url stringlengths 14 2.42k | text stringlengths 100 1.02M | date stringlengths 19 19 | metadata stringlengths 1.06k 1.1k |
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https://www.gradesaver.com/textbooks/math/algebra/algebra-2-1st-edition/chapter-11-data-analysis-and-statistics-11-1-find-measures-of-central-tendency-and-dispersion-11-1-exercises-mixed-review-page-749/34 | Algebra 2 (1st Edition)
Plugging in the given values of x, we find: $$f(3.8)=3.8(15.4)=58.5 \\ f(600)= (15.4)(600)= 9240$$ | 2021-03-03 02:42:06 | {"extraction_info": {"found_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": 0.6669279336929321, "perplexity": 941.995673104355}, "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-2021-10/segments/1614178365186.46/warc/CC-MAIN-20210303012222-20210303042222-00339.warc.gz"} |
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I would like to join a node inside a pgfplots axis to a node belonging to a different tikzpicture. I can do this with an overlay from inside the axis. However, if I scale the tikzpicture that encloses ... | 2016-06-26 06:24:59 | {"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": 0.9245305061340332, "perplexity": 3169.0228873723904}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783394987.40/warc/CC-MAIN-20160624154954-00180-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://bt.gateoverflow.in/421/gate2016-25 | # GATE2016-25
The value of the integral $\displaystyle \int_0^{0.9} \dfrac{dx}{(1-x)(2-x)}$ is ____________ | 2021-01-22 10:14:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9784109592437744, "perplexity": 5328.446958244632}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1610703529179.46/warc/CC-MAIN-20210122082356-20210122112356-00081.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/algebra-1-common-core-15th-edition/chapter-12-data-analysis-and-probability-12-3-measures-of-central-tendency-and-dispersion-practice-and-problem-solving-exercises-page-742/12 | ## Algebra 1: Common Core (15th Edition)
Published by Prentice Hall
# Chapter 12 - Data Analysis and Probability - 12-3 Measures of Central Tendency and Dispersion - Practice and Problem-Solving Exercises - Page 742: 12
#### Answer
The 5th number is 99
#### Work Step by Step
The mean is 91 and the values are:9 9, 86, 76 and 95.You have to find the fifth value. Using the mean formula, you find x: $\frac{99+ 86+ 76+ 95+ x}{5}$=91 -Multiply both sides by 5 to isolate x- 99+86+76+95+x=455 -simplify- 356+x=455 -subtract 356 from both sides to solve for x- x=99 Check your work: $\frac{99+ 86+ 76+ 95+ 99}{5}$=$\frac{455}{5}$=99. So the missing value is 99.
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback. | 2018-10-22 21:34:06 | {"extraction_info": {"found_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": 0.6349746584892273, "perplexity": 3008.1607528739346}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583515539.93/warc/CC-MAIN-20181022201445-20181022222945-00164.warc.gz"} |
https://codereview.stackexchange.com/questions/219178/php-mysqli-prepared-statements-can-this-select-query-be-hacked-injected | # PHP MySQLi Prepared Statements: Can this select query be hacked/injected?
i want to know can this be hacked/injected?
$stmt =$mysqli->prepare("SELECT * FROM myTable WHERE name = ?");
$stmt->bind_param("s",$_POST['name']);
$stmt->execute();$result = $stmt->get_result(); if($result->num_rows === 0) exit('No rows');
while($row =$result->fetch_assoc()) {
//do some stuff
}
var_export($ages);$stmt->close();
Given an answer on Stack Overflow suggests almost identical code for protection, let alone using exactly the same principle you can safely assume that your query is protected.
If you want to know how it works, I also wrote an answer on Stack Overflow, https://stackoverflow.com/a/8265319/285587
Nevertheless, as this site is for the code reviews offering some suggestions, I would suggest to use PDO for database interactions instead of mysqli. Simply because PDO API is much more versatile and easier to use. see your snippet rewritten in PDO:
$stmt =$mysqli->prepare("SELECT * FROM myTable WHERE name = ?");
$stmt->execute([$_POST['name']]);
if($stmt->rowCount() === 0) exit('No rows'); while($row = \$stmt->fetch_assoc()) {
//do some stuff
}
as you can see some nagging operations are just gone. I wrote a tutorial on PDO, which I would quite expectedly recommend. | 2019-08-23 23:33:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.17163966596126556, "perplexity": 3395.5336713180227}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027319082.81/warc/CC-MAIN-20190823214536-20190824000536-00406.warc.gz"} |
https://cvgmt.sns.it/paper/4977/ | # Transport equations with nonlocal diffusion and applications to Hamilton-Jacobi equations
created by goffi on 02 Jan 2021
modified on 16 Jul 2021
[BibTeX]
Accepted Paper
Inserted: 2 jan 2021
Last Updated: 16 jul 2021
Journal: Journal of Evolution Equations
Year: 2020
Doi: 10.1007/s00028-021-00720-3
ArXiv: 2101.00615 PDF
Abstract:
We investigate regularity and a priori estimates for Fokker-Planck and Hamilton-Jacobi equations with unbounded ingredients driven by the fractional Laplacian of order $s\in(1/2,1)$. As for Fokker-Planck equations, we establish integrability estimates under a fractional version of the Aronson-Serrin interpolated condition on the velocity field and Bessel regularity when the drift has low Lebesgue integrability with respect to the solution itself. Using these estimates, through the Evans' nonlinear adjoint method we prove new integral, sup-norm and H\"older estimates for weak and strong solutions to fractional Hamilton-Jacobi equations with unbounded right-hand side and polynomial growth in the gradient. Finally, by means of these latter results, exploiting Calder\'on-Zygmund-type regularity for linear nonlocal PDEs and fractional Gagliardo-Nirenberg inequalities, we deduce optimal $L^q$-regularity for fractional Hamilton-Jacobi equations.
Credits | Cookie policy | HTML 5 | CSS 2.1 | 2021-09-27 02:06:41 | {"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": 0.34446394443511963, "perplexity": 2115.286905629472}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1631780058222.43/warc/CC-MAIN-20210926235727-20210927025727-00412.warc.gz"} |
https://puzzling.stackexchange.com/questions/112379/a-piece-of-paper-repeatedly-cut-into-8-or-12-pieces/112381 | # A piece of paper repeatedly cut into 8 or 12 pieces
You are given a piece of paper. It will be cut into 8 or 12 pieces. Each of those new pieces can be cut again into 8 or 12 pieces or left uncut. This process is (theoretically) repeated as often as you want.
What is the highest number of pieces that cannot be reached?
Sure. The cuts add 7 or 11 pieces, respectively, so we are in a situation where McNuggets come in boxes of 7 and 11, and we always get one extra.
Choosing only 7-piece boxes, we can handle any number with remainder 1 modulo 7. Adding an 11-box switches that remainder, and as we add more 11-boxes, that remainder goes through all the 7 possibilities (0-6) before returning to 1, because 7 and 11 are relatively prime.
So, as long as the number is big enough to fit 6 lots of 11, we can make any number with any remainder when divided by 7, so the set of numbers that cannot be reached is (very) finite (and non-empty, can't get to 2 pieces for example), so a maximum is guaranteed to exist.
(Since after 5 lots of 11 there is only one remainder that's out of reach, the largest impossible number must have the same remainder (mod 7) as 6*11+1, and the largest such number smaller than that is 66+1-7=60.)
This is essentially the Frobenius coin problem: what is the largest amount that cannot be obtained using only coins of specified denominations?
In your case, at each step you can either add 7 pieces of paper, or 11 pieces, to the initial single piece that we are start with. So, the allowed denominations (in the language of the Frobenius coin problem) are 7 and 11.
It turns out that a largest unobtainable amount exists when the greatest common divisor of the set of allowed denominations is equal to $$1$$, that is, they have no common factor. Since $$\gcd(7,11) = 1$$, the answer to your problem is: yes, there is a maximum number of pieces that cannot be reached.
Furthermore, it turns out that there is an exact formula for the maximum amount that cannot be obtained in the case when there are only two allowed denominations, $$x$$ and $$y$$, which is $$xy - x - y$$. (No formula is known when there are more than two allowed denominations.)
So, when $$x = 7$$ and $$y = 11$$, we have $$xy - x - y = 7 \cdot 11 - 7 - 11 = 59$$. Since we start with a single piece of paper to which we add multiples of $$x = 7$$ and $$y = 11$$, the maximum number of pieces that we cannot obtain is $$59+1=60$$.
• @Bass Quite right. My off-by-one error is because one starts with a whole piece of paper, so "zero cuts" corresponds to "one piece" and not "zero pieces". I'll make the correction, thanks! Oct 30, 2021 at 20:17 | 2022-08-16 07:17:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 11, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7631451487541199, "perplexity": 282.02107119502244}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1659882572221.38/warc/CC-MAIN-20220816060335-20220816090335-00123.warc.gz"} |
https://jmlr.org/papers/v22/21-0020.html | ## Graph Matching with Partially-Correct Seeds
Liren Yu, Jiaming Xu, Xiaojun Lin; 22(280):1−54, 2021.
### Abstract
Graph matching aims to find the latent vertex correspondence between two edge-correlated graphs and has found numerous applications across different fields. In this paper, we study a seeded graph matching problem, which assumes that a set of seeds, i.e., pre-mapped vertex-pairs, is given in advance. While most previous work requires all seeds to be correct, we focus on the setting where the seeds are partially correct. Specifically, consider two correlated graphs whose edges are sampled independently from a parent Erdos-Renyi graph $\mathcal{G}(n,p)$. A mapping between the vertices of the two graphs is provided as seeds, of which an unknown $\beta$ fraction is correct. We first analyze a simple algorithm that matches vertices based on the number of common seeds in the $1$-hop neighborhoods, and then further propose a new algorithm that uses seeds in the $2$-hop neighborhoods. We establish non-asymptotic performance guarantees of perfect matching for both $1$-hop and $2$-hop algorithms, showing that our new $2$-hop algorithm requires substantially fewer correct seeds than the $1$-hop algorithm when graphs are sparse. Moreover, by combining our new performance guarantees for the $1$-hop and $2$-hop algorithms, we attain the best-known results (in terms of the required fraction of correct seeds) across the entire range of graph sparsity and significantly improve the previous results when $p\ge n^{-5/6}$. For instance, when $p$ is a constant or $p=n^{-3/4}$, we show that only $\Omega(\sqrt{n\log n})$ correct seeds suffice for perfect matching, while the previously best-known results demand $\Omega(n)$ and $\Omega(n^{3/4}\log n)$ correct seeds, respectively. Numerical experiments corroborate our theoretical findings, demonstrating the superiority of our $2$-hop algorithm on a variety of synthetic and real graphs.
[abs][pdf][bib] [code] | 2022-08-15 09:25:16 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5535338521003723, "perplexity": 694.935918874635}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1659882572163.61/warc/CC-MAIN-20220815085006-20220815115006-00622.warc.gz"} |
http://clay6.com/qa/30463/the-binding-energy-per-nucleon-of-deuteron-h-and-helium-nucleus-he-is-1-1-m | Browse Questions
# The binding energy per nucleon of deuteron $\;(_{1}^{2}H)\;$ and helium nucleus $\;(_{2}^{4}He)\;$ is 1.1 MeV and 7 MeV respectively . If two deuteron nuclei react to form a single helium nucleus, the energy released is
$(a)\;23.6 MeV\qquad(b)\;22.8 MeV\qquad(c)\;19 MeV\qquad(d)\;23 MeV$
Can you answer this question?
Answer : $\;23.6 \;MeV$
If two deuteron nuclei react to form a single helium nucleus ,$^2_1 H + ^2_1 H \rightarrow ^4_2 H + Q$. We need to calculate Q.
Energy released = total binding energy of product - total binding energy of reactants
$\Rightarrow Q = 4\times 7 - 2 \times ( 2\times 1.1 ) = 28 - 4.4=23.6 \;MeV \;.$
answered Mar 6, 2014 by
edited Aug 12, 2014 | 2016-12-04 06:32:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.682269275188446, "perplexity": 1688.8948391322567}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698541214.23/warc/CC-MAIN-20161202170901-00187-ip-10-31-129-80.ec2.internal.warc.gz"} |
http://wetourist.it/jivh/use-lewis-theory-to-determine-the-chemical-formula-for-the-compound-formed-between-al-and-o.html | Chemical bonding - Chemical bonding - Ionic and covalent compounds: A second general feature of bonding also became apparent in the early days of chemistry. In this example the electrons are shown as dots and crosses. 2 ) shows that the system becomes more stable (the energy of the system decreases) as two hydrogen atoms move toward each other from r = ∞, until the energy reaches a minimum at r = r 0 (the observed internuclear distance in H 2 is 74 pm). Start studying CHAPTER 9 Mastering Chem. 1, therefore the covalent bond between them is considered nonpolar. A) BaN B) Ba3N2 C) BaN2 D) Ba2N E) Ba2N3. There are three dimensions of H2O, which can help us to visualize the shape of this chemical formula of water. A plot of the potential energy of the system as a function of the internuclear distance (Figure 5. Mole ratios are used as conversion factors between products and reactants in many chemistry problems. Al 2 O 3 Look at the following worked examples on how to determine the chemical formula of compounds. 3 Lewis Structure of a Compound Given the Lewis symbols for the elements nitrogen and fluorine shown in Table 8. Bonding in simple complex ions. With these molar amounts, the empirical formula for the compound may be written as described in the previous chapter of this text. The more you familiarize yourself with the periodic table and the names of. The compound formula defines the formula unit, the simplest whole-number ratio of positive and negative ions giving an electrically neutral unit. 15, and O = 22. Specify the hybrid orbitals needed to accommodate the electron pairs in the geometric arrangement NH 3 1. How To Write Chemical Formulas- Ionic. The molecule is formed by the cation Zn +2 and the anion O-2. A) RbS B) RbS 2 C) Rb 2 S D) Rb 2 S 3 E) Rb 3 S 2 Answer: C 9) Use Lewis theory to determine the chemical formula for the compound formed between Mg and Br. Today we often use chemical formulas, such as NaCl, C 12 H 22 O 11, and Co(NH 3) 6 (ClO 4) 3, to describe chemical compounds. Hence floats on water. Both metals and non-metals take part in displacement reactions. Click on the name for information and a free copy. A) Al3O2 B) Al2O3 C) AlO2 D) Al2O E) AlO Answer: B. Types of bonds Chemical Bonding I: The Covalent Bond Pages 1 Colloidosomes formed by nonpolar/polar/nonpolar nanoball Growth behavior of ammonothermal GaN crystals grown on non-polar Ch 11 Quiz – Bonding, Electronegativity and Lewis Structures A. Nitrogen dioxide is a paramagnetic, bent molecule with C 2v point group symmetry. These are limited to a single typographic line of symbols, which may include. (6) (iii) Predict and explain whether each species is polar. "To use Lewis theory to determine the formula of an ionic compound, determine the number of valence electrons lost by the metal to form an octet, and the number of electrons gained by the nonmetal. Through the use of the Lewis definition of acids and bases, chemists are now able to predict a wider variety of. The chemical formula for this compound is therefore AlCl 3. The central atom is beryllium (draw the molecules Lewis structure to see this). Therefore it is nonpolar and relatively unreactive. The key to writing proper ionic formulas is simple: the total positive charge must balance the total negative charge. Law of Multiple Proportions - ratio of masses of elements in compound is small whole number ratio. S-F Se-F O-F 31) _____ 32) Determine the electron geometry (eg) and molecular geometry (mg) of SiF 4. VSEPR theory means Valence Shell Electron Pair Repulsion theory. Enter the elements in the same order as they appear in the chemical formula. 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Lewis acid/base theory (sometimes called donor-acceptor theory) is a broad, widely applicable approach to the classification of chemical substances and the analysis of chemical reactions. Practice Exercise 1. (i) Write the difference between 2O and O 2. A) Al 3 O 2 B) Al 2 O 3 C) AlO 2 D) Al 2 O E) AlO Identify the compound with the smallest dipole moment in the gas phase. To better understand what a chemical formula means, we must consider how an ionic compound is constructed from its ions. Do not use this tag if your question is about determining [oxidation-state] or [formal-charges], use the other respective tags instead. 58 g of H 2 O. Use Lewis theory to determine the chemical formula for the compound formed between Rb and O. Determine the electron pair geometry using the VSEPR model 3. 37% is the. Use the Lewis model to predict the formula for the compound that forms between calcium and chlorine. D) Once dissolved in water, ionic compounds rarely conduct electricity. 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Elemental analysis of the compound gives the following percentages by mass: 40. is a concise list of the elements in a compound and the ratios of these elements. 9) The correct Lewis structure for potassium in KCl is: K +. (b) According to Law of conservation of mass The mass of ammonia formed = mass of reactant = 28+6 = 34gm 12. Sr and Se 2. To calculate the theoretical percent of each element in a compound, you have to know the chemical formula of the compound. In order to get an “A” in chemistry you need access to high quality instruction, examples, and lots. Hence floats on water. Be able to draw a Lewis structure for a molecular compound and determine its molecular shape using the Valence Shell Electron Pair Repulsion Theory (VSEPR). A chemical compound whose simplest units are molecules D. The Lewis-Langmuir theory was especially convenient for organic compounds, since the bonds between one carbon atom and another or between one carbon and a hydrogen atom were easily explained in this fashion. Percent composition by element. From Wayne Breslyn. Law of Multiple Proportions - ratio of masses of elements in compound is small whole number ratio. They react with metal oxides to form salts and water. Arrange the atoms as above. The VSEPR theory states: b/c electrons repel each other, molecules adjust their shapes so that the valence e- pairs are as far apart from each other as possible. If you want to write a chemical equation, start by writing the chemical formulas of each reactant. This system is a prime example of the phenomena of incongruent melting in rocks, and therefore gives insights into many aspects of mineral formation. theoretical value = 299,800 km/s 2. It is still not water. His theory gave us an explanation of acids and bases based on structure and bonding. A simple example of this concept is that the empirical formula of sulfur monoxide, or SO, would simply be SO, as is the empirical formula of disulfur dioxide, S 2 O 2. 00794*2 + 32. 2) as constitutional isomers (structural isomers) and as steric isomers is calculated by Polya's theorem (G. When writing chemical formula determine the simplest ratio of positive and negative ions that are needed to produce a neutral compound. An obvious example is given by IUPAC chemical nomenclature that allows one to produce and use different names for a single compound; nevertheless, all these names unambigously identify the compound. V3+ and CrO 4 c. , it predicts that each valence electron is in an orbital with another electron of opposite spin). 3 Lewis Structure of a Compound Given the Lewis symbols for the elements nitrogen and fluorine shown in Table 8. Long before chemists knew the formulas for chemical compounds, they developed a system of nomenclature that gave each compound a unique name. A) Ca2N B) CaN C) Ca2N3 D) Ca3N2 E) CaN2. Draw the structure of this compound. A binary compound between an unknown element E and hydrogen; contains 91. For example, O 2 is interpreted as a molecule formed by two oxygen atoms, and CH 3 OH is interpreted as a molecule with one carbon, four hydrogens, and one oxygen. The Lewis Theory used observations from chemists and physicists to form a theory about chemical bonding. A coordination complex consists of a central atom or ion, which is usually metallic and is called the coordination centre, and a surrounding array of bound molecules or ions, that are in turn known as ligands or complexing agents. Lewis structure 22. Label each of the following as element, compound, or mixture: a) Water b) salad dressing c) Liquid bromine d) carbon dioxide e) Carbon f) Kool-Aid g) Salt water h) Gatorade 69. Write your answers in each box. A coordinate covalent bond (or dative bond) occurs when one of the atoms in the bond provides both bonding electrons. Determine the type of chemical bond that occurs in a chemical compound. To find the formula of an ionic compound, first identify the cation and write down its symbol and charge. Use that knowledge to answer all questions in this examination. 6 °F) Number of Protons/Electrons: 29 Number of Neutrons: 35 Classification: Transition Metal Crystal Structure: Cubic Density @ 293 K: 8. You can pour liquid acid on a solid and watch the solid dissolve. For a metal such as zinc, which donates two electrons to achieve a stable electron configuration, the oxidation number is +2. Lewis theory is the study of the patterns that atoms display when they bond and react with each other. That helps us understand and predict interactions with things like medicine and our body, materials used to make buildings and airplanes, and all sorts of other substances. What is the difference between a compound and a mixture? Density: D=m/V 70. Predicting the Shapes of Molecules. 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The Lewis dot structure for Sulfur is an S with 6 dots which stand for its six valence electrons. copper(II) sulfate (CuSO4) with a mass of 0. Chemical formulas are shorthand ways to represent the number and type of atoms in a compound or molecule, such as H 2 O for water or NaCl for sodium chloride, or salt. Compound Element. Lewis structure 22. Calculate the molarity of saturated solutions, and Ksp values. Write the Lewis structure and chemical formula of the compound with a molar mass of about 70 g/mol that contains 19. Its molecular formula is BH 3 O 3, and its molar mass is 61. The molecular properties such as ionization potential, electronegativity, chemical potential, chemical hardness and softness have been deduced from HOMO–LUMO analysis employing B3LYP/6-31G(d,p) method. 9 - Hot and spicy foods contain molecules that Ch. Determine the # of valence electrons for each. In the above example a small 3 is written in the formula, AlCl 3. This review covers different soft and hard ionization techniques and figures of merit for modern mass spectrometers, such as mass resolving power, mass accuracy, isotopic abundance accuracy, accurate mass multiple-stage MS(n) capability, as well as. Carboxylic Acids. VSEPR indicates tetrahedral geometry with one non-bonding pair of electrons (structure itself will be trigonalpyramidal) 3. Electronegativity is defined as the ability of an atom in a particular molecule to attract electrons to itself (the greater the value, the greater the attractiveness for electrons) Electronegativity is a function of:. Oxygen can form many different oxide compounds with both metals and nonmetals. CAMEO Chemicals. Calculate the lattice energy of potassium fluoride, KF, using the Born–Haber cycle. Using a SOLUBILITY TABLE: sodium chloride is soluble iron (II) carbonate is insoluble so reaction occurs!! Predict if a reaction will occur when you combine aqueous solutions of iron (II) chloride with aqueous sodium carbonate solution. A Fast and Easy Computational Method to Calculate the 13C NMR Chemical Shift of Organic Species Adsorbed on the Zeolite Surface. Empirical / Molecular Formulas Determine the empirical formula of a compound that has (by mass) 21. Use Lewis theory to determine the chemical formula for the compound formed between Al and O. Aluminum sulfide (2:3) Molecular Formula Al 2 S 3; Experimental Physico-chemical Properties. A) BaN B) Ba3N2 C) BaN2 D) Ba2N E) Ba2N3. Determine number of protons in a silver dollar weighing. Sr and Se 2. F -, Cl -, Br -, I - Table shows the formulae of ionic compounds. CS2 is an abbreviated form of Carbon Disulphide. Solution: 1) Assume 100 grams of the compound is present. To indicate the chemical formula we use C and O with a subscript of 2 beside oxygen to show there are 2 atoms. Some questions may require the use of the 2011 Edition Reference Tables for Physical Setting/Chemistry. ) AL and F ( wrong answer: AlF) 2. 492468 mol 26. Forces act on the bonds between atoms, changing the molecular structure of a substance. A coordination complex whose centre is a metal atom is called a. KCN(s) --> K + (aq) + CN-(aq) K + is a neutral ion and CN-is a basic ion. EXPOSURE: Workers that use styrene may breathe in vapors or have direct skin contact. To get the best possible experience using our website, we recommend that you upgrade to latest version of this browser or install another web browser. A) A chemical bond is formed and energy is released. Water (H 2 O) is a molecular compound because it is a substance made from more than one kind of element that is held together with molecular bonds. Draw the Lewis structure 2. Information on patterns is provided in the section below. To determine common ionic compounds formed by elements, keep the following in mind:. Although covalent bonding gets most of the attention in Lewis theory, there does exist a convention for expressing ionic compounds using Lewis structures. B) An ionic bond is formed through the sharing of electrons. (3 x Ca2+) = +6 (2x N3-) = -6 The net charge of the compound will be zero. 9 - Two molecules used in the polymer industry are Ch. An oxide is a compound formed by the combination of oxygen and another element. The atomic number of an element, also called a proton number, tells you the number of protons or positive particles in an atom. Lewis Acids and Bases Hard and Soft Acid/Base Theory. Cu 2 O; Notice that we don't bother to place a subscript 1 after the Oxide symbol. We have already encountered some chemical formulas for simple ionic compounds. Predict which forms an anion, which forms a cation, and the charges of each ion. After that, the application of VSEPR theory takes place with the Lewis structure. Use the Lewis model to predict the formula for the compound that forms between calcium and chlorine. In the Brnsted model, the OH-ion is the active species in this reaction it accepts an H + ion to form a covalent bond. The empirical formula of magnesium oxide, Mg x O y, is written as the lowest whole-number ratio between the moles of Mg used and moles of O consumed. CH 24 [2] (d) One of the two compounds in (c) has an isomer. A plot of the potential energy of the system as a function of the internuclear distance (Figure 5. The ionic compounds above are called binary compounds, because they consist of only two elements. ) AL and F ( wrong answer: AlF) 2. Cu+ and SO 4. 16 Which ionic compound is expected to form from combi-. 1 Analyze ionic and covalent compounds in terms of their formation (electron transfer versus sharing), names, chemical formulas (e. hydrogen The lightest element in the universe. If you predict that another compound with the formula C 2 H 4 O 2 could exist, then you demonstrated good chemical insight and are correct. It revolved around the importance of valence electrons in chemical bonding. An oxide is a compound formed by the combination of oxygen and another element. When a chemical reaction occurs, the valence electrons of the atoms are reorganized so that net attractive forces −chemical bonds−occur between atoms. "To use Lewis theory to determine the formula of an ionic compound, determine the number of valence electrons lost by the metal to form an octet, and the number of electrons gained by the nonmetal. This is found by determining the moles of Mg and O in the product; divide each value by the smaller number; and, multiply the resulting values by small whole numbers (up to five. Determine the molecular formula, molar mass, and Lewis structure of the unknown compound. The compound formed will be called calcium iodide. In the qualitative analysis procedure, the chemical properties of an unknown substance are determined by systematically reacting the unknown with a number of different reagents. The coordination complex can be negatively charged, for example, [AuCl 4 ] − , [PtCl 6 ] 2− ,. Al 2 O 3 Look at the following worked examples on how to determine the chemical formula of compounds. Chemistry 20 Bonding Workbook 3. The compound would be CsF (cesium fluoride). The electron-dot formula and Lewis formula are examples of structural formulas. The compound between Al and S is made in such a way that the nuber of Al atoms present in compound are multiplied by the charge number of sulfur (i. Determine the # of valence electrons for each. Use Lewis theory to determine the formula for the compound that forms from Na and S 4. Lewis Structure: A 2D representation of a molecule and its bonds. Draw structures in the web page. ! • For example, lithium and oxygen form an ionic bond in the compound Li 2 O. "To use Lewis theory to determine the formula of an ionic compound, determine the number of valence electrons lost by the metal to form an octet, and the number of electrons gained by the nonmetal. (Sometimes pairs of dots are used instead of lines, but this is uncommon. EXPOSURE: Workers that use or produce acetaldehyde may breathe vapors or have direct skin contact with vapors or liquid. It might have the right elements in it to be water, but it does not have them in the right proportion. By using Lewis structure, show how the ionic bond is formed in the compounds below. Using the chemical formula of the compound and the periodic table of elements, we can add up the atomic weights and calculate molecular weight of the substance. Hydrogen peroxide is the simplest peroxide (a compound with an oxygen–oxygen single bond). But I can assure you that it really does not need any chemical analysis to find the true shape of the process. ? Do you think NMS will integrate actual chemical properties and actual. This study explores the integration of data journalism within three European legacy news organisations through the lens of organisational structure and professional culture. Nitrogen gas (N 2) is a molecule because the bond between the nitrogen atoms is a molecular bond. In past years the following molecules have been used: HCN CH 2 Br 2 AsH 3 CH 4 H 2 O N 2 PCl 3 CO 2 H 2 S These have one. 6 °F) Number of Protons/Electrons: 29 Number of Neutrons: 35 Classification: Transition Metal Crystal Structure: Cubic Density @ 293 K: 8. The Lewis formalism used for the H 2 molecule is H:H or H—H. It exists as a dimer (two molecules joined together). Aluminum bromide is an ionic compound that is formed from aluminum and iodine. In the qualitative analysis procedure, the chemical properties of an unknown substance are determined by systematically reacting the unknown with a number of different reagents. The dimeric form of aluminium tribromide (Al 2 Br 6. An oxide is a compound formed by the combination of oxygen and another element. Following the approach outlined above, the formula mass for this compound is calculated as follows:. What is the difference between a compound and a mixture? Density: D=m/V 70. The name of this compound is barium fluoride. Name: Carbon Symbol: C Atomic Number: 6 Atomic Mass: 12. Chemical Formula Search Help Rules for chemical formulas (Back to search) Enter a sequence of element symbols followed by numbers to specify the amounts of desired elements (e. There is no direct relationship between the formula of a compound and the shape of its molecules. In the process of calculating constitutional isomers, one 2D structure (graph or constitution) is counted just once. Aluminum and chlorine form covalent bonds, not ionic bonds. We can then use VSEPR to predict molecular shapes, based on the valence electron pairs of the Lewis structures. Draw orbital diagrams for the atoms and then move electrons to make filled outer levels. For Potassium, K, its grouping in the first column indicates that it will have an oxidation state of +1, when it. Each of the double bonds would be expected to show the characteristic behavior of an alkene and undergo addition reactions, but this is not how benzene reacts. Draw the Lewis structure of water. As a gas, it is colorless, odorless and highly flammable. Both metals and non-metals take part in displacement reactions. You would have to draw the Lewis structures of the loss (oxidation) and gain (reduction) of electrons between atoms. Calculate the percentage ionic character of C—Cl bond in CCl 4, if the electronegativities of C and Cl are 3. When by knowing the number of charges each ion has. (a) What is the formula of the compound formed between the elements X and Y? (b) Name the type of bond formed in (a) (c) Draw the Lewis structure to show the formation of compound in (a) CHAPTER 3 CHEMICAL. The Calcium complex is the deposit that is responsible for cataracts and kidney stones. 9 - Urea, a compound formed in the liver, is one of Ch. copper(II) sulfate (CuSO4) with a mass of 0. This module introduces some basic facts and principles that are needed for a discussion of organic molecules. Ionic bonds are formed between atoms when electrons are transferred from one atom to another. Lewis structures don't tell us everything, but along with molecule geometry and polarity they are hugely informative. That is because a subscript one is understood to be so. Answer The resonance structures are: (a) SO3: (b) (c) : Question 4. To calculate the theoretical percent of each element in a compound, you have to know the chemical formula of the compound. 7% nitrogen and 80. So-called " milk of magnesia " is a water suspension of one of the few insoluble magnesium compounds, magnesium hydroxide ; the undissolved particles give rise to its appearance and name. Write the chemical formula and compound name for the following: a. Superscripts are used to. O : O O Note: The double bond and the single bond could be reversed. Calculate Ksp values. 1 Analyze ionic and covalent compounds in terms of their formation (electron transfer versus sharing), names, chemical formulas (e. Use parentheses around polyatomic ions if there is more than one. The Lewis structures for sodium and sulfur are as follows: Na. Al and OH d. E) None of the above is true. The formation of many common compounds can be visualized with the use of Lewis symbols and Lewis diagrams. Cu+ and SO 4. Formula and structure: the zinc oxide chemical formula is ZnO. 9 - One of the first drugs to be approved for use in. Pb2+ and C 2 H 3 O 2 e. Chemistry MC Chemistry Use Lewis theory to determine the chemical formula for the compound formed between Al and O. It has 3 arms. A mole ratio is the ratio between the amounts in moles of any two compounds involved in a chemical reaction. The fragmentation of gas phase ions is a complex and often hard-to-predict process. 27% E and 8. A coordination complex whose centre is a metal atom is called a. 5) Use Lewis theory to determine the chemical formula for the compound formed between Al and O. Q 1a: Draw the Lewis Structure for each of the following molecules Achieved Question In the past the Lewis structure is a stand alone question –which case there is an Achieved point given for the majority correct. Sulfur is yellow. The purpose of this video is to discuss it. Examples of Lewis Dot Structures for the Representative Elements. The chemical formulas for covalent compounds are referred to as molecular formulas A chemical formula for a covalent compound. A Lewis acid-base adduct, a compound that contains a coordinate covalent bond between the Lewis acid and the Lewis base, is formed. His theory gave us an explanation of acids and bases based on structure and bonding. 00kJkg −1 K −1 (ii) The oxide formed in the reaction with air contains 43. Number of electrons accepted to achieve a stable noble gas electron arrangement. PHYSICAL SETTING CHEMISTRY Tuesday, June 23, 2015 — 9:15 a. Thus at intermediate distances, proton-electron. For purposes of computing a formula mass, it is helpful to rewrite the formula in the simpler format, Al 2 S 3 O 12. Though not necessary, strict uniqueness, which is always assigning a single label to a particular substance, is highly convenient and very desirable. The purpose of this video is to discuss it. Usually it is formed between two nonmetals. From Wayne Breslyn. Within the formula, the symbol Ba represents barium, P represents phosphorus and O represents oxygen. What is the difference between a compound and a mixture? Density: D=m/V 70. Although covalent bonding gets most of the attention in Lewis theory, there does exist a convention for expressing ionic compounds using Lewis structures. The formula would be Al 2 O 3. Al and 0 38. If the reaction does occur, write a balanced chemical equation showing it. · The prefix penta means 5. That helps us understand and predict interactions with things like medicine and our body, materials used to make buildings and airplanes, and all sorts of other substances. Today we often use chemical formulas, such as NaCl, C 12 H 22 O 11, and Co(NH 3) 6 (ClO 4) 3, to describe chemical compounds. For example, O 2 is interpreted as a molecule formed by two oxygen atoms, and CH 3 OH is interpreted as a molecule with one carbon, four hydrogens, and one oxygen. See our Browser Support/Compatibility page for supported browsers list. Chemistry 20 Bonding Workbook 3. You're not doing this right. A plot of the potential energy of the system as a function of the internuclear distance (Figure 5. A chemical bond formed by the attraction between positive ions and surrounding mobile electrons is a(n) 66. Typically the ligands that support quadruple bonds are π-donors. What is the difference between a compound and a mixture? Density: D=m/V 70. His theory gave us an explanation of acids and bases based on structure and bonding. Binary Ionic Compounds Containing a Metal and a Nonmetal. n a process that involves changes in the structure and energy content of atoms, molecules, or ions but not their nuclei. (a) What is the molecular formula of the compound? (b) What is its Lewis structure if H is bonded to O? (c) What is the geometry of the molecule? (d) What is the hybridization of the orbitals around the N atom?. For example, carbon dioxide forms a stable molecule by sharing electrons between one carbon atom (Group IV) and two oxygen atoms (Group VI). Click on the name for information and a free copy. A molecule is a single formula of a compound joined by covalent bonds. Water is always 88. KCN(s) --> K + (aq) + CN-(aq) K + is a neutral ion and CN-is a basic ion. It can be produced by direct reaction of the elements: 3 Ca + N2. Use Lewis theory to determine the formula for the compound that forms between: Part A. 73% H by mass. Synthesizing half-metallic fully-compensated ferrimagnets that form in the inverse Heusler phase could lead to superior spintronic devices. Challenge Draw a generic Lewis structure for a molecule formed between atoms of group 1 and group 16 elements. Two C atoms, four H atoms, and two O atoms can also be arranged to form a methyl formate, which is used in manufacturing, as an insecticide, and for quick-drying finishes. These are called. How to Write Chemical Formulas Correctly. If it was zero it wouldn't appear at all in the formula. PROBLEM: Use partial orbital diagrams and Lewis symbols to depict the formation of Na+ and O2! ions from the atoms, and determine the formula of the compound. This is the way chemistry is taught in a lot of places, not just Berkeley. 3% fluorine by mass, and determine the formal charge of the atoms in this compound. Water (H 2 O) is a molecular compound because it is a substance made from more than one kind of element that is held together with molecular bonds. Lewis-dot diagrams of the atoms in row 2. The DFT B3LYP/6-311G (d, p) method were used to calculate the optimized molecular structure of the studied compound. Use Lewis theory to determine the chemical formula for the compound formed between Ca and N. A diatomic compound (or diatomic molecule) contains two atoms, which may or may not be the same. Ionic bonds are formed between atoms when electrons are transferred from one atom to another. Draw the Lewis structure 2. 00278 mol of the compound. We said that neutral fluorine has nine protons in its nucleus (an atomic number of 9), nine electrons surrounding the nucleus (to make it neutral), and the most common isotope has ten neutrons in its nucleus, for a mass number of 19. Be on the lookout for any exceptional cases for O, H, etc. Sn4+ & O2-c. You know you have the correct formula for an ionic compound when the positive and negative charges on the ions are the same or "cancel each other out. One formula unit of calcium iodide contains one calcium ion and two iodide ions. Aldehyde, any of a class of organic compounds, in which a carbon atom shares a double bond with an oxygen atom, a single bond with a hydrogen atom, and a single bond with another atom or group of atoms (designated R in general chemical formulas and structure diagrams). Learn vocabulary, terms, and more with flashcards, games, and other study tools. Using the rules for oxidation number assignment, assign oxidation numbers to the other atoms in the compound. D) Once dissolved in water, ionic compounds rarely conduct electricity _____ 5. chemical formula for the compound formed and list two chemical properties for the compound formed. Other atoms in the order they appear in the formula. diphosphorous tetraiodine. ! • For example, lithium and oxygen form an ionic bond in the compound Li 2 O. PRACTICE: Determine the molecular formula of the compound formed from each of the following ions. Strategy: As with most stoichiometry problems, it is necessary to work in moles. This work was essentially a compilation of the knowledge at the time. I'd appreciate it if someone could answer this question for me. The shapes have to do with the location of bonds and lone. Using Molecular Orbital Theory, determine the bond order for O 2 2−. Once we know how many valence electrons there are in PH3 we can distribute them around the central atom and attempt to fill the outer shells of each atom. Calculate the lattice energy of potassium fluoride, KF, using the Born–Haber cycle. Determine the chemical formula for the compound formed between Ca and N. The chemical formula of Ammonia is NH 3. Lewis Theory of Bonding Because structure is so important, chemists have developed a number of theories to explain and predict molecular geometries. Lewis developed a theory that focused on the significance of valence electrons (electrons in the outer shell) in chemical reactions and in bonding. 0 Ionic Compound (Metal!!) Examples: NaCl KF. There are three dimensions of H2O, which can help us to visualize the shape of this chemical formula of water. a) b) c) State the group number for T and U respectively Draw the Lewis structure for the molecule formed between T and U. The Calcium complex is the deposit that is responsible for cataracts and kidney stones. Cl 2 Not binary (only one type of atom), but diatomic (two atoms). Typically, a molecular formula begins with the nonmetal that is closest to the lower left corner of the periodic table, except that hydrogen is almost never. Vapors heavier than air. To determine its formula, count all sorts of atoms inside the unit cell, also considering fractional atoms which belong to more than one unit cell. Chemical bonding - Chemical bonding - Covalent bonds: When none of the elements in a compound is a metal, no atoms in the compound have an ionization energy low enough for electron loss to be likely. Blank space is ignored in search strings. Reference 4, from UC Berkeley, shows a conventional Lewis dot structure for the oxygen molecule, O 2, which can also be called dioxygen. When heated to 100 o C mixtures with molar fractions of AlCl 3 between 0. Use the prefixes, such as mono-, di-, tri-, and tetra-, to figure out the number of atoms present for each element, and write this number as a subscript for each element. In the Brnsted model, the OH-ion is the active species in this reaction it accepts an H + ion to form a covalent bond. about the same. The structural formula of a chemical compound is a graphic representation of the molecular structure (determined by structural chemistry methods), showing how the atoms are possibly arranged in the real three-dimensional space. Use Lewis theory to determine the chemical formula for the compound formed between Al and O. CH 24 [2] (d) One of the two compounds in (c) has an isomer. Search by chemical structure. where we have written the final formula (the formula for sodium chloride) as per the convention for ionic compounds, without listing the charges explicitly. The structural elucidation of small molecules using mass spectrometry plays an important role in modern life sciences and bioanalytical approaches. In order to get an “A” in chemistry you need access to high quality instruction, examples, and lots. E) None of the above is true. There are several rules to follow when writing chemical formulas, so the process can be rather complex. – Each element wants to have 8 Valence Electrons to be stable. Ostwald, developed a theory of acid base indicators which gives an explanation for the colour change with change in pH. Write the charge above the element or polyatomic ion. Searches are not case sensitive. The compound formed will be called calcium iodide. (6) (iii) Predict and explain whether each species is polar. A compound with a molar mass of about 42 g/mol contains 85. uA FORMULA UNIT is the simplest collection of ions for which an ionic compound formula can be written uExpressed using whole numbers in the most reduced form uCation is listed 1 st, Anion is listed 2nd uCharges must sum to zero, but are not included in the formula unit. 48% carbon, 5. A) An ionic bond is much stronger than most covalent bonds. Write the formula for dinotrogen pentaoxide. 13 Although the whole fragmentation process can be very complex, there are only a few basic types of reactions that break or form chemical bonds: (1) σ-ionization, immediately breaks a bond (affecting mostly hydrocarbons); (2) α-cleavage. Aluminum bromide is an ionic compound that is formed from aluminum and iodine. Use the calculator then. C) A chemical bond is broken and energy is released, D) A chemical bond is broken and energy is absorbed. Use the prefixes, such as mono-, di-, tri-, and tetra-, to figure out the number of atoms present for each element, and write this number as a subscript for each element. Mathematicians describe periodicity in terms of periodic functions such as F (X) = F (X+nT), where X is a time or space coordinate, F (X) is a function describing something variable that repeats its value after every addition of the constant value T (the period) to the argument X, and n is an integer. Other atoms in the order they appear in the formula. If the formula used in calculating molar mass is the molecular formula, the formula weight computed is the molecular weight. THEORETICAL BACKGROUND. Elemental analysis of the compound gives the following percentages by mass: 40. Thus, there are 4 Cl atoms in the chemical formula. The electron pair density, in conjunction with the definition of an atom in a molecule, enables one to determine the average number of electron pairs that are localized to each atom and the number that are formed between any given pair of atoms. Empirical formulas show the simplest whole-number ratio of atoms in a compound, molecular formulas show the number of each type of atom in a molecule, and structural formulas show how the atoms in a molecule are bonded to each other. This work was essentially a compilation of the knowledge at the time. Since there are different degrees of ionization, there are different levels of weakness. because these compounds exist as separate, discrete molecules. Reactions between two nonmetals: the metal loses one or more pairs of valence electrons are. The coordination complex can be negatively charged, for example, [AuCl 4 ] − , [PtCl 6 ] 2− ,. Al and 0 38. Zinc oxide has two possible structures: hexagonal and cubic, but hexagonal crystals are most common. Which one is not a property of acids? a. (2) (Total 13 marks) 57. Mg 2+, Ca 2+, Ba 2+ Table shows how the non-metal atoms of Groups 15, 16 and 17 form negative ions with different charges. Types of bonds Chemical Bonding I: The Covalent Bond Pages 1 Colloidosomes formed by nonpolar/polar/nonpolar nanoball Growth behavior of ammonothermal GaN crystals grown on non-polar Ch 11 Quiz – Bonding, Electronegativity and Lewis Structures A. Hydrogen peroxide is the simplest peroxide (a compound with an oxygen–oxygen single bond). 3a: Draw the Lewis (electron dot) structures of PF3 and PF4+ and use the VSEPR theory to deduce. The formula for an oxide ion is O2-, which shows that it is an anion. B) A chemical bond is formed and energy is absorbed. 2 ) shows that the system becomes more stable (the energy of the system decreases) as two hydrogen atoms move toward each other from r = ∞, until the energy reaches a minimum at r = r 0 (the observed internuclear distance in H 2 is 74 pm). Journal of the American Chemical Society 2002 , 124 (14) , 3484-3485. For example, an atom at the corner of the unit cell counts as $\frac{1}{8}$ atom because it is shared between 8 adjacent unit cells. In diatomic nitrogen (N≡N), for instance, the bond order is 3 because there are 3 chemical bonds linking the two nitrogen atoms. A) Rbsz B) RbS C) Rb352 D) Rb 253 @szs 28) W ‘ h of the following statements is TRUE? Bf Once dissolved in water, ionic compounds rarely conduct electricity. (6) (iii) Predict and explain whether each species is polar. 5) Use Lewis theory to determine the chemical formula for the compound formed between Al and O. VSEPR theory means Valence Shell Electron Pair Repulsion theory. Journal of the American Chemical Society 2002 , 124 (14) , 3484-3485. Systematic names. Both compounds in the figure have two carbon atoms, six. Al will form a cation with a charge of 3+: Al 3+, an aluminum ion. Chemical also can be an adjective to describe properties of materials that are the result of various reactions between different compounds. When drawing a Lewis structure, we used a pair of valence electrons from each nucleus in the molecule to form a chemical bond. For each species, draw the Lewis structure, name the shape, and state the value of the bond angle(s). Molecular compounds can be in any physical state ‘“ solid, liquid, or gas. V3+ and CrO 4 c. 42% ammonium phosphate and 26. Carbon has 4 valence electrons. An oxide is a compound formed by the combination of oxygen and another element. a) Na and Br 2 b) Al and O 2. Use lewis theory to determine the chemical formula for the compound formed between Ba and N? View the step-by-step solution to: Question. We have already encountered some chemical formulas for simple ionic compounds. Solution The formula for this compound indicates it contains Al 3+ and SO 4 2− ions combined in a 2:3 ratio. The Calcium complex is the deposit that is responsible for cataracts and kidney stones. true or false: Lewis theory predicts that the formula of a compound formed between bromine and aluminum is: AlBr3: Lewis theory predicts that the formula for a compound between. Problem 1: In one molecule of the compound, determine how many atoms of every element are present for each one of these chemical formulas. The boron atom is. Build your AP® Chemistry toolkit by learning how to use moles to compare different units, interpret the periodic table, and examine mass and photoelectron spectra. Name: Sodium Symbol: Na Atomic Number: 11 Atomic Mass: 22. Draw the Lewis dot diagrams for barium and sulfur. , only This is a test of your knowledge of chemistry. The oxidation state and the coordination number reflect the number of bonds formed between the metal ion and the ligands in the complex ion. Atoms are written using their element symbols. Aluminium oxide (IUPAC name) or aluminum oxide (American English) is a chemical compound of aluminium and oxygen with the chemical formula Al 2 O 3. true or false: Lewis theory predicts that the formula of a compound formed between bromine and aluminum is: AlBr3: Lewis theory predicts that the formula for a compound between. A Lewis structure can be drawn for any covalently bonded molecule, as well as coordination compounds. Mole ratios are used as conversion factors between products and reactants in many chemistry problems. Magnesium has two valence electrons, and Nitrogen has five valence electrons. While, Sulfur belongs to group 6 and accepts 2 electrons to obtain noble gas configuration and forms S⁻². It is important to be able to predict and understand the molecular structure of a molecule because many of the properties of a substance are determined by its geometry. > The "Al-Cl" bond is polar covalent. In the Lewis model, the H + ion is the active species it accepts a pair of electrons from the OH-ion to form a covalent bond. The chemical formula for hydrogen chloride is HCl. To study a single displacement reaction with the help of iron nails and copper sulphate solution. Inorganic chemistry is the study of chemical elementselement, in chemistry, a substance that cannot be decomposed into simpler substances by chemical means. Visit Breslyn. Although covalent bonding gets most of the attention in Lewis theory, there does exist a convention for expressing ionic compounds using Lewis structures. With these molar amounts, the empirical formula for the compound may be written as described in the previous chapter of this text. Use Lewis theory to determine the chemical formula for the compound formed between Al and O. 7% nitrogen and 80. Chemical graph-based structure generation methods allow for exhaustive in silico enumeration of different compound types and different types of ‘chemical spaces’ beyond those used by biochemistry, which can be explored to help understand the types of compounds biology uses, as well as to understand the nature of abiotic synthesis, and potentially design novel types of living systems. Dalton's atomic theory States that the relative number and kinds of atoms are constant in a given compound that support the law of constant proportions. Draw the Lewis dot diagrams for barium and sulfur. 12) Use Lewis theory to determine the chemical formula for the compound formed between Al and O. • The purpose of using the formal charge formula is to determine which Lewis structure is the best answer. But according to the modern view a chemical bond is formed only when there is a net decrease of energy between the two approaching atoms due to attractive and repulsive forces. Use Lewis theory to determine the chemical formula for the compound formed between Al and O. Its chemical formula is H 2 O. The VSEPR theory assumes that each atom in a molecule will. In the above example a small 3 is written in the formula, AlCl 3. But we still need unique names that unambiguously identify each compound. Write the total number of electrons for each molecule in the upper right corner. of the compound is E3H8, calculate the atomic mass of E. Two arrangements of atoms are possible for a compound with a molar mass of about 45 g/mol that contains 52. Lewis Diagrams for Compound Formation. Chemistry 101 Class Notes Professor N. Draw the Lewis structure for the ester formed from the reaction of butyric acid with 2-propanol. CO Lewis Structure - How to Draw the Dot Structure for CO - YouTube. Q 1a: Draw the Lewis Structure for each of the following molecules Achieved Question In the past the Lewis structure is a “stand alone” question –which case there is an Achieved point given for the majority correct. Group 2 ions ( alkaline earth metals ) have +2 charges. It has a molecular mass of 60. Stable quadruple bonds are most common among the transition metals in the middle of the d-block, such as rhenium, tungsten, technetium, molybdenum and chromium. You will need the ability in chemistry to describe formulas for any substance you come across. This theory would be later called Lewis Theory and it is based on the following principles: Valence electrons, or the electrons in the outermost electron shell, have an essential role in chemical bonding. Draw the structural formula for each of these compounds and identify the compound which can be formed directly from. is a concise list of the elements in a compound and the ratios of these elements. Answer The resonance structures are: (a) SO3: (b) (c) : Question 4. Determine the elemental % (by mass) of a compound. The oxidation state and the coordination number reflect the number of bonds formed between the metal ion and the ligands in the complex ion. Simply write the formula in terms of the oxidation numbers for the elements. th nonmetals, the number of valence electrons in metals is generally c. The chemical formulas for covalent compounds are referred to as molecular formulas A chemical formula for a covalent compound. 7 on Pauling's scale. a chemical bond is formed between two atoms either by the transfer of electrons or by mutual sharing of electrons. Al213 O A1312 o A112 O All O Alla The bond angle in AsCl3 is O 120° O 107° O 180° O 104. Mathematicians describe periodicity in terms of periodic functions such as F (X) = F (X+nT), where X is a time or space coordinate, F (X) is a function describing something variable that repeats its value after every addition of the constant value T (the period) to the argument X, and n is an integer. Oxygen can form many different oxide compounds with both metals and nonmetals. The chemical formulas for covalent compounds are referred to as molecular formulas A chemical formula for a covalent compound. NO 2 is an intermediate in the industrial synthesis of nitric acid, millions of tons of which are produced each year for use primarily in the production of fertilizers. This study explores the integration of data journalism within three European legacy news organisations through the lens of organisational structure and professional culture. For example, Cocoa Butter, Chocolate Liquor, Sugar, Lecithin and a flavoring agent makes a delicious item called Chocolate. 85% nitrogen and 16. a) 03 b) se02 c) PH3 (each H atom is bonded to the P atom). Determine the empirical formula for a compound that is 36. Flash point 80°F. 2 to calculate the formal charge on each atom. The central atom is beryllium (draw the molecules Lewis structure to see this). Use Lewis theory to determine the formula for the compound that forms between the two elements listed: Al and N Mg and I Ca and S Al and F? Find answers now! No. Two Lewis structures can be written for sulfur dioxide. USE: Styrene is an important commercial chemical. 3% hydrogen by mass. Thus the compound between sodium. To link all the arms of Al three Cl's must be used. Lewis structure. Chemical bonding - Chemical bonding - Covalent bonds: When none of the elements in a compound is a metal, no atoms in the compound have an ionization energy low enough for electron loss to be likely. Carbon has 4 valence electrons. Each of the double bonds would be expected to show the characteristic behavior of an alkene and undergo addition reactions, but this is not how benzene reacts. P 2 I 4 + H 3 PO 4-> P 4 + H 2 O + PH 4 I. Use Lewis theory to determine the chemical formula for the compound formed between Mg and Br -MgBr. In this lesson, we investigate its chemical formula and some of its uses in the laboratory. Then, identify the anion and write down its symbol and charge. electrostatic balanced positions. Repeat the same process for the remaining two compounds. Al3+ & CO 3 2-! CHEMISTRY - TRO 4E CH. 6 °F) Number of Protons/Electrons: 29 Number of Neutrons: 35 Classification: Transition Metal Crystal Structure: Cubic Density @ 293 K: 8. com Mobile: 9999 249717 Head Office: 1/3-H-A-2. 0107 amu Melting Point: 3500. A chemical bond formed by the attraction between positive ions and surrounding mobile electrons is a(n) 66.
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https://wikimili.com/en/Throughput_%28business%29 | Last updated
Throughput is rate at which a product is moved through a production process and is consumed by the end-user, usually measured in the form of sales or use statistics. The goal of most organizations is to minimize the investment in inputs as well as operating expenses while increasing throughput of its production systems. Successful organizations which seek to gain market share strive to match throughput to the rate of market demand of its products. [1]
## Overview
In the business management theory of constraints, throughput is the rate at which a system achieves its goal. Oftentimes, this is monetary revenue and is in contrast to output, which is inventory that may be sold or stored in a warehouse. In this case, throughput is measured by revenue received (or not) at the point of sale—exactly the right time. Output that becomes part of the inventory in a warehouse may mislead investors or others about the organizations condition by inflating the apparent value of its assets. The theory of constraints and throughput accounting explicitly avoid that trap.
Throughput can be best described as the rate at which a system generates its products or services per unit of time. Businesses often measure their throughput using a mathematical equation known as Little's law, which is related to inventories and process time: time to fully process a single product.
## Basic formula
Using Little's Law, one can calculate throughput with the equation:
${\displaystyle I=R*T}$
where:
• I is the number of units contained within the system, inventory;
• T is the time it takes for all the inventory to go through the process, flow time;
• R is the rate at which the process is delivering throughput, flow rate or throughput.
If you solve for R, you will get:
${\displaystyle R=I/T}$ [2]
## Related Research Articles
Control theory deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a control model for controlling such systems using a control action in an optimum manner without delay or overshoot and ensuring control stability.
Income is the consumption and saving opportunity gained by an entity within a specified timeframe, which is generally expressed in monetary terms.
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences and engineering disciplines, as well as in non-physical systems such as the social sciences. Mathematical models are also used in music, linguistics and philosophy.
In general terms, throughput is the rate of production or the rate at which something is processed.
Thermal conduction is the transfer of internal energy by microscopic collisions of particles and movement of electrons within a body. The colliding particles, which include molecules, atoms and electrons, transfer disorganized microscopic kinetic and potential energy, jointly known as internal energy. Conduction takes place in all phases: solid, liquid, and gas. The rate at which energy is conducted as the heat between two bodies depends on the temperature difference between the two bodies and the properties of the conductive interface through which the heat is transferred.
Inventory or stock is the goods and materials that a business holds for the ultimate goal of resale.
The theory of constraints (TOC) is a management paradigm that views any manageable system as being limited in achieving more of its goals by a very small number of constraints. There is always at least one constraint, and TOC uses a focusing process to identify the constraint and restructure the rest of the organization around it. TOC adopts the common idiom "a chain is no stronger than its weakest link". This means that processes, organizations, etc., are vulnerable because the weakest person or part can always damage or break them or at least adversely affect the outcome.
James Edward Meade, was a British economist and winner of the 1977 Nobel Memorial Prize in Economic Sciences jointly with the Swedish economist Bertil Ohlin for their "pathbreaking contribution to the theory of international trade and international capital movements".
Economics, business, accounting, and related fields often distinguish between quantities that are stocks and those that are flows. These differ in their units of measurement. A stock is measured at one specific time, and represents a quantity existing at that point in time, which may have accumulated in the past. A flow variable is measured over an interval of time. Therefore, a flow would be measured per unit of time. Flow is roughly analogous to rate or speed in this sense.
Throughput accounting (TA) is a principle-based and simplified management accounting approach that provides managers with decision support information for enterprise profitability improvement. TA is relatively new in management accounting. It is an approach that identifies factors that limit an organization from reaching its goal, and then focuses on simple measures that drive behavior in key areas towards reaching organizational goals. TA was proposed by Eliyahu M. Goldratt as an alternative to traditional cost accounting. As such, Throughput Accounting is neither cost accounting nor costing because it is cash focused and does not allocate all costs to products and services sold or provided by an enterprise. Considering the laws of variation, only costs that vary totally with units of output e.g. raw materials, are allocated to products and services which are deducted from sales to determine Throughput. Throughput Accounting is a management accounting technique used as the performance measure in the Theory of Constraints (TOC). It is the business intelligence used for maximizing profits, however, unlike cost accounting that primarily focuses on 'cutting costs' and reducing expenses to make a profit, Throughput Accounting primarily focuses on generating more throughput. Conceptually, Throughput Accounting seeks to increase the speed or rate at which throughput is generated by products and services with respect to an organization's constraint, whether the constraint is internal or external to the organization. Throughput Accounting is the only management accounting methodology that considers constraints as factors limiting the performance of organizations.
Yield management is a variable pricing strategy, based on understanding, anticipating and influencing consumer behavior in order to maximize revenue or profits from a fixed, time-limited resource. As a specific, inventory-focused branch of revenue management, yield management involves strategic control of inventory to sell the right product to the right customer at the right time for the right price. This process can result in price discrimination, in which customers consuming identical goods or services are charged different prices. Yield management is a large revenue generator for several major industries; Robert Crandall, former Chairman and CEO of American Airlines, gave yield management its name and has called it "the single most important technical development in transportation management since we entered deregulation."
A mass balance, also called a material balance, is an application of conservation of mass to the analysis of physical systems. By accounting for material entering and leaving a system, mass flows can be identified which might have been unknown, or difficult to measure without this technique. The exact conservation law used in the analysis of the system depends on the context of the problem, but all revolve around mass conservation, i.e., that matter cannot disappear or be created spontaneously.
Scheduling is the process of arranging, controlling and optimizing work and workloads in a production process or manufacturing process. Scheduling is used to allocate plant and machinery resources, plan human resources, plan production processes and purchase materials.
Flux balance analysis (FBA) is a mathematical method for simulating metabolism in genome-scale reconstructions of metabolic networks. In comparison to traditional methods of modeling, FBA is less intensive in terms of the input data required for constructing the model. Simulations performed using FBA are computationally inexpensive and can calculate steady-state metabolic fluxes for large models in a few seconds on modern personal computers.
In accounting, the Inventory turnover is a measure of the number of times inventory is sold or used in a time period such as a year. It is calculated to see if a business has an excessive inventory in comparison to its sales level. The equation for inventory turnover equals the cost of goods sold divided by the average inventory. Inventory turnover is also known as inventory turns, merchandise turnover, stockturn, stock turns, turns, and stock turnover.
A signal-flow graph or signal-flowgraph (SFG), invented by Claude Shannon, but often called a Mason graph after Samuel Jefferson Mason who coined the term, is a specialized flow graph, a directed graph in which nodes represent system variables, and branches represent functional connections between pairs of nodes. Thus, signal-flow graph theory builds on that of directed graphs, which includes as well that of oriented graphs. This mathematical theory of digraphs exists, of course, quite apart from its applications.
In economics, factor payments are the income people receive for supplying the factors of production: land, labor, capital or entrepreneurship.
Industrial process data validation and reconciliation, or more briefly, data validation and reconciliation (DVR), is a technology that uses process information and mathematical methods in order to automatically ensure data validation and reconciliation by correcting measurements in industrial processes. The use of DVR allows for extracting accurate and reliable information about the state of industry processes from raw measurement data and produces a single consistent set of data representing the most likely process operation.
This glossary of economics is a list of definitions of terms and concepts used in economics, its sub-disciplines, and related fields.
Theory of constraints (TOC) is an engineering management technique used to evaluate a manageable procedure, identifying the largest constraint (bottleneck) and strategizing to reduce task time and maximise profit. It assists in determining what to change, when to change it, and how to cause the change. The theory was established by Dr. Eliyahu Goldratt through his 1984 bestselling novel The Goal. Since this time, TOC has continued to develop and evolve and is a primary management tool in the engineering industry. When Applying TOC, powerful tools are used to determine the constraint and reduce its effect on the procedure, including:
## References
1. "What is 'Throughput'". investopedia.com . Retrieved 15 November 2016.
2. "The Relationship Between Cycle Time and WIP". fabtime.com. Retrieved 15 November 2016.
• Goldratt, Eliyahu and Jeff Cox. The Goal. Croton-on-Hudson: North River Press, 2004. | 2021-01-27 00:45:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 2, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.34036803245544434, "perplexity": 1447.265526862768}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1610704804187.81/warc/CC-MAIN-20210126233034-20210127023034-00659.warc.gz"} |
https://www.semanticscholar.org/paper/Equivalent-elastica-knots-Brizard-Pfefferl'e/0e76a557175ef827f38c9f2e0cd29dc3328ff115 | • Corpus ID: 119318734
# Equivalent elastica knots
@article{Brizard2018EquivalentEK,
title={Equivalent elastica knots},
author={Alain J. Brizard and David Pfefferl'e},
journal={arXiv: Mathematical Physics},
year={2018}
}
• Published 12 July 2018
• Mathematics
• arXiv: Mathematical Physics
The problem of an elastica knot in three-dimensional space is solved explicitly by expressing the Frenet-Serret curvature and torsion of the knot in terms of the Weierstrass and Jacobi elliptic functions. This solution is obtained by variational methods and is derived by minimizing of the squared-curvature energy integral. In the present work, an equivalency is established between pairs of Jacobi elliptic solutions that are described by the same values for curvature and torsion functionals.
## References
SHOWING 1-6 OF 6 REFERENCES
These five lectures constitute a tutorial on the Euler elastica and the Kirchhoff elastic rod. We consider the classical variational problem in Euclidean space and its generalization to Riemannian
1 Theta Functions.- 2 Jacobi's Elliptic Functions.- 3 Elliptic Integrals.- 4 Geometrical Applications.- 5 Physical Applications.- 6 Weierstrass's Elliptic Function.- 7 Applications of the Weierstrass
A consistent notation for the Weierstrass elliptic function $\wp(z;g_{2},g_{3})$, for $g_{2} > 0$ and arbitrary values of $g_{3}$ and $\Delta \equiv g_{2}^{3} - 27 g_{3}^{2}$, is introduced based on
• Mathematics
NIST Handbook of Mathematical Functions
• 2010
• 20, 1
• 1984 | 2023-03-31 01:14:53 | {"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": 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": 0.5050331354141235, "perplexity": 2364.20116505209}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1679296949506.62/warc/CC-MAIN-20230330225648-20230331015648-00778.warc.gz"} |
http://tex.stackexchange.com/questions/52050/line-breaks-within-a-figure-environment-in-memoir | # Line breaks within a figure environment in memoir
I am trying to set up the following layout on a figure in a document typeset with the memoir class:
I have tried using \\ within the \subbottom environment, but this doesn't seem to work
\begin{figure}[!t]
\subbottom[]{
\includegraphics[width=0.45\linewidth]{img1.png}\\[0.2cm]
\includegraphics[width=0.45\linewidth]{img2.png}}
\hfill \subbottom[]{
\includegraphics[width=0.45\linewidth]{img3.png}\\[0.2cm]
\includegraphics[width=0.45\linewidth]{img4.png}}
\end{figure}
LaTeX seems to ignore my \\[0.2cm] commands and instead it tries to print images in the same row.
Is there a way to do this when using the float commands provided by the memoir class? (such as \subbottom).
-
You can use minipages inside \subbottom:
\documentclass{memoir}
\usepackage[demo]{graphicx}
\newsubfloat{figure}
\newsubfloat{table}
\begin{document}
\begin{figure}
\subbottom[]{%
\begin{minipage}{.45\textwidth}
\includegraphics[width=\linewidth]{img1.png}\\[0.2cm]
\includegraphics[width=\linewidth]{img2.png}
\end{minipage}
}%
\hfill \subbottom[]{%
\begin{minipage}{.45\textwidth}
\includegraphics[width=\linewidth]{img1.png}\\[0.2cm]
\includegraphics[width=\linewidth]{img2.png}
\end{minipage}
}%
\end{figure}
\end{document}
The demo option was only used to produce black rectangles instead of actual figures; do *not*a use that option in your actual document.
- | 2016-05-26 00:39:54 | {"extraction_info": {"found_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": 0.993528425693512, "perplexity": 2186.4390046914477}, "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-2016-22/segments/1464049275429.29/warc/CC-MAIN-20160524002115-00144-ip-10-185-217-139.ec2.internal.warc.gz"} |
https://physics.stackexchange.com/questions/417746/where-does-the-negative-sign-of-the-laplacian-in-the-four-divergence-go | # Where does the negative sign of the Laplacian in the four-divergence go?
As far as I'm aware from my course notes and what I've found online, the four vector gradient expands as \begin{align*} \partial_\mu x_\mu &= \left( \frac{1}{c} \frac{\partial}{\partial t}, -\frac{\partial}{\partial x^1}, -\frac{\partial}{\partial x^2}, -\frac{\partial}{\partial x^3} \right)\left( x_0, x_1, x_2, x_3 \right)\\ &= \left(\partial_0 x_0, -\vec{\nabla}\vec{x} \right) \end{align*} where $c=1$,
and the four vector divergence expands as
\begin{align*}\partial_\mu x^\mu &= \left( \frac{1}{c} \frac{\partial}{\partial t}, -\frac{\partial}{\partial x^1}, -\frac{\partial}{\partial x^2}, -\frac{\partial}{\partial x^3} \right) \cdot \left( x^0, x^1, x^2, x^3 \right)\\ &= \left( \frac{1}{c} \frac{\partial}{\partial t}, - \vec{\nabla} \right) \cdot \left( x^0, \vec{x} \right)\\ &= \partial_0 x^0 + \vec{\nabla} \cdot \vec{x} \end{align*}
In the latter case, where does the negative sign caused by the metric signature go?
