2x}, e^{3x}\}$ is a basis of $V$.
(c) Prove that
$B’=\{e^x-2e^{3x}, e^x+e^{2x}+2e^{3x}, 3e^{2x}+e^{3x}\}$ is a basis for $V$.
## Proof.
### (a) Prove that $V$ is a subspace of $C[-1, 1]$.
Note that each function in the subset $V$ is a linear combination of the functions $e^x, e^{2x}, e^{3x}$.
Namely, we have
$V=\Span\{e^x, e^{2x}, e^{3x}\}$ and we know that the span is always a subspace. Hence $V$ is a subspace of $C[-1,1]$.
### (b) Prove that the set $B=\{e^x, e^{2x}, e^{3x}\}$ is a basis of $V$.
We noted in part (a) that $V=\Span(B)$. So it suffices to show that $B$ is linearly independent.
Consider the linear combination
$c_1e^x+c_2 e^{2x}+c_3 e^{3x}=\theta(x),$ where $\theta(x)$ is the zero function (the zero vector in $V$).
Taking the derivative, we get
$c_1e^x+2c_2 e^{2x}+3c_3 e^{3x}=\theta(x).$ Taking the derivative again, we obtain
$c_1e^x+4c_2 e^{2x}+9c_3 e^{3x}=\theta(x).$
Evaluating at $x=0$, we obtain the system of linear equations
\begin{align*}
c_1+c_2+c_3&=0\\
c_1+2c_2+3c_3&=0\\
c_1+4c_2+9c_3&=0.
\end{align*}
We reduce the augmented matrix for this system as follows:
\begin{align*}
\left[\begin{array}{rrr|r}
1 & 1 & 1 & 0 \\
1 &2 & 3 & 0 \\
1 & 4 & 9 & 0
\end{array} \right] \xrightarrow[R_3-R_1]{R_2-R_1}
\left[\begin{array}{rrr|r}
1 & 1 & 1 & 0 \\
0 &1 & 2 & 0 \\
0 & 3 & 8 & 0
\end{array} \right] \xrightarrow[R_3-3R_2]{R_1-R_2}\6pt] \left[\begin{array}{rrr|r} 1 & 0 & -1 & 0 \\ 0 &1 & 2 & 0 \\ 0 & 0 & 2 & 0 \end{array} \right] \xrightarrow{\frac{1}{2}R_3} \left[\begin{array}{rrr|r} 1 & 0 & -1 & 0 \\ 0 &1 & 2 & 0 \\ 0 & 0 & 1 & 0 \end{array} \right] \xrightarrow[R_2-2R_2]{R_1+R_3} \left[\begin{array}{rrr|r} 1 & 0 & 0 & 0 \\ 0 &1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array} \right]. \end{align*} It follows that the solution of the system is c_1=c_2=c_3=0. Hence the set B is linearly independent, and thus B is a basis for V. #### Anotehr approach. Alternatively, we can show that the coefficient matrix is nonsingular by using the Vandermonde determinant formula as follows. Observe that the coefficient matrix of the system is a Vandermonde matrix: \[A:=\begin{bmatrix} 1 & 1 & 1 \\ 1 &2 &3 \\ 1^2 & 2^2 & 3^2 \end{bmatrix}. The Vandermonde determinant formula yields that
$\det(A)=(3-1)(3-2)(2-1)=2\neq 0.$ Hence the coefficient matrix $A$ is nonsingular.
Thus we obtain the solution $c_1=c_2=c_3=0$.
### (c) Prove that $B’=\{e^x-2e^{3x}, e^x+e^{2x}+2e^{3x}, 3e^{2x}+e^{3x}\}$ is a basis for $V$.
We consider the coordinate vectors of vectors in $B’$ with respect to the basis $B$.
The coordinate vectors with respect to basis $B$ are
$[e^x-2e^{3x}]_B=\begin{bmatrix} 1 \\ 0 \\ -2 \end{bmatrix}, [e^x+e^{2x}+2e^{3x}]_B=\begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix}, [3e^{2x}+e^{3x}]_B=\begin{bmatrix} 0 \\ 3 \\ 1 \end{bmatrix}.$ Let $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ be these vectors and let $T=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$.
Then we know that $B’$ is a basis for $V$ if and only if $T$ is a basis for $\R^3$.
We claim that $T$ is linearly independent.
Consider the matrix whose column vectors are $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$:
\begin{align*}
\begin{bmatrix}
1 & 1 & 0 \\
0 &1 &3 \\
-2 & 2 & 1
\end{bmatrix}
\xrightarrow{R_3+2R_1}
\begin{bmatrix}
1 & 1 & 0 \\
0 &1 &3 \\
0 & 4 & 1
\end{bmatrix}
\xrightarrow[R_3-4R_1]{R_1-R_2}\6pt] \begin{bmatrix} 1 & 0 & -3 \\ 0 &1 &3 \\ 0 & 0 & -11 \end{bmatrix} \xrightarrow{-\frac{1}{11}R_3} \begin{bmatrix} 1 & 0 & -3 \\ 0 &1 &3 \\ 0 & 0 & 1 \end{bmatrix} \xrightarrow[R_2-3R_3]{R_1+3R_3} \begin{bmatrix} 1 & 0 & 0 \\ 0 &1 &0 \\ 0 & 0 & 1 \end{bmatrix}. \end{align*} Thus, the matrix is nonsingular and hence the column vectors \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 are linearly independent. As T consists of three linearly independent vectors in the three-dimensional vector space \R^3, we conclude that T is a basis for \R^3. Therefore, by the correspondence of the coordinates, we see that B’ is a basis for V. ## Related Question. If you know the Wronskian, then you may use the Wronskian to prove that the exponential functions e^x, e^{2x}, e^{3x} are linearly independent. See the post Using the Wronskian for Exponential Functions, Determine Whether the Set is Linearly Independent for the details. Try the next more general question. Problem. Let c_1, c_2,\dots, c_n be mutually distinct real numbers. Show that exponential functions \[e^{c_1x}, e^{c_2x}, \dots, e^{c_nx} are linearly independent over $\R$.
The solution is given in the post ↴
Exponential Functions are Linearly Independent
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##### Use Coordinate Vectors to Show a Set is a Basis for the Vector Space of Polynomials of Degree 2 or Less
Let $P_2$ be the vector space over $\R$ of all polynomials of degree $2$ or less. Let $S=\{p_1(x), p_2(x), p_3(x)\}$,...
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https://www.groundai.com/project/akari-infrared-observations-of-the-supernova-remnant-g292018-unveiling-circumstellar-medium-and-supernova-ejecta/ | AKARI Infrared Observations of G292.0+1.8
AKARI Infrared Observations of the Supernova Remnant G292.0+1.8: Unveiling Circumstellar Medium and Supernova Ejecta
Ho-Gyu Lee1 2 , Bon-Chul Koo3 , Dae-Sik Moon4 , Itsuki Sakon1 , Takashi Onaka1 , Woong-Seob Jeong5 , Hidehiro Kaneda6 , Takaya Nozawa7 , and Takashi Kozasa8
1affiliation: Department of Astronomy, Graduate School of Science, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan; [email protected] [email protected], [email protected]
2affiliationmark:
2affiliation: Department of Physics and Astronomy, Seoul National University, Seoul, 151-742, Korea; [email protected]
3affiliation: Department of Astronomy and Astrophysics, University of Toronto, Toronto, ON M5S 3H4, Canada; [email protected]
4affiliation: Korea Astronomy and Space Science Institute, 61-1, Whaam-dong, Yuseong-gu, Deajeon 305-348, Korea; [email protected]
5affiliation: Graduate School of Science, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan; [email protected]
6affiliation: Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Chiba 277-8568, Japan; [email protected]
7affiliation: Department of Cosmosciences, Graduate School of Science, Hokkaido University, Sapporo 060-0810, Japan; [email protected]
Abstract
We present the results of observations of the O-rich supernova remnant G292.01.8 using six IRC and four FIS bands covering 2.7–26.5 m and 50–180 m, respectively. The images show two prominent structures; a bright equatorial ring structure along the east-west direction and an outer elliptical shell structure surrounding the remnant. The equatorial ring structure is clumpy and incomplete with its western end opened. The outer shell is almost complete and slightly squeezed along the north-south direction. The central position of the outer shell is 1 northwest from the embedded pulsar and coincides with the center of the equatorial ring structure. In the northen and southwestern regions, there is also faint emission with a sharp boundary beyond the bright shell structure. The equatorial ring and the elliptical shell structures were partly visible in optical and/or X-rays, but they are much more clearly revealed in our images. There is no evident difference in infrared colors of the two prominent structures, which is consistent with the previous proposition that both structures are of circumstellar origin. However, we have detected faint infrared emission of a considerably high 15 to 24 m ratio associated with the supernova ejecta in the southeastern and northwestern areas. Our IRC spectra show that the high ratio is at least partly due to the emission lines from Ne ions in the supernova ejecta material. In addition we detect a narrow, elongated feature outside the SNR shell. We derive the physical parameters of the infrared-emitting dust grains in the shocked circumstellar medium and compare the result with model calculations of dust destruction by a SN shock. The results suggest that the progenitor was at the center of the infrared circumstellar shell in red supergiant stage and that the observed asymmetry in the SN ejecta could be a result of either a dense circumstellar medium in the equatorial plane and/or an asymmetric explosion.
ISM: individual(G292.0+1.8 (catalog )) — infrared: ISM — ISM: dust — shock waves — supernova remnants
slugcomment: Submitted: August 18, 2019
1 Introduction
Young core-collapse supernova remnants (SNRs) in the Galaxy provide an unique opportunity to study fine details of ejecta from supernova (SN) explosions as well as those of the circumstellar medium (CSM) produced over the final evolutionary stages of massive stars. An obvious group of such young core-collapse SNRs is O-rich SNRs that show optical spectra featuring strong O and Ne lines, with lines from lighter elements (e.g., He) either weak or absent. Since the ejecta from a progenitor and the swept-up CSM in the O-rich SNRs are not completely mixed together yet, observation of the O-rich SNRs is promising to study the ejecta and CSM related to the explosion of a progenitor and its late-stage evolution.
G292.01.8 (MSH11–54), together with Cassiopeia A and Puppis A, forms a rare group of O-rich SNRs in the Galaxy. The O-rich nature of G292.01.8 was discovered by the optical detection of fast-moving O-rich and Ne-rich ejecta (Goss at al., 1979; Murdin & Clark, 1979). Recent wide-field observations of the optical [O iii] line in G292.01.8 have revealed that the ejecta velocities range from 1400 km s to 1700 km s and also that distant ejecta knots are located primarily along the north-south direction (Ghavamian et al., 2005; Winkler & Long, 2006). The dynamical center of G292.01.8, based on the kinematics of the [O iii] line emission from the ejecta, is roughly coincident with the geometrical center of the radio emission, but shows an apparent offset from its pulsar position discovered at the southeast (Winkler et al., 2009, see below). Ghavamian et al. (2009) reported that the mid-infrared (MIR) spectrum of the ejecta is dominated by ionic lines and a broad bump around 17 m using the Space Telescope (). They proposed that the latter is produced either by Polycyclic Aromatic Hydrocarbons (PAHs) along the line of sight or newly-formed dust within the ejecta. Besides there is also an equatorial MIR continuum component from the dust associated with the shocked CSM partly overlapping with the ejecta emission. A previous infrared (IR) study based on the data obtained with Infrared Astronomical Satellite (), on the other hand, identified enhanced far-IR (FIR) emission in the southwestern part of the SNR, suggesting that the source has encountered adjacent clouds (Braun et al., 1986; Park et al., 2007).
In radio, G292.01.8 mainly consists of bright emission from a central pulsar wind nebula (PWN) of 4′ (in diameter) and relatively fainter outer plateau emission of 9′ (Lockhart et al., 1977; Braun et al., 1986; Gaensler & Wallace, 2003). The plateau emission declines sharply outward, and there is no apparent surrounding radio shell structure. The pulsar PSR J1124–5916, which is located 46″ from the dynamical center of the SNR in the southeast direction, was discovered by radio timing (Camilo et al., 2002), confirming the PWN nature of the central emission of the SNR. The pulsar and PWN were also identified in X-ray emission (Hughes et al., 2003), where the pulsar has a nearby jet and a torus of 5″ scale (Park et al., 2007). The jet axis is slightly tilted to the northeast direction (Park et al., 2007). There are two clear differences between the soft and hard X-ray emission of G292.01.8. First, while the former is produced by shocked gas of normal composition distributed as equatorial filaments, indicating it is likely from CSM, the latter is dominated by bright metallic lines from the ejecta (Park et al., 2002, 2004). Next, the X-ray emission is harder in the northwestern part than the southeastern part, implying density variation or asymmetric SN explosion (Park et al., 2007). The distance to G292.01.8, on the other hand, was estimated to be 6.2 kpc from H I absorption observations (Gaensler & Wallace, 2003).
In this paper we present an extensive IR study of G292.01.8 using multi-band imaging and spectroscopic data obtained with the space telescope (Murakami et al, 2007). We describe the details of our observations and present the results of the observations in §2 and 3, respectively. In §4 we discuss IR emission in G292.01.8 from circumstellar dust with a comparison to model calculations of dust destruction by a SN shock. We also discuss the IR emission from the ejecta and we suggest that there is a negligible amount of dust associated with the ejecta. Then, we make a comparison to the results of obserations. We finally discuss the SN explosion in G292.01.8 based on our observations followed by our conclusions in §5.
2 Observations
The multi-band imaging observations of G292.01.8 were carried out using its six near-IR (NIR) and MIR bands in the 2.7–26.5 m range as well as four FIR bands in the 50–180 m range on 2007 January 17 and 19. The NIR and MIR images were obtained with Infrared Camera (IRC) that has three simultaneously operating NIR, MIR-S, and MIR-L channels. The three channels have comparable field of views of . The NIR and MIR-S channels share the same pointing direction, while that of the MIR-L channel is 25′ away. The IRC observations were conducted in the two-filter mode that produced images of two bands for each channel (Onaka et al., 2007). The total on-source integration times were 178 s for the NIR (N3, N4) observations and 196 s for both the MIR-S (S7, S11) and MIR-L (L15, L24) observations. The basic calibration and data reduction, including dark subtraction, linearity correction, distortion correction, flat fielding, image stacking, and absolute position determination were performed by the standard IRC Imaging Data Reduction Pipeline (version 20071017). Tables 1 and 2 present a journal of our observations, including the spectroscopic observations, and the basic parameters of the imaging bands, respectively.
The FIR images were obtained with Far-Infrared Surveyor (FIS) in two round-trip scans using the cross-scan shift mode (Kawada et al., 2007). The scan speed and length were 15″ s and 240″, respectively, creating images of 40′ 12′ size elongated along the scanning direction. All the four FIS band (N60, Wide-S, Wide-L, and N160) images were obtained simultaneously by a single observing run (Table 1). The initial data calibration and reduction such as glitch detection, dark subtraction, flat fielding and flux calibration were processed with FIS Slow-Scan Toolkit (version 20070914), and the final image construction was performed with a refined sampling mechanism.
These IRC and FIS multi-band imaging observations were followed by NIR and MIR spectroscopic observations and L18W-band (13.9–25.6 m, combining both L15 and L24 bands) imaging observations carried out in 2007 July 20–22. The spectroscopic observations were conducted using four (NG, SG1, SG2, and LG2) grisms. Similar to the aforementioned imaging observations, the NG, SG1, and SG2 mode observations were conducted simultaneously, while LG2 mode observations were done separately (Ohyama et al., 2007). Table 3 lists the characteristics of the spectroscopic observations. Spectra from the peaks of the equatorial emission and ejecta identified in the ratio between the L15- and L24-band images (L15/L24 hereafter) were obtained together with that from a reference background position at the southeastern part of the source (Table 4). The data calibration and reduction were processed with IRC Spectroscopy Toolkit (version 20070913). The obtained flux was converted to the surface brightness based on the measured slit size. IRC spectroscopy is made at the slits located at the edge of the large imaging field-of-view (FoV). There is internal scattered light in the array of the MIR-S and the light diffusing from the imaging FoV affects SG1 and SG2 spectra to some extent. This effect has been corrected for according to the method given in Sakon et al. (2008). A similar scattered light is also recognized in LG2 spectra and has been corrected for in a similar way. In addition there is a contribution from the second order light from the edges of the imaging FoV in LG2 spectra, but we expect that it is not significant in the background-subtracted spectra.
These spectroscopic observations were simultaneously accompanied by supplemental short (49 s 3) L18W-band imaging observations of a field 5′ away from the central slit position in the southeast direction in order to check the pointing of the satellite. This supplemental, short-exposure L18W-band image of the southeast, which fortuitously revealed interesting structures beyond the SNR boundary (see § 3.4), was combined with deep (442 s) L18W-band image of the source to generate a final larger-area map. The L18W-band imaging data were processed with the same procedures that we used for other imaging data sets as described above.
3 Results
3.1 Multi-band Infrared Images of G292.0+1.8
Figure 1 presents our IRC multi-band images of G292.01.8 together with the ATCA 20 cm radio continuum image (Gaensler & Wallace, 2003), X-ray (0.3–8 keV) image (Park et al., 2002), and the background-subtracted S11-band image (S11–S7; see below) for comparison. The IR emission associated with the SNR is most apparent in the L15- and L24-band images as two prominent features: first, there is a ring-like structure composed of two clumpy, narrow, and long filaments crossing the central part of the SNR along the east-west direction – we name this “Equatorial Ring” (ER); secondly, the outer boundary of the SNR appears as an almost-complete shell structure with its eastern part opened – we name this “Outer Elliptical Shell” (OES).
The radius of the ER is 3 (or 5 pc). Its southern filament is brighter than the northern one by a factor of on average and has the brightest clump at (RA, decl.) = (, ) close to the filament center. The southern filament also shows prominent X-ray and [O iii] emission and has been called “equatorial bar” or “equatorial belt” (Park et al., 2002; Gonzalez & Safi-Harb, 2003; Ghavamian et al., 2005). The northern filament is 1′ away from the southern filament and both filaments are elongated roughly parallel to each other.
The OES shows an almost-complete ellipse of 7′ 6′ ( 12 pc 10 pc) with its major axis aligning roughly with the plane of the ER. The center of the ellipse determined by elliptical fits weighted by the 24 m surface brightness of the OES is (, ) located at the middle of two filaments of the ER. In the west the OES appears to be connected with the two filaments of the ER and its emission is enhanced there, especially at the southwest near the southern filament of the ER. In contrast the eastern part of the OES is open, and also the emission from the southern filament of the ER is truncated in the east before it reaches the boundary of the SNR.
In addition to the ER and OES, there appear to be at least two more clearly identifiable features in the images. First, the L15-band image shows a separate structure that extends 5 southward from the ER on the southeastern side of the SNR. It is elongated vertically with the bright portion located 0.5 below the southern equatorial filament. Next, there is faint emission extending beyond the northern and southeastern edges of the OES. This faint emission is contained within the outermost boundary of radio emission observed in G292.01.8 (Gaensler & Wallace, 2003).
As in Figure 1, G292.01.8 is not detected in the S7- and S11-band images: while some of the S11-band emission appears to arise from locations of strong L15- and L24-band emission, there is almost no corresponding emission in the S7-band image. This suggests that the S7-band emission is dominated by background emission, probably emission from PAHs in the line of sight interstellar medium (ISM). In the S11–S7 difference image (Figure 1) we subtract out scaled S7-band emission from the S11-band image and confirmed that the two filaments of the ER and southwestern part of the OES have appreciable S11-band emission. (Note that the three bright point-like sources close to the eastern end of the southern filament of the ER in the S11–S7 difference image are stellar sources.) The NIR N3- and N4-band images, on the other hand, show only stellar emission without any apparent feature brighter than 16 Jy at 3 m that might be associated with the SNR.
Table 5 presents IR measurements of G292.01.8. In the Table, we list the fluxes and ratios of whole remnant area and IR-Ejecta region (see befow for IR-Ejecta). In addition, we also list the peak intensities and ratios of the ER, the southweastern OES, and the IR-Ejecta.
3.2 Infrared Colors and Ejecta Identification
Figure 2 (left), which is a three-color (7, 15 and 24 m) MIR image of G292.01.8, shows that most of the prominent features have roughly similar MIR colors. One exception is the feature in the southeastern part of the SNR with significant excess in the shorter wavebands – we name this feature “IR-Ejecta” because it is believed to originate in shocked SN ejecta as we describe below. It is bright near the southern equatorial filament and stretching directly southward beyond the SNR boundary. This feature is clearly seen in Figure 2 (right) which shows the L15/L24 emission ratio. The ratio image was produced by dividing the L15-band image by the L24-band image after background subtraction. The background was estimated by fitting a slanted plane to the areas surrounding the SNR. The resulting background planes were almost flat, i.e., the brightness differences over the entire images were only 1.7 % and 2.4 % in L15 and L24, respectively. We applied a Gaussian convolution to the L15-band image in order to match the final spatial resolution to that of the L24-band image and masked out the pixels with small ( 30.2 MJy sr) L24-band intensities.
The resulting L15/L24 color image is significantly different from the original band images. In Figure 2, the L15/L24 ratio is almost uniform ( 0.25), and there is no feature corresponding to the prominent ER and OES. The most notable feature is the IR-Ejecta, where L15/L24 ratio raises to 0.8, which is in fact consistent with what we identified in the three-color image (left panel of Figure 2). It appears to be of a triangular shape with one of its vertices towards the direction to the SNR center. The southern vertex passes though the southern boundary of the OES and extends over the SNR boundary, while the northeastern vertex is located just above the ER. The region of high color ratio of the IR-Ejecta covers a large portion of the southeastern area of the SNR and positionally coincides with the O- and Ne-dominant ejecta identified by previous optical and X-ray observations (Ghavamian et al., 2005; Winkler & Long, 2006; Park et al., 2002). There is also a wispy patch of emission around (, ) far ( 1′) beyond the SNR boundary, which has the color ratio similar to that of the IR-Ejecta. It, however, is not shown in the [O iii] images (Winkler & Long, 2006). Besides the IR-Ejecta in the southeastern region, the northwestern region shows extended emission of the high color ratio, although it is not as significant as the southeastern region. Overall the two regions of the high color ratio are roughly symmetric with respect to the center of the OES.
Figure 3 compares the pixel values of the L24- and L15-band emission, where most of the points are distributed along the thick, solid line of a slope of 0.25. (Note that pixels of stars and pixels with small L24-band intensities are masked out.) If L24- and L15-band fluxes are from the same region and if they are both thermal dust emission, then the color ratio and the dust temperatures are related as
Iν(15)Iν(24)=κν(15)Bν(15,T)κν(24)Bν(24,T) , (1)
where is the surface brightness at (m) in Jy sr, is the Planck function, and is the dust opacity in cm g. The slope of 0.25 corresponds to the dust temperature of 126 K for a mixture of carbonaceous and silicate interstellar grain of = 3.1 (Draine, 2003). In Figure 3 there are two regions, where the data points show a high L15/L24 ratio compared to the thick, solid line: first, the thin line of the slope of 0.88 represents one group whose L24-band surface brightness is less than 33 MJy sr; secondly, there is another group of data points of higher L15-band emission whose L24-band surface brightness lies in the range of 33–40 MJy sr. The inset in the lower-right corner of Figure 3 shows that the data points with high L15/L24 ratios are all from the IR-Ejecta as expected. The ones in the second group, i.e., ones with high 24 m surface brightness, are superposed on the ER and therefore their emission is partly from the swept-up CSM while the data points in the first group should represent the emission only from the ejecta. The slope of 0.88 corresponds to a dust color temperature of 240 K; however the L15-band flux is largely from line emission at this position so that the physical dust temperature should be lower than this (see §3.4).
3.3 Mid-Infrared Spectroscopy of Ejecta and Equatorial Peak
Figure 4 shows the background-subtracted MIR grism spectra of the peak positions of the ER and the IR-Ejecta.444We only present the MIR spectra because NIR spectra were heavily contaminated by emission from nearby field stars. (Note that the LG2 spectra at 17 m were obtained from positions slightly shifted from those of the SG spectra of 5–14 m as we described in § 2.) The spectrum from the IR-Ejecta peak clearly shows the [Ne ii] line at 12.8 m, which confirms the nature of the radiatively shocked ejecta. No Ar lines (i.e., [Ar ii] at 7.0 m and [Ar iii] at 9.0 m) were detected, while neither [Ne iii] line at 15.6 m nor [O iv] line at 25.9 m was covered. Table 6 lists the flux of the [Ne ii] line and upper limits of the [Ar ii] and [Ar iii] lines. The flux of the [Ne ii] line at 12.8 m is 2.82.510 erg cm s sr, corresponding to the surface brightness of 0.240.21 MJy sr in the L15-band image. Given the L15-band surface brightness of 1.60.1 MJy sr, 15 % of the L15-band emission is due to the [Ne ii] line emission. For the S11 band, the observed [Ne ii] line emission flux corresponds to the S11-band surface brightness of 0.120.11 MJy sr. This is equivalent to 55 % contribution to the total S11-band emission, which is much larger than the case of the L15-band emission. The difference is because there is no strong emission line other than the [Ne ii] line in the S11 band while there is additional [Ne iii] 15.5 m line in the L15 band (see the spectral responses in Figure 4). (There could be some dust emission too. See § 4.3.) On the other hand, the spectrum at the ER peak is dominated by the continuum emission, although the LG2 spectra have a lower signal to noise ratio.
3.4 Large-scale Infrared Emission around G292.0+1.8
Figure 5 presents a combined L18W-band image (§ 2) covering both the SNR and an extended area in the southeast. Besides the features of the SNR that we already described in previous sections, there is a notable feature of the L18-band emission in the south. The elongated “Narrow tail” is located around (, ), 7′ apart from the center of the SNR. It is close to the IR-Ejecta in the southeastern part of the SNR and its elongation direction is roughly towards the center of the SNR. Its extent is or 2.6 pc. In addition, there is faint, diffuse emission toward the south and southeast too. The emission toward the southeast appers distinct - it appears to protrude from the open portion of the OES having similar outer boundary with radio plateau at the east and south, but extends southeast beyond the radio boundary of the SNR. We consider that this “Wide tail” could be associated with the SNR. There is large, diffuse H ii region superposed on G292.01.8 on the sky (RCW 54; Rodgers et al. 1960), but its H emission is elongated along the northeast-southwest direction without any emission corresponding to the Wide tail in the Southern H-Alpha Sky Survey Atlas (Gaustad et al., 2001). Therefore, the Wide tail is not associated with H ii regions and its association with the SNR is likely.
Figure 6 presents the FIS FIR-band images of G292.01.8. The images cover the entire SNR with a scan direction of northeast to southwest. The FIR emission of the SNR associated with the ER and OES is clearly detected in the N60- and Wide-S-band images with strong concentration in the southwestern part, although fine details are difficult to see because of their low spatial resolutions. The N160- and Wide-L-band images are, however, clearly different from images of the other bands. They show an elongated feature extended in the northwest-southeast direction which shows no correlation with the emission associated with the SNR. Also its peak position is located outside the boundary of the SNR. These indicate that the N160- and Wide-L-band emission is not directly associated with G292.01.8, although it is possible that the enhanced brightness in the SW part of OES is due to the interaction of the remnant with this extended structure (cf. Braun et al., 1986). The bottom panels of Figure 6 present the background-subtracted N60- and Wide-S-band images. The background emission was estimated by calculating scaling factors between the N60- and N160-band images and also between the Wide-S- and N160-band images from the areas outside the SNR. Compared to the N60- and Wide-S-band images of the top panels, the background-subtracted images show more clearly the emission associated with the SNR, including the northern part of the OES that is not clear in the orignial images.
4 Discussions
4.1 Destruction of Circumstellar Dust
4.1.1 Infrared Emission from Shocked Circumstellar Dust
The ER and OES are the most prominent features in our multi-band IR images (Figures 1 and 6). The ER is composed of two filaments, where the southern one is brighter than the northern counterpart. The southern filament appears to be of the normal composition in soft X-rays (Park et al., 2002) without any apparent radial motion in the optical (Ghavamian et al., 2005), which led the authors to conclude that it is CSM from the progenitor of the SN in G292.01.8. We suggest that the northern filament, which has been clearly found by observations, could be part of the same structure based on its similar distribution to the southern filament. We note that the bright clump at (, ) near the eastern end of the northern filament has a counterpart in the [O iii] image of Ghavamian et al. (2005). Its velocity is near zero, which supports that the northern and southern filaments form a single structure. Furthermore, it is most clearly visible at the 120 km s frame of the Rutgers Fabry-Perot velocity scan images and absent at the 0 km s frame where the southern filament is brightest (Figure 2 of Ghavamina et al. 2005). We interpret this velocity difference as an indication suggesting that the northern and southern filaments are parts of a tilted, expanding ring structure produced by the progenitor of the SN. The location of the center point of the OES at the middle of the two filaments also reconciles with the interpretation, given the CSM-nature of the OES (see below). We note that such a, but smaller, ring structure of the CSM was found in SN 1987A and also possibly in the Crab nebula (Bouchet et al., 2004; Green et al., 2004). The OES, on the other hand, is relatively fainter in X-rays than the ER (Park et al., 2002). The MIR brightness and color of the OES, however, are similar to those of the ER (Figure 2). Toward the southern filament of the ER, Ghavamian et al. (2005) reported the detection of the optical radiative lines produced by the partially radiative shocks starting to develop cooling zones. It implies that the X-ray emitting gas in the ER is cooler than that of the OES, while they have somewhat similar dust properties. And there is a possibility that the ER in S11-S7 difference image also contains the [Ne ii] line emission produced in the CSM region where the shock has started to cool down to 100,000 K. However, its contribution might be small, because we have not detected the [Ne ii] 12.8 m line at the ER.
In addition to the ER and OES, there are faint MIR emissions beyond the northern and southeastern edges of the OES (Figure 2). They have sharp outer boundaries, representing the current locations of the SN blast wave. Faint X-ray emission was detected in those areas too (Park et al., 2002). It is possible that the SN blast wave has overtaken the OES and produced those emission features while propagating into a more diffuse medium. On the other hand, it is also possible that the remnant has a front-back asymmetry and they are just projected boundaries of the more-extended shell. In any case, their asymmetric spatial distribution suggests that either the ambient density distribution or the mass ejection from progenitor was asymmetric.
As in Figure 4, the MIR emission of G292.01.8 is dominated by continuum emission, not by line emission. This implies that the IR emission is from shock-heated dust grains in the CSM. Figure 7 presents the spectral energy distribution (SED) of the SNR in Table 5. We applied modified blackbody fits composed of two dust components to the observed SED in order to obtain the best-fit dust temperatures. The dust model based on a mixture of carbonaceous and silicate interstellar grain of = 3.1 (Draine, 2003) gave dust temperatures of 103 K (warm dust) and 47 K (cold dust), corresponding to the mass of 4.5 0.9 10 M and 4.8 10 M, respectively. (Note that the lower limit of the cold dust temperature comes from the upper limit of the flux at 140 m.) For the case of the dust model based on the graphite and silicate grain of 0.001 to 0.1 m size (Draine & Lee, 1984; Laor & Draine, 1993), the derived total mass is in the range of (1.0–3.4) 10 M, comparable to the total mass obtained for the former model. The derived dust mass corresponds to the dust-to-gas ratio of 1.6 10, if we use the 30.5 M swept-up mass (at distance of 6.2 kpc) of the gas in the CSM obtained in X-ray observations (Gonzalez & Safi-Harb, 2003). This is lower than the ratio 6.2 10 found in the local ISM (Zubko et al., 2004). The low dust-to-gas ratios in the swept-up materials by the shock destruction were also obtained with Spitzer observations on the core-collapse SNRs in the Large Magellanic Cloud (Williams et al., 2006) and the Kepler (Blair et al., 2007). According to our result, 75 % of the dust in G292.01.8 might have been destroyed by the SN shock, or the initial dust-to-gas ratio surrounding G292.01.8 might be lower than the local value. It is comparable to the fraction derived in other SNRs, such as 64 % in Cas A, (Dwek et al., 1987) and 78 % in Kepler (Blair et al., 2007).
4.1.2 Model Calculations of Shock-heated Dust Emission
We perform model simulations for the destruction of dust by SN shock waves and the thermal emission from shock-processed dust. The physical processes of dust in shocks have been so far discussed in many works (e.g., Tielens et al., 1994; Vancura et al., 1994; Dwek et al., 1996; Jones, 2004). Once the circumstellar dust grains are swept up by the blast wave, they acquire high velocities relative to the gas and are eroded by kinetic and/or thermal sputtering in the shock-heated gas. Dust grains are also heated by collisions with energetic electrons in the hot gas and radiate thermal emission at IR wavelengths. Dynamics, erosion, and temperature of dust depend on the temperature and density of the gas as well as the chemical composition and size of dust grains.
We adopt the model of dust destruction calculation by Nozawa et al. (2006), in which the motion, destruction, and heating of dust in the shocked gas are treated in a self-consistent manner by following the time evolution of the temperature and density of the gas for the spherically symmetric shock wave. As the initial condition of the SN ejecta we consider the freely expanding ejecta with the velocity profile of and the density profile of at and at , where and are the velocity and radius of the outermost ejecta, respectively. Taking the kinetic energy of ergs and the ejecta mass of 8 (Gaensler & Wallace, 2003) for the density profile of core-collapse SNR of and (e.g., Chevalier, 1982; Matzner & McKee, 1999; Pittard et al., 2001), we obtain g cm, cm s, and cm at 10 yrs after explosion, when the simulations are started. Since most of the dust grains swept up by the forward shock during the later epoch of the evolution, the calculation results are not sensitive to the ejecta structure.
For the ambient medium we consider the constant hydrogen number density of 0.1, 0.5, 1, and 10 cm. The circumstellar dust is assumed to be amorphous carbon or silicate (forestrite) with the power-law size distributions () ranging from m to m (Mathis et al., 1977). The optical constants are taken from Edo (1983) and Semenov et al. (2003). Based on the time evolution of the size distribution and temperature of dust given by the simulation and the assumption of the dust grains being in thermal equilibrium, we calculate the IR SED by thermal emission from the shock-heated dust. The detailed description for calculating the IR SED from shocked dust will be given elsewhere (T. Nozawa et al. 2009 in preparation).
Figure 8 compares the observed fluxes of G292.01.8 with the calculated IR SEDs at 3,000 yrs for 0.1, 0.5, 1, and 10 cm. We present the results with the initial dust-to-gas mass ratio to best reproduce parts of the observed SED, for amorphous carbon (Figure 8a) and silicate (Figure 8b). The typical temperatures of dust are 35–55, 45–65, 50–70, and 60–80 K for 0.1, 0.5, 1, and 10 cm, respectively, and the resulting dust masses are in the range of (0.3–5) for carbon and (0.4–8) for silicate with the higher values for lower ambient density (thus lower temperature of dust). It can be seen that the results of the silicate grains with 0.5 cm, which coincides with the density estimated from X-ray observations (Gonzalez & Safi-Harb, 2003), can reasonably reproduce the overall shape of the IR SED for G292.01.8. In this case, the mass of grains radiating IR emission is . Note that the initial dust-to-gas mass ratio corresponding this result is , which is smaller than that in the local Galaxy. If the IR emission originates in the swept-up materials containing the mass-loss wind of RSG with the solar metallicity, the condensation of silicate is expected, and, according to our results, its condensation efficiency could be low.
It should be noted here that the simulation results for 10 cm significantly underestimates the flux at short (11 ) wavelength. However, the SED at shorter wavelengths could be resolved by including the effect of a stochastic heating of small grains; stochastically heated dust produces more emission at shorter wavelengths than the dust with equilibrium temperature, and it may also allow acceptable fits for lower initial density and/or different size distribution of dust than the current best fit. Alternatively, this disagreement may be caused by the difference in the assumed composition of dust. To gain deeper insights into the properties of dust in G292.01.8, we need further investigations by taking account of the stochastic heating and changing the composition and size distribution of dust as well as the density profile in the ambient medium.
4.2 Infrared Emission from Supernova Ejecta
4.2.1 Ejecta Neon Line Emission
We have detected MIR emission associated with the SN ejecta: IR-Ejecta. It is prominent in the 15/24 m ratio image by its high L15/L24 ratio (Figure 2), but is marginally seen in the total intensity maps of the MIR (11–24 m) and FIR (65–90 m) too. The emitting area coincides with the fast-moving ( km s), O-rich SN ejecta: the triangle-shaped bright portion near the equatorial filament was called “spur” and the extension to the south was called “streamers” by Ghavamian et al. (2005). In the high-resolution [O iii] 5007 image, the spur is crescent-shaped with a sharp boundary toward the SNR center, while the streamer appears to be composed of clumps embedded in diffuse emission. Recent measurement of their proper motions showed that they are expanding systematically from a point near the geometrical center of the OES (Winkler et al., 2009, see § 4.4 too).
According to our spectroscopic result, the IR-Ejecta shows [Ne ii] 12.8 m emission but no [Ar ii] 7.0 or [Ar iii] 9.0 m emission. Note that the latter lines from Ar ions were also detected at the metal-rich ejecta in young SNRs (Douvion et al., 2001; Smith et al., 2009; Williams et al., 2008). The lack of IR line emission from Ar ions in this area is consistent with the results of optical or X-ray studies which showed that there is no line emission from elements heavier than S in this area (Ghavamian et al. 2005 and references therein; Park et al. 2002). The absence of Ar lines supports the claim by Ghavamian et al. (2005) that we are not seeing the ejecta in the inner-most region accelerated by pulsar wind nebula but seeing the He-burning-synthesized, O-rich ejecta swept-up by reverse shock.
The IR-Ejecta shows high L15/L24 ratio, i.e., L15/L24=0.88 compared to 0.25 of the shocked CSM. The high L15/L24 ratio is at least partly due to Ne lines. (There is [O iv] 25.9 m line in the L24 band, but it is weak, i.e., % of the [Ne ii]+[Ne iii] lines. See § 4.3.) According to our estimation, [Ne ii] 12.8 m line contributes 15 % to the L15 flux. In the L15 band, there is another strong Ne line, [Ne iii] 15.6 m line. We may estimate the possible contribution of [Ne iii] 15.6 m line in G292.0+1.8 as follows.
The strength of a forbidden Ne line is given by where is the frequency of the line, is the column density of Ne or Ne ions in the upper state along the line of sight, and is the Einstein coefficient, so that the [Ne iii] 15.6 m/[Ne ii] 12.8 m ratio is given by
I15.6 μmI12.8 μm=0.57N(Ne++)N(Ne+)f(Ne++,3P1)f(Ne+,2P1/2) , (2)
where and are column densities of Ne and Ne ions, and are the fractions of Ne and Ne ions in the upper states of the corresponding emission lines (see Glassgold et al. 2007 for atomic parameters of these lines). Assuming statistical equilibrium, the last factor, , can be easily calculated (e.g., Glassgold at al., 2007), and varies from 1 to 2 as the electron density increases from a low-density limit to densities much higher than the critical density ( cm at 10,000 K). Therefore, the intensity ratio is unless , which happens at temperatures K in collisional equilibrium (e.g., Allen & Dupree, 1969). Ghavamian et al. (2005) suggested that ejecta undergoes radiative shocks with a velocity of 50–200 km s from the [O iii] 5007 line widths. The temperature of shocked ejecta gas immediately behind the shock of velocity should be high, i.e., K, assuming pure neon preshock gas, but since the shocked gas element cools fast due to enhanced heavy elements and the column density of the gas at might vary roughly with , we expect and therefore . Toward the SN ejecta in young core-collapse SNRs such as Crab, Cas A, the SNR B054069.3 in the LMC, or the SNR 1E010272.3 in the SMC, the [Ne iii]15.6 m/[Ne ii]12.8 m ratio varies from 0.3 to too (Douvion et al., 2001; Temin et al., 2006; Smith et al., 2009; Williams et al., 2008; Rho et al., 2009). We may conclude that the contribution of the [Ne iii]15.6 m line to the L15 band should be at most comparable to that of the [Ne ii]12.8 m line unless the shock is truncated at relatively high temperatures.
4.2.2 Dust Emission Associated with Ejecta
We have derived IR fluxes associated with the ejecta and the results are summarized in Table 5. It is difficult to extract the ejecta emission in IR observations, although our results unveiled the ejecta region unambiguously. The IR-Ejecta is clearly divided into two regions (Figures 2 and 3); one that coincides with the ejecta-only region and has a tight correlation in the L15 vs L24 plot and the other superposed on the ER and has a loose correlation in the L15 vs L24 plot. These IR-Ejecta regions are part of the structure defined as the [O iii] “Spur” by Ghavamian et al. (2005), with the ejecta-only region being the bottom of the Spur, and the ejecta superimposed on the ER being the top portion of the Spur. We measure the flux at the ejecta-only region masking the emission from stars. The area of the ejecta-only region is more than four times larger than the superposed ejecta region and it contains most of the flux. In order to remove the background emission, we subtract the average of nearby ( 2) low intensity region.
The ejecta spectrum in Figure 4 shows that the 11 and 15 m fluxes are not from dust continuum. We believe that the S11-band is mostly dominated by line emission, while the L15-band needs an additional component (see § 3.3). In the next paragraph, we show that some of the L15 flux is due to a bump at 15–25 m. On the other hand, FIR emission is certainly dominated by dust continuum. The L24-band contains the [O iv] 25.9 m line emission, whose wavelength is not covered with our spectroscopic observations. Its contribution in L24-band is expected to be less than 20 % in our estimation using the [O iv] 25.9 m line intensity in Ghavamian et al. (2009). At first, we fit 24–140 m band fluxes using a modified-blackbody for silicate and graphite grain models of 0.001–0.1 m size (Draine & Lee, 1984; Laor & Draine, 1993). The best fits give temperature of 64–88 K and dust mass of 2.0–8.2 10 M. If we exclude the L24-band data point, the temperature drops to 44–52 K and the dust mass increases by a factor of 5. This IR-emitting dust mass is much smaller ( 4 10) than the theoretically predicted dust mass (0.1–1 M) to be formed in the core-collapse SN explosion (e.g., Nozawa et al., 2003).
4.3 Comparison with Spitzer Spectroscopic Results
Recently the results from MIR observations of G292.01.8 with the satellite have been published (Ghavamian et al., 2009) based on low-resolution spectroscopy of the ejecta and the southern filament of the ER where the ejecta emission is superimposed on the emission from the CSM. The former is located 0′.5 north from our IR-Ejecta slit position, while the latter, which has a high L15/L24 ratio, is 1′ apart in the northeast direction. Their ejecta spectrum shows strong emission from the [Ne ii] 12.8 m, [Ne iii] 15.6 m, [O iv] 25.9 m lines and relatively weak [Ne v] 24.4 m and [Ne iii] 36.0 m lines, but no lines of Mg, Si, S, Ar or Fe are identified. (Note that the Si and S lines in their spectrum are not from the ejecta but from the background.) The observed ratio of [Ne ii] 12.8 m to [Ne iii] 15.6 m is 2.1–2.7. The non-detection of Ar lines and the observed ratio of Ne lines are consistent with our results and the prediction in § 4.2. The observed flux of [O iv] 25.9 m line is % of the [Ne ii] plus [Ne iii] line fluxes (Ghavamian et al., 2009). If the L15 and L24 fluxes are entirely due to lines, we estimate that the L15/L24 ratio should be 2.3–2.6 using the line fluxes in Ghavamian et al. (2009). (It becomes higher if we consider the 15-25 m bump in the next paragraph.) The observd ratio, however, is 0.25 for the ER and OES and 0.88 for the ejecta. Therefore, the contribution of [O iv] 25.9 m line flux to the observed L24-band flux should be small, particularly toward the ER where (§ 3.2). We, however, note that the IRS spectrum toward the ejecta in Ghavamian et al. (2009) does not show any obvious dust continuum emission between 24 and 36 m. We consider that it could be because there was dust continuum emission in the IRS background. In the 24 m image, there is faint filament coincident with the IRS background region (LL1 Sky in Ghavamian et al. 2009). This filament is associated with the OES and its 24 m emission might be dominated by dust continuum, so that, by subtracting its spectrum from the ejecta spectrum, the continuum feature could have been removed.
An interesting feature in the ejecta spectrum is a weak 15–25 m bump, which was suggested to be produced by newly-formed dust or swept-up PAHs along the line of sight. First of all, this bump explains the high L15/L24 ratio at the ejecta region described in § 4.2.1. The peak of the bump feature appears in the L15 band. Therefore, it contributes mainly to the 15 m band. Secondly, our observations show that the areas with a high L15/L24 ratio are coincident with those of the optical O-rich ejecta region. This supports the interpretation that the bump feature is related to the newly-formed dust in association with the SN ejecta, not to the swept-up PAHs.
According to Ghavamian et al. (2009), the spectrum from the southern filament of the ER is consistent with the emission from two dust components: a warm (or hot) component of 114 K and a cold component of 35 K. The temperature of the warm dust is higher than what we estimated with the data by 10 K, while that of the cold dust is lower than the temperature by a comparable amount. It is possible that the presence of the bump feature in the spectrum resulted in a higher temperature for the warm dust. For the cold dust temperature, as explained in Ghavamian et al. (2009), the absence of the longer wavelength ( 30 m) data in the spectrum is likely the primary reason of the lower temperature compared to the results. IR spectroscopic observations covering a broad band are necessary to clearly resolve the issues.
4.4 Supernova Explosion in G292.0+1.8
4.4.1 Circumstellar Shell and the Explosion Location
One interesting result that we obtained in this study is the difference ( 42″ 1 pc) between the center of the CSM and the dynamical center of the ejecta (Figure 2). The former corresponds to the center of the ER and OES that we determined with the results; the latter was determined by the distribution of the O-rich ejecta in the optical, which is close to the center of the SNR in the radio emission. In addition, the position of the pulsar is different from the both positions – it is shifted in the southeast direction from the dynamical center by 46″.
One possibility is that the progenitor star was at the center of OES during its RSG phase but exploded at the position of the dynamical center of optical knots. This is possible because the RSG wind could be confined by external pressure while the central star is moving. According to Chevalier (2005), the RSG wind from a 25–35 M star, which explodes as SN IIL/b, would be pressure confined while its outer radius becomes pc. The radius of OES (6 pc) is comparable to what the theory predicts. If the progenitor star was moving at 10 km s and exploded after 10 yrs of the pressure confinement of the shell, then the explosion center would be close to the dynamical center of optical knots.
Another possibility is that the SN exploded at the center of ER and OES, not at the dynamical center of the ejecta or the center of the radio emission. This is motivated by the fact that the OES shows a very well-defined shell structure surrounding the SNR, thereby the center of the OES may pinpoint to the real location of the progenitor, which later exploded as a SN. On the other hand, the centers of the radio emission and the O-rich ejecta motion could have been weighted toward the southeast: Firstly, the geometrical center of the radio nebula could be significantly weighted toward the southeast because of the existence of the bright PWN. Secondly, the dynamical center of optical knots is also weighted to the southeast because the bright optical knots in the southeastern area are moving relatively slowly and the center is derived using an assumption of an unhindered constant velocity since the explosion (Winkler et al., 2009).
If the SN explosion in G292.01.8 indeed occurred at the center of the OES, then the tangential velocity of the pulsar needs to be 1,000 km s. Although 1,000 km s is somewhat large as a pulsar kick velocity, it is still within the acceptable range (e.g., Ng & Romani, 2007), and the pulsar in another O-rich SNR Puppis A may also have such a high velocity (Hui & Becker, 2006).
4.4.2 Ejecta Distribution and Explosion Asymmetry
The optical and X-ray studies of the SN ejecta in G292.01.8 have shown that their spatial distribution and physical properties are not symmetric, including the northwest-southeast concentration of the O and Ne ejecta, and the distinctive difference of the X-ray plasma temperature between the northwestern and southeastern areas (Park et al., 2002; Ghavamian et al., 2005; Park et al., 2007). Also the ejecta in the northern and southern boundaries move faster than those in the eastern and western areas (Winkler et al., 2009). It is worth to note that the pulsar jet axis is also along the northeast-southwest direction (Park et al., 2007).
Our result on the Ne-line emitting ejecta material identified by their high L15/L24 ratio is consistent with the spatial distribution seen in [O iii] 5007: most of the / ions are distributed in the southeastern area called spur and streamers, several isolated ones coincide with the optical knots, and the other group of Ne-rich knots is distributed in the northwestern area and coincides with [O iii] optical emission knots (fast-moving knots, or FMKs, as reported by Ghavamian et al. (2005) and Winkler & Long (2006)). The northwest ejecta is less obvious in optical but it is easily recognized in the X-ray image of ionized O and Ne elements (Park et al., 2002). This type of bipolar ejecta distribution is also found in other young core-collapse SNRs, e.g., Cas A and G11.20.3 (Smith et al., 2009; Koo et al., 2007; Moon et al., 2009). Cas A shows a similar Ne ejecta distribution, which is almost perpendicular to the well-known narrow northeast-southwest jet, but is roughly aligned to the bipolar ionic ejecta, which sometimes is suggested be the major direction of explosion (Hwang et al., 2004; Wheeler et al., 2008; Smith et al., 2009). In G11.20.3, which is the remnant of the historical SN AD 386, the iron ejecta is found to be distributed mainly along northwest-southeast direction (Koo et al., 2007; Moon et al., 2009).
The symmetry axis in the spatial and kinematical distribution of ejecta in G292.01.8, therefore, is either along northwest-southeast or north-south, which is perpendicular to the plane of the ER. We consider that either the explosion was asymmetric and/or the CS wind was denser in the equatorial plane so that the ejecta expanding in this plane was slowed down more compared to those expanding to the other directions.
An interesting feature is the Narrow tail in the wide L18W-band image that extends from the end of the streamers to outside the remnant. A possible explanation is that the Narrow tail is a part of the O-rich streamer. Note that the Narrow tail is connected to the streamers by a southern patch of emission near the boundary of L15-band image. The distance from the center to the end of the Narrow tail is 7 (13 pc), which is 2 (3 pc) farther than the outermost O-rich clump in this area (Winkler et al., 2009). It, however, does not has an optical counterpart in the [O iii] image (Winkler et al., 2009). If we assume a constant velocity and adopt an age of 3,000 yrs (Ghavamian et al., 2005; Winkler et al., 2009), the transverse velocity of the Narrow tail is 4,000 km s. This gives a possibility that its radial velocity is also very large, beyond the velocity range of previous optical narrow-band imaging observations (e.g., 2,000 km s in Winkler & Long, 2006). An alternative explanation is that the Narrow tail is a reradiated IR light echo similar to the echoes identified in Cas A (Krause et al., 2005; Dwek & Arendt, 2008). The lack of counterpart in the optical [O iii] and X-ray images, together with the faint feature in the FIR images (Figure 6), could be consistent with the continuum origin of the IR emission. In case of a light echo, the location of the echo can be derived from the geometrical equation of ellipse whose two focuses are the SN and the observer (e.g., Couderc, 1939; Dwek & Arendt, 2008). Applying a projected radius of 13 pc and a time delay of 3,000 yrs, we obtain a location of the IR echo at 450 pc behind G292.01.8 and its angular offset of 2 from the line of sight. It appears that this alignment is too tight at a first glance, but it can be a selection effect caused by our limited imaging area. For example, Krause et al. (2005) discovered IR echoes along the scan direction in Cas A for the first time, but Dwek & Arendt (2008) reported that the echoes were distributed in many positions in large ( 2) area. More observations are definitely necessary to inspect the nature of the Narrow tail.
On the other hand, the sharp boundary of the SN blast wave was detected only toward the north and southwest (Figure 2). The absence of SN blast wave in the other directions implies that either the shock is trapped within the shell because that part of the shell has a larger column density or the SN blast wave has propagated far beyond because the ambient density is lower toward that direction. There is no indication in the images that the OES is denser, where the SN blast wave is missing. Instead those parts are fainter in the FIR images which indicates a lower column density. In this regard, the faint MIR emission that extends far beyond the bright shell to the southeast (Figure 5) is interesting because if it is part of the SNR, it indicates that the SN blast wave has propagated much further out toward this direction probably due to the lower ambient density. A deep radio observation could reveal faint features associated with this IR structure.
5 Conclusions
We have presented the NIR to FIR imaging and MIR spectroscopic observations of the O-rich SNR G292.01.8 using the IRC and FIS instruments aboard satellite. The almost continuous multiband imaging capability of covering wide IR wavelengths together its wide field of view enabled us to clearly see the distinct IR emission from the entire SNR and to derive its IR charactersitics. We derive the physical parameters of IR-emitting dust grains in the swept-up circumstellar medium and compare the result with the model calculations of dust destruction by a SN shock. The overall shape of the observed SED can be explained by a simple model using characteristic SNR parameters with a bit lower initial dust-to-gas ratio. At 11 m, the model flux is significantly smaller, which may indicate the importance of stochastic heating. We have not detected any signficant amount of freshly-formed dust associated with the SN ejecta.
The AKARI results in this paper give new insights into the explosion dynamics of G292.01.8. We have discovered an almost symmetric IR shell probably produced by the circumstellar wind from the progenitor star in the RSG phase. Its center is significantly offset from the previously suggested explosion centers. We consider that either OES represents the circumstellar shell pressure-confined by external medium or the SN exploded close to the center of the OES. In the latter case, the pulsar in G292.01.8 may be traveling at a speed of km s. At the same time, the ejecta distribution is unveiled by their high 15 to 24 m ratio. The ejecta are mainly distributed along the northwest-southeast direction. This symmetric pre-supernova structure and asymmetric ejecta distribution appear to be rather common in the remnants of SN IIL/b, which suffer strong mass-loss like Cas A (Hines et al., 2004; Smith et al., 2009). There is also a Narrow tail outside the SNR shell, which might be similar to the feature observed in Cas A. A detailed study is necessary to understand the nature of this feature.
We suggest that multi-band IR imaging observations are powerful tools to explore both the ejecta and CSM emission in young core-collapse SNRs. Especially the 15 and 24 m images are useful to reveal the detailed structure of IR features, which leads to better understanding of environments of the progenitor and the SN explosion.
This work is based on observations with , a JAXA project with the participation of ESA. We wish to thank all the members of the project. We also thank B. Gaensler for providing the ATCA 20 cm image and S. Park for providing the X-ray image. This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-357-C00052) and the Korea Science and Engineering Foundation (R01-2007-000-20336-0). This work was also supported in part by a Grant-in-Aid for Scientific Research for the Japan Society of Promotion of Science (18204014). T.N. has been supported in part by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, and by the Grant-in-Aid for Scientific Research of the Japan Society for the Promotion of Science (19740094).
Facility: Akari
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http://www.newworldencyclopedia.org/entry/Thermal_conductivity | # Thermal conductivity
Fire test used to test the heat transfer through firestops and penetrants used in construction listing and approval use and compliance.
In physics, thermal conductivity, $k$, is the property of a material that indicates its ability to conduct heat. It appears primarily in Fourier's Law for heat conduction.
Conduction is the most significant means of heat transfer in a solid. By knowing the values of thermal conductivities of various materials, one can compare how well they are able to conduct heat. The higher the value of thermal conductivity, the better the material is at conducting heat. On a microscopic scale, conduction occurs as hot, rapidly moving or vibrating atoms and molecules interact with neighboring atoms and molecules, transferring some of their energy (heat) to these neighboring atoms. In insulators the heat flux is carried almost entirely by phonon vibrations.
## Mathematical background
First, heat conduction can be defined by the formula:
$H=\frac{\Delta Q}{\Delta t}=k\times A\times\frac{\Delta T}{x}$
where $\frac{\Delta Q}{\Delta t}$ is the rate of heat flow, k is the thermal conductivity, A is the total surface area of conducting surface, ΔT is temperature difference and x is the thickness of conducting surface separating the two temperatures.
Thus, rearranging the equation gives thermal conductivity,
$k=\frac{\Delta Q}{\Delta t}\times\frac{1}{A}\times\frac{x}{\Delta T}$
(Note: $\frac{\Delta T}{x}$ is the temperature gradient)
In other words, it is defined as the quantity of heat, ΔQ, transmitted during time Δt through a thickness x, in a direction normal to a surface of area A, due to a temperature difference ΔT, under steady state conditions and when the heat transfer is dependent only on the temperature gradient.
Alternately, it can be thought of as a flux of heat (energy per unit area per unit time) divided by a temperature gradient (temperature difference per unit length)
$k=\frac{\Delta Q}{A\times{} \Delta t}\times\frac{x}{\Delta T}$
Typical units are SI: W/(m·K) and English units: Btu·ft/(h·ft²·°F). To convert between the two, use the relation 1 Btu·ft/(h·ft²·°F) = 1.730735 W/(m·K).[1]
## Examples
In metals, thermal conductivity approximately tracks electrical conductivity according to the Wiedemann-Franz law, as freely moving valence electrons transfer not only electric current but also heat energy. However, the general correlation between electrical and thermal conductance does not hold for other materials, due to the increased importance of phonon carriers for heat in non-metals. As shown in the table below, highly electrically conductive silver is less thermally conductive than diamond, which is an electrical insulator.
Thermal conductivity depends on many properties of a material, notably its structure and temperature. For instance, pure crystalline substances exhibit very different thermal conductivities along different crystal axes, due to differences in phonon coupling along a given crystal axis. Sapphire is a notable example of variable thermal conductivity based on orientation and temperature, for which the CRC Handbook reports a thermal conductivity of 2.6 W/(m·K) perpendicular to the c-axis at 373 K, but 6000 W/(m·K) at 36 degrees from the c-axis and 35 K (possible typo?).
Air and other gases are generally good insulators, in the absence of convection. Therefore, many insulating materials function simply by having a large number of gas-filled pockets which prevent large-scale convection. Examples of these include expanded and extruded polystyrene (popularly referred to as "styrofoam") and silica aerogel. Natural, biological insulators such as fur and feathers achieve similar effects by dramatically inhibiting convection of air or water near an animal's skin.
Thermal conductivity is important in building insulation and related fields. However, materials used in such trades are rarely subjected to chemical purity standards. Several construction materials' k values are listed below. These should be considered approximate due to the uncertainties related to material definitions.
The following table is meant as a small sample of data to illustrate the thermal conductivity of various types of substances. For more complete listings of measured k-values, see the references.
## List of thermal conductivities
This is a list of approximate values of thermal conductivity, k, for some common materials. Please consult the list of thermal conductivities for more accurate values, references and detailed information.
Material Thermal conductivity
W/(m·K)
Cement, portland [2] 0.29
Concrete, stone [2] 1.7
Air 0.025
Wood 0.04 - 0.4
Alcohols and oils 0.1 - 0.21
Silica Aerogel 0.004-0.03
Soil 1.5
Rubber 0.16
Epoxy (unfilled) 0.19
Epoxy (silica-filled) 0.30
Water (liquid) 0.6
Thermal grease 0.7 - 3
Thermal epoxy 1 - 4
Glass 1.1
Ice 2
Sandstone 2.4
Stainless steel[3] 12.11 ~ 45.0
Aluminum 237
Gold 318
Copper 401
Silver 429
Diamond 900 - 2320
LPG 0.23 - 0.26
## Measurement
Generally speaking, there are a number of possibilities to measure thermal conductivity, each of them suitable for a limited range of materials, depending on the thermal properties and the medium temperature. There can be made a distinction between steady-state and transient techniques.
In general the steady-state techniques perform a measurement when the temperature of the material that is measured does not change with time. This makes the signal analysis straight forward (steady state implies constant signals). The disadvantage generally is that it takes a well-engineered experimental setup. The Divided Bar (various types) is the most common device used for consolidated rock samples.
The transient techniques perform a measurement during the process of heating up. The advantage is that measurements can be made relatively quickly. Transient methods are usually carried out by needle probes (inserted into samples or plunged into the ocean floor).
For good conductors of heat, Searle's bar method can be used. For poor conductors of heat, Lees' disc method can be used. An alternative traditional method using real thermometers can be used as well. A thermal conductance tester, one of the instruments of gemology, determines if gems are genuine diamonds using diamond's uniquely high thermal conductivity.
### Standard Measurement Techniques
• IEEE Standard 442-1981, "IEEE guide for soil thermal resistivity measurements" see als soil_thermal_properties.[4]
• IEEE Standard 98-2002, "Standard for the Preparation of Test Procedures for the Thermal Evaluation of Solid Electrical Insulating Materials"[5]
• ASTM Standard D5470-06, "Standard Test Method for Thermal Transmission Properties of Thermally Conductive Electrical Insulation Materials"[6]
• ASTM Standard E1225-04, "Standard Test Method for Thermal Conductivity of Solids by Means of the Guarded-Comparative-Longitudinal Heat Flow Technique"[7]
• ASTM Standard D5930-01, "Standard Test Method for Thermal Conductivity of Plastics by Means of a Transient Line-Source Technique"[8]
• ASTM Standard D2717-95, "Standard Test Method for Thermal Conductivity of Liquids"[9]
## Difference between US and European notation
In Europe, the k-value of construction materials (e.g. window glass) is called λ-value.
U-value used to be called k-value in Europe, but is now also called U-value.
K-value (with capital k) refers in Europe to the total isolation value of a building. K-value is obtained by multiplying the form factor of the building (= the total inward surface of the outward walls of the building divided by the total volume of the building) with the average U-value of the outward walls of the building. K-value is therefore expressed as (m2.m-3).(W.K-1.m-2) = W.K-1.m-3. A house with a volume of 400 m³ and a K-value of 0.45 (the new European norm. It is commonly referred to as K45) will therefore theoretically require 180 W to maintain its interior temperature 1 degree K above exterior temperature. So, to maintain the house at 20°C when it is freezing outside (0°C), 3600 W of continuous heating is required.
## Related terms
The reciprocal of thermal conductivity is thermal resistivity, measured in kelvin-metres per watt (K·m·W−1).
When dealing with a known amount of material, its thermal conductance and the reciprocal property, thermal resistance, can be described. Unfortunately there are differing definitions for these terms.
### First definition (general)
For general scientific use, thermal conductance is the quantity of heat that passes in unit time through a plate of particular area and thickness when its opposite faces differ in temperature by one degree. For a plate of thermal conductivity k, area A and thickness L this is kA/L, measured in W·K−1 (equivalent to: W/°C). Thermal conductivity and conductance are analogous to electrical conductivity (A·m−1·V−1) and electrical conductance (A·V−1).
There is also a measure known as heat transfer coefficient: the quantity of heat that passes in unit time through unit area of a plate of particular thickness when its opposite faces differ in temperature by one degree. The reciprocal is thermal insulance. In summary:
• thermal conductance = kA/L, measured in W·K−1
• thermal resistance = L/kA, measured in K·W−1 (equivalent to: °C/W)
• heat transfer coefficient = k/L, measured in W·K−1·m−2
• thermal insulance = L/k, measured in K·m²·W−1.
The heat transfer coefficient is also known as thermal admittance
### Thermal Resistance
When thermal resistances occur in series, they are additive. So when heat flows through two components each with a resistance of 1 °C/W, the total resistance is 2 °C/W.
A common engineering design problem involves the selection of an appropriate sized heat sink for a given heat source. Working in units of thermal resistance greatly simplifies the design calculation. The following formula can be used to estimate the performance:
$R_{hs} = \frac {\Delta T}{P_{th}} - R_s$
where:
• Rhs is the maximum thermal resistance of the heat sink to ambient, in °C/W
• $\Delta T$ is the temperature difference (temperature drop), in °C
• Pth is the thermal power (heat flow), in Watts
• Rs is the thermal resistance of the heat source, in °C/W
For example, if a component produces 100 W of heat, and has a thermal resistance of 0.5 °C/W, what is the maximum thermal resistance of the heat sink? Suppose the maximum temperature is 125 °C, and the ambient temperature is 25 °C; then the $\Delta T$ is 100 °C. The heat sink's thermal resistance to ambient must then be 0.5 °C/W or less.
### Second definition (buildings)
When dealing with buildings, thermal resistance or R-value means what is described above as thermal insulance, and thermal conductance means the reciprocal. For materials in series, these thermal resistances (unlike conductances) can simply be added to give a thermal resistance for the whole.
A third term, thermal transmittance, incorporates the thermal conductance of a structure along with heat transfer due to convection and radiation. It is measured in the same units as thermal conductance and is sometimes known as the composite thermal conductance. The term U-value is another synonym.
In summary, for a plate of thermal conductivity k (the k value[10]), area A and thickness L:
• thermal conductance = k/L, measured in W·K−1·m−2;
• thermal resistance (R value) = L/k, measured in K·m²·W−1;
• thermal transmittance (U value) = 1/(Σ(L/k)) + convection + radiation, measured in W·K−1·m−2.
## Textile industry
In textiles, a tog value may be quoted as a measure of thermal resistance in place of a measure in SI units.
## Origins
The thermal conductivity of a system is determined by how atoms comprising the system interact. There are no simple, correct expressions for thermal conductivity. There are two different approaches for calculating the thermal conductivity of a system.
The first approach employs the Green-Kubo relations. Although this employs analytic expressions which in principle can be solved, in order to calculate the thermal conductivity of a dense fluid or solid using this relation requires the use of molecular dynamics computer simulation.
The second approach is based upon the relaxation time approach. Due to the anharmonicity within the crystal potential, the phonons in the system are known to scatter. There are three main mechanisms for scattering (Srivastava, 1990):
• Boundary scattering, a phonon hitting the boundary of a system;
• Mass defect scattering, a phonon hitting an impurity within the system and scattering;
• Phonon-phonon scattering, a phonon breaking into two lower energy phonons or a phonon colliding with another phonon and merging into one higher energy phonon.
## Notes
1. Perry's Chemical Engineers' Handbook, 7th ed., Table 1-4.
2. 2.0 2.1 Thermal Conductivity of some common Materials Retrieved May 26, 2008.
3. Thermal Conductivity of Metals Retrieved May 26, 2008.
4. IEEE guide for soil thermal resistivity measurements Retrieved May 26, 2008.
5. Standard for the Preparation of Test Procedures for the Thermal Evaluation of Solid Electrical Insulating Materials Retrieved May 26, 2008.
6. Standard Test Method for Thermal Transmission Properties of Thermally Conductive Electrical Insulation Materials Retrieved May 26, 2008.
7. Standard Test Method for Thermal Conductivity of Solids by Means of the Guarded-Comparative-Longitudinal Heat Flow Technique Retrieved May 26, 2008.
8. Standard Test Method for Thermal Conductivity of Plastics by Means of a Transient Line-Source Technique Retrieved May 26, 2008.
9. Standard Test Method for Thermal Conductivity of Liquids Retrieved May 26, 2008.
10. Definition of k value from Plastics New Zealand Retrieved May 26, 2008.
## References
• Baierlein, Ralph. 2003. Thermal Physics. Cambridge: Cambridge University Press. ISBN 0521658381
• Halliday, David, Robert Resnick, and Jearl Walker. 1997. Fundamentals of Physics, 5th ed. New York: Wiley. ISBN 0471105589
• Serway, Raymond A. and John W. Jewett. 2004. Physics for Scientists and Engineers. Belmont, CA: Thomson-Brooks/Cole. ISBN 0534408427
• Srivastava, G. P. 1990. The Physics of Phonons. Bristol: A. Hilger. ISBN 0852741537
• Young, Hugh D. and Roger A. Freedman. 2003. Physics for Scientists and Engineers. San Fransisco, CA: Pearson. ISBN 080538684X | {"extraction_info": {"found_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": 0.8980190753936768, "perplexity": 2070.3567190321073}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1438042990603.54/warc/CC-MAIN-20150728002310-00045-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://conhecoumlugar.com/vatel-online-ymnbbah/comparing-regression-coefficients-in-stata-f8e023 | As I don't know what these variables are, or even what discipline you are working in, I can't say whether that is the case here or not. Viewed 2k times 1. College Station, Texas: Stata Press. I divide the sample into two subsamples: male and female, and estimate two models on these two subsamples separately. X and Y) and 2) this relationship is additive (i.e. When two slope coefficients are different, a one-unit change in a predictor is associated with different mean changes in the response. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. The problem is that my regression suffers multicollinearity. How can I do this? Posted by 20 days ago. When you use software (like R, Stata, SPSS, etc.) Unlike linear models, the change in the coefficient of the variable of interest cannot be straightforwardly attributed to the inclusion of confounding variables. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. Using Heterogeneous Choice Models to Compare Logit and Probit Coefficients across Groups. what does the word 'edge' mean in this sentence from Sherlock Holmes? I want to show that the coefficient of "sranklow" is higher than the coefficent of "srankhigh". Close. Running such a regression in R with the lm or reg in stata will not make you happy, as you will need to invert a huge matrix. the coefficient on that interaction term will test whether the coefficient of interest are the same or not. However, the common practice of comparing the coefficients of a given variable across differently specified models fitted to the same sample does not warrant the same interpretation in logits and probits as in linear regression. Unstandardized coefficients are great for interpreting the relationship between an independent variable X and an outcome Y. y_{cist} = \alpha_{ci} + b_{sit} + \gamma_{it}+ X_{cist} '\beta + \epsilon_{cist} where . are state-time-industry fixed effects. The big point to remember is that… How do I create a fitted value with a subset of regression coefficients in place of all coefficients? Note that other statistical packages, such as SAS and Stata, omit the group of the dummy variable that is coded as zero. 17) andJohnson, Kemp, and Kotz(2005, chap. Press question mark to learn the rest of the keyboard shortcuts. Which fuels? In Stata … You can browse but not post. . I just want to prove that sranklow has an higher influence on the y-variable but as I mentioned both variables are not significant. which spacecraft? whether I can just estimate the model using the combined sample of males and females. I'm doing OLS fixed effects regression, and would like to test whether coefficients are the same between the two. Prior to this, the regression models will have to be stored first using the command est … Calculating maximum power transfer for given circuit. I am not aware of any Stata commands that test composite hypotheses such as this > that. Stack Overflow for Teams is a private, secure spot for you and Or are you thinking of a one-tailed test of sranklow = srankhigh? r/stata: Stata news, code tips and tricks, questions, and discussion! How does "quid causae" work grammatically? I wonder if that is possible to compare coefficients between two multivariate regression model? Login or. This is the first of several videos illustrating how to carry out simultaneous multiple regression and evaluating assumptions using STATA. Regression analysis is a form of inferential statistics. Do the confidence intervals for the estimates help? So, is there … 4) for information about the Poisson distribution. Would laser weapons have significant recoil? 8),Long and Freese(2014, chap. Dear all, I want to estimate a model with IV 2SLS method. • Williams, Richard. You can also form linear combinations of beta coefficients with the -lincom- command, but you have to ask yourself whether the meaning of the difference in the coefficients and its test answer your intended question. Your English is better than my <>. Asking for help, clarification, or responding to other answers. I found that 'suest ' of Stata is a very useful command for comparing regression coefficients between different (separated) regression models EASILY. For example, you might believe that the regression coefficient of height predicting weight would be higher for men than for women. your coworkers to find and share information. I have seen a guide to do that using Stata suest but only applies to one independent variable model. comparing the estimated coefficients of nested linear regression models. Regression Models for Categorical Dependent Variables Using Stata, 2nd Edition. A one-unit change in an independent variable is related to varying changes in the mean of the dependent variable depending on the condition or characteristic. | Stata FAQ. However, they are not useful for comparing the effect of an independent variable with another one in the model. One tailed tests are appropriate when the underlying science says that only one direction of difference is possible or meaningful. What is the extent of on-orbit refueling experience at the ISS? One can use the estimate to compare the effects of a particular covariate or a set of covariates across different subpopulations. 5.2 Confidence Intervals for Regression Coefficients. Then the (unstandardized) coefficient on an explanatory variable is equal to the change in … Stata commands. different x-variables, same y-variable). However, SPSS omits the group coded as one. One of the regressions has a different dependent variable than the other. You can get that just by dividing the p-value from the two-tailed test by two. 6. Active 5 years, 7 months ago. How does one promote a third queen in an over the board game? Press J to jump to the feed. SeeCameron and Trivedi(2013),Long(1997, chap. As described above, I would like to compare two correlation coefficients from two linear regression models that refer to the same dependent variable (i.e. Well, you have the results of testing whether they are equal. In the scatterplot below, it appears that a one-unit increase in Input is associated with a greater increase in Output in Condition B than in Condition A. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. The p-value for each independent variable tests the null hypothesis that the variable has no correlation with the dependent variable. you should run one single regression and interact everything with a black dummy variable. Also one of my favorite parts of Stata code that are sometimes tedious to replicate in other stat. Is there a way I can do it in Stata? User account menu. I can regress W on Q and get the predicted W, and then use it in the second-stage regression. The standard errors will, however, be incorrect. 9),McNeil(1996, chap. 6. Stata: comparing coefficients from different regressions (different dependent variables) Ask Question Asked 7 years ago. This is the case but both are statistically not significant. Many of my colleagues use Stata (note it is not STATA), and I particularly like it for various panel data models. But then I want to test whether all the coefficients in the two models based on the two subsamples are the same, i.e. Limitations of the unstandardized regression coefficients. An example in Stata, reg y x1 x2 est sto model1 reg y x1 x2 x3 est sto model2 lrtest model1 model2 The first model is the null model and the second model is … Often since regression coefficients (including those reported by Stata's fixed-effect routines) are in units of the dependent variable, one can say how much change a one unit change in an explanatory variable produces in a dependent variable. Hypothesis Tests for Comparing Regression Coefficients. Sociological Methods & Research 37(4): 531-559. Difference between drum sounds and melody sounds, Emitting signal when project property is changed using PyQGIS, Expectation of exponential of 3 correlated Brownian Motion, Effects of being hit by an object going at FTL speeds. One example, which we will use throughout this article, is the decomposition of total effects into direct and indirect effects. How to \futurelet the token after a space. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Thanks for contributing an answer to Stack Overflow! It just estimates OLS regression in the usual way, and then filters all the coefficients through this formula: βˆs j = βˆ j SD(x j) SD(Y) (see Eric Vittinghoff et al, Regression methodsin biostatistics: Linear, logistic, survival, and repeated measures models, Springer, 2005, p 75). You are not logged in. The p-values help determine whether the relationships that you observe in your sample also exist in the larger population. I am working on a course paper in which I need to compare several regression models and I would be very glad if I could make them nest within a single table like this one, from the estout Stata package.. A pre-publication version is available at Ho: B 1 = B 2 = B 3. where B 1 is the regression for the young, B 2 is the regression for the middle aged, and B 3 is the regression for senior citizens. The most … Sometimes your research may predict that the size of a regression coefficient should be bigger for one group than for another. 6), … Where can I travel to receive a COVID vaccine as a tourist? Comparing Regression Coefficients Between Models using Logit and Probit: A New Method Introduction Nonlinear probability models such as binary logit and probit models are widely used in quantitative sociological research. I mentioned both variables are not significant comparing the effect of an independent with. 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https://indico.cern.ch/event/73513/contributions/2078094/ | # ICHEP 2010
22-28 July 2010
Palais des Congrès de Paris
Europe/Paris timezone
## Azimuthal correlations of forward di-pions in d+Au collisions suppressed by saturation
22 Jul 2010, 14:40
12m
Salle 253
### Salle 253
Parallel Session Talk 08 - Heavy Ion Collisions and Soft Physics at Hadron Colliders
### Speaker
Dr Cyrille Marquet (Theory Division - CERN)
### Description
The STAR collaboration has recently measured the azimuthal correlation function of forward di-pions. The data show a disapearance of the away-side peak in central d+Au collisions, compared to p+p collisions. We argue that this effect, absent at mid-rapidity, is a consequence of the small-x evolution into the saturation regime of the Gold nucleus wave function, and we show that the data can be quantitavely described in the Color Glass Condensate framework. This confirmation that forward monojets are produced in central d+Au collision is a concrete evidence for parton saturation.
### Primary author
Dr Cyrille Marquet (Theory Division - CERN)
### Co-author
Dr Javier Albacete (IPhT - CEA/Saclay)
### Presentation Materials
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https://stats.stackexchange.com/questions/544711/is-wikipedias-page-on-the-sigmoid-function-incorrect | # Is Wikipedia's page on the sigmoid function incorrect?
Is Wikipedia's page on the sigmoid function incorrect?
It states that:
A common example of a sigmoid function is the logistic function
From my knowledge of machine learning, I thought that "the sigmoid function" is defined as the logistic function, $$\sigma(z) = \frac {1} {\left(1 + e^{-z}\right)}\text{.}$$ I have never seen or heard the phrasing that the logistic function is a type of sigmoid function.
Furthermore, that Wikipedia page says that other examples of a sigmoid function are the tanh and arctan functions. Again, I've never seen tanh nor arctan described as a type of sigmoid function.
These functions are considered to be peers, usually in a context like:
We can use various non-linear functions in this neural network, such as the sigmoid, tanh, and ReLU activation functions.
What am I missing here? Is the Wikipedia article correct or incorrect? I find that Wikipedia is usually accurate for math terms.
• tanh, if you scale and shift domain and range, is the same as the logistic function. Sep 15 at 10:19
• Usually, one applies linear transformations before and after the sigmoid function. In that context there is no difference to using tanh, only the parameters of the linear transforms shift somewhat. Sep 15 at 19:38
• If your primary exposure to math is dominated by current ML literature, than you might naturally associate "the sigmoid function" with the logistic sigmoid function. With a broader exposure to math, physics, chemistry, economics, etc., you'll learn that "sigmoid function" is a general concept, and the logistic sigmoid, tanh, x/(1+|x|), etc, are all examples of sigmoid functions. Sep 16 at 16:08
• Relevant aside: the logistic function is just a sigmoid function commonly used as a binary link for linear models. For example, there is also probit regression, and also complementary log-log regression, both also sigmoid functions. In fact, the CDF of many familiar probability distributions also form many other sigmoid functions. I suspect there is little reason beyond familiarity to label the logistic function the sigmoid function. Sep 16 at 16:37
• @stackoverflowuser2010 and statistics can be used in a non-machine learning context, which is still in the scope of this stack. Sep 17 at 10:03
The unsatisfying answer is "It depends who you ask." "Sigmoid", if you break it into parts, just means "S-shaped".
The logistic sigmoid function is so prevalent that people tend to gloss over the word "logistic". For machine learning folks, it's become the exemplar of the class, and most call it the sigmoid function. (Is it myopia to call it the sigmoid function?) Still, there are other communities that use S-shaped functions.
• I'm analytical chemist, and we use "sigmoid" in the more general S-shape sense without implying what function exactly. E.g. the rather typical detector behaviour that you get some roughly constant signal at very low concentrations, then the signal increases with analyte concentration (that's what we want to use) and finally, at high concentrations the signal becomes constant again, e.g. because of detector saturation is called sigmoid. Sep 15 at 12:20
• Re "is it myopia?": yes, definitely. This function has been around--with an established name ("logistic")--since the mid-1800's. A community that creates a new name for such an old, well-known object is actively rejecting its intellectual history.
– whuber
Sep 15 at 18:12
• Intellectual history does matter. Those who don't know it are doomed to repeat it, as the adage goes. It is difficult (practically impossible) to acquire a deep understanding of a concept or technique if you have to repeat for yourself centuries of investigation and discovery. In the present case, anybody who has taken a freshman college course in math, chemistry, physics, or biology has learned about logistic functions under that name, so ignorance is no excuse. Even Isaac Newton acknowledged that he "stood on the shoulders of giants." We, too, should take advantage of what precedes us.
– whuber
Sep 15 at 19:51
• My point is that ML people thinking that other (older) disciplines' names are wrong is hubris. (I say this as an ML person.) Just because we're a community with a large, loud online presence doesn't mean that we're the only truth out there. Norms in your field ≠ norms in other fields. So while your colleagues may understand "sigmoid function" to mean the logistic sigmoid function specifically, analytical chemists are also well-grounded in calling a broader class of functions "sigmoid functions". Sep 15 at 19:58
• @stackoverflowuser2010 There are lots of examples of machine learning/neural networks folks redefining terms. For instance, I know that when a NN paper writes about "cross entropy loss," they're almost certainly referring to "categorical cross entropy," even though you can write a cross entropy loss for other distributions.
– Sycorax
Sep 15 at 20:14
As Arya said, it depends who you ask, but this is not specific to Machine Learning, and even in Machine Learning the situation is not consistent (or not consistently bad). Bishop, for example, uses the term "logistic sigmoid function" and Jordan used "logistic function" already in 1995. In Statistical Mechanics, on the other hand, people are likely to call it the "Fermi-Dirac distribution/function". In some fields of biochemistry, including toxicology, you'll meet the same thing under the name "Hill equation". Etc.
It is IMHO important to remember that these are only names (words) used for describing a mathematical concept. Words is what people use to communicate, for example ideas and methods. As long as all participants of the communication understand what concept they are talking about, it doesn't really matter what words they use for it. Communities develop to a large part independently from each other (otherwise they would form a single community) and develop field-specific "dialects".
As a related example, the words "weight" and "bias", in the context of neural networks (and, through historical development, support vector machines) have completely different meanings from those used in statistics, but there is historical/field specific justification for using them.
Update: Actually, neural network pioneers commonly use "logistic function" or "logistic neuron": Hinton, Rumelhart and McClelland (also here), Sejnowski etc.
Update 2: Also, one might as well ask: "Is RBF just the Gaussian function?". For some reason, equating the two on CV doesn't seem to cause nearly as much commotion as your question.
It should be clear that the mentioned Wikipedia page has some terminology issues.
Wikipedia's statement
A common example of a sigmoid function is the logistic function
and assertions that these functions are examples of sigmoid functions
are confusing at best.
In machine learning, the sigmoid function is defined as $$\sigma(z) = \frac{1}{1 + e^{-z}}$$. Full stop. The tanh function should not be considered a type of sigmoid function. Using other terminology will at best confuse your ML peers and at worst get you fired.
Stanford's Andrew Ng states the terminology concisely in this video on neural network activation functions. This is the correct terminology to use if you are working in machine learning. Other fields may use their own idiosyncratic terms.
• I don't think you can make such a strong statement. Ng may use "sigmoid" and "logistic" as synonyms (jojo-m.cn/2021/01/07/machine%20learning-Andrew%20Ng-Stanford), but not all experts in the field do. In sklearn (scikit-learn.org/stable/modules/generated/…), the activation function can be 'tanh' or 'logistic', but not 'sigmoid'. Logistic function and tanh are closely related: $\tanh x = 2 \cdot$ logistic$(2x) - 1$. Whoever fires you for using a technically correct term is not worth working for. Sep 20 at 6:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 2, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7171326279640198, "perplexity": 1267.0389227686662}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1631780057589.14/warc/CC-MAIN-20210925021713-20210925051713-00548.warc.gz"} |
http://math.stackexchange.com/questions/61920/if-a-graph-has-no-cycles-of-odd-length-then-it-is-bipartite-is-my-proof-correc | If a graph has no cycles of odd length, then it is bipartite: is my proof correct?
I came up with a proof of
Graph $G$ has no cycles of odd length $\implies$ $G$ is bipartite.
like this:
Without loss of generality, let's only consider a connected component, because if every connected component of a graph is bipartite, then the whole graph is bipartite.
Pick up a random vertex $v$ in $G$, calculate the length of the shortest simple path from $v$ to any other node, call this value distance from $v$, and divide nodes into 2 groups according to the parity of their distance to $v$. If we can prove that nodes belong to the same group can not be adjacent, then we know that we actually get a partition of the $G$ that fulfill the definition of bipartite graph.
Now, to introduce contradiction, assume two nodes $x, y$ with both even or odd distance from $v$ are adjacent, then the shortest simple path $\langle v, x \rangle$, $\langle v, y \rangle$ and edge $\{x, y\}$ contains a cycle with odd length, which is contradictory to that $G$ has no cycles of odd length. In other words, nodes both with even or odd distance from $v$ can not be adjacent, which is exactly what we need.
So my question is, is my proof correct? And is there simpler method to prove the proposition?
Edit:
(to address comment from Srivatsan Narayanan)
To prove that $\langle v, x \rangle$ and $\langle v, y \rangle$, together with $\langle x, y \rangle$ contains a cycle with odd length is obvious when $\langle v, x \rangle$ and $\langle v, y\rangle$ are disjoint. When that's not the case, let's give the last node shared by $\langle v, x\rangle$ and $\langle v, y\rangle$ the name $v'$. So the three nodes $v', x, y$ forms a cycle with length $\newcommand{\len}{\operatorname{len}}$
$$L = \len(\langle v', x \rangle) + \len(\langle v', y \rangle) + 1 = \len(\langle v, x \rangle) + \len( \langle v, y \rangle) - 2 \cdot \len(\langle v, v'\rangle) + 1 .$$
where $\len()$ means the length of the shortest path.
As $\len(\langle v, x \rangle)$ and $\len(\langle v , y \rangle)$ are both even or odd, then $L$ must be odd. Therefore, in both cases, disjoint or not, $\langle v, x \rangle$, $\langle v, y \rangle$ and $\langle x, y\rangle$ contains a cycle with odd length.
Edit2
To see $\len(\langle v, x \rangle) = \len(\langle v, v'\rangle) + \len(\langle v', x \rangle)$, we can simply prove that both $\langle v, v' \rangle$ and $\langle v', x \rangle$ are both shortest path. And that's obvious, because if it's not the case, there exist a path shorter than $\langle v, v' \rangle$ from $v$ to $v'$, or there exist a path shorter than $\langle v', x\rangle$ from $v'$ to $x$, then $\langle v, x \rangle$ can not be a shortest path.
-
You will need to argue that the shortest paths from $v$ to $x$ and $y$, together with the edge $xy$, forms an odd cycle more carefully. This is clear when the shortest paths are disjoint; what would happen otherwise? (Also, a typo: your very first line should read connected graph, not a path.) – Srivatsan Sep 4 '11 at 22:38
Also: it should be "if every connected component", not "if any connected component" (I would understand the latter as saying that if at least one connected component is bipartite, then the graph is bipartite, that is clearly not what you mean to say). – Arturo Magidin Sep 4 '11 at 22:47
@ablmf: I don't think you are addressing Srivatsan's comment: just saying that the paths from $v$ to $x$, from $v$ to $y$, and the edge $[x,y]$ "contains a cycle of odd length" is not very informative. It's reasonably clear that they contain a cycle, and that if the paths $v\to x$ and $v\to y$ are disjoint, then the cycle will be of odd length; but you have to prove that it contains a cycle of odd length even if the paths are not disjoint, and that is not quite so obvious that you can get away with not saying anything about it. – Arturo Magidin Sep 5 '11 at 1:15
@ablmf I think that should do. One more nitpick: You still must justify that the distances from $v$ to $v'$ along the paths $\langle v, x \rangle$ and $\langle v,y \rangle$ are both equal to $len(v,v')$, the shortest distance from $v$ and $v'$. – Srivatsan Sep 5 '11 at 1:36
Thanks! This is my first proof on math.stackexchange.com Although it's answering my own question. – ablmf Sep 5 '11 at 1:40
I believe the question is resolved to the satisfaction of the OP. See the comments and the revisions to the question for the relevant discussions.
Here I present a different, and--in my mind--conceptually cleaner proof of the same fact.
Assume $G$ is a connected graph such that all of whose cycles are of even length. We generalize this slightly to the following
Proposition. Any closed walk in $G$ has even length.
Proof. Towards a contradiction, suppose not. Let $W$ be a closed walk of odd length such that the length of $W$ is as small as possible. By hypothesis, $W$ cannot be a cycle; i.e., $W$ visits some intermediate vertex at least twice. Hence we can write $W$ as the "concatenation" of two non-trivial closed walks $W_1$ and $W_2$, each of which is shorter than $W$. Further, $\len W_1 + \len W_2 = \len W$, which is odd. Thus at least one of $W_1$ and $W_2$ is of odd length, contradicting the minimality of $W$. Thus there cannot be any closed walk in $G$ of odd length. $\quad\quad \Box$
Partitioning the graph into even and odd vertices. Now, fix a vertex $v$, and define $E$ (resp. $O$) be the set of vertices $x$ in $G$ such that there is an even-length (resp. odd-length) walk from $v$ to $x$. The sets $E$ and $O$ partition $V$:
• Assuming $G$ is connected, then clearly $E \cup O = V$.
• We now show that $E \cap O = \emptyset$. To the contrary, suppose $x$ is in both $E$ and $O$. Then there is a $v$-$x$ walk $W_1$ of even length and another one $W_2$ of odd length. Then the walk $W_1 \circ \operatorname{reverse} (W_2)$ is a closed walk in $G$ of odd length, a contradiction.
Finally, we show that every edge crosses the cut $(E, O)$:
• Assume $x \in E$ and $xy$ is an edge. Then there exists a $v$-$x$ walk $W$ of even length. Therefore, $W \circ xy$ is a $v$-$y$ walk and it has odd length. Therefore, $y \in O$.
• Similarly, if $x \in O$ and $xy$ is an edge, we can show that $y$ is in $E$. This proof is similar to the above case.
This establishes that $G$ is bipartite, as desired.
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https://www.zbmath.org/?q=ra%3Adoumane.amina+ai%3Amaieli.roberto | ×
# zbMATH — the first resource for mathematics
Proof nets for multiplicative cyclic linear logic and Lambek calculus. (English) Zbl 1456.03092
Summary: This paper presents a simple and intuitive syntax for proof nets of the multiplicative cyclic fragment (McyLL) of linear logic (LL). The main technical achievement of this work is to propose a correctness criterion that allows for sequentialization (recovering a proof from a proof net) for all McyLL proof nets, including those containing cut links. This is achieved by adapting the idea of contractibility (originally introduced by Danos to give a quadratic time procedure for proof nets correctness) to cyclic LL. This paper also gives a characterization of McyLL proof nets for Lambek Calculus and thus a geometrical (i.e., non-inductive) way to parse phrases or sentences by means of Lambek proof nets.
##### MSC:
03F52 Proof-theoretic aspects of linear logic and other substructural logics 03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) 03F07 Structure of proofs
Full Text:
##### References:
[1] Abrusci, V. M., Classical conservative extensions of Lambek calculus, Studia Logica, 71, 3, 277-314, (2002) · Zbl 1019.03018 [2] Abrusci, V. M.; Maieli, R., Proceedings of the Conference WoLLIC 2015, 9160, Cyclic multiplicative proof nets of linear logic with an application to language parsing, 53-68, (2015), Bloomington, USA · Zbl 1365.03039 [3] Abrusci, V. M.; Maieli, R., Proceedings of the Conference FG 2015, 9804, Cyclic multiplicative and additive proof nets of linear logic with an application to language parsing, 43-59, (2015), Barcelona: Springer, Barcelona · Zbl 06658630 [4] Abrusci, V. M.; Ruet, P., Noncommutative logic I: the multiplicative fragment, Annals of Pure and Applied Logic, 101, 1, 29-64, (2000) · Zbl 0962.03054 [5] Andreoli, J.-M.; Pareschi, R., Proceedings of Workshop on Substructural Logic and Categorial Grammar, From Lambek Calculus to word-based parsing, (1991), CIS Munchen: Germany, CIS Munchen [6] Bagnol, M.; Doumane, A.; Saurin, A., Proceedings of the 18th International Conference, FoSSaCS 2015, On the dependencies of logical rules, 436-450, (2015), London, UK · Zbl 1367.03109 [7] Danos, V.; Regnier, L., The structure of multiplicatives, AML, 28, 181-203, (1989) · Zbl 0689.03013 [8] Danos, V., (1990) [9] de Naurois, P. J.; Mogbil, V., Proceedings of CSL 2007, 4646, Correctness of multiplicative (and exponential) proof structures is NL-complete, 435-450, (2007) · Zbl 1179.03063 [10] Girard, J.-Y., Linear logic, Theoretical Computer Science, 50, 1-102, (1987) · Zbl 0625.03037 [11] Girard, J.-Y.; Ursini, Agliano, Logic and Algebra, Proof nets: the parallel syntax for proof theory, (1995), New York: M. Dekker, New York [12] Guerrini, S., A linear algorithm for MLL proof net correctness and sequentialization, Theoretical Computer Science, 412, 20, 1958-1978, (2011) · Zbl 1222.03067 [13] Hughes, D.; van Glabbeek, R., Proceedings of the 18th IEEE Logic in Computer Science, Proof nets for unit-free multiplicative-additive linear logic, (2003), Los Alamitos [14] Lambek, J., The mathematics of sentence structure, The American Mathematical Monthly, 65, 3, 154-170, (1958) · Zbl 0080.00702 [15] Maieli, R., Archive for Mathematical Logic, 42, A new correctness criterion for multiplicative non-commutative proof-nets, 205-220, (2003), Berlin Heidelberg: Springer-Verlag, Berlin Heidelberg · Zbl 1025.03064 [16] Maieli, R., Proceedings of the 14th International Conference LPAR, 4790, Retractile proof nets of the purely multiplicative and additive fragment of linear logic, 363-377, (2007), Berlin Heidelberg: Springer-Verlag, Berlin Heidelberg · Zbl 1137.03324 [17] Maieli, R.; Dowek, G., Proceedings of the International Joint Conference RTA-TLCA, Vienna, 8560, Construction of retractile proof structures, 319-333, (2014), Switzerland: Springer International Publishing, Switzerland · Zbl 1417.03224 [18] Melliès, P.-A.; Ehrhard, T.; Girard, J.-Y.; Ruet, P.; Scott, P., Linear Logic in Computer Science, 316, A topological correctness criterion for multiplicative non-commutative logic, 283-321, (2004), Cambridge University Press [19] Mogbil, V., Proceedings of CSL 2001, Paris, France, 2142, Quadratic correctness criterion for noncommutative logic, 69-83, (2001), Berlin Heidelberg: Springer-Verlag, Berlin Heidelberg · Zbl 0999.03055 [20] Moot, R.; Retoré, Ch., The Logic of Categorial Grammars: A Deductive Account of Natural Language Syntax and Semantics, 6850, (2012), Berlin Heidelberg: Springer-Verlag, Berlin Heidelberg · Zbl 1261.03001 [21] Moot, R., (2002) [22] Retoré, C., Traitement Automatique des Langues, 37, 2, Calcul de Lambek et logique linéaire, 39-70, (1996) [23] Roorda, D., Proof nets for Lambek calculus, Journal of Logic and Computation, 2, 2, 21-233, (1992) · Zbl 0768.03035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. | {"extraction_info": {"found_math": 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": 0.858893096446991, "perplexity": 8879.360660004126}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1623487617599.15/warc/CC-MAIN-20210615053457-20210615083457-00616.warc.gz"} |
https://www.physicsforums.com/threads/hydrogen-atom.102374/ | # Hydrogen atom
1. Dec 2, 2005
### asdf1
hydrogen atoms in states of high quantum number have been created in the labortatory and observed in space. Find the quantum number of the Bohr orbit in a hydrogen atom whose radius is 0.01mm.
my problem:
n=(0.00001/(5.29*10^-11))^(1/2)=434.7
i think that n should be 434, because the electron doesn't have enough energy to move up to 435
but the correct answer is 435...
2. Dec 2, 2005
### alfredblase
I'm not sure but I think this may explain it. Orbits in the Bohr model are quantized. If they were not then the answer could be 434.7. Which means that the orbit was measured (inevitably) with some uncertainty as if the Bohr model is correct then it couldn't possibly be of 0.01mm radius. Now if we know that our value of 434.7 must be either 434 or 435, we look to see which "true" value our answer is closest to. And we find the value of n = 435. The point is we are not sure of the exact value of the atoms energy, but we know it is much more likely that our atom is in the 435 orbit than the 434 orbit.
Last edited: Dec 2, 2005
3. Dec 3, 2005
### asdf1
that's logical~ thank you very much!!! :)
Similar Discussions: Hydrogen atom | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9601314663887024, "perplexity": 639.7928318838523}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463612399.20/warc/CC-MAIN-20170529145935-20170529165935-00178.warc.gz"} |
http://mathoverflow.net/questions/69815/how-to-motivate-and-interpret-the-geometric-solutions-of-hamilton-jacobi-equatio/69821 | # How to motivate and interpret the geometric solutions of Hamilton-Jacobi equation?
Studying the Hamilton-Jacobi equation, I meet a generalization of the notion of its solutions, which is found already in the work of Sophus Lie.
For an H-J eqn, I mean a first order pde $H\circ dS=0$ in an unknown scalar function $S$ defined on a smooth manifold $M$, where $H\in C^\infty (T^\ast M,\mathbb{R})$.
If $S$ is a solution then the image $\Lambda$ of its differential $dS$ is included in $H^{-1}(0)$ and has the following properties:
1. $\Lambda$ is a lagrangian submanifold of $(T^\ast M,d\theta_M)$,
2. $\Lambda$ is transversal to the fibers of $\tau_M^{\ast}:T^\ast M\to M$,
3. the restriction of $\tau_M^{\ast}$ to $\Lambda$ is injective.
Conversely, if a submanifold $\Lambda$ of $T^\ast M$, included in $H^{-1}(0)$, satisfies the properties 1, 2, and 3, then it is equal to the image of the differential of a solution, unique up to a constant.
But if a submanifold $\Lambda$ of $T^\ast M$, included in $H^{-1}(0)$, satisfies only the conditions 1 and 2, then, around each of its points, it is again equal to the image of the differential of a solution, but this can fail to holds globally.
The idea of Sophus Lie was to give up both conditions 2 and 3.
Adopting this point of view, we define a generalized (or geometric) solution of $H\cic dS=0$ to be any lagrangian submanifold $\Lambda$ of $(T^\ast M,d\theta_M)$ which is included in $H^{-1}(0)$.
I don't think that this generalization is only due to the sake of abstractness. Infact, considering generalized solutions, it is possible, arguing with tecniques from symplectic geometry, to prove the local existence and uniqueness theorem, at the same time, for generalized and usual solutions.
But I am hoping to find "more" practical applications which illustrate the meaningfulness of geometric solutions. I would like to learn if ther is some physical or geometrical problem involving an H.-J. eqn, whose comprehension is sensibly augmented by the consideration of generalized solutions. So my question is:
What are the possible arguments and applications that motivate and help to interpret the notion of geometric solutions for an Hamilton-Jacobi equation?
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@Mathphysicist: I have merged two of your tags in one, hoping to be more descriptive of the content. – Giuseppe Tortorella Jul 9 '11 at 8:23
@Giuseppe: there's really no point for an geometric-theory-of-pdes tag. If you must, you should use the already existing geometric-analysis tag. – Willie Wong Jul 16 '11 at 2:04
A very interesting practical application is the problem of state estimation - for linear systems the answer is called the Kalman filter. Given a vector field $\dot{x} = a(x,v)$ and a measurement equation $y=c(x,w)$, compute the initial condition $x(t_0)$, the perturbation $v(t)$, and the measurement error $w(t)$ that minimize a cost function $J$. The cost is usually expressed as an integral over time of some function of $v$ and $w$.
Using Pontryagin's maximum principle or Bellman's dynamic programming, one arrives at a HJ equation which is used to find $v$. The additional step needed is to determine $x(t_0)$. It is a static minimization problem, which however needs to be repeated at each instant $t$ in the interval of interest. This is not a very practical answer. For linear systems with quadratic costs, the Kalman filter provides a recursive solution to the complete problem. In more general cases, the problem is much less studied either by engineers or by mathematicians. This is unlike the optimal control problem which has been studied extensively.
I think the geometry of the solutions is crucial. My understanding is that the filter equation is a particular symmetry of the Hamilton-Jacobi-Bellman partial differential equation - at least when everything is smooth. Meanwhile, the Hamiltonian vector field is a characteristic of the partial differential equation - also a particular symmetry, but not the one that gives a recursive solution to the estimation problem.
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Dear Pait, I appreciate very much your thoughtful responce. I have given a look at the corresponding sections of the book of Agrachev and Sachkov, but I have not found the lagrangian submanifolds not transversal to the fibers considered as solution generalized for the H.-J. eqn for the optimal cost. Where could I look for such objects in the context of control theory? I would like to learn if considering generalized solutions is possible obtain more information rather than using only usual solution. Thank you. – Giuseppe Tortorella Jul 9 '11 at 7:50
Would you mind helping me with (or pointing to) explanations for the terms "lagrangian submanifolds not transversal to the fibers" and "generalized solutions"? That would help me translate between the two sides of the literature, the mathematical and the engineering. Thanks! – Pait Jul 10 '11 at 21:38
I described this notion already in the text of my question. Please I woulde like to know the points of my question that are not enough clear, or are not written in proper english, so that I could correct them. Thank you in advance. – Giuseppe Tortorella Jul 11 '11 at 13:37
Given the HJ eqn $H\circ dS=0$ in the unknown function $S$ on the smooth manifold $M$, where $H\in C^\infty(T^\ast M)$. A generalized solution is defined to be a submanifold of $T^\ast M$, the cotangent space of $M$, which is included in $H^{−1}(0)$ and lagrangian w.r.t. the canonical symplectic form $dθ_M$. Here $θ_M$ is the tautological, or Liouville, 1-form on $T^\ast M$. – Giuseppe Tortorella Jul 11 '11 at 17:46
I think I wanted a reference, maybe to a book, with a more leisurely explanation. It's not that your text is in any way unclear, it's just that I have a different background and need to do my homework to learn the language better. – Pait Jul 13 '11 at 15:07
The famous KAM tori arose out of HJ considerations. They are Lagrangian torii. They were found by attempting to solve the HJ equation generally, and then finding one can only solve it when certain appropriately irrational frequency conditions hold. They occur in perturbations of integrable systems, or near `typical' linearly stable periodic orbits in a fixed Hamiltonian systems. You can read about them in an Appendix to Arnol'd's Classical Mechanics, and also get some idea from Chris Golé's book 'Symplectic Twist Maps', or from Siegel and Moser's 'Stable and Random Motion'.
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As you suspect these generalized solutions and their apparent singularities (=points of the Lagrangian submanifold where condition 2. fails) are unavoidable.
First observe that any Lagrangian submanifold contained in $H^{-1}(0)$ must be tangent to the Hamiltonian field $X_H$ (this is the method of characteristics). I assume here that $H^{-1}(0)$ is smoot and $2n-1$ dimensional. Now start with some non-characteristic classical initial data (= an $n-1$ dimensional submanifold in $H^{-1}(0)$ transversal to $\tau^*_M$ and transverslat to $X_M$). If you let the initial datum flow with $X_H$ this will swipe out the unique solution in $T^*M$. For short times this Lagrangian manifold will be transversal but at some point it can start to bend so that condition 2. fails. The projection to $M$ of points where transversality fails are called caustics in the literature.
Here's the classical physics example which you'll find for example in Arnolds books (his PDE course but I think also in his mechanics book): in the particle picture, light particles all move along straight lines with the same speed $c$ in possibly different directions (but they don't interact). An initial data would be given by a surface in the room and a direction field along this surface giving the initial direction of light rays. Initially the light rays don't intersect, but after some times they might start to intersect. The solution S(q) of HJ in this example describes the time after which the wave front arrives at a point q in space. If light rays intersect this function becomes multivalued.
By the way I'd be interested in the original source of Lie, could you add that to your question?
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Dear Michael, I stated that historical attribution not for having read the original papers of Lie on transformation groups written in the seventies of ninenteenth century, but I learned it from Ch.5 §2.2 "The Geometry of Differential Equations" in "Geometry I, EMS 28" of Alekseevskij, Vinogradov, Lychagin. – Giuseppe Tortorella Jul 9 '11 at 13:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9227256178855896, "perplexity": 386.79104889043845}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1409535917663.12/warc/CC-MAIN-20140901014517-00388-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://www.gradesaver.com/textbooks/science/chemistry/chemistry-molecular-science-5th-edition/chapter-2-chemical-compounds-questions-for-review-and-thought-topical-questions-page-90b/36a | ## Chemistry: The Molecular Science (5th Edition)
Incorrect, $AlCl_3$
We know that Aluminum has a charge of 3+, while chlorine has a charge of 1-. Thus, this ion would not have a neutral charge. To give it a neutral charge, we get: $AlCl_3$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7340227961540222, "perplexity": 1950.9342677062425}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1563195524502.23/warc/CC-MAIN-20190716035206-20190716061206-00296.warc.gz"} |
http://mathhelpforum.com/math-software/138724-differential-equations-maple.html | # Math Help - differential equations in maple
1. ## differential equations in maple
Problem Statement: In a catalytic reaction that follows a mechanism known as the Michaelis-Menten kinetics, the reactant (S) combines reversibly with a catalyst (E) to form a complex (ES) with a forward and reverse rate constant of k1 and ki, respectively. The complex then dissociates into product (P) with a reaction rate constant of k2 and the catalyst is regenerated.
k1 k2
E + S <---> ES --> E + P
ki
where
E = enzyme (i.e., catalyst)
S = substrate (i.e., reactant)
ES = enzyme-substrate complex')
P = product
k1, ki, k2 = elementary reaction rate constants
The following ODEs describe how the concentration of each of the four species (E, S, ES, P) changes with time.
dE/dt = -k1*E*S + ki*ES + k2*ES
dS/dt = -k1*E*S + ki*ES
dES/dt = k1*E*S - ki*ES - k2*ES
dP/dt = k2*ES
Integrate the above set of equations with the following rate constants and initial conditions:
k1 = 1
ki = 1
k2 = 10
E(0) = E0 = 0.1
S(0) = S0 = 1.
ES(0) = ES0 = 0.
P(0) = P0 = 0.
1) How do I solve those four in maple?? (tried searching in help, can't write equations correctly keep getting errors even with implicitdiff every example I try works fine.
2) I need to solve and find S(t)
3) How to express S(t) in matchematical rows in maple?? | {"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": 0.8539361953735352, "perplexity": 6131.086803725334}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1414637898226.7/warc/CC-MAIN-20141030025818-00068-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://grcs.uwseminars.com/pub/algebra.html | # Algebra
This is an online resource for instructors and students. While the material is designed to be taught to strong middle school students, these notes are written for instructors who are invited to guide and discuss topics with their students.
## Fundamentals
### The Integers
The first kinds of numbers discovered were the natural numbers: $0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, \dots$
There are (infinitely) many natural numbers, and there are many things that we can do with them. For example, we can add any two natural numbers to obtain another. It does not matter how big the numbers become: their sum is always still a natural number.
We may also multiply two natural numbers to obtain another. Again, it does not matter how big or small the numbers are: multiplication is something we can do with any natural numbers.
Natural numbers are great for representing “things”. In particular, they're great for representing “some number of things”. But they fall short when we want to represent less than nothing. But you can’t have less than nothing, so why would we ever want that?
The desire for a more general kind of number comes from the desire to represent change. If yesterday there were $20$ students in attendance, and today there are $24$ students in attendance, then we can say $4$ more people attended today than did yesterday. But what if tomorrow there will again be $24$ students in attendance? Then how many more students will have attended? Even worse, what if the day after tomorrow, there will be $20$ students in attendance again. Then what is the change in the number of students?
It is not that these problems cannot be solved with positive natural numbers. In fact, they certainly can. The number of students could increase by $4$, and we have a number for that. Or the number of students could decrease by $4$. We already have a number for that.
So to express a change in a quantity, we must convey two pieces of information: firstly, in what direction is the change; and secondly, by how much the quantity is changed. This seems quite complicated. It would be simpler to introduce a new kind of quantity that could represent these changes more conveniently and more compactly. This, of course, is the motivation for negative numbers.
Negative numbers do not “exist” in the real world. We cannot have a negative number of people in a class. But they provide an easy way to talk about change: if positive numbers represent increase, then negative numbers represent decrease.
The introduction of negative numbers allows us to expand our set of mathematical objects to the integers, $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$. As with natural numbers, we still have these two useful properties of integers, which we will call closure properties (think of a room withh closed door; if applying these operations inside the room of the integers, you never need to worry about anything outside the room):
• The sum of any two integers is again an integer.
• The product of any two integers is again an integer.
But in addition, we have a third closure property, which enables us to describe changes:
• The difference of any two integers is again an integer.
#### Exercise 1: Arithmetic with Integers
Evaluate each of the following expressions:
1. $1 + (-1) =$
2. $-1 - (-2) =$
3. $-3 \times 4 =$
4. $(-4) \times (-10) =$
##### Solution
1. $1 + (-1) = \boxed{0}$
2. $-1 - (-2) = \boxed{1}$
3. $-3 \times 4 = \boxed{-12}$
4. $(-4) \times (-10) = \boxed{40}$
### Variables
When we first learn arithmetic, we only have to worry about natural numbers. As we continue to develop new techniques, we find new kinds of numbers — like the integers mentioned above. Let’s take a step back and examine the philosophy behind what numbers really are.
We are familiar with everyday objects: things we can see, feel, or otherwise interact with. Mathematical objects are like those, but more abstract. In the mathematical world, you can interact with mathematical objects through statements of fact. A common example of such a statement is equality: $1 + 1 = 2$ is a statement of fact that the mathematical object referred to by $1 + 1$ is in fact the same as the mathematical object referred to by $2$.
Numbers are the most common mathematical objects, but as we will see later, they are not the only kind. Typically, we study mathematical objects because of some motivation from a real-world problem. In the real world, we may talk of quantity, location, a process, or any number of other things. To use mathematics to solve these problems, we must translate these real-world concepts into mathematical objects by using a model. Inside the model, we have many mathematical objects. Some of these objects represent the information given to us by the problem. Other objects might represent information we need to figure out — they might not be known from the start!
There are three ways we can describe a mathematical object in a simple model:
• A literal: for example, just a number, like $123$.
• A variable: a letter that we understand the meaning of, but we may or may not know the value of, like $x$. Essentially, we are naming a mathematical object. It can sometimes be useful to name known objects (for example, if they are complicated), but usually we only use variables for unknown objects.
• An expression: a combination of literals, variables, and operations, like $x - 123$.
For instance, a real world problem is as follows:
#### Exercise 2: More Lumber Is Required
Yahui and Zhen want to build a wooden treehouse. $100$ planks are required. Yahui has $33$ in her shed, and Zhen has $25$ in his shed. How many additional planks must they buy to complete the treehouse?
##### Solution
We can use whole numbers to model this problem. First, let’s give names (variables) to all mathematical objects (numbers) that the problem gives us or requests that we find. Let $y$ be the number of planks that Yahui has, $z$ be the number of planks that Zhen has, $r$ be the total number of planks required, and $x$ be the number of additional planks they must buy. The equation is: $y + z + x = r$
This equation is the governing equation of our model. We are given the values of $y$, $z$, and $r$, and we’re asked to find the value of $x$. (This is not always possible!) As mentioned above, we don’t need to use names for the known values, so we can substitute the literals: $33 + 25 + x = 100$
To find the unknown value we must add to $33 + 25 = 58$ to arrive at $100$, we can use subtraction. That is, $x = 100 - 58 = \boxed{42}$
Another way to understand what we have done in the above step is that $58 + x$ and $100$ refer to the same mathematical object, $100$. Therefore, we can subtract $58$ from both of them, and they will still be different names for the same object. That is, $58 + x - 58 = 100 - 58$. By using the properties of addition that we are familiar with, the left hand side is just another name for $x$ — and so we obtain the result seen above.
In this example, all of the variables we used represent specific mathematical objects. Three of them were immediately given to us in the question. The other, $x$, still represented a specific mathematical object, but we had to figure it out.
It is not always the case that variables represent specific mathematical objects — sometimes, we can attach a quantifier to a variable, to say that a statement is true of all mathematical objects of a certain type at once.
#### Exercise 3: Properties of Whole Number Addition
Give a concrete example for each of the following properties of addition of whole numbers:
1. For all integers $a$, $a + 0 = a$.
2. For all integers $a$ and $b$, $a + b = b + a$.
3. For all integers $a$, $b$ and $c$, $(a + b) + c = a + (b + c)$.
##### Solution
Solutions may vary.
1. Take $a = 2$. Then $2 + 0 = 2$.
2. Take $a = 3$ and $b = 7$. Then $3 + 7 = 10 = 7 + 3$.
3. Take $a = 1$, $b = 2$ and $c = 3$. Then $(1 + 2) + 3 = 3 + 3 = 6 = 1 + 5 = 1 + (2 + 3)$.
Note that these concrete examples are applications of, not justifications for, the properties in question. It can be difficult to give a formal justification for true properties that involve quantifiers. However, if a statement is false, it is often much easier: we can simply give a single concrete example which does not satisfy the statement.
#### Exercise 4: Counterexamples
Give a counterexample for each of the following incorrect statements about whole numbers:
1. For all integers $a$, $a + a > a$.
2. For all integers $a$ and $b$, $a - b = b - a$.
3. For all integers $a$, $b$ and $c$, $a + b = a + c$.
##### Solution
Solutions may vary.
1. Take $a = 0$. Then $a + a = 0 + 0 = 0 \not > 0 = a$.
2. Take $a = 1$ and $b = 0$. Then $a - b = 1 - 0 = 1 \ne -1 = 0 - 1 = b - a$.
3. Take $a = 0$, $b = 0$, and $c = 1$. Then $a + b = 0 + 0 = 0 \ne 1 = 0 + 1 = a + c$.
Note that even though each of the statements is false, they all have certain cases where they do hold. In the future, we will see techniques to find out exactly which cases the statement is true in.
## Rational Numbers
The motivation behind fractions is similar to that of expanding the whole numbers to the integers. Many things in the real world are divisible into portions. For instance, we can cut a cake into slices. Natural numbers are good at counting wholes, but bad at measuring parts of wholes.
To solve this problem, we introduce the concept of splitting a single whole into $n$ equal parts, where $n$ is a positive whole number. (Later, we will allow $n$ to be negative, but it cannot be zero — it is not possible to split something into zero equal parts.) We write this as $\frac{1}{n}$, and call each of the equal parts an $n$th part. (If $n := 10$, then they are called tenths.) We may sometimes require more than one such equal part. If we want $m$ parts where $m$ is any integer, we will write it as $\frac{m}{n}$.
All numbers that can be formed from fractions of integers are called rational numbers. We can talk about rational numbers as an extension to the integers, just like how we did with integers above, as an extension to the natural numbers. Each integer is still a rational number: for every integer $n$, and $n = \frac{n}{1}$.
There are various simple properties of fraction addition and subtraction, and multiplication, by counting the number of of $n$th parts:
1. For all integers $a$, $b$, and $c$, if $c > 0$, then $\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}$.
2. For all integers $a$, $b$, and $c$, if $c > 0$, then $\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}$.
3. For all integers $a$, $b$, and $c$, if $c > 0$, then $\frac{a}{c} \times b = \frac{ab}{c}$.
With the integers, we could not in general perform exact division on any two integers. Fractions cover the case of dividing almost any two integers (almost, since the denominator may not be $0$). But do we still have closure under addition, subtraction, and multiplication? Moreover, do we almost have closure under division (if we disallow a zero divisor)? We only saw some special cases of these operations above. We need to introduce stronger techniques — a general way to do addition, subtraction, and multiplication of fractions, to show that this closure is indeed the case.
### Fraction Multiplication
We saw above how to multiply a fraction by an integer. This has the same meaning as multiplying two integers; we can think of it as adding (maybe in the negative direction) repeated copies of the fraction. But it is harder to extend this idea toward multiplying two fractions: what do we mean when we say we want $\frac{1}{2}$ copies of $\frac{1}{3}$?
Luckily, there is in fact a single reasonable meaning for this. Recall that $\frac{m}{n}$ means to split a single whole into equal $n$th parts, then take $m$ copies of the $n$th parts. We can replace the single whole with something that is itself an arbitrary rational number: let $q \times \frac{m}{n}$ mean to split $q$ into $n$th parts, and then take $m$ of those. Then, $1 \times \frac{m}{n} = \frac{m}{n}$, as we would expect.
How do we split $\frac{a}{c}$ into $n$th parts? We can split each of the $a$ copies of the $b$th parts into $n$ parts, then only take one of every $n$. Each small part is a $b$th part split further into $n$ parts. In a single whole, there would be $c\times n$ equal such parts, so these small parts are in fact $\frac{1}{c\times n}$ each.
Therefore, we can justify the following fact (or definition, or a sort) about fraction multiplication:
#### Fact 1: Fraction Multiplication
Let $a$, $b$, $c$ and $d$ be integers with $c \ne 0 \ne d$. Then $\frac{a}{c} \times \frac{b}{d} = \frac{a \times b}{c \times d}$
### Equivalence Classes
It happens to be the case with fractions that distinct ordered pairs might represent the same quantity. For instance, $\frac{3}{6}$ and $\frac{7}{14}$ are different pairs of numbers, but they represent the same fraction: one half. All the fractions that represent the same particular quantity form a so-called equivalence class. In a sense, this is not very different from $2 - 5$ and $3 - 6$ representing the same change in quantity $-3$.
How can we decide whether two fractions represent the same quantity? That is, suppose that $\frac{a}{c}$ and $\frac{b}{d}$ are rational numbers. Are they equal? In the case where the denominator is the same, this is easy to answer: just compare the numerators. Rational numbers with the same denominator are equal if and only if the numerators are equal.
#### Fact 2: Equivalence of Fractions With Equal Denominator
Let $a$, $b$, and $c$ be integers, and $c \ne 0$. Then $\frac{a}{c} = \frac{b}{c} \iff a = b$
If the denominators are not the same, we need to rewrite the two fractions to have the same denominator. We can do this by first noticing the following fact, which we can obtain from our knowledge of fraction multiplication:
#### Fact 3: Common Factor of Numerator and Denominator
Let $a$, $b$ and $c$ be integers, and $c \ne 0 \ne b$. Then $\frac{a}{c} = \frac{a}{c} \times 1 = \frac{a}{c} \times \frac{b}{b} = \frac{a \times b}{c \times b}$
This fact allows us to rewrite $\frac{a}{c}$ as $\frac{a\times d}{c\times d}$, and $\frac{b}{d}$ as $\frac{b\times c}{d\times c}$. Now the denominators are the same (remember $c\times d = d\times c$ for any integers $c$ and $d$). So we can simply compare $a\times d$ with $b\times c$!
Visually, we are multiplying the top-left with the bottom-right, and the top-right with the bottom-left. This makes a cross shape, so one way to remember this technique is that it is often called “cross-multiplication”.
#### Fact 4: Cross-multiplication
Let $a$, $b$, $c$ and $d$ be integers, and $c \ne 0 \ne d$. Then $\frac{a}{b} = \frac{c}{d} \iff a\times d = b\times c$
We have seen above the example of rewriting fractions with a common denominator in order to compare them. But another use of fractions with a common denominator is that they are easy to add and subtract. We can apply the same technique:
#### Exercise 5: Fraction Addition and Subtraction
Rewrite the fractions using a common denominator in order to calculate:
1. $\displaystyle\frac{1}{2} + \frac{1}{3} =$
2. $\displaystyle\frac{1}{2} - \frac{1}{3} =$
3. $\displaystyle\frac{1}{3} - \frac{2}{-3} =$
##### Solution
1. $\displaystyle\frac{1}{2} + \frac{1}{3} = \frac{1\times 3}{2\times 3} + \frac{1\times 2}{3\times 2} = \frac{3}{6} + \frac{2}{6} = \boxed{\frac{5}{6}}$
2. $\displaystyle\frac{1}{2} - \frac{1}{3} = \frac{1\times 3}{2\times 3} - \frac{1\times 2}{3\times 2} = \frac{3}{6} - \frac{2}{6} = \boxed{\frac{1}{6}}$
3. $\displaystyle\frac{1}{3} - \frac{2}{-3} = \frac{1}{3} - \frac{2\times(-1)}{-3\times(-1)} = \frac{1}{3} - \frac{-2}{3} = \boxed{\frac{1}{3}}$
In general, we can derive formulas for addition and subtraction of fractions, but you should not memorize them. It is more useful to understand the process of arriving at the formulas.
#### Fact 5: Fraction Addition & Subtraction
Let $a$, $b$, $c$ and $d$ be integers with $c \ne 0 \ne d$. Then $\frac{a}{c} + \frac{b}{d} = \frac{a \times d}{c \times d} + \frac{b \times c}{d \times c} = \frac{a\times d + b\times c}{c \times d}$ and $\frac{a}{c} - \frac{b}{d} = \frac{a \times d}{c \times d} - \frac{b \times c}{d \times c} = \frac{a\times d - b\times c}{c \times d}$
Notice the resemblance to cross-multiplication. This is not accidental! From the subtraction formula, we see that if two fractions are equal, their difference is zero, and vice versa.
### Simplification
Often, given some fraction, we want to find the equivalent fraction with the smallest possible positive integer denominator. This is called the simplest form and is is useful for various reasons:
• Smaller positive integer denominators are easier for people to understand.
• Two fractions that are equal will have the same simplest form, so fractions in simplest form are easy to compare.
Reducing a fraction to simplest form is a matter of finding the largest common factor of both the numerator and denominator, and then dividing both the numerator and denominator by it.
#### Exercise 6: Fraction Operations and Simplification
Compute each of the following, then reduce it to simplest form.
1. $\displaystyle \frac{3}{8} \times \frac{2}{7} =$
2. $\displaystyle \frac{5}{9} \times \frac{2}{5} =$
##### Solution
1. $\displaystyle \frac{3}{8} \times \frac{2}{7} = \frac{6}{56} = \boxed{\frac{3}{28}}$
2. $\displaystyle \frac{5}{9} \times \frac{2}{5} = \frac{10}{45} = \boxed{\frac{2}{9}}$
#### Exercise 7: A Telescoping Product
Compute and reduce to simplest form: $\frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} \times \dots \times \frac{99}{100}$
##### Solution
Each fraction in this product, except for the last one, has a numerator which is the same as the denominator of the following fraction. These will cancel out if we multiply the fractions. For instance, $\frac{1}{2} \times \frac{2}{3} = \frac{1\times 2}{2 \times 3}$, and we can divide $2$ from both the numerator and the denominator to get $\frac{1}{3}$.
In this manner, all the numbers except for the first $1$ in the numerator and the last $100$ in the denominator will get cancelled out. So we are left with $\boxed{\frac{1}{100}}$.
## Linear Equations
### Ratios and Rates
Frequently, we may know certain quantities not in absolute terms, but only in relative terms. What does this mean? Let’s say you see two weights on the ground, labeled A and B. You might notice that B is twice as hard to lift up as A. Without a scale, it is hard for you to measure the actual weight of the objects, but you might be able to estimate the ratio of their weights.
In another example, suppose you are counting cars as they pass by on a highway. You might notice that for every $5$ personal cars you count, you see about one truck. It might be hard for you to estimate how many cars are passing by each minute, since it is hard to guess how long a minute is, but you could estimate the ratio of personal cars to trucks on this highway.
We will see a couple of word problems that involve known ratios, and try to determine the absolute quantities using additional information provided to us.
#### Exercise 8: Carcross Car Count
The community of Carcross, Yukon is quite small, with a population of only 301. Caroline counts the number of cars that passed her house over an hour and noticed that:
• There were $15$ cars that passed in total.
• All cars were either blue or silver.
• Twice as many cars were blue than silver.
How many blue cars passed by? How many silver cars?
##### Solution
Let $b$ and $s$ be integers representing the number of cars that passed her house. Then our equations are: \begin{aligned} b + s &= 15 \\ b &= 2s \\ \end{aligned}
We can substitute the second equation into the first equation, since $2s$ and $b$ refer to the same mathematical object. Thus: \begin{aligned} 2s + s &= 15 \\ 3s &= 15 \\ \textcolor{blue}{\frac{1}{3}} \times 3s &= \textcolor{blue}{\frac{1}{3}} \times 15 \\ s &= 5 \\ \end{aligned} so there were $\boxed{5}$ silver cars. Then we can substitute this back into that second equation $b = 2s$. So $b = 2 \times 5$, and so $b = 10$. Therefore there were $\boxed{10}$ blue cars.
#### Exercise 9: Produce Price Sum
A supermarket stocks four kinds of produce: apples, oranges, tomatos, and potatos. Apples cost twice as much as oranges, and oranges cost twice as much as tomatos. August bought $1\,\mathrm{kg}$ of each kind of produce, and the total price was $\20$.
Can we figure out what was the price of tomatos? If so, what was it?
##### Solution
Let $a$, $o$, $t$, and $p$ be rational numbers representing the prices of apples, oranges, tomatos, and potatos, all in dollars per kilogram. Then our equations are \begin{aligned} a &= 2o \\ o &= 2t \\ a + o + t + p &= 20 \\ \end{aligned}
If we substitute the first and second equations into the third, we get $4t + 2t + t + p = 20$ and thus $7t + p = 20$. But there is a problem: there are multiple solutions to this! There are even multiple integer solutions; for instance, maybe $t = 2$ and $p = 6$, or $t = 1$ and $p = 13$. So we do not have enough information to figure out the price of tomatos.
#### Exercise 10: An Unlikely Sprint?
Miran, Gosse, and Brayan participated in a $100\,\mathrm{m}$ sprint. Miran tells you that she won and was twice as fast as Brayan. Gosse agrees that Miran won, and says he was close behind with a time only $20\%$ higher than Miran’s. Brayan says that he came in last with a time $8\,\mathrm{s}$ longer than Gosse’s time.
You know, however, that sometimes Miran, Gosse, and Brayan aren’t the most reliable. Is it mathematically possible for all of these accounts to be accurate? If so, do we have enough information to determine what were each of their times? If so, calculate the times.
##### Solution
Let $x$ denote Miran’s time, $y$ denote Gosse’s time, and $z$ denote Brayan’s time. Based on what everyone said, the equations are: \begin{aligned} x &= \frac{1}{2} z \\ y &= 1.2 x \\ z &= y + 8\,\mathrm{s} \\ \end{aligned}
Substitute the expression for $x$ given by first equation into the second equation, to get $y = 1.2 \times \textcolor{blue}{\frac{1}{2} z} = \frac{3}{5} z$
Now substitute this into the third equation, to get \begin{aligned} z &= \textcolor{blue}{\frac{3}{5} z} + 8\,\mathrm{s} \\ \textcolor{blue}{-\frac{3}{5} z} + z &= \textcolor{blue}{-\frac{3}{5} z} + \frac{3}{5} z + 8\,\mathrm{s} \\ \frac{2}{5} z &= 8\,\mathrm{s} \\ \textcolor{blue}{\frac{5}{2}} \times \frac{2}{5} z &= \textcolor{blue}{\frac{5}{2}} \times 8\,\mathrm{s} \\ z &= 20\,\mathrm{s} \\ \end{aligned}
We can now substitute this back into $y = \frac{3}{5} z$ to get $y = 12\,\mathrm{s}$, and into $x = \frac{1}{2} z$ to get $x = 10\,\mathrm{s}$ (very fast indeed, maybe suspiciously so!).