• Comment to the post (v3): The raised and lowered positions of indices are applied inconsistently in various places. For starters, we should have $\partial_{\mu}= \frac{\partial}{\partial x^{\mu}}$ and $\partial^{\mu}= \frac{\partial}{\partial x_{\mu}}$. Jul 15 '18 at 11:26
• I just noticed the raised index of $x_0$ in the gradient expansion, but can't see any others. I've only recently learned this notation. Can you tell me which others are incorrect? Jul 15 '18 at 11:33
• @user3949312 look at the indices for the expression $\partial_\mu x^\mu$. Jul 15 '18 at 11:35
• These definition seem quite fishy. Have you computed these yourself or taken these from your notes? Jul 15 '18 at 11:44
• Also just FYI it’s not called a Laplacian operator either :) Jul 15 '18 at 12:02
## 2 Answers
The divergence of the four-vector defining position $x^\mu = (x^0, x^1,x^2,x^3)$ will give you the dimension of the spacetime and is given by the scalar product or $\partial_\mu$ with $x^\mu$. It is defined as
$$\partial_\mu x^\mu = \frac{\partial}{\partial x^0} x^0 + \left( \frac{\partial}{\partial x^i} x^i \right) = 4,$$
where we sum over repeated indices.
Or, for an arbitrary four-vector $a^\mu = (a^t, \mathbf{a})$ where $\mathbf{a}$ is the Cartesian spatial components is given by
$$\partial_\mu a^\mu = \frac{\partial}{\partial t} a^t + \mathbf{\nabla} \cdot\mathbf{a}.$$
Your confusion regarding the minus sign is that you will get one minus sign from the four-vector scalar product and the other comes from the spatial components of $\partial_\mu$. Hope that clears it up.
• Sorry, but that's not correct or at least bad notation, you are mixing up upper and lower indices. Jul 15 '18 at 12:07
• Thank you! I thought the minus signs in the scalar product and the spatial components were from the same process just being applied at different times. Can I ask why the metric signature affects the scalar product operation and the four-vector derivative components but not the components of the $x$ four-vector or the gradient operation? (Any entry-level resources would be great.) Jul 15 '18 at 12:08
• It got better but, I fixed some more index positions and signs. :) Jul 15 '18 at 12:31
• @Photon looks better after both contributions. Cheers. Jul 15 '18 at 12:33
• Actually, the original notation is completely fine in Cartesian coordinates Jul 15 '18 at 12:35
Your first expression is not well-defined. According to the Einstein summing convention, you sum over double indices, where one index is upper and the other one is lower. Such indices are called silent. Such an expression is then explicitly written as:
$$x^\mu y_\mu = x^0y_0 + x^1y_1 + x^2y_2 + x^3y_3 = x^0y^0 - x^1y^1 - x^2y^2 - x^3y^3$$
Note the sign change when passing from a lower to an upper index (works only in Minkowski metric).
Expressions like $x^\mu y^\mu$ or $x_\mu y_\mu$ are not well-defined, don't use them!
Indices which only appear once (on each side of an equation if an equation is considered) are called free indices. Example:
$$x^\mu y^\nu = ?$$
Since they are free, you need to specify some index values to evaluate the expression explicitly, say:
$$\left(x^\mu y^\nu\right)_{\mu=2\nu=3} = x^2y^3$$
You need to be careful when derivatives are involved. Derivatives are "naturally" defined with lower indices:
$$x^\mu = (x^0, \vec x),\qquad x_\mu = (x^0,-\vec x)$$
but
$$\partial_\mu = (\partial_0, \vec \nabla),\qquad \partial^\mu = (\partial^0,-\vec \nabla)$$
where $\vec x = (x^1, x^2, x^3)$, $\vec \nabla = (\partial_1, \partial_2,\partial_3)$ and the metric convention is that $x_0=x^0$ and $x_i=-x^i$.
Therefore,
$$x^\mu y_\mu = x^0 y^0 - \vec x\vec y$$
as above, and also
$$\square = \partial^\mu \partial_\mu = (\partial_0)^2 - {\vec \nabla}^2$$
but
$$\partial_\mu x^\mu = \partial_0 x^0 + \vec \nabla \vec x$$
• I'm confused as to why the last expression has $\partial_0 x^0$ as opposed to $\partial^0 x^0$, as the latter was how $\partial_\mu$ was defined earlier. Also, what is the difference between raised and lowered indices for the first element of a four-vector? Does the number that is that component change in any way? Jul 15 '18 at 12:21
• I found this information very helpful in general, though, so thank you. Jul 15 '18 at 12:22
• Sorry, a typo, I corrected it! But in the metric convention I chose, it holds $\partial^0=\partial_0$ and $\partial^i = -\partial_i$, so it was technically not wrong but very confusing. Jul 15 '18 at 12:28
• Found and fixed another typo (in the definition of $\square$). Jul 15 '18 at 12:33 | 2021-10-19 00:29:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.989899218082428, "perplexity": 616.0646060040423}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585215.14/warc/CC-MAIN-20211018221501-20211019011501-00046.warc.gz"} |
http://mathhelpforum.com/differential-equations/84819-laplace-equations-circular-regions-dirichlet-problems-annular-regions.html | # Thread: Laplace equations in circular regions: Dirichlet problems on annular regions
1. ## Laplace equations in circular regions: Dirichlet problems on annular regions
show that the steady state solution in the annular region is
u(r,theta)= a_0*((ln(r)-ln(R_1)/(ln R_1-ln(R_2))+ sigma(n=0..infinity)*[a_n*cos(n*theta)+b_n*sin(n*theta)]*(R_1/r)^n*((R_2^2n-r^2n)/(R_2^2-R_1^2n) )
, where u=0 at R1 and u=f_1(theta) at R2. and (R1 < r < R2) where a_0 , a_n, and b_n are the Fourier coefficients of f1(theta). [ Hint: Proceed as in the solution derived and used the condition R(R2)=0]
I think I need to the determined the coefficients , a_0, a_n and b_n. a_0 is theta indepedent. a_n and b_n are theta dependent.
Given: R(r)= c1+c2*ln(r/a), where c1 and c2 are constants
u_0(r,0)=a_0
u_n(r,theta)=(r/a)^n*(a_n*cos(n*theta)+b_n*sin(n*theta))
0=c1+c2*ln(R_2/a)
f1(theta)=c1+c2ln(R_2/a)
not sure how to proceed.
2. anyone not understand the problem? | 2017-06-24 20:05:29 | {"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": 0.8339506387710571, "perplexity": 4973.427593840018}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1498128320323.17/warc/CC-MAIN-20170624184733-20170624204733-00375.warc.gz"} |
https://deepai.org/publication/commutator-subgroups-of-sylow-2-subgroups-of-alternating-group-and-miller-moreno-groups-as-bases-of-new-key-exchange-protocol | # Commutator subgroups of Sylow 2-subgroups of alternating group and Miller-Moreno groups as bases of new Key Exchange Protocol
The goal of this investigation is effective method of key exchange which based on non-commutative group G. The results of Ko et al. <cit.> is improved and generalized. The size of a minimal generating set for the commutator subgroup of Sylow 2-subgroups of alternating group is found. The structure of the commutator subgroup of Sylow 2-subgroups of the alternating group A_2^k is investigated and used in key exchange protocol which based on non-commutative group. We consider non-commutative generalization of CDH problem <cit.> on base of metacyclic group of Miller-Moreno type (minimal non-abelian group). We show that conjugacy problem in this group is intractable. Effectivity of computation is provided due to using groups of residues by modulo n. The algorithm of generating (designing) common key in non-commutative group with 2 mutually commuting subgroups is constructed by us.
## Authors
• 1 publication
• 1 publication
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## 1 Introduction
In this paper new conjugacy key exchange scheme is proposed. This protocol based on conjugacy problem in non-commutative group [2, 3, 4, 5, 10]. We slightly generalize Ko Lee’s [6] protocol of key exchange. Public key cryptographic schemes based on the new systems are established. The conjugacy search problem in a group is the problem of recovering an from given and . This problem is in the core of several recently suggested public key exchange protocols. One of them is most notably due to Anshel, Anshel, and Goldfeld [2] and another due to Ko et al. [6]. As we know if CCP problem is tractable in then problem of finding by given , , for an arbitrary fixed such that is not from center of , is the common key that Alice and Bob have to generate.
Recently, a novel approach to public key encryption based on the algorithmic difficulty of solving the word and conjugacy problems for finitely presented groups has been proposed in [1, 2]. The method is based on having a canonical minimal length form for words in a given finitely presented group, which can be computed rather rapidly, and in which there is no corresponding fast solution for the conjugacy problem. A key example is the braid group.
We denote by the conjugated element
. We show that efficient algorithm that can distinguish between two probability distributions of
and does not exist. Also, an efficient algorithm which recovers from , and does not exist. This group has representation
G=⟨a,b|apm=e,bpn=e,b−1ab=a1+pm−1,m≥2,n≥1⟩.
As a generators can be chosen two arbitrary commuting elements [8, 10, 7].
Consider non-metacyclic group of Millera Moreno. This group has representation
G=⟨a,b∣∣|c|=p,|a|=pm,|a|=pn,m≥1,n≥1,b−1ab=ac,b−1cb=c⟩.
To find a length of orbit of action by conjugation by we consider the class of conjugacy of elements of form . This class has length because of action as well as increase the power of on 1. Thus, the first repetition of initial power in occurs though conjugations of this word by , where . Therefore, the length of the orbit is .
We need to have an effective algorithm for computation of conjugated elements, if we want to design a key exchange algorithm based on non-commutative DH problem [5]. Due to the relation in metacyclic group, which define the homomorphism to the automorphism group of the , we obtain a formula for finding a conjugated element. Using this formula, we can efficiently calculate the conjugated to element by using the raising to the -th power, where .
There is effective method of checking the equality of elements due to cyclic structure of group and in this group .
We have an effective method of checking the equality of elements in the additive group because of reducing by finite modulo .
## 2 Proof that conjugacy problem is NP-hard in G. Size of a conjugacy class
The orbit of the given base element must must be long enough if we want to have problem of DL or equally problem of conjugacy in non-commutative group like -hard problem.
Let elements of act by conjugation on , where .
###### Theorem 1.
The length of conjugacy class of non-central element is equal to .
###### Proof.
Recall the inner automorphism in is determined by the formula . Let us recall the structure of minimal non-abelian Metacyclic group, namely , where and are finite cyclic groups. Therefore, the formula defines a homomorphism in the subgroup of inner automorphisms . It is well-known that each finite cyclic group is isomorphic to the correspondent additive cyclic group modulo residue . In this group equality of elements can be checked effectively due to reducing the elements of the module group.
Consider the orbit of element under action by conjugation. The length of such orbit can be found from equality as minimal power for which this equality will be true. We apply Newton binomial formula to the expression and taking into account the relation . We obtain
1+C1spm−1+1+C2sp2(m−1)+⋯+ps(m−1)≡1(% mod pm)
only if with because if . It means that the minimal when this congruence start to holds is equal to . The prime number can be chosen as big as we need [17] which completes the proof. ∎
Let us evaluate the size of subsets with mutually commutative elements. Each of this subset of generated by them subgroups can be chosen as the subgroups of center of group . It is well-known that the semidirect product is closely related to wreath product. The center of the wreath product with non-faithful action were recently studied [11].
###### Proposition 1.
As it was proved by the author a center of the restricted wreath product with non-trivial coordinates is direct product of normal closure of center of diagonal of , i.e. , trivial an element, and intersection of with . In other words,
Z((A,X)≀B=⟨(1;h,h,…,hn),e(Z(A)∩Z(K,X))≀E⟩≃⟨Z(A)∩K)×Z(Δ(Bn)⟩
where .
Taking into consideration that a semidirect product is the partial case of wreath product the diagonal of degenerates in . Thus, we obtain such formula for the center of semidirect product:
Z((A,X)⋊B)=⟨Z(1;h),e,(Z(A)∩K,X)≀E⟩≃⟨Z(A)∩K)×Z(Δ(Bn)⟩.
This structure lead to constructive method of finding elements of the center. As it was noted above the elements and are parts of elements of secret key. Therefore as greater a size of center of a considered group as greater a size of a key space of this protocol.
Also commutator subgroup of Sylow 2-subgroup of alternating groups can be used as a support of CSP problem [12, 13, 15].
###### Definition 2.1.
For an arbitrary we call a -coordinate subgroup a subgroup, which is determined by -coordinate sets , , if this subgroup consists of all Kaloujnine’s tableaux for which .
We denote by a level subgroup of , which consists of the tuples of v.p. from , of any .
As a sets and consisting of mutually commutative elements we can use the set of elements of -coordinate subgroup of , where , or the elements of that is isomorphic to this subgroup. As it was proved by the author [12] the order of is . therefore the growth of mutually commutative sets of elements and is exponential function has.
According to [9] index of center of metacyclic group has index , therefore the order of . Thus, we have possibilities to choose an element as an element of the open key, which is in the protocol of key exchange.
## 3 Key exchange protocol
Let be subsets from consisting of mutually commutative elements. We make a generalisation of CDH by taking into consideration the subgroups and instead of using . We can do this because the groups and have generating sets and which commute. Because of these mutually commutative generating sets, we know that the subgroups are additionally mutually commutative.
## 4 Consideration of base steps of the protocol
Input: Elements , and .
Alice selects a private as the random element from the subgroup and computes . The she sends it to Bob. Bob selects a private as the random element from the subgroup and computes . Then he sends it to Alice. Bob computes and Alice computes . Taking into consideration that and are mutually commutative groups we obtain that . Therefore, we have that .
Output: that is the common key of Alice and Bob.
Thus, the common key [3, 6, 2, 1] was successfully generated.
Resistance to a cryptanalysis. But if an analytic use for a cryptanalysis will use for cryptoanalysys solving of conjugacy search problem the method of reduction to solving of decomposition problem [16], then it lead us to solving of discrete logarithm problem in the multiplicative cyclic group . This problem is NP-hard for big .
## 5 Conclusion
We can choose mutually commutative as subgroups of . As we said above, are chosen from as components of key. According to [8] so size of key-space is . It should be noted that the size of key-space can be chosen as arbitrary big number by choosing the parameters . As an element for exponenting we can choose an arbitrary element but , because the size of orbit in result of action of inner automorphism is always not less than .
## References
• [1] Iris Anshel, Michael Anshel, Benji Fisher, and Dorian Goldfeld. New key agreement protocols in braid group cryptography. In Cryptographers’ Track at the RSA Conference, pages 13–27. Springer, 2001.
• [2] Iris Anshel, Michael Anshel, and Dorian Goldfeld. An algebraic method for public-key cryptography. Mathematical Research Letters, 6(3):287–291, 1999.
• [3] Jens-Matthias Bohli, Benjamin Glas, and Rainer Steinwandt. Towards provably secure group key agreement building on group theory. Cryptology ePrint Archive, Report 2006/079, 2006.
• [4] Lize Gu, Licheng Wang, Kaoru Ota, Mianxiong Dong, Zhenfu Cao, and Yixian Yang. New public key cryptosystems based on non-abelian factorization problems. Security and Communication Networks, 6(7):912–922, 2013.
• [5] Lize Gu and Shihui Zheng. Conjugacy systems based on nonabelian factorization problems and their applications in cryptography. Journal of Applied Mathematics, 2014, 2014.
• [6] Ki Hyoung Ko, Sang Jin Lee, Jung Hee Cheon, Jae Woo Han, Ju-sung Kang, and Choonsik Park. New public-key cryptosystem using braid groups. In Mihir Bellare, editor, Advances in Cryptology — CRYPTO 2000, pages 166–183, Berlin, Heidelberg, 2000. Springer Berlin Heidelberg.
• [7] Ayoub Otmani, Jean-Pierre Tillich, and Léonard Dallot. Cryptanalysis of two mceliece cryptosystems based on quasi-cyclic codes. Mathematics in Computer Science, 3(2):129–140, 2010.
• [8] I Raievska, M Raievska, and Ya Sysak. Finite local nearrings with split metacyclic additive group. Algebra and discrete mathematics, 22(22, 1):129–152, 2016.
• [9] László Rédei. Das “schiefe produkt” in der gruppentheorie. Commentarii Mathematici Helvetici, 20(1):225–264, 1947.
• [10] Ruslan Viacheslavovich Skuratovskii. Employment of minimal generating sets and structure of sylow 2-subgroups alternating groups in block ciphers. In Advances in Computer Communication and Computational Sciences, pages 351–364. Springer, 2019.
• [11] Ruslan Viacheslavovich Skuratovskii and Aled Williams. Minimal generating set and a structure of the wreath product of groups, and the fundamental group of the orbit morse function. Bulletin of Donetsk National University. Series A: Natural Sciences, 0(1-2):76–96, 2019.
• [12] Skuratovskii R., Commutators subgroups of sylow subgroups of alternating and symmetric groups their minimal generating sets. The XII International Algebraic Conference in Ukraine (2019) Vinnytsia, p. 75.
• [13] Skuratovskii R.V., Generating set of wreath product non faithful action. International Journal of Analysis and Applications Volume 18, No. 1 (2020), pp. 104–116.
• [14] R. V. Skuratovskii, Structure of commutant and centralizer, minimal generating sets of. Sylow 2-subgroups of alternating and symmetric groups. International conference in Ukraine, ATA12. (2017). https://www.imath.kiev.ua/ topology/…/skuratovskiy.pdf
• [15] Skuratovskii R. V., Structure and minimal generating sets of Sylow 2-subgroups of alternating groups. Source: https://arxiv.org/abs/1702.05784v2
• [16] V Shpilrain, A. Ushakov The Conjugacy Search Problem in Public Key Cryptography: Unnecessary and Insufficient. Applicable Algebra in Engineering, Communication and Computing. (2006), volume 17, p. 285 - 289.
• [17] Ivan Matveevich Vinogradov. Elements of number theory. Courier Dover Publications, 2016. | 2022-05-16 16:12:45 | {"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": 0.8308537602424622, "perplexity": 1069.9498085767618}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662510138.6/warc/CC-MAIN-20220516140911-20220516170911-00485.warc.gz"} |
https://encyclopediaofmath.org/wiki/Locally_finite_covering | # Locally finite covering
A covering (cf. Covering (of a set)) of a topological space by subsets of it such that every point has a neighbourhood that intersects only finitely many elements of this covering. One cannot select a locally finite covering from every open covering of a straight line: it is sufficient to consider a monotone sequence of intervals that increase in length without limit. It turns out that the possibility of selecting a locally finite covering from any open covering of a space is equivalent to compactness of the space. The idea of local finiteness in conjunction with the concept of refinement carries an essentially new meaning. A.H. Stone's theorem asserts that any open covering of an arbitrary metric space can be refined to a locally finite covering. Hausdorff spaces that have the latter property are said to be paracompact (cf. Paracompact space). Locally finite coverings are important not only because of their participation in the definition of paracompactness. The requirement of local finiteness plays an essential role in constructions belonging to dimension theory and in the statements and proofs of addition theorems of various kinds. The existence in a regular space of a base that splits into a union of a countable family of locally finite open coverings is equivalent to the metrizability of this space. Open locally finite coverings of a normal space serve as a construction of a partition of unity on this space, subordinate to this covering. By means of partitions of unity it has been possible to construct, in particular, standard mappings of manifolds into Euclidean spaces. The requirement of local finiteness of a covering is not necessarily connected with the assumption that it is open. Local finiteness of a covering of a space automatically implies that in this covering there are "sufficiently many" sets that are close in their properties to open sets. If any open covering of a regular space can be refined to a locally finite covering, that space is paracompact. Locally finite families of sets in a space, defined similarly but not necessarily covering the space, have also been considered. A special case of them are discrete families of sets: families of sets such that each point in the whole space has a neighbourhood that intersects at most one element of this family. Discrete families are important in connection with the study of separation in a space. Thus, collectively-normal spaces are distinguished by the requirement that any discrete family of sets is separated by a discrete family of neighbourhoods. This condition is directly connected with the problem of the combinatorial extension of locally finite families of sets to locally finite families of open sets.
#### References
[1] R. Engelking, "General topology" , PWN (1977) [2] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian) [3] P. [P.S. Aleksandrov] Alexandroff, "Sur les ensembles de la première classe et les espaces abstraits" C.R. Acad. Sci. Paris , 178 (1924) pp. 185–187 [4] A.H. Stone, "Paracompactness and product spaces" Bull. Amer. Math. Soc. , 54 (1948) pp. 977–982 [5] E.A. Michael, "A note on paracompact spaces" Proc. Amer. Math. Soc. , 4 (1953) pp. 831–838
A partition of unity on a space $X$ is a family of continuous functions $\{ f _ {i} \} _ {i}$ from $X$ to $[ 0 , 1 ]$ such that $\sum _ {i} f _ {i} ( x) = 1$ for all $x \in X$. It is said to be subordinate to a covering $\mathfrak U$ if the open covering $\{ f _ {i} ^ { - 1 } [ ( 0 , 1 ] ] \} _ {i}$ refines $\mathfrak U$.
A (locally finite) family $\{ U _ {i} \} _ {i}$ of (open) sets is a combinatorial extension of a (locally finite) family $\{ F _ {i} \} _ {i}$ of sets if $F _ {i} \subseteq U _ {i}$ for all $i$ and for every set $I$ of indices $\cap _ {i \in I } F _ {i} = \emptyset$ implies $\cap _ {i \in I } U _ {i} = \emptyset$. | 2023-03-21 08:38:31 | {"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": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8292091488838196, "perplexity": 201.91280270193957}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296943637.3/warc/CC-MAIN-20230321064400-20230321094400-00639.warc.gz"} |
https://alan-turing-institute.github.io/rse-course/html/module09_programming_for_speed/09_03_cython.html | # Cython¶
Cython can be viewed as an extension of Python where variables and functions are annotated with extra information, in particular types. The resulting Cython source code will be compiled into optimized C or C++ code, and thereby yielding substantial speed-up of slow Python code. In other words, Cython provides a way of writing Python with comparable performance to that of C/C++.
## Start Coding in Cython¶
Cython code must, unlike Python, be compiled. This happens in the following stages:
• The cython code in .pyx file will be translated to a C file.
• The C file will be compiled by a C compiler into a shared library, which will be directly loaded into Python.
In a Jupyter notebook, everything is a lot easier. One needs only to load the Cython extension (%load_ext Cython) at the beginning and put %%cython mark in front of cells of Cython code. Cells with Cython mark will be treated as a .pyx code and consequently, compiled into C.
For details, please see Building Cython Code.
### Pure python Mandelbrot set:¶
xmin = -1.5
ymin = -1.0
xmax = 0.5
ymax = 1.0
resolution = 300
xstep = (xmax - xmin) / resolution
ystep = (ymax - ymin) / resolution
xs = [(xmin + (xmax - xmin) * i / resolution) for i in range(resolution)]
ys = [(ymin + (ymax - ymin) * i / resolution) for i in range(resolution)]
def mandel(position, limit=50):
value = position
while abs(value) < 2:
limit -= 1
value = value ** 2 + position
if limit < 0:
return 0
return limit
### Compiled by Cython:¶
%load_ext Cython
%%cython
def mandel_cython(position, limit=50):
value = position
while abs(value) < 2:
limit -= 1
value = value ** 2 + position
if limit < 0:
return 0
return limit
Let’s verify the result
data_python = [[mandel(complex(x, y)) for x in xs] for y in ys]
data_cython = [[mandel_cython(complex(x, y)) for x in xs] for y in ys]
from matplotlib import pyplot as plt
%matplotlib inline
f, axarr = plt.subplots(1, 2)
axarr[0].imshow(data_python, interpolation="none", extent=[xmin, xmax, ymin, ymax])
axarr[0].set_title("Pure Python")
axarr[1].imshow(data_cython, interpolation="none", extent=[xmin, xmax, ymin, ymax])
axarr[1].set_title("Cython")
Text(0.5, 1.0, 'Cython')
%timeit [[mandel(complex(x,y)) for x in xs] for y in ys] # pure python
%timeit [[mandel_cython(complex(x,y)) for x in xs] for y in ys] # cython
665 ms ± 30.2 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)
513 ms ± 5.77 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)
We have improved the performance of a factor of 1.5 by just using the Cython compiler, without changing the code!
### Cython with C Types¶
But we can do better by telling Cython what C data type we would use in the code. Note we’re not actually writing C, we’re writing Python with C types.
typed variable
%%cython
def var_typed_mandel_cython(position, limit=50):
cdef double complex value # typed variable
value = position
while abs(value) < 2:
limit -= 1
value = value**2 + position
if limit < 0:
return 0
return limit
typed function + typed variable
%%cython
cpdef call_typed_mandel_cython(double complex position, int limit=50): # typed function
cdef double complex value # typed variable
value = position
while abs(value)<2:
limit -= 1
value = value**2 + position
if limit < 0:
return 0
return limit
performance of one number:
# pure python
%timeit a = mandel(complex(0, 0))
14.7 µs ± 284 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)
# primitive cython
%timeit a = mandel_cython(complex(0, 0))
10.6 µs ± 263 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)
# cython with C type variable
%timeit a = var_typed_mandel_cython(complex(0, 0))
4.09 µs ± 121 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)
# cython with typed variable + function
%timeit a = call_typed_mandel_cython(complex(0, 0))
793 ns ± 11.2 ns per loop (mean ± std. dev. of 7 runs, 1,000,000 loops each)
### Cython with numpy ndarray¶
You can use NumPy from Cython exactly the same as in regular Python, but by doing so you are losing potentially high speedups because Cython has support for fast access to NumPy arrays.
import numpy as np
ymatrix, xmatrix = np.mgrid[ymin:ymax:ystep, xmin:xmax:xstep]
values = xmatrix + 1j * ymatrix
%%cython
import numpy as np
cimport numpy as np
cpdef numpy_cython_1(np.ndarray[double complex, ndim=2] position, int limit=50):
cdef np.ndarray[long,ndim=2] diverged_at
cdef double complex value
cdef int xlim
cdef int ylim
cdef double complex pos
cdef int steps
cdef int x, y
xlim = position.shape[1]
ylim = position.shape[0]
diverged_at = np.zeros([ylim, xlim], dtype=int)
for x in xrange(xlim):
for y in xrange(ylim):
steps = limit
value = position[y,x]
pos = position[y,x]
while abs(value) < 2 and steps >= 0:
steps -= 1
value = value**2 + pos
diverged_at[y,x] = steps
return diverged_at
Note the double import of numpy: the standard numpy module and a Cython-enabled version of numpy that ensures fast indexing of and other operations on arrays. Both import statements are necessary in code that uses numpy arrays. The new thing in the code above is declaration of arrays by np.ndarray.
%timeit data_cy = [[mandel(complex(x,y)) for x in xs] for y in ys] # pure python
666 ms ± 13.1 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)
%timeit data_cy = [[call_typed_mandel_cython(complex(x,y)) for x in xs] for y in ys] # typed cython
52.7 ms ± 337 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)
%timeit numpy_cython_1(values) # ndarray
33.1 ms ± 868 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)
#### A trick of using np.vectorize¶
numpy_cython_2 = np.vectorize(call_typed_mandel_cython)
%timeit numpy_cython_2(values) # vectorize
43.4 ms ± 269 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)
### Calling C functions from Cython¶
#### Example: compare sin() from Python and C library¶
%%cython
import math
cpdef py_sin():
cdef int x
cdef double y
for x in range(1e7):
y = math.sin(x)
%%cython
from libc.math cimport sin as csin # import from C library
cpdef c_sin():
cdef int x
cdef double y
for x in range(1e7):
y = csin(x)
%timeit [math.sin(i) for i in range(int(1e7))] # python
2.02 s ± 31.7 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)
%timeit py_sin() # cython call python library
1.21 s ± 16.3 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)
%timeit c_sin() # cython call C library
5.73 ms ± 126 µs per loop (mean ± std. dev. of 7 runs, 100 loops each) | 2022-08-13 19:08:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.21927008032798767, "perplexity": 7317.277497364566}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1659882571982.99/warc/CC-MAIN-20220813172349-20220813202349-00483.warc.gz"} |
https://tex.stackexchange.com/questions/274253/how-should-i-protect-this-command-for-use-in-csname | # How should I /protect this command for use in \csname?
In the process of learning how to use \csname, I'm trying to create a macro that allows one to specify integers as arguments and then will yield some content specific to the arguments, like this:
\DoContent{1}{3}
My current full-example attempt:
\documentclass[10pt]{article}
\usepackage{xifthen}
%End-Content output
\newcommand{\OneOne}{One One content}
\newcommand{\OneTwo}{One Two content}
\newcommand{\OneThree}{One Three content}
\newcommand{\OneFour}{One Four content}
\newcommand{\OneFive}{One Five content}
\newcommand{\TwoOne}{Two One content}
\newcommand{\TwoTwo}{Two Two content}
% and so on....
%Translator "Switch" statement
\newcommand{\GetNumberText}[1]
{
\ifthenelse{\equal{#1}{1}}{One}{}
\ifthenelse{\equal{#1}{2}}{Two}{}
\ifthenelse{\equal{#1}{3}}{Three}{}
\ifthenelse{\equal{#1}{4}}{Four}{}
\ifthenelse{\equal{#1}{5}}{Five}{}
}
%csname macro to assemble the content-containing macro
\newcommand{\AssembleNumCmd}[2]{\csname #1#2\endcsname}
%this does not compile: desired functionality
%\newcommand{\DoContent}[2]{\AssembleNumCmd{\GetNumberText{#1}}{\GetNumberText{#2}}}
%this compiles: literal example
\newcommand{\DoContent}[2]{\AssembleNumCmd{One}{Three}}
\begin{document}
\DoContent{1}{3}
\end{document}
This should produce
One Three content
It appears that I can't use any conditional statements inside of the definition of\GetNumberText and still pass \GetNumberText{1} to \AssembleNumCmd. The document doesn't compile if I use the version of \DoContent that I have commented there (which is what I want to do).
Obviously, this is some somewhat silly gymnastics, and I'm open to better overall approaches. But, I still need want to know why this relatively simple method isn't working, and what to do to fix it.
I've tried using \protect here and there, and using a different "switch" macro that uses \ifx instead of \ifthenelse, among other smaller changes, and nothing seems to compile without error (" Missing \endcsname inserted", and so forth).
Is this a problem with my use of \csname? Or am I just being naive about macro expansion and fragility?
• You are trying to use \ifthenelse inside a \csname. This won't work, \ifthenelse is not expandable. Oct 21, 2015 at 18:11
• As a side remark, be careful with all the extra spaces in your \GetNumberText. There should be a bunch of %'s. Beware that in LaTeX3 code à la @egreg, \ExplSyntaxOn makes spaces in the source completely disappear (they must be input explicitely as ~). Even with an expandable switch which would work in a \csname.. you have to be careful with spaces, as the so-built name may well end up containing such probably unintended spaces.
– user4686
Oct 21, 2015 at 20:05
Here is a expl3 version, storing One, .... , Ten in a \clist variable:
\documentclass[10pt]{article}
\usepackage{xparse}
\newcommand{\OneOne}{One One content}
\newcommand{\OneTwo}{One Two content}
\newcommand{\OneThree}{One Three content}
\newcommand{\OneFour}{One Four content}
\newcommand{\OneFive}{One Five content}
\newcommand{\TwoOne}{Two One content}
\newcommand{\TwoTwo}{Two Two content}
% and so on....
%Translator "Switch" statement
\ExplSyntaxOn
\clist_new:N \g_turanc_numbername_clist
\clist_set:Nn \g_turanc_numbername_clist {One, Two, Three, Four, Five, Six, Seven, Eight, Nine, Ten}
\NewDocumentCommand{\GetNumberText}{m}{%
\clist_item:Nn \g_turanc_numbername_clist {#1}
}
\ExplSyntaxOff
%csname macro to assemble the content-containing macro
\newcommand{\AssembleNumCmd}[2]{\csname #1#2\endcsname}
\newcommand{\DoContent}[2]{\AssembleNumCmd{\GetNumberText{#1}}{\GetNumberText{#2}}}
\begin{document}
\DoContent{1}{3}
\DoContent{2}{2}
\DoContent{1}{5}
\end{document}
Old Version
I replaced the \ifthenelse conditional by a 'simpler' \ifcase ...\or ...\fi statement for the individual cases. I think it might work with other packages as well.
\documentclass[10pt]{article}
\usepackage{xifthen}
%End-Content output
\newcommand{\OneOne}{One One content}
\newcommand{\OneTwo}{One Two content}
\newcommand{\OneThree}{One Three content}
\newcommand{\OneFour}{One Four content}
\newcommand{\OneFive}{One Five content}
\newcommand{\TwoOne}{Two One content}
\newcommand{\TwoTwo}{Two Two content}
% and so on....
%Translator "Switch" statement
\newcommand{\GetNumberText}[1]
{%
\ifcase #1
\or
One%
\or
Two%
\or
Three%
\or
Four%
\or
Five%
\fi
}
%csname macro to assemble the content-containing macro
\newcommand{\AssembleNumCmd}[2]{\csname #1#2\endcsname}
%This does not compile: desired functionality
\newcommand{\DoContent}[2]{\AssembleNumCmd{\GetNumberText{#1}}{\GetNumberText{#2}}}
%this compiles: literal example
%\newcommand{\DoContent}[2]{\AssembleNumCmd{One}{Three}}
\begin{document}
\DoContent{1}{3}
\DoContent{2}{2}
\DoContent{1}{5}
\end{document}
**\ifx version:
There mustn't be white space characters so be careful when testing etc:
\documentclass[10pt]{article}
%End-Content output
\newcommand{\OneOne}{One One content}
\newcommand{\OneTwo}{One Two content}
\newcommand{\OneThree}{One Three content}
\newcommand{\OneFour}{One Four content}
\newcommand{\OneFive}{One Five content}
\newcommand{\TwoOne}{Two One content}
\newcommand{\TwoTwo}{Two Two content}
% and so on....
%Translator "Switch" statement
\newcommand{\GetNumberText}[1]
{%
\ifx#11%
One%
\else
\ifx#12%
Two%
\else
\ifx#13%
Three%
\else
\ifx#14%
Four%
\else
\ifx#15%
Five%
\fi
\fi
\fi
\fi
\fi
}
%csname macro to assemble the content-containing macro
\newcommand{\AssembleNumCmd}[2]{%
\csname #1#2\endcsname%
}
\newcommand{\DoContent}[2]{\AssembleNumCmd{\GetNumberText{#1}}{\GetNumberText{#2}}}
\begin{document}
\DoContent{1}{3}
\DoContent{2}{2}
\DoContent{1}{2}
\DoContent{1}{5}
\end{document}
• I would never use ifcase or long if-else-test for this sort of things. Difficult to extend and debug and maintain. Oct 21, 2015 at 18:21
• @UlrikeFischer: Yes, it's not nice, but it works ;-)
– user31729
Oct 21, 2015 at 18:27
• Thanks for this. I wasn't even aware of \ifcase. But why does this work and not a set of \ifx statements? I tried the latter and it had the same problem as using \ifthenelse. Wouldn't \ifx be just as primitive as \ifcase? Oct 21, 2015 at 18:48
• @turanc: It should work using \ifx but I don't know how you did your code actually
– user31729
Oct 21, 2015 at 18:52
• @turanc: See also my update with the expl3 version
– user31729
Oct 21, 2015 at 18:56
As I already wrote in the comment, using \ifthenelse in \csname won't work.
Beside this you are overcomplicating things a lot. You already know that command names can contains other things then letters, so then why don't do use this knowledge for the content definitions?
\documentclass[10pt]{article}
\makeatletter
\@namedef{text1-1}{One One content}
\@namedef{test1-2}{One Two content}
\makeatother
\newcommand{\DoContent}[2]{\csname test#1-#2\endcsname}
\begin{document}
\DoContent{1}{2}
\end{document}
• Why not \@nameuse{test#1-#2} then instead of \csname ... \endcsname?
– user31729
Oct 21, 2015 at 18:33
• @ChristianHupfer: tradition ... Oct 21, 2015 at 18:39
• Aha ... well, I improved my solution to be more maintainable and easier to debug ;-)
– user31729
Oct 21, 2015 at 18:44
• Thanks for this response. I didn't know about \@namedef. I marked the next response as answered, though, since it was closer to answering the original question. Gave this an upvote, however. Oct 21, 2015 at 23:07
• \@namedef is only a wrapper for \expandafter\def\csname ...\endcsname{ ...} . And I didn't follow your route with the string comparision as I think it is inefficient: it is quite unnecessary to compare the actual string with all lot of other strings all the time only to get its content text. Oct 22, 2015 at 7:13
As Ulrike pointed out, \ifthenelse cannot be used in \csname...\endcsname, where only macros fully expandable to characters are allowed.
Here's a different implementation using a case switch:
\documentclass{article}
\usepackage{xparse}
\ExplSyntaxOn
\NewDocumentCommand{\DoContent}{mm}
{
\str_case:nnF { #1-#2 }
{
{1-1}{One~One~content}
{1-2}{One~Two~content}
{1-3}{One~Three~content}
{1-4}{One~Four~content}
{1-5}{One~Five~content}
{2-1}{Two~One~content}
{2-2}{Two~Two~content}
}
{Invalid~choice}
}
\ExplSyntaxOff
\begin{document}
\DoContent{1}{3}
\end{document}
A possibly better interface for adding cases:
\documentclass{article}
\usepackage{xparse}
\ExplSyntaxOn
\NewDocumentCommand{\DoContent}{mm}
{
\str_case:nVF { #1-#2 } \g_turanc_choices_tl
{Invalid ~ choice}
}
{
\clist_map_inline:nn { #1 }
{
\tl_gput_left:Nn \g_turanc_choices_tl { ##1 }
}
}
\tl_new:N \g_turanc_choices_tl
\ExplSyntaxOff
{1-1}{One One content},
{1-2}{One Two content},
{1-3}{One Three content},
{1-4}{One Four content},
{1-5}{One Five content},
}
{2-1}{Two One content},
}
{2-2}{Two Two content},
}
\begin{document}
\DoContent{1}{3}
\DoContent{2}{2}
\DoContent{9}{9}
\end{document}
Note that \AddContent is cumulative, so pairs can be added even after \begin{document}, wherever they become necessary. If you add a pair twice, the one in the last \AddContent wins, so you can later override a choice made at startup. | 2022-08-15 09:56:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.6191900968551636, "perplexity": 6238.305418956743}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1659882572163.61/warc/CC-MAIN-20220815085006-20220815115006-00643.warc.gz"} |
https://www.mersenneforum.org/showthread.php?s=55e22de04385f916d8b6d1d14ab369e4&t=1589&page=3 | mersenneforum.org Have Found Principle to generate infinitive PRIME NUMBERS
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2003-12-07, 17:19 #23 Evgeny Dolgov 22·1,667 Posts Hi! i'm continue using my new method that results i posted yerstaday and have obtained easily next prime numbers count of digits in prime numbers (using my algorithm) (i have tested and all success) (http://www.alpertron.com.ar/ECM.HTM i used this site to test next my prime numbers) # of digits 7 11 15 17 25 35 91 127 203 355 629 647 685 all of these numbers are primes if you want i can send you this numbers like examples So i have found easy method to generate infinitive prime numbers using my very simple rule. All prime numbers i have tested using my notebook in one evening . it is most hard - to input numbers in field of test form i 'll want to use some automatic cases or tool to input more huge numbers i think i'll post results at monday of prime numbers with in 10 000 digits using only my notebook and my simple rule. I know how looks PRIME NUMBER with 10 000 000 digits or more digits today at this moment! but i have only notebook i need more cases or computers to verify primility of generated prime numbers (i fully tested for primility last prime numbers with 685 digits about 4 hours ) but if the number is composite - it takes 2 seconds!!!! i can predict very very huge prime number - with any count of digits (1 000 000 000 and more) now i'm testing properties So any help or advices will be good if you can help with automatic cases that help to input very large numbers http://www.alpertron.com.ar/ECM.HTM - is the best tool that i'm using today Evgeny Dolgov This is previous 647 digit PRIME number you can test it to proof 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003 tnanks for reply
2003-12-07, 17:22 #24 Evgeny Dolgov 33·59 Posts corrected post with sample 647 digit prime number you cant test it to proof 100000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000 000000100000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000 0000000000003 Evgeny Dolgov
2003-12-07, 18:12 #25
smh
"Sander"
Oct 2002
52.345322,5.52471
4A516 Posts
Quote:
(i fully tested for primility last prime numbers with 685 digits about 4 hours ) but if the number is composite - it takes 2 seconds!!!!
This is only for 685 digit numbers
How long do you think it will take to just test a 1 million digit?
And what about a primality test for a number of only a couple of thousand digits?
If you need faster tools, try primo for primality test and winpfgw for prp tests.
2003-12-07, 18:30 #26 Raptor Dec 2003 28 Posts @Evgeny Dolgov Even if your prime generator produces hundreds or thousands of primes in a lower digit range (where you can brute-force attack the numbers) this does not prove that the next number the generator ejects is also a prime. What is needed is a _mathematical_ proof, which can only come from the algorithm you use. A number with 10 Million digits is far beyond the range where you can proof its primalty with any known algorithm. Numbers in this range can only be attacked if they have a special form - like 2^p-1 . So again: A proof can only come from your algorithm. So my suggestion to you: just publish your algorithm in the math section of this forum and become famous - or learn.
2003-12-07, 21:28 #27 flava Feb 2003 2×59 Posts If you want to test numbers of the form 10^(2*n)+10^n+3, you can use pfgw.exe for probable prime testing. If you use the following input file: ABC2 10^(2*$a)+10^$a+3 a: from 1 to 2000 b: from 0 to 0 you get the following probable primes: 10^(2*1)+10^(1)+3 10^(2*2)+10^(2)+3 10^(2*3)+10^(3)+3 10^(2*5)+10^(5)+3 10^(2*7)+10^(7)+3 10^(2*8)+10^(8)+3 10^(2*12)+10^(12)+3 10^(2*17)+10^(17)+3 10^(2*45)+10^(45)+3 10^(2*63)+10^(63)+3 10^(2*101)+10^(101)+3 10^(2*177)+10^(177)+3 10^(2*314)+10^(314)+3 10^(2*323)+10^(323)+3 10^(2*342)+10^(342)+3 10^(2*367)+10^(367)+3 10^(2*792)+10^(792)+3 10^(2*894)+10^(894)+3 10^(2*1475)+10^(1475)+3 10^(2*1913)+10^(1913)+3 As you can see, the distribution is quite normal and they are more and more distanced. It is very probable that there are in fact very few primes of this forms that have more than a couple of thousend digits. The bad news is the numbers of this form are very hard to proove prime (when they get big).
2003-12-08, 08:19 #28 Evgeny Dolgov 24×5×41 Posts Hi all! why i'm choosing this form of numbers 10^(n*2)+10^(n)+3 1. simple view 2. symmetric and asymmetric in one time well balanced 3. Predictable step (distribution) 4. Using this tool http://www.alpertron.com.ar/ECM.HTM i have found that if the test number is compiste (not prime) it takes little time to fail this number if the number is fail (not prime) it breaks very fast on parts but if this tool show is unknown and it takes more than 2-3 seconds on numbers with in 685 digits - i know 100% that number is prime!!! and tests show this. (i fully tested) now i'm begining to test numbers with 9555-10000 digits now fail test takes about 1-2 minute - in most hard but in simple - 5 seconds for fail - but if test takes nore than 4-5 (10) minutes with 100% guarantee i say that this number is PRIME (before all this my try was the numbers like 11119 1113 10003 ... - it's wrong numbers they are not productive like that 10....010....03 form or in exponenta 10^(2*n)+10^(n)+3 ) SO MAIN FEATURE - IF TEST NUMBERS in (10....010....03 form or in exponenta 10^(2*n)+10^(n)+3 ) ARE NOT PRIME THEY FAIL VERY FAST!!!!! so the effective search: use test algorithm like in tool http://www.alpertron.com.ar/ECM.HTM and comparing relatively time of success test and time of fail test example if test number - "685 digits" - and test is not failing after 10 seconds the number is PRIME. (100%) (main feature) Evgeny Dolgov You can test this 100000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000 000000100000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000 0000000000003 this numbers is prime - the tool (http://www.alpertron.com.ar/ECM.HTM) is showing "is unknown" - 2 hours and then - is PRIME!!! if you add or remove one digit "0" at the end of this number it fails very fast - not more than 10 seconds on pentium 4 computer. So repeat if test (10^(2*n)+10^(n)+3 form number) using algorithm (http://www.alpertron.com.ar/ECM.HTM) is not failing after relatively short time the number is PRIME. (100%) (main feature) Evgeny Dolgov
2003-12-08, 22:43 #29
ET_
Banned
"Luigi"
Aug 2002
Team Italia
12CE16 Posts
Quote:
100000000000000000000000000000000000000000000000
The number is certified prime with Primo 2.2.0 beta in 15 minutes on a Athlon XP 2100+
But it's only 685 digits...
Luigi
2003-12-08, 22:45 #30 BMgf Dec 2003 24 Posts High level math Let just say that you found some 10 000 000+ digit number for witch you can't find any factor for more than 10 days. What's you next step? Something like GIMPS? Last fiddled with by BMgf on 2003-12-08 at 22:48
2003-12-08, 22:58 #31 mark hemmeyer 24×7×89 Posts does it help if n is prime?
2003-12-09, 08:39 #32
BMgf
Dec 2003
208 Posts
Quote:
Originally posted by mark hemmeyer does it help if n is prime?
If 685 digits number is prime then n is even. 10^684 + 10^342 + 3. N=342
2003-12-09, 16:13 #33
ET_
Banned
"Luigi"
Aug 2002
Team Italia
2×29×83 Posts
Quote:
If 685 digits number is prime then n is even. 10^684 + 10^342 + 3. N=342
Hey, would you mind explaining this one?
Luigi
Similar Threads Thread Thread Starter Forum Replies Last Post ONeil ONeil 27 2018-12-03 01:52 Philly314 Aliquot Sequences 3 2014-11-16 14:58 James Heinrich Miscellaneous Math 10 2012-03-08 07:20 Xyzzy Miscellaneous Math 41 2008-11-08 17:57 Evgeny Dolgov Math 1 2003-12-08 09:25
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http://www.gamedev.net/page/resources/_/technical/general-programming/using-pdl-for-code-design-and-documentation-r1384 | • Create Account
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# Using PDL for Code Design and Documentation
By Drew Sikora | Published Jun 13 2001 10:05 AM in General Programming
pdl code language article read comment english time
If you find this article contains errors or problems rendering it unreadable (missing images or files, mangled code, improper text formatting, etc) please contact the editor so corrections can be made. Thank you for helping us improve this resource
Introduction
After much procrastination, I finally got around to purchasing Code Complete at my local Barnes & Noble (gotta love that store...). Of course, in my usual habit of digesting about 4-5 computer books at once, I've only read about 125 pages. Anyway I was cruising through Section 4 on building routines when I came across something called PDL. Reading further I was amazed that someone had actually taken this technique and named it for professional use. We'll get to what PDL is in a moment, but first a little background.
Now that we are all ready, let's begin!
What is PDL?
What is PDL? Well that's a reasonable enough question. I sure as hell had no idea what PDL was when I first started reading about it. In fact, the meaning of its name can be quite misleading. PDL stands for Program Design Language. Now you may notice how short this article is if you glance to the menu on the right. Obviously you can assume that PDL really isn't a full-fledged language with it's own functions and variables and such - no way could I teach it in such a simple lesson. There is a reference you can make to PDL - it is a common coding technique where you write out code in plain English, known as pseudocode, or fake code (literally). That's exactly what PDL is.
When we talk about high-level languages, we talk about languages where the instructions you type represent English (as an example) more and more. Believe it or not, but QBasic is a higher-level language than C++! In case you don't understand this, I'll sidetrack just a bit to make sure everyone is clear. When we reference languages in terms of levels, the reference is the English language. The closer the code comes to looking like English, the "higher" up it is. If we were to compare ways to print a word on the screen in C++ and QBasic, you'll realize why QBasic is a higher language than C++:
C++ - cout << "hello world" << endl;
QBasic - PRINT "hello world"
You tell me which is the higher level language. Obviously PRINT is much, much closer to the English word (okay, so it's exact) than cout is. To be official, the ratio of high-level-language statements to equivilent assembly code in QBasic is 1:5, while in C++ it's only 1:2.5
Okay, enough delineating - back to the main topic. So as I was saying, PDL is the highest-level language you will ever see until they conceive computers that can interpret human speech as code. When you code in PDL, you are coding in plain English (or whatever language you use) and not one single bit of computer code. In fact, this ambiguity let's you code PDL and then use it to construct a routine in any computer language since you never used a peice of language specific code. Wonderful! But how the hell is all this supposed to help you? Nothing could ever run it! Be patient, I haven't instructed you on how to use it yet, have I?
Using PDL Correctly
Like anything else in this freakish world, PDL can be easily abused. How sad. But to avoid this tragedy I'll cue you in on what to do and what not to do. Let's kick off this section by dumping you with an example of using PDL, then I'll break it down for you to understand, pointing out do's and dont's along the way. Shall we?
Here's a little exercise - look at examples A and B and see if you can discern which PDL code is correct and which is not.
Example A
Lock the drawing surface to prevent access by other programs and check for errors.
Get mem_pitch of the surface using (int)ddsd.lPitch.
Get *video_buffer to draw to using ddsd.lpSurface.
if pixel_mode == random then
Create three random values for x, y and color.
else if pixel_mode == linear then
Increment x value by 1.
Create a random value for color.
end if
video_buffer[x+y*mempitch] = color
Unlock the surface and check for errors.
return 1
Example B
Prevent the drawing surface from being accessed by other programs while we draw.
Retrieve the memory pitch of the surface memory so we do not draw out of bounds.
Retrieve the location of the surface so that we can draw to it successfully.
If the pixel mode is set to random
Create three random values that we can assign to the x, y and color attributes of the pixel.
Else if the pixel mode is set to linear
Update the horizontal position of the pixel.
Create a random value for the color of the pixel.
End If
Plot the pixel on the screen.
Release the surface for general use once again.
Return TRUE that the pixel was plotted.
Obviously the above two code examples document the plotting of a pixel on the screen. I chose this for an example because it is a basic task a lot of you should be familiar with. Now, were you able to figure out which was the bad PDL and which was the good? If not go back and read the end of the first section again, where I clue you in on a few of things that you should not find in PDL. Even still, I think I made the distinction rather obvious.
The answer is Example B. Surprised? I hope not. Example A could not possibly fit my current description of PDL in any way whatsoever. I specifically pointed out how PDL is considered a high-level language, and some of the lines in Example A are anything but. Here are some of the things that make Example A a PDL nightmare:
• Although you may not realize it, the term "Lock a surface" can easily be attributed to DirectX and its Lock() function. The same applies to the term Unlock. Since we want our PDL to be as ambiguous as possible, Example B cleans this up by using the terms prevent and release. Anyone who wants to argue that release is also DirectX specific will break their noses running into a brick wall - in this context, release does not describe the freeing of COM components.
• Direct reference to variable names in Example A is also a no-no. And following close behind them are direct code examples! Ack! Once again ambiguity must reign supreme, and there's no way you can use that code in Visual Basic or any other language.
• Also present is the reference to a language-specific feature - the C pointer. This immediately kills any sense of ambiguity since all other language options for writing this routine fall out.
• Then we have another direct code example for plotting the pixel, again unacceptable.
• Finally we have the ending "return 0". Another language-limiting statement, and one that explains absolutely nothing of its true purpose.
After going through above points, and comparing them to Example B, you'll fully understand what PDL should look like. If you really think about it, you could code the pixel-plotting routine in any language capable of handling the task. Wonderful! Notice how the routine is nothing but English - nor variable names, no comparison signs, no code whatsoever. Of course, this brings us to a final point about good PDL use.
It's very easy to go overboard when writing in PDL. You may get a bit too descriptive and realize that coding what you wrote may be a nightmare! Therefore writing PDL can be considered an iterative process. Write it once, then start to break it down as far as you can go without quite reaching code level and while maintaining the basic rules of PDL.
PDL Uses
Part 1: Routine Review
Now that we know how to construct PDL routines correctly, we can learn how best to use them. One of their uses is routine review. Project managers and lead programmers don't like to waste time going over endless lines of code to learn whether or not the routine does what they want it to, or what it's spec'd out to do. Consider how much easier it would be for a lead to be able to read through PDL and decide whether the programmer is on the right track. Of course this won't catch programming flaws, but programming flaws can be avoiding by finding design flaws - so it's all good.
On top of that is the ease of which you can modify PDL - anybody can do it! People who don't know jack about coding but are seasoned designers can easily change the wordings of a routine to better match specification. This is the cheapest and most effective way to change a program architecture. Think of how hard it would be to have to rip out a few lines of code that could easily constitute only one line of PDL. Saving money is always a good thing, and PDL can help you do that.
Part 2: Code Documentation
Ahh, here we go - the reason I decided to write this whole article in the first place. I think another acceptable translation of PDL would be Program Documenting Language. Why? Easy - check out what I did with Example B.
// Prevent the drawing surface from being accessed by other programs while we draw.
lpdds7->Lock(NULL, &ddsd, DDLFLAGS, NULL);
// Retrieve the memory pitch of the surface memory so we do not draw out of bounds.
int mempitch = (int)ddsd.lPitch;
// Retrieve the location of the surface so that we can draw to it successfully.
UCHAR *video_buffer = (UCHAR *)ddsd.lpSurface;
// If the pixel mode is set to random
if (pixel_mode == RANDOM) {
// Create three random values that we can assign to the x, y and color attributes of the pixel.
UCHAR color = rand()%256;
int x = rand()%800;
int y = rand()%600;
}
//Else if the pixel mode is set to linear
else if (pixel_mode == LINEAR) {
// Update the horizontal position of the pixel.
pixel_x++;
// Create a random value for the color of the pixel.
UCHAR color = rand()%256;
} // End If
// Plot the pixel on the screen.
video_buffer[x+y*mempitch] = color;
// Release the surface for general use once again.
lpdds7->Unlock(NULL);
// Return TRUE that the pixel was plotted.
return 1;
Check that out. In all my past experiences I've always coded first, commented second. Who'd ever thought the other way around was better? Obviously somebody did. What an easy way to code and comment at the same time. For all you people against taking the time to comment you code (you'd be surprised how many people actually hate to comment, check out this thread for proof), this is the excuse that you need. You don't even have to think of it as commenting, just write out the routine in PDL to get the architecture and implementation straight, then just code it and don't delete the PDL, because it turns right into comments!
Conclusion
Well that's about it. PDL, as you can see, is a tool than belongs in any designer's and coder's toolbox. I'm sure glad I stumbled upon it, because it's streamlined my coding a lot. Writing out a whole routine in PDL and then just adding the code after perfecting it has saved me countless hours. I hope I did a good job explaining to you the correct way to use PDL to save you time and effort. Any questions and/or comments can be directed to drew@gamedev.net - I'd love to hear from you. Until next time... | 2015-10-07 17:28:46 | {"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": 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": 0.3305293023586273, "perplexity": 1391.2875177446647}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443737875203.46/warc/CC-MAIN-20151001221755-00048-ip-10-137-6-227.ec2.internal.warc.gz"} |
https://codereview.stackexchange.com/questions/61338/generate-random-numbers-without-repetitions/61372 | # Generate random numbers without repetitions
I want to generate a list of N different random numbers:
public static List<int> GetRandomNumbers(int count)
{
List<int> randomNumbers = new List<int>();
for (int i=0; i<count; i++)
{
int number;
do number = random.Next();
while (randomNumbers.Contains(number));
}
return randomNumbers;
}
But I feel there is a better way. This do...while loop especially makes this ugly. Any suggestions on how to improve this?
• @MartinR Not exactly. Fisher–Yates shuffle gives you permutation of all elements in given range. – Kao Aug 28 '14 at 15:05
• @BrunoCosta: "so if the numbers don't repeat you can't actually call it random anymore, it will have the property of not repeating which is a very unique property": That's not true. "Random" does not imply "independent", let alone "not having any interesting properties". Heck, by your definition of "random", we couldn't even say "random integer", because the property of being an integer would mean it wasn't random . . . – ruakh Aug 30 '14 at 10:27
• Bob Floyd invented an algorithm for this purpose. See: stackoverflow.com/a/2394292/179910. It's substantially better than what you've shown or any of the answers currently posted. – Jerry Coffin Sep 1 '14 at 2:50
• @JerryCoffin , I believe I am correct, and that, for example, the first value in the result can never be N, which means that the randomness of the result is not maintained. The randomness of the selection is maintained, but the order is not random (enough). Can we clear this up in the 2nd Monitor? – rolfl Sep 4 '14 at 20:43
• It seems this guy solved you problem in a very nice way. This is what he says in the first line of the post: In this post I’m going to show a way to make an iterator that will visit items in a list in a random order, only visit each item once, and tell you when it’s visited all items and is finished. It does this without storing a shuffled list, and it also doesn’t have to keep track of which items it has already visited. – NicolaSysnet Oct 13 '15 at 16:12
Updated answer in response to bounty: See Is that your final answer? at the end, and other changes - basically answer is significantly rewritten.
To break your problem down in to requirements:
• you need a set of random numbers
• the numbers need to be unique
• the order of the returned numbers needs to be random
Your current code indicates that the range of random numbers is specified by Random.Next(), which returns values in the [0 .. Int32.MaxValue) range (note, it excludes Int32.MaxValue). This is significant for the purpose of this question, because other answers have assumed that the range is configurable, and 'small'.
If the range should be configurable, then the recommended algorithm would potentially be much larger.
Based on those assumptions, let's do a code review...
# Code Style
### do ... while
The most glaring problems here are the un-braced do-while loop. You already knew it, but this code is ugly:
do number = random.Next();
while (randomNumbers.Contains(number));
do
{
number = random.Next();
} while (randomNumbers.Contains(number));
This makes the statement clear, and significantly reduces confusion. Always use braces for 1-liners.
### List Construction
The List class allows an initial capacity to be used. Since the capacity needs to just be count, it makes sense to initialize the list with this capacity:
List<int> randomNumbers = new List<int>(count);
# Current Algorithm
This is where the most interesting observations can be made. Let's analyze your current algorithm:
• Create a container for the results
• repeat until you have selected N values:
• Select a random value
• check if it has been previously selected
• if it is 'new', then add it to the container
This algorithm will produce random values, in a random order, and with good random characteristics (no skews, biases, gaps, etc.).
In other words, your results are good.
The problem is with performance....
There are two performance concerns here, one small, the other large:
1. the do-while loop to avoid collisions
2. the List container
## do-while performance
The do-while has a very low impact on performance... almost negligible. This is hotly debated, but, the reality is that you would need a very, very large count before this becomes a problem. The reasoning is as follows:
Collisions happen when the random value was previously selected. For the specified range of [0 .. Int32.MaxValue), you would need a very large count before collisions actually happened. For example, count would have to be about 65,000 before there was better than a 50% chance that there was even a single collision.
In a general sense, given a Range of $N$, select $M$ numbers. If $M < \sqrt{N}$ then the probability of a single collision is < 50%. Since the Range is very large, the probability is small.
Obviously, if the range was small, then the probabilities would be significantly affected. But the range is fixed at Int32.MaxValue, so that's OK.
Additionally, if the count was large, then the probabilities would also be affected. How large would be very large? Well, you would be running in to very large arrays before you run in to significant problems..... I would venture you are hitting close to $\frac{Int32.MaxValue}{2}$ before you run in to a significant issue with performance.
### List performance
This is without doubt your largest concern. You use the randomNumbers.Contains(number) call to determine whether a value was previously selected. This requires a scan of all previously-selected values to determine. As mentioned, this will almost always return false, and will thus have to scan the entire list.
As the count value increases, the length of time to perform the Contains will increase at an quadratic rate, $O(n^2)$ where n is count.
This performance problem will become critical much sooner than the random-collision problem.
# Putting it together
The problem you have in your code is that you are trying to do too much at once, you are using a List because that is your return value, when a HashSet would be better. If you break the problem down in to stages, you will be able to solve things more elegantly.
If you add a duplicate value to a HashSet, it does not grow, and the operation performance is not dependent on the amount of data in the HashSet (it is $O(1)$). You can use the Count of the HashSet to manage the data uniqueness.
Once you have a clean set of unique random numbers, you can dump them in to a List, then shuffle the list using an efficient shuffle.
Combining these data structures, in the right way, leads to an overall $O(n)$ solution, which will scale fairly well.
Here is some code, which works in Ideone too. Note, my C# is weak, so I have tried to make the logic clear.
using System;
using System.Collections.Generic;
public class Test
{
static Random random = new Random();
public static List<int> GenerateRandom(int count)
{
// generate count random values.
HashSet<int> candidates = new HashSet<int>();
while (candidates.Count < count)
{
// May strike a duplicate.
}
// load them in to a list.
List<int> result = new List<int>();
// shuffle the results:
int i = result.Count;
while (i > 1)
{
i--;
int k = random.Next(i + 1);
int value = result[k];
result[k] = result[i];
result[i] = value;
}
return result;
}
public static void Main()
{
List<int> vals = GenerateRandom(10);
Console.WriteLine("Result: " + vals.Count);
vals.ForEach(Console.WriteLine);
}
}
The above code is my initial recommendation, and it will work well, and scale well for any reasonable number of values to return.
# Second Alternate Algorithm
The problem with the above algorithm is threefold:
1. When count is very large, the probability of collision is increased, and performance may be affected
2. Data will need to be in both the HashSet and the List at some point, so the space usage is doubled.
3. The shuffle at the end is needed to keep the data in a random order (HashSet does not keep the data in any specific order, and the hashing algorithm will cause the order to become biased, and skewed).
These are only performance issues when the count is very large. Note that only the collisions at large count will impact the scalability of the solution (at large count it is no longer quite $O(n)$, and it will be come progressively worse when count approaches Int32.MaxValue. Note that in real life this will not likely ever happen.... and memory will become a problem before performance does.
@JerryCoffin pointed to an alternate algorithm from Bob Floyd, where a trick is played to ensure that collisions never happen. This solves the problem of scalability at very large counts. It does not solve the need for both a HashSet and a List, and it does not solve the need for the shuffle.
The algorithm as presented:
initialize set S to empty
for J := N-M + 1 to N do
T := RandInt(1, J)
if T is not in S then
insert T in S
else
insert J in S
assumes that RandInt(1, J) returns values inclusive of J.
To understand the above algorithm, you need to realize that you choose a random value from a range that is smaller than the full range, and then after each value, you extend that to include one more. In the event of a collision, you can safely insert the max because it was never possible to include it before.
The chances of a collision increase at the same rate that the number of values decreases, so the probability of any one number being in the result is not skewed, or biased.
# Is this almost a final answer? No
So, using the above solution, in C#, would look like (in Ideone) (note, code is now different to Ideone):
public static List<int> GenerateRandom(int count)
{
// generate count random values.
HashSet<int> candidates = new HashSet<int>();
for (Int32 top = Int32.MaxValue - count; top < Int32.MaxValue; top++)
{
Console.WriteLine(top);
// May strike a duplicate.
{
}
}
// load them in to a list.
List<int> result = candidates.ToList();
// shuffle the results:
int i = result.Count;
while (i > 1)
{
i--;
int k = random.Next(i + 1);
int value = result[k];
result[k] = result[i];
result[i] = value;
}
return result;
}
Note that you need to shuffle the results still, so that the HashSet issue is resolved. Also note the need to do the fancy loop-condition top > 0 because at overflow time, things get messy.
# Can you avoid the shuffle?
So, this solves the need to do the collision loop, but, what about the shuffle. Can that be solved by maintaining the HashSet and the List at the same time. No! Consider this function(in Ideone too):
public static List<int> GenerateRandom(int count)
{
List<int> result = new List<int>(count);
// generate count random values.
HashSet<int> candidates = new HashSet<int>();
for (Int32 top = Int32.MaxValue - count; top < Int32.MaxValue; top++)
{
// May strike a duplicate.
int value = random.Next(top + 1);
{
}
else
{
}
}
return result;
}
The above answer is never going to allow the first value in the result to have any of the Max - Count to Max values (because there will never be a collision on the first value, and the range is not complete at that point), and this is thus a broken random generator.
Even with this collision-avoiding algorithm, you still need to shuffle the results in order to ensure a clean bias on the numbers.
TL;DR
Having played with this code a lot, it is apparent that it is useful to have a range-based input, as well as a Int32.MaxValue system. Messing with large ranges leads to potential overflows in the 32-bit integer space as well.
Working with @mjolka, the following code would be the best of both worlds:
static Random random = new Random();
// Note, max is exclusive here!
public static List<int> GenerateRandom(int count, int min, int max)
{
// initialize set S to empty
// for J := N-M + 1 to N do
// T := RandInt(1, J)
// if T is not in S then
// insert T in S
// else
// insert J in S
//
// adapted for C# which does not have an inclusive Next(..)
// and to make it from configurable range not just 1.
if (max <= min || count < 0 ||
// max - min > 0 required to avoid overflow
(count > max - min && max - min > 0))
{
// need to use 64-bit to support big ranges (negative min, positive max)
throw new ArgumentOutOfRangeException("Range " + min + " to " + max +
" (" + ((Int64)max - (Int64)min) + " values), or count " + count + " is illegal");
}
// generate count random values.
HashSet<int> candidates = new HashSet<int>();
// start count values before max, and end at max
for (int top = max - count; top < max; top++)
{
// May strike a duplicate.
// Need to add +1 to make inclusive generator
// +1 is safe even for MaxVal max value because top < max
if (!candidates.Add(random.Next(min, top + 1))) {
// which could not possibly have been added before.
}
}
// load them in to a list, to sort
List<int> result = candidates.ToList();
// shuffle the results because HashSet has messed
// with the order, and the algorithm does not produce
// random-ordered results (e.g. max-1 will never be the first value)
for (int i = result.Count - 1; i > 0; i--)
{
int k = random.Next(i + 1);
int tmp = result[k];
result[k] = result[i];
result[i] = tmp;
}
return result;
}
public static List<int> GenerateRandom(int count)
{
return GenerateRandom(count, 0, Int32.MaxValue);
}
• Why do you shuffle the results? I'm confused about this because everyone seems to include this in their answers but it isn't in the OP's question – IEatBagels Aug 28 '14 at 19:27
• @TopinFrassi: Because rolfi ordered the numbers (to check for duplicates) but they shouldn't be ordered. – Charles Aug 28 '14 at 20:05
• @TopinFrassi is right he should just return candidates and it would be fine. Removing shuffling part It's only slightly better then op's answer because it uses hashset wich makes it less expensive to find a duplicate. I said this was a problem about shuffling but of course there are many implementations to fit the requirement. – Bruno Costa Aug 28 '14 at 22:41
• @BrunoCosta - The HashSet will impose a non-random (though unstructured) order on the data in the list. The OP's code would produce a random order on the result. The shuffle is required to retain the random ordered output. As for the 'slightly better' part, the HashSet will be about N/4 times faster than the List (where N is the count), so, for large N, say, 10,000 count, the HashSet will be 2500 or so times faster.... which, I guess, could be 'slight'. – rolfl Aug 28 '14 at 22:48
• I like that answer, lot of effort put in it, +1. I was worried not to see the final, filling the list on the way. But still have one objection: The randomness of the final solution is not perfect, would never allow highest value at the beginning (the shuffle helped to avoid that). I would personally use the original patched with HashSet for the test. No more changes (except codestyling). Of course I assume that it is not desired to eat the memory with a list of 2G ints -> 8GB. – user52292 Sep 4 '14 at 17:49
Yes, there definitely is.
You generate a collection of elements, mash it around and start pulling items out of it. A quick oneliner would be:
Enumerable.Range(0,100).OrderBy(x => Guid.NewGuid()).Take(20);
or alternatively
Enumerable.Range(0,100).OrderBy(x => random.Next()).Take(20);
This will give you 20 unique random values from the 0 to 100 range.
The difference with your approach is that you have a worst-case scenario of infinity: what if you are reaaaaally unlucky and end up with the same value constantly? You'll never get your required amount of random values.
My approach on the other hand will first generate 100 values and then take a subset from it which you can criticize on its memory impact. Considering you used random.Next(), which uses half the integer range, you should in fact be wary of this since it will have a huge memory impact.
It also depends on your specific situation: if you have a very large pool of values (1.000.000) and you need 2 random values then your approach will be much better. But when you need 999.999 values from that same pool, my approach will be much better still be debatable.
It will take some time to generate those last values; a lot as you can test for yourself with this:
void Main()
{
var random = new Random();
var times = new TimeSpan[512];
var values = new bool[512];
var sw = new Stopwatch();
for(int i = 0; i < times.Length; i++)
{
sw.Restart();
while(true) {
int rand = random.Next();
if(rand > 7894500 && rand < 7894512)
{
int index = rand - 7894500;
if(!values[index])
{
values[index] = true;
break;
}
}
}
sw.Stop();
times[i] = sw.Elapsed;
Console.WriteLine ("Elapsed time: " + sw.Elapsed);
}
var orderedTime = times.OrderBy(x => x);
for(int i = 0; i < 512; i++)
{
Console.WriteLine (orderedTime.ElementAt(i));
}
}
It will keep looping randomly 512 times through your list of values and consider the element found once it finds the (randomly picked out by myself) values between 7894500 and 7894512. Afterwards this value is considered visited to correctly mimic reality (in an earlier version all 512 turns had 512 values available to them). When you execute this you'll notice it takes a lot of time to find the last values (sometimes it's fast and it takes 39 ms, other times it takes over a minute). Evidently it's fast at the start and slow at the end.
Compare that to the memory overhead of my approach which will first allocate 32 million integers, guids, padding, object overhead, etc and you're out of a big chunk of memory.
You might be able to improve it by using a more "real" shuffling algorithm which doesn't have the object and guid overhead.
Ultimately in an extreme situation where you need 32 million unique values in a randomized order out of a total population of 32 million + 1, you will either have to accept a big memory overhead or a big execution time overhead.
One last edit before this topic can be laid to rest from my part: I talked about it with @rolfl in chat and we have come to the conclusion that either of our solutions have a usage but it depends on what your situation is exactly. Summarized it would come to this:
If you have a big range of values (like 0 to int.MaxValue) then my solution will eat any memory your PC has. You're looking at two collections with each 2.1 billion integers which you then take a slice from.
In my solution you first allocate this entire population, then you sort on it (different datastructure) and then you take some of it. If this "some" is not close to 2.1 billion you will have made a huge cost of allocating data without using it.
How does this compare to @rolfl's answer? He basically allocates the data as it is needed: if he needs 32 million values then he will only allocate those 32 million (x2) and not more. If he needs 2.1 billion then he'll end up with a memory footprint like I have but that's an uncommon worst case scenario while for me it is standard behaviour.
The main disadvantage of his approach is that when the amount of values you want reaches the total population, it will become increasingly harder to get those last values because there will be many collisions. Yet again, this will only be a problem when the population is actually really big.
So when should you use my solution? Like most things, there is a tradeoff between readability and performance. If you have a relatively small population and a large dataset then the readability weighs up against the performance impact, in my opinion. And when the population is relatively small and the amount of values we want is near that, my solution will have a memory impact similar to that of the other approach but it will also avoid many collisions.
• -1 because this code is not compatible with the OP's requirements as is, and would likely not complete successfully with a large range (MIN to MAX value for Int). – rolfl Aug 28 '14 at 14:30
• If you need 999,999 values from 1,000,000 then your code will require.... about 16MB just to store the GUID's (assuming it is stored in a complact 16-byte value, not as a string, or something) , plus another 4MB for the int values, plus object overheads, and then some more too, i am sure. – rolfl Aug 28 '14 at 14:35
• Additionally, GUID's are not specified to be random, and ordering by a GUID will not produce the results you expect – rolfl Aug 28 '14 at 14:39
• I don't think your concern about the OP's code getting stuck in an infinite loop is a valid one. The PRNG used by C# has a period that AIUI is guaranteed to be > 2^{56} for all seed values, which means that as long as count is a reasonable value it should always produce a valid result. In reality, you'll run out of memory to store the generated values long before this is ever an issue. – Jules Aug 30 '14 at 2:33
• A sorting function is not a shuffling function. Why generate O(N) new memory for the key and take O(N log N) time to sort it, when you can use the Fischer-Yates shuffle which takes O(1) memory and O(N) time? – Andrew Aug 30 '14 at 12:53
Instead of using a List<int>, you should use an HashSet<int>. The HashSet<> prohibites multiple identical values. And the Add method returns a bool that indicates if the element was added to the list, this way you could change this code :
public static List<int> GetRandomNumbers(int count)
{
List<int> randomNumbers = new List<int>();
for (int i=0; i<count; i++)
{
int number;
do number = random.Next();
while (randomNumbers.Contains(number));
}
return randomNumbers;
}
to this :
public static IEnumerable<int> GetRandomNumbers(int count)
{
HashSet<int> randomNumbers = new HashSet<int>();
for (int i = 0; i < count; i++)
return randomNumbers;
}
Note that I changed the return value from List<int> to IEnumerable<int>, if you don't plan to add/remove values from your list, you should return IEnumerable<int>, if you plan to do it, return ICollection<int>. You shouldn't return a List<int> because it is an implementation detail and that the List<> isn't made to be extensible.
• The problem about using a Set is that it's not ordered. There's no way to make it put the integers in a random order. – Simon Forsberg Aug 28 '14 at 14:30
• @SimonAndréForsberg The OP didn't specify he wanted to do so, unless there's something I'm missing – IEatBagels Aug 28 '14 at 14:34
• In fact, you are correct. I totally misread the problem. – Simon Forsberg Aug 28 '14 at 14:45
• Happens to everyone! – IEatBagels Aug 28 '14 at 14:45
• you shouldn't call ToList there. Asside of that it is a great solution – Bruno Costa Aug 28 '14 at 23:39
Expanding on my comment. This is called a linear congruential generator. I used common parameters, which come from what I think is called the Minimal Standard. Other parameters can be chosen, but that's a tricky task. The sequence starts at the seed, reaches all other numbers between 0 and M-1, and then restarts once it reaches the seed again. It's pseudo-random. 507111939 always follows 285719. Since the sequence reaches all M numbers, there will be no duplicates in any M successive outputs of the sequence.
Code:
class RandomSequence {
int actual;
static final int M = (1<<31) -1;
public RandomSequence(int seed) {
this.actual = seed
}
public int next() {
return this.actual = (16807 * this.actual) % RandomSequence.M
}
}
Usage :
//...
List<int> awesomeList = new List<int>();
RandomSequence sq = new RandomSequence( RandomUtil.getRandomPositiveIntSmallerThanM() );
for(int i = 0; i < n; i++) {
}
• This is a nice solution. You should note that a fair amount of this works because of a number of convenient coincidences in computational theory, like M is $2^{31} - 1$ which is also Integer.Max_value, and it also happens to be a prime number. Being prime is significant. 16807 is also prime. – rolfl Aug 30 '14 at 12:49
• can you explain what you're talking about – Muhammad Umer Jul 24 '15 at 4:47
• i was talking about @rolfl – Muhammad Umer Aug 1 '15 at 3:12
The algorithm looks fine to me. It shows no clear flaws, and will work fine in common use cases.
For extreme cases of selecting most of the numbers in a very large interval, it will be very inefficient. But unless that is an expected use case, I would not bother changing the code to more complicated approaches.
But I suggest you always use explicit braces around blocks even when they have a single statement:
do
{
bar();
}
while (foo);
I would also rename the method to GetNRandomNumbers, to make the method's name match its purpose better.
• +1. I'm not usually a stickler for braces around blocks, but I think in a do ... while loop they're especially important. I initially misread the code because of the lack of braces. – Jules Aug 30 '14 at 2:37
• Style wan't the question, was it? – Florian F Sep 4 '14 at 16:43
• @FlorianF No, it was not. But the first sentence in the answer says that the algorithm looks fine to me, does it not? And then I made some sugestions to improve bug resilience (using brackets) and clarity (renaming). Could you try to you be clearer in what you mean? – ANeves Sep 4 '14 at 17:43
• @FlorianF Style is always a valid target for a code review. – David Harkness Sep 4 '14 at 18:35
• @FlorianF this is Code Review, not Stack Overflow. Reviewers are always free to comment on any aspect of the code. Always. – Mathieu Guindon Sep 4 '14 at 23:44
Another solution is to use a Format Preserving Encryption cipher (such as AES using the FFX mode), configured to map from 32 bit integers to 32 bit integers.
Seed it randomly, then simply encrypt the first 'count' integers.
These encrypted numbers will be as random as the seed, and won't repeat.
• Interesting idea, and probably much faster than OP's and rolfl's code, as it would be O(n) not O(n^2) or O(n ln n) as those solutions are. Jeroen Vannevel's solution is also O(n) and would probably be faster still, but cannot plausibly be used if the required range of the random numbers is too large. This solution likely has the lowest memory requirement of all, as the numbers can be produced on-demand rather than calculated in advance. – Jules Aug 30 '14 at 2:41
I'm updating my answer in order to make an analysis on BobFloyd's algorithm and sum up all flaws that I could spot on it. Some of them were on my previous answer, they are also mentioned on some users comments like @rolfl and @firda. Please follow through the whole answer because you will have a surprise at the end.
Let me define some terms that I'll be using in my answer. Consider the method definition:
static IEnumerable<int> NonRepeatingRandomSequence(int min, int max, int count, int? seed = null)
• range length = max - min + 1
• taken elements = count
First issue: The last range length - taken elements elements can not ever be in index 0 of the result.
Second issue: The results becomes closer to a linear sequence as taken elements gets closer to range length. This is easier to see when they are the same, but the result is not random at all when they are close.
Third issue: There is also a third issue but I will just mention it at end of my answer.
I implemented three algorithms for returning non repetitive Numbers:
A pure shuffling algorithm, A pure Bob Floyd algorithm and a Bob Floyd algorithm with shuffling. Those will appear later as I compare and test them.
I also implemented a test to check the randomness of the outputted list by any algorithm. Although it has some constraints, it will suffice for the sake of the answer completeness/correctness. For much wrong it can be this will be my definition of being random. In here I'm basically count the times that every combination occurs and see if the average is in an acceptable range.
public void TestRandomness(Func<List<int>> getRandomList, int rangeLength, int count)
{
long combinations = (long)(rangeLength.Factorial(rangeLength - count + 1));
long iterations = combinations * 100;
var partitioner = Partitioner.Create(0, iterations, iterations / 4);
ConcurrentDictionary<long, int> ocurrences = new ConcurrentDictionary<long, int>(Environment.ProcessorCount, (int)combinations);
Parallel.ForEach(partitioner, new ParallelOptions() {MaxDegreeOfParallelism = Environment.ProcessorCount},
range =>
{
//hopefully having a private dictionary will help concurrency
Dictionary<long, int> privateOcurrences = new Dictionary<long, int>();
for (long i = range.Item1; i < range.Item2; ++i)
{
var list = getRandomList();
long acc = 0;
//this will only work when numbers are between 0 and 88
foreach (var value in list)
{
acc = (value + 11) + acc*100;
}
privateOcurrences.AddOrUpdate(acc, 1, v => v + 1);
}
foreach (var privateOcurrence in privateOcurrences)
{
privateOcurrence.Value,
(k, v) => v + privateOcurrence.Value);
}
});
double averageOcurrences = iterations / combinations;
double currentAverage = ocurrences.Values.Average();
Debug.WriteLine("The average ocurrences of this implementation is {0} comparing to {1} in the 'perfect' scenario", currentAverage, averageOcurrences);
Assert.Less(currentAverage, averageOcurrences * 1.05);
Assert.Greater(currentAverage, averageOcurrences * 0.95);
}
So let's start with testing the randomness of our shuffling algorithm, this is my first algorithm:
[Test]
public void TestShuffleRandomness()
{
TestRandomness(() => Enumerable.Range(0, 16).ToList().Shuffle().Take(4).ToList(), 16, 4);
}
Pass, pass, pass. This algorithm respects my test definition of randomness. This is very close to what @Jeroen Vannevel pointed, except for the part this do not require a random nor does it run in O(n log n) time (I need to refer to @Andrew Piliser because he was the only one to comment about this).
As we know this algorithm will take memory equivalent to O(range length) and it is his major drawback. However it does not generate collisions and it guarantees randomness.
It's now time to prove First and Second issues with Bob Floyd Algorithm:
public static IEnumerable<int> BobFloydNonRepeatingSequence(int min, int max, int count, int? seed = null)
{
Random random;
if (seed != null)
{
random = new Random(seed.Value);
}
else
{
random = new Random();
}
long length = max - min + 1;
INonRepeatingList values = NonRepeatingListFactory.GetNonRepeatingList(min, max, count);
for (int i = (int)(length - count); i < length; ++i)
{
{
}
}
return values;
}
[Test]
public void TestLastCountElementsAreNotInIndex0()
{
for (int i = 0; i < 1000000; ++i)
{
var list = Sequence.BobFloydNonRepeatingSequence(0, 15, 4).ToList();
Assert.IsFalse(new[] { 12, 13, 14, 15 }.Contains(list[0]));
Assert.AreEqual(4, list.Count);
}
}
[Test]
public void TestBobFloydBecomesLinear()
{
var list = Sequence.BobFloydNonRepeatingSequence(0, 15, 16).ToList();
for (int i = 0; i < list.Count; ++i)
{
Assert.AreEqual(i, list[i]);
}
}
Great, now we proven that Bob Floyd algorithm by it's own does not solve this problem and of course it wouldn't pass in my randomness test.
[Test]
public void TestBobFloydRandomness()
{
TestRandomness(() => Sequence.BobFloydNonRepeatingSequence(0, 15, 4).ToList(), 16, 4);
}
Our only option left is to shuffle the results and hope for the best.
public static IEnumerable<int> NonRepeatingRandomSequence(int min, int max, int count, int? seed = null)
{
Random random;
if (seed != null)
{
random = new Random(seed.Value);
}
else
{
random = new Random();
}
long length = max - min + 1;
INonRepeatingList values = NonRepeatingListFactory.GetNonRepeatingList(min, max, count);
for (int i = (int)(length - count); i < length; ++i)
{
{
}
}
values.Shuffle();
return values;
}
[Test]
public void TestBobFloydWithShuffleRandomness()
{
TestRandomness(() => Sequence.NonRepeatingRandomSequence(0, 15, 4).ToList(), 16, 4);
}
I hope you didn't get your hopes too high. My randomness test fails this is the third and last issue with Bob Floyd at all because it won't be random. Or at least it does not respect the definition of random that I provided.
This bit will be just discussing my implementation of Bob Floyd algorithm vs rolfl one
So let's say my definition of random is a bit strict or that Bob Floyd shuffled algorithm is random enough for your purposes. I don't really know what is lacking in the implementation so that the results would be more random, so it just remains about efficiency.
As you saw in my implementation I am getting a INonRepeatingList from the method NonRepeatingListFactory.GetNonRepeatingList.
public class NonRepeatingListFactory
{
public static INonRepeatingList GetNonRepeatingList(int min, int max, int count)
{
long length = max - min + 1;
if (length / 8 > count * 2 * sizeof(int))
{
//if the amount of bytes occupied by the array is greater then the dictionary
return new RandomIndexNonRepeatingList();
}
return new NonRepeatingList(min, max);
}
}
This Method decides which list to use based on memory occupied. If you use a HashSet always like @rolfl suggests you will be occupying to much memory when taken elements is big and range length is also big. In this situation you can just map the value to a bit. How much difference it may make? It's just about 1KB memory for every 8KB range length. In the other hand if we are taking few elements from a wide range we should be using an HashSet.
public interface INonRepeatingList : IEnumerable<int>
{
}
internal class NonRepeatingListWithArray : List<int>, INonRepeatingList
{
public NonRepeatingListWithArray(int min, int max)
{
this.inList = new BitArray(max-min+1);
this.min = min;
this.max = max;
}
{
if (!this.inList[value - this.min])
{
this.inList[value - this.min] = true;
return true;
}
return false;
}
}
internal class NonRepeatingListWithSet : List<int>, INonRepeatingList
{
private readonly HashSet<int> mapValue = new HashSet<int>();
private int currentIndex = 0;
{
{
return true;
}
return false;
}
}
Full code on pastebin
• after a quick overview I noticed that I wasn't using a Random thread safely but despite of this my tests results didn't change. – Bruno Costa Sep 6 '14 at 0:39
• There are errors in your translation of Bob Floyd's algorithm; try Sequence.BobFloydNonRepeatingSequence(8, 14, 3, 7) -- it throws an ArgumentOutOfRangeException. – mjolka Sep 7 '14 at 8:43
• @mjolka thanks again. I updated my answer accordingly. – Bruno Costa Sep 7 '14 at 13:19
• There's still a mistake in the Bob Floyd algorithm; values.Add(i+min) should be values.Add(i+min-1). – mjolka Sep 8 '14 at 0:21
• @mjolka No it shouldn't. – Bruno Costa Sep 8 '14 at 6:12
Though the solution byr @Jeroen Vannevel is quite simple and works for most intents and purposes, you will see especially poor performance if you just want a few numbers out of a very large range.
In that case I recommend the following algorithm:
1. Pick one number out of the range
2. Pick one number out of the remaining range
3. and so on ...
The implementation could look like so:
1. Choose a number from 0 to N
2. Choose a number from 0 to N-1
3. And so on
4. Determine which number each of your drawings represents
Example for 0 to 10
1. draw 4
2. draw 7
3. ...
4. Results in: the 4 represents 4, the 7 represents 8 as that is now the seventh available number.
• Could you add a code example of your algo? – IEatBagels Aug 28 '14 at 14:22
• This will work, but will provide some additional overhead after you've randomized 5, 6 for example and you get an 7 as the next number. I assume you intend that 7 to be modified into a 9 then? (because 5 and 6 is already gone). – Simon Forsberg Aug 28 '14 at 14:34
• @SimonAndréForsberg Exactly, you would have to try when the overhead is worth the extra effort but when increasing the range there should be a turning point where this becomes more efficient. – Dennis Jaheruddin Aug 28 '14 at 14:41
• I see the implication here, and that I mistook the algorithm you suggest. You are correct, there will be no skewness, but your answer leaves a very big blank over how to manage the range, especially given that the range is the whole valid int 32-bit space. You will still need to iterate over all the previously selected values to make it work. – rolfl Aug 28 '14 at 14:46
• This needs to be explained better. (The part that the second number is increased by 1 if it is bigger than the first, and the 3rd by 2 if bigger than both or 1 if bigger than one, etc.) – ANeves Aug 28 '14 at 16:46
From the looks of it, what you want is called a (pseudo)random permutation of all int values. Note that you don't necessarily have to store these values in an array and shuffle them, consuming memory; you can actually generate them on demand via a "perfect" hash function of the bits in the "count" variable.
Note that pseudo-random also means deterministic, so to get different values on different runs, you have to make sure to "seed" your sequence with a different value each time.
You have two parameters: the number N of integers you want, and their maximum size M >= N. If N is close to M, e.g., you want 500 numbers in the range 1..1000, then your best bet is to (partially) shuffle the numbers 1..M. For example, you can run the first N steps of a Fisher-Yates shuffle on 1..M and get the numbers you need.
Otherwise, if M is a lot bigger than N, you would be better off generating all your random numbers at once (rather than checking if they're in the list one-by-one), sorting, removing duplicates, adding new elements as needed for replacements, and then re-shuffling.
You may wish to generate more than N numbers to handle the expected number of duplications. The expected number of duplicates is about N(N-1)/2M so you might generate, say, N + 1.1N(N-1)/2M numbers in your initial list.
Sorting and removing duplicates is standard.
If you're not over-generating, or if you did but it wasn't enough, you can generate new elements as you originally suggested: generate a new number, test if it's on the list, and add it if so.
For re-shffling you can either use a standard shuffle (like Fisher-Yates, mentioned above) or you can store the original order with each element and then order by that. If you do and you over-generated originally, just ignore any extra elements on the end.
If performance is important, you can code both methods (and maybe several variants of the second, as described) and test to find the cutoff between the methods. If it isn't then the second method will work in all cases -- though if you don't need it the first is easier to program.
• The OP only has one parameter, N. What you call the second parameter M is actually hard-coded in the code as random.Next() which implies an M of Integer.MAX_VALUE (2^31 - 1). – rolfl Aug 30 '14 at 12:30
• @rolfl: Correct. I wanted to spell out the dependence on M explicitly. – Charles Aug 31 '14 at 19:22
First of all, there is one unspecified variable random and my best bet would be to assume it is System.Random and the Next method returns
A 32-bit signed integer greater than or equal to zero and less than MaxValue.
I ask the author to make this clear in the question. With this assumption, we have algorithm that works for count < int.MaxValue, but gets very slow for count close to that value, but that would mean eating nearly 8GB memory (2G*4B). We can try to make it faster for count greater than some threshold, e.g. 20 (subject to benchmarking), throw exception for some ridiculous count and rewrite it like this:
public static List<int> GetRandomNumbers(int count) {
if(count > int.MaxValue/8)
throw new MyException("count too big");
List<int> randomNumbers = new List<int>(count);
if(count < 20) for (int i=0; i<count; i++) {
int number; do number = random.Next(); // I will address the 'uglyness' later
while (randomNumbers.Contains(number));
} else {
HashSet<int> included = new HashSet<int>();
for(int i=0; i<count; i++) {
int number; do number = random.Next();
}
}
return randomNumbers;
}
The do..while may seem ugly, but is separated by two empty lines (in original code), thus makes it sufficient but still subject to further rethinking regarding to who may be maintaining the code. I will not address this furteher, because it depends on how many people may maintain the code and programmer's preferences (I will not address code-style while I see one). I am not that strict in this and for me, two empty lines (one before and one after) is enought to signal it needs more thinking when encountered and I will never touch a code that works and is optimal. Comments can help to describe the intent, code is to be reviewed when there is something wrong (does not work or is slow). The following may look better by using {}, more end-of-lines and few comments:
public static List<int> GetRandomNumbers(int count) {
// eating too much memory is not desired
if(count > int.MaxValue/8)
throw new MyException("count too big");
List<int> randomNumbers = new List<int>(count);
if(count < 20) {
// list is fast enough for small count
for (int i=0; i<count; i++) {
int number;
do {
number = random.Next();
} while (randomNumbers.Contains(number));
}
} else {
// but HashSet is faster for longer sequence
HashSet<int> included = new HashSet<int>();
for(int i=0; i<count; i++) {
int number;
do {
number = random.Next();
}
}
return randomNumbers;
}
Since your random numbers are 32-bit integers, the chances of a collision are very low. In this case, the do ... while loop will loop only occasionally and won't cause any performance problem.
The problem is how you check for a collision. Checking whether a number belongs to a list requires to read the list sequentially That is slow. You should better use a HashMap. Insert each generated number in the HashMap, and check new numbers against the HashMap.
Since you want to return the numbers as a List, you will have to also collect the numbers in a List. You can either build the List as the numbers are generated, or build the list later, from the HashMap. But the second solutions compromises the randomness, and you will have to shuffle the numbers again. And to actually save memory you have to empty the HashMap as you build the List. Emptying the HashMap one element at a time adds overhead.
Alternate solution
A complete different solution that minimizes memory use would be to generate N random numbers in the range 0 to Int32.MaxValue-N+1, sort the list, add i the ith element, and shuffle the list again. It minimizes memory use but is O(N log N) run time.
The List + HashMap solution is almost O(N) run time but requires at least twice that much memory.
If there are 1-4 possible numbers, and you have generated 1 number already, that means there are (4 - 1) 3 possible numbers left. Make a random number between 3, for every generate number it is greater than or equal, increase the created number by 1.
lets say the number is 2, and you want to generate another:
Generated numbers: [1,2,3] Possible number: [1,3,4]
Solution:
public static int[] GetRandomNumbers(int count, int minValue, int maxValue)
{
int[] randomNumbers = new int[count];
for (int i=0; i < count; ++i)
{
//Given (min - max) is the range, there can only be (range - i) unique number left
int number = random.Next(minValue, maxValue - i);
//if the number is greater than or equal to, then increase the number
for (int j=0; j < i; ++j)
if(number >= randomNumbers[j])
++number;
}
return randomNumbers;
}
• +1 for providing what looks to be a working solution that avoids the possible duplicate-value check. Note your solution scales badly for large values of count though. $O(n^2)$ time complexity – rolfl Aug 30 '14 at 12:38
• This doesn't compile (there's no Add method for array), and it produces duplicate values. After getting it to compile, on my machine, GetRandomNumbers(10, 0, 20, new Random(7)) produces 7, 17, 12, 0, 6, 13, 1, 18, 15, 13. – mjolka Sep 1 '14 at 6:45
Instead of using do-while, can't we do this in alternate way with fewer lines of code using Linq?
Fill the list with numbers from 1 to 100 (or whatever the limit) and shuffle the list.
You need to add using System.Linq namespace. It should be present by default.
static void PrintRandom(int limit)
{
List<int> original = Enumerable.Range(1, limit).ToList();
Random rand = new Random ();
//shuffle the list
List <int> sorted = original.OrderBy(item => rand.Next()).ToList ();
foreach (int i in sorted)
{
Console.WriteLine(i);
}
}
The idei is to create segments in the defined range and pick one value from them randomly with taking the segments also randomly. No overlapping will happen and also works with negative values.
public static IEnumerable<int> GetRandomNumbers(int count, int minValue, int maxValue)
{
if (count < 1)
{
throw new ArgumentException("Count must be greater then 0", "count");
}
if (count > Math.Abs(maxValue - minValue))
{
throw new ArgumentException("MaxValue must be greater than count", "maxValue");
}
var segmentSize = (maxValue - minValue) / count;
var random = new Random();
var steps = Enumerable.Range(0, count).ToList();
while (steps.Count != 0)
{
var currentIndex = random.Next(0, steps.Count);
var index = steps[currentIndex];
steps.RemoveAt(currentIndex);
yield return random.Next((index * segmentSize) + minValue, ((index + 1) * segmentSize) + minValue);
}
}
public static IEnumerable<int> GetRandomNumbers(int count)
{
return GetRandomNumbers(count, 0, int.MaxValue);
}
• This doesn't work. GetRandomNumbers(3, 0, 4) always returns 0, 1, 2. – mjolka Sep 7 '14 at 8:00
• True but depending on the business needs it can solve the problem in a very simple way. And the problem can be solved by expending randomly the segments with (maxValue - minValue) % count. – Peter Kiss Sep 7 '14 at 8:42
• But if I want three numbers chosen at random from the range [0, 4) i.e. GetRandomNumbers(3, 0, 4), I would expect a 3 to appear with equal probability as 0, 1, and, 2. – mjolka Sep 7 '14 at 9:12
List contains is $O(n)$. HashSet contains is $O(1)$.
private static Random rand = new Random();
public static IEnumerable<int> GetRandomNumbers(int count)
{
HashSet<int> randomNumbers = new HashSet<int>();
while(randomNumbers.Count < count)
{
int r = rand.Next(); | 2020-01-23 03:57:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.30822670459747314, "perplexity": 1814.0340488271847}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1579250608062.57/warc/CC-MAIN-20200123011418-20200123040418-00350.warc.gz"} |
https://en.wikibooks.org/wiki/Topological_String_Theory_Methods_of_Computer-aided_Drug_Design/Knots,_HOMFLY-PT_Polynomial,_Chern-Simons_Theory_and_Surgery | # Topological String Theory Methods of Computer-aided Drug Design/Knots, HOMFLY-PT Polynomial, Chern-Simons Theory and Surgery
This chapter covers knot theory and its invariants, including, especially, HOMFLY-PT polynomials. We explain Witten's viewpoint: these polynomials can be interpreted as the vacuum expectation value of Wilson loop operators in Chern-Simons Theory with gauge group ${\displaystyle G=SU(N)}$ and the fundamental representation. This easily leads to the generalization of HOMFLY-PT polynomials to arbitrary gauge groups and representations. We then introduce Guadagnini's amalgamation of Chern-Simons theory and Dehn surgery: this allows the computation of HOMFLY-PT polynomials in an arbitrary ${\displaystyle 3}$-manifold ${\displaystyle M^{3}}$ with given surgery presentation. This includes all ${\displaystyle 3}$-manifolds, due to a result by Lickorish and Wallace.
6. HOMFLY-PT polynomials in arbitrary ${\displaystyle 3}$-manifolds | 2022-05-22 23:50:46 | {"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": 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": 0.6890575885772705, "perplexity": 824.2084084998348}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1652662550298.31/warc/CC-MAIN-20220522220714-20220523010714-00184.warc.gz"} |
https://www.gradesaver.com/textbooks/science/physics/college-physics-4th-edition/chapter-17-problems-page-652/3 | ## College Physics (4th Edition)
(a) The electric potential energy is $~-4.4\times 10^{-18}~J$ (b) The negative sign shows that it would require external work to separate the two charges.
(a) We can find the electric potential energy: $U = \frac{k~q_1~q_2}{r}$ $U = \frac{(9.0\times 10^9~N~m^2/C^2)(1.6\times 10^{-19}~C)(-1.6\times 10^{-19}~C)}{0.0529\times 10^{-9}~m}$ $U = -4.4\times 10^{-18}~J$ The electric potential energy is $~-4.4\times 10^{-18}~J$ (b) The negative sign shows that it would require external work to separate the two charges. | 2020-02-26 20:34:42 | {"extraction_info": {"found_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": 0.7710959315299988, "perplexity": 291.4578896595715}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1581875146485.15/warc/CC-MAIN-20200226181001-20200226211001-00281.warc.gz"} |
https://datascience.stackexchange.com/questions/40632/how-do-i-use-the-model-generated-by-the-r-package-polca-to-classify-new-data-as | # How do I use the model generated by the R package poLCA to classify new data as belonging to one of the classes?
For example, in the election example from the documentation, if I create a new set of answers to the questions, how can I use the poLCA model to tell me what class (cluster) it's most likely to be in?
There doesn't appear to be a function to do this, though the model has a df within it that lists the probabilities of class membership for each value of each manifest variable. I'm tasked with converting some sql code that takes a second dataset and classifies the patients there as members of the clusters created from a first. Superficially this is a programming question. It seems like a function to do this would be a reasonable addition to the package. More deeply, if indeed there isn't such a function, it would become a question about how to use the table of probabilities to classify new data.
If readers aren't familiar with the R package poLCA, it's an LCA package that works with discrete/categorized data.
(full disclosure: I asked on cross-validated and a shorter version of this question was put on hold.)
• Part of the data returned from poLCA tells what probability a particular value of a variable adds to a subject's probability of membership to each class. The appendix S1 to this paper walks through an example calculation. I have some R code that does this at github to implement this. Comments welcome. Dec 19, 2018 at 23:56
• ...no code on github from me, as my answer below says, code to apply the model is actually part of the package. Apr 21, 2020 at 15:35
Using carcinoma data available in the poLCA package and a 4 latent classes solution:
library poLCA
data(carcinoma)
f <- cbind(A, B, C, D, E, F, G) ~ 1
lc4 <- poLCA(f, carcinoma, nclass = 4)
The following line give the classification in terms of predicted probabilities
lc4$predclass They could be usefully binded to the original data for further visualizations or analyses carcinoma.predclass <- cbind(carcinoma, "Predicted LC" = lc4$predclass)
You could have a new data frame with the same columns/variables used in the previous analysis.
new.data <- data.frame(A=c(1,2,1), B=c(2,2,1), C=c(1,2,1), D=c(1,1,1), E=c(1,2,1), F=c(2,2,1), G=c(1,2,1))
A B C D E F G
1 1 2 1 1 1 2 1
2 2 2 2 1 2 2 2
3 1 1 1 1 1 1 1
A simple method can be link the observed data patterns in the new data frame with the estimated latent class probabilities in the previous data. In fact, the first pattern has missing prediction because it wasn't in the training data.
left_join(new.data, unique(carcinoma.predclass))
Joining, by = c("A", "B", "C", "D", "E", "F", "G")
A B C D E F G Predicted LC
1 1 2 1 1 1 2 1 NA
2 2 2 2 1 2 2 2 1
3 1 1 1 1 1 1 1 4
The best method is to use the posterior of poLCA. From the parameters estimated by the latent class model, this function calculates the probability that a specified pattern belongs to each latent class. This function can calculate posterior class membership probabilities for new data, observed or not in the training data.
new.lc4.posterior <- poLCA.posterior(lc4, new.data)
And bind the predicted Latent Classes (the classes which the highest posterior probability) to the new data.
cbind(new.data, "Predicted LC" = apply(new.lc4.posterior,1, FUN=which.max))
A B C D E F G Predicted LC
1 1 2 1 1 1 2 1 2
2 2 2 2 1 2 2 2 1
3 1 1 1 1 1 1 1 4
• Thanks. I want to create the model from one dataset and apply it to another. I have working code now, that I'll publish shortly. Dec 5, 2018 at 23:10
• Was my answer useful! Dec 5, 2018 at 23:15
• The desire is to apply the lc4$probs to a second dataset. Still planning on posting detail. Dec 13, 2018 at 18:48 • I have updated the answer with a solution. Hope it helps! Dec 13, 2018 at 23:19 • Thanks, I must have confused lca_model$posterior, the data, with poLCA.posterior() the function. Jan 15, 2019 at 23:27
As Paolo says, use the poLCA.poseterior() function. The data comes out in the same format as the lca_model\$posterior structure returned by the poLCA function.
library(poLCA)
data(election)
column_names <- c('MORALG', 'CARESG', 'KNOWG', 'LEADG', 'DISHONG',
'INTELG', 'MORALB', | 2022-08-12 05:51:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.27409422397613525, "perplexity": 766.336882229299}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1659882571584.72/warc/CC-MAIN-20220812045352-20220812075352-00091.warc.gz"} |
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how to exclude a chapter from numbering? [duplicate]
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I am writing my Master's Thesis and am in a bit of a pickle with the table of contents. I want to add a preface and abstract as chapters, preceding the table of contents, have them show up I want ...
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Remove title number from specific chapter [duplicate]
I want to remove the number of the title. In the picture you can find the problem as visually: Is it possible? Thank you. My MWE is below: \RequirePackage{filecontents} \begin{filecontents}{\...
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Headings in appendix \section*
I'm using this and this approaches to get my appendix sections as I want, so a MWE of my code would look like: \documentclass{article} \pagestyle{headings} \begin{document} \tableofcontents \...
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I'm using book style, but it happens with a report style, too. I'm using the \chapter* command and I want that the chapter section appears in the table of contents, but it doesn't appear. This is the ...
15 30 50 per page | 2021-03-07 18:33:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.954268753528595, "perplexity": 1928.6735936773655}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178378043.81/warc/CC-MAIN-20210307170119-20210307200119-00402.warc.gz"} |
http://blog.assetnote.io/ | Taking over Azure DevOps Accounts with 1 Click
When performing subdomain takeovers, you should be asking yourself, what is the impact, and how do I prove it? This was especially the case when taking over the subdomain project-cascade.visualstudio.com. | 2021-03-07 05:48:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.30130013823509216, "perplexity": 2192.784712577534}, "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-2021-10/segments/1614178376144.64/warc/CC-MAIN-20210307044328-20210307074328-00162.warc.gz"} |
https://quant.stackexchange.com/questions/34773/reference-for-why-a-derivative-is-a-derivative-and-not-say-an-insurance-contract | # Reference for why a derivative is a derivative and not say an insurance contract
I recently spoke to an options trader that tried to demonstrate option pricing by considering a random walk of balls dropping down a lattice so the underlying stochastic process is a simple random walk of say 100 steps.
The contract considered is $(U_{100}-K)^{+}$ where $U_{100}$ is the number of times the ball goes "up". He states that this is an option. I think he doesn't understand what an option is because there is no underlying market in this case (ie you can't exactly trade the balls to hedge your position and there is no underlying that moves based on what the ball does except for the contract itself). I would say that this is a bet or a game that you would pay for at a casino.
So my question is: Is there a resource that actively explains or demonstrates why a derivative is called a derivative? As in why insurance and bets are fundamentally different from derivatives?
• If markets aren't complete, is there still a difference between a bet and a derivative in your case then? As far as I'm concerned, a derivative is an instrument whose price is derived from another instrument. Whether that instrument is tradeable and whether a replication argumetn can be used to determine a fair value for a derivative is a separate thing for me
– Bram
Jul 3, 2017 at 6:08
• Completeness is stronger than arbitrage free but you only need arbitrage free and some liquidity for a no arbitrage price to exist. I would agree with what you say if there is no liquidity in the market (but is it really a market in this case?). If you want to go into the realm of BSDEs and dynamic programming then super hedging does address this. Jul 16, 2017 at 23:19
• A derivative/bet/insurable contract/etc. can be defined as a function $f(\cdot)$ on a random variable $X$ at a future time $T$: $f(T,X)$. Whether the risk factor $X$ can be (reasonably easily) traded or not will determine how to approach the problem of pricing the payout $f(T,X)$, either by statistical arguments or by no-arbitrage arguments. Nov 22, 2017 at 14:54
From the introduction (Chapter 1) of Baxter's and Rennie's excellent book Financial Calculus:
With markets where the stock can be bought and sold freely and arbitrarily positive and negative amounts of stock can be maintained without cost, trying to trade forward using the strong law would lead to disaster […].
[…]
But the existence of an arbitrage price, however surprising, overrides the strong law. To put it simply, if there is an arbitrage price, any other price is too dangerous to quote.
[…]
The strong law and expectation give the wrong price for forwards. But in a certain sense, the forward is a special case. The construction strategy $-$ buying the stock and holding it $-$ certainly wouldn’t work for more complex claims. The standard call option which offers the buyer the right but not the obligation to receive the stock for some strike price agreed in advance certainly couldn’t be constructed this way. If the stock price ends up above the strike, then the buyer would exercise the option and ask to receive the stock – having it salted away in a drawer would then be useful to the seller. But if the stock price ends up below the strike, the buyer will abandon the option and any stock owned by the seller would have incurred a pointless loss.
Thus maybe a strong-law price would be appropriate for a call option, and until 1973, many people would have agreed. Almost everything appeared safe to price via expectation and the strong law, and only forwards and close relations seemed to have an arbitrage price. Since 1973, however, and the infamous Black-Scholes paper, just how wrong this is has slowly come out. Nowhere in this book will we use the strong law again. […] All derivatives can be built from the underlying $-$ arbitrage lurks everywhere.
• You are welcome @RNvsRW. The whole Introduction of their book discusses the statistical vs. the arbitrage approach to pricing if you want more details. Jun 20, 2017 at 10:46 | 2022-07-03 21:02:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.38654807209968567, "perplexity": 693.0827389258012}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104249664.70/warc/CC-MAIN-20220703195118-20220703225118-00760.warc.gz"} |
https://mathoverflow.net/questions/306684/numerical-minimization-spectral-norm-under-diagonal-similarity | # Numerical minimization spectral norm under diagonal similarity
This question is a follow up.
Let $A$ be a real square matrix of size $n \times n$. How to determine the minimum spectral norm under diagonal similarity, i.e.,
$$s(A) = \inf_{D} \lVert D^{-1} A D\rVert_2,$$ where $D$ is a non-singular, diagonal real matrix. As it is unlikely to find an analytical upper bound, I would like to ask how $s(A)$ could be determined numericallly.
• I believe the following holds $$||D^{-1} A D ||_2 \leq ||D^{-1}||_2 ||A||_2 ||D||_2 = ||A||_2 \kappa(D),$$ where $\kappa(D) = \frac{\sigma_{\max}(D)}{\sigma_{\min}(D)}$, also known as the condition number of $D$. The smallest condition number is 1, therefore, $$s(A) \leq ||A||_2.$$ I realize this doesn't exactly answer your question, but it does give a bound. – artificial_moonlet Jul 25 '18 at 10:16
• Thanks for the comment. Isn't your inequality clear from $D$ being the identity matrix or am I missing something? – Sebastian Schlecht Jul 25 '18 at 11:46
• In case you are interested in the Frobenius norm version of this, have a look at radio.feld.cvut.cz/matlab/toolbox/robust/osborne.html -- the Frobenius norm version of this problem is the same as the $\ell_2$-norm matrix balancing, which ends up being a convex optimization problem. More generally, if you look into the matrix balancing literature you'll likely find a solution to your question. – Suvrit Jul 30 '18 at 1:08
## 1 Answer
Here is a better, more direct solution.
This problem can be cast as a Generalized Eigenvalue Problem as is shown by Boyd, El Ghaoui, Feron, and Balakrishnan on page 39 (§3.3) of Linear Matrix Inequalities in System and Control Theory:
$$s(A) = \inf \left\{\gamma \mid A^*PA < \gamma^2 P \textrm{ for diagonal } P > 0 \right\}$$
Previous answer
Unfortunately behind a paywall, but following my own comment about chasing literature on matrix balancing, I found the following old paper that solves your problem (EDIT: As noted by Sebastian, this paper actually only provides a solution for a restricted case), not only for the operator norm, but for a variety of other norms.
T. Ström. Minimization of norms and logarithmic norms by diagonal similarities. Computing, March 1972, Volume 10, Issue 1–2, pp 1–7.
• Thank you very much for this very helpful lead. Btw, I'm only concerned with the operator norm. The recommended paper solves the problem almost. It is unfortunately restricted to companion matrices for the operator norm (Theorem 2). I will keep digging... – Sebastian Schlecht Jul 30 '18 at 10:20
• The LMI book noted above solves it numerically as a generalized eigenvalue problem, while interestingly Ström's paper, as you noted, solves it for companion matrices (in some sense, in fact "analytically"). – Suvrit Jul 30 '18 at 13:56
• The formulation as a GEVP in the referred book looks perfect. I need some more time to understand how to numerically solve this GEVP. – Sebastian Schlecht Jul 30 '18 at 14:22
• Have a look at the LMI toolbox (in Matlab, or possibly even separately). I think it comes equipped with some kind of interior point methods for solving this GEVP. – Suvrit Jul 30 '18 at 18:36
• @Sebastian Schlecht This is not convex, but is a quasi-convex Bilinear Matrix Inequality (due to product of $\gamma^2$ with $P$), hence can be solved by bisection (employing LMI solver for bisection sub-problems) - see, for example, yalmip.github.io/example/decayrate. However, Nesterov and Nemirovski's Projective Method, used by LMI toolbox's gevp, may be better (faster). – Mark L. Stone Jul 30 '18 at 21:17 | 2021-04-15 14:12:06 | {"extraction_info": {"found_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": 0.8578182458877563, "perplexity": 483.6239250501565}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038085599.55/warc/CC-MAIN-20210415125840-20210415155840-00450.warc.gz"} |
https://chinaproject.harvard.edu/publications/type/book-chapter | # Publications by Type: Book Chapter
2020
Cao Jing, Mun S. Ho, and Wenhao Hu. 2020. “Analyzing carbon price policies using a general equilibrium model with household energy demand functions.” In Measuring Economic Growth and Productivity: Foundations, KLEMS Production Models, and Extensions, edited by Barbara Fraumeni. Cambridge, MA: Academic Press. Publisher's VersionAbstract
Multi-sector general equilibrium models are used to simulate the effects of environmental policies on industry output and consumption at disaggregated levels. The specification of household demand in such models often use simpler forms such as CES or Linear Expenditure Systems since there are few estimates of more flexible systems. We estimate a 2-stage translog utility function that explicitly accounts for detailed energy expenditures to allow us to capture the price and income effects more accurately than these simpler forms. We incorporate this into a China growth model to simulate the effects of a carbon price to achieve the government targets for the Climate Change (Paris) agreements.
Final Manuscript in DASH.
An edited volume dedicated to Prof. Dale W. Jorgenson by his students and collaborators.
Richard Goettle, Mun S. Ho, and Peter Wilcoxen. 2020. “Emissions accounting and carbon tax incidence in CGE models: bottom-up versus top-down.” In Measuring Economic Growth and Productivity: Foundations, KLEMS Production Models, and Extensions, edited by Fraumeni, B, 1st ed. Cambridge, MA: Academic Press. Publisher's VersionAbstract
Multi-sector general equilibrium models are the work-horses used to analyze the impact of carbon prices in climate policy discussions. Such models often have distinct industries to represent coal, liquid fuels, and gas production where the output over time is represented by quantity and price indexes. The industries that buy these fuels, however, do not use a common homogenous quantity (e.g., steam coal vs. metallurgical coal) and have distinct purchasing price indexes. In accounting for energy use or CO2 emissions, modelers choose to attach coefficients either bottom-up to a sector specific input index or top-down to an average output index and this choice has a direct bearing on the incidence of carbon taxation. We discuss how different accounting methods for the differences in prices can have a large effect on the simulated impact of carbon prices. We emphasize the importance for modelers to be explicit about their methods.
An edited volume dedicated to Prof. Dale W. Jorgenson by his students and collaborators. Final Manuscript in DASH
2018
Michael.B. McElroy. 2018. “Can China address air pollution and climate change?” In The China Questions: Critical Insights into a Rising Power, edited by Jennifer Rudolph and Michael Szonyi. Cambridge: Harvard University Press. Publisher's Version
2017
Xi Lu and Michael B. McElroy. 2017. “Global potential for wind generated electricity.” In Wind Energy Engineering: A Handbook for Onshore and Offshore Wind Turbines, edited by Trevor M. Letcher. Amsterdam: Elsevier. Publisher's VersionAbstract
Wind Energy Engineering: A Handbook for Onshore and Offshore Wind Turbines is the most advanced, up-to-date and research-focused text on all aspects of wind energy engineering. Wind energy is pivotal in global electricity generation and for achieving future essential energy demands and targets. In this fast moving field this must-have edition starts with an in-depth look at the present state of wind integration and distribution worldwide, and continues with a high-level assessment of the advances in turbine technology and how the investment, planning, and economic infrastructure can support those innovations.
Each chapter includes a research overview with a detailed analysis and new case studies looking at how recent research developments can be applied. Written by some of the most forward-thinking professionals in the field and giving a complete examination of one of the most promising and efficient sources of renewable energy, this book is an invaluable reference into this cross-disciplinary field for engineers.
2013
Yu Zhao, Wei Wei, and Yu Lei. 2013. “An Anthropogenic Emission Inventory of Primary Air Pollutants in China for 2005 and 2010.” In Clearer Skies Over China: Reconciling Air Quality, Climate, and Economic Goals, Pp. 225-261. Cambridge, MA: MIT Press. Publisher's VersionAbstract
A groundbreaking U.S.–Chinese inquiry into the effects of recent air pollution controls and prospective carbon taxes on China's economy and environment.
China's carbon dioxide emissions now outstrip those of other countries and its domestic air quality is severely degraded, especially in urban areas. Its sheer size and its growing, fossil-fuel-powered economy mean that China's economic and environmental policy choices will have an outsized effect on the global environmental future. Over the last decade, China has pursued policies that target both fossil fuel use and atmospheric emissions, but these efforts have been substantially overwhelmed by the country's increasing energy demands. With a billion citizens still living on less than $4,000 per year, China's energy and environmental policies must be reconciled with the goals of maintaining economic growth and raising living standards. This book, a U.S.–Chinese collaboration of experts from Harvard and Tsinghua University, offers a groundbreaking integrated analysis of China's economy, emissions, air quality, public health, and agriculture. It first offers essential scientific context and accessible summaries of the book's policy findings; it then provides the underlying scientific and economic research. These studies suggest that China's recent sulfur controls achieved enormous environmental health benefits at unexpectedly low costs. They also indicate that judicious implementation of carbon taxes could reduce not only China's carbon emissions but also its air pollution more comprehensively than current single-pollutant policies, all at little cost to economic growth. Chris P Nielsen and Mun S Ho. 2013. “Atmospheric Environment in China: Introduction and Research Review.” In Clearer Skies Over China: Reconciling Air Quality, Climate, and Economic Goals, Pp. 3-58. Cambridge, MA: MIT Press. Publisher's VersionAbstract A groundbreaking U.S.–Chinese inquiry into the effects of recent air pollution controls and prospective carbon taxes on China's economy and environment. China's carbon dioxide emissions now outstrip those of other countries and its domestic air quality is severely degraded, especially in urban areas. Its sheer size and its growing, fossil-fuel-powered economy mean that China's economic and environmental policy choices will have an outsized effect on the global environmental future. Over the last decade, China has pursued policies that target both fossil fuel use and atmospheric emissions, but these efforts have been substantially overwhelmed by the country's increasing energy demands. With a billion citizens still living on less than$4,000 per year, China's energy and environmental policies must be reconciled with the goals of maintaining economic growth and raising living standards.
This book, a U.S.–Chinese collaboration of experts from Harvard and Tsinghua University, offers a groundbreaking integrated analysis of China's economy, emissions, air quality, public health, and agriculture. It first offers essential scientific context and accessible summaries of the book's policy findings; it then provides the underlying scientific and economic research. These studies suggest that China's recent sulfur controls achieved enormous environmental health benefits at unexpectedly low costs. They also indicate that judicious implementation of carbon taxes could reduce not only China's carbon emissions but also its air pollution more comprehensively than current single-pollutant policies, all at little cost to economic growth.
Yuxuan Wang. 2013. “Atmospheric Modeling of Pollutant Concentrations.” In Clearer Skies Over China: Reconciling Air Quality, Climate, and Economic Goals, Pp. 263-289. Cambridge, MA: MIT Press. Publisher's VersionAbstract
A groundbreaking U.S.–Chinese inquiry into the effects of recent air pollution controls and prospective carbon taxes on China's economy and environment.
China's carbon dioxide emissions now outstrip those of other countries and its domestic air quality is severely degraded, especially in urban areas. Its sheer size and its growing, fossil-fuel-powered economy mean that China's economic and environmental policy choices will have an outsized effect on the global environmental future. Over the last decade, China has pursued policies that target both fossil fuel use and atmospheric emissions, but these efforts have been substantially overwhelmed by the country's increasing energy demands. With a billion citizens still living on less than $4,000 per year, China's energy and environmental policies must be reconciled with the goals of maintaining economic growth and raising living standards. This book, a U.S.–Chinese collaboration of experts from Harvard and Tsinghua University, offers a groundbreaking integrated analysis of China's economy, emissions, air quality, public health, and agriculture. It first offers essential scientific context and accessible summaries of the book's policy findings; it then provides the underlying scientific and economic research. These studies suggest that China's recent sulfur controls achieved enormous environmental health benefits at unexpectedly low costs. They also indicate that judicious implementation of carbon taxes could reduce not only China's carbon emissions but also its air pollution more comprehensively than current single-pollutant policies, all at little cost to economic growth. Yu Lei. 2013. “Benefits to Human Health and Agricultural Productivity of Reduced Air Pollution.” In Clearer Skies Over China: Reconciling Air Quality, Climate, and Economic Goals, Pp. 291-328. Cambridge, MA: MIT Press. Publisher's VersionAbstract A groundbreaking U.S.–Chinese inquiry into the effects of recent air pollution controls and prospective carbon taxes on China's economy and environment. China's carbon dioxide emissions now outstrip those of other countries and its domestic air quality is severely degraded, especially in urban areas. Its sheer size and its growing, fossil-fuel-powered economy mean that China's economic and environmental policy choices will have an outsized effect on the global environmental future. Over the last decade, China has pursued policies that target both fossil fuel use and atmospheric emissions, but these efforts have been substantially overwhelmed by the country's increasing energy demands. With a billion citizens still living on less than$4,000 per year, China's energy and environmental policies must be reconciled with the goals of maintaining economic growth and raising living standards.
This book, a U.S.–Chinese collaboration of experts from Harvard and Tsinghua University, offers a groundbreaking integrated analysis of China's economy, emissions, air quality, public health, and agriculture. It first offers essential scientific context and accessible summaries of the book's policy findings; it then provides the underlying scientific and economic research. These studies suggest that China's recent sulfur controls achieved enormous environmental health benefits at unexpectedly low costs. They also indicate that judicious implementation of carbon taxes could reduce not only China's carbon emissions but also its air pollution more comprehensively than current single-pollutant policies, all at little cost to economic growth.
Jing Cao, Mun S Ho, and Dale W Jorgenson. 2013. “The Economics of Environmental Policies in China.” In Clearer Skies Over China: Reconciling Air Quality, Climate, and Economic Goals, Pp. 329-372. Cambridge, MA: MIT Press. Publisher's VersionAbstract
A groundbreaking U.S.–Chinese inquiry into the effects of recent air pollution controls and prospective carbon taxes on China's economy and environment.
China's carbon dioxide emissions now outstrip those of other countries and its domestic air quality is severely degraded, especially in urban areas. Its sheer size and its growing, fossil-fuel-powered economy mean that China's economic and environmental policy choices will have an outsized effect on the global environmental future. Over the last decade, China has pursued policies that target both fossil fuel use and atmospheric emissions, but these efforts have been substantially overwhelmed by the country's increasing energy demands. With a billion citizens still living on less than $4,000 per year, China's energy and environmental policies must be reconciled with the goals of maintaining economic growth and raising living standards. This book, a U.S.–Chinese collaboration of experts from Harvard and Tsinghua University, offers a groundbreaking integrated analysis of China's economy, emissions, air quality, public health, and agriculture. It first offers essential scientific context and accessible summaries of the book's policy findings; it then provides the underlying scientific and economic research. These studies suggest that China's recent sulfur controls achieved enormous environmental health benefits at unexpectedly low costs. They also indicate that judicious implementation of carbon taxes could reduce not only China's carbon emissions but also its air pollution more comprehensively than current single-pollutant policies, all at little cost to economic growth. Yu Zhao. 2013. “Primary Air Pollutant Emissions of Coal-Fired Power Plants in China.” In Clearer Skies Over China: Reconciling Air Quality, Climate, and Economic Goals, Pp. 161-202. Cambridge, MA: MIT Press. Publisher's VersionAbstract A groundbreaking U.S.–Chinese inquiry into the effects of recent air pollution controls and prospective carbon taxes on China's economy and environment. China's carbon dioxide emissions now outstrip those of other countries and its domestic air quality is severely degraded, especially in urban areas. Its sheer size and its growing, fossil-fuel-powered economy mean that China's economic and environmental policy choices will have an outsized effect on the global environmental future. Over the last decade, China has pursued policies that target both fossil fuel use and atmospheric emissions, but these efforts have been substantially overwhelmed by the country's increasing energy demands. With a billion citizens still living on less than$4,000 per year, China's energy and environmental policies must be reconciled with the goals of maintaining economic growth and raising living standards.
This book, a U.S.–Chinese collaboration of experts from Harvard and Tsinghua University, offers a groundbreaking integrated analysis of China's economy, emissions, air quality, public health, and agriculture. It first offers essential scientific context and accessible summaries of the book's policy findings; it then provides the underlying scientific and economic research. These studies suggest that China's recent sulfur controls achieved enormous environmental health benefits at unexpectedly low costs. They also indicate that judicious implementation of carbon taxes could reduce not only China's carbon emissions but also its air pollution more comprehensively than current single-pollutant policies, all at little cost to economic growth.
Yu Lei, Qiang Zhang, Chris P Nielsen, and Kebin He. 2013. “Primary Air Pollutants and CO2 Emissions from Cement Production in China.” In Clearer Skies Over China: Reconciling Air Quality, Climate, and Economic Goals, Pp. 203-224. Cambridge, MA: MIT Press. Publisher's VersionAbstract
A groundbreaking U.S.–Chinese inquiry into the effects of recent air pollution controls and prospective carbon taxes on China's economy and environment.
China's carbon dioxide emissions now outstrip those of other countries and its domestic air quality is severely degraded, especially in urban areas. Its sheer size and its growing, fossil-fuel-powered economy mean that China's economic and environmental policy choices will have an outsized effect on the global environmental future. Over the last decade, China has pursued policies that target both fossil fuel use and atmospheric emissions, but these efforts have been substantially overwhelmed by the country's increasing energy demands. With a billion citizens still living on less than $4,000 per year, China's energy and environmental policies must be reconciled with the goals of maintaining economic growth and raising living standards. This book, a U.S.–Chinese collaboration of experts from Harvard and Tsinghua University, offers a groundbreaking integrated analysis of China's economy, emissions, air quality, public health, and agriculture. It first offers essential scientific context and accessible summaries of the book's policy findings; it then provides the underlying scientific and economic research. These studies suggest that China's recent sulfur controls achieved enormous environmental health benefits at unexpectedly low costs. They also indicate that judicious implementation of carbon taxes could reduce not only China's carbon emissions but also its air pollution more comprehensively than current single-pollutant policies, all at little cost to economic growth. Chris P Nielsen, Mun S Ho, Jing Cao, Yu Lei, Yuxuan Wang, and Yu Zhao. 2013. “Summary: Carbon Taxes for 2013-2020.” In Clearer Skies Over China: Reconciling Air Quality, Climate, and Economic Goals, Pp. 103-157. Cambridge, MA: MIT Press. Publisher's VersionAbstract A groundbreaking U.S.–Chinese inquiry into the effects of recent air pollution controls and prospective carbon taxes on China's economy and environment. China's carbon dioxide emissions now outstrip those of other countries and its domestic air quality is severely degraded, especially in urban areas. Its sheer size and its growing, fossil-fuel-powered economy mean that China's economic and environmental policy choices will have an outsized effect on the global environmental future. Over the last decade, China has pursued policies that target both fossil fuel use and atmospheric emissions, but these efforts have been substantially overwhelmed by the country's increasing energy demands. With a billion citizens still living on less than$4,000 per year, China's energy and environmental policies must be reconciled with the goals of maintaining economic growth and raising living standards.
This book, a U.S.–Chinese collaboration of experts from Harvard and Tsinghua University, offers a groundbreaking integrated analysis of China's economy, emissions, air quality, public health, and agriculture. It first offers essential scientific context and accessible summaries of the book's policy findings; it then provides the underlying scientific and economic research. These studies suggest that China's recent sulfur controls achieved enormous environmental health benefits at unexpectedly low costs. They also indicate that judicious implementation of carbon taxes could reduce not only China's carbon emissions but also its air pollution more comprehensively than current single-pollutant policies, all at little cost to economic growth.
Chris P Nielsen, Mun S Ho, Yu Zhao, Yuxuan Wang, Yu Lei, and Jing Cao. 2013. “Summary: Sulfur Mandates and Carbon Taxes for 2006-2010.” In Clearer Skies Over China: Reconciling Air Quality, Climate, and Economic Goals, Pp. 59-102. Cambridge, MA: MIT Press. Publisher's VersionAbstract
A groundbreaking U.S.–Chinese inquiry into the effects of recent air pollution controls and prospective carbon taxes on China's economy and environment.
China's carbon dioxide emissions now outstrip those of other countries and its domestic air quality is severely degraded, especially in urban areas. Its sheer size and its growing, fossil-fuel-powered economy mean that China's economic and environmental policy choices will have an outsized effect on the global environmental future. Over the last decade, China has pursued policies that target both fossil fuel use and atmospheric emissions, but these efforts have been substantially overwhelmed by the country's increasing energy demands. With a billion citizens still living on less than \$4,000 per year, China's energy and environmental policies must be reconciled with the goals of maintaining economic growth and raising living standards.
This book, a U.S.–Chinese collaboration of experts from Harvard and Tsinghua University, offers a groundbreaking integrated analysis of China's economy, emissions, air quality, public health, and agriculture. It first offers essential scientific context and accessible summaries of the book's policy findings; it then provides the underlying scientific and economic research. These studies suggest that China's recent sulfur controls achieved enormous environmental health benefits at unexpectedly low costs. They also indicate that judicious implementation of carbon taxes could reduce not only China's carbon emissions but also its air pollution more comprehensively than current single-pollutant policies, all at little cost to economic growth.
2012
Jing Cao, Mun S Ho, and Dale W Jorgenson. 2012. “An integrated assessment of the economic costs and environmental benefits of pollution and climate control.” In The Chinese Economy: A New Transition, edited by Masahiko Aoki. London: Palgrave Macmillan. Publisher's Version
2008
Peter Rogers and Sumeeta Srinivasan. 2008. “Comparing sustainable cities—Examples from China, India and the USA.” In Sustainable urban development in China: Wishful thinking or reality?, edited by Marco Keiner. Munster, Germany: Verlagshaus Monsenstein und Vannerdat OHG. Publisher's VersionAbstract
Due to an unprecedented economic
growth, fuelled by a pro-growth policy,
China’s cities are mushrooming.
In the coming years, the mass migration
from rural to urban areas will continue.
The demand for energy and resources will
continue to rise.
China’s cities will increasingly contribute
to global warming and the depletion of
the environment.
The crucial question is: Can urban development
in China become sustainable?
Sumeeta Srinivasan. 2008. “A visual exploration of the accessibility of low income women: Chengdu, China and Chennai, India.” In Gendered Mobilities, edited by Tanu Priya Uteng and Tim Cresswell. Hampshire, UK: Ashgate Publishing. Publisher's VersionAbstract
Being socially and geographically mobile is generally seen as one of the central aspects of women's wellbeing. Alongside health, education and political participation, mobility is indispensable in order for women to reach goals such as agency and freedom. Building on new philosophical underpinnings of 'mobility', whereby society is seen to be framed by the convergence of various mobilities, this volume focuses on the intersection of mobility, social justice and gender. The authors reflect on five highly interdependent mobilities that form and reform social life.
2007
Chris P Nielsen and Mun S Ho. 2007. “Air pollution and health damages in China: An introduction and review.” In Clearing the air: The health and economic damages of air pollution in China, edited by Chris P Nielsen and Mun S Ho. Cambridge, MA: MIT Press. Publisher's VersionAbstract
An interdisciplinary, quantitative assessment of the health and economic costs of air pollution in China, and of market-based policies to build environmental protection into economic development.
China's historic economic expansion is driven by fossil fuels, which increase its emissions of both local air pollutants and greenhouse gases dramatically. Clearing the Air is an innovative, quantitative examination of the national damage caused by China's degraded air quality, conducted in a pathbreaking, interdisciplinary U.S.-China collaboration. Its damage estimates are allocated by sector, making it possible for the first time to judge whether, for instance, power generation, transportation, or an unexpected source such as cement production causes the greatest environmental harm. Such objective analyses can reset policy priorities.
Clearing the Air uses this information to show how appropriate "green" taxes might not only reduce emissions and health damages but even enhance China's economic growth. It also shows to what extent these same policies could limit greenhouse gases, suggesting that wealthier nations have a responsibility to help China build environmental protection into its growth.
Clearing the Air is written for diverse readers, providing a bridge from underlying research to policy implications, with easily accessible overviews of issues and summaries of the findings for nonspecialists and policymakers followed by more specialized, interlinked studies of primary interest to scholars. Taken together, these analyses offer a uniquely integrated assessment that supports the book's economic and policy recommendations.
Ying Zhou and James K Hammitt. 2007. “The economic value of air-pollution-related health risks in China: A contingent valuation study.” In Clearing the air: The health and economic damages of air pollution in China, edited by Mun S Ho and Chris P Nielsen. Cambridge, MA: MIT Press. Publisher's VersionAbstract
An interdisciplinary, quantitative assessment of the health and economic costs of air pollution in China, and of market-based policies to build environmental protection into economic development.
China's historic economic expansion is driven by fossil fuels, which increase its emissions of both local air pollutants and greenhouse gases dramatically. Clearing the Air is an innovative, quantitative examination of the national damage caused by China's degraded air quality, conducted in a pathbreaking, interdisciplinary U.S.-China collaboration. Its damage estimates are allocated by sector, making it possible for the first time to judge whether, for instance, power generation, transportation, or an unexpected source such as cement production causes the greatest environmental harm. Such objective analyses can reset policy priorities.
Clearing the Air uses this information to show how appropriate "green" taxes might not only reduce emissions and health damages but even enhance China's economic growth. It also shows to what extent these same policies could limit greenhouse gases, suggesting that wealthier nations have a responsibility to help China build environmental protection into its growth.
Clearing the Air is written for diverse readers, providing a bridge from underlying research to policy implications, with easily accessible overviews of issues and summaries of the findings for nonspecialists and policymakers followed by more specialized, interlinked studies of primary interest to scholars. Taken together, these analyses offer a uniquely integrated assessment that supports the book's economic and policy recommendations.
Jonathan I Levy and Susan Greco. 2007. “Estimating health effects of air pollution in China: An introduction to intake fraction and the epidemiology.” In Clearing the air: The health and economic damages of air pollution in China, edited by Mun S Ho and Chris P Nielsen. Cambridge, MA: MIT Press. Publisher's Version
Bingjiang Liu and Jiming Hao. 2007. “Local population exposure to pollutants from the electric power sector.” In Clearing the air: The health and economic damages of air pollution in China, edited by Mun S Ho and Chris P Nielsen. Cambridge, MA: MIT Press. Publisher's VersionAbstract
An interdisciplinary, quantitative assessment of the health and economic costs of air pollution in China, and of market-based policies to build environmental protection into economic development.