We can check that these times match all three of the equations above, so it is mathematically possible and unique. This doesn’t mean that the statements were accurate, but they are not mathematically contradictory.
#### Exercise 11: Raccoon Population Growth
The number of raccoons in the city of Raccoonville is plotted on the following chart:
If the current trend continues, by what year will there be 180 raccoons in Raccoonville?
##### Solution
In this problem, we have to figure out the rate of increase of raccoons from the chart. The trend seems to be a straight line with an increase of $10$ raccoons every year. We can assume this trend will continue as the question asks us in that hypothetical. Let $x$ denote the number of years after $2019$. Then the number of raccoons will be $130 + 10x$. We want to solve: \begin{aligned} 130 + 10x &= 180 \\ \textcolor{blue}{-130} + 130 + 10x &= \textcolor{blue}{-130} + 180 \\ 10x &= 50 \\ \frac{10x}{\textcolor{blue}{10}} &= \frac{50}{\textcolor{blue}{10}} \\ x &= 5 \\ \end{aligned}
Since this means $5$ years after $2019$, the year that there will be $180$ raccoons in Raccoonville is $2019 + 5 = \boxed{2024}$.
### A General Approach
The examples above all have the same general form, where we have a number of equations of the form $ax = b$, where $a$, $b$, and $x$ are rational numbers, and we know $a$ and $b$ (but not $x$). Equations of this form are called “linear equations”. Why are they linear? Intuitively, one reason is that if we draw a line graph of the value of $ax$ as we increase the value of $x$, we will find a straight line:
The solution to $ax=b$, if one exists, is simply where this straight line reaches a vertical height of $b$. We saw a general technique to do this if $a\ne 0$: we can multiply both sides by $\frac{1}{a}$ (or equivalently, divide both sides by $a$). Thus the solution is $x = \frac{b}{a}$.
But what if $a = 0$? In this case, we cannot divide by $a$, since division by $0$ is meaningless. Instead, we have the flat blue line in the graph. Obviously, this line will never reach any vertical height except $0$! Therefore, there is no solution if $b \ne 0$. If $b = 0$, then we still have no information about $x$: any rational number will do. In this case, there are multiple solutions.
## Real Numbers
### Exponents
Recall that a positive exponent represents repeated multiplication, much like how a positive multiplier represents repeated addition. We can express this rule recursively using the following identity: $x^{n+1} = xx^n$ which says that if you increase the exponent by $1$ it is the same as multiply one more copy of the base.
#### Exercise 12: Positive Integer Exponents
Evaluate each expression. Write your answer as an integer in literal form.
1. $2^4=$
2. $3^2=$
3. $10^6=$
##### Solution
1. $2^4=2\times 2\times 2\times 2=\boxed{16}$
2. $3^2=3\times 3=\boxed{9}$
3. $10^6=10\times 10 \times 10\times 10\times 10\times 10=\boxed{1000000}$
There are a variety of facts about positive integer exponents that we can justify using the properties of multiplication. Here are a few. You do not need to memorize these, but it is helpful to understand why they are true.
#### Fact 6: Sum of Exponents
If $a$ is a rational number, and $n$ and $m$ are positive integers, then $a^m \times a^n = a^{m+n}$. That is: $a^m \times a^n = \underbrace{a\times\dots\times a}_{m\text{ times}} \times \underbrace{a\times\dots\times a}_{n\text{ times}} = \underbrace{a\times\dots\times a}_{m+n\text{ times}} = a^{m+n}$
(Notice how when $n=1$, this is just the recursive rule we discussed above.)
#### Fact 7: Product of Exponents
If $a$ is a rational number, and $n$ and $m$ are positive integers, then ${(a^m)}^n = a^{mn}$. That is: ${(a^m)}^n = \underbrace{\left(\underbrace{a\times\dots\times a}_{m\text{ times}}\right) \times \dots \times \left(\underbrace{a\times\dots\times a}_{m\text{ times}}\right)}_{n\text{ times}} = \underbrace{a\times\dots\times a}_{mn\text{ times}} = a^{mn}$
It is frequently useful to extend the system of exponents to non-positive numbers, which can be done by applying the recursive rule in the other direction. Thus we can derive that $x^0 = 1$ and that $x^{-1} = \frac{1}{x}$ for all non-zero values of $x$. We can check that this extension retains the sum and product rules of exponents that we mentioned above, which is a useful feature.
#### Exercise 13: Negative and Zero Exponents
Evaluate each expression. Write your answer in simplest form as a fraction or as an integer literal.
1. ${-1}^{-1}=\boxed{-1}$
2. $4^{-2}=\boxed{\frac{1}{16}}$
3. ${999}^0=\boxed{1}$
4. ${\left(\frac{-17}{4}\right)}^0=\boxed{1}$
A useful application of exponents is in shrinking large numbers to an more humanly understandable format. Indeed, we have a poor conception of how large certain numbers are. In science, it's common to see numbers way too large to count or way too small to visualize. Scientists have developed notation using exponents to make comparing such numbers easier. In scientific notation, a number $x$ is written as $y\times 10^n$, where $y$ is a number with exactly one non-zero decimal digit before the decimal point, and $n$ is a (positive, negative, or zero) exponent.
#### Exercise 14: Scientific Notation
Express in scientific notation.
1. $1234 = \boxed{1.234\times 10^3}$
2. $0.000987 = \boxed{9.87\times 10^{-4}}$
A natural question to ask after having defined negative exponents is: what about rational exponents? Could those be useful? In fact, for a positive rational number base, we may sometimes define rational exponents in a way that preserves both the sum and product laws of exponents mentioned above. The only way to do this is to ensure that ${\left(x^{\frac{1}{n}}\right)}^n = x$, that is, $x^{\frac{1}{n}}$ must be the $n$th root of $x$. We can also write that as $\sqrt[n]{x}$. With this definition and the product law, we can define $x^q$ for any positive rational base $x$ and any rational exponent $q$.
#### Exercise 15: Fractions, Exponents & Radicals
Evaluate each expression. Write your answer in simplest form as a fraction, or as an integer using the place value system.
1. $4^{\frac{1}{2}}=\boxed{2}$
2. $9^{\frac{3}{2}}=\boxed{27}$
3. ${\left(\frac{2}{3}\right)}^3=\boxed{\frac{8}{27}}$
4. $\sqrt{\frac{16}{25}}=\boxed{\frac{4}{5}}$
5. $\sqrt[4]{\frac{256}{81}}=\boxed{\frac{4}{3}}$
With integer exponents of rational numbers, we are always guaranteed that the result exists and is a rational number (since we compute these exponents by multiplying and dividing rational numbers, which are closed under these operations). As we will see later, when rational exponents are concerned, the result may not exist as a rational number.
### Definition of a Real Number
In previous sections, we were careful to only discuss square roots and other rational exponents when we knew that there was in fact a rational number that worked. In general, we cannot assume this is always the case.
We can give a proof that no rational number is equal to $\sqrt{3}$. One way to see this is a so-called proof by contradiction. In this kind of argument, we assume that $\sqrt{3}$ is in fact rational. That is, if $\sqrt{3} = \frac{p}{q}$ for some integers $p$ and $q\ne 0$ in simplest form, then $\frac{p^2}{q^2} = 3$, so $p^2 = 3q^2$. We see that $3$ is a factor of the right hand side, so it must also be a factor of the left hand side. But the left hand side is a square, so $p=3m$ for some integer $n$. Then $9k^2 = 3q^2$ so $3m^2 = q^2$. Now $3$ is a factor of the left hand side, so it should also be a factor of the right hand side. But the right hand side is a square, so $q=3n$ for some integer $n$. But then $\frac{p}{q} = \frac{3m}{3n}$ is clearly not in simplest form, so we have reached an absurd state — a contradiction. But of course, this is not possible, so something has gone wrong. Our argument is correct, so it must be our assumption that was wrong. The assumption we made was that $\sqrt{3}$ is a rational number.
Many of you will probably be uneasy with how we are talking about $\sqrt{3}$ as if it must exist, when we have already shown that no rational number squares to $3$. We have already complicated things by introducing fractions to the easier world of the integers! If we make the claim that $\sqrt{3}$ should exist as a number, then we run the risk of making things even more complicated and difficult. We don’t, in general, say that everything which doesn’t exist must be a new kind of number (we are happy to say that $\frac{1}{0}$ simply does not exist).
There is in fact a good reason, however, to suggest that $\sqrt{3}$ might be a useful number to have. The reason for this is that we can already get very very close to a potential square root of $3$! An example of this is the rational number $\frac{3900231685776981}{2251799813685248} \approx 2.9999999999999997$. In fact, we can get arbitrarily close. A way to visualize this is to see a graph that maps numbers to their squares.
We can start by plotting a point on some axes for integer values. The horizontal distance represents the number $x$, which we vary to take on the integer values we want to show. The vertical distance represents the square of that number, $x^2$.
Of course, we can also take the square of rational numbers. We can think of this as increasing the precision of our graph by plotting more points, for example, every $0.1$.
If we imagine that we continue this process, getting more and more precision, we would expect this curve to become continuous. We can see that it reaches a vertical value of $3$ at some point, and we saw earlier that no such rational number point exists. But the curve suggests that we can define a new kind of number that is on the number line, and we can get close to using rational numbers, but can’t get exactly there. This concept is called a real number.
There are many ways to formally define a real number, but it is not necessary to understand such a definition to understand what a real number is. Intuitively, we expect real numbers to plug the holes in continuous lines that we can draw. We can get as close as we want to a real number using rational numbers. In fact we don’t even need to use all rational numbers. All finite decimals are rational numbers, and by adding more decimal points, we can get closer and closer to any rereal number we are interested in. $\sqrt{3} \approx 1.73$, but an even better approximation is $\sqrt{3} \approx 1.732$. We can keep getting more and more precise, but we can never reach the number itself because it is not rational (hence not a decimal).
### Operations on Real Numbers
We are able to add, subtract, multiply, and divide (except by zero) real numbers, just as we can with rational numbers. (Remember that we can get as close as we want to a real number using rational numbers. So it makes sense that real numbers behave almost identically to rational numbers!)
However, unlike rational numbers, it is not always possible to write real numbers in a canonical simplest form. Instead, we can use algebraic techniques to make expressions look simpler from a human perspective.
#### Exercise 16: A Linear Equation using Real Numbers
Solve the following equation for real number $x$: $\sqrt{2} x = 2$
##### Solution
We divide both sides by $\sqrt{2}$, yielding: $\frac{\sqrt{2}}{\sqrt{2}} x = \frac{2}{\sqrt{2}}$
Then, simplifying, $x = \frac{2}{\sqrt{2}} = 2\cdot \frac{1}{\sqrt{2}} = 2\cdot \frac{1}{2^{\frac{1}{2}}} = 2^2\cdot 2^{-\frac{1}{2}} = 2^{1 - \frac{1}{2}} = 2^{\frac{1}{2}} = \boxed{\sqrt{2}}$
At this stage, it is helpful to understand some supplementary material on sets. This material is on a seperate page because it is not strictly related to what we are studying right now about algebra, but the notational conveniences of the material will be useful.
## Polynomials
### Linear Equations, Again
Recall earlier when we solved linear equations of the form $ax = b$, where $a$ and $b$ are known rational (or real) numbers, and $x$ is an unknown rational (or real) number. The solution is to divide both sides of the equation by $a$, which is valid because $ax$ and $b$ are different names for the same rational (or real) number. This results in the solution $x = \frac{b}{a}$.
Note that many equations that may look different are really linear equations after some rearranging. For example, $ax + b = 0$ can be rearranged into the linear equation $ax = -b$. $ax + b = c$ can be rearranged into the linear equation $ax = c - b$. We will say that a linear equation that looks like $ax + b = 0$ is in “standard form”.
Let’s now look at a graphical method to solve a linear equation in standard form. What we will do is rewrite the right hand of the equation from $0$ to another variable $y$. We will then draw a graph, similar to what we did earlier when we introduced real numbers. Let us first consider the linear equation $2x - 6 = 0$.
Note that the equation $2x - 6 = y$ is more general than $2x - 6 = 0$. If we set $y := 0$, then we get back the original equation $2x - 6 = 0$. Therefore, to solve this equation we can look on the graphical plot for all the values on the line corresponding to $y = 0$. We see that this is where the line intersects with the x-axis. This is called an x-intercept, or root, of the function $y = 2x - 6$. From the plot, we see that the only root is $x = 3$, which corresponds to the only solution to this equation.
We will now use the knowledge about plots and roots to solve equations which are not linear. Let us start with a simple example.
#### Exercise 17: Roots of $x^2 - 1$
Plot $y = x^2 - 1$, determine its roots, and use this information to solve the equation $x^2 = 1$.
##### Solution
Here is our plot:
The roots are marked. They are $x = -1$ and $x = 1$, which correspond to the solutions to our equation $x^2 = 1$.
#### Exercise 18: Roots of $(x-3)(x+2)$
Without using a plot, determine the roots of $y = (x-3)(x+2)$. These are also solutions to the standard form equation $x^2 - x - 6 = 0$; explain why.
##### Solution
The roots occur when $y = 0$, so we want to solve $0 = (x-3)(x+2)$. If the product of two numbers is $0$, that means either the first number is $0$ or the second number is $0$ (or both). Therefore, the set of solutions to the equation $0 = (x-3)(x+2)$ is the union of the set of solutions to $x - 3 = 0$ and the set of solutions to $x + 2 = 0$. If $x - 3 = 0$, this is a simple linear equation, where $x = 3$ is the only solution. If $x + 2 = 0$, this is also a simple linear equation, where $x = -2$ is the only solution.
Therefore, the roots are $x \in \{-2, 3\}$.
Note that by using the distributive property, we find that $(x-3)(x+2) = (x-3)x + (x-3) \cdot 2 = x^2 - 3x + 2x - 6 = x^2 - x - 6$. So in fact, $x^2 - x - 6 = (x-3)(x+2)$ for all values of $x$. Hence the solutions of the two equations must be the same!
Exercise 18 suggests a general approach to solving equations that involve $x$ and $x^2$ might be to decompose it into the product of two components which are both linear. This process is called factoring. An expression of the form $ax^2 + bx + c$, where $a\ne 0$, $b$, and $c$ are known, is called a quadratic polynomial, just as $ax + b$ where $a\ne 0$ and $b$ are known is called a linear polynomial.
Suppose we have $(sx - u)(tx - v) = 0$, where $s$, $t$, $u$, and $v$ are known real numbers with $s \ne 0 \ne t$. Then we know the solutions are $x \in \left\{\frac{u}{s}, \frac{v}{t}\right\}$. A quadratic polynomial written this way is easy to solve! Our goal is to take a polynomial in standard form, and convert it into this factored form.
It is easier to go backwards. From factored form, we can use distributivity to expand: $(sx - u)(tx - v) = st x^2 - (sv + tu)x + uv$. But what we want to figure out is how to turn standard form into factored form.
If we can write a polynomial in standard form to look like $stx^2 - (sv + tu)x + uv$, then we have found the solutions! The standard form is $ax^2 + bx + c$, $a \ne 0$, so we need: $a = st$, $b = - (sv + tu)$, $c = uv$. This is not easy to solve (in fact, it does not have a unique solution), so we need to make some simplifications first.
First of all, we need to to fix the fact that the solutions are not unique. We can factor, for example, $2x^2 - 2 = (2x-2)(x+1)$, but we can also factor it as $2x^2 - 2 = (x-1)(2x+2)$. The roots are of course the same, because the equations $x+1 = 0$ and $2x+2 = 0$ have the same solutions. But they do not quite look the same. In order to force the factored form to be unique, we need to enforce that the linear polynomials are monic, that is, they have no leading coefficient (multiplier for the $x$ term). Instead, we will pull out those coefficients into a single multiplier for the entire quadratic polynomial.
That is, given $(sx - u)(tx - v)$, we would like to turn this into $a(x-u')(x-v')$, where $u'$ and $v'$ are new coefficients. How do we calculate $a$, $u'$, and $v'$? Let’s focus on the two linear polynomials seperately. We know that $sx - u = s(x - \frac{u}{s})$, by distributivity. Similarly, $tx - v = t(x - \frac{v}{t})$. Therefore: $(sx - u)(tx - v) = s(x - \frac{u}{s})t(x - \frac{v}{t}) = st(x - \frac{u}{s})(x - \frac{v}{t})$
Therefore, we can set $a := st$ and $u' := \frac{u}{s}$ and $v' := \frac{v}{s}$. Note that the expansion of $a(x-u')(x-v')$ into standard form is $ax^2 - a(u' + v')x + au'v'$. This is not actually any different from what we had before, but it looks somewhat closer to what we need! We also now see why we have reused $a$ for the leading coefficient in both forms — in fact, this leading coefficient will be the same going from monic factored form to standard form.
Again, we have not really made much progress — the goal is not to go from a factored form to standard form, but actually the opposite! But it turns out the new system of equations is easier to solve. We need: $b = -a(u' + v')$, $c = au'v'$. Another way to write this is $\frac{b}{a} = -(u' + v')$, and $\frac{c}{a} = u'v'$. That is: after dividing by the leading coefficient, we want the constant coefficient to be the product of the solutions, and the $x$ coefficient to be the negative sum of the solutions. We can solve this problem with trial and error.
#### Exercise 19: Factoring a Quadratic via Trial and Error
Using trial and error, factor $25 + 5x - 6x^2$.
##### Solution
In standard form, this is $-6x^2 + 5x + 25$. We can divide by the leading coefficient to obtain $6 (x^2 - \frac{5}{6}x - \frac{25}{6})$. We want two numbers whose sum is $\frac{5}{6}$, and whose product is $-\frac{25}{6}$. One of them will need to be negative! Let’s guess that $u'$ and $v'$ will be rational numbers. Write them as $\frac{m}{p}$ and $\frac{n}{q}$. We would like $mn = -25$ and $pq = 6$. There are essentially two options for $p$ and $q$, such as $1$ and $6$, or $2$ and $3$ (all other options involve negatives or just switching around $p$ and $q$). There are then six options for $m$: $-25, -5, -1, 1, 5, 25$. The value of $n$ corresponding to each of those options would be $1, 5, 25, -25, -5, -1$. Now we just need to guess and check!
• If $p=1$, $q=6$, $m=-25$, $n=1$, then the sum is $\frac{-25}{1} + \frac{1}{6} = \frac{-149}{6} \ne \frac{5}{6}$.
• If $p=1$, $q=6$, $m=-5$, $n=5$, then the sum is $\frac{-5}{1} + \frac{5}{6} = \frac{-25}{6} \ne \frac{5}{6}$.
• If $p=1$, $q=6$, $m=-1$, $n=25$, then the sum is $\frac{-1}{1} + \frac{25}{6} = \frac{19}{6} \ne \frac{5}{6}$.
• If $p=1$, $q=6$, $m=1$, $n=-25$, then the sum is $\frac{1}{1} + \frac{-25}{6} = \frac{-19}{6} \ne \frac{5}{6}$.
• If $p=1$, $q=6$, $m=5$, $n=-5$, then the sum is $\frac{5}{1} + \frac{-5}{6} = \frac{25}{6} \ne \frac{5}{6}$.
• If $p=1$, $q=6$, $m=25$, $n=-1$, then the sum is $\frac{25}{1} + \frac{-1}{6} = \frac{149}{6} \ne \frac{5}{6}$.
• If $p=2$, $q=3$, $m=-25$, $n=1$, then the sum is $\frac{-25}{2} + \frac{1}{3} = \frac{-73}{6} \ne \frac{5}{6}$.
• If $p=2$, $q=3$, $m=-5$, $n=5$, then the sum is $\frac{-5}{2} + \frac{5}{3} = \frac{-5}{6} \ne \frac{5}{6}$.
• If $p=2$, $q=3$, $m=-1$, $n=25$, then the sum is $\frac{-1}{2} + \frac{25}{3} = \frac{47}{6} \ne \frac{5}{6}$.
• If $p=2$, $q=3$, $m=1$, $n=-25$, then the sum is $\frac{1}{2} + \frac{-25}{3} = \frac{-47}{6} \ne \frac{5}{6}$.
• If $p=2$, $q=3$, $m=5$, $n=-5$, then the sum is $\frac{5}{2} + \frac{-5}{3} = \frac{5}{6}$ — we’re finally done!
So the factored form is $-6 (x - \frac{5}{2})(x + \frac{5}{3})$.
You might have noticed that this exercise was really tedious. In fact, we will find that there is a more direct way to figure out the numbers we want. Nevertheless, this primitive method can be useful sometimes when the solutions are less difficult to find.
We will introduce an additional form, besides standard form and factored form, for a quadratic polynomial. This new form will be called vertex form.
The motivation for vertex form is that the graph of any quadratic polynomial will look like a parabola, either opening upwards or downwards based on the leading coefficient. All parabolas have either a minimum or a maximum $y$-value, which is attained at exactly one point. This point is called the vertex.
The vertex form of a quadratic polynomial is: $a(x - x_0)^2 + y_0$, where $x_0$ and $y_0$ are real numbers such that $(x_0, y_0)$ are the coordinates of the vertex.
#### Exercise 20: Vertex Form to Standard Form
Rewrite $4(x + 1)^2 - 9$ in standard form.
##### Solution
We expand the square using distributivity: \begin{aligned} 4(x+1)^2 - 9 &= 4(x+1)(x+1) - 9 \\ &= 4(x^2 + 2x + 1) - 9 \\ &= 4x^2 + 8x + 4 - 9 \\ &= 4x^2 + 8x - 5 \end{aligned} which is in standard form.
#### Exercise 21: Solving a Quadratic in Vertex Form
Solve $4(x + 1)^2 - 9 = 0$.
First, add $9$ to both sides: $4(x+1)^2 = 9$
Then, divide both sides by $2$: $(x+1)^2 = \frac{9}{4}$
We know that there are two possibilities for $x+1$, i.e. $x + 1 \in \left\{\frac{-3}{2}, \frac{3}{2}\right\}$
Therefore, the two possibilities for $x$ are $x \in \left\{\frac{-5}{2}, \frac{1}{2}\right\}$
Therefore, we can solve quadratic equations if we put them in vertex form. Based on whether the parabola opens upwards or downwards, and where the vertex is relative to the $x$-axis, the equation will either have $0$, $1$, or $2$ real solutions.
Now the question becomes: given a quadratic polynomial in standard form, can we put it in vertex form? Yes, we can! We just need to use distributivity in a clever way. We know that $(x - x_0)^2 = x^2 - 2x_0 x + {x_0}^2$. Therefore, we want to get something which looks like this by manipulating the standard form expression. Let’s start with a concrete example:
\begin{aligned} 4x^2 + 8x - 5 &= 4\left(x^2 + 2x\right) - 5 && \text{so we need }x_0 = 1 \\ &= 4\left(x^2 + 2x + \textcolor{blue}{1 - 1}\right) - 5 \\ &= 4\left(x^2 + 2x + 1\right) - 4 - 5 \\ &= 4{(x + 1)}^2 - 9 \end{aligned}
So in fact, it is possible to rewrite a standard form quadratic polynomial into vertex form, and this time we did not need any trial and error. In general:
\begin{aligned} ax^2 + bx + c &= a\left(x^2 + \frac{b}{a}x\right) + c \\ &= a\left(x^2 + 2\frac{b}{2a}x + \textcolor{blue}{\frac{b^2}{4a^2} - \frac{b^2}{4a^2}}\right) + c \\ &= a\left(x^2 + 2\frac{b}{2a}x + \frac{b^2}{4a^2}\right) - \frac{b^2}{4a} + c \\ &= a{\left(x + \frac{b}{2a}\right)}^2 - \frac{b^2}{4a} + c \\ &= a{\left(x - \frac{-b}{2a}\right)}^2 - \frac{b^2 - 4ac}{4a} \end{aligned}
We can furthermore find the roots of this quadratic polynomial in vertex form, using the method above:
\begin{aligned} ax^2 + bx + c &= 0 \\ a{\left(x - \frac{-b}{2a}\right)}^2 - \frac{b^2 - 4ac}{4a} &= 0 \\ a{\left(x - \frac{-b}{2a}\right)}^2 &= \frac{b^2 - 4ac}{4a} \\ {\left(x - \frac{-b}{2a}\right)}^2 &= \frac{b^2 - 4ac}{4a^2} \\ x - \frac{-b}{2a} &= \pm \frac{\sqrt{b^2 - 4ac}}{2a} \\ x &= \frac{-b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a} \\ x &= \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \\ \end{aligned}
This formula gives us a way to find the solutions of any quadratic polynomial in standard form. We will call it the quadratic formula.
### Polynomials
Let us first define two terms that will be useful. A monomial is a whole number power of a variable $x$ multiplied by some coefficient. For example, the following are monomials:
• $x$
• $3x^2$
• $7$
• $-\frac{x^{10}}{4}$
The following are not monomials:
• $2^x$
• $x + 2$
• $x^{-1}$
A polynomial in a single variable $x$ is an expression that involves the sum of some monomials (maybe just one). All monomials are also polynomials. For example, the following are polynomials:
• $0$
• $x + 2$
• $8 + x^2 - 4x^7$
The following are not polynomials:
• $\frac{x+1}{x-1}$
• $1 + 3^x$
A polynomial equation in a single variable $x$ is an equation that has polynomials on the left and right hand sides. For example, the following are polynomial equations:
• $7x = 3$
• $1 - x - x^2 = \frac{1}{5}$
• $x^2 = x^7$
In particular, all quadratic equations and linear equations (which we saw in the previous few weeks) are also polynomial equations.
The degree of a polynomial is the highest exponent (with a non-zero coefficient). For instance, the following polynomials correspond to degrees:
• $0x^2 + x$ → 1
• $5$ → 0
• $x^{99} - x^{199}$ → 199
We will define the degree of $0$ to be $-∞$, because there is no term with a a non-zero coefficient.
• $0$ → 1
The leading coefficient is the coefficient of the highest exponent term. The constant term is the coefficient of the $x^0$ term, i.e. the monomial which does not depend on $x$ (hence, constant).
• $0x^2 + x$ → Leading coefficient: $1$, constant term: $0$
• $5$ → Leading coefficient: $5$, coconstant term: $5$
• $x^{99} - x^{199}$ → Leading coeffcient: $-1$, coconstant term: $0$
### Factoring Polynomials
We have already seen how to solve polynomial equations of degrees $1$ and $2$ (they are linear equations and quadratic equations respectively). To solve polynomial equations with higher degrees, we would ideally want to write the equation in factored form (just like what we did with quadratic equations). There are a few tools we can use to do this.
The first tool we will need to learn is called long division. The idea behind this technique is that if we know one factor of a polynomial, we can get the other factor. This is similar to the concept of division for rational numbers that we are familiar with. In fact, the technique is exactly the same, except that we use the exponents of the variables instead of place values. The idea is to consider only the leading coefficient (coefficients corresponding to the highest degree monomial) each time. Let us do some examples of long division.
#### Exercise 22: Examples of Polynomial Long Division
Use polynomial long division to compute the following quotients.
1. $\frac{x^2 - 1}{x + 1}$
2. $\frac{x^3 - 27}{x^2 + 3x + 9}$
##### Solution
1. We write \begin{aligned} x^2 - 1 &= \textcolor{blue}{x(x + 1)} + x^2 - 1 - \textcolor{red}{(x^2 + x)} \\ &= x(x + 1) - x - 1 \\ &= x(x + 1) - \textcolor{blue}{1(x + 1)} - x - 1 - \textcolor{red}{(- x - 1)} \\ &= x(x + 1) - 1(x + 1) \\ &= (x - 1)(x + 1) \\ \end{aligned} and so $\frac{x^2 - 1}{x + 1} = x - 1$.
2. We write \begin{aligned} x^3 - 27 &= \textcolor{blue}{x(x^2 + 3x + 9)} + x^3 - 27 - \textcolor{red}{(x^3 + 3x^2 + 9x)} \\ &= x(x^2 + 3x + 9) - 3x^2 - 9x - 27 \\ &= x(x^2 + 3x + 9) - \textcolor{blue}{3(x^2 + 3x + 9)} - 3x^2 - 9x - 27 - \textcolor{red}{(- 3x^2 - 9x - 27)} \\ &= x(x^2 + 3x + 9) - 3(x^2 + 3x + 9) \\ &= (x - 3)(x^2 + 3x + 9) \\ \end{aligned} and so $\frac{x^3 - 27}{x^2 + 3x + 9} = x - 3$.
We now know what to do when we have a factor of the polynomial. How do we figure out The next tool that will be useful is called the remainder theorem. In fact, the remainder theorem is more general (it tells us more) than the version we will look at, but the version we will learn is enough for our purposes.
The remainder theorem states that if $p(x)$ is a polynomial, and $a$ is some number, then $p(a) = 0$ if and only if $(x - a)$ is a factor of $p(x)$. This means that if we can guess a root of a polynomial, then we can find at least one factor of it. Let us look at a few examples.
#### Exercise 23: Applications of the Remainder Theorem
Find a single factor of each polynomial by using the remainder theorem.
1. $x^7 + 1$
2. $x^4 + x^3 + x^2 + x$
3. $10x^{10} - 9x - 1$
##### Solution
1. ${(-1)}^7 + 1 = 0$, so by the remainder theorem, $x + 1$ is a factor.
2. $0^4 + 0^3 + 0^2 + 0 = 0$, so by the remainder theorem, $x$ is a factor.
3. $10\cdot 1^{10} - 9\cdot 1 - 1 = 0$, so by the remainder theorem, $x - 1$ is a factor.
By combining the remainder theorem with the technique of long division, we can factor some polynomials for which we can easily guess a root. But we will want yet a third tool to make guessing roots easier, if they are rational. This third tool is called the rational root theorem. It allows us to use trial and error to guess all the possible rational roots; if none of them work, then we know there are no rational roots. (Recall that we did something similar to find the rational roots of a quadratic equation.)
The rational root theorem states that if we have a polynomial $p(x)$ with integer coefficients, and the leading coefficient is $a$ while the constant term is $c$, then any rational root $q = \frac{m}{n}$ (in lowest form) must satisfy the following: $m$ is a factor of $c$, and $n$ is a factor of $a$. As a special case, if $a = 1$ (so that t | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 612, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.995243489742279, "perplexity": 223.40784437596284}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030334855.91/warc/CC-MAIN-20220926082131-20220926112131-00760.warc.gz"} |
http://www.cut-the-knot.org/Curriculum/Geometry/GeoGebra/TriangleThreeCirclesEllipse.shtml | # Ellipse Touching Sides of Triangle at Midpoints
Cõ Gẫng Lên suggested in a post at the CutTheKnotMath facebook page that the six points on the sides of a triangle equidistant from the respective midpoints lie on an ellipse. The statement below goes a step further, not requiring that all six distances be equal, but only for the points on the same side.
Assume $D,E,F$ are the midpoints of sides $BC,AC,AB$ of $\Delta ABC$. Points $G,H$ on $BC$; $I,J$ on $AC$, and $K,L$ on $AB$ are such that $DG=DH$, $EI=EJ$, $FK=FL$.
There is a unique ellipse through the six points $G,H,I,J,K,L.$ When the points coincide pairwise, the ellipse passes through and is tangent to the midpoints of the $\Delta ABC.$
(The applet below illustrates the configuration.)
Proof
Assume $D,E,F$ are the midpoints of sides $BC,AC,AB$ of $\Delta ABC$. Points $G,H$ on $BC$; $I,J$ on $AC$, and $K,L$ on $AB$ are such that $DG=DH$, $EI=EJ$, $FK=FL$.
There is a unique ellipse through the six points $G,H,I,J,K,L.$ When the points coincide pairwise, the ellipse passes through and is tangent to the midpoints of the $\Delta ABC.$
### Proof
For a proof, observe that affine transformations do not change the configuration: midpoints remain midpoints (of the sides of the triangle and of, say, segment $GH)$ and an ellipse remains ellipse. (You perform an affine transformation by moving any of the vertices of the triangle in the applet above.) It follows that we may choose a triangle to our liking, while the proof will be still valid in the general case.
So, let's set up a Cartesian coordinates system, with $B$ at the origin, $A(0,2)$ and $C(2,0).$ Then the midpoints also have simple coordinates, $D(1,0),$ $E(1,1)$, $F(0,1)$. For the other six points we have, $G,H=(1\pm h,0)$, $I,J=(1\pm i,1\mp i),$ $K,L=(0,1\pm j).$
We are going to look for a conic (if such exists) through the six points. A conic is given by a second degree equation:
$ax^{2}+2bxy+cy^{2}+2dx+2ey+f=0,$
with only $5$ coefficients independent (because the conic would not change after the equation was divided by one of non-zero coefficients.)
Let's substitute the coordinates of points $G,H,I,J,K,L$ into the equation:
$a(1\pm h)^{2}+2d(1\pm h)+f=0,$
$c(1\pm i)^{2}+2e(1\pm i)+f=0,$
$a(1\pm j)^{2}+2b(1-j^{2})+c(1\mp j)^{2}+2d(1\pm j)+2e(1\mp j)+f=0.$
Unfold the first equation into two:
$a(1+2h+h^{2})+2d(1+h)+f=0,$ and
$a(1-2h+h^{2})+2d(1-h)+f=0.$
Solving these for $a$ and $d$ gives
$\displaystyle a = -d = \frac{f}{h^{2}-1}.$
Similarly, unfolding the second pair of equations, we obtain
$\displaystyle c = -e = \frac{f}{i^{2}-1}.$
Note that $a+d=c+e=0.$
Adding and subtracting the two unfolded equations for $I$ and $J$ gives
$a(1+j^{2})+2b(1-j^{2})+c(1+j^{2})+2d+2e+f=0,$ and
$aj-cj+dj-ej=0,$
the latter of which holds automatically because, as we found, $a+d=c+e=0.$ From the former we can find the remaining coefficient $b$ in terms of $f.$ The six points $G,H,I,J,K,L$ determine a unique ellipse.
The cases we ignored where $h^{2}-1=0,$ or, $i^{2}-1=0,$ or, $j^{2}-1=0,$ correspond to the configurations where some of the points $G,H,I,J,K,L$ coincide with the vertices of the triangle. No ellipse, of cause, can pass through three collinear points.
Now, as $h\rightarrow 0$, $H$ and $G$ move toward $D$ and, at the limit, coincide with the latter. The resulting ellipse becomes tangent to $BC$ at $D$. Letting also $i\rightarrow 0$ and $j\rightarrow 0$ leads to an ellipse tangent to the sides of $\Delta ABC$ at the midpoints. Such ellipse is also unique. I do not see how its uniqueness follows from that of the ellipse through six points. But here is an independent proof.
Let's again make use of affine transformations. Let's preform a transformation (squashing or stretching) that converts one of the ellipses into a circle. In other words, let's assume without loss of generality that in $\Delta ABC$ the incircle is tangent to the sides at the midpoints $D,E,F.$ Let $S$ be the incenter. Then, on one hand, $DS=ES=FS,$ because these are radii of the incircle. Also, say, $BD=CD,$ making right triangles $BDS$ and $CDS$ equal; and the same is true for the other two pairs of right triangles. On the other hand, $BD=BF$ as two tangents from point $B$ to the same circle. It thus follows that all six right triangles are equal, making $\Delta ABC$ equilateral. The question is thus reduced to whether, besides the incircle, there is an ellipse in an equilateral triangle tangent to the midpoints of the sides. The answer is No because by the same device we would be able to convert that ellipse and the equilateral triangle it is inscribed into into a circle inscribed in another equilateral triangles. But an affine transformation (the inverse one) that maps an equilateral triangle onto an equilateral triangle also maps circles onto circles. This ellipse is known as Steiner's inellipse. Its equation in barycentric coordinates has been derived elsewhere.
Note that we chose the hard way to establish the existence of an ellipse tangent to the midpoints of a triangle. An affine transformation of an equilateral triangle onto a given one would map the incircle onto an ellipse immediately resolving the problem of existence. I believe the existence of an ellipse through the six points as discussed above is an independently curious feature.
The ellipse tangent to the midpoints has an exciting property that was designated by Dan Kalman as The Most Marvelous Theorem in Mathematics. The theorem is stated in the complex plane where points are identified with complex numbers:
The roots of the derivative of the polynomial $P(z)=(z-z_{1})(z-z_{2})(z-z_{3})$ are the foci of the ellipse tangent to the midpoints of $\Delta z_{1}z_{2}z_{3}.$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.924078106880188, "perplexity": 215.68750821861687}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202628.42/warc/CC-MAIN-20190322034516-20190322060516-00543.warc.gz"} |
http://econpapers.repec.org/article/sprsistpr/ | Statistical Inference for Stochastic Processes
1998 - 2014
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Volume 17, issue 3, 2014
On stationarity and second-order properties of bilinear random fields pp. 221-244
Abdelouahab Bibi and Karima Kimouche
Histograms for stationary linear random fields pp. 245-266
Michel Carbon
AIC type statistics for discretely observed ergodic diffusion processes pp. 267-282
Takayuki Fujii and Masayuki Uchida
Misparametrization subsets for penalized least squares model selection pp. 283-294
Xavier Guyon and Cécile Hardouin
On asymptotically distribution free tests with parametric hypothesis for ergodic diffusion processes pp. 295-319
Kleptsyna M. and Yu. Kutoyants
Volume 17, issue 2, 2014
Central limit theorems for empirical product densities of stationary point processes pp. 121-138
Lothar Heinrich and Stella Klein
On asymptotic distribution of parameter free tests for ergodic diffusion processes pp. 139-161
Yury Kutoyants
Truncated stochastic approximation with moving bounds: convergence pp. 163-179
Teo Sharia
Adaptive Bayes type estimators of ergodic diffusion processes from discrete observations pp. 181-219
Masayuki Uchida and Nakahiro Yoshida
Volume 17, issue 1, 2014
Change point testing for the drift parameters of a periodic mean reversion process pp. 1-18
Herold Dehling, Brice Franke, Thomas Kott and Reg Kulperger
Second-order continuous-time non-stationary Gaussian autoregression pp. 19-49
Lin N. and Lototsky S.
On goodness-of-fit testing for ergodic diffusion process with shift parameter pp. 51-73
Ilia Negri and Li Zhou
Parameter estimation for the stochastic SIS epidemic model pp. 75-98
Jiafeng Pan, Alison Gray, David Greenhalgh and Xuerong Mao
A least square-type procedure for parameter estimation in stochastic differential equations with additive fractional noise pp. 99-120
Andreas Neuenkirch and Samy Tindel
Volume 16, issue 3, 2013
Distributions of the maximum likelihood and minimum contrast estimators associated with the fractional Ornstein–Uhlenbeck process pp. 173-192
Katsuto Tanaka
On the asymptotic normality of frequency polygons for strongly mixing spatial processes pp. 193-206
Mohamed El Machkouri
A Cramér-von Mises test for symmetry of the error distribution in asymptotically stationary stochastic models pp. 207-236
Joseph Ngatchou-Wandji and Michel Harel
Maximum likelihood estimation for small noise multiscale diffusions pp. 237-266
Konstantinos Spiliopoulos and Alexandra Chronopoulou
Volume 16, issue 2, 2013
Asymptotic normality of recursive estimators under strong mixing conditions pp. 81-96
Aboubacar Amiri
Spectral characterization of the quadratic variation of mixed Brownian–fractional Brownian motion pp. 97-112
Ehsan Azmoodeh and Esko Valkeila
Predicting extinction or explosion in a Galton–Watson branching process pp. 113-125
Peter Guttorp and Michael Perlman
On rate-optimal nonparametric wavelet regression with long memory moving average errors pp. 127-145
Linyuan Li and Kewei Lu
Goodness-of-fit testing for fractional diffusions pp. 147-159
Mark Podolskij and Katrin Wasmuth
Local linear estimation for stochastic processes driven by $$\alpha$$ α -stable L $$\acute{\mathbf{e}}$$ e ´ vy motion pp. 161-171
Yunyan Wang and Lixin Zhang
Volume 16, issue 1, 2013
On the Cramér–von Mises test with parametric hypothesis for poisson processes pp. 1-13
Dabye A.
Improved estimation in a non-Gaussian parametric regression pp. 15-28
Evgeny Pchelintsev
On inference for fractional differential equations pp. 29-61
Alexandra Chronopoulou and Samy Tindel
Exact and approximate EM estimation of mutually exciting hawkes processes pp. 63-80
Jamie Olson and Kathleen Carley
Volume 15, issue 3, 2012
Non-parametric estimation of the diffusion coefficient from noisy data pp. 193-223
Emeline Schmisser
On large deviations in testing simple hypotheses for locally stationary Gaussian processes pp. 225-239
Inder Tecuapetla-Gómez and Michael Nussbaum
Multistage weighted least squares estimation of ARCH processes in the stable and unstable cases pp. 241-256
Abdelhakim Aknouche
Wavelet estimation in diffusions with periodicity pp. 257-284
Michael Diether
Volume 15, issue 2, 2012
Strong uniform consistency and asymptotic normality of a kernel based error density estimator in functional autoregressive models pp. 105-125
On the dependency for asymptotically independent estimates pp. 127-132
Design for estimation of the drift parameter in fractional diffusion systems pp. 133-149
Alexandre Brouste, Marina Kleptsyna and Alexandre Popier
Proving consistency of non-standard kernel estimators pp. 151-176
David Mason
A functional central limit theorem for empirical processes under a strong mixing condition pp. 177-192
Cristina Tone
Volume 15, issue 1, 2012
Confidence intervals for the Hurst parameter of a fractional Brownian motion based on finite sample size pp. 1-26
Jean-Christophe Breton and Jean-François Coeurjolly
Estimation of the instantaneous volatility pp. 27-59
Alexander Alvarez, Fabien Panloup, Monique Pontier and Nicolas Savy
Asymptotic inference of unstable periodic ARCH processes pp. 61-79
Abdelhakim Aknouche and Eid Al-Eid
On parameter estimation of threshold autoregressive models pp. 81-104
Ngai Chan and Yury Kutoyants
Volume 14, issue 3, 2011
Quasi-likelihood analysis for the stochastic differential equation with jumps pp. 189-229
Ogihara T. and Yoshida N.
A latent process model for time series of attributed random graphs pp. 231-253
Lee N. and Priebe C.
On compound Poisson processes arising in change-point type statistical models as limiting likelihood ratios pp. 255-271
Sergueï Dachian and Ilia Negri
On estimation of delay location pp. 273-305
Alexander Gushchin and Uwe Küchler
Volume 14, issue 2, 2011
Nonparametric estimation of trend for stochastic differential equations driven by fractional Brownian motion pp. 101-109
Mishra M. and Prakasa Rao B.
A branching particle approximation to a filtering micromovement model of asset price pp. 111-140
Jie Xiong and Yong Zeng
Non-uniform spacings processes pp. 141-175
Paul Deheuvels
Periodically correlated autoregressive Hilbertian processes pp. 177-188
Soltani A. and Hashemi M.
Volume 14, issue 1, 2011
Nonparametric signal detection with small type I and type II error probabilities pp. 1-19
Mikhail Ermakov
Sequential stochastic assignment under uncertainty: estimation and convergence pp. 21-46 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8283098936080933, "perplexity": 11346.630640099975}, "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-2015-11/segments/1424936462331.30/warc/CC-MAIN-20150226074102-00095-ip-10-28-5-156.ec2.internal.warc.gz"} |
http://maxspywareremover.com/standard-error/what-is-a-standard-error-of-difference.php | Home > Standard Error > What Is A Standard Error Of Difference
What Is A Standard Error Of Difference
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Why is the FBI making such a big deal out Hillary Clinton's private email server? However, different samples drawn from that same population would in general have different values of the sample mean, so there is a distribution of sampled means (with its own mean and Student approximation when σ value is unknown Further information: Student's t-distribution §Confidence intervals In many practical applications, the true value of σ is unknown. To do this, you have available to you a sample of observations $\mathbf{x} = \{x_1, \ldots, x_n \}$ along with some technique to obtain an estimate of $\theta$, $\hat{\theta}(\mathbf{x})$.
Standard Error Of Difference Calculator
As before, the problem can be solved in terms of the sampling distribution of the difference between means (girls - boys). How are they different and why do you need to measure the standard error? The notation for standard error can be any one of SE, SEM (for standard error of measurement or mean), or SE. Good estimators are consistent which means that they converge to the true parameter value.
This approximate formula is for moderate to large sample sizes; the reference gives the exact formulas for any sample size, and can be applied to heavily autocorrelated time series like Wall The sample mean will very rarely be equal to the population mean. The distribution of these 20,000 sample means indicate how far the mean of a sample may be from the true population mean. Sample Mean Difference Formula JSTOR2340569. (Equation 1) ^ James R.
My only comment was that, once you've already chosen to introduce the concept of consistency (a technical concept), there's no use in mis-characterizing it in the name of making the answer Standard Error Of Difference Definition Inferential statistics used in the analysis of this type of experiment depend on the sampling distribution of the difference between means. Using this convention, we can write the formula for the variance of the sampling distribution of the difference between means as: Since the standard error of a sampling distribution is the Notice that s x ¯ = s n {\displaystyle {\text{s}}_{\bar {x}}\ ={\frac {s}{\sqrt {n}}}} is only an estimate of the true standard error, σ x ¯ = σ n
Now let's look at an application of this formula. Standard Error Of The Difference In Sample Means Calculator Know These 9 Commonly Confused... Standard error of the mean (SEM) This section will focus on the standard error of the mean. A practical result: Decreasing the uncertainty in a mean value estimate by a factor of two requires acquiring four times as many observations in the sample.
Standard Error Of Difference Definition
School of Psychology, University of Bradford, UK Continue reading... http://link.springer.com/chapter/10.1007%2F978-94-011-7241-7_15 Standard Error of the Difference Between the Means of Two Samples The logic and computational details of this procedure are described in Chapter 9 of Concepts and Applications. Standard Error Of Difference Calculator The mean age was 23.44 years. Standard Error Of The Difference Between Means Calculator ISBN 0-521-81099-X ^ Kenney, J.
The standard error of $\hat{\theta}(\mathbf{x})$ (=estimate) is the standard deviation of $\hat{\theta}$ (=random variable). news Thus the probability that the mean of the sample from Species 1 will exceed the mean of the sample from Species 2 by 5 or more is 0.934. doi:10.2307/2340569. Because the age of the runners have a larger standard deviation (9.27 years) than does the age at first marriage (4.72 years), the standard error of the mean is larger for Standard Error Of Difference Between Two Proportions
B. The 5 cm can be thought of as a measure of the average of each individual plant height from the mean of the plant heights. The standard error estimated using the sample standard deviation is 2.56. have a peek at these guys The distribution of the mean age in all possible samples is called the sampling distribution of the mean.
B. Mean Difference Calculator The standard error (SE) is the standard deviation of the sampling distribution of a statistic,[1] most commonly of the mean. We calculate the mean of each of these samples and now have a sample (usually called a sampling distribution) of means.
Two sample variances are 80 or 120 (symmetrical).
A difference between means of 0 or higher is a difference of 10/4 = 2.5 standard deviations above the mean of -10. The graphs below show the sampling distribution of the mean for samples of size 4, 9, and 25. The SEM gets smaller as your samples get larger. Standard Error Formula If eight boys and eight girls were sampled, what is the probability that the mean height of the sample of girls would be higher than the mean height of the sample
The standard deviation of the age for the 16 runners is 10.23, which is somewhat greater than the true population standard deviation σ = 9.27 years. National Center for Health Statistics typically does not report an estimated mean if its relative standard error exceeds 30%. (NCHS also typically requires at least 30 observations – if not more In lieu of taking many samples one can estimate the standard error from a single sample. http://maxspywareremover.com/standard-error/what-does-the-standard-error-of-the-difference-tell-us.php B.
n is the size (number of observations) of the sample. But you can't predict whether the SD from a larger sample will be bigger or smaller than the SD from a small sample. (This is a simplification, not quite true. B. As you collect more data, you'll assess the SD of the population with more precision.
Correction for correlation in the sample Expected error in the mean of A for a sample of n data points with sample bias coefficient ρ. doi:10.4103/2229-3485.100662. ^ Isserlis, L. (1918). "On the value of a mean as calculated from a sample". If it is large, it means that you could have obtained a totally different estimate if you had drawn another sample. The confidence interval of 18 to 22 is a quantitative measure of the uncertainty – the possible difference between the true average effect of the drug and the estimate of 20mg/dL.
It makes them farther apart. Standard error is instead related to a measurement on a specific sample. Ecology 76(2): 628 – 639. ^ Klein, RJ. "Healthy People 2010 criteria for data suppression" (PDF). If values of the measured quantity A are not statistically independent but have been obtained from known locations in parameter space x, an unbiased estimate of the true standard error of
The concept of a sampling distribution is key to understanding the standard error. Because these 16 runners are a sample from the population of 9,732 runners, 37.25 is the sample mean, and 10.23 is the sample standard deviation, s. The mean of these 20,000 samples from the age at first marriage population is 23.44, and the standard deviation of the 20,000 sample means is 1.18. If people are interested in managing an existing finite population that will not change over time, then it is necessary to adjust for the population size; this is called an enumerative
The confidence interval is consistent with the P value. First, let's determine the sampling distribution of the difference between means. If symmetrical as variances, they will be asymmetrical as SD. The standard error of the mean (SEM) (i.e., of using the sample mean as a method of estimating the population mean) is the standard deviation of those sample means over all
Blackwell Publishing. 81 (1): 75–81. Dobson (4) Author Affiliations 3. We want to know whether the difference between sample means is a real one or whether it could be reasonably attributed to chance, i.e. If numerous samples were taken from each age group and the mean difference computed each time, the mean of these numerous differences between sample means would be 34 - 25 = | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9682376384735107, "perplexity": 419.48827788683633}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891816647.80/warc/CC-MAIN-20180225150214-20180225170214-00305.warc.gz"} |
http://mathhelpforum.com/advanced-statistics/113153-jack-jill-waiting-time-question-print.html | # Jack and Jill Waiting Time question.
• November 8th 2009, 05:34 AM
billym
Jack and Jill Waiting Time question.
Jack and Jill agree to meet at 1:30. If Jack arrives at a time uniformly distributed between 1:15 and 1:45, and if Jill independently arrives at a time uniformly distributed between 1:30 and 2,
1) What is the probability that Jack arrives first?
Straight forward enough. I figure it's 3/4.
2)What is the probability that Jack waits more that 15 minutes.
I think I'm stumped by this, maybe not. I set X to be the time after 1:15 that Jack arrives, and Y to be the time after 1:15 that Jill arrives. Am I correct in assuming that this is the way to solve it:
$\int_{15}^{45}\int_{0}^{y-15}f_{XY}(x,y)dxdy$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8111861944198608, "perplexity": 663.1323844705782}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422121934081.85/warc/CC-MAIN-20150124175214-00135-ip-10-180-212-252.ec2.internal.warc.gz"} |
http://physics.stackexchange.com/questions/52620/constant-pressure-and-temperature-mixing-of-2-different-ideal-gases-possible-w | Constant pressure and temperature mixing of 2 different ideal gases - possible work and heat?
A simple question I hope...
Initially, have two separate containers of 2 different ideal gases, 1.) N1, P, T, V1 and 2.) N2, P, T, V2.
After mixing, the pressure and temperature are still P and T, but the volume is additive. Assuming isolated system.
So I calculated the change in entropy with $\Delta\,S_i = n\,R\,\ln((V1+V2)/Vi)$. I found the change to be positive, which is to be expected.
Then when I look at the first law, $d\,U = \delta\,Q - \delta\,W$, I think that as the temperatures did not change then the internal energy of the gases did not change and $d\,U = 0$. Also, I think that there was no work done and so there is no heat transfer, $\delta\,Q$. *Is this right? * *Is the total energy change zero too? *
-
Let's assume that in your setup:
1. The combined system of gas 1 + gas 2 is thermally insulated from the environment (so that there is no heat exchange between the environment and the gases as they mix.
2. The original gas samples are separated by a partition, and to allow them to mix we simply remove the partition and allow them to freely expand into one another to fill the entire container.
In this case, one should be careful not to use the first law in differential form (which only holds for quasi-static processes). The first law is still true, however in the form $\Delta U = Q-W$ where $Q$ is the total heat transferred to the combined system, and $W$ is the total work done by the system. You are right that $Q=0$ and $W=0$ in such a situation and therefore $\Delta U = 0$ as you indicate.
Cheers!
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.917640209197998, "perplexity": 271.0482783432464}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1387345769117/warc/CC-MAIN-20131218054929-00096-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://myanalyticaldelights.blog/author/yohanceosborne/ | # An Integral for the Occasion: Happy New Year!
To celebrate the new year 2021, I’ve created an integral for the occasion which reads as follows.
2021 New Years Integral. Let $\alpha,\beta,$ and $\gamma$ be real-valued continuous functions defined in $\mathbb{R}^3$ such that
$\displaystyle 1<\gamma(x)<\frac{1+\sqrt{5}}{2},$
$\displaystyle 1<\alpha(x)<\frac{\gamma(x)}{1+\gamma(x)-\gamma(x)^2},$
and
$\displaystyle \beta(x):=\gamma(x)+\frac{1}{\alpha(x)}-\frac{1}{\gamma(x)},$
for all $x\in\mathbb{R}^3$. Furthermore, let $f,g:\mathbb{R}^3\to\mathbb{R}$ be functions given by
$\displaystyle f(x):=\pi^{-\frac{3}{2}}\exp\left(-\frac{x_1^2}{H^2}-\frac{x_2^2}{a^2p^2}-\frac{x_3^2}{p^2y^2}\right),$
and
$\displaystyle g(x):=\pi^{-\frac{3}{2}}\exp\left(-\frac{x_1^2}{4}-\frac{x_2^2}{2019!^2}-\frac{x_3^2}{2041210^2}\right),$
where $H,a,p,y>0$ are constants. If $G$ is a function on $\mathbb{R}^3$ defined by
$\displaystyle G(x):=\frac{(\alpha(x)\beta(x)\gamma(x))^2-1}{\alpha(x)\beta(x)\gamma(x)^2+\beta(x)\gamma(x)+1}\text{ }\text{ \emph{for all} }x\in\mathbb{R}^3,$
then the integral
$\displaystyle \int_{\mathbb{R}^3}G(x)\log\sqrt[\alpha(x)]{\exp\left({\frac{f(x)}{\gamma(x)}}\right)\sqrt[\beta(x)]{\exp({g(x)})\sqrt[\gamma(x)]{\exp\left({\frac{f(x)}{\gamma(x)}}\right) \sqrt[\alpha(x)]{\exp({g(x)})\sqrt[\beta(x)]{\cdots}}}}}\text{ }\mathrm{d}x$
evaluates to
$Happy+2021!$
Proof. Our first step to evaluating the integral is to simplify the nested radical appearing in the integrand. For this, we employ Theorem 6 found on my Nested Radicals page.
Theorem 6. Let $(a_n)_{n\in\mathbb{Z}_{\geq 1}}$ and $(b_n)_{n\in\mathbb{Z}_{\geq 1}}$ be periodic sequences of real numbers with periods $T_a,T_b\in \mathbb{Z}_{\geq 1}$ such that $a_n>0$, $a_n=a_{n+T_a}$ and $b_n=b_{n+T_b}$ for all $n\geq 1$. If $|b_1|,|b_2|,\cdots |b_{T_b-1}|,|b_{T_b}|>1$. Then
$\displaystyle \sqrt[b_1]{a_1\sqrt[b_2]{a_2\sqrt[b_3]{a_3\sqrt[b_4]{a_4\sqrt[b_5]{\cdots}}}}}=\prod_{n=1}^{T_a}a_n^{S(n)}$
where
$\displaystyle S(n)=\frac{(b_1\cdots b_{T_b})^{A}}{(b_1\cdots b_{T_b})^{A}-1}\sum_{k=0}^{B-1}\prod_{j=1}^{T_ak+n}b_j^{-1}$
for $n\in\mathbb{Z}_{\geq 1}$, with $A=\frac{l.c.m(T_a,T_b)}{T_b}$ and $B=\frac{l.c.m(T_a,T_b)}{T_a}$.
This result can be derived using the idea behind the proof of Theorem 2 found on my Nested Radicals page. For this it suffices to consider the pair of sequences $(\tilde{a}_n)$ and $(\tilde{b}_n)$ given by
• $\tilde{a}_n:=a_n$ for $1\leq n \leq T$, $\tilde{b}_n:=b_n$ for $1\leq n\leq T$ with $T=l.c.m(T_a,T_b)$, and
• $\tilde{a}_n=\tilde{a}_{n+T}$, and $\tilde{b}_n=\tilde{b}_{n+T}$ for $n\geq 1$.
Then one simplifies the resulting expressions to arrive at Theorem 6.
For our current problem, we let $(a_n)=(a_n(x))$ and $(b_n)=(b_n(x))$ be periodic sequences of functions defined on $\mathbb{R}^3$ where
$\displaystyle a_n=a_{n+2},\text{ }\text{ }b_{n}=b_{n+3}\text{ }\forall n\geq 1$
with
$\displaystyle a_1:=\exp\left({\frac{f(x)}{\gamma(x)}}\right),\text{ }a_2:=\exp(g(x)),$
$\displaystyle b_1:=\alpha(x),\text{ }b_2:=\beta(x),\text{ }\text{ and }\text{ }b_3:=\gamma(x).$
It is clear that, for each $x\in\mathbb{R}^3$, the sequence $(a_n(x))$ fits the assumptions of the theorem with $T_a=2$. But for $(b_n)$ we need to check that $|b_2|=|\beta(x)|>1$ for each $x\in\mathbb{R}^3$ since we already know that $|b_1|=\alpha(x)>1$ and $|b_3|=\gamma(x)>1$ for all $x\in\mathbb{R}^3$. From the definition of $\beta$ and the assumed bounds on $\alpha$ we find
$\displaystyle \beta(x)=\gamma(x)+\frac{1}{\alpha(x)}-\frac{1}{\gamma(x)}>\gamma(x)+\frac{1+\gamma(x)-\gamma(x)^2}{\gamma(x)}-\frac{1}{\gamma(x)}=1$
for all $x\in\mathbb{R}^3$. Noting that $T_a=2$ and $T_b=3$, we find in the context of the theorem that $A=2$ and $B=3$, giving
$\displaystyle \sqrt[b_1]{a_1\sqrt[b_2]{a_2\sqrt[b_3]{a_3\sqrt[b_4]{a_4\sqrt[b_5]{\cdots}}}}}=\prod_{n=1}^{2}a_n^{S(n)}$
where
$\displaystyle S(n)=\frac{(b_1b_2 b_{3})^2}{(b_1b_2 b_{3})^{2}-1}\sum_{k=0}^{2}\prod_{j=1}^{2k+n}b_j^{-1}$
for $n\in\{1,2\}$. Let’s rewrite $S(1)$ and $S(2)$ in terms of the functions $\alpha,\beta$ and $\gamma$. By periodicity of $(b_n)$ we find
$\displaystyle S(1)=\frac{(b_1b_2b_3)^2}{(b_1b_2b_3)^{2}-1}\sum_{k=0}^{2}\prod_{j=1}^{2k+1}b_j^{-1}=\frac{(b_1b_2b_3)^2}{(b_1b_2b_3)^{2}-1}\left(b_1^{-1}+b_1^{-1}b_2^{-1}b_3^{-1}+b_1^{-2}b_2^{-2}b_3^{-1}\right)$
or
$\displaystyle S(1)=\frac{\gamma(x)\left(\alpha(x)\beta(x)^2\gamma(x)+\alpha(x)\beta(x)+1\right)}{(\alpha(x)\beta(x)\gamma(x))^2-1},\text{ }\forall x\in\mathbb{R}^3.$
Similarly, we get
$\displaystyle S(2)=\frac{(b_1b_2 b_{3})^2}{(b_1b_2 b_{3})^{2}-1}\sum_{k=0}^{2}\prod_{j=1}^{2k+2}b_j^{-1}=\frac{\alpha(x)\beta(x)\gamma(x)^2+\beta(x)\gamma(x)+1}{(\alpha(x)\beta(x)\gamma(x))^2-1},$
for all $x\in\mathbb{R}^3.$
With these, we can simplify the expression for the function $l:\mathbb{R}^3\to\mathbb{R}$ given by
$\displaystyle l(x):=\log\sqrt[\alpha(x)]{\exp\left({\frac{f(x)}{\gamma(x)}}\right)\sqrt[\beta(x)]{\exp({g(x)})\sqrt[\gamma(x)]{\exp\left({\frac{f(x)}{\gamma(x)}}\right) \sqrt[\alpha(x)]{\exp({g(x)})\sqrt[\beta(x)]{\cdots}}}}}.$
From our application of Theorem 6, we get for each $x\in\mathbb{R}^3$
$\displaystyle l(x)=\log \prod_{n=1}^{2}a_n^{S(n)}=S(1)\log\left(\exp\left({\frac{f(x)}{\gamma(x)}}\right) \right)+S(2)\log( \exp({g(x)}))$
or
$\displaystyle l(x)=\frac{ (\alpha(x)\beta(x)^2\gamma(x)+\alpha(x)\beta(x)+1)f(x)+(\alpha(x)\beta(x)\gamma(x)^2+\beta(x)\gamma(x)+1)g(x)}{(\alpha(x)\beta(x)\gamma(x))^2-1}.$
Recalling the definition of the function $G$, our integrand thus takes the form
$\displaystyle G(x)l(x)=\frac{ (\alpha(x)\beta(x)^2\gamma(x)+\alpha(x)\beta(x)+1)f(x)+(\alpha(x)\beta(x)\gamma(x)^2+\beta(x)\gamma(x)+1)g(x)}{\alpha(x)\beta(x)\gamma(x)^2+\beta(x)\gamma(x)+1}$
or
$\displaystyle G(x)l(x)=\left(\frac{\alpha(x)\beta(x)^2\gamma(x)+\alpha(x)\beta(x)+1}{\alpha(x)\beta(x)\gamma(x)^2+\beta(x)\gamma(x)+1}\right)f(x)+g(x)\text{ }\forall x\in \mathbb{R}^3.$
Using the expression for $\beta$ we can simplify the coefficient of $f$ as follows: for each $x\in\mathbb{R}^3$
$\displaystyle \frac{\alpha(x)\beta(x)^2\gamma(x)+\alpha(x)\beta(x)+1}{\alpha(x)\beta(x)\gamma(x)^2+\beta(x)\gamma(x)+1}=\frac{\beta(x)^2\gamma(x)+\beta(x)+\frac{1}{\alpha(x)}}{\beta(x)\gamma(x)^2+\frac{\beta(x)\gamma(x)}{\alpha(x)}+\frac{1}{\alpha(x)}}$
$\displaystyle = \frac{\beta(x)^2\gamma(x)+\beta(x)+\beta(x)+\frac{1}{\gamma(x)}-\gamma(x)}{\beta(x)\gamma(x)^2+\beta(x)\gamma(x)\left(\beta(x)+\frac{1}{\gamma(x)}-\gamma(x)\right)+ \beta(x)+\frac{1}{\gamma(x)}-\gamma(x) }$
$\displaystyle =\frac{\beta(x)^2\gamma(x)+2\beta(x)+\frac{1}{\gamma(x)}-\gamma(x)}{\beta(x)^2\gamma(x)+ 2\beta(x)+\frac{1}{\gamma(x)}-\gamma(x) }=1.$
Let $I$ denote the integral that we are seeking to evaluate. With the above, our problem is now simplified massively: evaluate
$\displaystyle I=\int_{\mathbb{R}^3}(f(x)+g(x))\mathrm{d}x$
with
$\displaystyle f(x)=\pi^{-\frac{3}{2}}\exp\left(-\frac{x_1^2}{H^2}-\frac{x_2^2}{a^2p^2}-\frac{x_3^2}{p^2y^2}\right),$
and
$\displaystyle g(x)=\pi^{-\frac{3}{2}}\exp\left(-\frac{x_1^2}{4}-\frac{x_2^2}{2019!^2}-\frac{x_3^2}{2041210^2}\right).$
Apply Fubini’s theorem to find that
$\displaystyle \int_{\mathbb{R}^3}f(x)\mathrm{d}x=\pi^{-\frac{3}{2}}\left(\int_{\mathbb{R}}e^{-\frac{x_1^2}{H^2}}\mathrm{d}x_1\right)\left(\int_{\mathbb{R}}e^{-\frac{x_2^2}{(ap^2)}}\mathrm{d}x_2 \right)\left(\int_{\mathbb{R}}e^{-\frac{x_3^2}{(py)^2}}\mathrm{d}x_3\right),$
and
$\displaystyle \int_{\mathbb{R}^3}g(x)\mathrm{d}x=\pi^{-\frac{3}{2}}\left(\int_{\mathbb{R}}e^{-\frac{x_1^2}{4}}\mathrm{d}x_1\right)\left(\int_{\mathbb{R}}e^{-\frac{x_2^2}{2019!^2}}\mathrm{d}x_2 \right)\left(\int_{\mathbb{R}}e^{-\frac{x_3^2}{2041210^2}}\mathrm{d}x_3\right).$
The integrals in the above products are all of the form
$\displaystyle \int_{\mathbb{R}}e^{-\frac{x^2}{w^2}}\mathrm{d}x$
which evaluates to $\sqrt{\pi}w$ whenever $w>0$. Consequently,
$\displaystyle \int_{\mathbb{R}^3}f(x)\mathrm{d}x=Happy$
while
$\displaystyle \int_{\mathbb{R}^3}g(x)\mathrm{d}x=2\times 2019!\times 2041210.$
The conclusion then follows once we note that
$\displaystyle 2\times 2019!\times 2041210=2!(2021-2)!\times 2041210=2021!\frac{2041210}{\begin{pmatrix} 2021 \\ 2 \end{pmatrix}}$
and
$\displaystyle \begin{pmatrix} 2021 \\ 2 \end{pmatrix}=2041210,$
giving
$\displaystyle \int_{\mathbb{R}^3}g(x)\mathrm{d}x=2021!$
\\\\
# An Integral for the Occasion: Merry Christmas!
This post is dedicated to a proof of the following result.
Christmas Determinant Integral. Let $C,M,e,r,i,t,m,a,s,h,y>0$ be given constants such that $M^er^r\neq C^hr^is^tm^as!$ . Then, the determinant integral
$\displaystyle I:=\det\int_{0}^{\infty}\exp\begin{pmatrix}\log\left(\frac{\log^2\left(1+\frac{y}{x^2}\right)}{4\pi\log(2)}\right)-C^hr^is^tm^as!w & M^eC^hr^iw \\-s^tm^as!r^rw &\log\left(\frac{\log^2\left(1+\frac{y}{x^2}\right)}{4\pi\log(2)}\right)+M^er^rw\end{pmatrix}\mathrm{d}x$
where
$\displaystyle w:=\frac{\log\left(M^er^r\right)-\log\left(C^hr^is^tm^as!\right)}{M^er^r-C^hr^is^tm^as!},$
evaluates to
$\displaystyle I=\frac{M^er^ry}{C^hr^is^tm^as!}.$
Proof. Let $A$ be a 2×2 matrix
$\displaystyle A=\begin{pmatrix} \tilde{a} & \tilde{b} \\ \tilde{c} & \tilde{d} \end{pmatrix}$
that is diagonalisable, so that there exists an inveritable matrix $P$
$\displaystyle P=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$
and a diagonal matrix $D$
$\displaystyle D=\begin{pmatrix} g & 0 \\ 0 & f \end{pmatrix}$
satisfying $A=PDP^{-1}$ (where of course $ad\neq bc$). As such, we have
$\displaystyle A=\frac{1}{ad-bc} \begin{pmatrix} adg-bcf & abf-abg \\ cdg-cdf & adf-bcg \end{pmatrix}.$
With the above decomposition, it can be shown that the exponential of $A$ reads
$\displaystyle \exp(A)=P\begin{pmatrix}\exp(g) & 0 \\ 0 & \exp(f) \end{pmatrix} P^{-1}$
or
$\displaystyle \exp(A)=\frac{1}{\det(P)}\begin{pmatrix} ad\exp(g)-bc\exp(f) & ab\exp(f)-ab\exp(g) \\cd\exp(g)-cd\exp(f) & ad\exp(f)-bc\exp(g) \end{pmatrix}.$
Assuming that $a,b,c,d$ are constants, $\Omega\subset\mathbb{R}^n$ is given, and that $\exp(g)$ and $\exp(f)$ are integrable with respect to Lebesgue measure over $\Omega$, we have
$\displaystyle T(\exp(A))=\frac{1}{\det(P)}\begin{pmatrix} \int_{\Omega}(ad\exp(g)-bc\exp(f))\mathrm{d}x & \int_{\Omega}(ab\exp(f)-ab\exp(g))\mathrm{d}x \\ \int_{\Omega}(cd\exp(g)-cd\exp(f))\mathrm{d}x & \int_{\Omega}(ad\exp(f)-bc\exp(g))\mathrm{d}x \end{pmatrix}.$
Here, I’ve let $T(\exp(A))$ denote the integration of the matrix $\exp(A)$ entry-wise. This operation is discussed on my Determinant Integrals page.
Now, assume $a,b,c,d> 0$, and suppose $f:=\log\left(\frac{ad}{bc}\exp(g)\right)$. Then,
$\displaystyle T(\exp(A))=\frac{1}{\det(P)}\begin{pmatrix} 0 & \int_{\Omega}(ab\exp(f)-ab\exp(g))\mathrm{d}x \\ \int_{\Omega}(cd\exp(g)-cd\exp(f))\mathrm{d}x & \int_{\Omega}(ad\exp(f)-bc\exp(g))\mathrm{d}x\end{pmatrix}$
which implies
$\displaystyle \det(T(\exp(A)))=\frac{abcd}{(ad-bc)^2}\left(\int_{\Omega}(\exp(f)-\exp(g))\mathrm{d}x\right)^2$
or rather
(1)……. $\displaystyle \det(T(\exp(A)))=\frac{ad}{bc}\left(\int_{\Omega}\exp(g)\mathrm{d}x\right)^2.$
With the choice of $f$ made above, it can be shown that
$\displaystyle A=\begin{pmatrix} g-bc\frac{\log(ad)-\log(bc)}{ad-bc} & ab\frac{\log(ad)-\log(bc)}{ad-bc} \\ -cd\frac{\log(ad)-\log(bc)}{ad-bc} & g+ad\frac{\log(ad)-\log(bc)}{ad-bc} \end{pmatrix}.$
Setting $\Omega=(0,\infty)\subset\mathbb{R}$ and
$\displaystyle g:=\log\left(\frac{\log^2\left(1+\frac{y}{x^2}\right)}{4\pi\log(2)}\right), \text{ }x>0$
where $y>0$ is an arbitrary constant, it follows from (1) that
$\displaystyle \det(T(\exp(A)))=\frac{ad}{bc}\left(\int_{\Omega}\exp(g)\mathrm{d}x\right)^2=\frac{ad}{bc}\left(\int_{\Omega}\frac{\log^2\left(1+\frac{y}{x^2}\right)}{4\pi\log(2)} \mathrm{d}x\right)^2=\frac{ady}{bc}.$
Here, I have used the fact that
$\displaystyle \int_{0}^{\infty}\log^2\left(1+\frac{q}{x^2}\right)\mathrm{d}x=4\pi\sqrt{q}\log(2)$
holds for all $q>0$. This follows by using integration by parts and a trigonometric substitution to reduce the integral to the problem of evaluating $\int_0^{\pi/2}\frac{x}{\tan{x}}\mathrm{d}x$. Wolfram Alpha indicates that this latter integral is equal to $\frac{\pi}{2}\log{2}.$
We now have that $A$ takes the form
$\displaystyle A(x)=\begin{pmatrix} \log\left(\frac{\log^2\left(1+\frac{y}{x^2}\right)}{4\pi\log(2)}\right) -bc\frac{\log(ad)-\log(bc)}{ad-bc} & ab\frac{\log(ad)-\log(bc)}{ad-bc} \\ -cd\frac{\log(ad)-\log(bc)}{ad-bc} & \log\left(\frac{\log^2\left(1+\frac{y}{x^2}\right)}{4\pi\log(2)}\right)+ad\frac{\log(ad)-\log(bc)}{ad-bc} \end{pmatrix},$
for $x>0$, with
$\displaystyle \det(T(\exp(A)))=\frac{ady}{bc},\text{ }y>0.$
Setting $w$ to be the real number given by
$\displaystyle w:= \frac{\log(ad)-\log(bc)}{ad-bc},$
the proposed result follows by setting $a=M^e,$ $d=r^r$, $b=C^hr^i,$ and $c=s^tm^as!$, where $C,M,e,r,i,t,m,a,s,h>0$ are arbitrary constants such that $M^er^r\neq C^hr^is^tm^as!$. \\\\
# New Integral Corner Collection: Complex Analysis to the Rescue!
My latest addition the Integral Corner page is a collection of neat results that can be found using methods from Complex Analysis. Deriving these results by any other method isn’t particularly clear so I’ve called the collection Complex Analysis to the Rescue! Let’s visit the first few results.
The first of the collection asserts that,
1) For $a,b,c \in \mathbb{R}$ satisfying $|a|>|b|+|c|>0$, and given $n \in \mathbb{Z}_{\geq 0}$, there hold
$\displaystyle \int_0^{2\pi}\frac{\cos(nx)}{a+b\cos(x)+c\sin(x)}dx=2\pi \lambda (a,b,c,n) \cos(n\phi)$
and
$\displaystyle \int_0^{2\pi}\frac{\sin(nx)}{a+b\cos(x)+c\sin(x)}dx=2\pi \lambda (a,b,c,n) \sin(n\phi)$
where
$\displaystyle \lambda (a,b,c,n):=\frac{(\text{sgn}(a))^n\left(-|a|+\sqrt{a^2-b^2-c^2}\right)^n}{\sqrt{(a^2-b^2-c^2)(b^2+c^2)^n}}$
and
$\displaystyle \phi:=\text{arg}(b+ic)\in[-\pi,\pi).$
To derive these results one can rewrite each integral as a complex contour integral over the unit circle $|z|=1$ in $\mathbb{C}$ and employ the Residue Theorem after having checked the positions of poles of the integrand relative to the unit disc (which is where the inequality involving $a,b$ and $c$ comes in).
A similar approach can be used to tackle the second result in my collection which is a generalisation of the above that tells more about the kind of result to expect when integrating quotients of the form
$\displaystyle \frac{a_1\cos(nx)+a_2\sin(nx)+a_3}{a_4\cos(mx)+a_5\sin(mx)+a_6}$
over one period, provided $|a_6|>|a_5|+|a_4|>0$ and $n,m$ are nonnegative integers. It reads
2) For any $a,b,c,d,e,f\in\mathbb{R}$ and $m,n\in\mathbb{Z}_{\geq 0}$ such that $c>|a|+|b|>0$ and $m\geq 1$,
$\displaystyle \int_{0}^{2\pi}\frac{d\cos(nx)+e\sin(nx)+f}{a\cos(mx)+b\sin(mx)+c}\mathrm{d}x=\frac{2\pi }{m\sqrt{c^2-a^2-b^2}}\sum_{k=0}^{m-1}\left(f+\sqrt{d^2+e^2}R\cos\left(\beta-n\alpha_k\right)\right)$
where
$\displaystyle R:=\left(\frac{c-\sqrt{c^2-a^2-b^2}}{\sqrt{a^2+b^2}}\right)^{\frac{n}{m}},\text{ } \beta:=\text{arg}(d+ie)\in(-\pi,\pi],$
and, for $k\in\{0,\cdots,m-1\}$, we define
$\displaystyle \alpha_k:=\frac{\theta+(2k+1)\pi}{m} \text{ with } \theta=\text{arg}(a+bi) \in (-\pi,\pi].$
Using the first result above in 1), one can easily tackle the following third item of the collection
3) Given $|a|>|b|+|c|>0$ where $a,b,c\in\mathbb{R}$ and $m\in\mathbb{Z}_{\geq 0}$, for all $z\in\mathbb{R}$ there holds
$\displaystyle \int_0^{2\pi}\int_0^{2\pi}\frac{(\cos(mx)-\cos(my))(\cos(my)-\cos(mz))}{(a+b\cos(x)+c\sin(x))(a+b\cos(y)+c\sin(y))}\mathrm{d}x\mathrm{d}y=\frac{2\pi^2}{a^2-b^2-c^2}\left(\frac{(b^2+c^2)^m-\left(|a|+\sqrt{a^2-b^2-c^2}\right)^{2m}}{\left(|a|+\sqrt{a^2-b^2-c^2}\right)^{2m}}\right)$
Consequently, for all $n\in\mathbb{N}$ and $x_{2n+1}\in\mathbb{R}$ we have
$\displaystyle \int_0^{2\pi}\int_0^{2\pi}\cdots\int_0^{2\pi}\prod_{j=1}^{2n}\frac{\cos(mx_{j})-\cos(mx_{j+1})}{a+b\cos(x_{j})+c\sin(x_{j})}\mathrm{d}x_1\cdots\mathrm{d}x_{2n-1}\mathrm{d}x_{2n}= \frac{2^n\pi^{2n}}{(a^2-b^2-c^2)^n}\left(\frac{(b^2+c^2)^m-\left(|a|+\sqrt{a^2-b^2-c^2}\right)^{2m}}{\left(|a|+\sqrt{a^2-b^2-c^2}\right)^{2m}}\right)^n.$
Notice the first result in 3) does not depend on $z\in\mathbb{R}$, which allows us to deduce the second result by Fubini’s Theorem and induction on $n$. The same approach allows one to derive the following 4th result involving an odd number of terms in the product integrand instead.
4) Given $|a|>|b|+|c|>0$ where $a,b,c\in\mathbb{R}$ and $m,n\in\mathbb{Z}_{\geq 0}$, for all $x_{2n+2}\in\mathbb{R}$ there holds
$\displaystyle \int_0^{2\pi}\int_0^{2\pi}\cdots\int_0^{2\pi}\prod_{j=1}^{2n+1}\frac{\cos(mx_{j})-\cos(mx_{j+1})}{a+b\cos(x_{j})+c\sin(x_{j})}\mathrm{d}x_1\cdots\mathrm{d}x_{2n}\mathrm{d}x_{2n+1}= \Gamma(a,b,c,n,m)\left(\lambda(a,b,c,m)\cos(m\phi)-\lambda(a,b,c,0)\cos\left(mx_{2n+2}\right)\right)$
where
$\displaystyle \Gamma(a,b,c,n,m):=\frac{2^{n+1}\pi^{2n+1}}{(a^2-b^2-c^2)^n}\left(\frac{(b^2+c^2)^m-\left(|a|+\sqrt{a^2-b^2-c^2}\right)^{2m}}{\left(|a|+\sqrt{a^2-b^2-c^2}\right)^{2m}}\right)^n,$
$\displaystyle \lambda (a,b,c,w):=\frac{(\text{sgn}(a))^w\left(-|a|+\sqrt{a^2-b^2-c^2}\right)^w}{\sqrt{(a^2-b^2-c^2)(b^2+c^2)^w}},$
and $\phi:=\text{arg}(b+ic)\in[-\pi,\pi).$
Other problems I hope to consider involve integrands of the form
$\displaystyle \frac{f(a_1\cos(nx)+a_2\sin(nx)+a_3)}{a_4\cos(mx)+a_5\sin(mx)+a_6}$
where $f$ is an analytic function locally.
# A Picturesque Reduction of An Integral Determinant Equation 1
In this post we discuss example solutions of the following integral determinant equation that was derived on my Determinant Integrals page:
(1) …… $\displaystyle \left(\int_{\Omega}\det(A)\mathrm{d}x\right)\left(\int_{\Omega}\det(A^{-1})\mathrm{d}x\right)=1$
Let $\Omega \subset \mathbb{R}^m$ be a bounded open set. Given an invertible matrix $A\in\mathcal{M}_n(\Omega,\textbf{r})$ (see the Determinant Integral page for a description of the set $\mathcal{M}_n(\Omega,\textbf{r})$ if needed) we know that its determinant is integrable and
$\displaystyle \text{det}(A)=\frac{1}{\text{det}(A^{-1})}\text{ }\text{ a.e in }\Omega.$
Consequently, if equation (1) holds we have
$\displaystyle \left(\int_{\Omega}\text{det}(A)\text{ }\mathrm{d}x\right)\left(\int_{\Omega}\frac{1}{\text{det}(A)}\text{ }\mathrm{d}x\right)=1.$
This leads us to consider examples of bounded open sets $\Omega$ and scalar functions $f$ which satisfy
(2) …… $\displaystyle \left(\int_{\Omega}f\text{ }\mathrm{d}x\right)\left(\int_{\Omega}\frac{1}{f}\text{ }\mathrm{d}x\right)=1.$
Clearly, equation (2) is equation (1) in the case when $A$ is a $1\times 1$ matrix, so we’ve massively simplified the problem presented by equation (1). Given an integer $n\geq 2$, notice equation (1) on its own does not have a unique $n\times n$ matrix solution $A$ defined a.e in $\Omega$ if there exists a function $f$ satisfying equation (2). With a function $f$ known to satisfy equation (2), we could construct an uncountable number of matrices $A$ for which $\text{det}(A)=f$ a.e in $\Omega$. For example, consider triangular matrices $A$ whose main diagonals take the form $\text{diag}(1,\cdots,1,f,1,\cdots,1)\in\mathbb{R}^n$. We then have full freedom to determine the possibly non-degenerate off-diagonal entries of $A$, according to whether the matrix is upper or lower triangular.
We now present an example solution pair $(f,\Omega)$ to equation (2). Denote by $I[f,\Omega]$ the left-hand side of equation (2):
$\displaystyle I[f,\Omega]:= \left(\int_{\Omega}f\text{ }\mathrm{d}x\right)\left(\int_{\Omega}\frac{1}{f}\text{ }\mathrm{d}x\right).$
Let $f(x)=\cos(\alpha x)$ be defined over $\Omega=(a,b)$ with $\alpha \neq 0$ and $a, b\in\mathbb{R}$ such that $a\neq b$. I’ve allowed simply for the condition $a\neq b$ as opposed to the natural condition $a because equation (2) holds even if the limits of integration are interchanged, which allows for the case $a>b$. On the other hand, equation (2) can’t hold if $a=b$ or else we have $0=1$. Nonetheless, for suitable $a,b\in\mathbb{R}$ there holds
$\displaystyle I[f,\Omega]=\frac{1}{\alpha^2}\left(\sin(\alpha b)-\sin(\alpha a)\right)\log\left|\frac{\sec(\alpha b)+\tan(\alpha b)}{\sec(\alpha a)+\tan(\alpha a)}\right|.$
If we let
$z_{\alpha}(a,b)=\left(\sin(\alpha b)-\sin(\alpha a)\right)\log\left|\frac{\sec(\alpha b)+\tan(\alpha b)}{\sec(\alpha a)+\tan(\alpha a)}\right|$
equation (2) reads $z_{\alpha}(a,b)=\alpha^2$. Below we plot this equation implicitly in Python over $(a,b)\in\left[0,\frac{\pi}{2}\right]^2$ for $\alpha=1,2,3,4,5,$ and $9$.
In Figure 1 all curves of a given colour constitute the solution locus corresponding to one choice of $\alpha$. We describe this correspondence below:
• $\alpha=1\to$ Light Blue curves
• $\alpha=2\to$ Orange curves
• $\alpha=3 \to$ Red curves
• $\alpha=4\to$ Blue curves
• $\alpha=5\to$ Magenta curves
• $\alpha=9\to$ Green curves
In essence, for fixed $\alpha$ that generate curves in the plane of positive length, there are infinitely many choices of $\Omega$ that ensure equation (2) holds. In Figure 2 below we extend the plot to $[0,4\pi]^2$ which gives a picturesque pattern that would make for a neat wallpaper!
Between figures 1 and 2 we observe closed curves of decreasing diameter for $\alpha=1,2,3,4,$ and $9$, whereas for $\alpha=5$ we see small horizontal/vertical spikes which do not cross the red curves although they appear to be touching prior to zooming in. In Figure 2 we see a pattern repeating for any given coloured curve corresponding to a choice of $\alpha$. This is expected as $z_{\alpha}(a,b)$ is periodic with period $2\pi$ in both its arguments when $\alpha$ is an integer. It would be interesting then to see what family curves we observe when $\alpha$ is not an integer or irrational.
In this post I present my argument which proves integral no. 5 under the Radical Integrals section of The Integral Corner. The result in question reads as follows.
Radical Integral 5. Let $P$ and $Q$ real satisfy $0< P\leq Q$. Consider the function
$\displaystyle a(x):=1+e\gamma(x+1,1)$
for $x\geq 0$ where $\gamma(u,v)$ is the incomplete gamma function with integral representation
$\displaystyle \gamma(u,v)=\int_0^{v}t^{u-1}e^{-t}\mathrm{d}t.$
Then,
$\displaystyle \int_P^Q\Lambda(x)\log \sqrt[x+1]{a(x)^{x+2}\sqrt[x+2]{a(x)\sqrt[x+3]{a(x)\sqrt[x+4]{a(x)\sqrt[x+5]{\cdots}}}}}\text{ }\mathrm{d}x=\frac{1}{4e}\log\left(H(P,Q)\right)$
with
$\displaystyle \Lambda(x):=\int_0^{1}t^{x}e^{-t}\log(t)\text{ }\mathrm{d}t$
and
$\displaystyle H(P,Q):=\frac{a(Q)^{2a(Q)^2}}{a(P)^{2a(P)^2}}\exp(a(P)^2-a(Q)^2).$
Proof. If we let
$\displaystyle L(x):=\sqrt[x+1]{a(x)^{x+2}\sqrt[x+2]{a(x)\sqrt[x+3]{a(x)\sqrt[x+4]{a(x)\sqrt[x+5]{\cdots}}}}} \text{ }\text{ }(x\geq 0)$
we have for each $x>0$
$\displaystyle L(x)=a(x)^{1+\frac{1}{x+1}+\frac{1}{(x+1)(x+2)}+\frac{1}{(x+1)(x+2)(x+3)}+\cdots}.$
This can be further simplified to
$\displaystyle L(x)=a(x)^{1+e\gamma(x+1,1)}=a(x)^{a(x)}$
after establishing
$\displaystyle \sum_{k=0}^{\infty}\prod_{j=1}^{k+1}\frac{1}{x+j}=e\gamma(x+1,1)$
for $x\geq 0$. To see this, note that the “lower” incomplete gamma function $\gamma(s,z)$ is a holomorphic function with singularities at points $(s,z)$ where $z=0$ or $s$ is a non-positive integer (check out the Incomplete gamma function Wikipedia page). Moreover, it admits the representation
$\displaystyle \gamma(s,z)=z^s\Gamma(s)e^{-z}\sum_{k=0}^{\infty}\frac{z^k}{\Gamma(s+k+1)}.$
Setting $s=x+1$ and $z=1$, we find for each $x>0$
$\displaystyle \gamma(x+1,1)=e^{-1}\Gamma(x+1)\sum_{k=0}^{\infty}\frac{1}{\Gamma(x+k+2)}=e^{-1}\Gamma(x+1)\left(\frac{1}{\Gamma(x+2)}+\frac{1}{\Gamma(x+3)}+\cdots\right)$
Rearranging, we get
$\displaystyle \sum_{k=0}^{\infty}\frac{\Gamma(x+1)}{\Gamma(x+k+2)}=e\gamma(x+1,1).$
But notice that, formally,
$\prod_{j=1}^{k+1}\frac{1}{x+j}\equiv\frac{\Gamma(x+1)}{\Gamma(x+k+2)}.$
As such, we arrive at the desired identity for the infinite sum, justifying the identity $L(x)=a(x)^{a(x)}$ for $x>0$. Writing out $a$ as
$\displaystyle a(x)=1+e\int_0^{1}t^xe^{-t}\mathrm{d}t \text{ }(x\geq 0)$
we see that $a$ is differentiable over $(0,\infty)$ with derivative given by
$\displaystyle \frac{da}{dx}=e\int_0^1t^xe^{-t}\log(t)\mathrm{d}t=e\Lambda(x).$
Consequently, $a$ is monotone decreasing over $(0,\infty)$. Therefore, our proposed integral for given $0 can be evaluated as follows.
$\displaystyle \int_P^Q\Lambda(x)\log{L(x)}\mathrm{d}x=e^{-1}\int_P^Qa(x)\log(a(x))\frac{da}{dx}\mathrm{d}x=e^{-1}\int_{a(P)}^{a(Q)}v\log(v)\mathrm{d}v=e^{-1}\left[\frac{v^2}{2}\log(v)-\frac{1}{4}v^2\right]_{a(P)}^{a(Q)} =\frac{1}{4e}\left[\log\left((v^{2v^2}\exp(-v^{2})\right)\right]_{a(P)}^{a(Q)}.$
Simplification leads to the stated result:
$\displaystyle \int_P^Q\Lambda(x)\log\sqrt[x+1]{a(x)^{x+2}\sqrt[x+2]{a(x)\sqrt[x+3]{a(x)\sqrt[x+4]{a(x)\sqrt[x+5]{\cdots}}}}}\text{ }\mathrm{d}x=\frac{1}{4e}\log\left(H(P,Q)\right)$
with
$\displaystyle \Lambda(x):=\int_0^{1}t^{x}e^{-t}\log(t)\text{ }\mathrm{d}t$
and
$\displaystyle H(P,Q):=\frac{a(Q)^{2a(Q)^2}}{a(P)^{2a(P)^2}}\exp(a(P)^2-a(Q)^2).$
Featured
# Welcome!
Below are a few plots of mine for your viewing pleasure. These images depict the evolution of discrete-time dynamical systems that I devised following a coding exercise on Ikeda Maps. While you’re here, feel free to explore
• The Integral Corner: a place where I collect some of my favourite integrals, most of which I created for fun since August 2017;
• Nested Radicals: an account of my results on evaluating nested radicals through periodicity. This constitutes a short study that I undertook in July 2020;
• Determinant Integrals: my introduction to an interplay between matrix determinant and Lebesgue integration. I initiated this work in August 2020 and I look forward to developing it further;
• Sobolev Spaces: a discussion on fine properties of functions that are ubiquitous in the theory of partial differential equations. Some of my own work on PDE theory will be discussed here.
For more aesthetically pleasing plots, check out my Gallery! | {"extraction_info": {"found_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": 302, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9898619055747986, "perplexity": 301.8803473365022}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1627046149929.88/warc/CC-MAIN-20210723143921-20210723173921-00661.warc.gz"} |
https://www.investopedia.com/articles/stocks/11/5-ways-to-measure-money-managers.asp | The overall performance of your portfolio is the ultimate measure of success for your portfolio manager. However, total return cannot exclusively be used when determining whether or not your money manager is doing their job effectively.
For example, a 2% annual total portfolio return may initially seem small. However, if the market only increased by 1% during the same time interval, then the portfolio performed well compared to the universe of available securities. On the other hand, if this portfolio was exclusively focused on extremely risky micro-cap stocks, the 1% additional return over the market does not properly compensate the investor for risk exposure. To accurately measure performance, various ratios are used to determine the risk-adjusted return of an investment portfolio. We'll look at the five common ones in this article.
## Sharpe Ratio
$\frac{(\text{Expected Return}\ -\ \text{Risk Free Rate})}{\text{Portfolio Standard Deviation}}$
The Sharpe ratio, also known as the reward-to-variability ratio, is perhaps the most common portfolio management metric. The excess return of the portfolio over the risk-free rate is standardized by the standard deviation of the excess of the portfolio return. Hypothetically, investors should always be able to invest in government bonds and obtain the risk-free rate of return. The Sharpe ratio determines the expected realized return over that minimum. Within the risk-reward framework of portfolio theory, higher-risk investments should produce high returns. As a result, a high Sharpe ratio indicates superior risk-adjusted-performance. (For more, see: Understanding the Sharpe Ratio
Many of the ratios that follow are similar to the Sharpe in that a measure of return over a benchmark is standardized for the inherent risk of the portfolio, but each has a slightly different flavor that investors may find useful, depending on their situation.
## Roy's Safety-First Ratio
$\frac{(\text{Expected Return}\ -\ \text{Target Return})}{\text{Portfolio Standard Deviation}}$
Roy's safety-first ratio is similar to the Sharpe but introduces one subtle modification. Rather than comparing portfolio returns to the risk-free rate, the portfolio's performance is compared to a target return.
The investor will often specify the target return based on financial requirements to maintain a certain standard of living, or the target return can be another benchmark. In the former case, an investor may need $50,000 per year for spending purposes; the target return on a$1 million portfolio would then be 5%. In the latter scenario, the target return may be anything from the S&P 500 to annual gold performance – the investor would have to identify this target in the investment policy statement.
Roy's safety-first ratio is based on the safety-first rule, which states that a minimum portfolio return is required and that the portfolio manager must do everything they can in order to ensure this requirement is met.
## Sortino Ratio
$\frac{(\text{Expected Return}\ -\ \text{Target Return})}{\text{Downside Standard Deviation}}$
The Sortino ratio looks similar to the Roy's safety-first ratio – the difference being that, rather than standardizing the excess return over the standard deviation, only the downside volatility is used for the calculation. The previous two ratios penalize upward and downward variation; a portfolio that produced annual returns of +15%, +80%, and +10%, would be perceived as fairly risky, so the Sharpe and Roy's safety-first ratio would be adjusted downward.
The Sortino ratio, on the other hand, only includes the downside deviation. This means only the volatility that produces fluctuating returns below a specified benchmark is taken into consideration. Basically, only the left side of a normal distribution curve is considered as a risk indicator, so the volatility of excess positive returns are not penalized. That is, the portfolio manager's score isn't hurt by returning more than was expected.
## Treynor Ratio
$\frac{(\text{Expected Return}\ -\ \text{Risk Free Rate})}{\text{Portfolio Beta}}$
The Treynor ratio also calculates the additional portfolio return over the risk-free rate. However, beta is used as the risk measure to standardize performance instead of standard deviation. Thus, the Treynor ratio produces a result that reflects the number of excess returns attained by a strategy per unit of systematic risk. After Jack L. Treynor initially introduced this portfolio metric, it quickly lost some of its luster to the now more popular Sharpe ratio. However, Treynor will definitely not be forgotten. He studied under Italian economist Franco Modigliani and was one of the original researchers whose work paved the way for the capital asset pricing model.
Since the Treynor ratio bases portfolio returns on market risk, rather than portfolio-specific risk, it is usually combined with other ratios to give a more complete measure of performance.
## Information Ratio
$\frac{(\text{Portfolio Return}\ -\ \text{Benchmark Return})}{\text{Tracking Error}}$
The information ratio is slightly more complicated than the aforementioned metrics, yet it provides a greater understanding of the portfolio manager's stock-picking abilities. In contrast to passive investment management, active management requires regular trading to outperform the benchmark. While the manager may only invest in S&P 500 companies, he may attempt to take advantage of temporary security mispricing opportunities. The return above the benchmark is referred to as the active return, which serves as the numerator in the above formula.
In contrast to the Sharpe, Sortino and Roy's safety-first ratios, the information ratio uses the standard deviation of active returns as a measure of risk instead of the standard deviation of the portfolio. As the portfolio manager attempts to outperform the benchmark, they will sometimes exceed that performance and at other times fall short. The portfolio deviation from the benchmark is the risk metric used to standardize the active return.
## The Bottom Line
The above ratios essentially perform the same task: They help investors calculate the excess return per unit of risk. Differences arise when the formulas are adjusted to account for different kinds of risk and return. Beta, for example, is significantly different from tracking-error risk. It is always important to standardize returns on a risk-adjusted basis so investors understand that portfolio managers who follow a risky strategy are not more talented in any fundamental sense than low-risk managers – they are just following a different strategy.
Another important consideration regarding these metrics is that they can only be compared to one another directly. In other words, the Sortino ratio of one portfolio manager can only be compared to the Sortino ratio of another manager. The Sortino ratio of one manager cannot be compared to the information ratio of another. Fortunately, these five metrics can all be interpreted in the same manner: The higher the ratio, the greater the risk-adjusted-performance. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 5, "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": 0.728806734085083, "perplexity": 1582.5649059907091}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780060538.11/warc/CC-MAIN-20210928062408-20210928092408-00618.warc.gz"} |
https://brilliant.org/problems/logarithmic/ | # Logarithmic
Algebra Level 3
Let $$y$$ be a Real Number such that $$y = \lambda (\log (-x) + \log x)$$.
If $$\lambda = (\log (-x) + \log x)^2$$ provided that $$x \in \mathbb{R}$$.
Find the maximum value of $$y$$.
× | {"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": 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": 0.17715123295783997, "perplexity": 489.39535753196066}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988718284.75/warc/CC-MAIN-20161020183838-00195-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://edustatistics.org/nathanvan/setup/stat202/lectures/lecture9.html | # Stat 202: Lecture 9 (covers pp. 118-123)
Nathan VanHoudnos
10/13/2014
### Agenda
1. Homework comments
2. Checkpoint #10 results
3. Lecture 9 (covers pp. 118-123)
### Agenda
1. Homework comments
2. Checkpoint #10 results
3. Lecture 9 (covers pp. 118-123)
to fill in
### Checkpoint #10 Question 5
Dogs are inbred for such desirable characteristics as blue eye color; but an unfortunate by-product of such inbreeding can be the emergence of characteristics such as deafness. A 1992 study of Dalmatians (by Strain and others, as reported in The Dalmatians Dilemma) found the following:
(i) 31% of all Dalmatians have blue eyes.
(ii) 38% of all Dalmatians are deaf.
(iii) 42% of blue-eyed Dalmatians are deaf.
Based on the results of this study is “having blue eyes” independent of “being deaf”?
### Checkpoint #10 Question 5
Write this out:
(i) 31% of all Dalmatians have blue eyes.
$P(B) = .31$
(ii) 38% of all Dalmatians are deaf.
$P(D) = .38$
(iii) 42% of blue-eyed Dalmatians are deaf.
$P(D|B) \text{ or } P(B|D)$
$P(D|B) = .42$
### Checkpoint #10 Question 5
\begin{aligned} P(B) & = .31 & P(D) & = .38 \\ P(D|B) & = .42 \ \end{aligned}
Based on the results of this study is “having blue eyes” independent of “being deaf”?
• a) No, since .31 * .38 is not equal to .42.
• b) No, since .38 is not equal to .42.
• c) No, since .31 is not equal to .42.
Write out the symbols…
### Checkpoint #10 Question 5
\begin{aligned} P(B) & = .31 & P(D) & = .38 \\ P(D|B) & = .42 \ \end{aligned}
Based on the results of this study is “having blue eyes” independent of “being deaf”?
• a) No, since $$P(B)P(D)$$ is not equal to $$P(D|B)$$.
• b) No, since $$P(D)$$ is not equal to $$P(D|B)$$.
• c) No, since $$P(B)$$ is not equal to $$P(D|B)$$.
### Checkpoint #10 Question 5
\begin{aligned} P(B) & = .31 & P(D) & = .38 \\ P(D|B) & = .42 \ \end{aligned}
Based on the results of this study is “having blue eyes” independent of “being deaf”?
• a) No, since $$P(B)P(D)$$ is not equal to $$P(D|B)$$.
• b) No, since $$P(D)$$ is not equal to $$P(D|B)$$.
• c) No, since $$P(B)$$ is not equal to $$P(D|B)$$.
Independence if and only if: $P(D|B) = P(D)$ therefore b) is the correct choice.
### Agenda
1. Homework comments
2. Checkpoint #10 results
3. Lecture 9 (covers pp. 118-123)
### Probability Rules!
Let $$S$$ be the sample space, $$A$$ any event, $$A^c$$ its complement, and $$B$$ another event.
1. $$0 \le P(A) \le 1$$
2. $$P(S) = 1$$
3. $$P(A^c) = 1 - P(A)$$
4. $$P(A \text{ or } B ) = P(A) + P(B) - P(A \text{ and } B)$$
5. If and only if $$A$$ and $$B$$ are independent, then
$P(A \text{ and } B ) = P(A) \times P(B)$
### General multiplication rule
Recall the definition of conditional probability:
$P(A|B) = \frac{P(A \text{ and } B )}{P(B)}$
Therefore, for all events $$A$$ and $$B$$:
$P(A \text{ and } B )= P(A|B) \times P(B)$
### Why a generalization?
The general rule:
$P(A \text{ and } B ) = P(A|B) \times P(B)$
Recall that $$A$$ is independent of $$B$$ if and only if
$P(A|B) = P(A)$
Therefore, if $$A$$ is independent of $$B$$,
\begin{aligned} P(A \text{ and } B ) & = P(A|B) P(B) \\ & = P(A) P(B) \end{aligned}
which is rule #5.
### Probability Rules!
Let $$S$$ be the sample space, $$A$$ any event, $$A^c$$ its complement, and $$B$$ another event.
1. $$0 \le P(A) \le 1$$
2. $$P(S) = 1$$
3. $$P(A^c) = 1 - P(A)$$
4. $$P(A \text{ or } B ) = P(A) + P(B) - P(A \text{ and } B)$$
5. $$P(A \text{ and } B ) = P(A|B) P(B)$$
6. Independent: if and only if $$P(A|B) = P(A)$$.
### General multiplication rule
Note that: \begin{aligned} P(A|B) & = \frac{P(A \text{ and } B )}{P(B)} \\ P(B|A) & = \frac{P(A \text{ and } B )}{P(A)} \end{aligned}
Therefore:
$P(A \text{ and } B ) = P(A|B) P(B) = P(B|A) P(A)$
Both are correct.
### More than two events:
In later courses you will work with objects like this:
\begin{aligned} P(X, & \mu, \text{ and } \sigma) \\ & = P(X, \big\{ \mu \text{ and } \sigma \big\} ) \\ & = P(X|\big\{ \mu \text{ and } \sigma \big\}) P( \big\{ \mu \text{ and } \sigma \big\} ) \\ & = P(X| \mu, \sigma ) P(\mu|\sigma)P(\sigma) ) \end{aligned}
This chaining of the general multiplication rule is important for:
• Hierarchical Linear Models (HLM)
• All of Bayesian statistics
### An example
In a certain region, one in every thousand people (0.001) of all individuals are infected by the HIV virus that causes AIDS. Tests for presence of the virus are fairly accurate but not perfect. If someone actually has HIV, the probability of testing positive is 0.95.
Let $$H$$ denote the event of having HIV, and $$T$$ the event of testing positive.
\begin{aligned} P(H) & = ? & P(T) & = ? \\ P(H \text{ and } T) & = ? & P(H \text{ or } T) & = ? \\ P(H|T) & = ? & P(T|H) & = ? \end{aligned}
### An example
In a certain region, one in every thousand people (0.001) of all individuals are infected by the HIV virus that causes AIDS. Tests for presence of the virus are fairly accurate but not perfect. If someone actually has HIV, the probability of testing positive is 0.95.
Let $$H$$ denote the event of having HIV, and $$T$$ the event of testing positive.
\begin{aligned} P(H) & = 0.001 & P(T) & = ? \\ P(H \text{ and } T) & = ? & P(H \text{ or } T) & = ? \\ P(H|T) & = ? & P(T|H) & = 0.95 \end{aligned}
### An example
What is the probability that someone chosen at random tests has HIV and tests positive?
\begin{aligned} P(H) & = 0.001 & P(T) & = ? \\ P(H \text{ and } T) & = ? & P(H \text{ or } T) & = ? \\ P(H|T) & = ? & P(T|H) & = 0.95 \end{aligned}
We need: \begin{aligned} P(H \text{ and } T) & = P(T|H)P(H) \\ & = (0.95) (0.001) = 0.00095 \end{aligned}
implying that approximately 1/10 of 1% of people will have HIV and will test positive for it.
### A further example
A sales representative tells his friend that the probability of landing a major contract by the end of the week, resulting in a large commission, is .4. If the commission comes through, the probability that he will indulge in a weekend vacation in Bermuda is .9. Even if the commission doesn't come through, he may still go to Bermuda, but only with probability .3.
\begin{aligned} P(C) & = ? & P(V) & = ? \\ P(V|C) & = ? & P(V|\text{not } C) & = ? \\ \end{aligned}
### A further example
A sales representative tells his friend that the probability of landing a major contract by the end of the week, resulting in a large commission, is .4. If the commission comes through, the probability that he will indulge in a weekend vacation in Bermuda is .9. Even if the commission doesn't come through, he may still go to Bermuda, but only with probability .3.
\begin{aligned} P(C) & = 0.40 & P(V) & = ? \\ P(V|C) & = 0.90 & P(V|\text{not } C) & = 0.3 \\ \end{aligned}
### Probability Trees
“the probability of landing a major contract … is .4”
+----0.40
/
C
/
---<
\
not C
\
+----0.60
### Probability Trees
“If the commission comes through, the probability [of a] vacation … is .9.”
/-----0.90
V
/
+----0.40-----<
/ \
C not V
/ \-------------[
---<
\ /--------------[
not C V
\ /
+----0.60----<
\
not V
\--------------[
### Probability Trees
“If the commission comes through, the probability [of a] vacation … is .9.”
/-----0.90
V
/
+----0.40-----<
/ \
C not V
/ \-----0.10
---<
\ /--------------[
not C V
\ /
+----0.60----<
\
not V
\--------------[
### Probability Trees
“Even if the commission doesn't come through, he may still go … with probability .3.”
/-----0.90
V
/
+----0.40-----<
/ \
C not V
/ \-----0.10
---<
\ /------0.30
not C V
\ /
+----0.60----<
\
not V
\------0.70
### Read off conditional probabilities
/--P(V|C) = 0.90
V
P(C) /
+--0.40--<
/ \
C not V
/ \--P(not V|C) = 0.10
---<
\ /--P(V|not C) = 0.30
not C V
\ /
+--0.60--<
P(not C) \
not V
\--P(not V|not C) = 0.70
### A two-way probability table
| V | not V | Total
------|------|-------|------
C | | | 0.40
------|------|-------|------
not C | | | 0.60
------|------|-------|------
Total | | | 1
From the tree we have
• $$P(V|C)$$, $$P(\text{not } V|C)$$,
• $$P(V|\text{not } C)$$ and $$P(\text{not } V|\text{not } C)$$.
How do we get $$P(V \text{ and } C)$$ etc. to put in the table?
### E.g. Probability of a vacation?
There are two ways to take a vacation, with and without the commission:
$P(V) = P(V \text{ and } C) + P(V \text{ and not } C)$
By the general multiplication rule we have:
\begin{aligned} P(V \text{ and } C) & = P(V|C) P(C) \\ P(V \text{ and not } C) & = P(V|\text{not } C) P(\text{not } C) \end{aligned}
Therefore:
$P(V) = P(V|C) P(C) + P(V|\text{not } C) P(\text{not } C)$
### Probability of a vacation?
$$P(V) = P(V|C) P(C) + P(V|\text{not } C) P(\text{not } C)$$
+-0.90 # P(V|C)P(C) = 0.90 * 0.40
V = 0.36
|
+-0.40-+
| |
C not V
| +-0.10
<
| +-0.30
not C V
| |
+-0.60-+
|
not V
+-0.70
### Probability of a vacation?
$$P(V) = P(V|C) P(C) + P(V|\text{not } C) P(\text{not } C)$$
+-0.90 # P(V|C)P(C) = 0.90 * 0.40
V = 0.36
|
+-0.40-+
| |
C not V
| +-0.10
<
| +-0.30 # P(V|not C)P(not C)
not C V = 0.60 * 0.30
| | = 0.18
+-0.60-+
|
not V
+-0.70
### Probability of a vacation?
$$P(V) = P(V|C) P(C) + P(V|\text{not } C) P(\text{not } C)$$
+-0.90 # P(V|C)P(C) = 0.90 * 0.40
V = 0.36
|
+-0.40-+
| |
C not V
| +-0.10
<
| +-0.30 # P(V|not C)P(not C)
not C V = 0.60 * 0.30
| | = 0.18
+-0.60-+ ______________________
| { Therefore: }
not V { P(V) = 0.36 + 0.18 }
+-0.70 { = 0.54 }
### A two-way probability table
After multiplying out the tree:
| V | not V | Total
------|------|-------|------
C | 0.36 | | 0.40
------|------|-------|------
not C | 0.18 | | 0.60
------|------|-------|------
Total | 0.54 | | 1
### A two-way probability table
And finding the rest:
| V | not V | Total
------|------|-------|------
C | 0.36 | 0.04 | 0.40
------|------|-------|------
not C | 0.18 | 0.42 | 0.60
------|------|-------|------
Total | 0.54 | 0.46 | 1
### Summary thus far....
Two way tables (or Venn Diagrams)
• when the problem gives $$P(A \text{ and } B)$$ etc.
Probability trees
• when the problem gives $$P(A|B)$$ etc.
Can convert back-and-forth between them as needed.
### An exercise
Suppose the friend left for a week and came back to the office. When the friend returned, the salesman had left for Bermuda.
What is the probability that the salesman received a commission given that he is on vacation in Bermuda?
$P(C|V) = ?$
### From the probability table ...
$P(C|V) = ?$
| V | not V | Total
------|------|-------|------
C | 0.36 | 0.04 | 0.40
------|------|-------|------
not C | 0.18 | 0.42 | 0.60
------|------|-------|------
Total | 0.54 | 0.46 | 1
$P(C|V) = \frac{P(C \text{ and } V)}{P(V)} = \frac{.36}{.54} = .667$
implying that there is a 67% chance that he received the commission.
### Tree to table is an unsatisfying
The probability of landing a major contract by the end of the week is .4. If the commission comes through, the probability that he will vacation in Bermuda is .9. Even if the commission doesn't come through, he may still go to Bermuda, but only with probability .3.
\begin{aligned} P(C) & = 0.40 & P(V) & = ? \\ P(V|C) & = 0.90 & P(V|\text{not } C) & = 0.3 \\ \end{aligned}
What is the probability that the salesman received a commission given that he is on vacation in Bermuda? $P(C|V) = ?$
### A better way
Recall: \begin{aligned} P(A|B) & = \frac{P(A \text{ and } B )}{P(B)} \\ P(B|A) & = \frac{P(A \text{ and } B )}{P(A)} \\ P(A \text{ and } B ) & = P(A|B) P(B) = P(B|A) P(A) \end{aligned}
Therefore: $P(A|B) = \frac{P(B|A) P(A)}{P(B)}$
This is Bayes' Rule.
### Bayes' Rule and Total Probability
Bayes' Rule:
$P(A|B) = \frac{P(B|A) P(A)}{P(B)}$
• Allows you to reverse a conditional probability.
Law of Total Probability
\begin{aligned} P(B) & = P(B \text{ and } A) + P(B \text{ and not} A) \\ & = P(B|A)P(A) + P(B|\text{not }A)P(\text{not A}) \end{aligned}
• combine with Bayes' Rule to reverse a probability tree.
### Salesman Reprise
\begin{aligned} P(C) & = 0.40 & P(V) & = ? \\ P(V|C) & = 0.90 & P(V|\text{not } C) & = 0.3 \\ \end{aligned}
What is the probability that the salesman recieved a commission given that he is on vacation in Bermuda? $P(C|V) = ?$
\begin{aligned} P(C|V) & = \frac{P(V|C) P(C)}{P(V)} \\ P(V) & = P(V|C)P(C) + P(V|\text{not }C)P(\text{not C}) \end{aligned}
### Salesman Reprise
\begin{aligned} P(C) & = 0.40 & P(V) & = ? \\ P(V|C) & = 0.90 & P(V|\text{not } C) & = 0.3 \\ \end{aligned}
What is the probability that the salesman recieved a commission given that he is on vacation in Bermuda? $P(C|V) = ?$
\begin{aligned} P(C|V) & = \frac{ 0.90 * 0.40 }{P(V)} \\ P(V) & = .90 * 0.40 + 0.3 * (1 - 0.40) = 0.667 \end{aligned}
### Salesman Reprise
\begin{aligned} P(C) & = 0.40 & P(V) & = ? \\ P(V|C) & = 0.90 & P(V|\text{not } C) & = 0.3 \\ \end{aligned}
What is the probability that the salesman recieved a commission given that he is on vacation in Bermuda?
\begin{aligned} P(C|V) & = \frac{ 0.90 * 0.40 }{0.54} \\ & = 0.667 \end{aligned}
implying that there is a 67% chance that he recieved the comission.
### Summary
Recall that Stat 202 will let you solve probability problems your own way.
• brute force: (i) make a probability tree, then (ii) make a table, then (iii) find the relevant conditional probability.
• elegant: Bayes' Rule, the Law of Total Probability, and other probability rules.
### A spy example
Polygraph (lie-detector) tests are often routinely administered to employees or prospective employees in sensitive positions. Lie detector results are “better than chance, but well below perfection.” Typically, the test may conclude someone is a spy 80% of the time when he or she actually is a spy, but 16% of the time the test will conclude someone is a spy when he or she is not.
Let us assume that 1 in 1,000, or .001, of the employees in a certain highly classified workplace are actual spies.
### A spy example
Test may conclude someone is a spy 80% of the time when he or she actually is a spy, but 16% of the time the test will conclude someone is a spy when he or she is not. Assume that 1 in 1,000, or .001, are actual spies.
$P(S) = ?$
$P(S) = 0.001$
$P(D|S) \text{ or } P(S|D)?$
$P(D|S) = 0.80$
$P(D|\text{not }S) = ?$
$P(D|\text{not }S) = .16$
### A spy example
$P(S) = 0.001$
$P(D|S) = 0.80$
$P(D|\text{not }S) = .16$
If the polygraph detects a spy, are you convinced that the person is actually a spy?
$P(S|D) \text{ or } P(D|S)?$
$P(S|D) = \frac{P(D|S)P(S)}{P(D)}$
$P(D) = P(D|S)P(S) \\ + P(D|\text{not }S)P(\text{not }S)$
### A spy example
Law of total probability: \begin{aligned} P(D) & = P(D|S)P(S) + P(D|\text{not }S)P(\text{not }S) \\ & = .80 * .001 + .16 * (1 - .001) \\ & = .161 \end{aligned}
Bayes' Rule \begin{aligned} P(S|D) & = \frac{P(D|S)P(S)}{P(D)} \\ & = \frac{ 0.80 * 0.001 }{ .161} \\ & = 0.005 \end{aligned}
### A spy example
If the polygraph detects a spy, are you convinced that the person is actually a spy?
$P(S|D) = 0.005$
implying that about one half of one percent of “detections” are actual spies.
Are you convinced?
### $$P(S|D) \ne P(D|S) \text{reprise}$$
The order of conditioning matters:
$P(S|D) = 0.005$
implying that about one half of one percent of “detections” are actual spies.
$P(D|S) = 0.80$
implying that 80% of actual spies are dectected by the test.
Careful attention is required!
### Bayesian Statistics
c. 1701-1761
• Richard Price published Bayes' Rule after Bayes' death.
1749 - 1847
• Set the foundation for Bayesian Statististics | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 2, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9987303018569946, "perplexity": 3680.277738390379}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501172156.69/warc/CC-MAIN-20170219104612-00170-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://concise.org/notes/machine-learning/multivariate-linear-regression/multiple-features/ | ## Linear Regression: Multiple Features
Linear regression with multiple variables (features) is also known as "multivariate linear regression".
#### Hypothesis function for multiple features
The multivariable form of the hypothesis function:
$\hat{y} = h_\theta(x) = \theta_0 + \theta_1 x_1 + \theta_2 x_2 + ... + \theta_n x_n$
Denoting $x^{(i)}_0 = 1$ for $(i \in 1,2,...,m)$, we can rewrite above as:
$h_\theta(x) = \sum_{i=0}^n \theta_i x_i = \begin{bmatrix} \theta_0 \space \theta_1 \space ... \space \theta_n \end{bmatrix} \begin{bmatrix} x_0 \\ x_1 \\ \vdots \\ x_n \end{bmatrix} = \theta^T x$
where $\theta, x \in \mathbb{R}^{n+1}$
NOTE:
The above transformation of $h_\theta(x)$ from $\sum_{i=0}^n \theta_i x_i$ to $\theta^T x$ is an example of 'vectorization' technique which is used to speed-up computations using available optimized numerical linear algebra libraries.
Notations:
• $m$ : Number of training examples
• $n$ : Number of features
• $x^{(i)}_j$: Value of feature $j$ in the $i$th training example
• $x^{(i)}$: Input (features) of the $i$th training example; this is a vector
E.g.,
$x^{(i)} = \begin{bmatrix} x^{(i)}_0 \\ x^{(i)}_1 \\ \vdots \\ x^{(i)}_n \end{bmatrix} \in \mathbb{R}^{n+1}$
Vectorized Implementation: $h_\theta(X) = X\theta$
where
$X = \begin{bmatrix} ... \space (x^{(1)})^T \space ... \\ ... \space (x^{(2)})^T \space ... \\ \vdots \\ ... \space (x^{(m)})^T \space ... \end{bmatrix}$ $($m x (n+1) matrix$)$
$\theta = \begin{bmatrix} \theta_0 \\ \theta_1 \\ \vdots \\ \theta_n \end{bmatrix} \in \mathbb{R}^{n+1}$
Note that in this vectorized implementation, we calculate hypotheses of all $m$ training examples at once.
#### Cost function for multiple features
Recall that the cost function $J(\theta)$ is defined as:
$J(\theta) = \dfrac {1}{2m} \sum _{i=1}^m \left (h_\theta (x^{(i)}) - y^{(i)} \right)^2$
Vectorized implementation:
$J(\theta) = \dfrac {1}{2m} (X\theta - y)^T(X\theta - y)$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 25, "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": 0.7024679780006409, "perplexity": 2585.5588250018227}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107878633.8/warc/CC-MAIN-20201021205955-20201021235955-00211.warc.gz"} |
https://planetmath.org/mathitSL2F3 | $\mathit{SL}_{2}(F_{3})$
The special linear group over the finite field $\mathbbmss{F}_{3}$ is represented by $\mathit{SL}_{2}(\mathbbmss{F}_{3})$ and consists of the $2\times 2$ invertible matrices with determinant equal to $1$ and whose entries belong to $\mathbbmss{F}_{3}$.
Title $\mathit{SL}_{2}(F_{3})$ mathitSL2F3 2013-03-22 14:00:32 2013-03-22 14:00:32 drini (3) drini (3) 9 drini (3) Definition msc 20G40 SL(F3) FiniteField Field Group | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 7, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9144585132598877, "perplexity": 1918.6051790289782}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514572879.28/warc/CC-MAIN-20190916155946-20190916181946-00042.warc.gz"} |
http://fpish.net/topic/Some/0/59080 | Your code looks a bit mangled - I hope I got it right:
1
2
3
let rec split = function
| [] | [_] -> []
| x::xs -> ([x],xs) :: [for (ls,rs) in split xs -> x :: ls, rs]
By on 12/12/2009 4:24 AM () | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.1597762107849121, "perplexity": 19660.65951042068}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583739170.35/warc/CC-MAIN-20190120204649-20190120230649-00423.warc.gz"} |
http://support.river-valley.com/wiki/index.php?title=FAQ_-_elsarticle.cls | # FAQ - elsarticle.cls
## Preamble and Front matter
1. When I compile the pdf with elsarticle, I always get a blank first page which is numbered 1. The rest of the document starts at page 2. [Answer]
2. I have a problem with the document class elsarticle. In the final document, the first page is empty except for a comma, and the title. The abstract begins in the second page. With elsart.cls I did not encounter this problem. [Answer]
3. I am using elsarticle.cls and having problems with the abstract part. The abstract is missing from the PDF even if I code it using the {abstract} environment. [Answer]
4. The abstract is obtained as two successive lines only instead of a paragraph, even if it is coded in the {abstract} environment. [Answer]
5. I am using the lineno package to add line numbers to the manuscript before submission. This does not number the lines in the abstract. Is there a way around this? [Answer]
6. The title page appeared twice. How can I remove one? [Answer]
7. How can I typeset the title page and the main matter in separate pages? [Answer]
8. I have made numerous attempts to get the email addresses to appear on the title page of my article, but with no luck. [Answer]
9. How should I code a dedication? [Answer]
10. Whenever I try to add one of the options 1p, 3p, or 5p, I get the error" Package keyval Error: centering undefined.. What can be the problem [Answer]
11. I am writing on behalf of the IceCube collaboration, which has around 250 scientists and 39 institutions involved. When I put in all authors and institutions, I get the following error message when trying to compile with latex: ! LaTeX Error: Counter too large.. Could you please help? [Answer]
12. Some Elsevier journals require a Title page with only Title of the manuscript, Authors and Addresses, following with the main matter, including only Title and Abstract, beginning in a new page. I am wondering how can I do this in elsarticle class. [Answer]
## Main matter
1. I am preparing my paper using elsarticle.cls and LaTeX. Can you tell me how to compress lists of at least three consecutive numerical citations that occur together in the text? [Answer]
2. I am using the elsarticle.cls package for my paper submission. It really works very well, but I cannot use the natbib package with the sort&compress option; an option clash error appears when I use it. In such a case, the reference citation appears as [30, 40, 15, 31, 32, 4] rather than [4, 15, 30-32, 40] as does the ordinary article class. [Answer]
3. I am using elsarticle.cls, and I want to present two subfigures side by side, caption them separately and also would like to incorporate a global caption. [Answer]
4. When I use the review option, my tables become distorted due to double spacing. How can I prevent double spacing? Also, is there a way to prevent double spacing for a portion of the document? [Answer]
5. I have coded the bibliography for author–year citations. But still I obtain numbered citations. How can I format the citation in the author–year form? [Answer]
7. How can I create an acknowledgement section? [Answer]
8. When I use \Box, an error message appears — \Box not provided in base LaTeX2e'. [Answer]
## Back matter
1. I just cannot typeset the references and bibliography in harvard style (i.e., author–year format). [Answer]
2. I am using elsarticle-harv.bst' as a bibtex style file that is supposed to produce references in an author–year citation style. However, it produces the references in numerical style. Can you please tell me what went wrong? [Answer]
3. I have a problem with bibliography. The reference list does not appear. Also the ?' only appears where the references are cross-referred to. [Answer]
4. How do I get natbib to generate the references with journal abbreviations rather than full journal names. [Answer]
## General questions
1. I would like to sumbit a paper to Elsevier. Please advise me on how to proceed? [Answer]
2. I would like to know the status of my submited paper. [Answer]
3. I am preparing a paper for Chemical Engineering Science. I have downloaded the required LaTeX package. However, I am confused about which reference style I should use for this journal? [Answer]
4. I am using a template file (elsarticle-template-num). For the abstract part, the paper asks for an MSC code. Where do I find that code for my paper? [Answer]
6. How do I create the same line- and page-breaks as in the final print copy? [Answer]
7. I downloaded the elsarticle class file, put it in the appropriate directory, and when I try to compile the file using the simple template file provided by elsvier, it will not output. In fact, it gets hung up on the following line. I checked the path, and I do have the upsy.fd file installed on my computer, so I really am hung on this one. [Answer]
8. I use Latex at first time. I don't understand what I have do write from the beginning until \begin{frontmatter}. [Answer]
9. I have the following problem when compiling. What to do? [Answer]
“C:\Program Files\MiKTeX 2.8\tex\latex\psnfss\upsy.fd”))latex.exe: GUI framework cannot be initialized”
?` | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8605863451957703, "perplexity": 1408.8385662993794}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860121976.98/warc/CC-MAIN-20160428161521-00202-ip-10-239-7-51.ec2.internal.warc.gz"} |
http://arxiv.org/abs/hep-th/0312069 | hep-th
(what is this?)
(what is this?)
# Title: The M-theory 3-form and E8 gauge theory
Abstract: We give a precise formulation of the M-theory 3-form potential C in a fashion applicable to topologically nontrivial situations. In our model the 3-form is related to the Chern-Simons form of an E8 gauge field. This leads to a precise version of the Chern-Simons interaction of 11-dimensional supergravity on manifolds with and without boundary. As an application of the formalism we give a formula for the electric C-field charge, as an integral cohomology class, induced by self-interactions of the 3-form and by gravity. As further applications, we identify the M-theory Chern-Simons term as a cubic refinement of a trilinear form, we clarify the physical nature of Witten's global anomaly for 5-brane partition functions, we clarify the relation of M-theory flux quantization to K-theoretic quantization of RR charge, and we indicate how the formalism can be applied to heterotic M-theory.
Comments: 48pp. harvmac b-mode; v2: Several improvements added in response to the referee's comments. Typos fixed Subjects: High Energy Physics - Theory (hep-th) Report number: RUNHETC-2003-34 Cite as: arXiv:hep-th/0312069 (or arXiv:hep-th/0312069v2 for this version)
## Submission history
From: Gregory Moore [view email]
[v1] Fri, 5 Dec 2003 18:58:12 GMT (47kb)
[v2] Tue, 23 Mar 2004 17:12:43 GMT (50kb) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8434420228004456, "perplexity": 2452.2139924168077}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257828009.82/warc/CC-MAIN-20160723071028-00082-ip-10-185-27-174.ec2.internal.warc.gz"} |
http://mathoverflow.net/revisions/76765/list | A $2n\times 2n$ dimensional Hermitian matrix that can be diagonalized by a symplectic transformation can be viewed as an $n\times n$ matrix with elements consisting of $2\times 2$ blocks of the quaternion real form
${\bar{z}\;-\bar{w}}\choose{w\;\; z}$
so if you choose real $z$ and $w$ you have constructed a real symmetric matrix $M$ that can be diagonalized by a symplectic $S$; is S$. @Federico: this what you need?is the general form for matrices that commute,$MT=TM$, with$T=1_{N}\otimes{0\; 1}\choose{-1\; 0}K$($K$is the operator of complex conjugation); alternatively, one can take matrices that anticommute,$MT=-TM$; then the$2\times 2$blocks have the form${\bar{z}\;\bar{w}}\choose{w\;\; -z}$and again, for a real$M$one would choose real$w,z$. these two choices exhaust the possibilities. In applications to physical systems, the matrix$M$is a Hamiltonian and$T$is the operator of time reversal. Then only commuting matrices,$MT=TM$, are permitted. For a discussion in the physics context, see Section 1.4.2 of Forrester's book, online here: http://www.ms.unimelb.edu.au/~matpjf/b1.ps Post Undeleted by Carlo Beenakker 2 new attempt at an answer The condition is A$2n\times 2n$dimensional Hermitian matrix that the 2n eigenvalues can be diagonalized by a symplectic transformation can be viewed as an$n\times n$matrix with elements consisting of M should come in n twofold degenerate pairs. Then$2\times 2$blocks of the matrix S is both orthogonal quaternion real form${\bar{z}\;-\bar{w}}\choose{w\;\; z}$so if you choose real$z$and$w$you have constructed a real symmetric matrix$M$that can be diagonalized by a symplectic .$S\$; is this what you need? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.842527449131012, "perplexity": 435.73906664557677}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368706631378/warc/CC-MAIN-20130516121711-00022-ip-10-60-113-184.ec2.internal.warc.gz"} |
http://www.karlin.mff.cuni.cz/~pesta/NMFM404/mixed.html | # Mixed effects logistic regression
Mixed effects logistic regression is used to model binary outcome variables, in which the log odds of the outcomes are modeled as a linear combination of the predictor variables when data are clustered or there are both fixed and random effects.
library(ggplot2)
library(GGally)
library(reshape2)
library(lme4)
## Loading required package: Matrix
library(compiler)
library(parallel)
library(boot)
## Examples of mixed effects logistic regression
• Example 1: A researcher sampled applications to 40 different colleges to study factor that predict admittance into college. Predictors include student's high school GPA, extracurricular activities, and SAT scores. Some schools are more or less selective, so the baseline probability of admittance into each of the schools is different. School level predictors include whether the school is public or private, the current student-to-teacher ratio, and the school's rank.