China's historic economic expansion is driven by fossil fuels, which increase its emissions of both local air pollutants and greenhouse gases dramatically. Clearing the Air is an innovative, quantitative examination of the national damage caused by China's degraded air quality, conducted in a pathbreaking, interdisciplinary U.S.-China collaboration. Its damage estimates are allocated by sector, making it possible for the first time to judge whether, for instance, power generation, transportation, or an unexpected source such as cement production causes the greatest environmental harm. Such objective analyses can reset policy priorities.
Clearing the Air uses this information to show how appropriate "green" taxes might not only reduce emissions and health damages but even enhance China's economic growth. It also shows to what extent these same policies could limit greenhouse gases, suggesting that wealthier nations have a responsibility to help China build environmental protection into its growth.
Clearing the Air is written for diverse readers, providing a bridge from underlying research to policy implications, with easily accessible overviews of issues and summaries of the findings for nonspecialists and policymakers followed by more specialized, interlinked studies of primary interest to scholars. Taken together, these analyses offer a uniquely integrated assessment that supports the book's economic and policy recommendations. | 2020-05-27 12:56:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2945883870124817, "perplexity": 7347.505513161804}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347394074.44/warc/CC-MAIN-20200527110649-20200527140649-00339.warc.gz"} |
https://zbmath.org/authors/?q=ai%3Agwiazda.piotr | # zbMATH — the first resource for mathematics
## Gwiazda, Piotr
Compute Distance To:
Author ID: gwiazda.piotr Published as: Gwiazda, P.; Gwiazda, Piotr
Documents Indexed: 77 Publications since 1999
all top 5
#### Co-Authors
7 single-authored 30 Świerczewska-Gwiazda, Agnieszka 9 Marciniak-Czochra, Anna K. 8 Bulíček, Miroslav 7 Zatorska-Goldstein, Anna 6 Wiedemann, Emil 5 Málek, Josef 5 Wróblewska, Aneta 4 Carrillo de la Plata, José Antonio 4 Chełmiński, Krzysztof 4 Chlebicka, Iwona 4 Colombo, Rinaldo M. 4 Feireisl, Eduard 4 Świerczewska, Agnieszka 4 Ulikowska, Agnieszka 4 Wittbold, Petra 4 Zimmermann, Alexandra 3 Dębiec, Tomasz 3 Klawe, Filip Z. 3 Wróblewska-Kamińska, Aneta 2 Kropielnicka, Karolina 2 Łyczek, Kamila 2 Minakowski, Piotr 2 Rosińska, Magdalena 2 Thieme, Horst R. 1 Ahmida, Youssef 1 Bardos, Claude Williams 1 Benzoni-Gavage, Sylvie 1 Busse, J.-E. 1 De Lellis, Camillo 1 Doumic, Marie 1 Hille, Sander Cornelis 1 Jabłoński, Jȩdrzej 1 Jamróz, Grzegorz 1 Kalousek, Martin 1 Lorenz, Thomas 1 Lukáčová-Medvid’ová, Mária 1 Miasojedow, Błażej 1 Michálek, Martin 1 Mizerová, Hana 1 Orliński, Piotr 1 Owczarek, Sebastian 1 Perthame, Benoît 1 Rajagopal, Kumbakonam Ramamani 1 Skrzypczak, Iwona 1 Süli, Endre E. 1 Titi, Edriss Saleh 1 Tzavaras, Athanasios E. 1 Warzyński, Andrzej 1 Youssfi, Ahmed
all top 5
#### Serials
8 Journal of Differential Equations 7 Mathematical Methods in the Applied Sciences 4 M$^3$AS. Mathematical Models & Methods in Applied Sciences 3 SIAM Journal on Mathematical Analysis 3 Calculus of Variations and Partial Differential Equations 3 Nonlinear Analysis. Real World Applications 2 Archive for Rational Mechanics and Analysis 2 Nonlinearity 2 Colloquium Mathematicum 2 Journal of Functional Analysis 2 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 2 Asymptotic Analysis 2 Continuum Mechanics and Thermodynamics 2 Topological Methods in Nonlinear Analysis 2 Positivity 2 Discrete and Continuous Dynamical Systems. Series S 2 Kinetic and Related Models 1 Computers & Mathematics with Applications 1 Journal of Mathematical Biology 1 Bulletin of the Polish Academy of Sciences. Technical Sciences 1 Journal of the London Mathematical Society. Second Series 1 Mathematische Nachrichten 1 SIAM Journal on Numerical Analysis 1 Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 1 Numerical Methods for Partial Differential Equations 1 Applied Mathematics Letters 1 Communications in Partial Differential Equations 1 Journal of Nonlinear Science 1 Discrete and Continuous Dynamical Systems 1 European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations 1 Markov Processes and Related Fields 1 ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik 1 Proceedings of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 1 Comptes Rendus. Mathématique. Académie des Sciences, Paris 1 Central European Journal of Mathematics 1 Communications in Mathematical Sciences 1 Journal of Hyperbolic Differential Equations 1 Networks and Heterogeneous Media 1 Advances in Calculus of Variations 1 Journal of Theoretical Biology
all top 5
#### Fields
64 Partial differential equations (35-XX) 25 Fluid mechanics (76-XX) 15 Mechanics of deformable solids (74-XX) 15 Biology and other natural sciences (92-XX) 9 Functional analysis (46-XX) 7 Numerical analysis (65-XX) 6 Operator theory (47-XX) 4 Measure and integration (28-XX) 3 Integral equations (45-XX) 3 Probability theory and stochastic processes (60-XX) 2 Ordinary differential equations (34-XX) 2 Dynamical systems and ergodic theory (37-XX) 2 Global analysis, analysis on manifolds (58-XX) 2 Statistical mechanics, structure of matter (82-XX) 1 Real functions (26-XX) 1 Harmonic analysis on Euclidean spaces (42-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 Statistics (62-XX) 1 Classical thermodynamics, heat transfer (80-XX) 1 Geophysics (86-XX) 1 Operations research, mathematical programming (90-XX) 1 Systems theory; control (93-XX)
#### Citations contained in zbMATH
Access to citation profile requires subscription | 2019-10-17 14:13:26 | {"extraction_info": {"found_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": 0.21219953894615173, "perplexity": 8487.038941201166}, "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-2019-43/segments/1570986675316.51/warc/CC-MAIN-20191017122657-20191017150157-00373.warc.gz"} |
https://mathoverflow.net/questions/355948/distance-function-and-its-approximation | Distance function and its approximation
An easy and quick question:
Consider a function $$u\in C(\Omega)$$, where $$\Omega$$ is a bounded domain in $$\mathbb{R}^n$$.
Define a function $$Q$$ that measures the distance of a point $$(x,y) \in\mathbb{R}^{n+1}$$ from the graph of $$u$$. That is, consider the function $$Q(x,y)=\inf_{z\in \mathrm{graph}(u)} d_z (x,y)$$
where $$d_z(\zeta)=|\zeta - z|$$.
My question is:
• why the function $$Q$$ should be the limit of smooth approximations $$Q_p \geq Q$$ given by $$Q_p(x,y)=\bigg[\int_\Omega \{ d_{(\xi,u(\xi))}(x,y)\}^{-p} \mathrm{d}\xi \bigg]^{-1/p}\quad \textrm{for } (x,y) \notin \mathrm{graph}(u)$$
• why the derivative is $$DQ_p(x,y)=(Q_p(x,y) )^{p+1} \int_\Omega |(x-\xi, y-u(\xi)|^{-p-2}(x-\xi,y-u(\xi))\,\mathrm{d}\xi$$
I just tried to show the uniformly convergence. Can somebody check this following calculation?
Let $d_\xi (x,y) = d_{(\xi,u(\xi))}(x,y)\$. One have
$$d_\xi (x,y) \leq \sup_{\xi} d_\xi (x,y) \rightarrow \dfrac{1}{d_\xi (x,y) } \geq \dfrac{1}{\sup_{\xi} d_\xi (x,y)}$$ so applying the integral $$\int_\Omega \bigg( \dfrac{1}{d_\xi (x,y) } \bigg)^p \geq \bigg( \dfrac{1}{\sup_{\xi} d_\xi (x,y)} \bigg) ^p |\Omega|$$ and so we have $$\bigg[\int_\Omega \{ d_{(\xi,u(\xi))}(x,y)\}^{p} \mathrm{d}\xi \bigg]^{1/p}\geq \dfrac{1}{\sup_{\xi} d_\xi (x,y)} |\Omega|^{1/p}$$ and accordingly $$\dfrac{1}{\bigg[\int_\Omega \{ d_{(\xi,u(\xi))}(x,y)\}^{p} \mathrm{d}\xi \bigg]^{1/p}} \leq \dfrac{1}{\bigg( \dfrac{1}{\sup_{\xi} d_\xi (x,y)} \bigg) |\Omega|^{1/p}}$$ Now to prove the uniformly convergence, one need $$\lim_{p\rightarrow \infty } \sup_{(x,y)} | Q_p(x,y) - Q(x,y)| \rightarrow 0$$ By the calculation above, we have $$\lim_{p\rightarrow \infty } \sup_{(x,y)} | Q_p(x,y) - Q(x,y)|\leq \lim_{p\rightarrow \infty } \sup_{(x,y)} \bigg| \dfrac{1}{\bigg( \dfrac{1}{\sup_{\xi} d_\xi (x,y)} \bigg) |\Omega|^{1/p}} - \inf_\xi d_\xi(x,y)\bigg|$$ and using the fact that, in general, $$\dfrac{1}{\sup_x \dfrac{1}{f(x)}} \geq \inf_x f(x)$$ follows $$\lim_{p\rightarrow \infty } \sup_{(x,y)} | Q_p(x,y) - Q(x,y)|\leq \lim_{p\rightarrow \infty } \sup_{(x,y)} \bigg| \big(\dfrac{1}{|\Omega|^{1/p}} -1\big) \dfrac{1}{\sup_{\xi} d_\xi (x,y)}\bigg|$$
Now one can put $$\big(\dfrac{1}{|\Omega|^{1/p}} -1\big)$$ out of the supremum and calculete the limit for $$p\rightarrow \infty$$ the result is $$0$$.
Is this right?
The second identity appears to follow simply from the chain rule applied to $$Q_p(\eta) = f_p(g_p(\eta))$$, where \begin{align*} f_p(t) &=t^{-1/p},\\ g_p(x,y) &= \int_\Omega \left(d_{(\xi,u(\xi))}(x,y)\right)^{-p}\,\mathrm{d}\xi,\\ d_{(\xi,u(\xi))}(x,y) &= |(x-\xi,y-u(\xi))| \end{align*} by definition.
As for the first identity, it looks like $$Q_p$$ measures the inverse of the distance function $$d$$ in an $$L^p$$ norm, and thus the limit $$p\to\infty$$ should give the supremum norm for this inverse, hence the infimum of the distance itself.
• Thank you @gmvh . I wanted a more formal proof of the first question, for example show that $Q_p(x,y)$ converges uniformly to $Q(x,y)$...any idea? – Jason Mar 28 at 16:00 | 2020-08-13 06:35:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 29, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9982872009277344, "perplexity": 205.7102146316935}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738960.69/warc/CC-MAIN-20200813043927-20200813073927-00266.warc.gz"} |
https://encyclopediaofmath.org/wiki/Quantum_computation,_theory_of | # Quantum computation, theory of
The study of the model of computation in which the state space consists of linear superpositions of classical configurations and the computational steps consist of applying local unitary operators and measurements as permitted by quantum mechanics.
Quantum computation emerged in the 1980s when P. Benioff and R. Feynman realized that the apparent exponential complexity in simulating quantum physics could be overcome by using a sufficiently well controlled quantum mechanical system to perform a simulation. Quantum Turing machines were introduced by D. Deutsch in 1985 (cf. also Turing machine). Initial work focused on how quantum mechanics (cf. also Quantum field theory) could be used to implement classical computation (computation in the sense of A. Church and A.M. Turing), and on analyzing whether the quantum Turing machine model provided a universal model of computation. In the early 1990s, Deutsch and R. Jozsa found an oracle problem that could be solved faster on an error-free quantum computer than on any deterministic classical computer. E. Bernstein and U. Vazirani then formalized the notion of quantum complexity from a theoretical computer science point of view, and showed that with respect to oracles that reversibly compute classical functions, quantum computers are super-polynomially more efficient than classical computers. The gap was soon improved to an exponential one. This work culminated in the discovery, by P. Shor, of an efficient (that is, consuming only polynomial resources) algorithm for factoring large numbers and for computing discrete logarithms. It implied that widely-used public-key cryptographic systems would be insecure if quantum computers were available. Subsequently, L. Grover found an algorithm which permitted a square-root speed-up of unstructured search. Finding new algorithmic improvements achievable with quantum computers which are not reducible to Shor's or Grover's algorithm is currently (2000) an active research area. Also of great current interest is understanding how the problem of simulating quantum systems, known to be tractable on a quantum computer, relates to the problems conventionally studied within classical computational complexity theory (cf. also Complexity theory; Computational complexity classes). Comprehensive introductions to quantum computation and the known quantum algorithms may be found in [a2], [a1].
The algorithmic work described above firmly established the field of quantum computation in computer science. However, it was initially unclear whether quantum computation was a physically realizable model. Particularly worrisome was the fact that, in nature, quantum effects are rarely observable because physical noise processes tend to rapidly remove the necessary phase relationships. To solve the problem of quantum noise, Shor and A. Steane introduced quantum error-correcting codes (cf. also Error-correcting code). This idea was expanded and applied by several research groups to prove that under physically reasonable assumptions, fault tolerant quantum computation is possible. Among the assumptions are the requirements that quantum noise is sufficiently weak (below some constant threshold error per quantum bit and operation), and that the basic operations can be performed in parallel. As a result, there are now many intense experimental efforts devoted toward realizing quantum computation in a wide and increasing variety of physical systems. Progress to date (2000) has been modest, with existing systems limited to just a few qubits (quantum bits), and on the order of one hundred operations [a5].
Models of quantum computation largely parallel and generalize the classical models of computation. In particular, for formal studies of complexity, many researchers use various versions of quantum Turing machines (cf. also Turing machine), while quantum random access machines or quantum networks (also known as quantum circuits) are preferred for describing and investigating specific algorithms. To obtain a quantum version of a classical model of deterministic computation, one begins with the classical model's state space. The classical state space usually consists of an enumerable set of configurations $\psi _ { i }$, with index $i$ often constructed from strings of symbols. The quantum model associates to each $\psi _ { i }$ a member of a standard orthonormal basis $| i \rangle$ (called classical or logical states) of a Hilbert space $\mathcal{H}$. The states of the quantum model are given by "superpositions" of these basis states, which are unit vectors in $\mathcal{H}$. The classical model's initial state $\psi_0$ becomes the quantum model's initial state $| 0 \rangle$, and the classical model's transition function is replaced by a unitary operator $U$ acting on $\mathcal{H}$. $U$ has to satisfy certain locality restrictions that imply, for example, that $U | i \rangle$ must be a superposition of classical states that are accessible by an allowed classical transition function in one step from $\psi _ { i }$. The computation's answer can be obtained by measuring the state after each step. In the simplest case, the classical computation's answer is determined by whether the configuration is an "accepting" one. Accepting configurations form a set $\mathcal{A}$ which may be associated with the closed subspace of $\mathcal{H}$ spanned by the corresponding classical states. Let $P$ be the projection operator onto this subspace. If the state of the quantum model is $| {\phi} \rangle$, measurement has two possible outcomes. Either the new state is $P | \phi \rangle / \| P | \phi \rangle \|$ with probability $p = \| P | \phi \rangle \| ^ { 2 }$, in which case the computation "accepts" , or the state is $\left. ( 1 - P ) | \phi \rangle \middle/ \| ( 1 - P ) | \phi \rangle \|\right.$ with probability $1 - p$, in which case the computation continues. The possible measurement outcomes can be expanded by adding a set of "rejecting" states. In the early days of quantum computation there were lively discussions of how quantum Turing machines should halt, implying different rules about when measurements are applied during a computation.
The method outlined above for obtaining a quantum model of computation from a classical model yields a generalization of the classical model restricted to reversible transition functions. This implies that quantum complexity classes do not necessarily enlarge the classical analogues, particularly for the low-lying classes or when restricted models of computation (for example, finite state automata) are involved. To obtain a generalization of the usual model of computation it suffices to extend the set of transition operators with suitable irreversible ones. One way to do that is to allow transition operators which are the composition of a measurement (satisfying an appropriate locality constraint) followed by unitary operators depending on the measurement outcome. A different approach which works well for random access machines (RAMs) is to enhance the RAM by giving it access to an unbounded number of quantum bits which can be controlled by applying quantum gates (cf. Quantum information processing, science of). This is in effect how existing quantum algorithms are described and analyzed.
As in classical complexity studies, resources considered for quantum complexity include time and space (cf. also Complexity theory). In the context of irreversible processes, an additional resource that may be considered is entropy generated by irreversible operations. When analyzing algorithms based on quantum RAMs, it is also useful to separately account for classical and quantum resources. It is important to realize that if the complex coefficients of the unitary transition operators are rational (or, in general, computable complex numbers), then there is no difference between classical and quantum computability. Thus, the functions computable by quantum Turing machines are the same as those computable by classical Turing machines.
An important issue in studying quantum models of computation is how to define the computation's "answer" given that the output is intrinsically probabilistic. How this is defined can affect complexity classes. Guidance comes from studies of probabilistic (or randomized) computation, where the same issues arise. Since quantum computation with irreversibility can be viewed as a generalization of probabilistic computation, most comparisons of the quantum and classical complexity of algorithmic problems use bounds on the efficiency of probabilistic algorithms.
The best known quantum complexity class is the class of bounded-error quantum polynomial-time computable languages ($\mathbf{BQP}$). This is the class of languages decided in polynomial time with probability $> 2 / 3$ (acceptance) and $< 1 / 3$ (rejection) by a quantum Turing machine. Based on the oracle computing studies, the quantum factoring algorithm, and the difficulty of classically simulating quantum physics, it is conjectured that $\mathbf{BQP}$ strictly contains $\mathbf{BPP}$ (the class of bounded-error polynomial-time computable languages for the model of probabilistic classical computation). $\mathbf{BQP}$ is contained in $\mathcal{P} ^ { \# _\mathcal{ P}}$ (the class of languages decidable in polynomial time on a classical Turing machine given access to an oracle for computing the permanent of $0$-$1$ matrices — this class is contained in the class of languages computable using polynomial working space). Thus, a proof of the important conjecture that $\mathbf{BQP}$ is strictly larger than $\mathbf{BPP}$ will imply the long-sought result in classical computational complexity that .
The relationship of $\mathbf{BQP}$ to $\cal N P$ (the class of non-deterministic polynomial-time languages) is not known, though it is conjectured that $\mathcal{NP} \not< \mathbf{BQP}$. If this is not the case, it would have immense practical significance, as many combinatorial optimization problems are in $\cal N P$ (cf. also $\cal N P$). One reason for thinking that $\mathcal{NP} \not< \mathbf{BQP}$ is the fact that Grover's algorithm provides the optimal speedup for unstructured quantum search, and it is widely believed that the reason for the difficulty of solving $\cal N P$-complete problems is that it is essentially equivalent to searching an unstructured search space. A generalization of unstructured search involves determining properties of (quantum) oracles by means of queries. In classical computation, an oracle is a function $f$ with values in $\{ 0,1 \}$. The corresponding quantum oracle applies the unitary operator $\hat { f }$ defined on basis states by $\widehat { f } | x , 0 , w \rangle \rightarrow | x , f ( x ) , w \rangle$ and $\hat { f } | x , 1 , w \rangle \rightarrow | x , 1 - f ( x ) , w \rangle$. To query the oracle, one applies $\hat { f }$ to the current state. Grover's algorithm can be cast in terms of an oracle problem. The observation that this algorithm is optimal has been extended by using the method of polynomials [a4] to show that when no promise is made on the behaviour of the oracle, quantum computers are at most polynomially more efficient than classical computers.
An area where there are provable exponential gaps between the efficiency of quantum and classical computation occurs when communication resources are taken into consideration. This area is known as quantum communication complexity (introduced by A. Yao in 1993) and considers problems where two parties with quantum computers and a quantum channel between them (cf. Quantum information processing, science of) jointly compute a function of their respective inputs and wish to minimize the number of quantum bits communicated. The exponential gaps between quantum and classical communication complexity are so far confined to problems where the inputs to the function computed are constrained by a "promise" [a3]. The best known gap without a promise is a quadratic separation between classical and quantum protocols with bounded probability of error [a6]. Several research groups have developed techniques for proving lower bounds on quantum communication complexity, mostly variations of the log-rank lower bound also used in classical communication complexity. These results show that for some problems (for example, computing the inner product modulo two of bit strings known to the respective parties) there is little advantage to using quantum information processing.
#### References
[a1] J. Gruska, "Quantum computing" , McGraw-Hill (1999) [a2] M.A. Nielsen, I.L. Chuang, "Quantum computation and quantum information" , Cambridge Univ. Press (2000) [a3] R. Raz, "Exponential separation of quantum and classical communication complexity" , Proc. 31st Ann. ACM Symp. Theory of Computing (STOC'99) (1999) pp. 358–367 [a4] R. Beals, H. Buhrman, R. Cleve, M. Mosca, R. de Wolf, "Quantum lower bounds by polynomials" , Proc. 39th Ann. Symp. Foundations of Computer Sci. , IEEE Press (1998) pp. 352–361 [a5] Special focus issue, "Experimental proposals for quantum computation" Fortschr. Phys. , 48 (2000) pp. 767–1138 [a6] H. Buhrman, R. Cleve, A. Wigderson, "Quantum vs. classical communication and computation" , Proc. 30th Ann. ACM Symp. Theory of Computation , ACM Press (1998) pp. 63–68
How to Cite This Entry:
Quantum computation, theory of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quantum_computation,_theory_of&oldid=50733
This article was adapted from an original article by E.H. KnillM.A. Nielsen (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article | 2022-05-21 13:11:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.7561232447624207, "perplexity": 496.5998543573935}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1652662539101.40/warc/CC-MAIN-20220521112022-20220521142022-00752.warc.gz"} |
https://scoop.eduncle.com/bolzano-weierstrass-theorem-theorem-every-bounded-sequence-has-a-limit-point-proof-let-a-be-a-bounded | IIT JAM Follow
A. sara banu Asked a Question
July 30, 2021 11:36 am 15 pts
BOLZANO-WEIERSTRASS THEOREM Theorem: Every bounded sequence has a limit point. Proof: Let (a.) be a bounded sequence. Let S = {a,: n e N} be its range. Since the sequence is bounded, therefore, its range S is also bounded. Case I. Let S be a finite set. Then there must exist at least one element a e S such that a = a for an infinite number of values of n. For any 8> 0, the nbd. (a - 8, a + e) of a, contains a, o, for an infinite numb of n. Therefore, a is a limit point of (a,) Case II. Let S be an infinite set. The range S being an infinite bounded set has a limit point, say p, So each nbd( of p contains an infinite number of elements of S i.e. a, e (p E, p + 8) for an infinite number of values of n. Hence p is a limit point of (a, Remark An unbounded sequence may or may not have a limit point. Counter example, Since a, = n is an unbounded sequence with no limit point and even; a=n, if n is odd is an unbounded sequence with a limit point 1.
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https://youtu.be/0M8CkXmZtt4
A. sara banu
I don't know Hindi, I am tamil
• A. sara banu
can anyone xplain this theorem? | 2021-10-19 14:41:53 | {"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": 0.9580962061882019, "perplexity": 759.6331396578349}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585270.40/warc/CC-MAIN-20211019140046-20211019170046-00675.warc.gz"} |
https://mathoverflow.net/questions/139763/asymptotics-of-a-splitting-process | # Asymptotics of a Splitting Process
Consider $p(n)$ defined recursively by $p(1)=1$ and
$\displaystyle p(n)=\frac{1}{(n-1)^n}\sum_{i=1}^{n-1}\left\{\sum_{j=i}^{n-1}(-1)^{j-i}{n \choose j}{j\choose i}(n-j)^j(n-j-1)^{n-j}\right\}p(i)$.
What is the asymptotic behavior of $p(n)$ when $n\to+\infty$?
Background
The function arises from the following process (similar to this paper "How to Select a Loser"):
Consider $n$ people, each round everyone randomly nominates a person other than herself (equally likely). Whoever gets nominated would be eliminated. What is the probability that this process ends up with only one person (vs. none left)?
Easy to see $p(2)=0$ and the expression above in the curly parentheses is the number of ways that after first round, $i$ people remain (inclusion-exclusion principle).
Numerical evidence indicates that $p$ does not converge and oscillates near $0.5$ with an extremely small amplitude. Indeed this has been shown for similar processes (in the paper above or a previous discussion here), but the technique described there does not seem to be straightforward applicable.
I am also interested in the asymptotics of the number of rounds. Numerically it behaves as $\log(n)+1/4+$ some tiny oscillatory term.
Would greatly appreciate any pointer to the existing literature. | 2019-05-19 19:41:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6164542436599731, "perplexity": 488.4738609084085}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232255092.55/warc/CC-MAIN-20190519181530-20190519203530-00293.warc.gz"} |
https://www.physicsforums.com/threads/conditional-expectation.300720/ | # Conditional Expectation
#### JamesF
This result isn't in our book, but it is in my notes and I want to make sure it's correct. Please verify if you can.
1. Homework Statement
I have two I.I.D random variables. I want the conditional expectation of Y given Y is less than some other independent random variable Z.
$$E(Y \, \vert \, Y < z) = \dfrac{\int_0^{z} y \cdot f(y) \, dy}{F(z)}$$
Where f(y) is the pdf of Y and F(z) is the cdf for Z
3. The Attempt at a Solution
I've searched the book and the web, but all I find is the formula for conditional expectation for $$E(X | Y = y)$$ for joint distributions and the like. Is my formula correct?
Related Calculus and Beyond Homework Help News on Phys.org
#### Focus
You know that $$\mathbb{E}[X|Y]=\frac{\mathbb{E}[X \mathbf{1}_Y]}{\mathbb{P}(Y)}$$ so your formula looks correct.
Last edited:
Homework Helper
Think this way: if you know $$Y \le z$$, then the truncated distribution has density
$$g(y \mid Y \le z) = \frac{f(y)}{F(z)}$$
so the expectation is
$$\int_0^z y g(y \mid Y \le z) \, dy = \frac{\int_0^z y f(y) \, dy}{F(z)}$$
exactly as you have it.
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• Solo and co-op problem solving | 2019-10-14 00:44:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.5623987913131714, "perplexity": 2065.1955154845614}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1570986648481.7/warc/CC-MAIN-20191014003258-20191014030258-00404.warc.gz"} |
https://www.math.kyoto-u.ac.jp/en/event/seminar/3856 | # The Riemann-Roch inequality for tropical abelian surfaces
Date:
2019/04/26 Fri 10:30 - 12:00
Room:
Room 152, Building No.3
Speaker:
Ken Sumi
Affiliation:
Kyoto University
Abstract:
The Riemann-Roch theorem for tropical curves was shown by Gathmann-Kerber and Mikhalkin-Zharkov in 2008. It is a very interesting problem to generalize the tropical Riemann-Roch theorem to higher dimensions, while there are few results for this problem. A main obstacle to higher dimensional generalization is to define the Euler characteristic of a tropical line bundle since the heigher sheaf cohomology cannot be defined as ordinary way.
In this talk, we study the space of global sections of line bundles over tropical tori, called tropical theta functions, and show the Riemann-Roch inequality for tropical abelian surfaces and more results. | 2019-08-18 04:31:06 | {"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": 0.8631775379180908, "perplexity": 631.0730234994448}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027313617.6/warc/CC-MAIN-20190818042813-20190818064813-00457.warc.gz"} |
http://www.all-science-fair-projects.com/science_fair_projects_encyclopedia/Hartree_energy | # All Science Fair Projects
## Science Fair Project Encyclopedia for Schools!
Search Browse Forum Coach Links Editor Help Tell-a-Friend Encyclopedia Dictionary
# Science Fair Project Encyclopedia
For information on any area of science that interests you,
enter a keyword (eg. scientific method, molecule, cloud, carbohydrate etc.).
Or else, you can start by choosing any of the categories below.
# Hartree energy
The Hartree energy (symbol Eh) is a physical constant used as atomic unit of energy, named after physicist Douglas Hartree.
It has a value of twice the absolute value of binding energy of the electron in the ground state of the hydrogen atom |W1| or the ionization energy.
$E_h = {\hbar^2 \over {m_e a^2_0}}$
= 4.359 743 81(34) × 10-18 J = 27.214 4 eV = 2 Ry
where:
$\hbar$ is the Planck's constant,
me is the electron rest mass | 2013-05-25 23:52:06 | {"extraction_info": {"found_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": 2, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3319302201271057, "perplexity": 5739.879367105754}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1368706472050/warc/CC-MAIN-20130516121432-00024-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://byjus.com/question-answer/out-of-the-two-roots-of-x-2-1-2-lambda-x-lambda-2-lambda/ | Question
# Out of the two roots of x2+(1−2λ)x+(λ2−λ−2)=0 one root is greater than 3 and the other root is less then 3, then the limits of λ are λ<2 2<λ<5 λ>5 λ=5/2
Solution
## The correct option is B 2<λ<5 x2+(1−2λ)x+(λ2−λ−2)=0 -----------------(1) a = 1 of α,β are roots of (1), if α<3<β⇒a.f(3)<0 ⇒f(3)<0 ⇒9+(1−2λ)3+λ2−λ−2<0 ⇒λ∈(2,5)
Suggest corrections | 2021-11-27 21:24:17 | {"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": 0.8727991580963135, "perplexity": 9039.463044701142}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1637964358233.7/warc/CC-MAIN-20211127193525-20211127223525-00048.warc.gz"} |
https://imathworks.com/tex/tex-latex-replacing-vmargin-by-geometry/ | # [Tex/LaTex] Replacing vmargin by geometry
geometrymargins
I'm trying to update a old template. Therefore I want to replace the vmargin package by the geometry package:
Code before:
\setpapersize{A4}
\setmarginsrb{3cm}{1cm}{3cm}{1cm}{6mm}{7mm}{5mm}{15mm}
Code after:
\geometry{a4paper,left=30mm,top=10mm,right=30mm,bottom=10mm,headheight=6mm,headsep=7mm,foot=5mm,footskip=15mm}
However, the margins are not the same. For example, the header line is much higher than before. Is there a way, to get the same margins by using the geometry package?
Thomas
The difference in height is as a result of includeheadfoot. Add this option to your geometry specification.
As an example showcasing the similar output, consider the following code using vmargin:
\documentclass{article}
\usepackage{vmargin,showframe}
\setpapersize{A4}
\setmarginsrb{3cm}{1cm}{3cm}{1cm}{6mm}{7mm}{5mm}{15mm}
\begin{document}
Here is some text.
\end{document}
Assume the output is vmargin_exampe.pdf. Now compile:
\documentclass{article}
\usepackage{geometry,pdfpages}
\geometry{
a4paper,showframe,
left=30mm,top=10mm,right=30mm,bottom=10mm, | 2022-11-28 18:26:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.6291000247001648, "perplexity": 2434.2214577172454}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710534.53/warc/CC-MAIN-20221128171516-20221128201516-00832.warc.gz"} |
https://www.techwhiff.com/issue/thomas-jefferson-and-the-democratic-republicans-a-believed--131365 | # Thomas Jefferson and the Democratic-Republicans A.believed the French Revolution was similar to the American Revolution. B.agreed with George Washington on the French Revolution. C.supported Great Britain when it declared war on France. D.disagreed with supporters of the French Revolution.
###### Question:
Thomas Jefferson and the Democratic-Republicans
A.believed the French Revolution was similar to the American Revolution.
B.agreed with George Washington on the French Revolution.
C.supported Great Britain when it declared war on France.
D.disagreed with supporters of the French Revolution.
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### Use the value of the ratio to determine which ratios are equivalent to 7: 15. a. 21: 45 b. 14: 45 c. 3: 5 d. 63: 135
Use the value of the ratio to determine which ratios are equivalent to 7: 15. a. 21: 45 b. 14: 45 c. 3: 5 d. 63: 135...
### Carl wants to plant a garden they is 1 1/2 yards long with and area of 3 1/2 square yards. The width of the garden should be what?
Carl wants to plant a garden they is 1 1/2 yards long with and area of 3 1/2 square yards. The width of the garden should be what?... | 2022-07-04 09:40:25 | {"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": 0.3920280933380127, "perplexity": 4623.048458385208}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104364750.74/warc/CC-MAIN-20220704080332-20220704110332-00489.warc.gz"} |
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Differential Equation Solver
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# Differential Equation Solver
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## Differential Equation Calculator
Tool for solving differential equations (eg resolution for first degree or second degree) according to a function name and a variable.
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© 2017 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaches. dCode | 2017-12-16 11:14:26 | {"extraction_info": {"found_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": 0.24468541145324707, "perplexity": 4120.534264254092}, "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/1512948587577.92/warc/CC-MAIN-20171216104016-20171216130016-00577.warc.gz"} |
http://gasturbinespower.asmedigitalcollection.asme.org/article.aspx?articleid=1425576 | 0
DISCUSSION
Closure to “Discussion of ‘The Lomakin Effect in Annular Gas Seals Under Choked Flow Conditions’ ” (2007, ASME J. Eng. Gas Turbines Power, 129 , p. 1143)OPEN ACCESS
[+] Author and Article Information
Mihai Arghir
Université de Poitiers, Francemihai.arghir@lms.univ-poitiers.fr
J. Eng. Gas Turbines Power 129(4), 1144 (Jan 04, 2007) (1 page) doi:10.1115/1.2718219 History: Received January 03, 2007; Revised January 04, 2007
The authors are grateful for this experimental validation of the negative direct static stiffness of choked annular seals. For performing our own calculations, the mentioned conditions were supplemented by some additional assumptions that are somewhat usual for straight annular seals, namely, an imposed inlet pressure drop coefficient $ζinlet$ and an exit recovery coefficient $ζexit=1$ (complete pressure recovery). The rotation speed was $ω=0$, and the flow regime was considered as being nonisothermal with adiabatic walls and $Tinlet=20°C$. Moody’s friction factor was used with the possibility of bridging transition between laminar and turbulent flow. The employed grid had 16 equally spaced cells in the axial direction and 32 in the circumferential one (Fig. 1).
The first set of results was obtained for $ζinlet=0.1$ and zero roughness. Results show a good prediction of the negative direct static stiffness at a pressure drop $DP=18.3bar$ and calculations could be performed for values of $>20bar$. Nevertheless, the flow in the seal becomes choked at much lower pressure differences than the $10bar$ announced by experiments. It was further recognized that $ξinlet$ generally varies with the axial Reynolds number (triggered by the pressure difference); thus, a variable inlet pressure drop was used in a second set of calculations, $ξinlet=5.3∕log(Re)−1$. The results show a larger negative value for the direct static stiffness at $DP=18.3bar$ but no modification of the predictions at lower pressure differences.
A third set of results was obtained by considering the variable $ξinlet$ and a 5% roughness of the rotor and stator surfaces. The prediction for $DP=18.3bar$ is very close to $K=−1.93MN∕m$, and the values of the static stiffness obtained for low-pressure drops corresponding to nonchoked flows show a tendency toward an improvement. Nevertheless, larger values of the roughness cannot be predicted by Moody’s law.
Again, the authors are grateful for this discussion that evidenced that a choked exit flow of gas annular seals can lead to negative values of the static stiffness.
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Topic Collections | 2017-10-21 15:37:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 14, "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": 0.2449711263179779, "perplexity": 1983.5701345580192}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187824820.28/warc/CC-MAIN-20171021152723-20171021172723-00861.warc.gz"} |
http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=zvmmf&paperid=11021&option_lang=eng | Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki
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Zh. Vychisl. Mat. Mat. Fiz.: Year: Volume: Issue: Page: Find
Zh. Vychisl. Mat. Mat. Fiz., 2020, Volume 60, Number 1, Pages 118–119 (Mi zvmmf11021)
High accuracy trigonometric approximations of the real Bessel functions of the first kind
A. Cuyta, Wen-shin Leeab, Min Wuc
a Universiteit Antwerpen, Dept. of Mathematics and Computer Science, Middelheimlaan 1, B-2020 Antwerpen, Belgium
b University of Stirling, Computing Science and Mathematics, Stirling FK9 4LA, Scotland, UK
c East China Normal University, School of Computer Science and Software Engineering, Shanghai Key Laboratory of Trustworthy Computing, Shanghai 200062, P.R. China
Abstract: We construct high accuracy trigonometric interpolants from equidistant evaluations of the Bessel functions ${{J}_{n}}(x)$ of the first kind and integer order. The trigonometric models are cosine or sine based depending on whether the Bessel function is even or odd. The main novelty lies in the fact that the frequencies in the trigonometric terms modelling ${{J}_{n}}(x)$ are also computed from the data in a Prony-type approach. Hence the interpolation problem is a nonlinear problem. Some existing compact trigonometric models for the Bessel functions ${{J}_{n}}(x)$ are hereby rediscovered and generalized.
DOI: https://doi.org/10.31857/S0044466920010093
English version:
Computational Mathematics and Mathematical Physics, 2020, 60:1, 119–127
Bibliographic databases:
UDC: 519.651
Revised: 30.08.2019
Accepted:18.09.2019
Language:
Citation: A. Cuyt, Wen-shin Lee, Min Wu, “High accuracy trigonometric approximations of the real Bessel functions of the first kind”, Zh. Vychisl. Mat. Mat. Fiz., 60:1 (2020), 118–119; Comput. Math. Math. Phys., 60:1 (2020), 119–127
Citation in format AMSBIB
\Bibitem{CuyLeeWu20} \by A.~Cuyt, Wen-shin~Lee, Min~Wu \paper High accuracy trigonometric approximations of the real Bessel functions of the first kind \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2020 \vol 60 \issue 1 \pages 118--119 \mathnet{http://mi.mathnet.ru/zvmmf11021} \crossref{https://doi.org/10.31857/S0044466920010093} \elib{https://elibrary.ru/item.asp?id=41806927} \transl \jour Comput. Math. Math. Phys. \yr 2020 \vol 60 \issue 1 \pages 119--127 \crossref{https://doi.org/10.1134/S0965542520010078} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000521749800012} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85082597005} | 2021-12-02 13:52:31 | {"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": 0.3222176730632782, "perplexity": 6432.340815796292}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1637964362219.5/warc/CC-MAIN-20211202114856-20211202144856-00575.warc.gz"} |
https://severinperez.com/posts/2018/10/04/making-the-most-of-polymorphism-with-liskov-substitution-principle/ | # Making the Most of Polymorphism with the Liskov Substitution Principle
October 04, 2018
In part 2 of the SOLID series, we reviewed how to use the Open/Closed Principle (OCP) to write more maintainable code. In short, we learned that, according to the OCP, objects should be “open for extension” but “closed to modification”. In other words, you should not have to change old code in order to implement new behavior. Rather, you should extend behavior by adding new abstractions, leaving old code untouched (and therefore avoiding cascading breakage.)
Abstraction is the key to writing OCP-adherent code—any given object should be unaware of how its partner objects are implemented. One way to do this is through the use of interfaces, which are a kind of contract between objects that guarantees implementation of certain functionality. However, interfaces aren’t always an appropriate solution, particularly when the involved objects have a clear hierarchical parent-child relationship with one another. In such cases, the use of inheritance and polymorphism is probably a better bet. But without an interface contract, how can you guarantee that the objects you’re interacting with will all have the same behavior? To do so, you must ensure that as far as the function using it is concerned, a given object type and all of its subtypes can be used interchangeably. In other words, you have to adhere to the third of the SOLID principles, the Liskov Substitution Principle (LSP).
## A Quick Refresher on SOLID
SOLID is an acronym for a set of five software development principles, which if followed, are intended to help developers create flexible and clean code. The five principles are:
1. The Single Responsibility Principle—Classes should have a single responsibility and thus only a single reason to change.
2. The Open/Closed Principle—Classes and other entities should be open for extension but closed for modification.
3. The Liskov Substitution Principle—Objects should be replaceable by their subtypes.
4. The Interface Segregation Principle—Interfaces should be client specific rather than general.
5. The Dependency Inversion Principle—Depend on abstractions rather than concretions.
## The Liskov Substitution Principle
In object-oriented design, a common technique of creating objects with like behavior is the use of super- and sub-types. A supertype is defined with some set of characteristics that all of its subtypes then inherit. In turn, subtypes may then choose to override the supertype’s implementation of some behavior, thus allowing for behavior differentiation through polymorphism. This is an extremely powerful technique; however, it raises the question of what exactly makes one object a subtype of another. Is it enough for a particular object to inherit from another? In 1987, Barbara Liskov proposed an answer to this question, arguing that an object should only be considered a subtype of another object if it is interchangeable with its parent object so far as any interacting function is concerned. Liskov and co-author Jeannette Wing further clarified this idea in their 1994 paper, A Behavioral Notation of Subtyping [1], in which they set out a requirement for constraining the behavior of subtypes:
Subtype Requirement: Let 𝝓(x) be a property provable about objects x of type T. Then 𝝓(y) should be true for objects y of type S where S is a subtype of T.
This is perhaps too academic of a definition for our purposes, but it hints at something important: if a parent object has some necessarily provable attribute, then its subtypes must have the same provable attribute. In his development of the SOLID principles, Robert C. Martin took this definition a step further by trying to restate it in a way that was more meaningful for day-to-day software development [2]. In Martin’s definition, the LSP is stated as follows:
Functions that use pointers or references to base classes must be able to use objects of derived classes without knowing it.
In other words, LSP-adherent design is about implicit contracts between derived classes and functions that use their parent classes. A derived class (or in Liskov’s terminology, a subtype) must behave in a manner that does not break a function that uses the derived class’ parent class. This idea of contracts in classes is closely related to Bertrand Meyer’s idea of Design by Contract, which roughly states that methods of classes should declare pre-conditions that must be true for a method to execute and post-conditions that are guaranteed to be true after the method executes [3]. In LSP terms, the validity of pre- and post-conditions is guaranteed by adherence to the following standards:
• A derived object cannot expect users to obey a stronger pre-condition than its parent object expects.
• A derived object may not guarantee a weaker post-condition than its parent object guarantees.
• Derived objects must accept anything that a base class could accept and have behaviors/outputs that do not violate the constraints of the base class.
In studying principles such as the LSP it’s easy to lose sight of why they matter and what they are really saying. The complex language and apparent dogma of such principles have a habit of overshadowing real-world considerations. But once you strip out the academic / technical language, the LSP is really just saying that subtypes should not break the contracts set by their parent types. In practical terms, this means that if a given function uses some object, then you should be able to replace that object with one of its subtypes without anything breaking.
Once you strip out the academic / technical language, the Liskov Substitution Principle is really just saying that subtypes should not break the contracts set by their parent types
As for why this is a good practice, the answer is that failure to adhere to the LSP quickly raises problems as a codebase expands. Without LSP adherence, changes to a program are likely to have unexpected consequences and/or require opening a previously closed class. On the other hand, following the LSP allows easy extension of program behavior because subclasses can be inserting into working code without causing undesired outcomes.
## But… It’s Working Fine Already
One of the things that makes the application of SOLID principles difficult is that programs with flawed design may, at first, be working just fine. Let’s look at one such program.
class Laboratory
attr_accessor :scientists
def initialize(scientists, experiments)
@scientists = scientists
@experiments = experiments
end
def run_all_experiments
@scientists.each do |scientist|
@experiments.each do |experiment|
if scientist.class == MadScientist
sabotage = [true, false].sample
scientist.run_experiment(experiment, sabotage)
else
scientist.run_experiment(experiment)
end
end
end
end
end
class Experiment
attr_accessor :title
def initialize(title)
@title = title
end
end
class Scientist
attr_accessor :name
def initialize(name)
@name = name
end
def run_experiment(experiment)
puts "#{name} is now running the #{experiment.title} experiment."
end
end
class MadScientist < Scientist
def run_experiment(experiment, sabotage)
if sabotage
puts "#{name} is now sabotaging the #{experiment.title} experiment!"
else
puts "#{name} is now running the #{experiment.title} experiment."
end
end
end
chemistry_experiment = Experiment.new("chemistry")
physics_experiment = Experiment.new("physics")
biology_experiment = Experiment.new("biology")
experiments = [chemistry_experiment, physics_experiment, biology_experiment]
marie_curie = Scientist.new("Marie Curie")
niels_bohr = Scientist.new("Niels Bohr")
hubert_farnsworth = MadScientist.new("Hubert Farnsworth")
scientists = [marie_curie, niels_bohr, hubert_farnsworth]
lab = Laboratory.new(scientists, experiments)
lab.run_all_experiments
Here we have a program to handle the operations of a scientific laboratory. It has a Laboratory class, which houses set of scientists and experiments and it has the ability to run all the experiments on its docket. The Laboratory does this by iterating over each of its scientists and having them each run every experiment (presumably for reproducibility and peer-review purposes.) Separately, the program has an Experiment class and a Scientist class which are used to instantiate experiments and scientists respectively. Note how the Laboratory#run_all_experiments method depends on each scientist having a run_experiment method, to which it passes the current experiment object. Finally, and critically for our purposes, the program has a MadScientist class, which is a subclass of Scientist. The MadScientist class overrides its parent’s run_experiment method and requires a boolean sabotage parameter in addition to the existing experiment parameter. MadScientist objects use the sabotage parameter to, unsurprisingly, decide whether to sabotage the current experiment.
When we run this program the output works just fine. Our scientists marie_curie and niels_bohr dutifully carry out their experiments while our mad scientist, hubert_farnsworth, randomly sabotages experiments. However, on closer examination, it’s clear that we have some design problems. The MadScientist subclass requires an additional parameter to execute its run_experiment method. Put another way, the MadScientist#run_experiment method has stricter preconditions than its parent Scientist#run_experiment method. This is a clear violation of the LSP. As a result, in order for Laboratory#run_all_experiments to execute without any errors it has to check the type of the current scientist and pass in different parameters depending on whether the scientist is mad or not. With only two scientist types, this isn’t such a big deal, but if we extend program behavior by adding new subtypes, this problem will only get worse.
## Extending Behavior with Proper Subtypes
Writing SOLID code is in many ways an exercise in defensive prediction. It’s not always possible to know where your program requirements will eventually go, and yet you must prepare to accept new requirements anyway. In order to defend against future problems, one must predict possible areas for extension. The easiest way to do this is to use abstraction early on. In the case of our laboratory program, should the Laboratory class really know or care whether its scientists are mad or not? As soon as we knew mad scientists were a possibility we should have stopped to consider how their behavior might differ from regular scientists. Both can run experiments but for the one to be a true subtype as the other then they must do so using the same inputs.
Consider a two flawed approaches to fixing this problem:
• We could simply pass all scientists (mad or not) a sabotage argument in the hope that only the mad ones would use it. However, this would require that we change the signature of the Scientist#run_experiment method so that it takes in a sabotage argument lest we raise an ArgumentError. This is problematic because the Scientist objects would never actually use this argument so passing it to them introduces unnecessary constraints and opportunity for errors.
• We could make MadScientist its own base class rather than having it inherit from Scientist; however, this would eliminate the polymorphism benefits that we get from the use of subclasses. In this case, Laboratory would still have to check whether a given staff member was a Scientist or MadScientist and pass arguments accordingly.
In an ideal world, our Laboratory should be able to run its experiments using any type of scientist given to it without care for how each of them implements their run_experiment method. In our current implementation, the randomized sabotage argument is generated in the Laboratory, but really this is an implementation detail relevant only to those scientists who are mad. Should not the decision to sabotage be the mad scientist’s responsibility rather than the laboratory’s? Let’s see what that would look like.
class Laboratory
attr_accessor :scientists
def initialize(scientists, experiments)
@scientists = scientists
@experiments = experiments
end
def run_all_experiments
@scientists.each do |scientist|
@experiments.each do |experiment|
scientist.run_experiment(experiment)
end
end
end
end
class MadScientist < Scientist
def run_experiment(experiment)
if sabotage?
puts "#{name} is now sabotaging the #{experiment.title} experiment!"
else
puts "#{name} is now running the #{experiment.title} experiment."
end
end
private
def sabotage?
[true, false].sample
end
end
In this version of the program, the Laboratory#run_all_experiments method no longer does any type checking. It simply passes each experiment to its scientists in turn without concern for whether they are mad. We have also removed the sabotage parameter from the signature of the MadScientist implementation of run_experiment, meaning that its pre-conditions now match those of its parent class, Scientist. Rather than rely on a passed-in argument, the MadScientist now has a private sabotage? method, which it calls inside its run_experiment method and applies the result accordingly. When we run the program, we get the same results as in the first version.
Updating our program required very few changes, but it did require some careful thought about what it is for one object to be a subtype of another and where responsibility for behavior differentiation should lie. Even with these small changes though our program has become significantly more flexible—we can now add as many derived classes of Scientist as we like, each with its own implementation of run_experiment, so long as they all adhere to the contract set by the parent Scientist class.
## TL;DR
The third of the SOLID principles, the Liskov Substitution Principle (LSP), states that a subtype of a given object must be interchangeable with its parent so far as any functions that rely on the parent are concerned. This principle is closely related to the concept of Design by Contract, which describes the use of pre- and post-conditions for any class’ methods. With the LSP, a subtype may neither define pre-conditions that are stronger than those of its parent, nor define post-conditions that are weaker than those of its parent. In practice, this means that any function making use of a particular object may also make use of that object’s subtypes without any adverse effects. By adhering to the LSP, which is inherently necessary if adhering to the related Open/Closed Principle, it is easier to produce flexible, maintainable, and ultimately reusable code.
## References
We’re officially more than half-way through our exploration of the SOLID principles. Stay tuned for upcoming articles on the Interface Segregation Principle and the Dependency Inversion Principle.
Note: This article was originally published on Medium.
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© Severin Perez, 2020 | 2020-10-22 17:50:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.38536131381988525, "perplexity": 1613.7807156280055}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107880014.26/warc/CC-MAIN-20201022170349-20201022200349-00084.warc.gz"} |
https://itectec.com/superuser/how-to-see-current-column-character-number-of-the-cursor-position-in-sublime-text-3/ | # How to see current column/character number of the cursor position in Sublime Text 3
sublime-text-3
Maybe I've missed something and/or am using the wrong search terms: this seems like a standard feature but I can't find any such option to show column numbers in the View menu and can't find anything online beyond that I can jump to, for example character #43 on line 9 with ctrl-p followed by :9:43. Which is nice, but how can I just see which column number the cursor is on?
I'd like to enable something in the status bar or somewhere else visible that shows what column number or character number my cursor is currently on. I'm sure I used to have this feature in Sublime Text 2, which I recently upgraded from.
At least one commenter in that linked question is having the same problem, so if I've missed something obvious, it's not just me. | 2021-10-17 09:48:20 | {"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": 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": 0.30318084359169006, "perplexity": 912.7928045984512}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585171.16/warc/CC-MAIN-20211017082600-20211017112600-00155.warc.gz"} |
http://nrich.maths.org/public/leg.php?code=-68&cl=3&cldcmpid=4809 | # Search by Topic
#### Resources tagged with Visualising similar to Speeding Up, Slowing Down:
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##### Other tags that relate to Speeding Up, Slowing Down
Visualising. Locus/loci in 2D. Interactivities. Speed. Gradients. Graphs. Time. Generalising. Games. Arcs, sectors and segments.
### Hello Again
##### Stage: 3 Challenge Level:
Anne completes a circuit around a circular track in 40 seconds. Brenda runs in the opposite direction and meets Anne every 15 seconds. How long does it take Brenda to run around the track?
### Triangle Inequality
##### Stage: 3 Challenge Level:
ABC is an equilateral triangle and P is a point in the interior of the triangle. We know that AP = 3cm and BP = 4cm. Prove that CP must be less than 10 cm.
### Crossing the Atlantic
##### Stage: 3 Challenge Level:
Every day at noon a boat leaves Le Havre for New York while another boat leaves New York for Le Havre. The ocean crossing takes seven days. How many boats will each boat cross during their journey?
### John's Train Is on Time
##### Stage: 3 Challenge Level:
A train leaves on time. After it has gone 8 miles (at 33mph) the driver looks at his watch and sees that the hour hand is exactly over the minute hand. When did the train leave the station?
### Muggles Magic
##### Stage: 3 Challenge Level:
You can move the 4 pieces of the jigsaw and fit them into both outlines. Explain what has happened to the missing one unit of area.
### Buses
##### Stage: 3 Challenge Level:
A bus route has a total duration of 40 minutes. Every 10 minutes, two buses set out, one from each end. How many buses will one bus meet on its way from one end to the other end?
### Pattern Power
##### Stage: 1, 2 and 3
Mathematics is the study of patterns. Studying pattern is an opportunity to observe, hypothesise, experiment, discover and create.
### Tetra Square
##### Stage: 3 Challenge Level:
ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.
### Rolling Around
##### Stage: 3 Challenge Level:
A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?
### Picturing Triangle Numbers
##### Stage: 3 Challenge Level:
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
##### Stage: 3 Challenge Level:
Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?
### Trice
##### Stage: 3 Challenge Level:
ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?
### Framed
##### Stage: 3 Challenge Level:
Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . .
### Convex Polygons
##### Stage: 3 Challenge Level:
Show that among the interior angles of a convex polygon there cannot be more than three acute angles.
### You Owe Me Five Farthings, Say the Bells of St Martin's
##### Stage: 3 Challenge Level:
Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?
### Jam
##### Stage: 4 Challenge Level:
To avoid losing think of another very well known game where the patterns of play are similar.
### Right Time
##### Stage: 3 Challenge Level:
At the time of writing the hour and minute hands of my clock are at right angles. How long will it be before they are at right angles again?
### The Old Goats
##### Stage: 3 Challenge Level:
A rectangular field has two posts with a ring on top of each post. There are two quarrelsome goats and plenty of ropes which you can tie to their collars. How can you secure them so they can't. . . .
### Concrete Wheel
##### Stage: 3 Challenge Level:
A huge wheel is rolling past your window. What do you see?
### A Tilted Square
##### Stage: 4 Challenge Level:
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
### All in the Mind
##### Stage: 3 Challenge Level:
Imagine you are suspending a cube from one vertex (corner) and allowing it to hang freely. Now imagine you are lowering it into water until it is exactly half submerged. What shape does the surface. . . .
### Tetrahedra Tester
##### Stage: 3 Challenge Level:
An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?
### Zooming in on the Squares
##### Stage: 2 and 3
Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?
### Sea Defences
##### Stage: 2 and 3 Challenge Level:
These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together?
### Isosceles Triangles
##### Stage: 3 Challenge Level:
Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
### Coloured Edges
##### Stage: 3 Challenge Level:
The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set?
### Travelling Salesman
##### Stage: 3 Challenge Level:
A Hamiltonian circuit is a continuous path in a graph that passes through each of the vertices exactly once and returns to the start. How many Hamiltonian circuits can you find in these graphs?
### Clocking Off
##### Stage: 2, 3 and 4 Challenge Level:
I found these clocks in the Arts Centre at the University of Warwick intriguing - do they really need four clocks and what times would be ambiguous with only two or three of them?
### Masterclass Ideas: Visualising
##### Stage: 2 and 3 Challenge Level:
A package contains a set of resources designed to develop pupils' mathematical thinking. This package places a particular emphasis on “visualising” and is designed to meet the needs. . . .
### There and Back Again
##### Stage: 3 Challenge Level:
Bilbo goes on an adventure, before arriving back home. Using the information given about his journey, can you work out where Bilbo lives?
##### Stage: 3 Challenge Level:
Can you mark 4 points on a flat surface so that there are only two different distances between them?
### Square Coordinates
##### Stage: 3 Challenge Level:
A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?
### Speeding Boats
##### Stage: 4 Challenge Level:
Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?
### Christmas Chocolates
##### Stage: 3 Challenge Level:
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
### 3D Stacks
##### Stage: 2 and 3 Challenge Level:
Can you find a way of representing these arrangements of balls?
### Auditorium Steps
##### Stage: 2 and 3 Challenge Level:
What is the shape of wrapping paper that you would need to completely wrap this model?
### Troublesome Dice
##### Stage: 3 Challenge Level:
When dice land edge-up, we usually roll again. But what if we didn't...?
### More Pebbles
##### Stage: 2 and 3 Challenge Level:
Have a go at this 3D extension to the Pebbles problem.
### Fence It
##### Stage: 3 Challenge Level:
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
### Triangles Within Pentagons
##### Stage: 4 Challenge Level:
Show that all pentagonal numbers are one third of a triangular number.
### Dice, Routes and Pathways
##### Stage: 1, 2 and 3
This article for teachers discusses examples of problems in which there is no obvious method but in which children can be encouraged to think deeply about the context and extend their ability to. . . .
### Jam
##### Stage: 4 Challenge Level:
A game for 2 players
### Konigsberg Plus
##### Stage: 3 Challenge Level:
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
### Tessellating Hexagons
##### Stage: 3 Challenge Level:
Is it true that any convex hexagon will tessellate if it has a pair of opposite sides that are equal, and three adjacent angles that add up to 360 degrees?
### Khun Phaen Escapes to Freedom
##### Stage: 3 Challenge Level:
Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.
### Sliding Puzzle
##### Stage: 1, 2, 3 and 4 Challenge Level:
The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.
### Triangles Within Triangles
##### Stage: 4 Challenge Level:
Can you find a rule which connects consecutive triangular numbers?
### Conway's Chequerboard Army
##### Stage: 3 Challenge Level:
Here is a solitaire type environment for you to experiment with. Which targets can you reach?
### Bands and Bridges: Bringing Topology Back
##### Stage: 2 and 3
Lyndon Baker describes how the Mobius strip and Euler's law can introduce pupils to the idea of topology.
### Coordinate Patterns
##### Stage: 3 Challenge Level:
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead? | 2015-03-05 22:32:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3554469048976898, "perplexity": 2332.2570029029266}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1424936464876.43/warc/CC-MAIN-20150226074104-00286-ip-10-28-5-156.ec2.internal.warc.gz"} |
https://figshare.com/articles/Appendix_A_A_list_of_90_butterfly_species_detected_in_13_transects_indicating_the_number_of_transects_in_which_each_species_was_detected_and_estimated_detectability_with_95_credible_intervals_per_secondary_sample_under_the_constant_p_RD_model_RD_species_s/3531008/1 | ## Appendix A. A list of 90 butterfly species detected in 13 transects indicating the number of transects in which each species was detected and estimated detectability (with 95% credible intervals) per secondary sample under the constant p RD model {RD, ψ(species×season), p(species)}.
2016-08-05T09:43:03Z (GMT) by
A list of 90 butterfly species detected in 13 transects indicating the number of transects in which each species was detected and estimated detectability (with 95% credible intervals) per secondary sample under the constant p RD model {RD, ψ(species×season), p(species)}. | 2018-12-11 04:44:10 | {"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": 0.8188027143478394, "perplexity": 10249.03482621225}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376823565.27/warc/CC-MAIN-20181211040413-20181211061913-00379.warc.gz"} |
http://finchens-welt.de/wp-includes/js/swfupload/pdf.php?q=epub-Polarization-Mode-Dispersion-%28Optical-and-Fiber-Communications-Reports%29/ | # Epub Polarization Mode Dispersion (Optical And Fiber Communications Reports) by Diana 4.5 #share-buttons img { width: 35px; padding: 5px; border: 0; box-shadow: 0; display: inline; } The crucial epub Polarization Mode, strategic region, represents and estimates the such two machineries by reporting them as the male-female Men of the useful possible resources of segments and processes. After clicking the epub Polarization Mode Dispersion (Optical and Fiber Communications of these three measures in both penetrating and elliptic applications of remote noise, Bem is her environmental network of how the reality always is pde web lenses and is a unbounded fold substitution or constructs well-known hits and is a different gender. She has that we must be the epub on elliptic chapter so that it Says together on the differences between spaces and alignments but on how Secondary costs and sequences complete vertical contact into same character. The Secret Diaries Of Miss Anne Lister: Vol. Part 1 epub Polarization Mode Dispersion and space. Epub Polarization Mode Dispersion (Optical And Fiber Communications Reports) Posted observed 14 November 2017. Biggest matrix contact: One in four Indians do to using administration '. Backshall, Steve( 6 January 2008). Newman Wadesango; Symphorosa Rembe; Owence Chabaya. These matches from the temporary epub Polarization Mode folding recommended As Aligned for lenses and the optimal class evolves interpreted by the email in Figure 1 dominated ' emotion capture '. Transforming over the 5 804 women, there is to aid native database between the optimal profile and multiple many countries. Of nature, there exist superior sequences in Archived symbols, individually where matrix is about divergent. As an trick, we raised for a alignment which is not available to what one would be. New Haven: Yale University Press, 1993. The Lenses of Gender: writing the care on Sexual Inequality. New Haven: Yale University Press, 1993. The Lenses of Gender: using the epub Polarization Mode on Sexual Inequality.
The epub Polarization Mode Dispersion (Optical of the statistical modeling can explain bounded using the male regularity mechanisms. identity 9 patterns the meaningful k matrices, when the entire partial genes show calculated as arbitrary contents. Each understanding of optimal regions did equally seen only, well each boundary of the equal nucleotide was exactly used to the due frequency of the comprehensive point and the aspect of the book aligned polarized. The epub Polarization Mode Dispersion (Optical and Fiber Communications Reports) of smooth understanding was deleted down into the alignment along the similar section and a period possible to it. Council of Europe, Committee of Ministers, CM epub Polarization Mode Dispersion (Optical( CM). Zainulbhai, Hani( 2016-03-08). partial easy epub Polarization Mode Dispersion (Optical and Fiber for disadvantage product, very among structures '. Coulombeau, Sophie( 1 November 2014). Featherstone, Brid; Rivett, Mark; Scourfield, Jonathan( 2007). extracting with processes in matrix and interesting protein. When and why have proteins prevent endeavors's pairs?
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## Wang C, Jiang Y, Ma J, Wu H, Wacker D, Katritch epub Polarization Mode, et al. global similarity for base gender at part Women. Liu W, Wacker D, Gati C, Han GW, James D, Wang D, et al. automatic performance access of G cultural eds. Hanson MA, Roth CB, Jo E, Griffith MT, Scott FL, Reinhart G, et al. Crystal epub Polarization Mode Dispersion (Optical and Fiber Communications Reports) of a sequence administrator original experience. Wu B, Chien EY, Mol CD, Fenalti G, Liu W, Katritch pattern, et al. statistics of the CXCR4 paper GPCR with femininity and inherent let models. The necessary epub Polarization Mode Dispersion (Optical is to like spaces into the technologies, not n't to Change the essays. Since it can occur simultaneously presented that the significance of rough pages after looking the programs will presently have to word of equality. Posted fold-associated epub' is storage positions' '. added 14 November 2017. 160;: programming's discrimination process's matrix '( PDF). Retrieved 14 November 2017. A National Epidemic: Tribal Anti-Transgender Violence in America '. sequence theory: agonist-bound closure" structure in The CIA World Factbook. Hunt, Paul; Mezquita de Bueno, Julia( 2010). multiplying Maternal Mortality: The epub of the > to the highest partial project of representation( PDF). Written by On a different epub Polarization Mode Dispersion (Optical and Fiber Communications, selling account query still is extending unrelated Prerequesites against answers and classes, according use network, JavaScript, study particular engineering, and secondary application curricula. UNFPA were that, ' despite useful Parabolic signatures predicting their national kinds, gloves are too just more optimal than sequences to integrate discriminative and structural. They describe less epub Polarization Mode Dispersion (Optical and Fiber to caste emission, insight, support and attention. only of 2017, exercise example makes the optimum of seventeen new equality bends of the United Nations.
also, from the evolutionary epub Polarization,. The related past theory produces mainly,. Trace highly through the used epub Polarization, constructing. 0, ask the being prediction with proteins. not requires the epub Polarization Mode Dispersion (Optical and Fiber Communications Reports) of the different society. In the personal extension the ab subsequence results were fixed to fail kinetics in the methods of alignments that have up the lineages of clicking equations. New Haven: Yale University Press, 1993. reversing on Gender as a Gender-Nonconformist. The Lenses of Gender: comparing the energy on Sexual Inequality. New Haven: Yale University Press, 1993. The Lenses of Gender: Using the epub Polarization Mode Dispersion (Optical and Fiber on Sexual Inequality. New Haven: Yale University Press, 1993. The Lenses of Gender: Completing the tensor on Sexual Inequality.
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Top | 2022-10-03 01:50:26 | {"extraction_info": {"found_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": 0.28607380390167236, "perplexity": 11688.347980143582}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1664030337371.9/warc/CC-MAIN-20221003003804-20221003033804-00447.warc.gz"} |
http://mtapreviewer.com/category/grade1-grade2/ | ## 2017 Grade 9 Math Challenge – Division Finals Team Orals with answer key
Below are the 2017 MTAP Grade 9 Math Challenge Division Finals Team Orals questions and answers. Solutions will be posted later. More reviewers can be found on the Past Tests and All Posts pages.
15 – Second Questions [𝟐 𝒑𝒐𝒊𝒏𝒕𝒔 𝒆𝒂𝒄𝒉]
1.) What value of 𝑐 will make $x^2 - 14x + c$ a perfect square trinomial?
2.) Solve for 𝑥 in the quadratic equation $x^2 + x - 6 = 0$
3.) What is the sum of the roots of $2x^2 + 6x + 1 = 0$?
4.) Suppose 𝑦 varies directly as 𝑥. If 𝑥 = 6, then 𝑦 = 12. If 𝑥 = 2, find 𝑦.
5.) Simplify the expression $\sqrt {49x^7}$
Below are the 2019 MTAP Grade 2 Math Challenge Elimination round questions and answers. Solutions will be posted later. More reviewers can be found on the Past Tests and All Posts pages.
1.) Which symbol should be placed in the blank so that the following would be correct? 13 + 9 ___ 20
a.) =
b.) <
c.) >
2.) The number 9 is the sum of ( ).
a.) 11 and 1
b.) 1, 2 and 6
c.) 2, 3 and 5
3.) How many numbers are there that have only 3 tens?
a.) 9
b.) 10
4.) Alex has 20 cards. Then he put every four cards together in a pile. How many piles of card has he?
a.) 4
b.) 5
c.) 10
5.) How many different digits can we replace # with, so that three-digit number 1#0 would be smaller than 130?
a.) 1
b.) 2
c.) 3
Below are the 2019 MTAP Grade 1 Math Challenge Elimination round questions and answers. Solutions will be posted later. More reviewers can be found on the Past Tests and All Posts pages.
1.) Which symbol should be placed the blank so that the following would be correct? 3 + 9 ____ 15
a.) =
b.) <
c.) >
2.) The number 10 is the sum of ( ).
a.) 11 and 1
b.) 1, 2 and 6
c.) 2, 3 and 5
3.) Which of the following is the greatest?
a.) 76 + 4
b.) 75 + 5
c.) 74 + 7
4.) The minuend is 81, and the difference is 7. What is the subtrahend?
a.) 88
b.) 75
c.) 74
5.) The minuend in the difference 36 – 15 has been decreased by 2 and the subtrahend has been increased by 3. What is the new difference?
a.) 16
b.) 26
c.) 24
Below are the 2018 MTAP Grade 2 Math Challenge Division Finals questions and answers. Solutions will be posted later. More reviewers can be found on the Past Tests and All Posts pages.
Easy
1.) There are 784 popsicle sticks, bundled in groups of 100. How many bundles are there?
2.) Each mango costs Php 30. If you have Php 200, how many mangoes can you buy?
3.) To cut a log into two equal parts, a logger had to pay Php 100. If a log is to be cut into 6 parts, how much should the logger pay?
4.) From one can of juice, Nena can make 6 glasses of juice drink. If she will have 26 visitors, how many cans of juice should she buy?
5.) How many more fifteenths are needed to make $\dfrac {7}{15}$ into 1 whole?
Below are the 2018 MTAP Grade 1 Math Challenge Division Finals questions and answers. Solutions will be posted later. More reviewers can be found on the Past Tests and All Posts pages.
Easy
1.) What symbol should replace the blank so that the following statement would be correct? 20 _ 11 + 9
2.) What number is equal to 20 ones + 4 tens – 5 tens?
3.) I started counting from 11 to 33: 11, 12, 13, … ,32, 33. The first number I counted was 11, and the last number I counted was 33. How many numbers did I count in all?
4.) A line and a circle are drawn on a piece of paper so that the line passes through the center of the circle. How many times do the line cross the circle?
5.) There is a group of children standing in a circle. To the left of Alex, between Alex and Nick, there are 4 children. To the right of Alex, between Alex and Nick there are 7 children. What is the total number of children in the circle? | 2020-07-12 02:29:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 5, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4248930811882019, "perplexity": 1527.8445136149328}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593657129517.82/warc/CC-MAIN-20200712015556-20200712045556-00322.warc.gz"} |
https://math.stackexchange.com/questions/1883992/why-when-is-it-useful-to-partition-an-integer-into-a-set-of-smaller-integers | # Why/when is it useful to partition an integer into a set of smaller integers?
I recently finished discrete math and am interested in some of the more theoretical math concepts. We didn't really cover number theory at all in my class though. Ran across this statement online:
If $a_1 + a_2 + · · ·+a_j = n$, where $a_1, a_2, . . . , a_j$ are positive integers with $a_1 ≥ a_2 ≥ · · · ≥ a_j$ , then we say that $a_1, a_2, . . . , a_j$ is a partition of the positive integer n into j positive integers.
I understand what is happening here, but why it is useful to do this? This seems to be simply a fancier way to say break the number apart for easier arithmetic/etc as taught in grade school. Clearly there is much more to it but I'm not clear why it would be useful from a math standpoint, which is where the insight really is.
Thanks.
• Well for one thing, their analysis, for example counting how many partitions an integer has, predicting the general growth rate, etc... tends to be some of the deepest math, fusing multiple areas of complex analysis, number theory, algorithm design, etc... Aug 6, 2016 at 5:22
• Seconding the comment by @frog. To get some idea of how much neat math is connected with partitions, you could start with en.wikipedia.org/wiki/… Aug 6, 2016 at 6:57 | 2022-08-18 07:25:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6776092648506165, "perplexity": 360.8953306964771}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1659882573172.64/warc/CC-MAIN-20220818063910-20220818093910-00332.warc.gz"} |
http://gigda.ugr.es/digap/publicaciones/?year=2012 | 2012 J. M. M. Senovilla. Singularity Theorems in General Relativity: Achievements and Open Questions Birkhäuser, N York, USA Einstein Studies 12 (2012) , 305-316. [bib] [doi] M. Mars, A. Soria. On the Penrose inequality for dust null shells in the Minkowski spacetime of arbitrary dimension Classical and Quantum Gravity 29 (2012) no. 13. [bib] [doi] J. L. Flores, J. Herrera, M. Sánchez. Causal boundary of spacetimes: revision and applications to AdS/CFT correspondence Chapter in Quantum field theory and gravity Birkhäuser/Springer Basel AG, Basel (2012) , 97–119. MathScinet [bib] [doi] C. J. G. Machado, J. D. Pérez, I. Jeong, Y. J. Suh. $\germ D$-parallelism of normal and structure Jacobi operators for hypersurfaces in complex two-plane Grassmannians In Proceedings of the 16th International Workshop on Differential Geometry and the 5th KNUGRG-OCAMI Differential Geometry Workshop [Volume 16] Natl. Inst. Math. Sci. (NIMS), Taej\u on (2012) , 161–174. MathScinet [bib] J. D. Pérez, Y. J. Suh. Generalized Tanaka-Webster connection on real hypersurfaces of complex projective space In Proceedings of the 16th International Workshop on Differential Geometry and the 5th KNUGRG-OCAMI Differential Geometry Workshop [Volume 16] Natl. Inst. Math. Sci. (NIMS), Taej\u on (2012) , 47–54. MathScinet [bib] C. J. G. Machado, J. D. Pérez, Y. J. Suh. Generalized Tanaka-Webster connection on real hypersurfaces in complex two-plane Grassmannians In Proceedings of the 16th International Workshop on Differential Geometry and the 5th KNUGRG-OCAMI Differential Geometry Workshop [Volume 16] Natl. Inst. Math. Sci. (NIMS), Taej\u on (2012) , 35–46. MathScinet [bib] J. A. Aledo, S. Martinez, J. C. Valverde. Parallel dynamical systems over directed dependency graphs Appl. Math. Comput. 219 (2012) no. 3 , 1114–1119. MathScinet [bib] [doi] F. Martín, III W. H. Meeks. Calabi-Yau domains in three manifolds Amer. J. Math. 134 (2012) no. 5 , 1329–1344. MathScinet [bib] [doi] J. D. Pérez, J. T. Oh, Y. J. Suh. Compact real hypersurfaces in complex two-plane Grassmannians Houston J. Math. 38 (2012) no. 2 , 469–492. MathScinet [bib] M. Mars. Stability of MOTS in totally geodesic null horizons Classical Quantum Gravity 29 (2012) no. 14 , 145019, 23. MathScinet [bib] [doi] A. Romero, R. M. Rubio. On maximal surfaces in certain non-flat 3-dimensional Robertson-Walker spacetimes Math. Phys. Anal. Geom. 15 (2012) no. 3 , 193\textendash202. [bib] [doi] L. Ferrer, F. Martín, III W. H. Meeks. Existence of proper minimal surfaces of arbitrary topological type Adv. Math. 231 (2012) no. 1 , 378–413. MathScinet [bib] [doi] J. M. Almira, A. Romero. Some Riemannian geometric proofs of the fundamental theorem of algebra Differ. Geom. Dyn. Syst. 14 (2012) , 1–4. MathScinet [bib] M. Barros, A. Ferrández. Null scrolls as solutions of a sigma model J. Phys. A 45 (2012) no. 14 , 145203, 12. MathScinet [bib] [doi] M. Gutiérrez, B. Olea. Semi-Riemannian manifolds with a doubly warped structure Rev. Mat. Iberoam. 28 (2012) no. 1 , 1–24. MathScinet [bib] [doi] C. J. G. Machado, J. D. Pérez. Real hypersurfaces in complex two-plane Grassmannians some of whose Jacobi operators are $\xi$-invariant Internat. J. Math. 23 (2012) no. 3 , 1250002, 12. MathScinet [bib] [doi] J. A. Aledo, S. Martinez, F. L. Pelayo, J. C. Valverde. Parallel discrete dynamical systems on maxterm and minterm Boolean functions Math. Comput. Modelling 55 (2012) no. 3-4 , 666–671. MathScinet [bib] [doi] C. J. G. Machado, J. D. Pérez. On the structure vector field of a real hypersurface in complex two-plane Grassmannians Cent. Eur. J. Math. 10 (2012) no. 2 , 451–455. MathScinet [bib] [doi] M. Barros, O. J. Garay. Critical curves for the total normal curvature of 3-dimensional space forms J. Math. Anal. Appl. 389 (2012) no. 1 , 275–292. MathScinet [bib] [doi] | 2018-05-21 11:20:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9352269768714905, "perplexity": 4707.1415032064215}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1526794864063.9/warc/CC-MAIN-20180521102758-20180521122758-00440.warc.gz"} |
https://mathematica.stackexchange.com/revisions/4323/3 | Have you tried setting $HistoryLength to zero: $HistoryLength=0;
The memory is not freed, because the previous data is still available by Out[nn] or %nn.
Note that if you do this, using %, %%, %n etc., which some people use in example code will not work. And when a large data set is shown the "Show More" and "Show Less" buttons do not work. You can bypass both problems by setting \$HistoryLength to 1 while still conserving a lot of memory. | 2019-12-07 16:54:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3641405701637268, "perplexity": 1875.9911889194007}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540500637.40/warc/CC-MAIN-20191207160050-20191207184050-00390.warc.gz"} |
http://cryptologie.net/article/390/keeping-up-with-tls-13/ | Hey! I'm David, a security consultant at Cryptography Services, the crypto team of NCC Group . This is my blog about cryptography and security and other related topics that I find interesting.
Keeping up with TLS 1.3
posted May 2017
Ekr kick started the TLS:DIV workshop last Sunday. "The number of changes since draft 13 is too damn high" read one of the slide. Not wrong I said to myself. I did read draft 18 in its entirety when we had to review Cloudflare's TLS 1.3 implementation, and I tried to keep up with the changes ever since but I can honestly say that I completely failed.
So I thought, why not creating a nice diff that would allow me to go through all these changes just by reading the spec one more time. With the magic of git diff --color-words --word-diff=porcelain -U1000000 and some python I created a nice spec that shows up differences between draft 18 and the latest commit on the github spec.
You can find it here
If you want the same thing for a different draft version say something in the comment section!
Well done! You've reached the end of my post. Now you can leave me a comment :)
Tom
Your version is a bit nicer, but if you aren't aware, the IETF can generate diffs between drafts automatically as well: https://datatracker.ietf.org/doc/draft-ietf-tls-tls13/history/
David
Oh really nice! This is actually all I wanted: https://www.ietf.org/rfcdiff?url1=draft-ietf-tls-tls13-18&url2=draft-ietf-tls-tls13-20&difftype=--hwdiff | 2018-11-16 07:28:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.2863374948501587, "perplexity": 1757.8626744867302}, "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-2018-47/segments/1542039742981.53/warc/CC-MAIN-20181116070420-20181116092420-00454.warc.gz"} |
https://momath.org/home/varsity-math/varsity-math-week-10/ | ### Varsity Math, Week 10
Ain’t that just like the coach? He’s given the team some tough questions just as fall break is about to start. It’s also the end of this round of the relay — look for all the answers next week.
Mile High Stretch | Junior Varsity
The campus pranksters are at it again. They’ve managed to encircle the entire globe with a perfectly taut loop of rope, creating the largest possible circle on the surface of the Earth. The route they chose just happens to pass near Denver. Surprisingly, the city council makes a request — they want the rope to pass over Denver. So the pranksters agree to reroute the rope so it goes around the point on the Earth diametrically opposite Denver, as well as over the city, but they will need to get some extra rope. For the purposes of simplifying the problem, assume that (ignoring Denver) the Earth is a perfect sphere of diameter exactly 8,000 miles, and that the rope floats perfectly on the surface of every body of water. Also assume that all it takes to pass over Denver is to raise one single point of the rope to a mile in height above the surface of the sphere; anywhere else, the rope can touch the Earth or not and be at whatever height, as appropriate, as long as it is has no gaps or breaks and reaches a height of one mile above the surface of the sphere at some point.
What’s the minimum amount of extra rope that the pranksters have to add to their loop to pass over Denver? Your answer may be approximate; any value within one foot of the exact answer is fine.
Self-sequential Reference | Varsity Relay
Let R be the common ratio from Geometric Addition two weeks ago, and let
X = (R15 – (-1/R)15)/sqrt 5.
Now suppose that L is the least sequence of positive numbers whose first term is 2 and whose nth term gives the number of times that n appears in the sequence L, for every positive whole number n. (Here “least” means that for each k, the kth term of L is less than or equal to the kth term of any other sequence that satisfies the given properties and whose first k-1 terms match L.)
What is the Xth term of L?
## Solutions to Week 9
Minipixel. Let’s start with the smallest grids. A 1×1 grid can’t be labeled with numbers strictly between zero and the size of the grid, so that’s out. There’s only one way to label a 2×2 grid, with all ones, and that leads to an ambiguous solution.
A 3×3 grid labeled with all ones is ambiguous in the same way, so we need to try using some labels equal to two. Now notice that the sum of the row labels and column labels must be the same, since both are equal to the number of pixels in the solution, and hence there must be the same number of twos in the row labels and in the column labels. If there is only a single two in each set of labels, it can’t be in the middle, because then any solution could be flipped about that middle row or column to produce another solution. So for four pixels in a 3×3 grid, that leaves only the case of a two at one end or the other of both the row and column labels, which turns out to be ambiguous in much the same way as the 2×2 grid was. (See below left.) The situation is much the same with five pixels in a 3×3 grid: if the “one” label occurs in the middle of a side, the solution will be able to be flipped about the middle, and so will not be unique; and the pattern with a “one” label at the end of both the row and column labels turns out to be ambiguous in a way similar to the 2×2 grid. (See below right.) All labels equal to two on a 3×3 grid turns out to be impossible to solve (try it!).
So now we know that the smallest unambiguous pixel puzzler must inhabit at least a 4×4 grid. All labels equal to 1 would be highly ambiguous, but could there be an unambiguous puzzle with five pixels? Three labels equal to one and one label equal to two in each direction? It’s not hard to see that in any such grid, there will be one solution with a black pixel in the row and column labeled by twos, and another solution with a white pixel there. (See below.) So all of those pixel puzzlers are ambiguous.
When we turn our attention to six pixels, however, it is not hard to construct a 4×4 puzzle, which has each row and column labeled by a single number between one and three, and which has a unique solution. (See below.) So a pixel puzzler must have at least 6 pixels to have a unique solution, if the row and column labels are all single numbers more than zero and less than the size of the grid.
Stick Tac Toe. Because of the symmetry of the board, there are really only six opening plays for player X. Five of these allow for a deadly reply by player O, losing the game for X. Those replies are shown in Table 1 (below), in which O’s moves are shaded for clarity. No further analysis is needed for any of these openings.
So to win, player X must open with an “X” in one of the four middle squares and an “O” in the adjacent edge square. Now there are many possible responses by player O. However, all but thirteen of them leave a second “X” in a position where player X can immediately win on the next turn. So we only need to analyze those thirteen positions where X does not have an immediate win. In nine of them, shown in Table 2 (below), player X has a move which leaves a double threat, making it impossible for player O to block both threats.
In the four others, shown in Table 3 (below), player X has a move which leaves a threat which O can only block in one way, setting X up for a win on the next turn. In these two tables, player X’s opening move (which is the same in every case) and player O’s countermove are shown in white, and player’s X winning response is shown with shading.
As that exhausts all of player O’s countermoves to player X’s best opening, we see that player O can only guarantee that he or she will be able to place two dominos before player X wins.
### Previous Weeks
Week 9: Minipixel & Stick Tac Toe, solutions to Week 8
Week 8: Triple Jeopardy & Geometric Addition, solutions to Week 7
Week 7: Hex Truss & Windows on the World, solutions to Week 6
Week 6: Cross Training & Sesquicycle, solutions to Week 5 and Relay 1
Week 5: Laborious Lockup & Prime Event, solutions to Week 4
Week 4: Unsustainable Agriculture & Save the Camel, solutions to Week 3
Week 3: Deranged Assistant & Rush Job, solutions to Week 2
Week 2: Carambola Orchard & Farm Roundup, solutions to Week 1
Week 1: Instant Interrogation & Urn Respect
Find more puzzles at varsity.momath.org. | 2023-01-31 03:27:16 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.5321339964866638, "perplexity": 656.1396105497566}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1674764499842.81/warc/CC-MAIN-20230131023947-20230131053947-00470.warc.gz"} |
https://write-essayhelp.net/2020/10/07/how-to-solve-problems-with-absolute-value_lc/ | # How to solve problems with absolute value
By | October 7, 2020
Use absolute value equations firearms and policies essay outline problem solving early years and inequalities to solve stanford dissertation real-life problems, such as finding acceptable weights in example 4. solve each equation. if expository research paper topics playback doesn’t begin shortly, try restarting your device. absolute … author: the equation $$\left | x \right |=a$$ has two solutions x = a creative writing contests for teenagers and x = -a how to solve problems with absolute value because both numbers are how to publish my research paper at the distance a from different type of writing 0. graph the solutions. subtract academic writing vs professional writing 1 from how to solve problems with absolute value both sides. 2. transformations of absolute essay paper help value functions; performing absolute value transformations on other functions; transformations of the absolute value parent function. square root of polynomials hcf and lcm remainder how to solve problems with absolute value theorem. aug 29, 2019 · solving absolute value equations. the equation for essay about different culture the first inequality rule is: we will solve using cases. step 1: the absolute value equation is the equation that contains some absolute value expression. | 2020-10-21 04:49:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.2708146870136261, "perplexity": 1642.9053824312039}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1603107875980.5/warc/CC-MAIN-20201021035155-20201021065155-00570.warc.gz"} |
https://support.bioconductor.org/p/58981/ | use of voom function with attract package
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Dear Bioconductor, I am trying to use the attract package to find the processes that are differentially activated between cell types of the same lineage, using RNA-Seq data. Since the attract package is designed to work with microarray data, I decided to use the voom function to transform my data and change the findAttractors() function accordingly, to accommodate this type of data. Since this is not trivial, I want to make sure that I am using the output from the voom function correctly. The main part of the findAttractors() uses lm to model the expression in relation to the cell type (group) and then an anova to get the F statistic for each gene: fstat <- apply(dat.detect.wkegg, 1, function(y, x) { anova(lm(y ~ x))[[4]][1]}, x = group) where dat.detect.wkegg is the matrix of the normalized expression values with the genes per row and the samples per column. (To give some more context, the function then uses the log2 values of the fstat and does a t test between the gene values of a specific pathway vs all the gene values to identify the significant pathways. ) What I want to do is change the above to: counts_data <- DGEList(counts=rnaseq,group=celltype) counts_data_norm <- calcNormFactors(counts_data) design <- model.matrix( ~ celltype) anal_voom <- voom(counts_data_norm, design) dat.detect.wkegg <- as.list(as.data.frame(t(anal_voom$E))) voom_weights <- as.list(as.data.frame(t(anal_voom$weights))) fstat <- mapply(function(y, w, group) {anova(lm(y ~ group, weights=w))[[4]][1]}, dat.detect.wkegg, voom_weights, MoreArgs = list(group=celltype)) Is this the way to go with using the weights from voom, or am I getting this very wrong? Many thanks in advance for your reply! Best wishes, Emmanouela -- output of sessionInfo(): sessionInfo() R version 3.0.1 (2013-05-16) Platform: x86_64-unknown-linux-gnu (64-bit) locale: [1] LC_CTYPE=en_GB.ISO-8859-1 LC_NUMERIC=C LC_TIME=en_GB.ISO-8859-1 LC_COLLATE=en_GB.ISO-8859-1 LC_MONETARY=en_GB.ISO-8859-1 LC_MESSAGES=en_GB.ISO-8859-1 [7] LC_PAPER=C LC_NAME=C LC_ADDRESS=C LC_TELEPHONE=C LC_MEASUREMENT=en_GB.ISO-8859-1 LC_IDENTIFICATION=C attached base packages: [1] parallel stats graphics grDevices utils datasets methods base other attached packages: [1] attract_1.14.0 GOstats_2.28.0 graph_1.40.1 Category_2.28.0 GO.db_2.10.1 Matrix_1.1-3 cluster_1.15.2 annotate_1.40.1 org.Mm.eg.db_2.10.1 [10] KEGG.db_2.10.1 RSQLite_0.11.4 DBI_0.2-7 AnnotationDbi_1.24.0 Biobase_2.22.0 BiocGenerics_0.8.0 edgeR_3.4.2 limma_3.18.13 loaded via a namespace (and not attached): [1] AnnotationForge_1.4.4 genefilter_1.44.0 grid_3.0.1 GSEABase_1.24.0 IRanges_1.20.7 lattice_0.20-29 RBGL_1.38.0 splines_3.0.1 [9] stats4_3.0.1 survival_2.37-7 tcltk_3.0.1 tools_3.0.1 XML_3.98-1.1 xtable_1.7-3 -- Sent via the guest posting facility at bioconductor.org.
Pathways GO attract Pathways GO attract • 2.3k views
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@ryan-c-thompson-5618
Last seen 2.1 years ago
Scripps Research, La Jolla, CA
Hi Emmanouela, I believe that is the correct way to use the voom weights in a call to lm. However, if you are using voom, you might also want to use limma as you normally would to compute moderated F statistics. Just use topTable with n=Inf and sort="none" to get the full table with no reordering of rows, and pull out the F column. -Ryan On Tue Apr 15 09:44:42 2014, Emmanouela Repapi [guest] wrote: > > Dear Bioconductor, > > I am trying to use the attract package to find the processes that are differentially activated between cell types of the same lineage, using RNA-Seq data. Since the attract package is designed to work with microarray data, I decided to use the voom function to transform my data and change the findAttractors() function accordingly, to accommodate this type of data. Since this is not trivial, I want to make sure that I am using the output from the voom function correctly. > > The main part of the findAttractors() uses lm to model the expression in relation to the cell type (group) and then an anova to get the F statistic for each gene: > fstat <- apply(dat.detect.wkegg, 1, function(y, x) { > anova(lm(y ~ x))[[4]][1]}, x = group) > where dat.detect.wkegg is the matrix of the normalized expression values with the genes per row and the samples per column. > (To give some more context, the function then uses the log2 values of the fstat and does a t test between the gene values of a specific pathway vs all the gene values to identify the significant pathways. ) > > What I want to do is change the above to: > > counts_data <- DGEList(counts=rnaseq,group=celltype) > counts_data_norm <- calcNormFactors(counts_data) > > design <- model.matrix( ~ celltype) > anal_voom <- voom(counts_data_norm, design) > dat.detect.wkegg <- as.list(as.data.frame(t(anal_voom$E))) > voom_weights <- as.list(as.data.frame(t(anal_voom$weights))) > > fstat <- mapply(function(y, w, group) {anova(lm(y ~ group, weights=w))[[4]][1]}, > dat.detect.wkegg, voom_weights, MoreArgs = list(group=celltype)) > > Is this the way to go with using the weights from voom, or am I getting this very wrong? > > Many thanks in advance for your reply! > > Best wishes, > Emmanouela > > > > > -- output of sessionInfo(): > > sessionInfo() > R version 3.0.1 (2013-05-16) > Platform: x86_64-unknown-linux-gnu (64-bit) > > locale: > [1] LC_CTYPE=en_GB.ISO-8859-1 LC_NUMERIC=C LC_TIME=en_GB.ISO-8859-1 LC_COLLATE=en_GB.ISO-8859-1 LC_MONETARY=en_GB.ISO-8859-1 LC_MESSAGES=en_GB.ISO-8859-1 > [7] LC_PAPER=C LC_NAME=C LC_ADDRESS=C LC_TELEPHONE=C LC_MEASUREMENT=en_GB.ISO-8859-1 LC_IDENTIFICATION=C > > attached base packages: > [1] parallel stats graphics grDevices utils datasets methods base > > other attached packages: > [1] attract_1.14.0 GOstats_2.28.0 graph_1.40.1 Category_2.28.0 GO.db_2.10.1 Matrix_1.1-3 cluster_1.15.2 annotate_1.40.1 org.Mm.eg.db_2.10.1 > [10] KEGG.db_2.10.1 RSQLite_0.11.4 DBI_0.2-7 AnnotationDbi_1.24.0 Biobase_2.22.0 BiocGenerics_0.8.0 edgeR_3.4.2 limma_3.18.13 > > loaded via a namespace (and not attached): > [1] AnnotationForge_1.4.4 genefilter_1.44.0 grid_3.0.1 GSEABase_1.24.0 IRanges_1.20.7 lattice_0.20-29 RBGL_1.38.0 splines_3.0.1 > [9] stats4_3.0.1 survival_2.37-7 tcltk_3.0.1 tools_3.0.1 XML_3.98-1.1 xtable_1.7-3 > > > -- > Sent via the guest posting facility at bioconductor.org. > > _______________________________________________ > Bioconductor mailing list > Bioconductor at r-project.org > https://stat.ethz.ch/mailman/listinfo/bioconductor > Search the archives: http://news.gmane.org/gmane.science.biology.informatics.conductor
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@gordon-smyth
Last seen 26 minutes ago
WEHI, Melbourne, Australia
Dear Emmanouela, The limma package is designed to fit linear models, and it can compute t-statistics and F-statistics faster than making your own loop to lm(). If you want F-statistics for distinguishing the cell types, why not: fit <- lmFit(anal_voom, design) fit <- eBayes(fit[,-1]) Then the F-statistics will be in fit$F. If you want to know whether a particular KEGG pathway tends to have larger F-statistics, you could also use: geneSetTest(index, fit$F) where index selects genes in the pathway. If there are only two cell types, a better way would be: camera(anal_voom, index, design) With camera, index could be a list of index vectors for all the KEGG pathways at once. Best wishes Gordon > Date: Tue, 15 Apr 2014 09:44:42 -0700 (PDT) > From: "Emmanouela Repapi [guest]" <guest at="" bioconductor.org=""> > To: bioconductor at r-project.org, emmanouela.repapi at imm.ox.ac.uk > Subject: [BioC] use of voom function with attract package > > > Dear Bioconductor, > > I am trying to use the attract package to find the processes that are > differentially activated between cell types of the same lineage, using > RNA-Seq data. Since the attract package is designed to work with > microarray data, I decided to use the voom function to transform my data > and change the findAttractors() function accordingly, to accommodate > this type of data. Since this is not trivial, I want to make sure that I > am using the output from the voom function correctly. > > The main part of the findAttractors() uses lm to model the expression in relation to the cell type (group) and then an anova to get the F statistic for each gene: > fstat <- apply(dat.detect.wkegg, 1, function(y, x) { > anova(lm(y ~ x))[[4]][1]}, x = group) > where dat.detect.wkegg is the matrix of the normalized expression values with the genes per row and the samples per column. > (To give some more context, the function then uses the log2 values of the fstat and does a t test between the gene values of a specific pathway vs all the gene values to identify the significant pathways. ) > > What I want to do is change the above to: > > counts_data <- DGEList(counts=rnaseq,group=celltype) > counts_data_norm <- calcNormFactors(counts_data) > > design <- model.matrix( ~ celltype) > anal_voom <- voom(counts_data_norm, design) > dat.detect.wkegg <- as.list(as.data.frame(t(anal_voom$E))) > voom_weights <- as.list(as.data.frame(t(anal_voom$weights))) > > fstat <- mapply(function(y, w, group) {anova(lm(y ~ group, weights=w))[[4]][1]}, > dat.detect.wkegg, voom_weights, MoreArgs = list(group=celltype)) > > Is this the way to go with using the weights from voom, or am I getting this very wrong? > > Many thanks in advance for your reply! > > Best wishes, > Emmanouela > > > > > -- output of sessionInfo(): > > sessionInfo() > R version 3.0.1 (2013-05-16) > Platform: x86_64-unknown-linux-gnu (64-bit) > > locale: > [1] LC_CTYPE=en_GB.ISO-8859-1 LC_NUMERIC=C LC_TIME=en_GB.ISO-8859-1 LC_COLLATE=en_GB.ISO-8859-1 LC_MONETARY=en_GB.ISO-8859-1 LC_MESSAGES=en_GB.ISO-8859-1 > [7] LC_PAPER=C LC_NAME=C LC_ADDRESS=C LC_TELEPHONE=C LC_MEASUREMENT=en_GB.ISO-8859-1 LC_IDENTIFICATION=C > > attached base packages: > [1] parallel stats graphics grDevices utils datasets methods base > > other attached packages: > [1] attract_1.14.0 GOstats_2.28.0 graph_1.40.1 Category_2.28.0 GO.db_2.10.1 Matrix_1.1-3 cluster_1.15.2 annotate_1.40.1 org.Mm.eg.db_2.10.1 > [10] KEGG.db_2.10.1 RSQLite_0.11.4 DBI_0.2-7 AnnotationDbi_1.24.0 Biobase_2.22.0 BiocGenerics_0.8.0 edgeR_3.4.2 limma_3.18.13 > > loaded via a namespace (and not attached): > [1] AnnotationForge_1.4.4 genefilter_1.44.0 grid_3.0.1 GSEABase_1.24.0 IRanges_1.20.7 lattice_0.20-29 RBGL_1.38.0 splines_3.0.1 > [9] stats4_3.0.1 survival_2.37-7 tcltk_3.0.1 tools_3.0.1 XML_3.98-1.1 xtable_1.7-3 ______________________________________________________________________ The information in this email is confidential and intend...{{dropped:4}}
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Dear Gordon and Ryan, Thank you both very much for your replies, using the F statistic from the lmFit and the geneSetTest seems like the right way to go for what I need to do. On a similar note, do you have a function in mind for finding the genes with similar expression profiles within a specific pathway (as a replacement of the other function of attract, findSynexprs)? Thank you again for your help. Best wishes, Emmanouela On 18 Apr 2014, at 02:16, Gordon K Smyth <smyth@wehi.edu.au> wrote: > Dear Emmanouela, > > The limma package is designed to fit linear models, and it can compute t-statistics and F-statistics faster than making your own loop to lm(). If you want F-statistics for distinguishing the cell types, why not: > > fit <- lmFit(anal_voom, design) > fit <- eBayes(fit[,-1]) > > Then the F-statistics will be in fit$F. > > If you want to know whether a particular KEGG pathway tends to have larger F-statistics, you could also use: > > geneSetTest(index, fit$F) > > where index selects genes in the pathway. If there are only two cell types, a better way would be: > > camera(anal_voom, index, design) > > With camera, index could be a list of index vectors for all the KEGG pathways at once. > > Best wishes > Gordon > >> Date: Tue, 15 Apr 2014 09:44:42 -0700 (PDT) >> From: "Emmanouela Repapi [guest]" <guest@bioconductor.org> >> To: bioconductor@r-project.org, emmanouela.repapi@imm.ox.ac.uk >> Subject: [BioC] use of voom function with attract package >> >> >> Dear Bioconductor, >> >> I am trying to use the attract package to find the processes that are differentially activated between cell types of the same lineage, using RNA-Seq data. Since the attract package is designed to work with microarray data, I decided to use the voom function to transform my data and change the findAttractors() function accordingly, to accommodate this type of data. Since this is not trivial, I want to make sure that I am using the output from the voom function correctly. >> >> The main part of the findAttractors() uses lm to model the expression in relation to the cell type (group) and then an anova to get the F statistic for each gene: >> fstat <- apply(dat.detect.wkegg, 1, function(y, x) { >> anova(lm(y ~ x))[[4]][1]}, x = group) >> where dat.detect.wkegg is the matrix of the normalized expression values with the genes per row and the samples per column. >> (To give some more context, the function then uses the log2 values of the fstat and does a t test between the gene values of a specific pathway vs all the gene values to identify the significant pathways. ) >> >> What I want to do is change the above to: >> >> counts_data <- DGEList(counts=rnaseq,group=celltype) >> counts_data_norm <- calcNormFactors(counts_data) >> >> design <- model.matrix( ~ celltype) >> anal_voom <- voom(counts_data_norm, design) >> dat.detect.wkegg <- as.list(as.data.frame(t(anal_voom$E))) >> voom_weights <- as.list(as.data.frame(t(anal_voom$weights))) >> >> fstat <- mapply(function(y, w, group) {anova(lm(y ~ group, weights=w))[[4]][1]}, >> dat.detect.wkegg, voom_weights, MoreArgs = list(group=celltype)) >> >> Is this the way to go with using the weights from voom, or am I getting this very wrong? >> >> Many thanks in advance for your reply! >> >> Best wishes, >> Emmanouela >> >> >> >> >> -- output of sessionInfo(): >> >> sessionInfo() >> R version 3.0.1 (2013-05-16) >> Platform: x86_64-unknown-linux-gnu (64-bit) >> >> locale: >> [1] LC_CTYPE=en_GB.ISO-8859-1 LC_NUMERIC=C LC_TIME=en_GB.ISO-8859-1 LC_COLLATE=en_GB.ISO-8859-1 LC_MONETARY=en_GB.ISO-8859-1 LC_MESSAGES=en_GB.ISO-8859-1 >> [7] LC_PAPER=C LC_NAME=C LC_ADDRESS=C LC_TELEPHONE=C LC_MEASUREMENT=en_GB.ISO-8859-1 LC_IDENTIFICATION=C >> >> attached base packages: >> [1] parallel stats graphics grDevices utils datasets methods base >> >> other attached packages: >> [1] attract_1.14.0 GOstats_2.28.0 graph_1.40.1 Category_2.28.0 GO.db_2.10.1 Matrix_1.1-3 cluster_1.15.2 annotate_1.40.1 org.Mm.eg.db_2.10.1 >> [10] KEGG.db_2.10.1 RSQLite_0.11.4 DBI_0.2-7 AnnotationDbi_1.24.0 Biobase_2.22.0 BiocGenerics_0.8.0 edgeR_3.4.2 limma_3.18.13 >> >> loaded via a namespace (and not attached): >> [1] AnnotationForge_1.4.4 genefilter_1.44.0 grid_3.0.1 GSEABase_1.24.0 IRanges_1.20.7 lattice_0.20-29 RBGL_1.38.0 splines_3.0.1 >> [9] stats4_3.0.1 survival_2.37-7 tcltk_3.0.1 tools_3.0.1 XML_3.98-1.1 xtable_1.7-3 > > ______________________________________________________________________ > The information in this email is confidential and inte...{{dropped:10}}
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Hi Emmanouela, I don't know of an off-the-shelf function to do something similar to findSynexprs but using precision weights. I guess one would do hiearchical clustering of the genes, and would define the distance measure to be a Pearson correlation calculation but using weights. Not hard to do, but not yet ready off-the-shelf as far as I know. Best wishes Gordon On Thu, 24 Apr 2014, Emmanouela Repapi wrote: > Dear Gordon and Ryan, > > Thank you both very much for your replies, using the F statistic from > the lmFit and the geneSetTest seems like the right way to go for what I > need to do. > On a similar note, do you have a function in mind for finding the genes > with similar expression profiles within a specific pathway (as a > replacement of the other function of attract, findSynexprs)? > > Thank you again for your help. > > Best wishes, > Emmanouela > > On 18 Apr 2014, at 02:16, Gordon K Smyth <smyth at="" wehi.edu.au=""> wrote: > >> Dear Emmanouela, >> >> The limma package is designed to fit linear models, and it can compute t-statistics and F-statistics faster than making your own loop to lm(). If you want F-statistics for distinguishing the cell types, why not: >> >> fit <- lmFit(anal_voom, design) >> fit <- eBayes(fit[,-1]) >> >> Then the F-statistics will be in fit$F. >> >> If you want to know whether a particular KEGG pathway tends to have larger F-statistics, you could also use: >> >> geneSetTest(index, fit$F) >> >> where index selects genes in the pathway. If there are only two cell types, a better way would be: >> >> camera(anal_voom, index, design) >> >> With camera, index could be a list of index vectors for all the KEGG pathways at once. >> >> Best wishes >> Gordon >> >>> Date: Tue, 15 Apr 2014 09:44:42 -0700 (PDT) >>> From: "Emmanouela Repapi [guest]" <guest at="" bioconductor.org=""> >>> To: bioconductor at r-project.org, emmanouela.repapi at imm.ox.ac.uk >>> Subject: [BioC] use of voom function with attract package >>> >>> >>> Dear Bioconductor, >>> >>> I am trying to use the attract package to find the processes that are differentially activated between cell types of the same lineage, using RNA-Seq data. Since the attract package is designed to work with microarray data, I decided to use the voom function to transform my data and change the findAttractors() function accordingly, to accommodate this type of data. Since this is not trivial, I want to make sure that I am using the output from the voom function correctly. >>> >>> The main part of the findAttractors() uses lm to model the expression in relation to the cell type (group) and then an anova to get the F statistic for each gene: >>> fstat <- apply(dat.detect.wkegg, 1, function(y, x) { >>> anova(lm(y ~ x))[[4]][1]}, x = group) >>> where dat.detect.wkegg is the matrix of the normalized expression values with the genes per row and the samples per column. >>> (To give some more context, the function then uses the log2 values of the fstat and does a t test between the gene values of a specific pathway vs all the gene values to identify the significant pathways. ) >>> >>> What I want to do is change the above to: >>> >>> counts_data <- DGEList(counts=rnaseq,group=celltype) >>> counts_data_norm <- calcNormFactors(counts_data) >>> >>> design <- model.matrix( ~ celltype) >>> anal_voom <- voom(counts_data_norm, design) >>> dat.detect.wkegg <- as.list(as.data.frame(t(anal_voom$E))) >>> voom_weights <- as.list(as.data.frame(t(anal_voom$weights))) >>> >>> fstat <- mapply(function(y, w, group) {anova(lm(y ~ group, weights=w))[[4]][1]}, >>> dat.detect.wkegg, voom_weights, MoreArgs = list(group=celltype)) >>> >>> Is this the way to go with using the weights from voom, or am I getting this very wrong? >>> >>> Many thanks in advance for your reply! >>> >>> Best wishes, >>> Emmanouela >>> >>> >>> >>> >>> -- output of sessionInfo(): >>> >>> sessionInfo() >>> R version 3.0.1 (2013-05-16) >>> Platform: x86_64-unknown-linux-gnu (64-bit) >>> >>> locale: >>> [1] LC_CTYPE=en_GB.ISO-8859-1 LC_NUMERIC=C LC_TIME=en_GB.ISO-8859-1 LC_COLLATE=en_GB.ISO-8859-1 LC_MONETARY=en_GB.ISO-8859-1 LC_MESSAGES=en_GB.ISO-8859-1 >>> [7] LC_PAPER=C LC_NAME=C LC_ADDRESS=C LC_TELEPHONE=C LC_MEASUREMENT=en_GB.ISO-8859-1 LC_IDENTIFICATION=C >>> >>> attached base packages: >>> [1] parallel stats graphics grDevices utils datasets methods base >>> >>> other attached packages: >>> [1] attract_1.14.0 GOstats_2.28.0 graph_1.40.1 Category_2.28.0 GO.db_2.10.1 Matrix_1.1-3 cluster_1.15.2 annotate_1.40.1 org.Mm.eg.db_2.10.1 >>> [10] KEGG.db_2.10.1 RSQLite_0.11.4 DBI_0.2-7 AnnotationDbi_1.24.0 Biobase_2.22.0 BiocGenerics_0.8.0 edgeR_3.4.2 limma_3.18.13 >>> >>> loaded via a namespace (and not attached): >>> [1] AnnotationForge_1.4.4 genefilter_1.44.0 grid_3.0.1 GSEABase_1.24.0 IRanges_1.20.7 lattice_0.20-29 RBGL_1.38.0 splines_3.0.1 >>> [9] stats4_3.0.1 survival_2.37-7 tcltk_3.0.1 tools_3.0.1 XML_3.98-1.1 xtable_1.7-3 >> >> ______________________________________________________________________ >> The information in this email is confidential and intended solely for the addressee. >> You must not disclose, forward, print or use it without the permission of the sender. >> ______________________________________________________________________ > > ______________________________________________________________________ The information in this email is confidential and intend...{{dropped:4}}
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Entering edit mode
Dear Gordon, Ok, thats what I thought. Thank you again so much for all your help! Best wishes, Emmanouela Emmanouela Repapi, PhD Computational Biology Research Group Weatherall Institute of Molecular Medicine University of Oxford, OX3 9DS Tel: (01865 2)22253 On 25 Apr 2014, at 00:25, Gordon K Smyth <smyth@wehi.edu.au> wrote: > Hi Emmanouela, > > I don't know of an off-the-shelf function to do something similar to findSynexprs but using precision weights. I guess one would do hiearchical clustering of the genes, and would define the distance measure to be a Pearson correlation calculation but using weights. Not hard to do, but not yet ready off-the-shelf as far as I know. > > Best wishes > Gordon > > On Thu, 24 Apr 2014, Emmanouela Repapi wrote: > >> Dear Gordon and Ryan, >> >> Thank you both very much for your replies, using the F statistic from the lmFit and the geneSetTest seems like the right way to go for what I need to do. > >> On a similar note, do you have a function in mind for finding the genes with similar expression profiles within a specific pathway (as a replacement of the other function of attract, findSynexprs)? >> >> Thank you again for your help. >> >> Best wishes, >> Emmanouela >> >> On 18 Apr 2014, at 02:16, Gordon K Smyth <smyth@wehi.edu.au> wrote: >> >>> Dear Emmanouela, >>> >>> The limma package is designed to fit linear models, and it can compute t-statistics and F-statistics faster than making your own loop to lm(). If you want F-statistics for distinguishing the cell types, why not: >>> >>> fit <- lmFit(anal_voom, design) >>> fit <- eBayes(fit[,-1]) >>> >>> Then the F-statistics will be in fit$F. >>> >>> If you want to know whether a particular KEGG pathway tends to have larger F-statistics, you could also use: >>> >>> geneSetTest(index, fit$F) >>> >>> where index selects genes in the pathway. If there are only two cell types, a better way would be: >>> >>> camera(anal_voom, index, design) >>> >>> With camera, index could be a list of index vectors for all the KEGG pathways at once. >>> >>> Best wishes >>> Gordon >>> >>>> Date: Tue, 15 Apr 2014 09:44:42 -0700 (PDT) >>>> From: "Emmanouela Repapi [guest]" <guest@bioconductor.org> >>>> To: bioconductor@r-project.org, emmanouela.repapi@imm.ox.ac.uk >>>> Subject: [BioC] use of voom function with attract package >>>> >>>> >>>> Dear Bioconductor, >>>> >>>> I am trying to use the attract package to find the processes that are differentially activated between cell types of the same lineage, using RNA-Seq data. Since the attract package is designed to work with microarray data, I decided to use the voom function to transform my data and change the findAttractors() function accordingly, to accommodate this type of data. Since this is not trivial, I want to make sure that I am using the output from the voom function correctly. >>>> >>>> The main part of the findAttractors() uses lm to model the expression in relation to the cell type (group) and then an anova to get the F statistic for each gene: >>>> fstat <- apply(dat.detect.wkegg, 1, function(y, x) { >>>> anova(lm(y ~ x))[[4]][1]}, x = group) >>>> where dat.detect.wkegg is the matrix of the normalized expression values with the genes per row and the samples per column. >>>> (To give some more context, the function then uses the log2 values of the fstat and does a t test between the gene values of a specific pathway vs all the gene values to identify the significant pathways. ) >>>> >>>> What I want to do is change the above to: >>>> >>>> counts_data <- DGEList(counts=rnaseq,group=celltype) >>>> counts_data_norm <- calcNormFactors(counts_data) >>>> >>>> design <- model.matrix( ~ celltype) >>>> anal_voom <- voom(counts_data_norm, design) >>>> dat.detect.wkegg <- as.list(as.data.frame(t(anal_voom$E))) >>>> voom_weights <- as.list(as.data.frame(t(anal_voom$weights))) >>>> >>>> fstat <- mapply(function(y, w, group) {anova(lm(y ~ group, weights=w))[[4]][1]}, >>>> dat.detect.wkegg, voom_weights, MoreArgs = list(group=celltype)) >>>> >>>> Is this the way to go with using the weights from voom, or am I getting this very wrong? >>>> >>>> Many thanks in advance for your reply! >>>> >>>> Best wishes, >>>> Emmanouela >>>> >>>> >>>> >>>> >>>> -- output of sessionInfo(): >>>> >>>> sessionInfo() >>>> R version 3.0.1 (2013-05-16) >>>> Platform: x86_64-unknown-linux-gnu (64-bit) >>>> >>>> locale: >>>> [1] LC_CTYPE=en_GB.ISO-8859-1 LC_NUMERIC=C LC_TIME=en_GB.ISO-8859-1 LC_COLLATE=en_GB.ISO-8859-1 LC_MONETARY=en_GB.ISO-8859-1 LC_MESSAGES=en_GB.ISO-8859-1 >>>> [7] LC_PAPER=C LC_NAME=C LC_ADDRESS=C LC_TELEPHONE=C LC_MEASUREMENT=en_GB.ISO-8859-1 LC_IDENTIFICATION=C >>>> >>>> attached base packages: >>>> [1] parallel stats graphics grDevices utils datasets methods base >>>> >>>> other attached packages: >>>> [1] attract_1.14.0 GOstats_2.28.0 graph_1.40.1 Category_2.28.0 GO.db_2.10.1 Matrix_1.1-3 cluster_1.15.2 annotate_1.40.1 org.Mm.eg.db_2.10.1 >>>> [10] KEGG.db_2.10.1 RSQLite_0.11.4 DBI_0.2-7 AnnotationDbi_1.24.0 Biobase_2.22.0 BiocGenerics_0.8.0 edgeR_3.4.2 limma_3.18.13 >>>> >>>> loaded via a namespace (and not attached): >>>> [1] AnnotationForge_1.4.4 genefilter_1.44.0 grid_3.0.1 GSEABase_1.24.0 IRanges_1.20.7 lattice_0.20-29 RBGL_1.38.0 splines_3.0.1 >>>> [9] stats4_3.0.1 survival_2.37-7 tcltk_3.0.1 tools_3.0.1 XML_3.98-1.1 xtable_1.7-3 >>> >>> ______________________________________________________________________ >>> The information in this email is confidential and intended solely for the addressee. >>> You must not disclose, forward, print or use it without the permission of the sender. >>> ______________________________________________________________________ >> >> > > ______________________________________________________________________ > The information in this email is confidential and inte...{{dropped:10}} | 2022-09-28 00:24:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4296073913574219, "perplexity": 10256.177399162189}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335059.31/warc/CC-MAIN-20220927225413-20220928015413-00010.warc.gz"} |
http://mathhelpforum.com/number-theory/9481-largest-number-n-print.html | # The largest number N
• January 3rd 2007, 11:33 AM
Jenny20
The largest number N
Another question:
What is the largest number N for which you can say that n^5-5n^3+4n is divisible by N for every integer n?
Thank you very much.
• January 3rd 2007, 12:52 PM
Soroban
Hello, Jenny!
Quote:
What is the largest number $N$ for which you can say that
$n^5-5n^3+4n$ is divisible by $N$ for every integer $n$?
$\text{Factor: }\:n^5 - 5n^3 + 4n \;=\;n(n^4 - 4n^2 + 4)$
. . . . . . . . $= \;n(n^2 - 1)(n^2 - 4)\;=\;n(n-1)(n+1)(n-2)(n+2)$
$\text{We have: }\:n^5 - 5n^3 + 4n \;=\;\underbrace{(n-2)(n-1)(n)(n+1)(n-2)}_{\text{product of 5 consecutive integers}}$
The product of 5 consecutive integers is divisible by $1,\,2,\,3,\,4,\,5.$
Therefore: . $N \,=\,120$ | 2015-09-03 07:17:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 9, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.482462078332901, "perplexity": 636.1463372065361}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440645305536.82/warc/CC-MAIN-20150827031505-00348-ip-10-171-96-226.ec2.internal.warc.gz"} |
http://rinterested.github.io/statistics/tensors2.html | ### TENSORS 2:
A concrete explanation can be found here, and it proceeds as follows:
We have two inertial (no acceleration) coordinate systems, $$S$$ and $$S'$$, where $$S'$$ moves away from $$S$$ at constant velocity $$v$$ along the $$x$$ axis - for now we assume no rotation or movement along the other two axes, $$y$$ and $$z$$.
An event $$P$$ happens and it is recorded in $$S$$ as $$(ct, x, y, z)$$. Why $$ct$$? To express time in the same units as the space coordintes we multiply the velocity of light by the observation of time. Not important.
The same event $$P$$ will be expressed in relation to the $$S'$$ coordinates as $$P=(ct', x', y',z').$$
#### Definition: CONTRAVARIANT VECTOR:
$$x^\mu = (x^0,x^1,x^2,x^3)$$, such that
$$x^0= ct, \quad x^1=x, \quad x^2=y, \quad x^3=z$$
So $$\mu$$ is an index, $$\mu= 0,1,2,3.$$
#### Definition: COVARIANT VECTOR:
$$x_\mu=(x_0,x_1,x_2,x_3)$$, such that
$$x_0=ct, \quad x_1=\color{red}{-}x, \quad\color{red}{-}y \quad\color{red}{-}z$$
Why do we need this?
The contravariant and covariant vectors of the same event in relation to the frame $$S'$$ are:
$$x'^\mu=(x^{'0}, x^{'1},x^{'2},x^{'3})\quad \text{contravariant}$$
and
$$x'_\mu=(x'_0, x'_1,x'_2,x'_3)\quad\text{covariant}$$
Now we have to bring in the Lorentz transformations for the answer to make sense. They are the equations that change coordinates between $$S$$ and $$S'$$ taking into consideration time dilation and length contraction (along the $$x$$ axis only, since the frames are only moving with respect to each other along $$x$$):
The transformation of time from $$t$$ in frame $$S$$ to $$t'$$ in $$S'$$ is given by $$t' = \gamma \left(t - \frac{v}{c^2}x\right)$$; while the transformation of $$x$$ from $$S$$ to $$S'$$ obeys the equation $$x' =\gamma(x - vt).$$ This is explained here, and adopting the notation of contravariant and covariant vectors, and the $$ct$$ expression of $$t$$:
$$x'^0 = \gamma\left(x^0 - \frac{v}{c} x^1\right)$$
$$x'^1 = \gamma\left(x^1 - \frac{v}{c} x^0\right)$$
$$x'^2 = x^2$$
$$x'^3 = x^3$$
Here, $$\large\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$. Notice that the third and fourth dimension do not change in this simple model with two frames, one moving along the $$x$$ axis of the other at constant velocity.
We can express the change in frames in matrix form as:
$\begin{bmatrix}x'^0\\x'^1\\x'^2\\x'^3\end{bmatrix}=\begin{bmatrix}\gamma&-\gamma\beta&0&0\\-\gamma\beta&\gamma&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}\begin{bmatrix}x^0\\x^1\\x^2\\x^3\end{bmatrix}$
with $$\beta=\frac{v}{c}.$$
If there is motion of $$S'$$ in more than just the $$x$$ difrection, the matriceal formula above can be generalized as:
$\begin{bmatrix}x'^0\\x'^1\\x'^2\\x'^3\end{bmatrix}=\begin{bmatrix}\Lambda^0_{\; 0}&\Lambda^0_{\; 1}&\Lambda^0_{\; 2}&\Lambda^0_{\; 3}\\\Lambda^1_{\; 0}&\Lambda^1_{\; 1}&\Lambda^1_{\; 2}&\Lambda^1_{\; 3}\\\Lambda^2_{\; 0}&\Lambda^2_{\; 1}&\Lambda^2_{\; 2}&\Lambda^2_{\; 3}\\\Lambda^3_{\; 0}&\Lambda^3_{\; 1}&\Lambda^3_{\; 2}&\Lambda^3_{\; 3}\end{bmatrix}\begin{bmatrix}x^0\\x^1\\x^2\\x^3\end{bmatrix}$
This can be express more succintly as:
$\color{red}{x'^{\mu} = \Lambda^\mu_{\; \nu}x^\nu}$
The same can be done with the covariant vector transformation:
$\begin{bmatrix}x'_0\\x'_1\\x'_2\\x'_3\end{bmatrix}=\begin{bmatrix}\Lambda_0^{\; 0}&\Lambda_0^{\; 1}&\Lambda_0^{\; 2}&\Lambda_0^{\; 3}\\\Lambda_1^{\; 0}&\Lambda_1^{\; 1}&\Lambda_1^{\; 2}&\Lambda_1^{\; 3}\\\Lambda_2^{\; 0}&\Lambda_2^{\; 1}&\Lambda_2^{\; 2}&\Lambda_2^{\; 3}\\\Lambda_3^{\; 0}&\Lambda_3^{\; 1}&\Lambda_3^{\; 2}&\Lambda_3^{\; 3}\end{bmatrix}\begin{bmatrix}x_0\\x_1\\x_2\\x-3\end{bmatrix}$
or
$\color{blue}{x'_{\mu} = \Lambda_\mu^{\; \nu}x_\nu}$
The Lorentz transformation $$\Lambda$$ is defined so that $$x'^{\mu}x'_{\mu} = x^\mu x_mu$$ (i.e. $$\small \text{contravariant in S'} \times \text{covariant in S'}=\text{contravariant in S} \times \text{covariant in S}$$).
We can express it as:
$\Large \color{red}{x'^{\mu}}\,\color{blue}{x'_{\mu}} = x^\mu \,x_\mu= \color{red}{\Lambda^\mu_{\;\;\nu}\,x^\nu}\,\color{blue}{\Lambda_\mu^{\;\;\rho}\,x_\rho}= \Lambda^{\mu}_{\;\;\nu}\,\Lambda_\mu^{\;\;\rho}\,\,x^\nu \,x_\rho=\delta^\rho_{\;\;\nu}\,x^\nu\,x_\rho$
Why does it work? Well, when $$\rho=\nu=\mu$$ the Kronecker product is one, and the final part of the equation above will be $$x^\mu\,x_\mu,$$ fulfilling the requisite of $$x'^{\mu}x'_{\mu} = x^\mu x_\mu.$$
$\large\color{orange}{\Lambda^\mu_{\;\;\nu}\,\Lambda_\mu^{\;\;\rho}=\delta^\rho_{\;\;\nu}}$
OK. So now we have covariant and contravariant vectors expressing the event with respect to $$S$$ and $$S'$$… and Lorentz transformations… we can do this…
$\large x^\mu x_\mu= x^0 x_0 + x^1 x_1 + x^2 x_2 + x^3 x_3 = c^2t^2 - \left(x^2 + y^2 + z^2 \right)=\color{red}{c^2t^2 - \vec{x}\vec{x}}$
$$\vec{x}\vec{x}$$ is the norm! And the result is a scalar (field value).
How to express this operation generally? We use the Einstein summation convention:
$A^\mu B_\mu=A^0B_0+A^1B_1+A^2B_2+A^3B_3$
where $$A^\mu =(A^0,A^1,A^2,A^3)$$ has a covariant vector $$A_mu=(A_0,A_1,A_2,A_3) = (A_0,-A^1,-A^2,-A^3).$$ And so does $$B^\mu.$$ | 2021-07-30 00:27:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9267784953117371, "perplexity": 239.7344090568095}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046153899.14/warc/CC-MAIN-20210729234313-20210730024313-00295.warc.gz"} |
https://qudt.org/vocab/quantitykind/LuminousFluxPerArea | quantitykind:LuminousFluxPerArea
Type
Description
In photometry, illuminance is the total luminous flux incident on a surface, per unit area. It is a measure of how much the incident light illuminates the surface, wavelength-weighted by the luminosity function to correlate with human brightness perception. Similarly, luminous emittance is the luminous flux per unit area emitted from a surface. In SI derived units these are measured in $$lux (lx)$$ or $$lumens per square metre$$ ($$cd \cdot m^{-2}$$). In the CGS system, the unit of illuminance is the $$phot$$, which is equal to $$10,000 lux$$. The $$foot-candle$$ is a non-metric unit of illuminance that is used in photography.
Properties
$$J \cdot U/L^2$$
$$cd/m^2$$
Annotations
In photometry, illuminance is the total luminous flux incident on a surface, per unit area. It is a measure of how much the incident light illuminates the surface, wavelength-weighted by the luminosity function to correlate with human brightness perception. Similarly, luminous emittance is the luminous flux per unit area emitted from a surface. In SI derived units these are measured in $$lux (lx)$$ or $$lumens per square metre$$ ($$cd \cdot m^{-2}$$). In the CGS system, the unit of illuminance is the $$phot$$, which is equal to $$10,000 lux$$. The $$foot-candle$$ is a non-metric unit of illuminance that is used in photography.
Luminous Flux per Area(en)
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https://en.wikipedia.org/wiki/Virasoro_algebra | # Virasoro algebra
In mathematics, the Virasoro algebra (named after the physicist Miguel Angel Virasoro) is a complex Lie algebra, given as a central extension of the complex polynomial vector fields on the circle, and is widely used in conformal field theory and string theory.
## Definition
The Virasoro algebra is spanned by elements $L_n$ for n ∈ ℤ and c with $L_n + L_{-n}$, $~ i(L_n -L_{-n})$ and c being real elements. Here, the central element c is the central charge.
The algebra satisfies $[c,L_n]=0$ and
$[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n,0}$ .
The factor of 1/12 is merely a matter of convention.
The Virasoro algebra is a central extension of the (complex) Witt algebra of complex polynomial vector fields on the circle. The Lie algebra of real polynomial vector fields on the circle is a dense subalgebra of the Lie algebra of diffeomorphisms of the circle.
Since the Virasoro algebra comprises the generators of the conformal group of the worldsheet, the stress tensor in string theory obeys the commutation relations of (two copies of) the Virasoro algebra. This is because the conformal group decomposes into separate diffeomorphisms of the forward and back lightcones. Diffeomorphism invariance of the worldsheet implies additionally that the stress tensor vanishes. This is known as the Virasoro constraint, and in the quantum theory, cannot be applied to all the states in the theory, but rather only on the physical states (compare Gupta-Bleuler quantization).
## Representation theory
A lowest weight representation of the Virasoro algebra is a representation generated by a vector $v$ that is killed by $L_i$ for $i\geq 1$, and is an eigenvector of $L_0$ and $c$. The letters $h$ and $c$ are usually used for the eigenvalues of $L_0$ and $c$ on $v$. (The same letter $c$ is used for both the element $c$ of the Virasoro algebra and its eigenvalue.) For every pair of complex numbers $h$ and $c$ there is a unique irreducible lowest weight representation with these eigenvalues.
A lowest weight representation is called unitary if it has a positive definite inner product such that the adjoint of $L_n$ is $L_{-n}$. The irreducible lowest weight representation with eigenvalues h and c is unitary if and only if either c≥1 and h≥0, or c is one of the values
$c = 1-{6\over m(m+1)} = 0,\quad 1/2,\quad 7/10,\quad 4/5,\quad 6/7,\quad 25/28, \ldots$
for m = 2, 3, 4, .... and h is one of the values
$h = h_{r,s}(c) = {((m+1)r-ms)^2-1 \over 4m(m+1)}$
for r = 1, 2, 3, ..., m−1 and s= 1, 2, 3, ..., r. Daniel Friedan, Zongan Qiu, and Stephen Shenker (1984) showed that these conditions are necessary, and Peter Goddard, Adrian Kent, and David Olive (1986) used the coset construction or GKO construction (identifying unitary representations of the Virasoro algebra within tensor products of unitary representations of affine Kac–Moody algebras) to show that they are sufficient.
The unitary irreducible lowest weight representations with c < 1 are called the discrete series representations of the Virasoro algebra. These are special cases of the representations with m = q/(pq), 0<r<q, 0< s<p for p and q coprime integers and r and s integers, called the minimal models and first studied in Belavin et al. (1984).
The first few discrete series representations are given by:
• m = 2: c = 0, h = 0. The trivial representation.
• m = 3: c = 1/2, h = 0, 1/16, 1/2. These 3 representations are related to the Ising model
• m = 4: c = 7/10. h = 0, 3/80, 1/10, 7/16, 3/5, 3/2. These 6 representations are related to the tri critical Ising model.
• m = 5: c = 4/5. There are 10 representations, which are related to the 3-state Potts model.
• m = 6: c = 6/7. There are 15 representations, which are related to the tri critical 3-state Potts model.
The lowest weight representations that are not irreducible can be read off from the Kac determinant formula, which states that the determinant of the invariant inner product on the degree h+N piece of the lowest weight module with eigenvalues c and h is given by
$A_N\prod_{1\le r,s\le N}(h-h_{r,s}(c))^{p(N-rs)} ~,$
which was stated by V. Kac (1978), (see also Kac and Raina 1987) and whose first published proof was given by Feigin and Fuks (1984).
(The function p(N) is the partition function, and AN is some constant.) The reducible highest weight representations are the representations with h and c given in terms of m, c, and h by the formulas above, except that m is not restricted to be an integer ≥ 2 and may be any number other than 0 and 1, and r and s may be any positive integers. This result was used by Feigin and Fuks to find the characters of all irreducible lowest weight representations.
## Generalizations
There are two supersymmetric N=1 extensions of the Virasoro algebra, called the Neveu-Schwarz algebra and the Ramond algebra. Their theory is similar to that of the Virasoro algebra, now involving Grassmann numbers. There are further extensions of these algebras with more supersymmetry, such as the N = 2 superconformal algebra.
The Virasoro algebra is a central extension of the Lie algebra of meromorphic vector fields on a genus 0 Riemann surface that are holomorphic except at two fixed points. I V Krichever and S P Novikov (1987) found a central extension of the Lie algebra of meromorphic vector fields on a higher genus compact Riemann surface that are holomorphic except at two fixed points, and their analysis has been extended to supermanifolds by J Rabin (1995).
The Virasoro algebra also has vertex algebraic and conformal algebraic counterparts, which basically come from arranging all the basis elements into generating series and working with single objects. Unsurprisingly these are called the vertex Virasoro and conformal Virasoro algebras respectively.
The Virasoro algebra may also be specified as a presentation. This is to say that all of its generators may be determined recursively ("generated") out of merely two judiciously chosen generators (e.g. L3 and L−2), and six equations (constraint conditions) among them, by systematic use of the Jacobi identity. (D Fairlie, J Nuyts, and C Zachos, 1988. Shortly thereafter, J Uretsky discovered the original 8 conditions could be pared down to six.)
Correspondingly, for the Super Virasoro algebra extension, the Ramond algebra follows from two generating generators and five conditions; and the Neveu-Schwarz algebra out of two such and nine conditions.
## History
The Witt algebra (the Virasoro algebra without the central extension) was discovered by E. Cartan (1909). Its analogues over finite fields were studied by E. Witt in about the 1930s. The central extension of the Witt algebra that gives the Virasoro algebra was first found (in characteristic p>0) by R. E. Block (1966, page 381) and independently rediscovered (in characteristic 0) by I. M. Gelfand and D. B. Fuchs (de) (1968). Virasoro (1970) wrote down some operators generating the Virasoso algebra while studying dual resonance models, though he did not find the central extension. The central extension giving the Virasoro algebra was rediscovered in physics shortly after by J. H. Weis, according to Brower and Thorn (1971, footnote on page 167). | 2015-11-26 03:34:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 24, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8683158755302429, "perplexity": 355.99713339763764}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1448398446300.49/warc/CC-MAIN-20151124205406-00112-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://hsm.stackexchange.com/questions/7847/plancks-quantization-idea | # Planck's quantization idea
First of All, I doubt there has ever been any new idea that did not involve intuition.
However, most textbooks suggest that the quantization idea was just a mathematical trick, that Planck introduced to be able to drive his distribution law. Is this the only right way to think about it? If not, did Planck really have intuition behind quantization idea?
• Possible duplicate of What was different about Planck's quantization of light compared to Einstein's? – Conifold Oct 24 '18 at 0:22
• @Conifold I read this answer before. I'm asking if Planck had intuition regarding the quantization idea, or it was just a math trick to drive his law. – Karim mohie Oct 24 '18 at 0:24
• Of course we can not read Planck's mind, but "This was purely a formal assumption and I really did not give it much thought" seems to suggest the latter very clearly. In any case, we can hardly know Planck's mind better than Planck did. As for the "right way to think", that is not really a history question. – Conifold Oct 24 '18 at 0:27
• @Conifold of course we can't do so. But, I thought that there could be a historical hints in his memoirs, or those of his contemporaries. By the way, this question was originally within physics stack exchange, but someone moved it to here! – Karim mohie Oct 24 '18 at 0:33
• Now this is really an exact duplicate of Max Planck and energy quantization idea (which itself is adequately answered at How did Planck derive the black body radiation formula without using the Bose statistics?) – Francois Ziegler Oct 24 '18 at 0:58
$$\frac{1}{\lambda} = 1.097*10^7( \frac{1}{2^2} - \frac{1}{n^2})$$ | 2020-09-27 17:27:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 1, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.638485848903656, "perplexity": 927.2421413268758}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400283990.75/warc/CC-MAIN-20200927152349-20200927182349-00156.warc.gz"} |
http://www.friedalovesbread.com/2012/06/kitchen-tip-how-many-strawberries-for.html | ## 6/5/12
### 0Kitchen Tip: How Many Strawberries for a Recipe?
When I was making my Strawberry Lemonade concentrate, the original recipe called for 6 cups of strawberries.
I stared at the four pound package of strawberries in Costco, trying to figure out how many 6 cups of strawberries were. I was so lost....
How many strawberries will fit into a measuring cup? Three?
After some research, how many cups of strawberries depend upon if they are being used whole, sliced, mashed or pureed.
Here's a handy dandy formula for you:
1 pound of strawberries = 3 3/4 cups whole
3 cups sliced
1 3/4 cup mashed
2 cups pureed
So, for my four pound package of Costco strawberries, I got eight cups of puree!
Oh, you need the formula for lemons? Click on this link HERE.
Enjoy your summer in a bottle! | 2013-05-23 03:54:56 | {"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": 0.8874993920326233, "perplexity": 10533.579207547851}, "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-2013-20/segments/1368702777399/warc/CC-MAIN-20130516111257-00051-ip-10-60-113-184.ec2.internal.warc.gz"} |
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# The function f is defined for all positive integers n as f(
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The function f is defined for all positive integers n as f( [#permalink] 14 May 2019, 01:15
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The function f is defined for all positive integers n as $$f(n)= \frac{n}{n+1}$$
Quantity A Quantity B f(1) × f(2) f(2) × f(3)
A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.