• Example 2: A large HMO wants to know what patient and physician factors are most related to whether a patient's lung cancer goes into remission after treatment as part of a larger study of treatment outcomes and quality of life in patients with lunger cancer.
• Example 3: A television station wants to know how time and advertising campaigns affect whether people view a television show. They sample people from four cities for six months. Each month, they ask whether the people had watched a particular show or not in the past week. After three months, they introduced a new advertising campaign in two of the four cities and continued monitoring whether or not people had watched the show.
# Lung cancer data
In this example, we are going to explore Example 2 about lung cancer using a simulated dataset, which we have posted online. A variety of outcomes were collected on patients, who are nested within doctors, who are in turn nested within hospitals. There are also a few doctor level variables, such as Experience that we will use in our example.
hdp <- read.csv("http://www.karlin.mff.cuni.cz/~pesta/prednasky/NMFM404/Data/hdp.csv")
hdp <- within(hdp, {
Married <- factor(Married, levels = 0:1, labels = c("no", "yes"))
DID <- factor(DID)
HID <- factor(HID)
})
Now we are going to graph our continuous predictor variables. Visualizing data can help us understand the distributions, catch coding errors (e.g., we know a variable only takes values from 0 to 7, but we see a 999 in the graph), and give us a sense of the relationship among our variables. For example, we might see that two predictors are highly correlated and decide we only want to include one in the model, or we might note a curvilinear relation between two variables. Data visualization is a fast, intuitive way to check all of this at once. If most your predictors appear independent of each other, that is fine. It shapes your expectations of the model. For example, if they are independent, the estimate for one predictor should not change much when you enter another predictor (although the standard error and significance tests may). We can get all of this information and intuition about what and how to model are data by simply viewing it.
ggpairs(hdp[, c("IL6", "CRP", "LengthofStay", "Experience")])
There do not seem to be any strong linear relations among our continuous predictors. Let us look at the distributions of our variables by CancerStage. Because LengthofStay is coded discretely in days, we can examine how CancerStage is associated with it using bubble plots. The area of each bubble is proportional to the number of observations with those values. For the continuous predictors, we use violin plots with jittered data values. All of the raw data is presented separated by CancerStage. To alleviate overplotting and see the values better, we add a small amount of random noise (primarily to the x axis) as well as set the alpha transparency. Although the jittered dots are helpful for seeing the raw data, it can be difficult to get a precise sense of the distribution. For that, we add violin plots. Violin plots are just kernel density plots reflected around the plotting axis. We plot the violin plots on top of the jittered points with a transparency so that you can stil see the raw data, but the violin plots are dominant. Because both IL6 and CRP tend to have skewed distributions, we use a square root scale on the y axis. The distributions look fairly normal and symmetric, although you can still see the long right tail, even using a square root scale (note that only the scale was shifted, the values themselves are not transformed, which is important because this lets you see and interpret the actual scores, rather than the square root of the scores).
ggplot(hdp, aes(x = CancerStage, y = LengthofStay)) +
stat_sum(aes(size = ..n.., group = 1)) +
scale_size_area(max_size=10)
tmp <- melt(hdp[, c("CancerStage", "IL6", "CRP")], id.vars="CancerStage")
ggplot(tmp, aes(x = CancerStage, y = value)) +
geom_jitter(alpha = .1) +
geom_violin(alpha = .75) +
facet_grid(variable ~ .) +
scale_y_sqrt()
Because it is difficult to see how binary variables change over levels of continuous variables, we can flip the problem around and look at the distribution of continuous variables at each level of the binary outcome.
tmp <- melt(hdp[, c("remission", "IL6", "CRP", "LengthofStay", "Experience")],
id.vars="remission")
ggplot(tmp, aes(factor(remission), y = value, fill=factor(remission))) +
geom_boxplot() +
facet_wrap(~variable, scales="free_y")
## Analysis methods you might consider
Below is a list of analysis methods you may have considered.
• Mixed effects probit regression is very similar to mixed effects logistic regression, but it uses the normal CDF instead of the logistic CDF. Both model binary outcomes and can include fixed and random effects.
• Fixed effects logistic regression is limited in this case because it may ignore necessary random effects and/or non independence in the data.
• Fixed effects probit regression is limited in this case because it may ignore necessary random effects and/or non independence in the data.
• Logistic regression with clustered standard errors. These can adjust for non independence but does not allow for random effects.
• Probit regression with clustered standard errors. These can adjust for non independence but does not allow for random effects.
# GLMM for binary outcome
Below we use the glmer command to estimate a mixed effects logistic regression model with Il6, CRP, and LengthofStay as patient level continuous predictors, CancerStage as a patient level categorical predictor (I, II, III, or IV), Experience as a doctor level continuous predictor, and a random intercept by DID, doctor ID.
Estimating and interpreting generalized linear mixed models (GLMMs, of which mixed effects logistic regression is one) can be quite challenging.
# estimate the model and store results in m
m <- glmer(remission ~ IL6 + CRP + CancerStage + LengthofStay + Experience +
(1 | DID), data = hdp, family = binomial, nAGQ = 10)
## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = ## control$checkConv, : Model failed to converge with max|grad| = 0.00107021
## (tol = 0.001, component 2)
# print the mod results without correlations among fixed effects
print(m, corr = FALSE)
## Generalized linear mixed model fit by maximum likelihood (Adaptive
## Gauss-Hermite Quadrature, nAGQ = 10) [glmerMod]
## Family: binomial ( logit )
## Formula:
## remission ~ IL6 + CRP + CancerStage + LengthofStay + Experience +
## (1 | DID)
## Data: hdp
## AIC BIC logLik deviance df.resid
## 7397.276 7460.733 -3689.638 7379.276 8516
## Random effects:
## Groups Name Std.Dev.
## DID (Intercept) 2.015
## Number of obs: 8525, groups: DID, 407
## Fixed Effects:
## (Intercept) IL6 CRP CancerStageII
## -2.05164 -0.05677 -0.02148 -0.41398
## CancerStageIII CancerStageIV LengthofStay Experience
## -1.00340 -2.33682 -0.12119 0.12003
The first part tells us the estimates are based on an adaptive Gaussian Hermite approximation of the likelihood. In particular we used 10 integration points. As we use more integration points, the approximation becomes more accurate converging to the ML estimates; however, more points are more computationally demanding and can be extremely slow or even intractable with today's technology.
The next section gives us basic information that can be used to compare models, followed by the random effect estimates. This represents the estimated variability in the intercept on the logit scale. Had there been other random effects, such as random slopes, they would also appear here. The top section concludes with the total number of observations, and the number of level 2 observations. In our case, this includes the total number of patients (8,525) and doctors (407).
The last section is a table of the fixed effects estimates. For many applications, these are what people are primarily interested in. The estimates represent the regression coefficients. These are unstandardized and are on the logit scale. The estimates are followed by their standard errors (SEs). As is common in GLMs, the SEs are obtained by inverting the observed information matrix (negative second derivative matrix). However, for GLMMs, this is again an approximation. The approximations of the coefficient estimates likely stabilize faster than do those for the SEs. Thus if you are using fewer integration points, the estimates may be reasonable, but the approximation of the SEs may be less accurate. The Wald tests, Estimate/SE, rely on asymptotic theory, here referring to as the highest level unit size converges to infinity, these tests will be normally distributed, and from that, p values (the probability of obtaining the observed estimate or more extreme, given the true estimate is 0).
It can be nice to get confidence intervals (CIs). We can get rough estimates using the SEs.
se <- sqrt(diag(vcov(m)))
# table of estimates with 95% CI
(tab <- cbind(Est = fixef(m), LL = fixef(m) - 1.96 * se, UL = fixef(m) + 1.96 *
se))
## Est LL UL
## (Intercept) -2.05164301 -3.09327975 -1.010006262
## IL6 -0.05677289 -0.07934881 -0.034196973
## CRP -0.02148351 -0.04151146 -0.001455563
## CancerStageII -0.41397971 -0.56247659 -0.265482839
## CancerStageIII -1.00340497 -1.19603829 -0.810771638
## CancerStageIV -2.33681678 -2.64659735 -2.027036207
## LengthofStay -0.12118784 -0.18710918 -0.055266504
## Experience 0.12002708 0.06622201 0.173832158
If we wanted odds ratios instead of coefficients on the logit scale, we could exponentiate the estimates and CIs.
exp(tab)
## Est LL UL
## (Intercept) 0.12852357 0.04535296 0.3642167
## IL6 0.94480862 0.92371767 0.9663811
## CRP 0.97874562 0.95933834 0.9985455
## CancerStageII 0.66101436 0.56979617 0.7668356
## CancerStageIII 0.36662895 0.30238982 0.4445149
## CancerStageIV 0.09663476 0.07089202 0.1317253
## LengthofStay 0.88586754 0.82935318 0.9462329
## Experience 1.12752739 1.06846390 1.1898558
# Multilevel bootstrapping
Inference from GLMMs is complicated. Except for cases where there are many observations at each level (particularly the highest), assuming that EstimateSE is normally distributed may not be accurate. A variety of alternatives have been suggested including Monte Carlo simulation, Bayesian estimation, and bootstrapping. Each of these can be complex to implement. We are going to focus on a small bootstrapping example.
Bootstrapping is a resampling method. It is by no means perfect, but it is conceptually straightforward and easy to implement in code. One downside is that it is computationally demanding. For large datasets or complex models where each model takes minutes to run, estimating on thousands of bootstrap samples can easily take hours or days. In the example for this page, we use a very small number of samples, but in practice you would use many more. Perhaps 1,000 is a reasonable starting point.
For single level models, we can implement a simple random sample with replacement for bootstrapping. With multilevel data, we want to resample in the same way as the data generating mechanism. We start by resampling from the highest level, and then stepping down one level at a time. In our case, we first will sample from doctors, and then within each doctor sampled, we will sample from their patients. To do this, we first need to write a function to resample at each level. The Biostatistics Department at Vanderbilt has a nice page describing the idea http://biostat.mc.vanderbilt.edu/wiki/Main/HowToBootstrapCorrelatedData.
sampler <- function(dat, clustervar, replace = TRUE, reps = 1) {
cid <- unique(dat[, clustervar[1]])
ncid <- length(cid)
recid <- sample(cid, size = ncid * reps, replace = TRUE)
if (replace) {
rid <- lapply(seq_along(recid), function(i) {
cbind(NewID = i, RowID = sample(which(dat[, clustervar] == recid[i]),
replace = TRUE))
})
} else {
rid <- lapply(seq_along(recid), function(i) {
cbind(NewID = i, RowID = which(dat[, clustervar] == recid[i]))
})
}
dat <- as.data.frame(do.call(rbind, rid))
dat$Replicate <- factor(cut(dat$NewID, breaks = c(1, ncid * 1:reps), include.lowest = TRUE,
labels = FALSE))
dat$NewID <- factor(dat$NewID)
return(dat)
}
Now we will resample our data and take 100 replicates. Again in practice you would probably take thousands. We set the seed so that our results are reproducible. It is also likely that you will need to sample more replicates than you ultimately want because many samples may not converge so you do not get estimates from them.
set.seed(20)
tmp <- sampler(hdp, "DID", reps = 100)
bigdata <- cbind(tmp, hdp[tmp$RowID, ]) Next we refit the model on the resampled data. First we store the estimates from our original model, which we will use as start values for the bootstrap models. Then we make a local cluster with 4 nodes (the number of processors on our machine; set to the number of processors you have on yours). Next, we export the data and load the lme4 package on the cluster. Finally, we write a function to fit the model and return the estimates. The call to glmer() is wrapped in try because not all models may converge on the resampled data. This catches the error and returns it, rather than stopping processing. f <- fixef(m) r <- getME(m, "theta") cl <- makeCluster(4) clusterExport(cl, c("bigdata", "f", "r")) clusterEvalQ(cl, require(lme4)) ## [[1]] ## [1] TRUE ## ## [[2]] ## [1] TRUE ## ## [[3]] ## [1] TRUE ## ## [[4]] ## [1] TRUE myboot <- function(i) { object <- try(glmer(remission ~ IL6 + CRP + CancerStage + LengthofStay + Experience + (1 | NewID), data = bigdata, subset = Replicate == i, family = binomial, nAGQ = 1, start = list(fixef = f, theta = r)), silent = TRUE) if (class(object) == "try-error") return(object) c(fixef(object), getME(object, "theta")) } Now that we have the data, the local cluster, and the fitting function setup, we are ready to actually do the bootstrapping. To do this, we use the parLapplyLB function, which loops through every replicate, giving them out to each node of the cluster to estimate the models. The "LB" stands for load balancing, which means replicates are distributed as a node completes its current job. This is valuable because not all replicates will converge, and if there is an error and it happens early on, one node may be ready for a new job faster than another node. There is some extra communication overhead, but this is small compared to the time it takes to fit each model. The results from all nodes are aggregated back into a single list, stored in the object res. Once that is done, we can shut down the local cluster, which terminates the additional R instances and frees memory. start <- proc.time() res <- parLapplyLB(cl, X = levels(bigdata$Replicate), fun = myboot)
end <- proc.time()
# shut down the cluster
stopCluster(cl)
Now that we have the bootstrap results, we can summarize them. First, we calculate the number of models that successfully converged. We do this by checking whether a particular result is numeric or not. Errors are not numeric, so they will be skipped. We can calculate the mean of the successes to see the proportion of replicates that converged and that we have results for.
# calculate proportion of models that successfully converged
success <- sapply(res, is.numeric)
mean(success)
## [1] 1
Next we convert the list of bootstrap results into a matrix, and then calculate the 2.5th and 97.5th percentiles for each parameter. Finally, we can make a table of the results, including the original estimates and standard errors, the mean bootstrap estimate (which is asymptotically equivalent to the original results, but may be biased for a small number of replicates, as in our case), and the bootstrapped confidence intervals. With these data, you could also calculate bias-corrected bootstrap confidence intervals if you wanted, although we only show the percentile CIs.
# combine successful results
bigres <- do.call(cbind, res[success])
# calculate 2.5th and 97.5th percentiles for 95% CI
(ci <- t(apply(bigres, 1, quantile, probs = c(0.025, 0.975))))
## 2.5% 97.5%
## (Intercept) -3.63944293 -1.007201621
## IL6 -0.08743904 -0.030617950
## CRP -0.05018041 0.003039692
## CancerStageII -0.59388626 -0.242818869
## CancerStageIII -1.30051865 -0.753841874
## CancerStageIV -2.92950112 -2.029114134
## LengthofStay -0.22031001 -0.044977250
## Experience 0.06794416 0.207263575
## NewID.(Intercept) 2.04852634 2.461902813
# All results
finaltable <- cbind(Est = c(f, r), SE = c(se, NA), BootMean = rowMeans(bigres),
ci)
# round and print
round(finaltable, 3)
## Est SE BootMean 2.5% 97.5%
## (Intercept) -2.052 0.531 -2.211 -3.639 -1.007
## IL6 -0.057 0.012 -0.059 -0.087 -0.031
## CRP -0.021 0.010 -0.023 -0.050 0.003
## CancerStageII -0.414 0.076 -0.417 -0.594 -0.243
## CancerStageIII -1.003 0.098 -1.043 -1.301 -0.754
## CancerStageIV -2.337 0.158 -2.458 -2.930 -2.029
## LengthofStay -0.121 0.034 -0.143 -0.220 -0.045
## Experience 0.120 0.027 0.129 0.068 0.207
## DID.(Intercept) 2.015 NA 2.260 2.049 2.462
# Predicted probabilities and graphing
These results are great to put in the table or in the text of a research manuscript; however, the numbers can be tricky to interpret. Visual presentations are helpful to ease interpretation and for posters and presentations. As models become more complex, there are many options. We will discuss some of them briefly and give an example how you could do one.
In a logistic model, the outcome is commonly on one of three scales:
• Log odds (also called logits), which is the linearized scale
• Odds ratios (exponentiated log odds), which are not on a linear scale
• Probabilities, which are also not on a linear scale
For tables, people often present the odds ratios. For visualization, the logit or probability scale is most common. There are some advantages and disadvantages to each. The logit scale is convenient because it is linearized, meaning that a 1 unit increase in a predictor results in a coefficient unit increase in the outcome and this holds regardless of the levels of the other predictors (setting aside interactions for the moment). A downside is the scale is not very interpretable. It is hard for readers to have an intuitive understanding of logits. Conversely, probabilities are a nice scale to intuitively understand the results; however, they are not linear. This means that a one unit increase in the predictor, does not equal a constant increase in the probability - the change in probability depends on the values chosen for the other predictors. In ordinary logistic regression, you could just hold all predictors constant, only varying your predictor of interest. However, in mixed effects logistic models, the random effects also bear on the results. Thus, if you hold everything constant, the change in probability of the outcome over different values of your predictor of interest are only true when all covariates are held constant and you are in the same group, or a group with the same random effect. The effects are conditional on other predictors and group membership, which is quite narrowing. An attractive alternative is to get the average marginal probability. That is, across all the groups in our sample (which is hopefully representative of your population of interest), graph the average change in probability of the outcome across the range of some predictor of interest.
We are going to explore an example with average marginal probabilities. These take more work than conditional probabilities, because you have to calculate separate conditional probabilities for every group and then average them. It is also not easy to get confidence intervals around these average marginal effects in a frequentist framework (although they are trivial to obtain from Bayesian estimation).
We create $${\bf X}_i$$ by taking $${\bf X}$$ and setting a particular predictor of interest, say in column $$j$$, to a constant. If we only cared about one value of the predictor, $$i\in\{1\}$$. However, more commonly, we want a range of values for the predictor in order to plot how the predicted probability varies across its range. We can do this by taking the observed range of the predictor and taking k samples evenly spaced within the range. For example, suppose our predictor ranged from 5 to 10, and we wanted 6 samples, (10-5)/(6-1)=1, so each sample would be 1 apart from the previous and they would be: $$\{5,6,7,8,9,10\}$$. Then we create $$k$$ different $${\bf X}_i$$ where $$i\in\{1,\ldots,k\}$$ where in each case, the jth column is set to some constant. Then we calculate: ${\boldsymbol\eta}_i={\bf X}_i{\boldsymbol\beta}+{\bf Z}_i{\boldsymbol\gamma}$ These are all the different linear predictors. Finally, we take $$h({\boldsymbol\eta})$$, which gives us $${\boldsymbol\mu}_i$$, which are the conditional expectations on the original scale, in our case, probabilities. We can then take the expectation of each $${\boldsymbol\mu}_i$$ and plot that against the value our predictor of interest was held at. We could also make boxplots to show not only the average marginal predicted probability, but also the distribution of predicted probabilities.
You may have noticed that a lot of variability goes into those estimates. We are using $${\bf X}$$ only holding our predictor of interest at a constant, which allows all the other predictors to take on values in the original data. Also, we have left $${\bf Z}{\boldsymbol\gamma}$$ as in our sample, which means some groups are more or less represented than others. If we had wanted, we could have re-weighted all the groups to have equal weight. We chose to leave all these things as-is in this example based on the assumption that our sample is truly a good representative of our population of interest. Rather than attempt to pick meaningful values to hold covariates at (even the mean is not necessarily meaningful, particularly if a covariate as a bimodal distribution, it may be that no participant had a value at or near the mean), we used the values from our sample. This also suggests that if our sample was a good representation of the population, then the average marginal predicted probabilities are a good representation of the probability for a new random sample from our population.
Now that we have some background and theory, let's see how we actually go about calculating these things. We get a summary of LengthofStay, our predictor of interest, and then get 100 values across its range to use in prediction. We make a copy of our data so we can fix the values of one of the predictors and then use the predict function to calculate the predicted values. All random effects are included by default, see ?predict.merMod for more details. Note that the predict method for mixed effects models is new and currently is only in the development version of lme4, so make sure that you have that installed.
# temporary data
tmpdat <- hdp[, c("IL6", "CRP", "CancerStage", "LengthofStay", "Experience",
"DID")]
summary(hdp$LengthofStay) ## Min. 1st Qu. Median Mean 3rd Qu. Max. ## 1.000 5.000 5.000 5.492 6.000 10.000 jvalues <- with(hdp, seq(from = min(LengthofStay), to = max(LengthofStay), length.out = 100)) # calculate predicted probabilities and store in a list pp <- lapply(jvalues, function(j) { tmpdat$LengthofStay <- j
predict(m, newdata = tmpdat, type = "response")
})
Now that we have all the predicted probabilities, we can work on displaying them. For example, we could look at the average marginal predicted probability at a handful of different lengths of stay. We can also plot all of them.
# average marginal predicted probability across a few different Lengths of
# Stay
sapply(pp[c(1, 20, 40, 60, 80, 100)], mean)
## [1] 0.3652354 0.3366378 0.3075494 0.2796343 0.2530078 0.2277651
# get the means with lower and upper quartiles
plotdat <- t(sapply(pp, function(x) {
c(M = mean(x), quantile(x, c(0.25, 0.75)))
}))
# add in LengthofStay values and convert to data frame
plotdat <- as.data.frame(cbind(plotdat, jvalues))
# better names and show the first few rows
colnames(plotdat) <- c("PredictedProbability", "Lower", "Upper", "LengthofStay")
## PredictedProbability Lower Upper LengthofStay
## 1 0.3652354 0.08489961 0.6155674 1.000000
## 2 0.3637090 0.08404758 0.6129569 1.090909
## 3 0.3621849 0.08320332 0.6103400 1.181818
## 4 0.3606630 0.08236679 0.6077167 1.272727
## 5 0.3591434 0.08153791 0.6050872 1.363636
## 6 0.3576261 0.08071664 0.6024515 1.454545
# plot average marginal predicted probabilities
ggplot(plotdat, aes(x = LengthofStay, y = PredictedProbability)) + geom_line() +
ylim(c(0, 1))
We could also add the lower and upper quartiles. This information shows us the range in which 50 percent of the predicted probabilities fell.
ggplot(plotdat, aes(x = LengthofStay, y = PredictedProbability)) + geom_linerange(aes(ymin = Lower,
ymax = Upper)) + geom_line(size = 2) + ylim(c(0, 1))
This is just the beginning of what can be done. For plots, it is useful to add more information. We could make the same average marginal predicted probabilities, but in addition to varying LengthofStay we could do it for each level of CancerStage.
# calculate predicted probabilities and store in a list
biprobs <- lapply(levels(hdp$CancerStage), function(stage) { tmpdat$CancerStage[] <- stage
lapply(jvalues, function(j) {
tmpdat$LengthofStay <- j predict(m, newdata = tmpdat, type = "response") }) }) # get means and quartiles for all jvalues for each level of CancerStage plotdat2 <- lapply(biprobs, function(X) { temp <- t(sapply(X, function(x) { c(M=mean(x), quantile(x, c(.25, .75))) })) temp <- as.data.frame(cbind(temp, jvalues)) colnames(temp) <- c("PredictedProbability", "Lower", "Upper", "LengthofStay") return(temp) }) # collapse to one data frame plotdat2 <- do.call(rbind, plotdat2) # add cancer stage plotdat2$CancerStage <- factor(rep(levels(hdp$CancerStage), each = length(jvalues))) # show first few rows head(plotdat2) ## PredictedProbability Lower Upper LengthofStay CancerStage ## 1 0.4474697 0.1547464 0.7328397 1.000000 I ## 2 0.4458035 0.1533108 0.7306772 1.090909 I ## 3 0.4441385 0.1518862 0.7285036 1.181818 I ## 4 0.4424747 0.1504724 0.7263191 1.272727 I ## 5 0.4408123 0.1490695 0.7241237 1.363636 I ## 6 0.4391511 0.1476774 0.7219174 1.454545 I # graph it ggplot(plotdat2, aes(x = LengthofStay, y = PredictedProbability)) + geom_ribbon(aes(ymin = Lower, ymax = Upper, fill = CancerStage), alpha = .15) + geom_line(aes(colour = CancerStage), size = 2) + ylim(c(0, 1)) + facet_wrap(~ CancerStage) Things look fairly bleak for the chances of a Stage IV lung cancer patient who was in the hospital 10 days having cancer in remission (please remember that these are simulated data). It also looks like the distribution is skewed. We can examine the distribution of predicted probabilities just for that group. ggplot(data.frame(Probs = biprobs[[4]][[100]]), aes(Probs)) + geom_histogram() + scale_x_sqrt(breaks = c(0.01, 0.1, 0.25, 0.5, 0.75)) ## stat_bin: binwidth defaulted to range/30. Use 'binwidth = x' to adjust this. Even using a square root scale that stretches out the lower values, it is still extremely skewed. The vast majority are estimated to have less than a .1 probability of being in remission. # Three level mixed effects logistic regression We have looked at a two level logistic model with a random intercept in depth. This is the simplest mixed effects logistic model possible. Now we are going to briefly look at how you can add a third level and random slope effects as well as random intercepts. Below we estimate a three level logistic model with a random intercept for doctors and a random intercept for hospitals. In this examples, doctors are nested within hospitals, meaning that each doctor belongs to one and only one hospital. The alternative case is sometimes called "cross classified" meaning that a doctor may belong to multiple hospitals, such as if some of the doctor's patients are from hospital A and others from hospital B. In glmer you do not need to specify whether the groups are nested or cross classified, R can figure it out based on the data. We use the same (1 | ID) general syntax to indicate the intercept (1) varying by some ID. For models with more than a single scalar random effect, glmer only supports a single integration point, so we use nAGQ=1. # estimate the model and store results in m m3a <- glmer(remission ~ Age + LengthofStay + FamilyHx + IL6 + CRP + CancerStage + Experience + (1 | DID) + (1 | HID), data = hdp, family = binomial, nAGQ=1) ## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl =
## control$checkConv, : Model failed to converge with max|grad| = 0.0640339 ## (tol = 0.001, component 3) # print the mod results without correlations among fixed effects print(m3a, corr=FALSE) ## Generalized linear mixed model fit by maximum likelihood (Laplace ## Approximation) [glmerMod] ## Family: binomial ( logit ) ## Formula: ## remission ~ Age + LengthofStay + FamilyHx + IL6 + CRP + CancerStage + ## Experience + (1 | DID) + (1 | HID) ## Data: hdp ## AIC BIC logLik deviance df.resid ## 7199.094 7283.703 -3587.547 7175.094 8513 ## Random effects: ## Groups Name Std.Dev. ## DID (Intercept) 1.9523 ## HID (Intercept) 0.5487 ## Number of obs: 8525, groups: DID, 407; HID, 35 ## Fixed Effects: ## (Intercept) Age LengthofStay FamilyHxyes ## -1.68828 -0.01493 -0.04467 -1.30662 ## IL6 CRP CancerStageII CancerStageIII ## -0.05686 -0.02216 -0.31846 -0.85692 ## CancerStageIV Experience ## -2.13751 0.12686 The output tells us the family (binomial for binary outcomes) and the link (logit). Followed by usual fit indices and the variance of the random effects. In this case the variability in the intercept (on the log odds scale) between doctors and between hospitals. The standard deviation is also displayed (simply the square root of the variance, not the standard error of the estimate of the variance). We also get the number of unique units at each level. Last are the fixed effects, as before. It can also be useful to look at the distribution of the conditional modes, which we do with caterpillar polots below. The blue dots are the conditional models with error bars. We do this for both doctors and hospitals. For example for doctors, we can see a bit of a long right tail in that there are more extreme positive than negative values. For the doctors, we suppress their IDs (using the scales=list(y = list(alternating=0)) argument) because there are so many, but we leave them in for the hospitals. dotplot(ranef(m3a, which = "DID", postVar = TRUE), scales = list(y = list(alternating = 0))) ## Warning in ranef.merMod(m3a, which = "DID", postVar = TRUE): 'postVar' is ## deprecated: please use 'condVar' instead ##$DID
dotplot(ranef(m3a, which = "HID", postVar = TRUE))
## Warning in ranef.merMod(m3a, which = "HID", postVar = TRUE): 'postVar' is
## $HID We can easily add random slopes to the model as well, and allow them to vary at any level. We are just going to add a random slope for LengthofStay that varies between doctors. As in regular R formulae, we use the + operator to "add" an effect, and we do it in the section for doctor random effects. All terms in one group of parentheses use an unstructured covariance matrix, you can get a diagonal covariance structure by splitting the grouping into separate pieces. Between groupings is assumed indepedent. # estimate the model and store results in m m3b <- glmer(remission ~ Age + LengthofStay + FamilyHx + IL6 + CRP + CancerStage + Experience + (1 + LengthofStay | DID) + (1 | HID), data = hdp, family = binomial, nAGQ = 1) ## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl =
## control$checkConv, : Model failed to converge with max|grad| = 0.120807 ## (tol = 0.001, component 1) # print the mod results without correlations among fixed effects print(m3b, corr = FALSE) ## Generalized linear mixed model fit by maximum likelihood (Laplace ## Approximation) [glmerMod] ## Family: binomial ( logit ) ## Formula: ## remission ~ Age + LengthofStay + FamilyHx + IL6 + CRP + CancerStage + ## Experience + (1 + LengthofStay | DID) + (1 | HID) ## Data: hdp ## AIC BIC logLik deviance df.resid ## 7147.761 7246.471 -3559.880 7119.761 8511 ## Random effects: ## Groups Name Std.Dev. Corr ## DID (Intercept) 0.5056 ## LengthofStay 0.3715 -0.11 ## HID (Intercept) 0.7301 ## Number of obs: 8525, groups: DID, 407; HID, 35 ## Fixed Effects: ## (Intercept) Age LengthofStay FamilyHxyes ## -0.54702 -0.01527 -0.18974 -1.34010 ## IL6 CRP CancerStageII CancerStageIII ## -0.05864 -0.02103 -0.29421 -0.86536 ## CancerStageIV Experience ## -2.29502 0.10458 dotplot(ranef(m3b, which = "DID", postVar = TRUE), scales = list(y = list(alternating = 0))) ## Warning in ranef.merMod(m3b, which = "DID", postVar = TRUE): 'postVar' is ## deprecated: please use 'condVar' instead ##$DID
dotplot(ranef(m3b, which = "HID", postVar = TRUE),
scales = list(y = list(alternating = 0)))
## Warning in ranef.merMod(m3b, which = "HID", postVar = TRUE): 'postVar' is
## \$HID
## Things to consider
• Sample size: Often the limiting factor is the sample size at the highest unit of analysis. For example, having 500 patients from each of ten doctors would give you a reasonable total number of observations, but not enough to get stable estimates of doctor effects nor of the doctor-to-doctor variation. 10 patients from each of 500 doctors (leading to the same total number of observations) would be preferable.
• Parameter estimation: Because there are not closed form solutions for GLMMs, you must use some approximation. Three are fairly common.
• Quasi-likelihood approaches use a Taylor series expansion to approximate the likelihood. Thus parameters are estimated to maximize the quasi-likelihood. That is, they are not true maximum likelihood estimates. A Taylor series uses a finite set of differentiations of a function to approximate the function, and power rule integration can be performed with Taylor series. With each additional term used, the approximation error decreases (at the limit, the Taylor series will equal the function), but the complexity of the Taylor polynomial also increases. Early quasi-likelihood methods tended to use a first order expansion, more recently a second order expansion is more common. In general, quasi-likelihood approaches are the fastest (although they can still be quite complex), which makes them useful for exploratory purposes and for large datasets. Because of the bias associated with them, quasi-likelihoods are not preferred for final models or statistical inference.
• The true likelihood can also be approximated using numerical integration. Quadrature methods are common, and perhaps most common among these use the Gaussian quadrature rule, frequently with the Gauss-Hermite weighting function. It is also common to incorporate adaptive algorithms that adaptively vary the step size near points with high error. The accuracy increases as the number of integration points increases. Using a single integration point is equivalent to the so-called Laplace approximation. Each additional integration point will increase the number of computations and thus the speed to convergence, although it increases the accuracy. Adaptive Gauss-Hermite quadrature might sound very appealing and is in many ways. However, the number of function evaluations required grows exponentially as the number of dimensions increases. A random intercept is one dimension, adding a random slope would be two. For three level models with random intercepts and slopes, it is easy to create problems that are intractable with Gaussian quadrature. Consequently, it is a useful method when a high degree of accuracy is desired but performs poorly in high dimensional spaces, for large datasets, or if speed is a concern.
• A final set of methods particularly useful for multidimensional integrals are Monte Carlo methods including the famous Metropolis-Hastings algorithm and Gibbs sampling which are types of Markov chain Monte Carlo (MCMC) algorithms. Although Monte Carlo integration can be used in classical statistics, it is more common to see this approach used in Bayesian statistics.
• Complete or quasi-complete separation: Complete separation means that the outcome variable separate a predictor variable completely, leading perfect prediction by the predictor variable. Particularly if the outcome is skewed, there can also be problems with the random effects. For example, if one doctor only had a few patients and all of them either were in remission or were not, there will be no variability within that doctor.
## References
• UCLA: IDRE (Institute for Digital Research and Education). Data Analysis Examples. from http://www.ats.ucla.edu/stat/dae/ (accessed January 31, 2014)
• Agresti, A. (2013). Categorical Data Analysis (3rd Ed.), Hoboken, New Jersey: John Wiley & Sons. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 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": 0.754307746887207, "perplexity": 1519.6911932428893}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1510934804518.38/warc/CC-MAIN-20171118021803-20171118041803-00052.warc.gz"} |
https://arxiver.wordpress.com/2017/03/14/rapid-formation-of-massive-black-holes-in-close-proximity-to-embryonic-proto-galaxies-ga/ | # Rapid Formation of Massive Black Holes in close proximity to Embryonic Proto-Galaxies [GA]
The Direct Collapse Black Hole (DCBH) scenario provides a solution for forming the massive black holes powering bright quasars observed in the early Universe. A prerequisite for forming a DCBH is that the formation of (much less massive) Population III stars be avoided – this can be achieved by destroying H$_2$ via Lyman-Werner (LW) radiation (E$_{\rm{LW}}$ = 12.6 eV). We find that two conditions must be met in the proto-galaxy that will host the DCBH. First, prior star formation must be delayed; this can be achieved with a background LW flux of J$_{\rm BG} \gtrsim 100\ J_{21}$. Second, an intense burst of LW radiation from a neighbouring star-bursting proto-galaxy is required, just before the gas cloud undergoes gravitational collapse, to finally suppress star formation completely. We show here for the first time using high-resolution hydrodynamical simulations, including full radiative transfer, that this low-level background, combined with tight synchronisation and irradiation of a secondary proto-galaxy by a primary proto-galaxy, inevitably moves the secondary proto-galaxy onto the isothermal atomic cooling track, without the deleterious effects of either photo-evaporating the gas or polluting it by heavy elements. These, atomically cooled, massive proto-galaxies are expected to ultimately form a DCBH of mass $10^4 – 10^5 M_{\odot}$.
J. Regan, E. Visbal, J. Wise, et. al.
Tue, 14 Mar 17
48/74
Comments: Published in Nature Astronomy, March 13th 2017 | {"extraction_info": {"found_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": 0.7268918752670288, "perplexity": 4533.34738253942}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463607704.68/warc/CC-MAIN-20170523221821-20170524001821-00036.warc.gz"} |
http://education.lms.ac.uk/2016/04/ | # Three PhD studentships at Loughborough
The Mathematics Education Centre at Loughborough University has three fully-funded PhD studentships available to start in October 2016. Each project is full time for three years.
# Tony Gardiner: “The Man Who Knew Infinity”
The film The Man Who Knew Infinity goes on UK general release from 8th April.
It is a compressed, and beautifully dramatised version of the theme treated more fully in Robert Kanigel’s double biography of the same name – which treats Ramanujan alongside a partial portrait of G.H.Hardy.
Mathematicians can be remarkably unforgiving about attempts to present mathematics to a general audience. And Ramanujan’s story could so easily be cheapened – with awkward aspects being trivialised, in order to pander to current prejudices. The Good News is that, not only has this been avoided, but the film manages to incorporate much of the detail and spirit of what we know, while using its dramatic freedom to confront important issues that are often either treated too tritely, or passed over in silence. The project may have taken 10 years in the making, but the result has been worth it.
As someone who does not usually watch movies, I simply encourage everyone to see it
(perhaps several times), to encourage others to see it, and to use it to discuss the issues which it raises.
A film is not meant to be a reflection of reality. This film would seem to be a fairly faithful representation of what we know in those areas where fidelity matters. In other respects it exercises flexibility. In contrast to Ramanujan, Dev Patel is slim and beautifully formed; yet he manages to capture an essential seriousness and devotion which is entirely plausible. His wife is portrayed as older and I suspect much more beautiful than the real Janaki; yet her portrayal of profound simplicity is moving in a way that seems entirely appropriate (whether or not it is documented).
In his review for the February issue of the Notices of the AMS
George Andrews suggested that the film will help students appreciate the importance of “proofs”. In fact, the struggle between proof and intuition, between Hardy and Ramanujan, is not so cleanly resolved, and there is a danger that the film may leave many strengthened in their belief in mathematical invention as “magical intuition”. So the film should be used to actively encourage a deeper discussion of the relative importance of proof, and what is too often simply labelled “intuition” (as if it were not susceptible to, any further explanation).
Here is a chance to grapple with the often neglected interplay between
(a) technical, or formal, training in universal methods – whereby my individual “mental
universe” is disciplined to fit with yours (or with some imaginary “Platonic ideal”),
and
(b) our individual, idiosyncratic way of thinking about these shared objects and processes – whereby my thoughts avoid being mechanical replicas of everyone else’s, and so provide scope for originality.
Without the second, we are little better than machines. And without the first, we are almost bound to go astray.
Almost all students need a significant dose of (a) before their (b)-type thoughts can become fruitful. But some individuals’ (b)-type thoughts flourish – mostly unerringly – with relatively little (a)-type formalism. One thinks of Euler, or Schubert, or 19th century Italian algebraic geometers, or Feynman, or Thurston, or … . The problem is then how to check the resulting claimed insights, to embed them within mathematics as a whole, and to make the methods available to the rest of us. By neglecting such delicate matters we leave a vacuum that is too easily filled by half-truths.
Tony Gardiner
# Tony Gardiner Receives the 2016 Award for Excellence in Mathematics Education
Citation for the 2016 Award for Excellence in Mathematics Education to
Dr. Anthony David Gardiner
It is with great pleasure that the Award Committee hereby announces that the 2016 Award is given to Dr. Anthony D. Gardiner, currently retired from University of Birmingham, United Kingdom, in recognition of his more than forty years of sustained and multiple major contributions to enhancing the problem-solving skills of generations of mathematics students in the United Kingdom (UK) and beyond.
Gardiner’s major achievements include:
• orchestrating teams of volunteers from many constituencies, including teachers, mathematics educators and university mathematicians, to create a portfolio of mathematics contests, leading eventually to the creation of the UK Mathematics Trust, which creates problem-solving challenges taken by well over half a million students per year;
• creating structures that dramatically increased and broadened participation in mathematics competitions and other activities supporting UK participation in the International Mathematics Olympiad;
• leading the UK IMO team (1990 – 95);
• creating problem solving journals for school students (including grading thousands of solutions personally), leading eventually to the Problem Solving Journal for Secondary Students (edited by Dr. Gardiner since 2003, with a circulation over 5,000);
• authoring 15 books on mathematical thinking and mathematical problem solving, including Understanding Infinity, Discovering Mathematics: the art of investigation, Mathematical Puzzling (all reprinted by Dover Publications), the four volume series Extension Mathematics (Oxford), and the recent Teaching mathematics at secondary level (Open Book Publishers).
In addition, Gardiner’s expertise on the problem-solving abilities of English schoolchildren, and his insights into omissions in UK mathematics education has led to his being consulted by multiple UK Ministers of State for Education, and have influenced significant changes in the UK mathematics curriculum. Gardiner has also served in multiple high level leadership positions in mathematics education both in the UK and internationally, including Council of the London Mathematical Society, and member of the Education Committee (1990s), Presidency of the (UK) Mathematical Association in 1997-98, chair of the Education Committee of the European Mathematical Society (2000-04), and Senior Vice President of the World Federation of National Mathematics Competitions (2004-08). He has addressed major teacher conferences in more than 10 countries, and he was an Invited Lecturer at the 10th International Congress of Mathematics Education in 2004. He has organized many meetings and programs to support mathematics education, teacher professional development, and to promote problem solving. He has contributed numerous articles to newspapers and magazines to communicate the goals of successful mathematics education to a broader public. Both the extent and impact of Gardiner’s efforts are remarkable. He provides an inspiring example of how a mathematician can have a positive impact on mathematics education; he is a most worthy recipient of the Texas A&M Award for Excellence in Mathematics Education.
Gardiner received his doctorate in 1973 from the University of Warwick, UK. He taught at the University of East Africa from 1968-69, University of Birmingham from 1974 to 2012. During that time he worked at the Free University of Berlin on a fellowship, and held numerous visiting positions including at the University of Bielefeld in Germany, University of Waterloo, the University of Melbourne and the University of Western Australia.
This award is established at the Texas A&M University to recognize works of lasting significance and impact in advancing mathematics education as an interdisciplinary field that links mathematics, educational studies and practices. In particular, the award recognizes major contributions to new knowledge and scholarship as well as exemplary contributions in promoting interdisciplinary collaboration in mathematics
education.
This is an annual award that consists of a commemorative plaque and a cash prize ($3000). A recipient will be selected yearly and will be invited to give a keynote talk, with all travel expenses covered, at a workshop dedicated to advancing mathematics education. Moreover, subject to the availability of the recipient, a housing allowance and a$5000 stipend will also be provided to the recipient to spend two weeks in residence at Texas A&M University interacting with students and faculty in seminars and informal mentoring sessions. | {"extraction_info": {"found_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": 0.21135546267032623, "perplexity": 4866.772015370255}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1563195526359.16/warc/CC-MAIN-20190719202605-20190719224605-00511.warc.gz"} |
https://rd.springer.com/article/10.1186%2Fs13638-018-1190-6 | # An improved localization scheme based on IMDV-hop for large-scale wireless mobile sensor aquaculture networks
• Jianping Zhu
• Chunfeng Lv
• Zhengsu Tao
Open Access
Research
## Abstract
With the hasteful development of ocean economy and the increasing exploiture of ocean resources, offshore water is contaminated seriously. Ocean ecological environment is unprecedentedly faced to threat and destruction. Moreover, the desire for aquatic and marine products is increasing consumedly according to people’s health attention rising. It is extremely urgent to establish and maintain low-cost and high-efficient transmission and localization schemes for real-time large-scale aquaculture surveillance systems. Localization scheme IMDV-hop (Intermittent Mobile DV-hop) embedded in WLS (weighted least square) method, accompanying with HTC (Hidden Terminal Couple), is proposed in this work for the purpose of environment surveillance, object localization for early warning, rescue operations, and restructuring plan, etc. Two critical parameters, correction coefficient kc and weighted coefficient $${w}_{N_x,i}$$, are introduced into IMDV-hop scheme for large-scale aquaculture monitoring and localization mobile sensor systems to evaluate the influence on localization behaviors, and subsequently guarantee localization accuracy and time-critical performance. And localization error, delay, and consumption are predicted by comprehensive NS-2 simulations. Besides, performance comparisons of IMDV-hop scheme with other DV-hop-based schemes and MCL-based scheme are also proposed. Analysis and comparison results show that delay behavior of IMDV-hop is improved largely relative to other schemes, while accuracy and energy consumption performance is improved in some cases of more node density and lower mobile velocity.
## Keywords
DV-hop Wireless sensor networks IMDV-hop WLS HTC
## Abbreviations
AOA
Angle of arrival
APIT
Approximate Point in Triangle
COMCL
Constraint rules Optimized Monte Carlo Localization Scheme
CPE
Convex position estimation
CSMA/CA
Carrier Sense Multiple Access with Collision Avoidance
DV-hop
Distance Vector-hop
GPS
Global Positioning System
HDV-hop
Hybrid DV-hop
HTC
Hidden Terminal Couple
IMCL
Improved MCL
IMDV-hop
Intermittent Mobile DV-hop
IWC-DV-hop
Improved weighted centroid DV-hop
MCB
Monte Carlo localization boxed
MCL
Monte Carlo localization
NLLS
Non-linear least square
O-WSN
Hybrid optical WSN
SMC
Sequential MCL
SSW
Sub-square weighted
TDOA
Time difference of arrival
TOA
Time of arrival
UWSNs
Underwater wireless sensor networks
WLS
Weighted least square
WMCL
Weighted MCL
WSN
Wireless sensor network
## 1 Introduction
In recent years, wireless sensor networks (WSNs) have revolutionized the world of distributed systems and enabled many new applications. WSNs play more and more decisive roles in various aspects such as environmental and habitat monitoring, precision agriculture, animal tracking, disaster rescue and almost touch upon all aspects of our life. In many applications, it is essential for nodes to know their position information [1, 2, 3, 4], that is, measurement data or information exchanges happened in WSNs without location information are meaningless. For example, locations must be known in environmental monitoring applications such as bush fire surveillance, water quality monitoring, and precision agriculture. And sensor positions can also help to facilitate the network with an overall point of view, such as routing or connectivity for WSNs. Therefore, localizations of sensors or events have become fundamental elements in WSNs.
And also, growing close attentions are paid to underwater WSNs (UWSNs) [5, 6] during the last couple of years. Ocean explorations and a multitude of underwater applications such as oceanographic data collection, warning systems for natural disasters, ecological applications, military under water surveillance, and industrial applications are provoking people to focus on UWSNs. And people’s growing needs for aquatic organisms to be rich in proteinic substance and other trace elements can be fulfilled or relieved through aquaculture farm. Our aquaculture farm cultivates the Chinese soft-shell turtle, which is a kind of precious and costly aquatic product for farmers. Turtles are equipped sensor nodes for farmers to obtain real-time positions of them and, subsequently, are executed on-time nurses if they encounter disease or inactivity, to increase survival ratio. And also, turtle positions can also be used for routing and networking purposes in such a large-scale surveillance WSNs. Of course, the straightforward method for localization in WSNs is to use Global Positioning System (GPS) [7], attached on every sensor node. However, it is infeasible for large-scale aquaculture monitoring WSNs equipped GPS receiver on every node, which brings about so much high cost or inconvenience.
Depending on whether absolute range measurements are used or not, localization schemes can be roughly classified into two categories [1, 2, 3, 4]: range-based and range-free. Range-based algorithms measure the exact distance or angle of pending localization nodes adopting techniques such as TOA (time of arrival) [8, 9], TDOA (time difference of arrival) [10], RSSI (received signal strength indicator) [11, 12, 13, 14], or AOA (angle of arrival) [15].
Acknowledging that high cost of hardware required by range-based solutions may be inappropriate in relation to the required location precision, researchers have sought alternate range-free solutions to localization problems in WSNs. Typical range-free algorithms [16, 17, 18, 19] include centroid [20, 21, 22], CPE (convex position estimation) [23], APIT (Approximate Point in Triangle) [18, 24], and DV-hop (Distance Vector-hop) [25, 26]. Among range-free localization methods, centroid and CPE are relatively simple, having low complexity, but they require a normal node to have at least three neighboring anchors. DV-hop algorithm can handle the case where a normal node has less than three neighbor anchors, and computation complexity is relatively low which saves lots of energy consumption. Considering these interesting and attractive advantages of DV-hop algorithm, we prefer to localization methods based on DV-hop algorithm in our aquaculture WSNs.
The rest of this paper is structured as follows: Section 2 gives a summary of related works and analysis premise of our model. A novel localization algorithm IMDV-hop is proposed in Section 3, after presenting the inferiors of original DV-hop algorithm. Performance analysis models of IMDV-hop accompanying with HTC scheme are also presented in Section 3. In Section 4, accurate analyses and validations of localization error, localization delay, and energy consumption are presented using NS-2 simulator, and performance comparisons of our model with other models are also proposed. Finally, concluding remarks and future work are presented in Section 5.
## 2 Related works
Literature reviews presented here are twofold: (1) references related to localization schemes based on DV-hop algorithm for WSNs and (2) references related to localization schemes for mobile WSNs.
### 2.1 DV-hop localization schemes
DV-hop scheme is an attractive and low-energy consumption localization scheme, which is the most general range-free localization scheme, basing on connectivity information between nodes. Many algorithms based on DV-hop have been proposed these years. Not similar to other range-free localization schemes, DV-hop algorithm can handle the case where a node has less three neighbor anchors. Hence, for computation simplicity and sustainable localization accuracy of DV-hop scheme, our work focuses on DV-hop-based scheme.
Three steps, localization information exchange phase for obtaining hop counters, average hop distance computation phase for every anchor, and estimated position phase using trilateration or maximum likelihood estimation method, are firstly proposed by Niculescu and Nath in [25, 26]. An improved DV-hop algorithm is proposed in [27] to reduce location error accumulated over multiple hops by using a differential error correction scheme. Difference or error between estimated distance of two anchors and actual distance of these anchors is calculated, and this error can be generalized to calculate estimated error of distance between unknown node and its nearest anchor. DV-Loc algorithm is proposed in [28], which is designed by use of Voronoi diagrams to limit the scope of flooding and error of computed positions to improve localization accuracy, that is, through improving accuracy of hop count. Anchors are previously divided into several levels. A Voronoi diagram is built based on position information of the first-level anchors firstly. Then, the second-level anchors compute average size of a hop like DV-hop. Other levels of anchors repeated these two steps until all levels of anchors are used. DDV-hop [29] and RDV-hop [30] localization schemes are proposed based on DV-hop algorithm, using a weighting method to determine a weighted distance-per-hop value for each normal node, which consumes additional energy for obtaining differential error or network topology separately. An advanced DV-hop (ADV-hop) localization scheme [31] uses the hop size of anchors, from which unknown node measures distance between anchors or between unknown node and anchors. Inherent error in estimated distance between anchor and unknown node is reduced in the third step of ADV-hop, and WLS algorithm is used, in which weight factor is set as the inverse of the minimum number of hops between unknown node and an anchor. And locations are refined by using extraneous information obtained by solving mathematical equations. A threshold M is introduced in [32], which uses weighted average hop distance of anchors within M hops, not all anchors in networks of original DV-hop scheme, to calculate average hop distance of unknown nodes, and location results are corrected of this DV-hop based scheme. HDV-hop (hybrid DV-hop) is proposed in [33] to obtain high localization accuracy and minimize flooding, and then reduce energy consumption, in which anchors are deployed only on the perimeter of WSN and not inside it. In the first step of HDV-hop, sensor S maintains HopCounti in a table called the HopCountsTable, and S maintains ID of one-hop neighbor from which it has received the minimum HopCount value, referring to this minimum value as GlobalMinimum, the neighbor that delivered this value to S as GlobalMinimumNeighbor. In the second step of HDV-hop, energy-expensive flooding phase is eliminated and each anchor sends its calculated average hop length to the base station via the RingofAnchors. A trilateration algorithm is executed by a powerful base station instead of individual sensors, and an iterative method to solve NLLS problem. Because of anchors are located on the perimeter of network, unknown node can consume lots of energy to establish or maintain DV-hop calculation chains.
Two refined localization algorithms, that is, hyperbolic-DV-hop localization algorithm and improved weighted centroid DV-hop localization scheme (IWC-DV-hop) are proposed in [34]. Instead of taking average hop size of the anchor nearest to unknown node, hyperbolic-DV-hop scheme chooses average hop sizes among all anchors as average hop size of unknown node. And also, selecting appropriate anchors and centroid scheme instead of maximum likelihood estimation scheme can improve accuracy for IWC-DV-hop algorithm.
Quad DV-hop and other two DV-hop-based schemes, iDV-hop1 and iDV-hop2, are proposed in [35] to improve localization accuracy. Quad DV-hop formulates localization problems as bounded least squares problems, to be solved by quadratic programming. Checkout DV-hop scheme and selected-3-anchor DV-hop are proposed in [36]. The former one adjusts position of a normal node based on its distance to the nearest anchor neighbor, in which a checkout step is added to change estimated position from NDV − hop to a new one called NCheckout, a relative accuracy value for computing the distance between unknown node and each anchor. The other one chooses the best three anchors based on connectivity parameters. Mostly, three anchors can sufficiently localize a normal sensor, rather than involving in all available anchors in the network. But how to choose appropriate three anchors to improve localization accuracy requires taking network topology into account. Especially, non-slotted CSMA/CA scheme are embedded in Checkout DV-hop scheme and selected-3-anchor scheme to solve problems of frame collisions and link congestion, which however frequently happen during the broadcast of position frames and distance-per-hop frames, almost not considered in original DV-hop algorithm or other DV-hop-based schemes. Sub-square weighted (SSW) DV-hop algorithm is proposed and applied with rectangular topology in O-WSN (hybrid optical WSN) [37], and various factors that affect localization accuracy of DV-hop-based algorithm in WSNs are investigated, including communication radius of a node, number of beacon nodes, and number of total nodes.
Most DV-hop-based schemes are designed for localizing static WSNs. But DV-hop-based localization scheme can also be used for mobile WSNs such as selected-3-anchor scheme in [36], and we can adopt DV-hop-based scheme to mobile networks in this work. Mobility behaviors of DV-hop-based scheme are then analyzed elaborately, and localization performance such as error and delay is also compared with MCL (Monte Carlo localization)-based scheme. Impacts of mobility models on DV-hop-based localization in mobile WSNs are presented in [38].
Moreover, there are many measurement means for DV-hop-based localization schemes for WSNs. In the third step of the original DV-hop scheme, calculation of estimated location is achieved using trilateration method or maximum likelihood estimation method. Most of the localization schemes based on DV-hop algorithm adopt these two calculation methods, such as [36]. And normal sensor computes hop size based on all hop size values it receives from anchors, instead of just taking the first received hop size value, that is, hop size value of the nearest anchor in [38]. So, positions of normal nodes can be calculated by using WLS method. As related above, ADV-hop in [31] also uses WLS algorithm to calculate positions of unknown nodes. And also, hyperbolic location algorithm is also used to obtain locations of normal nodes related in [34, 39].
### 2.2 Localization schemes for mobile WSNs
Most of the schemes related above focus on the localization for static sensor networks. However, mobile sensors are inevitably required in some applications such that target tracking, floats for sea environmental monitoring. Recently, many localization schemes are proposed for mobile sensor networks, and most of these algorithms are usually based on MCL method [40]. MCL algorithm has been extensively used in robotics [41], in which a robot estimates its location based on its motion, perception, and possibly a pre-learned map of its environment. Each step of MCL algorithm is divided into three steps: an initialization phase, a prediction phase, and an update phase. A node makes a movement and uncertainty of its position increases in prediction phase, and new measurements are incorporated to filter and update data in update phase. SMC (sequential MCL) algorithm is firstly used for mobile sensor networks in [42], without a predefined map or special hardware, assuming that time is divided into discrete time units. Localization process of SMC for WSNs is briefly related as follows. Each node localizes itself in each time interval. A sensor randomly chooses a set of N samples $${L}_0=\left\{{l}_0^0,{l}_0^1,\dots, {l}_0^{N-1}\right\}$$, treating as localizations, within deployment area during localization initial phase. Then, SMC algorithm executes two steps to locate unknown nodes, the same as algorithm of MCL, prediction and filtering. A sensor generates a new set of localizations Lt based on previous set Lt − 1 at random time t during the prediction. Random location $${l}_t^i$$ is randomly chosen from the disk of radius vmax round $${l}_{t-1}^i$$ on the premise of given location $${l}_{t-1}^i$$ from Lt − 1, and vmax is the maximum speed of nodes. During filtering phase, all impossible locations $${l}_t^i$$ are deleted among new localizations Lt, using position information obtained from both one-hop and two-hop anchors. If a node cannot hear from its neighbor anchors, or not enough samples can be obtained, or too samples are neglected, localization error can be introduced in SMC. MCB (Monte Carlo localization boxed) is proposed in [42] to improve energy behaviors in SMC, in which drawing samples can easily swallow up lots of energy. In MCB scheme, to filter redundant samples, a node that heard one-hop or two-hop neighbor anchors builds a box which covers the region where anchors’ radio range overlap. Location of a node can be not only constrained from anchors within one-hop and two-hop but also from its one-hop normal nodes in [43], rather than localization information is constrained only from anchors within one-hop or two-hop, such as SMC schemes related above. And also, impossible position samples are filtered out, through predicting moving direction, in which three mobility models are introduced in IMCL (improved MCL) scheme. An energy-efficient algorithm WMCL (weighted MCL), a further reducing the size of bounding box used, is proposed in [44] to achieve both high sampling efficiency and high localization accuracy, which most SMC schemes suffer from. Moreover, WMCL can achieve high localization accuracy no matter static networks or mobile networks, especially for relatively high mobile velocity. COMCL (Constraint rules Optimized Monte Carlo Localization Scheme) is proposed in [45] to improve accuracy or reduce bounding box further. A set of rigorous constraint conditions is generated after the upper and lower bounds of optimized constraint rules have been solved for each neighbor seed of the node. And then, a bounding box could be built as sampling area according to these constraint conditions. Constraint conditions can also be used in filtering phase for the proposed scheme so that COMCL adapts more strict filtering conditions than other previous SMC-based localization schemes.
All MCL-based localization schemes, there are two main inferiors generally. The first is that sampling and filtering iteratively consume lots of energy. The other is that higher localization accuracy is achieved through increasing the number of anchors. These bring out extraordinary consequences, especially for WSNs, energy-constraint and economic-constraint applications. Thus, there are some improved spaces for MCL-based localization for mobile WSNs. Besides MCL methods used for mobile WSNs, there are several other localization methods for mobile WSNs, such as localization scheme based on convex method, localization based on geometric constraints, and localization based on perpendicular bisector of a chord [46, 47, 48].
In this paper, an improved cross-layer DV-hop localization scheme IMDV-hop is proposed for real-time large-scale aquaculture monitoring and localization system, accompanying with HTC scheme [49] in mobile WSN. At first, inferiorities of original DV-hop scheme are denoted elaborately after brought briefly about it. Then, an improved localization scheme embedded in WLS is proposed based on mitigating these inferiorities from four aspects. Firstly, HTC scheme is combined into IMDV-hop scheme. Secondly, hop count hopi decreases sharply for adopting HTC algorithm, consequently increases accurate of average distance per hop dphi. Thirdly, distances between known node and anchors adopting dphi of each anchor, rather than adopting dphnear which refer to average distance per hop of the nearest anchor. Finally, two critical parameters, correction coefficient kc and weighted coefficient $${w}_{N_x,i}$$, are introduced into IMDV-hop scheme to improve localization performance. And then, localization behaviors such as error, delay, and energy consumption are validated adopting NS-2 simulations, taking parameters describing the network into account, such as ratio of anchors (p), node density (λ), velocity of nodes (v), localization window (LW), and transmission range (R). Moreover, comprehensive performance comparisons between IMDV-hop and other DV-hop-based schemes and performance comparisons between IMDV-hop and MCL-based schemes are proposed.
The main contributions in this paper are threefold. Firstly, cross layer time-critical localization scheme which combines HTC scheme with IMDV-hop localization scheme embedded a modification of WLS method is proposed. IMDV-hop localization scheme attenuates inferiors of original DV-hop scheme from four aspects. Secondly, comprehensive simulations are presented to validate performance evaluations taking account of p, λ, v, LW, R. Finally, comparisons between IMDV-hop scheme and other DV-hop-based schemes or MCL-based mobile localization schemes are proposed to validate delay superiority of this time-critical scheme IMDV-hop for static or mobility monitoring networks.
## 3 Methods/experimental
Firstly, we briefly explain the original DV-hop scheme as well as some pending improvements in this scheme. Then, we present our improved localization scheme IMDV-hop-based DV-hop algorithm, accompanying with HTC algorithm for mobile WSNs. The aim of IMDV-hop is to improve the delay behavior for large-scale mobile localization networks, without reducing localization accuracy and energy-saving.
### 3.1 The original DV-hop algorithm
DV-hop localization algorithm, a range-free-based localization scheme, was firstly proposed by Niculescu [25, 26]. It is a suitable solution for normal nodes having three or less neighbor anchors. There are two types of nodes in DV-hop scheme, and one type is anchor nodes equipped with GPS which can obtain their location information. The other is normal nodes pending to be localized. As shown in Fig. 1, although the pending localized node Nx has no direct neighbor or reachable anchor, Nx can use DV-hop algorithm to be localized. DV-hop scheme consists of following three steps:
Step 1: First, each anchor Ai broadcasts a message throughout network containing position of Ai and a hop count field hopi set to 0. Value of hopi increases with the hop during the message broadcasting, which means, hop count value hopi in the message will be incremented as soon as a node receives this message. Every node N (either anchor or normal node) records position of Ai and initializes value of hopi as hop count value in the message. And hopi is the minimum hop count between N and Ai. If the same message is received again, node N maintains hopi, and if this received message contains a lower hop count value than hopi, N will update hopi with that lower value and relay the message. Otherwise, N will ignore the message. Through this mechanism, each node can obtain the minimum hop count hopi between each anchor and it separately.
Step 2: Second, when an anchor Ai receives positions of other anchors as well as the minimum hop counts to other anchors, Ai can calculate its average distance per hop, which is denoted as dphi. dphi is calculated as Eq. 1.
$${dph}_i=\frac{\sum_{j=1,j\ne i}^M\sqrt{{\left({x}_i-{x}_j\right)}^2+{\left({\mathrm{y}}_i-{\mathrm{y}}_j\right)}^2}}{\sum_{j=1,j\ne i}^M{hop}_{i,j}}$$
(1)
In Eq. 1, M is the number of anchors in the network, and node j refers to other anchor different from anchor i. hopi, j is the minimum distance between anchor i and anchor j measured by hops. (xi, yi) and (xj, yj) refer to coordinates of anchor i and anchor j, respectively. Once dphi is calculated, it will be broadcasted by Ai. And then, all anchors can obtain all other anchors’ dphi, but unknown nodes Nx can only maintain value of dphi broadcasted by the nearest anchor Anear (either one-hop or higher hop neighbor anchor).
When receiving dphi of Anear, Nx can obtain distances to each anchor Ai (including Anear), which multiplies $${hop}_{i,{N}_x}$$ (its distance to Ai by hop count) by dphnear (the average distance one hop dphi for Anear). This distance is denoted as $${d}_{i,{N}_x}$$. Then, M distances are obtained by node Nx, which refers to $${d}_{1,{N}_x}$$, $${d}_{2,{N}_x}$$, $${d}_{i,{N}_x}$$, till $${d}_{M,{N}_x}$$.
$${d}_{i,{N}_x}={dph}_{near}\times {hop}_{i,{N}_x}$$
(2)
Note that all multiple factor dphnear is the same value in Eq. 2, which is the average distance one hop for Anear from Nx.
For example, anchor A1 is 40 m and two hops away from anchor A2, 100 m and six hops away from anchorA3, and 40 m and three hops away from anchor A4 in Fig. 1. A1 can calculate dph1 using Eq. 1. Hence, dph1 is equal to (40 + 100 + 40)/(2 + 6 + 3) = 16.36 m. In the similar way, A2, A3, and A4 calculate the value of dph2, dph3, anddph4, as (40 + 75 + 60)/(2 + 5 + 3) = 17.5 m, (100 + 75 + 95)/(6 + 5 + 5) = 16.88 m, and(40 + 60 + 95)/(3 + 3 + 5) = 17.73 m, respectively. Then, each anchor Ai(i = 1, 2, 3, 4)broadcasts its dphi in the network, so other anchors and normal sensors receive it. Unknown node Nx will maintain the value of dph2, for A2 is the nearest anchor to it, and calculate the distance away from A1, A2, A3, and A4, as 17.5 × 3 = 52.5 m, 17.5 × 2 = 35 m, 17.5 × 3 = 52.5 m, and 17.5 × 2 = 35 m, respectively.
Step 3: Third, when receiving the distance of Nx and Ai, unknown node Nx can calculate its estimated position by trilateration or other arithmetic methods as follows: (xi, yi) in Eqs. 3 and 4 is the coordinate of anchor Ai.
$$\left\{\begin{array}{l}{\left({x}_1-x\right)}^2+{\left({y}_1-y\right)}^2={d}_1^2\\ {}\dots \\ {}{\left({x}_M-x\right)}^2+{\left({y}_M-y\right)}^2={d}_M^2\end{array}\right.$$
(3)
$$A=\left[\begin{array}{l}2\left({x}_1-{x}_M\right)\kern1.25em 2\left({y}_1-{y}_M\right)\\ {}\dots \\ {}2\left({x}_{M-1}-{x}_M\right)\kern0.75em 2\left({y}_{M-1}-{y}_M\right)\end{array}\right],B=\left[\begin{array}{l}{x}_1^2-{x}_M^2+{y}_1^2-{y}_M^2+{d}_M^2-{d}_1^2\\ {}\dots \\ {}{x}_{M-1}^2-{x}_M^2+{y}_{M-1}^2-{y}_M^2+{d}_M^2-{d}_{M-1}^2\end{array}\right],X=\left[\begin{array}{l}x\\ {}y\end{array}\right]$$
Coordinate (x, y) of unknown node Nx can be obtained through standard minimum mean variance estimation method as:
$$X={\left({A}^TA\right)}^{-1}{A}^TB$$
(4)
### 3.2 The motivations for the improved algorithms
DV-hop algorithm, range-free localization scheme as related above, can localize unknown nodes with less than three neighbor anchors and obtain relative satisfied localization errors with less complexity and less additional hardware. But there are some defects for original DV-hop scheme, and localization performance does not come up to expectations.
Firstly, hop count value hopi can be incremented with message broadcasting if its received hop count is less than the former maintained one, which is related in step 1 of the original DV-hop scheme. Hop count is incremented so long as the message is broadcasted once, no matter if the next node is in the transmission range or not, no matter how node density is. This leads to over-estimate hop count value hopi if node density is relatively high, and subsequently, average hop distance dphi is underestimated adopting Eq. 1. For example, hop count between anchor A1 and anchor A3 is 6, and geometrical distance between A1 and A3 is 100 m, while hop count between anchor A4 and anchor A3 is 5, and geometrical distance between A4 and A3 is 95 m. If hop count is increased with broadcasting, real distance between A1 and A3is underestimated.
Secondly, it cannot decide to select which anchor to calculate value of dphnear if there are several anchors with the same distance away from Nx. In step 2 of the original DV-hop scheme, node Nx maintains dphi, which is the nearest anchor Anear to it, and calculates distances away from all anchors with this value of dphi (dphnear). That is to say, dphnear is an important factor in localization performance. In Eq. 2, distances of Nx away from other anchors can be obtained through hop distance multiplying by dphnear, and all anchors adopt the same value of dphi no matter which distance away from this normal Nx. However, this uniformdphi can bring out a plenty of localization errors. For example, Nx has two two-hop neighbor anchors A2 and A4 in Fig. 1, and it cannot decide which one it maintains dph2or dph4. If Nx maintains dph2, distance ofNxaway fromA1, A2, A3, andA4 is 17.5 multiplied by 3, 2, 3, and 2, respectively. While Nx maintains dph4, distance of Nx away from A1, A2, A3, and A4 is the value of 17.73 multiplied by 3, 2, 3 and 2, respectively. Thus, distance $${d}_{2,{N}_x}$$ is relatively accurate, and other distances are relatively inaccurate.
Thirdly, network connectivity also plays an important role in DV-hop localization scheme. Distance between two nodes (either anchor or normal node) is represented by hops, rather than by geometrical distance. Network connectivity exerts a tremendous influence on hop distance, which means hop distance can bring about greater inaccuracy if network connectivity is relatively low. For example, hop distance between anchor A4 and anchor A1 is three hops through broadcasting in Fig. 1, but geometrical distance is only 40 m in fact, almost two hops away from each other.
Finally, unknown node can obtain its location through the method of trilateration, which need three known anchors at least, no matter where anchors located. Hops can be enlarged if anchor density is relatively low, which brings about value of dphnear lower than factual one. And also, unknown node tries to seek farther anchors through two or more hop relay transmission, which can bring about a plenty of unnecessary energy consumption.
These inconveniences can be overcome through several modified method. Firstly, hop counts between nodes, either between an anchor and unknown node or between an anchor and another anchor, are decreased through adopting HTC scheme. Secondly, localization messages can be exchanged in one-hop or two-hop transmissions, which decreases energy consumed in plenty of broadcast. Then, network connectivity can be improved through increasing node density. And also, model’s accuracy is improved largely through involving in two parameters kc and $${w}_{N_x,i}$$. Lastly, $${d}_{i,{N}_x}={dph}_i\times {hop}_{i,{N}_x}$$ instead of $${d}_{i,{N}_x}={dph}_{near}\times {hop}_{i,{N}_x}$$ is used to calculate the distance of Nx and Ai, accompanying with WLS method.
### 3.3 Method
Several assumptions are presented before deriving IMDV-hop models:
1. 1.
Nodes, normal nodes (Nx denoted as in IMDV-hop) or anchors (A denoted as in IMDV-hop), are randomly located in circle plane, according to a two-dimensional Poisson distribution with a density of λ. Parameters λ and p denote the density of Poisson distribution for nodes including anchors and unknown nodes and the ratio of anchors, respectively. We assume identical range R of communication, interference, and carrier sensing. That is, the number of anchors can be denoted as Nanchor(hop = 1) = λpπR2 in one-hop transmission range, and the number of normal nodes can be denoted asNx(hop = 1) = λ(1 − p)πR2.
2. 2.
Time is divided into discrete intervals, and a sensor node localizes itself in each localization window LW, which is constituted by several intervals. Environment parameters can be collected and transmitted periodically, that is, at the beginning of each LW. Nodes can also execute localization scheme IMDV-hop at the beginning slot of each LW. And so, we can simplify mobile localization as relatively static localization process in each LW, and mobile patterns in eachLW can be ignored, which are related elaborately in [37].
3. 3.
In our monitoring aquaculture network, we pay close attention to environmental information such as culture temperature, PH value, or environment oxygen content and exact locations which these information are brought about. Antennas of nodes are floating upon the water, and information collecting parts of them are soaking under the water, which can be presented in Fig. 3. Thus, all signal transmissions happen upon the water, without taking signal attenuation into account.
4. 4.
Nodes including anchors regardless of locations are bestowed on the fair chance to transmit messages with CSMA/CA scheme to its neighbors through one-hop or two-hop transmission heterogeneously and non-preemptively, as similar to the HTC algorithm [49].
And now, IMDV-hop scheme accompanying with HTC algorithm is presented elaborately.
#### 3.3.1 Step 1: localization request
First, LW and hopi are initialized, which are set to LW0 and 0, respectively. After network initialization completing, each node (either anchor or unknown node) can be aware which hop belongs to this time slot (that is, in this LW period). Also, all nodes can establish their one-hop neighbor list and update it every slot of LW. As localization request phase starts up, an unknown node Nx in its one-hop circle (d < R, which can be seen from Fig. 2) can confirm its one-hop anchor neighbors A1i (anchors in one-hop) in its transmission range just in one-hop circle. And then, node Nx will send a HTC frame to anchors A1i, and HTC frame contains the value of LW and all one-hop neighbors including one-hop anchor neighbors A1i of Nx. If the number of A1i is greater than or equal to 3, node Nx can directly go to execute DV-hop scheme as original DV-hop scheme. If the number of A1i is less than 3, it needs to spread its HTC frame to two-hop neighbors. And also, two-hop neighbors including anchor neighbors A2i (anchors in two-hop of Nx) are added to this frame. If the sum of A1i and A2i is greater than or equal to 3, localization scheme can be promoted. Otherwise, this localization can be a failure.
It is to be noted that we execute IMDV-hop scheme in two-hop networks (either two-hop of Nx or two-hop of anchors) for several reasons. Firstly, HTC algorithm can be presented the most of the advantage in two-hop networks for avoiding transmission collisions. Secondly, node density of this work is relatively high, which can satisfy the condition of the number of Ai is greater than or equal to 3 in two-hop circle. The last and foremost reason is that node Nx or anchor Ai need not to establish or maintain routings to transmit information of hop count, hop distance, or environmental parameters, which can save overwhelming majority of energy in WSNs. After this request phase accomplishes, unknown node Nx can obtain its one-hop and two-hop neighbors including anchor neighbors, and its one-hop and two-hop neighbor anchors can obtain its HTC frame.
#### 3.3.2 Step 2: hop information exchange
Unlike broadcasting HopCount message to obtain the minimum hop count as original DV-hop scheme, IMDV-hop fusing HTC algorithm can obtain hop count as follows: After localization request phase finishes, one-hop anchor neighbor A1i broadcasts a HTC frame containing location of A1i, ID of A1i, circumstance parameters such as culture temperature, PH value, and environment oxygen content, and also a hop count field hopi. There are three cases for obtaining hopi. The first one is that the number of anchor nodes in the intersection area for one-hop neighbors of A1i and one-hop neighbors of Nx is greater than or equal to 3, each minimum hop count hopi is set to 1, and the sum ∑hopij(i ≠ j) is set to the number of other anchors (case 1). If the number of anchors in the intersection area for one-hop neighbors of A1i and one-hop neighbors of Nx is less than 3, hop count hopi is set to 1 if other anchors in one-hop of A1i, and set to 2 if other anchors in two-hop of A1i (anchors in one-hop). Sum ∑hopij(i ≠ j) is set to $$\left({N}_{A_{1i}}+2{N}_{A_{2i}}-1\ \right)$$ (case 2). Of course, if number of anchors in one-hop of Nx is less than 3, IMDV-hop can calculate the location of Nx by expanding to anchors A2i in two-hop of Nx, and ∑hopij(i ≠ j) is the same as the second one as related above (case 3). And also, each anchor involved in localization calculation can be a router to transmit monitoring information, which performs real-time environmental monitoring in our aquaculture sensor networks.
$$\sum {hop}_{ij}\left(\mathrm{i}\ne \mathrm{j}\right)=\left\{\begin{array}{l}{N}_{A_{1i}}-1\kern8.5em \mathrm{case}\ 1\\ {}{N}_{A_{1i}}+2{N}_{A_{2i}}-1\kern5.5em \mathrm{case}\ 2,3\ \end{array}\right.$$
(5)
$$\left\{\begin{array}{l}{N}_{A_{1i}}=4 p\lambda {\int}_{d/2}^{2R}\sqrt{4{R}^2-{x}^2} dx\kern12.75em \\ {}{N}_{A_{2i}}= p\lambda \pi {R}^2-4 p\lambda {\int}_{d/2}^{2R}\sqrt{4{R}^2-{x}^2} dx-\\ {}\kern2.5em 2 p\lambda \left({\int}_{\left(3{R}^2-{d}^2\right)/2d}^R\sqrt{R^2-{x}^2} dx-{\int}_{d+\left(3{R}^2-{d}^2\right)/2d}^{2R}\sqrt{4{R}^2-{x}^2} dx\right)\ \Big)\kern1.5em \end{array}\right.$$
#### 3.3.3 Step 3: distance calculations
Like the calculation of distances between anchors and pending localized nodes, anchors in IMDV-hop fusing HTC algorithm can obtain distances as follows: After A1i or A2i receive positions of other anchors as well as the minimum hop count hopi, anchor A1i or A2i can calculate its average distance per hop, which is denoted as dphi. In Eq. 6, (xi, yi) and (xj, yj) refers to coordinates of anchor i and anchor j, respectively. Once dphi is calculated, it will be broadcasted by A1i or A2i, and then, all anchor nodes can obtain all other anchors’ dphi as Eq. 6.
$${dph}_i=\frac{\sum_{i\ne j}\sqrt{{\left({x}_i-{x}_j\right)}^2+{\left({\mathrm{y}}_i-{\mathrm{y}}_j\right)}^2}}{\sum_{i\ne j}{hop}_{ij}}$$
(6)
Unknown node Nx can maintain dphi broadcasted by all anchors A1i or A2i, not similar to the value of dphi in original DV-hop for node Nx. When receiving dphi, unknown node Nxcan calculate the distance to each anchor A1i or A2i, which is denoted as $${dph}_{N_x,i}$$. Distance $${dph}_{N_x,i}$$ is the value of dph1i if anchor A1i is in one-hop of Nx, and 2dph2i if anchor A2i is in two-hop of Nx. That is to say, value $${dph}_{N_x,i}$$ is the multiplication of dphi (distance per hop for anchor Ai) and parameter 1 or 2 (its distance to Ai by hop count), which is not similar to the original DV-hop.
$${dph}_{N_x,i}=\left\{\begin{array}{l}{dph}_{1i}\kern4em \mathrm{case}\ 1\\ {}{2}^{\ast }{dph}_{2i}\kern2.75em \mathrm{case}\ 2,3\kern1em \end{array}\right.$$
(7)
#### 3.3.4 Step 4: location calculations
In this algorithm, we can introduce a correction coefficient kc to improve localization accuracy of distances between unknown node Nx and anchors, which is shown in Eq. 8.
$${k}_c=\frac{dph_{est}^{i,j}-{dph}_{true}^{i,\mathrm{j}}}{dph_{true}^{i,j}}\left|{}_{i\ne j}\right.$$
(8)
In Eq. 8, actual distance $${dph}_{true}^{i,j}$$ of each anchor can be calculated using geometric method involving actual coordinates, $${dph}_{true}^{i,j}=\sqrt{{\left({x}_i-{x}_j\right)}^2+{\left({y}_i-{y}_j\right)}^2}$$ and estimation distance $${dph}_{est}^{i,j}$$ is calculated through Eq. 1 or Eq. 6. In the same way, distance of Nx and anchors can be calculated using Eq. 9. Actual distance $${dph}_{true}^{N_x,i}$$ and estimation distance $${dph}_{est}^{N_x,i}$$ of Nx and anchors can be calculated as related above. We can consider the difference between $${dph}_{true}^{N_x,i}$$ and $${dph}_{est}^{N_x,i}$$ as the similar to difference between $${dph}_{true}^{i,j}$$ and $${dph}_{est}^{i,j}$$ for our uniform randomly networks.
$${dph}_{true}^{N_x,i}=\frac{dph_{est}^{N_x,i}}{1+{k}_c}$$
(9)
If the coordinate of unknown node Nx is denoted as (x, y), the location of Nx is then calculated by using following system of Eqs. 10 and 11, in which $${dph}_{true}^{N_x,i}$$ can be presented as a simple form di. And n is the number of anchors, that is, the simple form for $${N}_{A_i}$$.
$$\left.\begin{array}{l}\sqrt{{\left(x-{x}_1\right)}^2+{\left(y-{y}_1\right)}^2}={d}_1\\ {}\sqrt{{\left(x-{x}_2\right)}^2+{\left(y-{y}_2\right)}^2}={d}_2\\ {}\dots \\ {}\sqrt{{\left(x-{x}_n\right)}^2+{\left(y-{y}_n\right)}^2}={d}_n\end{array}\right\}$$
(10)
$$\left.\begin{array}{l}\sqrt{{\left(x-{x}_1\right)}^2+{\left(y-{y}_1\right)}^2}-\sqrt{{\left(x-{x}_n\right)}^2+{\left(y-{y}_n\right)}^2}={d}_1-{d}_n\\ {}\sqrt{{\left(x-{x}_2\right)}^2+{\left(y-{y}_2\right)}^2}-\sqrt{{\left(x-{x}_n\right)}^2+{\left(y-{y}_n\right)}^2}={d}_2-{d}_n\\ {}\dots \\ {}\sqrt{{\left(x-{x}_{n-1}\right)}^2+{\left(y-{y}_{n-1}\right)}^2}-\sqrt{{\left(x-{x}_n\right)}^2+{\left(y-{y}_n\right)}^2}={d}_{n-1}-{d}_n\end{array}\right\}$$
(11)
Squaring both sides and simplifying Eq. 11, we can obtain Eq. 12 as the same as [30].
$$\left.\begin{array}{l}-2x\left({x}_1+{x}_n\right)-2y\left({y}_1+{y}_n\right)+2\left({x}^2+{y}^2\right)={d_1}^2+{d_n}^2-\left({x}_1^2+{y}_1^2+{x}_n^2+{y}_n^2\right)\\ {}-2x\left({x}_2+{x}_n\right)-2y\left({y}_2+{y}_n\right)+2\left({x}^2+{y}^2\right)={d_2}^2+{d_n}^2-\left({x}_2^2+{y}_2^2+{x}_n^2+{y}_n^2\right)\\ {}\dots \\ {}-2x\left({x}_{n-1}+{x}_n\right)-2y\left({y}_{n-1}+{y}_n\right)+2\left({x}^2+{y}^2\right)={d_{n-1}}^2+{d_n}^2-\left({x}_{n-1}^2+{y}_{n-1}^2+{x}_n^2+{y}_n^2\right)\end{array}\right\}$$
(12)
$$Q=\left[\begin{array}{l}-2x\left({x}_1+{x}_n\right)\kern1em -2y\left({y}_1+{y}_n\right)\kern1.5em 1\\ {}-2x\left({x}_2+{x}_n\right)\kern1em -2y\left({y}_2+{y}_n\right)\kern1.25em 1\ \\ {}\dots \\ {}-2x\left({x}_{n-1}+{x}_n\right)\kern0.5em -2y\left({y}_{n-1}+{y}_n\right)\kern0.75em 1\end{array}\right]$$,$$H=\left[\begin{array}{l}{d_1}^2+{d_n}^2-\left({x}_1^2+{y}_1^2+{x}_n^2+{y}_n^2\right)\\ {}{d_2}^2+{d_n}^2-\left({x}_2^2+{y}_2^2+{x}_n^2+{y}_n^2\right)\ \\ {}\dots \\ {}{d_{n-1}}^2+{d_n}^2-\left({x}_{n-1}^2+{y}_{n-1}^2+{x}_n^2+{y}_n^2\right)\ \end{array}\right]$$$$, Z=\left[\begin{array}{l}x\\ {}y\\ {}k\end{array}\right]$$
$$QZ=H$$
(13)
WLS algorithm is adopted to solve the coordinate of Nx in our IMDV-hop scheme, to improve location accuracy. In WLS method, unknown parameters in Eq. 14 can be presented as:
$$Z={\left({Q}^{\hbox{'}}{W}^{\hbox{'}} WQ\right)}^{-1}{Q}^{\hbox{'}}{W}^{\hbox{'}} WH$$
(14)
In which, W is the weighted matrix which presents the influence of distance between anchors and unknown node Nx, transmission range, and distances between an anchor and other anchors. For example, an anchor is farther away from unknown node Nx, greater error is brought in localization for Nx. Thus, the weight of this anchor can be set to a lower value. And also, transmission range and distance between an anchor and other anchors play an important role in localization error.
$$W=\left[\begin{array}{l}{w}_{N_x,1}\kern1em 0\kern1em \dots \kern1.5em 0\kern1em \\ {}0\kern1.5em {w}_{N_x,2}\kern0.5em \dots \kern1.5em 0\kern0.75em \\ {}\dots \\ {}0\kern2em 0\kern1em \dots \kern1.25em {w}_{N_x,n-1}\ \end{array}\right]$$
(15)
Weight $${w}_{N_x,i}$$ in Eq. 15 is taken as the inverse of the minimum number of hops between each anchor and Nx in [31, 50]. Not only the minimum number of hops as [31, 50], the weight $${w}_{N_x,i}$$ should also take transmission range R and hopij into account in this work, which is demonstrated as Eq. 16.
$${w}_{N_x,i}=\frac{1}{hop_{N_x,i}}\frac{1}{N_{A_i}-1}{\sum}_{k=1}^{N_{A_i}-1}\left(1-\frac{{hop_{ik}}^2}{R^2}\right)$$
(16)
In Eq. 16, the number of anchors can be presented as $${N}_{A_i}$$ as shown in Eq. 5. $${hop}_{N_x,i}$$ and hopik denote distance between Nx and anchor i and distance between anchor i and other anchor k, respectively. Weight values influenced by these three parameters play important roles on localization accuracy, which can be analyzed through simulations in Section 4. It is to be noted that hopik can be denoted as 1 if anchork in one-hop of anchor i or 2 if anchor k in two-hop of anchori, respectively.
IMDV-hop scheme can be illustrated as flow chart in Fig. 3. When an unknown node Nx is being in the state of pending localization, it will execute four steps complying with the step of Fig. 3. In IMDV-hop scheme, Nx initializes and maintains LWand HTC table, and anchors initialize and maintain their coordinates, IDs, hopi, LW and HTC table. At first, Nx initializes LW = LW0 and HTC table (steps 1–3). After localization request finishes (step 4), IMDV-hop goes to localization stage. Anchors involving one-hop and two-hop broadcast and exchange their initialization information containing their coordinates, IDs, hopi, and HTC table (steps 5–6). After obtaining hop counts, anchors can calculate its distance of each hop (step 7). Receiving hopi broadcasting by each anchor, unknown node Nx calculates the distance away from each anchor, which multiples value hopi of each anchor by the hop away from this anchor, involving kc (step 8). Then, node Nx executes WLS method to obtain its location (step 9). If LW is decreasing to 0, localization of this period fails.
### 3.4 Experimental
As assumptions related in Section 3.3, nodes’ structural representation in our aquaculture WSNs are illustrated in Fig. 4a, with antennas fixing upon the float, which floats upon water. Perceptive part is soaking under water, which collects circumstance information. Sensor nodes collect environmental parameters, such as culture temperature, PH value, or environment oxygen content, and transmit this information to sinks shown in Fig. 4b. Simultaneously, locations of normal nodes can also be obtained through executing IMDV-hop scheme.
Physical node in this aquaculture network is illustrated in Fig. 5, with the antennas fixing upon the float, and sensors immerge in the water to apperceive and collect circumstance parameters. And sensors including anchors and unknown nodes are floating randomly in aquaculture farm, without taking water currents and waves into account.
We present extensive simulations to validate the accuracy of evaluated metrics for localization error, delay, and energy consumption of IMDV-hop scheme using NS-2 simulator according to previous analyses. NS-2 is a popular discrete-event simulator which was originally designed for wired networks and has been subsequently extended to support wireless simulations. Localization accuracy, delay, and energy consumption are validated through extensive comprehensive simulations which are derived based on analyses of different parameters such as p, λ, v, and LW. And also, comprehensive behavior comparisons between IMDV-hop scheme and DV-hop-based localization schemes are proposed to validate some delay superiority of this time-critical scheme IMDV-hop. And performance comparisons of IMDV-hop with MCL-based localization scheme are also involved in a certain extent, according to velocity of nodes.
Nodes are randomly located in the circle according to a two-dimensional Poisson distribution as Fig. 6, and the case of three-dimensional distribution, such as the localization scheme of [6, 51], cannot take account in this paper. Experimental setups of NS-2 simulator used to conduct validations are similar to presentations in [49] in detail, and parameters involved in CSMA/CA scheme, such as UnitSuperframeSlot, BaseSlotDuration, and packet length, are also similar with those of [49], and other specific parameters touched upon simulations, such as LW and v, are listed on the head of result figures. It is noted that LW can be scaled by the number of UnitSuperframeSlot. Propagation delay can be ignored in our scheme simulations. We validate performance of our IMDV-hop firstly. Then, we compare performance of our scheme with that of previous schemes such as ADV-hop [31], HDV-hop scheme [33], and Selective-3-anchor scheme [36]. And also, mobility performance of IMDV-hop scheme is compared with that of WMCL scheme [44]. Our simulation results are the mean values derived from 30 experience values for each scenario.
## 4 Results and discussion
### 4.1 Localization error validations
As related in Section 3.3 above, localization error in IMDV-hop is related to correction coefficient kc and weight factor $${w}_{N_x,i}$$, which ultimately is related to the minimum hop count, hop size, number of anchors, and LW (in fact, intermittent mobile period). And also, we take v into account for localization error. Symbols of “sim” and “ana” in all figures are denoted as abbreviations of simulation results and analysis results, respectively.
With p increasing, error decreases as shown in Fig. 7a. Either HTC algorithm or IMDV-hop scheme, high anchor ratio means that anchors in one-hop of Nx are enough for localizing unknown node, dispensing with spreading HTC table to two-hop neighbors for normal nodes in localization stage and broadcasting hop information to two-hop anchor neighbors for anchors in information exchange stage. And also, with p increasing, weight $${w}_{N_x,i}$$ of each anchor decreases, each distance between an anchor and unknown node can only play an insignificant role on localization accuracy. Moreover, with anchor number increasing, accuracy of correction coefficient kc increases, consequently location for Nx has more accuracy shown in Eqs. 8 and 16.
Node density plays the similar role on localization behaviors as that of anchor ratio shown as Fig. 7b. As LW is increasing, the error decreases. If each localization period is not enough to transmit localization information, unknown node or anchors need to retransmit these messages or decide localization failure in this period. And also, nodes maybe are wasted time after localized when LW is too high. For example, error of LW = 40 is higher than that of LW = 20and LW = 10. With anchor ratio increasing, error for LW = 10 varies sharply in Fig. 7b.
We only consider low velocity or intermittent mobility for nodes in our aquaculture networks. Velocity is more than 5, error increases sharply, and then stays in a relatively high state shown in Fig. 7c. And error performance presents some independent behaviors with the increasing of node or anchor density, mobility velocity is shown in Fig. 7a–c. For example, error maintains a stable value when anchor ratio is higher than 20.4% at λ = 0.04.
### 4.2 Delay validations
Delay is the most important character in our time-critical localization system, and we always attempt to improve behavior of delay in order to obtain real-time monitoring and localizing, ensuring the health of culture objects. As anchor ratio is increasing, localization delay decreases. One-hop anchors in one-hop of Nx are greater than or equal to 3; Nx needs to transmit its HTC table or LW only once, which saves plenty of time shown in Fig. 8a.
If node density increases, more nodes including anchors will access the channel simultaneously, which leads to retransmission increasing. Consequently, delay behavior will become inferior which is demonstrated in Fig. 8b. And also, relative optimal value of LW is between 10 and 20, that is, LW is less than 10, unknown node has no enough time to localize itself in one period LW. On the other hand, unknown node need plenty of time to wait next intermittent mobile period to be localized if LW is too large, such as LW = 40 in Fig. 8b.
With v increasing, localization delay increases. When v is more than 5, delay increases sharply, especially when anchor ratio is relatively low as shown in Fig. 8c. IMDV-hop scheme presents relatively weak mobility behaviors as shown in Fig. 8c. And, delay largely depend on the value of v, especially when v > 5 m/s.
### 4.3 Energy validations
Energy consumption is an important metric in WSNs, and we also analyze it elaborately. With anchor ratio increasing, localization energy decreases. Shown in Fig. 9a, energy consumption in one LW presents an optimal behavior for λ = 0.04 when anchor ratio is less than 12.56%.
### 4.4 Preformation comparisons
Analysis and simulation results shown above are comprehensive for applications, and we can compare performance metrics of IMDV-hop mechanism with those of other DV-hop schemes. IMDV-hop is used for time-critical monitoring and detection application, in which minimized delay is the most important target. Through comprehensive comparisons, we can derive that delay performance metrics of IMDV-hop scheme accompanying with HTC algorithm are obviously improved over other schemes such as [31, 33, 36], while localization error and energy efficiency are improved over others on the conditions of more node density and less mobility velocity.
ADV-hop in [31] reduces localization errors using WLS method and other improved methods. ADV-hop scheme takes the weight as the inverse of the minimum number of hop count between unknown node and each anchor. In fact, weight factor has also business with anchor number, transmission range and hop counts among anchors. So, ADV-hop can present some inferior performance for some occasions, especially for delay behavior.
Three estimated distance values away from three different anchors are sufficient for unknown node to calculate its location related in selective 3-anchor DV-hop (selective-3-anchor) [36]. Based on the first two steps of original DV-hop, an unknown node can obtain a group of candidates to calculate its location. Then, it chooses the best 3-anchor group to establish estimated position using iterative method. Of course, this best 3-anchor group can also include two-hop neighbor anchors or higher hop anchors to participate in the trilateral method. And also, computational complexity of searching out this best 3-anchor group using iterative method is O(m3), in which m is the number of anchors. Relatively high computational complexity consumes plenty of energy, which is unsuitable for WSNs and is also incomparable to IMDV-hop.
Hybrid DV-hop (HDV-hop) in [33] is suitable for localizing events in hostile environments, in which anchors are deployed on perimeters of networks rather than scattering them inside hostile terrain. Consequently, unknown node which localizes itself will transmit localized messages, including hop count, hop size and the distance of this unknown node and each anchor, transverse two or more hops. This consumes plenty of energy which is intractable to normal monitoring WSNs.
We can set parameters of these three schemes as similar to IMDV-hop scheme in order to compare behaviors between IMDV-hop and other three schemes. For example, in HDV-hop scheme, anchors are deployed on the perimeters of networks, and locations of unknown nodes are similar to those of IMDV-hop. And also, those three localization schemes are designed for static localization, but they also can be adopted for mobile localization, such as mobile scenarios in Section 4.2.4 of [36]. We can set the same mobile environments for all these schemes as IMDV-hop. Localization for all these three schemes and IMDV-hop can executed at the beginning slot of each LW. Mobile behaviors are compared for cases of low velocity.
Localization error compares demonstrate that localization error presents the prior behaviors when anchor ratio is higher than 14.65% and less than 9.45% shown in Fig. 10a. IMDV-hop shows better localization accuracy with the variety of node density shown in Fig. 10b. Nodes including anchors increasing, unknown node can execute HTC scheme in two-hop sensor networks, which increases the accuracy of hop count, hop size and distance of unknown node and each anchor. IMDV-hop scheme is designed for mobile WSNs on the condition that nodes including anchors move intermittently. We take low mobility velocity into account in IMDV-hop, and localization accuracy shows better behaviors for low velocities shown in Fig. 10c.
We always devote in improving delay behavior for IMDV-hop, a time-critical monitoring and localization scheme for WSNs. Shown in Fig. 11, delay compares demonstrate that delay of it presents prior behaviors when anchor ratio is higher than 7.23%, yet shows less priority for higher mobility velocities such as v ≥ 8.12m/s. Energy consumption comparisons are similar to that of delay behavior.
Nearly all of DV-hop-based schemes are designed for localizing static WSNs, but these schemes can also be executed for localizing mobility WSNs with relatively low velocity. Performance comparisons illustrated in Figs. 10, 11 and 12, IMDV-hop presents not fully up to expectations for relatively high velocity. But, it is noted that IMDV-hop is designed to collect circumstance parameter periodically and localize intermittent mobile aquaculture objects, and low velocity of v ≤ 8m/s is sufficient for our monitoring and localizing aquaculture networks.
Moreover, performance compares are also proposed between IMDV-hop and WMCL, which designs specially for mobile WSNs. In WMCL scheme, bounding-box is used to improve sampling efficiency by reducing the scope of selecting of candidate samples, accompanying with two-hop beacon neighbors’ negative effects and sensor neighbors’ estimated position information to reduce the bounding-box. Moreover, WMCL scheme can be also used for static localization, rather than be only used for mobility localization for other MCL schemes. But iterative seeking for candidate samples consume plenty of energy for WMCL scheme, and also, localization error of WMCL increases sharply when number of anchors decreasing.
Localization error increases in low velocity for WMCL scheme shown in Fig. 13a, such as v ≤ 6.15 m/s, and tends to steady values when the velocity is higher than 6.2 m/s. On the other hand, error for IMDV-hop increases smoothly in low velocity such as v ≤ 4.02 m/s, while increases sharply v ≥ 4.1 m/s, and error differentiation with WMCL scheme is growing bigger and bigger. Similarly, delay for IMDV-hop scheme presents the prior behavior when the velocity is lower than v ≤ 13.04 m/s shown in Fig. 13b.
## 5 Conclusions
In this paper, we have presented range-free cross-layer localization scheme IMDV-hop based on DV-hop algorithm in intermittent mobile WSNs, accompanying HTC algorithm. At first, inferiorities of original DV-hop scheme are denoted elaborately after brought briefly about it. Then, an improved localization scheme IMDV-hop embedded in WLS method is proposed, and two critical parameters, correction coefficient kc and weighted coefficient $${w}_{N_x,i}$$, are introduced into it to improve localization performance. And then, localization performance is presented taking parameters describing about the network into account, such as anchor’s number, node density, localization window, node’s velocity, and transmission range. Moreover, comprehensive performance comparisons between IMDV-hop algorithm and other DV-hop-based schemes, between IMDV-hop and MCL-based scheme are proposed. Comprehensive NS-2 simulations according to these parameters demonstrate that analysis results of these models match well with simulation results, especially for lower mobile velocity and relative higher node density. Besides, analysis and comparison results show that delay behaviors of this low-velocity, time-critical scheme IMDV-hop is improved largely relative to other schemes, and localization accuracy is improved in some cases of more node density and lower mobile velocity.
For determining the location of nodes in mobile WSNs, anchor’s ratio has significant impact on localization behaviors. But more anchors can bring out much energy and economic consumption. And if the propagation model is taken into account, the distance between unknown node and anchors plays greater role in localization performance. Moreover, mobility velocity of nodes including anchors and unknown nodes, either periodically or randomly moving, can also be questions in localization. In the recent future, we can also extend our analysis to these pending problems.
## Notes
### Funding
The experiments of this research are funded by the National Natural Science Foundation of China under grant number 61362017 and Building Foundation for Teaching Team of Shanghai Ocean University under grant number A1-0201-00-0322062.
### Authors’ contributions
JZ carried out the IMDV-hop localization scheme, accompanying with HTC transmission scheme, and helped to draft the manuscript. CLv carried out most of the simulation experiments. ZT carried out the defects of original DV-hop scheme and drafted the manuscript. All authors contributed to the work. All authors read and approved the final manuscript.
### Authors’ information
Jianping Zhu received Ph.D. degree in the Department of Electronic, Information and Electrical Engineering of Shanghai Jiaotong University, China. Currently, she is an associate professor of SOU College of Engineering Science and Technology in Shanghai Ocean University. Zhu’s main research interests include wireless sensor networks, digital communication systems analysis and design, and information theory. Chunfeng Lv received Ph.D. degrees in the Department of Electronic, Information and Electrical Engineering of Shanghai Jiaotong University, China. Currently, he is a lecturer of SOU College of Engineering Science and Technology in Shanghai Ocean University. Lv’s main research interests include detection technique using WSNs, digital communication systems analysis and design, and information theory. Zhengsu Tao received a Ph.D. degree in the Department of Electronics and Information Engineering from Hongkong University of Science & Technology. Currently, he is a professor of the Department of Electronic, Information and Electrical Engineering of Shanghai Jiaotong University. His research interests include detection technique using WSNs.
### Competing interests
The authors declare that they have no competing interests.
### Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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ML Jiang, JY Luo, XK Zou, Research on algorithm of three-dimensional wireless sensor networks node localization. J. Sensors. 5, 1–9 (2016)Google Scholar | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.6821487545967102, "perplexity": 3787.2017816271814}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1537267156524.34/warc/CC-MAIN-20180920155933-20180920180333-00466.warc.gz"} |
http://physics.stackexchange.com/questions/58127/strings-and-qft-particles-moving-backward-in-time | # Strings and QFT: particles moving backward in time?
New question: In string theory and QFT, do particles travel back in time? Not related to antimatter: Do they travel back and forth in time in reality or are these just interpretations of mathematical formulas used to make sense of calculations?
This is a different question (from this one) as it concerns strings and not antimatter...
-
It really is the same question. The "going backward in time" notion is a alternate way of thinking about anti-matter. It is intrinsically quantum mechanical, and strings are quantum mechanical objects. You can't duck this, but I am going to leave it to one of my fellow moderators to close (or not close) this as they see fit. – dmckee Mar 26 '13 at 19:27
– dmckee Mar 26 '13 at 19:28
Finally, you can merge your several accounts using physics.stackexchange.com/help/user-merge – dmckee Mar 26 '13 at 19:29
The "motion backwards in time" is often mentioned in popular texts/shows about physics but it can't be understood literally. All processes in the Universe are taking place forward in time. An essential subtlety in the previous sentence is the tense – "are -ing" (present progressive), in this case – which automatically includes the information about what is happening with time during the process described by the word: it is increasing from $t$ to $t+dt\gt t$, and so on.
We may look at the history of a process backwards in time, but that's something else than that the process is actually happening. It is always "happening" while time is doing the same thing: going forward.
Instead, what this "back in time" stuff means in physics is that certain objects – in particular, antiparticles – may be related to other objects – particles. And the relation is such that the processes involving antiparticles are naturally the time-reversed (back-in-time interpreted) processes involving the original particles.
This map may be literally applied to Feynman diagrams. In Feynman diagrams, antiparticles are given by the same line as the particles except that their presence in the initial state must be represented by the original particle's presence in the final state and vice versa (and energy has to flip the sign). This trick is related to the Dirac sea for fermions: a positron is a hole in the Dirac sea of negative-energy states, and a hole/positron that disappears from A and appears in B is the same change as an electron that disappears from B and appears in A (because the occupation numbers are reverted).
These relationships between particles and their antiparticles hold in string theory, too. A string moving backwards in time is described by locally the same pieces of strings, the same world sheet, but the orientation may get reflected. For example, when a string is wound around a circle $w$ times, its antiobject has the opposite charge $-w$, so it's wound in the opposite direction. The winding number is an example of a charge. If we revert the orientation of time inside the world sheet, it switches the overall orientation of the world sheet which may be compensated by another switch of the orientation of the spatial direction, too – and that's what changes $+w$ to $-w$ and replaces particles by antiparticles etc.
So physics doesn't support any other "real and impressive" interpretation of "motion back in time" except for its being a good interpretation of the motion of antiparticles that makes some of the properties of antimatter – why they're related to properties of matter – manifest. It's really a way to visualize a certain symmetry but the particles and antiparticles are still doing the same thing and the only right way to "look" what's going on is to look at everything as time goes forward.
-
So rons comment as in the other thread that states that point particles can reproduce the quantum field if they are allowed to go backwards doesn't mean the point particles go backwards in time? – Peter Mar 26 '13 at 19:55
Lubos or other you there? – Peter Mar 26 '13 at 23:30
The statement "something goes backward in time" only makes sense if you parameterize the motion of "something" by yet another time that isn't the same time as the time in the sentence "backward in time". So the world lines of these objects in spacetime may be parameterized by $\tau$, a different coordinate along the lines than the $t$ coordinate in the spacetime, and the relationship between $t$ and $\tau$ may be either increasing or decreasing. The decreasing case means "backward in time" and it's useful when one describes antiparticles. – Luboš Motl Mar 27 '13 at 7:08
But if one only talks about one time $t$, the phrase "something is happening back in time" is meaningless. What's happening is always described as some functions of $t$ and higher $t$ means future, lower $t$ means past, and the right interpretation of any sentence about a process is that it starts with a lower $t$ and goes to higher $t$: this is really a rule of grammar. There's no way to qualitatively change any process so that it would happen back in time. Whatever the process is, it's a qualitatively the same function of $t$ that should be read from low $t$ to high $t$ when we "live it". – Luboš Motl Mar 27 '13 at 7:10
Thank you lubos so I now understand it that particles don't literally have to go back in time to recreate the quantum field as Ron says – Peter Mar 27 '13 at 7:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.547947883605957, "perplexity": 347.1120273820655}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1432207929561.98/warc/CC-MAIN-20150521113209-00239-ip-10-180-206-219.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/73551/smoothness-along-rays-sufficient-for-global-smoothness | # Smoothness along rays sufficient for global smoothness
Hi,
Suppose I have a function $f:\mathbb{R}^d \to \mathbb{R}$ and I know that $f$ is smooth ($C^\infty$) along each ray $t \mapsto f(td)$ on $t \in [-\epsilon, \epsilon]$ and all directions $d \in \mathbb{R}^d$.
Is smoothness along these rays sufficient for $f$ to be smooth around $0$ as a multivariate function (all partial derivatives exist)?
Thanks.
-
No. Just take a non-smooth function $g \colon S^{d-1} \to \mathbb{R}$ with $g(-p) = -g(p)$ and define $f(v) = \|v\|g(v/\|v\|)$. This is smooth since on each ray it is just $t \mapsto k t$ for some $k$, but the non-smoothness of $g$ means that $f$ is not smooth. The closest to this that I know of is if you take all smooth curves (not just rays). Then $f$ is smooth. This is due to Jan Boman. – Andrew Stacey Aug 24 '11 at 10:13
That is indeed a nice counterexample. Would you happen to have a reference of this result of Jan Boman? In addition, would smoothness along all real analytic curves be sufficient too? – Bart Aug 24 '11 at 12:15
The result Andrew Stacey refers to is this paper of Boman (MathSciNet link): ams.org/mathscinet-getitem?mr=237728 The article doesn't look like it is available online. In the MathSciNet review the following counterexample is mentioned: there exists a non-continuous function $f$ from $\mathbb{R}^2\to\mathbb{R}$ such that $f\circ u \in C^{\infty}(\mathbb{R},\mathbb{R})$ for every quasianalytic $u$. (Though it is not clear which quasianalytic class the example applies to.) This should give at least a partial answer to your follow-up question. – Willie Wong Aug 24 '11 at 13:12
Thanks Willie. Looking at the Boman paper, demanding smoothness along only analytic curves is indeed not sufficient. It turns out that when $f \circ u$ is real analytic for every real analytic curve $u$, that $f$ is real analytic. See "An Ontology of Directional Regularity Implying Joint Regularity" published in Real Analysis Exchange, available at math.wustl.edu/~sk/joint.pdf . – Bart Aug 24 '11 at 13:35
Apparently, in the published version there is the condition that the k-th partial derivative of $f \circ u$ needs to be smaller than $C k! / r^k$ for some $r>0$. – Bart Aug 24 '11 at 14:19
show 1 more comment | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.807005763053894, "perplexity": 520.4350206103924}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1387345777253/warc/CC-MAIN-20131218054937-00033-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://support.airkit.com/reference/exp | The function EXP take some number n and returns en.
This function takes a Number as input. It returns another Number: the numerical constant e taken to the power of the given Number.
### Declaration
``````EXP(n) -> number
``````
Parameters
n (required, type: `Number`)
Any Number; this is what e will be taken to the power of.
### Return Values
number (type: `Number`)
The numerical estimation of e raised to the nth power, or en.
### Examples
The following example calculates a numerical estimation of e1, or simply e. Note that e is an irrational number; the output of this function is an approximation of the value of e rather than an infinite decimal. This should be sufficient for most practical applications, though it can be the source of small rounding errors:
``````EXP(1) -> 2.718281828459045
``````
The EXP function does not require input in whole numbers. The following example calculates a numerical estimation of e0.5 (otherwise known as the square root of e):
``````EXP(0.5) -> 1.6487212707001282
``````
### Discussion
The function EXP only takes the constant e to a given power. To calculate any number to a given power, use the POWER function. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8094032406806946, "perplexity": 1260.0529660103073}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710417.25/warc/CC-MAIN-20221127173917-20221127203917-00248.warc.gz"} |
https://www.physicsforums.com/threads/conceptual-understanding-of-derivates-question.768488/ | # Conceptual understanding of Derivates Question
1. Sep 1, 2014
### RJLiberator
Question:A bicyclist starts from home and rides back and forth along a straight East/West path. Her (instantaneous) velocity as a function of time is given by v(t), where time t is measured in minutes. Consider:
$d_{1} = \int^{30}_{0} v(t) dt$ and $d_{2} = \int^{30}_{0} |v(t)| dt$
Choose what it represents from the following:
(a) the total distance the bicyclist rode in 30 minutes
(b) the bicyclist's average velocity over 30 minutes
(c) the bicyclist's distance from the home after 30 minutes
(d) none of the above.
My thinking: Allright well, the two integrals (d1 and d2) are essentially the same thing other than the absolute value sign. This leads me to believe that d1 is (c) and d2 is (a). This is kind of a logical guess on my part. Can anyone explain what the equation is actually saying?
To me, d1 is saying: The area under the curve from 0minutes to 30 minutes is represented by v'(t). This would mean to me that (c) should be the correct answer. While, d2 is saying the absolute value (negative and positive area combined) of v'(t) from 0mintues to 30 minutes. This would be the total distance traveled aka (a).
If my conceptual understanding/writing is wrong, please do point it out to me.
Is my thinking correct?
Thank you all for your help.
2. Sep 1, 2014
### mal4mac
Do you mean, "Choose what each represents from the following"? What's v'(t)?
General tip - draw (rough) graphs of v against t and |v| against t and remember the general definition of integration = "area under the curve".
3. Sep 1, 2014
### RJLiberator
Yes, I do mean that. Thank you.
Hm. The only problem with graphing is, I don't have a function to graph.
So this would be a purely conceptual question. I suppose.
Maybe I can use a dummy function and try to graph that to help my conceptual understanding. I will try this.
4. Sep 1, 2014
### HallsofIvy
Staff Emeritus
Yes, you are correct- (d2) is the total distance ridden, (d1) is the distance from home. You can check that by taking a simple example. Suppose she rides directly away from home at constant velocity v (meters per minute)> 0 for 15 minutes, then rides directly back home with velocity -v for 15 minutes. After 15 minutes she will be 15v meters from home, then she turns around and after another 15 minutes is back home: 0 meters from home. But she rode a total of 30v meters.
$$d1= \int_0^{30} v(t)dt= \int_0^{15} v dt+ \int_{15}^{30}(-v)dt= v(15- 0)+ (-v)(30-15)= 15v- 15v= 0$$
$$d2= \int_0^{30} |v(t)|dt= \int_0^{30} v dt= 30v$$
Last edited: Sep 1, 2014
5. Sep 1, 2014
### RJLiberator
HallsofIvy, thank you for explaining the conceptual side AND the mathematical side of the problem to me. My hunch of logic has been verified.
My kindest regards to you for helping me understand this.
6. Sep 1, 2014
### RJLiberator
One last Calculus question:
So I understand that the integral of e^t^2 is not a normal integral and cannot be easily calculated.
But in situation where
the integral from 0 to sqrt(x) of e^t^2
Is stated and then it asks me to Compute f'(x)
Can I accomplish this?
My thinking is that, Sure the Integral is not really possible for me to computer, by does f'(x) = the integral? I don't believe so.
7. Sep 1, 2014
### WWGD
Look into the Fundamental Theorem of Calculus, e.g.,http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus .
WWGD :What Would Gauss Do?
8. Sep 1, 2014
### RJLiberator
Okay, so I am trying to work on this.
I plug the integral into a calculator (for integrals) and receive the answer of:
e$^{x}F\sqrt{x}$
I'm just not quite sure what this is telling me. This seems to be telling me the area under the curve from 0 to sqrt(x) of the initial function.
I am trying to find f'(x)
So I've been supplied (post above) with the fundamental theorem of Calculus. This seems to help, but I'm not putting everything together.
So, since f(x) = e^t^2 dt is f'(x) = 2t*e^t^2
Is it that simple?
9. Sep 1, 2014
### Ray Vickson
No. I hope you realize that what you wrote is nonsense: you have x on one side and t on the other. On the other hand, maybe you do not actually mean what you wrote. Originally you had
$$f(x) =\int_0^{\sqrt{x}} e^{t^2} \, dt \,$$
although you wrote something that could be interpreted as $(e^t)^2$, which is very different. Use parentheses, like this: e^(t^2).
Also: if you are just a beginner at this material, I would recommend that you avoid using the calculator, except for numerics; just do things directly, by hand, and reason it out step-by-step, taking as long as necessary and using as much paper as you need. With practice you will get better and faster---but likely only if you avoid the calculator.
10. Sep 1, 2014
### mal4mac
Suggestion on writing equations in physics forums - look up to the right of the editor and you will see an editing symbol that looks like X2. Click on that to write expressions like et2
Draft saved Draft deleted
Similar Discussions: Conceptual understanding of Derivates Question | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8843520879745483, "perplexity": 1187.295418108658}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886106754.4/warc/CC-MAIN-20170820131332-20170820151332-00690.warc.gz"} |
https://docs.microsoft.com/en-us/dotnet/standard/design-guidelines/exception-throwing | # Exception Throwing
Exception-throwing guidelines described in this section require a good definition of the meaning of execution failure. Execution failure occurs whenever a member cannot do what it was designed to do (what the member name implies). For example, if the OpenFile method cannot return an opened file handle to the caller, it would be considered an execution failure.
Most developers have become comfortable with using exceptions for usage errors such as division by zero or null references. In the Framework, exceptions are used for all error conditions, including execution errors.
❌ DO NOT return error codes.
Exceptions are the primary means of reporting errors in frameworks.
✔️ DO report execution failures by throwing exceptions.
✔️ CONSIDER terminating the process by calling System.Environment.FailFast (.NET Framework 2.0 feature) instead of throwing an exception if your code encounters a situation where it is unsafe for further execution.
❌ DO NOT use exceptions for the normal flow of control, if possible.
Except for system failures and operations with potential race conditions, framework designers should design APIs so users can write code that does not throw exceptions. For example, you can provide a way to check preconditions before calling a member so users can write code that does not throw exceptions.
The member used to check preconditions of another member is often referred to as a tester, and the member that actually does the work is called a doer.
There are cases when the Tester-Doer Pattern can have an unacceptable performance overhead. In such cases, the so-called Try-Parse Pattern should be considered (see Exceptions and Performance for more information).
✔️ CONSIDER the performance implications of throwing exceptions. Throw rates above 100 per second are likely to noticeably impact the performance of most applications.
✔️ DO document all exceptions thrown by publicly callable members because of a violation of the member contract (rather than a system failure) and treat them as part of your contract.
Exceptions that are a part of the contract should not change from one version to the next (i.e., exception type should not change, and new exceptions should not be added).
❌ DO NOT have public members that can either throw or not based on some option.
❌ DO NOT have public members that return exceptions as the return value or an out parameter.
Returning exceptions from public APIs instead of throwing them defeats many of the benefits of exception-based error reporting.
✔️ CONSIDER using exception builder methods.
It is common to throw the same exception from different places. To avoid code bloat, use helper methods that create exceptions and initialize their properties.
Also, members that throw exceptions are not getting inlined. Moving the throw statement inside the builder might allow the member to be inlined.
❌ DO NOT throw exceptions from exception filter blocks.
When an exception filter raises an exception, the exception is caught by the CLR, and the filter returns false. This behavior is indistinguishable from the filter executing and returning false explicitly and is therefore very difficult to debug.
❌ AVOID explicitly throwing exceptions from finally blocks. Implicitly thrown exceptions resulting from calling methods that throw are acceptable. | {"extraction_info": {"found_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": 0.1694808006286621, "perplexity": 2227.338032338199}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250607407.48/warc/CC-MAIN-20200122191620-20200122220620-00380.warc.gz"} |
https://iacr.org/cryptodb/data/author.php?authorkey=11045 | ## CryptoDB
#### Publications
Year
Venue
Title
2021
CRYPTO
In 2012, Aaronson and Christiano introduced the idea of hidden subspace states to build public-key quantum money [STOC '12]. Since then, this idea has been applied to realize several other cryptographic primitives which enjoy some form of unclonability. In this work, we propose a generalization of hidden subspace states to hidden coset states. We study different unclonable properties of coset states and several applications: * We show that, assuming indistinguishability obfuscation (iO), hidden coset states possess a certain direct product hardness property, which immediately implies a tokenized signature scheme in the plain model. Previously, a tokenized signature scheme was known only relative to an oracle, from a work of Ben-David and Sattath [QCrypt '17]. * Combining a tokenized signature scheme with extractable witness encryption, we give a construction of an unclonable decryption scheme in the plain model. The latter primitive was recently proposed by Georgiou and Zhandry [ePrint '20], who gave a construction relative to a classical oracle. * We conjecture that coset states satisfy a certain natural monogamy-of-entanglement property. Assuming this conjecture is true, we remove the requirement for extractable witness encryption in our unclonable decryption construction. As potential evidence in support of the conjecture, we prove a weaker version of this monogamy property, which we believe will still be of independent interest. * Finally, we give the first construction of a copy-protection scheme for pseudorandom functions (PRFs) in the plain model. Our scheme is secure either assuming iO and extractable witness encryption, or iO, LWE and the conjectured monogamy property mentioned above. This is the first example of a copy-protection scheme with provable security in the plain model for a class of functions that is not evasive.
2021
CRYPTO
We prove that quantum-hard one-way functions imply simulation-secure quantum oblivious transfer (QOT), which is known to suffice for secure computation of arbitrary quantum functionalities. Furthermore, our construction only makes black-box use of the quantum-hard one-way function. Our primary technical contribution is a construction of extractable and equivocal quantum bit commitments based on the black-box use of quantum-hard one-way functions in the standard model. Instantiating the Crépeau-Kilian (FOCS 1988) framework with these commitments yields simulation-secure quantum oblivious transfer.
2021
CRYPTO
We construct the first constant-round protocols for secure quantum computation in the two-party (2PQC) and multi-party (MPQC) settings with security against malicious adversaries. Our protocols are in the common random string (CRS) model. - Assuming two-message oblivious transfer (OT), we obtain (i) three-message 2PQC, and (ii) five-round MPQC with only three rounds of online (input-dependent) communication; such OT is known from quantum-hard Learning with Errors (QLWE). - Assuming sub-exponential hardness of QLWE, we obtain (i) three-round 2PQC with two online rounds and (ii) four-round MPQC with two online rounds. - When only one (out of two) parties receives output, we achieve minimal interaction (two messages) from two-message OT; classically, such protocols are known as non-interactive secure computation (NISC), and our result constitutes the first maliciously-secure quantum NISC. Additionally assuming reusable malicious designated-verifier NIZK arguments for NP (MDV-NIZKs), we give the first MDV-NIZK for QMA that only requires one copy of the quantum witness. Finally, we perform a preliminary investigation into two-round secure quantum computation where each party must obtain output. On the negative side, we identify a broad class of simulation strategies that suffice for classical two-round secure computation that are unlikely to work in the quantum setting. Next, as a proof-of-concept, we show that two-round secure quantum computation exists with respect to a quantum oracle.
2020
CRYPTO
We initiate the study of non-interactive zero-knowledge (NIZK) arguments for languages in QMA. Our first main result is the following: if Learning With Errors (LWE) is hard for quantum computers, then any language in QMA has an NIZK argument with preprocessing. The preprocessing in our argument system consists of (i) the generation of a CRS and (ii) a single (instance-independent) quantum message from verifier to prover. The instance-dependent phase of our argument system involves only a single classical message from prover to verifier. Importantly, verification in our protocol is entirely classical, and the verifier needs not have quantum memory; its only quantum actions are in the preprocessing phase. Our second contribution is to extend the notion of a classical proof of knowledge to the quantum setting. We introduce the notions of arguments and proofs of quantum knowledge (AoQK/PoQK), and we show that our non-interactive argument system satisfies the definition of an AoQK. In particular, we explicitly construct an extractor which can recover a quantum witness from any prover which is successful in our protocol. Finally, we show that any language in QMA has an (interactive) proof of quantum knowledge.
2019
EUROCRYPT
The problem of reliably certifying the outcome of a computation performed by a quantum device is rapidly gaining relevance. We present two protocols for a classical verifier to verifiably delegate a quantum computation to two non-communicating but entangled quantum provers. Our protocols have near-optimal complexity in terms of the total resources employed by the verifier and the honest provers, with the total number of operations of each party, including the number of entangled pairs of qubits required of the honest provers, scaling as $O(g\log g)$ for delegating a circuit of size g. This is in contrast to previous protocols, whose overhead in terms of resources employed, while polynomial, is far beyond what is feasible in practice. Our first protocol requires a number of rounds that is linear in the depth of the circuit being delegated, and is blind, meaning neither prover can learn the circuit or its input. The second protocol is not blind, but requires only a constant number of rounds of interaction.Our main technical innovation is an efficient rigidity theorem which allows a verifier to test that two entangled provers perform measurements specified by an arbitrary m-qubit tensor product of single-qubit Clifford observables on their respective halves of m shared EPR pairs, with a robustness that is independent of m. Our two-prover classical-verifier delegation protocols are obtained by combining this rigidity theorem with a single-prover quantum-verifier protocol for the verifiable delegation of a quantum computation, introduced by Broadbent.
#### Coauthors
James Bartusek (2)
Alex B. Grilo (1)
Stacey Jeffery (1)
Dakshita Khurana (2)
Jiahui Liu (1)
Qipeng Liu (1)
Fermi Ma (2)
Thomas Vidick (2)
Mark Zhandry (1)
Tina Zhang (1) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7324022650718689, "perplexity": 2167.7565284918874}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320302740.94/warc/CC-MAIN-20220121071203-20220121101203-00181.warc.gz"} |
https://www.clutchprep.com/physics/practice-problems/41064/a-small-rock-block-with-mass-0-400-kg-is-placed-against-a-compressed-spring-at-t-1 | Springs & Elastic Potential Energy Video Lessons
Example
# Problem: A small rock block with mass 0.400 kg is placed against a compressed spring at the bottom of a 37.0° incline. The compressed spring has 50.0 J of elastic potential energy stored in it. The spring is released and the block moves a distance of 12.0 m along the incline before momentarily coming to rest. How much work does the friction force do on the block during the motion? What is the coefficient of kinetic frinction μk between the block and the incline?
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###### Problem Details
A small rock block with mass 0.400 kg is placed against a compressed spring at the bottom of a 37.0° incline. The compressed spring has 50.0 J of elastic potential energy stored in it. The spring is released and the block moves a distance of 12.0 m along the incline before momentarily coming to rest. How much work does the friction force do on the block during the motion? What is the coefficient of kinetic frinction μk between the block and the incline? | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9300209283828735, "perplexity": 400.70499419770994}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107897022.61/warc/CC-MAIN-20201028073614-20201028103614-00292.warc.gz"} |
http://comprensivobovalino.it/jktg/norm-in-latex-overleaf.html | Use dollar symbols around the text. You just need to escape characters that have special meaning: # $% & ~ _ ^ \ { }. The last point is the norm of a vector, the square root from the sum of its components. aa aah aahed aahing aahs aal aalii aaliis aals aardvark aardvarks aardwolf aardwolves aargh aas aasvogel aasvogels aba abaca abacas abaci aback abacus abacuses abaft. There are currently two packages providing international language support, namely, Polyglossia and Babel. I suggest to define new commands (again with amsmath ) in the preamble for an easier workflow. Number sets (prime, natural, integer, rational, real and complex) in LaTeX. x 0 doit être placé sous x -- 0 qui. Number sets such as natural numbers or complex numbers are not provided by default by LaTeX. It's widely used throughout the academic world to publish technical. This online tool is a wrapper script for latexdiff, which is the LaTeX alternative to Track Changes in Microsoft Word. Such basics can be found in introductions like lshort. The main idea of CS is, by exploiting the sparsity nature of the signal (in any domain), we can reconstruct the signal from very fewer samples than required by Shannon-Nyquist sampling theorem. 예를 들어 em이나 px. d3 is the first move in \variation{} Open an example of the skak package in ShareLaTeX. - texasbruce Jan 7 '13 at 20:29. d4 is the last move before analysing a variation, hence 3. I usually add a LaTeX comment that mentions the lipsum package and \lipsum command simply generate filler text. Using prod Edit. Posted: (18 days ago) Beamer - Overleaf, Online LaTeX Editor. Uppercase Greek Letters Binary operation symbols. This tag should be used for questions that concern the backend; questions concerning use of the web service are probably better asked on tex. Open an example in Overleaf. The Normal distribution requires two parameters, the mean and the standard deviation. For unordered lists, LaTeX provides the itemize environment and for ordered lists there is the enumerate environment. commentcamarche. RESPONSE XS90 P0240031 XS90 package £2,999. Here are a few examples:. - 48 - 198. The command \variation{} helps to analyse variations of a move. Tutorial de Overleaf para LaTeX en español. 5 Different LaTeX Template Tutorials using OverLeaf. Integer and sum limits improvement. ) Ihres Buches in unser Literatursuchsystem und unser System generiert Ihnen in Kürze die Literaturangabe in Ihre gewünschte Zitierform. What are Green3, Blue3 or Red3? #2 Brent Longborough, October 8, 2013 at 3:10 p. edited Sep 13 '14 at 19:51. LaTeX files usually have a. Thus from Euclid's five postulates we have arrived at our final result, known as the Pythagorean theorem: $$A^2+B^2=C^2$$. See images here : Overleaf v2$\endgroup$– Frobenius Jul 16 '18 at 21:25 |.$\begingroup@knzhou So, I was lucky to find an online editor : Overleaf (first v1 and later v2). Author and date are the key components in the in-text citation of the APA referencing style. Similar is for limit expressions. Add a New Comment. sourceforge. \left[ \begin {array}{ccc} 9&13&17\\ oalign{\medskip} 14&18&22 \end {array} \right] If you put this code inside a LaTeX displaymath environment, you will get the matrix typeset. References are not included in the 10-page. Um editor de LaTeX online fácil de usar. Formatting Optimization Problems with LaTeX. Prebiotic foods like garlic, onions and avocado help to reduce stress, as well as food high in Omega 3. Starting at step (2-4) there is a single sentence that continues to the end of p60. Para escribir el valor absoluto o la norma en LaTeX es conveniente crear un comando antes que usar la barra vertical "|". I think that providing compilable documents makes it lot easier for the reader to test the code to see if it has desired effect. ISBN 0-8176-3805-9 (acid-free paper) (pbk. It doesn't mean that LaTeX doesn't know those sets, or more importantly their symbols…. Redefines \maketitle to save space. 480 Topics 2547 Posts Last post Re: Clickable url by willithefox 06. Bold, italics and underlining Simple text formatting helps to highlight important concepts within a document and make it more readable. All Stanford students, faculty and staff have access to a free Overleaf Pro account. Open a text editor like Notepad and create a new LaTeX document by typing: \begin {document} \end {document} Type the following between the "begin" and "end" commands to create your bullet point list: Video of the Day. LaTeX is a powerful tool to typeset math. The Normal distribution requires two parameters, the mean and the standard deviation. Subscripts & Superscripts. The following table shows the whole Greek alphabet along with the commands in a nice table. Citations are an abomination, as are numbering equations; random crashes, corruption of files that contain a lot of graphics etc. \begin {document} View which changes have been added and removed. Similar is for limit expressions. Some of these commands are LATEX macros, while oth- ers belong to plain TEX; no attempt to di eren-. Open an example in ShareLaTeX. x \pod a x (a) Use \equiv to get the congruence symbol (three…. Compressive Sensing is a Signal Processing technique, which gave a break through in 2004. If you want the limits of an integral/sum/product to be specified above and below the symbol in inline math mode, use the \limits command before limits specification. The Möbius function is a more complicated example of a three-part definition. % Specifies … Continue reading "LaTeX - bold vectors and arrow vectors". 1 Beginning a document ndocumentclassfarticleg nusepackagefgraphicx, amssymbg nbeginfdocumentg ntextwidth 6. For example, Authorea has tools to handle the annoying parts of writing like automated reference insert and formatting, and export in different journal formats. Integral\int_ {a}^ {b} x^2 dxinside. Code: [Select all] [Expand/Collapse] [Download] (untitled. Add a New Comment. In a similar manner to the summation symbol, the product symbol can easily be added to a LaTeX document using the \prod notation. Specifically a state diagram describes the behavior of a single object in response to a series of events in a system. LaTeX is a high-quality typesetting system; it includes features designed for the production of technical and scientific documentation. 常用英语词汇6000_英语考试_外语学习_教育专区。常用英语词汇6000. Information used to produce the title is obtained from the following declarations, which should precede the \maketitle command. They'll be productive from day one and be able to pick up small amounts of LaTeX as they go. Overleaf is a collaborative cloud based LaTeX tool which makes the process of writing, editing and publishing scientific papers faster and easier. This is the basic introduction to Matlab. 4A65G69 1995 95-36881 688. The main idea of CS is, by exploiting the sparsity nature of the signal (in any domain), we can reconstruct the signal from very fewer samples than required by Shannon-Nyquist sampling theorem. We appreciate Lindsay being able to svolcano at our inaugural Raleigh FuturePub and share the work Research Square is doing in the scholarly issueing and collaborative survey space. here 1 is subscript of C. Sign in to make your opinion count. For more information about the Encyclopedia, see the Welcome page. d3 is the first move in \variation{} Open an example of the skak package in ShareLaTeX. Reduces page margins to 0. A rendered preview of all letters is shown alongside all commands in a nice table. Foreign symbols. com , or follow them on twitter at @Overleaf. 93 silver badges. If you do want full latex, try Overleaf. tab samples azimuth azimuth_dec_correction citations description dip geologic_classes geologic_types lat lithologies lon method_codes orientation_quality sample site 260 0 This study Archived samples from 1965, 66 expeditions. You can help LaTeX Wiki by expanding it. cursive 'l' Post by abhimanyulad ». Each provides a table for expressions, aligned in rows and columns. Para escribir el valor absoluto o la norma en LaTeX es conveniente crear un comando antes que usar la barra vertical "|". Step 2: Enter author information. If you have any enquiries about this website or the content on it, please contact: [email protected] 1 DeclareRobustCommand. Eine gute Möglichkeit, Zitate einzufügen und zu formatieren, ist BibTeX (). \end {document}. The sybmol can be compressed to fit on one line (useful for small equations displayed within a text block), or enlarged to make it more readable. The command \variation{} helps to analyse variations of a move. Nobody knows if funeral prices will go up or down in future. Removes page numbers. Sometimes you find yourself wanting to put something over another symbol, e. Here's how to easily switch between a bold vector and an arrow vector. What are Green3, Blue3 or Red3? #2 Brent Longborough, October 8, 2013 at 3:10 p. d4 is the last move before analysing a variation, hence 3. You can use one of the predefined layouts (pseudocode, pascal and c and others), with or without modifications,. Relational Operators (math mode). Here’s how to use the \$$\\LaTeX \$$ template from Springer for a Computer Science conference publication. In this post, I’m not so much interested in. (Since matrices are large, they are almost always set as displays. The literature argues that the cause of norm death is widespread violation. I will be using OverLeaf to run these written tutorials where we shall unleash a new part of LaTeX every week. As you can see, there are three basic commands and they can be nested to get combined effects. Student t Distributed Linear Value-at-Risk December 2, 2015 by Pawel One of the most underestimated feature of the financial asset distributions is their kurtosis. 75in for more space. They consist of plain text interspersed with some LaTeX commands. The LaTeX for Physicists Header has the following features: Sets font size to 11pt. The sybmol can be compressed to fit on one line (useful for small equations displayed within a text block), or enlarged to make it more readable. Sublime Text. ShareLaTeX is so easy to get started with that you'll be able to invite your non-LaTeX colleagues to contribute directly to your LaTeX documents. tex extension. HTML The icon in HTML, if it is defined as a named mark. Designed with a LinkedIn-like layout, this template reliefs the burden of worrying about customization and rather focuses on increasing productivity by filling out your CV's content in proper sectioned files; that can be later imported to the main. You edit and the compilation is direct and online showing you your probable mistakes. 85 Trachyte 166. This is an information page. d3 is the first move in \variation{} Open an example of the skak package in ShareLaTeX. Embed formulas in your text by surrounding them with dollar signs The equation environment is used to typeset one formula. In combination with a text editor and distributed version control, both reproducibility and simultaneous collaboration are improved by these plain text mark-up languages. at midnight than anyone else. 2 New environments. I think that providing compilable documents makes it lot easier for the reader to test the code to see if it has desired effect. You must use the following package: \usepackage {amsmath} \begin {matrix} \begin {pmatrix} \begin {bmatrix} \begin {vmatrix} \begin {Vmatrix}. Specifically a state diagram describes the behavior of a single object in response to a series of events in a system. Open an example in Overleaf. In LaTeX, I use a \qquad for this. To get exp to appear as a superscript, you type ^{exp}. Our team is passionate about delivering a flawless customer experience as we play matchmaker between great hotels with unsold rooms and the on-the-go people who want to book them. Miscellaneous symbols. The main idea of CS is, by exploiting the sparsity nature of the signal (in any domain), we can reconstruct the signal from very fewer samples than required by Shannon-Nyquist sampling theorem. In a three-part series for Weekend, Fiona Cairns reveals how to make a simplified version of the cake she made for the Duke and Duchess of Cambridge's Royal Wedding and other treats. By using this tool you avoid the command line and having to install Perl. Some preferred LaTeX/Overleaf, while others preferred the R/Knitr/RMarkdown/Sweave ecosystem. The tombstone is sometimes open: (hollow black square). My mental model is that the best we can hope for, the best attainable policy is the projection into the manifold of expertise-informed policies subject to the contraints of pressure from vested interests allowed at the time we chose that policy and available attention. The frequently used left delimiters include (, [ and {, which are obtained by typing (, [and \{respectively. It even does the right thing when something has both a subscript and a superscript. If you want the limits of an integral/sum/product to be specified above and below the symbol in inline math mode, use the \limits command before limits specification. It is a way of bridging the Wikipedia–academia gap by enabling academics, scholars and professionals to contribute expert knowledge to the Wikimedia movement in the familiar academic. Using a 30 day notice letter sample can give you a guide to work with and ensure that your letter of notice says on track. 3 Declare commands within new environment. bonsoir à tous j'aimerai écrire en latex limite quand x tend vers 0, x 0. November 2013 at 18:28. LaTeX template for the CFM 2017 article submission Instructions: Comments in the file below indicate where the following steps have to be performed. Since anti-HIV tests became commercially available in 1985 they have been widely used in diagnostic and transfusion laboratories in the developed world. Languages: English Shqip العربية Bangla Български Català 中文 ( 正體字 , 简化字 (1) , 简化字 (2)) Hrvatski Čeština Dansk Nederlands Esperanto Eesti فارسی Suomi Français Deutsch Ελληνικά ગુજરાતી עברית. sty file that contains these commands. This is a simple step, if you use LaTeX frequently surely you already know this. Thanks for contributing an answer to Stack Overflow! Please be sure to answer the question. See also % command used to denote a comment This article is a stub. As you are aware, there are commands to put a bar or a tilde over a symbol in math mode in LaTeX. com Knowledge base dedicated to Linux and applied mathematics. Basic LaTeX Julie Mitchell This resource was adapted from notes provided by Jerry Marsden 1 Basic Formatting 1. By default Latex justifies all your text so that it lines up on both the left and right margins. improve this answer. When the compiler processes your base file and reaches one of the commands \input or \include, it reads. 1 LaTeX LaTeX is a free document markup system speci cally for technical and scienti c writing. asd - Free ebook download as Text File (. The word command may sound scary. UnicodeMath resembles real mathematical notation the most in comparison to all of the math linear formats, and it is the most concise linear format, though some may prefer editing in the LaTeX input over UnicodeMath since that is widely used in academia. Computerized typesetting. Computerized typesetting. LaTeX symbols in MediaWiki. Bold, italics and underlining Simple text formatting helps to highlight important concepts within a document and make it more readable. In this article, I will walk you through the 5 different LaTeX templates that you will use one day or another and provide a simple overview of how to use them and what they mean. Area between Curves Calculator - eMathHelp. Overleaf es un programa online que nos permite editar y compilar desde nuestro navegador un documento de LaTeX. Relational Operators (math mode). The main idea of CS is, by exploiting the sparsity nature of the signal (in any domain), we can reconstruct the signal from very fewer samples than required by Shannon-Nyquist sampling theorem. The usage is pretty easy, you can basically type the name of the letter and put a backslash in front of it. When two maths elements appear on either side of the sign, it is assumed to be a binary operator, and as such, allocates some space to either side of the sign. Ce document intitulé « LaTeX - Table de caractères » issu de Comment Ça Marche (www. Consider using overleaf. The Bullet Material CV is a minimalist template with a professional feel. with manual sizing (by using two g's) and the unneeded \left. 574 silver badges. These are not guaranteed to work in MathJax but are a good place to start. So LaTeX with its strengths in math typesetting is a very good choice for writers. IEEE Access received an impact factor of 4. -57 Extrusive:Igneous Lava Flow -77. Juli 2014 um 10. Some other constructions. In the preamble of the document include the code: \usepackage{amsmath} Open an example of the amsmath package in Overleaf. Delimiters. Using lists in LaTeX is pretty straightforward and doesn't require you do add any additional packages. In Excel, there are multiple ways to draw this function:. RESPONSE XS90 P0240031 XS90 package £2,999. Literaturverzeichnis erstellen - Schreiben Sie einfach den Titel (Autor, Verlag, etc. First we must take a quick look at LaTeX syntax. The effect should be almost the same with $#^a$ except that a is on the top instead of top right of #. Thanks for contributing an answer to Mathematica Stack Exchange! Please be sure to answer the question. Labels are a necessary part of typesetting as they are efficient pointers to information. Um editor de LaTeX online fácil de usar. Vous pouvez. Boston, MA. cond(A) berechnet die Konditionszahl von A in 2-Norm. Document history. 1 3 ing the instruments that generated some of the data. There is a wealth of guides on how to do this available on. Carol JVF Burns's page of. 폰트 크기 (font size) 문서에서 폰트 크기는 10 pt, 11 pt, 12 pt 따위가 가장 널리 쓰인다. See also % command used to denote a comment This article is a stub. I think I'd lean towards the latter option, just because collectively edited answers tend to see more action and "peer review" than tag wikis: Post edits displayed at the top of the main (meta) page, whereas a tag wiki edit is only (?) visible in the suggested edits. Electrical units are easily written complying to standards using the siunitx package. The \quad command adds in some additional spacing between the definition and the inequalities. Input: two revisions of one LaTeX document. The colour of a second block of text, delimited by { and }, is set to red with the command \color{red}, then a. The template is in accordance with the latest version of the Norms for the printing of thesis/dissertations of the UNICAMP (CCPG Nº 001/2019). To get an expression exp to appear as a subscript, you just type _{exp}. The amsmath package provides commands to typeset matrices with different delimiters. Related MATLAB, Maple, Mathematica, LaTeX News on Phys. Similar is for limit expressions. Das Literaturverzeichnis Nicht erst seit den Rücktritten der Bundesminister Gutenberg und Schavan sollte allen Studierenden klar sein, dass korrekte Zitate und ein vollständiges Literaturverzeichnis für eine wissenschaftliche Arbeit essenziell sind 📚☝️ – egal ob Hausarbeit, Bachelor- oder Masterarbeit. Sublime Text. Open an example in Overleaf. Integral expressions are formed from the use of sub- and superscript, the judicious use of spacing, and simply writing out the differential. This post summarizes symbols used in complex number theory. Lyx is a nice mix of the professional layout and typesetting provided by Latex, with a nice visual interface that provides immediate feedback to the user. New to LaTeX? This tool will help you learn the basics and collaborate with others. August 2007 by tom 42 Comments. In combination with a text editor and distributed version control, both reproducibility and simultaneous collaboration are improved by these plain text mark-up languages. x \bmod a : there will be a short gap between x and mod. This is another way to generate a PDF of your entire write-up. LaTeX symbols have either names (denoted by backslash) or special characters. Compressive Sensing is a Signal Processing technique, which gave a break through in 2004. 825 bronze badges. Phone Number Information; 781-723-2024: Esnaider Mcclone - Shadow Valley Dr, Malden, MA: 781-723-5549: Dorrian Disla - Adams Rd, Malden, MA: 781-723-6157. HotelTonight is a pioneer in mobile commerce. To learn more, see our tips on writing great. These notations describe the limiting behavior of a function in mathematics or classify algorithms in computer science according to their complexity / processing time. dictionary book. Any text in between \begin{flushleft}\end{flushleft} will be aligned with the left-hand margin, but have a ragged right-hand edge. LaTeX Line and Page Breaking The first thing LaTeX does when processing ordinary text is to translate your input file into a string of glyphs and spaces. Uppgifter utan källhänvisning kan ifrågasättas och tas bort utan att det behöver diskuteras på diskussionssidan. cond(A) berechnet die Konditionszahl von A in 2-Norm. Many script-languages use backslash "\" to denote special commands. Step 3: Enter key words. LyX is a graphical interface, nearly WYSIWYG, to the LaTeX word processing package. The segment is called How to LaTeX with OverLeaf. Some examples from the MathJax demos site are reproduced below, as well as the Markdown+TeX source. The Best Latex How To Make A Table Free Download PDF And Video. Whether researchers occasionally turn to Bayesian statistical methods out of convenience or whether they firmly subscribe to the Bayesian paradigm for philosophical reasons: The use of Bayesian statistics in the social sciences is becoming increasingly widespread. By using this tool you avoid the command line and having to install Perl. The math environment is for formulas that appear right in the text. Mucho más que documentos. Hi all - Do you know of any research that compares the typesetting of LaTeX, MS Word, and LibreOffice? I'm especially interested in work that compares the justification algorithms, kerning, and microtypography features, using modern versions of these applications (e. Refer to the external references at the end of this article for more information. \author \date. We've documented and categorized hundreds of macros!. at midnight than anyone else. LaTeX formats mathematics the way it's done in mathematics texts. Step 5: Enter references, e. Hi everyone! I'm working on a proof by cases in Overleaf LaTeX formatter and I'm having trouble formatting the cases within the proof. Da es praktisch unmöglich ist, alle jemals in der Mathematik verwendeten Symbole aufzuführen, werden in dieser Liste nur diejenigen Symbole angegeben, die häufig im Mathematikunterricht oder im Mathematikstudium auftreten. We've documented and categorized hundreds of macros!. View Larger Preview. Posted: (18 days ago) Beamer - Overleaf, Online LaTeX Editor. , are commonly used for inverse hyperbolic trigonometric functions (area hyperbolic functions), even though they are misnomers, since the prefix arc is the abbreviation for arcus, while the prefix ar stands for area. 202544536–dc20 CIP Printed on acid-free paper °c Birkh. Tags: cap, cup, document, intersection, LaTeX, layout, math, maths, SET, sets, union. Add a New Comment. Word 2019 or 2016, LibreOffice 6. Um editor de LaTeX online fácil de usar. Fortunately, there's a tool that can greatly simplify the search for the command for a specific symbol. Some might say that the resulting norm "fences" in the example above are a bit too large and thus threaten to dominate visually the rest of the math stuff. Math symbols defined by LaTeX package «fourier» CapitalGreeklettersdonotchangeshapeinmathalphabets. ISBN -8176-3805-9 (acid-free paper) (pbk. How to use Overleaf with IEEE Collabratec™ - your quick guide to getting started Posted by John on December 15, 2015 NOTE: This article shows screenshots of the integration of IEEE Collabratec™ with the Overleaf v1 platform, but the process is largely the same in the new Overleaf v2 platform. A list of LaTEX Math mode symbols. Overleaf is so easy to get started with that you'll be able to invite your non-LaTeX colleagues to contribute directly to your LaTeX documents. r o u g h g u i d e s. Free essays, homework help, flashcards, research papers, book reports, term papers, history, science, politics. My mental model is that the best we can hope for, the best attainable policy is the projection into the manifold of expertise-informed policies subject to the contraints of pressure from vested interests allowed at the time we chose that policy and available attention. They consist of plain text interspersed with some LaTeX commands. Open an example in Overleaf. Home; Domestic appliances; Large home appliances; Washing machines; Instructions for installation and use WASHER-DRYER Contents. Such basics can be found in introductions like lshort. Letters are printed in italics, with more space left in-between, spaces are ignored. \begin {document} View which changes have been added and removed. What they have in common is that they process the contents of filename. 常用数学符号的 LaTeX 表示方法 （以下内容主要摘自“一份不太简短的 LATEX2e 介绍”） 1、指数和下标可以用^和_后加相应字符来实现。比如： 2、平方根（square root）的输入命令为：\sqrt，n 次方根相应地为: \sqrt[n]。方根符号的大小由LATEX自动加以调整。. sty file that contains these commands. The shaft is not flexible below the electrodes, but it is. % Minimal latex example: % Shows how to switch between bold and arrow vectors. The template and relevant files are found at this page: Information for Authors of Computer Science Publications I remember this being pretty confusing the first time I looked at how to do this. We've documented and categorized hundreds of macros!. Other options here include c, for center-aligned, and r for right-aligned. To insert a citation where label is the label of a bibliographic entry in a. Keine Installation notwendig, Zusammenarbeit in Echtzeit, Versionskontrolle, Hunderte von LaTeX-Vorlagen und mehr. 636-46-05-15 VICEPRESIDENTA: CONCEPCIÓN PÉREZ GONZÁLEZ. Integer and sum limits improvement. This keeps the main body of text concise. Greek letters []. @CharlesStewart Thanks! If others find this useful, we could either use it as a tag wiki, or link here from the tag wiki. d3 is the first move in \variation{} Open an example of the skak package in ShareLaTeX. Domicilio Fiscal: C/ Melíes, nº 50, Urbanización Santa María - 08800 - Vila Nova i la Geltrú - BARCELONA. beginner woodwork. TeX has \\int as the integral sign. , there are no licence fees, etc. 85 Trachyte 166. De forma muy simple antes del cuerpo del documento basta añadir las dos instrucciones siguientes: \providecommand{\abs}[1]{\lvert#1\rvert} \providecommand{ orm}[1]{\lVert#1\rVert} Estos comandos se usan de la siguiente manera:. The mathematical symbol is produced using \partial. LaTeX equations always start with \ ( and end with \). Compressive Sensing is a Signal Processing technique, which gave a break through in 2004. This is a simple step, if you use LaTeX frequently surely you already know this. Manhattan: Take the sum of the absolute values of the differences of the coordinates. Open, and a variety of correspondence (postal) tournaments, and (4) officially represents the. As you can see, there are three basic commands and they can be nested to get combined effects. IEEE article templates let you quickly format your article and prepare a draft for peer review. 098 in the 2018 JCR release. stackexchange. latex2exp is an R package that parses and converts LaTeX math formulas to R's plotmath expressions. The differences between these two ways to include files is explained below. Mathematical modes. \begin {document} View which changes have been added and removed. Table 238: fge Math-mode Accents. An online LaTeX editor that's easy to use. The usage is pretty easy, you can basically type the name of the letter and put a backslash in front of it. Open an example in ShareLaTeX. Readbag users suggest that JRCALC Clinical Practice Guidelines 2004 is worth reading. Embed formulas in your text by surrounding them with dollar signs The equation environment is used to typeset one formula. norm(X,p) berechnet die p-Norm des Vektors X. This document is also listed in a special topic for beginners [1]. Formatting Optimization Problems with LaTeX. Open a text editor like Notepad and create a new LaTeX document by typing: \begin {document} \end {document} Type the following between the "begin" and "end" commands to create your bullet point list: Video of the Day. answered May 24 '10 at 4:21. Part 1 | Part 2 | Part 3 | Part 4 | Part 5. Hyperbolic functions The abbreviations arcsinh, arccosh, etc. To learn more about Overleaf, visit their website: www. Each column ends with an ampersand (&). Hat and underscore are used for superscripts and subscripts. bmatrix Latex matrix pmatrix vmatrix. A state diagram shows the behavior of classes in response to external stimuli. tex) [1] LaTeX Resources for Beginners. This typically indicates Rd problems. This is the basic introduction to Matlab. The calculator will find the principal unit normal vector of the vector-valued function at the given point, with steps shown. The first one is used to write formulas that are part of a text. 9f November19,2018 Thegeneral-purposedrawingpackageTikZcanbeusedtotypesetcommutativediagramsandotherkinds. I made report in LaTeX during my six weeks training. How exactly you format such citations then depends on the citation style that you are being asked to use (e. No installation, real-time collaboration, version control, hundreds of LaTeX templates, and more. Arrow symbols. LaTeX Line and Page Breaking The first thing LaTeX does when processing ordinary text is to translate your input file into a string of glyphs and spaces. Juli 2014 um 10. Add text to the graph that contains an integral expression using LaTeX markup. In a three-part series for Weekend, Fiona Cairns reveals how to make a simplified version of the cake she made for the Duke and Duchess of Cambridge's Royal Wedding and other treats. L a T e X allows two writing modes for mathematical expressions: the inline mode and the display mode. (although it still helps to know how to code maths in Latex). GB GB IT DE Contents English,1 Italiano,13 Deutsch,25 Installation, 2-3 Unpacking and levelling Connecting the electricity and water supplies Technical data Care and maintenance, 4 Cutting off the water and electricity supplies Cleaning the machine Cleaning the detergent dispenser. My view is that as with any service where there is increased competition, prices will stagnate and potentially come down in future. Variable-sized symbols. How do I change my cursor back to vertical? I accidentally pressed a key, and it seemed to have changed my vertical cursor to a horizontal/underscore type cursor!. All results are completely general, numerically sound, and based on general realizations allowing for poles at infinity. Personally I use ShareLaTeX, although I can see how Overleaf would be easier for a non-LaTeX field since you can edit directly in a WYSIWYG format or in the LaTeX format. Amphibian study shows stress increases vulnerability to virus; Mutations in SARS-CoV-2 offer insights into virus evolution. tex) files, or else just remain in the LyX domain (. Similar is for limit expressions. Hey guys, I'm writing a beamer presentation on LaTex and I'm facing a problem I can't seem to solve. LaTeX deals with the + and − signs in two possible ways. The leading scientists behind eLife are committed to rapid, fair, and constructive. LaTeX is a very flexible program for typesetting math, but sometimes figuring out how to get the effect you want can be tricky. Math symbols defined by LaTeX package «fourier» CapitalGreeklettersdonotchangeshapeinmathalphabets. com , or follow them on twitter at @Overleaf. Literaturverzeichnis erstellen - Schreiben Sie einfach den Titel (Autor, Verlag, etc. Use \left\lVert before the expression and \right\rVert after it. Greek letters. If you want different spacing, LaTeX provides the following four commands for use in math mode: \; - a thick space \: - a medium space \, - a thin space. The usage is pretty easy, you can basically type the name of the letter and put a backslash in front of it. Log-like symbols. Writing congruence relations in latex. 1 DeclareRobustCommand. November 2013 by tom 8 Comments. 예를 들어 em이나 px. Fortunately, there are alternative commands that do the same task differently that we can try and there are…. Juli 2014 um 10. Step 2: Enter author information. Um editor de LaTeX online fácil de usar. Permanent Link Edit Delete. It even does the right thing when something has both a subscript and a superscript. Overleaf /LaTex Not sure students need to know too much latex anymore… markdown/r-md is a lot simpler and using it with css and html bits is very flexible. No author: when author information is not available, use the source title to replace the author's position. Letters are printed in italics, with more space left in-between, spaces are ignored. ) The names of certain standard functions and abbreviations are obtained by typing a backlash \ before the name. Bachelor-, Diplom- oder Doktorarbeiten sind in der Regel mit einem enormen Aufwand verbunden. Many script-languages use backslash "\" to denote special commands. This is a simple step, if you use LaTeX frequently surely you already know this. Citations are an abomination, as are numbering equations; random crashes, corruption of files that contain a lot of graphics etc. Once you have loaded \usepackage {amsmath} in your preamble, you can use the following environments in your math environments: If you need to create matrices with different delimiters, you can add them manually to a plain matrix. Sometimes, the output doesn't come out the way some of us might expect or want. the course of thirty-nine editions, Hart's Rules has grown to be the standard work in itsfield,explaining subject by subject each major aspect of punctuation, capitalization, italics, hyphenation. 7 posts • Page 1 of 1. Area between Curves Calculator - eMathHelp. What are LaTeX “environments” While TeX makes direct provision for commands, LaTeX adds a concept of “environment”; environments perform an action on a block (of something or other) rather than than just doing something at one place in your document. aa aah aahed aahing aahs aal aalii aaliis aals aardvark aardvarks aardwolf aardwolves aargh aas aasvogel aasvogels aba abaca abacas abaci aback abacus abacuses abaft. Easy-to-use symbol, keyword, package, style, and formatting reference for LaTeX scientific publishing markup language. 예를 들어 em이나 px. LaTeX forum ⇒ Math & Science ⇒ Norm symbol to correspond argument height Topic is solved Information and discussion about LaTeX's math and science related features (e. To learn more, see our tips on writing great. Right now I have \theoremstyle{case} \newtheorem{case}{Case} at the beginning of my document, and \begin{case} within my proof. com , or follow them on twitter at @Overleaf. texblog because LaTeX matters. Convert Latex equations into beautiful, transparency-correct PNGs. The next chapter will focus on Plain TeX and will explain advanced techniques for programming. Any text in between \begin{flushleft}\end{flushleft} will be aligned with the left-hand margin, but have a ragged right-hand edge. formulas, graphs). Diese Liste mathematischer Symbole zeigt eine Auswahl der gebräuchlichsten Symbole, die in moderner mathematischer Notation innerhalb von Formeln verwendet werden. SaxLove Recommended for you. Integral\int_ {a}^ {b} x^2 dxinside. Languages: English Shqip العربية Bangla Български Català 中文 ( 正體字 , 简化字 (1) , 简化字 (2)) Hrvatski Čeština Dansk Nederlands Esperanto Eesti فارسی Suomi Français Deutsch Ελληνικά ગુજરાતી עברית. Such basics can be found in introductions like lshort. here 1 is subscript of C. Document history. There is a simple way to add "normal text" fragments in. Step 1: Enter abstract title. A Normal distribution with a mean of zero and a standard deviation of 1 is also known as the Standard Normal Distribution (m =0, s =1) as in Figure 1. Plotmath expressions are used to enter mathematical formulas and symbols to be rendered as text, axis labels, etc. The Mac app is finally stable enough. A list of LaTEX Math mode symbols. Creation of vectors is included with a few basic operations. In LaTeX backslash is used to generate a special symbol or a command. LaTeX The LaTeX command that creates the icon. Dictionary - Free ebook download as Text File (. pro woodwork projects. you want to put a small piece of text in a specific type style, you can do it as follows: If you want to put larger amounts of text into these type styles, you can use {{\begin}} and {{\end}} commands; i. Includes index. The main idea of CS is, by exploiting the sparsity nature of the signal (in any domain), we can reconstruct the signal from very fewer samples than required by Shannon-Nyquist sampling theorem. NUESTRA JUNTA DIRECTIVA ESTÁ FORMADA POR: PRESIDENTA: FRANCISCA GIL QUINTANA-- TELF. improve this answer. HOME: Next: Arrow symbols (amssymb) Last: Relation symbols (amssymb) Top: Index Page Index Page. Relation symbols. Consider using overleaf. The amsmath package provides commands \lvert, \rvert, \lVert, \rVert which change size dynamically. Home > Latex > FAQ > Latex - FAQ > LateX Derivatives, Limits, Sums, Products and Integrals. (Refer to bio-contamination data overleaf) Suspension Systems 'Blue Tongue' Aluminium Clean Room Flat Faced Tee Grid & PeakForm Prelude Steel 24mm Tee above. 9f November19,2018 Thegeneral-purposedrawingpackageTikZcanbeusedtotypesetcommutativediagramsandotherkinds. It even does the right thing when something has both a subscript and a superscript. The amsmath The amsmath package comes standard with most L A TEX distributions and is loaded by physics for your convenience. The appropriate LaTeX command is \overset{annotation}{symbol}. Use MathJax to format equations. Computerized typesetting. Type H for immediate help. LaTeX is a fairly high-level language compared to Plain TeX and thus is more limited. Was ist 9+4 ? Oh, die ist schwer. Integer and sum limits improvement. Today I tried to write the solution of a differential equation in LaTeX. Similar is for limit expressions. Miscellaneous symbols. Other options here include c, for center-aligned, and r for right-aligned. This is another way to generate a PDF of your entire write-up. This tag should be used for questions that concern the backend; questions concerning use of the web service are probably better asked on tex. journals) on Instagram: “made it to college! super late spread but more are headin' your way <3 - #bujo #bulletjournal. Part 1 | Part 2 | Part 3 | Part 4 | Part 5. The amsmath package provides commands \lvert, \rvert, \lVert, \rVert which change size dynamically. A key transformed subproblem involving K„i, x„i, d„i appears in the middle. Newer Post Older Post Home. No extra packages are required to use these symbols. Similar is for limit expressions. 75in for more space. This is the template for LaTeX submissions to eLife. Using lists in LaTeX is pretty straightforward and doesn't require you do add any additional packages. Diese können Sie mit help funcname erfahren. Other tests depend on the binding of a fluorescein or chemiluminescent conjugate, or the visible agglutination of HIV-coated gelatin or latex particles.\begingroup$@knzhou So, I was lucky to find an online editor : Overleaf (first v1 and later v2). In addition to the actual “math mode” environments, wherein math symbols and structures are the norm and text is the exception, you may also want environments in which the content is primarily textual, but which contain logical constructs, such as algorithms, answers, assertions and axioms (and that’s just the A’s!). The displaymath environment is for formulas that appear on their own line. x \mod a : there will be a long gap between x and mod. Den här artikeln behöver källhänvisningar för att kunna verifieras. Here I'd figured I must have been doing something wrong. Easy-to-use symbol, keyword, package, style, and formatting reference for LaTeX scientific publishing markup language. improve this answer. Das Literaturverzeichnis Nicht erst seit den Rücktritten der Bundesminister Gutenberg und Schavan sollte allen Studierenden klar sein, dass korrekte Zitate und ein vollständiges Literaturverzeichnis für eine wissenschaftliche Arbeit essenziell sind 📚☝️ – egal ob Hausarbeit, Bachelor- oder Masterarbeit. Kurz: Es ist praktisch und ich kann von dort aus entscheiden, ob ich länger in Sublime weiterschreibe oder ein PDF oder eine LaTeX-Datei generiere. , there are no licence fees, etc. Overleaf, Online LaTeX Editor. \end {document}. Grade ]?ercentage of f. au/whatson/academic 1497947400 2017 6 20 Tuesday 16:30 1497951000 2017 6 20 Tuesday 17:30. \begin {document} View which changes have been added and removed. You can use Overleaf to write and collaborate online in LaTeX using the template. sty file that contains these commands. Previous ones: Basics and overview Use of mathematical symbols in formulas and equations Many of the examples shown here were adapted from the Wikipedia article Displaying a formula, which is actually about formulas in Math Markup. Using the C++ programming language as an example, one can find nearly every citation for the C++ standards in BibTeX format. cond(A) berechnet die Konditionszahl von A in 2-Norm. Spacing in Math Mode. 098 in the 2018 JCR release. Hi everyone! I'm working on a proof by cases in Overleaf LaTeX formatter and I'm having trouble formatting the cases within the proof. pdf) or read book online for free. In LaTeX backslash is used to generate a special symbol or a command. Select, ver Figura 13. Dictionary - Free ebook download as Text File (. All the versions of this article: < français > Here are few examples to write quickly matrices. Im Bildungs- und Erziehungsauftrag des Gymnasiums wird gefordert, dass das Gymnasium den Schüler „[] auch dazu befähigt, den Anforderungen einer modernen Berufs- und Arbeitswelt gewachsen zu sein" ([4], S. The main things used in it are: Fractions : These can be written as: \frac{x/y} Subscripts: These are wriiten as. Overleaf es un programa online que nos permite editar y compilar desde nuestro navegador un documento de LaTeX. In combination with a text editor and distributed version control, both reproducibility and simultaneous collaboration are improved by these plain text mark-up languages. Compressive Sensing is a Signal Processing technique, which gave a break through in 2004. com JavaScript MIT 17 5 4 2 Updated Apr 23, 2020. tex before continuing with the rest of the base file (the file that contains these statements). Eine gute Möglichkeit, Zitate einzufügen und zu formatieren, ist BibTeX (). The following table shows the whole Greek alphabet along with the commands in a nice table. It's widely used throughout the academic world to publish technical. Tags: document, LaTeX, multiplication, prod, product, symbol. formulas, graphs). For Microsoft Word, and other word processors, you can choose PDF inside of the “File !Save As” menu. No installation, real-time collaboration, version control, hundreds of LaTeX templates, and more. Some other constructions. nach dem Autor oder dem Alphabet) - an einer gewünschten. Embed formulas in your text by surrounding them with dollar signs$ The equation environment is used to typeset one formula. Markdown into LaTeX with Style Post-publication update (13 May, 2017): We are grateful to Vít Novotný, the author/maintainer of the markdown package, for writing to us with some helpful feedback concerning the original article. 2 posts • Page 1 of 1. If you want different spacing, LaTeX provides the following four commands for use in math mode:. \bibliographystyle {bstfilename} To choose a BibTeX bibliographic style file with the extension. HOME: Next: Arrow symbols (amssymb) Last: Relation symbols (amssymb) Top: Index Page Index Page. Quotation Marks and Dashes. LaTeX files usually have a. Overleaf es un programa online que nos permite editar y compilar desde nuestro navegador un documento de LaTeX. Math into LaTeX : an introduction to LaTeX and AMS-LaTeX / George Gr¨atzer p. I'm taking a matrix algebra course this term and thought I'd LaTeX my homework assignments just for practice. » You can assign values to patterns involving Integrate to give results for new classes of integrals. Dictionary - Free ebook download as Text File (. Most of the stock math commands are written for typesetting math or computer science papers for academic journals, so you might need to dig deeper into LaTeX commands to get the vector notation styles that are common in physics textbooks and articles. This puts the annotation in a smaller type size directly above the symbol. Brackets and Norms The frequently used left delimiters include (, [ and {, which are obtained by typing ( , [ and \{ respectively. \frac {d} {dt} \bigg|_ {t=0} f (t) achieves a similar effect. Vous pouvez. 3 posts • Page 1 of 1. Was ist 9+4 ? Oh, die ist schwer. If you want the limits above and below, place the \limits command after the sum command as follows: $\sum\limits_{k=1}^n k$. This is another way to generate a PDF of your entire write-up. ShareLaTeX is so easy to get started with that you'll be able to invite your non-LaTeX colleagues to contribute directly to your LaTeX documents. To produce a printed document, this string must be broken into lines, and these lines must be broken into pages. Relation symbols. 8, AUGUST 2015 1 How to Use the IEEEtran LATEX Class Michael Shell, Member, IEEE (Invited Paper) Abstract—This article describes how to use the IEEEtran class. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. Im Bildungs- und Erziehungsauftrag des Gymnasiums wird gefordert, dass das Gymnasium den Schüler „[] auch dazu befähigt, den Anforderungen einer modernen Berufs- und Arbeitswelt gewachsen zu sein" ([4], S. Amphibian study shows stress increases vulnerability to virus; Mutations in SARS-CoV-2 offer insights into virus evolution. No extra packages are required to use these symbols. The usage is pretty easy, you can basically type the name of the letter and put a backslash in front of it. Poetica – Get clear feedback, wherever you’re writing. GitHub is home to over 40 million developers working together to host and review code, manage projects, and build software together. Delimiters. Arrow symbols. It even does the right thing when something has both a subscript and a superscript. The command \variation{} helps to analyse variations of a move. \end {document}. \end {document}. The USCF is the national chess organization of the United States. 09 layered on T X v2. For example, one obtains by typing $\cos(\theta + \phi) = \cos \theta \cos \phi - \sin \theta \sin \phi$ The following standard functions are represented by control sequences defined in LaTeX:. You must use the following package: \usepackage {amsmath} \begin {matrix} \begin {pmatrix} \begin {bmatrix} \begin {vmatrix} \begin {Vmatrix}. Embed formulas in your text by surrounding them with dollar signs \$ The equation environment is used to typeset one formula. LaTeX is available as free software. {tikzcd} CommutativediagramswithTikZ Version0. As you see, the way the equations are displayed depends on the delimiter, in this case and . tex file via the \input{filename} command. The version I have is on a compilation of Martin Hannett produced tracks called “And Here Is The Young Man”. In case you are working with LaTeX, there are two very good (free!) sites offering collaboration functionality: Overleaf and ShareLaTeX. edited Feb 23 '15 at 10:57. Previous ones: Basics and overview Use of mathematical symbols in formulas and equations Many of the examples shown here were adapted from the Wikipedia article Displaying a formula, which is actually about formulas in Math Markup. , are commonly used for inverse hyperbolic trigonometric functions (area hyperbolic functions), even though they are misnomers, since the prefix arc is the abbreviation for arcus, while the prefix ar stands for area.
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https://mathshistory.st-andrews.ac.uk/Biographies/Ricci_Giovanni/ | # Giovanni Ricci
### Quick Info
Born
17 August 1904
Florence, Italy
Died
9 September 1973
Milan, Italy
Summary
Giovanni Ricci was an Italian mathematician who worked in number theory, differential geometry, mathematical analysis and theory of series.
### Biography
Giovanni Ricci was brought up in Florence where he underwent his school education. He went to Pisa where he studied mathematics at the Scuola Normale Superiore which is associated with the University of Pisa. He was only 21 years old when he submitted his doctoral thesis Le transformazioni de Christoffel e di Darboux per le superficie rotonde,coniche e cilindriche. Alcune generalizioni, per rotolamento del cono e del cilindro di rotazion and was awarded a doctorate on 15 December 1925.
After graduating, Ricci went to Rome where he was appointed as a lecturer. He returned to Pisa in 1928 when he was appointed as a professor at the Scuola Normale Superiore [1]:-
The time he spent at the Scuola Normale Superiore was very fruitful in scientific research and gave birth to some papers which stand out among his publications.
In fact Ricci held this position for eight years. During that time he published work on the Goldbach conjecture which concerns writing numbers as the sum of primes, and also on Hilbert's Seventh Problem which asks whether or not aı was transcendental when a and b are algebraic. Hilbert himself remarked that he expected this Seventh Problem to be harder than the solution of the Riemann conjecture. However Riemann's intuition was not too good here for it was solved independently by Gelfond and Schneider in 1934. As examples of the papers Ricci published during his eight years as a professor in Pisa we mention: Sulle funzioni simmetriche delle radici dell'unità secondo un modulo composto (1931), Sui grandi divisori primi delle coppie di interi in posti corrispondenti di due progressioni aritmetiche. Applicazione del metodo di Brun (1933), Sulle serie di potenze che rappresentano funzioni razionali a coefficienti razionali e con i poli appartenenti ad una progressione geometrica (1934), Su un teorema di Tchebychef-Nagel (1934), and Sui teoremi Tauberiani (1934). Let us also note that, in 1951, Ricci wrote a paper on the school of mathematics in Pisa La scuola matematica pisana dal 1848 al 1948 . This covered a period of one hundred years, including the time that he both studied there and worked there as a professor.
In 1936 Ricci moved to Milan where he was appointed as Professor of Mathematical Analysis at the University. Of course this was a difficult period in Italy. In 1935 the League of Nations had imposed sanctions against Italy following its invasion of Abyssinia. The following year Mussolini and Hitler declared the Rome-Berlin Axis and in 1937 Italy left the League of Nations. Although World War II began September 1939, it was not until June 1940 that Italy declared war on Britain and France. Ricci was unable to continue his research with the vigour that he had in Pisa and was more involved in teaching and administration. Between his taking up the position in Milan and the end of World War II, a period of nine years, he published only two works. Sull'irrazionalità del rapporto della circonferenza al diametro (1942) was the published version of a lecture he gave at the Second Italian Mathematical Congress in Bologna in 1940. The other paper Problemi secolari e risposte recenti nel campo dell'aritmetica (1945) was again the published version of a lecture. In it Ricci discussed the estimation of exponential sums, the distribution of $f (x)$ mod 1, Waring's problem and Goldbach's problem.
Ricci held the chair of Mathematical Analysis in Milan for over 36 years. He published many excellent expository and historical articles from 1948 onwards, and also did some interesting research on the distribution of primes. However his research never returned to the output of his days in Pisa, either in quantity nor depth. We will look briefly, however, at some of his interesting expository articles: Figure, reticoli e computo di nodi (1948) discussed classical lattice-point problems and puts these in historical perspective; La differenza di numeri primi consecutivi (1952) looks at the the history of the various problems concerned with the difference between consecutive primes; and Aritmetica additiva: aspetti e problemi (1954) is an expository account of classical and modern results in the additive theory of numbers.
We mentioned Ricci's research on the distribution of primes. In the papers Sul coefficiente di Viggo Brun (1953) and Sull'andamento della differenza di numeri primi consecutivi (1954) he presented many results on the difference between consecutive primes. We should not, however, give the impression that all Ricci's work was in number theory for he also made significant contributions to the theory of functions of a complex variable.
Cugiani, who knew Ricci from 1937 onwards, writes in [1] about his abilities as a teacher:-
He was a born teacher, partly owing to his mastery of the language, since he came from Toscana, the region where the best Italian is spoken; but, most of all, he was an excellent teacher because of his extreme clarity of mind and his profound intellectual honesty, which forced him to present topics in such a way that their logical connections became absolutely obvious; those who listened to him often thought that what he had explained was a chain of trivial statements (as he used to say). Of great help was also his wide mathematical knowledge, deriving both from the school he had followed and from his continuous interest in research.
As to his character and personality, Cugiani writes:-
His profound aesthetic sense was quite an important aspect of his personality, and it seemed to be related to his fondness for library organisation, his love of books, which he always regarded as objects of high aesthetic value. He once said to me: "Opening a book neatly printed on good paper and properly bound is one of the pleasures allowed to man". To his everyday work, his aesthetic sense brought clarity and a sober and balanced language, both in research and in teaching. Of course it also had other, more evident, manifestations, for instance his love of music, which he often combined with the pleasures of mathematical studies through a subtle transposition of activities which he felt to be very close to each other.
Finally we mention that Ricci's most famous doctoral student during his time in Milan was Enrico Bombieri.
### References (show)
1. M Cugiani, Giovanni Ricci (1904-1973), Acta Arith. 46 (4) (1986), 303-311.
2. M Cugiani, Commemoration of Giovanni Ricci (Italian), Geometry of Banach spaces and related topics, Milan, 1983, Rend. Sem. Mat. Fis. Milano 53 (1983), 11-15.
3. M Cugiani, Commemorazione di Giovanni Ricci, Rend. Sem. Mat. Fis. Milano 43 (1973), 7-23.
4. F G Tricomi, Giovanni Ricci (1904-1973) (Italian), Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 108 (3-4) (1974), 585-589.
### Additional Resources (show)
Other websites about Giovanni Ricci:
Written by J J O'Connor and E F Robertson
Last Update July 2007 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5611138343811035, "perplexity": 3822.8827602531733}, "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-2021-31/segments/1627046155322.12/warc/CC-MAIN-20210805032134-20210805062134-00023.warc.gz"} |
https://zbmath.org/?q=an:1114.60084 | ×
On multidimensional branching random walks in random environment.(English)Zbl 1114.60084
Summary: We study branching random walks in random i.i.d. environment in $$\mathbb{Z}^d$$, $$d\geq 1$$. For this model, the population size cannot decrease, and a natural ural definition of recurrence is introduced. We prove a dichotomy for recurrence/transience, depending only on the support of the environmental law. We give sufficient conditions for recurrence and for transience. In the recurrent case, we study the asymptotics of the tail of the distribution of the hitting times and prove a shape theorem for the set of lattice sites which are visited up to a large time.
MSC:
60K37 Processes in random environments 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
Full Text:
References:
[1] Alves, O. S. M., Machado, F. P. and Popov, S. Yu. (2002). The shape theorem for the frog model. Ann. Appl. Probab. 12 533–546. · Zbl 1013.60081 [2] Athreya, K. B. and Ney, P. E. (1972). Branching Processes . Springer, New York. · Zbl 0259.60002 [3] Baillon, J.-B., Clément, P., Greven, A. and den Hollander, F. (1993). A variational approach to branching random walk in random environment. Ann. Probab . 21 290–317. · Zbl 0770.60088 [4] Biggins, J. (1978). The asymptotic shape of the branching random walk. Adv. in Appl. Probab . 10 62–84. JSTOR: · Zbl 0383.60078 [5] Bramson, M. and Griffeath, D. (1980). On the Williams–Bjerknes tumor growth model. II. Math. Proc. Cambridge Philos. Soc. 88 339–357. · Zbl 0459.92013 [6] Comets, F., Menshikov, M. V. and Popov, S. Yu. (1998). One-dimensional branching random walk in random environment: A classification. Markov Process. Related Fields 4 465–477. · Zbl 0938.60081 [7] Dembo, A., Peres, Y. and Zeitouni, O. (1996). Tail estimates for one-dimensional random walk in random environment. Comm. Math. Phys. 181 667–683. · Zbl 0868.60058 [8] Devulder, A. (2005). A branching system of random walks in random environment. Available at http://www.proba.jussieu.fr/mathdoc/textes/PMA-834.pdf. · Zbl 1138.60341 [9] Durrett, R. and Griffeath, D. (1982). Contact processes in several dimensions. Z. Wahrsch. Verw. Gebiete 59 535–552. · Zbl 0483.60089 [10] Engländer, J. (2005). Branching Brownian motion with ‘mild’ Poissonian obstacles. Available at http://arxiv.org/math.PR/0508585. [11] Fayolle, G., Malyshev, V. A. and Menshikov, M. V. (1995). Topics in the Constructive Theory of Countable Markov Chains. Cambridge Univ. Press. · Zbl 0823.60053 [12] Gantert, N. and Müller, S. (2005). The critical branching random walk is transient. Available at http://arxiv.org/math.PR/0510556. · Zbl 1115.60077 [13] Gantert, N. and Zeitouni, O. (1998). Quenched sub-exponential tail estimates for one-dimensional random walk in random environment. Comm. Math. Phys. 194 177–190. · Zbl 0982.60037 [14] Greven, A. and den Hollander, F. (1992). Branching random walk in random environment: Phase transitions for local and global growth rates. Probab. Theory Related Fields 91 195–249. · Zbl 0744.60079 [15] den Hollander, F., Menshikov, M. V. and Popov, S. Yu. (1999). A note on transience versus recurrence for a branching random walk in random environment. J. Statist. Phys. 95 587–614. · Zbl 0933.60089 [16] Kingman, J. F. C. (1973). Subadditive ergodic theory. Ann. Probab. 1 883–909. JSTOR: · Zbl 0311.60018 [17] Lawler, G. F. (1983). A discrete stochastic integral inequality and balanced random walk in a random environment. Duke Math. J. 50 1261–1274. · Zbl 0569.60071 [18] Liggett, T. M. (1985). An improved subadditive ergodic theorem. Ann. Probab. 13 1279–1285. · Zbl 0579.60023 [19] Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York. · Zbl 0559.60078 [20] Machado, F. P. and Popov, S. Yu. (2000). One-dimensional branching random walk in a Markovian random environment. J. Appl. Probab. 37 1157–1163. · Zbl 0995.60070 [21] Machado, F. P. and Popov, S. Yu. (2003). Branching random walk in random environment on trees. Stochastic Process. Appl. 106 95–106. · Zbl 1075.60570 [22] Nagaev, S. V. (1979). Large deviations of sums of independent random variables. Ann. Probab. 7 745–789. · Zbl 0418.60033 [23] Pisztora, A. and Povel, T. (1999). Large deviation principle for random walk in a quenched random environment in the low speed regime. Ann. Probab. 27 1389–1413. · Zbl 0964.60056 [24] Pisztora, A., Povel, T. and Zeitouni, O. (1999). Precise large deviation estimates for a one-dimensional random walk in a random environment. Probab. Theory Related Fields 113 191–219. · Zbl 0922.60059 [25] Sznitman, A.-S. (1999). Slowdown and neutral pockets for a random walk in random environment. Probab. Theory Related Fields 115 287–323. · Zbl 0947.60095 [26] Sznitman, A.-S. (2000). Slowdown estimates and central limit theorem for random walks in random environment. J. Eur. Math. Soc. (JEMS) 2 93–143. · Zbl 0976.60097 [27] Sznitman, A.-S. (2002). An effective criterion for ballistic behavior of random walks in random environment. Probab. Theory Related Fields 122 509–544. · Zbl 0995.60097 [28] Sznitman, A.-S. (2003). On new examples of ballistic random walks in random environment. Ann. Probab. 31 285–322. · Zbl 1017.60104 [29] Sznitman, A.-S. and Zerner, M. (1999). A law of large numbers for random walks in random environment. Ann. Probab. 27 1851–1869. · Zbl 0965.60100 [30] Varadhan, S. R. S. (2003). Large deviations for random walks in a random environment. Comm. Pure Appl. Math. 56 1222–1245. · Zbl 1042.60071 [31] Volkov, S. (2001). Branching random walk in random environment: Fully quenched case. Markov Process. Related Fields 7 349–353. · Zbl 0991.60073 [32] Zeitouni, O. (2004). Random walks in random environment. Lecture Notes in Math. 1837 190–312. Springer, Berlin. · Zbl 1060.60103 [33] Zerner, M. (1998). Lyapounov exponents and quenched large deviations for multidimensional random walk in random environment. Ann. Probab. 26 1446–1476. · Zbl 0937.60095 [34] Zerner, M. (2002). A non-ballistic law of large numbers for random walks in i.i.d. random environment. Electron. Comm. Probab. 7 191–197. · Zbl 1008.60107
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https://www.sawaal.com/aptitude-reasoning/verbal-reasoning-mental-ability/series-questions-and-answers.htm?page=2&sort= | # Number Series Questions
Q:
Find the wrong number in the series.
3, 8, 15, 24, 34, 48, 63
A) 15 B) 24 C) 34 D) 48
Explanation:
The difference between consecutive numbers of the given series are respectively 5, 7, 9, 11, 13, and 15.
Therefore, 24+11=35 But in your problem it is given as 34.so 34 is wrong number
491 74871
Q:
Find the missing number in the given number series?
625, 625, 600, ?, 475, 875
A) 545 B) 700 C) 675 D) 725
Explanation:
Here the given number series 625, 625, 600, ?, 475, 875 follows a pattern that
625
625 + (0 x 0) = 625
625 - (5 x 5) = 625 - 25 = 600
600 + (10 x 10) = 600 + 100 = 700
700 - (15 x 15) = 700 - 225 = 475
475 + (20 x 20) = 475 + 400 = 875
Hence, the missing number in the given number series is 700.
126 60965
Q:
Find the missing number in the following series?
2, 3, 10, 39, ?, 885
A) 128 B) 156 C) 172 D) 189
Explanation:
Take a Look at the below sereies
$2×1+12=3$
$3×2+22=10$
$10×3+32=39$
$39×4+42=172$
$172×5+52=885$
218 57453
Q:
8, 15, 28, 53, ?, 199
A) 101 B) 102 C) 103 D) 104
Explanation:
Here the series of the form is
53 x 2 - 4 = 106 - 4 = 102
500 54292
Q:
7, 8, 18, 57, ?, 1165, 6996
A) 228 B) 542 C) 232 D) 415
Explanation:
Here 2nd number = (1st number x 1 )+1
3rd number = (2nd number x 2 +2)
4th number = (3rd number x 3 )+3 and so on..
Therefore, 5th number = (4th number x 4) +4 =57 x 4 + 4 =232.
294 50028
Q:
Find the wrong number in the series
7, 28, 63, 124, 215, 342, 511
A) 28 B) 124 C) 215 D) 342
Explanation:
Here the number follows the given rule
But 28 has been given in problem series.
so 28 is wrong number.
226 49279
Q:
What will be the next term in
BKS, DJT, FIU, HHV, __
A) IJX B) IGX C) JGW D) JGU
Explanation:
Here the first letter of the group is moved with a gap of one letter, the second letter of the group is backwardness and the third letter of the group is forwardness proceeding like this we get the letter group "JGW"
97 47317
Q:
Which fraction comes next in the sequence
A) 1/27 B) 11/278 C) 9/48 D) 7/123
Explanation:
Clearly, the numerators of the fractions in the given sequence follow the series of 2, 3, 5, 7,... and the denominator follows 3, 9, 27, 81, 243, ...
Then, the next term in the given sequence is . | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 5, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8524905443191528, "perplexity": 1610.4939222849302}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337338.11/warc/CC-MAIN-20221002150039-20221002180039-00067.warc.gz"} |
https://forums.spybot.info/showthread.php?10014-svchost-exe-spawns-iexplore-exe&s=3fbe819aa93c4cebd81e13e52c564272&p=61674&highlight= | # Thread: svchost.exe spawns iexplore.exe
1. ## svchost.exe spawns iexplore.exe
Hi All, First thanks for the assistance on this issue. I'm all out of options, and the last and final step would be to reformat the drive.
I have a problem where the system would call svchost.exe and it would kickoff a iexplore.exe. This would start happening the minute I log into windows XP Home. The iexplore.exe would just grind at 99%, and cause me not to be able to use the system. It would keep starting iexplore.exe processes. I had to disable iexplore.exe by renaming the entire folder.
I had some other trojans that I recently removed, but apparently not clean enough. I removed windhcp.ocx, Lineage.
When I go to msconfig I would see an entry on there that is 'checked' in startup. The startup item and command are both 'junk characters'. The location is
When I go to that location using regedit, the values look blank. But when I display binary data, there's actually something in there. I can overwrite that value. If I reboot in safe mode, my value sticks in there. But if I reboot in normal mode, something is re-populating that registry entry. I can see it in msconfig everytime. I put an entry to load c:\hjt\hijackthis.exe in there, but if i load in normal mode, it will call my entry, then overwrite it with the trojan.
My guess is that something is starting in normal mode that is not started in safe mode. But i can't figure out what that 'something' is.
On top of all that, I removed my Norton Antivirus but can't install it again, so I can't re-scan. (althouth the first scan came up empty)
Here's the programs that I've used to scan and none reported anything:
Spybot
Norton Antivirus
CA eTrust Antivirus Online
F-Secure Online scanner.
AVG Anti-Spyware
all of them using the latest definitions.
I think i'm 99% of the way there, but that last 1% is what's killing the system.
Here's my hijacklog:
Logfile of HijackThis v1.99.1
Scan saved at 12:14:52 PM, on 12/28/2006
Platform: Windows XP SP2 (WinNT 5.01.2600)
MSIE: Internet Explorer v6.00 SP2 (6.00.2900.2180)
Running processes:
C:\WINDOWS\System32\smss.exe
C:\WINDOWS\system32\winlogon.exe
C:\WINDOWS\system32\services.exe
C:\WINDOWS\system32\lsass.exe
C:\WINDOWS\system32\svchost.exe
C:\WINDOWS\System32\svchost.exe
C:\WINDOWS\system32\spoolsv.exe
C:\WINDOWS\Explorer.EXE
C:\WINDOWS\system32\ctfmon.exe
C:\WINDOWS\system32\cisvc.exe
C:\WINDOWS\System32\CTsvcCDA.exe
C:\Program Files\Common Files\Microsoft Shared\VS7Debug\mdm.exe
C:\WINDOWS\system32\nvsvc32.exe
C:\WINDOWS\System32\svchost.exe
C:\WINDOWS\System32\MsPMSPSv.exe
C:\WINDOWS\system32\cidaemon.exe
C:\Program Files\Mozilla Firefox\firefox.exe
C:\Program Files\Internet Explorer2\iexplore.exe
C:\WINDOWS\regedit.exe
C:\HJT\scanner.exe
F3 - REG:win.ini: load=c:\hjt\hijackthis.exe
O2 - BHO: Adobe PDF Reader Link Helper - {06849E9F-C8D7-4D59-B87D-784B7D6BE0B3} - C:\Program Files\Adobe\Acrobat 7.0\ActiveX\AcroIEHelper.dll
O2 - BHO: NTIECatcher Class - {C56CB6B0-0D96-11D6-8C65-B2868B609932} - C:\Program Files\Xi\NetTransport 2\NTIEHelper.dll
O4 - HKLM\..\Run: [nwiz] nwiz.exe /install
O4 - HKLM\..\Run: [NvMediaCenter] RUNDLL32.EXE C:\WINDOWS\system32\NvMcTray.dll,NvTaskbarInit
O4 - HKLM\..\Run: [NvCplDaemon] RUNDLL32.EXE C:\WINDOWS\system32\NvCpl.dll,NvStartup
O4 - HKCU\..\Run: [ctfmon.exe] C:\WINDOWS\system32\ctfmon.exe
O8 - Extra context menu item: Download all by Net Transport - C:\Program Files\Xi\NetTransport 2\NTAddList.html
O8 - Extra context menu item: E&xport to Microsoft Excel - res://C:\PROGRA~1\MICROS~2\Office10\EXCEL.EXE/3000
O9 - Extra button: (no name) - {08B0E5C0-4FCB-11CF-AAA5-00401C608501} - C:\WINDOWS\System32\msjava.dll
O9 - Extra 'Tools' menuitem: Sun Java Console - {08B0E5C0-4FCB-11CF-AAA5-00401C608501} - C:\WINDOWS\System32\msjava.dll
O9 - Extra button: Create Mobile Favorite - {2EAF5BB1-070F-11D3-9307-00C04FAE2D4F} - C:\Program Files\Microsoft ActiveSync\INETREPL.DLL
O9 - Extra button: (no name) - {2EAF5BB2-070F-11D3-9307-00C04FAE2D4F} - C:\Program Files\Microsoft ActiveSync\INETREPL.DLL
O9 - Extra 'Tools' menuitem: Create Mobile Favorite... - {2EAF5BB2-070F-11D3-9307-00C04FAE2D4F} - C:\Program Files\Microsoft ActiveSync\INETREPL.DLL
O9 - Extra button: Research - {92780B25-18CC-41C8-B9BE-3C9C571A8263} - C:\PROGRA~1\MICROS~2\OFFICE11\REFIEBAR.DLL
O9 - Extra button: AIM - {AC9E2541-2814-11d5-BC6D-00B0D0A1DE45} - C:\Program Files\AIM95\aim.exe
O9 - Extra button: Yahoo! Messenger - {E5D12C4E-7B4F-11D3-B5C9-0050045C3C96} - C:\Program Files\Yahoo!\Messenger\YahooMessenger.exe
O9 - Extra 'Tools' menuitem: Yahoo! Messenger - {E5D12C4E-7B4F-11D3-B5C9-0050045C3C96} - C:\Program Files\Yahoo!\Messenger\YahooMessenger.exe
O9 - Extra button: Messenger - {FB5F1910-F110-11d2-BB9E-00C04F795683} - C:\Program Files\Messenger\msmsgs.exe
O9 - Extra 'Tools' menuitem: Windows Messenger - {FB5F1910-F110-11d2-BB9E-00C04F795683} - C:\Program Files\Messenger\msmsgs.exe
O16 - DPF: {26FCCDF9-A7E1-452A-A73D-7BF7B4D0BA6C} (AOL Pictures Uploader Class) - http://pictures.aol.com/ap/Resources...s.10.4.0.3.cab
O16 - DPF: {7B297BFD-85E4-4092-B2AF-16A91B2EA103} (WScanCtl Class) - http://www3.ca.com/securityadvisor/v...fo/webscan.cab
O16 - DPF: {9D190AE6-C81E-4039-8061-978EBAD10073} (F-Secure Online Scanner 3.0) - http://support.f-secure.com/ols/fscax.cab
O16 - DPF: {E06E2E99-0AA1-11D4-ABA6-0060082AA75C} (GpcContainer Class) - https://magicsoftware.webex.com/clie...ex/ieatgpc.cab
O20 - Winlogon Notify: PCANotify - C:\WINDOWS\SYSTEM32\PCANotify.dll
O23 - Service: AVG Anti-Spyware Guard - Anti-Malware Development a.s. - C:\Program Files\Grisoft\AVG Anti-Spyware 7.5\guard.exe
O23 - Service: pcAnywhere Host Service (awhost32) - Symantec Corporation - C:\Program Files\Symantec\pcAnywhere\awhost32.exe
O23 - Service: Creative Service for CDROM Access - Creative Technology Ltd - C:\WINDOWS\System32\CTsvcCDA.exe
O23 - Service: Cisco Systems, Inc. VPN Service (CVPND) - Cisco Systems, Inc. - C:\Program Files\Cisco Systems\VPN Client\cvpnd.exe
O23 - Service: Diskeeper - Executive Software International, Inc. - C:\Program Files\Executive Software\DiskeeperLite\DKService.exe
O23 - Service: InstallDriver Table Manager (IDriverT) - Macrovision Corporation - C:\Program Files\Common Files\InstallShield\Driver\11\Intel 32\IDriverT.exe
O23 - Service: iPodService - Apple Computer, Inc. - C:\Program Files\iPod\bin\iPodService.exe
O23 - Service: Norton AntiVirus Auto-Protect Service (navapsvc) - Unknown owner - C:\Program Files\Norton AntiVirus\navapsvc.exe (file missing)
O23 - Service: Intel(R) NMS (NMSSvc) - Intel Corporation - C:\WINDOWS\System32\NMSSvc.exe
O23 - Service: NVIDIA Display Driver Service (NVSvc) - NVIDIA Corporation - C:\WINDOWS\system32\nvsvc32.exe
O23 - Service: Pml Driver HPZ12 - HP - C:\WINDOWS\System32\HPZipm12.exe
O23 - Service: SPBBCSvc - Symantec Corporation - C:\Program Files\Common Files\Symantec Shared\SPBBC\SPBBCSvc.exe
*************************
Here's the startup list:
StartupList report, 12/28/2006, 12:16:50 PM
StartupList version: 1.52.2
Started from : C:\HJT\scanner.EXE
Detected: Windows XP SP2 (WinNT 5.01.2600)
Detected: Internet Explorer v6.00 SP2 (6.00.2900.2180)
* Using default options
==================================================
Running processes:
C:\WINDOWS\System32\smss.exe
C:\WINDOWS\system32\winlogon.exe
C:\WINDOWS\system32\services.exe
C:\WINDOWS\system32\lsass.exe
C:\WINDOWS\system32\svchost.exe
C:\WINDOWS\System32\svchost.exe
C:\WINDOWS\system32\spoolsv.exe
C:\WINDOWS\Explorer.EXE
C:\WINDOWS\system32\ctfmon.exe
C:\WINDOWS\system32\cisvc.exe
C:\WINDOWS\System32\CTsvcCDA.exe
C:\Program Files\Common Files\Microsoft Shared\VS7Debug\mdm.exe
C:\WINDOWS\system32\nvsvc32.exe
C:\WINDOWS\System32\svchost.exe
C:\WINDOWS\System32\MsPMSPSv.exe
C:\WINDOWS\system32\cidaemon.exe
C:\Program Files\Mozilla Firefox\firefox.exe
C:\Program Files\Internet Explorer2\iexplore.exe
C:\DOCUME~1\Jim\LOCALS~1\Temp\OnlineScanner\Anti-Virus\fsgk32.exe
C:\DOCUME~1\Jim\LOCALS~1\Temp\OnlineScanner\Anti-Virus\fssm32.exe
C:\WINDOWS\regedit.exe
C:\HJT\scanner.exe
--------------------------------------------------
Checking Windows NT UserInit:
[HKLM\Software\Microsoft\Windows NT\CurrentVersion\Winlogon]
UserInit = C:\WINDOWS\system32\userinit.exe,
--------------------------------------------------
Autorun entries from Registry:
HKLM\Software\Microsoft\Windows\CurrentVersion\Run
nwiz = nwiz.exe /install
NvMediaCenter = RUNDLL32.EXE C:\WINDOWS\system32\NvMcTray.dll,NvTaskbarInit
NvCplDaemon = RUNDLL32.EXE C:\WINDOWS\system32\NvCpl.dll,NvStartup
--------------------------------------------------
Autorun entries from Registry:
HKCU\Software\Microsoft\Windows\CurrentVersion\Run
ctfmon.exe = C:\WINDOWS\system32\ctfmon.exe
--------------------------------------------------
Load/Run keys from C:\WINDOWS\WIN.INI:
Load/Run keys from Registry:
HKLM\..\Windows NT\CurrentVersion\Windows: AppInit_DLLs=
--------------------------------------------------
Shell & screensaver key from C:\WINDOWS\SYSTEM.INI:
Shell & screensaver key from Registry:
Shell=explorer.exe
SCRNSAVE.EXE=C:\WINDOWS\System32\logon.scr
Policies Shell key:
--------------------------------------------------
Enumerating Browser Helper Objects:
(no name) - C:\Program Files\Adobe\Acrobat 7.0\ActiveX\AcroIEHelper.dll - {06849E9F-C8D7-4D59-B87D-784B7D6BE0B3}
(no name) - C:\Program Files\Xi\NetTransport 2\NTIEHelper.dll - {C56CB6B0-0D96-11D6-8C65-B2868B609932}
--------------------------------------------------
Enumerating Task Scheduler jobs:
1-Click Maintenance.job
E2F079C3ABBBF193.job
FRU Task #Hewlett-Packard#hp psc 2170 series#1083718745.job
ISP signup reminder 1.job
Symantec NetDetect.job
{3CF9F8D9-420F-4C21-96C8-B13BBA9E7A11}_DELL_4550_Jim.job
{E1E55917-E2F4-49B5-A0DD-5D2B74416E71}_DELL_4550_Jim.job
{F5114CCB-18A9-42C9-A470-B84F791188FE}_DELL_4550_Jim.job
--------------------------------------------------
[AOL Pictures Uploader Class]
InProcServer32 = C:\Program Files\AOL Pictures\10_4_0_3a\aolpUploader.dll
CODEBASE = http://pictures.aol.com/ap/Resources...s.10.4.0.3.cab
[WScanCtl Class]
[F-Secure Online Scanner 3.0]
CODEBASE = http://support.f-secure.com/ols/fscax.cab
[Shockwave Flash Object]
InProcServer32 = C:\WINDOWS\system32\Macromed\Flash\Flash9.ocx
[GpcContainer Class]
CODEBASE = https://magicsoftware.webex.com/clie...ex/ieatgpc.cab
--------------------------------------------------
Enumerating Windows NT logon/logoff scripts:
*No scripts set to run*
Windows NT checkdisk command:
BootExecute = autocheck autochk *
Windows NT 'Wininit.ini':
PendingFileRenameOperations: C:\WINDOWS\system32\VundoFix.exe||C:\DOCUME~1\Jim\LOCALS~1\Temp\GLB1A2B.EXE||C:\DOCUME~1\Jim\LOCALS~1\Temp\aol5A.tmp
--------------------------------------------------
PostBootReminder: C:\WINDOWS\system32\SHELL32.dll
CDBurn: C:\WINDOWS\system32\SHELL32.dll
WebCheck: C:\WINDOWS\System32\webcheck.dll
SysTray: C:\WINDOWS\System32\stobject.dll
--------------------------------------------------
End of report, 6,615 bytes
Report generated in 0.031 seconds
Command line options:
/verbose - to add additional info on each section
/complete - to include empty sections and unsuspicious data
/full - to include several rarely-important sections
/force9x - to include Win9x-only startups even if running on WinNT
/forcent - to include WinNT-only startups even if running on Win9x
/forceall - to include all Win9x and WinNT startups, regardless of platform
/history - to list version history only
*****************************************
Thanks so much for the assistance!
2. Hi uimagine, Welcome.
Remove Hijackthis from that load value
Your not connecting to the internet while in safe mode with networking are you ?
Do you have any items on Hijackthis ignorlist ? if so remove them, we need to see all of it.
Did windhcp.ocx stay deleted ?
do you have this service ?
http://vil.nai.com/vil/content/v_141038.htm
WinDHCPsvc
Since you have removed Norton antivirus i suggest you install a differant av and cleanup after Norton
Install update and do a full scan with (only one) of the free av's mentioned here
after that
Symantec Removal: http://basconotw.mvps.org/SymRem.htm
Post a new hijackthis log taken not while in safe mode and with no items in its ignore list (if there were any)
3. Hi Lonny,
I was actually able to find the problem and fixed it. But in response to your comments:
I actually removed windhcp.ocx successfully previously. I DID have a winDHCP service which I removed.
I used all the virus scanners and was not able to find my problem.
Trend Micro, F-Secure, I even used AVG after your reply, and still no luck.
Eventually, I looked up all the locations that windows would possibly start applications, and found a registry entry at the following location:
HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\Policies\Explorer\Run
It had the value:
"twin"="C:\windows\system32\twunk32.exe"
I deleted the registry value, and also deleted the twunk32.exe file.
Rebooted, and problem solved.
Just wanted to share with everybody on this board because this virus was NOT located with any of the anti-virus software, nor any of the adware/malware programs. My hijackthis log that was posted did not have anything in the ignorelist and it STILL did not find the culprit. The problem is that it puts entries in the startup list and then just uses svchost.exe programs and iexplorer.exe so it is almost impossible to detect.
I was able to clean things up without reformatting, and I hope this helps somebody solve their issues in the future.
4. Do you still have the file ? perhaps in the recycle bin
One of our experts is curious what these Task Scheduler jobs are
{3CF9F8D9-420F-4C21-96C8-B13BBA9E7A11}_DELL_4550_Jim.job
{E1E55917-E2F4-49B5-A0DD-5D2B74416E71}_DELL_4550_Jim.job
{F5114CCB-18A9-42C9-A470-B84F791188FE}_DELL_4550_Jim.job
are you on xp pro or home ?
5. sorry but I no longer have the file. I deleted it from my recycle bin.
those 3 jobs you mentioned are probably PCAnywhere related. After I removed Norton Anti-Virus, the only other Symantec product I have is PCAnywhere.
I'm actually on XP Home.
6. Hi
Next time try sending it to your av vendor first.
Are they visible in Scheduled Tasks ? if so see what they point to
you could delete the norton job if it's is still there.
What av did you decide on ?
7. Lonny, Good idea on sending it to the AV vendor. I was just so frustrated, that I was just happy to rid of it. next time i'll copy it off somewhere.
The 3 jobs you mentioned do not appear anywhere in scheduled tasks. I actually uninstalled PCAnywhere and it is no longer there.
I ended up with AVGFree just because it is less intrusive. I may end up with Norton 2006 again, but have been quite disappointed that no virus scanner was able to pick up anything. Trend Micro was the only one that reported the winDHCP AFTER i had a clean norton run.
I'm sure that they all have good and bad points, so I'll stick to AVG for now.
thanks for your attention!
8. So no current problems ?
Think Prevention: Put in place a good hosts file
http://www.mvps.org/winhelp2002/hosts.htm
How To Download and Extract the HOSTS file:
http://www.mvps.org/winhelp2002/hosts2.htm
Repeat that proccess about once or twice a month
To help avoid reinfection see "So how did I get infected in the first place?"
9. I neglected to ask you if you have renamed the Internet Explorer folder back ?
If you havent yet do rename it back to normal
Go start run copy/paste in
"C:\WINDOWS\Offline Web Pages"
press enter, tell us what you see there ?
Any offline web page present you did not set uo ?
Another start run command
start run type in | {"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": 0.9188588261604309, "perplexity": 21077.947213849322}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964362619.23/warc/CC-MAIN-20211203091120-20211203121120-00049.warc.gz"} |
http://mathoverflow.net/questions/97015/hilton-eckmann-dual-of-the-steenrod-algebra | # Hilton-Eckmann dual of the Steenrod Algebra
In essence my question can be stated as follows: fill in the analogy
$$\text{cup product} \qquad\qquad \leftrightarrow \qquad \text{Samelson product}$$
$$\updownarrow \qquad\qquad \qquad\qquad \qquad\qquad\updownarrow$$
$$\quad \quad \text{cup}_i \text{-product} \quad \qquad \leftrightarrow \qquad \qquad \quad\qquad\text{?}\qquad \qquad\quad\quad$$ It is known that Samelson products (for a loop space) are Hilton-Eckmann dual to cup products (see e.g., Arkowitz, Martin: Commutators and cup products. Illinois J. Math. 8 1964 571–581. )
Taking this a bit further, the construction of the Steenrod algebra uses the fact that the reduced diagonal $X \to X\wedge X$ is $\Bbb Z/2$-equivariant.
It is not hard to show that the Samelson product also has an equivariance. This suggests to me that the graded Lie algebra structure on the homotopy groups of a space can be refined to take this into account.
Any ideas?
-
Your upper $\leftrightarrow$ is Koszul-Quillen duality between commutative and Lie algebras, therefore, since the $\smile_i$-products form an $E_\infty$-algebra, the lower one should be the Koszul-Quillen duality between $E_\infty$-algebras and $L_\infty$-algebras. Have a look at any reference book on rational homotopy theory, where this theory works simplest. – Fernando Muro May 15 '12 at 15:38
@Fernando: In my understanding in the rational case, we can take a simplicial group model for the loop space, say $G$, and form its group ring over $\Bbb Q$ to get a simlpicial Hopf algebra. The degree-wise primitives then give a differential graded Lie algebra, which is a strict version of an $L_\infty$-algebra. However, if we work integrally (or maybe mod 2), are you saying that we get a non-strict $L_\infty$-algebra? Is this written down anywhere? – John Klein May 15 '12 at 17:18
Not really. In fact, the characteristic $>0$ case is not yet well understood or developed, but since you just asked about analogies I offered you the characteristic $0$ analogue ;-) – Fernando Muro May 16 '12 at 7:35
@Fernando, can we take your suggestion a little further and ask whether there are interesting analogues of Steenrod $p$'th powers coming from the $\mathbb{Z}/p$-equivariant homology of the p-th term $L_{\infty}(p)$ of the $L_{\infty}$-operad? – Craig Westerland May 16 '12 at 12:48
Looking for something else I've come up with this paper: Smirnov, V. A. E∞-structures on homotopy groups. (Russian) Mat. Zametki 61 (1997), no. 1, 152--156; translation in Math. Notes 61 (1997), no. 1-2, 127–130. It seems to contain an appropriate notion of $L_\infty$-algebras over $\mathbb{F}_p$. Unforturately the paper is very short and gives no proofs, and I haven't found references where the claims in this paper are proven. – Fernando Muro May 25 '12 at 23:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9190407395362854, "perplexity": 505.0155080026783}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936464303.77/warc/CC-MAIN-20150226074104-00069-ip-10-28-5-156.ec2.internal.warc.gz"} |
https://worldwidescience.org/topicpages/f/field+theory+applications.html | #### Sample records for field theory applications
1. Neural fields theory and applications
CERN Document Server
Graben, Peter; Potthast, Roland; Wright, James
2014-01-01
With this book, the editors present the first comprehensive collection in neural field studies, authored by leading scientists in the field - among them are two of the founding-fathers of neural field theory. Up to now, research results in the field have been disseminated across a number of distinct journals from mathematics, computational neuroscience, biophysics, cognitive science and others. Starting with a tutorial for novices in neural field studies, the book comprises chapters on emergent patterns, their phase transitions and evolution, on stochastic approaches, cortical development, cognition, robotics and computation, large-scale numerical simulations, the coupling of neural fields to the electroencephalogram and phase transitions in anesthesia. The intended readership are students and scientists in applied mathematics, theoretical physics, theoretical biology, and computational neuroscience. Neural field theory and its applications have a long-standing tradition in the mathematical and computational ...
2. Closed superstring field theory and its applications
Science.gov (United States)
de Lacroix, Corinne; Erbin, Harold; Kashyap, Sitender Pratap; Sen, Ashoke; Verma, Mritunjay
2017-10-01
We review recent developments in the construction of heterotic and type II string field theories and their various applications. These include systematic procedures for determining the shifts in the vacuum expectation values of fields under quantum corrections, computing renormalized masses and S-matrix of the theory around the shifted vacuum and a proof of unitarity of the S-matrix. The S-matrix computed this way is free from all divergences when there are more than 4 noncompact space-time dimensions, but suffers from the usual infrared divergences when the number of noncompact space-time dimensions is 4 or less.
3. Holographic applications of logarithmic conformal field theories
NARCIS (Netherlands)
Grumiller, D.; Riedler, W.; Rosseel, J.; Zojer, T.
2013-01-01
We review the relations between Jordan cells in various branches of physics, ranging from quantum mechanics to massive gravity theories. Our main focus is on holographic correspondences between critically tuned gravity theories in anti-de Sitter space and logarithmic conformal field theories in
4. Introduction to conformal field theory. With applications to string theory
International Nuclear Information System (INIS)
Blumenhagen, Ralph; Plauschinn, Erik
2009-01-01
Based on class-tested notes, this text offers an introduction to Conformal Field Theory with a special emphasis on computational techniques of relevance for String Theory. It introduces Conformal Field Theory at a basic level, Kac-Moody algebras, one-loop partition functions, Superconformal Field Theories, Gepner Models and Boundary Conformal Field Theory. Eventually, the concept of orientifold constructions is explained in detail for the example of the bosonic string. In providing many detailed CFT calculations, this book is ideal for students and scientists intending to become acquainted with CFT techniques relevant for string theory but also for students and non-specialists from related fields. (orig.)
5. Higgs effective field theories. Systematics and applications
Energy Technology Data Exchange (ETDEWEB)
Krause, Claudius G.
2016-07-28
Researchers of the Large Hadron Collider (LHC) at the European Organization for Nuclear Research (CERN) announced on July 4th, 2012, the observation of a new particle. The properties of the particle agree, within the relatively large experimental uncertainties, with the properties of the long-sought Higgs boson. Particle physicists around the globe are now wondering, ''Is it the Standard Model Higgs that we observe; or is it another particle with similar properties?'' We employ effective field theories (EFTs) for a general, model-independent description of the particle. We use a few, minimal assumptions - Standard Model (SM) particle content and a separation of scales to the new physics - which are supported by current experimental results. By construction, effective field theories describe a physical system only at a certain energy scale, in our case at the electroweak-scale v. Effects of new physics from a higher energy-scale, Λ, are described by modified interactions of the light particles. In this thesis, ''Higgs Effective Field Theories - Systematics and Applications'', we discuss effective field theories for the Higgs particle, which is not necessarily the Higgs of the Standard Model. In particular, we focus on a systematic and consistent expansion of the EFT. The systematics depends on the dynamics of the new physics. We distinguish two different consistent expansions. EFTs that describe decoupling new-physics effects and EFTs that describe non-decoupling new-physics effects. We briefly discuss the first case, the SM-EFT. The focus of this thesis, however, is on the non-decoupling EFTs. We argue that the loop expansion is the consistent expansion in the second case. We introduce the concept of chiral dimensions, equivalent to the loop expansion. Using the chiral dimensions, we expand the electroweak chiral Lagrangian up to next-to-leading order, O(f{sup 2}/Λ{sup 2})=O(1/16π{sup 2}). Further, we discuss how different
6. Noether's theorems applications in mechanics and field theory
CERN Document Server
2016-01-01
The book provides a detailed exposition of the calculus of variations on fibre bundles and graded manifolds. It presents applications in such area's as non-relativistic mechanics, gauge theory, gravitation theory and topological field theory with emphasis on energy and energy-momentum conservation laws. Within this general context the first and second Noether theorems are treated in the very general setting of reducible degenerate graded Lagrangian theory.
7. Cosmological applications of algebraic quantum field theory in curved spacetimes
CERN Document Server
Hack, Thomas-Paul
2016-01-01
This book provides a largely self-contained and broadly accessible exposition on two cosmological applications of algebraic quantum field theory (QFT) in curved spacetime: a fundamental analysis of the cosmological evolution according to the Standard Model of Cosmology; and a fundamental study of the perturbations in inflation. The two central sections of the book dealing with these applications are preceded by sections providing a pedagogical introduction to the subject. Introductory material on the construction of linear QFTs on general curved spacetimes with and without gauge symmetry in the algebraic approach, physically meaningful quantum states on general curved spacetimes, and the backreaction of quantum fields in curved spacetimes via the semiclassical Einstein equation is also given. The reader should have a basic understanding of General Relativity and QFT on Minkowski spacetime, but no background in QFT on curved spacetimes or the algebraic approach to QFT is required.
8. Probabilistic theory of mean field games with applications
CERN Document Server
Carmona, René
2018-01-01
This two-volume book offers a comprehensive treatment of the probabilistic approach to mean field game models and their applications. The book is self-contained in nature and includes original material and applications with explicit examples throughout, including numerical solutions. Volume I of the book is entirely devoted to the theory of mean field games without a common noise. The first half of the volume provides a self-contained introduction to mean field games, starting from concrete illustrations of games with a finite number of players, and ending with ready-for-use solvability results. Readers are provided with the tools necessary for the solution of forward-backward stochastic differential equations of the McKean-Vlasov type at the core of the probabilistic approach. The second half of this volume focuses on the main principles of analysis on the Wasserstein space. It includes Lions' approach to the Wasserstein differential calculus, and the applications of its results to the analysis of stochastic...
9. Application of KBc Subalgebra in String Field Theory
Science.gov (United States)
Zeze, S.
Recently, a classical solution of open cubic string field theory (CSFT) which corresponds to the closed string vacuum is found by Erler and Schnabl. In their work, a very simple subalgebra of open string star algebra --- called K B c subalgebra --- plays a crucial role. In this talk, we demonstrate two applications of the K B c subalgebra. One is evaluation of classical and effective tachyon potential. It turns out that the level expansion in the K B c subalgebra terminates at a certain level, so that analytic evaluation of effective potential is available. The other application is regularization of the identity based solutions. It is demonstrated that the Okawa-Erler-Schnabl type solution naturally includes gauge invariant regularization of identity based solutions.
10. Conformal field theory and its application to strings
International Nuclear Information System (INIS)
Verlinde, E.P.
1988-01-01
Conformal field theories on Riemann surfaces are considered and the result is applied to study the loop amplitudes for bosonic strings. It is shown that there is a close resemblance between the loop amplitudes for φ 3 -theory and the expressions for string multi-loop amplitudes. The similarity between φ 3 -amplitudes in curved backgrounds and the analytic structure of string amplitudes in backgrounds described by conformal field theories is also pointed out. 60 refs.; 5 figs.; 200 schemes
11. Post Modernity Theory and Its Educational Applications in School Fields
Science.gov (United States)
El-Baz, Maaly Bent Mohamed Saleh
2017-01-01
This paper aims to identify the fundamental principles on which the post modernity theory is based and to notice this in the field of Education, since this theory deals with two basic rules on which the postmodernist orientation is based, one of them denies on the absolute truth on Ontology level (related to the existence nature), and the other…
12. Field theory
CERN Multimedia
1999-11-08
In these lectures I will build up the concept of field theory using the language of Feynman diagrams. As a starting point, field theory in zero spacetime dimensions is used as a vehicle to develop all the necessary techniques: path integral, Feynman diagrams, Schwinger-Dyson equations, asymptotic series, effective action, renormalization etc. The theory is then extended to more dimensions, with emphasis on the combinatorial aspects of the diagrams rather than their particular mathematical structure. The concept of unitarity is used to, finally, arrive at the various Feynman rules in an actual, four-dimensional theory. The concept of gauge-invariance is developed, and the structure of a non-abelian gauge theory is discussed, again on the level of Feynman diagrams and Feynman rules.
13. Bound state quantum field theory application to atoms and ions
CERN Document Server
Sapirstein, Jonathan
2019-01-01
Two aspects of the book should appeal to a wide audience. One aspect would be the comprehensive coverage on the latest updates and developments this book provides, besides Bethe and Salpeter's handbook on hydrogen and helium, which is still widely regarded as useful. The other aspect would be that a major part of the book uses effective field theory, a way of including quantum electrodynamics (QED) that starts with the familiar Schrödinger equation, and then adds perturbing operators derived in a rather simple manner that incorporates QED. Effective field theory is used in a number of fields including particle physics and nuclear physics, and readership is targeted at these communities too.Additionally, students using this book in conjunction with Peskin's textbook could learn to carry out fairly sophisticated calculations in QED in order to learn the technique, as this book comes with practical calculations.Also included is a very clear exposition of the BetheSalpeter equation, which is simply either ...
14. The application of mean field theory to image motion estimation.
Science.gov (United States)
Zhang, J; Hanauer, G G
1995-01-01
Previously, Markov random field (MRF) model-based techniques have been proposed for image motion estimation. Since motion estimation is usually an ill-posed problem, various constraints are needed to obtain a unique and stable solution. The main advantage of the MRF approach is its capacity to incorporate such constraints, for instance, motion continuity within an object and motion discontinuity at the boundaries between objects. In the MRF approach, motion estimation is often formulated as an optimization problem, and two frequently used optimization methods are simulated annealing (SA) and iterative-conditional mode (ICM). Although the SA is theoretically optimal in the sense of finding the global optimum, it usually takes many iterations to converge. The ICM, on the other hand, converges quickly, but its results are often unsatisfactory due to its "hard decision" nature. Previously, the authors have applied the mean field theory to image segmentation and image restoration problems. It provides results nearly as good as SA but with much faster convergence. The present paper shows how the mean field theory can be applied to MRF model-based motion estimation. This approach is demonstrated on both synthetic and real-world images, where it produced good motion estimates.
15. Applicability of self-consistent mean-field theory
International Nuclear Information System (INIS)
Guo Lu; Sakata, Fumihiko; Zhao Enguang
2005-01-01
Within the constrained Hartree-Fock (CHF) theory, an analytic condition is derived to estimate whether a concept of the self-consistent mean field is realized in the level repulsive region. The derived condition states that an iterative calculation of the CHF equation does not converge when the quantum fluctuations coming from two-body residual interaction and quadrupole deformation become larger than a single-particle energy difference between two avoided crossing orbits. By means of numerical calculation, it is shown that the analytic condition works well for a realistic case
16. Test-particle motion in Einstein's unified field theory. I. General theory and application to neutral test particles
International Nuclear Information System (INIS)
Johnson, C.R.
1985-01-01
We develop a method for finding the exact equations of structure and motion of multipole test particles in Einstein's unified field theory: the theory of the nonsymmetric field. The method is also applicable to Einstein's gravitational theory. Particles are represented by singularities in the field. The method is covariant at each step of the analysis. We also apply the method and find both in Einstein's unified field theory and in Einstein's gravitational theory the equations of structure and motion of neutral pole-dipole test particles possessing no electromagnetic multipole moments. In the case of Einstein's gravitational theory the results are the well-known equations of structure and motion of a neutral pole-dipole test particle in a given background gravitational field. In the case of Einstein's unified field theory the results are the same, providing we identify a certain symmetric second-rank tensor field appearing in Einstein's theory with the metric and gravitational field. We therefore discover not only the equations of structure and motion of a neutral test particle in Einstein's unified field theory, but we also discover what field in Einstein's theory plays the role of metric and gravitational field
17. Algebric generalization of symmetry Dirac bracket. Application to field theory
International Nuclear Information System (INIS)
Rocha Filho, T.M. da.
1987-01-01
The A set of observable of a physical system with finite e infinite number of degrees of freedom and submitted to certain constraint conditions, is considered. Using jordan algebra structure on A in relation to bymmetric Poisson bracket obtained by Droz-Vincent, a jordan product is obtained on the A/I quocient set with regard to I ideal generated by constraints of second class. It is shown that this product on A/I corresponds to symmetric Dirac bracket. The developed formulation is applied to a system corresponding to harmonic oscillators, non relativistic field, Rarita-Schwinger field and the possibility of its utilization in fermionic string theories is discussed. (M.C.K.)
18. Field theories with subcanonical fields
International Nuclear Information System (INIS)
Bigi, I.I.Y.
1976-01-01
The properties of quantum field theories with spinor fields of dimension less than the canonical value of 3/2 are studied. As a starting point for the application of common perturbation theory we look for the linear version of these theories. A gange-interaction is introduced and with the aid of power counting the renormalizability of the theory is shown. It follows that in the case of a spinor-field with negative dimension renormalization can only be attained if the interaction has a further symmetry. By this symmetry the theory is determined in an unequivocal way. The gange-interaction introduced in the theory leads to a spontaneous breakdown of scale invariance whereby masses are produced. At the same time the spinor-field operators can now be separated in two orthogonal sections with opposite norm. It is proposed to use the section with negative (positive) norm to describe hadrons (leptons) respectively. (orig./WL) [de
19. Extremes in random fields a theory and its applications
CERN Document Server
Yakir, Benjamin
2013-01-01
Presents a useful new technique for analyzing the extreme-value behaviour of random fields Modern science typically involves the analysis of increasingly complex data. The extreme values that emerge in the statistical analysis of complex data are often of particular interest. This book focuses on the analytical approximations of the statistical significance of extreme values. Several relatively complex applications of the technique to problems that emerge in practical situations are presented. All the examples are difficult to analyze using classical methods, and as a result, the author pr
20. Probabilistic theory of mean field games with applications I mean field FBSDEs, control, and games
CERN Document Server
Carmona, René
2018-01-01
This two-volume book offers a comprehensive treatment of the probabilistic approach to mean field game models and their applications. The book is self-contained in nature and includes original material and applications with explicit examples throughout, including numerical solutions. Volume I of the book is entirely devoted to the theory of mean field games without a common noise. The first half of the volume provides a self-contained introduction to mean field games, starting from concrete illustrations of games with a finite number of players, and ending with ready-for-use solvability results. Readers are provided with the tools necessary for the solution of forward-backward stochastic differential equations of the McKean-Vlasov type at the core of the probabilistic approach. The second half of this volume focuses on the main principles of analysis on the Wasserstein space. It includes Lions' approach to the Wasserstein differential calculus, and the applications of its results to the analysis of stochastic...
1. Superspace conformal field theory
International Nuclear Information System (INIS)
Quella, Thomas
2013-07-01
Conformal sigma models and WZW models on coset superspaces provide important examples of logarithmic conformal field theories. They possess many applications to problems in string and condensed matter theory. We review recent results and developments, including the general construction of WZW models on type I supergroups, the classification of conformal sigma models and their embedding into string theory.
2. Superspace conformal field theory
Energy Technology Data Exchange (ETDEWEB)
Quella, Thomas [Koeln Univ. (Germany). Inst. fuer Theoretische Physik; Schomerus, Volker [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany)
2013-07-15
Conformal sigma models and WZW models on coset superspaces provide important examples of logarithmic conformal field theories. They possess many applications to problems in string and condensed matter theory. We review recent results and developments, including the general construction of WZW models on type I supergroups, the classification of conformal sigma models and their embedding into string theory.
3. Conformal field theory
CERN Document Server
Ketov, Sergei V
1995-01-01
Conformal field theory is an elegant and powerful theory in the field of high energy physics and statistics. In fact, it can be said to be one of the greatest achievements in the development of this field. Presented in two dimensions, this book is designed for students who already have a basic knowledge of quantum mechanics, field theory and general relativity. The main idea used throughout the book is that conformal symmetry causes both classical and quantum integrability. Instead of concentrating on the numerous applications of the theory, the author puts forward a discussion of the general
4. Microwave field-efffect transistors theory, design, and application
CERN Document Server
Pengelly, Raymond
1994-01-01
This book covers the use of devices in microwave circuits and includes such topics as semiconductor theory and transistor performance, CAD considerations, intermodulation, noise figure, signal handling, S-parameter mapping, narrow- and broadband techniques, packaging and thermal considerations.
5. Engineering field theory
CERN Document Server
2014-01-01
Engineering Field Theory focuses on the applications of field theory in gravitation, electrostatics, magnetism, electric current flow, conductive heat transfer, fluid flow, and seepage.The manuscript first ponders on electric flux, electrical materials, and flux function. Discussions focus on field intensity at the surface of a conductor, force on a charged surface, atomic properties, doublet and uniform field, flux tube and flux line, line charge and line sink, field of a surface charge, field intensity, flux density, permittivity, and Coulomb's law. The text then takes a look at gravitation
6. Shielding Flowers Developing under Stress: Translating Theory to Field Application
Directory of Open Access Journals (Sweden)
Noam Chayut
2014-07-01
Full Text Available Developing reproductive organs within a flower are sensitive to environmental stress. A higher incidence of environmental stress during this stage of a crop plants’ developmental cycle will lead to major breaches in food security. Clearly, we need to understand this sensitivity and try and overcome it, by agricultural practices and/or the breeding of more tolerant cultivars. Although passion fruit vines initiate flowers all year round, flower primordia abort during warm summers. This restricts the season of fruit production in regions with warm summers. Previously, using controlled chambers, stages in flower development that are sensitive to heat were identified. Based on genetic analysis and physiological experiments in controlled environments, gibberellin activity appeared to be a possible point of horticultural intervention. Here, we aimed to shield flowers of a commercial cultivar from end of summer conditions, thus allowing fruit production in new seasons. We conducted experiments over three years in different settings, and our findings consistently show that a single application of an inhibitor of gibberellin biosynthesis to vines in mid-August can cause precocious flowering of ~2–4 weeks, leading to earlier fruit production of ~1 month. In this case, knowledge obtained on phenology, environmental constraints and genetic variation, allowed us to reach a practical solution.
7. Effective field theories
CERN Document Server
Petrov, Alexey A
2016-01-01
This book is a broad-based text intended to help the growing student body interested in topics such as gravitational effective theories, supersymmetric effective theories, applications of effective theory techniques to problems in condensed matter physics (superconductivity) and quantum chromodynamics (such as soft-collinear effective theory). It begins with a review of the use of symmetries to identify the relevant degrees of freedom in a problem, and then presents a variety of methods that can be used to solve physical problems. A detailed discussion of canonical examples of effective field theories with increasing complexity is then conducted. Special cases such as supersymmetry and lattice EFT are discussed, as well as recently-found applications to problems in gravitation and cosmology. An appendix includes various factoids from group theory and other topics that are used throughout the text, in an attempt to make the book self-contained.
8. String theory or field theory?
International Nuclear Information System (INIS)
Marshakov, A.V.
2002-01-01
The status of string theory is reviewed, and major recent developments - especially those in going beyond perturbation theory in the string theory and quantum field theory frameworks - are analyzed. This analysis helps better understand the role and place of experimental phenomena, it is emphasized that there are some insurmountable problems inherent in it - notably the impossibility to formulate the quantum theory of gravity on its basis - which prevent it from being a fundamental physical theory of the world of microscopic distances. It is this task, the creation of such a theory, which string theory, currently far from completion, is expected to solve. In spite of its somewhat vague current form, string theory has already led to a number of serious results and greatly contributed to progress in the understanding of quantum field theory. It is these developments, which are our concern in this review [ru
9. String theory or field theory?
International Nuclear Information System (INIS)
Marshakov, Andrei V
2002-01-01
The status of string theory is reviewed, and major recent developments - especially those in going beyond perturbation theory in the string theory and quantum field theory frameworks - are analyzed. This analysis helps better understand the role and place of string theory in the modern picture of the physical world. Even though quantum field theory describes a wide range of experimental phenomena, it is emphasized that there are some insurmountable problems inherent in it - notably the impossibility to formulate the quantum theory of gravity on its basis - which prevent it from being a fundamental physical theory of the world of microscopic distances. It is this task, the creation of such a theory, which string theory, currently far from completion, is expected to solve. In spite of its somewhat vague current form, string theory has already led to a number of serious results and greatly contributed to progress in the understanding of quantum field theory. It is these developments which are our concern in this review. (reviews of topical problems)
10. Quantum field theory of material properties. Its application to models of Rashba spin splitting
International Nuclear Information System (INIS)
Schober, Giulio Albert Heinrich
2016-01-01
In this thesis, we argue that microscopic field theories - which as such are already scientifically established - have emerged as a new paradigm in materials physics. We hence seek to elaborate on such field theories which underlie modern ab initio calculations, and we apply them to the bismuth tellurohalides (BiTeX with X=I,Br,Cl) as a prototypical class of spin-based materials. For this purpose, we begin by constructing tight-binding models which approximately describe the spin-split conduction bands of BiTeI. Following this, we derive the theory of temperature Green functions systematically from their fundamental equations of motion. This in turn enables us to develop a combined functional renormalization and mean-field approach which is suitable for application to multiband models. For the Rashba model including an attractive, local interaction, this approach yields an unconventional superconducting phase with a singlet gap function and a mixed singlet-triplet order parameter. We further investigate the unusual electromagnetic response of BiTeI, which is caused by the Rashba spin splitting and which includes, in particular, an orbital paramagnetism. Finally, we conclude by summarizing the Functional Approach to electrodynamics of media as a microscopic field theory of electromagnetic material properties which sits in accordance with ab initio physics.
11. Theory of interacting quantum fields
International Nuclear Information System (INIS)
Rebenko, Alexei L.
2012-01-01
This monograph is devoted to the systematic presentation of foundations of the quantum field theory. Unlike numerous monographs devoted to this topic, a wide range of problems covered in this book are accompanied by their sufficiently clear interpretations and applications. An important significant feature of this monograph is the desire of the author to present mathematical problems of the quantum field theory with regard to new methods of the constructive and Euclidean field theory that appeared in the last thirty years of the 20 th century and are based on the rigorous mathematical apparatus of functional analysis, the theory of operators, and the theory of generalized functions. The monograph is useful for students, post-graduate students, and young scientists who desire to understand not only the formality of construction of the quantum field theory but also its essence and connection with the classical mechanics, relativistic classical field theory, quantum mechanics, group theory, and the theory of path integral formalism.
12. Advances in dynamic and mean field games theory, applications, and numerical methods
CERN Document Server
Viscolani, Bruno
2017-01-01
This contributed volume considers recent advances in dynamic games and their applications, based on presentations given at the 17th Symposium of the International Society of Dynamic Games, held July 12-15, 2016, in Urbino, Italy. Written by experts in their respective disciplines, these papers cover various aspects of dynamic game theory including mean-field games, stochastic and pursuit-evasion games, and computational methods for dynamic games. Topics covered include Pedestrian flow in crowded environments Models for climate change negotiations Nash Equilibria for dynamic games involving Volterra integral equations Differential games in healthcare markets Linear-quadratic Gaussian dynamic games Aircraft control in wind shear conditions Advances in Dynamic and Mean-Field Games presents state-of-the-art research in a wide spectrum of areas. As such, it serves as a testament to the continued vitality and growth of the field of dynamic games and their applications. It will be of interest to an interdisciplinar...
13. Application of self-consistent field theory to self-assembled bilayer membranes
International Nuclear Information System (INIS)
Zhang Ping-Wen; Shi An-Chang
2015-01-01
Bilayer membranes self-assembled from amphiphilic molecules such as lipids, surfactants, and block copolymers are ubiquitous in biological and physiochemical systems. The shape and structure of bilayer membranes depend crucially on their mechanical properties such as surface tension, bending moduli, and line tension. Understanding how the molecular properties of the amphiphiles determine the structure and mechanics of the self-assembled bilayers requires a molecularly detailed theoretical framework. The self-consistent field theory provides such a theoretical framework, which is capable of accurately predicting the mechanical parameters of self-assembled bilayer membranes. In this mini review we summarize the formulation of the self-consistent field theory, as exemplified by a model system composed of flexible amphiphilic chains dissolved in hydrophilic polymeric solvents, and its application to the study of self-assembled bilayer membranes. (topical review)
14. The application of Regge calculus to quantum gravity and quantum field theory in a curved background
International Nuclear Information System (INIS)
Warner, N.P.
1982-01-01
The application of Regge calculus to quantum gravity and quantum field theory in a curved background is discussed. A discrete form of exterior differential calculus is developed, and this is used to obtain Laplacians for p-forms on the Regge manifold. To assess the accuracy of these approximations, the eigenvalues of the discrete Laplacians were calculated for the regular tesselations of S 2 and S 3 . The results indicate that the methods obtained in this paper may be used in curved space-times with an accuracy comparing with that obtained in lattice gauge theories on a flat background. It also becomes evident that Regge calculus provides particularly suitable lattices for Monte-Carlo techniques. (author)
15. Field theory and strings
International Nuclear Information System (INIS)
Bonara, L.; Cotta-Ramusino, P.; Rinaldi, M.
1987-01-01
It is well-known that type I and heterotic superstring theories have a zero mass spectrum which correspond to the field content of N=1 supergravity theory coupled to supersymmetric Yang-Mills theory in 10-D. The authors study the field theory ''per se'', in the hope that simple consistency requirements will determine the theory completely once one knows the field content inherited from string theory. The simplest consistency requirements are: N=1 supersymmetry; and absence of chiral anomalies. This is what the authors discuss in this paper here leaving undetermined the question of the range of validity of the resulting field theory. As is known, a model of N=1 supergravity (SUGRA) coupled to supersymmetric Yang-Mills (SYM) theory was known in the form given by Chapline and Manton. The coupling of SUGRA to SYM was determined by the definition of the ''field strength'' 3-form H in this paper
16. Quantum field theory
International Nuclear Information System (INIS)
Ryder, L.H.
1985-01-01
This introduction to the ideas and techniques of quantum field theory presents the material as simply as possible and is designed for graduate research students. After a brief survey of particle physics, the quantum theory of scalar and spinor fields and then of gauge fields, is developed. The emphasis throughout is on functional methods, which have played a large part in modern field theory. The book concludes with a bridge survey of ''topological'' objects in field theory and assumes a knowledge of quantum mechanics and special relativity
17. String field theory
International Nuclear Information System (INIS)
Kaku, M.
1987-01-01
In this article, the authors summarize the rapid progress in constructing string field theory actions, such as the development of the covariant BRST theory. They also present the newer geometric formulation of string field theory, from which the BRST theory and the older light cone theory can be derived from first principles. This geometric formulation allows us to derive the complete field theory of strings from two geometric principles, in the same way that general relativity and Yang-Mills theory can be derived from two principles based on global and local symmetry. The geometric formalism therefore reduces string field theory to a problem of finding an invariant under a new local gauge group they call the universal string group (USG). Thus, string field theory is the gauge theory of the universal string group in much the same way that Yang-Mills theory is the gauge theory of SU(N). The geometric formulation places superstring theory on the same rigorous group theoretical level as general relativity and gauge theory
18. Algebraic conformal field theory
International Nuclear Information System (INIS)
Fuchs, J.; Nationaal Inst. voor Kernfysica en Hoge-Energiefysica
1991-11-01
Many conformal field theory features are special versions of structures which are present in arbitrary 2-dimensional quantum field theories. So it makes sense to describe 2-dimensional conformal field theories in context of algebraic theory of superselection sectors. While most of the results of the algebraic theory are rather abstract, conformal field theories offer the possibility to work out many formulae explicitly. In particular, one can construct the full algebra A-bar of global observables and the endomorphisms of A-bar which represent the superselection sectors. Some explicit results are presented for the level 1 so(N) WZW theories; the algebra A-bar is found to be the enveloping algebra of a Lie algebra L-bar which is an extension of the chiral symmetry algebra of the WZW theory. (author). 21 refs., 6 figs
19. Field theory approach to gravitation
International Nuclear Information System (INIS)
Yilmaz, H.
1978-01-01
A number of authors considered the possibility of formulating a field-theory approach to gravitation with the claim that such an approach would uniquely lead to Einstein's theory of general relativity. In this article it is shown that the field theory approach is more generally applicable and uniqueness cannot be claimed. Theoretical and experimental reasons are given showing that the Einsteinian limit appears to be unviable
20. Field theory and particle physics
International Nuclear Information System (INIS)
Eboli, O.J.P.; Gomes, M.; Santoro, A.
1990-01-01
This book contains the proceedings of the topics covered during the fifth Jorge Andre Swieca Summer School. The first part of the book collects the material devoted to quantum field theory. There were four courses on methods in Field Theory; H. O. Girotti lectured on constrained dynamics, R. Jackiw on the Schrodinger representation in Field Theory, S.-Y. Pi on the application of this representation to quantum fields in a Robertson-Walker spacetime, and L. Vinet on Berry Connections. There were three courses on Conformal Field Theory: I. Todorov focused on the problem of construction and classification of conformal field theories. Lattice models, two-dimensional S matrices and conformal field theory were looked from the unifying perspective of the Yang-Baxter algebras in the lectures given by M. Karowski. Parasupersymmetric quantum mechanics was discussed in the lectures by L. Vinet. Besides those courses, there was an introduction to string field theory given by G. Horowitz. There were also three seminars: F. Schaposnik reported on recent applications of topological methods in field theory, P. Gerbert gave a seminar on three dimensional gravity and V. Kurak talked on two dimensional parafermionic models. The second part of this proceedings is devoted to phenomenology. There were three courses on Particle Physics: Dan Green lectured on collider physics, E. Predrazzi on strong interactions and G. Cohen-Tanoudji on the use of strings in strong interactions
1. Electron traps in polar liquids. An application of the formalism of the random field theory
International Nuclear Information System (INIS)
Hilczer, M.; Bartczak, W.M.
1992-01-01
The potential energy surface in a disordered medium is described, using the concepts of the mathematical theory of random fields. The statistics of trapping sites (the regions of an excursion of the random field) is obtained for liquid methanol as a numerical example of the theory. (author). 15 refs, 4 figs
2. Finite discrete field theory
International Nuclear Information System (INIS)
Souza, Manoelito M. de
1997-01-01
We discuss the physical meaning and the geometric interpretation of implementation in classical field theories. The origin of infinities and other inconsistencies in field theories is traced to fields defined with support on the light cone; a finite and consistent field theory requires a light-cone generator as the field support. Then, we introduce a classical field theory with support on the light cone generators. It results on a description of discrete (point-like) interactions in terms of localized particle-like fields. We find the propagators of these particle-like fields and discuss their physical meaning, properties and consequences. They are conformally invariant, singularity-free, and describing a manifestly covariant (1 + 1)-dimensional dynamics in a (3 = 1) spacetime. Remarkably this conformal symmetry remains even for the propagation of a massive field in four spacetime dimensions. We apply this formalism to Classical electrodynamics and to the General Relativity Theory. The standard formalism with its distributed fields is retrieved in terms of spacetime average of the discrete field. Singularities are the by-products of the averaging process. This new formalism enlighten the meaning and the problem of field theory, and may allow a softer transition to a quantum theory. (author)
3. Geophysical Field Theory
International Nuclear Information System (INIS)
Eloranta, E.
2003-11-01
The geophysical field theory includes the basic principles of electromagnetism, continuum mechanics, and potential theory upon which the computational modelling of geophysical phenomena is based on. Vector analysis is the main mathematical tool in the field analyses. Electrostatics, stationary electric current, magnetostatics, and electrodynamics form a central part of electromagnetism in geophysical field theory. Potential theory concerns especially gravity, but also electrostatics and magnetostatics. Solid state mechanics and fluid mechanics are central parts in continuum mechanics. Also the theories of elastic waves and rock mechanics belong to geophysical solid state mechanics. The theories of geohydrology and mass transport form one central field theory in geophysical fluid mechanics. Also heat transfer is included in continuum mechanics. (orig.)
4. Nonlocal continuum field theories
CERN Document Server
2002-01-01
Nonlocal continuum field theories are concerned with material bodies whose behavior at any interior point depends on the state of all other points in the body -- rather than only on an effective field resulting from these points -- in addition to its own state and the state of some calculable external field. Nonlocal field theory extends classical field theory by describing the responses of points within the medium by functionals rather than functions (the "constitutive relations" of classical field theory). Such considerations are already well known in solid-state physics, where the nonlocal interactions between the atoms are prevalent in determining the properties of the material. The tools developed for crystalline materials, however, do not lend themselves to analyzing amorphous materials, or materials in which imperfections are a major part of the structure. Nonlocal continuum theories, by contrast, can describe these materials faithfully at scales down to the lattice parameter. This book presents a unif...
5. Group theory and general relativity representations of the Lorentz group and their applications to the gravitational field
CERN Document Server
Carmeli, Moshe
2000-01-01
This is the only book on the subject of group theory and Einstein's theory of gravitation. It contains an extensive discussion on general relativity from the viewpoint of group theory and gauge fields. It also puts together in one volume many scattered, original works, on the use of group theory in general relativity theory.There are twelve chapters in the book. The first six are devoted to rotation and Lorentz groups, and their representations. They include the spinor representation as well as the infinite-dimensional representations. The other six chapters deal with the application of groups
6. An application of random field theory to analysis of electron trapping sites in disordered media
International Nuclear Information System (INIS)
Hilczer, M.; Bartczak, W.M.
1993-01-01
The potential energy surface in a disordered medium is considered a random field and described using the concepts of the mathematical theory of random fields. The preexisting traps for excess electrons are identified with certain regions of excursion (extreme regions) of the potential field. The theory provides an analytical method of statistical analysis of these regions. Parameters of the cavity-averaged potential field, which are provided by computer simulation of a given medium, serve as input data for the analysis. The statistics of preexisting traps are obtained for liquid methanol as a numerical example of the random field method. 26 refs., 6 figs
7. Hyperfunction quantum field theory
International Nuclear Information System (INIS)
Nagamachi, S.; Mugibayashi, N.
1976-01-01
The quantum field theory in terms of Fourier hyperfunctions is constructed. The test function space for hyperfunctions does not contain C infinitely functios with compact support. In spite of this defect the support concept of H-valued Fourier hyperfunctions allows to formulate the locality axiom for hyperfunction quantum field theory. (orig.) [de
8. Quantum field theory
CERN Document Server
Mandl, Franz
2010-01-01
Following on from the successful first (1984) and revised (1993) editions, this extended and revised text is designed as a short and simple introduction to quantum field theory for final year physics students and for postgraduate students beginning research in theoretical and experimental particle physics. The three main objectives of the book are to: Explain the basic physics and formalism of quantum field theory To make the reader proficient in theory calculations using Feynman diagrams To introduce the reader to gauge theories, which play a central role in elementary particle physic
9. Analytical Thermal Field Theory Applicable to Oil Hydraulic Fluid Film Lubrication
DEFF Research Database (Denmark)
Johansen, Per; Roemer, Daniel Beck; Pedersen, Henrik C.
2014-01-01
An analytical thermal field theory is derived by a perturbation series expansion solution to the energy conservation equation. The theory is valid for small values of the Brinkman number and the modified Peclet number. This condition is sufficiently satisfied for hydraulic oils, whereby...... expansion of the thermal field. The series solution is truncated at first order in order to obtain a closed form approximation. Finally a numerical thermohydrodynamic simulation of a piston-cylinder interface is presented, and the results are used for a comparison with the analytical theory in order...
10. Algebraic quantum field theory
International Nuclear Information System (INIS)
Foroutan, A.
1996-12-01
The basic assumption that the complete information relevant for a relativistic, local quantum theory is contained in the net structure of the local observables of this theory results first of all in a concise formulation of the algebraic structure of the superselection theory and an intrinsic formulation of charge composition, charge conjugation and the statistics of an algebraic quantum field theory. In a next step, the locality of massive particles together with their spectral properties are wed for the formulation of a selection criterion which opens the access to the massive, non-abelian quantum gauge theories. The role of the electric charge as a superselection rule results in the introduction of charge classes which in term lead to a set of quantum states with optimum localization properties. Finally, the asymptotic observables of quantum electrodynamics are investigated within the framework of algebraic quantum field theory. (author)
11. Closed string field theory
International Nuclear Information System (INIS)
Strominger, A.
1987-01-01
A gauge invariant cubic action describing bosonic closed string field theory is constructed. The gauge symmetries include local spacetime diffeomorphisms. The conventional closed string spectrum and trilinear couplings are reproduced after spontaneous symmetry breaking. The action S is constructed from the usual ''open string'' field of ghost number minus one half. It is given by the associator of the string field product which is non-vanishing because of associativity anomalies. S does not describe open string propagation because open string states associate and can thereby be shifted away. A field theory of closed and open strings can be obtained by adding to S the cubic open string action. (orig.)
12. Lectures on matrix field theory
CERN Document Server
2017-01-01
These lecture notes provide a systematic introduction to matrix models of quantum field theories with non-commutative and fuzzy geometries. The book initially focuses on the matrix formulation of non-commutative and fuzzy spaces, followed by a description of the non-perturbative treatment of the corresponding field theories. As an example, the phase structure of non-commutative phi-four theory is treated in great detail, with a separate chapter on the multitrace approach. The last chapter offers a general introduction to non-commutative gauge theories, while two appendices round out the text. Primarily written as a self-study guide for postgraduate students – with the aim of pedagogically introducing them to key analytical and numerical tools, as well as useful physical models in applications – these lecture notes will also benefit experienced researchers by providing a reference guide to the fundamentals of non-commutative field theory with an emphasis on matrix models and fuzzy geometries.
13. Interpolating string field theories
International Nuclear Information System (INIS)
Zwiebach, B.
1992-01-01
This paper reports that a minimal area problem imposing different length conditions on open and closed curves is shown to define a one-parameter family of covariant open-closed quantum string field theories. These interpolate from a recently proposed factorizable open-closed theory up to an extended version of Witten's open string field theory capable of incorporating on shell closed strings. The string diagrams of the latter define a new decomposition of the moduli spaces of Riemann surfaces with punctures and boundaries based on quadratic differentials with both first order and second order poles
14. Application of the random field theory in PET imaging - injection dose optimization
Czech Academy of Sciences Publication Activity Database
Dvořák, Jiří; Boldyš, Jiří; Skopalová, M.; Bělohlávek, O.
2013-01-01
Roč. 49, č. 2 (2013), s. 280-300 ISSN 0023-5954 R&D Projects: GA MŠk 1M0572 Institutional support: RVO:67985556 Keywords : random field theory * Euler characteristic * PET imaging * PET image quality Subject RIV: BD - Theory of Information Impact factor: 0.563, year: 2013 http://library.utia.cas.cz/separaty/2013/ZOI/boldys-0397176.pdf
15. 2D fractional supersymmetry for rational conformal field theory: application for third-integer spin states
International Nuclear Information System (INIS)
Perez, A.; Simon, P.
1996-01-01
A 2D fractional supersymmetry theory is algebraically constructed. The Lagrangian is derived using an adapted superspace including, in addition to a scalar field, two fields with spins 1/3,2/3. This theory turns out to be a rational conformal field theory. The symmetry of this model goes beyond the super-Virasoro algebra and connects these third-integer spin states. Besides the stress-momentum tensor, we obtain a supercurrent of spin 4/3. Cubic relations are involved in order to close the algebra; the basic algebra is no longer a Lie or a super-Lie algebra. The central charge of this model is found to be 5/3. Finally, we analyze the form that a local invariant action should take. (orig.)
16. Axiomatic conformal field theory
International Nuclear Information System (INIS)
Gaberdiel, M.R.; Goddard, P.
2000-01-01
A new rigourous approach to conformal field theory is presented. The basic objects are families of complex-valued amplitudes, which define a meromorphic conformal field theory (or chiral algebra) and which lead naturally to the definition of topological vector spaces, between which vertex operators act as continuous operators. In fact, in order to develop the theory, Moebius invariance rather than full conformal invariance is required but it is shown that every Moebius theory can be extended to a conformal theory by the construction of a Virasoro field. In this approach, a representation of a conformal field theory is naturally defined in terms of a family of amplitudes with appropriate analytic properties. It is shown that these amplitudes can also be derived from a suitable collection of states in the meromorphic theory. Zhu's algebra then appears naturally as the algebra of conditions which states defining highest weight representations must satisfy. The relationship of the representations of Zhu's algebra to the classification of highest weight representations is explained. (orig.)
17. Quantum theory of fields
CERN Document Server
Wentzel, Gregor
1949-01-01
A prominent figure in twentieth-century physics, Gregor Wentzel made major contributions to the development of quantum field theory, first in Europe and later at the University of Chicago. His Quantum Theory of Fields offers a knowledgeable view of the original literature of elementary quantum mechanics and helps make these works accessible to interested readers.An introductory volume rather than an all-inclusive account, the text opens with an examination of general principles, without specification of the field equations of the Lagrange function. The following chapters deal with particular
18. Theoretical physics. Field theory
International Nuclear Information System (INIS)
Landau, L.; Lifchitz, E.
2004-01-01
This book is the fifth French edition of the famous course written by Landau/Lifchitz and devoted to both the theory of electromagnetic fields and the gravity theory. The talk of the theory of electromagnetic fields is based on special relativity and relates to only the electrodynamics in vacuum and that of pointwise electric charges. On the basis of the fundamental notions of the principle of relativity and of relativistic mechanics, and by using variational principles, the authors develop the fundamental equations of the electromagnetic field, the wave equation and the processes of emission and propagation of light. The theory of gravitational fields, i.e. the general theory of relativity, is exposed in the last five chapters. The fundamentals of the tensor calculus and all that is related to it are progressively introduced just when needed (electromagnetic field tensor, energy-impulse tensor, or curve tensor...). The worldwide reputation of this book is generally allotted to clearness, to the simplicity and the rigorous logic of the demonstrations. (A.C.)
19. Application of Stochastic Unsaturated Flow Theory, Numerical Simulations, and Comparisons to Field Observations
DEFF Research Database (Denmark)
Jensen, Karsten Høgh; Mantoglou, Aristotelis
1992-01-01
A stochastic unsaturated flow theory and a numerical simulation model have been coupled in order to estimate the large-scale mean behavior of an unsaturated flow system in a spatially variable soil. On the basis of the theoretical developments of Mantoglou and Gelhar (1987a, b, c), the theory...... unsaturated flow equation representing the mean system behavior is solved using a finite difference numerical solution technique. The effective parameters are evaluated from the stochastic theory formulas before entering them into the numerical solution for each iteration. The stochastic model is applied...... to a field site in Denmark, where information is available on the spatial variability of soil parameters and variables. Numerical simulations have been carried out, and predictions of the mean behavior and the variance of the capillary tension head and the soil moisture content have been compared to field...
20. Introduction to gauge field theory
International Nuclear Information System (INIS)
Bailin, David; Love, Alexander
1986-01-01
The book is intended as an introduction to gauge field theory for the postgraduate student of theoretical particle physics. The topics discussed in the book include: path integrals, classical and quantum field theory, scattering amplitudes, feynman rules, renormalisation, gauge field theories, spontaneous symmetry breaking, grand unified theory, and field theories at finite temperature. (UK)
1. Gauge field theory
International Nuclear Information System (INIS)
Aref'eva, I.Ya.; Slavnov, A.A.
1981-01-01
This lecture is devoted to the discussion of gauge field theory permitting from the single point of view to describe all the interactions of elementary particles. The authors used electrodynamics and the Einstein theory of gravity to search for a renormgroup fixing a form of Lagrangian. It is shown that the gauge invariance added with the requirement of the minimum number of arbitraries in Lagrangian fixes unambigously the form of the electromagnetic interaction. The generalization of this construction for more complicate charge spaces results in the Yang-Mills theory. The interaction form in this theory is fixed with the relativity principle in the charge space. A quantum scheme of the Yang-Mills fields through the explicit separation of true dynamic variables is suggested. A comfortable relativistically invariant diagram technique for the calculation of a producing potential for the Green functions is described. The Ward generalized identities have been obtained and a procedure of the elimination of ultraviolet and infrared divergencies has been accomplished. Within the framework of QCD (quantum-chromodynamic) the phenomenon of the asymptotic freedom being the most successful prediction of the gauge theory of strong interactions was described. Working methods with QCD outside the framework of the perturbation theory have been described from a coupling constant. QCD is represented as a single theory possessing both the asymptotical freedom and the freedom retaining quarks [ru
2. Theory of electromagnetic fields
CERN Document Server
Wolski, Andrzej
2011-01-01
We discuss the theory of electromagnetic fields, with an emphasis on aspects relevant to radiofrequency systems in particle accelerators. We begin by reviewing Maxwell's equations and their physical significance. We show that in free space, there are solutions to Maxwell's equations representing the propagation of electromagnetic fields as waves. We introduce electromagnetic potentials, and show how they can be used to simplify the calculation of the fields in the presence of sources. We derive Poynting's theorem, which leads to expressions for the energy density and energy flux in an electromagnetic field. We discuss the properties of electromagnetic waves in cavities, waveguides and transmission lines.
3. Applications of Canonical transformations and nontrivial vacuum solutions to flavor mixing and critical phenomena in quantum field theory
Energy Technology Data Exchange (ETDEWEB)
Mishchenko, Yuriy [North Carolina State Univ., Raleigh, NC (United States)
2004-12-01
MISHCHENKO, YURIY. Applications of Canonical Transformations and Nontrivial Vacuum Solutions to flavor mixing and critical phenomena in Quantum Field Theory. (Under the direction of Chueng-Ryong Ji.) In this dissertation we consider two recent applications of Bogoliubov Transformation to the phenomenology of quantum mixing and the theory of critical phenomena. In recent years quantum mixing got in the focus of the searches for New Physics due to its unparalleled sensitivity to SM parameters and indications of neutrino mixing. It was recently suggested that Bogoliubov Transformation may be important in proper definition of the flavor states that otherwise results in problems in perturbative treatment. As first part of this dissertation we investigate this conjecture and develop a complete formulation of such a mixing field theory involving introduction of general formalism, analysis of space-time conversion and phenomenological implications. As second part of this dissertati
4. Applications of the renormalization group approach to problems in quantum field theory
International Nuclear Information System (INIS)
Renken, R.L.
1985-01-01
The presence of fluctuations at many scales of length complicates theories of quantum fields. However, interest is often focused on the low-energy consequences of a theory rather than the short distance fluctuations. In the renormalization-group approach, one takes advantage of this by constructing an effective theory with identical low-energy behavior, but without short distance fluctuations. Three problems of this type are studied here. In chapter 1, an effective lagrangian is used to compute the low-energy consequences of theories of technicolor. Corrections to weak-interaction parameters are found to be small, but conceivably measurable. In chapter 2, the renormalization group approach is applied to second order phase transitions in lattice gauge theories such as the deconfining transition in the U(1) theory. A practical procedure for studying the critical behavior based on Monte Carlo renormalization group methods is described in detail; no numerical results are presented. Chapter 3 addresses the problem of computing the low-energy behavior of atoms directly from Schrodinger's equation. A straightforward approach is described, but is found to be impractical
5. Euclidean quantum field theory
International Nuclear Information System (INIS)
Jaffe, A.
1985-01-01
In four seminal papers, written from 1963 to 1968, Kurt Symanzik laid the foundations for his euclidean quantum field theory program (EQFT). His original goal was to use EQFT as a tool to approach the existence question for interacting quantum fields. In 1968, when other methods appeared better suited for the existence question, Symanzik abandoned this heroic attempt and redirected his research toward different questions. (orig./HSI)
6. Quantum Field Theory
CERN Document Server
Zeidler, Eberhard
This is the first volume of a modern introduction to quantum field theory which addresses both mathematicians and physicists ranging from advanced undergraduate students to professional scientists. The book tries to bridge the existing gap between the different languages used by mathematicians and physicists. For students of mathematics it is shown that detailed knowledge of the physical background helps to motivate the mathematical subjects and to discover interesting interrelationships between quite different mathematical topics. For students of physics, fairly advanced mathematics is presented, which is beyond the usual curriculum in physics. It is the author's goal to present the state of the art of realizing Einstein's dream of a unified theory for the four fundamental forces in the universe (gravitational, electromagnetic, strong, and weak interaction). From the reviews: "… Quantum field theory is one of the great intellectual edifices in the history of human thought. … This volume differs from othe...
7. Microcontinuum field theories
CERN Document Server
Eringen, A Cemal
1999-01-01
Microcontinuum field theories constitute an extension of classical field theories -- of elastic bodies, deformations, electromagnetism, and the like -- to microscopic spaces and short time scales. Material bodies are here viewed as collections of large numbers of deformable particles, much as each volume element of a fluid in statistical mechanics is viewed as consisting of a large number of small particles for which statistical laws are valid. Classical continuum theories are valid when the characteristic length associated with external forces or stimuli is much larger than any internal scale of the body under consideration. When the characteristic lengths are comparable, however, the response of the individual constituents becomes important, for example, in considering the fluid or elastic properties of blood, porous media, polymers, liquid crystals, slurries, and composite materials. This volume is concerned with the kinematics of microcontinua. It begins with a discussion of strain, stress tensors, balanc...
8. Covariant density functional theory beyond mean field and applications for nuclei far from stability
International Nuclear Information System (INIS)
Ring, P
2010-01-01
Density functional theory provides a very powerful tool for a unified microscopic description of nuclei all over the periodic table. It is not only successful in reproducing bulk properties of nuclear ground states such as binding energies, radii, or deformation parameters, but it also allows the investigation of collective phenomena, such as giant resonances and rotational excitations. However, it is based on the mean field concept and therefore it has its limits. We discuss here two methods based based on covariant density functional theory going beyond the mean field concept, (i) models with an energy dependent self energy allowing the coupling to complex configurations and a quantitative description of the width of giant resonances and (ii) methods of configuration mixing between Slater determinants with different deformation and orientation providing are very successful description of transitional nuclei and quantum phase transitions.