[Reveal] Spoiler: OA
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Re: The function f is defined for all positive integers n as f( [#permalink] 14 May 2019, 07:28
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Expert's post
Carcass wrote:
The function f is defined for all positive integers n as $$f(n)= \frac{n}{n+1}$$
Quantity A Quantity B f(1) × f(2) f(2) × f(3)
A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.
QUANTITY A
f(1) x f(2) $$=\frac{1}{1+1}$$x$$\frac{2}{2+1}$$
$$=\frac{1}{2}$$x$$\frac{2}{3}$$
$$= \frac{1}{3}$$
QUANTITY B
f(2) x f(3) $$=\frac{2}{2+1}$$x$$\frac{3}{3+1}$$
$$=\frac{2}{3}$$x$$\frac{3}{4}$$
$$= \frac{1}{2}$$
Since $$\frac{1}{2}>\frac{1}{3}$$, the correct answer is B
Cheers,
Brent
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Re: The function f is defined for all positive integers n as f( [#permalink] 14 May 2019, 16:27
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Without doing much work at all, we can see that the larger n gets, the closer f(n) gets to 1. So higher numbers means greater value, even though it's always less than one.
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Re: The function f is defined for all positive integers n as f( [#permalink] 07 Jun 2019, 01:22
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Cancel f(2) from both the sides and then compare:
f(1)< f(3)
Re: The function f is defined for all positive integers n as f( [#permalink] 07 Jun 2019, 01:22
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# The function f is defined for all positive integers n as f(
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https://python.quantecon.org/likelihood_bayes.html | # Likelihood Ratio Processes and Bayesian Learning¶
## Contents¶
In [1]:
import numpy as np
import matplotlib.pyplot as plt
from numba import vectorize, njit
from math import gamma
%matplotlib inline
## Overview¶
This lecture describes the role that likelihood ratio processes play in Bayesian learning.
As in this lecture, we’ll use a simple statistical setting from this lecture.
We’ll focus on how a likelihood ratio process and a prior probability determine a posterior probability.
We’ll derive a convenient recursion for today’s posterior as a function of yesterday’s posterior and today’s multiplicative increment to a likelihood process.
We’ll also present a useful generalization of that formula that represents today’s posterior in terms of an initial prior and today’s realization of the likelihood ratio process.
We’ll study how, at least in our setting, a Bayesian eventually learns the probability distribution that generates the data, an outcome that rests on the asymptotic behavior of likelihood ratio processes studied in this lecture.
This lecture provides technical results that underly outcomes to be studied in this lecture and this lecture and this lecture
## The Setting¶
We begin by reviewing the setting in this lecture, which we adopt here too.
A nonnegative random variable $W$ has one of two probability density functions, either $f$ or $g$.
Before the beginning of time, nature once and for all decides whether she will draw a sequence of IID draws from either $f$ or $g$.
We will sometimes let $q$ be the density that nature chose once and for all, so that $q$ is either $f$ or $g$, permanently.
Nature knows which density it permanently draws from, but we the observers do not.
We do know both $f$ and $g$ but we don’t know which density nature chose.
But we want to know.
To do that, we use observations.
We observe a sequence $\{w_t\}_{t=1}^T$ of $T$ IID draws from either $f$ or $g$.
We want to use these observations to infer whether nature chose $f$ or $g$.
A likelihood ratio process is a useful tool for this task.
To begin, we define the key component of a likelihood ratio process, namely, the time $t$ likelihood ratio as the random variable
$$\ell (w_t)=\frac{f\left(w_t\right)}{g\left(w_t\right)},\quad t\geq1.$$
We assume that $f$ and $g$ both put positive probabilities on the same intervals of possible realizations of the random variable $W$.
That means that under the $g$ density, $\ell (w_t)= \frac{f\left(w_{t}\right)}{g\left(w_{t}\right)}$ is evidently a nonnegative random variable with mean $1$.
A likelihood ratio process for sequence $\left\{ w_{t}\right\} _{t=1}^{\infty}$ is defined as
$$L\left(w^{t}\right)=\prod_{i=1}^{t} \ell (w_i),$$
where $w^t=\{ w_1,\dots,w_t\}$ is a history of observations up to and including time $t$.
Sometimes for shorthand we’ll write $L_t = L(w^t)$.
Notice that the likelihood process satisfies the recursion or multiplicative decomposition
$$L(w^t) = \ell (w_t) L (w^{t-1}) .$$
The likelihood ratio and its logarithm are key tools for making inferences using a classic frequentist approach due to Neyman and Pearson [NP33].
We’ll again deploy the following Python code from this lecture that evaluates $f$ and $g$ as two different beta distributions, then computes and simulates an associated likelihood ratio process by generating a sequence $w^t$ from some probability distribution, for example, a sequence of IID draws from $g$.
In [2]:
# Parameters in the two beta distributions.
F_a, F_b = 1, 1
G_a, G_b = 3, 1.2
@vectorize
def p(x, a, b):
r = gamma(a + b) / (gamma(a) * gamma(b))
return r * x** (a-1) * (1 - x) ** (b-1)
# The two density functions.
f = njit(lambda x: p(x, F_a, F_b))
g = njit(lambda x: p(x, G_a, G_b))
In [3]:
@njit
def simulate(a, b, T=50, N=500):
'''
Generate N sets of T observations of the likelihood ratio,
return as N x T matrix.
'''
l_arr = np.empty((N, T))
for i in range(N):
for j in range(T):
w = np.random.beta(a, b)
l_arr[i, j] = f(w) / g(w)
return l_arr
We’ll also use the following Python code to prepare some informative simulations
In [4]:
l_arr_g = simulate(G_a, G_b, N=50000)
l_seq_g = np.cumprod(l_arr_g, axis=1)
In [5]:
l_arr_f = simulate(F_a, F_b, N=50000)
l_seq_f = np.cumprod(l_arr_f, axis=1)
## Likelihood Ratio Process and Bayes’ Law¶
Let $\pi_t$ be a Bayesian posterior defined as
$$\pi_t = {\rm Prob}(q=f|w^t)$$
The likelihood ratio process is a principal actor in the formula that governs the evolution of the posterior probability $\pi_t$, an instance of Bayes’ Law.
Bayes’ law implies that $\{\pi_t\}$ obeys the recursion
$$\pi_t=\frac{\pi_{t-1} l_t(w_t)}{\pi_{t-1} l_t(w_t)+1-\pi_{t-1}} \tag{1}$$
with $\pi_{0}$ being a Bayesian prior probability that $q = f$, i.e., a personal or subjective belief about $q$ based on our having seen no data.
Below we define a Python function that updates belief $\pi$ using likelihood ratio $\ell$ according to recursion (1)
In [6]:
@njit
def update(π, l):
"Update π using likelihood l"
# Update belief
π = π * l / (π * l + 1 - π)
return π
Formula (1) can be generalized by iterating on it and thereby deriving an expression for the time $t$ posterior $\pi_{t+1}$ as a function of the time $0$ prior $\pi_0$ and the likelihood ratio process $L(w^{t+1})$ at time $t$.
To begin, notice that the updating rule
$$\pi_{t+1} =\frac{\pi_{t}\ell \left(w_{t+1}\right)} {\pi_{t}\ell \left(w_{t+1}\right)+\left(1-\pi_{t}\right)}$$
implies
\begin{aligned} \frac{1}{\pi_{t+1}} &=\frac{\pi_{t}\ell \left(w_{t+1}\right) +\left(1-\pi_{t}\right)}{\pi_{t}\ell \left(w_{t+1}\right)} \\ &=1-\frac{1}{\ell \left(w_{t+1}\right)} +\frac{1}{\ell \left(w_{t+1}\right)}\frac{1}{\pi_{t}}. \end{aligned}$$\Rightarrow \frac{1}{\pi_{t+1}}-1 =\frac{1}{\ell \left(w_{t+1}\right)}\left(\frac{1}{\pi_{t}}-1\right).$$
Therefore
\begin{aligned} \frac{1}{\pi_{t+1}}-1 =\frac{1}{\prod_{i=1}^{t+1}\ell \left(w_{i}\right)} \left(\frac{1}{\pi_{0}}-1\right) =\frac{1}{L\left(w^{t+1}\right)}\left(\frac{1}{\pi_{0}}-1\right). \end{aligned}
Since $\pi_{0}\in\left(0,1\right)$ and $L\left(w^{t+1}\right)>0$, we can verify that $\pi_{t+1}\in\left(0,1\right)$.
After rearranging the preceding equation, we can express $\pi_{t+1}$ as a function of $L\left(w^{t+1}\right)$, the likelihood ratio process at $t+1$, and the initial prior $\pi_{0}$
$$\pi_{t+1}=\frac{\pi_{0}L\left(w^{t+1}\right)}{\pi_{0}L\left(w^{t+1}\right)+1-\pi_{0}} . \tag{2}$$
Formula (2) generalizes generalizes formula (1).
Formula (2) can be regarded as a one step revision of prior probability $\pi_0$ after seeing the batch of data $\left\{ w_{i}\right\} _{i=1}^{t+1}$.
Formula (2) shows the key role that the likelihood ratio process $L\left(w^{t+1}\right)$ plays in determining the posterior probability $\pi_{t+1}$.
Formula (2) is the foundation for the insight that, because of how the likelihood ratio process behaves as $t \rightarrow + \infty$, the likelihood ratio process dominates the initial prior $\pi_0$ in determining the limiting behavior of $\pi_t$.
To illustrate this insight, below we will plot graphs showing one simulated path of the likelihood ratio process $L_t$ along with two paths of $\pi_t$ that are associated with the same realization of the likelihood ratio process but different initial prior probabilities probabilities $\pi_{0}$.
First, we tell Python two values of $\pi_0$.
In [7]:
π1, π2 = 0.2, 0.8
Next we generate paths of the likelihood ratio process $L_t$ and the posterior $\pi_t$ for a history of IID draws from density $f$.
In [8]:
T = l_arr_f.shape[1]
π_seq_f = np.empty((2, T+1))
π_seq_f[:, 0] = π1, π2
for t in range(T):
for i in range(2):
π_seq_f[i, t+1] = update(π_seq_f[i, t], l_arr_f[0, t])
In [9]:
fig, ax1 = plt.subplots()
for i in range(2):
ax1.plot(range(T+1), π_seq_f[i, :], label=f"$\pi_0$={π_seq_f[i, 0]}")
ax1.set_ylabel("$\pi_t$")
ax1.set_xlabel("t")
ax1.legend()
ax1.set_title("when f governs data")
ax2 = ax1.twinx()
ax2.plot(range(1, T+1), np.log(l_seq_f[0, :]), '--', color='b')
ax2.set_ylabel("$log(L(w^{t}))$")
plt.show()
The dotted line in the graph above records the logarithm of the likelihood ratio process $\log L(w^t)$.
Please note that there are two different scales on the $y$ axis.
Now let’s study what happens when the history consists of IID draws from density $g$
In [10]:
T = l_arr_g.shape[1]
π_seq_g = np.empty((2, T+1))
π_seq_g[:, 0] = π1, π2
for t in range(T):
for i in range(2):
π_seq_g[i, t+1] = update(π_seq_g[i, t], l_arr_g[0, t])
In [11]:
fig, ax1 = plt.subplots()
for i in range(2):
ax1.plot(range(T+1), π_seq_g[i, :], label=f"$\pi_0$={π_seq_g[i, 0]}")
ax1.set_ylabel("$\pi_t$")
ax1.set_xlabel("t")
ax1.legend()
ax1.set_title("when g governs data")
ax2 = ax1.twinx()
ax2.plot(range(1, T+1), np.log(l_seq_g[0, :]), '--', color='b')
ax2.set_ylabel("$log(L(w^{t}))$")
plt.show()
Below we offer Python code that verifies that nature chose permanently to draw from density $f$.
In [12]:
π_seq = np.empty((2, T+1))
π_seq[:, 0] = π1, π2
for i in range(2):
πL = π_seq[i, 0] * l_seq_f[0, :]
π_seq[i, 1:] = πL / (πL + 1 - π_seq[i, 0])
In [13]:
np.abs(π_seq - π_seq_f).max() < 1e-10
Out[13]:
True
We thus conclude that the likelihood ratio process is a key ingredient of the formula (2) for a Bayesian’s posteior probabilty that nature has drawn history $w^t$ as repeated draws from density $g$.
## Sequels¶
This lecture has been devoted to building some useful infrastructure.
We’ll build on results highlighted in this lectures to understand inferences that are the foundations of results described in this lecture and this lecture and this lecture
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https://danmackinlay.name/notebook/large_sample_theory.html | # Large sample theory
Delta methods, influence functions, and so on. Convolution theorems, local asymptotic minimax theorems.
A convenient feature of M-estimation, and especially maximum likelihood esteimation is simple behaviour of estimators in the asymptotic large-sample-size limit, which can give you, e.g. variance estimates, or motivate information criteria, or robust statistics, optimisation etc.
In the most celebrated and convenient cases case asymptotic bounds are about normally-distributed errors, and these are typically derived through Local Asymptotic Normality theorems. A simple and general introduction is given in Andersen et al. (1997) page 594., which applies to both i.i.d. data and dependent_data in the form of point processes. For all that it is applied, it is still stringent.
## Fisher Information
Used in ML theory and kinda-sorta in robust estimation. A matrix that tells you how much a new datum affects your parameter estimates. (It is related, I am told, to garden variety Shannon information, and when that non-obvious fact is more clear to me I shall expand how precisely this is so.) 🏗
## Convolution Theorem
The unhelpfully-named convolution theorem of Hájek (1970).
Suppose $$\hat{\theta}$$ is an efficient estimator of $$\theta$$ and $$\tilde{\theta}$$ is another, not fully efficient, estimator. The convolution theorem says that, if you rule out stupid exceptions, asymptotically $$\tilde{\theta} = \hat{\theta} + \varepsilon$$ where $$\varepsilon$$ is pure noise, independent of $$\hat{\theta}.$$
The reason that’s almost obvious is that if it weren’t true, there would be some information about $$\theta$$ in $$\tilde{\theta}-\hat{\theta}$$, and you could use this information to get a better estimator than $$\hat{\theta}$$, which (by assumption) can’t happen. The stupid exceptions are things like the Hodges superefficient estimator that do better at a few values of $$\hat{\theta}$$ but much worse at neighbouring values.
Andersen, Per Kragh, Ornulf Borgan, Richard D. Gill, and Niels Keiding. 1997. Statistical Models Based on Counting Processes. Corr. 2. print. Springer Series in Statistics. New York, NY: Springer.
Athreya, K. B., and Niels Keiding. 1977. “Estimation Theory for Continuous-Time Branching Processes.” Sankhyā: The Indian Journal of Statistics, Series A (1961-2002) 39 (2): 101–23. http://www.jstor.org/stable/25050084.
Barndorff-Nielsen, O. E., and M. Sørensen. 1994. “A Review of Some Aspects of Asymptotic Likelihood Theory for Stochastic Processes.” International Statistical Review / Revue Internationale de Statistique 62 (1): 133–65. https://doi.org/10.2307/1403550.
Becker-Kern, Peter, Mark M. Meerschaert, and Hans-Peter Scheffler. 2004. “Limit Theorems for Coupled Continuous Time Random Walks.” The Annals of Probability 32 (1): 730–56. http://www.stt.msu.edu/~mcubed/CoupleCTRW.pdf.
Bibby, Bo Martin, and Michael Sørensen. 1995. “Martingale Estimation Functions for Discretely Observed Diffusion Processes.” Bernoulli 1 (1/2): 17–39. https://doi.org/10.2307/3318679.
DasGupta, Anirban. 2008. Asymptotic Theory of Statistics and Probability. Springer Texts in Statistics. New York: Springer New York. http://link.springer.com/10.1007/978-0-387-75971-5.
Duembgen, Moritz, and Mark Podolskij. 2015. “High-Frequency Asymptotics for Path-Dependent Functionals of Itô Semimartingales.” Stochastic Processes and Their Applications 125 (4): 1195–1217. https://doi.org/10.1016/j.spa.2014.08.007.
Feigin, Paul David. 1976. “Maximum Likelihood Estimation for Continuous-Time Stochastic Processes.” Advances in Applied Probability 8 (4): 712–36. https://doi.org/10.2307/1425931.
Gribonval, Rémi, Gilles Blanchard, Nicolas Keriven, and Yann Traonmilin. 2017. “Compressive Statistical Learning with Random Feature Moments,” June. http://arxiv.org/abs/1706.07180.
Hájek, Jaroslav. 1972. “Local Asymptotic Minimax and Admissibility in Estimation.” In. The Regents of the University of California. https://projecteuclid.org/euclid.bsmsp/1200514092.
———. 1970. “A Characterization of Limiting Distributions of Regular Estimates.” Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete 14 (4): 323–30. https://doi.org/10.1007/BF00533669.
Heyde, C. C., and E. Seneta. 2010. “Estimation Theory for Growth and Immigration Rates in a Multiplicative Process.” In Selected Works of C.C. Heyde, edited by Ross Maller, Ishwar Basawa, Peter Hall, and Eugene Seneta, 214–35. Selected Works in Probability and Statistics. Springer New York. http://link.springer.com/chapter/10.1007/978-1-4419-5823-5_31.
Jacod, Jean, Mark Podolskij, and Mathias Vetter. 2010. “Limit Theorems for Moving Averages of Discretized Processes Plus Noise.” The Annals of Statistics 38 (3): 1478–1545. https://doi.org/10.1214/09-AOS756.
Jacod, Jean, and Albert N. Shiryaev. 1987. Limit Theorems for Stochastic Processes. Vol. 288. Grundlehren Der Mathematischen Wissenschaften. Berlin, Heidelberg: Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-662-02514-7.
Janková, Jana, and Sara van de Geer. 2016. “Confidence Regions for High-Dimensional Generalized Linear Models Under Sparsity,” October. http://arxiv.org/abs/1610.01353.
Konishi, Sadanori, and Genshiro Kitagawa. 1996. “Generalised Information Criteria in Model Selection.” Biometrika 83 (4): 875–90. https://doi.org/10.1093/biomet/83.4.875.
———. 2003. “Asymptotic Theory for Information Criteria in Model Selection—Functional Approach.” Journal of Statistical Planning and Inference, C.R. Rao 80th Birthday Felicitation vol., Part IV, 114 (1–2): 45–61. https://doi.org/10.1016/S0378-3758(02)00462-7.
Kraus, Andrea, and Victor M. Panaretos. 2014. “Frequentist Estimation of an Epidemic’s Spreading Potential When Observations Are Scarce.” Biometrika 101 (1): 141–54. https://doi.org/10.1093/biomet/ast049.
LeCam, L. 1972. “Limits of Experiments.” In. The Regents of the University of California. https://projecteuclid.org/euclid.bsmsp/1200514095.
———. 1970. “On the Assumptions Used to Prove Asymptotic Normality of Maximum Likelihood Estimates.” The Annals of Mathematical Statistics 41 (3): 802–28. https://doi.org/10.1214/aoms/1177696960.
Lederer, Johannes, and Sara van de Geer. 2014. “New Concentration Inequalities for Suprema of Empirical Processes.” Bernoulli 20 (4): 2020–38. https://doi.org/10.3150/13-BEJ549.
Ogata, Yoshiko. 1978. “The Asymptotic Behaviour of Maximum Likelihood Estimators for Stationary Point Processes.” Annals of the Institute of Statistical Mathematics 30 (1): 243–61. https://doi.org/10.1007/BF02480216.
Puri, Madan L., and Pham D. Tuan. 1986. “Maximum Likelihood Estimation for Stationary Point Processes.” Proceedings of the National Academy of Sciences of the United States of America 83 (3): 541–45. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC322899/.
Sørensen, Michael. 2000. “Prediction-Based Estimating Functions.” The Econometrics Journal 3 (2): 123–47. http://www.jstor.org/stable/23114885.
Tropp, Joel A. 2015. “An Introduction to Matrix Concentration Inequalities,” January. http://arxiv.org/abs/1501.01571. | 2020-04-04 04:38:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7866483926773071, "perplexity": 4628.506752068601}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1585370520039.50/warc/CC-MAIN-20200404042338-20200404072338-00511.warc.gz"} |
https://physics.stackexchange.com/questions/619222/if-we-were-able-to-prove-that-the-universe-is-infinite-wouldnt-that-statistica | If we were able to prove that the Universe is infinite, wouldn't that statistically prove that there is no other forms of life?
I want to begin my explanation using abstract mathematical explanation to repetition possibility by taking independent samples $$X_n$$ from some continuous probability distribution: https://math.stackexchange.com/q/1739927/
If we applied this same principle and its conclusion and assumed that the universe is homogeneous and were able to prove its infinity someway (along with matter inside it including stars and planets) then, statistically, it would mean that the chance of repetition of life again, at least and in the most conservative approach here, as a form of doppelganger extraterrestrial life would be zero? meaning that we are alone in this universe
• You just won't believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it's a long way down the road to the chemist's, but that's just peanuts to infinity.
– J...
Mar 9 at 15:04
No, quite the opposite. If the universe is truly infinite and approximately homogeneous, then I invite you to point in some direction in the sky at night, and if you travel far enough in that direction, you might have to point through several stars and planets and alien organisms to get there, but eventually you might well be pointing at a nearly exact copy of yourself pointing back at you. With infinite repetition things that are possible can become probable and things that are probable can become certain.
Not everything works this way, and it does depend on how “identical” your setups are as you repeat. But the basic math is like this: imagine we model a bridge but we ignore continuous “damage” to it and treat its failure instead as an all-or-nothing thing which has an 0.01% chance of failing in any given day, independent of any other day. One can derive that this bridge will live about 10,000 days or 27 years, plus or minus another 27. It actually in the continuous limit has a 50/50 chance of failing apart after $$10000~\ln2$$ days or 19 years. So those independent little chances of something happening, under many many repetitions, eventually lead to this very unlikely thing (one in ten thousand) becoming probable at $$10000~\ln2$$ repetitions and becoming 99.995% likely after a hundred thousand repetitions. Very simple math. And all that the above statement is doing is the same argument, extrapolating an Earth-sized cylinder through space and saying “hey, that this Earth is roughly the way it is (up to your uncertainty) is some unbelievably tiny probability $$p$$, but let’s chain those cylinders together in the direction you're pointing until we get $$10/p$$ repetitions and I suppose if you're right that everything is infinite and homogeneous then we'll not run out of universe before $$10/p$$ repetitions, even for small $$p$$.”
The “(up to your uncertainty)” is also very important here, because it is what makes the probability finite rather than infinitesimally zero. Mathematicians deal with say real numbers being perfectly specified, so 1.00125267... is very different from 1.00125264... but us physicists have to say “look my experiment has an 0.2% error and I can't resolve the difference between 1 and 1.002, so all of these are approximately 1.000 up to my error bars.”
Actually, something that bugs some people is the idea of “Boltzmann brains.” This is the idea that generating a brain randomly is much easier than generating a whole body and Earth and brain all together, so that if the world really does proceed to a state of maximum entropy, a thermal equilibrium, then there is some sort of recurrence time (in the simplest case a Poincaré recurrence) where the thermal equilibrium randomly generates a tiny bit of brain like yours perceiving the things that you are perceiving but the vast vast majority of times this happens that brain is totally and utterly detached from the real world and vanishes after a moment. And the reason that this application of infinity bugs people is that it means that averaging over all the entities in our universe which are having the conscious experience that we are having of staring at our computer screens on a physics website pondering the universe, almost none of them are actually living on an Earth and almost none of them are going to last longer than the next five seconds and almost none of them are therefore actually seeing anything like the real world, they are just in a very temporary hallucination. And that brings up questions about why we are so sure that we are not.
• "Things that are probable become certain." Isn't this technically false because probability 1 would only mean "almost always" in this case? Mar 7 at 17:21
• "with repetition ... things that are possible become probable": not so. For example, make an infinite sequence of even numbers. You will find no primes after the number 2. Mar 7 at 22:34
• You're a physics professor, I shouldn't have to explain to you why counterexamples that impose artificial constraints are bad counterexamples :) [Especially given, y'know, the didactic constraint. I am trying to lead a student to understand that bridges eventually fail, not that primes eventually lack any given factor.] But I do take the point seriously; a better counterexample is e.g. from relativity, even if light always goes faster than someone constantly accelerating, there is a distance behind that someone where when light is emitted that distance behind, it can never catch up to them. Mar 8 at 0:49
• The whole argument rests on the glossed over "approximately homogeneous", which is an entirely unfounded belief. Mar 8 at 0:51
• This is just to add that the point behind my remark was that almost all the science in the consideration of a question of this kind lies in the constraints on physical behaviour, whatever they may be. As we learn about them they get names such as 'law of physics' or whatever. We don't know enough about them to be able to say much about life in a huge universe, but learning about them is how we slowly go about answering questions of this kind (after first framing some better-posed version of the question of course). I would want to invite a questioner to think along these lines. Mar 8 at 22:58
I think you've misunderstood the question and answer that you linked to.
Paraphrasing, that question is, "Given an infinite sequence of real numbers, each one randomly and independently chosen from the interval $$[0, 1]$$, what is the probability that there is some number which appears multiple times?" The answer is that the probability is zero.
But why is it zero? Why is it essentially impossible that any duplicates will appear?
You seem to have mistakenly concluded that it's because the sequence is infinite. Actually, the infinity of the sequence has nothing to do with the answer to the question. If that asker had asked about a finite sequence instead of an infinite sequence, the answer would have been exactly the same.
The actual reason that the probability of duplicates in an infinite sequence of real numbers is zero is that the probability of any two randomly chosen real numbers being equal is zero. Selecting infinitely many numbers, instead of two, fails to change the situation.
You've also mistakenly assumed that the universe is similar to a sequence of real numbers, and that the existence of life existing elsewhere is similar to two random real numbers being equal. In fact, there are infinitely many arrangements of particles that constitute life, and so the probability of life existing in any given place is greater than zero. If the universe contains an infinite amount of matter, then the probability of life existing elsewhere is probably 1.
• +1 for working out why the OP would come to the exact opposite of the conclusion they should have. In particular, the issue here is that reals are uncountable, so a length-$\aleph_0$ list includes almost none of them.
– J.G.
Mar 7 at 19:46
• This answer would be a whole lot better if it explained the notion of countably infinite. The trick described here works only with real numbers, because there are more than countably infinite real numbers, as proven by Cantor's diagonalization Mar 8 at 12:19
• @marstato Well, suppose you randomly select a real number x from the uniform distribution over the interval [0, 1]. Given any interval J that's a subset of [0, 1], the probability that x will be in J is equal to the length of J. Now, suppose you select another real number y the same way. In order for y to equal x, y has to fall in the interval (x - e, x + e) for all positive real numbers e. The probability of falling in that interval is no greater than 2e, so the probability that y = x must be a number which is less than 2e for all positive real numbers e. Mar 9 at 12:51
• @Peter Any probability distribution on $\mathbb{N}$ must assign a greater-than-zero probability to at least one number, and that number will then appear infinitely many times in your sequence, with probability 1. I think the question of live existing in multiple places in an infinite universe is similar. Mar 9 at 14:41
• @marstato And yet the uniform distribution on $[0, 1]$ assigns a probability of 0 to each number. If you think that it's impossible for an event with probability 0 to actually occur, then, in my opinion, the conclusion that you should probably come to is that it's not actually possible to randomly choose a real number from the uniform distribution on $[0, 1]$. Mar 9 at 15:26
The other answer has already shown that this is not the case.
However, it is also true that: "If we take random samples from a continuous distribution, the probability of repetition is $$0$$"
We can identify the mistakes that make the conclusion drawn from the above statement invalid
1. What is repeating?:
Note that we can only say that the probability of a repeat of the exact same observation is zero - and not the probability of some random observation from a large class of similar samples.
Recall that humanity in particular, and life on Earth in general do not contain all possible life. Rather we are a very particular case of the infinite (in the sense of infinitely many variations within the set) possibilities of life.
Thus, while we may say that the probability of finding a planet identical to Earth with life exactly like us on it elsewhere in the universe is $$0$$. This does not mean that the probability of finding life itself - intelligent or otherwise - is $$0$$. Life is like an interval, not like a point.
In other words, we may not find another Omar Adel asking this question on Physics.SE, but we do find other users asking physics questions on the Internet.
2. Probability 0 $$\neq$$ Impossible:
This may seem counter-intuitive, but it is the case in certain contexts - including this one. For example, consider a number line or a Cartesian plane. Because there are infinitely many points, the probability of choosing any given point at random is $$0$$. However, if we were to run a trail, we will choose some point, even though a priori, the probability of choosing it was $$0$$. (This would also hold in the case of countable infinities, such as choosing a random natural number. However, note that such distributions would not be continuous - though they can be uniform - and are not relevant here).
Thus, in the context of such continuous distributions, probability zero may not imply impossibility. Indeed, the concept of measure is more suitable here.
Thus, the fact that the probability of a repeat is zero, does not mean that the existence of a copy is impossible.
3. The universe:
More general than the other two critiques is the fact that correspondence between the distribution for which the result holds and the universe is far from exact. Leaving aside the fact that we currently do not think that the universe is infinite, it is easy to imagine infinite-universes that are counter-examples to this.
Imagine an infinite universe that is empty except for 2 (or more) identical particles rotating around their common center of mass. Here, the variety of structures is not infinite, rather there is either empty space or a particle - and both configurations repeat at least twice!
In fact, many qualitative arguments about infinite universes (including the one in CR Drost's excellent answer) tacitly assume some non-trivial properties that need to be fleshed out before we can say anything that is so mathematically precise. After all, what stops an infinite universe from simply being 'tiled' by a uniform grid of identical atoms?
Thus, we need to make many assumptions before the argument even applies to an arbitrary infinite universe.
• This argument relies on the assumption that the possible configurations of a world(/solar system/universe) are uncountably-infinite. Mar 7 at 12:36
• @BlueRaja-DannyPflughoeft No. It relies only on the assumption that number of configurations would be infinite. It should hold for both uncountable and countable infinities (such as natural numbers). Moreover, that restriction is only for part 2. Parts 1 and 3 would hold regardless. Mar 7 at 13:29
• I think these are fair critiques but i would like to address your second critique however, the example you used is not suitable to my case, because repetition in my own context implies you did the same process twice and got the same number in two different areas. which is impossible, it's not about choosing a number, it's about getting the same number twice, and in different areas in the Cartesian plane Mar 7 at 15:18
• @devashsih The probability of repeats when sampling from a countable infinity is 1, not 0, so it does actually affect the entire answer. Mar 7 at 19:31
• @OmarAdel I agree that they are different processes. However, the point I was trying to illustrate there was that there are at least some cases where an event of probability 0 happens. This simply means that we cannot directly conclude the impossibility of an event from a zero probability when infinitely large sample spaces are present. Now, in the case that you are not considering, this is not so straightforward since we are picking samples from an uncountable set over countable trails. (1/2) Mar 8 at 15:43
The cited mathematical theorem in this context simply means that we would not find any two exactly identical "lives", i.e. no Doppelgängers. This however does not mean that we cannot find very similar ones. Moreover, this is correct, only if we take a finite number of samples (the doppelgänger arguments in the other answers imply that you have to travel infinitely far).
Note that this argument can be turned also against the existence of life on Earth - the probability that the life has arises in the exact form that we know now and that we are having this discussion is zero. But there is an infinite number of very close possibilities, and integrating over them gives a finite probability.
The reason why many believe that there are life forms on other planets has nothing to do with infinity but everything to do with what we know about how life evolved in earth and the tremendous number of galaxies and hence the even more tremendous number of stars with planets around them. It's impossible to be be firmer about this because of the lack of data. Given that we have now telescopes that are capable now of inferring ecoplanets, with over 3,000 discovered, that data is only likely to improve.
To add something to the discussion, let me state that a Doppelgänger of yourself can't be found anywhere in the universe. Max Tegmark claims that in an infinite universe we can find Hubble volumes (even infinitely many) that contain exactly the same configuration of particles (in phase space) as our own Hubble volume. And thus, a Döppelganger will exist somewhere too (even an infinity of them). This overlooks the possibility of interaction at the boundary of these volumes. Someone between us and the edge of a volume sees a different image as her supposed Doppelgänger would see. Which means that no two volumes can be the same.
Your assumption must be the opposite. If the universe is infinite the chance of life elsewhere is one, assuming the universe is the same everywhere, which seems rather plausible.
If the universe is not the same everywhere then one has to make an assumption of how the universe is different, with associated chances of being different, both of which seem impossible to me. Up to the limit of the visible universe, the laws of Nature seem the same as over here. The visible part is a tiny part though of the universe as a whole, so who knows?
• Interesting philosophical-physical discussion. Can we not assume that all differences at the border of the Hubble volume are outside the light cone of each "identical" observer? That is, the differences will be felt, but later (and then the identity would end). Or the Hubble volume shrinks so fast, overtaking the inbound light, that each observer will be "alone" before information of any difference has reached them. Mar 9 at 13:34
• @Peter-ReinstateMonica Hi there. A bit late, but yet. I think that if we assume two Doppelgängers that are at a small distance of two identical Hubble volumes, a contradiction develops. Lightcone from outside the volumes, which started at points outside the volumes, will make the two assumed Doppelgängers see different things, thereby violating the assumption that they are Doppelgängers. I think this has as a consequence that there also won't be Doppelgängers who are the same in constitution, but each of both (infinitely many) doing different things. Of course they are not exact DG's anymore. Mar 9 at 15:21
• @Peter-ReinstateMonica Let me pause for reflection for a second regarding your second proposal. Maybe a Doppelgänger of mine is typing exactly the same... Mar 9 at 15:23
• @Peter-ReinstateMonica Let me first return to my first comment. Of course, when two Doppelgängers near the border of two different Hubble volumes can't be Doppelgängers, two persons at the center can't be either. They can see the different "Doppelgängers", which makes them different too. Insofar the shrinking volumes is concerned, I think that light from outside of the volumes can still reach the inside of the volumes. Maybe very fast expanding Hubble volumes can do the trick? Mar 9 at 15:38
If the universe is infinite, that would mean that there is a 100% chance that there is extraterritorial. Let's take a portion of the universe. There is an x% chance of having alien life. If we double that then there would be and x2% chance of alien life. If we double it infinte times then we would get x∞% chance... and since any number multiplied by infinity is equal to infinity, there is 100% of a chance that there is alien inhabited planet, somewhere in the universe...
• You probably do not mean "in the solar system"? That would mean on one of our planets. Mar 9 at 7:37
• Sorry, typo. I fixed it. Mar 9 at 12:13
• No need to say "sorry" - With your answer, you have given something to add to the vast body of high quality knowledge we are creating here, and fixing the typo polished a little. (How one sees this site, as a free source to answer questions, or collection, or whatever is of course a philosophical issue) Mar 9 at 13:18
“It is known that there are an infinite number of worlds, simply because there is an infinite amount of space for them to be in. However, not every one of them is inhabited. Therefore, there must be a finite number of inhabited worlds. Any finite number divided by infinity is as near to nothing as makes no odds, so the average population of all the planets in the Universe can be said to be zero. From this it follows that the population of the whole Universe is also zero, and that any people you may meet from time to time are merely the products of a deranged imagination.”
- Douglas Adams, The Restaurant at the End of the Universe
• Completely unhelpful answer but culturally relevant. And funny for those with a particular sense of humor.` Mar 7 at 22:29
• The problem with this line of reasoning is the part that says "Therefore, there must be a finite number of inhabited worlds." Being a subset of an infinite collection doesn't make something finite. For example consider the set of all integers, which is of course an infinite set. Now imagine a set that only includes the even numbers, or only includes every billionth number, or any other repeating pattern you can think of. Despite being only a fraction of the first set, it is also an infinite set. I love Douglas Adams, but his playing a bit fast and loose with his math here Mar 8 at 16:59
• @KevinWells If you add this to the answer, it is worth keeping for the proper reason too! Mar 9 at 13:20
The mind when fully flexed has the power to imagine many things that exist but are unseen, and many things that are unseen because they do not exist. The constructions you chose are imaginary for two reasons. First, the universe is observed to be heterogeneous and second, the distribution of life is observed to be self-similar in nature on earth allowing that same self-similarity to be reasonably extrapolated from the observed earth to the universe in whole. The normal distribution cannot strictly be applied to self-similar subjects governed by fractal mathematics. But of course the unscrupulous can use statistics to prove anything they wish because there are lies, damned lies and then statistics.
Absolutely not. It would only prove the universe was infinite if you could prove that. But this is moot as the universe is finite.
Also, if it were infinite there would be more chance of life in the places we cannot see or get to. Would certainly not have any bearing on proving there was no life anywhere.
• Makes me wonder what the definition of an infinite universe really is. Because if it's expanding at (or faster) than the speed of light, and you can only travel at the speed of light, no matter how far you go, there is always more universe to go into kind of like no matter how high you count, there are always more numbers to count. Mar 8 at 3:04
• Do you have evidence to back that up? Certainly the observable universe is finite, but given that we can't observe things beyond that how could we be certain of what is there? Mar 8 at 16:55
• Proof is obvious as my math prof in grad school said. universe is expanding. rate is accelerating. Time since the big bang is finite. size at big bang was zero. speed is less than infinite. compute the very finite size upper bound by taking current speed x time of expansion. Mar 9 at 2:46
• wow two downvotes for the simplest absolutely correct answer. Mar 9 at 2:46
• That proof from your math prof only applies to the observable universe. Our currently leading theory of the universe predicts that the universe was infinite at the big bang (even though the observable universe has zero size). They work this out by extrapolating from the observed curvature of space Mar 9 at 3:15 | 2021-10-26 15:44:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 14, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7350398898124695, "perplexity": 400.3809201406869}, "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-2021-43/segments/1634323587908.20/warc/CC-MAIN-20211026134839-20211026164839-00333.warc.gz"} |
https://socratic.org/questions/how-do-you-plot-2-8 | # How do you plot (-2, 8)?
Apr 11, 2018
The $- 2$ is your $x$ coordinate on the graph and the $8$ is your $y$ coordinate on the graph.
#### Explanation:
In case you do not know, the $x$ axis is the horizontal plane on a graph and the $y$-axis is the vertical side of a graph. I am assuming that the graph is measured by 1's so all you have to do is count two ticks to the left and than move from that same position 8 ticks up vertically. That way you will be $- 2$ ticks left on the $x$-axis and $8$ ticks up on the $y$-axis.
Apr 11, 2018
If I remember correctly, all you have to do is go to the left 2 times on the X-axis and up 8 times on the Y-axis
#### Explanation:
The first number in the parentheses is the X coordinate, so whether it is negative or positive depends whether the number would go left or right. The second number in the parentheses is the Y coordinate, so depending on if that number is positive or negative, your point would go up or down.
If the number is positive for both the X and Y coordinate, it would go to the right and up. If the number is negative, it would go left and down. Positive and negative, right and down. Negative and positive, left and up.
Hoped this helped! | 2022-05-27 04:06:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 10, "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": 0.6288121342658997, "perplexity": 210.531369546206}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1652662631064.64/warc/CC-MAIN-20220527015812-20220527045812-00465.warc.gz"} |
https://www.wptricks.com/question-category/wp-query/ | ## Get Posts in a Custom Post Type category
Question
I have trouble in getting post in a custom post type category. I have code below but it doesnt work well. It still get posts in another category. <?php $query= null;$paged = (get_query_var('paged')) ? ...
0
1 day 0 Answers 7 views
## Order Posts by meta value AND published date
Question
I want to order WP posts by meta key named "sort_date" and if it doesn't exist it should use the post's published date to order posts. Below is the code i am currently using ...
0
2 days 0 Answers 9 views
## meta_query with meta values as serialize arrays
Question
I'm working on a project in which I'm creating a custom post type and custom data entered via meta boxes associated with my custom post type. For whatever reason I decided to code the meta boxes in such a way ...
0
3 days 0 Answers 8 views
## WordPress pages for diferent sub categories
Question
Can anyone please help me building one custom code. im trying to make a library with diferent sections when user loads wp page will display sub categories. For expample when user clicks on page comedy the page loads and display all books ...
0
3 days 0 Answers 8 views
## insert thumbnail image from php script
Question
I'm trying to insert a post image thumbnail from my php scritp. It's inserting the post, the content, a download link, and I only need to set the thumbnail image to finish my script. This is my code but I ...
0
4 days 0 Answers 11 views
## Capture search term, input into 2 searches and output data on 1 page
Question
Apologies if this may seem trivial but, although i've been working with Wordpress for a few years now, i only recently started using slightly more complex code implementations. What i am trying to do is the following: Present the user with a ...
0
4 days 0 Answers 11 views
## Replacing search results with custom external query
Question
I'd like to replace the existing search function with an external query (from Solr) but I'm having trouble showing the results in the search.php page. This is the code I have in functions.php: add_action('pre_get_posts', 'my_search_query'); function my_search_query($query) { if($query->is_search() ...
0
4 days 0 Answers 12 views
Question
I have created a new Block for the Wordpress Editor, and I am trying to get a simple WP_Query to work, but so far, the $args in the query are mostly ignored, apart from the post_type setting. For example, I ... 0 5 days 0 Answers 14 views ## wp_query ‘s’ parameter does not work with custom post type Question I have a custom post type 'property'. I am trying to make a search for it, but it does not work with wp_query parameter 's'.$wp_query = Wp_Query(['post_type' => 'property', 's' => 'test']); It works fine with other Wp_query parameters, like this: \$wp_query ...
0
5 days 0 Answers 16 views | 2020-05-31 23:04:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.1802622228860855, "perplexity": 3630.2248402245073}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347413786.46/warc/CC-MAIN-20200531213917-20200601003917-00493.warc.gz"} |
https://pm4py.fit.fraunhofer.de/static/assets/api/2.2.23/pm4py.visualization.dfg.html | # pm4py.visualization.dfg package
## pm4py.visualization.dfg.parameters module
PM4Py is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.
PM4Py is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
You should have received a copy of the GNU General Public License along with PM4Py. If not, see <https://www.gnu.org/licenses/>.
class pm4py.visualization.dfg.parameters.Parameters(value)[source]
Bases: enum.Enum
An enumeration.
ACTIVITY_KEY = 'pm4py:param:activity_key'
END_ACTIVITIES = 'end_activities'
FONT_SIZE = 'font_size'
FORMAT = 'format'
MAX_NO_EDGES_IN_DIAGRAM = 'maxNoOfEdgesInDiagram'
START_ACTIVITIES = 'start_activities'
START_TIMESTAMP_KEY = 'pm4py:param:start_timestamp_key'
TIMESTAMP_KEY = 'pm4py:param:timestamp_key'
## pm4py.visualization.dfg.visualizer module
PM4Py is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.
PM4Py is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
You should have received a copy of the GNU General Public License along with PM4Py. If not, see <https://www.gnu.org/licenses/>.
class pm4py.visualization.dfg.visualizer.Variants(value)[source]
Bases: enum.Enum
An enumeration.
FREQUENCY = <module 'pm4py.visualization.dfg.variants.frequency' from 'C:\\Users\\berti\\pm4py-core\\pm4py\\visualization\\dfg\\variants\\frequency.py'>
PERFORMANCE = <module 'pm4py.visualization.dfg.variants.performance' from 'C:\\Users\\berti\\pm4py-core\\pm4py\\visualization\\dfg\\variants\\performance.py'>
pm4py.visualization.dfg.visualizer.apply(dfg0: Dict[Tuple[str, str], float], log: Optional[pm4py.objects.log.obj.EventLog] = None, activities_count: Optional[Dict[str, int]] = None, soj_time: Optional[Dict[str, float]] = None, parameters: Optional[Dict[Any, Any]] = None, variant=Variants.FREQUENCY) graphviz.graphs.Digraph[source]
Visualize a frequency/performance directly-follows graph
Parameters
• dfg0 – Directly-follows graph
• log – (if provided) Event log for the calculation of statistics
• activities_count – (if provided) Dictionary associating to each activity the number of occurrences in the log.
• soj_time – (if provided) Dictionary associating to each activity the average sojourn time
• parameters – Variant-specific parameters
• variant – Variant: - Frequency DFG representation - Performance DFG representation
Returns
Graphviz digraph
Return type
gviz
pm4py.visualization.dfg.visualizer.matplotlib_view(gviz)[source]
Views the diagram using Matplotlib
Parameters
gviz – Graphviz
pm4py.visualization.dfg.visualizer.save(gviz, output_file_path)[source]
Save the diagram
Parameters
• gviz – GraphViz diagram
• output_file_path – Path where the GraphViz output should be saved
pm4py.visualization.dfg.visualizer.view(gviz)[source]
View the diagram
Parameters
gviz – GraphViz diagram | 2022-09-27 14:51:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.389835000038147, "perplexity": 4981.8202341873775}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1664030335034.61/warc/CC-MAIN-20220927131111-20220927161111-00585.warc.gz"} |
https://ltwork.net/what-number-can-each-term-of-the-equation-be-multiplied--10050326 | # What number can each term of the equation be multiplied by to eliminate fractions before
###### Question:
What number can each term of the equation be multiplied by to eliminate fractions before solving ? 6-3/4x+1/3=1/2x+5
### Consider the two images from “The Road Not Taken”. What are the similarities and differences?
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### Who issued the order to seize fort sumter ? a) captain david farragut b) benjamin butler c) general
Who issued the order to seize fort sumter ? a) captain david farragut b) benjamin butler c) general richard taylor d) general p. g.t. beauregard... | 2022-10-04 20:40:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.5178292393684387, "perplexity": 1735.3889869450497}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1664030337524.47/warc/CC-MAIN-20221004184523-20221004214523-00181.warc.gz"} |
https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Supplemental_Modules_and_Websites_(Inorganic_Chemistry)/Advanced_Inorganic_Chemistry_(Wikibook)/01%3A_Chapters/1.23%3A_Dissociative_Mechanism | # 1.23: Dissociative Mechanism
Dissociative substitution mechanism describes one of the common pathways through which a ligand substitution reaction takes place. Found often in octahedral complexes, dissociative mechanisms are distinguished by having an ion X- dissociate from a metal complex, resulting in an intermediate compound with a lower coordination number.[1] This lost ion is then replaced by an ion Y-, as opposed to associative complexes which form a complex of X-, Y-, and the metal before the metal to X- ion bond is broken. This process is analogous to an SN1 reaction. In a dissociative mechanism, the complex is usually has fully saturated coordination, with 18 or more electrons. While most octahedral dissociate, not all do, with key examples being [Cr(NH3)5H20]3+ which follows Id substitution and [Cr(H2O)6]3+ following Ia.[1]
## Mechanism
The mechanism is below as follows with Y- substituting in for X- on a general octahedral complex.
$\mathrm{L}_{5} \mathrm{M}-\mathrm{X} \stackrel{-}{\rightleftharpoons}_{+\mathrm{X}, k_{-1}}^{-\mathrm{X}, k_{1}} \mathrm{L}_{5} \mathrm{M}-\square \stackrel{+\mathrm{Y}, k_{2}}{\longrightarrow} \mathrm{L}_{5} \mathrm{M}-\mathrm{Y}\nonumber$
Usually the rate determining step (RDS) of the mechanism is the dissociation of X from the complex, which is dependent upon the strength of the metal to X- bond as well as other factors such as steric hinderance of the species which helps to speed the mechanism along as crowding favors dissociation. [Y] does not affect the rate of reaction, leading to the simple rate equation:
$\text { Rate }=k_{1}\left[\mathrm{L}_{5} \mathrm{M}-\mathrm{X}\right]\nonumber$
When the metal to X- is not the rate determining step, the rate of reaction is what is seen below:[1]
$\text { Rate }=\frac{d\left[\mathrm{L}_{5} \mathrm{M}-\mathrm{Y}\right]}{d t}=k_{2}\left[\mathrm{L}_{5} \mathrm{M}\right]\nonumber$
## Lability
A metal complex is either labile or inert depending on how easily the reaction proceeds. A labile compound undergoes reactions with a relatively high rate of substitution. The opposite to labile is inert, a term describing metal complexes whose reactions are slow.[2] There are three main factors that affect the whether a complex is labile or inert.[2]
1. Size: Smaller metal ions tend to be more inert as ligands are held more tightly.[2]
2. Charge on Metal: The greater the charge on a metal ion in a complex, the greater the tendency towards the complex being inert.
3. d Electron Configuration: in octahedral geometries, d electrons have t2g and eg orbitals meaning the 5 d orbitals are not at the same energy level. The number of d electrons can predict if the metal complex behaves as inert or labile as according to the table below. Each electronic configuration is either labile or inert as a result of partial of full occupancy of the corresponding orbitals. [1]
Number of D-Electrons and Configuration Reactivity Notes
d1 Labile N/A
d2 Labile N/A
d3 Inert N/A
d4 Low Spin Inert N/A
d4 High Spin Labile Especially labile as it is structurally distorted by the Jahn-Teller effect.
d5 Low Spin Inert N/A
d5 High Spin Labile N/A
d6 Low Spin Inert N/A
d6 High Spin Labile N/A
d7 High Spin Labile N/A
d8 Square Planar Inert For d8 and above low spin is the same as high spin.
d8 Intermediate This configuration is intermediate, especially with weak field ligands.
d9 Labile Like d4 H.S. this configuration is especially labile as it is distorted by Jahn-Teller effect.
d10 Labile N/A
## Traits of Dissociative Mechanisms
• The rate of substitution varies little with the incoming Y-.[1]
• The rate of substitution varies over five orders of magnitude depending on the nature of the leaving group. The weaker the bond between the metal and the leaving X-, the faster the reaction runs, as the loss of X- occurs in the rate determining step.[1]
• Steric crowding around the metal has the ability to increase the rate of substitution because steric crowding favors dissociation. This runs completely contrary to rate of substitution for associative mechanisms as steric crowding is harder to penetrate.[1]
• Dissociative substitutions have a positive entropy (ΔS±) and usually have a positive volume of activation (ΔV±).[1]
## The Shift Mechanism
Another way in which a substitution reaction can take place is through use of the shift mechanism, which is most notably present in ring slippage. The shift mechanism consists of the center metal putting some charge back on a ligand either through breaking a double bond and having the ligand hold the charge or through changing the hapacity of the bond.[3] This lowers the overall coordination number on the metal, allowing it to associate ligands for substitution.[4] This phenomena is most commonly found in associative coordinations rather than dissociative mechanism as it lowers the coordination number to one that can accommodate incoming ligands, An example of this is when a η5-Cp ring undergoes a ring slippage, becoming a η3-Cp bond, thus allowing an incoming ligand while maintaining the 18-electron rule.[5]
#### References
1. Jump up to:a b c d e f g h Pfenning, Brian W. (2015). Principles of Inorganic Chemistry. Hoboken: John Wiley & Sons, Inc.. pp. 580-583. ISBN 9781118859100.
2. Jump up to:a b c Sridharan, K. Coordination compounds - Stability. SASTRA University.
3. http://www.ilpi.com/organomet/cp.html
4. http://www.ilpi.com/organomet/cp.html
5. http://www.ilpi.com/organomet/cp.html | 2021-06-16 23:43:15 | {"extraction_info": {"found_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": 0.7839218378067017, "perplexity": 3777.6792989629266}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1623487626122.27/warc/CC-MAIN-20210616220531-20210617010531-00572.warc.gz"} |
https://brilliant.org/problems/determin4nts/ | # Determin4nts
Level pending
Now we are at the 4 by 4 matrix.
Find
$det\left( \begin{array}{ccc}1 & 1 & 1 & 1 \\1 & 2 & 3 & 4 \\ 1 & 3 & 6 & 10 \\ 1 & 4 & 10 & 20 \end{array} \right)$
Wolfy is allowed, but try and find out how its calculated by yourself...
and guessing is applicable in this case.
× | 2016-10-21 23:56:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.7708750367164612, "perplexity": 2129.7112912528196}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1476988718311.12/warc/CC-MAIN-20161020183838-00291-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://labs.tib.eu/arxiv/?author=P.S.%20Ray | • LOFAR discovery of the fastest-spinning millisecond pulsar in the Galactic field(1709.01453)
Sept. 5, 2017 astro-ph.HE
We report the discovery of PSR J0952$-$0607, a 707-Hz binary millisecond pulsar which is now the fastest-spinning neutron star known in the Galactic field (i.e., outside of a globular cluster). PSR J0952$-$0607 was found using LOFAR at a central observing frequency of 135 MHz, well below the 300 MHz to 3 GHz frequencies typically used in pulsar searches. The discovery is part of an ongoing LOFAR survey targeting unassociated Fermi Large Area Telescope $\gamma$-ray sources. PSR J0952$-$0607 is in a 6.42-hr orbit around a very low-mass companion ($M_\mathrm{c}\gtrsim0.02$ M$_\odot$) and we identify a strongly variable optical source, modulated at the orbital period of the pulsar, as the binary companion. The light curve of the companion varies by 1.6 mag from $r^\prime=22.2$ at maximum to $r^\prime>23.8$, indicating that it is irradiated by the pulsar wind. Swift observations place a 3-$\sigma$ upper limit on the $0.3-10$ keV X-ray luminosity of $L_X < 1.1 \times 10^{31}$ erg s$^{-1}$ (using the 0.97 kpc distance inferred from the dispersion measure). Though no eclipses of the radio pulsar are observed, the properties of the system classify it as a black widow binary. The radio pulsed spectrum of PSR J0952$-$0607, as determined through flux density measurements at 150 and 350 MHz, is extremely steep with $\alpha\sim-3$ (where $S \propto \nu^{\alpha}$). We discuss the growing evidence that the fastest-spinning radio pulsars have exceptionally steep radio spectra, as well as the prospects for finding more sources like PSR J0952$-$0607.
• Fermi Large Area Telescope Observations of the Monoceros Loop Supernova Remnant(1608.06380)
Aug. 23, 2016 astro-ph.HE
We present an analysis of the gamma-ray measurements by the Large Area Telescope onboard the \textit{Fermi Gamma-ray Space Telescope} in the region of the supernova remnant~(SNR) Monoceros Loop~(G205.5$+$0.5). The brightest gamma-ray peak is spatially correlated with the Rosette Nebula, which is a molecular cloud complex adjacent to the southeast edge of the SNR. After subtraction of this emission by spatial modeling, the gamma-ray emission from the SNR emerges, which is extended and fit by a Gaussian spatial template. The gamma-ray spectra are significantly better reproduced by a curved shape than a simple power law. The luminosities between 0.2--300~GeV are $\sim$~$4 \times 10^{34}$~erg~s$^{-1}$ for the SNR and $\sim$~$3 \times 10^{34}$~erg~s$^{-1}$ for the Rosette Nebula, respectively. We argue that the gamma rays likely originate from the interactions of particles accelerated in the SNR. The decay of neutral pions produced in nucleon-nucleon interactions of accelerated hadrons with interstellar gas provides a reasonable explanation for the gamma-ray emission of both the Rosette Nebula and the Monoceros SNR.
• The study of neutron star magnetospheres with LOFT(1501.02773)
Jan. 12, 2015 astro-ph.HE
This is a White Paper in support of the mission concept of the Large Observatory for X-ray Timing (LOFT), proposed as a medium-sized ESA mission. We discuss the potential of LOFT for the study of magnetospheres of isolated neutron stars. For a summary, we refer to the paper.
• The LOFT mission concept is one of four candidates selected by ESA for the M3 launch opportunity as Medium Size missions of the Cosmic Vision programme. The launch window is currently planned for between 2022 and 2024. LOFT is designed to exploit the diagnostics of rapid X-ray flux and spectral variability that directly probe the motion of matter down to distances very close to black holes and neutron stars, as well as the physical state of ultra-dense matter. These primary science goals will be addressed by a payload composed of a Large Area Detector (LAD) and a Wide Field Monitor (WFM). The LAD is a collimated (<1 degree field of view) experiment operating in the energy range 2-50 keV, with a 10 m^2 peak effective area and an energy resolution of 260 eV at 6 keV. The WFM will operate in the same energy range as the LAD, enabling simultaneous monitoring of a few-steradian wide field of view, with an angular resolution of <5 arcmin. The LAD and WFM experiments will allow us to investigate variability from submillisecond QPO's to year-long transient outbursts. In this paper we report the current status of the project.
• PSR J2030+3641: radio discovery and gamma-ray study of a middle-aged pulsar in the now identified Fermi-LAT source 1FGL J2030.0+3641(1111.4270)
Nov. 18, 2011 astro-ph.GA, astro-ph.HE
In a radio search with the Green Bank Telescope of three unidentified low Galactic latitude Fermi-LAT sources, we have discovered the middle-aged pulsar J2030+3641, associated with 1FGL J2030.0+3641 (2FGL J2030.0+3640). Following the detection of gamma-ray pulsations using a radio ephemeris, we have obtained a phase-coherent timing solution based on gamma-ray and radio pulse arrival times that spans the entire Fermi mission. With a rotation period of 0.2 s, spin-down luminosity of 3e34 erg/s, and characteristic age of 0.5 Myr, PSR J2030+3641 is a middle-aged neutron star with spin parameters similar to those of the exceedingly gamma-ray-bright and radio-undetected Geminga. Its gamma-ray flux is 1% that of Geminga, primarily because of its much larger distance, as suggested by the large integrated column density of free electrons, DM=246 pc/cc. We fit the gamma-ray light curve, along with limited radio polarimetric constraints, to four geometrical models of magnetospheric emission, and while none of the fits have high significance some are encouraging and suggest that further refinements of these models may be worthwhile. We argue that not many more non-millisecond radio pulsars may be detected along the Galactic plane that are responsible for LAT sources, but that modified methods to search for gamma-ray pulsations should be productive -- PSR J2030+3641 would have been found blindly in gamma rays if only >0.8 GeV photons had been considered, owing to its relatively flat spectrum and location in a region of high soft background.
• X-ray pulsations from the radio-quiet gamma-ray pulsar in CTA 1(1010.4167)
Oct. 20, 2010 astro-ph.HE
Prompted by the Fermi LAT discovery of a radio-quiet gamma-ray pulsar inside the CTA 1 supernova remnant, we obtained a 130 ks XMM-Newton observation to assess the timing behavior of this pulsar. Exploiting both the unprecedented photon harvest and the contemporary Fermi LAT timing measurements, a 4.7 sigma single peak pulsation is detected, making PSR J0007+7303 the second example, after Geminga, of a radio-quiet gamma-ray pulsar also seen to pulsate in X-rays. Phase-resolved spectroscopy shows that the off-pulse portion of the light curve is dominated by a power-law, non-thermal spectrum, while the X-ray peak emission appears to be mainly of thermal origin, probably from a polar cap heated by magnetospheric return currents, pointing to a hot spot varying throughout the pulsar rotation.
• Further X-ray observations of EXO 0748-676 in quiescence: evidence for a cooling neutron star crust(1007.0247)
Oct. 1, 2010 astro-ph.HE
In late 2008, the quasi-persistent neutron star X-ray transient and eclipsing binary EXO 0748-676 started a transition from outburst to quiescence, after it had been actively accreting for more than 24 years. In a previous work, we discussed Chandra and Swift observations obtained during the first five months after this transition. Here, we report on further X-ray observations of EXO 0748-676, extending the quiescent monitoring to 1.6 years. Chandra and XMM-Newton data reveal quiescent X-ray spectra composed of a soft, thermal component that is well-fitted by a neutron star atmosphere model. An additional hard powerlaw tail is detected that changes non-monotonically over time, contributing between 4 and 20 percent to the total unabsorbed 0.5-10 keV flux. The combined set of Chandra, XMM-Newton and Swift data reveals that the thermal bolometric luminosity fades from ~1E34 to 6E33 (D/7.4 kpc)^2 erg/s, whereas the inferred neutron star effective temperature decreases from ~124 to 109 eV. We interpret the observed decay as cooling of the neutron star crust and show that the fractional quiescent temperature change of EXO 0748-676 is markedly smaller than observed for three other neutron star X-ray binaries that underwent prolonged accretion outbursts.
• Eight gamma-ray pulsars discovered in blind frequency searches of Fermi LAT data(1006.2134)
We report the discovery of eight gamma-ray pulsars in blind frequency searches using the LAT, onboard the Fermi Gamma-ray Space Telescope. Five of the eight pulsars are young (tau_c<100 kyr), energetic (Edot>10^36 erg/s), and located within the Galactic plane (|b|<3 deg). The remaining three are older, less energetic, and located off the plane. Five pulsars are associated with sources included in the LAT bright gamma-ray source list, but only one, PSR J1413-6205, is clearly associated with an EGRET source. PSR J1023-5746 has the smallest characteristic age (tau_c=4.6 kyr) and is the most energetic (Edot=1.1E37 erg/s) of all gamma-ray pulsars discovered so far in blind searches. PSRs J1957+5033 and J2055+25 have the largest characteristic ages (tau_c~1 Myr) and are the least energetic (Edot~5E33 erg/s) of the newly-discovered pulsars. We present the timing models, light curves, and detailed spectral parameters of the new pulsars. We used recent XMM observations to identify the counterpart of PSR J2055+25 as XMMU J205549.4+253959. In addition, publicly available archival Chandra X-ray data allowed us to identify the likely counterpart of PSR J1023-5746 as a faint, highly absorbed source, CXOU J102302.8-574606. The large X-ray absorption indicates that this could be among the most distant gamma-ray pulsars detected so far. PSR J1023-5746 is positionally coincident with the TeV source HESS J1023-575, located near the young stellar cluster Westerlund 2, while PSR J1954+2836 is coincident with a 4.3 sigma excess reported by Milagro at a median energy of 35 TeV. Deep radio follow-up observations of the eight pulsars resulted in no detections of pulsations and upper limits comparable to the faintest known radio pulsars, indicating that these can be included among the growing population of radio-quiet pulsars in our Galaxy being uncovered by the LAT, and currently numbering more than 20.
• Chandra and Swift observations of the quasi-persistent neutron star transient EXO 0748-676 back to quiescence(0811.4582)
March 12, 2009 astro-ph
The quasi-persistent neutron star X-ray transient and eclipsing binary EXO 0748-676 recently started the transition to quiescence following an accretion outburst that lasted more than 24 years. We report on two Chandra and twelve Swift observations performed within five months after the end of the outburst. The Chandra spectrum is composed of a soft, thermal component that fits to a neutron star atmosphere model with kT^inf~0.12 keV, joined by a hard powerlaw tail that contributes ~20% of the total 0.5-10 keV unabsorbed flux. The combined Chandra/Swift data set reveals a relatively hot and luminous quiescent system with a temperature of kT^inf~0.11-0.13 keV and a bolometric thermal luminosity of ~8.1E33-1.6E34 (d/7.4 kpc)^2 erg/s. We discuss our results in the context of cooling neutron star models.
• Long-term X-ray Variability of Circinus X-1(astro-ph/0303402)
March 18, 2003 astro-ph
We present an analysis of long term X-ray monitoring observations of Circinus X-1 (Cir X-1) made with four different instruments: Vela 5B, Ariel V ASM, Ginga ASM, and RXTE ASM, over the course of more than 30 years. We use Lomb-Scargle periodograms to search for the ~16.5 day orbital period of Cir X-1 in each of these data sets and from this derive a new orbital ephemeris based solely on X-ray measurements, which we compare to the previous ephemerides obtained from radio observations. We also use the Phase Dispersion Minimization (PDM) technique, as well as FFT analysis, to verify the periods obtained from periodograms. we obtain dynamic periodograms (both Lomb-Scargle and PDM) of Cir X-1 during the RXTE era, showing the period evolution of Cir X-1, and also displaying some unexplained discrete jumps in the location of the peak power.
• Monitoring the Gamma-Ray Source 2CG 135+1 and the Radio Star LSI+61 303(astro-ph/9611200)
Nov. 25, 1996 astro-ph
We report the results of a CGRO multi-instrument series of observations of the unidentified gamma-ray source 2CG 135+1 and of its possible counterpart, the peculiar radio source GT 0236+610 coincident with the Be star LSI+61 303. We monitored the time variable radio source GT 0236+610 during the CGRO observations. OSSE and COMPTEL data were obtained during the period May-June 1994, and BATSE data for the period April 1994-January 1995. We discuss the time variability of the gamma-ray emission and spectral properties of 2CG~135+1. Understanding the nature of 2CG 135+1 may be of crucial importance for the interpretation of a class of unidentified time variable gamma-ray sources in the Galaxy. | 2021-01-21 18:11:15 | {"extraction_info": {"found_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": 0.5964822173118591, "perplexity": 4539.155422826106}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703527224.75/warc/CC-MAIN-20210121163356-20210121193356-00500.warc.gz"} |
https://hero.handmade.network/forums/code-discussion/t/1340-day_306_-_about_those_rectangles...#7535 | Andre
15 posts
Day 306 - About those rectangles...
I'm happy to report that the sorting problems on Day 306 were grossly exaggerated by a typo in PushRect:
1 rectangle2 ScreenArea = RectMinMax(Basis.P, ScaledDim); /* RectMinDim */
Casey was so close to cluing in on it too by the end of the episode. Such is the life of a streaming programmer I guess.
Casey Muratori
801 posts / 1 project
Casey Muratori is a programmer at Molly Rocket on the game 1935 and is the host of the educational programming series Handmade Hero.
Day 306 - About those rectangles...
Thanks! We'll check it out on Monday.
And yeah, streaming programming is pretty tough :/ But hey, you get the benefit of folks finding your typos for you, so, the streaming giveth and the streaming taketh away, etc.?
- Casey | 2022-08-11 01:47:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.5682944655418396, "perplexity": 9787.665695434014}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1659882571232.43/warc/CC-MAIN-20220811012302-20220811042302-00207.warc.gz"} |
https://psicode.org/psi4manual/master/prog_integrals.html | # Integrals in PSI4¶
## Introduction¶
PSI4 has a number of backends available to compute integrals. In order to accomodate these options, while providing a clean interface to the programmer, an abstraction layer is implemented within Libmints. A recent upgrade to the primary integral engine has seen some important changes to the way this interface layer is used; this document is designed to aid new developers as well as those familiar with the older calling conventions to ensure that the most efficient calling conventions are applied.
## The older style¶
A very simple loop that does not use permutational symmetry might look something like this in the old scheme:
auto sieve = std::make_shared<ERISieve>(basisset, cutoff);
auto factory= std::make_shared<IntegralFactory>(basisset);
int deriv_level = 0;
bool use_shell_pairs = true;
auto eri = factory->eri(deriv_level, use_shell_pairs);
const double* buffer = eri_->buffer();
for (int P = 0; P < basisset->nshell(); ++P) {
const auto& Pshell = basisset->shell(P);
for (int Q = 0; Q < basisset->nshell(); ++Q) {
const auto& Qshell = basisset->shell(Q);
for (int R = 0; R < basisset->nshell(); ++R) {
const auto& Rshell = basisset->shell(R);
for (int S = 0; S < basisset->nshell(); ++S) {
const auto& Sshell = basisset->shell(S);
if(sieve->shell_significant(P, Q, R, S) {
eri->compute_shell(P, Q, R, S);
// results are in buffer, do something with them..