9. Affine field theories
International Nuclear Information System (INIS)
1989-01-01
The author constructs a non-Abelian field theory by gauging a Kac-Moody algebra, obtaining an infinite tower of interacting vector fields and associated ghosts, that obey slightly modified Feynman rules. She discusses the spontaneous symmetry breaking of such theory via the Higgs mechanism. If the Higgs particle lies in the Cartan subalgebra of the Kac-Moody algebra, the previously massless vectors acquire a mass spectrum that is linear in the Kac-Moody index and has additional fine structure depending on the associated Lie algebra. She proceeds to show that there is no obstacle in implementing the affine extension of supersymmetric Yang-Mills theories. The result is valid in four, six and ten space-time dimensions. Then the affine extension of supergravity is investigated. She discusses only the loop algebra since the affine extension of the super-Poincare algebra appears inconsistent. The construction of the affine supergravity theory is carried out by the group manifold method and leads to an action describing infinite towers of spin 2 and spin 3/2 fields that interact subject to the symmetries of the loop algebra. The equations of motion satisfy the usual consistency check. Finally, she postulates a theory in which both the vector and scalar fields lie in the loop algebra of SO(3). This theory has an expanded soliton sector, and corresponding to the original 't Hooft-Polyakov solitonic solutions she now finds an infinite family of exact, special solutions of the new equations. She also proposes a perturbation method for obtaining an arbitrary solution of those equations for each level of the affine index
10. Slave Boson Theory of Orbital Differentiation with Crystal Field Effects: Application to UO2
Science.gov (United States)
Lanatà, Nicola; Yao, Yongxin; Deng, Xiaoyu; Dobrosavljević, Vladimir; Kotliar, Gabriel
2017-03-01
We derive an exact operatorial reformulation of the rotational invariant slave boson method, and we apply it to describe the orbital differentiation in strongly correlated electron systems starting from first principles. The approach enables us to treat strong electron correlations, spin-orbit coupling, and crystal field splittings on the same footing by exploiting the gauge invariance of the mean-field equations. We apply our theory to the archetypical nuclear fuel UO2 and show that the ground state of this system displays a pronounced orbital differentiation within the 5 f manifold, with Mott-localized Γ8 and extended Γ7 electrons.
11. Slave Boson Theory of Orbital Differentiation with Crystal Field Effects: Application to UO_{2}.
Science.gov (United States)
Lanatà, Nicola; Yao, Yongxin; Deng, Xiaoyu; Dobrosavljević, Vladimir; Kotliar, Gabriel
2017-03-24
We derive an exact operatorial reformulation of the rotational invariant slave boson method, and we apply it to describe the orbital differentiation in strongly correlated electron systems starting from first principles. The approach enables us to treat strong electron correlations, spin-orbit coupling, and crystal field splittings on the same footing by exploiting the gauge invariance of the mean-field equations. We apply our theory to the archetypical nuclear fuel UO_{2} and show that the ground state of this system displays a pronounced orbital differentiation within the 5f manifold, with Mott-localized Γ_{8} and extended Γ_{7} electrons.
12. Bucharest PhD Training School : Modern Aspects of Quantum Field Theory and Applications
CERN Document Server
2015-01-01
Bucharest 2015 – Modern Aspects of Quantum Field Theory is part of the CERN – SEENET-MTP PhD Training Program, which consists of a number of seminars in theoretical high energy Physics. This is the second seminar organized by this Program. Here are some photos from this event held in Bucharest between 8-14 November 2015. The previous seminar was organized in Belgrade, under the name Belgrade 2015 - Supergravity.
13. Probabilistic theory of mean field games with applications II mean field games with common noise and master equations
CERN Document Server
Carmona, René
2018-01-01
This two-volume book offers a comprehensive treatment of the probabilistic approach to mean field game models and their applications. The book is self-contained in nature and includes original material and applications with explicit examples throughout, including numerical solutions. Volume II tackles the analysis of mean field games in which the players are affected by a common source of noise. The first part of the volume introduces and studies the concepts of weak and strong equilibria, and establishes general solvability results. The second part is devoted to the study of the master equation, a partial differential equation satisfied by the value function of the game over the space of probability measures. Existence of viscosity and classical solutions are proven and used to study asymptotics of games with finitely many players. Together, both Volume I and Volume II will greatly benefit mathematical graduate students and researchers interested in mean field games. The authors provide a detailed road map t...
14. Neural field theory of synaptic metaplasticity with applications to theta burst stimulation.
Science.gov (United States)
Fung, P K; Robinson, P A
2014-01-07
Transcranial magnetic stimulation (TMS) is characterized by strong nonlinear plasticity effects. Experimental results that highlight such nonlinearity include continuous and intermittent theta-burst stimulations (cTBS and iTBS, respectively), where depression is induced in the continuous case, but insertion of an off period of around 8s for every 2s of stimulation changes the induced plasticity to potentiation in the intermittent case. Another nonlinearity is that cTBS and iTBS exhibit dosage dependency, where doubling of the stimulation duration changes the direction of induced plasticity. Guided by previous experimental results, this study postulates on the characteristics of metaplasticity and formulates a physiological system-level plasticity theory to predict TMS experiments. In this theory, plasticity signaling induces plasticity in NMDA receptors to modulate further plasticity signals, and is followed by a signal transduction delayed plasticity expression. Since this plasticity in NMDA receptor affects subsequent plasticity induction, it is a form of metaplasticity. Incorporating this metaplasticity into a recent neural field theory of calcium dependent plasticity gives a physiological basis for the theory of Bienenstock, Cooper, Munro (1982), where postsynaptic intracellular calcium level becomes the measure of temporal averaged postsynaptic activity, and converges to the plasticity threshold to give homeostatic effects. Simulations of TMS protocol responses show that intracellular calcium oscillations around the threshold predicts the aforementioned nonlinearities in TMS-induced plasticity, as well as the interpersonal TBS response polarity found experimentally, where the same protocol may induce opposite plasticity effect for different subjects. Thereby, recommendations for future experiments and TMS protocol optimizations are made. Input selectivity via spatially extended, mean field neural dynamics is also explored. © 2013 Elsevier Ltd. All rights
15. Theory and applications of internal photoemission in the MOS system at low electric fields
Science.gov (United States)
Przewlocki, Henryk M.
2001-08-01
A new theory is presented of the photoelectric phenomena, which take place in UV illuminated MOS structures, in the presence of weak electric fields (|E|photoelectric measurement methods of the MOS system parameters. Two of such methods are shortly presented. The first is the measurement method of the φMS factor of the MOS system, which has already been fully verified experimentally and has been shown to be the most accurate of the existing methods of this parameter determination. The second is the method to determine trapping properties of the dielectric in the MOS system, which is currently being optimized and verified experimentally.
16. Introduction to string field theory
International Nuclear Information System (INIS)
Horowitz, G.T.
1989-01-01
A light cone gauge superstring field theory is constructed. The BRST approach is described discussing generalizations to yield gauge invariant free superstring field theory and interacting theory for superstrings. The interaction term is explicitly expressed in terms of first quantized oscillators. A purily cubic action for superstring field theory is also derived. (author)
17. On the application of the field-redefinition theorem to the heterotic superstring theory
Science.gov (United States)
Pollock, M. D.
2015-05-01
The ten-dimensional effective action which defines the heterotic superstring theory at low energy is constructed by hypothesis in such a way that the resulting classical equation of motion for the space-time metric simultaneously implies the vanishing of the beta-function for the N = 1 supersymmetric non-linear sigma-model on the world sheet. At four-loop order it was found by Grisaru and Zanon (see also Freeman et al.) that the effective Lagrangian so constructed differs in the numerical coefficient of the term from that obtained directly from the four-point gravitational scattering amplitude. The two expressions can be related via a metric field redefinition , activation of which, however, results in the appearance of ghosts at higher gravitational order , n > 4, as shown by Lawrence. Here, we prove, after reduction of to the physical dimensionality D = 4, that the corresponding field redefinition yields the identity g' ij = g ij , signified by L 3/ R = 0, in a Friedmann space-time generated by a perfect-fluid source characterized by adiabatic index γ ≡ 1 + p/ ρ, where p is the pressure and ρ is the energy density, if, and only if, κ 6 ρ 3 γ 2( γ - 1) = 0. That is, the theory remains free of ghosts in Minkowski space ρ = 0, in a maximally symmetric space-time γ = 0, or in a dust Universe γ = 1. Further aspects of ghost freedom and dimensional reduction, especially to D = 4, are discussed.
18. Superstring field theory
International Nuclear Information System (INIS)
Green, M.B.
1984-01-01
Superstring field theories are formulated in terms of light-cone-gauge superfields that are functionals of string coordinates chi(sigma) and theta(sigma). The formalism used preserves only the manifest SU(4) symmetry that corresponds to rotations among six of the eight transverse directions. In type I theories, which have one ten-dimensional supersymmetry and describe both open and closed strings, there are five interaction terms of two basic kinds. One kind is a breaking or joining interaction, which is a string generalization of a cubic Yang-Mills coupling. It is relevant to both the three open-string vertex and the open-string to closed-string transition vertex. The other kind is an exchange or crossing-over interaction, which is a string generalization of a cubic gravitational coupling. All the interactions can be uniquely determined by requiring continuity of the coordinates chi(sigma) and theta(sigma) (which implies local conservation of the conjugate momenta) and by imposing the global supersymmetry algebra. Specific local operators are identified for each of the two kinds of interactions. In type II theories, which have two ten-dimensional supersymmetries and contain closed strings only, the entire interaction hamiltonian consists of a single cubic vertex. The higher-order contact terms of the N=8 supergravity theory that arises in the low-energy limit give an effective description of the exchange of massive string modes. (orig.)
19. A superstring field theory for supergravity
Science.gov (United States)
Reid-Edwards, R. A.; Riccombeni, D. A.
2017-09-01
A covariant closed superstring field theory, equivalent to classical tendimensional Type II supergravity, is presented. The defining conformal field theory is the ambitwistor string worldsheet theory of Mason and Skinner. This theory is known to reproduce the scattering amplitudes of Cachazo, He and Yuan in which the scattering equations play an important role and the string field theory naturally incorporates these results. We investigate the operator formalism description of the ambitwsitor string and propose an action for the string field theory of the bosonic and supersymmetric theories. The correct linearised gauge symmetries and spacetime actions are explicitly reproduced and evidence is given that the action is correct to all orders. The focus is on the NeveuSchwarz sector and the explicit description of tree level perturbation theory about flat spacetime. Application of the string field theory to general supergravity backgrounds and the inclusion of the Ramond sector are briefly discussed.
20. Beyond mean field theory: statistical field theory for neural networks.
Science.gov (United States)
Buice, Michael A; Chow, Carson C
2013-03-01
Mean field theories have been a stalwart for studying the dynamics of networks of coupled neurons. They are convenient because they are relatively simple and possible to analyze. However, classical mean field theory neglects the effects of fluctuations and correlations due to single neuron effects. Here, we consider various possible approaches for going beyond mean field theory and incorporating correlation effects. Statistical field theory methods, in particular the Doi-Peliti-Janssen formalism, are particularly useful in this regard.
1. Media Accountability Online in Israel. An application of Bourdieu’s field theory
Directory of Open Access Journals (Sweden)
Ronja Kniep
2015-12-01
Full Text Available Due to structural changes in journalism, such as deregulation, privatisation and the influence of new technologies, it has become increasingly important to study media accountability (MA. By applying Bourdieu’s theory of social fields, this paper proposes a new approach to do so: MA is defined as a function of both journalistic autonomy and influence in the media field. Here, online communication potentially widens the scope of action for media’s transparency, responsiveness as well as the articulation of media criticism by a variety of actors. In Israel, media criticism is driven by the agent’s struggle for interpretive authority over public discourse in a politically polarized society. Semi-structured interviews with Israeli journalists, media activists and experts suggest that journalistic agents who have yet to earn credibility and reputation exploit online communication to its full potential, while agents in the field of power tend to dismiss online criticism. The influence of the audience’s media criticism is not solely dependent on the technical ability of connecting and hearing the voices of the masses; it has to be in combination with symbolic or political capital. However, the demand for media’s social responsibility is also related to being more careful and less critical, which is very evident in Israel. Thus, it is important to critically reflect on what happens when media accountability practices become more efficient and a stronger sense for “being watched” develops.
2. Thermal Field Theory in Equilibrium
OpenAIRE
Andersen, Jens O.
2000-01-01
In this talk, I review recent developments in equilibrium thermal field theory. Screened perturbation theory and hard-thermal-loop perturbation theory are discussed. A self-consistent $\\Phi$-derivable approach is also briefly reviewed.
3. Class field theory
CERN Document Server
Artin, Emil
2009-01-01
This classic book, originally published in 1968, is based on notes of a year-long seminar the authors ran at Princeton University. The primary goal of the book was to give a rather complete presentation of algebraic aspects of global class field theory, and the authors accomplished this goal spectacularly: for more than 40 years since its first publication, the book has served as an ultimate source for many generations of mathematicians. In this revised edition, two mathematical additions complementing the exposition in the original text are made. The new edition also contains several new foot
4. Higgs Effective Field Theories
CERN Document Server
2016-01-01
The main focus of this meeting is to present new theoretical advancements related to effective field theories, evaluate the impact of initial results from the LHC Run2, and discuss proposals for data interpretation/presentation during Run2. A crucial role of the meeting is to bring together theorists from different backgrounds and with different viewpoints and to extend bridges towards the experimental community. To this end, we would like to achieve a good balance between senior and junior speakers, enhancing the visibility of younger scientists while keeping some overview talks.
5. Finite volume method for self-consistent field theory of polymers: Material conservation and application
Science.gov (United States)
Yong, Daeseong; Kim, Jaeup U.
2017-12-01
For the purpose of checking material conservation of various numerical algorithms used in the self-consistent-field theory (SCFT) of polymeric systems, we develop an algebraic method using matrix and bra-ket notation, which traces the Hermiticity of the product of the volume and evolution matrices. Algebraic tests for material conservation reveal that the popular pseudospectral method in the Cartesian grid conserves material perfectly, while the finite-volume method (FVM) is the proper tool when real-space SCFT with the Crank-Nicolson method is adopted in orthogonal coordinate systems. We also find that alternating direction implicit methods combined with the FVM exhibit small mass errors in the SCFT calculation. By introducing fractional cells in the FVM formulation, accurate SCFT calculations are performed for systems with irregular geometries and the results are consistent with previous experimental and theoretical works.
6. Studies in quantum field theory
International Nuclear Information System (INIS)
Bender, C.M.; Mandula, J.E.; Shrauner, J.E.
1982-01-01
Washington University is currently conducting research in many areas of high energy theoretical and mathematical physics. These areas include: strong-coupling approximation; classical solutions of non-Abelian gauge theories; mean-field approximation in quantum field theory; path integral and coherent state representations in quantum field theory; lattice gauge calculations; the nature of perturbation theory in large orders; quark condensation in QCD; chiral symmetry breaking; the l/N expansion in quantum field theory; effective potential and action in quantum field theories, including QCD
7. Quantum field theory
International Nuclear Information System (INIS)
Mancini, F.
1986-01-01
Theoretical physicists, coming from different countries, working on different areas, gathered at Positano: the Proceedings contain all the lectures delivered as well as contributed papers. Many areas of physics are represented, elementary particles in high energy physics, quantum relativity, quantum geometry, condensed matter physics, statistical mechanics; but all works are concerned with the use of the methods of quantum field theory. The first motivation of the meeting was to pay homage to a great physicist and a great friend; it was also an occasion in which theoretical physicists got together to discuss and to compare results in different fields. The meeting was very intimate; the relaxed atmosphere allowed constructive discussions and contributed to a positive exchange of ideas. (orig.)
8. Statistical field theory with constraints: Application to critical Casimir forces in the canonical ensemble.
Science.gov (United States)
Gross, Markus; Gambassi, Andrea; Dietrich, S
2017-08-01
The effect of imposing a constraint on a fluctuating scalar order parameter field in a system of finite volume is studied within statistical field theory. The canonical ensemble, corresponding to a fixed total integrated order parameter (e.g., the total number of particles), is obtained as a special case of the theory. A perturbative expansion is developed which allows one to systematically determine the constraint-induced finite-volume corrections to the free energy and to correlation functions. In particular, we focus on the Landau-Ginzburg model in a film geometry (i.e., in a rectangular parallelepiped with a small aspect ratio) with periodic, Dirichlet, or Neumann boundary conditions in the transverse direction and periodic boundary conditions in the remaining, lateral directions. Within the expansion in terms of ε=4-d, where d is the spatial dimension of the bulk, the finite-size contribution to the free energy of the confined system and the associated critical Casimir force are calculated to leading order in ε and are compared to the corresponding expressions for an unconstrained (grand canonical) system. The constraint restricts the fluctuations within the system and it accordingly modifies the residual finite-size free energy. The resulting critical Casimir force is shown to depend on whether it is defined by assuming a fixed transverse area or a fixed total volume. In the former case, the constraint is typically found to significantly enhance the attractive character of the force as compared to the grand canonical case. In contrast to the grand canonical Casimir force, which, for supercritical temperatures, vanishes in the limit of thick films, in the canonical case with fixed transverse area the critical Casimir force attains for thick films a negative value for all boundary conditions studied here. Typically, the dependence of the critical Casimir force both on the temperaturelike and on the fieldlike scaling variables is different in the two ensembles.
9. Digestible quantum field theory
CERN Document Server
Smilga, Andrei
2017-01-01
This book gives an intermediate level treatment of quantum field theory, appropriate to a reader with a first degree in physics and a working knowledge of special relativity and quantum mechanics. It aims to give the reader some understanding of what QFT is all about, without delving deep into actual calculations of Feynman diagrams or similar. The author serves up a seven‐course menu, which begins with a brief introductory Aperitif. This is followed by the Hors d'oeuvres, which set the scene with a broad survey of the Universe, its theoretical description, and how the ideas of QFT developed during the last century. In the next course, the Art of Cooking, the author recaps on some basic facts of analytical mechanics, relativity, quantum mechanics and also presents some nutritious “extras” in mathematics (group theory at the elementary level) and in physics (theory of scattering). After these preparations, the reader should have a good appetite for the Entrées ‐ the central par t of the book where the...
10. Control theory in physics and other fields of science concepts, tools and applications
CERN Document Server
Schulz, Michael
2006-01-01
This book covers systematically and in a simple language the mathematical and physical foundations of controlling deterministic and stochastic evolutionary processes in systems with a high degree of complexity. Strong emphasis is placed on concepts, methods and techniques for modelling, assessment and the solution or estimation of control problems in an attempt to understand the large variability of these problems in several branches of physics, chemistry and biology as well as in technology and economics. The main focus of the book is on a clear physical and mathematical understanding of the dynamics and kinetics behind several kinds of control problems and their relation to self-organizing principles in complex systems. The book is a modern introduction and a helpful tool for researchers, engineers as well as post-docs and graduate students interested in an application oriented control theory and related topics.
11. Application of optimal control theory to laser heating of a plasma in a solenoidal magnetic field
International Nuclear Information System (INIS)
Neal, R.D.
1975-01-01
Laser heating of a plasma column confined by a solenoidal magnetic field is studied via modern optimal control techniques. A two-temperature, constant pressure model is used for the plasma so that the temperature and density are functions of time and location along the plasma column. They are assumed to be uniform in the radial direction so that refraction of the laser beam does not occur. The laser intensity used as input to the column at one end is taken as the control variable and plasma losses are neglected. The localized behavior of the plasma heating dynamics is first studied and conventional optimal control theory applied. The distributed parameter optimal control problem is next considered with minimum time to reach a specified final ion temperature criterion as the objective. Since the laser intensity can only be directly controlled at the input end of the plasma column, a boundary control situation results. The problem is unique in that the control is the boundary value of one of the state variables. The necessary conditions are developed and the problem solved numerically for typical plasma parameters. The problem of maximizing the space-time integral of neutron production rate in the plasma is considered for a constant distributed control problem where the laser intensity is assumed fixed at maximum and the external magnetic field is taken as a control variable
12. Study of the convergence of the nuclear field theory and its application on the lead isotopes
International Nuclear Information System (INIS)
Scoccola, N.N.
1985-01-01
It is shown that highly satisfactory results can be obtained not only in schematic problems (four particles in a degenerate j-shell), but in realistic ones (low lying 204 Pb spectrum), provided second order diagrams and/or diagonalization procedures are used. In both cases energies and two-body transfer amplitudes are calculated and compared with exact and other approximate results. In the second part, the electromagnetic emission of the giant quadrupole resonance (GQR) in 208 Pb after its excitation by inelastic scattering of 17 O to 380 MeV is studied. As the GQR is unstable with respect to the decay to compound nucleous, the reaction mechanism is carefully analized. A formalism is proposed in which the emission probability is factorized in three independent contributions: one due to the electromagnetic field, another to the nuclear reaction and the third to the nuclear structure. The last one is carefully studied in the lowest order of the nuclear field theory, taking into account the mixture of the different isospin states. The results are consistent with the upper experimental limit of the ratio between the transition populating the 3 - (2.62 MeV) state and the one that populates the ground state. However, they failed to reproduce the strong dipole transition to the 3 - (4.97 MeV) state. (Author) [es
13. Correlation functions in finite temperature field theories: formalism and applications to quark-gluon plasma
International Nuclear Information System (INIS)
Gelis, Francois
1998-12-01
The general framework of this work is thermal field theory, and more precisely the perturbative calculation of thermal Green's functions. In a first part, I consider the problems closely related to the formalism itself. After two introductory chapters devoted to set up the framework and the notations used afterwards, a chapter is dedicated to a clarification of certain aspects of the justification of the Feynman rules of the real time formalism. Then, I consider in the chapter 4 the problem of cutting rules in the real time formalisms. In particular, after solving a controversy on this subject, I generalize these cutting rules to the 'retarded-advanced' version of this formalism. Finally, the last problem considered in this part is that of the pion decay into two photons in a thermal bath. I show that the discrepancies found in the literature are due to peculiarities of the analytical properties of the thermal Green's functions. The second part deals with the calculations of the photons or dilepton (virtual photon) production rate by a quark gluon plasma. The framework of this study is the effective theory based on the resummation of hard thermal loops. The first aspects of this study is related to the production of virtual photons, where we show that important contributions arise at two loops, completing the result already known at one loop. In the case of real photon production, we show that extremely strong collinear singularities make two loop contributions dominant compared to one loop ones. In both cases, the importance of two loop contributions can be interpreted as weaknesses of the hard thermal loop approximation. (author)
14. Topics in quantum field theory
International Nuclear Information System (INIS)
Svaiter, N.F.
2006-11-01
This paper presents some important aspects on quantum field theory, covering the following aspects: the triumph and limitations of the quantum field theory; the field theory in curved spaces - Hawking and Unruh-Davies effects; the problem of divergent theory of the zero-point; the problem of the spinning detector and the Trocheries-Takeno vacuum; the field theory at finite temperature - symmetry breaking and phase transition; the problem of the summability of the perturbative series and the perturbative expansion for the strong coupling; quantized fields in presence of classical macroscopic structures; the Parisi-Wu stochastic quantization method
15. Fractional Stochastic Field Theory
Science.gov (United States)
Honkonen, Juha
2018-02-01
Models describing evolution of physical, chemical, biological, social and financial processes are often formulated as differential equations with the understanding that they are large-scale equations for averages of quantities describing intrinsically random processes. Explicit account of randomness may lead to significant changes in the asymptotic behaviour (anomalous scaling) in such models especially in low spatial dimensions, which in many cases may be captured with the use of the renormalization group. Anomalous scaling and memory effects may also be introduced with the use of fractional derivatives and fractional noise. Construction of renormalized stochastic field theory with fractional derivatives and fractional noise in the underlying stochastic differential equations and master equations and the interplay between fluctuation-induced and built-in anomalous scaling behaviour is reviewed and discussed.
16. Chameleon field theories
International Nuclear Information System (INIS)
Khoury, Justin
2013-01-01
Chameleons are light scalar fields with remarkable properties. Through the interplay of self-interactions and coupling to matter, chameleon particles have a mass that depends on the ambient matter density. The manifestation of the fifth force mediated by chameleons therefore depends sensitively on their environment, which makes for a rich phenomenology. In this paper, we review two recent results on chameleon phenomenology. The first result a pair of no-go theorems limiting the cosmological impact of chameleons and their generalizations: (i) the range of the chameleon force at cosmological density today can be at most ∼Mpc; (ii) the conformal factor relating Einstein- and Jordan-frame scale factors is essentially constant over the last Hubble time. These theorems imply that chameleons have negligible effect on the linear growth of structure, and cannot account for the observed cosmic acceleration except as some form of dark energy. The second result pertains to the quantum stability of chameleon theories. We show how requiring that quantum corrections be small, so as to allow reliable predictions of fifth forces, leads to an upper bound of m −3 ) 1/3 eV for gravitational strength coupling, whereas fifth force experiments place a lower bound of m > 0.0042 eV. An improvement of less than a factor of 2 in the range of fifth force experiments could test all classical chameleon field theories whose quantum corrections are well-controlled and couple to matter with nearly gravitational strength regardless of the specific form of the chameleon potential. (paper)
17. An introduction to conformal field theory
International Nuclear Information System (INIS)
Zuber, J.B.
1995-01-01
The aim of these lectures is to present an introduction at a fairly elementary level to recent developments in two dimensional field theory, namely in conformal field theory. We shall see the importance of new structures related to infinite dimensional algebras: current algebras and Virasoro algebra. These topics will find physically relevant applications in the lectures by Shankar and Ian Affeck. (author)
18. Quantum field theory of fluids.
Science.gov (United States)
Gripaios, Ben; Sutherland, Dave
2015-02-20
The quantum theory of fields is largely based on studying perturbations around noninteracting, or free, field theories, which correspond to a collection of quantum-mechanical harmonic oscillators. The quantum theory of an ordinary fluid is "freer", in the sense that the noninteracting theory also contains an infinite collection of quantum-mechanical free particles, corresponding to vortex modes. By computing a variety of correlation functions at tree and loop level, we give evidence that a quantum perfect fluid can be consistently formulated as a low-energy, effective field theory. We speculate that the quantum behavior is radically different from both classical fluids and quantum fields.
19. Relativistic quantum mechanics and field theory
CERN Document Server
Gross, Franz
1999-01-01
An accessible, comprehensive reference to modern quantum mechanics and field theory.In surveying available books on advanced quantum mechanics and field theory, Franz Gross determined that while established books were outdated, newer titles tended to focus on recent developments and disregard the basics. Relativistic Quantum Mechanics and Field Theory fills this striking gap in the field. With a strong emphasis on applications to practical problems as well as calculations, Dr. Gross provides complete, up-to-date coverage of both elementary and advanced topics essential for a well-rounded understanding of the field.
20. Worked examples in engineering field theory
CERN Document Server
1976-01-01
Worked Examples in Engineering Field Theory is a product of a lecture course given by the author to first-year students in the Department of Engineering in the University of Leicester. The book presents a summary of field theory together with a large number of worked examples and solutions to all problems given in the author's other book, Engineering Field Theory. The 14 chapters of this book are organized into two parts. Part I focuses on the concept of flux including electric flux. This part also tackles the application of the theory in gravitation, ideal fluid flow, and magnetism. Part II d
1. Advanced number theory with applications
CERN Document Server
Mollin, Richard A
2009-01-01
Algebraic Number Theory and Quadratic Fields Algebraic Number Fields The Gaussian Field Euclidean Quadratic Fields Applications of Unique Factorization Ideals The Arithmetic of Ideals in Quadratic Fields Dedekind Domains Application to Factoring Binary Quadratic Forms Basics Composition and the Form Class Group Applications via Ambiguity Genus Representation Equivalence Modulo p Diophantine Approximation Algebraic and Transcendental Numbers Transcendence Minkowski's Convex Body Theorem Arithmetic Functions The Euler-Maclaurin Summation Formula Average Orders The Riemann zeta-functionIntroduction to p-Adic AnalysisSolving Modulo pn Introduction to Valuations Non-Archimedean vs. Archimedean Valuations Representation of p-Adic NumbersDirichlet: Characters, Density, and Primes in Progression Dirichlet Characters Dirichlet's L-Function and Theorem Dirichlet DensityApplications to Diophantine Equations Lucas-Lehmer Theory Generalized Ramanujan-Nagell Equations Bachet's Equation The Fermat Equation Catalan and the A...
2. Naturality in conformal field theory
International Nuclear Information System (INIS)
Moore, G.; Seiberg, N.
1989-01-01
We discuss constraints on the operator product coefficients in diagonal and nondiagonal rational conformal field theories. Nondiagonal modular invariants always arise from automorphisms of the fusion rule algebra or from extensions of the chiral algebra. Moreover, when the chiral algebra has been maximally extended a strong form of the naturality principle of field theory can be proven for rational conformal field theory: operator product coefficients vanish if and only if the corresponding fusion rules vanish; that is, if and only if the vanishing can be understood in terms of a symmetry. We illustrate these ideas with several examples. We also generalize our ideas about rational conformal field theories to a larger class of theories: 'quasi-rational conformal field theories' and we explore some of their properties. (orig.)
3. An Empirical Study on the Application of Theme Theory in the Field of Writing Pedagogy
Science.gov (United States)
Jingxia, Liu; Li, Liu
2013-01-01
English writing instruction is an important part in college English pedagogy. Traditional way of teaching English writing lays much emphasis on word, grammar and sentence rather than the level of discourse. Under the traditional way, the students have difficulties to yield well-organized and coherent compositions. Theme Theory provides a…
4. Conformal field theories and tensor categories. Proceedings
Energy Technology Data Exchange (ETDEWEB)
Bai, Chengming [Nankai Univ., Tianjin (China). Chern Institute of Mathematics; Fuchs, Juergen [Karlstad Univ. (Sweden). Theoretical Physics; Huang, Yi-Zhi [Rutgers Univ., Piscataway, NJ (United States). Dept. of Mathematics; Kong, Liang [Tsinghua Univ., Beijing (China). Inst. for Advanced Study; Runkel, Ingo; Schweigert, Christoph (eds.) [Hamburg Univ. (Germany). Dept. of Mathematics
2014-08-01
First book devoted completely to the mathematics of conformal field theories, tensor categories and their applications. Contributors include both mathematicians and physicists. Some long expository articles are especially suitable for beginners. The present volume is a collection of seven papers that are either based on the talks presented at the workshop ''Conformal field theories and tensor categories'' held June 13 to June 17, 2011 at the Beijing International Center for Mathematical Research, Peking University, or are extensions of the material presented in the talks at the workshop. These papers present new developments beyond rational conformal field theories and modular tensor categories and new applications in mathematics and physics. The topics covered include tensor categories from representation categories of Hopf algebras, applications of conformal field theories and tensor categories to topological phases and gapped systems, logarithmic conformal field theories and the corresponding non-semisimple tensor categories, and new developments in the representation theory of vertex operator algebras. Some of the papers contain detailed introductory material that is helpful for graduate students and researchers looking for an introduction to these research directions. The papers also discuss exciting recent developments in the area of conformal field theories, tensor categories and their applications and will be extremely useful for researchers working in these areas.
5. Topological quantum field theory and four manifolds
CERN Document Server
Marino, Marcos
2005-01-01
The present book is the first of its kind in dealing with topological quantum field theories and their applications to topological aspects of four manifolds. It is not only unique for this reason but also because it contains sufficient introductory material that it can be read by mathematicians and theoretical physicists. On the one hand, it contains a chapter dealing with topological aspects of four manifolds, on the other hand it provides a full introduction to supersymmetry. The book constitutes an essential tool for researchers interested in the basics of topological quantum field theory, since these theories are introduced in detail from a general point of view. In addition, the book describes Donaldson theory and Seiberg-Witten theory, and provides all the details that have led to the connection between these theories using topological quantum field theory. It provides a full account of Witten’s magic formula relating Donaldson and Seiberg-Witten invariants. Furthermore, the book presents some of the ...
6. Mean Field Theory of a Coupled Heisenberg Model and Its Application to an Organic Antiferromagnet with Magnetic Anions
Science.gov (United States)
Ito, Kazuhiro; Shimahara, Hiroshi
2016-02-01
We examine the mean field theory of a uniaxial coupled Heisenberg antiferromagnet with two subsystems, one of which consists of strongly interacting small spins and the other consists of weakly interacting large spins. We reanalyze the experimental data of specific heat and magnetic susceptibility obtained by previous authors for the organic compound λ-(BETS)2FeCl4 at low temperatures, where BETS stands for bis(ethylenedithio)tetraselenafulvalene. The model parameters for this compound are evaluated, where the applicability of the theory is checked. As a result, it is found that J1 ≫ J12 ≫ J2, where J1, J2, and J12 denote the exchange coupling constant between π spins, that between 3d spins, and that between π and 3d spins, respectively. At the low-temperature limit, both sublattice magnetizations of the 3d and π spins are saturated, and the present model is reduced to the Schottky model, which successfully explains experimental observations in previous studies. As temperature increases, fluctuations of 3d spins increase, while π spins remain almost saturated. Near the critical temperature, both spins fluctuate significantly, and thus the mean field approximation breaks down. It is revealed that the magnetic anisotropy, which may be crucial to the antiferromagnetic long-range order, originates from J12 rather than from J2 and that the angle between the magnetic easy-axis and the crystal c-axis is approximately 26-27° in the present effective model.
7. Class field theory from theory to practice
CERN Document Server
Gras, Georges
2003-01-01
Global class field theory is a major achievement of algebraic number theory, based on the functorial properties of the reciprocity map and the existence theorem. The author works out the consequences and the practical use of these results by giving detailed studies and illustrations of classical subjects (classes, idèles, ray class fields, symbols, reciprocity laws, Hasse's principles, the Grunwald-Wang theorem, Hilbert's towers,...). He also proves some new or less-known results (reflection theorem, structure of the abelian closure of a number field) and lays emphasis on the invariant (/cal T) p, of abelian p-ramification, which is related to important Galois cohomology properties and p-adic conjectures. This book, intermediary between the classical literature published in the sixties and the recent computational literature, gives much material in an elementary way, and is suitable for students, researchers, and all who are fascinated by this theory. In the corrected 2nd printing 2005, the author improves s...
8. Broken symmetries in field theory
NARCIS (Netherlands)
Kok, Mark Okker de
2008-01-01
The thesis discusses the role of symmetries in Quantum Field Theory. Quantum Field Theory is the mathematical framework to describe the physics of elementary particles. A symmetry here means a transformation under which the model at hand is invariant. Three types of symmetry are distinguished: 1.
9. Renormalization and effective field theory
CERN Document Server
Costello, Kevin
2011-01-01
This book tells mathematicians about an amazing subject invented by physicists and it tells physicists how a master mathematician must proceed in order to understand it. Physicists who know quantum field theory can learn the powerful methodology of mathematical structure, while mathematicians can position themselves to use the magical ideas of quantum field theory in "mathematics" itself. The retelling of the tale mathematically by Kevin Costello is a beautiful tour de force. --Dennis Sullivan This book is quite a remarkable contribution. It should make perturbative quantum field theory accessible to mathematicians. There is a lot of insight in the way the author uses the renormalization group and effective field theory to analyze perturbative renormalization; this may serve as a springboard to a wider use of those topics, hopefully to an eventual nonperturbative understanding. --Edward Witten Quantum field theory has had a profound influence on mathematics, and on geometry in particular. However, the notorio...
10. Introductory lectures on quantum field theory
International Nuclear Information System (INIS)
Alvarez-Gaume, L.; Vasquez-Mozo, M.A.
2011-01-01
In these lectures we present a few topics in quantum field theory in detail. Some of them are conceptual and some more practical. They have been selected because they appear frequently in current applications to particle physics and string theory. (author)
11. Perturbation series at large orders in quantum mechanics and field theories: application to the problem of resummation
International Nuclear Information System (INIS)
Zinn-Justin, J.; Freie Univ. Berlin
1981-01-01
In this review I present a method to estimate the large order behavior of perturbation theory in quantum mechanics and field theory. The basic idea, due to Lipatov, is to relate the large order behavior to (in general complex) instanton contributions to the path integral representation of Green's functions. I explain the method first in the case of a simple integral and of the anharmonic oscillator and recover the results of Bender and Wu. I apply it then to the PHI 4 field theory. I study general potentials and boson field theories. I show, following Parisi, how the method can be generalized to theories with fermions. Finally I outline the implications of these results for the summability of the series. In particular I explain a method to sum divergent series based on a Borel transformation. In a last section I compare the larger order behavior predictions to actual series calculation. I present also some numerical examples of series summation. (orig.)
12. Acoustic array systems theory, implementation, and application
CERN Document Server
Bai, Mingsian R; Benesty, Jacob
2013-01-01
Presents a unified framework of far-field and near-field array techniques for noise source identification and sound field visualization, from theory to application. Acoustic Array Systems: Theory, Implementation, and Application provides an overview of microphone array technology with applications in noise source identification and sound field visualization. In the comprehensive treatment of microphone arrays, the topics covered include an introduction to the theory, far-field and near-field array signal processing algorithms, practical implementations, and common applic
13. Unified field theory from the classical wave equation: Preliminary application to atomic and nuclear structure
Science.gov (United States)
Múnera, Héctor A.
2016-07-01
It is postulated that there exists a fundamental energy-like fluid, which occupies the flat three-dimensional Euclidean space that contains our universe, and obeys the two basic laws of classical physics: conservation of linear momentum, and conservation of total energy; the fluid is described by the classical wave equation (CWE), which was Schrödinger's first candidate to develop his quantum theory. Novel solutions for the CWE discovered twenty years ago are nonharmonic, inherently quantized, and universal in the sense of scale invariance, thus leading to quantization at all scales of the universe, from galactic clusters to the sub-quark world, and yielding a unified Lorentz-invariant quantum theory ab initio. Quingal solutions are isomorphic under both neo-Galilean and Lorentz transformations, and exhibit nother remarkable property: intrinsic unstability for large values of ℓ (a quantum number), thus limiting the size of each system at a given scale. Unstability and scale-invariance together lead to nested structures observed in our solar system; unstability may explain the small number of rows in the chemical periodic table, and nuclear unstability of nuclides beyond lead and bismuth. Quingal functions lend mathematical basis for Boscovich's unified force (which is compatible with many pieces of evidence collected over the past century), and also yield a simple geometrical solution for the classical three-body problem, which is a useful model for electronic orbits in simple diatomic molecules. A testable prediction for the helicoidal-type force is suggested.
14. An application of modular inclusion to quantum field theory in curved space-time
International Nuclear Information System (INIS)
Summers, S.J.; Verch, R.
1993-09-01
Applying recent results by Borchers connecting geometric modular action, modular inclusion and the spectrum condition, earlier results by Kay and Wald concerning the temperature of physically significant states of the linear Hermitean scalar field propagating in the background of a space-time with a bifurcate Killing horizon are generalized. (orig.)
15. Semiclassical methods in field theories
International Nuclear Information System (INIS)
Ventura, I.
1978-10-01
A new scheme is proposed for semi-classical quantization in field theory - the expansion about the charge (EAC) - which is developed within the canonical formalism. This method is suitable for quantizing theories that are invariant under global gauge transformations. It is used in the treatment of the non relativistic logarithmic theory that was proposed by Bialynicki-Birula and Mycielski - a theory we can formulate in any number of spatial dimensions. The non linear Schroedinger equation is also quantized by means of the EAC. The classical logarithmic theories - both, the non relativistic and the relativistic one - are studied in detail. It is shown that the Bohr-Sommerfeld quantization rule(BSQR) in field theory is, in many cases, equivalent to charge quantization. This rule is then applied to the massive Thirring Model and the logarithmic theories. The BSQR can be see as a simplified and non local version of the EAC [pt
16. Austerity and geometric structure of field theories
International Nuclear Information System (INIS)
Kheyfets, A.
1986-01-01
The relation between the austerity idea and the geometric structure of the three basic field theories - electrodynamics, Yang-Mills theory, and general relativity - is studied. One of the most significant manifestations of the austerity idea in field theories is thought to be expressed by the boundary of a boundary principle (BBP). The BBP says that almost all content of the field theories can be deduced from the topological identity of delta dot produced with delta = 0 used twice, at the 1-2-3-dimensional level (providing the homogeneous field equations), and at the 2-3-4-dimensional level (providing the conservation laws for the source currents). There are some difficulties in this line of thought due to the apparent lack of universality in application of the BBP to the three basic modern field theories above. This dissertation: (a) analyzes the difficulties by means of algebraic topology, integration theory, and modern differential geometry based on the concepts of principal bundles and Ehresmann connections: (b) extends the BBP to the unified Kaluza-Klein theory; (c) reformulates the inhomogeneous field equations and the BBP in terms of E. Cartan moment of rotation, in the way universal for the three theories and compatible with the original austerity idea; and (d) underlines the important role of the soldering structure on spacetime, and indicates that the future development of the austerity idea would involve the generalized theories
17. Lectures on quantum field theory
CERN Document Server
Das, Ashok
2008-01-01
This book consists of the lectures for a two-semester course on quantum field theory, and as such is presented in a quite informal and personal manner. The course starts with relativistic one-particle systems, and develops the basics of quantum field theory with an analysis of the representations of the Poincaré group. Canonical quantization is carried out for scalar, fermion, Abelian and non-Abelian gauge theories. Covariant quantization of gauge theories is also carried out with a detailed description of the BRST symmetry. The Higgs phenomenon and the standard model of electroweak interactio
18. Introduction to quantum field theory
International Nuclear Information System (INIS)
Kazakov, D.I.
1988-01-01
The lectures appear to be a continuation to the introduction to elementary principles of the quantum field theory. The work is aimed at constructing the formalism of standard particle interaction model. Efforts are made to exceed the limits of the standard model in the quantum field theory context. Grand unification models including strong and electrical weak interactions, supersymmetric generalizations of the standard model and grand unification theories and, finally, supergravitation theories including gravitation interaction to the universal scheme, are considered. 3 refs.; 19 figs.; 2 tabs
19. Bayesian theory and applications
CERN Document Server
Dellaportas, Petros; Polson, Nicholas G; Stephens, David A
2013-01-01
The development of hierarchical models and Markov chain Monte Carlo (MCMC) techniques forms one of the most profound advances in Bayesian analysis since the 1970s and provides the basis for advances in virtually all areas of applied and theoretical Bayesian statistics. This volume guides the reader along a statistical journey that begins with the basic structure of Bayesian theory, and then provides details on most of the past and present advances in this field. The book has a unique format. There is an explanatory chapter devoted to each conceptual advance followed by journal-style chapters that provide applications or further advances on the concept. Thus, the volume is both a textbook and a compendium of papers covering a vast range of topics. It is appropriate for a well-informed novice interested in understanding the basic approach, methods and recent applications. Because of its advanced chapters and recent work, it is also appropriate for a more mature reader interested in recent applications and devel...
20. A landscape of field theories
Energy Technology Data Exchange (ETDEWEB)
Maxfield, Travis [Enrico Fermi Institute, University of Chicago,Chicago, IL 60637 (United States); Robbins, Daniel [George P. and Cynthia W. Mitchell Institute for Fundamental Physics and Astronomy,Texas A& M University,College Station, TX 77843-4242 (United States); Sethi, Savdeep [Enrico Fermi Institute, University of Chicago,Chicago, IL 60637 (United States)
2016-11-28
Studying a quantum field theory involves a choice of space-time manifold and a choice of background for any global symmetries of the theory. We argue that many more choices are possible when specifying the background. In the context of branes in string theory, the additional data corresponds to a choice of supergravity tensor fluxes. We propose the existence of a landscape of field theory backgrounds, characterized by the space-time metric, global symmetry background and a choice of tensor fluxes. As evidence for this landscape, we study the supersymmetric six-dimensional (2,0) theory compactified to two dimensions. Different choices of metric and flux give rise to distinct two-dimensional theories, which can preserve differing amounts of supersymmetry.
1. Embedding classical fields in quantum field theories
International Nuclear Information System (INIS)
Blaha, S.
1978-01-01
We describe a procedure for quantizing a classical field theory which is the field-theoretica analog of Sudarshan's method for embedding a classical-mechanical system in a quantum-mechanical system. The essence of the difference between our quantization procedure and Fock-space quantization lies in the choice of vacuum states. The key to our choice of vacuum is the procedure we outline for constructing Lagrangians which have gradient terms linear in the field varialbes from classical Lagrangians which have gradient terms which are quadratic in field variables. We apply this procedure to model electrodynamic field theories, Yang-Mills theories, and a vierbein model of gravity. In the case of electrodynamics models we find a formalism with a close similarity to the coherent-soft-photon-state formalism of QED. In addition, photons propagate to t = + infinity via retarded propagators. We also show how to construct a quantum field for action-at-a-distance electrodynamics. In the Yang-Mills case we show that a previously suggested model for quark confinement necessarily has gluons with principle-value propagation which allows the model to be unitary despite the presence of higher-order-derivative field equations. In the vierbein-gravity model we show that our quantization procedure allows us to treat the classical and quantum parts of the metric field in a unified manner. We find a new perturbation scheme for quantum gravity as a result
2. Topological field theories and duality
International Nuclear Information System (INIS)
Stephany, J.; Universidad Simon Bolivar, Caracas
1996-05-01
Topologically non trivial effects appearing in the discussion of duality transformations in higher genus manifold are discussed in a simple example, and their relation with the properties of Topological Field Theories is established. (author). 16 refs
3. Renormalization in classical field theory
International Nuclear Information System (INIS)
Corbo, Guido
2010-01-01
We discuss simple examples in which renormalization is required in classical field theory. The presentation is accessible to undergraduate students with a knowledge of the basic notions of classical electromagnetism. (letters and comments)
4. Games, theory and applications
CERN Document Server
Thomas, L C
2011-01-01
Anyone with a knowledge of basic mathematics will find this an accessible and informative introduction to game theory. It opens with the theory of two-person zero-sum games, two-person non-zero sum games, and n-person games, at a level between nonmathematical introductory books and technical mathematical game theory books. Succeeding sections focus on a variety of applications - including introductory explanations of gaming and meta games - that offer nonspecialists information about new areas of game theory at a comprehensible level. Numerous exercises appear with full solutions, in addition
5. Finite-temperature field theory
International Nuclear Information System (INIS)
Kapusta, J.I.; Landshoff, P.V.
1989-01-01
Particle number is not conserved in relativistic theories although both lepton and baryon number are. Therefore when discussing the thermodynamics of a quantum field theory one uses the grand canonical formalism. The entropy S is maximised, keeping fixed the ensemble averages E and N of energy and lepton number. Two lagrange multipliers are introduced. (author)
6. Supersymmetric field theories and generalized cohomology
OpenAIRE
Teichner, Peter; Stolz, Stephan
2011-01-01
This survey discusses our results and conjectures concerning supersymmetric field theories and their relationship to cohomology theories. A careful definition of supersymmetric Euclidean field theories is given, refining Segal's axioms for conformal field theories. We state and give an outline of the proof of various results relating field theories to cohomology theories.
7. Octonionic methods in field theory
International Nuclear Information System (INIS)
Duendarer, A.R.
1987-01-01
Some applications of octonion algebra and octonionic analysis to group theory and higher dimensional field theories are presented. To this end an eight dimensional covariant treatment of the octonion algebra is needed. The existing formulations which are covariant only in seven dimensions are reviewed. In this work the eight dimensional formulation is developed through the introduction of fourth rank tensors f abcd and f' abcd in eight dimensions that generalize the octonionic structure constants. The seven octonion units e α are generalized to an 8-vector e a and two second rank tensors e ab and e' ab . Higher rank tensors associated with e α are also introduced. Chirality and duality properties of the structure tensors, f,f' and the octonionic tensors e a , e ab , etc. are discussed and various new identities relating these quantities are derived. New vector products for two, three and four octonions are introduced and their duality properties with respect to the eight-dimensional Levi-Civita tensor as well as their orthogonality properties are studied
8. Introduction to classical and quantum field theory
International Nuclear Information System (INIS)
Ng, Tai-Kai
2009-01-01
This is the first introductory textbook on quantum field theory to be written from the point of view of condensed matter physics. As such, it presents the basic concepts and techniques of statistical field theory, clearly explaining how and why they are integrated into modern quantum (and classical) field theory, and includes the latest developments. Written by an expert in the field, with a broad experience in teaching and training, it manages to present such substantial topics as phases and phase transitions or solitons and instantons in an accessible and concise way. Divided into three parts, the first part covers fundamental physics and the mathematics background needed by students in order to enter the field, while the second part introduces more advanced concepts and techniques. Part III discusses applications of quantum field theory to a few basic problems. The emphasis here lies on how modern concepts of quantum field theory are embedded in these approaches, and also on the limitations of standard quantum field theory techniques in facing, 'real' physics problems. Throughout there are numerous end-of-chapter problems, and a free solutions manual is available for lecturers. (orig.)
9. Dimensional continuation in field theory
International Nuclear Information System (INIS)
Lee, T.
1988-01-01
The continuation of space-time dimension to an arbitrary complex number is discussed. The ultra-violet and infra-red divergences are simply regularized by analytically continuing to some proper dimension n. Combined with functional integral quantization, it provides a simple and elegant description of quantum field theory. Two well known field theories are discussed. Scalar field theory and quantum electrodynamics. In the scalar theory, the focus is on the operator product expansion. It is showed that a renormalization scheme (minimal subtraction) clearly defines the operator product expansion. In the quantum electrodynamics, it is shown that BRS symmetry can simplify the renormalization process. Composite operators are the renormalized and renormalized stress-energy tensor is formed
10. [Studies in quantum field theory
International Nuclear Information System (INIS)
1990-01-01
During the period 4/1/89--3/31/90 the theoretical physics group supported by Department of Energy Contract No. AC02-78ER04915.A015 and consisting of Professors Bender and Shrauner, Associate Professor Papanicolaou, Assistant Professor Ogilvie, and Senior Research Associate Visser has made progress in many areas of theoretical and mathematical physics. Professors Bender and Shrauner, Associate Professor Papanicolaou, Assistant Professor Ogilvie, and Research Associate Visser are currently conducting research in many areas of high energy theoretical and mathematical physics. These areas include: strong-coupling approximation; classical solutions of non-Abelian gauge theories; mean-field approximation in quantum field theory; path integral and coherent state representations in quantum field theory; lattice gauge calculations; the nature of perturbation theory in large order; quark condensation in QCD; chiral symmetry breaking; the 1/N expansion in quantum field theory; effective potential and action in quantum field theories, including OCD; studies of the early universe and inflation, and quantum gravity
11. Field theory of strings
International Nuclear Information System (INIS)
Ramond, P.
1987-01-01
We review the construction of the free equations of motion for open and closed strings in 26 dimensions, using the methods of the Florida Group. Differing from previous treatments, we argue that the constraint L 0 -anti L 0 =0 should not be imposed on all the fields of the closed string in the gauge invariant formalism; we show that it can be incorporated in the gauge invariant formalism at the price of being unable to extract the equations of motion from a Langrangian. We then describe our purely algebraic method to introduce interactions, which works equally well for open and closed strings. Quartic interactions are absent except in the Physical Gauge. Finally, we speculate on the role of the measure of the open string path functional. (orig.)
12. Field theory of strings
International Nuclear Information System (INIS)
Ramond, P.
1986-01-01
We review the construction of the free equations of motion for open and closed strings in 26 dimensions, using the methods of the Florida Group. Differing from previous treatments, we argue that the constraint L 0 - L 0 -bar = 0 should not be imposed on all the fields of the closed string in the gauge invariant formalism: we show that it can be incorporated in the invariant formalism at the price of being unable to extract the equations of motion from a Lagrangian. We then describe our purely algebraic method to introduce interactions, which works equally well for open and closed strings. Quartic interactions are absent except in the Physical Gauge. Finally, we speculate on the role of the measure of the open string path functional. 20 refs
13. Probability theory and applications
CERN Document Server
Hsu, Elton P
1999-01-01
This volume, with contributions by leading experts in the field, is a collection of lecture notes of the six minicourses given at the IAS/Park City Summer Mathematics Institute. It introduces advanced graduates and researchers in probability theory to several of the currently active research areas in the field. Each course is self-contained with references and contains basic materials and recent results. Topics include interacting particle systems, percolation theory, analysis on path and loop spaces, and mathematical finance. The volume gives a balanced overview of the current status of probability theory. An extensive bibliography for further study and research is included. This unique collection presents several important areas of current research and a valuable survey reflecting the diversity of the field.
14. Classical field theory with fermions
International Nuclear Information System (INIS)
Borsanyi, Sz.; Hindmarsh, M.
2009-01-01
Classical field theory simulations have been essential for our understanding of non-equilibrium phenomena in particle physics. In this talk we discuss the possible extension of the bosonic classical field theory simulations to include fermions. In principle we use the inhomogeneous mean field approximation as introduced by Aarts and Smit. But in practice we turn from their deterministic technique to a stochastic approach. We represent the fermion field as an ensemble of pairs of spinor fields, dubbed male and female. These c-number fields solve the classical Dirac equation. Our improved algorithm enables the extension of the originally 1+1 dimensional analyses and is suitable for large-scale inhomogeneous settings, like defect networks.
15. Wavelet theory and its applications
Energy Technology Data Exchange (ETDEWEB)
Faber, V.; Bradley, JJ.; Brislawn, C.; Dougherty, R.; Hawrylycz, M.
1996-07-01
This is the final report of a three-year, Laboratory-Directed Research and Development (LDRD) project at the Los Alamos National Laboratory (LANL). We investigated the theory of wavelet transforms and their relation to Laboratory applications. The investigators have had considerable success in the past applying wavelet techniques to the numerical solution of optimal control problems for distributed- parameter systems, nonlinear signal estimation, and compression of digital imagery and multidimensional data. Wavelet theory involves ideas from the fields of harmonic analysis, numerical linear algebra, digital signal processing, approximation theory, and numerical analysis, and the new computational tools arising from wavelet theory are proving to be ideal for many Laboratory applications. 10 refs.
16. Conformal techniques in string theory and string field theory
International Nuclear Information System (INIS)
Giddings, S.B.
1987-01-01
The application of some conformal and Riemann surface techniques to string theory and string field theory is described. First a brief review of Riemann surface techniques and of the Polyakov approach to string theory is presented. This is followed by a discussion of some features of string field theory and of its Feynman rules. Specifically, it is shown that the Feynman diagrams for Witten's string field theory respect modular invariance, and in particular give a triangulation of moduli space. The Polyakov formalism is then used to derive the Feynman rules that should follow from this theory upon gauge-fixing. It should also be possible to apply this derivation to deduce the Feynman rules for other gauge-fixed string field theories. Following this, Riemann surface techniques are turned to the problem of proving the equivalence of the Polyakov and light-cone formalisms. It is first shown that the light-cone diagrams triangulate moduli space. Then the Polyakov measure is worked out for these diagrams, and shown to equal that deduced from the light-cone gauge fixed formalism. Also presented is a short description of the comparison of physical states in the two formalisms. The equivalence of the two formalisms in particular constitutes a proof of the unitarity of the Polyakov framework for the closed bosonic string
17. Application of effective field theory on nuclear matter and neutron matter; Anwendung effektiver Feldtheorie auf Kernmaterie und Neutronenmaterie
Energy Technology Data Exchange (ETDEWEB)
Saviankou, Pavel
2009-05-15
In the thesis the effective field theory in NLO and NNLO order is applied. The order NLO still knows no three-particle forces. The theory yields however already in this order the saturation behaviour of nuclear matter. This is due to the fact that in the NLO order the scattering phases are qualitatively correctly reproduced, especially the scattering phases {sup 1}S{sub 0} and {sup 3}S{sub 1} are for energies above 200 MeV negative, which is in all potentials by a so called hard core represented. In the NNLO orde three-particle forces occur, which lead to a larger improvement of the saturation curve, however the saturation point lies still at too high densities. A correction of the low-energy constants by scarcely three percent of the value in the vacuum generates however a saturation curve, which reproduces the empirical binding energy per particle, the density and the compressibility of nuclear matter. About the equation of state of neutron matter is empirically few known. At small densities of neutron matter (k{sub f}<1 fm{sup -1}) the NLO and NNLO orders scarcely differ, but indeed from the free Fermi gas. For applications in finite nuclei a simplified parametrization of the nucleon-nucleon interactions was developed, which reproduces both the known scattering phases with an NLO-comparable accuracy and the empirical saturation behaviour. [German] In der Arbeit wird die Effektive Feldtheorie in der Ordnung NLO und NNLO angewandt. Die Ordnung NLO kennt noch keine Dreiteilchenkraefte. Die Theorie liefert jedoch bereits in dieser Ordnung das Saettigungsverhalten von Kernmaterie. Dies liegt daran, dass bereits in der Ordnung NLO die Streuphasen qualitativ korrekt reproduziert werden, insbesondere sind die Streuphasen {sup 1}S{sub 0} und {sup 3}S{sub 1} fuer Energien oberhalb 200 MeV negativ, was in allen Potentialen durch einen sogenannten ''hard core'' dargestellt wird. In der Ordnung NNLO treten Dreiteilchenkraefte auf, die zu einer grossen
18. Electromagnetic field theories for engineering
CERN Document Server
Salam, Md Abdus
2014-01-01
A four year Electrical and Electronic engineering curriculum normally contains two modules of electromagnetic field theories during the first two years. However, some curricula do not have enough slots to accommodate the two modules. This book, Electromagnetic Field Theories, is designed for Electrical and Electronic engineering undergraduate students to provide fundamental knowledge of electromagnetic fields and waves in a structured manner. A comprehensive fundamental knowledge of electric and magnetic fields is required to understand the working principles of generators, motors and transformers. This knowledge is also necessary to analyze transmission lines, substations, insulator flashover mechanism, transient phenomena, etc. Recently, academics and researches are working for sending electrical power to a remote area by designing a suitable antenna. In this case, the knowledge of electromagnetic fields is considered as important tool.
19. Generalized field theory of gravitation
International Nuclear Information System (INIS)
Yilmaz, H.
1976-01-01
It is shown that if, on empirical grounds, one rules out the existence of cosmic fields of Dicke-Brans (scalar) and Will Nordvedt (vector, tensor) type, then the most general experimentally viable and theoretically reasonable theory of gravitation seems to be a LAMBDA-dependent generalization of Einstein and Yilmez theories, which reduces to the former for LAMBDA=0 and to the latter for LAMBDA=1
20. Towards the mathematics of quantum field theory
CERN Document Server
Paugam, Frédéric
2014-01-01
The aim of this book is to introduce mathematicians (and, in particular, graduate students) to the mathematical methods of theoretical and experimental quantum field theory, with an emphasis on coordinate-free presentations of the mathematical objects in play. This should in turn promote interaction between mathematicians and physicists by supplying a common and flexible language for the good of both communities, even if the mathematical one is the primary target. This reference work provides a coherent and complete mathematical toolbox for classical and quantum field theory, based on categorical and homotopical methods, representing an original contribution to the literature. The first part of the book introduces the mathematical methods needed to work with the physicists' spaces of fields, including parameterized and functional differential geometry, functorial analysis, and the homotopical geometric theory of non-linear partial differential equations, with applications to general gauge theories. The second...
1. Renormalization of topological field theory
International Nuclear Information System (INIS)
Birmingham, D.; Rakowski, M.; Thompson, G.
1988-11-01
One loop corrections to topological field theory in three and four dimensions are presented. By regularizing determinants, we compute the effective action and β-function in four dimensional topological Yang-Mills theory and find that the BRST symmetry is preserved. Moreover, the minima of the effective action still correspond to instanton configurations. In three dimensions, an analysis of the Chern-Simons theory shows that the topological nature of the theory is also preserved to this order. In addition, we find that this theory possesses an extra supersymmetry when quantized in the Landau gauge. Using dimensional regularization, we then study the Ward identities of the extended BRST symmetry in the three dimensional topological Yang-Mills-Higgs model. (author). 22 refs
2. Gauge and supergauge field theories
International Nuclear Information System (INIS)
Slavnov, A.
1977-01-01
The most actual problems concerning gauge fields are reviwed. Theoretical investigations conducted since as early as 1954 are enclosed. Present status of gauge theories is summarized, including intermediate vector mesons, heavy leptons, weak interactions of hadrons, V-A structure, universal interaction, infrared divergences in perturbation theory, particle-like solutions in gauge theories, spontaneous symmetry breaking. Special emphasis is placed on strong interactions, or more precisely, on the alleged unobservability of ''color'' objects (quark confinement). Problems dealing with the supersymmetric theories invariant under gauge transformations and spontaneous breaking of supersymmetry are also discussed. Gauge theories are concluded to provide self-consistent apparatus for weak and electromagnetic interactions. As to strong interactions such models are still to be discovered
3. Embedded mean-field theory.
Science.gov (United States)
Fornace, Mark E; Lee, Joonho; Miyamoto, Kaito; Manby, Frederick R; Miller, Thomas F
2015-02-10
We introduce embedded mean-field theory (EMFT), an approach that flexibly allows for the embedding of one mean-field theory in another without the need to specify or fix the number of particles in each subsystem. EMFT is simple, is well-defined without recourse to parameters, and inherits the simple gradient theory of the parent mean-field theories. In this paper, we report extensive benchmarking of EMFT for the case where the subsystems are treated using different levels of Kohn-Sham theory, using PBE or B3LYP/6-31G* in the high-level subsystem and LDA/STO-3G in the low-level subsystem; we also investigate different levels of density fitting in the two subsystems. Over a wide range of chemical problems, we find EMFT to perform accurately and stably, smoothly converging to the high-level of theory as the active subsystem becomes larger. In most cases, the performance is at least as good as that of ONIOM, but the advantages of EMFT are highlighted by examples that involve partitions across multiple bonds or through aromatic systems and by examples that involve more complicated electronic structure. EMFT is simple and parameter free, and based on the tests provided here, it offers an appealing new approach to a multiscale electronic structure.
4. Geometry of lattice field theory
International Nuclear Information System (INIS)
Honan, T.J.
1986-01-01
Using some tools of algebraic topology, a general formalism for lattice field theory is presented. The lattice is taken to be a simplicial complex that is also a manifold and is referred to as a simplicial manifold. The fields on this lattice are cochains, that are called lattice forms to emphasize the connections with differential forms in the continuum. This connection provides a new bridge between lattice and continuum field theory. A metric can be put onto this simplicial manifold by assigning lengths to every link or I-simplex of the lattice. Regge calculus is a way of defining general relativity on this lattice. A geometric discussion of Regge calculus is presented. The Regge action, which is a discrete form of the Hilbert action, is derived from the Hilbert action using distribution valued forms. This is a new derivation that emphasizes the underlying geometry. Kramers-Wannier duality in statistical mechanics is discussed in this general setting. Nonlinear field theories, which include gauge theories and nonlinear sigma models are discussed in the continuum and then are put onto a lattice. The main new result here is the generalization to curved spacetime, which consists of making the theory compatible with Regge calculus
5. Bosonic colored group field theory
Energy Technology Data Exchange (ETDEWEB)
Ben Geloun, Joseph [Universite Paris XI, Laboratoire de Physique Theorique, Orsay Cedex (France); University of Abomey-Calavi, Cotonou (BJ). International Chair in Mathematical Physics and Applications (ICMPA-UNESCO Chair); Universite Cheikh Anta Diop, Departement de Mathematiques et Informatique, Faculte des Sciences et Techniques, Dakar (Senegal); Magnen, Jacques [Ecole Polytechnique, Centre de Physique Theorique, Palaiseau Cedex (France); Rivasseau, Vincent [Universite Paris XI, Laboratoire de Physique Theorique, Orsay Cedex (France)
2010-12-15
Bosonic colored group field theory is considered. Focusing first on dimension four, namely the colored Ooguri group field model, the main properties of Feynman graphs are studied. This leads to a theorem on optimal perturbative bounds of Feynman amplitudes in the ''ultraspin'' (large spin) limit. The results are generalized in any dimension. Finally, integrating out two colors we write a new representation, which could be useful for the constructive analysis of this type of models. (orig.)
6. Gravitation and bilocal field theory
International Nuclear Information System (INIS)
Vollendorf, F.
1975-01-01
The starting point is the conjecture that a field theory of elementary particles can be constructed only in a bilocal version. Thus the 4-dimensional space time has to be replaced by the 8-dimensional manifold R 8 of all ordered pairs of space time events. With special reference to the Schwarzschild metric it is shown that the embedding of the time space into the manifold R 8 yields a description of the gravitational field. (orig.) [de
7. Computers for lattice field theories
International Nuclear Information System (INIS)
Iwasaki, Y.
1994-01-01
Parallel computers dedicated to lattice field theories are reviewed with emphasis on the three recent projects, the Teraflops project in the US, the CP-PACS project in Japan and the 0.5-Teraflops project in the US. Some new commercial parallel computers are also discussed. Recent development of semiconductor technologies is briefly surveyed in relation to possible approaches toward Teraflops computers. (orig.)
8. Dimensional analysis in field theory
International Nuclear Information System (INIS)
Stevenson, P.M.
1981-01-01
Dimensional Transmutation (the breakdown of scale invariance in field theories) is reconciled with the commonsense notions of Dimensional Analysis. This makes possible a discussion of the meaning of the Renormalisation Group equations, completely divorced from the technicalities of renormalisation. As illustrations, I describe some very farmiliar QCD results in these terms
9. Informetrics theory, methods and applications
CERN Document Server
Qiu, Junping; Yang, Siluo; Dong, Ke
2017-01-01
This book provides an accessible introduction to the history, theory and techniques of informetrics. Divided into 14 chapters, it develops the content system of informetrics from the theory, methods and applications; systematically analyzes the six basic laws and the theory basis of informetrics and presents quantitative analysis methods such as citation analysis and computer-aided analysis. It also discusses applications in information resource management, information and library science, science of science, scientific evaluation and the forecast field. Lastly, it describes a new development in informetrics- webometrics. Providing a comprehensive overview of the complex issues in today's environment, this book is a valuable resource for all researchers, students and practitioners in library and information science.
10. Neuronal coupling by endogenous electric fields: Cable theory and applications to coincidence detector neurons in the auditory brainstem
OpenAIRE
Goldwyn, Joshua H.; Rinzel, John
2015-01-01
The ongoing activity of neurons generates a spatially- and time-varying field of extracellular voltage ($V_e$). This $V_e$ field reflects population-level neural activity, but does it modulate neural dynamics and the function of neural circuits? We provide a cable theory framework to study how a bundle of model neurons generates $V_e$ and how this $V_e$ feeds back and influences membrane potential ($V_m$). We find that these "ephaptic interactions" are small but not negligible. The model neur...
11. Text Mining Applications and Theory
CERN Document Server
Berry, Michael W
2010-01-01
Text Mining: Applications and Theory presents the state-of-the-art algorithms for text mining from both the academic and industrial perspectives. The contributors span several countries and scientific domains: universities, industrial corporations, and government laboratories, and demonstrate the use of techniques from machine learning, knowledge discovery, natural language processing and information retrieval to design computational models for automated text analysis and mining. This volume demonstrates how advancements in the fields of applied mathematics, computer science, machine learning
12. Hydrodynamics, fields and constants in gravitational theory
International Nuclear Information System (INIS)
Stanyukovich, K.P.; Mel'nikov, V.N.
1983-01-01
Results of original inveatigations into problems of standard gravitation theory and its generalizations are presented. The main attention is paid to the application of methods of continuous media techniques in the gravitation theory; to the specification of the gravitation role in phenomena of macro- and microworld, accurate solutions in the case, when the medium is the matter, assigned by hydrodynamic energy-momentum tensor; and to accurate solutions for the case when the medium is the field. GRT generalizations are analyzed, such as the new cosmologic hypothesis which is based on the gravitation vacuum theory. Investigations are performed into the quantization of cosmological models, effects of spontaneous symmetry violation and particle production in cosmology. Graeity theory with fundamental Higgs field is suggested in the framework of which in the atomic unit number one can explain possible variations of the effective gravitational bonds, and in the gravitation bond, variations of masses of all particles
13. Double field theory: a pedagogical review
International Nuclear Information System (INIS)
Aldazabal, Gerardo; Marqués, Diego; Núñez, Carmen
2013-01-01
Double field theory (DFT) is a proposal to incorporate T-duality, a distinctive symmetry of string theory, as a symmetry of a field theory defined on a double configuration space. The aim of this review is to provide a pedagogical presentation of DFT and its applications. We first introduce some basic ideas on T-duality and supergravity in order to proceed to the construction of generalized diffeomorphisms and an invariant action on the double space. Steps towards the construction of a geometry on the double space are discussed. We then address generalized Scherk–Schwarz compactifications of DFT and their connection to gauged supergravity and flux compactifications. We also discuss U-duality extensions and present a brief parcours on worldsheet approaches to DFT. Finally, we provide a summary of other developments and applications that are not discussed in detail in the review. (topical review)
14. Growing up with field theory
International Nuclear Information System (INIS)
Vajskopf, V.F.
1982-01-01
The article deals with the history of the development of quantum electrodynamics since the date of publishing the work by P.A.M. Dirac ''The Quantum Theory of the Emission and Absorption of Radiation''. Classic ''before-Dirac'' electrodynamics related with the names of Maxwell, Lorenz, Hertz, is outlined. Work of Bohr and Rosenfeld is shown to clarify the physical sense of quantized field and to reveal the existence of uncertainties between the strengths of different fields. The article points to the significance of the article ''Quantum theory of radiation'' by E. Fermi which clearly describes the Dirac theory of radiation, relativistic wave equation and fundamentals of quantum electrodynamics. Shown is work on elimination of troubles related with the existence of states with negative kinetic energy or with negative mass. Hypothesis on the Dirac filled-in vacuum led to understanding of the existence of antiparticles and two unknown till then fundamental processes - pair production and annihilation. Ways of fighting against the infinite quantities in quantum electrodynamics are considered. Renormalization of the theory overcame all the infinities and gave a pattern for calculation of any processes of electron interactions with electromagnetic field to any desired accuracy
15. Cutkosky rules for superstring field theory
International Nuclear Information System (INIS)
Pius, Roji; Sen, Ashoke
2016-01-01
Superstring field theory expresses the perturbative S-matrix of superstring theory as a sum of Feynman diagrams each of which is manifestly free from ultraviolet divergences. The interaction vertices fall off exponentially for large space-like external momenta making the ultraviolet finiteness property manifest, but blow up exponentially for large time-like external momenta making it impossible to take the integration contours for loop energies to lie along the real axis. This forces us to carry out the integrals over the loop energies by choosing appropriate contours in the complex plane whose ends go to infinity along the imaginary axis but which take complicated form in the interior navigating around the various poles of the propagators. We consider the general class of quantum field theories with this property and prove Cutkosky rules for the amplitudes to all orders in perturbation theory. Besides having applications to string field theory, these results also give an alternative derivation of Cutkosky rules in ordinary quantum field theories.
16. Cutkosky rules for superstring field theory
Energy Technology Data Exchange (ETDEWEB)
Pius, Roji [Perimeter Institute for Theoretical Physics,Waterloo, ON N2L 2Y5 (Canada); Sen, Ashoke [Harish-Chandra Research Institute,Chhatnag Road, Jhusi, Allahabad 211019 (India)
2016-10-06
Superstring field theory expresses the perturbative S-matrix of superstring theory as a sum of Feynman diagrams each of which is manifestly free from ultraviolet divergences. The interaction vertices fall off exponentially for large space-like external momenta making the ultraviolet finiteness property manifest, but blow up exponentially for large time-like external momenta making it impossible to take the integration contours for loop energies to lie along the real axis. This forces us to carry out the integrals over the loop energies by choosing appropriate contours in the complex plane whose ends go to infinity along the imaginary axis but which take complicated form in the interior navigating around the various poles of the propagators. We consider the general class of quantum field theories with this property and prove Cutkosky rules for the amplitudes to all orders in perturbation theory. Besides having applications to string field theory, these results also give an alternative derivation of Cutkosky rules in ordinary quantum field theories.
17. Effective field theory for triaxially deformed nuclei
Energy Technology Data Exchange (ETDEWEB)
Chen, Q.B. [Technische Universitaet Muechen, Physik-Department, Garching (Germany); Peking University, State Key Laboratory of Nuclear Physics and Technology, School of Physics, Beijing (China); Kaiser, N. [Technische Universitaet Muechen, Physik-Department, Garching (Germany); Meissner, Ulf G. [Universitaet Bonn, Helmholtz-Institut fuer Strahlen- und Kernphysik and Bethe Center for Theoretical Physics, Bonn (Germany); Institute for Advanced Simulation, Institut fuer Kernphysik, Juelich Center for Hadron Physics and JARA-HPC, Forschungszentrum Juelich, Juelich (Germany); Meng, J. [Peking University, State Key Laboratory of Nuclear Physics and Technology, School of Physics, Beijing (China); Beihang University, School of Physics and Nuclear Energy Engineering, Beijing (China); University of Stellenbosch, Department of Physics, Stellenbosch (South Africa)
2017-10-15
Effective field theory is generalized to investigate the rotational motion of triaxially deformed even-even nuclei. The Hamiltonian for the triaxial rotor is obtained up to next-to-leading order within the effective field theory formalism. Its applicability is examined by comparing with a five-dimensional rotor-vibrator Hamiltonian for the description of the energy spectra of the ground state and γ band in Ru isotopes. It is found that by taking into account the next-to-leading order corrections, the ground state band in the whole spin region and the γ band in the low spin region are well described. The deviations for high-spin states in the γ bands point towards the importance of including vibrational degrees of freedom in the effective field theory formulation. (orig.)
18. Exceptional field theory: SL(5)
International Nuclear Information System (INIS)
Musaev, Edvard T.
2016-01-01
In this work the exceptional field theory formulation of supergravity with SL(5) gauge group is considered. This group appears as a U-duality group of D=7 maximal supergravity. In the formalism presented the hidden global duality group is promoted into a gauge group of a theory in dimensions 7+number of extended directions. This work is a continuation of the series of works for E 8,7,6 ,SO(5,5) and SL(3)×SL(2) duality groups.
19. Perturbative coherence in field theory
International Nuclear Information System (INIS)
Aldrovandi, R.; Kraenkel, R.A.
1987-01-01
A general condition for coherent quantization by perturbative methods is given, because the basic field equations of a fild theory are not always derivable from a Lagrangian. It's seen that non-lagrangian models way have well defined vertices, provided they satisfy what they call the 'coherence condition', which is less stringent than the condition for the existence of a Lagrangian. They note that Lagrangian theories are perturbatively coherent, in the sense that they have well defined vertices, and that they satisfy automatically that condition. (G.D.F.) [pt
20. Einstein's theory of unified fields
CERN Document Server
Tonnelat, Marie Antoinette
2014-01-01
First published in1966, here is presented a comprehensive overview of one of the most elusive scientific speculations by the pre-eminent genius of the 20th century. The theory is viewed by some scientists with deep suspicion, by others with optimism, but all agree that it represents an extreme challenge. As the author herself affirms, this work is not intended to be a complete treatise or 'didactic exposition' of the theory of unified fields, but rather a tool for further study, both by students and professional physicists. Dealing with all the major areas of research whic
1. A course in field theory
CERN Document Server
Baal, Pierre Van
2014-01-01
""… a pleasant novelty that manages the impossible: a full course in field theory from a derivation of the Dirac equation to the standard electroweak theory in less than 200 pages. Moreover, the final chapter consists of a careful selection of assorted problems, which are original and either anticipate or detail some of the topics discussed in the bulk of the chapters. Instead of building a treatise out of a collection of lecture notes, the author took the complementary approach and constructed a course out of a number of well-known and classic treatises. The result is fresh and useful. … the
2. Introduction to quantum field theory
CERN Document Server
Chang, Shau-Jin
1990-01-01
This book presents in a short volume the basics of quantum field theory and many body physics. The first part introduces the perturbative techniques without sophisticated apparatus and applies them to numerous problems including quantum electrodynamics (renormalization), Fermi and Bose gases, the Brueckner theory of nuclear system, liquid Helium and classical systems with noise. The material is clear, illustrative and the important points are stressed to help the reader get the understanding of what is crucial without overwhelming him with unnecessary detours or comments. The material in the s
3. Multifractals theory and applications
CERN Document Server
Harte, David
2001-01-01
Although multifractals are rooted in probability, much of the related literature comes from the physics and mathematics arena. Multifractals: Theory and Applications pulls together ideas from both these areas using a language that makes them accessible and useful to statistical scientists. It provides a framework, in particular, for the evaluation of statistical properties of estimates of the Renyi fractal dimensions.The first section provides introductory material and different definitions of a multifractal measure. The author then examines some of the various constructions for describing multifractal measures. Building from the theory of large deviations, he focuses on constructions based on lattice coverings, covering by point-centered spheres, and cascades processes. The final section presents estimators of Renyi dimensions of integer order two and greater and discusses their properties. It also explores various applications of dimension estimation and provides a detailed case study of spatial point patte...
4. The Global Approach to Quantum Field Theory
International Nuclear Information System (INIS)
Folacci, Antoine; Jensen, Bruce
2003-01-01
is rather difficult to read because of its great breadth. From the start he is faithful to his own view of field theory by developing a powerful formalism which permits him to discuss broad general features common to all field theories. He demands a considerable effort from the reader to penetrate his formalism, and a reading of Appendix A which presents the basics of super-analysis is a prerequisite. To keep the reader on course, DeWitt offers a series of exercises on applications of global formalism in Part 8, nearly 200 pages worth. The exercises are to be worked in parallel with reading the text, starting from the beginning. Before concluding, some criticisms. DeWitt has anticipated some criticism himself in the Preface, where he warns the reader that 'this book is in no sense a reference book on quantum field theory and its application to particle physics. The selection of topics is idiosyncratic. But the reviewers should add a few more remarks: (1) There are very few references. Of course, this is because the work is largely original. Even where the work of other researchers is presented, it has mostly been transformed by the DeWittian point of view. (2) There are very few diagrams, which sometimes hinders the exposition. In summary, in our opinion, this is one of the best books dealing with quantum field theory existing today. It will be of great interest for graduate and postgraduate students as well as workers in the domains of quantum field theory in flat and in curved spacetime and string theories. But we believe that the reader must have previously studied standard textbooks on quantum field theory and general relativity. Even with this preparation, it is by no means an easy book to read. However, the reward is to be able to share the deep and unique vision of the quantum theory of fields and its formalism by one of its greatest expositors. (book review)
5. Plasmonics theory and applications
CERN Document Server
Shahbazyan, Tigran V
2014-01-01
This contributed volume summarizes recent theoretical developments in plasmonics and its applications in physics, chemistry, materials science, engineering, and medicine. It focuses on recent advances in several major areas of plasmonics including plasmon-enhanced spectroscopies, light scattering, many-body effects, nonlinear optics, and ultrafast dynamics. The theoretical and computational methods used in these investigations include electromagnetic calculations, density functional theory calculations, and nonequilibrium electron dynamics calculations. The book presents a comprehensive overview of these methods as well as their applications to various current problems of interest.
6. Graphs Theory and Applications
CERN Document Server
Fournier, Jean-Claude
2008-01-01
This book provides a pedagogical and comprehensive introduction to graph theory and its applications. It contains all the standard basic material and develops significant topics and applications, such as: colorings and the timetabling problem, matchings and the optimal assignment problem, and Hamiltonian cycles and the traveling salesman problem, to name but a few. Exercises at various levels are given at the end of each chapter, and a final chapter presents a few general problems with hints for solutions, thus providing the reader with the opportunity to test and refine their knowledge on the
7. Weak gravity conjecture and effective field theory
Science.gov (United States)
Saraswat, Prashant
2017-01-01
The weak gravity conjecture (WGC) is a proposed constraint on theories with gauge fields and gravity, requiring the existence of light charged particles and/or imposing an upper bound on the field theory cutoff Λ . If taken as a consistency requirement for effective field theories (EFTs), it rules out possibilities for model building including some models of inflation. I demonstrate simple models which satisfy all forms of the WGC, but which through Higgsing of the original gauge fields produce low-energy EFTs with gauge forces that badly violate the WGC. These models illustrate specific loopholes in arguments that motivate the WGC from a bottom-up perspective; for example the arguments based on magnetic monopoles are evaded when the magnetic confinement that occurs in a Higgs phase is accounted for. This indicates that the WGC should not be taken as a veto on EFTs, even if it turns out to be a robust property of UV quantum gravity theories. However, if the latter is true, then parametric violation of the WGC at low energy comes at the cost of nonminimal field content in the UV. I propose that only a very weak constraint is applicable to EFTs, Λ ≲(log 1/g )-1 /2Mpl , where g is the gauge coupling, motivated by entropy bounds. Remarkably, EFTs produced by Higgsing a theory that satisfies the WGC can saturate but not violate this bound.
8. Neuronal coupling by endogenous electric fields: cable theory and applications to coincidence detector neurons in the auditory brain stem.
Science.gov (United States)
Goldwyn, Joshua H; Rinzel, John
2016-04-01
The ongoing activity of neurons generates a spatially and time-varying field of extracellular voltage (Ve). This Ve field reflects population-level neural activity, but does it modulate neural dynamics and the function of neural circuits? We provide a cable theory framework to study how a bundle of model neurons generates Ve and how this Ve feeds back and influences membrane potential (Vm). We find that these "ephaptic interactions" are small but not negligible. The model neural population can generate Ve with millivolt-scale amplitude, and this Ve perturbs the Vm of "nearby" cables and effectively increases their electrotonic length. After using passive cable theory to systematically study ephaptic coupling, we explore a test case: the medial superior olive (MSO) in the auditory brain stem. The MSO is a possible locus of ephaptic interactions: sounds evoke large (millivolt scale)Vein vivo in this nucleus. The Ve response is thought to be generated by MSO neurons that perform a known neuronal computation with submillisecond temporal precision (coincidence detection to encode sound source location). Using a biophysically based model of MSO neurons, we find millivolt-scale ephaptic interactions consistent with the passive cable theory results. These subtle membrane potential perturbations induce changes in spike initiation threshold, spike time synchrony, and time difference sensitivity. These results suggest that ephaptic coupling may influence MSO function. Copyright © 2016 the American Physiological Society.
9. Euler-Poincare reduction for discrete field theories
International Nuclear Information System (INIS)
Vankerschaver, Joris
2007-01-01
In this note, we develop a theory of Euler-Poincare reduction for discrete Lagrangian field theories. We introduce the concept of Euler-Poincare equations for discrete field theories, as well as a natural extension of the Moser-Veselov scheme, and show that both are equivalent. The resulting discrete field equations are interpreted in terms of discrete differential geometry. An application to the theory of discrete harmonic mappings is also briefly discussed
10. Variational methods for field theories
International Nuclear Information System (INIS)
Ben-Menahem, S.
1986-09-01
Four field theory models are studied: Periodic Quantum Electrodynamics (PQED) in (2 + 1) dimensions, free scalar field theory in (1 + 1) dimensions, the Quantum XY model in (1 + 1) dimensions, and the (1 + 1) dimensional Ising model in a transverse magnetic field. The last three parts deal exclusively with variational methods; the PQED part involves mainly the path-integral approach. The PQED calculation results in a better understanding of the connection between electric confinement through monopole screening, and confinement through tunneling between degenerate vacua. This includes a better quantitative agreement for the string tensions in the two approaches. Free field theory is used as a laboratory for a new variational blocking-truncation approximation, in which the high-frequency modes in a block are truncated to wave functions that depend on the slower background modes (Boron-Oppenheimer approximation). This ''adiabatic truncation'' method gives very accurate results for ground-state energy density and correlation functions. Various adiabatic schemes, with one variable kept per site and then two variables per site, are used. For the XY model, several trial wave functions for the ground state are explored, with an emphasis on the periodic Gaussian. A connection is established with the vortex Coulomb gas of the Euclidean path integral approach. The approximations used are taken from the realms of statistical mechanics (mean field approximation, transfer-matrix methods) and of quantum mechanics (iterative blocking schemes). In developing blocking schemes based on continuous variables, problems due to the periodicity of the model were solved. Our results exhibit an order-disorder phase transition. The transfer-matrix method is used to find a good (non-blocking) trial ground state for the Ising model in a transverse magnetic field in (1 + 1) dimensions
11. Variational methods for field theories
Energy Technology Data Exchange (ETDEWEB)
Ben-Menahem, S.
1986-09-01
Four field theory models are studied: Periodic Quantum Electrodynamics (PQED) in (2 + 1) dimensions, free scalar field theory in (1 + 1) dimensions, the Quantum XY model in (1 + 1) dimensions, and the (1 + 1) dimensional Ising model in a transverse magnetic field. The last three parts deal exclusively with variational methods; the PQED part involves mainly the path-integral approach. The PQED calculation results in a better understanding of the connection between electric confinement through monopole screening, and confinement through tunneling between degenerate vacua. This includes a better quantitative agreement for the string tensions in the two approaches. Free field theory is used as a laboratory for a new variational blocking-truncation approximation, in which the high-frequency modes in a block are truncated to wave functions that depend on the slower background modes (Boron-Oppenheimer approximation). This ''adiabatic truncation'' method gives very accurate results for ground-state energy density and correlation functions. Various adiabatic schemes, with one variable kept per site and then two variables per site, are used. For the XY model, several trial wave functions for the ground state are explored, with an emphasis on the periodic Gaussian. A connection is established with the vortex Coulomb gas of the Euclidean path integral approach. The approximations used are taken from the realms of statistical mechanics (mean field approximation, transfer-matrix methods) and of quantum mechanics (iterative blocking schemes). In developing blocking schemes based on continuous variables, problems due to the periodicity of the model were solved. Our results exhibit an order-disorder phase transition. The transfer-matrix method is used to find a good (non-blocking) trial ground state for the Ising model in a transverse magnetic field in (1 + 1) dimensions.
12. Types of two-dimensional N = 4 superconformal field theories
Superconformal field theory; free field realization; string theory; AdS-CFT correspon- dence. PACS Nos 11.25.Hf; 11.25.-w; 11.30.Ly; 11.30.Pb. Conformal symmetries in two space-time dimensions have been very extensively studied owing to their applications both in string theory and two-dimensional statistical systems.
13. Theory of field reversed configurations
International Nuclear Information System (INIS)
Steinhauer, L.C.
1990-01-01
This final report surveys the results of work conducted on the theory of field reversed configurations. This project has spanned ten years, beginning in early 1980. During this period, Spectra Technology was one of the leading contributors to the advances in understanding FRC. The report is organized into technical topic areas, FRC formation, equilibrium, stability, and transport. Included as an appendix are papers published in archival journals that were generated in the course of this report. 33 refs
14. Renormalization and Interaction in Quantum Field Theory
International Nuclear Information System (INIS)
RATSIMBARISON, H.M.
2008-01-01
This thesis works on renormalization in quantum field theory (QFT), in order to show the relevance of some mathematical structures as C*-algebraic and probabilistic structures. Our work begins with a study of the path integral formalism and the Kreimer-Connes approach in perturbative renormalization, which allows to situate the statistical nature of QFT and to appreciate the ultra-violet divergence problem of its partition function. This study is followed by an emphasis of the presence of convolution products in non perturbative renormalisation, through the construction of the Wilson effective action and the Legendre effective action. Thanks to these constructions and the definition of effective theories according J. Polchinski, the non perturbative renormalization shows in particular the general approach of regularization procedure. We begin the following chapter with a C*-algebraic approach of the scale dependence of physical theories by showing the existence of a hierarchy of commutative spaces of states and its compatibility with the fiber bundle formulation of classical field theory. Our Hierarchy also allows us to modelize the notion of states and particles. Finally, we develop a probabilistic construction of interacting theories starting from simple model, a Bernoulli random processes. We end with some arguments on the applicability of our construction -such as the independence between the free and interacting terms and the possibility to introduce a symmetry group wich will select the type of interactions in quantum field theory. [fr
15. Dynamic random walks theory and applications
CERN Document Server
2006-01-01
The aim of this book is to report on the progress realized in probability theory in the field of dynamic random walks and to present applications in computer science, mathematical physics and finance. Each chapter contains didactical material as well as more advanced technical sections. Few appendices will help refreshing memories (if necessary!).· New probabilistic model, new results in probability theory· Original applications in computer science· Applications in mathematical physics· Applications in finance
16. Electricity markets theories and applications
CERN Document Server
Lin, Jeremy
2017-01-01
Electricity Markets: Theories and Applications offers students and practitioners a clear understanding of the fundamental concepts of the economic theories, particularly microeconomic theories, as well as information on some advanced optimization methods of electricity markets. The authors--noted experts in the field--cover the basic drivers for the transformation of the electricity industry in both the United States and around the world and discuss the fundamentals of power system operation, electricity market design and structures, and electricity market operations. The text also explores advanced topics of power system operations and electricity market design and structure including zonal versus nodal pricing, market performance and market power issues, transmission pricing, and the emerging problems electricity markets face in smart grid and micro-grid environments. The authors also examine system planning under the context of electricity market regime. They explain the new ways to solve problems with t...
17. On incompleteness of classical field theory
OpenAIRE
Sardanashvily, G.
2009-01-01
Classical field theory is adequately formulated as Lagrangian theory on fibre bundles and graded manifolds. One however observes that non-trivial higher stage Noether identities and gauge symmetries of a generic reducible degenerate Lagrangian field theory fail to be defined. Therefore, such a field theory can not be quantized.
18. Holography for field theory solitons
Science.gov (United States)
Domokos, Sophia K.; Royston, Andrew B.
2017-07-01
We extend a well-known D-brane construction of the AdS/dCFT correspondence to non-abelian defects. We focus on the bulk side of the correspondence and show that there exists a regime of parameters in which the low-energy description consists of two approximately decoupled sectors. The two sectors are gravity in the ambient spacetime, and a six-dimensional supersymmetric Yang-Mills theory. The Yang-Mills theory is defined on a rigid AdS4 × S 2 background and admits sixteen supersymmetries. We also consider a one-parameter deformation that gives rise to a family of Yang-Mills theories on asymptotically AdS4 × S 2 spacetimes, which are invariant under eight supersymmetries. With future holographic applications in mind, we analyze the vacuum structure and perturbative spectrum of the Yang-Mills theory on AdS4 × S 2, as well as systems of BPS equations for finite-energy solitons. Finally, we demonstrate that the classical Yang-Mills theory has a consistent truncation on the two-sphere, resulting in maximally supersymmetric Yang-Mills on AdS4.
19. Generalized filtering of laser fields in optimal control theory: application to symmetry filtering of quantum gate operations
International Nuclear Information System (INIS)
Schroeder, Markus; Brown, Alex
2009-01-01
We present a modified version of a previously published algorithm (Gollub et al 2008 Phys. Rev. Lett.101 073002) for obtaining an optimized laser field with more general restrictions on the search space of the optimal field. The modification leads to enforcement of the constraints on the optimal field while maintaining good convergence behaviour in most cases. We demonstrate the general applicability of the algorithm by imposing constraints on the temporal symmetry of the optimal fields. The temporal symmetry is used to reduce the number of transitions that have to be optimized for quantum gate operations that involve inversion (NOT gate) or partial inversion (Hadamard gate) of the qubits in a three-dimensional model of ammonia.
20. Braided quantum field theories and their symmetries
International Nuclear Information System (INIS)
Sasai, Yuya; Sasakura, Naoki
2007-01-01
Braided quantum field theories, proposed by Oeckl, can provide a framework for quantum field theories that possess Hopf algebra symmetries. In quantum field theories, symmetries lead to non-perturbative relations among correlation functions. We study Hopf algebra symmetries and such relations in the context of braided quantum field theories. We give the four algebraic conditions among Hopf algebra symmetries and braided quantum field theories that are required for the relations to hold. As concrete examples, we apply our analysis to the Poincare symmetries of two examples of noncommutative field theories. One is the effective quantum field theory of three-dimensional quantum gravity coupled to spinless particles formulated by Freidel and Livine, and the other is noncommutative field theory on the Moyal plane. We also comment on quantum field theory in κ-Minkowski spacetime. (author)
1. Causality Constraints in Conformal Field Theory
CERN Multimedia
CERN. Geneva
2015-01-01
Causality places nontrivial constraints on QFT in Lorentzian signature, for example fixing the signs of certain terms in the low energy Lagrangian. In d-dimensional conformal field theory, we show how such constraints are encoded in crossing symmetry of Euclidean correlators, and derive analogous constraints directly from the conformal bootstrap (analytically). The bootstrap setup is a Lorentzian four-point function corresponding to propagation through a shockwave. Crossing symmetry fixes the signs of certain log terms that appear in the conformal block expansion, which constrains the interactions of low-lying operators. As an application, we use the bootstrap to rederive the well known sign constraint on the (∂φ)4 coupling in effective field theory, from a dual CFT. We also find constraints on theories with higher spin conserved currents. Our analysis is restricted to scalar correlators, but we argue that similar methods should also impose nontrivial constraints on the interactions of spinni...
2. Classification of networks of automata by dynamical mean field theory
International Nuclear Information System (INIS)
Burda, Z.; Jurkiewicz, J.; Flyvbjerg, H.
1990-01-01
Dynamical mean field theory is used to classify the 2 24 =65,536 different networks of binary automata on a square lattice with nearest neighbour interactions. Application of mean field theory gives 700 different mean field classes, which fall in seven classes of different asymptotic dynamics characterized by fixed points and two-cycles. (orig.)
3. Progress on The GEMS (Gravity Electro-Magnetism-Strong) Theory of Field Unification and Its Application to Space Problems
International Nuclear Information System (INIS)
Brandenburg, J. E.
2008-01-01
Progress on the GEMS (Gravity Electro-Magnetism-Strong), theory is presented as well as its application to space problems. The GEMS theory is now validated through the Standard Model of physics. Derivation of the value of the Gravitation constant based on the observed variation of α with energy: results in the formula G congruent with (ℎ/2π)c/M ηc 2 exp(-1/(1.61α)), where α is the fine structure constant,(ℎ/2π), is Planck's constant, c, is the speed of light, and M ηc is the mass of the η cc Charmonium meson that is shown to be identical to that derived from the GEM postulates. Covariant formulation of the GEM theory is now possible through definition of the spacetime metric tensor as a portion of the EM stress tensor normalized by its own trace: g ab = 4(F c a F cb )/(F ab F ab ), it is found that this results in a massless ground state vacuum and a Newtonian gravitation potential φ = 1/2 E 2 /B 2 . It is also found that a Lorentz or flat-space metric is recovered in the limit of a full spectrum ZPF
4. Large-signal model of the bilayer graphene field-effect transistor targeting radio-frequency applications: Theory versus experiment
Energy Technology Data Exchange (ETDEWEB)
Pasadas, Francisco, E-mail: [email protected]; Jiménez, David [Departament d' Enginyeria Electrònica, Escola d' Enginyeria, Universitat Autònoma de Barcelona, 08193 Bellaterra (Spain)
2015-12-28
Bilayer graphene is a promising material for radio-frequency transistors because its energy gap might result in a better current saturation than the monolayer graphene. Because the great deal of interest in this technology, especially for flexible radio-frequency applications, gaining control of it requires the formulation of appropriate models for the drain current, charge, and capacitance. In this work, we have developed them for a dual-gated bilayer graphene field-effect transistor. A drift-diffusion mechanism for the carrier transport has been considered coupled with an appropriate field-effect model taking into account the electronic properties of the bilayer graphene. Extrinsic resistances have been included considering the formation of a Schottky barrier at the metal-bilayer graphene interface. The proposed model has been benchmarked against experimental prototype transistors, discussing the main figures of merit targeting radio-frequency applications.
5. Group theory and lattice gauge fields
International Nuclear Information System (INIS)
Creutz, M.
1988-09-01
Lattice gauge theory, formulated in terms of invariant integrals over group elements on lattice bonds, benefits from many group theoretical notions. Gauge invariance provides an enormous symmetry and powerful constraints on expectation values. Strong coupling expansions require invariant integrals over polynomials in group elements, all of which can be evaluated by symmetry considerations. Numerical simulations involve random walks over the group. These walks automatically generate the invariant group measure, avoiding explicit parameterization. A recently proposed overrelaxation algorithm is particularly efficient at exploring the group manifold. These and other applications of group theory to lattice gauge fields are reviewed in this talk. 17 refs
6. Inverse bootstrapping conformal field theories
Science.gov (United States)
Li, Wenliang
2018-01-01
We propose a novel approach to study conformal field theories (CFTs) in general dimensions. In the conformal bootstrap program, one usually searches for consistent CFT data that satisfy crossing symmetry. In the new method, we reverse the logic and interpret manifestly crossing-symmetric functions as generating functions of conformal data. Physical CFTs can be obtained by scanning the space of crossing-symmetric functions. By truncating the fusion rules, we are able to concentrate on the low-lying operators and derive some approximate relations for their conformal data. It turns out that the free scalar theory, the 2d minimal model CFTs, the ϕ 4 Wilson-Fisher CFT, the Lee-Yang CFTs and the Ising CFTs are consistent with the universal relations from the minimal fusion rule ϕ 1 × ϕ 1 = I + ϕ 2 + T , where ϕ 1 , ϕ 2 are scalar operators, I is the identity operator and T is the stress tensor.
7. About Applications of the Fixed Point Theory
Directory of Open Access Journals (Sweden)
Bucur Amelia
2017-06-01
Full Text Available The fixed point theory is essential to various theoretical and applied fields, such as variational and linear inequalities, the approximation theory, nonlinear analysis, integral and differential equations and inclusions, the dynamic systems theory, mathematics of fractals, mathematical economics (game theory, equilibrium problems, and optimisation problems and mathematical modelling. This paper presents a few benchmarks regarding the applications of the fixed point theory. This paper also debates if the results of the fixed point theory can be applied to the mathematical modelling of quality.
8. Regularization of quantum field theories
International Nuclear Information System (INIS)
Rayski, J.
1985-01-01
General idea of regularization and renormalization in quantum field theory is presented. It is postulated that it is possible not to go to infinity with the auxiliary masses of regularization but to attach to them a certain physical meaning, but it is equivalent with a violation of unitarity of the operator of evolution in time. It may be achieved in two different ways: it might be simply assumed that only the direction but not the length of the state vector possesses a physical meaning and that not all possible physical events are predictable. 3 refs., 1 fig. (author)
9. Quantum field theory of point particles and strings
CERN Document Server
Hatfield, Brian
1992-01-01
The purpose of this book is to introduce string theory without assuming any background in quantum field theory. Part I of this book follows the development of quantum field theory for point particles, while Part II introduces strings. All of the tools and concepts that are needed to quantize strings are developed first for point particles. Thus, Part I presents the main framework of quantum field theory and provides for a coherent development of the generalization and application of quantum field theory for point particles to strings.Part II emphasizes the quantization of the bosonic string.
10. The Global Approach to Quantum Field Theory
Energy Technology Data Exchange (ETDEWEB)
Folacci, Antoine; Jensen, Bruce [Faculte des Sciences, Universite de Corse (France); Department of Mathematics, University of Southampton (United Kingdom)
2003-12-12
be noted that DeWitt's book is rather difficult to read because of its great breadth. From the start he is faithful to his own view of field theory by developing a powerful formalism which permits him to discuss broad general features common to all field theories. He demands a considerable effort from the reader to penetrate his formalism, and a reading of Appendix A which presents the basics of super-analysis is a prerequisite. To keep the reader on course, DeWitt offers a series of exercises on applications of global formalism in Part 8, nearly 200 pages worth. The exercises are to be worked in parallel with reading the text, starting from the beginning. Before concluding, some criticisms. DeWitt has anticipated some criticism himself in the Preface, where he warns the reader that 'this book is in no sense a reference book on quantum field theory and its application to particle physics. The selection of topics is idiosyncratic. (book review)[abstract truncated
11. Twisted conformal field theories and Morita equivalence
Energy Technology Data Exchange (ETDEWEB)
Marotta, Vincenzo [Dipartimento di Scienze Fisiche, Universita di Napoli ' Federico II' and INFN, Sezione di Napoli, Compl. universitario M. Sant' Angelo, Via Cinthia, 80126 Napoli (Italy); Naddeo, Adele [CNISM, Unita di Ricerca di Salerno and Dipartimento di Fisica ' E.R. Caianiello' , Universita degli Studi di Salerno, Via Salvador Allende, 84081 Baronissi (Italy); Dipartimento di Scienze Fisiche, Universita di Napoli ' Federico II' , Compl. universitario M. Sant' Angelo, Via Cinthia, 80126 Napoli (Italy)], E-mail: [email protected]
2009-04-01
The Morita equivalence for field theories on noncommutative two-tori is analysed in detail for rational values of the noncommutativity parameter {theta} (in appropriate units): an isomorphism is established between an Abelian noncommutative field theory (NCFT) and a non-Abelian theory of twisted fields on ordinary space. We focus on a particular conformal field theory (CFT), the one obtained by means of the m-reduction procedure [V. Marotta, J. Phys. A 26 (1993) 3481; V. Marotta, Mod. Phys. Lett. A 13 (1998) 853; V. Marotta, Nucl. Phys. B 527 (1998) 717; V. Marotta, A. Sciarrino, Mod. Phys. Lett. A 13 (1998) 2863], and show that it is the Morita equivalent of a NCFT. Finally, the whole m-reduction procedure is shown to be the image in the ordinary space of the Morita duality. An application to the physics of a quantum Hall fluid at Jain fillings {nu}=m/(2pm+1) is explicitly discussed in order to further elucidate such a correspondence and to clarify its role in the physics of strongly correlated systems. A new picture emerges, which is very different from the existing relationships between noncommutativity and many body systems [A.P. Polychronakos, arXiv: 0706.1095].
12. Topics in low-dimensional field theory
Energy Technology Data Exchange (ETDEWEB)
Crescimanno, M.J.
1991-04-30
Conformal field theory is a natural tool for understanding two- dimensional critical systems. This work presents results in the lagrangian approach to conformal field theory. The first sections are chiefly about a particular class of field theories called coset constructions and the last part is an exposition of the connection between two-dimensional conformal theory and a three-dimensional gauge theory whose lagrangian is the Chern-Simons density.
13. Number theory arising from finite fields analytic and probabilistic theory
CERN Document Server
Knopfmacher, John
2001-01-01
""Number Theory Arising from Finite Fields: Analytic and Probabilistic Theory"" offers a discussion of the advances and developments in the field of number theory arising from finite fields. It emphasizes mean-value theorems of multiplicative functions, the theory of additive formulations, and the normal distribution of values from additive functions. The work explores calculations from classical stages to emerging discoveries in alternative abstract prime number theorems.
14. Perturbative study in quantum field theory at finite temperature, application to lepton pair production from a quark-gluon plasma
International Nuclear Information System (INIS)
Altherr, T.
1989-12-01
The main topic of this thesis is a perturbative study of Quantum Field Theory at Finite Temperature. The real-time formalism is used throughout this work. We show the cancellation of infrared and mass singularities in the case of the first order QCD corrections to lepton pair production from a quark-gluon plasma. Two methods of calculation are presented and give the same finite result in the limit of vanishing quark mass. These finite terms are analysed and give small corrections in the region of interest for ultra-relativistic heavy ions collisions, except for a threshold factor. Specific techniques for finite temperature calculations are explicited in the case of the fermionic self-energy in QED [fr
15. Vertex operator algebras and conformal field theory
International Nuclear Information System (INIS)
Huang, Y.Z.
1992-01-01
This paper discusses conformal field theory, an important physical theory, describing both two-dimensional critical phenomena in condensed matter physics and classical motions of strings in string theory. The study of conformal field theory will deepen the understanding of these theories and will help to understand string theory conceptually. Besides its importance in physics, the beautiful and rich mathematical structure of conformal field theory has interested many mathematicians. New relations between different branches of mathematics, such as representations of infinite-dimensional Lie algebras and Lie groups, Riemann surfaces and algebraic curves, the Monster sporadic group, modular functions and modular forms, elliptic genera and elliptic cohomology, Calabi-Yau manifolds, tensor categories, and knot theory, are revealed in the study of conformal field theory. It is therefore believed that the study of the mathematics involved in conformal field theory will ultimately lead to new mathematical structures which would be important to both mathematics and physics
16. On quantum field theory in gravitational background
International Nuclear Information System (INIS)
Haag, R.; Narnhofer, H.; Stein, U.
1984-02-01
We discuss Quantum Fields on Riemannian space-time. A principle of local definitness is introduced which is needed beyond equations of motion and commutation relations to fix the theory uniquely. It also allows to formulate local stability. In application to a region with a time-like Killing vector field and horizons it yields the value of the Hawking temperature. The concept of vacuum and particles in a non stationary metric is treated in the example of the Robertson-Walker metric and some remarks on detectors in non inertial motion are added. (orig.)
17. Light front field theory: an advanced primer
International Nuclear Information System (INIS)
Martinovic, L.
2007-01-01
We present an elementary introduction to quantum field theory formulated in terms of Dirac's light front variables. In addition to general principles and methods, a few more specific topics and approaches based on the author's work will be discussed. Most of the discussion deals with massive two-dimensional models formulated in a finite spatial volume starting with a detailed comparison between quantization of massive free fields in the usual field theory and the light front (LF) quantization. We discuss basic properties such as relativistic invariance and causality. After the LF treatment of the soluble Federbush model, a LF approach to spontaneous symmetry breaking is explained and a simple gauge theory - the massive Schwinger model in various gauges is studied. A LF version of bosonization and the massive Thirring model are also discussed. A special chapter is devoted to the method of discretized light cone quantization and its application to calculations of the properties of quantum solitons. The problem of LF zero modes is illustrated with the example of the two/dimensional Yukawa model. Hamiltonian perturbation theory in the LF formulation is derived and applied to a few simple processes to demonstrate its advantages. As a byproduct, it is shown that the LF theory cannot be obtained as a 'light-like' limit of the usual field theory quantized on a initial space-like surface. A simple LF formulation of the Higgs mechanism is then given Since our intention was to provide a treatment of the light front quantization accessible to postgradual students, an effort was made to discuss most of the topics pedagogically and number of technical details and derivations are contained in the appendices (Author)
18. Quantum Field Theory in (0 + 1) Dimensions
Science.gov (United States)
Boozer, A. D.
2007-01-01
We show that many of the key ideas of quantum field theory can be illustrated simply and straightforwardly by using toy models in (0 + 1) dimensions. Because quantum field theory in (0 + 1) dimensions is equivalent to quantum mechanics, these models allow us to use techniques from quantum mechanics to gain insight into quantum field theory. In…
19. Application of the nuclear field theory to monopole interactions which include all the vertices of a general force
International Nuclear Information System (INIS)
Bes, D.R.; Dussel, G.G.; Liotta, R.J.; Sofia, H.M.; Broglia, R.A.
1976-01-01
The field treatment is applied to the monopole pairing and monopole particle-hole interactions in a two-level model. All the vertices of realistic interactions appear, and the problems treated here have most of the complexities of real nuclei. Yet, the model remains sufficiently simple, so that a close comparison with the results of a (conventional) treatment in which only the fermion degrees of freedom are considered is possible. The applicability to actual physical situations appears to be feasible, both for schematic or realistic forces. The advantage of including the exchange components of the interaction in the construction of the phonon is discussed. (Auth.)
20. Gaussian processes and constructive scalar field theory
International Nuclear Information System (INIS)
Benfatto, G.; Nicolo, F.
1981-01-01
The last years have seen a very deep progress of constructive euclidean field theory, with many implications in the area of the random fields theory. The authors discuss an approach to super-renormalizable scalar field theories, which puts in particular evidence the connections with the theory of the Gaussian processes associated to the elliptic operators. The paper consists of two parts. Part I treats some problems in the theory of Gaussian processes which arise in the approach to the PHI 3 4 theory. Part II is devoted to the discussion of the ultraviolet stability in the PHI 3 4 theory. (Auth.)
1. Effective Field Theory on Manifolds with Boundary
Science.gov (United States)
Albert, Benjamin I.
In the monograph Renormalization and Effective Field Theory, Costello made two major advances in rigorous quantum field theory. Firstly, he gave an inductive position space renormalization procedure for constructing an effective field theory that is based on heat kernel regularization of the propagator. Secondly, he gave a rigorous formulation of quantum gauge theory within effective field theory that makes use of the BV formalism. In this work, we extend Costello's renormalization procedure to a class of manifolds with boundary and make preliminary steps towards extending his formulation of gauge theory to manifolds with boundary. In addition, we reorganize the presentation of the preexisting material, filling in details and strengthening the results.
2. Logarithmic conformal field theory: beyond an introduction
Science.gov (United States)
Creutzig, Thomas; Ridout, David
2013-12-01
of the underlying chiral algebra and the modular data pertaining to the characters of the representations. Each of the archetypal logarithmic conformal field theories is studied here by first determining its irreducible spectrum, which turns out to be continuous, as well as a selection of natural reducible, but indecomposable, modules. This is followed by a detailed description of how to obtain character formulae for each irreducible, a derivation of the action of the modular group on the characters, and an application of the Verlinde formula to compute the Grothendieck fusion rules. In each case, the (genuine) fusion rules are known, so comparisons can be made and favourable conclusions drawn. In addition, each example admits an infinite set of simple currents, hence extended symmetry algebras may be constructed and a series of bulk modular invariants computed. The spectrum of such an extended theory is typically discrete and this is how the triplet model \\mathfrak {W} (1,2) arises, for example. Moreover, simple current technology admits a derivation of the extended algebra fusion rules from those of its continuous parent theory. Finally, each example is concluded by a brief description of the computation of some bulk correlators, a discussion of the structure of the bulk state space, and remarks concerning more advanced developments and generalizations. The final part gives a very short account of the theory of staggered modules, the (simplest class of) representations that are responsible for the logarithmic singularities that distinguish logarithmic theories from their rational cousins. These modules are discussed in a generality suitable to encompass all the examples met in this review and some of the very basic structure theory is proven. Then, the important quantities known as logarithmic couplings are reviewed for Virasoro staggered modules and their role as fundamentally important parameters, akin to the three-point constants of rational conformal field
3. Negative power spectra in quantum field theory
International Nuclear Information System (INIS)
Hsiang, Jen-Tsung; Wu, Chun-Hsien; Ford, L.H.
2011-01-01
We consider the spatial power spectra associated with fluctuations of quadratic operators in field theory, such as quantum stress tensor components. We show that the power spectrum can be negative, in contrast to most fluctuation phenomena where the Wiener-Khinchin theorem requires a positive power spectrum. We show why the usual argument for positivity fails in this case, and discuss the physical interpretation of negative power spectra. Possible applications to cosmology are discussed. -- Highlights: → Wiener-Khinchin theorem usually implies a positive power spectrum of fluctuations. → We show this is not always the case in quantum field theory. → Quantum stress tensor fluctuations can have a negative power spectrum. → Negative power interchanges correlations and anticorrelations.
4. PPP mode’s applications motivation in the field of water conservancy project - based on the “money service” theory of Milton Friedman
Science.gov (United States)
Chen, Zurong; Feng, Jingchun; Wang, Yuting; Xue, Song
2017-06-01
We study on PPP mode’s applications motivation in the field of water conservancy project, on the basis of analyzing Friedman’s “money service” theory, for the disadvantages of traditional investment mode in water conservancy project field. By analyzing the way of government and social capital spending money in PPP projects, we get conclusion that both of which are the way of “spending their own money to do their own thing”, which fully reflects that the two sides are a win-win partnership in PPP mode. From the application motivation, PPP mode can not only compensate for the lack of local funds, improve the investment efficiency of the government, but also promote marketization and the supply-side structural reforms.
5. N=1 field theory duality from M theory
International Nuclear Information System (INIS)
Schmaltz, M.; Sundrum, R.
1998-01-01
We investigate Seiberg close-quote s N=1 field theory duality for four-dimensional supersymmetric QCD with the M-theory 5-brane. We find that the M-theory configuration for the magnetic dual theory arises via a smooth deformation of the M-theory configuration for the electric theory. The creation of Dirichlet 4-branes as Neveu-Schwarz 5-branes are passed through each other in type IIA string theory is given an elegant derivation from M theory. copyright 1998 The American Physical Society
6. Quantum Field Theory A Modern Perspective
CERN Document Server
Parameswaran Nair, V
2005-01-01
Quantum field theory, which started with Paul Dirac’s work shortly after the discovery of quantum mechanics, has produced an impressive and important array of results. Quantum electrodynamics, with its extremely accurate and well-tested predictions, and the standard model of electroweak and chromodynamic (nuclear) forces are examples of successful theories. Field theory has also been applied to a variety of phenomena in condensed matter physics, including superconductivity, superfluidity and the quantum Hall effect. The concept of the renormalization group has given us a new perspective on field theory in general and on critical phenomena in particular. At this stage, a strong case can be made that quantum field theory is the mathematical and intellectual framework for describing and understanding all physical phenomena, except possibly for a quantum theory of gravity. Quantum Field Theory: A Modern Perspective presents Professor Nair’s view of certain topics in field theory loosely knit together as it gr...
7. Quantum field theory of universe
International Nuclear Information System (INIS)
Hosoya, Akio; Morikawa, Masahiro.
1988-08-01
As is well-known, the wave function of universe dictated by the Wheeler-DeWitt equation has a difficulty in its probabilistic interpretation. In order to overcome this difficulty, we explore a theoretical possibility of the second quantization of universe, following the same passage historically taken for the Klein-Gordon particles and the Nambu-Goto strings. It turns out that multiple production of universes is an inevitable consequence even if the initial state is nothing. The problematical interpretation of wave function of universe is circumvented by introducing an internal comoving model detector, which is an analogue of the DeWitt-Unruh detector in the quantum field theory in curved space-time. (author)
8. Families and degenerations of conformal field theories
Energy Technology Data Exchange (ETDEWEB)
Roggenkamp, D.
2004-09-01
In this work, moduli spaces of conformal field theories are investigated. In the first part, moduli spaces corresponding to current-current deformation of conformal field theories are constructed explicitly. For WZW models, they are described in detail, and sigma model realizations of the deformed WZW models are presented. The second part is devoted to the study of boundaries of moduli spaces of conformal field theories. For this purpose a notion of convergence of families of conformal field theories is introduced, which admits certain degenerated conformal field theories to occur as limits. To such a degeneration of conformal field theories, a degeneration of metric spaces together with additional geometric structures can be associated, which give rise to a geometric interpretation. Boundaries of moduli spaces of toroidal conformal field theories, orbifolds thereof and WZW models are analyzed. Furthermore, also the limit of the discrete family of Virasoro minimal models is investigated. (orig.)
9. Dependence logic theory and applications
CERN Document Server
Kontinen, Juha; Väänänen, Jouko; Vollmer, Heribert
2016-01-01
In this volume, different aspects of logics for dependence and independence are discussed, including both the logical and computational aspects of dependence logic, and also applications in a number of areas, such as statistics, social choice theory, databases, and computer security. The contributing authors represent leading experts in this relatively new field, each of whom was invited to write a chapter based on talks given at seminars held at the Schloss Dagstuhl Leibniz Center for Informatics in Wadern, Germany (in February 2013 and June 2015) and an Academy Colloquium at the Royal Netherlands Academy of Arts and Sciences (March 2014). Altogether, these chapters provide the most up-to-date look at this developing and highly interdisciplinary field and will be of interest to a broad group of logicians, mathematicians, statisticians, philosophers, and scientists. Topics covered include a comprehensive survey of many propositional, modal, and first-order variants of dependence logic; new results concerning ...
10. Evolution operator equation: Integration with algebraic and finite difference methods. Applications to physical problems in classical and quantum mechanics and quantum field theory
Energy Technology Data Exchange (ETDEWEB)
Dattoli, Giuseppe; Torre, Amalia [ENEA, Centro Ricerche Frascati, Rome (Italy). Dipt. Innovazione; Ottaviani, Pier Luigi [ENEA, Centro Ricerche Bologna (Italy); Vasquez, Luis [Madris, Univ. Complutense (Spain). Dept. de Matemateca Aplicado
1997-10-01
The finite-difference based integration method for evolution-line equations is discussed in detail and framed within the general context of the evolution operator picture. Exact analytical methods are described to solve evolution-like equations in a quite general physical context. The numerical technique based on the factorization formulae of exponential operator is then illustrated and applied to the evolution-operator in both classical and quantum framework. Finally, the general view to the finite differencing schemes is provided, displaying the wide range of applications from the classical Newton equation of motion to the quantum field theory.
11. BRST field theory of relativistic particles
International Nuclear Information System (INIS)
Holten, J.W. van
1992-01-01
A generalization of BRST field theory is presented, based on wave operators for the fields constructed out of, but different from the BRST operator. The authors discuss their quantization, gauge fixing and the derivation of propagators. It is shown, that the generalized theories are relevant to relativistic particle theories in the Brink-Di Vecchia-Howe-Polyakov (BDHP) formulation, and argue that the same phenomenon holds in string theories. In particular it is shown, that the naive BRST formulation of the BDHP theory leads to trivial quantum field theories with vanishing correlation functions. (author). 22 refs
12. Large N Field Theories, String Theory and Gravity
CERN Document Server
Aharony, O; Maldacena, J M; Ooguri, H; Oz, Y
2000-01-01
We review the holographic correspondence between field theories and string/M theory, focusing on the relation between compactifications of string/M theory on Anti-de Sitter spaces and conformal field theories. We review the background for this correspondence and discuss its motivations and the evidence for its correctness. We describe the main results that have been derived from the correspondence in the regime that the field theory is approximated by classical or semiclassical gravity. We focus on the case of the N=4 supersymmetric gauge theory in four dimensions, but we discuss also field theories in other dimensions, conformal and non-conformal, with or without supersymmetry, and in particular the relation to QCD. We also discuss some implications for black hole physics.
13. Stochastic Gravity: Theory and Applications
Directory of Open Access Journals (Sweden)
Hu Bei Lok
2008-05-01
Full Text Available Whereas semiclassical gravity is based on the semiclassical Einstein equation with sources given by the expectation value of the stress-energy tensor of quantum fields, stochastic semiclassical gravity is based on the Einstein–Langevin equation, which has, in addition, sources due to the noise kernel. The noise kernel is the vacuum expectation value of the (operator-valued stress-energy bitensor, which describes the fluctuations of quantum-matter fields in curved spacetimes. A new improved criterion for the validity of semiclassical gravity may also be formulated from the viewpoint of this theory. In the first part of this review we describe the fundamentals of this new theory via two approaches: the axiomatic and the functional. The axiomatic approach is useful to see the structure of the theory from the framework of semiclassical gravity, showing the link from the mean value of the stress-energy tensor to the correlation functions. The functional approach uses the Feynman–Vernon influence functional and the Schwinger–Keldysh closed-time-path effective action methods. In the second part, we describe three applications of stochastic gravity. First, we consider metric perturbations in a Minkowski spacetime, compute the two-point correlation functions of these perturbations and prove that Minkowski spacetime is a stable solution of semiclassical gravity. Second, we discuss structure formation from the stochastic-gravity viewpoint, which can go beyond the standard treatment by incorporating the full quantum effect of the inflaton fluctuations. Third, using the Einstein–Langevin equation, we discuss the backreaction of Hawking radiation and the behavior of metric fluctuations for both the quasi-equilibrium condition of a black-hole in a box and the fully nonequilibrium condition of an evaporating black hole spacetime. Finally, we briefly discuss the theoretical structure of stochastic gravity in relation to quantum gravity and point out
14. On magnetohydrodynamic gauge field theory
Science.gov (United States)
Webb, G. M.; Anco, S. C.
2017-06-01
Clebsch potential gauge field theory for magnetohydrodynamics is developed based in part on the theory of Calkin (1963 Can. J. Phys. 41 2241-51). It is shown how the polarization vector {P} in Calkin’s approach naturally arises from the Lagrange multiplier constraint equation for Faraday’s equation for the magnetic induction {B} , or alternatively from the magnetic vector potential form of Faraday’s equation. Gauss’s equation, (divergence of {B} is zero) is incorporated in the variational principle by means of a Lagrange multiplier constraint. Noether’s theorem coupled with the gauge symmetries is used to derive the conservation laws for (a) magnetic helicity, (b) cross helicity, (c) fluid helicity for non-magnetized fluids, and (d) a class of conservation laws associated with curl and divergence equations which applies to Faraday’s equation and Gauss’s equation. The magnetic helicity conservation law is due to a gauge symmetry in MHD and not due to a fluid relabelling symmetry. The analysis is carried out for the general case of a non-barotropic gas in which the gas pressure and internal energy density depend on both the entropy S and the gas density ρ. The cross helicity and fluid helicity conservation laws in the non-barotropic case are nonlocal conservation laws that reduce to local conservation laws for the case of a barotropic gas. The connections between gauge symmetries, Clebsch potentials and Casimirs are developed. It is shown that the gauge symmetry functionals in the work of Henyey (1982 Phys. Rev. A 26 480-3) satisfy the Casimir determining equations.
15. A philosophical approach to quantum field theory
CERN Document Server
Öttinger, Hans Christian
2015-01-01
This text presents an intuitive and robust mathematical image of fundamental particle physics based on a novel approach to quantum field theory, which is guided by four carefully motivated metaphysical postulates. In particular, the book explores a dissipative approach to quantum field theory, which is illustrated for scalar field theory and quantum electrodynamics, and proposes an attractive explanation of the Planck scale in quantum gravity. Offering a radically new perspective on this topic, the book focuses on the conceptual foundations of quantum field theory and ontological questions. It also suggests a new stochastic simulation technique in quantum field theory which is complementary to existing ones. Encouraging rigor in a field containing many mathematical subtleties and pitfalls this text is a helpful companion for students of physics and philosophers interested in quantum field theory, and it allows readers to gain an intuitive rather than a formal understanding.
16. A philosophical approach to quantum field theory
CERN Document Server
Öttinger, Hans Christian
2017-01-01
This text presents an intuitive and robust mathematical image of fundamental particle physics based on a novel approach to quantum field theory, which is guided by four carefully motivated metaphysical postulates. In particular, the book explores a dissipative approach to quantum field theory, which is illustrated for scalar field theory and quantum electrodynamics, and proposes an attractive explanation of the Planck scale in quantum gravity. Offering a radically new perspective on this topic, the book focuses on the conceptual foundations of quantum field theory and ontological questions. It also suggests a new stochastic simulation technique in quantum field theory which is complementary to existing ones. Encouraging rigor in a field containing many mathematical subtleties and pitfalls this text is a helpful companion for students of physics and philosophers interested in quantum field theory, and it allows readers to gain an intuitive rather than a formal understanding.
17. Generalized locally Toeplitz sequences theory and applications
CERN Document Server
Garoni, Carlo
2017-01-01
Based on their research experience, the authors propose a reference textbook in two volumes on the theory of generalized locally Toeplitz sequences and their applications. This first volume focuses on the univariate version of the theory and the related applications in the unidimensional setting, while the second volume, which addresses the multivariate case, is mainly devoted to concrete PDE applications. This book systematically develops the theory of generalized locally Toeplitz (GLT) sequences and presents some of its main applications, with a particular focus on the numerical discretization of differential equations (DEs). It is the first book to address the relatively new field of GLT sequences, which occur in numerous scientific applications and are especially dominant in the context of DE discretizations. Written for applied mathematicians, engineers, physicists, and scientists who (perhaps unknowingly) encounter GLT sequences in their research, it is also of interest to those working in the fields of...
18. Particles, fields and quantum theory
International Nuclear Information System (INIS)
Bongaarts, P.J.M.
1982-01-01
The author gives an introduction to the development of gauge theories of the fundamental interactions. Starting from classical mechanics and quantum mechanics the development of quantum electrodynamics and non-abelian gauge theories is described. (HSI)
19. Toward a gauge field theory of gravity.
Science.gov (United States)
Yilmaz, H.
Joint use of two differential identities (Bianchi and Freud) permits a gauge field theory of gravity in which the gravitational energy is localizable. The theory is compatible with quantum mechanics and is experimentally viable.
20. Towards weakly constrained double field theory
Directory of Open Access Journals (Sweden)
Kanghoon Lee
2016-08-01
Full Text Available We show that it is possible to construct a well-defined effective field theory incorporating string winding modes without using strong constraint in double field theory. We show that X-ray (Radon transform on a torus is well-suited for describing weakly constrained double fields, and any weakly constrained fields are represented as a sum of strongly constrained fields. Using inverse X-ray transform we define a novel binary operation which is compatible with the level matching constraint. Based on this formalism, we construct a consistent gauge transform and gauge invariant action without using strong constraint. We then discuss the relation of our result to the closed string field theory. Our construction suggests that there exists an effective field theory description for massless sector of closed string field theory on a torus in an associative truncation.
1. Gauge field theories: various mathematical approaches
OpenAIRE
Jordan, François; Serge, Lazzarini; Thierry, Masson
2014-01-01
This paper presents relevant modern mathematical formulations for (classical) gauge field theories, namely, ordinary differential geometry, noncommutative geometry, and transitive Lie algebroids. They provide rigorous frameworks to describe Yang-Mills-Higgs theories or gravitation theories, and each of them improves the paradigm of gauge field theories. A brief comparison between them is carried out, essentially due to the various notions of connection. However they reveal a compelling common...
2. Stochastic Gravity: Theory and Applications
Directory of Open Access Journals (Sweden)
Hu Bei Lok
2004-01-01
Full Text Available Whereas semiclassical gravity is based on the semiclassical Einstein equation with sources given by the expectation value of the stress-energy tensor of quantum fields, stochastic semiclassical gravity is based on the Einstein-Langevin equation, which has in addition sources due to the noise kernel. The noise kernel is the vacuum expectation value of the (operator-valued stress-energy bi-tensor which describes the fluctuations of quantum matter fields in curved spacetimes. In the first part, we describe the fundamentals of this new theory via two approaches: the axiomatic and the functional. The axiomatic approach is useful to see the structure of the theory from the framework of semiclassical gravity, showing the link from the mean value of the stress-energy tensor to their correlation functions. The functional approach uses the Feynman-Vernon influence functional and the Schwinger-Keldysh closed-time-path effective action methods which are convenient for computations. It also brings out the open systems concepts and the statistical and stochastic contents of the theory such as dissipation, fluctuations, noise, and decoherence. We then focus on the properties of the stress-energy bi-tensor. We obtain a general expression for the noise kernel of a quantum field defined at two distinct points in an arbitrary curved spacetime as products of covariant derivatives of the quantum field's Green function. In the second part, we describe three applications of stochastic gravity theory. First, we consider metric perturbations in a Minkowski spacetime. We offer an analytical solution of the Einstein-Langevin equation and compute the two-point correlation functions for the linearized Einstein tensor and for the metric perturbations. Second, we discuss structure formation from the stochastic gravity viewpoint, which can go beyond the standard treatment by incorporating the full quantum effect of the inflaton fluctuations. Third, we discuss the backreaction
3. Quantum field theory in gravitational background
International Nuclear Information System (INIS)
Narnhofer, H.
1986-01-01
The author suggests ignoring the influence of the quantum field on the gravitation as the first step to combine quantum field theory and gravitation theory, but to consider the gravitational field as fixed and thus study quantum field theory on a manifold. This subject evoked interest when thermal radiation of a black hole was predicted. The author concentrates on the free quantum field and can split the problem into two steps: the Weyl-algebra of the free field and the Wightman functional on the tangent space
4. The general theory of quantized fields in the 1950s
International Nuclear Information System (INIS)
Wightman, A.S.
1989-01-01
This review describes developments in theoretical particle physics in the 1950s which were important in the race to develop a putative general theory of quantized fields, especially ideas that offered a mathematically rigorous theory. Basic theoretical concepts then available included the Hamiltonian formulation of quantum dynamics, canonical quantization, perturbative renormalization theory and the theory of distributions. Following a description of various important theoretical contributions of this era, the review ends with a summary of the most important contributions of axiomatic field theory to concrete physics applications. (UK)
5. Analytic aspects of rational conformal field theories
International Nuclear Information System (INIS)
Kiritsis, E.B.; Lawrence Berkeley Lab., CA
1990-01-01
The problem of deriving linear differential equations for correlation functions of Rational Conformal Field Theories is considered. Techniques from the theory of fuchsian differential equations are used to show that knowledge of the central charge, dimensions of primary fields and fusion rules are enough to fix the differential equations for one- and two-point functions on the tours. Any other correlation function can be calculated along similar lines. The results settle the issue of 'exact solution' of rational conformal field theories. (orig.)
6. Nonequilibrium molecular dynamics theory, algorithms and applications
CERN Document Server
Todd, Billy D
2017-01-01
Written by two specialists with over twenty-five years of experience in the field, this valuable text presents a wide range of topics within the growing field of nonequilibrium molecular dynamics (NEMD). It introduces theories which are fundamental to the field - namely, nonequilibrium statistical mechanics and nonequilibrium thermodynamics - and provides state-of-the-art algorithms and advice for designing reliable NEMD code, as well as examining applications for both atomic and molecular fluids. It discusses homogenous and inhomogenous flows and pays considerable attention to highly confined fluids, such as nanofluidics. In addition to statistical mechanics and thermodynamics, the book covers the themes of temperature and thermodynamic fluxes and their computation, the theory and algorithms for homogenous shear and elongational flows, response theory and its applications, heat and mass transport algorithms, applications in molecular rheology, highly confined fluids (nanofluidics), the phenomenon of slip and...
7. Nonlinear boundary value problems in quantum field theory
International Nuclear Information System (INIS)
1989-01-01
We discuss the general structure of a quantum field theory which is free in the interior of a bounded set B of R n . It is shown how to recover the field theory in the interior of B from a certain quantum field theory on the boundary. With an application to string theory in mind, we discuss the example where B equals an interval and the boundary value problem is given in terms of a euclidean functional integral with a P(var phi) interaction restricted to the boundary. copyright 1989 Academic Press, Inc
8. Alternative approaches to maximally supersymmetric field theories
International Nuclear Information System (INIS)
Broedel, Johannes
2010-01-01
The central objective of this work is the exploration and application of alternative possibilities to describe maximally supersymmetric field theories in four dimensions: N=4 super Yang-Mills theory and N=8 supergravity. While twistor string theory has been proven very useful in the context of N=4 SYM, no analogous formulation for N=8 supergravity is available. In addition to the part describing N=4 SYM theory, twistor string theory contains vertex operators corresponding to the states of N=4 conformal supergravity. Those vertex operators have to be altered in order to describe (non-conformal) Einstein supergravity. A modified version of the known open twistor string theory, including a term which breaks the conformal symmetry for the gravitational vertex operators, has been proposed recently. In a first part of the thesis structural aspects and consistency of the modified theory are discussed. Unfortunately, the majority of amplitudes can not be constructed, which can be traced back to the fact that the dimension of the moduli space of algebraic curves in twistor space is reduced in an inconsistent manner. The issue of a possible finiteness of N=8 supergravity is closely related to the question of the existence of valid counterterms in the perturbation expansion of the theory. In particular, the coefficient in front of the so-called R 4 counterterm candidate has been shown to vanish by explicit calculation. This behavior points into the direction of a symmetry not taken into account, for which the hidden on-shell E 7(7) symmetry is the prime candidate. The validity of the so-called double-soft scalar limit relation is a necessary condition for a theory exhibiting E 7(7) symmetry. By calculating the double-soft scalar limit for amplitudes derived from an N=8 supergravity action modified by an additional R 4 counterterm, one can test for possible constraints originating in the E 7(7) symmetry. In a second part of the thesis, the appropriate amplitudes are calculated
9. Strings - Links between conformal field theory, gauge theory and gravity
International Nuclear Information System (INIS)
Troost, J.
2009-05-01
String theory is a candidate framework for unifying the gauge theories of interacting elementary particles with a quantum theory of gravity. The last years we have made considerable progress in understanding non-perturbative aspects of string theory, and in bringing string theory closer to experiment, via the search for the Standard Model within string theory, but also via phenomenological models inspired by the physics of strings. Despite these advances, many deep problems remain, amongst which a non-perturbative definition of string theory, a better understanding of holography, and the cosmological constant problem. My research has concentrated on various theoretical aspects of quantum theories of gravity, including holography, black holes physics and cosmology. In this Habilitation thesis I have laid bare many more links between conformal field theory, gauge theory and gravity. Most contributions were motivated by string theory, like the analysis of supersymmetry preserving states in compactified gauge theories and their relation to affine algebras, time-dependent aspects of the holographic map between quantum gravity in anti-de-Sitter space and conformal field theories in the bulk, the direct quantization of strings on black hole backgrounds, the embedding of the no-boundary proposal for a wave-function of the universe in string theory, a non-rational Verlinde formula and the construction of non-geometric solutions to supergravity
10. Singularity theory and N = 2 superconformal field theories
International Nuclear Information System (INIS)
Warner, N.P.
1989-01-01
The N = 2 superconformal field theories that appear at the fixed points of the renormalization group flows of Landau-Ginsburg models are discussed. Some of the techniques of singularity theory are employed to deduce properties of these superconformal theories. These ideas are then used to deduce the relationship between Calabi-Yau compactifications and tensored discrete series models. The chiral rings of general N = 2 superconformal theories are also described. 14 refs
11. Advanced electromagnetism foundations, theory and applications
CERN Document Server
Barrett, Terence W
1995-01-01
Advanced Electromagnetism: Foundations, Theory and Applications treats what is conventionally called electromagnetism or Maxwell's theory within the context of gauge theory or Yang-Mills theory. A major theme of this book is that fields are not stand-alone entities but are defined by their boundary conditions. The book has practical relevance to efficient antenna design, the understanding of forces and stresses in high energy pulses, ring laser gyros, high speed computer logic elements, efficient transfer of power, parametric conversion, and many other devices and systems. Conventional electro
12. Algebraic quantum field theory, perturbation theory, and the loop expansion
International Nuclear Information System (INIS)
Duetsch, M.; Fredenhagen, K.
2001-01-01
The perturbative treatment of quantum field theory is formulated within the framework of algebraic quantum field theory. We show that the algebra of interacting fields is additive, i.e. fully determined by its subalgebras associated to arbitrary small subregions of Minkowski space. We also give an algebraic formulation of the loop expansion by introducing a projective system A (n) of observables ''up to n loops'', where A (0) is the Poisson algebra of the classical field theory. Finally we give a local algebraic formulation for two cases of the quantum action principle and compare it with the usual formulation in terms of Green's functions. (orig.)
13. Operator algebras and conformal field theory
International Nuclear Information System (INIS)
Gabbiani, F.; Froehlich, J.
1993-01-01
We define and study two-dimensional, chiral conformal field theory by the methods of algebraic field theory. We start by characterizing the vacuum sectors of such theories and show that, under very general hypotheses, their algebras of local observables are isomorphic to the unique hyperfinite type III 1 factor. The conformal net determined by the algebras of local observables is proven to satisfy Haag duality. The representation of the Moebius group (and presumably of the entire Virasoro algebra) on the vacuum sector of a conformal field theory is uniquely determined by the Tomita-Takesaki modular operators associated with its vacuum state and its conformal net. We then develop the theory of Mebius covariant representations of a conformal net, using methods of Doplicher, Haag and Roberts. We apply our results to the representation theory of loop groups. Our analysis is motivated by the desire to find a 'background-independent' formulation of conformal field theories. (orig.)
14. Fundamental number theory with applications
CERN Document Server
Mollin, Richard A
2008-01-01
An update of the most accessible introductory number theory text available, Fundamental Number Theory with Applications, Second Edition presents a mathematically rigorous yet easy-to-follow treatment of the fundamentals and applications of the subject. The substantial amount of reorganizing makes this edition clearer and more elementary in its coverage. New to the Second Edition Removal of all advanced material to be even more accessible in scope New fundamental material, including partition theory, generating functions, and combinatorial number theory Expa
15. Classical gravity and quantum matter fields in unified field theory
Science.gov (United States)
von Borzeszkowski, Horst-Heino; Treder, Hans-Jürgen
1996-01-01
The Einstein-Schrödinger purely affine field theory of the non-symmetric field provides canonical field equations without constraints. These equations imply the Heisenberg-Pauli commutation rules of quantum field theory. In the Schrödinger gauging of the Einstein field coordinatesU {/kl i }=Γ{/kl i }-δ{/l i }Γ{/km m }, this unified geometric field theory becomes a model of the coupling between a quantized Maxwellian field in a medium and classical gravity. Therefore, independently of the question as to the physical truth of this model, its analysis performed in the present paper demonstrates that, in the framework of a quantized unified field theory, gravity can appear as a genuinely classical field.
16. Applications of polyfold theory I
CERN Document Server
Hofer, H; Zehnder, E
2017-01-01
In this paper the authors start with the construction of the symplectic field theory (SFT). As a general theory of symplectic invariants, SFT has been outlined in Introduction to symplectic field theory (2000), by Y. Eliashberg, A. Givental and H. Hofer who have predicted its formal properties. The actual construction of SFT is a hard analytical problem which will be overcome be means of the polyfold theory due to the present authors. The current paper addresses a significant amount of the arising issues and the general theory will be completed in part II of this paper. To illustrate the polyfold theory the authors use the results of the present paper to describe an alternative construction of the Gromov-Witten invariants for general compact symplectic manifolds.
17. Singularity Theory and its Applications
CERN Document Server
Stewart, Ian; Mond, David; Montaldi, James
1991-01-01
A workshop on Singularities, Bifuraction and Dynamics was held at Warwick in July 1989, as part of a year-long symposium on Singularity Theory and its applications. The proceedings fall into two halves: Volume I mainly on connections with algebraic geometry and volume II on connections with dynamical systems theory, bifurcation theory and applications in the sciences. The papers are original research, stimulated by the symposium and workshop: All have been refereed and none will appear elsewhere. The main topic of volume II is new methods for the study of bifurcations in nonlinear dynamical systems, and applications of these.
18. Mathematical aspects of quantum field theories
CERN Document Server
Strobl, Thomas
2015-01-01
Despite its long history and stunning experimental successes, the mathematical foundation of perturbative quantum field theory is still a subject of ongoing research. This book aims at presenting some of the most recent advances in the field, and at reflecting the diversity of approaches and tools invented and currently employed. Both leading experts and comparative newcomers to the field present their latest findings, helping readers to gain a better understanding of not only quantum but also classical field theories. Though the book offers a valuable resource for mathematicians and physicists alike, the focus is more on mathematical developments. This volume consists of four parts: The first Part covers local aspects of perturbative quantum field theory, with an emphasis on the axiomatization of the algebra behind the operator product expansion. The second Part highlights Chern-Simons gauge theories, while the third examines (semi-)classical field theories. In closing, Part 4 addresses factorization homolo...
19. Lectures on classical and quantum theory of fields
CERN Document Server
Arodz, Henryk
2017-01-01
This textbook addresses graduate students starting to specialize in theoretical physics. It provides didactic introductions to the main topics in the theory of fields, while taking into account the contemporary view of the subject. The student will find concise explanations of basic notions essential for applications of the theory of fields as well as for frontier research in theoretical physics. One third of the book is devoted to classical fields. Each chapter contains exercises of varying degree of difficulty with hints or solutions, plus summaries and worked examples as useful. It aims to deliver a unique combination of classical and quantum field theory in one compact course.
20. New results in topological field theory and Abelian gauge theory
International Nuclear Information System (INIS)
Thompson, G.
1995-10-01
These are the lecture notes of a set of lectures delivered at the 1995 Trieste summer school in June. I review some recent work on duality in four dimensional Maxwell theory on arbitrary four manifolds, as well as a new set of topological invariants known as the Seiberg-Witten invariants. Much of the necessary background material is given, including a crash course in topological field theory, cohomology of manifolds, topological gauge theory and the rudiments of four manifold theory. My main hope is to wet the readers appetite, so that he or she will wish to read the original works and perhaps to enter this field. (author). 41 refs, 5 figs
1. Theory and computation of the matrix elements of the full interaction of the electromagnetic field with an atomic state: Application to the Rydberg and the continuous spectrum
International Nuclear Information System (INIS)
Komninos, Yannis; Mercouris, Theodoros; Nicolaides, Cleanthes A.
2002-01-01
We develop practical formulas for the calculation of the matrix elements of the interaction of the electromagnetic field with an atomic state, beyond the long-wavelength approximation. The atom-plus-field Hamiltonian is chosen to have the multipolar form, containing the electric, paramagnetic, and diamagnetic operators. The final workable expressions include the interactions to all orders and are derived by first expanding the fields in partial waves. The electric-field operator reaches a constant value as the radial variable becomes large, contrary to the result of the electric-dipole approximation (EDA) where the value of the corresponding operator increases indefinitely. Applications are given for Rydberg states of hydrogen up to n=50 and for free-free transitions in a Coulomb potential. Such matrix elements are relevant to a number of real and virtual processes occurring during laser-atom interactions. The computation is done numerically, using a combination of analytic with numerical techniques. By comparing the results of the EDA with those of the exact treatment, it is shown that the former is inadequate in such cases. This finding has repercussions on the theory and understanding of the physics of quantum systems in high-lying Rydberg levels and wave packets or in scattering states
2. Introduction to algebraic quantum field theory
International Nuclear Information System (INIS)
Horuzhy, S.S.
1990-01-01
This volume presents a systematic introduction to the algebraic approach to quantum field theory. The structure of the contents corresponds to the way the subject has advanced. It is shown how the algebraic approach has developed from the purely axiomatic theory of observables via superselection rules into the dynamical formalism of fields and observables. Chapter one discusses axioms and their consequences -many of which are now classical theorems- and deals, in general, with the axiomatic theory of local observable algebras. The absence of field concepts makes this theory incomplete and, in chapter two, superselection rules are shown to be the key to the reconstruction of fields from observables. Chapter three deals with the algebras of Wightman fields, first unbounded operator algebras, then Von Neumann field algebras (with a special section on wedge region algebras) and finally local algebras of free and generalised free fields. (author). 447 refs.; 4 figs
3. Using field theory in hadron physics
International Nuclear Information System (INIS)
Abarbanel, H.D.I.
1978-03-01
Topics are covered on the connection of field theory and hadron physics. The renormalization group and infrared and ultraviolet limits of field theory, in particular quantum chromodynamics, spontaneous mass generation, color confinement, instantons, and the vacuum state in quantum chromodynamics are treated. 21 references
4. Calculations in perturbative string field theory
International Nuclear Information System (INIS)
Thorn, C.B.
1987-01-01
The author discusses methods for evaluating the Feynman diagrams of string field theory, with particular emphasis on Witten's version of open string field theory. It is explained in some detail how the rules states by Giddings and Martinec for relating a given diagram to a Polyakov path integral emerge from the Feynman rules
5. Using field theory in hadron physics
Energy Technology Data Exchange (ETDEWEB)
Abarbanel, H.D.I.
1978-03-01
Topics are covered on the connection of field theory and hadron physics. The renormalization group and infrared and ultraviolet limits of field theory, in particular quantum chromodynamics, spontaneous mass generation, color confinement, instantons, and the vacuum state in quantum chromodynamics are treated. 21 references. (JFP)
6. Semiclassical Quantization of Classical Field Theories
NARCIS (Netherlands)
Cattaneo, A.; Mnev, P.; Reshetikhin, N.; Calaque, D.; Strobi, Th.
2015-01-01
Abstract These lectures are an introduction to formal semiclassical quantization of classical field theory. First we develop the Hamiltonian formalism for classical field theories on space time with boundary. It does not have to be a cylinder as in the usual Hamiltonian framework. Then we outline
7. Neoclassical Theory and Its Applications
Energy Technology Data Exchange (ETDEWEB)
Shaing, Ker-Chung [Univ. of Wisconsin, Madison, WI (United States)
2015-11-20
The grant entitled Neoclassical Theory and Its Applications started on January 15 2001 and ended on April 14 2015. The main goal of the project is to develop neoclassical theory to understand tokamak physics, and employ it to model current experimental observations and future thermonuclear fusion reactors. The PI had published more than 50 papers in refereed journals during the funding period.
8. On the interplay between string theory and field theory
International Nuclear Information System (INIS)
Brunner, I.
1998-01-01
In this thesis, we have discussed various aspects of branes in string theory and M-theory. In chapter 2 we were able to construct six-dimensional chiral interacting eld theories from Hanany-Witten like brane setups. The field theory requirement that the anomalies cancel was reproduced by RR-charge conservation in the brane setup. The data of the Hanany-Witten setup, which consists of brane positions, was mapped to instanton data. The orbifold construction can be extended to D and E type singularities. In chapter 3 we discussed a matrix conjecture, which claims that M-theory in the light cone gauge is described by the quantum mechanics of D0 branes. Toroidal compactifications of M-theory have a description in terms of super Yang-Mills theory an the dual torus. For more than three compactified dimensions, more degrees of freedom have to be added. In some sense, the philosophy in this chapter is orthogonal to the previous chapter: Here, we want to get M-theory results from eld theory considerations, whereas in the previous chapter we obtained eld theory results by embedding the theories in string theory. Our main focus was on the compactification on T 6 , which leads to complications. Here, the Matrix model is again given by an eleven dimensional theory, not by a lower dimensional field theory. Other problems and possible resolutions of Matrix theory are discussed at the end of chapter 3. In the last chapter we considered M- and F-theory compactifications on Calabi-Yau fourfolds. After explaining some basics of fourfolds, we showed that the web of fourfolds is connected by singular transitions. The two manifolds which are connected by the transition are different resolutions of the same singular manifold. The resolution of the singularities can lead to a certain type of divisors, which lead to non-perturbative superpotentials, when branes wrap them. The vacua connected by the transitions can be physically very different. (orig.)
9. Quantum Field Theory in a Semiotic Perspective
CERN Document Server
Günter Dosch, Hans; Sieroka, Norman
2005-01-01
Viewing physical theories as symbolic constructions came to the fore in the middle of the nineteenth century with the emancipation of the classical theory of the electromagnetic field from mechanics; most notably this happened through the work of Helmholtz, Hertz, Poincaré, and later Weyl. The epistemological problems that nourished this development are today highlighted within quantum field theory. The present essay starts off with a concise and non-technical outline of the firmly based aspects of relativistic quantum field theory, i.e. the very successful description of subnuclear phenomena. The particular methods, by which these different aspects have to be accessed, then get described as distinct facets of quantum field theory. The authors show how these different facets vary with respect to the relation between quantum fields and associated particles. Thus, by emphasising the respective role of various basic concepts involved, the authors claim that only a very general epistemic approach can properly ac...
10. Introduction to field theory of strings
International Nuclear Information System (INIS)
Kikkawa, K.
1987-01-01
The field theory of bosonic string is reviewed. First, theory is treated in a light-cone gauge. After a brief survey of the first quantized theory of free string, the second quantization is discussed. All possible interactions of strings are introduced based on a smoothness condition of work sheets swept out by strings. Perturbation theory is developed. Finally a possible way to the manifest covariant formalism is discussed
11. Variational Wigner-Kirkwood approach to relativistic mean field theory
International Nuclear Information System (INIS)
Del Estal, M.; Centelles, M.; Vinas, X.
1997-01-01
The recently developed variational Wigner-Kirkwood approach is extended to the relativistic mean field theory for finite nuclei. A numerical application to the calculation of the surface energy coefficient in semi-infinite nuclear matter is presented. The new method is contrasted with the standard density functional theory and the fully quantal approach. copyright 1997 The American Physical Society
12. Constrained variational calculus for higher order classical field theories
International Nuclear Information System (INIS)
Campos, Cedric M; De Leon, Manuel; De Diego, David MartIn
2010-01-01
We develop an intrinsic geometrical setting for higher order constrained field theories. As a main tool we use an appropriate generalization of the classical Skinner-Rusk formalism. Some examples of applications are studied, in particular to the geometrical description of optimal control theory for partial differential equations.
13. Schrodinger representation in renormalizable quantum field theory
International Nuclear Information System (INIS)
Symanzik, K.
1983-01-01
The problem of the Schrodinger representation arose from work on the Nambu-Goto Ansatz for integration over surfaces. Going beyond semiclassical approximation leads to two problems of nonrenormalizibility and of whether Dirichlet boundary conditions can be imposed on a ''Euclidean'' quantum field theory. The Schrodinger representation is constructed in a way where the principles of general renormalization theory can be refered to. The Schrodinger function of surface terms is studied, as well as behaviour at the boundary. The Schrodinger equation is derived. Completeness, unitarity, and computation of expectation values are considered. Extensions of these methods into other Bose field theories such as Fermi fields and Marjorana fields is straightforward
14. Local algebras in Euclidean quantum field theory
International Nuclear Information System (INIS)
Guerra, Francesco.
1975-06-01
The general structure of the local observable algebras of Euclidean quantum field theory is described, considering the very simple examples of the free scalar field, the vector meson field, and the electromagnetic field. The role of Markov properties, and the relations between Euclidean theory and Hamiltonian theory in Minkowski space-time are especially emphasized. No conflict appears between covariance (in the Euclidean sense) and locality (in the Markov sense) on one hand and positive definiteness of the metric on the other hand [fr
15. Random measures, theory and applications
CERN Document Server
Kallenberg, Olav
2017-01-01
Offering the first comprehensive treatment of the theory of random measures, this book has a very broad scope, ranging from basic properties of Poisson and related processes to the modern theories of convergence, stationarity, Palm measures, conditioning, and compensation. The three large final chapters focus on applications within the areas of stochastic geometry, excursion theory, and branching processes. Although this theory plays a fundamental role in most areas of modern probability, much of it, including the most basic material, has previously been available only in scores of journal articles. The book is primarily directed towards researchers and advanced graduate students in stochastic processes and related areas.
16. Mathematical aspects of quantum field theory
CERN Document Server
de Faria, Edson
2010-01-01
Over the last century quantum field theory has made a significant impact on the formulation and solution of mathematical problems and inspired powerful advances in pure mathematics. However, most accounts are written by physicists, and mathematicians struggle to find clear definitions and statements of the concepts involved. This graduate-level introduction presents the basic ideas and tools from quantum field theory to a mathematical audience. Topics include classical and quantum mechanics, classical field theory, quantization of classical fields, perturbative quantum field theory, renormalization, and the standard model. The material is also accessible to physicists seeking a better understanding of the mathematical background, providing the necessary tools from differential geometry on such topics as connections and gauge fields, vector and spinor bundles, symmetries and group representations.
17. Structural aspects of quantum field theory and noncommutative geometry
CERN Document Server
Grensing, Gerhard
2013-01-01
This book is devoted to the subject of quantum field theory. It is divided into two volumes. The first can serve as a textbook on the main techniques and results of quantum field theory, while the second treats more recent developments, in particular the subject of quantum groups and noncommutative geometry, and their interrelation. The first volume is directed at graduate students who want to learn the basic facts about quantum field theory. It begins with a gentle introduction to classical field theory, including the standard model of particle physics, general relativity, and also supergravity. The transition to quantized fields is performed with path integral techniques, by means of which the one-loop renormalization of a self-interacting scalar quantum field, of quantum electrodynamics, and the asymptotic freedom of quantum chromodynamics is treated. In the last part of the first volume, the application of path integral methods to systems of quantum statistical mechanics is covered. The book ends with a r...
18. Quantum scattering from classical field theory
International Nuclear Information System (INIS)
Gould, T.M.; Poppitz, E.R.
1995-01-01
We show that scattering amplitudes between initial wave packet states and certain coherent final states can be computed in a systematic weak coupling expansion about classical solutions satisfying initial-value conditions. The initial-value conditions are such as to make the solution of the classical field equations amenable to numerical methods. We propose a practical procedure for computing classical solutions which contribute to high energy two-particle scattering amplitudes. We consider in this regard the implications of a recent numerical simulation in classical SU(2) Yang-Mills theory for multiparticle scattering in quantum gauge theories and speculate on its generalization to electroweak theory. We also generalize our results to the case of complex trajectories and discuss the prospects for finding a solution to the resulting complex boundary value problem, which would allow the application of our method to any wave packet to coherent state transition. Finally, we discuss the relevance of these results to the issues of baryon number violation and multiparticle scattering at high energies. ((orig.))
19. Supersymmetry and Duality in Field Theory and String Theory
CERN Document Server
Kiritsis, Elias B
1999-01-01
This is a set of lectures given at the 99' Cargese Summer School, "Particle Physics : Ideas and Recent Developments". They contain a pedestrian exposition of recent theoretical progress in non-perturbative field theory and string theory based on ideas of duality.
20. Introduction to conformal field theory and string theory
International Nuclear Information System (INIS)
Dixon, L.J.
1989-12-01
These lectures are meant to provide a brief introduction to conformal field theory (CFT) and string theory for those with no prior exposure to the subjects. There are many excellent reviews already available, and most of these go in to much more detail than I will be able to here. 52 refs., 11 figs
1. Introduction to conformal field theory and string theory
Energy Technology Data Exchange (ETDEWEB)
Dixon, L.J.
1989-12-01
These lectures are meant to provide a brief introduction to conformal field theory (CFT) and string theory for those with no prior exposure to the subjects. There are many excellent reviews already available, and most of these go in to much more detail than I will be able to here. 52 refs., 11 figs.
2. Application of the weak-field asymptotic theory to the analysis of tunneling ionization of linear molecules
DEFF Research Database (Denmark)
Madsen, Lars Bojer; Tolstikhin, Oleg I.; Morishita, Toru
2012-01-01
Hartree-Fock wave functions for the diatomics, and a Hartree-Fock quantum chemistry wave function for CO2. The structure factors are expanded in terms of standard functions and the associated structure coefficients, allowing the determination of the ionization rate for any orientation of the molecule...... with respect to the field, are tabulated. Our results, which are exact in the weak-field limit for H2+ and, in addition, under the Hartree-Fock approximation for the diatomics, are compared with results from the recent literature....
3. String field theory in curved space
International Nuclear Information System (INIS)
Kikkawa, Keiji; Maeno, Masahiro; Sawada, Shiro
1988-01-01
The purely cubic action in the string field theory is shown to provide a set of equations of motion for background fields which agree to those obtained by the vanishing condition of β-functions in the non-linear sigma model. Using the sigma model as an auxiliary tool, a systematic method for solving the string field theory in curved space is proposed. (author)
4. Classical field theory on electrodynamics, non-Abelian gauge theories and gravitation
CERN Document Server
Scheck, Florian
2012-01-01
The book describes Maxwell's equations first in their integral, directly testable form, then moves on to their local formulation. The first two chapters cover all essential properties of Maxwell's equations, including their symmetries and their covariance in a modern notation. Chapter 3 is devoted to Maxwell theory as a classical field theory and to solutions of the wave equation. Chapter 4 deals with important applications of Maxwell theory. It includes topical subjects such as metamaterials with negative refraction index and solutions of Helmholtz' equation in paraxial approximation relevant for the description of laser beams. Chapter 5 describes non-Abelian gauge theories from a classical, geometric point of view, in analogy to Maxwell theory as a prototype, and culminates in an application to the U(2) theory relevant for electroweak interactions. The last chapter 6 gives a concise summary of semi-Riemannian geometry as the framework for the classical field theory of gravitation. The chapter concludes wit...
5. A Conceptual Application of Attachment Theory and Research to the Social Work Student-Field Instructor Supervisory Relationship
Science.gov (United States)
Bennett, Susanne; Saks, Loretta Vitale
2006-01-01
This article conceptualizes an attachment-based model of the student-field instructor relationship, based on empirical research concerning internal working models of attachment, which continue into adulthood and serve as templates for life-long relating. Supportive relationships within a noncritical context are salient for effective supervision;…
6. Representations of classical groups on the lattice and its application to the field theory on discrete space-time
OpenAIRE
Lorente, M.
2003-01-01
We explore the mathematical consequences of the assumption of a discrete space-time. The fundamental laws of physics have to be translated into the language of discrete mathematics. We find integral transformations that leave the lattice of any dimension invariant and apply these transformations to field equations.
7. Light-front quantization of field theory
Energy Technology Data Exchange (ETDEWEB)
Srivastava, Prem P. [Universidade do Estado, Rio de Janeiro, RJ (Brazil). Inst. de Fisica]|[Centro Brasileiro de Pesquisas Fisicas (CBPF), Rio de Janeiro, RJ (Brazil)
1996-07-01
Some basic topics in Light-Front (LF) quantized field theory are reviewed. Poincare algebra and the LF spin operator are discussed. The local scalar field theory of the conventional framework is shown to correspond to a non-local Hamiltonian theory on the LF in view of the constraint equations on the phase space, which relate the bosonic condensates to the non-zero modes. This new ingredient is useful to describe the spontaneous symmetry breaking on the LF. The instability of the symmetric phase in two dimensional scalar theory when the coupling constant grows is shown in the LF theory renormalized to one loop order. Chern-Simons gauge theory, regarded to describe excitations with fractional statistics, is quantized in the light-cone gauge and a simple LF Hamiltonian obtained which may allow us to construct renormalized theory of anyons. (author). 20 refs.
8. Field theory of relativistic strings: I. Trees
International Nuclear Information System (INIS)
Kaku, M.; Kikkawa, K.
1985-01-01
The authors present an entirely new kind of field theory, a field theory quantized not at space-time points, but quantized along an extended set of multilocal points on a string. This represents a significant departure from the usual quantum field theory, whose free theory represents a definite set of elementary particles, because the field theory on relativistic strings can accommodate an infinite set of linearly rising Regge trajectories. In this paper, the authors (1) present canonical quantization and the Green's function of the free string, (2) introduce three-string interactions, (3) resolve the question of multiple counting, (4) complete the counting arguments for all N-point trees, and (5) introduce four-string interactions which yield a Yang-Mills structure when the zero-slope limit is taken
9. A Field Theory with Curvature and Anticurvature
Directory of Open Access Journals (Sweden)
M. I. Wanas
2014-01-01
Full Text Available The present work is an attempt to construct a unified field theory in a space with curvature and anticurvature, the PAP-space. The theory is derived from an action principle and a Lagrangian density using a symmetric linear parameterized connection. Three different methods are used to explore physical contents of the theory obtained. Poisson’s equations for both material and charge distributions are obtained, as special cases, from the field equations of the theory. The theory is a pure geometric one in the sense that material distribution, charge distribution, gravitational and electromagnetic potentials, and other physical quantities are defined in terms of pure geometric objects of the structure used. In the case of pure gravity in free space, the spherical symmetric solution of the field equations gives the Schwarzschild exterior field. The weak equivalence principle is respected only in the case of pure gravity in free space; otherwise it is violated.
10. Mass corrections in string theory and lattice field theory
International Nuclear Information System (INIS)
Del Debbio, Luigi; Kerrane, Eoin; Russo, Rodolfo
2009-01-01
Kaluza-Klein (KK) compactifications of higher-dimensional Yang-Mills theories contain a number of 4-dimensional scalars corresponding to the internal components of the gauge field. While at tree level the scalar zero modes are massless, it is well known that quantum corrections make them massive. We compute these radiative corrections at 1 loop in an effective field theory framework, using the background field method and proper Schwinger-time regularization. In order to clarify the proper treatment of the sum over KK modes in the effective field theory approach, we consider the same problem in two different UV completions of Yang-Mills: string theory and lattice field theory. In both cases, when the compactification radius R is much bigger than the scale of the UV completion (R>>√(α ' ), a), we recover a mass renormalization that is independent of the UV scale and agrees with the one derived in the effective field theory approach. These results support the idea that the value of the mass corrections is, in this regime, universal for any UV completion that respects locality and gauge invariance. The string analysis suggests that this property holds also at higher loops. The lattice analysis suggests that the mass of the adjoint scalars appearing in N=2, 4 super Yang-Mills is highly suppressed, even if the lattice regularization breaks all supersymmetries explicitly. This is due to an interplay between the higher-dimensional gauge invariance and the degeneracy of bosonic and fermionic degrees of freedom.
11. Fermion boson metamorphosis in field theory
International Nuclear Information System (INIS)
Ha, Y.K.
1982-01-01
In two-dimensional field theories many features are especially transparent if the Fermi fields are represented by non-local expressions of the Bose fields. Such a procedure is known as boson representation. Bilinear quantities appear in the Lagrangian of a fermion theory transform, however, as simple local expressions of the bosons so that the resulting theory may be written as a theory of bosons. Conversely, a theory of bosons may be transformed into an equivalent theory of fermions. Together they provide a basis for generating many interesting equivalences between theories of different types. In the present work a consistent scheme for constructing a canonical Fermi field in terms of a real scalar field is developed and such a procedure is valid and consistent with the tenets of quantum field theory is verified. A boson formulation offers a unifying theme in understanding the structure of many theories. This is illustrated by the boson formulation of a multifermion theory with chiral and internal symmetries. The nature of dynamical generation of mass when the theory undergoes boson transmutation and the preservation of continuous chiral symmetry in the massive case are examined. The dynamics of the system depends to a great extent on the specific number of fermions and different models of the same system can have very different properties. Many unusual symmetries of the fermion theory, such as hidden symmetry, duality and triality symmetries, are only manifest in the boson formulation. The underlying connections between some models with U(N) internal symmetry and another class of fermion models built with Majorana fermions which have O(2N) internal symmetry are uncovered
12. Elements of the theory of Markov processes and their applications
CERN Document Server
Bharucha-Reid, A T
2010-01-01
This graduate-level text and reference in probability, with numerous applications to several fields of science, presents nonmeasure-theoretic introduction to theory of Markov processes. The work also covers mathematical models based on the theory, employed in various applied fields. Prerequisites are a knowledge of elementary probability theory, mathematical statistics, and analysis. Appendixes. Bibliographies. 1960 edition.
13. Abelian gauge theories with tensor gauge fields
International Nuclear Information System (INIS)
Kapuscik, E.
1984-01-01
Gauge fields of arbitrary tensor type are introduced. In curved space-time the gravitational field serves as a bridge joining different gauge fields. The theory of second order tensor gauge field is developed on the basis of close analogy to Maxwell electrodynamics. The notion of tensor current is introduced and an experimental test of its detection is proposed. The main result consists in a coupled set of field equations representing a generalization of Maxwell theory in which the Einstein equivalence principle is not satisfied. (author)
14. Quantum field theory in curved space-times: with an application to the reduced model of deSitter universe
International Nuclear Information System (INIS)
Peter, I. J.
1995-06-01
The work deals with space-times with fixed background metric. The topics were arranged in a straight course, the first chapter collects basic facts on Lorentzian manifolds as time-orientability, causal structure, ... Further free neutral scalar fields and spinor fields described by the Klein-Gordon equation resp. the Dirac equation are dealt with. Having in mind the construction of the Weyl algebra and the Fermi algebra in the second chapter, it was put emphasis on the structure of the spaces of solutions of these equations: In the first case the space of solutions is a symplectic vector space in a canonical manner, in the second case a Hilbert space. It was made some effort to stay as general as possible. Most of the material in the second chapter already exists for several years, but it is largely scattered over various journal articles. In the third chapter the construction of a vacuum on the special example of deSitter universe is described. A close investigation of a recent work by J. Bros and U. Moschella made it possible to refine a result concerning temperature felt by an accelerated observer in deSitter space. The last part of this thesis is concerned with vacua for spinor fields on the two-dimensional deSitter universe. A procedure introduced by R. Haag, H. Narnhofer and U. Stein for four dimensional space-times does not seem to work in two dimensions. (author)
15. Effective theories of single field inflation when heavy fields matter
CERN Document Server
Achucarro, Ana; Hardeman, Sjoerd; Palma, Gonzalo A; Patil, Subodh P
2012-01-01
We compute the low energy effective field theory (EFT) expansion for single-field inflationary models that descend from a parent theory containing multiple other scalar fields. By assuming that all other degrees of freedom in the parent theory are sufficiently massive relative to the inflaton, it is possible to derive an EFT valid to arbitrary order in perturbations, provided certain generalized adiabaticity conditions are respected. These conditions permit a consistent low energy EFT description even when the inflaton deviates off its adiabatic minimum along its slowly rolling trajectory. By generalizing the formalism that identifies the adiabatic mode with the Goldstone boson of this spontaneously broken time translational symmetry prior to the integration of the heavy fields, we show that this invariance of the parent theory dictates the entire non-perturbative structure of the descendent EFT. The couplings of this theory can be written entirely in terms of the reduced speed of sound of adiabatic perturbat...
16. [Studies in quantum field theory]: Progress report
International Nuclear Information System (INIS)
Polmar, S.K.
1988-01-01
The theoretical physics group at Washington University has been devoted to the solution of problems in theoretical and mathematical physics. All of the personnel on this task have a similar approach to their research in that they apply sophisticated analytical and numerical techniques to problems primarily in quantum field theory. Specifically, this group has worked on quantum chromodynamics, classical Yang-Mills fields, chiral symmetry breaking condensates, lattice field theory, strong-coupling approximations, perturbation theory in large order, nonlinear waves, 1/N expansions, quantum solitons, phase transitions, nuclear potentials, and early universe calculations
17. Effective field theory for NN interactions
International Nuclear Information System (INIS)
Tran Duy Khuong; Vo Hanh Phuc
2003-01-01
The effective field theory of NN interactions is formulated and the power counting appropriate to this case is reviewed. It is more subtle than in most effective field theories since in the limit that the S-wave NN scattering lengths go to infinity. It is governed by nontrivial fixed point. The leading two body terms in the effective field theory for nucleon self interactions are scale invariant and invariant under Wigner SU(4) spin-isospin symmetry in this limit. Higher body terms with no derivatives (i.e. three and four body terms) are automatically invariant under Wigner symmetry. (author)
18. Clifford algebra in finite quantum field theories
International Nuclear Information System (INIS)
Moser, M.
1997-12-01
We consider the most general power counting renormalizable and gauge invariant Lagrangean density L invariant with respect to some non-Abelian, compact, and semisimple gauge group G. The particle content of this quantum field theory consists of gauge vector bosons, real scalar bosons, fermions, and ghost fields. We assume that the ultimate grand unified theory needs no cutoff. This yields so-called finiteness conditions, resulting from the demand for finite physical quantities calculated by the bare Lagrangean. In lower loop order, necessary conditions for finiteness are thus vanishing beta functions for dimensionless couplings. The complexity of the finiteness conditions for a general quantum field theory makes the discussion of non-supersymmetric theories rather cumbersome. Recently, the F = 1 class of finite quantum field theories has been proposed embracing all supersymmetric theories. A special type of F = 1 theories proposed turns out to have Yukawa couplings which are equivalent to generators of a Clifford algebra representation. These algebraic structures are remarkable all the more than in the context of a well-known conjecture which states that finiteness is maybe related to global symmetries (such as supersymmetry) of the Lagrangean density. We can prove that supersymmetric theories can never be of this Clifford-type. It turns out that these Clifford algebra representations found recently are a consequence of certain invariances of the finiteness conditions resulting from a vanishing of the renormalization group β-function for the Yukawa couplings. We are able to exclude almost all such Clifford-like theories. (author)
19. Bayreuther festspiele as a field for application of Peter Berger’s and Thomas Luckmann’s Theory of social construction of reality
Directory of Open Access Journals (Sweden)
Jeremić-Molnar Dragana
2006-01-01
Full Text Available The Stage festival in Bauyeuth (Bayreuther Festspiele, established in 1876. by German composer Richard Wagner (1813-1883, is, even nowadays, a complex and unique phenomenon which continually attracts the attention of scholars from various (mainly humanistic and social scientific fields. In many different methodological approaches to Bayreuther Festspiele, including those made by social scientists, one can not find the application of the sociological theory of Peter Berger and Thomas Luckmann. However, one has to bare in mind the important fact that Richard Wagner founded his completely innovative festive institution mainly in order to carry out and to spread his regenerative Weltanschauung - already formulated in his numerous theoretical writings and incorporated into his musical dramas. The fact that Wagner’s Weltanschauung was based on the idea of changing the reality of everyday life by constructing the new reality, is of equal importance. Considering all this, it becomes appropriate to explain Wagner’s motivation for establishing the stage festival, as well as his idea of festival, from the standpoint of the theory of social construction of reality.
20. Polynomial field theories and nonintegrability
International Nuclear Information System (INIS)
Euler, N.; Steeb, W.H.; Cyrus, K.
1990-01-01
The nonintegrability of the nonlinear field equation v ηξ = v 3 is studied with the help of the Painleve test. The condition at the resonance is discussed in detail. Particular solutions are given. (orig.)
1. Workshop on Thermal Field Theory to Neural Networks
CERN Document Server
Veneziano, Gabriele; Aurenche, Patrick
1996-01-01
Tanguy Altherr was a Fellow in the Theory Division at CERN, on leave from LAPP (CNRS) Annecy. At the time of his accidental death in July 1994, he was only 31.A meeting was organized at CERN, covering the various aspects of his scientific interests: thermal field theory and its applications to hot or dense media, neural networks and its applications to high energy data analysis. Speakers were among his closest collaborators and friends.
2. Graph theory with applications
CERN Document Server
Vasudev, C
2006-01-01
Salient Features Over 1500 problems are used to illustrate concepts, related to different topics, and introduce applications. Over 1000 exercises in the text with many different types of questions posed. Precise mathematical language is used without excessive formalism and abstraction. Care has been taken to balance the mix of notation and words in mathematical statements. Problem sets are stated clearly and unambiguously, and all are carefully graded for various levels of difficulty. This text has been carefully designed for flexible use.
3. Metric quantum field theory: A preliminary look
International Nuclear Information System (INIS)
Watson, W.N.
1988-01-01
Spacetime coordinates are involved in uncertainty relations; spacetime itself appears to exhibit curvature. Could the continua associated with field variables exhibit curvature? This question, as well as the ideas that (a) difficulties with quantum theories of gravitation may be due to their formulation in an incorrect analogy with other quantum field theories, (b) spacetime variables should not be any more basic than others for describing physical phenomena, and (c) if field continua do not exhibit curvature, the reasons would be of interest, motivated the formulation of a theory of variable curvature and torsion in the electromagnetic four-potential's reciprocal space. Curvature and torsion equation completely analogous to those for a gauge theory of gravitation (the Einstein-Cartan-Sciama-Kibble theory) are assumed for this continuum. The interaction-Hamiltonian density of this theory, to a first approximation, implies that in addition to the Maxwell-Dirac field interaction of ordinary quantum electrodynamics, there should also be an interaction between Dirac-field vector and pseudovector currents unmediated by photons, as well as other interactions involving two or three Dirac-field currents interacting with the Maxwell field at single spacetime events. Calculations expressing Bhabha-scattering cross sections for incident beams with parallel spins differ from those of unmodified quantum electrodynamics by terms of first order in the gravitational constant of the theory, but the corresponding cross section for unpolarized incident beams differs from that of the unmodified theory only by terms of higher order in that constant. Undesirable features of the present theory include its nonrenormalizability, the obscurity of the meaning of its inverse field operator, and its being based on electrodynamics rather than electroweak dynamics
4. Lectures on classical and quantum theory of fields
Energy Technology Data Exchange (ETDEWEB)
Arodz, Henryk; Hadasz, Leszek [Jagiellonian Univ., Krakow (Poland). Inst. Physics
2010-07-01
This textbook on classical and quantum theory of fields addresses graduate students starting to specialize in theoretical physics. It provides didactic introductions to the main topics in the theory of fields, while taking into account the contemporary view of the subject. The student will find concise explanations of basic notions essential for applications of the theory of fields as well as for frontier research in theoretical physics. One third of the book is devoted to classical fields. Each chapter contains exercises of varying degree of difficulty with hints or solutions, plus summaries and worked examples as useful. The textbook is based on lectures delivered to students of theoretical physics at Jagiellonian University. It aims to deliver a unique combination of classical and quantum field theory in one compact course. (orig.)
5. Rheology v.3 theory and applications
CERN Document Server
Eirich, Frederick
1960-01-01
Rheology: Theory and Applications, Volume 3 is a collection of articles contributed by experts in the field of rheology - the science of deformation and flow. This volume is composed of specialized chapters on the application of normal coordinate analysis to the theory of high polymers; principles of rheometry; and the rheology of cross-linked plastics, poly electrolytes, latexes, inks, pastes, and clay. Also included are a series of technological articles on lubrication, spinning, molding, extrusion, and adhesion and a survey of the general features of industrial rheology. Materials scientist
6. Solitons and their interactions in classical field theory
International Nuclear Information System (INIS)
Belova, T.I.; Kudryavtsev, A.E.
1997-01-01
Effects of nonlinearity in the classical field theory for non-integrated systems are considered, such as soliton scattering, soliton bound states, the fractal nature of resonant structures, kink scattering by inhomogeneities, and domain bladder collapse. The results are presented in both (1 + 1) and higher dimensions. Both neutral and charged scalar fields are considered. Possible applications areas for the nonlinearity effects are discussed
7. Global integrability of field theories. Proceedings
International Nuclear Information System (INIS)
Calmet, J.; Seiler, W.M.; Tucker, R.W.
2006-01-01
The GIFT 2006 workshop covers topics related to the Global Integration of Field Theories. These topics span several domains of science including Mathematics, Physics and Computer Science. It is indeed an interdisciplinary event and this feature is well illustrated by the diversity of papers presented at the workshop. Physics is our main target. A simple approach would be to state that we investigate systems of partial differential equations since it is widely believed that they provide a fair description of our world. The questions whether this world is Einsteinian or not, is described by String Theory or not are not however on our agenda. At this stage we have defined what we mean with field theories. To assess what global integrability means we surf on the two other domains of our interest. Mathematics delivers the main methodologies and tools to achieve our goal. It is a trivial remark to say that there exists several approaches to investigate the concept of integrability. Only selected ones are to be found in these proceedings. We do not try to define precisely what global integrability means. Instead, we only suggest two tracks. The first one is by analogy with the design of algorithms, in Computer Algebra or Computer Science, to solve systems of differential equations. The case of ODEs is rather well understood since a constructive methodology exists. Although many experts claim that numerous results do exist to solve systems of PDEs, no constructive decision method exists. This is our first track. The second track follows directly since the real world is described by systems of PDEs, which are mainly non-linear ones. To be able to decide in such a case of the existence of solutions would increase immediately the scope of new technologies applicable to indus trial problems. It is this latter remark that led to the European NEST project with the same name. The GIFT project aims at making progresses in the investigation of field theories through the use of very
8. The conceptual framework of quantum field theory
CERN Document Server
Duncan, Anthony
2012-01-01
The book attempts to provide an introduction to quantum field theory emphasizing conceptual issues frequently neglected in more "utilitarian" treatments of the subject. The book is divided into four parts, entitled respectively "Origins", "Dynamics", "Symmetries", and "Scales". The emphasis is conceptual - the aim is to build the theory up systematically from some clearly stated foundational concepts - and therefore to a large extent anti-historical, but two historical Chapters ("Origins") are included to situate quantum field theory in the larger context of modern physical theories. The three remaining sections of the book follow a step by step reconstruction of this framework beginning with just a few basic assumptions: relativistic invariance, the basic principles of quantum mechanics, and the prohibition of physical action at a distance embodied in the clustering principle. The "Dynamics" section of the book lays out the basic structure of quantum field theory arising from the sequential insertion of quan...
9. Factorization algebras in quantum field theory
CERN Document Server
Costello, Kevin
2017-01-01
Factorization algebras are local-to-global objects that play a role in classical and quantum field theory which is similar to the role of sheaves in geometry: they conveniently organize complicated information. Their local structure encompasses examples like associative and vertex algebras; in these examples, their global structure encompasses Hochschild homology and conformal blocks. In this first volume, the authors develop the theory of factorization algebras in depth, but with a focus upon examples exhibiting their use in field theory, such as the recovery of a vertex algebra from a chiral conformal field theory and a quantum group from Abelian Chern-Simons theory. Expositions of the relevant background in homological algebra, sheaves and functional analysis are also included, thus making this book ideal for researchers and graduates working at the interface between mathematics and physics.
10. Magnetic fields, special relativity and potential theory elementary electromagnetic theory
CERN Document Server
Chirgwin, B H; Kilmister, C W
1972-01-01
Magnetic Fields, Special Relativity and Potential Theory is an introduction to electromagnetism, special relativity, and potential theory, with emphasis on the magnetic field of steady currents (magnetostatics). Topics covered range from the origin of the magnetic field and the magnetostatic scalar potential to magnetization, electromagnetic induction and magnetic energy, and the displacement current and Maxwell's equations. This volume is comprised of five chapters and begins with an overview of magnetostatics, followed by a chapter on the methods of solving potential problems drawn from elec
11. The Mie Theory Basics and Applications
CERN Document Server
Wriedt, Thomas
2012-01-01
This book presents in a concise way the Mie theory and its current applications. It begins with an overview of current theories, computational methods, experimental techniques, and applications of optics of small particles. There is also some biographic information on Gustav Mie, who published his famous paper on the colour of Gold colloids in 1908. The Mie solution for the light scattering of small spherical particles set the basis for more advanced scattering theories and today there are many methods to calculate light scattering and absorption for practically any shape and composition of particles. The optics of small particles is of interest in industrial, atmospheric, astronomic and other research. The book covers the latest developments in divers fields in scattering theory such as plasmon resonance, multiple scattering and optical force.
12. Thermo field dynamics: a quantum field theory at finite temperature
International Nuclear Information System (INIS)
Mancini, F.; Marinaro, M.; Matsumoto, H.
1988-01-01
A brief review of the theory of thermo field dynamics (TFD) is presented. TFD is introduced and developed by Umezawa and his coworkers at finite temperature. The most significant concept in TFD is that of a thermal vacuum which satisfies some conditions denoted as thermal state conditions. The TFD permits to reformulate theories at finite temperature. There is no need in an additional principle to determine particle distributions at T ≠ 0. Temperature and other macroscopic parameters are introduced in the definition of the vacuum state. All operator formalisms used in quantum field theory at T=0 are preserved, although the field degrees of freedom are doubled. 8 refs
13. Repeated sprints, high-intensity interval training, small-sided games: theory and application to field sports.
Science.gov (United States)
Hoffmann, James J; Reed, Jacob P; Leiting, Keith; Chiang, Chieh-Ying; Stone, Michael H
2014-03-01
Due to the broad spectrum of physical characteristics necessary for success in field sports, numerous training modalities have been used develop physical preparedness. Sports like rugby, basketball, lacrosse, and others require athletes to be not only strong and powerful but also aerobically fit and able to recover from high-intensity intermittent exercise. This provides coaches and sport scientists with a complex range of variables to consider when developing training programs. This can often lead to confusion and the misuse of training modalities, particularly in the development of aerobic and anaerobic conditioning. This review outlines the benefits and general adaptations to 3 commonly used and effective conditioning methods: high-intensity interval training, repeated-sprint training, and small-sided games. The goals and outcomes of these training methods are discussed, and practical implementations strategies for coaches and sport scientists are provided.
14. String theory inspired deformations of quantum field theories
Science.gov (United States)
Chiou, Dah-Wei
In this dissertation, some extensions on field theories with deformations inspired by string theory are explored and their implications are investigated. These are: (i) noncommutative dipole field theory (DFT) and unitarity; (ii) three dimensional super Yang-Mills theory and mini-twistor string theory; (iii) massive super Yang-Mills theory and twistor string theory; and (iv) a deformation of twistor space and N = 4 super Yang-Mills theory with a chiral mass term. The DFT with fixed spacetime vectors ("dipole-vectors") is formulated for gauge theory coupled with a scalar field of adjoint charge. The argument for the violation of unitarity in field theories on a noncommutative spacetime is extended to the case of DFT: with a timelike dipole vector, 1-loop amplitudes are shown not to obey the optical theorem and thus violate unitarity. Likewise, a simple 0 + 1D quantum mechanical system with nonlocal potential of finite extent in time also gives violation of unitarity. Associated with D = 3 super Yang-Mills theory, the topological B-model is constructed for the twistor string theory, of which the target space is the (super-)mini-twistor space. As the D = 4 twistor space can be considered as a fibration over D = 3 mini-twistor space, the dimensional reduction from D = 4 to D = 3 is conducted to obtain the scattering amplitudes for D = 3 super Yang-Mills theory. The result shows that, analogous to the D = 4 case, the twistor transformed D = 3 amplitudes are supported on holomorphic curves in the (super-)mini-twistor space. Another alternative twistor description---Berkovits's open string theory---is also analyzed. By the prescription which interrelates Witten's B-model and Berkovits's open string theory, the dimensional reduction can be made for Berkovits's model as well, in which the enhanced R-symmetry Spin(7) is recognized, whereas only the subgroup SU(4) is manifest in the B-model. The extension of the twistor string theory by adding mass terms is then proposed and
15. Functional analysis theory and applications
CERN Document Server
Edwards, RE
2011-01-01
""The book contains an enormous amount of information - mathematical, bibliographical and historical - interwoven with some outstanding heuristic discussions."" - Mathematical Reviews.In this massive graduate-level study, Emeritus Professor Edwards (Australian National University, Canberra) presents a balanced account of both the abstract theory and the applications of linear functional analysis. Written for readers with a basic knowledge of set theory, general topology, and vector spaces, the book includes an abundance of carefully chosen illustrative examples and excellent exercises at the
16. Lectures on interacting string field theory
International Nuclear Information System (INIS)
Jevicki, A.
1986-09-01
We give a detailed review of the current formulations of interacting string field theory. The historical development of the subject is taken beginning with the old dual resonance model theory. The light cone approach is reviewed in some detail with emphasis on conformal mapping techniques. Witten's covariant approach is presented. The main body of the lectures concentrates on developing the operator formulation of Witten's theory. 38 refs., 22 figs., 5 tabs
17. Dynamical symmetry breaking in quantum field theories
CERN Document Server
1993-01-01
The phenomenon of dynamical symmetry breaking (DSB) in quantum field theory is discussed in a detailed and comprehensive way. The deep connection between this phenomenon in condensed matter physics and particle physics is emphasized. The realizations of DSB in such realistic theories as quantum chromodynamics and electroweak theory are considered. Issues intimately connected with DSB such as critical phenomenona and effective lagrangian approach are also discussed.
18. Recent progress in reggeon field theory
International Nuclear Information System (INIS)
Sugar, R.L.
1977-01-01
The present status of the pomeron theory in the reggeon field theory is summarized. For α 0 ( 0 -a bare intercept, αsub(oc) - a certain critical value) the theory is in a very good shape. It appears to satisfy both S and t-channel unitarity, and to avoid all of the decreases which plagued the simple pole model of the pomeron. For α 0 >αsub(oc) the situation is less clear
19. Pure field theories and MACSYMA algorithms
Science.gov (United States)
Ament, W. S.
1977-01-01
A pure field theory attempts to describe physical phenomena through singularity-free solutions of field equations resulting from an action principle. The physics goes into forming the action principle and interpreting specific results. Algorithms for the intervening mathematical steps are sketched. Vacuum general relativity is a pure field theory, serving as model and providing checks for generalizations. The fields of general relativity are the 10 components of a symmetric Riemannian metric tensor; those of the Einstein-Straus generalization are the 16 components of a nonsymmetric. Algebraic properties are exploited in top level MACSYMA commands toward performing some of the algorithms of that generalization. The light cone for the theory as left by Einstein and Straus is found and simplifications of that theory are discussed.
20. Axion topological field theory of topological superconductors
Science.gov (United States)
Qi, Xiao-Liang; Witten, Edward; Zhang, Shou-Cheng
2013-04-01
Topological superconductors are gapped superconductors with gapless and topologically robust quasiparticles propagating on the boundary. In this paper, we present a topological field theory description of three-dimensional time-reversal invariant topological superconductors. In our theory the topological superconductor is characterized by a topological coupling between the electromagnetic field and the superconducting phase fluctuation, which has the same form as the coupling of “axions” with an Abelian gauge field. As a physical consequence of our theory, we predict the level crossing induced by the crossing of special “chiral” vortex lines, which can be realized by considering s-wave superconductors in proximity with the topological superconductor. Our theory can also be generalized to the coupling with a gravitational field.
1. An introduction to relativistic quantum field theory
CERN Document Server
Schweber, Silvan S
1961-01-01
Complete, systematic, and self-contained, this text introduces modern quantum field theory. "Combines thorough knowledge with a high degree of didactic ability and a delightful style." - Mathematical Reviews. 1961 edition.
2. Mathematical game theory and applications
CERN Document Server
2014-01-01
An authoritative and quantitative approach to modern game theory with applications from diverse areas including economics, political science, military science, and finance. Explores areas which are not covered in current game theory texts, including a thorough examination of zero-sum game.Provides introductory material to game theory, including bargaining, parlour games, sport, networking games and dynamic games.Explores Bargaining models, discussing new result such as resource distributions, buyer-seller instructions and reputation in bargaining models.Theoretical results are presented along
3. Quantum field theory with infinite component local fields as an alternative to the string theories
International Nuclear Information System (INIS)
Krasnikov, N.V.
1987-05-01
We show that the introduction of the infinite component local fields with higher order derivatives in the interaction makes the theory completely ultraviolet finite. For the γ 5 -anomalous theories the introduction of the infinite component field makes the theory renormalizable or superrenormalizable. (orig.)
4. The conceptual basis of Quantum Field Theory
NARCIS (Netherlands)
Hooft, G. 't
2005-01-01
Relativistic Quantum Field Theory is a mathematical scheme to describe the sub-atomic particles and forces. The basic starting point is that the axioms of Special Relativity on the one hand and those of Quantum Mechanics on the other, should be combined into one theory. The fundamental
5. Renormalizability of effective scalar field theory
CERN Document Server
Ball, R D
1994-01-01
We present a comprehensive discussion of the consistency of the effective quantum field theory of a single $Z_2$ symmetric scalar field. The theory is constructed from a bare Euclidean action which at a scale much greater than the particle's mass is constrained only by the most basic requirements; stability, finiteness, analyticity, naturalness, and global symmetry. We prove to all orders in perturbation theory the boundedness, convergence, and universality of the theory at low energy scales, and thus that the theory is perturbatively renormalizable in the sense that to a certain precision over a range of such scales it depends only on a finite number of parameters. We then demonstrate that the effective theory has a well defined unitary and causal analytic S--matrix at all energy scales. We also show that redundant terms in the Lagrangian may be systematically eliminated by field redefinitions without changing the S--matrix, and discuss the extent to which effective field theory and analytic S--matrix theory...
6. Klein Topological Field Theories from Group Representations
Directory of Open Access Journals (Sweden)
Sergey A. Loktev
2011-07-01
Full Text Available We show that any complex (respectively real representation of finite group naturally generates a open-closed (respectively Klein topological field theory over complex numbers. We relate the 1-point correlator for the projective plane in this theory with the Frobenius-Schur indicator on the representation. We relate any complex simple Klein TFT to a real division ring.
7. Electromagnetic Field Theory A Collection of Problems
CERN Document Server
Mrozynski, Gerd
2013-01-01
After a brief introduction into the theory of electromagnetic fields and the definition of the field quantities the book teaches the analytical solution methods of Maxwell’s equations by means of several characteristic examples. The focus is on static and stationary electric and magnetic fields, quasi stationary fields, and electromagnetic waves. For a deeper understanding, the many depicted field patterns are very helpful. The book offers a collection of problems and solutions which enable the reader to understand and to apply Maxwell’s theory for a broad class of problems including classical static problems right up to waveguide eigenvalue problems. Content Maxwell’s Equations - Electrostatic Fields - Stationary Current Distributions – Magnetic Field of Stationary Currents – Quasi Stationary Fields: Eddy Currents - Electromagnetic Waves Target Groups Advanced Graduate Students in Electrical Engineering, Physics, and related Courses Engineers and Physicists Authors Professor Dr.-Ing. Gerd Mrozynski...
8. Field theories with multiple fermionic excitations
International Nuclear Information System (INIS)
Crawford, J.P.
1978-01-01
The reason for the existence of the muon has been an enigma since its discovery. Since that time there has been a continuing proliferation of elementary particles. It is proposed that this proliferation of leptons and quarks is comprehensible if there are only four fundamental particles, the leptons ν/sub e/ and e - , and the quarks u and d. All other leptons and quarks are imagined to be excited states of these four fundamental entities. Attention is restricted to the charged leptons and the electromagnetic interactions only. A detailed study of a field theory in which there is only one fundamental charged fermionic field having two (or more) excitations is made. When the electromagnetic interactions are introduced and the theory is second quantized, under certain conditions this theory reproduces the S matrix obtained from usual OED. In this case no electromagnetic transitions are allowed. A leptonic charge operator is defined and a superselection rule for this leptonic charge is found. Unfortunately, the mass spectrum cannot be obtained. This theory has many renormalizable generalizations including non-abelian gauge theories, Yukawa-type theories, and Fermi-type theories. Under certain circumstances the Yukawa- and Fermi-type theories are finite in perturbation theory. It is concluded that there are no fundamental objections to having fermionic fields with more than one excitation
9. Simple recursion relations for general field theories
International Nuclear Information System (INIS)
Cheung, Clifford; Shen, Chia-Hsien; Trnka, Jaroslav
2015-01-01
On-shell methods offer an alternative definition of quantum field theory at tree-level, replacing Feynman diagrams with recursion relations and interaction vertices with a handful of seed scattering amplitudes. In this paper we determine the simplest recursion relations needed to construct a general four-dimensional quantum field theory of massless particles. For this purpose we define a covering space of recursion relations which naturally generalizes all existing constructions, including those of BCFW and Risager. The validity of each recursion relation hinges on the large momentum behavior of an n-point scattering amplitude under an m-line momentum shift, which we determine solely from dimensional analysis, Lorentz invariance, and locality. We show that all amplitudes in a renormalizable theory are 5-line constructible. Amplitudes are 3-line constructible if an external particle carries spin or if the scalars in the theory carry equal charge under a global or gauge symmetry. Remarkably, this implies the 3-line constructibility of all gauge theories with fermions and complex scalars in arbitrary representations, all supersymmetric theories, and the standard model. Moreover, all amplitudes in non-renormalizable theories without derivative interactions are constructible; with derivative interactions, a subset of amplitudes is constructible. We illustrate our results with examples from both renormalizable and non-renormalizable theories. Our study demonstrates both the power and limitations of recursion relations as a self-contained formulation of quantum field theory.
10. Phase-field-crystal dynamics for binary systems: Derivation from dynamical density functional theory, amplitude equation formalism, and applications to alloy heterostructures.
Science.gov (United States)
Huang, Zhi-Feng; Elder, K R; Provatas, Nikolas
2010-08-01
The dynamics of phase field crystal (PFC) modeling is derived from dynamical density functional theory (DDFT), for both single-component and binary systems. The derivation is based on a truncation up to the three-point direct correlation functions in DDFT, and the lowest order approximation using scale analysis. The complete amplitude equation formalism for binary PFC is developed to describe the coupled dynamics of slowly varying complex amplitudes of structural profile, zeroth-mode average atomic density, and system concentration field. Effects of noise (corresponding to stochastic amplitude equations) and species-dependent atomic mobilities are also incorporated in this formalism. Results of a sample application to the study of surface segregation and interface intermixing in alloy heterostructures and strained layer growth are presented, showing the effects of different atomic sizes and mobilities of alloy components. A phenomenon of composition overshooting at the interface is found, which can be connected to the surface segregation and enrichment of one of the atomic components observed in recent experiments of alloying heterostructures.
11. Classical theory of electric and magnetic fields
CERN Document Server
Good, Roland H
1971-01-01
Classical Theory of Electric and Magnetic Fields is a textbook on the principles of electricity and magnetism. This book discusses mathematical techniques, calculations, with examples of physical reasoning, that are generally applied in theoretical physics. This text reviews the classical theory of electric and magnetic fields, Maxwell's Equations, Lorentz Force, and Faraday's Law of Induction. The book also focuses on electrostatics and the general methods for solving electrostatic problems concerning images, inversion, complex variable, or separation of variables. The text also explains ma
12. Best matching theory & applications
CERN Document Server
2017-01-01
Mismatch or best match? This book demonstrates that best matching of individual entities to each other is essential to ensure smooth conduct and successful competitiveness in any distributed system, natural and artificial. Interactions must be optimized through best matching in planning and scheduling, enterprise network design, transportation and construction planning, recruitment, problem solving, selective assembly, team formation, sensor network design, and more. Fundamentals of best matching in distributed and collaborative systems are explained by providing: § Methodical analysis of various multidimensional best matching processes § Comprehensive taxonomy, comparing different best matching problems and processes § Systematic identification of systems’ hierarchy, nature of interactions, and distribution of decision-making and control functions § Practical formulation of solutions based on a library of best matching algorithms and protocols, ready for direct applications and apps development. Design...
13. Unified-field theory: yesterday, today, tomorrow
International Nuclear Information System (INIS)
Bergman, P.G.
1982-01-01
Beginning with the expounding of Einstein understanding of advantages and disadvantages of general relativity theory, the authors proceed to consideration of what the complete unified theory have to be according to Einstein. The four theories which can be considered as ''unified'', namely weyl and Calutsa ones, worked out a half of century ago, and twistor twisting and supersymmetry theories, nowadays attracting attention, are briefly described and discussed. The authors come to a conclusion that achievements in elementary-particle physics have to affect any future theory, that this theory has to explain the principle contradictions between classical and quantum field theories, and that finally it can lead to change of the modern space-time model as a four-dimensional variety
14. Quantum field theory in a semiotic perspective
International Nuclear Information System (INIS)
Dosch, H.G.
2005-01-01
Viewing physical theories as symbolic constructions came to the fore in the middle of the nineteenth century with the emancipation of the classical theory of the electromagnetic field from mechanics; most notably this happened through the work of Helmholtz, Hertz, Poincare, and later Weyl. The epistemological problems that nourished this development are today highlighted within quantum field theory. The present essay starts off with a concise and non-technical outline of the firmly based aspects of relativistic quantum field theory, i.e. the very successful description of subnuclear phenomena. The particular methods, by which these different aspects have to be accessed, then get described as distinct facets of quantum field theory. The authors show how these different facets vary with respect to the relation between quantum fields and associated particles. Thus, by emphasising the respective role of various basic concepts involved, the authors claim that only a very general epistemic approach can properly account for this diversity - an account they trace back to the philosophical writings of the aforementioned physicists and mathematicians. Finally, what they call their semiotic perspective on quantum field theory gets related to recent discussions within the philosophy of science and turns out to act as a counterbalance to, for instance, structural realism. (orig.)
15. Quantum field theory in a semiotic perspective
Energy Technology Data Exchange (ETDEWEB)
Dosch, H.G. [Heidelberg Univ. (Germany). Inst. fuer Theoretische Physik; Mueller, V.F. [Technische Univ. Kaiserslautern (Germany). Fachbereich Physik; Sieroka, N. [Zurich Univ. (Switzerland)
2005-07-01
Viewing physical theories as symbolic constructions came to the fore in the middle of the nineteenth century with the emancipation of the classical theory of the electromagnetic field from mechanics; most notably this happened through the work of Helmholtz, Hertz, Poincare, and later Weyl. The epistemological problems that nourished this development are today highlighted within quantum field theory. The present essay starts off with a concise and non-technical outline of the firmly based aspects of relativistic quantum field theory, i.e. the very successful description of subnuclear phenomena. The particular methods, by which these different aspects have to be accessed, then get described as distinct facets of quantum field theory. The authors show how these different facets vary with respect to the relation between quantum fields and associated particles. Thus, by emphasising the respective role of various basic concepts involved, the authors claim that only a very general epistemic approach can properly account for this diversity - an account they trace back to the philosophical writings of the aforementioned physicists and mathematicians. Finally, what they call their semiotic perspective on quantum field theory gets related to recent discussions within the philosophy of science and turns out to act as a counterbalance to, for instance, structural realism. (orig.)
16. Superstring field theory equivalence: Ramond sector
International Nuclear Information System (INIS)
Kroyter, Michael
2009-01-01
We prove that the finite gauge transformation of the Ramond sector of the modified cubic superstring field theory is ill-defined due to collisions of picture changing operators. Despite this problem we study to what extent could a bijective classical correspondence between this theory and the (presumably consistent) non-polynomial theory exist. We find that the classical equivalence between these two theories can almost be extended to the Ramond sector: We construct mappings between the string fields (NS and Ramond, including Chan-Paton factors and the various GSO sectors) of the two theories that send solutions to solutions in a way that respects the linearized gauge symmetries in both sides and keeps the action of the solutions invariant. The perturbative spectrum around equivalent solutions is also isomorphic. The problem with the cubic theory implies that the correspondence of the linearized gauge symmetries cannot be extended to a correspondence of the finite gauge symmetries. Hence, our equivalence is only formal, since it relates a consistent theory to an inconsistent one. Nonetheless, we believe that the fact that the equivalence formally works suggests that a consistent modification of the cubic theory exists. We construct a theory that can be considered as a first step towards a consistent RNS cubic theory.
17. Quantum groups, quantum categories and quantum field theory
CERN Document Server
Fröhlich, Jürg
1993-01-01
This book reviews recent results on low-dimensional quantum field theories and their connection with quantum group theory and the theory of braided, balanced tensor categories. It presents detailed, mathematically precise introductions to these subjects and then continues with new results. Among the main results are a detailed analysis of the representation theory of U (sl ), for q a primitive root of unity, and a semi-simple quotient thereof, a classfication of braided tensor categories generated by an object of q-dimension less than two, and an application of these results to the theory of sectors in algebraic quantum field theory. This clarifies the notion of "quantized symmetries" in quantum fieldtheory. The reader is expected to be familiar with basic notions and resultsin algebra. The book is intended for research mathematicians, mathematical physicists and graduate students.
18. Magnetic Catalysis in Graphene Effective Field Theory.
Science.gov (United States)
DeTar, Carleton; Winterowd, Christopher; Zafeiropoulos, Savvas
2016-12-23
We report on the first calculation of magnetic catalysis at zero temperature in a fully nonperturbative simulation of the graphene effective field theory. Using lattice gauge theory, a nonperturbative analysis of the theory of strongly interacting, massless, (2+1)-dimensional Dirac fermions in the presence of an external magnetic field is performed. We show that in the zero-temperature limit, a nonzero value for the chiral condensate is obtained which signals the spontaneous breaking of chiral symmetry. This result implies a nonzero value for the dynamical mass of the Dirac quasiparticle.
19. Supersymmetric gauge theories, quantization of Mflat, and conformal field theory
International Nuclear Information System (INIS)
Teschner, J.; Vartanov, G.S.
2013-02-01
We propose a derivation of the correspondence between certain gauge theories with N=2 supersymmetry and conformal field theory discovered by Alday, Gaiotto and Tachikawa in the spirit of Seiberg-Witten theory. Based on certain results from the literature we argue that the quantum theory of the moduli spaces of flat SL(2,R)-connections represents a nonperturbative ''skeleton'' of the gauge theory, protected by supersymmetry. It follows that instanton partition functions can be characterized as solutions to a Riemann-Hilbert type problem. In order to solve it, we describe the quantization of the moduli spaces of flat connections explicitly in terms of two natural sets of Darboux coordinates. The kernel describing the relation between the two pictures represents the solution to the Riemann Hilbert problem, and is naturally identified with the Liouville conformal blocks.
20. Random light beams theory and applications
CERN Document Server
Korotkova, Olga
2013-01-01
Random Light Beams: Theory and Applications contemplates the potential in harnessing random light. This book discusses light matter interactions, and concentrates on the various phenomena associated with beam-like fields. It explores natural and man-made light fields and gives an overview of recently introduced families of random light beams. It outlines mathematical tools for analysis, suggests schemes for realization, and discusses possible applications. The book introduces the essential concepts needed for a deeper understanding of the subject, discusses various classes of deterministic par
1. Infrared problems in field perturbation theory
International Nuclear Information System (INIS)
David, Francois.
1982-12-01
The work presented mainly covers questions related to the presence of ''infrared'' divergences in perturbation expansions of the Green functions of certain massless field theories. It is important to determine the mathematical status of perturbation expansions in field theory in order to define the region in which they are valid. Renormalization and the symmetry of a theory are important factors in infrared problems. The main object of this thesis resides in the mathematical techniques employed: integral representations of the Feynman amplitudes; methods for desingularization, regularization and dimensional renormalization. Nonlinear two dimensional space-time sigma models describing Goldstone's low energy boson dynamics associated with a breaking of continuous symmetry are studied. Random surface models are then investigated followed by infrared divergences in super-renormalizable theories. Finally, nonperturbation effects in massless theories are studied by expanding the two-dimensional nonlinear sigma model in 1/N [fr
2. Proceedings of the 5. Jorge Andre Swieca Summer School Field Theory and Particle Physics
International Nuclear Information System (INIS)
Eboli, O.J.P.; Gomes, M.; Santoro, A.
1989-01-01
Lectures on quantum field theories and particle physics are presented. The part of quantum field theories contains: constrained dynamics; Schroedinger representation in field theory; application of this representation to quantum fields in a Robertson-Walker space-time; Berry connection; problem of construction and classification of conformal field theories; lattice models; two-dimensional S matrices and conformal field theory for unifying perspective of Yang-Baxter algebras; parasupersymmetric quantum mechanics; introduction to string field theory; three dimensional gravity and two-dimensional parafermionic model. The part of particle physics contains: collider physics; strong interactions and use of strings in strong interactions. (M.C.K.)
3. Towards field theory in spaces with multivolume junctions
CERN Document Server
Fomin, P I
2002-01-01
We consider a spacetime formed by several pieces with common timelike boundary which plays the role of a junction between them. We establish junction conditions for fields of various spins and derive the resulting laws of wave propagation through the junction, which turn out to be quite similar for fields of all spins. As an application, we consider the case of multivolume junctions in four-dimensional spacetime that may arise in the context of the theory of quantum creation of a closed universe on the background of a big mother universe. The theory developed can also be applied to braneworld models and to the superstring theory.
4. Theory of semigroups and applications
CERN Document Server
Sinha, Kalyan B
2017-01-01
The book presents major topics in semigroups, such as operator theory, partial differential equations, harmonic analysis, probability and statistics and classical and quantum mechanics, and applications. Along with a systematic development of the subject, the book emphasises on the explorations of the contact areas and interfaces, supported by the presentations of explicit computations, wherever feasible. Designed into seven chapters and three appendixes, the book targets to the graduate and senior undergraduate students of mathematics, as well as researchers in the respective areas. The book envisages the pre-requisites of a good understanding of real analysis with elements of the theory of measures and integration, and a first course in functional analysis and in the theory of operators. Chapters 4 through 6 contain advanced topics, which have many interesting applications such as the Feynman–Kac formula, the central limit theorem and the construction of Markov semigroups. Many examples have been given in...
5. Sinusoids theory and technological applications
CERN Document Server
Kythe, Prem K
2014-01-01
A Complete Treatment of Current Research Topics in Fourier Transforms and Sinusoids Sinusoids: Theory and Technological Applications explains how sinusoids and Fourier transforms are used in a variety of application areas, including signal processing, GPS, optics, x-ray crystallography, radioastronomy, poetry and music as sound waves, and the medical sciences. With more than 200 illustrations, the book discusses electromagnetic force and sychrotron radiation comprising all kinds of waves, including gamma rays, x-rays, UV rays, visible light rays, infrared, microwaves, and radio waves. It also covers topics of common interest, such as quasars, pulsars, the Big Bang theory, Olbers' paradox, black holes, Mars mission, and SETI.The book begins by describing sinusoids-which are periodic sine or cosine functions-using well-known examples from wave theory, including traveling and standing waves, continuous musical rhythms, and the human liver. It next discusses the Fourier series and transform in both continuous and...
6. Gravitation Field Dynamics in Jeans Theory
2016-01-27
Jan 27, 2016 ... Closed system of time equations for nonrelativistic gravitation field and hydrodynamic medium was obtained by taking into account binary correlations of the field, which is the generalization of Jeans theory. Distribution function of the systemwas built on the basis of the Bogolyubov reduced description ...
7. Field theory of polar continua
International Nuclear Information System (INIS)
Heinz, C.
1988-01-01
A Lagrangian density in the polar space X 1+3+3 depending of the potentials and their derivativs and of the fluxes is introduced. The potentials are then the mechanical and electromagnetic potentials, the potentials of gravity and in the polar space X 1+3+3 the components of affine connection. The fluxes are essentially the tangential motors of the mechanical and electromagnetic world-lines multiplied with the density of mass and electric charge. The Hamilton principle gives, with the in variational calculus usual integrations by part, here done via the theorem of Gauss, the equations of motion and the field equations. The conditions of integrability for these equations are discussed. (author)
8. Mean-field magnetohydrodynamics and dynamo theory
CERN Document Server
Krause, F
2013-01-01
Mean-Field Magnetohydrodynamics and Dynamo Theory provides a systematic introduction to mean-field magnetohydrodynamics and the dynamo theory, along with the results achieved. Topics covered include turbulence and large-scale structures; general properties of the turbulent electromotive force; homogeneity, isotropy, and mirror symmetry of turbulent fields; and turbulent electromotive force in the case of non-vanishing mean flow. The turbulent electromotive force in the case of rotational mean motion is also considered. This book is comprised of 17 chapters and opens with an overview of the gen
9. Circuit complexity in quantum field theory
Science.gov (United States)
Jefferson, Robert A.; Myers, Robert C.
2017-10-01
Motivated by recent studies of holographic complexity, we examine the question of circuit complexity in quantum field theory. We provide a quantum circuit model for the preparation of Gaussian states, in particular the ground state, in a free scalar field theory for general dimensions. Applying the geometric approach of Nielsen to this quantum circuit model, the complexity of the state becomes the length of the shortest geodesic in the space of circuits. We compare the complexity of the ground state of the free scalar field to the analogous results from holographic complexity, and find some surprising similarities.
10. Dark Matter, Elko Fields and Weinberg's Quantum Field Theory Formalism
Science.gov (United States)
2012-02-01
The Elko quantum field was introduced by Ahluwalia and Grumiller, who proposed it as a candidate for dark matter. We study the Elko field in Wemberg's formalism for quantum field theory. We prove that if one takes the symmetry group to be the full Pomcaré group then the Elko field is not a quantum field in the sense of Weinberg. This confirms results of Ahluwalia, Lee and Schritt, who showed using a different approach that the Elko field does not transform covariantly under rotations and hence has a preferred axis.
11. Coadjoint orbits and conformal field theory
Energy Technology Data Exchange (ETDEWEB)
Taylor, IV, Washington [Univ. of California, Berkeley, CA (United States)
1993-08-01
This thesis is primarily a study of certain aspects of the geometric and algebraic structure of coadjoint orbit representations of infinite-dimensional Lie groups. The goal of this work is to use coadjoint orbit representations to construct conformal field theories, in a fashion analogous to the free-field constructions of conformal field theories. The new results which are presented in this thesis are as follows: First, an explicit set of formulae are derived giving an algebraic realization of coadjoint orbit representations in terms of differential operators acting on a polynomial Fock space. These representations are equivalent to dual Verma module representations. Next, intertwiners are explicitly constructed which allow the construction of resolutions for irreducible representations using these Fock space realizations. Finally, vertex operators between these irreducible representations are explicitly constructed as chain maps between the resolutions; these vertex operators allow the construction of rational conformal field theories according to an algebraic prescription.
12. Applications of the complex-mass renormalization scheme in effective field theory; Anwendungen des Komplexe-Masse-Renormierungsschemas in effektiver Feldtheorie
Energy Technology Data Exchange (ETDEWEB)
Bauer, Torsten
2012-07-11
In the first part of the this doctoral thesis the perturbative unitarity in the complex-mass scheme (CMS) is analysed. To that end a procedure for calculating cutting rules for loop integrals containing propagators with finite widths is presented. A toy-model Lagrangian describing the interaction of a heavy vector boson with a light fermion is used to demonstrate that the CMS respects unitarity order by order in perturbation theory provided that the renormalized coupling constant remains real. The second part of the thesis deals with various applications of the CMS to chiral effective field theory (EFT). In particular, mass and width of the delta resonance, elastic electromagnetic form factors of the Roper resonance, form factors of the nucleon-to-Roper transition, pion-nucleon scattering, and pion photo- and electroproduction for center-of-mass energies in the region of the Roper mass are calculated. By choosing appropriate renormalization conditions, a consistent chiral power counting scheme for EFT with resonant degrees of freedom can be established. This allows for a systematic investigation of the above processes in terms of an expansion in small quantities. The obtained results can be applied to the extrapolation of corresponding simulations in the context of lattice QCD to the physical value of the pion mass. Therefore, in addition to the Q{sup 2} dependence of the form factors, also the pion-mass dependence of the magnetic moment and electromagnetic radii of the Roper resonance is explored. Both a partial wave decomposition and a multipole expansion are performed for pion-nucleon scattering and pion photo- and electroproduction, respectively. In this connection the P11 partial wave as well as the M{sub 1-} and S{sub 1-} multipoles are fitted via non-linear regression to empirical data.
13. The space-time operator product expansion in string theory duals of field theories
International Nuclear Information System (INIS)
Aharony, Ofer; Komargodski, Zohar
2008-01-01
We study the operator product expansion (OPE) limit of correlation functions in field theories which possess string theory duals, from the point of view of the string worldsheet. We show how the interesting ('single-trace') terms in the OPE of the field theory arise in this limit from the OPE of the worldsheet theory of the string dual, using a dominant saddle point which appears in computations of worldsheet correlation functions in the space-time OPE limit. The worldsheet OPE generically contains only non-physical operators, but all the non-physical contributions are resummed by the saddle point to a contribution similar to that of a physical operator, which exactly matches the field theory expectations. We verify that the OPE limit of the worldsheet theory does not have any other contributions to the OPE limit of space-time correlation functions. Our discussion is completely general and applies to any local field theory (conformal at high energies) that has a weakly coupled string theory dual (with arbitrary curvature). As a first application, we compare our results to a proposal of R. Gopakumar for the string theory dual of free gauge theories
14. Tephra: field, theory and application
Science.gov (United States)
Pouget, Solene
In this work we briefly introduced the current state of the art for plume dynamics and plume modelling (chapters 1 and 2). From these, it was found that several questions remained unanswered. One of them what about adding some quantitative methodology to tephra identification when using geochemistry. Using discontinuous two tephra layers discovered at Burney Spring Mountain, northern California, this aspect was explored. Stratigraphic relationships suggest that they are two distinct tephras. Binary plots and standard similarity coefficients of electron probe microanalysis data have been supplemented with principal component analysis in log-ratio transformed data to correlate the two tephra layers to known regional tephras. Using principal component analysis, we are furthermore able to bound our uncertainty in the correlation of the two tephra layers (chapter 3). After removal of outliers, within the 95% prediction interval, we can say that one tephra layer is likely the Rockland tephra, aged 565-610 ka, and the second layer is likely from Mt Mazama, the Trego Hot Springs tephra, aged ~29 ka. Using cluster analysis on several vectors of chemical elements another quantitative methodology was explored (chapter 4). It was found that in most cases, geochemical analysis of a tephra layer will be assign to a single cluster, however in some cases the analysis are spread over several clusters. This spreading is a direct result of mixing and reworking happening in the tephra layer. The dynamics of volcanic plumes were also investigated. We introduce a new method to estimate mass eruption rate (MER) and mass loading from the growth of a volcanic umbrella cloud or downwind plume using satellite images, or photographs where ground-based observations are available with a gravity current model (chapter 5). The results show a more fully characterised MER as a function of time than do the results given by pre-existing methods, and allow a faster, remote assessment of the mass eruption rate, even for volcanoes that are difficult to study. A new gravity current model for umbrella cloud was tested which allows to transition from one regime to another against measured data from several eruptions (chapter 6). Once the model was proved to be accurate, the different variables were tested to observed their impact on the spreading of the umbrella cloud. As a result it was found that the evolution of the radius changes not only in power-law with time but also indicates transitions in regimes. Chapter 7 is an end-to-end framework to probabilistic forecasting of volcanic ash transport and improved eruption source parameters. In summary, this dissertation demonstrates four main contributions to volcanology: 1. The importance of bringing quantitative methods to tephra identification and how these methods can help in characterization of tephra. 2. The importance of the spread of volcanic cloud in the atmosphere as a gravity current. Particularly for prediction of the ash dispersal since the spreading as a gravity current happens over large distances from the volcano and even upwind. But also because it can help in getting a first and fast estimation of the mass eruption rate of an eruption which can be followed with time. 3. The importance of studying the structures and features on volcanic plume as they can reveal information about the dynamics of spreading and improve the estimation of regime transitions. 4. The need for the different communities working on tephra to communicate and understand each others approach fo better collaboration and multi-approach work.
15. Distributed hash table theory, platforms and applications
CERN Document Server
Zhang, Hao; Xie, Haiyong; Yu, Nenghai
2013-01-01
This SpringerBrief summarizes the development of Distributed Hash Table in both academic and industrial fields. It covers the main theory, platforms and applications of this key part in distributed systems and applications, especially in large-scale distributed environments. The authors teach the principles of several popular DHT platforms that can solve practical problems such as load balance, multiple replicas, consistency and latency. They also propose DHT-based applications including multicast, anycast, distributed file systems, search, storage, content delivery network, file sharing and c
16. Knots, topology and quantum field theories
International Nuclear Information System (INIS)
Lusanna, L.
1989-01-01
The title of the workshop, Knots, Topology and Quantum Field Theory, accurate reflected the topics discussed. There have been important developments in mathematical and quantum field theory in the past few years, which had a large impact on physicist thinking. It is historically unusual and pleasing that these developments are taking place as a result of an intense interaction between mathematical physicists and mathematician. On the one hand, topological concepts and methods are playing an increasingly important lead to novel mathematical concepts: for instance, the study of quantum groups open a new chapter in the deformation theory of Lie algebras. These developments at present will lead to new insights into the theory of elementary particles and their interactions. In essence, the talks dealt with three, broadly defined areas of theoretical physics. One was topological quantum field theories, the other the problem of quantum groups and the third one certain aspects of more traditional field theories, such as, for instance, quantum gravity. These topics, however, are interrelated and the general theme of the workshop defies rigid classification; this was evident from the cross references to be found in almo all the talks
17. On the unitary transformation between non-quasifree and quasifree state spaces and its application to quantum field theory on curved spacetimes
International Nuclear Information System (INIS)
Gottschalk, Hanno; Hack, Thomas-Paul
2009-12-01
Using *-calculus on the dual of the Borchers-Uhlmann algebra endowed with a combinatorial co-product, we develop a method to calculate a unitary transformation relating the GNS representations of a non-quasifree and a quasifree state of the free hermitian scalar field. The motivation for such an analysis and a further result is the fact that a unitary transformation of this kind arises naturally in scattering theory on non-stationary backgrounds. Indeed, employing the perturbation theory of the Yang-Feldman equations with a free CCR field in a quasifree state as an initial condition and making use of extended Feynman graphs, we are able to calculate the Wightman functions of the interacting and outgoing fields in a φ p -theory on arbitrary curved spacetimes. A further examination then reveals two major features of the aforementioned theory: firstly, the interacting Wightman functions fulfil the basic axioms of hermiticity, invariance, spectrality (on stationary spacetimes), perturbative positivity, and locality. Secondly, the outgoing field is free and fulfils the CCR, but is in general not in a quasifree state in the case of a non-stationary spacetime. In order to obtain a sensible particle picture for the outgoing field and, hence, a description of the scattering process in terms of particles (in asymptotically flat spacetimes), it is thus necessary to compute a unitary transformation of the abovementioned type. (orig.)
18. Experimental signature of scaling violation implied by field theories
International Nuclear Information System (INIS)
Tung, W.
1975-01-01
Renormalizable field theories are found to predict a surprisingly specific pattern of scaling violation in deep inelastic scattering. Comparison with experiments is discussed. The feasibility of distinguishing asymptotically free field theories from conventional field theories is evaluated
19. Nonequilibrium statistical field theory for classical particles: Basic kinetic theory.
Science.gov (United States)
Viermann, Celia; Fabis, Felix; Kozlikin, Elena; Lilow, Robert; Bartelmann, Matthias
2015-06-01
Recently Mazenko and Das and Mazenko [Phys. Rev. E 81, 061102 (2010); J. Stat. Phys. 149, 643 (2012); J. Stat. Phys. 152, 159 (2013); Phys. Rev. E 83, 041125 (2011)] introduced a nonequilibrium field-theoretical approach to describe the statistical properties of a classical particle ensemble starting from the microscopic equations of motion of each individual particle. We use this theory to investigate the transition from those microscopic degrees of freedom to the evolution equations of the macroscopic observables of the ensemble. For the free theory, we recover the continuity and Jeans equations of a collisionless gas. For a theory containing two-particle interactions in a canonical perturbation series, we find the macroscopic evolution equations to be described by the Born-Bogoliubov-Green-Kirkwood-Yvon hierarchy with a truncation criterion depending on the order in perturbation theory. This establishes a direct link between the classical and the field-theoretical approaches to kinetic theory that might serve as a starting point to investigate kinetic theory beyond the classical limits.
20. Further studies in aesthetic field theory
International Nuclear Information System (INIS)
1984-01-01
We study different facets of Aesthetic Field Theory. First, we have found within the complex version of the theory a bounded particle system which has more structure than what we have previously observed. The particle is built from planar 3 maxima--minima confluence regions. The confluence region closes in 3 spatial dimensions, so once again we have a ''topological'' particle. If we characterize bound stats by the number of large magnitude regions in close proximity, then the simplest interpretation of what we are seeing is that of a 3 particle bound system. Secondly, again within the framework of complex Aesthetic Field Theory, but using a more symmetric system of equations, we observe a confluence type topological particle spontaneously arising out of the vacuum (creation effect). The particle again has a loop shape. The extended particles thus far found in 4 dimensional Aesthetic Field Theory have always had problems with the spreading of the particle system as time went on. Thirdly, we found a bounded confluence particle system, which in addition to confinement and non attenuation shows the desirable property of not spreading in time. In this case, we work exclusively with real fields. The particle system has a dipole looking shape. We also studied complex null Aesthetic Field Theory in 8 dimensions having a 4 direct-sum 4 structure. We were not able to find a bound to our particle system here
1. Massive deformations of Type IIA theory within double field theory
Science.gov (United States)
Çatal-Özer, Aybike
2018-02-01
We obtain massive deformations of Type IIA supergravity theory through duality twisted reductions of Double Field Theory (DFT) of massless Type II strings. The mass deformation is induced through the reduction of the DFT of the RR sector. Such reductions are determined by a twist element belonging to Spin+(10, 10), which is the duality group of the DFT of the RR sector. We determine the form of the twists and give particular examples of twists matrices, for which a massive deformation of Type IIA theory can be obtained. In one of the cases, requirement of gauge invariance of the RR sector implies that the dilaton field must pick up a linear dependence on one of the dual coordinates. In another case, the choice of the twist matrix violates the weak and the strong constraints explicitly in the internal doubled space.
2. Recent Developments in D=2 String Field Theory
OpenAIRE
Kaku, Michio
1994-01-01
In this review article, we review the recent developments in constructing string field theories that have been proposed, all of which correctly reproduce the correlation functions of two-dimensional string theory. These include: (a) free fermion field theory (b) collective string field theory (c) temporal gauge string field theory (d) non-polynomial string field theory. We analyze discrete states, the $w(\\infty)$ symmetry, and correlation functions in terms of these different string field the...
3. A non-linear field theory
International Nuclear Information System (INIS)
Skyrme, T.H.R.
1994-01-01
A unified field theory of mesons and their particle sources is proposed and considered in its classical aspects. The theory has static solutions of a singular nature, but finite energy, characterized by spin directions; the number of such entities is a rigorously conserved constant of motion; they interact with an external meson field through a derivative-type coupling with the spins, akin to the formalism of strong-coupling meson theory. There is a conserved current identifiable with isobaric spin, and another that may be related to hypercharge. The postulates include one constant of the dimensions of length, and another that is conjecture necessarily to have the value (h/2π)c, or perhaps 1/2(h/2π)c, in the quantized theory. (author). 5 refs
4. Field theory of the Eulerian perfect fluid
Science.gov (United States)
Ariki, Taketo; Morales, Pablo A.
2018-01-01
The Eulerian perfect-fluid theory is reformulated from its action principle in a pure field-theoretic manner. Conservation of the convective current is no longer imposed by Lin’s constraints, but rather adopted as the central idea of the theory. Our formulation, for the first time, successfully reduces redundant degrees of freedom promoting one half of the Clebsch variables to true dynamical fields. Interactions on these fields allow for the exchange of the convective current of quantities such as mass and charge, which are uniformly understood as the breaking of the underlying symmetry of the force-free fluid. The Clebsch fields play the essential role of exchanging angular momentum with the force field producing vorticity.
5. A general field-covariant formulation of quantum field theory
International Nuclear Information System (INIS)
Anselmi, Damiano
2013-01-01
In all nontrivial cases renormalization, as it is usually formulated, is not a change of integration variables in the functional integral, plus parameter redefinitions, but a set of replacements, of actions and/or field variables and parameters. Because of this, we cannot write simple identities relating bare and renormalized generating functionals, or generating functionals before and after nonlinear changes of field variables. In this paper we investigate this issue and work out a general field-covariant approach to quantum field theory, which allows us to treat all perturbative changes of field variables, including the relation between bare and renormalized fields, as true changes of variables in the functional integral, under which the functionals Z and W=lnZ behave as scalars. We investigate the relation between composite fields and changes of field variables, and we show that, if J are the sources coupled to the elementary fields, all changes of field variables can be expressed as J-dependent redefinitions of the sources L coupled to the composite fields. We also work out the relation between the renormalization of variable-changes and the renormalization of composite fields. Using our transformation rules it is possible to derive the renormalization of a theory in a new variable frame from the renormalization in the old variable frame, without having to calculate it anew. We define several approaches, useful for different purposes, in particular a linear approach where all variable changes are described as linear source redefinitions. We include a number of explicit examples. (orig.)
6. Neutrix calculus and finite quantum field theory
International Nuclear Information System (INIS)
Ng, Y Jack; Dam, H van
2005-01-01
In general, quantum field theories (QFT) require regularizations and infinite renormalizations due to ultraviolet divergences in their loop calculations. Furthermore, perturbation series in theories like quantum electrodynamics are not convergent series, but are asymptotic series. We apply neutrix calculus, developed in connection with asymptotic series and divergent integrals, to QFT, obtaining finite renormalizations. While none of the physically measurable results in renormalizable QFT is changed, quantum gravity is rendered more manageable in the neutrix framework. (letter to the editor)
7. Quantum field theory in generalised Snyder spaces
International Nuclear Information System (INIS)
Meljanac, S.; Meljanac, D.; Mignemi, S.; Štrajn, R.
2017-01-01
We discuss the generalisation of the Snyder model that includes all possible deformations of the Heisenberg algebra compatible with Lorentz invariance and investigate its properties. We calculate perturbatively the law of addition of momenta and the star product in the general case. We also undertake the construction of a scalar field theory on these noncommutative spaces showing that the free theory is equivalent to the commutative one, like in other models of noncommutative QFT.
8. Staircase Models from Affine Toda Field Theory
CERN Document Server
Dorey, P; Dorey, Patrick; Ravanini, Francesco
1993-01-01
We propose a class of purely elastic scattering theories generalising the staircase model of Al. B. Zamolodchikov, based on the affine Toda field theories for simply-laced Lie algebras g=A,D,E at suitable complex values of their coupling constants. Considering their Thermodynamic Bethe Ansatz equations, we give analytic arguments in support of a conjectured renormalisation group flow visiting the neighbourhood of each W_g minimal model in turn.
9. Quantum field theory in generalised Snyder spaces
Energy Technology Data Exchange (ETDEWEB)
Meljanac, S.; Meljanac, D. [Rudjer Bošković Institute, Bijenička cesta 54, 10002 Zagreb (Croatia); Mignemi, S., E-mail: [email protected] [Dipartimento di Matematica e Informatica, Università di Cagliari, viale Merello 92, 09123 Cagliari (Italy); INFN, Sezione di Cagliari, Cittadella Universitaria, 09042 Monserrato (Italy); Štrajn, R. [Dipartimento di Matematica e Informatica, Università di Cagliari, viale Merello 92, 09123 Cagliari (Italy); INFN, Sezione di Cagliari, Cittadella Universitaria, 09042 Monserrato (Italy)
2017-05-10
We discuss the generalisation of the Snyder model that includes all possible deformations of the Heisenberg algebra compatible with Lorentz invariance and investigate its properties. We calculate perturbatively the law of addition of momenta and the star product in the general case. We also undertake the construction of a scalar field theory on these noncommutative spaces showing that the free theory is equivalent to the commutative one, like in other models of noncommutative QFT.
10. Magnetic monopoles in field theory and cosmology.
Science.gov (United States)
Rajantie, Arttu
2012-12-28
The existence of magnetic monopoles is predicted by many theories of particle physics beyond the standard model. However, in spite of extensive searches, there is no experimental or observational sign of them. I review the role of magnetic monopoles in quantum field theory and discuss their implications for particle physics and cosmology. I also highlight their differences and similarities with monopoles found in frustrated magnetic systems.
11. Catastrophe theory and its application status in mechanical engineering
Directory of Open Access Journals (Sweden)
Jinge LIU
Full Text Available Catastrophe theory is a kind of mathematical method which aims to apply and interpret the discontinuous phenomenon. Since its emergence, it has been widely used to explain a variety of emergent phenomena in the fields of natural science, social science, management science and some other science and technology fields. Firstly, this paper introduces the theory of catastrophe in several aspects, such as its generation, radical principle, basic characteristics and development. Secondly, it summarizes the main applications of catastrophe theory in the field of mechanical engineering, focusing on the research progress of catastrophe theory in revealing catastrophe of rotor vibration state, analyzing friction and wear failure, predicting metal fracture, and so on. Finally, it advises that later development of catastrophe theory should pay more attention to the combination of itself with other traditional nonlinear theories and methods. This paper provides a beneficial reference to guide the application of catastrophe theory in mechanical engineering and related fields for later research.
12. Adsorption refrigeration technology theory and application
CERN Document Server
Wang, Ruzhu; Wu, Jingyi
2014-01-01
Gives readers a detailed understanding of adsorption refrigeration technology, with a focus on practical applications and environmental concerns Systematically covering the technology of adsorption refrigeration, this book provides readers with a technical understanding of the topic as well as detailed information on the state-of-the-art from leading researchers in the field. Introducing readers to background on the development of adsorption refrigeration, the authors also cover the development of adsorbents, various thermodynamic theories, the design of adsorption systems and adsorption refri
13. Wireless network security theories and applications
CERN Document Server
Chen, Lei; Zhang, Zihong
2013-01-01
Wireless Network Security Theories and Applications discusses the relevant security technologies, vulnerabilities, and potential threats, and introduces the corresponding security standards and protocols, as well as provides solutions to security concerns. Authors of each chapter in this book, mostly top researchers in relevant research fields in the U.S. and China, presented their research findings and results about the security of the following types of wireless networks: Wireless Cellular Networks, Wireless Local Area Networks (WLANs), Wireless Metropolitan Area Networks (WMANs), Bluetooth
14. Effective field theory for magnetic compactifications
Science.gov (United States)
Buchmuller, Wilfried; Dierigl, Markus; Dudas, Emilian; Schweizer, Julian
2017-04-01
Magnetic flux plays an important role in compactifications of field and string theories in two ways, it generates a multiplicity of chiral fermion zero modes and it can break supersymmetry. We derive the complete four-dimensional effective action for N = 1 supersymmetric Abelian and non-Abelian gauge theories in six dimensions compactified on a torus with flux. The effective action contains the tower of charged states and it accounts for the mass spectrum of bosonic and fermionic fields as well as their level-dependent interactions. This allows us to compute quantum corrections to the mass and couplings of Wilson lines. We find that the one-loop corrections vanish, contrary to the case without flux. This can be traced back to the spontaneous breaking of symmetries of the six-dimensional theory by the background gauge field, with the Wilson lines as Goldstone bosons.
15. Effective field theory for magnetic compactifications
Energy Technology Data Exchange (ETDEWEB)
Buchmuller, Wilfried; Dierigl, Markus; Schweizer Julian [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany); Dudas, Emilian [Univ. Paris-Saclay, Palaiseau (France). Ecole Polytechnique
2016-12-15
Magnetic flux plays an important role in compactifications of field and string theories in two ways, it generates a multiplicity of chiral fermion zero modes and it can break supersymmetry. We derive the complete four-dimensional effective action for N=1 supersymmetric Abelian and non-Abelian gauge theories in six dimensions compactified on a torus with flux. The effective action contains the tower of charged states and it accounts for the mass spectrum of bosonic and fermionic fields as well as their level-dependent interactions. This allows us to compute quantum corrections to the mass and couplings of Wilson lines. We find that the one-loop corrections vanish, contrary to the case without flux. This can be traced back to the spontaneous breaking of symmetries of the six-dimensional theory by the background gauge field, with the Wilson lines as Goldstone bosons.
16. Effective field theory for magnetic compactifications
Energy Technology Data Exchange (ETDEWEB)
Buchmuller, Wilfried; Dierigl, Markus [Deutsches Elektronen-Synchrotron DESY,22607 Hamburg (Germany); Dudas, Emilian [Centre de Physique Théorique, École Polytechnique, CNRS, Université Paris-Saclay,F-91128 Palaiseau (France); Schweizer, Julian [Deutsches Elektronen-Synchrotron DESY,22607 Hamburg (Germany)
2017-04-10
Magnetic flux plays an important role in compactifications of field and string theories in two ways, it generates a multiplicity of chiral fermion zero modes and it can break supersymmetry. We derive the complete four-dimensional effective action for N=1 supersymmetric Abelian and non-Abelian gauge theories in six dimensions compactified on a torus with flux. The effective action contains the tower of charged states and it accounts for the mass spectrum of bosonic and fermionic fields as well as their level-dependent interactions. This allows us to compute quantum corrections to the mass and couplings of Wilson lines. We find that the one-loop corrections vanish, contrary to the case without flux. This can be traced back to the spontaneous breaking of symmetries of the six-dimensional theory by the background gauge field, with the Wilson lines as Goldstone bosons.
17. Group theory for chemists fundamental theory and applications
CERN Document Server
Molloy, K C
2010-01-01
The basics of group theory and its applications to themes such as the analysis of vibrational spectra and molecular orbital theory are essential knowledge for the undergraduate student of inorganic chemistry. The second edition of Group Theory for Chemists uses diagrams and problem-solving to help students test and improve their understanding, including a new section on the application of group theory to electronic spectroscopy.Part one covers the essentials of symmetry and group theory, including symmetry, point groups and representations. Part two deals with the application of group theory t
18. Classical field theory on electrodynamics, non-abelian gauge theories and gravitation
CERN Document Server
Scheck, Florian
2018-01-01
Scheck’s successful textbook presents a comprehensive treatment, ideally suited for a one-semester course. The textbook describes Maxwell's equations first in their integral, directly testable form, then moves on to their local formulation. The first two chapters cover all essential properties of Maxwell's equations, including their symmetries and their covariance in a modern notation. Chapter 3 is devoted to Maxwell's theory as a classical field theory and to solutions of the wave equation. Chapter 4 deals with important applications of Maxwell's theory. It includes topical subjects such as metamaterials with negative refraction index and solutions of Helmholtz' equation in paraxial approximation relevant for the description of laser beams. Chapter 5 describes non-Abelian gauge theories from a classical, geometric point of view, in analogy to Maxwell's theory as a prototype, and culminates in an application to the U(2) theory relevant for electroweak interactions. The last chapter 6 gives a concise summary...
19. Conformal field theory with gauge symmetry
CERN Document Server
Ueno, Kenji
2008-01-01
This book presents a systematic approach to conformal field theory with gauge symmetry from the point of view of complex algebraic geometry. After presenting the basic facts of the theory of compact Riemann surfaces and the representation theory of affine Lie algebras in Chapters 1 and 2, conformal blocks for pointed Riemann surfaces with coordinates are constructed in Chapter 3. In Chapter 4 the sheaf of conformal blocks associated to a family of pointed Riemann surfaces with coordinates is constructed, and in Chapter 5 it is shown that this sheaf supports a projective flat connection-one of
20. Reggeon field theory for large Pomeron loops
International Nuclear Information System (INIS)
Altinoluk, Tolga; Kovner, Alex; Levin, Eugene; Lublinsky, Michael
2014-01-01
We analyze the range of applicability of the high energy Reggeon Field Theory H RFT derived in http://dx.doi.org/10.1088/1126-6708/2009/03/109. We show that this theory is valid as long as at any intermediate value of rapidity η throughout the evolution at least one of the colliding objects is dilute. Importantly, at some values of η the dilute object could be the projectile, while at others it could be the target, so that H RFT does not reduce to either H JIMWLK or H KLWMIJ . When both objects are dense, corrections to the evolution not accounted for in http://dx.doi.org/10.1088/1126-6708/2009/03/109 become important. The same limitation applies to other approaches to high energy evolution available today, such as for example (http://dx.doi.org/10.1103/PhysRevD.78.054019; http://dx.doi.org/10.1103/PhysRevD.78.054020 and http://dx.doi.org/10.1016/S0370-2693(00)00571-2; http://dx.doi.org/10.1140/epjc/s2003-01565-9; http://dx.doi.org/10.1016/j.physletb.2005.10.054). We also show that, in its regime of applicability H RFT can be simplified. We derive the simpler version of H RFT and in the large N c limit rewrite it in terms of the Reggeon creation and annihilation operators. The resulting H RFT is explicitly self dual and provides the generalization of the Pomeron calculus developed in (http://dx.doi.org/10.1016/S0370-2693(00)00571-2; http://dx.doi.org/10.1140/epjc/s2003-01565-9; http://dx.doi.org/10.1016/j.physletb.2005.10.054) by including higher Reggeons in the evolution. It is applicable for description of ‘large’ Pomeron loops, namely Reggeon graphs where all the splittings occur close in rapidity to one dilute object (projectile), while all the merging close to the other one (target). Additionally we derive, in the same regime expressions for single and double inclusive gluon production (where the gluons are not separated by a large rapidity interval) in terms of the Reggeon degrees of freedom
1. Supergauge Field Theory of Covariant Heterotic Strings
OpenAIRE
Michio, KAKU; Physics Department, Osaka University : Physics Department, City College of the City University of New York
1986-01-01
We present the gauge covariant second quantized field theory for free heterotic strings, which is leading candidate for a unified theory of all known particles. Our action is invariant under the semi-direct product of the super Virasoro and the Kac-Moody E_8×E_8 or Spin(32)/Z_2 group. We derive the covariant action by path integrals in the same way that Feynman originally derived the Schrodinger equation. By adding an infinite number of auxiliary fields, we can also make the action explicitly...
2. Field theory a path integral approach
CERN Document Server
Das, Ashok
2006-01-01
This unique book describes quantum field theory completely within the context of path integrals. With its utility in a variety of fields in physics, the subject matter is primarily developed within the context of quantum mechanics before going into specialized areas.Adding new material keenly requested by readers, this second edition is an important expansion of the popular first edition. Two extra chapters cover path integral quantization of gauge theories and anomalies, and a new section extends the supersymmetry chapter, where singular potentials in supersymmetric systems are described.
3. A geometric formulation of exceptional field theory
Energy Technology Data Exchange (ETDEWEB)
Bosque, Pascal du [Arnold Sommerfeld Center for Theoretical Physics,Department für Physik, Ludwig-Maximilians-Universität München,Theresienstraße 37, 80333 München (Germany); Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, Föhringer Ring 6, 80805 München (Germany); Hassler, Falk [Department of Physics and Astronomy, University of North Carolina, Phillips Hall, CB #3255, 120 E. Cameron Ave., Chapel Hill, NC 27599-3255 (United States); City University of New York, The Graduate Center, 365 Fifth Avenue, New York, NY 10016 (United States); Department of Physics, Columbia University, Pupin Hall, 550 West 120th St., New York, NY 10027 (United States); Lüst, Dieter [Arnold Sommerfeld Center for Theoretical Physics,Department für Physik, Ludwig-Maximilians-Universität München,Theresienstraße 37, 80333 München (Germany); Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, Föhringer Ring 6, 80805 München (Germany); Malek, Emanuel [Arnold Sommerfeld Center for Theoretical Physics,Department für Physik, Ludwig-Maximilians-Universität München,Theresienstraße 37, 80333 München (Germany)
2017-03-01
We formulate the full bosonic SL(5) exceptional field theory in a coordinate-invariant manner. Thereby we interpret the 10-dimensional extended space as a manifold with SL(5)×ℝ{sup +}-structure. We show that the algebra of generalised diffeomorphisms closes subject to a set of closure constraints which are reminiscent of the quadratic and linear constraints of maximal seven-dimensional gauged supergravities, as well as the section condition. We construct an action for the full bosonic SL(5) exceptional field theory, even when the SL(5)×ℝ{sup +}-structure is not locally flat.
4. Statistical field theory of futures commodity prices
Science.gov (United States)
Baaquie, Belal E.; Yu, Miao
2018-02-01
The statistical theory of commodity prices has been formulated by Baaquie (2013). Further empirical studies of single (Baaquie et al., 2015) and multiple commodity prices (Baaquie et al., 2016) have provided strong evidence in support the primary assumptions of the statistical formulation. In this paper, the model for spot prices (Baaquie, 2013) is extended to model futures commodity prices using a statistical field theory of futures commodity prices. The futures prices are modeled as a two dimensional statistical field and a nonlinear Lagrangian is postulated. Empirical studies provide clear evidence in support of the model, with many nontrivial features of the model finding unexpected support from market data.
5. From topological quantum field theories to supersymmetric gauge theories
International Nuclear Information System (INIS)
Bossard, G.
2007-10-01
This thesis contains 2 parts based on scientific contributions that have led to 2 series of publications. The first one concerns the introduction of vector symmetry in cohomological theories, through a generalization of the so-called Baulieu-Singer equation. Together with the topological BRST (Becchi-Rouet-Stora-Tyutin) operator, this symmetry gives an off-shell closed sub-sector of supersymmetry that permits to determine the action uniquely. The second part proposes a methodology for re-normalizing supersymmetric Yang-Mills theory without assuming a regularization scheme which is both supersymmetry and gauge invariance preserving. The renormalization prescription is derived thanks to the definition of 2 consistent Slavnov-Taylor operators for supersymmetry and gauge invariance, whose construction requires the introduction of the so-called shadow fields. We demonstrate the renormalizability of supersymmetric Yang-Mills theories. We give a fully consistent, regularization scheme independent, proof of the vanishing of the β function and of the anomalous dimensions of the one half BPS operators in maximally supersymmetric Yang-Mills theory. After a short introduction, in chapter two, we give a review of the cohomological Yang-Mills theory in eight dimensions. We then study its dimensional reductions in seven and six dimensions. The last chapter gives quite independent results, about a geometrical interpretation of the shadow fields, an unpublished work about topological gravity in four dimensions, an extension of the shadow formalism to superconformal invariance, and finally the solution of the constraints in a twisted superspace. (author)
6. Uncertainty quantification theory, implementation, and applications
CERN Document Server
Smith, Ralph C
2014-01-01
The field of uncertainty quantification is evolving rapidly because of increasing emphasis on models that require quantified uncertainties for large-scale applications, novel algorithm development, and new computational architectures that facilitate implementation of these algorithms. Uncertainty Quantification: Theory, Implementation, and Applications provides readers with the basic concepts, theory, and algorithms necessary to quantify input and response uncertainties for simulation models arising in a broad range of disciplines. The book begins with a detailed discussion of applications where uncertainty quantification is critical for both scientific understanding and policy. It then covers concepts from probability and statistics, parameter selection techniques, frequentist and Bayesian model calibration, propagation of uncertainties, quantification of model discrepancy, surrogate model construction, and local and global sensitivity analysis. The author maintains a complementary web page where readers ca...
7. Effective field theory and the quark model
International Nuclear Information System (INIS)
Durand, Loyal; Ha, Phuoc; Jaczko, Gregory
2001-01-01
We analyze the connections between the quark model (QM) and the description of hadrons in the low-momentum limit of heavy-baryon effective field theory in QCD. By using a three-flavor-index representation for the effective baryon fields, we show that the 'nonrelativistic' constituent QM for baryon masses and moments is completely equivalent through O(m s ) to a parametrization of the relativistic field theory in a general spin-flavor basis. The flavor and spin variables can be identified with those of effective valence quarks. Conversely, the spin-flavor description clarifies the structure and dynamical interpretation of the chiral expansion in effective field theory, and provides a direct connection between the field theory and the semirelativistic models for hadrons used in successful dynamical calculations. This allows dynamical information to be incorporated directly into the chiral expansion. We find, for example, that the striking success of the additive QM for baryon magnetic moments is a consequence of the relative smallness of the non-additive spin-dependent corrections
8. On space of integrable quantum field theories
Directory of Open Access Journals (Sweden)
F.A. Smirnov
2017-02-01
Full Text Available We study deformations of 2D Integrable Quantum Field Theories (IQFT which preserve integrability (the existence of infinitely many local integrals of motion. The IQFT are understood as “effective field theories”, with finite ultraviolet cutoff. We show that for any such IQFT there are infinitely many integrable deformations generated by scalar local fields Xs, which are in one-to-one correspondence with the local integrals of motion; moreover, the scalars Xs are built from the components of the associated conserved currents in a universal way. The first of these scalars, X1, coincides with the composite field (TT¯ built from the components of the energy–momentum tensor. The deformations of quantum field theories generated by X1 are “solvable” in a certain sense, even if the original theory is not integrable. In a massive IQFT the deformations Xs are identified with the deformations of the corresponding factorizable S-matrix via the CDD factor. The situation is illustrated by explicit construction of the form factors of the operators Xs in sine-Gordon theory. We also make some remarks on the problem of UV completeness of such integrable deformations.
9. Singular traces theory and applications
CERN Document Server
Sukochev, Fedor; Zanin, Dmitriy
2012-01-01
This text is the first complete study and monograph dedicated to singular traces. For mathematical readers the text offers, due to Nigel Kalton's contribution, a complete theory of traces on symmetrically normed ideals of compact operators. For mathematical physicists and other users of Connes' noncommutative geometry the text offers a complete reference to Dixmier traces and the deeper mathematical features of singular traces. An application section explores the consequences of these features, which previously were not discussed in general texts on noncommutative geometry.
10. Asymptotic functions and their application in quantum theory
International Nuclear Information System (INIS)
Khristov, Kh.Ya.; Damyanov, B.P.
1979-01-01
An asymptotic function introduced as a limit for a certain class of successions has been determined. The basic properties of the functions are given: continuity, differentiability, integrability. The fields of application of the asymptotic functions in the quantum field theory are presented. The shortcomings and potentialities of further development of the theory are enumerated
11. Wavelets theory, algorithms, and applications
CERN Document Server
Montefusco, Laura
2014-01-01
Wavelets: Theory, Algorithms, and Applications is the fifth volume in the highly respected series, WAVELET ANALYSIS AND ITS APPLICATIONS. This volume shows why wavelet analysis has become a tool of choice infields ranging from image compression, to signal detection and analysis in electrical engineering and geophysics, to analysis of turbulent or intermittent processes. The 28 papers comprising this volume are organized into seven subject areas: multiresolution analysis, wavelet transforms, tools for time-frequency analysis, wavelets and fractals, numerical methods and algorithms, and applicat
12. Diatomic interaction potential theory applications
CERN Document Server
Goodisman, Jerry
2013-01-01
Diatomic Interaction Potential Theory, Volume 2: Applications discusses the variety of applicable theoretical material and approaches in the calculations for diatomic systems in their ground states. The volume covers the descriptions and illustrations of modern calculations. Chapter I discusses the calculation of the interaction potential for large and small values of the internuclear distance R (separated and united atom limits). Chapter II covers the methods used for intermediate values of R, which in principle means any values of R. The Hartree-Fock and configuration interaction schemes des
13. The nonlinearity of the scalar field in a relativistic mean-field theory of the nucleus
International Nuclear Information System (INIS)
Reinhard, P.G.
1987-10-01
The form of the nonlinear selfcoupling of the scalar meson field in a nuclear relativistic mean-field theory is investigated. The conventional ansatz is shown to produce instabilities in critical applications. A modified selfcoupling is proposed which guarantees stability under all conditions. (orig.)
14. On the general theory of quantized fields
International Nuclear Information System (INIS)
Fredenhagen, K.
1991-10-01
In my lecture I describe the present stage of the general theory of quantized fields on the example of 5 subjects. They are ordered in the direction from large to small distances. The first one is the by now classical problem of the structure of superselection sectors. It involves the behavior of the theory at spacelike infinity and is directly connected with particle statistics and internal symmetries. It has become popular in recent years by the discovery of a lot of nontrivial models in 2d conformal-field theory, by connections to integrable models and critical behavior in statistical mechanics and by the relations to the Jones' theory of subfactors in von Neumann algebras and to the corresponding geometrical objects (braids, knots, 3d manifolds, ...). At large timelike distances the by far most important feature of quantum field theory is the particle structure. This will be the second subject of my lecture. It follows the technically most involved part which is concerned with the behavior at finite distances. Two aspets, nuclearity which emphasizes the finite density of states in phase space, and the modular structure which relies on the infinite number of degrees of freedom present even locally, and their mutual relations will be treated. The next point, involving the structure at infinitesimal distances, is the connection between the Haag-Kastler framework of algebras of local and the framework of Wightman fields. Finally, problems in approaches to quantum gravity will be discussed, as far as they are accessible by the methods of the general theory of quantized fields. (orig.)
15. On the History of Unified Field Theories
Directory of Open Access Journals (Sweden)
Goenner Hubert F.M.
2004-01-01
Full Text Available This article is intended to give a review of the history of the classical aspects of unified field theories in the 20th century. It includes brief technical descriptions of the theories suggested, short biographical notes concerning the scientists involved, and an extensive bibliography. The present first installment covers the time span between 1914 and 1933, i.e., when Einstein was living and working in Berlin - with occasional digressions into other periods. Thus, the main theme is the unification of the electromagnetic and gravitational fields augmented by short-lived attempts to include the matter field described by Schrödinger's or Dirac's equations. While my focus lies on the conceptual development of the field, by also paying attention to the interaction of various schools of mathematicians with the research done by physicists, some prosopocraphical remarks are included.
16. Maslow's theory and its application to librarianship
OpenAIRE
Sridhar, M. S.
1981-01-01
Explains the basis for Maslow’s theory, enumerates Maslow’s hierarchy of needs, describes implications of the theory and finally presents application of Maslow’s theory to librarianship with suitable examples and illustrations.
17. The path integral method in quantum field theory
International Nuclear Information System (INIS)
Burden, C.J.
1990-01-01
Richard Feynman is reputed to have once the that in his whole life he had only ever had two really clever ideas. As it has turned out, these two ideas, the path integral formulation of quantum mechanics and the diagrammatic representation of perturbation theory have become the cornerstone of modern quantum field theory and particle physics. The path integral, first hinted at by Dirac in the thirties but developed fully by Feynman in the late 40's provides us with an alternative, though equivalent, description of quantum mechanics to canonical quantization. It was not until the seventies that it found broad application in quantum field theory (QFT), leading of gauge theories. It has also lead to the invention of non-perturbative techniques such as lattice gauge theory and has revealed an intimate connection between QFT and statistical mechanics. This paper, the author develops the basic of QFT form the path integral formalism and introduce the idea of Feynman diagrams for calculating perturbation expansions. I will concentrate mainly on the example of φ 4 theory since it is probably the simplest example of an interacting field theory, though the methods generalize readily to more sophisticated theories
18. Symmetry analysis for anisotropic field theories
International Nuclear Information System (INIS)
Parra, Lorena; Vergara, J. David
2012-01-01
The purpose of this paper is to study with the help of Noether's theorem the symmetries of anisotropic actions for arbitrary fields which generally depend on higher order spatial derivatives, and to find the corresponding current densities and the Noether charges. We study in particular scale invariance and consider the cases of higher derivative extensions of the scalar field, electrodynamics and Chern-Simons theory.
19. Anomalies in Witten's NSR superstring field theory
International Nuclear Information System (INIS)
Aref'eva, I.Ya.; Medvedev, P.B.
1988-01-01
The action of Witten's NSR superstring field theory if shown to depend on the regularization being choosen to define its value on non-smooth states that are generated by supertransformation. The necessity of additional regularization originates from the appearance of products of picture-changing operators in coincident points. Two different regularization are described, one corresponding to Witten's scheme and the other to the scheme based on the notion of truncated fields
20. Proceedings of quantum field theory, quantum mechanics, and quantum optics
International Nuclear Information System (INIS)
Dodonov, V.V.; Man; ko, V.I.
1991-01-01
This book contains papers presented at the XVIII International Colloquium on Group Theoretical Methods in Physics held in Moscow on June 4-9, 1990. Topics covered include; applications of algebraic methods in quantum field theory, quantum mechanics, quantum optics, spectrum generating groups, quantum algebras, symmetries of equations, quantum physics, coherent states, group representations and space groups
1. Grassmann methods in lattice field theory and statistical mechanics
International Nuclear Information System (INIS)
Bilgici, E.; Gattringer, C.; Huber, P.
2006-01-01
Full text: In two dimensions models of loops can be represented as simple Grassmann integrals. In our work we explore the generalization of these techniques to lattice field theories and statistical mechanic systems in three and four dimensions. We discuss possible strategies and applications for representations of loop and surface models as Grassmann integrals. (author)
2. Compositional Data Analysis Theory and Applications
CERN Document Server
Pawlowsky-Glahn, Vera
2011-01-01
This book presents the state-of-the-art in compositional data analysis and will feature a collection of papers covering theory, applications to various fields of science and software. Areas covered will range from geology, biology, environmental sciences, forensic sciences, medicine and hydrology. Key features:Provides the state-of-the-art text in compositional data analysisCovers a variety of subject areas, from geology to medicineWritten by leading researchers in the fieldIs supported by a website featuring R code
3. Integrable structures in quantum field theory
International Nuclear Information System (INIS)
Negro, Stefano
2016-01-01
This review was born as notes for a lecture given at the Young Researchers Integrability School (YRIS) school on integrability in Durham, in the summer of 2015. It deals with a beautiful method, developed in the mid-nineties by Bazhanov, Lukyanov and Zamolodchikov and, as such, called BLZ. This method can be interpreted as a field theory version of the quantum inverse scattering, also known as the algebraic Bethe ansatz. Starting with the case of conformal field theories (CFTs) we show how to build the field theory analogues of commuting transfer T matrices and Baxter Q -operators of integrable lattice models. These objects contain the complete information of the integrable structure of the theory, viz. the integrals of motion, and can be used, as we will show, to derive the thermodynamic Bethe ansatz and nonlinear integral equations. This same method can be easily extended to the description of integrable structures of certain particular massive deformations of CFTs; these, in turn, can be described as quantum group reductions of the quantum sine-Gordon model and it is an easy step to include this last theory in the framework of BLZ approach. Finally we show an interesting and surprising connection of the BLZ structures with classical objects emerging from the study of classical integrable models via the inverse scattering transform method. This connection goes under the name of ODE/IM correspondence and we will present it for the specific case of quantum sine-Gordon model only. (topical review)
4. Dual field theories of quantum computation
International Nuclear Information System (INIS)
Vanchurin, Vitaly
2016-01-01
Given two quantum states of N q-bits we are interested to find the shortest quantum circuit consisting of only one- and two- q-bit gates that would transfer one state into another. We call it the quantum maze problem for the reasons described in the paper. We argue that in a large N limit the quantum maze problem is equivalent to the problem of finding a semiclassical trajectory of some lattice field theory (the dual theory) on an N+1 dimensional space-time with geometrically flat, but topologically compact spatial slices. The spatial fundamental domain is an N dimensional hyper-rhombohedron, and the temporal direction describes transitions from an arbitrary initial state to an arbitrary target state and so the initial and final dual field theory conditions are described by these two quantum computational states. We first consider a complex Klein-Gordon field theory and argue that it can only be used to study the shortest quantum circuits which do not involve generators composed of tensor products of multiple Pauli Z matrices. Since such situation is not generic we call it the Z-problem. On the dual field theory side the Z-problem corresponds to massless excitations of the phase (Goldstone modes) that we attempt to fix using Higgs mechanism. The simplest dual theory which does not suffer from the massless excitation (or from the Z-problem) is the Abelian-Higgs model which we argue can be used for finding the shortest quantum circuits. Since every trajectory of the field theory is mapped directly to a quantum circuit, the shortest quantum circuits are identified with semiclassical trajectories. We also discuss the complexity of an actual algorithm that uses a dual theory prospective for solving the quantum maze problem and compare it with a geometric approach. We argue that it might be possible to solve the problem in sub-exponential time in 2 N , but for that we must consider the Klein-Gordon theory on curved spatial geometry and/or more complicated (than N
5. String amplitudes: from field theories to number theory
CERN Multimedia
CERN. Geneva
2017-01-01
In a variety of recent developments, scattering amplitudes hint at new symmetries of and unexpected connections between physical theories which are otherwise invisible in their conventional description via Feynman diagrams or Lagrangians. Yet, many of these hidden structures are conveniently accessible to string theory where gauge interactions and gravity arise as the low-energy excitations of open and closed strings. In this talk, I will give an intuitive picture of gravity as a double copy of gauge interactions and extend the web of relations to scalar field theories including chiral Lagrangians for Goldstone bosons. The string corrections to gauge and gravity amplitudes beyond their point-particle limit exhibit elegant mathematical structures and offer a convenient laboratory to explore modern number-theoretic concepts in a simple context. As a common theme with Feynman integrals, string amplitudes introduce a variety of periods and special functions including multiple zeta values and polylogarithms, orga...