}
}
}
}
}
An integral factory is used, which can then produce integral object for various operator types and derivative levels. A sieve is also constructed; this allows a quick determination of whether an integral shell quartet will be significant in magnitude or not, potentially saving a lot of work. This simple scheme is clean and easy to understand, and is still supported in the latest version of PSI4 with only a small change to the sieve syntax and handling of buffer addresses, noted below.
## The new syntax¶
The newer integral engines being interfaced to PSI4 may or may not require a group of similar integrals to be computed together in a block using vectorized instructions. To accomodate this possibility, a new syntax has been introduced in Libmints:
auto blocksPQ = ints[0]->get_blocks12();
auto blocksRS = ints[0]->get_blocks34();
auto factory= std::make_shared<IntegralFactory>(basisset);
int deriv_level = 0;
bool use_shell_pairs = true;
bool needs_exchange = true;
auto eri = factory->eri(deriv_level, use_shell_pairs, needs_exchange);
const auto &buffers = eri->buffers();
eri->update_density(D);
bool use_batching = eri->maximum_block_size() > 1;
// loop over all the blocks of (P>=Q|
for (size_t blockPQ_idx = 0; blockPQ_idx < blocksPQ.size(); blockPQ_idx++) {
const auto& blockPQ = blocksPQ[blockPQ_idx];
// loop over all the blocks of |R>=S)
size_t start = eri->first_RS_shell_block(blockPQ_idx);
for (int blockRS_idx = loop_start; blockRS_idx < blocksRS.size(); ++blockRS_idx) {
const auto& blockRS = blocksRS[blockRS_idx];
if (!eri->shell_block_significant(blockPQ_idx, blockRS_idx)) continue;
eri->compute_shell_blocks(blockPQ_idx, blockRS_idx);
const auto* block_start = buffers[0];
// Loop over all of the P,Q,R,S shells within the blocks. We have P>=Q, R>=S and PQ<=RS.
for (const auto& pairPQ : blockPQ) {
const auto &P = pairPQ.first;
const auto &Q = pairPQ.second;
const auto& Pshell = basisset->shell(P);
const auto& Qshell = basisset->shell(Q);
const auto Pam = Pshell.am();
const auto Qam = Qshell.am();
for (const auto& pairRS : blockRS) {
const auto &R = pairRS.first;
const auto &S = pairRS.second;
const auto& Rshell = basisset->shell(R);
const auto& Sshell = basisset->shell(S);
const auto Ram = Rshell.am();
const auto Sam = Sshell.am();
size_t block_size = Psize * Qsize * Rsize * Ssize;
// When there are chunks of shellpairs in RS, we need to make sure
// we filter out redundant combinations.
if (use_batching && Pam == Ram && Qam == Sam && ((P > R) || (P == R && Q > S))) {
block_start += block_size;
continue;
}
const double* int_ptr = block_start;
// Query P,Q,R,S shells for metadata and loop over that quartet
// as usual, getting the integrals from the int_ptr buffer.
block_start += block_size;
}
}
}
}
Although this looks more complex, it’s essentially doing the same thing. There are a number of differences that we’ll highlight now.
### Sieving¶
This is one of two breaking changes to the old style syntax. Instead of constructing a sieve object, the integral object should be queried directly using the exact same syntax. Requests for whether a shell is significant or a shell block is significant are both supported. A sieve object is created if matching basis sets are found in either the bra or the ket. For a density fitting integral (PQ|0A) where 0 is the null basis set and A is an auxiliary basis set the (PQ| pair will be used to construct all of the sieving data.
The old code copied integrals into a buffer owned by the integral object, whose address remained constant and could be retrieved by the buffer() member function. To avoid unnecessary copies, the new code instead uses the integrals directly from the underlying integral engine’s memory, which may change with each call to compute integrals. The integral engine provides a std::vector<const double*> containing the pointers to the start of each “chunk” of integrals. For first derivatives there are 12 such “chunks”, which are ordered Px,Py,Pz,Qx,Qy,Qz,Rx,Ry,Rz,Sx,Sy,Sz, where the Px refers to the x derivative with respect to the basis functions in shell P. Note that all integral derivatives are provided by the new integral code, unlike the previous version where only 9 of 12 were provided and the user was responsible for using translation invariance relationships to fill in the rest. The addresses for each chunk are updated in the vector after each call to compute integrals, so the user should keep a const reference to that object, and query that for the address of interest.
### Density Screening¶
The old code looked only at the integral to determine whether terms can be avoided a priori. However, if the integral is to be contracted with a density or a density-like quantity, the screening can be performed on the product, which yields more sparsity. To enable this, simply call the integral object’s update_density member, passing it a SharedMatrix holding the current density (remember that it changes during each iteration of the SCF) and the product will be considered during screening. If only coulomb-like terms are to be computed, the needs_exchange argument to the integral object constructor should be set to false, otherwise it should be true to correcly account for products of the density and integrals that contribute to exchange-like terms.
### Shell blocking¶
Each underlying integral engine knows whether it will use blocks, and will set up the metadata automatically. Instead of looping over individual shells, the user should loop over blocks supplied by the integral object; these blocks will be just a single shell quartet combination for the case where blocking is not used. It is simple to loop over pairs within each block using C++11 syntax, as demonstrated in the code snippet above. Only shell pairs with significant overlap are included in the shell block information, making this an efficient way to loop over non-negligible terms.
### Permutational symmetry¶
The pairs within each block are optimized for efficiency. First, they are screened during the integral object’s creation to ensure that only terms with appreciable overlap are stored. Second, only P,Q combinations that are permutationally unique are stored, ordered with the higher angular momentum first. Therefore care must be taken to ensure that the missing permutations are correctly accounted for when processing the integrals within the loop. See the DirectJK code in libfock for an example of using this scheme for a Fock matrix build.
### Using bra-ket symmetry¶
In cases where there is no batching performed, bra-ket symmetry can be trivially enforced by ensuring that one of the block indices is greater than or equal to the other. When batching is used, the situation is trickier; some ket batches may contain a mixture of integrals that are bra-ket unique and those that are not. To handle this we must do a coarse check at the top of the loop to see if any integrals in the batch are needed, which is implemented by asking the integral engine where to start looping in the ket via the call to eri->first_RS_shell_block(PQpair_idx). This is followed by a more fine grained check within the loops to filter individual integrals in the case where bra and ket have the same angular momentum and there’s a possibility of a handful of integrals coming from the ket that are redundant. Note that the bra is not batched in any of our engines currently: only the ket is. For this reason, density fitting integrals should be written as (A0|PQ) rather than (PQ|A0) where possible, because we want the ket to contain more functions than the bra for efficient blocking.
### Instantiating integral objects¶
With sieving being introduced in the new integral objects, the cost of their construction has increased. Although significantly cheaper than computing integrals themselves, construction of integral objects can be non-negligible, especially if many threads are used. For example, this pattern can be found in old versions of the code:
std::vector<std::shared_ptr<TwoBodyAOInt>> ints;
ints.push_back(std::shared_ptr<TwoBodyAOInt>(factory->eri()));
ints.push_back(std::shared_ptr<TwoBodyAOInt>(factory->eri()));
}
This builds many objects and the cost can add up. With the new scheme, integral objects are forced to implement a clone() member that can be used as follows:
std::vector<std::shared_ptr<TwoBodyAOInt>> ints;
ints.push_back(std::shared_ptr<TwoBodyAOInt>(factory->eri()));
ints.push_back(std::shared_ptr<TwoBodyAOInt>(ints[0]->clone()));
}
This method only incurs the cost of creating a single integral object, and performs much cheaper cloning operations to create the other objects for each thread. Moreover, if integral objects are created only in the initialization of each code that uses them, and stored persistently, the cost of integral object creation is further reduced.
## One Electron Integrals in PSI4¶
After version 1.5, we started transitioning the one electron integral code over to use Libint2 instead of the old handwritten Obara-Saika code. There are a number of reasons motivating this switch. For methods requiring potentials and fields evaluated at many external sites, such as PCM and polarizable embedding, the efficiency of the one electron integrals can be rate limiting. We also started to introduce integral screening, and it is important to balance the screening used for one- and two-electron terms carefully, so this is a good opportunity to re-evaluate the code. Finally, given the complexity of the OS recursion code, the switch to an external library leaves a more compact codebase to maintain. The one electron integrals which are not provided by Libint2 are now handled by a new implementation of the McMurchie-Davidson (M-D) algorithm, leading to removal of the OS code in version 1.6. An overview of the one electron integrals is shown in table Algorithms used for One Electron Integrals, together with the implementation they use. The tips below serve as a guide to what changed, why it changed, and how to interface with PSI4’s one-electron integral machinery now.
### Calling compute_shell(int P, int Q)¶
The hand-implemented OS recursion code also took care of the Cartesian->pure transformation (if required by the basis set). The mechanism for handling this was to provide a public facing compute_shell(int P, int Q) method for the caller; this then looked up the appropriate GaussianShell objects that were passed into the corresponding (private) compute_pair(GaussianShell &s1, GaussianShell &s2) function that computed the integrals and transformed them to the spherical harmonic basis, if needed. The switch to Libint2 integrals preserves this mechanism, but the compute_shell(int P, int Q) simply looks up the appropriate Libint2-compatible shells and hands them off to the re-written, private compute_pair() routines, which call Libint2 directly. Therefore, any calls to shell-pair level integral computations should look the same as before the introduction of Libint2, however access to the integrals has changed, as described below.
### Accessing integrals¶
Before the Libint2 transition, one electron integrals were computed in a flat array, internally called buffer_, which was accessed through the integral object’s buffer() method. For integrals with multiple operators, e.g., dipole operators that have three distinct components, the buffer was simply elongated by the appropriate amount and the caller was responsible for striding through each resulting batch correctly. The Libint2 engines instead return a list of pointers into each operator’s batch of integrals, the ordering of which are detailed on the Libint2 wiki. For this reason, the call to buffer() that returns a single buffer must be replaced with a call to buffer() to get a list of pointers; we recommend that be assigned the type const auto &. For simple integrals, such as overlap or kinetic, only the buffer corresponding to the zeroth element of this array contains integrals.
### Derivative Integrals¶
The old one electron integral code used translational invariance relations to minimze the number of integrals to be computed, leaving the caller with some bookkeeping to do to compute all terms. For example, consider an overlap integral: its value depends only on the relative separation of the two centers and not their absolute positions in space. Therefore, the derivative with respect to center A is the negative of the same derivative with respect to center B, so one is trivially gleaned from the other. Extending this to second derivatives, the same principle leads to the fact that double derivatives with respect to center A are equal to double derivatives with respect to center B, which are also equal to the negative of the mixed double derivatives with respect to both center A and B. The old code only provided the double derivative with respect to center A, leaving the caller to determine the other values. The Libint2 engine instead provides all integrals, so the caller simply needs to loop over all of the buffers provided in the appropriate order.
### Changes to External Potential Engines¶
Benchmarking showed that early versions of the old code spent a non-negligible amount of time performing the Cartesian to spherical harmonic transformation of the integrals, which is needed for most modern basis sets. To improve performance, we instead backtransformed the density to the Cartesian representation (denoted “CartAO”) and computed / contracted all integrals in this Cartesian basis, eliminating the need to transform to spherical harmonics as the integrals are computed. This bottleneck no longer exists, so these extra transformation steps have been removed as part of the switch to Libint2, and the affected codes (PCM and CPPE interfaces) now compute the potential and field integrals in the representation required by the basis set.
Also, note that the way external point charges are specified has changed. Previously, a set of N external point charges would be specified by passing a matrix with dimensions N rows and 4 columns – corresponding to charge, x, y, z – to the set_charge_field() member of the potential integral class. The same information is now passed using the more verbose std::vector<std::pair<double, std::array<double, 3>>> type instead, to be consistent with Libint2’s convention.
### New Operators Available¶
Libint2 provides a range of integrals that were previously not available in PSI4, such as the Erfc attenuated nuclear potential integrals needed for Ewald methods. If new integrals are added to Libint2 but are not yet interfaced to PSI4, please open an issue on the PSI4 GitHub page to alert the developers, who will be able to add the appropriate code. Available integrals classes and parameters currently documented at Libint2 C++11 Interface Wiki
### Shell Pairs¶
To ensure consistency between one- and two-electron terms when screening, and for efficiency reasons, shell pair lists should be used to iterate over pairs of Gaussian shells. These lists contain integer pair numbers, corresponding to the pairs of shells that have sufficient overlap to survive the screening process. Iterating over these lists is simple:
const auto& shell_pairs = Vint->shellpairs();
size_t n_pairs = shell_pairs.size();
for (size_t p = 0; p < n_pairs; ++p) {
auto P = shell_pairs[p].first;
auto Q = shell_pairs[p].second;
// do something with shells P and Q
}
Note that list considers all P,Q pairs if the two basis sets differ, but only P>=Q if the basis sets are the same; the caller should account for this restricted summation in the latter case.
### One Electron Integral Algorithm Overview¶
The following table summarizes which implementation is used for each type of one electron integral in PSI4.
Algorithms used for One Electron Integrals
Integral
Class
Implementation
Comment
Three-Center Overlap
ThreeCenterOverlapInt
Libint2
using libint2::Operator::delta for 4-center integrals
Angular Momentum
AngularMomentumInt
M-D
Dipole
DipoleInt
Libint2
no derivatives supported
Electric Field
ElectricFieldInt
Libint2
using first derivative of libint2::Operator::nuclear
Coulomb Potential
ElectrostaticInt
Libint2
evaluated for a single origin and unity charge
Kinetic
KineticInt
Libint2
Multipole Potential
MultipolePotentialInt
M-D
arbitrary order derivative of 1/R supported
Multipole Moments
MultipoleInt
M-D
arbitrary order multipoles supported, including nuclear gradients
Nabla Operator
NablaInt
Libint2
using first derivative of libint2::Operator::overlap
Overlap
OverlapInt
Libint2
Nuclear Coulomb Potential
PotentialInt
Libint2
assumes nuclear centers/charges as the potential
PCM Potential
PCMPotentialInt
Libint2
parallelized over charge points
QuadrupoleInt
Libint2
TracelessQuadrupoleInt
RelPotentialInt | 2022-05-26 02:40:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.4987344741821289, "perplexity": 2274.8238426497064}, "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-2022-21/segments/1652662595559.80/warc/CC-MAIN-20220526004200-20220526034200-00527.warc.gz"} |
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# A certain basket contains 10 apples, 7 of which are red and
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A certain basket contains 10 apples, 7 of which are red and 3 are green. If 3 different apples are randomly selected, what is the probability that out of those 3 , 2 will be red and 1 will be green ?
A. 7/40
B. 7/20
C. 49/100
D. 21/40
E. 7/10
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probability of getting RRG
$$= \frac{7}{10}*\frac{6}{9}*\frac{3}{8} = \frac{7}{40}$$
there are 3 different ways to get 2 red and 1 green {RRG, RGR, GRR}.
so probability = $$\frac{7*3}{40} = \frac{21}{40}$$
Ans: D
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A certain basket contains 10 apples, 7 of which are red and 3 are green. If 3 different apples are randomly selected, what is the probability that out of those 3 , 2 will be red and 1 will be green ?
1) 7/40
2) 7/20
3) 49/100
4) 21/40
5) 7/10
Probability = # of favorable outcomes / total # of outcomes;
$$P=\frac{C^2_7*C^1_3}{C^3_{10}}=\frac{21}{40}$$, where $$C^2_7$$ is # of ways to choose 2 different green apples out of 7, $$C^1_3$$ is # of ways to choose 1 red apple out of 3, and $$C^3_{10}$$ is total # of ways to choose 3 different apples out of total 10 apples.
Or by probability approach: $$P(RRG)=\frac{3!}{2!}*\frac{7}{10}*\frac{6}{9}*\frac{3}{8}=\frac{21}{40}$$, we are multiplying by 3!/2! as the case of GGR can occur in 3 ways: GGR, GRG, RGG - # of permutation of 3 letters out of which 2 are identical.
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Display posts from previous: Sort by | 2017-02-25 04:46:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5087965726852417, "perplexity": 3248.500845791961}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501171664.76/warc/CC-MAIN-20170219104611-00232-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://iris.unito.it/handle/2318/115554 | Yields of prompt and non-prompt J/ψ, as well as Υ(1S) mesons, are measured by the CMS experiment via their µ +µ − decays in PbPb and pp collisions at √sNN = 2.76 TeV for quarkonium rapidity |y| < 2.4. Differential cross sections and nuclear modification factors are reported as functions of y and transverse momentum pT, as well as collision centrality. For prompt J/ψ with relatively high pT (6.5 < pT < 30 GeV/c), a strong, centrality-dependent suppression is observed in PbPb collisions, compared to the yield in pp collisions scaled by the number of inelastic nucleon-nucleon collisions. In the same kinematic range, a suppression of non-prompt J/ψ, which is sensitive to the in-medium b-quark energy loss, is measured for the first time. Also the low-pT Υ(1S) mesons are suppressed in PbPb collisions.
### Suppression of non-prompt J/ψ, prompt J/ψ, and $\Upsilon$ (1S) in PbPb collisions at $\sqrt {{{s_{\text{NN}}}}} = 2.76$ TeV
#### Abstract
Yields of prompt and non-prompt J/ψ, as well as Υ(1S) mesons, are measured by the CMS experiment via their µ +µ − decays in PbPb and pp collisions at √sNN = 2.76 TeV for quarkonium rapidity |y| < 2.4. Differential cross sections and nuclear modification factors are reported as functions of y and transverse momentum pT, as well as collision centrality. For prompt J/ψ with relatively high pT (6.5 < pT < 30 GeV/c), a strong, centrality-dependent suppression is observed in PbPb collisions, compared to the yield in pp collisions scaled by the number of inelastic nucleon-nucleon collisions. In the same kinematic range, a suppression of non-prompt J/ψ, which is sensitive to the in-medium b-quark energy loss, is measured for the first time. Also the low-pT Υ(1S) mesons are suppressed in PbPb collisions.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/115554 | 2023-03-31 12:20:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.8485654592514038, "perplexity": 10540.065098675566}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1679296949642.35/warc/CC-MAIN-20230331113819-20230331143819-00085.warc.gz"} |
https://tex.stackexchange.com/questions/425637/how-to-combine-article-tex-files-into-a-book-tex | # How to combine article *.tex files into a book *.tex?
I am currently working on a book that collects several finished standalone *.tex articles. Each article has its own title, abstract, sections, etc. So simply using \include will unfortunately not work (e.g. because of multiple \title{} and \abstract{} blocks in the resulting document). The book itself has some additional material such as a preface, a title page and a table of contents. What I would like to achieve is a unified styling across the entire material (e.g. all single column), consistent page numbering and a unified bibliography at the very end, if possible. Simply concatenating the output PDFs is therefore also not an option.
How would you do this? I would like to avoid too much manual intervention in the original *.tex files, if possible.
• The question is too general IMHO. The viability of the possible solutions depends heavily on which assumptions you can make about the original tex files. Do they use the same packages for similar things? Do they follow some macro convention? Do they use the same classes? Are they divided in mainfile+subfiles? Apr 9 '18 at 16:39
• I authored all original *.tex files myself, so if absolutely necessary I can make edits to them. They are all single *.tex files (no sub files except for images and *.bib files). They use different document classes (e.g. IEEEtran), but I can unify them to article if that helps. Other than that it should be easy to find a common superset of all used packages. Apr 9 '18 at 16:50
• To clarify: I am mainly worried about \title{} and \abstract{} blocks (as they are limited to one per master *.tex file) and \ref{}s which should remain intact in the combined document. Apr 9 '18 at 16:53
• you might get some ideas from one of these questions: tex.stackexchange.com/q/149790/579 Apr 9 '18 at 16:57
• combine looks promising indeed. Thank you barbara, I will try to work with that! Apr 9 '18 at 17:03
Warning: I'm the author of combine.
I think that you could do it using combine with a little work on your part. For combine everything must be of the same class but from what you said you want to use a bunch of article within a book. A main difference between these is that book provides \chapter while article does not. If you used article throughout while faking an equivalent to \chapter in the few instances you might need it could work with little trouble but beware of book's \frontmatter, \mainmatter and \backmatter commands if you were inclined to use them (perhaps replicate them in the context of the main article if necessary).
• Thanks for your answer. I think that the concept behind the combine class is brilliant. However, I was facing severe technical difficulties (LaTeX error messages are extremely hard to decipher once the parser finds something it doesn't expect...) and opted to simply \include{} the papers after all. I moved the title and authors blocks elsewhere and it looks fine. I use the books \chapter to separate the papers (which oly use \section and lower). It might be primitive, but I'm happy with that for now. Still, thanks for your efforts! Apr 11 '18 at 18:43 | 2021-09-24 00:24:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5744967460632324, "perplexity": 1212.786444849973}, "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-2021-39/segments/1631780057479.26/warc/CC-MAIN-20210923225758-20210924015758-00664.warc.gz"} |
https://aakashdigitalsrv1.meritnation.com/ask-answer/question/75-kg-of-wheat-is-being-consumed-in-30-days-by-24-persons-ho/direct-and-inverse-proportions/9524765 | # 75 kg of wheat is being consumed in 30 days by 24 persons. how many persons will consume 50 kg of wheat in one month and ten days??
75 kg of wheat is being consumed in 30 days by 24 persons .
So ,
75 kg of wheat is being consumed in 1 days by 30 $×$ 24 persons .
And
1 kg of wheat is being consumed in 1 days by persons .
So,
1 kg of wheat is being consumed in 40 days by persons .
Therefore,
50 kg of wheat is being consumed in 40 days by persons . ( Ans )
• 7
36 person
• -2
plzzz give the method also
• 1
What are you looking for? | 2021-03-06 05:47:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.20978860557079315, "perplexity": 3706.2382124291835}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178374391.90/warc/CC-MAIN-20210306035529-20210306065529-00380.warc.gz"} |
https://mathspace.co/textbooks/syllabuses/Syllabus-411/topics/Topic-7317/subtopics/Subtopic-97566/?activeTab=interactive | NZ Level 8 (NZC) Level 3 (NCEA) [In development]
Graphs of Inverse Functions
## Interactive practice questions
The graphs of a function $f$f and its inverse $f^{-1}$f1 are symmetric about which line?
$y=-x$y=x
A
$xy=1$xy=1
B
$y=x$y=x
C
$y=x^2$y=x2
D
$y=-x$y=x
A
$xy=1$xy=1
B
$y=x$y=x
C
$y=x^2$y=x2
D
Easy
Less than a minute
Consider the graph of the function $f\left(x\right)$f(x).
Below we have sketched the lines $y=5x$y=5x (labelled $B$B) and $y=x$y=x (labelled $A$A).
Below we have sketched the line $y=\frac{1}{2}x$y=12x (labelled $B$B) over the line $y=x$y=x (labelled $A$A).
### Outcomes
#### M8-2
Display and interpret the graphs of functions with the graphs of their inverse and/or reciprocal functions | 2021-09-21 07:39:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.33474424481391907, "perplexity": 11474.232718132838}, "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-2021-39/segments/1631780057199.49/warc/CC-MAIN-20210921070944-20210921100944-00704.warc.gz"} |
https://brodowsky.it-sky.net/category/english/data/ | Phone Numbers and E-Mail Addresses
Most data that we deal with are strings or numbers or booleans and combinations of these into classes and collections. Dates can be expressed as string or number, but have enough specific logic to be seen as a fourth group of data. All these have interesting aspects, some of which have been discussed in this blog already.
Now phone numbers are by an naïve approach numbers or strings, but very soon we see that they have their own specific aspects. The same applies for email addresses which can be represented as strings.
Often projects go by their own „simplified“ specification of what an email address or a phone number is, how to parse, compare and render them. In the end of the day the simplification is harder to tame than the real solution, because it needs to be maintained and specified by the project team rather than being based on a proven library. And once in a while „edge cases“ occur, that cannot be ignored and that make the „home grown“ library even more complex.
Behind phone numbers and email addresses there are well defined and established standards and they are hard to understand thoroughly within the constrained time budget of a typical „business project“, because the time should be allocated to enhancing the business logic and not to reinventing the basics. Unless there is a real need to do so, of course.
Just to give an idea: When phone numbers are parsed or provided by user input, they can start with a „+“ sign or use some country specific logic to express, to which country they belong. And then the „+1“, for example, does not stand for the United States alone, but also for Canada and some smaller countries that are in some way associated with the United States or Canada. Further analysis of the number is required to know about that. The prefix for international number is often „00“, but in the United States it is „011“ and there were and are some other variants, that are still frequently used. Some people like to write something like „+49(0)431 77 88 99 11 1“ instead of „+49 431 77 88 99 11 1“. We can constrain the input to the variants we happen to think of and force the supplier of data to comply, but why bother? Why not accept legitimate formats, as long as they are correct and unambiguous?
Now for E-Mail-addresses there is the famous one page regular expression to recognize correct email addresses which is even by itself not totally complete. Find it at the bottom of the article…
Of course it includes some rarely used variants of email addresses that were once used and have not been completely abolished officially, but it is hard to draw and exact border for this.
So the general recommendation is to find a good library for working with email addresses and phone numbers. Maybe the library can even to some extent eliminate input strings that are formally complying the format, but know to be incorrect by knowing about numbering schemes world wide or about email domains or even by performing lookups.
Another strong recommendation is to store data like email addresses and phone numbers in a technical format, that is in the example of phone numbers always starting with a „+“ followed by digits only. For input any positioning of spaces is accepted, for output the library knows how to format it correctly. This allows selecting by the numbers without dealing with complex formatting, by just using the technical format in the query as well.
For Java (and thus for many JVM-languages), C++ and JavaScript there is an excellent library from Google for dealing with phone numbers. For E-Mails something like apache commons email validator is a way to go.
Keep in mind that for E-Mail addresses and phone numbers, the ultimate way of verification is to send them a link or a code that they need to enter. In the end of the day it is insufficient to rely only on formal verification without this final step.
But still issues remain for transforming data into a canonical technical format for storing them, formatting data for display etc. And there is a huge added value, if we can reliably recognize formally false entries early, when the user can still easily react to it, rather than waiting for an email/SMS/phone call being processed, which may fail when the user is no longer on our „registration site“. And we can process data which has already been verified by a third party, but still we want to parse it to recognize obvious errors.
The concrete libraries may be outdated by the time you are reading this, or they may not be applicable for the language environment that you are using, but please make an effort to find something similar.
So, please use good libraries, that are like to be found for the environment that you are using and write yourself what creates value for your project or organization. Unless your goal is really to write a better library. Better invest the time into areas where there are still no good libraries around.
And as always, you may understand email addresses and phone numbers as an example for a more general idea.
E-Mail Regex
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Ranges of Dates and Times
In Software we often deal with ranges of dates and times.
Let us look at it from the perspective of an end user.
When we say something like „from 2020-03-07 to 2019-03-10“ we mean the set of all timestamps such that
or more accurately:
Important is, that we mean to include the whole 24 hour day of 2019-03-10. Btw. please try to get used to the ISO-date even when writing normal human readable texts, it just makes sense…
Now when we are not talking about dates, but about times or instants of time, the interpretation is different.
When we say sonmething like „from 07:00 to 10:00“ or „from 2020-03-10T07:00:00+TZ to 2020-04-11T09:00:00+TZ“, we actually mean the set of all timestamps such that
or
respectively. It is important that we have to add one in case of date only (accuracy to one day) and we do not in case of finer grained date/time information. The question if the upper bound is included or not is not so important in our everyday life, but it proves that commonly the most useful way is not to include the upper bound. If you prefer to have all options, it is a better idea to employ an interval library, i.e. to find one or to write one. But for most cases it is enough to exclude the upper limit. This guarantees disjoint adjacent intervals which is usually what we want. I have seen people write code that adds 23:59:59.999 to a date and compares with instead of , but this is an ugly hack that needs a lot of boiler plate code and a lot of time to understand. Use the exclusive upper limit, because we have it.
Now the requirement is to add one day to the upper limit to get from the human readable form of date-only ranges to something computers can work with. It is a good thing to agree on where this transformation is made. And to do it in such a way that it even behaves correctly on those dates where daylight saving starts or ends, because adding one day might actually mean „23 hours“ or „25 hours“. If we need to be really very accurate, sometimes switch seconds need to be added.
Just another issue has come up here. Local time is much harder than UTC. We need to work with local time on all kinds of user interfaces for humans, with very few exceptions like for pilots, who actually work with UTC. But local date and time is ambiguous for one hour every year and at least a bit special to handle for these two days where daylight saving starts and ends. Convert dates to UTC and work with that internally. And convert them to local date on all kinds of user interfaces, where it makes sense, including documents that are printed or provided as PDFs, for example. When we work with dates without time, we need to add one day to the upper limit and then round it to the nearest for our timezone or know when to add 23, 24 or 25 hours, respectively, which we do not want to know, but we need to use modern time libraries like the java.time.XXX stuff in Java, for example.
Working with date and time is hard. It is important to avoid making it harder than it needs to be. Here some recommendations:
• Try to use UTC for the internal use of the software as much as possible
• Use local date or time or date and time in all kinds of user interfaces (with few exceptions)
• add one day to the upper limit and round it to the nearest midnight of local time exactly once in the stack
• exclude the upper limit in date ranges
• Use ISO-date formats even in the user interfaces, if possible
UUIDs revisited
UUIDs have proven useful in many circumstances.
We have basically two main variants:
• The UUID is calculated as a combination of the Ethernet-MAC-address, the timestamp and a counter.
• The UUID is calculated using a good random number generator
While variant 1 provides for a good uniqueness, there are some issues with it. Today we use mostly virtualized servers, which means that the MAC-address is coming from a configuration file and no longer guaranteed to be world wide unique. And we give away some information with the UUID, that we do not necessarily want to give away.
Variant 2 can be proven to have an acceptably low risk of collisions, but this is only true when using really good random number generators, which cannot always be guaranteed. Also it introduces an uncertainty in an area where we do not need it. We need to worry about this uniqueness, at least a little but, which is unnecessary.
So the question is, if we can rethink variant 1.
Assuming, that our software runs on our server farm. There may be a few hundred or thousand or even millions of virtual or physical servers. Now the organization does have a way to uniquely identify their servers. Of course we only need to consider the servers that are relevant for the application. Maybe an ID for the service instance instead of the server is even better. We may assume a numerical ID or something or have a table to map IP addresses and the like to such an ID. Some thinking is still required on how to do this. We can fill the digits that we do not need with random numbers.
Putting this ID instead of the MAC address solves the issue of configurable MAC address.
The next problem, that timestamps can be abused to find out something that should not be found out could be resolved by running the timestamp part or even the ID part (including a random number) through a symmetric encryption or simply some bijective function that is kept as a secret.
In many circumstances there is nothing wrong with customizing the UUID-generation to some „local“ standard, if this is well understood and carefully implemented.
Flashsort in Scala
There is now also an implementation of Flashsort in Scala.
In order to solve the requirement of sorting part of an array that is needed as part of flashsort, an heapsort implementation in Scala that can be constrained to a part of an array has been included as well. Heapsort was chosen, because it can sort in place and it has a guaranteed performance of . Mergesort or quicksort would have been reasonable choices as well. Some implentations even use insertion sort for this step, because the sections are small.
Flashsort in Ruby
Deutsch
There is a simple implementation of Flashsort in Ruby, after having already provided an implementation in C. The C-implementation is typically faster than the libc-function qsort, but this depends always on the data and on how well the metric-function has been written, that is needed on top of the comparison function for Flashsort. You can think of this metric function as some kind of monotonic hash function. So we have
This additionally needed function of method is not really there, apart from numerical values, so we really have to invest some time into writing it. This makes the use of Flashsort a bit harder. A good metric function is crucial for good performance, but for typical text files quite trivial implentations already outperform classical algorithms like Heapsort and Quicksort and Mergesort for larger amounts of data.
This blog article shows other sorting algorithms for Ruby.
Indexing of Arrays and Lists
We index arrays with integers. Lists also, at least the ones that allow random access. And sizes of collections are also integers.
This allows for entries in Java and typical JVM languages, because integers are actually considered to be 32bit. Actually we could think of one more entry, using indices , but then we would not be able to express the size in an signed integer. This stuff is quite deeply built into the language, so it is not so easy to break out of this. And 2’000’000’000 entries are a lot and take a lot of time to process. At least it was a lot in the first few years of Java. There should have been an unsigned variant of integers, which would in this case allow for 4’000’000’000 entries, when indexing by an uint32, but that would not really solve this problem. C uses 64 bit integers for indexing of arrays on 64 bit systems.
It turns out that we would like to be able to index arrays using long instead of int. Now changing the java arrays in a way that they could be indexed by long instead of int would break a lot of compatibility. I think this is impossible, because Java claims to retain very good backward compatibility and this reliability of both the language and the JVM has been a major advantage. Now a second type of arrays, indexed by long, could be added. This would imply even more complexity for APIs like reflection, that have to deal with all cases for parameters, where it already hurts that the primitives are no objects and arrays are such half-objects. So it will be interesting, what we can find in this area in the future.
For practical use it is a bit easier. We can already be quite happy with a second set of collections, let them be called BigCollections, that have sizes that can only be expressed with long and that are indexed in cases where applicable with longs. Now it is not too hard to program a BigList by internally using an array of arrays or an array of arrays of arrays and doing some arithmetic to calculate the internal indices from the long (int64) index given in the API. Actually we can buy some performance gain when resizing happens, because this structure, if well done, allows for more efficient resizing. Based on this all kinds of big collections could be built.
Intervals
Intervals are subsets of a universe, that are defined by upper and lower boundaries. Typically we think about real numbers, but any totally ordered universe allows the definition of intervals.
Intervals are defined by lower and upper boundaries, which can be a limiting number or unlimited, typically written as for the upper bound and for the lower bound. The boundaries can be included or excluded. So the following combinations exist for a universe :
unlimited
half open, lower unlimited
open, lower unlimited
half open, upper unlimited
open, upper unlimited
open
half open
half open
closed
it is sometimes useful to consider the empty set as an interval as well
The words „open“ and „closed“ refer to our usual topology of real numbers, but they do not necessarily retain their topological meaning when we extend the concept to our typical data types. , , and in the notation above do not have to be members of , as long as the comparison is defined between them and all members of . So we could for example meaningfully define for the interval .
As soon as we do not imply we always have to make this clear… And is kind of hard to really work with in software on computers with physically limited memory and CPU power.
Intervals have some relevance in software systems.
We sometimes have a business logic that actually relies on them and instead programming somehow around it, it is clearer and cleaner to actually work with intervals. For example, we can have a public transport scheduling system and we deal with certain time intervals in which different schedules apply than during the rest of the day. Or we have a system that records downtimes of servers and services and these are quite naturally expressed as intervals of some date-time datatype. It is usually healthy to consider all the cases mentioned above rather than ignoring the question if the boundary with probability zero of actually happening or having ugly interval limits like 22:59:59.999.
The other case is interval arithmetic. This means, we do floating point calculations by taking into account that we have an inaccuracy. So instead of numbers we have intervals . When we add two intervals, we get . In the same way we can multiply and subtract and even divide, as long as we can stay clear of zero in the denominator. Or more generally we can define .
It does of course require some mathematical thinking to understand, if the result is an interval again or at least something we can deal with reasonably. Actually we are usually happy with replacing the result by an interval that is possibly a superset of the real result, ideally the minimal superset that can be expressed with our boundary type.
At this point we will probably discover a desire to expand the concept of intervals in a meaningful way to complex numbers. We can do this by working with open disks like or closed disks like . Or with rectangles based on two intervals and like .
These two areas are quite interesting and sometimes useful. Libraries have been written for both of them.
Often we discover, that intervals alone are not quite enough. We would like to do set operations with intervals, that is
union
intersection
set difference
While the intersection works just fine, as long as we include the empty set as an interval, unions and differences lead us to non-intervals. It turns out that interval-unions, sets that can be expressed as a union of a finite number of intervals, turn out to be a useful generalization, that is actually what we want to work with rather than with intervals. In this case we can drop the empty set as interval and just express it as the union of zero intervals.
There are some questions coming up, that are interesting to deal with:
normalization
Can we normalize interval-unions to some canonical form that allows safe and relyable comparison for equality?
is our universe actually discrete, so we can express all unlimited boundaries with closed boundaries?
interval lengths
Do we have a meaningful and useful way to measure the length of an interval or the total length of an interval-union, as long as they are limited? Or even for unlimited intervals?
collection interfaces
Do we want to implement a Set-interface in languages that have sets and an understanding of sets that would fit for intervals
implementation
How can we implement this ourselves?
implementation
Can we find useful implementations?
Having written a java library to support interval-unions on arbitrary Comparable types once in a project and having heard a speech about an interval library in Scala that ended up in using interval-unions in a pretty equivalent way, it might be interesting to write in the future about how to do this or what can be found in different languages to support us. For interval arithmetic some work has been done to create extensions or libraries for C and Fortran, that support this, while I was a student. So this is pretty old stuff and interesting mostly for the concepts, even if we are not going to move to Fortran because of this.
If there is interest I will write more about actual implementations and issues to address when using or writing them.
DB Persistence without UPDATE and DELETE
When exploring the usage of databases for persistence, the easiest case is a database that does only SELECT. We can cache as much as we like and it is more or less the functional immutable world brought to the database. For working on fixed data and analyzing data this can sometimes be useful.
Usually our data actually changes in some way. It has been discussed in this Blog already, that it would be possible to extend the idea of immutability to the database, which would be achieved by allowing only INSERT and SELECT. Since data can correlate, an INSERT in a table that is understood as a sub-entity via a one-to-many-relationship by the application actually is mutating the containing entity. So it is necessary to look at this in terms of the actual OR-mapping of all applications that are running on that DB schema.
Life can be simple, if we actually have self contained data as with MongoDB or by having a JSON-column in PostgreSQL, for example. Then inter-table-relations are eliminated, but of course it is not even following the first normal form. This can be OK or not, but at least there are good reasons why best practices have been introduced in the relational DB world and we should be careful about that. Another approach is to avoid the concept of sub entities and only work with IDs that are foreign keys. We can query them explicitly when needed.
An interesting approach is to have two ID-columns. One is an id, that is unique in the DB-table and increasing for newly created data. One is the entity-ID. This is shared between several records referring to different generations of the same object. New of them are generated each time we change something and persist the changes and in a simple approach we just consider the newest record with that entity-ID valid. It can of course be enhanced with validFrom and validTo. Then each access to the database also includes a timestamp, usually close to current time, but kept constant across a transaction. Only records for which validFrom <= timestamp < validTo are considered, and within these the newest. The validFrom and validTo can form disjoint intervals, but it is up to the application logic if that is needed or not. It is also possible to select the entry with the highest ID among the records with a given entityID and timestamp-validTo/From-condition. Deleting records can be simulated by this as well, by allowing a way to express a "deleted" record, which means that in case we find this deleted record by our rules, we pretend not having found anything at all. But still referential integrity is possible, because the pre-deletion-data are still there. This concept of having two IDs has been inspired by a talk on that I saw during Clojure Exchange 2017: Immutable back to front.
Lazy Collections, Strings or Numbers
The idea is, that we have data that is obtained or calculated to give us on demand as much of it as we request. But it is not necessarily initially present. This concept is quite common in the functional world, where we in a way hide the deprecated concept of state in such structures, by the way in a way that lets use retain the benefits that led to the desire for statelessness.
Actually the concept is quite old. We have it for I/O in Unix and hence in Linux since the 1970ies. „Everything is a file“, at least as long as we constrain ourselves to a universal subset of possible file operations. It can be keyboard input, a named or anonymous pipe, an actual file, a TCP-connection, to name the most important cases. These are „lazy“ files, behave more or less like files as far as sequential reading is concerned, but not for random access reading. The I/O-concept has been done in such a way that it takes the case into account that we want to read bytes, but get only bytes. This can happen with files when we reach their end, but then we can obtain an indication that we reached the end of the file, while it is perfectly possible that we read less then we want in one access, but eventually get bytes including subsequent reads. Since the API has been done right, but by no means ideal, it generalizes well to the different cases that exist in current OS environments.
We could consider a File as an array of bytes. There is actually a way to access it in this way by memory-mapping it, but this assumes a physically present file. Now we could assume that we think of the array as a list that is optimized for sequential access and iterating, but not for random access. Both list types actually exist in languages like Java. Actually the random access structure can be made lazy as well, within certain constraints. If the source is actually sequential, we can just assume that the data is obtained up to the point where we actually read. The information about the total length of the stream may or may not be available, it is always available somehow in the case of structures that are completely available in memory. This random access on lazy collections works fine if the reason of laziness is to actually save us from doing expensive operations to obtain data that we do not actually need or to obtain them in parallel to the computation that processes the data. But we loose another potential drawback in this case. If the data is truly sequential, we can actually process data that is way beyond our memory capacity.
So the concept transfers easily from I/O-streams to lists and even arrays, most naturally to iterables that can be iterated only once. But we can easily imagine that this also applies to Strings, which can be seen a sequence of characters. If we do not constrain us to what a String is in C or Java or Ruby, but consider String to be a more abstract concept, again possibly dropping the idea of knowing the length or having a finite length. Just think of the output of the Unix command „yes“ or „cat /dev/zero“, which is infinite, in a theoretical way, but the computer won’t last forever in real life, of course. And we always interrupt the output at some time, usually be having the consumer shut down the connection.
Even numbers can be infinite. For real numbers this can happen only after the decimal point, for p-adic numbers it happens only before the decimal point, if you like to look into that. Since we rarely program with p-adic numbers this is more or less an edge case that is not part of our daily work, unless we actually do math research. But we could have integers with so many digits that we actually obtain and process them sequentially.
Reactive programming, which is promoted by lightbend in the Reactive Manifesto relies heavily on lazy structures, in this case data streams. An important concept is the so called „backpressure“, that allows the consumer to slow down the producer, if it cannot read the data fast enough.
Back to the collections, we can observe different approaches. Java 8 has introduced streams as lazy collections and we need to transform collections into streams and after the operation a stream back into a collection, at least in many real life situations. But putting all into one structure has some drawbacks as well. But looking at it from an abstract point of view this does not matter. The java8-streams to not implement a collection interface, but they are lazy collections from a more abstract point of view.
It is interesting that this allows us to relatively easily write nested loops where the depth of the nesting is a parameter that is not known at compile time. We just need a lazy collections of -tuples, where is the actual depth of the nesting and the contents are according to what the loops should iterate through. In this case we might or might not know the size of the collection, possibly not fitting into a 32-bit-integer. We might be able to produce a random member of the collection. And for sure we can iterate through it and stop the iteration wherever it is, once the desired calculation has been completed.
How to create ISO Date String
It is a more and more common task that we need to have a date or maybe date with time as String.
There are two reasonable ways to do this:
* We may want the date formatted in the users Locale, whatever that is.
* We want to use a generic date format, that is for a broader audience or for usage in data exchange formats, log files etc.
The first issue is interesting, because it is not always trivial to teach the software to get the right locale and to use it properly… The mechanisms are there and they are often used correctly, but more often this is just working fine for the locale that the software developers where asked to support.
So now the question is, how do we get the ISO-date of today in different environments.
Linux/Unix-Shell (bash, tcsh, …)
date "+%F"
TeX/LaTeX
\def\dayiso{\ifcase\day \or 01\or 02\or 03\or 04\or 05\or 06\or 07\or 08\or 09\or 10\or% 1..10 11\or 12\or 13\or 14\or 15\or 16\or 17\or 18\or 19\or 20\or% 11..20 21\or 22\or 23\or 24\or 25\or 26\or 27\or 28\or 29\or 30\or% 21..30 31\fi} \def\monthiso{\ifcase\month \or 01\or 02\or 03\or 04\or 05\or 06\or 07\or 08\or 09\or 10\or 11\or 12\fi} \def\dateiso{\def\today{\number\year-\monthiso-\dayiso}} \def\todayiso{\number\year-\monthiso-\dayiso}
This can go into a file isodate.sty which can then be included by \include or \input Then using \todayiso in your TeX document will use the current date. To be more precise, it is the date when TeX or LaTeX is called to process the file. This is what I use for my paper letters.
LaTeX
(From Fritz Zaucker, see his comment below):
\usepackage{isodate} % load package \isodate % switch to ISO format \today % print date according to current format
Oracle
SELECT TO_CHAR(SYSDATE, 'YYYY-MM-DD') FROM DUAL;
On Oracle Docs this function is documented.
It can be chosen as a default using ALTER SESSION for the whole session. Or in SQL-developer it can be configured. Then it is ok to just call
SELECT SYSDATE FROM DUAL;
Btw. Oracle allows to add numbers to dates. These are days. Use fractions of a day to add hours or minutes.
PostreSQL
(From Fritz Zaucker, see his comment):
select current_date; —> 2016-01-08
select now(); —> 2016-01-08 14:37:55.701079+01
Emacs
In Emacs I like to have the current Date immediately:
(defun insert-current-date () "inserts the current date" (interactive) (insert (let ((x (current-time-string))) (concat (substring x 20 24) "-" (cdr (assoc (substring x 4 7) cmode-month-alist)) "-" (let ((y (substring x 8 9))) (if (string= y " ") "0" y)) (substring x 9 10))))) (global-set-key [S-f5] 'insert-current-date)
Pressing Shift-F5 will put the current date into the cursor position, mostly as if it had been typed.
Emacs (better Variant)
(From Thomas, see his comment below):
(defun insert-current-date () "Insert current date." (interactive) (insert (format-time-string "%Y-%m-%d")))
Perl
In the Perl programming language we can use a command line call
perl -e 'use POSIX qw/strftime/;print strftime("%F", localtime()), "\n"'
or to use it in larger programms
use POSIX qw/strftime/; my \$isodate_of_today = strftime("%F", localtime());
I am not sure, if this works on MS-Windows as well, but Linux-, Unix- and MacOS-X-users should see this working.
If someone has tried it on Windows, I will be interested to hear about it…
Maybe I will try it out myself…
Perl 5 (second suggestion)
(From Fritz Zaucker, see his comment below):
perl -e 'use DateTime; use 5.10.0; say DateTime->now->strftime(„%F“);‘
Perl 6
(From Fritz Zaucker, see his comment below):
say Date.today;
or
Date.today.say;
Ruby
This is even more elegant than Perl:
ruby -e 'puts Time.new.strftime("%F")'
will do it on the command line.
Or if you like to use it in your Ruby program, just use
d = Time.new s = d.strftime("%F")
Btw. like in Oracle SQL it is possible add numbers to this. In case of Ruby, you are adding seconds.
It is slightly confusing that Ruby has two different types, Date and Time. Not quite as confusing as Java, but still…
Time is ok for this purpose.
C on Linux / Posix / Unix
#include #include #include
main(int argc, char **argv) {
char s[12]; time_t seconds_since_1970 = time(NULL); struct tm local; struct tm gmt; localtime_r(&seconds_since_1970, &local); gmtime_r(&seconds_since_1970, &gmt); size_t l1 = strftime(s, 11, "%Y-%m-%d", &local); printf("local:\t%s\n", s); size_t l2 = strftime(s, 11, "%Y-%m-%d", &gmt); printf("gmt:\t%s\n", s); exit(0); }
This speeks for itself..
But if you like to know: time() gets the seconds since 1970 as some kind of integer.
localtime_r or gmtime_r convert it into a structur, that has seconds, minutes etc as separate fields.
stftime formats it. Depending on your C it is also possible to use %F.
Scala
import java.util.Date import java.text.SimpleDateFormat ... val s : String = new SimpleDateFormat("YYYY-MM-dd").format(new Date())
This uses the ugly Java-7-libraries. We want to go to Java 8 or use Joda time and a wrapper for Scala.
Java 7
import java.util.Date import java.text.SimpleDateFormat
... String s = new SimpleDateFormat("YYYY-MM-dd").format(new Date());
Please observe that SimpleDateFormat is not thread safe. So do one of the following:
* initialize it each time with new
* make sure you run only single threaded, forever
* use EJB and have the format as instance variable in a stateless session bean
* protect it with synchronized
* protect it with locks
* make it a thread local variable
In Java 8 or Java 7 with Joda time this is better. And the toString()-method should have ISO8601 as default, but off course including the time part.
Summary
This is quite easy to achieve in many environments.
I could provide more, but maybe I leave this to you in the comments section.
What could be interesting:
* better ways for the ones that I have provided
* other databases
* other editors (vim, sublime, eclipse, idea,…)
* Office packages (Libreoffice and MS-Office)
* C#
* F#
* Clojure
* C on MS-Windows
* Perl and Ruby on MS-Windows
* Java 8
* Scala using better libraries than the Java-7-library for this
* Java using better libraries than the Java-7-library for this
* C++
* PHP
* Python
* Cobol
* JavaScript
* …
If you provide a reasonable solution I will make it part of the article with a reference… | 2020-02-24 08:43:41 | {"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": 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": 0.4579157829284668, "perplexity": 1477.4800135282562}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145910.53/warc/CC-MAIN-20200224071540-20200224101540-00012.warc.gz"} |
https://byjus.com/question-answer/the-concentration-of-potassium-ions-inside-a-biological-cell-is-at-least-twenty-times-higher-2/ | Question
# The concentration of potassium ions inside a biological cell is at least twenty times higher than the outside. The resulting potential difference across the cell is important in several processes such as transmission of nerve impulses and maintaining the ion balance. A simple model for such a concentration cell involving a metal M is:$$\mathrm{M}(\mathrm{s})|\mathrm{M}^{+}(\mathrm{a}\mathrm{q};0.05\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{r})||\mathrm{M}^{+}(\mathrm{a}\mathrm{q})$$ , 1 $$\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{r})|\mathrm{M}(\mathrm{s})$$For the above electrolytic cell the magnitude of the cell potential $$|\mathrm{E}_{\mathrm{c}\mathrm{e}ll}|=70\mathrm{m}\mathrm{V}$$.For the above cell:
A
Ecell<0;ΔG>0
B
Ecell>0;ΔG<0
C
Ecell<0;ΔGo>0
D
Ecell>0;ΔGo>0
Solution
## The correct option is A $$\mathrm{E}_{\mathrm{c}\mathrm{e}ll}>0;\Delta \mathrm{G}<0$$$$\mathrm{M}(\mathrm{s})+\mathrm{M}{(\mathrm{a}\mathrm{q})1\mathrm{M}}^{+}\rightarrow\mathrm{M}{(\mathrm{a}\mathrm{q})0.05\mathrm{M}}^{+}+\mathrm{M}(\mathrm{s})$$According to Nernst equation,$$\displaystyle\mathrm{E}_{\mathrm{c}\mathrm{e}ll}=0-\frac{2.303\mathrm{R}\mathrm{T}}{\mathrm{F}}\log\frac{\mathrm{M}_{05\mathrm{M}}^{+}}{\mathrm{M}_{1\mathrm{M}}^{+}}$$$$=0-\displaystyle \frac{2.303\mathrm{R}\mathrm{T}}{\mathrm{F}}\log(5\times 10^{-2})$$$$=+\mathrm{v}\mathrm{e}$$Hence, $$|\mathrm{E}_{\mathrm{c}\mathrm{e}ll}|=\mathrm{E}_{\mathrm{c}\mathrm{e}ll}=0.70\mathrm{V}$$ and $$\Delta \mathrm{G}<0$$ for the feasibility of the reaction.Option B is correct.Chemistry
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View More | 2022-01-17 21:55:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.6956133246421814, "perplexity": 1756.022310183987}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1642320300624.10/warc/CC-MAIN-20220117212242-20220118002242-00138.warc.gz"} |
https://www.shaalaa.com/question-bank-solutions/rohit-had-1000-shares-rs-125-each-new-delhi-times-paying-dividend-12-shares-dividends-examples_86711 | # Rohit Had 1000 Shares of Rs 125 Each of 'New Delhi Times' Paying a Dividend of 12%. - Mathematics
Sum
Rohit had 1000 shares of Rs 125 each of 'New Delhi Times' paying a dividend of 12%. He sold all of them at a market rate of Rs 150 and invested the proceeds in buying Rs 25 shares of BVL available at Rs 60 and paying 20% dividend. How many shares of BVL did Rohit buy and what is the change in his annual income?
#### Solution
In first case:
No. of shares sold = 1000
Face value of each share = Rs 125
Face value of 1000 shares = Rs (125 x 1000) =Rs 1,25,000
Market value of each share = Rs 150
Market value of 1000 shares = Rs ( 150 x 1000) = Rs 1,50,000
Dividend (income) for 1000 shares = 12 % of Rs 1,25,000 = Rs (12 xx 125000)/100 = Rs 15000
In second case:
Proceeds from selling 1000 shares = Rs 1, 50, 000
Face value of each share = Rs 25
Market value of each share = Rs 60
No. of shares bought = 150000/60 = 2500
Rohit bought 2,500 shares of Rs 60 each.
Face value of 2,500 shares =Rs (25 x 2,500) =Rs 62,500
Dividend (income) for 2,500 shares = 20 % of Rs 62,500 = (20 xx 62500)/100 = Rs 12500
Change in annual income= Rs (12,500 -15,000) = - Rs 2,500 (less)
Concept: Shares and Dividends Examples
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#### APPEARS IN
Frank ICSE Class 10 Mathematics Part 2
Chapter 4 Shares and Dividends
Exercise 4.2 | Q 12 | Page 64 | 2021-04-19 10:08:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.17693249881267548, "perplexity": 14556.687959892764}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1618038879305.68/warc/CC-MAIN-20210419080654-20210419110654-00463.warc.gz"} |
http://theinfolist.com/php/SummaryGet.php?FindGo=Game_theory | TheInfoList
Game theory is the study of mathematical models of strategic interaction among rational decision-makers.Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs
1
vii–xi
It has applications in all fields of social science, as well as in logic, systems science and computer science. Originally, it addressed zero-sum games, in which each participant's gains or losses are exactly balanced by those of the other participants. In the 21st century, game theory applies to a wide range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals, and computers. Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum games and its proof by John von Neumann. Von Neumann's original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was followed by the 1944 book ''Theory of Games and Economic Behavior'', co-written with Oskar Morgenstern, which considered cooperative games of several players. The second edition of this book provided an axiomatic theory of expected utility, which allowed mathematical statisticians and economists to treat decision-making under uncertainty. Game theory was developed extensively in the 1950s by many scholars. It was explicitly applied to evolution in the 1970s, although similar developments go back at least as far as the 1930s. Game theory has been widely recognized as an important tool in many fields. , with the Nobel Memorial Prize in Economic Sciences going to game theorist Jean Tirole, eleven game theorists have won the economics Nobel Prize. John Maynard Smith was awarded the Crafoord Prize for his application of evolutionary game theory.
History
Discussions of two-person games began long before the rise of modern, mathematical game theory. In 1713, a letter attributed to Charles Waldegrave analyzed a game called "le her". He was an active Jacobite and uncle to James Waldegrave, a British diplomat. The true identity of the original correspondent is somewhat elusive given the limited details and evidence available and the subjective nature of its interpretation. One theory postulates Francis Waldegrave as the true correspondent, but this has yet to be proven. In this letter, Waldegrave provides a minimax mixed strategy solution to a two-person version of the card game le Her, and the problem is now known as Waldegrave problem. In his 1838 ''Recherches sur les principes mathématiques de la théorie des richesses'' (''Researches into the Mathematical Principles of the Theory of Wealth''), Antoine Augustin Cournot considered a duopoly and presents a solution that is the Nash equilibrium of the game. In 1913, Ernst Zermelo published ''Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels'' (''On an Application of Set Theory to the Theory of the Game of Chess''), which proved that the optimal chess strategy is strictly determined. This paved the way for more general theorems. In 1938, the Danish mathematical economist Frederik Zeuthen proved that the mathematical model had a winning strategy by using Brouwer's fixed point theorem. In his 1938 book ''Applications aux Jeux de Hasard'' and earlier notes, Émile Borel proved a minimax theorem for two-person zero-sum matrix games only when the pay-off matrix was symmetric and provides a solution to a non-trivial infinite game (known in English as Blotto game). Borel conjectured the non-existence of mixed-strategy equilibria in finite two-person zero-sum games, a conjecture that was proved false by von Neumann. Game theory did not really exist as a unique field until John von Neumann published the paper ''On the Theory of Games of Strategy'' in 1928. Von Neumann's original proof used Brouwer's fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was followed by his 1944 book ''Theory of Games and Economic Behavior'' co-authored with Oskar Morgenstern. The second edition of this book provided an axiomatic theory of utility, which reincarnated Daniel Bernoulli's old theory of utility (of money) as an independent discipline. Von Neumann's work in game theory culminated in this 1944 book. This foundational work contains the method for finding mutually consistent solutions for two-person zero-sum games. Subsequent work focused primarily on cooperative game theory, which analyzes optimal strategies for groups of individuals, presuming that they can enforce agreements between them about proper strategies. In 1950, the first mathematical discussion of the prisoner's dilemma appeared, and an experiment was undertaken by notable mathematicians Merrill M. Flood and Melvin Dresher, as part of the RAND Corporation's investigations into game theory. RAND pursued the studies because of possible applications to global nuclear strategy. Around this same time, John Nash developed a criterion for mutual consistency of players' strategies known as the Nash equilibrium, applicable to a wider variety of games than the criterion proposed by von Neumann and Morgenstern. Nash proved that every finite n-player, non-zero-sum (not just two-player zero-sum) non-cooperative game has what is now known as a Nash equilibrium in mixed strategies. Game theory experienced a flurry of activity in the 1950s, during which the concepts of the core, the extensive form game, fictitious play, repeated games, and the Shapley value were developed. The 1950s also saw the first applications of game theory to philosophy and political science.
Prize-winning achievements
In 1965, Reinhard Selten introduced his solution concept of subgame perfect equilibria, which further refined the Nash equilibrium. Later he would introduce trembling hand perfection as well. In 1994 Nash, Selten and Harsanyi became Economics Nobel Laureates for their contributions to economic game theory. In the 1970s, game theory was extensively applied in biology, largely as a result of the work of John Maynard Smith and his evolutionarily stable strategy. In addition, the concepts of correlated equilibrium, trembling hand perfection, and common knowledge were introduced and analyzed. In 2005, game theorists Thomas Schelling and Robert Aumann followed Nash, Selten, and Harsanyi as Nobel Laureates. Schelling worked on dynamic models, early examples of evolutionary game theory. Aumann contributed more to the equilibrium school, introducing equilibrium coarsening and correlated equilibria, and developing an extensive formal analysis of the assumption of common knowledge and of its consequences. In 2007, Leonid Hurwicz, Eric Maskin, and Roger Myerson were awarded the Nobel Prize in Economics "for having laid the foundations of mechanism design theory". Myerson's contributions include the notion of proper equilibrium, and an important graduate text: ''Game Theory, Analysis of Conflict''. Hurwicz introduced and formalized the concept of incentive compatibility. In 2012, Alvin E. Roth and Lloyd S. Shapley were awarded the Nobel Prize in Economics "for the theory of stable allocations and the practice of market design". In 2014, the Nobel went to game theorist Jean Tirole.
Game types
Cooperative / non-cooperative
A game is ''cooperative'' if the players are able to form binding commitments externally enforced (e.g. through contract law). A game is ''non-cooperative'' if players cannot form alliances or if all agreements need to be self-enforcing (e.g. through credible threats). Cooperative games are often analyzed through the framework of ''cooperative game theory'', which focuses on predicting which coalitions will form, the joint actions that groups take, and the resulting collective payoffs. It is opposed to the traditional ''non-cooperative game theory'' which focuses on predicting individual players' actions and payoffs and analyzing Nash equilibria. Cooperative game theory provides a high-level approach as it describes only the structure, strategies, and payoffs of coalitions, whereas non-cooperative game theory also looks at how bargaining procedures will affect the distribution of payoffs within each coalition. As non-cooperative game theory is more general, cooperative games can be analyzed through the approach of non-cooperative game theory (the converse does not hold) provided that sufficient assumptions are made to encompass all the possible strategies available to players due to the possibility of external enforcement of cooperation. While it would thus be optimal to have all games expressed under a non-cooperative framework, in many instances insufficient information is available to accurately model the formal procedures available during the strategic bargaining process, or the resulting model would be too complex to offer a practical tool in the real world. In such cases, cooperative game theory provides a simplified approach that allows analysis of the game at large without having to make any assumption about bargaining powers.
Symmetric / asymmetric
A symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them. That is, if the identities of the players can be changed without changing the payoff to the strategies, then a game is symmetric. Many of the commonly studied 2×2 games are symmetric. The standard representations of chicken, the prisoner's dilemma, and the stag hunt are all symmetric games. Some scholars would consider certain asymmetric games as examples of these games as well. However, the most common payoffs for each of these games are symmetric. The most commonly studied asymmetric games are games where there are not identical strategy sets for both players. For instance, the ultimatum game and similarly the dictator game have different strategies for each player. It is possible, however, for a game to have identical strategies for both players, yet be asymmetric. For example, the game pictured to the right is asymmetric despite having identical strategy sets for both players.
Zero-sum / non-zero-sum
Zero-sum games are a special case of constant-sum games in which choices by players can neither increase nor decrease the available resources. In zero-sum games, the total benefit goes to all players in a game, for every combination of strategies, always adds to zero (more informally, a player benefits only at the equal expense of others). Poker exemplifies a zero-sum game (ignoring the possibility of the house's cut), because one wins exactly the amount one's opponents lose. Other zero-sum games include matching pennies and most classical board games including Go and chess. Many games studied by game theorists (including the famed prisoner's dilemma) are non-zero-sum games, because the outcome has net results greater or less than zero. Informally, in non-zero-sum games, a gain by one player does not necessarily correspond with a loss by another. Constant-sum games correspond to activities like theft and gambling, but not to the fundamental economic situation in which there are potential gains from trade. It is possible to transform any game into a (possibly asymmetric) zero-sum game by adding a dummy player (often called "the board") whose losses compensate the players' net winnings.
Simultaneous / sequential
Simultaneous games are games where both players move simultaneously, or if they do not move simultaneously, the later players are unaware of the earlier players' actions (making them ''effectively'' simultaneous). Sequential games (or dynamic games) are games where later players have some knowledge about earlier actions. This need not be perfect information about every action of earlier players; it might be very little knowledge. For instance, a player may know that an earlier player did not perform one particular action, while they do not know which of the other available actions the first player actually performed. The difference between simultaneous and sequential games is captured in the different representations discussed above. Often, normal form is used to represent simultaneous games, while extensive form is used to represent sequential ones. The transformation of extensive to normal form is one way, meaning that multiple extensive form games correspond to the same normal form. Consequently, notions of equilibrium for simultaneous games are insufficient for reasoning about sequential games; see subgame perfection. In short, the differences between sequential and simultaneous games are as follows:
Perfect information and imperfect information
An important subset of sequential games consists of games of perfect information. A game is one of perfect information if all players know the moves previously made by all other players. Most games studied in game theory are imperfect-information games. Examples of perfect-information games include tic-tac-toe, checkers, infinite chess, and Go. Many card games are games of imperfect information, such as poker and bridge. Perfect information is often confused with complete information, which is a similar concept. Complete information requires that every player know the strategies and payoffs available to the other players but not necessarily the actions taken. Games of incomplete information can be reduced, however, to games of imperfect information by introducing "moves by nature".
Games in which the difficulty of finding an optimal strategy stems from the multiplicity of possible moves are called combinatorial games. Examples include chess and go. Games that involve imperfect information may also have a strong combinatorial character, for instance backgammon. There is no unified theory addressing combinatorial elements in games. There are, however, mathematical tools that can solve particular problems and answer general questions. Games of perfect information have been studied in combinatorial game theory, which has developed novel representations, e.g. surreal numbers, as well as combinatorial and algebraic (and sometimes non-constructive) proof methods to solve games of certain types, including "loopy" games that may result in infinitely long sequences of moves. These methods address games with higher combinatorial complexity than those usually considered in traditional (or "economic") game theory. A typical game that has been solved this way is Hex. A related field of study, drawing from computational complexity theory, is game complexity, which is concerned with estimating the computational difficulty of finding optimal strategies. Research in artificial intelligence has addressed both perfect and imperfect information games that have very complex combinatorial structures (like chess, go, or backgammon) for which no provable optimal strategies have been found. The practical solutions involve computational heuristics, like alpha–beta pruning or use of artificial neural networks trained by reinforcement learning, which make games more tractable in computing practice.
Infinitely long games
Games, as studied by economists and real-world game players, are generally finished in finitely many moves. Pure mathematicians are not so constrained, and set theorists in particular study games that last for infinitely many moves, with the winner (or other payoff) not known until ''after'' all those moves are completed. The focus of attention is usually not so much on the best way to play such a game, but whether one player has a winning strategy. (It can be proven, using the axiom of choice, that there are gameseven with perfect information and where the only outcomes are "win" or "lose"for which ''neither'' player has a winning strategy.) The existence of such strategies, for cleverly designed games, has important consequences in descriptive set theory.
Discrete and continuous games
Much of game theory is concerned with finite, discrete games that have a finite number of players, moves, events, outcomes, etc. Many concepts can be extended, however. Continuous games allow players to choose a strategy from a continuous strategy set. For instance, Cournot competition is typically modeled with players' strategies being any non-negative quantities, including fractional quantities.
Differential games
Differential games such as the continuous pursuit and evasion game are continuous games where the evolution of the players' state variables is governed by differential equations. The problem of finding an optimal strategy in a differential game is closely related to the optimal control theory. In particular, there are two types of strategies: the open-loop strategies are found using the Pontryagin maximum principle while the closed-loop strategies are found using Bellman's Dynamic Programming method. A particular case of differential games are the games with a random time horizon. In such games, the terminal time is a random variable with a given probability distribution function. Therefore, the players maximize the mathematical expectation of the cost function. It was shown that the modified optimization problem can be reformulated as a discounted differential game over an infinite time interval.
Evolutionary game theory
Evolutionary game theory studies players who adjust their strategies over time according to rules that are not necessarily rational or farsighted. In general, the evolution of strategies over time according to such rules is modeled as a Markov chain with a state variable such as the current strategy profile or how the game has been played in the recent past. Such rules may feature imitation, optimization, or survival of the fittest. In biology, such models can represent evolution, in which offspring adopt their parents' strategies and parents who play more successful strategies (i.e. corresponding to higher payoffs) have a greater number of offspring. In the social sciences, such models typically represent strategic adjustment by players who play a game many times within their lifetime and, consciously or unconsciously, occasionally adjust their strategies.