6. Correlation functions in finite temperature field theories: formalism and applications to quark-gluon plasma; Fonctions de correlations en theorie des champs a temperature finie: aspects formels et applications au plasma de quarks et de gluons
Energy Technology Data Exchange (ETDEWEB)
Gelis, Francois [Savoie Univ., 73 - Chambery (France)
1998-12-01
The general framework of this work is thermal field theory, and more precisely the perturbative calculation of thermal Greens functions. In a first part, I consider the problems closely related to the formalism itself. After two introductory chapters devoted to set up the framework and the notations used afterwards, a chapter is dedicated to a clarification of certain aspects of the justification of the Feynman rules of the real time formalism. Then, I consider in the chapter 4 the problem of cutting rules in the real time formalisms. In particular, after solving a controversy on this subject, I generalize these cutting rules to the retarded-advanced version of this formalism. Finally, the last problem considered in this part is that of the pion decay into two photons in a thermal bath. I show that the discrepancies found in the literature are due to peculiarities of the analytical properties of the thermal Greens functions. The second part deals with the calculations of the photons or dilepton (virtual photon) production rate by a quark gluon plasma. The framework of this study is the effective theory based on the resummation of hard thermal loops. The first aspects of this study is related to the production of virtual photons, where we show that important contributions arise at two loops, completing the result already known at one loop. In the case of real photon production, we show that extremely strong collinear singularities make two loop contributions dominant compared to one loop ones. In both cases, the importance of two loop contributions can be interpreted as weaknesses of the hard thermal loop approximation. (author) 366 refs., 109 figs.
7. Cross Sections From Scalar Field Theory
Science.gov (United States)
Norbury, John W.; Dick, Frank; Norman, Ryan B.; Nasto, Rachel
2008-01-01
A one pion exchange scalar model is used to calculate differential and total cross sections for pion production through nucleon- nucleon collisions. The collisions involve intermediate delta particle production and decay to nucleons and a pion. The model provides the basic theoretical framework for scalar field theory and can be applied to particle production processes where the effects of spin can be neglected.
8. Gravitational descendants in symplectic field theory
NARCIS (Netherlands)
Fabert, O.
2011-01-01
It was pointed out by Y. Eliashberg in his ICM 2006 plenary talk that the rich algebraic formalism of symplectic field theory leads to a natural appearance of quantum and classical integrable systems, at least in the case when the contact manifold is the prequantization space of a symplectic
9. Quantum field theory with soliton conservation laws
CERN Document Server
Schrör, B
1978-01-01
Field theories with soliton conservation laws are the most promising candidates for explicitly constructable models. The author exemplifies in the case of the massive Thirring model how the old S matrix bootstrap idea, supplemented with a soliton factorization property, may be used as a systematic starting point for the construction of the S matrix, form factors and (hopefully) correlation functions. (34 refs).
10. Covariant field theory of closed superstrings
International Nuclear Information System (INIS)
Siopsis, G.
1989-01-01
The authors construct covariant field theories of both type-II and heterotic strings. Toroidal compactification is also considered. The interaction vertices are based on Witten's vertex representing three strings interacting at the mid-point. For closed strings, the authors thus obtain a bilocal interaction
11. Fusion rules in conformal field theory
International Nuclear Information System (INIS)
Fuchs, J.
1993-06-01
Several aspects of fusion rings and fusion rule algebras, and of their manifestations in two-dimensional (conformal) field theory, are described: diagonalization and the connection with modular invariance; the presentation in terms of quotients of polynomial rings; fusion graphs; various strategies that allow for a partial classification; and the role of the fusion rules in the conformal bootstrap programme. (orig.)
12. The quantum symmetry of rational field theories
International Nuclear Information System (INIS)
Fuchs, J.
1993-12-01
The quantum symmetry of a rational quantum field theory is a finite-dimensional multi-matrix algebra. Its representation category, which determines the fusion rules and braid group representations of superselection sectors, is a braided monoidal C*-category. Various properties of such algebraic structures are described, and some ideas concerning the classification programme are outlined. (orig.)
13. Reconstructing bidimensional scalar field theory models
International Nuclear Information System (INIS)
Flores, Gabriel H.; Svaiter, N.F.
2001-07-01
In this paper we review how to reconstruct scalar field theories in two dimensional spacetime starting from solvable Scrodinger equations. Theree different Schrodinger potentials are analyzed. We obtained two new models starting from the Morse and Scarf II hyperbolic potencials, the U (θ) θ 2 In 2 (θ 2 ) model and U (θ) = θ 2 cos 2 (In(θ 2 )) model respectively. (author)
14. General relativity invariance and string field theory
International Nuclear Information System (INIS)
Aref'eva, I.Ya.; Volovich, I.V.
1987-04-01
The general covariance principle in the string field theory is considered. The algebraic properties of the string Lie derivative are discussed. The string vielbein and spin connection are introduced and an action invariant under general co-ordinate transformation is proposed. (author). 18 refs
15. Construction of topological field theories using BV
NARCIS (Netherlands)
Jonghe, F. de; Vandoren, S.
1993-01-01
We discuss in detail the construction of topological field theories us- ing the Batalin–Vilkovisky (BV) quantisation scheme. By carefully examining the dependence of the antibracket on an external metric, we show that differentiating with respect to the metric and the BRST charge do not commute
16. Wilson lines in quantum field theory
CERN Document Server
Cherednikov, Igor O; Veken, Frederik F van der
2014-01-01
The objective of this book is to get the reader acquainted with theoretical and mathematical foundations of the concept of Wilson loops in the context of modern quantum field theory. It teaches how to perform independently with some elementary calculations on Wilson lines, and shows the recent development of the subject in different important areas of research.
17. Translationally invariant self-consistent field theories
International Nuclear Information System (INIS)
Shakin, C.M.; Weiss, M.S.
1977-01-01
We present a self-consistent field theory which is translationally invariant. The equations obtained go over to the usual Hartree-Fock equations in the limit of large particle number. In addition to deriving the dynamic equations for the self-consistent amplitudes we discuss the calculation of form factors and various other observables
18. Causality and analyticity in quantum fields theory
International Nuclear Information System (INIS)
Iagolnitzer, D.
1992-01-01
This is a presentation of results on the causal and analytical structure of Green functions and on the collision amplitudes in fields theories, for massive particles of one type, with a positive mass and a zero spin value. (A.B.)
19. Asymptotic mass degeneracies in conformal field theories
International Nuclear Information System (INIS)
Kani, I.; Vafa, C.
1990-01-01
By applying a method of Hardy and Ramanujan to characters of rational conformal field theories, we find an asymptotic expansion for degeneracy of states in the limit of large mass which is exact for strings propagating in more than two uncompactified space-time dimensions. Moreover we explore how the rationality of the conformal theory is reflected in the degeneracy of states. We also consider the one loop partition function for strings, restricted to physical states, for arbitrary (irrational) conformal theories, and obtain an asymptotic expansion for it in the limit that the torus degenerates. This expansion depends only on the spectrum of (physical and unphysical) relevant operators in the theory. We see how rationality is consistent with the smoothness of mass degeneracies as a function of moduli. (orig.)
20. Superconformal quantum field theories in string. Gauge theory dualities
Energy Technology Data Exchange (ETDEWEB)
Wiegandt, Konstantin
2012-08-14
In this thesis aspects of superconformal field theories that are of interest in the so-called AdS/CFT correspondence are investigated. The AdS/CFT correspondence states a duality between string theories living on Anti-de Sitter space and superconformal quantum field theories in Minkowski space. In the context of the AdS/CFT correspondence the so-called Wilson loop/amplitude duality was discovered, stating the equality of the finite parts of n-gluon MHV amplitudes and n-sided lightlike polygonal Wilson loops in N=4 supersymmetric Yang-Mills (SYM) theory. It is the subject of the first part of this thesis to investigate the Wilson loop side of a possible similar duality in N=6 superconformal Chern-Simons matter (ABJM) theory. The main result is, that the expectation value of n-sided lightlike polygonal Wilson loops vanishes at one-loop order and at two-loop order is identical in its functional form to the Wilson loop in N=4 SYM theory at one-loop order. Furthermore, an anomalous conformal Ward identity for Wilson loops in Chern-Simons theory is derived. Related developments and symmetries of amplitudes and correlators in ABJM theory are discussed as well. In the second part of this thesis we calculate three-point functions of two protected operators and one twist-two operator with arbitrary even spin j in N=4 SYM theory. In order to carry out the calculations, the indices of the spin j operator are projected to the light-cone and the correlator is evaluated in a soft-limit where the momentum coming in at the spin j operator becomes zero. This limit largely simplifies the perturbative calculation, since all three-point diagrams effectively reduce to two-point diagrams and the dependence on the one-loop mixing matrix drops out completely. The result is in agreement with the analysis of the operator product expansion of four-point functions of half-BPS operators by Dolan and Osborn in 2004.
1. A periodic table of effective field theories
Energy Technology Data Exchange (ETDEWEB)
Cheung, Clifford [Walter Burke Institute for Theoretical Physics,California Institute of Technology,Pasadena, CA (United States); Kampf, Karol; Novotny, Jiri [Institute of Particle and Nuclear Physics,Faculty of Mathematics and Physics, Charles University,Prague (Czech Republic); Shen, Chia-Hsien [Walter Burke Institute for Theoretical Physics,California Institute of Technology,Pasadena, CA (United States); Trnka, Jaroslav [Center for Quantum Mathematics and Physics (QMAP),Department of Physics, University of California,Davis, CA (United States)
2017-02-06
We systematically explore the space of scalar effective field theories (EFTs) consistent with a Lorentz invariant and local S-matrix. To do so we define an EFT classification based on four parameters characterizing 1) the number of derivatives per interaction, 2) the soft properties of amplitudes, 3) the leading valency of the interactions, and 4) the spacetime dimension. Carving out the allowed space of EFTs, we prove that exceptional EFTs like the non-linear sigma model, Dirac-Born-Infeld theory, and the special Galileon lie precisely on the boundary of allowed theory space. Using on-shell momentum shifts and recursion relations, we prove that EFTs with arbitrarily soft behavior are forbidden and EFTs with leading valency much greater than the spacetime dimension cannot have enhanced soft behavior. We then enumerate all single scalar EFTs in d<6 and verify that they correspond to known theories in the literature. Our results suggest that the exceptional theories are the natural EFT analogs of gauge theory and gravity because they are one-parameter theories whose interactions are strictly dictated by properties of the S-matrix.
2. Noncommutative gravity and quantum field theory on noncummutative curved spacetimes
International Nuclear Information System (INIS)
Schenkel, Alexander
2011-01-01
quantum field theory at short distances, i.e. in the ultraviolet. In the third part we develop elements of a more powerful, albeit more abstract, mathematical approach to noncommutative gravity. The goal is to better understand global aspects of homomorphisms between and connections on noncommutative vector bundles, which are fundamental objects in the mathematical description of noncommutative gravity. We prove that all homomorphisms and connections of the deformed theory can be obtained by applying a quantization isomorphism to undeformed homomorphisms and connections. The extension of homomorphisms and connections to tensor products of modules is clarified, and as a consequence we are able to add tensor fields of arbitrary type to the noncommutative gravity theory of Wess et al. As a nontrivial application of the new mathematical formalism we extend our studies of exact noncommutative gravity solutions to more general deformations.
3. Noncommutative gravity and quantum field theory on noncummutative curved spacetimes
Energy Technology Data Exchange (ETDEWEB)
Schenkel, Alexander
2011-10-24
noncommutative quantum field theory at short distances, i.e. in the ultraviolet. In the third part we develop elements of a more powerful, albeit more abstract, mathematical approach to noncommutative gravity. The goal is to better understand global aspects of homomorphisms between and connections on noncommutative vector bundles, which are fundamental objects in the mathematical description of noncommutative gravity. We prove that all homomorphisms and connections of the deformed theory can be obtained by applying a quantization isomorphism to undeformed homomorphisms and connections. The extension of homomorphisms and connections to tensor products of modules is clarified, and as a consequence we are able to add tensor fields of arbitrary type to the noncommutative gravity theory of Wess et al. As a nontrivial application of the new mathematical formalism we extend our studies of exact noncommutative gravity solutions to more general deformations.
4. Quantum tunneling and field electron emission theories
CERN Document Server
Liang, Shi-Dong
2013-01-01
Quantum tunneling is an essential issue in quantum physics. Especially, the rapid development of nanotechnology in recent years promises a lot of applications in condensed matter physics, surface science and nanodevices, which are growing interests in fundamental issues, computational techniques and potential applications of quantum tunneling. The book involves two relevant topics. One is quantum tunneling theory in condensed matter physics, including the basic concepts and methods, especially for recent developments in mesoscopic physics and computational formulation. The second part is the f
5. A symplectic framework for field theories
International Nuclear Information System (INIS)
Kijowski, J.; Tulczyjew, W.M.
1979-01-01
These notes are concerned with the formulation of a new conceptual framework for classical field theories. Although the formulation is based on fairly advanced concepts of symplectic geometry these notes cannot be viewed as a reformulation of known structures in more rigorous and elegant torns. Our intention is rather to communicate to theoretical physicists a set of new physical ideas. We have chosen for this purpose the language of local coordinates which is more elementary and more widely known than the abstract language of modern differntial geometry. Our emphasis is directed more to physical intentions than to mathematical vigour. We start with a symplectic analysis of staties. Both discrete and continuous systems are considered on a largely intuitive level. The notion of reciprocity and potentiality of the theory is discussed. Chapter II is a presentation of particle dynamics together with more rigorous definitions of the geometric structure. Lagrangian-Submanifolds and their generating function 3 are defined and the time evolution of particle states is studied. Chapter II form the main part of these notes. Here we describe the construction of canonical momenta and discuss the field dynamics in finite domains of space-time. We also establish the relation between our symplectic framework and the geometric formulation of the calculus of variations of multiple integrals. In the following chapter we give a few examples of field theories selected to illustrate various features of the new approach. A new formulation of the theory of gravity consists of using the affine connection in space-time as the field configuration. In the past section we present an analysis of hydrodynamics within our framework which reveals a formal analogy with electrodynamics. The discovery of potentials for hydrodynamics and the subsequent formulation of a variational principle provides an excellent example for the fruitfulness of the new approach to field theory. A short review of
6. Management applications of discontinuity theory
Science.gov (United States)
Angeler, David G.; Allen, Craig R.; Barichievy, Chris; Eason, Tarsha; Garmestani, Ahjond S.; Graham, Nicholas A.J.; Granholm, Dean; Gunderson, Lance H.; Knutson, Melinda; Nash, Kirsty L.; Nelson, R. John; Nystrom, Magnus; Spanbauer, Trisha; Stow, Craig A.; Sundstrom, Shana M.
2015-01-01
Human impacts on the environment are multifaceted and can occur across distinct spatiotemporal scales. Ecological responses to environmental change are therefore difficult to predict, and entail large degrees of uncertainty. Such uncertainty requires robust tools for management to sustain ecosystem goods and services and maintain resilient ecosystems.We propose an approach based on discontinuity theory that accounts for patterns and processes at distinct spatial and temporal scales, an inherent property of ecological systems. Discontinuity theory has not been applied in natural resource management and could therefore improve ecosystem management because it explicitly accounts for ecological complexity.Synthesis and applications. We highlight the application of discontinuity approaches for meeting management goals. Specifically, discontinuity approaches have significant potential to measure and thus understand the resilience of ecosystems, to objectively identify critical scales of space and time in ecological systems at which human impact might be most severe, to provide warning indicators of regime change, to help predict and understand biological invasions and extinctions and to focus monitoring efforts. Discontinuity theory can complement current approaches, providing a broader paradigm for ecological management and conservation.
7. Quantum field theory in topology changing spacetimes
International Nuclear Information System (INIS)
Bauer, W.
2007-03-01
The goal of this diploma thesis is to present an overview of how to reduce the problem of topology change of general spacetimes to the investigation of elementary cobordisms. In the following we investigate the possibility to construct quantum fields on elementary cobordisms, in particular we discuss the trousers topology. Trying to avoid the problems occuring at spacetimes with instant topology change we use a model for simulating topology change. We construct the algebra of observables for a free scalar field with the algebraic approach to quantum field theory. Therefore we determine a fundamental solution of the eld equation. (orig.)
8. Continuous and distributed systems theory and applications
CERN Document Server
2014-01-01
In this volume, the authors close the gap between abstract mathematical approaches, such as abstract algebra, number theory, nonlinear functional analysis, partial differential equations, methods of nonlinear and multi-valued analysis, on the one hand, and practical applications in nonlinear mechanics, decision making theory and control theory on the other. Readers will also benefit from the presentation of modern mathematical modeling methods for the numerical solution of complicated engineering problems in hydromechanics, geophysics and mechanics of continua. This compilation will be of interest to mathematicians and engineers working at the interface of these field. It presents selected works of the open seminar series of Lomonosov Moscow State University and the National Technical University of Ukraine “Kyiv Polytechnic Institute”. The authors come from Germany, Italy, Spain, Russia, Ukraine, and the USA.
9. Scattering of decuplet baryons in chiral effective field theory
Energy Technology Data Exchange (ETDEWEB)
Haidenbauer, J. [Institut fuer Kernphysik, Institute for Advanced Simulation and Juelich Center for Hadron Physics, Juelich (Germany); Petschauer, S.; Kaiser, N.; Weise, W. [Technische Universitaet Muenchen, Physik Department, Garching (Germany); Meissner, Ulf G. [Institut fuer Kernphysik, Institute for Advanced Simulation and Juelich Center for Hadron Physics, Juelich (Germany); Universitaet Bonn, Helmholtz-Institut fuer Strahlen- und Kernphysik and Bethe Center for Theoretical Physics, Bonn (Germany)
2017-11-15
A formalism for treating the scattering of decuplet baryons in chiral effective field theory is developed. The minimal Lagrangian and potentials in leading-order SU(3) chiral effective field theory for the interactions of octet baryons (B) and decuplet baryons (D) for the transitions BB → BB, BB <-> DB, DB → DB, BB <-> DD, DB <-> DD, and DD → DD are provided. As an application of the formalism we compare with results from lattice QCD simulations for ΩΩ and NΩ scattering. Implications of our results pertinent to the quest for dibaryons are discussed. (orig.)
10. Symmetry aspects of nonholonomic field theories
Energy Technology Data Exchange (ETDEWEB)
Vankerschaver, Joris [Control and Dynamical Systems, California Institute of Technology, MC 107-81, Pasadena, CA 91125 (United States); Diego, David MartIn de [Instituto de Matematicas y Fisica Fundamental, Consejo Superior de Investigaciones CientIficas, Serrano 123, 28006 Madrid (Spain)
2008-01-25
The developments in this paper are concerned with nonholonomic field theories in the presence of symmetries. Having previously treated the case of vertical symmetries, we now deal with the case where the symmetry action can also have a horizontal component. As a first step in this direction, we derive a new and convenient form of the field equations of a nonholonomic field theory. Nonholonomic symmetries are then introduced as symmetry generators whose virtual work is zero along the constraint submanifold, and we show that for every such symmetry, there exists a so-called momentum equation, describing the evolution of the associated component of the momentum map. Keeping up with the underlying geometric philosophy, a small modification of the derivation of the momentum lemma allows us to also treat generalized nonholonomic symmetries, which are vector fields along a projection. Such symmetries arise for example in practical examples of nonholonomic field theories such as the Cosserat rod, for which we recover both energy conservation (a previously known result) and a modified conservation law associated with spatial translations.
11. Mean fields and self consistent normal ordering of lattice spin and gauge field theories
International Nuclear Information System (INIS)
Ruehl, W.
1986-01-01
Classical Heisenberg spin models on lattices possess mean field theories that are well defined real field theories on finite lattices. These mean field theories can be self consistently normal ordered. This leads to a considerable improvement over standard mean field theory. This concept is carried over to lattice gauge theories. We construct first an appropriate real mean field theory. The equations determining the Gaussian kernel necessary for self-consistent normal ordering of this mean field theory are derived. (orig.)
12. Interaction vertices in reduced string field theories
International Nuclear Information System (INIS)
Embacher, F.
1989-01-01
In contrast to previous expectations, covariant overlap vertices are not always suitable for gauge-covariant formulations of bosonic string field theory with a reduced supplementary field content. This is demonstrated for the version of the theory suggested by Neveu, Schwarz and West. The method to construct the interaction, as formulated by Neveu and West, fails at one level higher than these authors have considered. The condition for a general vertex to describe formally a local gauge-invariant interaction is derived. The solution for the action functional and the gauge transformation law is exhibited for all fields at once, to the first order in the coupling constant. However, all these vertices seem to be unphysical. 21 refs. (Author)
13. Extending Gurwitsch's field theory of consciousness.
Science.gov (United States)
Yoshimi, Jeff; Vinson, David W
2015-07-01
Aron Gurwitsch's theory of the structure and dynamics of consciousness has much to offer contemporary theorizing about consciousness and its basis in the embodied brain. On Gurwitsch's account, as we develop it, the field of consciousness has a variable sized focus or "theme" of attention surrounded by a structured periphery of inattentional contents. As the field evolves, its contents change their status, sometimes smoothly, sometimes abruptly. Inner thoughts, a sense of one's body, and the physical environment are dominant field contents. These ideas can be linked with (and help unify) contemporary theories about the neural correlates of consciousness, inattention, the small world structure of the brain, meta-stable dynamics, embodied cognition, and predictive coding in the brain. Published by Elsevier Inc.
14. Quantum field theory the why, what and how
CERN Document Server
2016-01-01
This book describes, in clear terms, the Why, What and the How of Quantum Field Theory. The raison d'etre of QFT is explained by starting from the dynamics of a relativistic particle and demonstrating how it leads to the notion of quantum fields. Non-perturbative aspects and the Wilsonian interpretation of field theory are emphasized right from the start. Several interesting topics such as the Schwinger effect, Davies-Unruh effect, Casimir effect and spontaneous symmetry breaking introduce the reader to the elegance and breadth of applicability of field theoretical concepts. Complementing the conceptual aspects, the book also develops all the relevant mathematical techniques in detail, leading e.g., to the computation of anomalous magnetic moment of the electron and the two-loop renormalisation of the self-interacting scalar field. It contains nearly a hundred problems, of varying degrees of difficulty, making it suitable for both self-study and classroom use.
15. A Generalized Field Theory: Charged Spherical Symmetric Solution
Science.gov (United States)
Wanas, M. I.
1985-06-01
Three solutions with spherical symmetry are obtained for the field equations of the generalized field theory established recently by Mikhail and Wanas. The solutions found are in agreement with classical known results. The solution representing a generalized field, outside a spherical symmetric charged body, is found to have an extra term compared with the Reissner-Nordström metric. The space used for application is of type FIGI, so the solutions obtained correspond to a field in a matter-free space. A brief comparison between the solutions obtained and those given by other field theories is given. Two methods have been used to get physical results: the first is the type analysis, and the second is the comparison with classical known results by writing down the metric of the associated Riemannian space.
16. Cosmological field theory for observational astronomers
International Nuclear Information System (INIS)
Zel'Dovich, Y.B.
1987-01-01
Theories of the very early Universe that use scalar fields (i.e., the so-called inflationary models of the Universe) have now come into wide use. The inflationary universe approach may perhaps solve some of the most difficult enigmas about the Universe as a whole. The inflationary universe forms a good bridge between the quantum theory of the birth of the Universe (which is still in the initial stages of development) and the standard hot Big Bang theory (which is well established, at least qualitatively). Therefore, an understanding of the basic ideas of inflation is a must for astronomers interested in the broad picture of the science. Astronomers are mathematically oriented enough (via celestial mechanics, electromagnetic theory, magnetohydrodynamics, nuclear reactions,etc.) that there is no negative attitude towards formulae in general. What the astronomer lacks is a knowledge of recent developments in particle physics and field theory. The astronomer should not be blamed for this, because these branches of physics are developing in a very peculiar fashion: some subfields of it are progressing comparatively slowly, with experimental verifications at each and every step, while other subfields progress rapidly
17. Copula Theory and Its Applications
CERN Document Server
Jaworski, Piotr; Hardle, Wolfgang Karl; Rychlik, Tomasz
2010-01-01
Copulas are mathematical objects that fully capture the dependence structure among random variables and hence offer great flexibility in building multivariate stochastic models. Since their introduction in the early 50's, copulas have gained considerable popularity in several fields of applied mathematics, such as finance, insurance and reliability theory. Today, they represent a well-recognized tool for market and credit models, aggregation of risks, portfolio selection, etc. This book is divided into two main parts: Part I - 'Surveys' contains 11 chapters that provide an up-to-date account o
18. Relating the archetypes of logarithmic conformal field theory
International Nuclear Information System (INIS)
Creutzig, Thomas; Ridout, David
2013-01-01
Logarithmic conformal field theory is a rich and vibrant area of modern mathematical physics with well-known applications to both condensed matter theory and string theory. Our limited understanding of these theories is based upon detailed studies of various examples that one may regard as archetypal. These include the c=−2 triplet model, the Wess–Zumino–Witten model on SL(2;R) at level k=−1/2 , and its supergroup analogue on GL(1|1). Here, the latter model is studied algebraically through representation theory, fusion and modular invariance, facilitating a subsequent investigation of its cosets and extended algebras. The results show that the archetypes of logarithmic conformal field theory are in fact all very closely related, as are many other examples including, in particular, the SL(2|1) models at levels 1 and −1/2 . The conclusion is then that the archetypal examples of logarithmic conformal field theory are practically all the same, so we should not expect that their features are in any way generic. Further archetypal examples must be sought
19. Relating the archetypes of logarithmic conformal field theory
Energy Technology Data Exchange (ETDEWEB)
Creutzig, Thomas, E-mail: [email protected] [Department of Physics and Astronomy, University of North Carolina, Phillips Hall, CB 3255, Chapel Hill, NC 27599-3255 (United States); Fachbereich Mathematik, Technische Universität Darmstadt, Schloßgartenstraße 7, 64289 Darmstadt (Germany); Ridout, David, E-mail: [email protected] [Department of Theoretical Physics, Research School of Physics and Engineering, Australian National University, Canberra, ACT 0200 (Australia); Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200 (Australia)
2013-07-21
Logarithmic conformal field theory is a rich and vibrant area of modern mathematical physics with well-known applications to both condensed matter theory and string theory. Our limited understanding of these theories is based upon detailed studies of various examples that one may regard as archetypal. These include the c=−2 triplet model, the Wess–Zumino–Witten model on SL(2;R) at level k=−1/2 , and its supergroup analogue on GL(1|1). Here, the latter model is studied algebraically through representation theory, fusion and modular invariance, facilitating a subsequent investigation of its cosets and extended algebras. The results show that the archetypes of logarithmic conformal field theory are in fact all very closely related, as are many other examples including, in particular, the SL(2|1) models at levels 1 and −1/2 . The conclusion is then that the archetypal examples of logarithmic conformal field theory are practically all the same, so we should not expect that their features are in any way generic. Further archetypal examples must be sought.
20. Noncommutative analysis, operator theory and applications
CERN Document Server
Cipriani, Fabio; Colombo, Fabrizio; Guido, Daniele; Sabadini, Irene; Sauvageot, Jean-Luc
2016-01-01
This book illustrates several aspects of the current research activity in operator theory, operator algebras and applications in various areas of mathematics and mathematical physics. It is addressed to specialists but also to graduate students in several fields including global analysis, Schur analysis, complex analysis, C*-algebras, noncommutative geometry, operator algebras, operator theory and their applications. Contributors: F. Arici, S. Bernstein, V. Bolotnikov, J. Bourgain, P. Cerejeiras, F. Cipriani, F. Colombo, F. D'Andrea, G. Dell'Antonio, M. Elin, U. Franz, D. Guido, T. Isola, A. Kula, L.E. Labuschagne, G. Landi, W.A. Majewski, I. Sabadini, J.-L. Sauvageot, D. Shoikhet, A. Skalski, H. de Snoo, D. C. Struppa, N. Vieira, D.V. Voiculescu, and H. Woracek.
1. Social signals: from theory to applications.
Science.gov (United States)
Poggi, Isabella; D'Errico, Francesca; Vinciarelli, Alessandro
2012-10-01
The Special Issue Editorial introduces the research milieu in which Social Signal Processing originates, by merging computer scientists and social scientists and giving rise to this field in parallel with Human-Computer Interaction, Affective Computing, and Embodied Conversational Agents, all similarly characterized by high interdisciplinarity, stress on multimodality of communication, and the continuous loop from theory to simulation and application. Some frameworks of the cognitive and social processes underlying social signals are identified as reference points (Theory of Mind and Intersubjectivity, mirror neurons, and the ontogenesis and phylogenesis of communication), while three dichotomies (automatic vs. controlled, individualistic vs. intersubjective, and meaning vs. influence) are singled out as leads to navigate within the theoretical and applicative studies presented in the Special Issue.
2. BRST quantization of topological field theories
International Nuclear Information System (INIS)
Birmingham, D.; Rakowski, M.; Thompson, G.
1988-07-01
We consider in detail the construction of a variety of topological quantum field theories through BRST quantization. In particular, we show that supersymmetric quantum mechanics on an arbitrary Riemannian manifold can be obtained as the BRST quantization of a purely bosonic theory. The introduction of a new local symmetry allows for the possibility of different gauge choices, and we show how this freedom can simplify the evaluation of the Witten index in certain cases. Topological sigma models are also constructed via the same mechanism. In three dimensions, we consider a Yang-Mills-Higgs model related to the four dimensional TQFT of Witten. (author). 24 refs
3. Thermo field theory versus imaginary time formalism
International Nuclear Information System (INIS)
Fujimoto, Y.; Nishino, H.; Grigjanis, R.
1983-11-01
We calculate a two-loop diagram at finite temperature to compare Thermo Field Theory (=Th.F.Th.) with the conventional imaginary time formalism (=Im.T.F.). The summation over the Matsubara frequency in Im.T.F. is carried out at two-loop level, and the result is shown to coincide with that of Th.F.Th. We confirm that in Im.T.F. the temperature dependent divergences cancel out at least in the calculation of effective potential of phi 4 theory, as in Th.F.Th. (author)
4. Recursion equations in gauge field theories
International Nuclear Information System (INIS)
Migdal, A.A.
1975-01-01
An approximate recursive equation describing scale transformation of the effective action of a gauging field has been formulated. The equation becomes exact in the two-dimensional space-time. In the four-dimensional theory it reproduces the asymptotic freedom with an accuracy of 30% in β-function coefficients. In the region of strong coupling β-function remains negative, that leads to an asymptotic ''prison'' in the infrared range. Some possible generalizations and appendices to the colour quark-gluon gauging theory are being discussed
5. Twistors and supertwistors for exceptional field theory
Energy Technology Data Exchange (ETDEWEB)
Cederwall, Martin [Dept. of Fundamental Physics, Chalmers University of Technology, Gothenburg, SE 412 96 (Sweden)
2015-12-18
As a means of examining the section condition and its possible solutions and relaxations, we perform twistor transforms related to versions of exceptional field theory with Minkowski signature. The spinor parametrisation of the momenta naturally solves simultaneously both the mass-shell condition and the (weak) section condition. It is shown that the incidence relations for multi-particle twistors force them to share a common section, but not to be orthogonal. The supersymmetric extension contains additional scalar fermionic variables shown to be kappa-symmetry invariants. We speculate on some implications, among them a possible relation to higher spin theory.
6. Biometrics Theory, Methods, and Applications
CERN Document Server
Boulgouris, N V; Micheli-Tzanakou, Evangelia
2009-01-01
An in-depth examination of the cutting edge of biometrics. This book fills a gap in the literature by detailing the recent advances and emerging theories, methods, and applications of biometric systems in a variety of infrastructures. Edited by a panel of experts, it provides comprehensive coverage of:. Multilinear discriminant analysis for biometric signal recognition;. Biometric identity authentication techniques based on neural networks;. Multimodal biometrics and design of classifiers for biometric fusion;. Feature selection and facial aging modeling for face recognition;. Geometrical and
7. Quantile regression theory and applications
CERN Document Server
Davino, Cristina; Vistocco, Domenico
2013-01-01
A guide to the implementation and interpretation of Quantile Regression models This book explores the theory and numerous applications of quantile regression, offering empirical data analysis as well as the software tools to implement the methods. The main focus of this book is to provide the reader with a comprehensivedescription of the main issues concerning quantile regression; these include basic modeling, geometrical interpretation, estimation and inference for quantile regression, as well as issues on validity of the model, diagnostic tools. Each methodological aspect is explored and
8. Surface chemistry theory and applications
CERN Document Server
Bikerman, J J
2013-01-01
Surface Chemistry Theory and Applications focuses on liquid-gas, liquid-liquid, solid-gas, solid-liquid, and solid-solid surfaces. The book first offers information on liquid-gas surfaces, including surface tension, measurement of surface tension, rate of capillarity rise, capillary attraction, bubble pressure and pore size, and surface tension and temperature. The text then ponders on liquid-liquid and solid-gas surfaces. Discussions focus on surface energy of solids, surface roughness and cleanness, adsorption of gases and vapors, adsorption hysteresis, interfacial tension, and interfacial t
9. Self-consistent normal ordering of gauge field theories
International Nuclear Information System (INIS)
Ruehl, W.
1987-01-01
Mean-field theories with a real action of unconstrained fields can be self-consistently normal ordered. This leads to a considerable improvement over standard mean-field theory. This concept is applied to lattice gauge theories. First an appropriate real action mean-field theory is constructed. The equations determining the Gaussian kernel necessary for self-consistent normal ordering of this mean-field theory are derived. (author). 4 refs
10. Protective relaying theory and applications
CERN Document Server
Elmore, Walter A
2003-01-01
Targeting the latest microprocessor technologies for more sophisticated applications in the field of power system short circuit detection, this revised and updated source imparts fundamental concepts and breakthrough science for the isolation of faulty equipment and minimization of damage in power system apparatus. The Second Edition clearly describes key procedures, devices, and elements crucial to the protection and control of power system function and stability. It includes chapters and expertise from the most knowledgeable experts in the field of protective relaying, and describes micropro
11. Effective field theory for halo nuclei
International Nuclear Information System (INIS)
Hagen, Philipp Robert
2014-01-01
We investigate properties of two- and three-body halo systems using effective field theory. If the two-particle scattering length a in such a system is large compared to the typical range of the interaction R, low-energy observables in the strong and the electromagnetic sector can be calculated in halo EFT in a controlled expansion in R/ vertical stroke a vertical stroke. Here we focus on universal properties and stay at leading order in the expansion. Motivated by the existence of the P-wave halo nucleus 6 He, we first set up an EFT framework for a general three-body system with resonant two-particle P-wave interactions. Based on a Lagrangian description, we identify the area in the effective range parameter space where the two-particle sector of our model is renormalizable. However, we argue that for such parameters, there are two two-body bound states: a physical one and an additional deeper-bound and non-normalizable state that limits the range of applicability of our theory. With regard to the three-body sector, we then classify all angular-momentum and parity channels that display asymptotic discrete scale invariance and thus require renormalization via a cut-off dependent three-body force. In the unitary limit an Efimov effect occurs. However, this effect is purely mathematical, since, due to causality bounds, the unitary limit for P-wave interactions can not be realized in nature. Away from the unitary limit, the three-body binding energy spectrum displays an approximate Efimov effect but lies below the unphysical, deep two-body bound state and is thus unphysical. Finally, we discuss possible modifications in our halo EFT approach with P-wave interactions that might provide a suitable way to describe physical three-body bound states. We then set up a halo EFT formalism for two-neutron halo nuclei with resonant two-particle S-wave interactions. Introducing external currents via minimal coupling, we calculate observables and universal correlations for such
12. Automata theory and its applications
CERN Document Server
2001-01-01
The theory of finite automata on finite stings, infinite strings, and trees has had a dis tinguished history. First, automata were introduced to represent idealized switching circuits augmented by unit delays. This was the period of Shannon, McCullouch and Pitts, and Howard Aiken, ending about 1950. Then in the 1950s there was the work of Kleene on representable events, of Myhill and Nerode on finite coset congruence relations on strings, of Rabin and Scott on power set automata. In the 1960s, there was the work of Btichi on automata on infinite strings and the second order theory of one successor, then Rabin's 1968 result on automata on infinite trees and the second order theory of two successors. The latter was a mystery until the introduction of forgetful determinacy games by Gurevich and Harrington in 1982. Each of these developments has successful and prospective applications in computer science. They should all be part of every computer scientist's toolbox. Suppose that we take a computer scientist's ...
13. Quantum field theory and critical phenomena
CERN Document Server
Zinn-Justin, Jean
1996-01-01
Over the last twenty years quantum field theory has become not only the framework for the discussion of all fundamental interactions except gravity, but also for the understanding of second-order phase transitions in statistical mechanics. This advanced text is based on graduate courses and summer schools given by the author over a number of years. It approaches the subject in terms of path and functional intergrals, adopting a Euclidean metric and using the language of partition and correlation functions. Renormalization and the renormalization group are examined, as are critical phenomena and the role of instantons. Changes for this edition 1. Extensive revision to eliminate a few bugs that had survived the second edition and (mainly) to improve the pedagogical presentation, as a result of experience gathered by lecturing. 2. Additional new topics; holomorphic or coherent state path integral; functional integral and representation of the field theory S-matrix in the holomorphic formalis; non-relativistic li...
14. Propositional systems in local field theories
International Nuclear Information System (INIS)
Banai, M.
1980-07-01
The authors investigate propositional systems for local field theories, which reflect intrinsically the uncertainties of measurements made on the physical system, and satisfy the isotony and local commutativity postulates of Haag and Kastler. The spacetime covariance can be implemented in natural way in these propositional systems. New techniques are introduced to obtain these propositional systems: the lattice-valued logics. The decomposition of the complete orthomodular lattice-valued logics shows that these logics are more general than the usual two-valued ones and that in these logics there is enough structure to characterize the classical and quantum, non relativistic and relativistic local field theories in a natural way. The Hilbert modules give the natural inner product ''spaces'' (modules) for the realization of the lattice-valued logics. (author)
15. Propositional systems in local field theories
Energy Technology Data Exchange (ETDEWEB)
Banai, M.
1981-03-01
We investigate propositional systems for local field theories, which reflect intrinsically the uncertainties of measurements made on the physical system, and satisfy the isotony and local commutativity postulates of Haag and Kastler. The space-time covariance can be implemented in a natural way in these propositional systems. New techniques are introduced to obtain these propositional systems: the lattice-valued logics. The decomposition of the complete orthomodular lattice-valued logics shows that these logics are more general than the usual two-valued ones and that in these there is enough structure to characterize the classical and quantum, nonrelativistic and relativistic local field theories in a natural way. The Hilbert modules give the natural inner product ''spaces'' (modules) for the realization of the lattice-valued logics.
16. Magnetic fields and density functional theory
International Nuclear Information System (INIS)
Salsbury, Freddie Jr.
1999-01-01
A major focus of this dissertation is the development of functionals for the magnetic susceptibility and the chemical shielding within the context of magnetic field density functional theory (BDFT). These functionals depend on the electron density in the absence of the field, which is unlike any other treatment of these responses. There have been several advances made within this theory. The first of which is the development of local density functionals for chemical shieldings and magnetic susceptibilities. There are the first such functionals ever proposed. These parameters have been studied by constructing functionals for the current density and then using the Biot-Savart equations to obtain the responses. In order to examine the advantages and disadvantages of the local functionals, they were tested numerically on some small molecules
17. New ideas about unified field theory
International Nuclear Information System (INIS)
Gleiser, M.
1986-01-01
An outline of the physical concepts evolution is given from the ancient philosophers to the present time. With qualitative explanations about the meaning of the theories that is the milestones of these concepts evolution, it mentions the ideas which lead the studies to the conception of a unified field theory. Chronologically, it has brief information about the ideas of Laplace (mechanical determinism), Maxwell (the field concept), Einsten (the space-time structure), Heisenberg and Schroedinger (the quantum mechanics), Dirac (the relativistic quantum and the antiparticles), Gell-Mann (the quarks), Weinberg-Salam (Weak interactions and eletromagnetic unification), H. Georgi and S. Glashon (strong interactions plus Weinberg-Salam), Kaluza-Klein (a fifth space-time coordinate), and Zumino-Weiss (supersymmetry and supergravity). (G.D.F.) [pt
18. Discrete Curvature Theories and Applications
KAUST Repository
Sun, Xiang
2016-08-25
Discrete Di erential Geometry (DDG) concerns discrete counterparts of notions and methods in di erential geometry. This thesis deals with a core subject in DDG, discrete curvature theories on various types of polyhedral surfaces that are practically important for free-form architecture, sunlight-redirecting shading systems, and face recognition. Modeled as polyhedral surfaces, the shapes of free-form structures may have to satisfy di erent geometric or physical constraints. We study a combination of geometry and physics { the discrete surfaces that can stand on their own, as well as having proper shapes for the manufacture. These proper shapes, known as circular and conical meshes, are closely related to discrete principal curvatures. We study curvature theories that make such surfaces possible. Shading systems of freeform building skins are new types of energy-saving structures that can re-direct the sunlight. From these systems, discrete line congruences across polyhedral surfaces can be abstracted. We develop a new curvature theory for polyhedral surfaces equipped with normal congruences { a particular type of congruences de ned by linear interpolation of vertex normals. The main results are a discussion of various de nitions of normality, a detailed study of the geometry of such congruences, and a concept of curvatures and shape operators associated with the faces of a triangle mesh. These curvatures are compatible with both normal congruences and the Steiner formula. In addition to architecture, we consider the role of discrete curvatures in face recognition. We use geometric measure theory to introduce the notion of asymptotic cones associated with a singular subspace of a Riemannian manifold, which is an extension of the classical notion of asymptotic directions. We get a simple expression of these cones for polyhedral surfaces, as well as convergence and approximation theorems. We use the asymptotic cones as facial descriptors and demonstrate the
19. Consistency relations in effective field theory
Energy Technology Data Exchange (ETDEWEB)
Munshi, Dipak; Regan, Donough, E-mail: [email protected], E-mail: [email protected] [Astronomy Centre, School of Mathematical and Physical Sciences, University of Sussex, Brighton BN1 9QH (United Kingdom)
2017-06-01
The consistency relations in large scale structure relate the lower-order correlation functions with their higher-order counterparts. They are direct outcome of the underlying symmetries of a dynamical system and can be tested using data from future surveys such as Euclid. Using techniques from standard perturbation theory (SPT), previous studies of consistency relation have concentrated on continuity-momentum (Euler)-Poisson system of an ideal fluid. We investigate the consistency relations in effective field theory (EFT) which adjusts the SPT predictions to account for the departure from the ideal fluid description on small scales. We provide detailed results for the 3D density contrast δ as well as the scaled divergence of velocity θ-bar . Assuming a ΛCDM background cosmology, we find the correction to SPT results becomes important at k ∼> 0.05 h/Mpc and that the suppression from EFT to SPT results that scales as square of the wave number k , can reach 40% of the total at k ≈ 0.25 h/Mpc at z = 0. We have also investigated whether effective field theory corrections to models of primordial non-Gaussianity can alter the squeezed limit behaviour, finding the results to be rather insensitive to these counterterms. In addition, we present the EFT corrections to the squeezed limit of the bispectrum in redshift space which may be of interest for tests of theories of modified gravity.
20. New framework for gauge field theories
International Nuclear Information System (INIS)
Blaha, S.
1979-01-01
Gauge theories are formulated within the framework of a generalization of quantum field theory. In particular, models of electrodynamics and of Yang-Mills theories, we discuss a model of the strong interaction with higher-order derivatives and quark confinement and a renormalizable model of pure quantum gravity with Einstein Lagrangian. In the case of electrodynamics it is shown that two models are possible: one with predictions which are identical to QED and one which is a quantum action-at-a-distance model of electrodynamics. In the case of Yang-Mills theories a model is constructed which is identical in predictions to any conventional model, or a quantum action-at-a-distance model. In the second case it is possible to eliminate all loops of Yang-Mills particles (in all gauges) in a manner consistent with unitarity. A variation of Yang-Mills models exists in this formulation which has higher-order derivative field equations. It is unitary and has positive probabilities. It can be used to construct a model of the strong interactions which has a linear potential and manifest quark confinement. Finally, it is shown how to construct an action-at-a-distance model of pure quantum gravity (whose classical limit is the dynamics of the Einstein Lagrangian) coupled to an external classical source. The model is trivially renormalizable. (author)
1. Why are tensor field theories asymptotically free?
Science.gov (United States)
Rivasseau, V.
2015-09-01
In this pedagogic letter we explain the combinatorics underlying the generic asymptotic freedom of tensor field theories. We focus on simple combinatorial models with a 1/p2 propagator and quartic interactions and on the comparison between the intermediate field representations of the vector, matrix and tensor cases. The transition from asymptotic freedom (tensor case) to asymptotic safety (matrix case) is related to the crossing symmetry of the matrix vertex, whereas in the vector case, the lack of asymptotic freedom (“Landau ghost”), as in the ordinary scalar φ^44 case, is simply due to the absence of any wave function renormalization at one loop.
2. Un-reduction in field theory.
Science.gov (United States)
Arnaudon, Alexis; López, Marco Castrillón; Holm, Darryl D
2018-01-01
The un-reduction procedure introduced previously in the context of classical mechanics is extended to covariant field theory. The new covariant un-reduction procedure is applied to the problem of shape matching of images which depend on more than one independent variable (for instance, time and an additional labelling parameter). Other possibilities are also explored: nonlinear [Formula: see text]-models and the hyperbolic flows of curves.
3. Two problems in thermal field theory
F can be calculated perturbatively as a sum of vacuum ... F / F id eal d c b a. Figure 4. Results of the screened perturbative expansion for the free energy as a func- tion of the coupling constant in scalar field theory [8]. (a) and (b): first ... for the pressure of a SU(3) Yang–Mills gas just by introducing a mass in the propagator.
4. Special relativity and classical field theory
CERN Document Server
Susskind, Leonard
2017-01-01
Physicist Leonard Susskind and data engineer Art Friedman are back. This time, they introduce readers to Einstein's special relativity and Maxwell's classical field theory. Using their typical brand of real math, enlightening drawings, and humor, Susskind and Friedman walk us through the complexities of waves, forces, and particles by exploring special relativity and electromagnetism. It's a must-read for both devotees of the series and any armchair physicist who wants to improve their knowledge of physics' deepest truths.
5. Multibrane solutions in open string field theory
Czech Academy of Sciences Publication Activity Database
Murata, Masaki; Schnabl, Martin
2012-01-01
Roč. 2012, č. 7 (2012), 1-26 ISSN 1126-6708 R&D Projects: GA MŠk(CZ) LH11106 Grant - others:EUROHORC and ESF(XE) EYI/07/E010 Institutional research plan: CEZ:AV0Z10100502 Keywords : string field theory * tachyon condensation Subject RIV: BF - Elementary Particles and High Energy Physics Impact factor: 5.618, year: 2012 http://link.springer.com/article/10.1007%2FJHEP07%282012%29063
6. Topics on field theories at finite temperature
International Nuclear Information System (INIS)
Eboli, O.J.P.
1985-01-01
The dynamics of a first order phase transition through the study of the decay rate of the false vacuum in the high temperature limit are analysed. An alternative approach to obtain the phase diagram of a field theory which is based on the study of the free energy of topological defects, is developed the behavior of coupling constants with the help of the Dyson-Schwinger equations at finite temperature, is evaluated. (author) [pt
7. Numerical studies of gauge field theories
International Nuclear Information System (INIS)
Creutz, M.
1981-06-01
Monte Carlo simulation of statistical systems is a well established technique of the condensed matter physicist. In the last few years, particle theorists have rediscovered this method and are having a marvelous time applying it to quantized gauge field theories. The main result has been strong numerical evidence that the standard SU(3) non-Abelian gauge theory of the strong interaction is capable of simultaneously confining quarks into the physical hadrons and exhibiting asymptotic freedom, the phenomenon of quark interactions being small at short distances. In four dimensions, confinement is a non-perturbative phenomenon. Essentially all models of confinement tie widely separated quarks together with strings of gauge field flux. This gives rise to a linear potential at long distances. A Monte Carlo program generates a sequence of field configuration by a series of random changes of the fields. The algorithm is so constructed that ultimately the probability density for finding any given configuration is proportional to the Boltzmann weighting. We bring our lattices into thermal equilibrium with a heat bath at a temperature specified by the coupling constant. Thus we do computer experiments with four-dimensional crystals stored in a computer memory. As the entire field configuration is stored, we have access to any correlation function desired. These lectures describe the kinds of experiments being done and the implications of these results for strong interaction physics
8. Heavy Quarks, QCD, and Effective Field Theory
Energy Technology Data Exchange (ETDEWEB)
Thomas Mehen
2012-10-09
The research supported by this OJI award is in the area of heavy quark and quarkonium production, especially the application Soft-Collinear E ective Theory (SCET) to the hadronic production of quarkonia. SCET is an e ffective theory which allows one to derive factorization theorems and perform all order resummations for QCD processes. Factorization theorems allow one to separate the various scales entering a QCD process, and in particular, separate perturbative scales from nonperturbative scales. The perturbative physics can then be calculated using QCD perturbation theory. Universal functions with precise fi eld theoretic de nitions describe the nonperturbative physics. In addition, higher order perturbative QCD corrections that are enhanced by large logarithms can be resummed using the renormalization group equations of SCET. The applies SCET to the physics of heavy quarks, heavy quarkonium, and similar particles.
9. Nonlinear analysis approximation theory, optimization and applications
CERN Document Server
2014-01-01
Many of our daily-life problems can be written in the form of an optimization problem. Therefore, solution methods are needed to solve such problems. Due to the complexity of the problems, it is not always easy to find the exact solution. However, approximate solutions can be found. The theory of the best approximation is applicable in a variety of problems arising in nonlinear functional analysis and optimization. This book highlights interesting aspects of nonlinear analysis and optimization together with many applications in the areas of physical and social sciences including engineering. It is immensely helpful for young graduates and researchers who are pursuing research in this field, as it provides abundant research resources for researchers and post-doctoral fellows. This will be a valuable addition to the library of anyone who works in the field of applied mathematics, economics and engineering.
10. Tachyon condensation in superstring field theory
International Nuclear Information System (INIS)
Berkovits, Nathan; Sen, Ashoke; Zwiebach, Barton
2000-01-01
It has been conjectured that at the stationary point of the tachyon potential for the D-brane-anti-D-brane pair or for the non-BPS D-brane of superstring theories, the negative energy density cancels the brane tensions. We study this conjecture using a Wess-Zumino-Witten-like open superstring field theory free of contact term divergences and recently shown to give 60% of the vacuum energy by condensation of the tachyon field alone. While the action is non-polynomial, the multiscalar tachyon potential to any fixed level involves only a finite number of interactions. We compute this potential to level three, obtaining 85% of the expected vacuum energy, a result consistent with convergence that can also be viewed as a successful test of the string field theory. The resulting effective tachyon potential is bounded below and has two degenerate global minima. We calculate the energy density of the kink solution interpolating between these minima finding good agreement with the tension of the D-brane of one lower dimension
11. Regularity Theory for Mean-Field Game Systems
KAUST Repository
Gomes, Diogo A.
2016-09-14
Beginning with a concise introduction to the theory of mean-field games (MFGs), this book presents the key elements of the regularity theory for MFGs. It then introduces a series of techniques for well-posedness in the context of mean-field problems, including stationary and time-dependent MFGs, subquadratic and superquadratic MFG formulations, and distinct classes of mean-field couplings. It also explores stationary and time-dependent MFGs through a series of a-priori estimates for solutions of the Hamilton-Jacobi and Fokker-Planck equation. It shows sophisticated a-priori systems derived using a range of analytical techniques, and builds on previous results to explain classical solutions. The final chapter discusses the potential applications, models and natural extensions of MFGs. As MFGs connect common problems in pure mathematics, engineering, economics and data management, this book is a valuable resource for researchers and graduate students in these fields.
12. Regularity theory for mean-field game systems
CERN Document Server
Gomes, Diogo A; Voskanyan, Vardan
2016-01-01
Beginning with a concise introduction to the theory of mean-field games (MFGs), this book presents the key elements of the regularity theory for MFGs. It then introduces a series of techniques for well-posedness in the context of mean-field problems, including stationary and time-dependent MFGs, subquadratic and superquadratic MFG formulations, and distinct classes of mean-field couplings. It also explores stationary and time-dependent MFGs through a series of a-priori estimates for solutions of the Hamilton-Jacobi and Fokker-Planck equation. It shows sophisticated a-priori systems derived using a range of analytical techniques, and builds on previous results to explain classical solutions. The final chapter discusses the potential applications, models and natural extensions of MFGs. As MFGs connect common problems in pure mathematics, engineering, economics and data management, this book is a valuable resource for researchers and graduate students in these fields.
13. Lattice theory special topics and applications
CERN Document Server
Wehrung, Friedrich
George Grätzer's Lattice Theory: Foundation is his third book on lattice theory (General Lattice Theory, 1978, second edition, 1998). In 2009, Grätzer considered updating the second edition to reflect some exciting and deep developments. He soon realized that to lay the foundation, to survey the contemporary field, to pose research problems, would require more than one volume and more than one person. So Lattice Theory: Foundation provided the foundation. Now we complete this project with Lattice Theory: Special Topics and Applications, written by a distinguished group of experts, to cover some of the vast areas not in Foundation. This first volume is divided into three parts. Part I. Topology and Lattices includes two chapters by Klaus Keimel, Jimmie Lawson and Ales Pultr, Jiri Sichler. Part II. Special Classes of Finite Lattices comprises four chapters by Gabor Czedli, George Grätzer and Joseph P. S. Kung. Part III. Congruence Lattices of Infinite Lattices and Beyond includes four chapters by Friedrich W...
14. Topics in string theory and quantum field theory
Science.gov (United States)
Giombi, Simone
In this dissertation we study several topics in string theory and quantum field theory, which we collect into three main parts. The first part contains some studies in the context of twistor string theory. Witten proposed that the perturbative expansion of N = 4 super Yang-Mills theory has a dual formulation in terms of a topological string theory on the supertwistor space CP3|4 . We discuss extensions of this construction in two directions. First, we make some preliminary considerations on the possibility of having a similar twistor approach to perturbative gravity. Then we extend the construction to theories with lower supersymmetry by taking orbifolds in the fermionic directions of CP3|4 . We consider N = 1 and N = 2 superconformal quiver gauge theories as specific examples. In the second part of the dissertation we study worldline methods in curved space. In particular, we use the N = 2 spinning particle to describe antisymmetric tensors of arbitrary rank propagating in a curved background. The path integral quantization of the N = 2 particle produces a novel and compact representation of the one loop effective action for generic differential p-forms, including the vector field as a special example. We study both the massless and massive case, and show that the worldline representation of the one loop effective action can be used to efficiently study various quantum effects for antisymmetric tensor fields of arbitrary rank in arbitrary dimension. In the last and final part we study some topics in the context of the AdS/CFT correspondence. We start by investigating the recently discovered description of half-BPS supergravity backgrounds in terms of one-dimensional free fermions. We study a generalization of this construction obtained by considering free fermions at non-zero temperature. The ADM mass of the corresponding supergravity background is shown to agree with the fermion thermal energy, and we propose a way to qualitatively match the entropy in the two
15. Undergraduate Lecture Notes in Topological Quantum Field Theory
OpenAIRE
Ivancevic, Vladimir G.; Ivancevic, Tijana T.
2008-01-01
These third-year lecture notes are designed for a 1-semester course in topological quantum field theory (TQFT). Assumed background in mathematics and physics are only standard second-year subjects: multivariable calculus, introduction to quantum mechanics and basic electromagnetism. Keywords: quantum mechanics/field theory, path integral, Hodge decomposition, Chern-Simons and Yang-Mills gauge theories, conformal field theory
16. Particle versus field structure in conformal quantum field theories
International Nuclear Information System (INIS)
Schroer, Bert
2000-06-01
I show that a particle structure in conformal field theory is incompatible with interactions. As a substitute one has particle-like excitations whose interpolating fields have in addition to their canonical dimension an anomalous contribution. The spectra of anomalous dimension is given in terms of the Lorentz invariant quadratic invariant (compact mass operator) of a conformal generator R μ with pure discrete spectrum. The perturbative reading of R o as a Hamiltonian in its own right, associated with an action in a functional integral setting naturally leads to the Ad S formulation. The formal service role of Ad S in order to access C QFT by a standard perturbative formalism (without being forced to understand first massive theories and then taking their scale-invariant limit) vastly increases the realm of conventionally accessible 4-dim. C QFT beyond those for which one had to use Lagrangians with supersymmetry in order to have a vanishing Beta-function. (author)
17. On the background independence of string field theory
International Nuclear Information System (INIS)
Sen, A.
1990-01-01
Given a solution Ψ cl of the classical equations of motion in either closed or open string field theory formulated around a given conformal field theory background, we can construct a new operator Q B in the corresponding two-dimensional field theory such that (Q B ) 2 =0. It is shown that in the limit when the background field Ψ cl is weak, Q B can be identified with the BRST charge of a new local conformal field theory. This indicates that the string field theories formulated around these two different conformal field theories are actually the same theory, and that these two conformal field theories may be regarded as different classical solutions of this string field theory. (orig.)
18. An application of Anthony Giddens' structuration theory
African Journals Online (AJOL)
The aim of this article is to discuss the structuration theory of Anthony Giddens with regard to its applicability to translation studies. Key concepts of Giddens' sociological theory as agent, agency, structure, system and structuration will be explored in terms of their applicability to translation. In this article, structuration theory ...
19. Theory of singlet-ground-state magnetism. Application to field-induced transitions in CsFeCl3 and CsFeBr3
DEFF Research Database (Denmark)
Lindgård, P.-A.; Schmid, B.
1993-01-01
In the singlet ground-state systems CsFeCl3 and CsFeBr3 a large single-ion anisotropy causes a singlet ground state and a doubly degenerate doublet as the first excited states of the Fe2+ ion. In addition the magneteic interaction is anisotropic being much larger along the z axis than perpendicular...... to it. Therefore, these quasi-one-dimensional magnetic model systems are ideal to demonstrate unique correlation effects. Within the framework of the correlation theory we derive the expressions for the excitation spectrum. When a magnetic field is applied parallel to the z axis both substances have...
20. (Non-)decoupled supersymmetric field theories
International Nuclear Information System (INIS)
Pietro, Lorenzo Di; Dine, Michael; Komargodski, Zohar
2014-01-01
We study some consequences of coupling supersymmetric theories to (super)gravity. To linear order, the couplings are determined by the energy-momentum supermultiplet. At higher orders, the couplings are determined by contact terms in correlation functions of the energy-momentum supermultiplet. We focus on the couplings of one particular field in the supergravity multiplet, the auxiliary field M. We discuss its linear and quadratic (seagull) couplings in various supersymmetric theories. In analogy to the local renormalization group formalism (http://dx.doi.org/10.1016/0370-2693(89)90729-6; http://dx.doi.org/10.1016/0550-3213(90)90584-Z; http://dx.doi.org/10.1016/0550-3213(91)80030-P), we provide a prescription for how to fix the quadratic couplings. They generally arise at two-loops in perturbation theory. We check our prescription by explicitly computing these couplings in several examples such as mass-deformed N=4 and in the Coulomb phase of some theories. These couplings affect the Lagrangians of rigid supersymmetric theories in curved space. In addition, our analysis leads to a transparent derivation of the phenomenon known as Anomaly Mediation. In contrast to previous approaches, we obtain both the gaugino and scalar masses of Anomaly Mediation by relying just on classical, minimal supergravity and a manifestly local and supersymmetric Wilsonian point of view. Our discussion naturally incorporates the connection between Anomaly Mediation and supersymmetric AdS 4 Lagrangians. This note can be read without prior familiarity with Anomaly Mediated Supersymmetry Breaking (AMSB)
1. Motion of small bodies in classical field theory
International Nuclear Information System (INIS)
Gralla, Samuel E.
2010-01-01
I show how prior work with R. Wald on geodesic motion in general relativity can be generalized to classical field theories of a metric and other tensor fields on four-dimensional spacetime that (1) are second-order and (2) follow from a diffeomorphism-covariant Lagrangian. The approach is to consider a one-parameter-family of solutions to the field equations satisfying certain assumptions designed to reflect the existence of a body whose size, mass, and various charges are simultaneously scaled to zero. (That such solutions exist places a further restriction on the class of theories to which our results apply.) Assumptions are made only on the spacetime region outside of the body, so that the results apply independent of the body's composition (and, e.g., black holes are allowed). The worldline 'left behind' by the shrinking, disappearing body is interpreted as its lowest-order motion. An equation for this worldline follows from the 'Bianchi identity' for the theory, without use of any properties of the field equations beyond their being second-order. The form of the force law for a theory therefore depends only on the ranks of its various tensor fields; the detailed properties of the field equations are relevant only for determining the charges for a particular body (which are the ''monopoles'' of its exterior fields in a suitable limiting sense). I explicitly derive the force law (and mass-evolution law) in the case of scalar and vector fields, and give the recipe in the higher-rank case. Note that the vector force law is quite complicated, simplifying to the Lorentz force law only in the presence of the Maxwell gauge symmetry. Example applications of the results are the motion of 'chameleon' bodies beyond the Newtonian limit, and the motion of bodies in (classical) non-Abelian gauge theory. I also make some comments on the role that scaling plays in the appearance of universality in the motion of bodies.
2. Fuzzy neural network theory and application
CERN Document Server
Liu, Puyin
2004-01-01
This book systematically synthesizes research achievements in the field of fuzzy neural networks in recent years. It also provides a comprehensive presentation of the developments in fuzzy neural networks, with regard to theory as well as their application to system modeling and image restoration. Special emphasis is placed on the fundamental concepts and architecture analysis of fuzzy neural networks. The book is unique in treating all kinds of fuzzy neural networks and their learning algorithms and universal approximations, and employing simulation examples which are carefully designed to he
3. Modeling and Optimization : Theory and Applications Conference
CERN Document Server
Terlaky, Tamás
2015-01-01
This volume contains a selection of contributions that were presented at the Modeling and Optimization: Theory and Applications Conference (MOPTA) held at Lehigh University in Bethlehem, Pennsylvania, USA on August 13-15, 2014. The conference brought together a diverse group of researchers and practitioners, working on both theoretical and practical aspects of continuous or discrete optimization. Topics presented included algorithms for solving convex, network, mixed-integer, nonlinear, and global optimization problems, and addressed the application of deterministic and stochastic optimization techniques in energy, finance, logistics, analytics, healthcare, and other important fields. The contributions contained in this volume represent a sample of these topics and applications and illustrate the broad diversity of ideas discussed at the meeting.
4. Modeling and Optimization : Theory and Applications Conference
CERN Document Server
Terlaky, Tamás
2017-01-01
This volume contains a selection of contributions that were presented at the Modeling and Optimization: Theory and Applications Conference (MOPTA) held at Lehigh University in Bethlehem, Pennsylvania, USA on August 17-19, 2016. The conference brought together a diverse group of researchers and practitioners, working on both theoretical and practical aspects of continuous or discrete optimization. Topics presented included algorithms for solving convex, network, mixed-integer, nonlinear, and global optimization problems, and addressed the application of deterministic and stochastic optimization techniques in energy, finance, logistics, analytics, health, and other important fields. The contributions contained in this volume represent a sample of these topics and applications and illustrate the broad diversity of ideas discussed at the meeting.
5. Constraints on Interacting Scalars in 2T Field Theory and No Scale Models in 1T Field Theory
CERN Document Server
Bars, Itzhak
2010-01-01
In this paper I determine the general form of the physical and mathematical restrictions that arise on the interactions of gravity and scalar fields in the 2T field theory setting, in d+2 dimensions, as well as in the emerging shadows in d dimensions. These constraints on scalar fields follow from an underlying Sp(2,R) gauge symmetry in phase space. Determining these general constraints provides a basis for the construction of 2T supergravity, as well as physical applications in 1T-field theory, that are discussed briefly here, and more detail elsewhere. In particular, no scale models that lead to a vanishing cosmological constant at the classical level emerge naturally in this setting.
6. Triboluminescence theory, synthesis, and application
CERN Document Server
Okoli, Okenwa; Fontenot, Ross; Hollerman, William
2016-01-01
This book expounds on progress made over the last 35 years in the theory, synthesis, and application of triboluminescence for creating smart structures. It presents in detail the research into utilization of the triboluminescent properties of certain crystals as new sensor systems for smart engineering structures, as well as triboluminescence-based sensor systems that have the potential to enable wireless, in-situ, real time and distributed (WIRD) structural health monitoring of composite structures. The sensor component of any structural health monitoring (SHM) technology — measures the effects of the external load/event and provides the necessary inputs for appropriate preventive/corrective action to be taken in a smart structure — sits at the heart of such a system. This volume explores advances in materials properties and structural behavior underlying creation of smart composite structures and sensor systems for structural health monitoring of critical engineering structures, such as bridges, aircraf...
7. Conformal Field Theory, Automorphic Forms and Related Topics
CERN Document Server
Weissauer, Rainer; CFT 2011
2014-01-01
This book, part of the series Contributions in Mathematical and Computational Sciences, reviews recent developments in the theory of vertex operator algebras (VOAs) and their applications to mathematics and physics. The mathematical theory of VOAs originated from the famous monstrous moonshine conjectures of J.H. Conway and S.P. Norton, which predicted a deep relationship between the characters of the largest simple finite sporadic group, the Monster, and the theory of modular forms inspired by the observations of J. MacKay and J. Thompson. The contributions are based on lectures delivered at the 2011 conference on Conformal Field Theory, Automorphic Forms and Related Topics, organized by the editors as part of a special program offered at Heidelberg University that summer under the sponsorship of the MAThematics Center Heidelberg (MATCH).
8. An invitation to quantum field theory
International Nuclear Information System (INIS)
Alvarez-Gaume, Luis; Vazquez-Mozo, Miguel A.
2012-01-01
This book provides an introduction to Quantum Field Theory (QFT) at an elementary level - with only special relativity, electromagnetism and quantum mechanics as prerequisites. For this fresh approach to teaching QFT, based on numerous lectures and courses given by the authors, a representative sample of topics has been selected containing some of the more innovative, challenging or subtle concepts. They are presented with a minimum of technical details, the discussion of the main ideas being more important than the presentation of the typically very technical mathematical details necessary to obtain the final results. Special attention is given to the realization of symmetries in particle physics: global and local symmetries, explicit, spontaneously broken, and anomalous continuous symmetries, as well as discrete symmetries. Beyond providing an overview of the standard model of the strong, weak and electromagnetic interactions and the current understanding of the origin of mass, the text enumerates the general features of renormalization theory as well as providing a cursory description of effective field theories and the problem of naturalness in physics. Among the more advanced topics the reader will find are an outline of the first principles derivation of the CPT theorem and the spin-statistics connection. As indicated by the title, the main aim of this text is to motivate the reader to study QFT by providing a self-contained and approachable introduction to the most exciting and challenging aspects of this successful theoretical framework. (orig.)
9. An invitation to quantum field theory
Energy Technology Data Exchange (ETDEWEB)
Alvarez-Gaume, Luis [CERN, Geneva (Switzerland). Physics Dept.; Vazquez-Mozo, Miguel A. [Salamanca Univ. (Spain). Dept. de Fisica Fundamental
2012-07-01
This book provides an introduction to Quantum Field Theory (QFT) at an elementary level - with only special relativity, electromagnetism and quantum mechanics as prerequisites. For this fresh approach to teaching QFT, based on numerous lectures and courses given by the authors, a representative sample of topics has been selected containing some of the more innovative, challenging or subtle concepts. They are presented with a minimum of technical details, the discussion of the main ideas being more important than the presentation of the typically very technical mathematical details necessary to obtain the final results. Special attention is given to the realization of symmetries in particle physics: global and local symmetries, explicit, spontaneously broken, and anomalous continuous symmetries, as well as discrete symmetries. Beyond providing an overview of the standard model of the strong, weak and electromagnetic interactions and the current understanding of the origin of mass, the text enumerates the general features of renormalization theory as well as providing a cursory description of effective field theories and the problem of naturalness in physics. Among the more advanced topics the reader will find are an outline of the first principles derivation of the CPT theorem and the spin-statistics connection. As indicated by the title, the main aim of this text is to motivate the reader to study QFT by providing a self-contained and approachable introduction to the most exciting and challenging aspects of this successful theoretical framework. (orig.)
10. Monoidal categories and topological field theory
CERN Document Server
2017-01-01
This monograph is devoted to monoidal categories and their connections with 3-dimensional topological field theories. Starting with basic definitions, it proceeds to the forefront of current research. Part 1 introduces monoidal categories and several of their classes, including rigid, pivotal, spherical, fusion, braided, and modular categories. It then presents deep theorems of Müger on the center of a pivotal fusion category. These theorems are proved in Part 2 using the theory of Hopf monads. In Part 3 the authors define the notion of a topological quantum field theory (TQFT) and construct a Turaev-Viro-type 3-dimensional state sum TQFT from a spherical fusion category. Lastly, in Part 4 this construction is extended to 3-manifolds with colored ribbon graphs, yielding a so-called graph TQFT (and, consequently, a 3-2-1 extended TQFT). The authors then prove the main result of the monograph: the state sum graph TQFT derived from any spherical fusion category is isomorphic to the Reshetikhin-Turaev surgery gr...
11. Introduction to soliton theory applications to mechanics
CERN Document Server
Munteanu, Ligia
2005-01-01
This monograph provides the application of soliton theory to solve certain problems selected from the fields of mechanics. The work is based of the authors' research, and on some specified, significant results existing in the literature. The present monograph is not a simple translation of its predecessor appeared in Publishing House of the Romanian Academy in 2002. Improvements outline the way in which the soliton theory is applied to solve some engineering problems. The book addresses concrete resolution methods of certain problems such as the motion of thin elastic rod, vibrations of initial deformed thin elastic rod, the coupled pendulum oscillations, dynamics of left ventricle, transient flow of blood in arteries, the subharmonic waves generation in a piezoelectric plate with Cantor-like structure, and some problems related to Tzitzeica surfaces. This comprehensive study enables the readers to make connections between the soliton physical phenomenon and some partical, engineering problems.
12. Relativistic mean field theory for unstable nuclei
International Nuclear Information System (INIS)
Toki, Hiroshi
2000-01-01
We discuss the properties of unstable nuclei in the framework of the relativistic mean field (RMF) theory. We take the RMF theory as a phenomenological theory with several parameters, whose form is constrained by the successful microscopic theory (RBHF), and whose values are extracted from the experimental values of unstable nuclei. We find the outcome with the newly obtained parameter sets (TM1 and TMA) is promising in comparison with various experimental data. We calculate systematically the ground state properties of even-even nuclei up to the drip lines; about 2000 nuclei. We find that the neutron magic shells (N=82, 128) at the standard magic numbers stay at the same numbers even far from the stability line and hence provide the feature of the r-process nuclei. However, many proton magic numbers disappear at the neutron numbers far away from the magic numbers due to the deformations. We discuss how to describe giant resonances for the case of the non-linear coupling terms for the sigma and omega mesons in the relativistic RPA. We mention also the importance of the relativistic effect on the spin observables as the Gamow-Teller strength and the longitudinal and transverse spin responses. (author)
13. GNSS remote sensing theory, methods and applications
CERN Document Server
Jin, Shuanggen; Xie, Feiqin
2014-01-01
This book presents the theory and methods of GNSS remote sensing as well as its applications in the atmosphere, oceans, land and hydrology. It contains detailed theory and study cases to help the reader put the material into practice.
14. Game Theory: An Application to Tanners and 'Pomo' Wholesalers in ...
African Journals Online (AJOL)
Game Theory: An Application to Tanners and 'Pomo' Wholesalers in Hides Marketing Competition in Nigeria. ... Agricultural economics is an applied social science so game theory was applied practically from a field survey to determine the level of competition between tanners and 'pomo' wholesalers in Nigeria. All the ...
15. Bare coupling constants and asymptotic behaviour in reggeon field theory
International Nuclear Information System (INIS)
Baig, M.
1983-01-01
A relation between the values of bare coupling constants and the asymptotic behaviour of the reggeon field theory (RFT) is discussed. It is shown how the numerical values of bare coupling constants fix the starting point of renormalization group trajectories which, in turn, determine the asymptotic behaviour of the RFT. Applications to a pure pomeron theory and a pomeron plus f-pole model are discussed. Some nontrivial phenomenological information concerning the values of bare triple-Regge pomeron-f-pole coupling constants is obtained
16. Yang-Baxter algebra - Integrable systems - Conformal quantum field theories
International Nuclear Information System (INIS)
Karowski, M.
1989-01-01
This series of lectures is based on investigations [1,2] of finite-size corrections for the six-vertex model by means of Bethe ansatz methods. In addition a review on applications of Yang-Baxter algebras and an introduction to the theory of integrable systems and the algebraic Bethe ansatz is presented. A Θ-vacuum like angle appearing in the RSOS-models is discussed. The continuum limit in the critical case of these statistical models is performed to obtain the minimal models of conformal quantum field theory. (author)
17. Theory of field-reversed configurations
International Nuclear Information System (INIS)
Steinhauer, L.C.
1993-01-01
This report summarizes results from the theoretical program on field reversed configurations (FRC) at STI Optronics. The program, which has spanned the last 13 years, has included analytical as well as computational components. It has led to published papers on every major topic of FRC theory. The report is outlined to summarize results from each of these topic areas: formation, equilibrium, stability, and confinement. Also briefly described are Steinhauer's activities as Compact Toroid Theory Listening Post. Appendix A is a brief listing of the major advances achieved in this program. Attached at the back of this report is a collection of technical papers in archival journals that resulted from work in this program. The discussion within each subsection is given chronologically in order to give a historical sense of the evolution of understanding of FRC physics
18. Working Group Report: Lattice Field Theory
Energy Technology Data Exchange (ETDEWEB)
Blum, T.; et al.,
2013-10-22
This is the report of the Computing Frontier working group on Lattice Field Theory prepared for the proceedings of the 2013 Community Summer Study ("Snowmass"). We present the future computing needs and plans of the U.S. lattice gauge theory community and argue that continued support of the U.S. (and worldwide) lattice-QCD effort is essential to fully capitalize on the enormous investment in the high-energy physics experimental program. We first summarize the dramatic progress of numerical lattice-QCD simulations in the past decade, with some emphasis on calculations carried out under the auspices of the U.S. Lattice-QCD Collaboration, and describe a broad program of lattice-QCD calculations that will be relevant for future experiments at the intensity and energy frontiers. We then present details of the computational hardware and software resources needed to undertake these calculations.
19. A Cahn-Hilliard-type phase-field theory for species diffusion coupled with large elastic deformations: Application to phase-separating Li-ion electrode materials
Science.gov (United States)
Di Leo, Claudio V.; Rejovitzky, Elisha; Anand, Lallit
2014-10-01
We formulate a unified framework of balance laws and thermodynamically-consistent constitutive equations which couple Cahn-Hilliard-type species diffusion with large elastic deformations of a body. The traditional Cahn-Hilliard theory, which is based on the species concentration c and its spatial gradient ∇c, leads to a partial differential equation for the concentration which involves fourth-order spatial derivatives in c; this necessitates use of basis functions in finite-element solution procedures that are piecewise smooth and globally C1-continuous. In order to use standard C0-continuous finite-elements to implement our phase-field model, we use a split-method to reduce the fourth-order equation into two second-order partial differential equations (pdes). These two pdes, when taken together with the pde representing the balance of forces, represent the three governing pdes for chemo-mechanically coupled problems. These are amenable to finite-element solution methods which employ standard C0-continuous finite-element basis functions. We have numerically implemented our theory by writing a user-element subroutine for the widely used finite-element program Abaqus/Standard. We use this numerically implemented theory to first study the diffusion-only problem of spinodal decomposition in the absence of any mechanical deformation. Next, we use our fully coupled theory and numerical-implementation to study the combined effects of diffusion and stress on the lithiation of a representative spheroidal-shaped particle of a phase-separating electrode material.
20. Vortex operators in gauge field theories
International Nuclear Information System (INIS)
Polchinski, J.G.
1980-01-01
We study several related aspects of the t Hooft vortex operator. The first chapter reviews the current picture of the vacuum of quantum chromodynamics, the idea of dual field theories, and the idea of the vortex operator. The second chapter deals with the Abelian vortex operator written in terms of elementary fields and with the calculation of its Green's functions. The Dirac veto problem appears in a new guise. We present a two dimensional solvable model of a Dirac string. This leads us to a new solution of the veto problem; we discuss its extension to four dimensions. We then show how the Green's functions can be expressed more neatly in terms of Wu and Yang's geometrical idea of sections. In the third chapter we discuss the dependence of the Green's functions of the Wilson and t Hooft operators on the nature of the vacuum. In the fourth chapter we consider systems which have fields in the fundamental representation, so that there are no vortex operators. When these fields enter only weakly into the dynamics, as is the case in QCD and in real superconductors, we would expect to be able to define a vortex-like operator. We show that any such operator can no longer be local looplike, but must have commutators at long range. We can still find an operator with useful properties, its cluster property, though more complicated than that of the usual vortex operator, still appears to distinguish Higgs, confining and perturbative phases. To test this, we consider a U(1) lattice gauge theory with two matter fields, one singly charged (fundamental) and one doubly charged (adjoint)
1. The Supersymmetric Effective Field Theory of Inflation
Energy Technology Data Exchange (ETDEWEB)
Delacrétaz, Luca V.; Gorbenko, Victor [Stanford Institute for Theoretical Physics, Stanford University,Stanford, CA 94306 (United States); Senatore, Leonardo [Stanford Institute for Theoretical Physics, Stanford University,Stanford, CA 94306 (United States); Kavli Institute for Particle Astrophysics and Cosmology, Stanford University and SLAC,Menlo Park, CA 94025 (United States)
2017-03-10
We construct the Supersymmetric Effective Field Theory of Inflation, that is the most general theory of inflationary fluctuations when time-translations and supersymmetry are spontaneously broken. The non-linear realization of these invariances allows us to define a complete SUGRA multiplet containing the graviton, the gravitino, the Goldstone of time translations and the Goldstino, with no auxiliary fields. Going to a unitary gauge where only the graviton and the gravitino are present, we write the most general Lagrangian built out of the fluctuations of these fields, invariant under time-dependent spatial diffeomorphisms, but softly-breaking time diffeomorphisms and gauged SUSY. With a suitable Stückelberg transformation, we introduce the Goldstone boson of time translation and the Goldstino of SUSY. No additional dynamical light field is needed. In the high energy limit, larger than the inflationary Hubble scale for the Goldstino, these fields decouple from the graviton and the gravitino, greatly simplifying the analysis in this regime. We study the phenomenology of this Lagrangian. The Goldstino can have a non-relativistic dispersion relation. Gravitino and Goldstino affect the primordial curvature perturbations at loop level. The UV modes running in the loops generate three-point functions which are degenerate with the ones coming from operators already present in the absence of supersymmetry. Their size is potentially as large as corresponding to f{sub NL}{sup equil.,orthog.}∼1 or, for particular operators, even ≫1. The non-degenerate contribution from modes of order H is estimated to be very small.
2. Quantitative graph theory mathematical foundations and applications
CERN Document Server
Dehmer, Matthias
2014-01-01
The first book devoted exclusively to quantitative graph theory, Quantitative Graph Theory: Mathematical Foundations and Applications presents and demonstrates existing and novel methods for analyzing graphs quantitatively. Incorporating interdisciplinary knowledge from graph theory, information theory, measurement theory, and statistical techniques, this book covers a wide range of quantitative-graph theoretical concepts and methods, including those pertaining to real and random graphs such as:Comparative approaches (graph similarity or distance)Graph measures to characterize graphs quantitat
3. Towers of algebras in rational conformal field theories
International Nuclear Information System (INIS)
Gomez, C.; Sierra, G.
1991-01-01
This paper reports on Jones fundamental construction applied to rational conformal field theories. The Jones algebra which emerges in this application is realized in terms of duality operations. The generators of the algebra are an open version of Verlinde's operators. The polynomial equations appear in this context as sufficient conditions for the existence of Jones algebra. The ADE classification of modular invariant partition functions is put in correspondence with Jones classification of subfactors
4. Bookshelf (The Quantum Theory of Fields, La lumiere des neutrinos)
International Nuclear Information System (INIS)
Anon.
1995-01-01
The Quantum Theory of Fields Volume 1: Foundations by Steven Weinberg, Cambridge University Press, 1995: Steven Weinberg is celebrated for his many contributions to quantum field theory and its applications to elementary particle physics - most notably, for developing the electroweak theory, the unified model of the electromagnetic and weak forces that forms part of the Standard Model that has explained essentially all accelerator data on the behaviour of elementary particles. This is the culmination of the developments in quantum field theory that started in the early days of quantum mechanics and came to maturity with the development of quantum electrodynamics in the late 1940s. Quantum field theory is the basic theoretical framework for research in particle physics as well as in many areas of condensed matter physics. No wonder the community has been waiting with anticipation for Weinberg's exposition of the subject in the form of a two-volume textbook - the more so since, despite the existence of many textbooks, few of them are written with the insight and detail that are needed for a thorough understanding. The community will not be disappointed, at least on the basis of this first volume - Volume 2 is due to appear next year. Volume 1 is 600 pages of meticulous exposition of the fundamentals of the subject, starting from a historical introduction and the canonical formulation of quantum field theory to modern path integral methods applied to the quantization of electrodynamics and a first discussion of renormaiization. In addition to a superb treatment of all the conventional topics there are numerous sections covering areas that are not normally emphasized, such as the subject of field redefinitions, higher-rank tensor fields and an unusually clear and thorough treatment of infrared effects. This is only the basics - Volume 2 promises to develop the subjects at the cutting edge of modern research such as Yang-Mills theory, the renormalization group
5. Basics of thermal field theory a tutorial on perturbative computations
CERN Document Server
Laine, Mikko
2016-01-01
This book presents thermal field theory techniques, which can be applied in both cosmology and the theoretical description of the QCD plasma generated in heavy-ion collision experiments. It focuses on gauge interactions (whether weak or strong), which are essential in both contexts. As well as the many differences in the physics questions posed and in the microscopic forces playing a central role, the authors also explain the similarities and the techniques, such as the resummations, that are needed for developing a formally consistent perturbative expansion. The formalism is developed step by step, starting from quantum mechanics; introducing scalar, fermionic and gauge fields; describing the issues of infrared divergences; resummations and effective field theories; and incorporating systems with finite chemical potentials. With this machinery in place, the important class of real-time (dynamic) observables is treated in some detail. This is followed by an overview of a number of applications, ranging from t...
6. Coarse grainings and irreversibility in quantum field theory
International Nuclear Information System (INIS)
Anastopoulos, C.
1997-01-01
In this paper we are interested in studying coarse graining in field theories using the language of quantum open systems. Motivated by the ideas of Hu and Calzetta on correlation histories we employ the Zwanzig projection technique to obtain evolution equations for relevant observables in self-interacting scalar field theories. Our coarse-graining operation consists in concentrating solely on the evolution of the correlation functions of degree less than n, a treatment which corresponds to the familiar truncation of the BBKGY hierarchy at the nth level. We derive the equations governing the evolution of mean-field and two-point functions thus identifying the terms corresponding to dissipation and noise. We discuss possible applications of our formalism, the emergence of classical behavior, and the connection to the decoherent histories framework. copyright 1997 The American Physical Society
7. Mathematical analysis, approximation theory and their applications
CERN Document Server
Gupta, Vijay
2016-01-01
Designed for graduate students, researchers, and engineers in mathematics, optimization, and economics, this self-contained volume presents theory, methods, and applications in mathematical analysis and approximation theory. Specific topics include: approximation of functions by linear positive operators with applications to computer aided geometric design, numerical analysis, optimization theory, and solutions of differential equations. Recent and significant developments in approximation theory, special functions and q-calculus along with their applications to mathematics, engineering, and social sciences are discussed and analyzed. Each chapter enriches the understanding of current research problems and theories in pure and applied research.
8. Field theory approach to quantum hall effect
International Nuclear Information System (INIS)
Cabo, A.; Chaichian, M.
1990-07-01
The Fradkin's formulation of statistical field theory is applied to the Coulomb interacting electron gas in a magnetic field. The electrons are confined to a plane in normal 3D-space and also interact with the physical 3D-electromagnetic field. The magnetic translation group (MTG) Ward identities are derived. Using them it is shown that the exact electron propagator is diagonalized in the basis of the wave functions of the free electron in a magnetic field whenever the MTG is unbroken. The general tensor structure of the polarization operator is obtained and used to show that the Chern-Simons action always describes the Hall effect properties of the system. A general proof of the Streda formula for the Hall conductivity is presented. It follows that the coefficient of the Chern-Simons terms in the long-wavelength approximation is exactly given by this relation. Such a formula, expressing the Hall conductivity as a simple derivative, in combination with diagonal form of the full propagator allows to obtain a simple expressions for the filling factor and the Hall conductivity. Indeed, these results, after assuming that the chemical potential lies in a gap of the density of states, lead to the conclusion that the Hall conductivity is given without corrections by σ xy = νe 2 /h where ν is the filling factor. In addition it follows that the filling factor is independent of the magnetic field if the chemical potential remains in the gap. (author). 21 ref, 1 fig
9. Theory of microemulsions in a gravitational field
Science.gov (United States)
Jeng, J. F.; Miller, Clarence A.
1989-01-01
A theory of microemulsions developed previously is extended to include the effect of a gravitational field. It predicts variation with position of drop size, drop volume fraction, and area per molecule in the surfactant films within a microemulsion phase. Variation in volume fraction is greatest and occurs in such a way that oil content increases with increasing elevation, as has been found experimentally. Large composition variations are predicted within a middle phase microemulsion near optimal conditions because inversion from the water-continuous to the oil-continuous arrangement occurs with increasing elevation. Generally speaking, gravity reduces solubilization within microemulsions and promotes separation of excess phases.
10. A matrix model from string field theory
Directory of Open Access Journals (Sweden)
Syoji Zeze
2016-09-01
Full Text Available We demonstrate that a Hermitian matrix model can be derived from level truncated open string field theory with Chan-Paton factors. The Hermitian matrix is coupled with a scalar and U(N vectors which are responsible for the D-brane at the tachyon vacuum. Effective potential for the scalar is evaluated both for finite and large N. Increase of potential height is observed in both cases. The large $N$ matrix integral is identified with a system of N ZZ branes and a ghost FZZT brane.
11. The Effective Field Theory of nonsingular cosmology
International Nuclear Information System (INIS)
Cai, Yong; Wan, Youping; Li, Hai-Guang; Qiu, Taotao; Piao, Yun-Song
2017-01-01
In this paper, we explore the nonsingular cosmology within the framework of the Effective Field Theory (EFT) of cosmological perturbations. Due to the recently proved no-go theorem, any nonsingular cosmological models based on the cubic Galileon suffer from pathologies. We show how the EFT could help us clarify the origin of the no-go theorem, and offer us solutions to break the no-go. Particularly, we point out that the gradient instability can be removed by using some spatial derivative operators in EFT. Based on the EFT description, we obtain a realistic healthy nonsingular cosmological model, and show the perturbation spectrum can be consistent with the observations.
12. Purely cubic action for string field theory
Science.gov (United States)
Horowitz, G. T.; Lykken, J.; Rohm, R.; Strominger, A.
1986-01-01
It is shown that Witten's (1986) open-bosonic-string field-theory action and a closed-string analog can be written as a purely cubic interaction term. The conventional form of the action arises by expansion around particular solutions of the classical equations of motion. The explicit background dependence of the conventional action via the Becchi-Rouet-Stora-Tyutin operator is eliminated in the cubic formulation. A closed-form expression is found for the full nonlinear gauge-transformation law.
13. Exact integrability in quantum field theory
International Nuclear Information System (INIS)
Thacker, H.B.
1980-08-01
The treatment of exactly integrable systems in various branches of two-dimensional classical and quantum physics has recently been placed in a unified framework by the development of the quantum inverse method. This method consolidates a broad range of developments in classical nonlinear wave (soliton) physics, statistical mechanics, and quantum field theory. The essential technique for analyzing exactly integrable quantum systems was invested by Bethe in 1931. The quantum-mechanical extension of the inverse scattering method and its relationship to the methods associated with Bethe's ansatz are examined here
14. Field theory approaches to new media practices
DEFF Research Database (Denmark)
Hartley, Jannie Møller; Willig, Ida; Waltorp, Karen
2015-01-01
In this article introducing the theme of the special issue we argue that studies of new media practices might benefit from especially Pierre Bourdieu’s research on cultural production. We introduce some of the literature, which deals with the use of digital media, and which have taken steps...... on more studies within a field theory framework, as the ability of the comprehensive theoretical work and the ideas of a reflexive sociology is able to trigger the good questions, more than it claims to offer a complete and self-sufficient sociology of media and inherent here also new media....
15. Spectral theory and nonlinear analysis with applications to spatial ecology
CERN Document Server
Cano-Casanova, S; Mora-Corral , C
2005-01-01
This volume details some of the latest advances in spectral theory and nonlinear analysis through various cutting-edge theories on algebraic multiplicities, global bifurcation theory, non-linear Schrödinger equations, non-linear boundary value problems, large solutions, metasolutions, dynamical systems, and applications to spatial ecology. The main scope of the book is bringing together a series of topics that have evolved separately during the last decades around the common denominator of spectral theory and nonlinear analysis - from the most abstract developments up to the most concrete applications to population dynamics and socio-biology - in an effort to fill the existing gaps between these fields.
16. A simple proof of orientability in colored group field theory.
Science.gov (United States)
Caravelli, Francesco
2012-01-01
Group field theory is an emerging field at the boundary between Quantum Gravity, Statistical Mechanics and Quantum Field Theory and provides a path integral for the gluing of n-simplices. Colored group field theory has been introduced in order to improve the renormalizability of the theory and associates colors to the faces of the simplices. The theory of crystallizations is instead a field at the boundary between graph theory and combinatorial topology and deals with n-simplices as colored graphs. Several techniques have been introduced in order to study the topology of the pseudo-manifold associated to the colored graph. Although of the similarity between colored group field theory and the theory of crystallizations, the connection between the two fields has never been made explicit. In this short note we use results from the theory of crystallizations to prove that color in group field theories guarantees orientability of the piecewise linear pseudo-manifolds associated to each graph generated perturbatively. Colored group field theories generate orientable pseudo-manifolds. The origin of orientability is the presence of two interaction vertices in the action of colored group field theories. In order to obtain the result, we made the connection between the theory of crystallizations and colored group field theory.
17. Quantum field theory lectures of Sidney Coleman
CERN Document Server
Derbes, David; Griffiths, David; Hill, Brian; Sohn, Richard; Ting, Yuan-Sen
2018-01-01
Sidney Coleman was a physicist's physicist. He is largely unknown outside of the theoretical physics community, and known only by reputation to the younger generation. He was an unusually effective teacher, famed for his wit, his insight and his encyclopedic knowledge of the field to which he made many important contributions. There are many first-rate quantum field theory books (the ancient Bjorken and Drell, the more modern Itzykson and Zuber, the now-standard Peskin and Schroder, and the recent Zee), but the immediacy of Prof. Coleman's approach and his ability to present an argument simply without sacrificing rigor makes his book easy to read and ideal for the student. Part of the motivation in producing this book is to pass on the work of this outstanding physicist to later generations, a record of his teaching that he was too busy to leave himself.
18. Advanced concepts in particle and field theory
CERN Document Server
Hübsch, Tristan
2015-01-01
Uniting the usually distinct areas of particle physics and quantum field theory, gravity and general relativity, this expansive and comprehensive textbook of fundamental and theoretical physics describes the quest to consolidate the basic building blocks of nature, by journeying through contemporary discoveries in the field, and analysing elementary particles and their interactions. Designed for advanced undergraduates and graduate students and abounding in worked examples and detailed derivations, as well as including historical anecdotes and philosophical and methodological perspectives, this textbook provides students with a unified understanding of all matter at the fundamental level. Topics range from gauge principles, particle decay and scattering cross-sections, the Higgs mechanism and mass generation, to spacetime geometries and supersymmetry. By combining historically separate areas of study and presenting them in a logically consistent manner, students will appreciate the underlying similarities and...
19. Field theory approaches to new media practices
DEFF Research Database (Denmark)
Willig, Ida; Waltorp, Karen; Hartley, Jannie Møller
2015-01-01
This special issue of MedieKultur specifically addresses new media practices and asks how field theory approaches can help us understand how culture is (prod)used via various digital platforms. In this article introducing the theme of the special issue, we argue that studies of new media practices...... could benefit particularly from Pierre Bourdieu’s research on cultural production. We introduce some of the literature that concerns digital media use and has been significant for field theory’s development in this context. We then present the four thematic articles in this issue and the articles...... of a reflexive sociology are capable of prompting important questions without necessarily claiming to offer a complete and self-sufficient sociology of media, including new media....
20. Histories and observables in covariant field theory
Science.gov (United States)
Paugam, Frédéric
2011-09-01
Motivated by DeWitt's viewpoint of covariant field theory, we define a general notion of a non-local classical observable that applies to many physical Lagrangian systems (with bosonic and fermionic variables), by using methods that are now standard in algebraic geometry. We review the methods of local functional calculus, as they are presented by Beilinson and Drinfeld, and relate them to our construction. We partially explain the relation of these with Vinogradov's secondary calculus. The methods present here are all necessary to understand mathematically properly, and with simple notions, the full renormalization of the standard model, based on functional integral methods. Our approach is close in spirit to non-perturbative methods since we work with actual functions on spaces of fields, and not only formal power series. This article can be seen as an introduction to well-grounded classical physical mathematics, and as a good starting point to study quantum physical mathematics, which make frequent use of non-local functionals, like for example in the computation of Wilson's effective action. We finish by describing briefly a coordinate-free approach to the classical Batalin-Vilkovisky formalism for general gauge theories, in the language of homotopical geometry. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8130489587783813, "perplexity": 983.3647510646471}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794866938.68/warc/CC-MAIN-20180525024404-20180525044404-00538.warc.gz"} |
https://www.physicsforums.com/threads/pendulum-maximum-displacement.85042/ | # Pendulum- Maximum displacement?
• Thread starter yourmom98
• Start date
• #1
yourmom98
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0
what is the maximum displacement of a pendulum i don't know what it is. is it the distance from the central point to the end of the arm? and how do i solve it if i am given a periodic Sin function?
## Answers and Replies
• #2
Homework Helper
43,021
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The "maximum displacement" is usually the distance, measured along the arc, from the lowest point of the pendulum to the highest. If you are given a function of the form s= A sin(ωt), then the displacement is the amplitude, A.
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http://alexkritchevsky.com/2019/12/22/many-worlds.html | A possible way to get the Born Rule in Many Worlds
[December 22, 2019]
The Many-Worlds Interpretation (MWI) of quantum mechanics is probably roughly correct. There is no reason to think that the rules of atomic phenomena would stop applying at larger scales when an experimenter becomes entangled with their experiment (kooky interjections about consciousness notwithstanding…).
However, MWI has the problem that it does not easily explain why quantum randomness leads to the probabilities that we observe. The Born Rule says that if a system is in a state $\alpha \| 0 \> + \beta \| 1 \>$, upon ‘measurement’ (in which we entangle with one or the other outcome), we measure the eigenvalue associated with the state $\| 0 \>$ with probability
The Born Rule is normally included as an additional postulate in MWI, and this is somewhat unsatisfying. Or at least, it is apparently difficult to justify, given that I’ve read a bunch of attempts, each of which talks about how there haven’t been any other satisfactory attempts. I think it would be unobjectionable to say that there is not a consensus on how to motivate the Born Rule from MWI without any other assumptions.
Anyway here’s an argument that I find somewhat compelling? See what you think.
1. A classical coin
First let’s think about classical probability, but write it in a notation suggestive of quantum mechanics. Suppose we’re flipping a biased coin that gets heads with probability $P[H] = p$ and $P[T] = q$. Let’s call its states $\| H \>$ and $\| T \>$, so the results of a coin flip are written as $p \| H \> + q \| T \>$ with $p + q= 1$. Upon $n$ iterations of classical coin-flipping we end up in state
Where $\| H^k T^{n-k} \>$ means a state in which we have observed $k$ heads and $n-k$ tails (in any order).
Now suppose this whole experiment is being performed by a poor experimenter who’s, like, locked in a box or something. The experimenter does the experiment, writes down what they think the probability of heads is, and then transmits that to us, outside of the box. So the only value we end up seeing is the value of their measurement of $P[H] = p$, which we’ll call $\hat{P}[H]$. The best estimate that the experimenter can give, of course, is their observed frequency $\frac{k}{n}$, so we might say that the resulting system’s states are identified by the probability perceived by the experimenter:
If you let $n$ get very large, the system with $\hat{P}[H] = p$ will end up having the highest-magnitude amplitude, and so we expect to end up in a ‘universe’ where the measurement of the probability $p$ converges on the true value of $p$. This is easily seen, because for large $n$ the binomial distribution $B(n, p, q)$ converges to a normal distribution $\mathcal{N}(np, npq)$ with mean $np$. So, asymptotically, the state $\| \hat{P}[H] = \frac{np}{n} = p \>$ becomes increasingly high-amplitude relative to all of the others. This is a way of phrasing the law of large numbers.
I think this is as good an explanation as any as to what probability ‘is’. Instead of trying to figure out what it means for us to experience an infinite number of events and observe a probability, let’s just let an experimenter who’s locked in a box figure it out for us, and then just have them send us their results! Unsurprisingly, the experimenter does a good job of recovering classical probability.
2. A quantum coin
Now let’s try it for a quantum coin (okay, a qubit). The individual experiment runs are now given by $\alpha \| 0 \> + \beta \| 1 \>$ where $\alpha, \beta$ are probability amplitudes with $\| \alpha \|^2 + \| \beta \|^2 = 1$. Note that normalizing these to sum to 1 doesn’t predetermine what the experienced probabilities are, and as we will see the normalization isn’t necessary.
As before we generate a state that’s something like:
Where are things going to go differently? A potential problem is that each of the measurement results that comprise a $\| P = \frac{k}{n} \>$ macrostate could have different phases, and there is no reason to think that they will add up neatly – there could be interference between different ways of getting the same result. I’m not totally sure this is reasonable, but it leads to an interesting result, so let’s assume it is.
Consider running the experiment twice, but letting each $\| 0 \>$ state have a different have a different phase $\alpha_j = \| \alpha \| e^{i \theta_j}$. (We can ignore the $\beta$ phase without loss of generality by treating it as an overall coefficient to the entire wave function)
The state we generate will be:
This is no longer a clean binomial distribution. Writing $a = \| \alpha \|$ and $b = \| \beta \|$ for clarity, the two-iteration wave function is:
And $ab (e^{i \theta_1} + e^{i \theta_2}) \| 0^1 1^1 \>$ only has the same magnitude as $2ab \| 0^1 1^1 \>$ when $\theta_1 = \theta_2$.
3. Random Walks in Phase Space
Now let’s consider what this looks like as $n \ra \infty$.
For a state with $k$ $\alpha\| 0 \>$ terms, we end up with a sum of exponentials with $k$ phases in them:
Here $S_{k,n}$ is the set of $k$-element subsets of $n$ elements. For instance if $k=2, n=3$:
Our wave function for $n$ iterations of the experiment is given by
The classical version of this is a binomial distribution because $E_{k, n}$ is replaced with ${n \choose k}$. The quantum version observes some cancellation. We want to know: as $n \ra \infty$, what value of $k$ dominates?
We don’t know anything the phases themselves, so we’ll treat them as classical independent random variables. This means that $\bb{E}[e^{i \theta}] = 0$ and therefore $\bb{E}[E_{k, n}] = 0$ for all $k$. But the expected magnitude is not 0. The sum of all of these random vectors forms a random walk in the complex plane, and the expected amplitude of a random walk is given by $\bb{E}[ \| E_{1, n} \|^2 ] = n$.
Briefly: this comes from the fact that
This means that the magnitude of the $k=1$ term for our quantum coin is proportional to $\sqrt{n}$, rather than the classical value of $n$.
For $k > 1$, the same argument applies (it’s still basically a random walk), except that there are ${ n \choose k }$ terms in the sum, so in every case we get an expected amplitude $\bb{E} [ \| E_{k, n} \|^2 ] = { n \choose k }$.
4. The Born Rule
These don’t tell us the constant of proportionality, since $\bb{E}[ \| E_{k, n} \|^2] \neq \bb{E}[ \| E_{k, n} \|]^2$, but fortunately we only need to compute the value of $k$ at the peak, and we can find that using $\| \psi \|^2$, which is easy to work with:
This is a binomial distribution $B(n, a^2, b^2) = B(n, \|\alpha\|^2, \| \beta \|^2)$, which asymptotically looks like a normal distribution $\mathcal{N}(n \| \alpha \|^2, n \| \alpha \|^2 \| \beta \|^2)$ with maximum $k = n \| \alpha \|^2$, which means that the highest-amplitude state measures is:
Thus we conclude that the observed probability of measuring $\| 0 \>$ when interacting with a system in state $\alpha \| 0 \> + \beta \| 1 \>$ is centered around $\| \alpha \|^2$, as reported by an experimenter in a box who runs the measurement many times, which is what we probably are anyway. And that’s the Born Rule.
Ultimately this seems to be because different ways of seeing the same result interfere with each other, suppressing the amplitudes of seeing less uniform results by a factor of the square root of their multiplicity.
(Note that this argument should still work if $\|\alpha \|^2 + \| \beta \|^2 \neq 1$; the resulting asymptotic normal distribution will end up having mean $\frac{n \| \alpha \|^2}{\| \alpha \|^2 + \| \beta \|^2}$.)
So that’s interesting.
I find the argument that “random walks in phase space might lead to a peak amplitude that matches the Born Rule” to be suspiciously clean, and therefore compelling, but I don’t any confidence that I’ve correctly identified what might actually lead to the random interference in this experiment. Is it the experimental apparatus interfering with itself? Is it hidden degrees of freedom in the experiment itself? Or maybe it’s all of reality, from the point of view of an observer trying to make sense of all historical evidence for the Born Rule. And it’s unclear to me how carefully isolated an experiment would have to be for different orderings of its results to interfere with each other. Presumably the answer is “a lot”, but what if it isn’t?
Suffice to say I would love to know a) what’s wrong with this argument (maybe it’s circular, but I haven’t figured out how), or b) if it exists in the literature somewhere, cause I haven’t found anything, although admittedly I didn’t look very hard.
I can think of some strange implications of this argument but I don’t want to get ahead of myself.
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http://sk.sagepub.com/books/doing-cultural-theory | # Doing Cultural Theory
Books
### David Walton
• Chapters
• Front Matter
• Back Matter
• Subject Index
• ## Copyright
View Copyright Page
## Acknowledgements
While this book carries my name, books are always, in varying degrees, collaborative efforts and I would like to offer my thanks to a number of colleagues and friends.
Thanks go to Patricia Bickers from Art Monthly for contacting Hans Haacke on my behalf and to Hans Haacke himself for giving me permission to use his photos of his ‘MetroMobilitan’ installation (Figures 14.1 and 14.2). Also, thanks go to Bas Beentjes for permission to use his EO photo (Figure 15.1) and to John Harris who was very generous in terms of sending me valuable feedback on the photo he took of Lesley Boulton at the ‘Battle of Orgreave’.
Figure 14.1 Hans Haacke's MetroMobilitan shown at the John Weber Gallery (1985) © Hans Haacke/Artists Rights Society
Figure 14.2 Photomural of a funeral procession depicting victims shot by South African Police (detail)
Figure 15.1 Culture jamming in action
I would also like to thank the anonymous readers at SAGE for their valuable comments. I did not always agree with the points they made but they undoubtedly helped me to refine my ideas and reconsider a number of features. Thanks must go to Elizabeth Ezra who generously offered to read the opening chapters at a very early stage. Her comments, advice and encouragement have been very important in terms of giving me the energy to complete the book. Nuria Urzaiz not only gave me encouragement but kindly offered to give very detailed feedback on the first drafts of a number of chapters from the point of view of the kind of reader I was aiming at. Many thanks, too, to Verónica Morales who offered me her support by reading some of the later drafts of the final chapters. My biggest debt is to Mila Steele, my editor at SAGE, for her very constructive criticism, encouragement, support and advice. Mila helped me to perfect the overall structure and content, and has been there at every stage of the book's development. I would also like to thank Sarah-Jayne Boyd (Mila's assistant) for her advice and help and efficiency, especially in the final stages of completing the book.
Finally, I would like to recognize the support and encouragement of my family and my many friends and colleagues who have helped in a million small (and not so small) ways – sometimes without knowing it. Special thanks go to Cathy Staveley, Liz Murphy, María Reyes, Juan Antonio Suárez, María González, Asensio López, Raquel González and Andy Sotiriou and, last but not least, Dan Walter. Without them the writing of this book would have been a much more arduous task. As it is, the process has been thoroughly enjoyable. Finally, thanks must go to my students who have often helped me to refine my ideas by showing what works and, often, what does not.
## Introduction: How to Use This Book
Who Is This Book Written for?
You may have come across terms like semiological systems, signification, the problematic, symptomatic reading, deconstruction, logocentrism, the big Other, anti-essentialism and the postmodern subject, but not have been sure about what they mean or how they might be used in practice. If this is the case this book should be able to help you. This book, then, is aimed at readers who already have some knowledge of cultural studies but who want to get a firmer grasp of the way cultural theory relates to practice. However, those who have little or no knowledge of cultural analysis, but who feel they have the academic experience and confidence to tackle cultural theory and practice at a higher level, will also find it useful. With this in mind, the first chapter is designed to take account of different possible readerships. On the one hand, it helps to show how the theories and ideas explained and illustrated in this book fit into a larger historical picture. On the other hand, it can serve as valuable revision for those readers who are already familiar with the area or a basic introduction for those with no previous knowledge.
General Aims and Approach of This Book
This book can be seen as akin to the viaduct – a conduit which carries something or someone from one place to another (a kind of ‘theory-duct’). One aim of this book is to describe the different theories, however complex, in an accessible style. However, I have not avoided the use of complex terms and have tried to describe them as clearly as possible in relation to how they relate to various forms of cultural analysis. This raises the problem of up to what point a technical vocabulary is necessary. As Lawrence Grossberg has said, while scientists who describe the physical world are expected ‘to use languages not available to most people’, those who explore social reality are often expected to write so that anyone can understand. Yet ‘human reality’ is not necessarily any less complex than the world of subatomic particles. Thus, ‘sometimes we need complex and nonobvious explanations of what's going on’ (1992: 30–31). We can try to explain quantum mechanics or computer science in ordinary terms but to get to the finer subtleties it is more often than not necessary to learn specific concepts and ways of thinking.
The specialized vocabulary introduced in this book, then, is not an elitist attempt to put ourselves above others but part of an effort to immerse ourselves in the language of a particular discipline. The understanding of theory is constantly explained with an eye to how it may function in practice. In this way its application is a little like a voyage of discovery, where the world is experienced in new and perhaps surprising ways through the assimilation, adaptation and refinement of concepts previously unknown (or vague) to us. The general approach adopted in this book assumes that interpretation and analysis are always a product of, and dependent on, very particular shaping strategies, something which makes the critic a maker or ‘fashioner’. One of the other main aims of the book, then, is to help the reader to become articulate in these theoretical ‘languages’.
I would describe most of the theories introduced in this book as ‘radical’ in the sense that most of them have been used to challenge fixed ideas and have often been used in efforts to change the world for the better. Marx's point that philosophers ‘have only interpreted the world, in various ways’ but the point is ‘to change it’ (Marx, 1845/1976: 65) can be seen as one of the most important ethical tenets underpinning much cultural studies. I have tried to reflect something of this ethical concern (and utopian project) in the various theories I have introduced, regardless of how they have been labelled. Finally, this book has been written in the belief that learning to understand theory (and putting it into practice) may be challenging but it does not have to be a chore and that cultural studies can be a bold, stimulating and even exciting enterprise.
Strategies for Using This Book
As indicated above, the opening chapter will familiarize the reader with something of the history of the cultural studies tradition and reflect on key movements, definitions and strategies. Readers who are already familiar with definitions of culture and the (mainly British) cultural studies tradition (including the work of Matthew Arnold, the Leavises, the Frankfurt School, the ‘culturalist’ writers like Hoggart, Thompson and Williams, and the early work of the Birmingham Centre for Contemporary Cultural Studies) can skim (or even skip) this chapter. Readers who want to begin experimenting with concepts without any preamble can start with the chapters dedicated to structuralism. Ideally, the chapters should be read chronologically because I introduce important concepts in the early chapters which I develop and refine later on in the book. Readers who read systematically will realize that a number of different thematic threads are developed which provide narrative strands which weave in and out of particular chapters to provide multiple points of comparison, cohesion and a sense of continuity and development. Of course, this does not preclude more creative approaches where readers can dip into the chapters that most interest them.
What Can I Realistically Expect from a Careful Reading of This Book?
A careful reading of this book should help to provide a detailed knowledge of some of the key theoretical trends that have shaped much thinking and interpretation within cultural studies. It should also help readers to develop their interpretive skills and knowledge because many of the concepts are not only described and explained but illustrated with practical examples. Where possible, advice has been given on further practice to aid interpretive independence – the main idea of the book being to help readers get to that ‘other place’ of the specialist.
What Are the Chief Pedagogical Features of This Book?
• Each chapter begins with an explanation of its content, the main learning goals and a list of the key concepts that will be explained and illustrated.
• All key concepts are introduced in bold type to help readers navigate through the chapters.
• Regular help files, practice sections and summaries are introduced to consolidate learning and aid practice.
• Ideas are often clarified with reference to examples drawn from the world of popular culture (although not exclusively) and there are some playful, creative sections designed to aid further understanding of complex ideas (for example, in one chapter the idea of postmodernism is expressed in both the form and the content of the section).
• The chapters progress in terms of difficulty so readers can build on their knowledge of previous examples.
• Theories and individual concepts are always related to the ways they may help to elucidate various forms of culture.
• Each chapter is concluded with a brief summary of the main points and followed by sections on further reading.
• There is a full glossary of key terms at the end of the book to help in the assimilation of the material.
The Organization of Chapters and Basic Content
Following a brief introduction to (British) cultural studies the content can be described as going from structuralism and poststructuralism to postmodernism and beyond. This strategy encompasses the major theorists and includes approaches which have been, and continue to be, of interest to cultural studies scholars concerned with questions of things like gender, class, sexuality, race/ethnicity, ideology, identity politics, post-colonialism, discourse, popular culture, history, media, consumerism, commodification, globalization, new social movements and neoliberalism.
The ‘beyond’ mentioned above describes the content of the last two chapters which consider ways that cultural analysis can complement or supersede approaches focused on the dominant themes developed within structuralism, poststructuralism and postmodernism. In fact, one of the ways this book attempts to depart from most books dedicated to theory and practice in cultural studies is in the way it helps readers to map out their place within the multinational, corporate world of late capitalism. As stated above,Chapter 1 will go into more detail about the structure of the book and how it relates to general trends within (mainly British) cultural studies.
• ## Glossary: A Mini-Dictionary of Key Cultural Concepts and Terms Used in This Book
Words printed in bold refer to other entries in the Glossary to allow cross-referencing.
• Anchorage: A term used by Roland Barthes to describe how the signifieds of a linguistic message help to limit the possible meanings generated by images.
• Anti-essentialism: A term commonly found in poststructuralism and postmodernism that indicates an approach to knowledge which questions the idea that the discourses of the sciences can discover and reflect objective, essential truths which are not products of those discourses.
• Arbitrary closure: A notion recommended by Stuart Hall where the critic uses deconstruction but subordinates its more radical potential to strategic political ends (and avoids becoming a slave to it).
• Arborescence: A concept used by Deleuze and Guattari which describes vertical tree-like structures that fix order and can be used to describe more traditional political structures that rely on a vertical model where power centres, connected to hierarchies, dominate the functioning of the party. See rhizome.
• Big Other: A highly ambivalent concept found in Jacques Lacan's work which very generally refers to the place of the Symbolic (with all its rules and structures) or, more specifically, can be experienced as benign or malicious hidden agencies controlling things behind the scenes (like the Christian God or a horrifying paranoiac agency).
• Binary oppositions: In structuralism cultures are seen to make sense of the world through distinguishing between foundational opposites like life/death, good/evil, freedom/repression, etc., where one term is often taken to have positive connotations and the other negative.
• Breakdown of the high verses low culture distinction: Drawing on the work of Jean Baudrillard, critics like Fredric Jameson and George Ritzer argue that this is one of the fundamental premises of postmodernism.
• Code: In structuralism, signifiers and signifieds can only produce meanings if they are organized by a code (like a system of grammar).
• Coded iconic message: According to Roland Barthes this is where meanings are created for images through deliberate patterning and manipulation, as opposed to the non-coded iconic message.
• Cognitive mapping: Given the deconstructive tendencies of postmodern thought and cultures, Fredric Jameson argues that there is a need for a more globalized form of politics to resist global capitalism and a new political art that is capable of formulating new forms and strategies that might offer special insights into our place within multinational capitalism. See high-tech paranoia and the homeopathic strategy.
• Commodity fetishism: A concept drawn from the writings of Marx which Fredric Jameson argues is fundamental to postmodernism. For Marx, once an object becomes a commodity it becomes ‘transcendent’ and is treated in such a way that the labour that went into producing it becomes invisible. In this way it is fetishized. See late captitalism.
• Communicative rationality: In his essay on modernity as an incomplete project Jürgen Habermas rejected the basic premises of poststructuralism preferring the idea of ‘communicative rationality’ which posits that individuals, with a will to communicate and understand, can reach consensus through reasoned debate. Jean-François Lyotard rejected these ideas because he believed they were only another way of reformulating the grand narratives and the old totalities and certainties that he rejected.
• Connotation: This describes meanings produced through suggestion or association as opposed to the explicit literal meaning of denotation.
• Consumer society: Jean Baudrillard argued that advanced capitalist societies were at the point where consumption was invading every aspect of life and that the act of consumption was not just about consuming commodities but messages (signs, brand icons, etc.) in such a way that late capitalism was becoming increasingly a question of not who we are but what brands we consume and what lifestyles we adopt. In this way postmodern identity is not something outside consumption but negotiated within it.
• Crisis of (or incredulity towards) metanarratives: This is a phrase used by Jean-François Lyotard to describe a condition where society no longer believes in the ethical, philosophical, social, political metanarratives that were once thought of as providing the justification for education, learning and the production of knowledge. See the postmodern condition, grand narrative and legitimation.
• Critique of reason: A common notion found in poststructuralism and postmodernism (but not exclusive to them) and given emphasis in the work of Michel Foucault who did not reject reason per se but posed thoroughgoing questions about its nature, limits, historical effects and dangers. See knowledge, discourse, truth and power.
• Culturalism: A simplifying label that has been used to describe the work of Richard Hoggart, E. P. Thompson and Raymond Williams which emphasizes the lives, cultures, experiences and resistance of ordinary people and their capacity to be active agents of change, rather than dupes of history.
• Cultural logic of late capitalism: Fredric Jameson sees this cultural ‘logic’ at work when art and culture are fully complicit with the values of late capitalism (unlike the products of high modernism). This is part of postmodernism as he understands it.
• Cultural schizophrenia: Adapting ideas from the work of Jacques Lacan, Fredric Jameson argues that in postmodern cultural forms there is a loss of progressive temporality where the experience of ‘existential time’ and ‘deep memory’ (of high modernism) gives way to dislocation, fragmentation and the loss of ‘coherent experience’. See depthlessness, parody and pastiche and waning of affect.
• Cultural studies: This is a loose miscellany of self-reflexive, inter-disciplinary approaches (spread across many nations) which Grossberg et al. (1992) claim has no precise methodology – practitioners typically drawing on whatever discipline is necessary in order to produce the knowledge required for a given project. Cultural studies often politicizes the understanding of culture (understood in its widest sense) by exploring how cultural products and practices relate to concepts like class, race/ethnicity, gender, sexuality, ideology, representation and relations of power. All these features complicate the identity of cultural studies, even while they help to establish dominant ways of thinking about and understanding it.
• Culture: A notoriously difficult word to define but which, according to Raymond Williams (1983), describes processes of human development with relation to the cultivation of the mind, behaviour or society. In the most general sense it describes language, art, knowledge and belief but also things like law, ethics and customs. Williams emphasized that (in its modern sense) it should be located in the social and political changes brought about by industrial capitalism and linked to ‘a whole way of life’ and include not only ‘high’ culture but the understanding of institutions, the organization of production, social practices, sport, entertainment and everyday behaviour. Many contemporary critics stress the importance of understanding of signifying practices, and things like consumption habits and relations of power as a means to understand culture. However, while these lists of possibilities are very useful at a more general and abstract level, any attempt to limit the definition at the level of particular objects of analysis is futile because as the world changes new possibilities (or domains of interest) for the understanding of cultures are constantly appearing.
• Culture and civilization tradition: This sums up nineteenth- and twentieth-century writers like Matthew Arnold and F. R. and Queenie Leavis who tended to privilege high cultural forms (and especially literature as the best that had been thought and written) as the means by which civilization could be defined, preserved, propagated and assessed. See minority culture.
• Culture industry: This is a term used by Theodor Adorno and other Frankfurt critics to refer to forms of mass culture produced for profit and associated with the rise of mass entertainment and mass communications within industrial capitalism. Adorno saw mass culture as depoliticizing and pacifying the exploited masses, while acclimatizing them to the degrading conditions of their lives, while impoverishing them materially, emotionally and intellectually.
• Culture jamming: This is a strategy of semiotic warfare where activists or ‘subvertizers’ sabotage, alter or deface advertisements (or intercept radio and TV programmes) in the interests of questioning the messages they transmit, often as a way of challenging the abuses of the major corporations. See new social movements.
• Death of the author: An idea taken from Roland Barthes’ writings which challenges the idea of limiting a text's meaning to the author's intentions. For Barthes, once the text is written the author is effectively dead in the sense that s/he cannot serve as a basis for grounding or controlling interpretation.
• Death of the subject: A notion used by Fredric Jameson to convey the idea that in contemporary postmodern theory there is a loss of the centred subject, a demise in the capacity to feel (once expressed in angst) and a breakdown of the unique style associated with the great writers and artists of high modernism. See the waning of affect and depthlessness.
• Decoding: Within semiology decoding describes how messages are deciphered by receivers with relation to pre-established codes within a given medium. See encoding.
• Deconstruction: A complex system of thought initiated by Jacques Derrida which questions and undermines the basis of Western forms of thought through a series of (anti)concepts, wordplay, stylistic inventiveness and tortuous argumentation. See poststructuralism, intertextuality, différance, difference, deferral, trace, structure, transcendental signified, logocentrism and phonocentrism.
• Deferral: Together with difference this term forms part of Derrida's deconstructive (anti)concept of différance which undermines the possibilities of fixed meanings. See poststructuralism.
• Delegitimation: Within the context of performativity, Jean-François Lyotard argued that the crisis of grand narratives results ‘from the ends of action to its means’ (1987: 37) in such a way that science no longer relies on these grand narratives to legitimate it – hence the term delegitimation. In the postmodern condition each science establishes its own criteria for acceptance, through the consensus of experts, rather than referring to criteria outside themselves. See the crisis of (or incredulity towards) metanarratives.
• Denotation: This describes the literal meaning of words as opposed to the associative meaning associated with connotation.
• Depthlessness: A term used by Fredric Jameson to define the cultural logic of late capitalism. See postmodernism.
• Desire: A multifaceted term used by Jacques Lacan which is seen to be the product of the subject's entry into the Symbolic, rather than something which pre-exists or stands outside inter-subjective relations. See need.
• Destratification: A concept used by Deleuze and Guattari that can be used to describe how things or spaces can be de-colonized and transformed by new powers or forces. See territorialization and reterritorialization.
• Difference: Along with deferral this term forms part of Derrida's deconstructive (anti-)concept of différance which challenges the possibilities of stable meanings. See poststructuralism.
• Différance: This is a key (anti)concept within Derridean deconstruction which plays on the concepts of difference and deferral in such a way that meaning in discourse is constantly divided and indefinitely postponed. In this way no text has an intrinsic meaning.
• Disaster capitalism: An idea put forward by Naomi Klein where governments (aided and abetted by free-market economists) push through emergency measures when the public is vulnerable and severely disoriented. It consists in waiting for (or provoking) a major crisis then deregulating the markets, breaking down trade barriers and radically reducing public spending as part of a massive effort to sell-off national assets which tend to favour global capitalists and the multinationals at the expense of national interests. See the shock doctrine.
• Discourse: Michel Foucault emphasized that knowledge and truth cannot be understood outside the discourses (branches of knowledge) which produce them and which are, in turn, regulated by key institutions. It is in these discourses that society generates its regimes or general politics of truth which exercise power over subjects. See also Panopticon and gender as performative.
• Double reading: In Althusserian Marxism a double reading takes account of, on the one hand, the gaps and silences that reveal the ideological limits of the discourse (the problematic) and, on the other, the way texts inadvertently answer questions they never pose. See symptomatic reading.
• Empire: A notion found in the work of Michael Hardt and Antonio Negri which understands the world economy as dominated by global capitalism and the multinationals (linked to the interests of the US and its allies) and supported by organizations like the G8, NATO the World Trade Organization and the International Monetary Fund. This is the (Roman-like) ‘Empire’ in late capitalism that guarantees the economic inequalities that maintain huge accumulations of capital and the perpetuation of poverty.
• Encoding: Within semiology encoding describes how messages are structured into a particular form with relation to a given medium. See decoding.
• First- and second-order semiological systems: These describe Roland Barthes’ approach to understanding how images are read in semiological terms. The first-order system (the simple recognition of the image) functions as the signifier of the second-order system, which describes the culturally loaded meaning(s) of the image (which corresponds to connotation). See modern mythololgies.
• Frankfurt School: This is the name given to the Institute for Social Research at the University of Frankfurt to which important Marxist critics like Thoedor Adorno, Max Horkheimer and Walter Benjamin were affiliated. They helped to fuse Marxist thinking with other important approaches like, in the case of Adorno and Horkheimer, psychoanalysis to create trenchant criticisms of contemporary culture and society. See the culture industry.
• Genealogy: A Nietzschean term used by Michel Foucault to construct an approach to the writing of history that challenges more traditional accounts which assume it pre-exists historical writing and can be reduced to more or less ‘objective’ narratives, origins and ends. Foucault's approach is anti-essentialist and questions the neutral disinterested view that seeks continuities, forces and universal laws.
• Gender: A widespread term used in much cultural criticism, especially that focused on feminism and gender studies. More specifically, it is theorized in the work of Michel Foucault and Judith Butler, the latter defining it against concepts like sex and sexuality. For Butler gender describes the characteristics that a given culture understands as masculine or feminine, dependent on social interactions and the assimilation of social norms. However, gender is seen as a cultural construction, not in terms of something given by nature.
• Gender as performative: When discussing gender, Judith Butler uses this term to argue that carefully selected features associated with femininity or masculinity within normative social discourses are used to signify some interior essence. However, this inner essence, which seems to be described by the discourses, is actually a product of them. The implication of this for Butler is that gender is conceived of as the repeated performance of the chosen traits which stand for being woman or man and are made to perform gender. See gender, sex, sexuality and gender subversion.
• Gender subversion: Judith Butler argues that parodic and hyperbolic practices like drag can serve to ‘denaturalize’ the body in such a way that what are commonly taken as the ‘natural’ characteristics of gender are exposed as the performative cultural constructs they are. See gender as performative.
• Grand narrative: Jean-François Lyotard argued that, traditionally, science legitimated itself with relation to philosophical or political metanarratives (or grand narratives). The production of knowledge was justified with reference to criteria outside itself: that it is a good in itself or in the best interests of the people or the nation. However, the postmodern condition marks a point of the crisis of (or incredulity towards) metanarratives. See performativity and delegitimation.
• Hegemony: This term derives from the work of the Italian Marxist Antonio Gramsci who used it to describe how, in modern democracies, political power and leadership, while ultimately backed up by force, depend on alliances and the winning of consent through compromise and negotiation. This challenges more simplistic notions of ideology that assume that ideas and values can be linked in deterministic ways to particular classes. These ideas have been used to describe all kinds of power relations, including those which govern the (de)valuation of different forms of culture.
• High culture: This concept is used by critics, like those of the culture and civilization tradition, to refer to the most worthy and valuable cultural forms in opposition to what are seen as trivial and debasing products of popular (or mass, industrial) culture.
• High modernism: A term Fredric Jameson uses to distinguish what he regards as the generally inferior cultural products of postmodernism from those of the most distinguished modernists. For him high modernism refers to movements like abstract expressionism in painting, existentialism in philosophy, and writers like Eliot, Stevens, Joyce, Woolf, Kafka and Faulkner (in poetry and the novel), or the auteur directors like Bergman, Kurosawa, Hitchcock and Fellini (in cinema). See the cultural logic of late capitalism.
• High-tech paranoia: Within the writings of Fredric Jameson this refers to contemporary narratives (like cyberpunk) in which the impossibly complex circuits of global computer networks are linked to the convoluted intrigues of rival information agencies. For Jameson, these narratives can help to understand ‘the impossible totality of the contemporary world system’. See cognitive mapping and homeopathic strategy.
• Historiography: This idea is often used to describe the writing of more traditional historians who assume that historical narratives can more or less reflect the past in objective ways with relation to periods and movements. See poststructuralism, postmodernism and genealogy.
• History of sexuality: This refers to one of Michel Foucault's major projects where he showed how sexual practices had been gradually reduced to a set of social and scientific discourses that could produce truths and forms of knowledge and power which would enable the social control and regulation of the sexualized body (while opening up discursive spaces for non-normative sexualities).
• Homeopathic strategy: Within his conception of cognitive mapping Fredric Jameson recommends this tactic where an artist uses the very thing which is considered corrupt (like advertising images) to criticize the institutions in which they circulate. Thus, the corrupt form is used as a means to critique it. See cognitive mapping.
• Hyperreality: Jean Baudrillard used this concept, which is often seen as a key notion defining postmodernism, to describe the condition in late capitalism where one-to-one relations between signs and the referential world are lost and individuals find themselves adrift in the infinite proliferation of media images, information and advertising (simulacra). This is hyperreality because it indicates simulated reality in excess where reality is confused with its simulation – it is simulation.
• Ideological state apparatuses: Louis Althusser associated these apparatuses with politics, education, religion, the law, family, trade unionism, communications and culture (in general) which were conceived of as having a secondary repressive function – that is, secondary to the repressive state apparatuses.
• Ideology: A notoriously difficult concept fundamental to Marxist analyses that, at the simplest level, describes the dominant ideas, beliefs and values that govern individuals, groups, processes or political parties or economic policies. It is sometimes contrasted with Gramscian hegemony theory. Roland Barthes discussed the notion with relation to mass culture seeing collective representations as sign systems that transform (petit-bourgeois) culture into nature. See also modern mythologies, and Althusser's notions of ideological and state apparatuses, material and imaginary aspect of ideology, overdetermination and interpellation.
• Imaginary: A concept used within Lacanian psychoanalysis which deals with how subjects construct and misrecognize themselves (and identify with others) through images (or fantasies) that give the illusion of a unified identity. From this point of view the self is structured on a series of illusions; however, the Imaginary also establishes the basic coordinates of the self that enable the subject's functioning in the world. See master signifiers.
• Imaginary aspect of ideology: This refers to Louis Althusser's idea that in ideology subjects do not express their ‘real’ relation to their conditions of existence but the way they live that relation. See material aspect of ideology.
• Interpellation: In Althusserian Marxism this describes how ideology functions to shape individuals into ‘subjects’ subject to capitalist society and culture.
• Intertextuality: A complex term coined by Julia Kristeva and used by Roland Barthes and often associated with poststructuralism. At its simplest level it refers to how texts incorporate others into themselves (through quotations, references, etc.). At a more complex level language itself can be seen as radically intertextual and thus any utterance or text is already implicated in complex networks of linguistic usages, sayings, tags, etc. before anything is formally cited or referred to.
• Knowledge: In the writings of Michel Foucault knowledge is linked to regimes of truth. From this point of view it is not divorced from the exercise of multiple networks of power. See truth and discourse.
• Langue: In structuralism this refers to a pre-established (but negotiable) system of common rules and conventions (which make up a code) which, within a given community, can be used to produce meaningful utterances (or paroles).
• Late capitalism: Fredric Jameson argued that this characterized the post-industrial ‘multinational’ stage of capitalism associating it with postmodernism. This is where the whole world is dominated by the narrow interests of the ruling, capitalist classes whose activities transcend the nation state. See the cultural logic of late capitalism.
• Legitimation: Jean-François Lyotard, when discussing the postmodern condition, claimed that in modern societies science does not only seek the truth of things but is also under an obligation to legitimate ‘the rules of its own game’. These justifications, which are political or philosophical, are what Lyotard thinks of as discourses of legitimation which contain metanarratives. See crisis of (or incredulity towards) metanarratives.
• Little narratives: For Jean-François Lyotard the little narrative avoids the universalizing principles and overarching claims of grand narratives which he felt, within the postmodern condition, were bankrupt and could no longer be justified. See crisis of (or incredulity toward) metanarratives and delegitimation.
• Logocentrism: A term Jacques Derrida used to define a characteristic of Western thinking where meaning is grounded in the metaphysics of presence, and where presence (and speech) are privileged over absence (and writing). See deconstruction and phonocentrism.
• Mass culture: This term is normally used by those who regard mass-produced culture for commercial ends as trivial and dangerous to those without the appropriate intellectual training to see through their dubious allurements. See the culture and civilization tradition and the culture industry.
• Master signifiers: A term used by Jacques Lacan to indicate those signifiers which function to provide the basic coordinates of identity (gender, nationality, religious belief, class, sexual preferences, etc.) but only give the illusion of meaning because they are all dispersed through other signifiers. However, if a person does not rely on the master signifiers and internalize the basic rules laid down in the Symbolic (the rules symbolized by the Name-of-the-Father) the result would be psychosis. See the subject and the Imaginary.
• Material aspect of ideology: This refers to Louis Althusser's idea that ideology is not just a question of thoughts, values and beliefs but is manifested in the official practices of everyday behaviour governed by rituals. See imaginary aspect of ideology.
• Message without a code: Roland Barthes referred to non-coded iconic messages in this way meaning that the simple, literal (iconic) recognition of objects is a form of communication although there is no underlying code – unlike in the case of coded-iconic messages.
• Military–industrial complex: This describes the intricate relations between arms manufacture and central government. The concern is that if national economies are dependent on the production and sale of armaments (and the sector employs significant numbers of people) those representing the armament industries may have an unfair and inordinate influence on the decisions taken by politicians, thus compromising the democratic process.
• Minority culture: this is associated with the culture and civilization tradition where writers argued that narrow cannons of ‘high culture’ (the great poets, novelists, dramatists, etc.) needed to be preserved by the few enlightened minds capable of assessing and deciding what counts as culture. See mass culture.
• Mirror Stage: Within Lacanian psychoanalysis this describes how a child, at the pre-linguistic Imaginary stage, undergoes identification with ‘itself’ on the way to becoming a subject within the Symbolic.
• Modern mythologies: Roland Barthes used this term when analysing the products of mass culture and treating ‘collective representations’ as sign systems in such a way that he was able to uncover the mystifications that transform ‘petit-bourgeois culture into universal nature’ (Barthes, 1957/1972:11). See ideology and semioclasm.
• Motivation: From Roland Barthes’ point of view the modern myth (or ideological message) is drawn from the common stock of pre-existing meanings. These meanings are what ‘motivate’ the meanings of modern mythologies. See also first- and second-order semiological systems.
• Name-of-the-Father: A concept found in Lacanian psychoanalysis that does not refer to an actual father (or the particular image of him that individuals may have) but is purely symbolic of the laws that govern cultures that provide the necessary structures for the Symbolic order so necessary to the structure and stability of the psyche.
• Need: In Lacanian psychoanalysis the origin of organic need is found in the Real but the expression of need in the Symbolic (as desire) leads to constant deferral, frustration and lack. This is because the satisfaction of need is constantly deferred by its transmission through signifiers.
• Negotiated readings: When discussing television discourse Stuart Hall used this to describe how audiences, while influenced by dominant codes of interpretation established by the producers of the message, may resist (or fail to recognize) aspects of the hegemonic preferred reading. See encoding, decoding, and oppositional readings.
• New social movements: This term describes activists who form alliances (or networks) to keep a constant eye on things like human rights abuses and the environment. Rather than rely on well-established political parties with their formal organization and overarching ideologies (and members) they prefer to be independent of the political structures they see as antiquated, bankrupt and corrupt. These are often thought of as a postmodern approach to politics. See culture jamming and rhizome.
• Non-coded iconic message: In Roland Barthes’ work this describes how objects are recognized as images before any secondary meanings are attributed to them, as opposed to coded iconic messages.
• Objets petit a: A complex and ambiguous term coined by Jacques Lacan which is short for objet petit autre (‘the little object of the other’) that Lacan refused to define in any strict way. One way of understanding the idea is to associate it with the importance given to things that are experienced as parts of, or complements to, the self and not understood or experienced as separate from it.
• Oligopoly: This term is related to monopoly and describes a tendency in late capitalism where large corporations and conglomerates progressively gain greater control over markets by buying up the competition and monopolizing the manufacture and supply of particular products.
• Oppositional readings: With reference to television discourse Stuart Hall used this concept to describe how audiences, while recognizing the meanings produced by the dominant codes of interpretation established by the producers of the message, resist aspects of the hegemonic preferred reading. In this way the audiences deliberately interrupt the relations between encoding and decoding. See encoding, decoding, and negotiated readings.
• Organic intellectuals: This idea derives from Antonio Gramsci's writings and describes a class of intellectuals who could theorize culture and communicate counter-hegemonic ideals to a new revolutionary class.
• Overdetermination: When discussing repressive and ideological state apparatuses, Louis Althusser adapted this Freudian concept to explain the complexity and relative autonomy of ideology (rather than rely on a more simplistic notion where the material base determines the ideological superstructure).
• Panopticon: An idea created by Jeremy Bentham that Michel Foucault adapted to argue that the modern state, through pervasive surveillance mechanisms, produces subjects characterized by self-regulation.
• Parody and pastiche: Fredric Jameson makes a distinction between these two notions to emphasize the differences between modernism and postmodernism. While they both imitate a recognizable style, Jameson values parody (associated with modernism) over pastiche (associated with the postmodern) because the former contains a ‘satiric impulse’ which is lost in the latter, which merely evokes previous styles. See pastness and depthlessness.
• Parole: in structuralismparoles are meaningful utterances which result from langue (systems of pre-given rules and conventions which make them possible).
• Pastness: Fredric Jameson argues that postmodern cultures manifest a crisis in history in the way that they no longer evoke historical referents (something real that actually happened with actual historical content) but only construct the past through nostalgic references, which means that the past is merely a question of ‘stylistic connotation’. See depthlessness and the cultural logic of late capitalism.
• Patriarchy: A disputed term referring to power relations where women are subordinated (partially or wholly) by men in the interests of preserving male domination.
• Penis envy: A controversial and contested term (especially in feminist discourses) which is found in Freud's theory of psychosexual development where girls, on discovering that they lacks a penis, envy it and all that it represents in terms of power and authority. This discovery brings with it self-loathing and rejection of the mother. See Phallus and the Name-of-the-Father.
• Performativity: For Jean-François Lyotard this term describes the tendency in the postmodern condition for society to become increasingly dominated by a skills approach to knowledge, which is ruled by performance and efficiency without reference to its emancipatory or speculative values. This is where input is minimized and output maximized and where the ultimate goal of the backers of research is power. See crisis of (or incredulity towards) metanarratives and deligitimation.
• Phallus: This is a controversial and ambiguous term found in Lacanian psychoanalysis which does not refer to the genital organ, as such, but symbolizes things like authority, power, security and wholeness of being. While the concept is rooted in patriarchal notions of male power, either sex can aspire to what the Phallus represents. See penis envy and the Name-of-the-Father.
• Phonocentrism: Jacques Derrida used this term to indicate that in Western thinking there is a tendency to privilege speech (as closer to the origin of production) over writing (understood as being secondary). Thus, speech is given priority over the written word because it is seen as more authentic. See deconstruction and logocentrism.
• Photogenia: A term coined by Roland Barthes to describe the way aesthetic factors like posing, placement, lighting and trick effects influence the interpretation of images. See photographic message, anchorage and coded iconic messages.
• Photographic message: A term used by Roland Barthes to describe the relations between texts and images (especially in the press) where headlines, captions and the article influence the way an image is understood. See anchorage.
• Popular culture: a term often linked to mass culture and the culture industry (see the Frankfurt School) which usually describes the entertainment, tastes and choices of ordinary people and is contrasted with high culture.
• Post-colonialism: This refers to a complex set of discourses which focus on the historical, economic, political and other cultural legacies of imperialist expansion.
• Post-industrial society: A term popularized by the sociologist Daniel Bell which is used to describe societies founded on computers and telecommunications where knowledge replaces material goods as the most important commodity for production and exchange, as opposed to industrial and pre-industrial societies. See the postmodern condition.
• Postmodern condition: For Jean-François Lyotard, this term defines society when it has developed its technologies to the point where information itself is the central commodity and society no longer believes in the ethical, philosophical, social, political narratives that were once thought of as providing the justification for education, learning and the production of knowledge. Lyotard's basic approach was to position these transformations in the context of what he called the crisis of (or incredulity towards) metanarratives. See legitimation, grand narrative, performativity and little narratives.
• Postmodern feminism: Susan Bordo argues that when gender is meticulously fragmented by things like class, race, ‘historical particularity’ and subjected to difference in such a way that its meaning is ‘constantly deferred’ then we are in the presence of postmodern feminism.
• Postmodern identity: This is associated with Jean Baudrillard's idea that identity in late capitalism is not be understood as something outside consumption but negotiated within it. See consumer society, simulation, simulacra and hyperreality.
• Postmodernism: A complex set of (sometimes contradictory) discourses which attempt to characterize the cultures, thought and experiences of late capitalist societies. See post-industrial society, postmodern condition, postmodern feminism, postmodern identity, late capitalism, hyperreality, simulation, simulacra, crisis of (or incredulity towards) metanarratives, cultural logic of late capitalism, depthlessness, the waning of affect, death of the subject, parody and pastiche, pastness, cultural schizophrenia, new social movements and high-tech paranoia.
• Poststructuralism: Generally speaking the term refers to the work of writers who use concepts taken from Saussurean linguistics, but who subject them to radical questioning. However, sometimes writers, like Michel Foucault, who do not use structuralist concepts in a systematic way, are related to the concept because their work shares certain thematic concerns with those deemed poststructuralist. See deconstruction, différance, difference, deferral, trace, structure, transcendental signified, logocentrism and phonocentrism.
• Power: In Michel Foucault's writings he constantly stressed the relations between truth, knowledge and power. Power is a complex notion not conceived of as exclusive to dominant groups but something exercised at every level, moving in complex ways with relation to multiple contexts. See discourse, self-regulation and surveillance.
• Preferred reading: When discussing television discourse Stuart Hall used this to designate an interpretation that corresponds to the intentions of those who encode a message. See encoding, decoding, and negotiated and oppositional readings.
• Problematic: Within Louis Althusser's work this describes the theoretical or ideological limits of a text. Another variation is the ‘invisible problematic’ which is a symptom manifested by (and repressed in) the original problematic which is revealed by symptomatic reading. See double reading.
• Real: Very simply put, in Lacanian psychoanalysis the Real (to describe this in Althusserian terms) is an absent cause and that upon which the Symbolic works. It can be understood as defining a baby's being prior to its identifications in the Imaginary and its symbolization and stratification in the Symbolic. See need and desire.
• Relay: A term used by Roland Barthes to describe how words like ‘earlier’, ‘later’, etc. can be attached to a series of images to convey a sense of temporal progression.
• Repressive state apparatuses: By this phrase Louis Althusser referred to things like the legal system (linked to the police, the courts and the prisons) and the army; apparatuses which are understood as key instruments of direct social control backed up by the secondary ideological state apparatuses.
• Reterritorialization: A concept used by Deleuze and Guattari that can be used to describe how things or spaces can be re-colonized and re-stratified after a process of destratification, thereby transforming existing powers or forces. See territorialization.
• Rhizome: A concept used by Deleuze and Guattari which can be used as a metaphor to describe any process which spreads and disperses power horizontally rather than rely on the more traditional vertical power structures associated with what they call arborescence. See new social movements.
• Self-regulation: A concept found in Michel Foucault's work where he associates the rise of the modern state with an increasing reliance on the internalization of values by subjects who are characterized by self-control. See surveillance and Panopticon.
• Semioclasm: A neologism coined by Roland Barthes where he combined the terms semiotics and iconoclasm to describe his general method when discussing modern mythologies. Semioclasm exposes the bourgeois class interests camouflaged in the culture of everyday life where history or culture is constantly dressed up as nature. See ideology.
• Semiology: In structuralism the study of the relations between the signifier and the signified with relation to a code (otherwise known as semiotics).
• Sex: A pervasive term used in much cultural criticism, especially that focused on feminism and gender studies. More particularly, it features strongly in the work of Michel Foucault and Judith Butler, the latter defining it against concepts like gender and sexuality. For Butler it functions within discourses to create debatable distinctions between males and females by emphasizing things like biological differences, chromosomes, hormonal characteristics, internal and external reproductive/sex organs, etc.
• Sexuality: An extensive term used in much cultural criticism, especially that found in feminism and gender studies. More specifically, it is theorized in the work of Michel Foucault and Judith Butler, the latter defining it against concepts like sex and gender. Butler emphasizes that sexuality concerns how individuals are categorised with relation to sexual attitudes, choices and behaviour, which are often used to define what is properly or intrinsically male of female.
• Shock doctrine: An idea put forward by Naomi Klein that holds that the narrow interests of powerful capitalists are, and have been, imposed all over the world through shock tactics where economists, on behalf of governments, plan austere privatization measures which often attack civil liberties. These strategies are applied to all kinds of crises whether they be ‘natural’, provoked by wars, or by economic meltdowns. See disaster capitalism.
• Sign: In Saussurean linguistics the sign is made up of the signifier and the signified. It is fundamental to an understanding of structuralism.
• Signification: Within structuralism this refers to the study of the relations between signifiers and signifieds to show how they produce meaning with relation to codes.
• Signified: In the structuralist theory of the sign the signified designates the concept or idea attached to a signifier.
• Signifier: Within structuralism this is the form a sign takes – this can be a word or anything that that is capable of conveying meaning and is related to its counterpart the signified.
• Simulacra: This term, often seen as a key to the definition of postmodernism, is used by Jean Baudrillard when theorizing his notion of hyperreality. Simulacra describe images which, in advanced capitalism, cannot be traced back to an origin or cause in such a way that truth, knowledge and reference to a primary reality are no longer possible. See simulation.
• Simulation: Jean Baudrillard used this concept to explain a key idea with relation to his theory of hyperreality, an idea that has become important to some definitions of postmodernism. To simulate is to pretend to have something that you have not got, meaning there is nothing behind appearance (as opposed to dissimulation, which is where someone pretends not to have something they have, in fact, got and, thus, there is something behind mere appearance). See simulacra.
• Structuralism: This is associated with the Swiss linguist Ferdinand de Saussure's theory of the sign as a combination of signifiers and signifieds. From this point of view meaning is not produced because words refer to things but because pre-established codes allow the organization of paradigmatic elements to generate meaningful statements. Thus, structuralism is not referential but relational. These ideas have helped to develop a general theory of the sign and have been applied beyond language to many aspects of culture. See semiology, syntagmatic and paradigmatic relations, binary oppositions, langue, parole and poststructuralism.
• Structure: This term is used in a very particular way by Jacques Derrida who argued that within Western philosophical thinking it tends to provide a centre or a stabilizing point of reference within a discourse and thereby limit its ability to signify indefinitely. Deconstruction challenges this. See différance, transcendental signified and logocentrism.
• Subject: Within structuralism and poststructuralism (particularly in the work of Jacques Lacan) the term is generally used to challenge the idea that the self is prior to the entry into symbolic and cultural networks. Here the subject-self is a product of these things. See subjectivity and desire as an inter-subjective phenomenon.
• Subjectivity: Within structuralism and poststructuralism (and particularly in the work of Jacques Lacan) the term indicates that what is understood as the self is the product of language and it is through it that humans are constituted as a subject.
• Subject supposed to know: A term used in Lacanian psychoanalysis which indicates a living person (or an imaginary figure) who provides the grounds for belief, thereby displacing the authority for faith or knowledge onto others. See the Name-of-the-Father.
• Surveillance: Michel Foucault saw surveillance as a key part of the modern state where subjects exercise self-regulation and control because all kinds of apparatuses create an atmosphere of permanent visibility (a surveillance society) which ensures the automatic functioning of power. See Panopticon.
• Symbolic: In Lacanian psychoanalysis this concept refers to language, culture or any system through which meanings can be produced. Once immersed in the Symbolic the subject comes into existence and is effectively cut off from the Real because it is now mediated through signifiers and subject to signification.
• Symptomatic reading: This describes part of Louis Althusser's double reading which involves acknowledging the gaps and silences that underlie the structure of a text. See problematic.
• Syntagmatic and paradigmatic relations: In structuralism this refers to how (syntagmatic) combinations of elements (paradigms) create meanings (paroles).
• Territorialization: A concept used in the work of Deleuze and Guattari that can be used to describe how things or spaces are taken control of or colonized. See reterritorialization and destratification.
• The end of this or that: Referring to this idea Fredric Jameson argues that it characterizes an important strand of thought within postmodernism. This is where, since the Second World War, all kinds of ‘ends’ have been announced, like end of ideology, social democracy, ‘man’, social class, the welfare state, art, communism, etc.
• Trace: An important term used by Jacques Derrida which posits that signs always mark an absence of a presence and indicate the absence of an authorial origin. See death of the author and deconstruction.
• Transcendental signified: A term coined by Jacques Derrida within deconstruction that refers to something outside a system that would suspend the indefinite play of signification. See différance, logocentrism and phonocentrism.
• Truth: According to Michel Foucault truth is not something other worldly or beyond politics or above the complex workings of power but something produced by them in discourses which create the conditions for its production. See critique of reason.
• Two interpretations of interpretation: This refers to Jacques Derrida's point that interpreters can be grouped into those who dream of recovering truths and origins to discover fixed meanings and those who affirm textual play and renounce full presence and fixed identities (like the practitioners of deconstruction).
• Waning of affect: A concept used by Fredric Jameson to express the idea that feeling, emotion and subjectivity (consciousness) are ebbing away in postmodern culture. It is important to understand that it is a lessening, not a complete disappearance. See cultural schizophrenia, depthlessness, death of the subject and logic of late capitalism.
• Youth subculture: An important area of study within cultural studies that focuses on things like the social background, groupings, practices, values, concerns, styles, behaviour and consumption patterns, and meaning-making activities of young people. Other key notions are the production of generational differences and rebellion.
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• Loading... | {"extraction_info": {"found_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": 0.395506352186203, "perplexity": 14001.466456471318}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891815843.84/warc/CC-MAIN-20180224152306-20180224172306-00197.warc.gz"} |
https://dsp.stackexchange.com/questions/43227/rnn-as-posterior-probability-estimation-in-speech-recognition-with-htk | # RNN as posterior probability estimation in speech recognition with HTK
I'm new to speech recognition and deep learning and in a learning phase.
I'm trying to follow this paper to learn how to use RNN as posterior probability estimation in an HTK environment. The paper proposes RNN-HMM hybrid system, so for the HMM part I need to use HTK platform.
The problem is that I couldn't even start from anywhere. I have a sample code which uses HMM to recognize digits, but I'm unable to solve at which part should I insert RNN to the code.
If there are any ideas, I would be glad.
The system in the paper is shown as follows:
I have codes in python environment and HMM is applied using HTK. After converting data to MFCC format, I should use RNN, but after using RNN, which steps of HMM should I apply to generate RNN-HMM acoustic system. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8090977072715759, "perplexity": 1164.9659827231972}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107882581.13/warc/CC-MAIN-20201024110118-20201024140118-00356.warc.gz"} |
https://deepai.org/publication/a-reductions-approach-to-fair-classification | # A Reductions Approach to Fair Classification
We present a systematic approach for achieving fairness in a binary classification setting. While we focus on two well-known quantitative definitions of fairness, our approach encompasses many other previously studied definitions as special cases. Our approach works by reducing fair classification to a sequence of cost-sensitive classification problems, whose solutions yield a randomized classifier with the lowest (empirical) error subject to the desired constraints. We introduce two reductions that work for any representation of the cost-sensitive classifier and compare favorably to prior baselines on a variety of data sets, while overcoming several of their disadvantages.
## Authors
• 53 publications
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• ### Classification with Fairness Constraints: A Meta-Algorithm with Provable Guarantees
Developing classification algorithms that are fair with respect to sensi...
06/15/2018 ∙ by L. Elisa Celis, et al. ∙ 0
• ### Fairness Under Composition
Much of the literature on fair classifiers considers the case of a singl...
06/15/2018 ∙ by Cynthia Dwork, et al. ∙ 0
• ### Fair Decision Rules for Binary Classification
In recent years, machine learning has begun automating decision making i...
07/03/2021 ∙ by Connor Lawless, et al. ∙ 0
• ### Leveraging Labeled and Unlabeled Data for Consistent Fair Binary Classification
We study the problem of fair binary classification using the notion of E...
06/12/2019 ∙ by Evgenii Chzhen, et al. ∙ 0
• ### Fairness Sample Complexity and the Case for Human Intervention
With the aim of building machine learning systems that incorporate stand...
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• ### Fairmandering: A column generation heuristic for fairness-optimized political districting
The American winner-take-all congressional district system empowers poli...
03/21/2021 ∙ by Wes Gurnee, et al. ∙ 0
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## 1 Introduction
Over the past few years, the media have paid considerable attention to machine learning systems and their ability to inadvertently discriminate against minorities, historically disadvantaged populations, and other protected groups when allocating resources (e.g., loans) or opportunities (e.g., jobs). In response to this scrutiny—and driven by ongoing debates and collaborations with lawyers, policy-makers, social scientists, and others
(e.g., Barocas & Selbst, 2016)—machine learning researchers have begun to turn their attention to the topic of “fairness in machine learning,” and, in particular, to the design of fair classification and regression algorithms.
In this paper we study the task of binary classification subject to fairness constraints with respect to a pre-defined protected attribute, such as race or sex. Previous work in this area can be divided into two broad groups of approaches.
The first group of approaches incorporate specific quantitative definitions of fairness into existing machine learning methods, often by relaxing the desired definitions of fairness, and only enforcing weaker constraints, such as lack of correlation (e.g., Woodworth et al., 2017; Zafar et al., 2017; Johnson et al., 2016; Kamishima et al., 2011; Donini et al., 2018). The resulting fairness guarantees typically only hold under strong distributional assumptions, and the approaches are tied to specific families of classifiers, such as SVMs.
The second group of approaches eliminate the restriction to specific classifier families and treat the underlying classification method as a “black box,” while implementing a wrapper that either works by pre-processing the data or post-processing the classifier’s predictions (e.g., Kamiran & Calders, 2012; Feldman et al., 2015; Hardt et al., 2016; Calmon et al., 2017). Existing pre-processing approaches are specific to particular definitions of fairness and typically seek to come up with a single transformed data set that will work across all learning algorithms, which, in practice, leads to classifiers that still exhibit substantial unfairness (see our evaluation in Section 4). In contrast, post-processing allows a wider range of fairness definitions and results in provable fairness guarantees. However, it is not guaranteed to find the most accurate fair classifier, and requires test-time access to the protected attribute, which might not be available.
We present a general-purpose approach that has the key advantage of this second group of approaches—i.e., the underlying classification method is treated as a black box—but without the noted disadvantages. Our approach encompasses a wide range of fairness definitions, is guaranteed to yield the most accurate fair classifier, and does not require test-time access to the protected attribute. Specifically, our approach allows any definition of fairness that can be formalized via linear inequalities on conditional moments, such as
demographic parity or
equalized odds
(see Section 2.1). We show how binary classification subject to these constraints can be reduced to a sequence of cost-sensitive classification problems. We require only black-box access to a cost-sensitive classification algorithm, which does not need to have any knowledge of the desired definition of fairness or protected attribute. We show that the solutions to our sequence of cost-sensitive classification problems yield a randomized classifier with the lowest (empirical) error subject to the desired fairness constraints.
Corbett-Davies et al. (2017) and Menon & Williamson (2018) begin with a similar goal to ours, but they analyze the Bayes optimal classifier under fairness constraints in the limit of infinite data. In contrast, our focus is algorithmic, our approach applies to any classifier family, and we obtain finite-sample guarantees. Dwork et al. (2018)
also begin with a similar goal to ours. Their approach partitions the training examples into subsets according to protected attribute values and then leverages transfer learning to jointly learn from these separate data sets. Our approach avoids partitioning the data and assumes access only to a classification algorithm rather than a transfer learning algorithm.
A preliminary version of this paper appeared at the FAT/ML workshop (Agarwal et al., 2017), and led to extensions with more general optimization objectives (Alabi et al., 2018) and combinatorial protected attributes (Kearns et al., 2018).
In the next section, we formalize our problem. While we focus on two well-known quantitative definitions of fairness, our approach also encompasses many other previously studied definitions of fairness as special cases. In Section 3, we describe our reductions approach to fair classification and its guarantees in detail. The experimental study in Section 4 shows that our reductions compare favorably to three baselines, while overcoming some of their disadvantages and also offering the flexibility of picking a suitable accuracy–fairness tradeoff. Our results demonstrate the utility of having a general-purpose approach for combining machine learning methods and quantitative fairness definitions.
## 2 Problem Formulation
We consider a binary classification setting where the training examples consist of triples , where
is a feature vector,
is a protected attribute, and is a label. The feature vector can either contain the protected attribute as one of the features or contain other features that are arbitrarily indicative of . For example, if the classification task is to predict whether or not someone will default on a loan, each training example might correspond to a person, where represents their demographics, income level, past payment history, and loan amount; represents their race; and represents whether or not they defaulted on that loan. Note that might contain their race as one of the features or, for example, contain their zipcode—a feature that is often correlated with race. Our goal is to learn an accurate classifier from some set (i.e., family) of classifiers
, such as linear threshold rules, decision trees, or neural nets, while satisfying some definition of fairness. Note that the classifiers in
do not explicitly depend on .
### 2.1 Fairness Definitions
We focus on two well-known quantitative definitions of fairness that have been considered in previous work on fair classification; however, our approach also encompasses many other previously studied definitions of fairness as special cases, as we explain at the end of this section.
The first definition—demographic (or statistical) parity—can be thought of as a stronger version of the US Equal Employment Opportunity Commission’s “four-fifths rule,” which requires that the “selection rate for any race, sex, or ethnic group [must be at least] four-fifths (4/5) (or eighty percent) of the rate for the group with the highest rate.”111See the Uniform Guidelines on Employment Selection Procedures, 29 C.F.R. §1607.4(D) (2015).
###### Definition 1 (Demographic parity—DP).
A classifier satisfies demographic parity under a distribution over if its prediction is statistically independent of the protected attribute —that is, if for all , . Because , this is equivalent to for all .
The second definition—equalized odds—was recently proposed by Hardt et al. (2016) to remedy two previously noted flaws with demographic parity (Dwork et al., 2012). First, demographic parity permits a classifier which accurately classifies data points with one value , such as the value with the most data, but makes random predictions for data points with
as long as the probabilities of
match. Second, demographic parity rules out perfect classifiers whenever is correlated with . In contrast, equalized odds suffers from neither of these flaws.
###### Definition 2 (Equalized odds—EO).
A classifier satisfies equalized odds under a distribution over if its prediction is conditionally independent of the protected attribute given the label —that is, if for all , , and . Because , this is equivalent to for all , .
We now show how each definition can be viewed as a special case of a general set of linear constraints of the form
Mμ(h)≤c, (1)
where matrix and vector describe the linear constraints, each indexed by , and is a vector of conditional moments of the form
where and is an event defined with respect to . Crucially, depends on , while cannot depend on in any way.
###### Example 1 (Dp).
In a binary classification setting, demographic parity can be expressed as a set of equality constraints, each of the form . Letting , for all , , and , where refers to the event encompassing all points in the sample space, each equality constraint can be expressed as .222Note that . Finally, because each such constraint can be equivalently expressed as a pair of inequality constraints of the form
μa(h)−μ⋆(h) ≤0 −μa(h)+μ⋆(h) ≤0,
demographic parity can be expressed as equation (1), where , , , , , and . Expressing each equality constraint as a pair of inequality constraints allows us to control the extent to which each constraint is enforced by positing for some (or all) .
###### Example 2 (Eo).
In a binary classification setting, equalized odds can be expressed as a set of equality constraints, each of the form . Letting , for all , , and , each equality constraint can be equivalently expressed as
μ(a,y)(h)−μ(⋆,y)(h) ≤0 −μ(a,y)(h)+μ(⋆,y)(h) ≤0.
As a result, equalized odds can be expressed as equation (1), where , , , , , and . Again, we can posit for some (or all) to allow small violations of some (or all) of the constraints.
Although we omit the details, we note that many other previously studied definitions of fairness can also be expressed as equation (1). For example, equality of opportunity (Hardt et al., 2016) (also known as balance for the positive class; Kleinberg et al., 2017), balance for the negative class (Kleinberg et al., 2017), error-rate balance (Chouldechova, 2017), overall accuracy equality (Berk et al., 2017), and treatment equality (Berk et al., 2017) can all be expressed as equation (1); in contrast, calibration (Kleinberg et al., 2017) and predictive parity (Chouldechova, 2017) cannot because to do so would require the event to depend on . We note that our approach can also be used to satisfy multiple definitions of fairness, though if these definitions are mutually contradictory, e.g., as described by Kleinberg et al. (2017), then our guarantees become vacuous.
### 2.2 Fair Classification
In a standard (binary) classification setting, the goal is to learn the classifier with the minimum classification error: . However, because our goal is to learn the most accurate classifier while satisfying fairness constraints, as formalized above, we instead seek to find the solution to the constrained optimization problem333We consider misclassification error for concreteness, but all the results in this paper apply to any error of the form , where .
minh∈Herr(h)subject toMμ(h)≤c. (2)
Furthermore, rather than just considering classifiers in the set , we can enlarge the space of possible classifiers by considering randomized classifiers that can be obtained via a distribution over . By considering randomized classifiers, we can achieve better accuracy–fairness tradeoffs than would otherwise be possible. A randomized classifier makes a prediction by first sampling a classifier from and then using to make the prediction. The resulting classification error is and the conditional moments are (see Appendix A for the derivation). Thus we seek to solve
minQ∈Δerr(Q)subject toMμ(Q)≤c, (3)
where is the set of all distributions over .
In practice, we do not know the true distribution over and only have access to a data set of training examples . We therefore replace and in equation (3) with their empirical versions and . Because of the sampling error in , we also allow errors in satisfying the constraints by setting for all , where . After these modifications, we need to solve the empirical version of equation (3):
minQ∈Δˆerr(Q)subject toMˆμ(Q)≤ˆc. (4)
## 3 Reductions Approach
We now show how the problem (4) can be reduced to a sequence of cost-sensitive classification problems. We further show that the solutions to our sequence of cost-sensitive classification problems yield a randomized classifier with the lowest (empirical) error subject to the desired constraints.
### 3.1 Cost-sensitive Classification
We assume access to a cost-sensitive classification algorithm for the set . The input to such an algorithm is a data set of training examples , where and denote the losses—costs in this setting—for predicting the labels or , respectively, for . The algorithm outputs
argminh∈Hn∑i=1h(Xi)C1i+(1−h(Xi))C0i. (5)
This abstraction allows us to specify different costs for different training examples, which is essential for incorporating fairness constraints. Moreover, efficient cost-sensitive classification algorithms are readily available for several common classifier representations (e.g., Beygelzimer et al., 2005; Langford & Beygelzimer, 2005; Fan et al., 1999). In particular, equation (5) is equivalent to a weighted classification problem, where the input consists of labeled examples with and , and the goal is to minimize the weighted classification error . This is equivalent to equation (5) if we set and .
### 3.2 Reduction
To derive our fair classification algorithm, we rewrite equation (4) as a saddle point problem. We begin by introducing a Lagrange multiplier for each of the constraints, summarized as , and form the Lagrangian
L(Q,λ) =ˆerr(Q)+λ⊤(Mˆμ(Q)−ˆc).
Thus, equation (4) is equivalent to
For computational and statistical reasons, we impose an additional constraint on the norm of and seek to simultaneously find the solution to the constrained version of (6) as well as its dual, obtained by switching min and max:
Because is linear in and and the domains of and are convex and compact, both problems have solutions (which we denote by and ) and the minimum value of (P) and the maximum value of (D) are equal and coincide with . Thus, is the saddle point of (Corollary 37.6.2 and Lemma 36.2 of Rockafellar, 1970).
We find the saddle point by using the standard scheme of Freund & Schapire (1996), developed for the equivalent problem of solving for an equilibrium in a zero-sum game. From game-theoretic perspective, the saddle point can be viewed as an equilibrium of a game between two players: the -player choosing and the -player choosing . The Lagrangian specifies how much the -player has to pay to the -player after they make their choices. At the saddle point, neither player wants to deviate from their choice.
Our algorithm finds an approximate equilibrium in which neither player can gain more than by changing their choice (where is an input to the algorithm). Such an approximate equilibrium corresponds to a -approximate saddle point of the Lagrangian, which is a pair , where
L(ˆQ,ˆλ) ≤L(Q,ˆλ)+ν for all Q∈Δ, L(ˆQ,ˆλ) ≥L(ˆQ,λ)−ν for all λ∈R|K|+, ∥λ∥1≤B.
We proceed iteratively by running a no-regret algorithm for the -player, while executing the best response of the -player. Following Freund & Schapire (1996), the average play of both players converges to the saddle point. We run the exponentiated gradient algorithm (Kivinen & Warmuth, 1997) for the -player and terminate as soon as the suboptimality of the average play falls below the pre-specified accuracy . The best response of the -player can always be chosen to put all of the mass on one of the candidate classifiers , and can be implemented by a single call to a cost-sensitive classification algorithm for the set .
Algorithm 1 fully implements this scheme, except for the functions and , which correspond to the best-response algorithms of the two players. (We need the best response of the -player to evaluate whether the suboptimality of the current average play has fallen below .) The two best response functions can be calculated as follows.
#### \textscBestλ(Q): the best response of the λ-player.
The best response of the -player for a given is any maximizer of over all valid s. In our setting, it can always be chosen to be either or put all of the mass on the most violated constraint. Letting and letting denote the vector of the standard basis, returns
{0if ˆγ(Q)≤ˆc,Bek∗otherwise, where k∗=argmaxk[ˆγk(Q)−ˆck].
#### \textscBesth(λ): the best response of the Q-player.
Here, the best response minimizes over all s in the simplex. Because is linear in , the minimizer can always be chosen to put all of the mass on a single classifier . We show how to obtain the classifier constituting the best response via a reduction to cost-sensitive classification. Letting be the empirical event probabilities, the Lagrangian for which puts all of the mass on a single is then
L(h,λ)=ˆerr(h)+λ⊤(Mˆμ(h)−ˆc) =ˆE[1{h(X)≠Y}]−λ⊤ˆc+∑k,jMk,jλkˆμj(h) =−λ⊤ˆc+ˆE[1{h(X)≠Y}] +∑k,jMk,jλkpjˆE[gj(X,A,Y,h(X))1{(X,A,Y)∈Ej}].
Assuming a data set of training examples , the minimization of over then corresponds to cost-sensitive classification on with costs444For general error, , the costs and contain, respectively, the terms and instead of and .
C0i =1{Yi≠0} +∑k,jMk,jλkpjgj(Xi,Ai,Yi,0)1{(Xi,Ai,Yi)∈Ej} C1i =1{Yi≠1} +∑k,jMk,jλkpjgj(Xi,Ai,Yi,1)1{(Xi,Ai,Yi)∈Ej}.
###### Theorem 1.
Letting , Algorithm 1 satisfies the inequality
νt≤Blog(|K|+1)ηt+ηρ2B.
Thus, for , Algorithm 1 will return a -approximate saddle point of in at most iterations.
This theorem, proved in Appendix B, bounds the suboptimality of the average play , which is equal to its suboptimality as a saddle point. The right-hand side of the bound is optimized by , leading to the bound . This bound decreases with the number of iterations and grows very slowly with the number of constraints . The quantity is a problem-specific constant that bounds how much any single classifier can violate the desired set of fairness constraints. Finally, is the bound on the -norm of , which we introduced to enable this specific algorithmic scheme. In general, larger values of will bring the problem (P) closer to (6), and thus also to (4), but at the cost of needing more iterations to reach any given suboptimality. In particular, as we derive in the theorem, achieving suboptimality may need up to iterations.
###### Example 3 (Dp).
Using the matrix for demographic parity as described in Section 2, the cost-sensitive reduction for a vector of Lagrange multipliers uses costs
C0i=1{Yi≠0},C1i=1{Yi≠1}+λAipAi−∑a∈Aλa,
where and , effectively replacing two non-negative Lagrange multipliers by a single multiplier, which can be either positive or negative. Because for all , . Furthermore, because all empirical moments are bounded in , we can assume , which yields the bound . Thus, Algorithm 1 terminates in at most iterations.
###### Example 4 (Eo).
For equalized odds, the cost-sensitive reduction for a vector of Lagrange multipliers uses costs
C0i =1{Yi≠0}, C1i =1{Yi≠1}+λ(Ai,Yi)p(Ai,Yi)−∑a∈Aλ(a,Yi)p(⋆,Yi),
where , , and . If we again assume , then we obtain the bound . Thus, Algorithm 1 terminates in at most iterations.
### 3.3 Error Analysis
Our ultimate goal, as formalized in equation (3), is to minimize the classification error while satisfying fairness constraints under a true but unknown distribution over . In the process of deriving Algorithm 1, we introduced three different sources of error. First, we replaced the true classification error and true moments with their empirical versions. Second, we introduced a bound on the magnitude of . Finally, we only run the optimization algorithm for a fixed number of iterations, until it reaches suboptimality level . The first source of error, due to the use of empirical rather than true quantities, is unavoidable and constitutes the underlying statistical error. The other two sources of error, the bound and the suboptimality level , stem from the optimization algorithm and can be driven arbitrarily small at the cost of additional iterations. In this section, we show how the statistical error and the optimization error affect the true accuracy and the fairness of the randomized classifier returned by Algorithm 1—in other words, how well Algorithm 1 solves our original problem (3).
To bound the statistical error, we use the Rademacher complexity of the classifier family , which we denote by , where is the number of training examples. We assume that for some and . We note that in the vast majority of classifier families, including norm-bounded linear functions (see Theorem 1 of Kakade et al., 2009
), neural networks (see Theorem 18 of
Bartlett & Mendelson, 2002), and classifier families with bounded VC dimension (see Lemma 4 and Theorem 6 of Bartlett & Mendelson, 2002).
Recall that in our empirical optimization problem we assume that , where are error bounds that account for the discrepancy between and . In our analysis, we assume that these error bounds have been set in accordance with the Rademacher complexity of .
###### Assumption 1.
There exists and such that and , where is the number of data points that fall in ,
nj\coloneqq∣∣{i:(Xi,Ai,Yi)∈Ej}∣∣.
The optimization error can be bounded via a careful analysis of the Lagrangian and the optimality conditions of (P) and (D). Combining the three different sources of error yields the following bound, which we prove in Appendix C.
###### Theorem 2.
Let Assumption 1 hold for , where . Let be any -approximate saddle point of , let minimize subject to , and let . Then, with probability at least , the distribution satisfies
err(ˆQ) ≤err(Q⋆)+2ν+˜O(n−α), γk(ˆQ) ≤ck+1+2νB+∑j∈J|Mk,j|˜O(n−αj) for all k,
where suppresses polynomial dependence on . If for all , then, for all ,
γk(ˆQ)≤ck+1+2νB+∑j∈J|Mk,j|˜O((np⋆j)−α).
In other words, the solution returned by Algorithm 1 achieves the lowest feasible classification error on the true distribution up to the optimization error, which grows linearly with , and the statistical error, which grows as . Therefore, if we want to guarantee that the optimization error does not dominate the statistical error, we should set . The fairness constraints on the true distribution are satisfied up to the optimization error
and up to the statistical error. Because the statistical error depends on the moments, and the error in estimating the moments grows as
, we can set to guarantee that the optimization error does not dominate the statistical error. Combining this reasoning with the learning rate setting of Theorem 1 yields the following theorem (proved in Appendix C).
###### Theorem 3.
Let . Let Assumption 1 hold for , where . Let minimize subject to . Then Algorithm 1 with , and terminates in iterations and returns , which with probability at least satisfies
err(ˆQ) ≤err(Q⋆)+˜O(n−α), γk(ˆQ) ≤ck+∑j∈J|Mk,j|˜O(n−αj) for all k.
###### Example 5 (Dp).
If denotes the number of training examples with , then Assumption 1 states that we should set and Theorem 3 then shows that for a suitable setting of , , , and , Algorithm 1 will return a randomized classifier with the lowest feasible classification error up to while also approximately satisfying the fairness constraints
∣∣E[h(X)|A=a]−E[h(X)]∣∣≤˜O(n−αa)for all a,
where is with respect to as well as .
###### Example 6 (Eo).
Similarly, if denotes the number of examples with and and denotes the number of examples with , then Assumption 1 states that we should set and Theorem 3 then shows that for a suitable setting of , , , and , Algorithm 1 will return a randomized classifier with the lowest feasible classification error up to while also approximately satisfying the fairness constraints
∣∣E[h(X)|A=a,Y=y]−E[h(X)|Y=y]∣∣≤˜O(n−α(a,y))
for all , . Again, includes randomness under the true distribution over as well as .
### 3.4 Grid Search
In some situations, it is preferable to select a deterministic classifier, even if that means a lower accuracy or a modest violation of the fairness constraints. A set of candidate classifiers can be obtained from the saddle point . Specifically, because is a minimizer of and is linear in , the distribution puts non-zero mass only on classifiers that are the -player’s best responses to . If we knew , we could retrieve one such best response via the reduction to cost-sensitive learning introduced in Section 3.2.
We can compute using Algorithm 1, but when the number of constraints is very small, as is the case for demographic parity or equalized odds with a binary protected attribute, it is also reasonable to consider a grid of values , calculate the best response for each value, and then select the value with the desired tradeoff between accuracy and fairness.
###### Example 7 (Dp).
When the protected attribute is binary, e.g., , then the grid search can in fact be conducted in a single dimension. The reduction formally takes two real-valued arguments and , and then adjusts the costs for predicting by the amounts
δa=λapa−λa−λa′andδa′=λa′pa′−λa−λa′,
respectively, on the training examples with and . These adjustments satisfy , so instead of searching over and , we can carry out the grid search over alone and apply the adjustment to the protected attribute value .
With three attribute values, e.g., , we similarly have , so it suffices to conduct grid search in two dimensions rather than three.
###### Example 8 (Eo).
If , we obtain the adjustment
δ(a,y)=λ(a,y)p(a,y)−λ(a,y)+λ(a′,y)p(⋆,y)
for an example with protected attribute value and label , and similarly for protected attribute value . In this case, separately for each , the adjustments satisfy
p(a,y)δ(a,y)+p(a′,y)δ(a′,y)=0,
so it suffices to do the grid search over and and set the parameters for to .
## 4 Experimental Results
We now examine how our exponentiated-gradient reduction performs at the task of binary classification subject to either demographic parity or equalized odds. We provide an evaluation of our grid-search reduction in Appendix D.
We compared our reduction with the score-based post-processing algorithm of Hardt et al. (2016), which takes as its input any classifier, (i.e., a standard classifier without any fairness constraints) and derives a monotone transformation of the classifier’s output to remove any disparity with respect to the training examples. This post-processing algorithm works with both demographic parity and equalized odds, as well as with binary and non-binary protected attributes.
For demographic parity, we also compared our reduction with the reweighting and relabeling approaches of Kamiran & Calders (2012). Reweighting can be applied to both binary and non-binary protected attributes and operates by changing importance weights on each example with the goal of removing any statistical dependence between the protected attribute and label.666Although reweighting was developed for demographic parity, the weights that it induces are achievable by our grid search, albeit the grid search for equalized odds rather than demographic parity. Relabeling was developed for binary protected attributes. First, a classifier is trained on the original data (without considering fairness). The training examples close to the decision boundary are then relabeled to remove all disparity while minimally affecting accuracy. The final classifier is then trained on the relabeled data.
As the base classifiers for our reductions, we used the weighted classification implementations of logistic regression and gradient-boosted decision trees in scikit-learn
(Pedregosa et al., 2011). In addition to the three baselines described above, we also compared our reductions to the “unconstrained” classifiers trained to optimize accuracy only.
We used four data sets, randomly splitting each one into training examples (75%) and test examples (25%):
• [nosep]
• The adult income data set (Lichman, 2013)
(48,842 examples). Here the task is to predict whether someone makes more than \$50k per year, with gender as the protected attribute. To examine the performance for non-binary protected attributes, we also conducted another experiment with the same data, using both gender and race (binarized into white and non-white) as the protected attribute. Relabeling, which requires binary protected attributes, was therefore not applicable here.
• ProPublica’s COMPAS recidivism data (7,918 examples). The task is to predict recidivism from someone’s criminal history, jail and prison time, demographics, and COMPAS risk scores, with race as the protected attribute (restricted to white and black defendants).
• Law School Admissions Council’s National Longitudinal Bar Passage Study (Wightman, 1998) (20,649 examples). Here the task is to predict someone’s eventual passage of the bar exam, with race (restricted to white and black only) as the protected attribute.
• The Dutch census data set (Dutch Central Bureau for Statistics, 2001) (60,420 examples). Here the task is to predict whether or not someone has a prestigious occupation, with gender as the protected attribute.
While all the evaluated algorithms require access to the protected attribute at training time, only the post-processing algorithm requires access to at test time. For a fair comparison, we included in the feature vector , so all algorithms had access to it at both the training time and test time.
We used the test examples to measure the classification error for each approach, as well as the violation of the desired fairness constraints, i.e., and for demographic parity and equalized odds, respectively.
We ran our reduction across a wide range of tradeoffs between the classification error and fairness constraints. We considered and for each value ran Algorithm 1 with across all . As expected, the returned randomized classifiers tracked the training Pareto frontier (see Figure 2 in Appendix D). In Figure 1, we evaluate these classifiers alongside the baselines on the test data.
For all the data sets, the range of classification errors is much smaller than the range of constraint violations. Almost all the approaches were able to substantially reduce or remove disparity without much impact on classifier accuracy. One exception was the Dutch census data set, where the classification error increased the most in relative terms.
Our reduction generally dominated or matched the baselines. The relabeling approach frequently yielded solutions that were not Pareto optimal. Reweighting yielded solutions on the Pareto frontier, but often with substantial disparity. As expected, post-processing yielded disparities that were statistically indistinguishable from zero, but the resulting classification error was sometimes higher than achieved by our reduction under a statistically indistinguishable disparity. In addition, and unlike the post-processing algorithm, our reduction can achieve any desired accuracy–fairness tradeoff, allows a wider range of fairness definitions, and does not require access to the protected attribute at test time.
Our grid-search reduction, evaluated in Appendix D, sometimes failed to achieve the lowest disparities on the training data, but its performance on the test data very closely matched that of our exponentiated-gradient reduction. However, if the protected attribute is non-binary, then grid search is not feasible. For instance, for the version of the adult income data set where the protected attribute takes on four values, the grid search would need to span three dimensions for demographic parity and six dimensions for equalized odds, both of which are prohibitively costly.
## 5 Conclusion
We presented two reductions for achieving fairness in a binary classification setting. Our reductions work for any classifier representation, encompass many definitions of fairness, satisfy provable guarantees, and work well in practice.
Our reductions optimize the tradeoff between accuracy and any (single) definition of fairness given training-time access to protected attributes. Achieving fairness when training-time access to protected attributes is unavailable remains an open problem for future research, as does the navigation of tradeoffs between accuracy and multiple fairness definitions.
## Acknowledgements
We would like to thank Aaron Roth, Sam Corbett-Davies, and Emma Pierson for helpful discussions.
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## Appendix A Error and Fairness for Randomized Classifiers
Let denote the distribution over triples . The accuracy of a classifier is measured by 0-1 error, , which for a randomized classifier becomes
err(Q)\coloneqqP(X,A,Y)∼D,h∼Q[h(X)≠Y]=∑h∈HQ(h)err(h).
The fairness constraints on a classifier are . Recall that . For a randomized classifier we define its moment as
μj(Q)\coloneqqE(X,A,Y)∼D,h∼Q[gj(X,A,Y,h(X))∣∣Ej]=∑h∈HQ(h)μj(h),
where the last equality follows because is independent of the choice of .
## Appendix B Proof of Theorem 1
The proof follows immediately from the analysis of Freund & Schapire (1996) applied to the Exponentiated Gradient (EG) algorithm (Kivinen & Warmuth, 1997), which in our specific case is also equivalent to Hedge (Freund & Schapire, 1997).
Let and . We associate any with the that is equal to on coordinates through and puts the remaining mass on the coordinate .
Consider a run of Algorithm 1. For each , let be the associated element of . Let and let be equal to on coordinates through and put zero on the coordinate . Thus, for any and the associated , we have, for all ,
λ⊤rt=(λ′)⊤r′t, (7)
and, in particular,
λ⊤t(Mˆ | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8228172063827515, "perplexity": 1585.2439617203968}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780056392.79/warc/CC-MAIN-20210918093220-20210918123220-00505.warc.gz"} |
http://corochann.com/category/machine-learning/chainer | # Write predict code using concat_examples
This tutorial corresponds to 03_custom_dataset_mlp folder in the source code. We have trained the model with own dataset, MyDataset, in previous post, let’s write predict code. Source code: predict_custom_dataset1.py predict_custom_dataset2.py Prepare test data It is not difficult for the model to fit to the train data, so we will check how the model is fit to the test data.
I used the same seed (=13) to extract the train and test data used in the training phase. Load trained model
The procedure to load the trained model is Instantiate the model (which is a subclass of Chain: here, it is MyMLP) Send the parameters to GPU if […] | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.312521368265152, "perplexity": 2616.291430671263}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320077.32/warc/CC-MAIN-20170623170148-20170623190148-00426.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/introductory-algebra-for-college-students-7th-edition/chapter-7-section-7-1-rational-expressions-and-their-simplification-exercise-set-page-491/3 | ## Introductory Algebra for College Students (7th Edition)
$x=8$
A rational expression is undefined when the value of its denominator is zero. Write and solve an expression where the denominator is zero. $x-8=0$ Add 8 to both sides. $x-8+8=0+8$ Simplify. $x=8$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.980858325958252, "perplexity": 517.4313179634551}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676592636.68/warc/CC-MAIN-20180721145209-20180721165209-00369.warc.gz"} |
https://mklimenko.github.io/english/2018/06/19/write-only-variables/ | # Write-only variables
Earlier this week we’ve had a nice chat with my colleague about an interesting task, which I’ve found quite entertaining and, most important, a nice interview topic.
In plain C and C++, we use memory-mapped I/O to access registers, which means that every register has a dedicated address by which we can access it. Here’s an example of a (partial) memory layout from the documentation.
There are basically three types of registers:
• Write-only
There are no problems in defining both read-write and write-only registers via references or pointers, although I’m not a big fan of a bare register access (as I’ve discussed this earlier, I prefer to add a couple of layers of indirection to make my code less error-prone):
volatile auto &read_write = *reinterpret_cast<std::uint32_t*>(register_address);
To make variable read-only we simply add the const-qualifier to the reference declaration. During the discussion of the previous article, a couple of people were puzzled about the reinterpret_cast part. This operator is the only way to convert address (as a fixed number from the documentation) to the pointer. Via the pointer, we will access the underlying data.
But there is no qualifier in the language to perform the write-only access. And this is where we unleash the true power of the C++: custom types or classes. But before we dive into solving this task, let’s measure the initial approach performance. I’ll compile the code with the -O2 optimization and -std=c++17 flag:
/// Code:
void foo() {
}
/// Disassembly:
foo():
mov eax, DWORD PTR ds:123456
ret
Only two instructions, which is hard to beat, but we’re not afraid. Let’s take the advantage of the known register address, so we can use constexpr in our task. First of all, we will declare a class with a private reference. Read and write operations may be implemented as a Get() and Set() methods:
class Register {
private:
static volatile inline std::uint32_t &ref = *reinterpret_cast<std::uint32_t*>(register_address);
public:
static std::uint32_t Get(){
return ref;
}
static void Set(std::uint32_t val){
ref = val;
}
};
There are a lot of qualifiers for the internal reference, to make it clear:
1. static makes this reference independent from the structure object
2. volatile is there to indicate that the value of the reference may change during the program execution and the compiler should not cache it
3. inline. A great C++17 feature, allowing to declare static variables without the need for an external .cpp file. Have you ever tried to initialize a std::map inside a class? Now you can do it without any additional fuss.
And here’s the compiler output:
/// Code:
void RegGet() {
auto dst = Register::Get();
}
void RegSet(std::uint32_t val) {
Register::Set(val);
}
/// Disassembly:
RegGet():
mov eax, DWORD PTR ds:123456
ret
RegSet(unsigned int):
mov DWORD PTR ds:123456, edi
ret
Exactly the same disassembly and performance, yay! Although, there might be a problem with an old codebase you’ll try to update because it is quite difficult to search and replace all of the assignments to the Get() methods. I don’t know any C++ refactoring tool which is capable of doing such a thing, and, most importantly, I even have no idea how one might be implemented.
To solve this we will add a couple of overloads to our class:
1. Assignment operator for the write operations
2. std::uint32_t casting operator for the read operations
3. T casting operator to raise a compile-time error when you try to read register into some type other than std::uint32_t
class Register {
// ...
Register& operator=(std::uint32_t val){
ref = val;
return *this;
}
template <typename T>
operator T() const{
static_assert(std::is_same_v<T, std::uint32_t>, "You should assign this register to the std::uint32_t value");
return T();
}
operator std::uint32_t() const {
return ref;
}
} reg;
Since the assignment and casting operators cannot be static, we have to create an object of our class to use it:
/// Code
void RegAssign() {
std::uint32_t dst = reg;
}
void RegGetCast(std::uint32_t val) {
reg = val;
}
/// Disassembly:
RegAssign():
mov eax, DWORD PTR ds:123456
ret
RegGetCast(unsigned int):
mov DWORD PTR ds:123456, edi
ret
And we still get the same performance with the familiar syntax! There is one more tweak we should do to make this class ready for use: get rid of the hard-coded address in the reference. Bear in mind, that we should make it compile-time friendly. This may be done with the help of the templates. A couple of changes:
template <std::size_t address>
class Register {
private:
static volatile inline std::uint32_t &ref = *reinterpret_cast<std::uint32_t*>(address);
//...
};
And we’re done for today. You may fiddle with the code here. It really looks to me like a nice little task to chat about.
Update:
Here’s the write-only part:
template <std::size_t address>
class Register {
private:
static volatile inline std::uint32_t &ref = *reinterpret_cast<std::uint32_t*>(address);
public:
static void Set(std::uint32_t val){
ref = val;
}
Register& operator=(std::uint32_t val){
ref = val;
return *this;
}
};
template <std::size_t address>
class Register {
private:
static volatile inline std::uint32_t &ref = *reinterpret_cast<std::uint32_t*>(address);
public:
static std::uint32_t Get(){
return ref;
}
template <typename T>
operator T() const{
static_assert(std::is_same_v<T, std::uint32_t>, "You should assign this register to the std::uint32_t value");
return T();
}
operator std::uint32_t() const {
return ref;
}
};
Update2:
A couple of days has passed and I has found an even better (more flexible and universal) solution. We will create the base class with all of the operations (read and write) declared protected. The code is here to play.
template <std::size_t address>
class RegisterBase {
private:
static volatile inline std::uint32_t &ref = *reinterpret_cast<std::uint32_t*>(address);
protected:
static std::uint32_t Get(){
return ref;
}
static void Set(std::uint32_t val){
ref = val;
}
RegisterBase& operator=(std::uint32_t val){
ref = val;
return *this;
}
template <typename T>
operator T() const{
static_assert(std::is_same_v<T, std::uint32_t>, "You should assign this register to the std::uint32_t value");
return T();
}
operator std::uint32_t() const {
return ref;
}
};
To get read-only and write-only we will inherit from the base class and change the visibility of certain methods:
template <std::size_t address>
public:
static std::uint32_t Get(){
}
template <typename T>
operator T() const{
}
operator std::uint32_t() const{
}
};
class RegisterWrite : public RegisterBase<address> {
public:
static void Set(std::uint32_t val){
}
RegisterWrite& operator=(std::uint32_t val){
}
};
Note the static_cast<> part. Because RegisterBase<address>::operator= returns a reference to the base class, we should cast it to the reference to the derived class object.
The following code is pretty much the same:
RegisterRead<register_address> reg_read;
void foo() {
}
void RegAssign() {
}
void RegGet() {
}
void RegSet(std::uint32_t val) {
reg_write.Set(val);
}
void RegGetCast(std::uint32_t val) {
reg_write = val;
} | {"extraction_info": {"found_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": 0.21220901608467102, "perplexity": 9281.675089066897}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676592387.80/warc/CC-MAIN-20180721051500-20180721071500-00159.warc.gz"} |
http://export.arxiv.org/abs/1810.08253 | physics.optics
(what is this?)
# Title: Infrared laser magnetometry with a NV doped diamond intracavity etalon
Abstract: We propose an hybrid laser system consisting of a semiconductor external cavity laser associated to an intra-cavity diamond etalon doped with nitrogen-vacancy color centers. We consider laser emission tuned to the infrared absorption line that is enhanced under the magnetic field dependent nitrogen-vacancy electron spin resonance and show that this architecture leads to a compact solid-state magnetometer that can be operated at room-temperature. The sensitivity to the magnetic field limited by the photon shot-noise of the output laser beam is estimated to be around $250~\mathrm{fT/\sqrt{Hz}}$. Unlike usual NV center infrared magnetometry, this method would not require an external frequency stabilized laser. Since the proposed system relies on the competition between the laser threshold and an intracavity absorption, such laser-based optical sensor could be easily adapted to a broad variety of physical systems.
Subjects: Optics (physics.optics); Quantum Physics (quant-ph) DOI: 10.1364/OE.27.001706 Cite as: arXiv:1810.08253 [physics.optics] (or arXiv:1810.08253v1 [physics.optics] for this version)
## Submission history
From: Yannick Dumeige [view email]
[v1] Thu, 18 Oct 2018 19:28:29 GMT (317kb)
Link back to: arXiv, form interface, contact. | {"extraction_info": {"found_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": 0.6090856194496155, "perplexity": 6280.544731714164}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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-20210421130234-20210421160234-00107.warc.gz"} |
https://rmonat.fr/ | # Recent Publications
• Precise Thread-Modular Abstract Interpretation of Concurrent Programs using Relational Interference Abstractions
# Recent Posts
### Binary slides number in Beamer
One day, a friend asked if it was possible to have binary slides number in beamer. This is totally useless but the question was still fun :D. | {"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": 0.9246318936347961, "perplexity": 5333.5300280666}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934808972.93/warc/CC-MAIN-20171124214510-20171124234510-00690.warc.gz"} |
https://nigerianscholars.com/tutorials/metals-metalloids-nonmetals/reactions-2/ | Chemistry » Metals, Metalloids, and Nonmetals » Occurrence, Preparation, and Compounds of Oxygen
# Reactions
## Reactions
Elemental oxygen is a strong oxidizing agent. It reacts with most other elements and many compounds.
## Reaction with Elements
Oxygen reacts directly at room temperature or at elevated temperatures with all other elements except the noble gases, the halogens, and few second- and third-row transition metals of low reactivity (those with higher reduction potentials than copper). Rust is an example of the reaction of oxygen with iron. The more active metals form peroxides or superoxides. Less active metals and the nonmetals give oxides. Two examples of these reactions are:
$$\text{2Mg}(s)+{\text{O}}_{2}(g)\;⟶\;\text{2MgO}(s)$$
$${\text{P}}_{4}(s)+5{\text{O}}_{2}(g)\;⟶\;{\text{P}}_{4}{\text{O}}_{10}(s)$$
The oxides of halogens, at least one of the noble gases, and metals with higher reduction potentials than copper do not form by the direct action of the elements with oxygen.
## Reaction with Compounds
Elemental oxygen also reacts with some compounds. If it is possible to oxidize any of the elements in a given compound, further oxidation by oxygen can occur. For example, hydrogen sulfide, H2S, contains sulfur with an oxidation state of 2−. Because the sulfur does not exhibit its maximum oxidation state, we would expect H2S to react with oxygen. It does, yielding water and sulfur dioxide. The reaction is:
$$2{\text{H}}_{2}\text{S}(g)+3{\text{O}}_{2}(g)\;⟶\;2{\text{H}}_{2}\text{O}(l)+2{\text{SO}}_{2}(g)$$
It is also possible to oxidize oxides such as CO and P4O6 that contain an element with a lower oxidation state. The ease with which elemental oxygen picks up electrons is mirrored by the difficulty of removing electrons from oxygen in most oxides. Of the elements, only the very reactive fluorine can oxidize oxides to form oxygen gas. | {"extraction_info": {"found_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": 0.6031176447868347, "perplexity": 3244.506852804554}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875146160.21/warc/CC-MAIN-20200225202625-20200225232625-00336.warc.gz"} |
https://www.lmfdb.org/knowledge/show/g2c.num_rat_pts | show · g2c.num_rat_pts all knowls · up · search:
By a celebrated theorem of Faltings, the number of rational points on a curve of genus $g\ge 2$ defined over a number field is finite. Faltings' theorem is unfortunately ineffective, and computing this finite set is a difficult problem, in general.
For curves of genus $g\ge 2$ in the LMFDB, we store all known rational points, and we indicate cases where this is provably all rational points.
Authors:
Knowl status:
• Review status: reviewed
• Last edited by Jennifer Paulhus on 2019-04-20 15:09:59
Referred to by:
History:
Differences | {"extraction_info": {"found_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": 0.7201837301254272, "perplexity": 1466.167105855732}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988758.74/warc/CC-MAIN-20210506144716-20210506174716-00258.warc.gz"} |
https://uk.arxiv.org/list/cs.NA/pastweek?show=69 | Numerical Analysis
Authors and titles for recent submissions
[ total of 65 entries: 1-65 ]
[ showing up to 69 entries per page: fewer | more ]
Fri, 24 Sep 2021
[1]
Title: Piecewise Padé-Chebyshev Approximation of Bivariate Piecewise smooth Functions
Authors: Akansha Singh
Comments: 25 pages, 15 figures. arXiv admin note: text overlap with arXiv:1910.10385 by other authors
Subjects: Numerical Analysis (math.NA)
[2]
Title: Deep Neural Network Algorithms for Parabolic PIDEs and Applications in Insurance Mathematics
Subjects: Numerical Analysis (math.NA); Probability (math.PR); Computational Finance (q-fin.CP); Machine Learning (stat.ML)
[3]
Title: A new block diagonal preconditioner for a class of $3\times 3$ block saddle point problems
Comments: 14 pages, 2021, to appear in Mediterranean Journal of Mathematics
Subjects: Numerical Analysis (math.NA)
[4]
Title: Two-phase segmentation for intensity inhomogeneous images by the Allen-Cahn Local Binary Fitting Model
Subjects: Numerical Analysis (math.NA)
[5]
Title: A note on perturbation analysis for T-product based tensor singular values
Subjects: Numerical Analysis (math.NA)
[6]
Title: Carleman contraction mapping for a 1D inverse scattering problem with experimental time-dependent data
Subjects: Numerical Analysis (math.NA); Analysis of PDEs (math.AP)
[7]
Title: A discontinuous Galerkin pressure correction scheme for the incompressible Navier-Stokes equations: stability and convergence
Subjects: Numerical Analysis (math.NA)
[8] arXiv:2109.11502 (cross-list from math.OC) [pdf, other]
Title: Inequality Constrained Stochastic Nonlinear Optimization via Active-Set Sequential Quadratic Programming
Subjects: Optimization and Control (math.OC); Machine Learning (cs.LG); Numerical Analysis (math.NA); Machine Learning (stat.ML)
[9] arXiv:2109.11354 (cross-list from cs.LG) [pdf, other]
Title: Arbitrary-Depth Universal Approximation Theorems for Operator Neural Networks
Subjects: Machine Learning (cs.LG); Classical Analysis and ODEs (math.CA); Numerical Analysis (math.NA)
Thu, 23 Sep 2021
[10]
Title: Simple exponential acceleration of the power iteration algorithm
Comments: 7 pages, 1 figure, 2 tables, 3 Python files
Subjects: Numerical Analysis (math.NA); Disordered Systems and Neural Networks (cond-mat.dis-nn)
[11]
Title: A modified block alternating splitting iteration method for solving a class of two-by-two block complex linear systems
Subjects: Numerical Analysis (math.NA)
[12]
Title: Evaluation of mechanical and energy properties for the phase field modeling of failure
Subjects: Numerical Analysis (math.NA)
[13]
Title: Error bounds of fourth-order compact finite difference methods for the Dirac equation in the massless and nonrelativistic regime
Authors: Yue Feng, Ying Ma
Subjects: Numerical Analysis (math.NA)
[14]
Title: Improved variants of the Hutch++ algorithm for trace estimation
Subjects: Numerical Analysis (math.NA)
[15]
Title: Numerical analysis of a finite element formulation of the P2D model for Lithium-ion cells
Authors: Rodolfo Bermejo
Subjects: Numerical Analysis (math.NA)
[16]
Title: Filtered integration rules for finite Hilbert transforms
Subjects: Numerical Analysis (math.NA)
[17]
Title: Relative-error stability of numerical algorithms
Subjects: Numerical Analysis (math.NA)
[18]
Title: Numerical Continued Fraction Interpolation
Subjects: Numerical Analysis (math.NA)
[19]
Title: Modewise Operators, the Tensor Restricted Isometry Property, and Low-Rank Tensor Recovery
Subjects: Numerical Analysis (math.NA)
[20] arXiv:2109.10864 (cross-list from eess.SP) [pdf, other]
Title: Reliable Linearized Phase Retrieval for Near-Field Antenna Measurements with Truncated Measurement Surfaces
Comments: 7 pages, 5 figures, submitted to IEEE Transaction and Antennas and Propagation
Subjects: Signal Processing (eess.SP); Numerical Analysis (math.NA); Optimization and Control (math.OC)
[21] arXiv:2109.10765 (cross-list from physics.flu-dyn) [pdf, other]
Title: An artificial neural network approach to bifurcating phenomena in computational fluid dynamics
Subjects: Fluid Dynamics (physics.flu-dyn); Machine Learning (cs.LG); Numerical Analysis (math.NA); Computational Physics (physics.comp-ph)
[22] arXiv:2109.10636 (cross-list from math.AP) [pdf, ps, other]
Title: Weak-strong Uniqueness for Heat Conducting non-Newtonian Incompressible Fluids
Subjects: Analysis of PDEs (math.AP); Numerical Analysis (math.NA)
[23] arXiv:2109.10484 (cross-list from cs.IT) [pdf, other]
Title: Numerically Stable Binary Coded Computations
Comments: 25 pages, 4 figures, 1 table
Subjects: Information Theory (cs.IT); Numerical Analysis (math.NA)
Wed, 22 Sep 2021
[24]
Title: A semi-Lagrangian scheme for Hamilton-Jacobi-Bellman equations with oblique boundary conditions
Subjects: Numerical Analysis (math.NA)
[25]
Title: Gramian-based model reduction for unstable stochastic systems
Subjects: Numerical Analysis (math.NA); Probability (math.PR)
[26]
Title: Analytical travelling vortex solutions of hyperbolic equations for validating very high order schemes
Subjects: Numerical Analysis (math.NA)
[27]
Title: Virtual element method for the system of time dependent nonlinear convection-diffusion-reaction equation
Subjects: Numerical Analysis (math.NA)
[28]
Title: Nonlinear and Linearised Primal and Dual Initial Boundary Value Problems: When are they Bounded? How are they Connected?
Authors: Jan Nordström
Subjects: Numerical Analysis (math.NA)
[29]
Title: High order direct parametrisation of invariant manifolds for model order reduction of finite element structures: application to large amplitude vibrations and uncovering of a folding point
Comments: 43 pages, 11 figures, 3 tables
Subjects: Numerical Analysis (math.NA); Computational Engineering, Finance, and Science (cs.CE)
[30]
Title: Conditioning of a Hybrid High-Order scheme on meshes with small faces
Subjects: Numerical Analysis (math.NA)
[31] arXiv:2109.10175 (cross-list from cs.CE) [pdf, other]
Title: A spatially adaptive phase-field model of fracture
Subjects: Computational Engineering, Finance, and Science (cs.CE); Numerical Analysis (math.NA)
[32] arXiv:2109.10096 (cross-list from cs.LG) [pdf, ps, other]
Title: Transferability of Graph Neural Networks: an Extended Graphon Approach
Subjects: Machine Learning (cs.LG); Numerical Analysis (math.NA)
[33] arXiv:2109.10003 (cross-list from cs.CE) [pdf, other]
Title: Geometrically exact static isogeometric analysis of an arbitrarily curved spatial Bernoulli-Euler beam
Subjects: Computational Engineering, Finance, and Science (cs.CE); Numerical Analysis (math.NA)
Tue, 21 Sep 2021
[34]
Title: A Multivariate Spline based Collocation Method for Numerical Solution of Partial Differential Equations
Subjects: Numerical Analysis (math.NA); Analysis of PDEs (math.AP)
[35]
Title: Convergence analysis of an operator-compressed multiscale finite element method for Schrödinger equations with multiscale potentials
Subjects: Numerical Analysis (math.NA)
[36]
Title: Energy-stable discretization of two-phase flows in deformable porous media with frictional contact at matrix-fracture interfaces
Subjects: Numerical Analysis (math.NA)
[37]
Title: Neural Networks with Inputs Based on Domain of Dependence and A Converging Sequence for Solving Conservation Laws, Part I: 1D Riemann Problems
Subjects: Numerical Analysis (math.NA); Fluid Dynamics (physics.flu-dyn)
[38]
Title: A monotone discretization for integral fractional Laplacian on bounded Lipschitz domains: Pointwise error estimates under Hölder regularity
Subjects: Numerical Analysis (math.NA)
[39]
Title: Explicit Solutions of the Singular Yang--Baxter-like Matrix Equation and Their Numerical Computation
Comments: 13 pages, 2 figures, This work has been accepted for publication in Mediterranean Journal of Mathematics and it will be published by April 2022
Subjects: Numerical Analysis (math.NA)
[40]
Title: An adaptive stochastic Galerkin method based on multilevel expansions of random fields: Convergence and optimality
Subjects: Numerical Analysis (math.NA)
[41]
Title: A link between the steepest descent method and fixed-point iterations
Authors: Pascal Heid
Subjects: Numerical Analysis (math.NA)
[42]
Title: On the convergence and applications of the inertial-like method for null-point problems
Journal-ref: SIAM J. Imaging Sci.(2014);SIAM J. Optim.(2014);Math. Program.(2018);J. Math. Anal. Appl.(2007);Numerical Algorithms(2019)
Subjects: Numerical Analysis (math.NA); Functional Analysis (math.FA)
[43]
Title: Sampling discretization of integral norms and its application
Authors: F. Dai, V. Temlyakov
Subjects: Numerical Analysis (math.NA); Classical Analysis and ODEs (math.CA); Functional Analysis (math.FA)
[44]
Title: Mean square stability of stochastic theta method for stochastic differential equations driven by fractional Brownian motion
Subjects: Numerical Analysis (math.NA); Probability (math.PR)
[45]
Title: Graph-Theoretical Based Algorithms for Structural Optimization
Subjects: Numerical Analysis (math.NA)
[46]
Title: Improved uniform error bounds for the time-splitting methods for the long-time dynamics of the Schrödinger/nonlinear Schrödinger equation
Subjects: Numerical Analysis (math.NA)
[47]
Title: Quadrature by fundamental solutions: kernel-independent layer potential evaluation for large collections of simple objects
Comments: 41 pages, 12 figures; submitted to Adv. Comput. Math. (topical collection on integral equations)
Subjects: Numerical Analysis (math.NA); Computational Physics (physics.comp-ph)
[48]
Title: Unbiased Bregman-Risk Estimators: Application to Regularization Parameter Selection in Tomographic Image Reconstruction
Subjects: Numerical Analysis (math.NA); Optimization and Control (math.OC)
[49]
Title: Mapping of coherent structures in parameterized flows by learning optimal transportation with Gaussian models
Subjects: Numerical Analysis (math.NA)
[50] arXiv:2109.09703 (cross-list from math.DS) [pdf, other]
Title: Learning to Forecast Dynamical Systems from Streaming Data
Comments: 30 pages, 3 tables, 8 figures
Subjects: Dynamical Systems (math.DS); Machine Learning (cs.LG); Numerical Analysis (math.NA)
[51] arXiv:2109.09697 (cross-list from physics.flu-dyn) [pdf, other]
Title: How to train your solver: A method of manufactured solutions for weakly-compressible SPH
Subjects: Fluid Dynamics (physics.flu-dyn); Numerical Analysis (math.NA); Computational Physics (physics.comp-ph)
[52] arXiv:2109.09673 (cross-list from math.ST) [pdf, other]
Title: A Stochastic Covariance Shrinkage Approach to Particle Rejuvenation in the Ensemble Transform Particle Filter
Subjects: Statistics Theory (math.ST); Numerical Analysis (math.NA)
[53] arXiv:2109.09444 (cross-list from cs.LG) [pdf, other]
Title: When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?
Subjects: Machine Learning (cs.LG); Dynamical Systems (math.DS); Numerical Analysis (math.NA); Machine Learning (stat.ML)
[54] arXiv:2109.09367 (cross-list from cs.LG) [pdf, other]
Title: Network Clustering by Embedding of Attribute-augmented Graphs
Comments: 29 pages, 14 figures, preprint
Subjects: Machine Learning (cs.LG); Numerical Analysis (math.NA); Statistics Theory (math.ST)
[55] arXiv:2109.09306 (cross-list from cs.FL) [pdf, other]
Title: Abelian Repetition Threshold Revisited
Subjects: Formal Languages and Automata Theory (cs.FL); Numerical Analysis (math.NA)
[56] arXiv:2109.09218 (cross-list from math.AT) [pdf, other]
Title: Instability of the Betti Sequence for Persistent Homology and a Stabilized Version of the Betti Sequence
Subjects: Algebraic Topology (math.AT); Computational Geometry (cs.CG); Numerical Analysis (math.NA)
[57] arXiv:2109.08909 (cross-list from cs.CV) [pdf, other]
Title: Measuring the rogue wave pattern triggered from Gaussian perturbations by deep learning
Subjects: Computer Vision and Pattern Recognition (cs.CV); Image and Video Processing (eess.IV); Numerical Analysis (math.NA)
[58] arXiv:2109.08738 (cross-list from q-fin.CP) [pdf, ps, other]
Title: SINH-acceleration for B-spline projection with Option Pricing Applications
Subjects: Computational Finance (q-fin.CP); Computational Engineering, Finance, and Science (cs.CE); Numerical Analysis (math.NA)
Mon, 20 Sep 2021
[59]
Title: Risk Assessment for Performance-Driven Building Design with BIM-Based Parametric Methods
Subjects: Numerical Analysis (math.NA)
[60]
Title: Hyperbolic balance laws: residual distribution, local and global fluxes
Subjects: Numerical Analysis (math.NA)
[61]
Title: $h-$ and $r-$ adaptation on simplicial meshes using MMG tools
Subjects: Numerical Analysis (math.NA); Computational Physics (physics.comp-ph)
[62]
Title: Mixed virtual volume methods for elliptic problems
Subjects: Numerical Analysis (math.NA)
[63]
Title: On Overcoming the Transverse Boundary Error of the SU/PG Scheme for Moving Conductor Problems | {"extraction_info": {"found_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": 0.6361881494522095, "perplexity": 12616.805042365162}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057830.70/warc/CC-MAIN-20210926053229-20210926083229-00508.warc.gz"} |
https://flamechallenge.authorea.com/doi/full/10.1002/essoar.10510480.2 | • Laura Sammon,
• William F McDonough
Laura Sammon
University of Maryland, University of Maryland
Corresponding Author:[email protected]
Author Profile
William F McDonough
University of Maryland, University of Maryland
Author Profile
## Abstract
Earth’s internal heat drives its dynamic engine, causing mantle convection, plate tectonics, and the geodynamo. These renewing and protective processes, which make Earth habitable, are fueled by a primordial (kinetic) and radiogenic heat. For the past two decades, particle physicists have measured the flux of geoneutrinos, electron antineutrinos emitted during β − decay. These ghost-like particles provide a direct measure of the amount of heat producing elements (HPE: Th & U) in the Earth and in turn define the planet’s absolute concentration of the refractory elements. The geoneutrino flux has contributions from the lithosphere and mantle. Detector sensitivity follows a 1/r 2 (source detector separation distance) dependence. Accordingly, an accurate geologic model of the Near-Field Lithosphere (NFL, closest 500 km) surrounding each experiment is required to define the mantle’s contribution. Because of its proximity to the detector and enrichment in HPEs, the local lithosphere contributes ∼50% of the signal and has the greatest effect on interpreting the mantle’s signal. We re-analyzed the upper crustal compositional model used by Agostini et al. (2020) for the Borexino experiment. We documented the geology of the western Near-Field region as rich in potassic volcanism, including some centers within 50 km of the detector. In contrast, the Agostini study did not include these lithologies and used only a HPE-poor, carbonate-rich, model for upper crustal rocks in the surrounding ∼150 km of the Borexino experiment. Consequently, we report 3× higher U content for the local upper crust, which produces a 200% decrease in Earth’s radiogenic heat budget, when compared to their study. Results from the KamLAND and Borexino geoneutrino experiments are at odds with one another and predict mantle compositional heterogeneity that is untenable. Combined analyses of the KamLAND and Borexino experiments using our revised local models strongly favor an Earth with ∼20 TW present-day total radiogenic power. The next generation of geoneutrino detectors (SNO+, counting; and JUNO, under construction) will better constrain the HPE budget of the Earth. | {"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": 0.9121394157409668, "perplexity": 8071.642156748277}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296943746.73/warc/CC-MAIN-20230321193811-20230321223811-00423.warc.gz"} |
https://slideplayer.com/slide/3062426/ | # Lecture 3 Universal TM. Code of a DTM Consider a one-tape DTM M = (Q, Σ, Γ, δ, s). It can be encoded as follows: First, encode each state, each direction,
## Presentation on theme: "Lecture 3 Universal TM. Code of a DTM Consider a one-tape DTM M = (Q, Σ, Γ, δ, s). It can be encoded as follows: First, encode each state, each direction,"— Presentation transcript:
Lecture 3 Universal TM
Code of a DTM Consider a one-tape DTM M = (Q, Σ, Γ, δ, s). It can be encoded as follows: First, encode each state, each direction, and each symbol into a natural number (code(B) = 0, code(R) =1, code(L) = 2, code(s)=3, code(h)=4,... ). Then encode each transition δ(q, a) = (p, b, D) into a string 0 10 10 10 10 qapbD
The code of M is obtained by combining all codes code i of transitions together: 111code 1 11code 2 11∙∙∙11code m 111. Remark: Each TM has many codes. All codes of TMs form a Turing-decidable language.
Universal DTM One can design a three-tape DTM M* which behaves as follows: On input, M* first decodes M on the second tape and then simulates M on the output tape. Clearly, L(M*) = { | x ε L(M)}. Thus, Theorem 1. { | x ε L(M)} is Turing- acceptable.
Next, we prove Theorem 2. { | x ε L(M)} isn't Turing-decidable. To do so, we consider A = { M | M accepts M} and prove Lemma. A isn't Turing-decidable.
Barber cuts his own hair Class 1: {Barber | he can cut his own hair} Class 2: {Barber | he cannot cut own hair} Question: Is there a barber who cuts hair of everybody in class 2, but not cut hair of anybody in class 1. Answer: No!!! Proof. Suppose such a barber exists. If he cuts his own hair, then he is in class 1 and hence he cannot cut his own hair, a contradiction.
If he cannot cut his own hair, then he belongs to class 2 and hence he can cut his own hair, a contradiction. This argument is called diagonalization. hair barber
Example. There exists an irrational number. Proof. Consider all rational numbers in (0,1). They are countable, a1, a2, …. Now, we construct a number such that its i-th digit is different from the i-th digit of ai. Then this number is not rational. a1 a2 digits
Proof. For contradiction, suppose that A is accepted by a one-tape DTM M’. We look at M’ on input M’. If M’ accepts M’, then M’ is in A, which means that M’ rejects M’, a contradiction. If M’ rejects M’, then M’ isn’t in A which means that M’ accepts M’, a contradiction.
Many-one reduction Consider two sets A c Σ* and B c Γ*. If there exists a Turing-computable total function f : Σ* → Γ* such that x ε A iff f(x) ε B, then we say that A is many-one reducible to B, and write A ≤ m B.
A = { M | M accepts M} B = { | M accepts x} Claim. A ≤ m B. Proof. Define f(M) =. M ε A iff M accepts M iff ε B
Theorem. A ≤ m B, B ≤ m C imply A ≤ m C. (This means that ≤ m is a partial ordering.) Theorem. If A ≤ m B and B is Turing- decidable, then A is Turing-decidable. By this theorem, { | M accepts x} isn’t Turing-decidable.
Complete in r. e. An r. e. set A is complete in r. e. if for every r. e. set B, B ≤ m A.
Halting problem Theorem. K = { | M accepts x } is complete in r. e.. Proof. (1) K is a r. e. set. (2) For any r. e. set A, there exists a DTM M A such that A = L(M A ). For every input x of M A, define f(x) =. Then x ε A iff f(x) ε K.
Halting problem Theorem. K = { | M accepts x } is complete in r. e.. Proof. (1) K is a r. e. set. (2) For any r. e. set A, there exists a DTM M A such that A = L(M A ). For every input x of M A, define f(x) =. Then x ε A iff f(x) ε K.
Nonempty Nonempty = {M | L(M) ≠ Φ } is complete in r. e. Proof. (1) Nonempty is a r. e. set. Construct a DTM M* as follows: For each M, we may try every input of M, one by one. If M accepts an input, then M is accepted by M*.
(2) K ≤ m Nonempty. Suppose M’ is a DTM accepting every input. For each input of K, we define f( ) = M where M is a DTM working as follows: on an input y, Step 1. M simulates M on input x. If M accepts x, then go to Step 2. Step 2. M simulates M’ on input y
Therefore, ε K => M accepts x => M accepts every input y => f( ) = M ε Nonempty not in K => M doesn’t halt on x => M doesn’t halt on y => L(M ) = Φ => f( ) not in Nonempty
r. e. –hard A set B is r. e.-hard if for every r. e. set A, A ≤ m B Remark Every complete set is r. e.-hard. However, not every r. e.-hard set is complete. Every r. e.-hard set is not recursive.
All = {M | M accepts all inputs} All is r. e. hard. All is not r. e. All is not complete.
All = {M | M accepts all inputs} All is r. e. hard. All is not r. e. All is not complete.
r. e. property A subset P of TM codes is called a r. e. property if M ε P and L(M’) = L(M) imply M’ ε P. e.g., Nonempty, Empty, All are r. e. properties. Question: Give an example which is a subsets of TM codes, but not a r. e. property.
Nontrivial A r. e. property is trivial if either it is empty or it contains all r. e. set.
Rice Theorem 1 Every nontrivial r. e. property is not recursive.
Proof Let P be a nontrivial r. e. property. For contradiction. Suppose P is a recursive set. So is its complement. Note that either P or its complement P does not contains the empty set. Without loss of generality, assume that P does not contains the empty set.
Since P is nontrivial, P contains a nonempty r. e. set A. Let Ma be a TM accepting A, i.e., A=L(Ma). We want to prove K ≤m P. For each input of K, we define f( ) = M where M is a DTM working as follows. For each input y of M, it first goes to Step 1.
Step 1. M simulates M on input x of M. If M accepts x, then go to Step 2. Step 2. M simulates Ma on y. If Ma accepts y, then M accepts y. Therefore, if ε K then L(M ) = L(Ma) = A ε P, and if not in K, then L(M ) = Φ not in P
Since K is not recursive and K ≤ m P, we obtain a contradiction. Recursive = {M | L(M) is recursive} is not recursive. RE = {M | L(M) is r. e.} is trivial.
Question: Is K an r. e. property? Is every r. e. property complete? Is it true that for any r. e. property, either it or its complement is complete?
Rice Theorem 2 A r. e. property P is r. e. iff the following three conditions hold: (1)If A ε P and A c B for some r. e. set B, then B ε P. (2) If A is an infinite set in P, then A has a finite subset in P. (3) The set of finite languages in P is enumerable, in the sense that there is a TM that generates the (possibly) infinite string code1#code2# …, where code i is a code for the ith finite languages in P.
The code for the finite language {w 1, w 2, …, w n } is [w 1,w 2,…,w n]. In other words, there exists an r. e. set B that is a subset of codes of finite languages in P such that for every finite language F in P, B contains at least one code of F.
Examples All is not r. e. because All does not satisfy condition (2). The complement of ALL is not r. e. because it does not satisfy condition (1). Empty is not r. e. because it does not satisfy (1) Nonempty is r. e. because it satisfies (1), (2) and (3).
Undecidable Problems
Given TMs M and M’, is it true that L(M)=L(M’)? This problem is undecidable, i.e., A = { | L(M) = L(M’)} is not recursive. Proof. Empty ≤ m A. Let M o be a fixed TM such that L(M o ) = Φ. Define f(M) =. Then, M ε Empty iff ε A.
Let A and B be two nonempty proper subsets of Σ*. If A B and B A are recursive, then A ≤ m B. Proof. Let y ε B and z ε B. Define y if x ε A B f(x) = z if x ε B A x, otherwise
Research Problem For a DFA M=(Q, Σ, δ, s, F), L(M) = L(M*) where M* = (Q, Σ, δ, s, Q-F). Given a DTM M, could we have an algorithm to compute a DTM M* such that L(M*) = L(M) when L(M) is regular, and M* will not halt when L(M) is not regular?
Proof of Hierarchy Theorems Diagonalization
Proof of Hierarchy Theorems
Space-constructible function s(n) is fully space-constructible if there exists a DTM M such that for sufficiently large n and any input x with |x|=n, Space M (x) = s(n).
Space Hierarchy If s 2 (n) is a fully space-constructible function, s 1 (n)/s 2 (n) → 0 as n → infinity, s 1 (n) > log n, then DSPACE(s 2 (n)) DSPACE(s 1 (n)) ≠ Φ
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https://groups.oist.jp/representations/liron-speyer/papers | # Papers
All of my papers are also available on my arXiv page. Click titles of papers to view them. Alternatively, hover over a title for more details.
## Published
1. Decomposable Specht modules indexed by bihooks II (with Robert Muth and Louise Sutton). Algebr. Represent. Theory, to appear, DOI.
2. Strong Gelfand subgroups of $$F\wr S_n$$ (with Mahir Can and Yiyang She). Internat. J. Math., 32 (2021), no. 2, 2150010, DOI.
3. Decomposable Specht modules indexed by bihooks (with Louise Sutton). Pacific J. Math. 304 (2020), no. 2, 655–711, DOI.
4. An analogue of row removal for diagrammatic Cherednik algebras (with Chris Bowman). Math. Z. 293 (2019), no. 3, 935–955, DOI.
5. Specht modules for quiver Hecke algebras of type C (with Susumu Ariki and Euiyong Park). Publ. Res. Inst. Math. Sci. 55 (2019), no. 3, 565–626, DOI.
6. On bases of some simple modules of symmetric groups and Hecke algebras (with Melanie de Boeck, Anton Evseev and Sinéad Lyle). Transform. Groups 23 (2018), no. 3, 631–669, DOI. This paper also refers to some GAP code, available here.
7. On the semisimplicity of the cyclotomic quiver Hecke algebra of type C. Proc. Amer. Math. Soc. 146 (2018), no. 5, 1845–1857, DOI.
8. Kleshchev's decomposition numbers for diagrammatic Cherednik algebras (with Chris Bowman). Trans. Amer. Math. Soc. 370 (2018), no. 5, 3551–3590, DOI.
9. A family of graded decomposition numbers for diagrammatic Cherednik algebras (with Chris Bowman and Anton Cox). Int. Math. Res. Not. IMRN 2017 (2017), no. 9, 2686–2734, DOI.
10. Generalised column removal for graded homomorphisms between Specht modules (with Matthew Fayers). J. Algebraic Combin. 44 (2016), no. 2, 393–432, DOI.
11. Decomposable Specht modules for the Iwahori–Hecke algebra $$\mathscr{H}_{\mathbb{F},-1}(\mathfrak{S}_n)$$. J. Algebra 418 (2014), 227–264, DOI. | {"extraction_info": {"found_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": 0.7393164038658142, "perplexity": 1776.664804224049}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1652662558015.52/warc/CC-MAIN-20220523101705-20220523131705-00470.warc.gz"} |
https://www.physicsforums.com/threads/tricky-mean-life-question-potassium.63074/ | # Tricky mean life question (potassium)
1. Feb 7, 2005
### JamesJames
Natural potassiu, has an atomic weight of 39.089 and contains 0.0118 atomic percent of the isotope 19 K 40, which has 2 decay modes:
19 K 40 -> 20 Ca 40 + beta particle + neutrino (nu bar)
19 K 40 + e- -> 18 Ar 40 * + beta particle + neutrino (nu no bar)
where 18 Ar 40 * means an excited state of 18 Ar 40. In this case, this excited state decays to the fround state by emitting a single gamma ray. The total intensity of beta particles emitted is 2.7*10^4 kg^-1 . s^-1 of natural potassium and on average there are 12 gamma rays emitted to every 100 beta particles emitted. Estimate the mean life of 19 K 40.
If someone can set up the question for me, I will take it from there. I am very confused right now as to how the intensity can be used to compute the lifetime. Can someone please explain this using equations?
James
2. Feb 7, 2005
### JamesJames
Please guys..any equations you can write to help me would be greatly appreciated.
James
3. Feb 8, 2005
### JamesJames
Come on guys...someone msut be able to help me...please
JAmes
4. Feb 8, 2005
### HallsofIvy
Staff Emeritus
I'm no expert on this but you are told how many, on average, beta particles are emitted per second. Since each Argon atom has to emit a beta particle when it disintegrates, that gives you the the average number of Argon atoms that disintegrate per second (atoms/second) which should be the reciprocal of the average lifetime (seconds/atom).
Any suggestions, Doc Al?
5. Feb 8, 2005
### Gamma
I agree with HallsofIvy. Problem can be solved with some mathematics. However, since there are two modes of decay, which one do we use to find the mean life? For the first mode of decay, one needs to consider the number of beta particles emitted per second.(mean life is the reciprocal of that as HallsofIvy said)
regards.
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https://www.physicsforums.com/threads/finding-general-solution-of-radical-equation.639250/ | # Finding general solution of Radical Equation
• #1
I_am_learning
686
16
Before trying to find out the general solution of a radical equation; I would first like to know if it can be found?
For example I have a equation of the form
$\text{A1}+\text{A2} x + \text{A3}\sqrt{\text{B1}+\text{B2} x+\text{B3} x^{\frac{3}{2}}+\text{B4}\sqrt{x}+\text{B5} x^2}+ \text{A4}\sqrt{\text{C1}+\text{C2} x+\text{C3} x^{\frac{3}{2}}+\text{C4}\sqrt{x}+\text{C5} x^2}=0$
Can I find x in terms of the Constants A1,A2 etc?
What is the general view on deciding whether a general solution to radical equation exist or not?
I tried searching, but couldn't find out the answer regarding radical equation.
For polynomial equation though, I learned that a general solution doesn't exist for polynomials of degree 5 or higher.
http://en.wikipedia.org/wiki/Abel–Ruffini_theorem
• #2
Homework Helper
43,021
971
1. Isolate one of the square roots.
2. Square both sides which will leave a single square root.
3. Isolate that square root.
4. Now you will have an equation involving powers of x of 2 3/2, 1, and 1/2. Let y= x1/2 so that you have a polynomial involving y4, y3, y2, and y.
(There cannot be a "general solution for a polynomial equation" of degee greater than 5 involving only powers and roots of the coefficients because they may have solution that cannot be written in terms of roots.)
• #3
I_am_learning
686
16
1. Isolate one of the square roots.
2. Square both sides which will leave a single square root.
3. Isolate that square root and Square both sides
4. Now you will have an equation involving powers of x of 2 3/2, 1, and 1/2. Let y= x1/2 so that you have a polynomial involving y4, y3, y2, and y.
(There cannot be a "general solution for a polynomial equation" of degee greater than 5 involving only powers and roots of the coefficients because they may have solution that cannot be written in terms of roots.)
(I made a little amendments). Thanks.
1.$\text{A1}+\text{A2} x + \text{A3}\sqrt{\text{B1}+\text{B2} x+\text{B3} x^{\frac{3}{2}}+\text{B4}\sqrt{x}+\text{B5} x^2}= \text{A4}\sqrt{\text{C1}+\text{C2} x+\text{C3} x^{\frac{3}{2}}+\text{C4}\sqrt{x}+\text{C5} x^2}$
2.$\left(\text{A1}+\text{A2} x + \text{A3}\sqrt{\text{B1}+\text{B2} x+\text{B3} x^{\frac{3}{2}}+\text{B4}\sqrt{x}+\text{B5} x^2}\right)^2= \left(\text{A4}\sqrt{\text{C1}+\text{C2} x+\text{C3} x^{\frac{3}{2}}+\text{C4}\sqrt{x}+\text{C5} x^2}\right)^2$
$\left(\text{A1}^2+\text{A3}^2 \text{B1}+\text{A3}^2 \text{B4} \sqrt{x}+2 \text{A1} \text{A2} x+\text{A3}^2 \text{B2} x+\text{A3}^2 \text{B3} x^{3/2}+\text{A2}^2 x^2+\text{A3}^2 \text{B5} x^2+\text{A3} (2 \text{A1}+2 \text{A2} x) \sqrt{\text{B1}+\text{B4} \sqrt{x}+\text{B2} x+\text{B3} x^{3/2}+\text{B5} x^2}\right)=\text{A4}^2 \text{C1}+\text{A4}^2 \text{C4} \sqrt{x}+\text{A4}^2 \text{C2} x+\text{A4}^2 \text{C3} x^{3/2}+\text{A4}^2 \text{C5} x^2$
3.$\text{A3} (2 \text{A1}+2 \text{A2} x) \sqrt{\text{B1}+\text{B4} \sqrt{x}+\text{B2} x+\text{B3} x^{3/2}+\text{B5} x^2}=\left(\text{A4}^2 \text{C1}+\text{A4}^2 \text{C4} \sqrt{x}+\text{A4}^2 \text{C2} x+\text{A4}^2 \text{C3} x^{3/2}+\text{A4}^2 \text{C5} x^2\right)-\left(\text{A1}^2+\text{A3}^2 \text{B1}+\text{A3}^2 \text{B4} \sqrt{x}+2 \text{A1} \text{A2} x+\text{A3}^2 \text{B2} x+\text{A3}^2 \text{B3} x^{3/2}+\text{A2}^2 x^2+\text{A3}^2 \text{B5} x^2\right)$
$\left(\text{A3} (2 \text{A1}+2 \text{A2} x) \sqrt{\text{B1}+\text{B4} \sqrt{x}+\text{B2} x+\text{B3} x^{3/2}+\text{B5} x^2}\right)^2=\left(\left(\text{A4}^2 \text{C1}+\text{A4}^2 \text{C4} \sqrt{x}+\text{A4}^2 \text{C2} x+\text{A4}^2 \text{C3} x^{3/2}+\text{A4}^2 \text{C5} x^2\right)-\left(\text{A1}^2+\text{A3}^2 \text{B1}+\text{A3}^2 \text{B4} \sqrt{x}+2 \text{A1} \text{A2} x+\text{A3}^2 \text{B2} x+\text{A3}^2 \text{B3} x^{3/2}+\text{A2}^2 x^2+\text{A3}^2 \text{B5} x^2\right)\right)^2$
4.$4 \text{A1}^2 \text{A3}^2 \text{B1}+4 \text{A1}^2 \text{A3}^2 \text{B4} \sqrt{x}+8 \text{A1} \text{A2} \text{A3}^2 \text{B1} x+4 \text{A1}^2 \text{A3}^2 \text{B2} x+4 \text{A1}^2 \text{A3}^2 \text{B3} x^{3/2}+8 \text{A1} \text{A2} \text{A3}^2 \text{B4} x^{3/2}+4 \text{A2}^2 \text{A3}^2 \text{B1} x^2+8 \text{A1} \text{A2} \text{A3}^2 \text{B2} x^2+4 \text{A1}^2 \text{A3}^2 \text{B5} x^2+8 \text{A1} \text{A2} \text{A3}^2 \text{B3} x^{5/2}+4 \text{A2}^2 \text{A3}^2 \text{B4} x^{5/2}+4 \text{A2}^2 \text{A3}^2 \text{B2} x^3+8 \text{A1} \text{A2} \text{A3}^2 \text{B5} x^3+4 \text{A2}^2 \text{A3}^2 \text{B3} x^{7/2}+4 \text{A2}^2 \text{A3}^2 \text{B5} x^4=\text{A1}^4+2 \text{A1}^2 \text{A3}^2 \text{B1}+\text{A3}^4 \text{B1}^2-2 \text{A1}^2 \text{A4}^2 \text{C1}-2 \text{A3}^2 \text{A4}^2 \text{B1} \text{C1}+\text{A4}^4 \text{C1}^2+2 \text{A1}^2 \text{A3}^2 \text{B4} \sqrt{x}+2 \text{A3}^4 \text{B1} \text{B4} \sqrt{x}-2 \text{A3}^2 \text{A4}^2 \text{B4} \text{C1} \sqrt{x}-2 \text{A1}^2 \text{A4}^2 \text{C4} \sqrt{x}-2 \text{A3}^2 \text{A4}^2 \text{B1} \text{C4} \sqrt{x}+2 \text{A4}^4 \text{C1} \text{C4} \sqrt{x}+4 \text{A1}^3 \text{A2} x+4 \text{A1} \text{A2} \text{A3}^2 \text{B1} x+2 \text{A1}^2 \text{A3}^2 \text{B2} x+2 \text{A3}^4 \text{B1} \text{B2} x+\text{A3}^4 \text{B4}^2 x-4 \text{A1} \text{A2} \text{A4}^2 \text{C1} x-2 \text{A3}^2 \text{A4}^2 \text{B2} \text{C1} x-2 \text{A1}^2 \text{A4}^2 \text{C2} x-2 \text{A3}^2 \text{A4}^2 \text{B1} \text{C2} x+2 \text{A4}^4 \text{C1} \text{C2} x-2 \text{A3}^2 \text{A4}^2 \text{B4} \text{C4} x+\text{A4}^4 \text{C4}^2 x+2 \text{A1}^2 \text{A3}^2 \text{B3} x^{3/2}+2 \text{A3}^4 \text{B1} \text{B3} x^{3/2}+4 \text{A1} \text{A2} \text{A3}^2 \text{B4} x^{3/2}+2 \text{A3}^4 \text{B2} \text{B4} x^{3/2}-2 \text{A3}^2 \text{A4}^2 \text{B3} \text{C1} x^{3/2}-2 \text{A3}^2 \text{A4}^2 \text{B4} \text{C2} x^{3/2}-2 \text{A1}^2 \text{A4}^2 \text{C3} x^{3/2}-2 \text{A3}^2 \text{A4}^2 \text{B1} \text{C3} x^{3/2}+2 \text{A4}^4 \text{C1} \text{C3} x^{3/2}-4 \text{A1} \text{A2} \text{A4}^2 \text{C4} x^{3/2}-2 \text{A3}^2 \text{A4}^2 \text{B2} \text{C4} x^{3/2}+2 \text{A4}^4 \text{C2} \text{C4} x^{3/2}+6 \text{A1}^2 \text{A2}^2 x^2+2 \text{A2}^2 \text{A3}^2 \text{B1} x^2+4 \text{A1} \text{A2} \text{A3}^2 \text{B2} x^2+\text{A3}^4 \text{B2}^2 x^2+2 \text{A3}^4 \text{B3} \text{B4} x^2+2 \text{A1}^2 \text{A3}^2 \text{B5} x^2+2 \text{A3}^4 \text{B1} \text{B5} x^2-2 \text{A2}^2 \text{A4}^2 \text{C1} x^2-2 \text{A3}^2 \text{A4}^2 \text{B5} \text{C1} x^2-4 \text{A1} \text{A2} \text{A4}^2 \text{C2} x^2-2 \text{A3}^2 \text{A4}^2 \text{B2} \text{C2} x^2+\text{A4}^4 \text{C2}^2 x^2-2 \text{A3}^2 \text{A4}^2 \text{B4} \text{C3} x^2-2 \text{A3}^2 \text{A4}^2 \text{B3} \text{C4} x^2+2 \text{A4}^4 \text{C3} \text{C4} x^2-2 \text{A1}^2 \text{A4}^2 \text{C5} x^2-2 \text{A3}^2 \text{A4}^2 \text{B1} \text{C5} x^2+2 \text{A4}^4 \text{C1} \text{C5} x^2+4 \text{A1} \text{A2} \text{A3}^2 \text{B3} x^{5/2}+2 \text{A3}^4 \text{B2} \text{B3} x^{5/2}+2 \text{A2}^2 \text{A3}^2 \text{B4} x^{5/2}+2 \text{A3}^4 \text{B4} \text{B5} x^{5/2}-2 \text{A3}^2 \text{A4}^2 \text{B3} \text{C2} x^{5/2}-4 \text{A1} \text{A2} \text{A4}^2 \text{C3} x^{5/2}-2 \text{A3}^2 \text{A4}^2 \text{B2} \text{C3} x^{5/2}+2 \text{A4}^4 \text{C2} \text{C3} x^{5/2}-2 \text{A2}^2 \text{A4}^2 \text{C4} x^{5/2}-2 \text{A3}^2 \text{A4}^2 \text{B5} \text{C4} x^{5/2}-2 \text{A3}^2 \text{A4}^2 \text{B4} \text{C5} x^{5/2}+2 \text{A4}^4 \text{C4} \text{C5} x^{5/2}+4 \text{A1} \text{A2}^3 x^3+2 \text{A2}^2 \text{A3}^2 \text{B2} x^3+\text{A3}^4 \text{B3}^2 x^3+4 \text{A1} \text{A2} \text{A3}^2 \text{B5} x^3+2 \text{A3}^4 \text{B2} \text{B5} x^3-2 \text{A2}^2 \text{A4}^2 \text{C2} x^3-2 \text{A3}^2 \text{A4}^2 \text{B5} \text{C2} x^3-2 \text{A3}^2 \text{A4}^2 \text{B3} \text{C3} x^3+\text{A4}^4 \text{C3}^2 x^3-4 \text{A1} \text{A2} \text{A4}^2 \text{C5} x^3-2 \text{A3}^2 \text{A4}^2 \text{B2} \text{C5} x^3+2 \text{A4}^4 \text{C2} \text{C5} x^3+2 \text{A2}^2 \text{A3}^2 \text{B3} x^{7/2}+2 \text{A3}^4 \text{B3} \text{B5} x^{7/2}-2 \text{A2}^2 \text{A4}^2 \text{C3} x^{7/2}-2 \text{A3}^2 \text{A4}^2 \text{B5} \text{C3} x^{7/2}-2 \text{A3}^2 \text{A4}^2 \text{B3} \text{C5} x^{7/2}+2 \text{A4}^4 \text{C3} \text{C5} x^{7/2}+\text{A2}^4 x^4+2 \text{A2}^2 \text{A3}^2 \text{B5} x^4+\text{A3}^4 \text{B5}^2 x^4-2 \text{A2}^2 \text{A4}^2 \text{C5} x^4-2 \text{A3}^2 \text{A4}^2 \text{B5} \text{C5} x^4+\text{A4}^4 \text{C5}^2 x^4$
But now there are $x^4,x^{\frac{7}{2}},x^3,x^{\frac{5}{2}},x^2,x^{\frac{3}{2}},x,\sqrt{x}$
If I replace x = y^2 then I will have polynomial of degree 8.
So it appears the equation won't have general solution. (Atleast not in terms of roots and powers as you said). But I wonder in what form I might get the solution, if at all.
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https://wiki.contextgarden.net/index.php?title=Programming_in_LuaTeX&oldid=10138&printable=yes | # Programming in LuaTeX
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
NOTE: This is a wikified version of this TugBoat article . Feel free to modify it.
In this article, I explain how to use lua to write macros in LuaTeX. I give some examples of macros that are complicated in PdfTeX, but can be defined easily using lua in luaTeX. These examples include macros that do arithmetic on their arguments, use loops, and parse their arguments.
# Introduction
TeX is getting a new engine—luaTeX. As its name suggests, luaTeX adds lua, a programming language, to TeX, the typesetter. I cannot overemphasize the significance of being able to program TeX in a high-level programming language. For example, consider a TeX macro that divides two numbers. Such a macro is provided by the fp package and also by pgfmath library of the TikZ package. The following comment is from the fp package
\def\FP@div#1#2.#3.#4\relax#5.#6.#7\relax{%
% [...] algorithmic idea (for x>0, y>0)
% - %determine \FP@shift such that
% y*10^\FP@shift < 100000000
% <=y*10^(\FP@shift+1)
% - %determine \FP@shift' such that
% x*10^\FP@shift'< 100000000
% <=x*10^(\FP@shift+1)
% - x=x*\FP@shift'
% - y=y*\FP@shift
% - \FP@shift=\FP@shift-\FP@shift'
% - res=0
% - while y>0 %fixed-point representation!
% - \FP@times=0
% - while x>y
% - \FP@times=\FP@times+1
% - x=x-y
% - end
% - y=y/10
% - res=10*res+\FP@times/1000000000
% - end
% - %shift the result according to \FP@shift
The pgfmath library implements the macro in a similar way, but limits the number of shifts that it does. These macros highlight the state of affairs in writing TeX macros. Even simple things like multiplying two numbers are hard; you either have to work extremely hard to circumvent the programming limitations of TeX, or, more frequently, hope that someone else has done the hard work for you. In luaTeX, such a function can be written using the / operator (I will explain the details later):
\def\DIVIDE#1#2{\directlua{tex.print(#1/#2)}}
Thus, with luaTeX ordinary users can write simple macros; and, perhaps more importantly, can read and understand macros written by TeX wizards.
Since the luaTeX project started it has been actively supported by ConTeXt. [1] These days, the various How do I write such a macro questions on the ConTeXt mailing list are answered by a solution that uses lua. I present a few such examples in this article. I have deliberately avoided examples about fonts and non-Latin languages. There is already quite a bit of documentation about them. In this article, I want to highlight how to use luaTeX to write macros that require some flow control: randomized outputs, loops, and parsing.
# Interaction between TeX and lua
To a first approximation, the interaction between TeX and lua is straightforward. When TeX (i.e., the luaTeX engine) starts, it loads the input file in memory and processes it token by token. When TeX encounters \directlua, it stops reading the file in memory, {\em fully expands the argument of\/ \directlua}, and passes the control to a lua instance. The lua instance, which runs with a few preloaded libraries, processes the expanded arguments of \directlua. This lua instance has a special output stream which can be accessed using tex.print(...). The function tex.print(...) is just like the lua function print(...) except that tex.print(...) prints to a TeX stream rather than to the standard output. When the lua instance finishes processing its input, it passes the contents of the TeX stream back to TeX.[2] TeX then inserts the contents of the TeX stream at the current location of the file that it was reading; expands the contents of the TeX stream; and continues. If TeX encounters another \directlua, the above process is repeated.
As an exercise, imagine what happens when the following input is processed by luaTeX. [3]
\directlua%
{tex.print("Depth 1
\\directlua{tex.print('Depth 2')}")}
On top of these luaTeX primitives, ConTeXt provides a higher level interface. There are two ways to call lua from ConTeXt. The first is a macro \ctxlua (read as ConTeXt lua), which is similar to \directlua. (Aside: It is possible to run the lua instance under different name spaces. \ctxlua is the default name space; other name spaces are explained later.) \ctxlua is good for calling small snippets of lua. The argument of \ctxlua is parsed under normal TeX catcodes (category codes), so the end of line character has the same catcode as a space. This can lead to surprises. For example, if you try to use a lua comment, everything after the comment gets ignored.
\ctxlua
{-- A lua comment
tex.print("This is not printed")}
This can be avoided by using a TeX comment instead of a lua comment. However, working under normal TeX catcodes poses a bigger problem: special TeX characters like \letterampersand, \letterhash, \letterdollar, \{, \}, etc., need to be escaped. For example, \letterhash\ has to be escaped with \string to be used in \ctxlua.
\ctxlua
{local t = {1,2,3,4}
tex.print("length " .. \string#t)}
As the argument of \ctxlua is fully expanded, escaping characters can sometimes be tricky. To circumvent this problem, ConTeXt defines a environment called \startluacode ... \stopluacode. This sets the catcodes to what one would expect in lua. Basically only \ has its usual TeX meaning, the catcode of everything else is set to other. So, for all practical purposes, we can forget about catcodes inside \startluacode ... \stopluacode. The above two examples can be written as
\startluacode
-- A lua comment
tex.print("This is printed.")
local t = {1,2,3,4}
tex.print("length " .. #t)
\stopluacode
This environment is meant for moderately sized code snippets. For longer lua code, it is more convenient to write the code in a separate lua file and then load it using lua's dofile(...) function.
ConTeXt also provides a lua function to conveniently write to the TeX stream. The function is called context(...) and it is equivalent to tex.print(string.format(...)).
Using the above, it is easy to define TeX macros that pass control to lua, do some processing in lua, and then pass the result back to TeX. For example, a macro to convert a decimal number to hexadecimal can be written simply, by asking lua to do the conversion.
\def\TOHEX#1{\ctxlua{context("\%X",#1)}}
\TOHEX{35}
The percent sign had to be escaped because \ctxlua assumes TeX catcodes. Sometimes, escaping arguments can be difficult; instead, it can be easier to define a lua function inside \startluacode ... \stopluacode and call it using \ctxlua. For example, a macro that takes a comma separated list of strings and prints a random item can be written as
\startluacode
userdata = userdata or {}
math.randomseed( os.time() )
function userdata.random(...)
context(arg[math.random(1, #arg)])
end
\stopluacode
\def\CHOOSERANDOM#1%
{\ctxlua{userdata.random(#1)}}
\CHOOSERANDOM{"one", "two", "three"}
I could have written a wrapper so that the function takes a list of words and chooses a random word among them. For an example of such a conversion, see the sorting a list of tokens page on the Sort_a_token_list luaTeX wiki
In the above, I created a name space called userdata and defined the function random in that name space. Using a name space avoids clashes with the lua functions defined in luaTeX and ConTeXt.
In order to avoid name clashes, ConTeXt also defines independent name spaces of lua instances. They are
user a private user instance third third party module instance module ConTeXt module instance isolated an isolated instance
Thus, for example, instead of \ctxlua and \type{\startluacode ... \stopluacode}, the user instance can be accessed via the macros \usercode and \startusercode ... \stopusercode. In instances other than isolated, all the lua functions defined by ConTeXt (but not the inbuilt lua functions) are stored in a global name space. In the isolated instance, all lua functions defined by ConTeXt are hidden and cannot be accessed. Using these instances, we could write the above \CHOOSERANDOM macro as follows
\startusercode
math.randomseed( global.os.time() )
function random(...)
global.context(arg[math.random(1, #arg)])
end
\stopusercode
\def\CHOOSERANDOM#1%
{\usercode{random(#1)}}
Since I defined the function random in the user instance of lua, I did not bother to use a separate name space for the function. The lua functions os.time, which is defined by a luaTeX library, and context, which is defined by ConTeXt, needed to be accessed through a global name space. On the other hand, the math.randomseed function, which is part of lua, could be accessed as is.
A separate lua instance also makes debugging slightly easier. With \ctxlua the error message starts with
! LuaTeX error <main ctx instance>:
With \usercode the error message starts with
! LuaTeX error <private user instance>:
This makes it easier to narrow down the source of error.
Normally, it is best to define your lua functions in the user name space. If you are writing a module, then define your lua functions in the third instance and in a name space which is the name of your module. In this article, I will simply use the default lua instance, but take care to define all my lua functions in a userdata name space.
Now that we have some idea of how to work with luaTeX, let's look at some examples.
# Arithmetic without using a abacus
Doing simple arithmetic in TeX can be extremely difficult, as illustrated by the division macro in the introduction. With lua, simple arithmetic becomes trivial. For example, if you want a macro to find the cosine of an angle (in degrees), you can write
\def\COSINE#1%
{\ctxlua(context(math.cos(#1*2*pi/360))}
The built-in math.cos function assumes that the argument is specified in radians, so we convert from degrees to radians on the fly. If you want to type the value of $\pi$ in an article, you can simply say
$\pi = \ctxlua{context(math.pi)}$
or if you want less precision (notice the percent sign is escaped)
$\pi = \ctxlua{context("\%.6f", math.pi)}$
# Loops without worrying about expansion
Loops in TeX are tricky because macro assignments and macro expansion interact in strange ways. For example, suppose we want to typeset a table showing the sum of the roll of two dice and want the output to look like this
\setupcolors[state=start]
\setupTABLE[each][each][width=2em,height=2em,align={middle,middle}]
\setupTABLE[r][1][background=color,backgroundcolor=gray]
\setupTABLE[c][1][background=color,backgroundcolor=gray]
\bTABLE
\bTR \bTD $(+)$ \eTD \bTD 1 \eTD \bTD 2 \eTD
\bTD 3 \eTD \bTD 4 \eTD \bTD 5 \eTD \bTD 6 \eTD \eTR
\bTR \bTD 1 \eTD \bTD 2 \eTD \bTD 3 \eTD
\bTD 4 \eTD \bTD 5 \eTD \bTD 6 \eTD \bTD 7 \eTD \eTR
\bTR \bTD 2 \eTD \bTD 3 \eTD \bTD 4 \eTD
\bTD 5 \eTD \bTD 6 \eTD \bTD 7 \eTD \bTD 8 \eTD \eTR
\bTR \bTD 3 \eTD \bTD 4 \eTD \bTD 5 \eTD
\bTD 6 \eTD \bTD 7 \eTD \bTD 8 \eTD \bTD 9 \eTD \eTR
\bTR \bTD 4 \eTD \bTD 5 \eTD \bTD 6 \eTD
\bTD 7 \eTD \bTD 8 \eTD \bTD 9 \eTD \bTD 10 \eTD \eTR
\bTR \bTD 5 \eTD \bTD 6 \eTD \bTD 7 \eTD
\bTD 8 \eTD \bTD 9 \eTD \bTD 10 \eTD \bTD 11 \eTD \eTR
\bTR \bTD 6 \eTD \bTD 7 \eTD \bTD 8 \eTD
\bTD 9 \eTD \bTD 10 \eTD \bTD 11 \eTD \bTD 12 \eTD \eTR
\eTABLE
The tedious (but faster!) way to achieve this is to simply type the whole table by hand.
It is however natural to want to write this table as a loop, and compute the values. A first ConTeXt implementation using the recursion level might be:
\bTABLE
\bTR
\bTD $(+)$ \eTD
\dorecurse{6}
{\bTD \recurselevel \eTD}
\eTR
\dorecurse{6}
{\bTR
\bTD \recurselevel \eTD
\edef\firstrecurselevel{\recurselevel}
\dorecurse{6}
{\bTD
\the\numexpr\firstrecurselevel+\recurselevel
\eTD}%
\eTR}
\eTABLE
However, this does not work as expected, yielding all zeros.
A natural table stores the contents of all the cells, before typesetting it. But it does not expand the contents of its cell before storing them. So, at the time the table is actually typeset, TeX has already finished the \dorecurse and \recurselevel is set to 0.
The solution is to place \expandafter at the correct location(s) to coax TeX into expanding the \recurselevel macro before the natural table stores the cell contents. The difficult part is figuring out the exact location of \expandafters. Here is a solution that works:
\bTABLE
\bTR
\bTD $(+)$ \eTD
\dorecurse{6}
{\expandafter \bTD \recurselevel \eTD}
\eTR
\dorecurse{6}
{\bTR
\edef\firstrecurselevel{\recurselevel}
\expandafter\bTD \recurselevel \eTD
\dorecurse{6}
{\expandafter\bTD
\the\numexpr\firstrecurselevel+\recurselevel
\relax
\eTD}
\eTR}
\eTABLE
We only needed to add three \expandafters to make the naive loop work. Nevertheless, finding the right location of \expandafter can be frustrating, especially for a non-expert.
By contrast, in luaTeX writing loops is easy. Once a lua instance starts, TeX does not see anything until the lua instance exits. So, we can write the loop in lua, and simply print the values that we would have typed to the TeX stream. When the control is passed to TeX, TeX sees the input as if we had typed it by hand. Consequently, macro expansion is no longer an issue. For example, we can get the above table by:
\startluacode
context.bTABLE()
context.bTR()
context.bTD() context("$(+)$") context.eTD()
for j=1,6 do
context.bTD() context(j) context.eTD()
end
context.eTR()
for i=1,6 do
context.bTR()
context.bTD() context(i) context.eTD()
for j=1,6 do
context.bTD() context(i+j) context.eTD()
end
context.eTR()
end
context.eTABLE()
\stopluacode
The lua functions such as context.bTABLE() and context.bTR() are just abbreviations for running context ("\\bTABLE"), context("\\bTR"), etc. See the ConTeXt lua document manual for more details about such functions. The rest of the code is a simple nested for-loop that computes the sum of two dice. We do not need to worry about macro expansion at all!
# Parsing input without exploding your~head
In order to get around the weird rules of macro expansion, writing a parser in TeX involves a lot of macro jugglery and catcode trickery. It is a black art, one of the biggest mysteries of TeX for ordinary users.
As an example, let's consider typesetting chemical molecules in TeX. Normally, molecules should be typeset in text mode rather than math mode. For example, , can be input as H\low{2SO\lohi{4}{--}}. Typing so much markup can be cumbersome. Ideally, we want a macro such that we type \molecule{H_2SO_4^-} and the macro translates this into H\low{2SO\lohi{4}{--}}. Such a macro can be written in TeX as follows.
\newbox\chemlowbox
\def\chemlow#1%
{\setbox\chemlowbox
\hbox{{\switchtobodyfont[small]#1}}}
\def\chemhigh#1%
{\ifvoid\chemlowbox
\high{{\switchtobodyfont[small]#1}}%
\else
\lohi{\box\chemlowbox}
{{\switchtobodyfont[small]#1}}
\fi}
\def\finishchem%
{\ifvoid\chemlowbox\else
\low{\box\chemlowbox}
\fi}
\unexpanded\def\molecule%
{\bgroup
\catcode\_=\active \uccode\~=\_
\uppercase{\let~\chemlow}%
\catcode\^=\active \uccode\~=\^
\uppercase{\let~\chemhigh}%
\dostepwiserecurse {65}{90}{1}
{\catcode \recurselevel = \active
\uccode\~=\recurselevel
\uppercase{\edef~{\noexpand\finishchem
\rawcharacter{\recurselevel}}}}%
\catcode\-=\active \uccode\~=\-
\uppercase{\def~{--}}%
\domolecule }%
\def\domolecule#1{#1\finishchem\egroup}
This monstrosity is a typical TeX parser. Appropriate characters need to be made active; occasionally, \lccode and \uccode need to be set; signaling tricks are needed (for instance, checking if \chemlowbox is void); and then magic happens (or so it seems to a flabbergasted user). More sophisticated parsers involve creating finite state automata, which look even more monstrous.
With luaTeX, things are different. luaTeX includes a general parser based on PEG (parsing expression grammar) called lpeg. This makes writing parsers in TeX much more comprehensible. For example, the above \molecule macro can be written as
\startluacode
userdata = userdata or {}
local lowercase = lpeg.R("az")
local uppercase = lpeg.R("AZ")
local backslash = lpeg.P("\\")
local csname = backslash * lpeg.P(1)
* (1-backslash)^0
local plus = lpeg.P("+") / "\\textplus "
local minus = lpeg.P("-") / "\\textminus "
local digit = lpeg.R("09")
local sign = plus + minus
local cardinal = digit^1
local integer = sign^0 * cardinal
local leftbrace = lpeg.P("{")
local rightbrace = lpeg.P("}")
local nobrace = 1 - (leftbrace + rightbrace)
local nested = lpeg.P {leftbrace
* (csname + sign + nobrace
+ lpeg.V(1))^0 * rightbrace}
local any = lpeg.P(1)
local subscript = lpeg.P("_")
local superscript = lpeg.P("^")
local somescript = subscript + superscript
local content = lpeg.Cs(csname + nested
+ sign + any)
local lowhigh = lpeg.Cc("\\lohi{%s}{%s}")
* subscript * content
* superscript * content
/ string.format
local highlow = lpeg.Cc("\\hilo{%s}{%s}")
* superscript * content
* subscript * content
/ string.format
local low = lpeg.Cc("\\low{%s}")
* subscript * content
/ string.format
local high = lpeg.Cc("\\high{%s}")
* superscript * content
/ string.format
local justtext = (1 - somescript)^1
local parser = lpeg.Cs((csname + lowhigh
+ highlow + low
+ high + sign + any)^0)
userdata.moleculeparser = parser
function userdata.molecule(str)
return parser:match(str)
end
\stopluacode
\def\molecule#1%
{\ctxlua{userdata.molecule("#1")}}
This is more verbose than the TeX solution, but is easier to read and write. With a proper parser, I do not have to use tricks to check if either one or both _ and ^ are present. More importantly, anyone (once they know the lpeg syntax) can read the parser and easily understand what it does. This is in contrast to the implementation based on TeX macro jugglery which require you to implement a TeX interpreter in your head to understand.
# Conclusion
luaTeX is removing many TeX barriers: using system fonts, reading and writing Unicode files, typesetting non-Latin languages, among others. However, the biggest feature of luaTeX is the ability to use a high-level programming language to program TeX. This can potentially lower the learning curve for programming TeX.
In this article, I have mentioned only one aspect of programming TeX: macros that manipulate their input and output some text to the main TeX stream. Many other kinds of manipulations are possible: luaTeX provides access to TeX boxes, token lists, dimensions, glues, catcodes, direction parameters, math parameters, etc. The details can be found in the luaTeX manual.
1. Not surprising, as two of luaTeX's main developers\Dash Taco Hoekwater and Hans Hagen\Dash are also the main ConTeXt developers.
2. The output of tex.print(...) is buffered and not passed to TeX until the lua instance has stopped.
3. In this example, I used two different kinds of quotations to avoid escaping quotes. Escaping quotes inside \directlua is tricky. The above was a contrived example; if you ever need to escape quotes, you can use the \startluacode ... \stopluacode syntax explained later. | {"extraction_info": {"found_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": 0.9763116240501404, "perplexity": 5085.153622227738}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1642320300289.37/warc/CC-MAIN-20220117031001-20220117061001-00018.warc.gz"} |
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