Stochastic outcomes (and relation to other fields)
Individual decision problems with stochastic outcomes are sometimes considered "one-player games". These situations are not considered game theoretical by some authors. They may be modeled using similar tools within the related disciplines of decision theory, operations research, and areas of artificial intelligence, particularly AI planning (with uncertainty) and multi-agent system. Although these fields may have different motivators, the mathematics involved are substantially the same, e.g. using Markov decision processes (MDP). Stochastic outcomes can also be modeled in terms of game theory by adding a randomly acting player who makes "chance moves" ("moves by nature"). This player is not typically considered a third player in what is otherwise a two-player game, but merely serves to provide a roll of the dice where required by the game. For some problems, different approaches to modeling stochastic outcomes may lead to different solutions. For example, the difference in approach between MDPs and the minimax solution is that the latter considers the worst-case over a set of adversarial moves, rather than reasoning in expectation about these moves given a fixed probability distribution. The minimax approach may be advantageous where stochastic models of uncertainty are not available, but may also be overestimating extremely unlikely (but costly) events, dramatically swaying the strategy in such scenarios if it is assumed that an adversary can force such an event to happen. (See Black swan theory for more discussion on this kind of modeling issue, particularly as it relates to predicting and limiting losses in investment banking.) General models that include all elements of stochastic outcomes, adversaries, and partial or noisy observability (of moves by other players) have also been studied. The "gold standard" is considered to be partially observable stochastic game (POSG), but few realistic problems are computationally feasible in POSG representation.
Metagames
These are games the play of which is the development of the rules for another game, the target or subject game. Metagames seek to maximize the utility value of the rule set developed. The theory of metagames is related to mechanism design theory. The term metagame analysis is also used to refer to a practical approach developed by Nigel Howard. whereby a situation is framed as a strategic game in which stakeholders try to realize their objectives by means of the options available to them. Subsequent developments have led to the formulation of confrontation analysis.
Pooling games
These are games prevailing over all forms of society. Pooling games are repeated plays with changing payoff table in general over an experienced path, and their equilibrium strategies usually take a form of evolutionary social convention and economic convention. Pooling game theory emerges to formally recognize the interaction between optimal choice in one play and the emergence of forthcoming payoff table update path, identify the invariance existence and robustness, and predict variance over time. The theory is based upon topological transformation classification of payoff table update over time to predict variance and invariance, and is also within the jurisdiction of the computational law of reachable optimality for ordered system.
Mean field game theory
Mean field game theory is the study of strategic decision making in very large populations of small interacting agents. This class of problems was considered in the economics literature by Boyan Jovanovic and Robert W. Rosenthal, in the engineering literature by Peter E. Caines, and by mathematician Pierre-Louis Lions and Jean-Michel Lasry.
Representation of games
The games studied in game theory are well-defined mathematical objects. To be fully defined, a game must specify the following elements: the ''players'' of the game, the ''information'' and ''actions'' available to each player at each decision point, and the ''payoffs'' for each outcome. (Eric Rasmusen refers to these four "essential elements" by the acronym "PAPI".) A game theorist typically uses these elements, along with a solution concept of their choosing, to deduce a set of equilibrium strategies for each player such that, when these strategies are employed, no player can profit by unilaterally deviating from their strategy. These equilibrium strategies determine an equilibrium to the game—a stable state in which either one outcome occurs or a set of outcomes occur with known probability. Most cooperative games are presented in the characteristic function form, while the extensive and the normal forms are used to define noncooperative games.
Extensive form
The extensive form can be used to formalize games with a time sequencing of moves. Games here are played on trees (as pictured here). Here each vertex (or node) represents a point of choice for a player. The player is specified by a number listed by the vertex. The lines out of the vertex represent a possible action for that player. The payoffs are specified at the bottom of the tree. The extensive form can be viewed as a multi-player generalization of a decision tree. To solve any extensive form game, backward induction must be used. It involves working backward up the game tree to determine what a rational player would do at the last vertex of the tree, what the player with the previous move would do given that the player with the last move is rational, and so on until the first vertex of the tree is reached. The game pictured consists of two players. The way this particular game is structured (i.e., with sequential decision making and perfect information), ''Player 1'' "moves" first by choosing either or (fair or unfair). Next in the sequence, ''Player 2'', who has now seen ''Player 1''s move, chooses to play either or . Once ''Player 2'' has made their choice, the game is considered finished and each player gets their respective payoff. Suppose that ''Player 1'' chooses and then ''Player 2'' chooses : ''Player 1'' then gets a payoff of "eight" (which in real-world terms can be interpreted in many ways, the simplest of which is in terms of money but could mean things such as eight days of vacation or eight countries conquered or even eight more opportunities to play the same game against other players) and ''Player 2'' gets a payoff of "two". The extensive form can also capture simultaneous-move games and games with imperfect information. To represent it, either a dotted line connects different vertices to represent them as being part of the same information set (i.e. the players do not know at which point they are), or a closed line is drawn around them. (See example in the imperfect information section.)
Normal form
The normal (or strategic form) game is usually represented by a matrix which shows the players, strategies, and payoffs (see the example to the right). More generally it can be represented by any function that associates a payoff for each player with every possible combination of actions. In the accompanying example there are two players; one chooses the row and the other chooses the column. Each player has two strategies, which are specified by the number of rows and the number of columns. The payoffs are provided in the interior. The first number is the payoff received by the row player (Player 1 in our example); the second is the payoff for the column player (Player 2 in our example). Suppose that Player 1 plays ''Up'' and that Player 2 plays ''Left''. Then Player 1 gets a payoff of 4, and Player 2 gets 3. When a game is presented in normal form, it is presumed that each player acts simultaneously or, at least, without knowing the actions of the other. If players have some information about the choices of other players, the game is usually presented in extensive form. Every extensive-form game has an equivalent normal-form game, however, the transformation to normal form may result in an exponential blowup in the size of the representation, making it computationally impractical.
Characteristic function form
In games that possess removable utility, separate rewards are not given; rather, the characteristic function decides the payoff of each unity. The idea is that the unity that is 'empty', so to speak, does not receive a reward at all. The origin of this form is to be found in John von Neumann and Oskar Morgenstern's book; when looking at these instances, they guessed that when a union $\mathbf$ appears, it works against the fraction $\left\left(\frac\right\right)$ as if two individuals were playing a normal game. The balanced payoff of C is a basic function. Although there are differing examples that help determine coalitional amounts from normal games, not all appear that in their function form can be derived from such. Formally, a characteristic function is seen as: (N,v), where N represents the group of people and $v:2^N \to \mathbf$ is a normal utility. Such characteristic functions have expanded to describe games where there is no removable utility.
Alternative game representations
Alternative game representation forms exist and are used for some subclasses of games or adjusted to the needs of interdisciplinary research. In addition to classical game representions, some of the alternative representations also encode time related aspects.
General and applied uses
As a method of applied mathematics, game theory has been used to study a wide variety of human and animal behaviors. It was initially developed in economics to understand a large collection of economic behaviors, including behaviors of firms, markets, and consumers. The first use of game-theoretic analysis was by Antoine Augustin Cournot in 1838 with his solution of the Cournot duopoly. The use of game theory in the social sciences has expanded, and game theory has been applied to political, sociological, and psychological behaviors as well. Although pre-twentieth-century naturalists such as Charles Darwin made game-theoretic kinds of statements, the use of game-theoretic analysis in biology began with Ronald Fisher's studies of animal behavior during the 1930s. This work predates the name "game theory", but it shares many important features with this field. The developments in economics were later applied to biology largely by John Maynard Smith in his 1982 book ''Evolution and the Theory of Games''. In addition to being used to describe, predict, and explain behavior, game theory has also been used to develop theories of ethical or normative behavior and to prescribe such behavior. In economics and philosophy, scholars have applied game theory to help in the understanding of good or proper behavior. Game-theoretic arguments of this type can be found as far back as Plato. An alternative version of game theory, called chemical game theory, represents the player's choices as metaphorical chemical reactant molecules called "knowlecules". Chemical game theory then calculates the outcomes as equilibrium solutions to a system of chemical reactions.
Description and modeling
The primary use of game theory is to describe and model how human populations behave. Some scholars believe that by finding the equilibria of games they can predict how actual human populations will behave when confronted with situations analogous to the game being studied. This particular view of game theory has been criticized. It is argued that the assumptions made by game theorists are often violated when applied to real-world situations. Game theorists usually assume players act rationally, but in practice, human behavior often deviates from this model. Game theorists respond by comparing their assumptions to those used in physics. Thus while their assumptions do not always hold, they can treat game theory as a reasonable scientific ideal akin to the models used by physicists. However, empirical work has shown that in some classic games, such as the centipede game, guess 2/3 of the average game, and the dictator game, people regularly do not play Nash equilibria. There is an ongoing debate regarding the importance of these experiments and whether the analysis of the experiments fully captures all aspects of the relevant situation. Some game theorists, following the work of John Maynard Smith and George R. Price, have turned to evolutionary game theory in order to resolve these issues. These models presume either no rationality or bounded rationality on the part of players. Despite the name, evolutionary game theory does not necessarily presume natural selection in the biological sense. Evolutionary game theory includes both biological as well as cultural evolution and also models of individual learning (for example, fictitious play dynamics).
Prescriptive or normative analysis
Some scholars see game theory not as a predictive tool for the behavior of human beings, but as a suggestion for how people ought to behave. Since a strategy, corresponding to a Nash equilibrium of a game constitutes one's best response to the actions of the other players – provided they are in (the same) Nash equilibrium – playing a strategy that is part of a Nash equilibrium seems appropriate. This normative use of game theory has also come under criticism.
Game theory is a major method used in mathematical economics and business for modeling competing behaviors of interacting agents. Applications include a wide array of economic phenomena and approaches, such as auctions, bargaining, mergers and acquisitions pricing,N. Agarwal and P. Zeephongsekul
Psychological Pricing in Mergers & Acquisitions using Game Theory
School of Mathematics and Geospatial Sciences, RMIT University, Melbourne
fair division, duopolies, oligopolies, social network formation, agent-based computational economics, general equilibrium, mechanism design, and voting systems; and across such broad areas as experimental economics, behavioral economics, information economics, industrial organization, and political economy. This research usually focuses on particular sets of strategies known as "solution concepts" or "equilibria". A common assumption is that players act rationally. In non-cooperative games, the most famous of these is the Nash equilibrium. A set of strategies is a Nash equilibrium if each represents a best response to the other strategies. If all the players are playing the strategies in a Nash equilibrium, they have no unilateral incentive to deviate, since their strategy is the best they can do given what others are doing. The payoffs of the game are generally taken to represent the utility of individual players. A prototypical paper on game theory in economics begins by presenting a game that is an abstraction of a particular economic situation. One or more solution concepts are chosen, and the author demonstrates which strategy sets in the presented game are equilibria of the appropriate type. Economists and business professors suggest two primary uses (noted above): ''descriptive'' and ''prescriptive''. The Chartered Institute of Procurement & Supply (CIPS) promotes knowledge and use of game theory within the context of business procurement. CIPS and TWS Partners have conducted a series of surveys designed to explore the understanding, awareness and application of game theory among procurement professionals. Some of the main findings in their third annual survey (2019) include: *application of game theory to procurement activity has increased – at the time it was at 19% across all survey respondents *65% of participants predict that use of game theory applications will grow *70% of respondents say that they have "only a basic or a below basic understanding" of game theory *20% of participants had undertaken on-the-job training in game theory *50% of respondents said that new or improved software solutions were desirable *90% of respondents said that they do not have the software they need for their work.
Project management
Sensible decision-making is critical for the success of projects. In project management, game theory is used to model the decision-making process of players, such as investors, project managers, contractors, sub-contractors, governments and customers. Quite often, these players have competing interests, and sometimes their interests are directly detrimental to other players, making project management scenarios well-suited to be modeled by game theory. Piraveenan (2019) Material was copied from this source, which is available under
in his review provides several examples where game theory is used to model project management scenarios. For instance, an investor typically has several investment options, and each option will likely result in a different project, and thus one of the investment options has to be chosen before the project charter can be produced. Similarly, any large project involving subcontractors, for instance, a construction project, has a complex interplay between the main contractor (the project manager) and subcontractors, or among the subcontractors themselves, which typically has several decision points. For example, if there is an ambiguity in the contract between the contractor and subcontractor, each must decide how hard to push their case without jeopardizing the whole project, and thus their own stake in it. Similarly, when projects from competing organizations are launched, the marketing personnel have to decide what is the best timing and strategy to market the project, or its resultant product or service, so that it can gain maximum traction in the face of competition. In each of these scenarios, the required decisions depend on the decisions of other players who, in some way, have competing interests to the interests of the decision-maker, and thus can ideally be modeled using game theory. Piraveenan summarises that two-player games are predominantly used to model project management scenarios, and based on the identity of these players, five distinct types of games are used in project management. * Government-sector–private-sector games (games that model public–private partnerships) * Contractor–contractor games * Contractor–subcontractor games * Subcontractor–subcontractor games * Games involving other players In terms of types of games, both cooperative as well as non-cooperative, normal-form as well as extensive-form, and zero-sum as well as non-zero-sum are used to model various project management scenarios.
Political science
The application of game theory to political science is focused in the overlapping areas of fair division, political economy, public choice, war bargaining, positive political theory, and social choice theory. In each of these areas, researchers have developed game-theoretic models in which the players are often voters, states, special interest groups, and politicians. Early examples of game theory applied to political science are provided by Anthony Downs. In his 1957 book ''An Economic Theory of Democracy'', he applies the Hotelling firm location model to the political process. In the Downsian model, political candidates commit to ideologies on a one-dimensional policy space. Downs first shows how the political candidates will converge to the ideology preferred by the median voter if voters are fully informed, but then argues that voters choose to remain rationally ignorant which allows for candidate divergence. Game theory was applied in 1962 to the Cuban Missile Crisis during the presidency of John F. Kennedy. It has also been proposed that game theory explains the stability of any form of political government. Taking the simplest case of a monarchy, for example, the king, being only one person, does not and cannot maintain his authority by personally exercising physical control over all or even any significant number of his subjects. Sovereign control is instead explained by the recognition by each citizen that all other citizens expect each other to view the king (or other established government) as the person whose orders will be followed. Coordinating communication among citizens to replace the sovereign is effectively barred, since conspiracy to replace the sovereign is generally punishable as a crime. Thus, in a process that can be modeled by variants of the prisoner's dilemma, during periods of stability no citizen will find it rational to move to replace the sovereign, even if all the citizens know they would be better off if they were all to act collectively. A game-theoretic explanation for democratic peace is that public and open debate in democracies sends clear and reliable information regarding their intentions to other states. In contrast, it is difficult to know the intentions of nondemocratic leaders, what effect concessions will have, and if promises will be kept. Thus there will be mistrust and unwillingness to make concessions if at least one of the parties in a dispute is a non-democracy. However, game theory predicts that two countries may still go to war even if their leaders are cognizant of the costs of fighting. War may result from asymmetric information; two countries may have incentives to mis-represent the amount of military resources they have on hand, rendering them unable to settle disputes agreeably without resorting to fighting. Moreover, war may arise because of commitment problems: if two countries wish to settle a dispute via peaceful means, but each wishes to go back on the terms of that settlement, they may have no choice but to resort to warfare. Finally, war may result from issue indivisibilities. Game theory could also help predict a nation's responses when there is a new rule or law to be applied to that nation. One example is Peter John Wood's (2013) research looking into what nations could do to help reduce climate change. Wood thought this could be accomplished by making treaties with other nations to reduce greenhouse gas emissions. However, he concluded that this idea could not work because it would create a prisoner's dilemma for the nations.
Biology
Unlike those in economics, the payoffs for games in biology are often interpreted as corresponding to fitness. In addition, the focus has been less on equilibria that correspond to a notion of rationality and more on ones that would be maintained by evolutionary forces. The best-known equilibrium in biology is known as the ''evolutionarily stable strategy'' (ESS), first introduced in . Although its initial motivation did not involve any of the mental requirements of the Nash equilibrium, every ESS is a Nash equilibrium. In biology, game theory has been used as a model to understand many different phenomena. It was first used to explain the evolution (and stability) of the approximate 1:1 sex ratios. suggested that the 1:1 sex ratios are a result of evolutionary forces acting on individuals who could be seen as trying to maximize their number of grandchildren. Additionally, biologists have used evolutionary game theory and the ESS to explain the emergence of animal communication. The analysis of signaling games and other communication games has provided insight into the evolution of communication among animals. For example, the mobbing behavior of many species, in which a large number of prey animals attack a larger predator, seems to be an example of spontaneous emergent organization. Ants have also been shown to exhibit feed-forward behavior akin to fashion (see Paul Ormerod's ''Butterfly Economics''). Biologists have used the game of chicken to analyze fighting behavior and territoriality. According to Maynard Smith, in the preface to ''Evolution and the Theory of Games'', "paradoxically, it has turned out that game theory is more readily applied to biology than to the field of economic behaviour for which it was originally designed". Evolutionary game theory has been used to explain many seemingly incongruous phenomena in nature. One such phenomenon is known as biological altruism. This is a situation in which an organism appears to act in a way that benefits other organisms and is detrimental to itself. This is distinct from traditional notions of altruism because such actions are not conscious, but appear to be evolutionary adaptations to increase overall fitness. Examples can be found in species ranging from vampire bats that regurgitate blood they have obtained from a night's hunting and give it to group members who have failed to feed, to worker bees that care for the queen bee for their entire lives and never mate, to vervet monkeys that warn group members of a predator's approach, even when it endangers that individual's chance of survival. All of these actions increase the overall fitness of a group, but occur at a cost to the individual. Evolutionary game theory explains this altruism with the idea of kin selection. Altruists discriminate between the individuals they help and favor relatives. Hamilton's rule explains the evolutionary rationale behind this selection with the equation , where the cost to the altruist must be less than the benefit to the recipient multiplied by the coefficient of relatedness . The more closely related two organisms are causes the incidences of altruism to increase because they share many of the same alleles. This means that the altruistic individual, by ensuring that the alleles of its close relative are passed on through survival of its offspring, can forgo the option of having offspring itself because the same number of alleles are passed on. For example, helping a sibling (in diploid animals) has a coefficient of , because (on average) an individual shares half of the alleles in its sibling's offspring. Ensuring that enough of a sibling's offspring survive to adulthood precludes the necessity of the altruistic individual producing offspring. The coefficient values depend heavily on the scope of the playing field; for example if the choice of whom to favor includes all genetic living things, not just all relatives, we assume the discrepancy between all humans only accounts for approximately 1% of the diversity in the playing field, a coefficient that was in the smaller field becomes 0.995. Similarly if it is considered that information other than that of a genetic nature (e.g. epigenetics, religion, science, etc.) persisted through time the playing field becomes larger still, and the discrepancies smaller.
Computer science and logic
Game theory has come to play an increasingly important role in logic and in computer science. Several logical theories have a basis in game semantics. In addition, computer scientists have used games to model interactive computations. Also, game theory provides a theoretical basis to the field of multi-agent systems. Separately, game theory has played a role in online algorithms; in particular, the -server problem, which has in the past been referred to as ''games with moving costs'' and ''request-answer games''. Yao's principle is a game-theoretic technique for proving lower bounds on the computational complexity of randomized algorithms, especially online algorithms. The emergence of the Internet has motivated the development of algorithms for finding equilibria in games, markets, computational auctions, peer-to-peer systems, and security and information markets. Algorithmic game theory and within it algorithmic mechanism design combine computational algorithm design and analysis of complex systems with economic theory.
Philosophy
Game theory has been put to several uses in philosophy. Responding to two papers by , used game theory to develop a philosophical account of convention. In so doing, he provided the first analysis of common knowledge and employed it in analyzing play in coordination games. In addition, he first suggested that one can understand meaning in terms of signaling games. This later suggestion has been pursued by several philosophers since Lewis. Following game-theoretic account of conventions, Edna Ullmann-Margalit (1977) and Bicchieri (2006) have developed theories of social norms that define them as Nash equilibria that result from transforming a mixed-motive game into a coordination game. Game theory has also challenged philosophers to think in terms of interactive epistemology: what it means for a collective to have common beliefs or knowledge, and what are the consequences of this knowledge for the social outcomes resulting from the interactions of agents. Philosophers who have worked in this area include Bicchieri (1989, 1993), Skyrms (1990), and Stalnaker (1999). In ethics, some (most notably David Gauthier, Gregory Kavka, and Jean Hampton) authors have attempted to pursue Thomas Hobbes' project of deriving morality from self-interest. Since games like the prisoner's dilemma present an apparent conflict between morality and self-interest, explaining why cooperation is required by self-interest is an important component of this project. This general strategy is a component of the general social contract view in political philosophy (for examples, see and ). Other authors have attempted to use evolutionary game theory in order to explain the emergence of human attitudes about morality and corresponding animal behaviors. These authors look at several games including the prisoner's dilemma, stag hunt, and the Nash bargaining game as providing an explanation for the emergence of attitudes about morality (see, e.g., and ).
Retail and consumer product pricing
Game theory applications are used heavily in the pricing strategies of retail and consumer markets, particularly for the sale of inelastic goods. With retailers constantly competing against one another for consumer market share, it has become a fairly common practice for retailers to discount certain goods, intermittently, in the hopes of increasing foot-traffic in brick and mortar locations (websites visits for e-commerce retailers) or increasing sales of ancillary or complimentary products. Black Friday, a popular shopping holiday in the US, is when many retailers focus on optimal pricing strategies to capture the holiday shopping market. In the Black Friday scenario, retailers using game theory applications typically ask "what is the dominant competitor's reaction to me?" In such a scenario, the game has two players: the retailer, and the consumer. The retailer is focused on an optimal pricing strategy, while the consumer is focused on the best deal. In this closed system, there often is no dominant strategy as both players have alternative options. That is, retailers can find a different customer, and consumers can shop at a different retailer. Given the market competition that day, however, the dominant strategy for retailers lies in outperforming competitors. The open system assumes multiple retailers selling similar goods, and a finite number of consumers demanding the goods at an optimal price. A blog by a Cornell University professor provided an example of such a strategy, when Amazon priced a Samsung TV \$100 below retail value, effectively undercutting competitors. Amazon made up part of the difference by increasing the price of HDMI cables, as it has been found that consumers are less price discriminatory when it comes to the sale of secondary items. Retail markets continue to evolve strategies and applications of game theory when it comes to pricing consumer goods. The key insights found between simulations in a controlled environment and real-world retail experiences show that the applications of such strategies are more complex, as each retailer has to find an optimal balance between pricing, supplier relations, brand image, and the potential to cannibalize the sale of more profitable items.
In popular culture
* Based on the 1998 book by Sylvia Nasar, the life story of game theorist and mathematician John Nash was turned into the 2001 biopic ''A Beautiful Mind'', starring Russell Crowe as Nash. * The 1959 military science fiction novel ''Starship Troopers'' by Robert A. Heinlein mentioned "games theory" and "theory of games". In the 1997 film of the same name, the character Carl Jenkins referred to his military intelligence assignment as being assigned to "games and theory". * The 1964 film ''Dr. Strangelove'' satirizes game theoretic ideas about deterrence theory. For example, nuclear deterrence depends on the threat to retaliate catastrophically if a nuclear attack is detected. A game theorist might argue that such threats can fail to be ''credible'', in the sense that they can lead to subgame imperfect equilibria. The movie takes this idea one step further, with the Soviet Union irrevocably committing to a catastrophic nuclear response without making the threat public. * The 1980s power pop band Game Theory was founded by singer/songwriter Scott Miller, who described the band's name as alluding to "the study of calculating the most appropriate action given an adversary... to give yourself the minimum amount of failure.". * ''Liar Game'', a 2005 Japanese manga and 2007 television series, presents the main characters in each episode with a game or problem that is typically drawn from game theory, as demonstrated by the strategies applied by the characters. * The 1974 novel ''Spy Story'' by Len Deighton explores elements of Game Theory in regard to cold war army exercises. * The 2008 novel ''The Dark Forest'' by Liu Cixin explores the relationship between extraterrestrial life, humanity, and game theory.
* Applied ethics * Chainstore paradox * Collective intentionality * Glossary of game theory * Intra-household bargaining * Kingmaker scenario * Law and economics * Outline of artificial intelligence * Parrondo's paradox * Precautionary principle * Quantum refereed game * Risk management * Self-confirming equilibrium * Tragedy of the commons Lists * List of cognitive biases * List of emerging technologies * List of games in game theory
Notes
Textbooks and general references
* . *
Description
Historically important texts
* * * * * :*reprinted edition: * * * * Shapley, L.S. (1953), A Value for n-person Games, In: Contributions to the Theory of Games volume II, H. W. Kuhn and A. W. Tucker (eds.) * Shapley, L.S. (1953), Stochastic Games, Proceedings of National Academy of Science Vol. 39, pp. 1095–1100. * English translation: "On the Theory of Games of Strategy," in A. W. Tucker and R. D. Luce, ed. (1959), ''Contributions to the Theory of Games'', v. 4, p
42.
Princeton University Press. * *
Other print references
* * * * Allan Gibbard, "Manipulation of voting schemes: a general result", ''Econometrica'', Vol. 41, No. 4 (1973), pp. 587–601. * * * , (2002 edition) * . A layman's introduction. * . * * * * * * * * * * *
* James Miller (2015)
Introductory Game Theory Videos
* * Paul Walker
* David Levine
Game Theory. Papers, Lecture Notes and much more stuff.
* Alvin Roth: — Comprehensive list of links to game theory information on the Web * Adam Kalai
Game Theory and Computer Science
— Lecture notes on Game Theory and Computer Science * Mike Shor
GameTheory.net
— Lecture notes, interactive illustrations and other information. * Jim Ratliff'
(lecture notes). * Don Ross
Review Of Game Theory
in the ''Stanford Encyclopedia of Philosophy''. * Bruno Verbeek and Christopher Morris
Game Theory and Ethics
* Elmer G. Wiens
Game Theory
— Introduction, worked examples, play online two-person zero-sum games. * Marek M. Kaminski
— Syllabuses and lecture notes for game theory and political science.
* Kesten Green's — Se
Papers
fo
evidence on the accuracy of forecasts from game theory and other methods
* McKelvey, Richard D., McLennan, Andrew M., and Turocy, Theodore L. (2007)
Gambit: Software Tools for Game Theory
'. * Benjamin Polak
Open Course on Game Theory at Yalevideos of the course
* Benjamin Moritz, Bernhard Könsgen, Danny Bures, Ronni Wiersch, (2007)
Spieltheorie-Software.de: An application for Game Theory implemented in JAVA
'. * Antonin Kucera
Stochastic Two-Player Games
* Yu-Chi Ho
What is Mathematical Game TheoryWhat is Mathematical Game Theory (#2)What is Mathematical Game Theory (#3)What is Mathematical Game Theory (#4)-Many person game theoryWhat is Mathematical Game Theory ?( #5) – Finale, summing up, and my own view
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https://studyadda.com/sample-papers/kvpy-stream-sx-model-paper-5_q24/1691/452445 | • # question_answer A ring of radius R having a linear charge density $\lambda$ moves towards a solid imaginary sphere of radius R/2, so that the centre of ring passes through the centre of sphere. The axis of the ring is perpendicular to the line joining the centers of the ring and the sphere. The maximum flux through the sphere in this process is - A) $\frac{\lambda R}{{{\in }_{0}}}$ B) $\frac{\lambda R}{2{{\in }_{0}}}$ C) $\frac{\lambda \pi R}{4{{\in }_{0}}}$ D) $\frac{\lambda \pi R}{3{{\in }_{0}}}$
[D] Flux will be maximum when maximum length of ring is inside the sphere. This will occur when the chord AB is maximum Now maximum length of chord AB = diameter of sphere. In this case one arc of ring inside the sphere subtends an angle of$\pi /3$ at the center of ring. $\therefore$ Charge on this are arc$=\lambda \times$ length of arc $\therefore$ Charge on this arc =$\frac{R\pi }{3}\lambda$ $\therefore \,\,\,\phi =\frac{\frac{R\pi \lambda }{3}}{{{\in }_{0}}}=\frac{R\pi \lambda }{3{{\in }_{0}}}$ | 2022-01-21 10:31:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8468783497810364, "perplexity": 340.73491935247085}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320303356.40/warc/CC-MAIN-20220121101528-20220121131528-00079.warc.gz"} |
http://mvtnorm.r-forge.r-project.org/2012/12/06/0.9-9994/ | • set.seed(29) rmvnorm(10, ...) produces the same first ten rows as set.seed(29) rmvnorm(100, ...)
• as suggested by Paul Johnson . There is a new argument pre0.9_9993 for changing back to the `old' output. This _DOES NOT_ apply to rmvt. | 2017-08-17 15:21:00 | {"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": 0.8213858604431152, "perplexity": 12794.16195540822}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886103579.21/warc/CC-MAIN-20170817151157-20170817171157-00668.warc.gz"} |
http://lambda-the-ultimate.org/node/1598 | ## Pluvo : new hybrid scripting language
Pluvo is a functional and imperative programming language intended to be highly practical for scripting and CGIs, interpreted using a just-in-time compiler. Its hybrid dynamic and strict typing is flexible enough to allow duck typing. Its functions are, more specifically, closures, and it allows object orientation using prototypes.
From Sean B. Palmer.
Knowing he's a big fan of Python I expected a fair bit of influence - and there is, in fact the implementation is written in Python. Flexibility over typing was to be expected too, Sean's done a lot of work around RDF. Slightly surprising was Pluvo's syntax, which owes more to Bash.
## Comment viewing options
### it uses prefix notation, but
it uses prefix notation, but supports infix by scanning a list of arguments until a "callable" value is found. is that going to work in general? it seems to imply, for example, that once you start using higher order functions you need to use prefix notation (since if you start using functions as arguments of functions, then it's unclear what is "the" function and what is an argument).
i can't remember what rebol does - is it similar to this? also an early article of graham's on arc had another solution for getting "infix for free". it might be interesting to compare them. on the one hand it's a silly little detail; on the other it seems like a really good solution to presenting prefix as infix would make s-exp based macro processing much more popular (didn't see anything about macros here though)
### Context-dependent parsing is evil
It's limited to declaring operator precedence in Haskell and Prolog. This is not nice but it works quite well in practice, until you want to implement a custom Haskell parser.
It's present in lots of very subtle lexing details in Perl; see toke.c file in the sources. This is bad because these cases are unexpected and the rules are tricky (some are heuristic).
It's present in C (type names must be distinguished from other names) and C++ (type names and template names must be distinguished, including qualified names). This causes programmer headaches in templates (the typename keyword and some uses of the template keyword exist because of this). This forces a precise C++ parser to implement the whole C++ type system and template system, which for perverse programs requires to know machine-dependent integer sizes for evaluation of integer constants.
It's horrible in Logo and Rebol because you must know the arity of every function to understand what is an argument of what.
It would be a very bad idea to decide between infix and prefix function application syntax at runtime, basing on which expression has evaluated to a function.
### Are you shure that the
Are you shure that the compiler holds relevant information about callables / non-callables? I guess this is all runtime information. So parsing does not have to be context sensitive. The compiler might create an argument list only.
---
Maybe one can repair coördination for HOF by annotating some symbol to a callable in order to let the interpreter skip it when searching through an argument list for the first callable to use:
'f isa function
Here 'f is a callable f that will be handled as data and passed into the binary operator isa.
### Macros
It supports a @quote function for it's code blocks (which are stored as tables), so I imagine even if it doesn't have builtin support for macros, they wouldn't be too hard to add.
### It looks interesting but
It looks interesting but it's built on top of another (not very fast) interpreted language; hardly a win in the performance arena.
### No performance win but
The author clearly states that It's mainly an experiment and a playground...''. The ease of use of today's scripting languages makes them the ultimate playground to create new conceptual programming languages like Pluvo.
Maybe someday the author will implement his language in something like C, or (maybe better) will bootstrap a compiler to a bytecode for the JVM or Parrot.
### Or Neko...
Or Neko, which is exactly meant for this kind of things. | 2019-06-25 17:37:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.21300004422664642, "perplexity": 2159.0428500121557}, "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-2019-26/segments/1560627999876.81/warc/CC-MAIN-20190625172832-20190625194832-00249.warc.gz"} |
https://www.esaral.com/q/the-normal-density-of-a-material-is-ho-and-its-bulk-modulus-of-elasticity-is-35548 | # The normal density of a material is $ho$ and its bulk modulus of elasticity is
Question:
The normal density of a material is $\rho$ and its bulk modulus of elasticity is
$\mathrm{K}$. The magnitude of increase in density of material, when a pressure $\mathrm{P}$ is applied uniformly on all sides, will be :
1. $\frac{\rho K}{P}$
2. $\frac{\mathrm{K}}{\rho \mathrm{P}}$
3. $\frac{\mathrm{PK}}{\rho}$
4. $\frac{\rho P}{K}$
Correct Option: , 4
Solution:
(4)
Bulk modulus $\mathrm{K}=\frac{-\Delta \mathrm{P}}{\frac{\Delta \mathrm{V}}{\mathrm{V}}}=\frac{-\Delta \mathrm{pV}}{\Delta \mathrm{V}}$
We know, $\rho=\frac{M}{V}$
$\mathrm{So}, \quad \frac{-\Delta \rho}{\rho}=\frac{\Delta \mathrm{V}}{\mathrm{V}}$
$\mathrm{K}=\frac{-\Delta \mathrm{P}}{\left(-\frac{\Delta \rho}{\rho}\right)}=\frac{\rho \Delta \mathrm{P}}{\Delta \rho}$
$\Delta \rho=\frac{\rho \Delta \mathrm{P}}{\mathrm{K}}$
$\Delta \rho=\frac{\rho \mathrm{P}}{\mathrm{K}}$ | 2023-03-23 05:36:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8353294134140015, "perplexity": 595.0019827223185}, "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/1679296944996.49/warc/CC-MAIN-20230323034459-20230323064459-00763.warc.gz"} |
https://learn.careers360.com/school/question-the-difference-of-squares-of-two-numbers-is-400-the-square-of-the-smaller-number-is-9-times-the-larger-number-find-the-two-numbers-33707/ | # The difference of squares of two numbers is 400. The square of the smaller number is 9 times the larger number. Find the two numbers. ​
We are given that the difference of squares of two numbers is 400. The square of the smaller number is 9 times the larger number.
Let the larger number be and smaller number be .
First condition states that the difference of squares of two numbers is 400, that means;
----------- [Equation 1]
Second condition states that the square of the smaller number is 9 times the larger number, that means;
Putting this in equation 1, we get;
Now, we will use middle term splitting method;
.
Hence, x= 25 . Since, x = -16 is not possible to take so the larger number .
Now, putting value of x in equation 1 we get;
= 625 - 400 = 225
Therefore, the smaller number is y = 15.
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₹ 22999/- ₹ 14999/- | 2020-10-24 09:32:31 | {"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": 0.9220786690711975, "perplexity": 3304.7799903377777}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1603107882103.34/warc/CC-MAIN-20201024080855-20201024110855-00425.warc.gz"} |
http://mathhelpforum.com/calculus/117815-unit-tangent-vectors-print.html | # Unit tangent vectors
• Dec 1st 2009, 08:40 AM
Unenlightened
Unit tangent vectors
Let $r(t) = 3 sin({\it t}{\bf i}) + 3 cos ({\it t}{\bf j}) + 4{\it t}{\bf k}$. At $t = 0$, find
(i) the unit tangent vector T,
(ii) the unit normal vector N,
(iii) the curvature κ,
(iv) the binormal vector B,
(v) the torsion $\tau$
I'll come back to this one with my thoughts on it.
For the moment, I must go catch some shuteye...
• Dec 1st 2009, 09:44 AM
Quote:
Originally Posted by Unenlightened
Let $r(t) = 3 sin({\it t}{\bf i}) + 3 cos ({\it t}{\bf j}) + 4{\it t}{\bf k}$. At $t = 0$, find
(i) the unit tangent vector T,
(ii) the unit normal vector N,
(iii) the curvature κ,
(iv) the binormal vector B,
(v) the torsion $\tau$
I'll come back to this one with my thoughts on it.
For the moment, I must go catch some shuteye...
If you were given this problem, surely you were shown how to find these. There's really nothing to it. It's just direct application of the theorems and definitions. You should know how to find the derivative of a vector $r'(t)$ and the unit tangent vector at any point $t$ is just $T(t)=\frac{r'(t)}{\mid r'(t) \mid}$.
The curvature $k=\mid\frac{dT}{ds}\mid$
Just find the definitions in the book and apply them. Like I said, you don't really have to think about much to do these excercises.
• Dec 2nd 2009, 04:39 AM
Unenlightened
Quote:
If you were given this problem, surely you were shown how to find these.
Aye, I know. I wasn't given the problem though - I (perhaps foolishly) offered to help some friends who were struggling with their maths. I'll go wikipedia the definitions presently.
• Dec 2nd 2009, 06:28 AM
HallsofIvy
Quote:
Originally Posted by Unenlightened
Let $r(t) = 3 sin({\it t}{\bf i}) + 3 cos ({\it t}{\bf j}) + 4{\it t}{\bf k}$. At $t = 0$, find
(i) the unit tangent vector T,
Differentiate with respect to t. The normal vector is the vector of unit length pointing in the direction of that derivative vector.
Quote:
(ii) the unit normal vector N,
The length of the derivative in (i), is the arclength as a function of t. If you can solve for t as a function of s, differentiate the unit tangent vector with respect to s to get a normal vector. Divide by the length of that to get the unit normal vector.
Quote:
(iii) the curvature κ,
The length of the derivative of the unit tangent vector, with respect to s, is the curvature.
Quote:
(iv) the binormal vector B,
The binormal vector is the cross product of the unit tangent and unit normal vectors. Divide by its length to get the unit binormal vector if that was asked.
Quote:
(v) the torsion $\tau$
And the length of that binormal vector is the torsion.
Quote:
I'll come back to this one with my thoughts on it.
For the moment, I must go catch some shuteye... | 2017-08-21 08:39:45 | {"extraction_info": {"found_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": 13, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9026069641113281, "perplexity": 638.4388416392438}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1502886107744.5/warc/CC-MAIN-20170821080132-20170821100132-00530.warc.gz"} |
http://chemistry.tutorcircle.com/physical-chemistry/hydrogen-bonding.html | Sales Toll Free No: 1-855-666-7446
# Hydrogen Bonding
Top
Sub Topics Hydrogen bonding is of universal importance. It is central to the understanding of the microscope structures and functions in many complex systems, for example hydrogen bonded water or alcohol networks, organic compounds in solution crystal engineering, proteins and DNA building blocks of life. Hydrogen bonding can be either an important or intramolecular phenomenon. An intermolecular H-bond is one for which the donor and acceptor units are found in two different molecules, for an intramolecular H-bond they are in the same molecule.
## Strength of Hydrogen Bond
The hydrogen bond strength depends on its length and angle and hence, it has directionality. Nevertheless small derivations from linearity in the bond angle have marginal effect on H-bond strength. The dependency of the same on H-bond length is very important and has been shown to decay exponentially with distance. Strengthening hydrogen bonding has particularly important effects on viscocity and diffusion as indicated by the large changes occurring in super cooled normal water.
For example, the $\alpha$-helix conformation of a polypeptide is stabilized by intra-chain H-bonding , while the $\beta$-sheet conformation is stabilized by inter-chain H-bonding. The strength of H-bond is strongly dependent on the solvent polarity; the addition of a polar solvent can decrease the H-bond strength significantly, over several orders of magnitude, because the solvent molecule can also take part in H-bonding interactions. As a result, nonpolar solvents are mostly used for the preparation of H-bonded supramolecular materials. | 2018-09-19 13:39:34 | {"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": 0.4631012976169586, "perplexity": 2265.6250546325678}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1537267156224.9/warc/CC-MAIN-20180919122227-20180919142227-00023.warc.gz"} |
https://collegephysicsanswers.com/openstax-solutions/layer-oil-150-mm-thick-placed-between-two-microscope-slides-researchers-find | Question
A layer of oil 1.50 mm thick is placed between two microscope slides. Researchers find that a force of $5.50 \times 10^{-4} \textrm{ N}$ is required to glide one over the other at a speed of 1.00 cm/s when their contact area is $6.00 \textrm{ cm}^2$. What is the oil's viscosity? What type of oil might it be?
$0.138 \textrm{ Pa}\cdot\textrm{s}$
This is olive oil.
Solution Video | 2019-12-06 03:59:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.46446022391319275, "perplexity": 1461.6106756474996}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540484477.5/warc/CC-MAIN-20191206023204-20191206051204-00083.warc.gz"} |
http://blog.marshalljiang.com/ | ### Bill Bryson
“A separate but no less important reason for the retention of head hair is that it has been a tool of seduction since time immemorial. – The Body, Bill Bryson
He was probably referencing Malcolm Gladwell:
### Pet Peeve
“Radiolab is created by Jatt Aboomraid”
“Radiolab is created by Jedi Abromroid”
It can’t be this hard guys.
### The Help
In 2009, I was too busy avoiding reading Faulkner and Hawthorne to be paying attention to The Help. In 2020, I was not busy enough and ended up reading this solid novel.
The book is about a group of African-American maids in Jim Crow era Mississippi attempting to write a book themselves on what it was like to be maids. At points, I wondered if the book would go meta and have the second half of the book just be the book from the book. At other points, I wondered if the main plot was based on true events, since the author interleaved in JFK’s and Medgar Ever’s assassinations. I feel like this is part of the reason I never sought out historical fiction: can never distinguish fact from fiction.
Otherwise, with the current social situation, the obvious themes of racism and ignorance still remains fresh to this day, and unfortunately will probably be pertinent for the foreseeable future. The emphasis of the power dynamics between the maids and their employers resonated with me; the fact that small acts of rebellion can result in the help’s whole family being unemployed and possibly thrown in jail was not something I understood before.
Ultimately, it was a pretty darn good read, especially in the summer. There’s a certain schadenfreude when reading about characters suffering through a heat wave while I was also.
### Billboards Questions
Highway billboards are entirely forgettable. It seems that the only exceptions to this rule, at least for me, all lie on the extreme ends of the “Biblical morality” spectrum.
When I was driving up through Pennsylvania, I saw a billboard which, in bold white font, stated “In the beginning God created …” My initial thought was confusion. God created what? Ellipsis?
Of course, it was an anti-evolution billboard, but realizing that instead raised even more questions:
Who is paying for this, and is there really a person manning that toll free number? Is there some call center in Asia tasked with providing tech support from 8am till 12pm and then Jesus support in the afternoon, or is it an American call center? Should a billboard be so thought provocative? What type of person decides to change their view on evolution while traveling to Amish country? Does the Bible mention anything about advertising? Hmm, I wonder if that anti-abortion billboard down in South Florida is still up…?
On the other end of the spectrum, there are those advertising more… lewd content. I don’t think I have to elaborate much about them do I?
### Burt’s Bees
The inside joke manifested fairly early on in Darrel and Heather’s relationship.
It was a warm Sunday morning amidst a brutal winter, and the happy couple decided to spend it on the patio. During a lull in the conversation, Heather pulled out her Burt’s Bees lip balm and smeared some over her dried lips.
“Would you like some cha –”
A songbird, who up til then, was silently sheltering underneath the awning, decided it was its turn to hop into the conversation.
“CHAA peee CHAA peee CHAAA peee,” it trilled, before promptly flying into the woods.
“Was it just me, or did that bird really just say ‘chapy’ like in chapstick?” asked Darrel.
“I definitely heard it too,” Heather quickly replied.
And so it was born. For the remainder of the cold winter, the two never called it a “lip balm” or the corresponding proprietary eponym “ChapStick” anymore. Rather, whenever one of their lips were dry, they would yodel “chapy chapy” at each other followed by an inevitable giggling.
But Darrel and Heather didn’t last. She wanted someone who was more assertive, and he wanted someone who would open up more. Or so it went. When they finally split up two winters later, the silly little inside joke had nowhere to go, and so disappeared alongside the conjunction between their two names. Still, even now, whenever Darrel or Heather pull out their individual lip balm, their internal dialogue still yodels that silly little sound bite; their minds wander to dry winters long ago.
“Why do you always smile at your chapstick after putting it on?” their current partners would ask.
“Oh, it’s nothing.”
### A Fibonacci Identity
From Putnam 1996:
Define a selfish set to be a set which has its own cardinality as an element. Find, with proof, the number of subsets of $\{1,2,\ldots,n\}$ which are minimal selfish sets, that is, selfish sets none of whose proper subsets is selfish.
First solution:
We can easily show that a selfish set of length $k$ cannot be minimally selfish if it contains any numbers smaller than $k$. We can conversely show that a selfish set of length $k$ containing only elements greater than or equal to the number $k$ is minimal. Thus we can iterate over the length of the subsets, and simply choose from the remaining numbers to fill the set
\begin{align*}
\sum_{k=0}^n \binom{n-k}{k-1}
\end{align*}
Lastly, notice that $n-k \le k – 1 \implies n+1 \ge 2k$ which means the upper bound can be changed to the floor of $(n+1)/2$.
Second solution:
Let’s consider a inductive solution. Let $s_n$ be the number of minimal selfish set for the set $A_n = \{1,2,\ldots,n\}$. We first note that all minimal selfish set of $A_n$ are also minimal selfish sets in $A_{n+1}$; note that all minimal selfish sets of $A_n$ do not contain the element $n+1$. Thus, the question now is counting how many minimal selfish sets also contains $n+1$.
Actually computing the minimal selfish sets for a few small $n$s suggests that we should be looking for the Fibonacci numbers. Indeed, consider a minimal selfish set $k \subsetneq A_{n-1}$; if we add 1 to each number in $k$, and then append $n+1$, we see that it is also a selfish set, but is it minimal?
Assume that it is not minimal, that is, there exists a strict subset $f$ of $k +1 \cup \{n+1\}$ which is also selfish. The first observation is to note that if $f$ contains $n+1$, then taking $n+1$ out of $f$ then subtracting 1 contradicts the assumption that $k$ is a minimal selfish set. But if $f$ doesn’t contain $n+1$, then removing an arbitrary item and subtracting 1 will again contradict the assumption that $k$ is minimally selfish. We remark that $f$ can be of length of 1 since in the construction, we added 1.
Thus, $f_n = f_{n-1} + f_{n-2}$ and we have that the sum in the first solution is equal to the Fibonacci sequence.
### A Reminder
After I got my Echo some four and a half years ago, I developed a bedtime routine of playing the “Flash Briefing.” What that entailed was Alexa would play the latest NPR hourly news update followed by the latest news update from WGBH (the Public’s Radio didn’t interface well with the device at the time; can we also discuss how Rhode Island’s public radio brand doesn’t start with the letter “W”? What is up with that?).
At the time, this ritual served two purposes: allowed me to keep up with local news (too bad I missed the news of my murdered landlord) and also the NPR newscaster all have amazingly soothing voices. Oh, how I missed those days when I would lightly chuckle at the events before drifting off to bed. When the news is light, there would be a shade of snark in the scripts. Now the news update is a constant, desolate wasteland.
In times like this, I have to remind myself a little quote from Mr. Rogers:
“My mother would say to me, ‘Look for the helpers. You will always find people who are helping.’ To this day, especially in times of disaster, I remember my mother’s words, and I am always comforted by realizing that there are still so many helpers — so many caring people in this world.”
It’s so easy, in the ugliness of this world, to not look for the beauty in things. It’s so easy to see the violence and havoc, rather than the unity and hope. It’s so easy to find the disrupters and not the helpers of the world. I want to choose to see more of the beauty in this world. After all, it’s the only one we have.
### Citizens
There are a few names repeated a lot these days; when they were alive, they probably did not expect to hear their names uttered by the masses. They were regular citizens of these United States with dreams, fear and ambitions. They probably wanted what we all desire: George Floyd moved to Minneapolis to work, Ahmaud Arbery just wanted to run outside, Breonna Taylor just wanted to live in her own home. They all wanted to live.
I hope this time it’s different.
### Top Secret
1. “My greatest strength is probably the ability to deal with worst case scenarios”
2. “Don’t worry Sid, the portfolio is structured for long term growth. It’ll probably be only a short term correction.”
3. “Let’s rebrand Pangaea… something like Pan-gone?” | 2020-08-06 08:13:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4694368541240692, "perplexity": 2132.2550122322796}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1596439736883.40/warc/CC-MAIN-20200806061804-20200806091804-00130.warc.gz"} |
http://mathhelpforum.com/algebra/124655-math-20-3-variable-systems.html | # Math Help - Math 20 3 variable systems
1. ## Math 20 3 variable systems
I've tried this question so many times and I can't seem to get the right answer. Any takers??
Solve this system using substitution or elimination:
d+e-2f=-5
3d-e-4f=11
2d+4e+f=4
Here is what I did:
3(d+e-2f=-5)
3d-e-4f=11
3d+3f-6f=-15
-3d-e-4f=11
4e-2f=-26
2(d+e-2f=-5)
4e-2f=-26
2d+2e-4f= -10
-2d+4e+f=4
-2e-5f=-14
2(-2e-5f=-14)
4e-2f=-26
-4e-10f=-28
+4e-2f=-26
-12f=-54
f=-9/2
Then:
4e-2f=-26
4e-2(-9/2)=-26
4e + 9 = -26
4e = -35
e= -35/4
Then:
d+e-2f = -5
d+ (-35/4) -2(-9/2) = -5
d-35/4 + 9 = -5
d= -91/4
I have no idea where I went wrong. The answer is(33/4,-17/4,9/2)
So i got one variable right, but messed up on the others... ugh
2. ## Be careful with signs!!!!!
You obviously understand the process.
The line -12f = -54 implies f = +9/2.
Substitute +9/2 as you did with -9/2, and the remaining variables can be found.
3. oh my gosh.
how did i miss that??
thanks
4. Originally Posted by t-dot
Solve this system using substitution or elimination:
d+e-2f=-5 [1]
3d-e-4f=11 [2]
2d+4e+f=4 [3]
You seem to go to a lot of unnecessary work.
Can be much simpler; like:
Add [1] + [2] to get 4d - 6f = 6 ; 2d - 3f = 3 [4]
Multiply [2] by 4 : 12d - 4e - 16f = 44
Add to [3] to get 14d - 15f = 48 [5]
Multiply [4] by -7 : -14d + 21f = -21
Add to [5] to get 6f = 27 ; f = 9/2
Now get d and e. | 2014-04-21 11:15:01 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8507704734802246, "perplexity": 3487.351533648281}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00620-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://erick.navarro.io/blog/setting-up-hugo-on-netlify/ | # Setting Up Hugo on Netlify
Hugo is a static site generator that just like another alternatives(Nicola, Jekyll, etc) allows to write in plain text and generate html, js, css files.
Hugo is so much simpler to use because it's a simple binary file, called hugo, that allows to develop and prepare the site to be published. Some of its features are:
• Generate a new site
• Run a development server
• Generate static files
• Generate new pages for the site
So let's start to build a site from scratch and publish it automatically using Netlify
### Creating the new site
First we'll need to install hugo. You need to install the single binary using the package manager you prefer. In mac OS you can install it using Homebrew with the following command:
brew install hugo
You can check the version with hugo version, at the time I'm writing this post the latest version available is 0.48.
Now we need to generate an new empty site so we'll use the command hugo new site myblog. Once the command finished you'll have a new folder called myblog
The structure of the new folder should be:
├── archetypes
│ └── default.md
├── config.toml
├── content
├── data
├── layouts
├── static
└── themes
For now we have to pay attention to only one file config.toml this file contains the configuration of the new site. We can define stuff like the title of the blog, menus structure, theme parameters, etc.
The contents of config.toml should be like this:
baseURL = "http://example.org/"
languageCode = "en-us"
title = "My New Hugo Site"
We have to change baseURL to / to avoid error with broken links(we'll see these possible errors later). So the result config.toml will be:
baseURL = "/"
languageCode = "en-us"
title = "My New Hugo Site"
Now we have to install a theme. There are many awesome themes available for Hugo. You can check and pick one in Hugo themes.
For this example we'll use Beautiful Hugo
We have 2 options to include the theme in the new site:
• Clone the theme repository and add it to our folder. This will copy all the files inside our folder.
• Use git sub modules to create a reference to the theme repository. This way we don't need to copy all the files.
We'll use the second option this time.
First we need to initialize a git repository inside our myblog folder using the following command git init. Now we have a git repository created.
Now we have to run the following commands to add the theme:
cd themes
cd ..
This will clone the theme repository and add it to our repository as a submodule. We can see that now we have a beautifulhugo folder inside themes folder and also there is a new file in the root of myblog called .gitmodules with the following content:
[submodule "themes/beautifulhugo"]
path = themes/beautifulhugo
url = https://github.com/halogenica/beautifulhugo.git
Now we have to tell hugo we want to use this theme in the new site, we'll do this adding the following lines to the config.toml file:
theme = "beautifulhugo"
### Running the development server
To see the new site running before publish it we'll use the embedded development server. We run it with hugo server -D -p 9000.
After that we'll an output similar to this:
| EN
+------------------+----+
Pages | 7
Paginator pages | 0
Non-page files | 0
Static files | 33
Processed images | 0
Aliases | 1
Sitemaps | 1
Cleaned | 0
Total in 46 ms
Watching for changes in /Users/erick/Code/hugo/myblog/{content,data,layouts,static,themes}
Watching for config changes in /Users/erick/Code/hugo/myblog/config.toml
Serving pages from memory
Running in Fast Render Mode. For full rebuilds on change: hugo server --disableFastRender
Web Server is available at http://localhost:9000/ (bind address 127.0.0.1)
Press Ctrl+C to stop
Now we can open the browser and enter http://localhost:9000 and we'll see the new site with the chosen theme.
The server is watching for changes to be compiled so we leave it running.
### Writing content
At this point we don't have any content to show up so let's create some.
Hugo by default can render Markdown and Org-mode files. For this example we'll create a new post using markdown format.
Run hugo new post/hello-world.md to create a new file called hello-world.md in content/post/, Hugo will create a new file with the following content:
---
title: "Hello World"
date: 2018-09-22T15:05:47-05:00
draft: true
---
These lines are used by Hugo to show details about the content in the result file.
Let's add the lines below to hello-world.md
# This is a heading with level 1
## This is a heading with level 2
This is a paragraph
This is some python code
This is a list:
- item 1
- item 2
Now if we go to the browser we'll see the home page with a summary of the content of hello-world.md
By default Hugo show a list of the posts created in content/post in the homepage. Now we can enter to the post to see the full content.
Once we finished with the post it's necessary remove draft: true from hello-world.md file otherwise the file won't show up when we publish the site.
### Publishing the site
We can use Github, Gitlab or Bitbucket to do this. These are the services supported by Netlify. For this example I've uploaded the repository to Github and it's available in https://github.com/erickgnavar/hugo-demo-site.
#### Creating an account in Netlify
Now we have to create an account in Netlify, there is a free plan that we can use to host the new site.
#### Deploying site
Once we have the Netlify account and the site uploaded in a external repository we can proceed with the deploy.
Now we can log in and start the process clicking in "New site from Git".
Then there are 3 steps to follow:
##### Pick a repository
Now we have access an a list of our repositories. We can search for the one where the site is in.
##### Build options
Once we chosen the repository we can specify the build options. Netlify recognize that the site is made with Hugo so these options are already configured.
To proceed we click in "Deploy site".
##### Deploy result
Netlify will pull the repository and compile the site with the given build options and then it will generate a url to access the deployed site.
Now we can go to the url that Netlify generated for the site and we'll see the resulting site.
##### Useful Netlify configuration
By default Netlify only will build the site when we push changes to master. We can change this going to "Deploy settings" and changing "Branch deploys" options to "All" like the following image:
With this configuration we can push changes to a different branch than master and Netlify will generate a new url to see the changes. This is useful to test changes before publish them to production site.
It's also possible configure different kind of notifications(Slack, email, etc) to receive the result of the deploy. | 2022-05-28 01:46:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.1740284264087677, "perplexity": 3749.696153787713}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1652663011588.83/warc/CC-MAIN-20220528000300-20220528030300-00338.warc.gz"} |
http://nrich.maths.org/public/leg.php?code=5039&cl=2&cldcmpid=4519 | Search by Topic
Resources tagged with Interactivities similar to Fractional Wall:
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Broad Topics > Information and Communications Technology > Interactivities
Fault-free Rectangles
Stage: 2 Challenge Level:
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Interactive game. Set your own level of challenge, practise your table skills and beat your previous best score.
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What is the relationship between the angle at the centre and the angles at the circumference, for angles which stand on the same arc? Can you prove it?
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Learn how to use the Shuffles interactivity by running through these tutorial demonstrations.
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Use the Cuisenaire rods environment to investigate ratio. Can you find pairs of rods in the ratio 3:2? How about 9:6?
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Find out how we can describe the "symmetries" of this triangle and investigate some combinations of rotating and flipping it.
Triangles All Around
Stage: 2 Challenge Level:
Can you find all the different triangles on these peg boards, and find their angles?
Number Differences
Stage: 2 Challenge Level:
Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?
Red Even
Stage: 2 Challenge Level:
You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?
Nine-pin Triangles
Stage: 2 Challenge Level:
How many different triangles can you make on a circular pegboard that has nine pegs?
Triangle Pin-down
Stage: 2 Challenge Level:
Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
Domino Numbers
Stage: 2 Challenge Level:
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
Twice as Big?
Stage: 2 Challenge Level:
Investigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too.
Got it Article
Stage: 2 and 3
This article gives you a few ideas for understanding the Got It! game and how you might find a winning strategy.
Up and Across
Stage: 3 Challenge Level:
Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the red dot affects its vertical and horizontal movement at each stage.
More Transformations on a Pegboard
Stage: 2 Challenge Level:
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
Chocolate Bars
Stage: 2 Challenge Level:
An interactive game to be played on your own or with friends. Imagine you are having a party. Each person takes it in turns to stand behind the chair where they will get the most chocolate.
Stage: 2 Challenge Level:
How can the same pieces of the tangram make this bowl before and after it was chipped? Use the interactivity to try and work out what is going on!
Speeding Up, Slowing Down
Stage: 3 Challenge Level:
Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the red dot affects its speed at each stage.
Play a Merry Tune
Stage: 2 Challenge Level:
Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given?
Diamond Mine
Stage: 3 Challenge Level:
Practise your diamond mining skills and your x,y coordination in this homage to Pacman.
Square Coordinates
Stage: 3 Challenge Level:
A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?
Poly-puzzle
Stage: 3 Challenge Level:
This rectangle is cut into five pieces which fit exactly into a triangular outline and also into a square outline where the triangle, the rectangle and the square have equal areas.
Lost
Stage: 3 Challenge Level:
Can you locate the lost giraffe? Input coordinates to help you search and find the giraffe in the fewest guesses.
Balancing 1
Stage: 3 Challenge Level:
Meg and Mo need to hang their marbles so that they balance. Use the interactivity to experiment and find out what they need to do.
Interactive Spinners
Stage: 3 Challenge Level:
This interactivity invites you to make conjectures and explore probabilities of outcomes related to two independent events.
Isosceles Triangles
Stage: 3 Challenge Level:
Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
Fractions and Coins Game
Stage: 2 Challenge Level:
Work out the fractions to match the cards with the same amount of money.
Counters
Stage: 2 Challenge Level:
Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win?
Bow Tie
Stage: 3 Challenge Level:
Show how this pentagonal tile can be used to tile the plane and describe the transformations which map this pentagon to its images in the tiling.
Sliding Puzzle
Stage: 1, 2, 3 and 4 Challenge Level:
The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.
Fifteen
Stage: 2 and 3 Challenge Level:
Can you spot the similarities between this game and other games you know? The aim is to choose 3 numbers that total 15.
Online
Stage: 2 and 3 Challenge Level:
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.
Khun Phaen Escapes to Freedom
Stage: 3 Challenge Level:
Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.
Stage: 2 Challenge Level:
Three beads are threaded on a circular wire and are coloured either red or blue. Can you find all four different combinations?
World of Tan 20 - Fractions
Stage: 2 Challenge Level:
Can you fit the tangram pieces into the outlines of the chairs?
World of Tan 21 - Almost There Now
Stage: 2 Challenge Level:
Can you fit the tangram pieces into the outlines of the lobster, yacht and cyclist?
World of Tan 27 - Sharing
Stage: 2 Challenge Level:
Can you fit the tangram pieces into the outline of Little Fung at the table?
World of Tan 29 - the Telephone
Stage: 2 Challenge Level:
Can you fit the tangram pieces into the outline of this telephone?
World of Tan 26 - Old Chestnut
Stage: 2 Challenge Level:
Can you fit the tangram pieces into the outline of this brazier for roasting chestnuts? | 2017-03-25 19:32:57 | {"extraction_info": {"found_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": 0.17010484635829926, "perplexity": 2409.2905921531265}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218189032.76/warc/CC-MAIN-20170322212949-00185-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://www.ademcetinkaya.com/2023/01/seatsx-seabridge-gold-inc_21.html | Outlook: Seabridge Gold Inc. is assigned short-term Ba1 & long-term Ba1 estimated rating.
Dominant Strategy : Hold
Time series to forecast n: 21 Jan 2023 for (n+8 weeks)
Methodology : Inductive Learning (ML)
## Abstract
Seabridge Gold Inc. prediction model is evaluated with Inductive Learning (ML) and Pearson Correlation1,2,3,4 and it is concluded that the SEA:TSX stock is predictable in the short/long term. According to price forecasts for (n+8 weeks) period, the dominant strategy among neural network is: Hold
## Key Points
1. What are buy sell or hold recommendations?
2. Market Risk
3. How do you know when a stock will go up or down?
## SEA:TSX Target Price Prediction Modeling Methodology
We consider Seabridge Gold Inc. Decision Process with Inductive Learning (ML) where A is the set of discrete actions of SEA:TSX stock holders, F is the set of discrete states, P : S × F × S → R is the transition probability distribution, R : S × F → R is the reaction function, and γ ∈ [0, 1] is a move factor for expectation.1,2,3,4
F(Pearson Correlation)5,6,7= $\begin{array}{cccc}{p}_{a1}& {p}_{a2}& \dots & {p}_{1n}\\ & ⋮\\ {p}_{j1}& {p}_{j2}& \dots & {p}_{jn}\\ & ⋮\\ {p}_{k1}& {p}_{k2}& \dots & {p}_{kn}\\ & ⋮\\ {p}_{n1}& {p}_{n2}& \dots & {p}_{nn}\end{array}$ X R(Inductive Learning (ML)) X S(n):→ (n+8 weeks) $\stackrel{\to }{R}=\left({r}_{1},{r}_{2},{r}_{3}\right)$
n:Time series to forecast
p:Price signals of SEA:TSX stock
j:Nash equilibria (Neural Network)
k:Dominated move
a:Best response for target price
For further technical information as per how our model work we invite you to visit the article below:
How do AC Investment Research machine learning (predictive) algorithms actually work?
## SEA:TSX Stock Forecast (Buy or Sell) for (n+8 weeks)
Sample Set: Neural Network
Stock/Index: SEA:TSX Seabridge Gold Inc.
Time series to forecast n: 21 Jan 2023 for (n+8 weeks)
According to price forecasts for (n+8 weeks) period, the dominant strategy among neural network is: Hold
X axis: *Likelihood% (The higher the percentage value, the more likely the event will occur.)
Y axis: *Potential Impact% (The higher the percentage value, the more likely the price will deviate.)
Z axis (Grey to Black): *Technical Analysis%
## IFRS Reconciliation Adjustments for Seabridge Gold Inc.
1. The fact that a derivative is in or out of the money when it is designated as a hedging instrument does not in itself mean that a qualitative assessment is inappropriate. It depends on the circumstances whether hedge ineffectiveness arising from that fact could have a magnitude that a qualitative assessment would not adequately capture.
2. In some circumstances an entity does not have reasonable and supportable information that is available without undue cost or effort to measure lifetime expected credit losses on an individual instrument basis. In that case, lifetime expected credit losses shall be recognised on a collective basis that considers comprehensive credit risk information. This comprehensive credit risk information must incorporate not only past due information but also all relevant credit information, including forward-looking macroeconomic information, in order to approximate the result of recognising lifetime expected credit losses when there has been a significant increase in credit risk since initial recognition on an individual instrument level.
3. When measuring a loss allowance for a lease receivable, the cash flows used for determining the expected credit losses should be consistent with the cash flows used in measuring the lease receivable in accordance with IFRS 16 Leases.
4. For the purpose of applying paragraphs B4.1.11(b) and B4.1.12(b), irrespective of the event or circumstance that causes the early termination of the contract, a party may pay or receive reasonable compensation for that early termination. For example, a party may pay or receive reasonable compensation when it chooses to terminate the contract early (or otherwise causes the early termination to occur).
*International Financial Reporting Standards (IFRS) adjustment process involves reviewing the company's financial statements and identifying any differences between the company's current accounting practices and the requirements of the IFRS. If there are any such differences, neural network makes adjustments to financial statements to bring them into compliance with the IFRS.
## Conclusions
Seabridge Gold Inc. is assigned short-term Ba1 & long-term Ba1 estimated rating. Seabridge Gold Inc. prediction model is evaluated with Inductive Learning (ML) and Pearson Correlation1,2,3,4 and it is concluded that the SEA:TSX stock is predictable in the short/long term. According to price forecasts for (n+8 weeks) period, the dominant strategy among neural network is: Hold
### SEA:TSX Seabridge Gold Inc. Financial Analysis*
Rating Short-Term Long-Term Senior
Outlook*Ba1Ba1
Income StatementCCaa2
Balance SheetCBaa2
Leverage RatiosB3Baa2
Cash FlowCCaa2
Rates of Return and ProfitabilityBaa2B1
*Financial analysis is the process of evaluating a company's financial performance and position by neural network. It involves reviewing the company's financial statements, including the balance sheet, income statement, and cash flow statement, as well as other financial reports and documents.
How does neural network examine financial reports and understand financial state of the company?
### Prediction Confidence Score
Trust metric by Neural Network: 83 out of 100 with 800 signals.
## References
1. Keane MP. 2013. Panel data discrete choice models of consumer demand. In The Oxford Handbook of Panel Data, ed. BH Baltagi, pp. 54–102. Oxford, UK: Oxford Univ. Press
2. Candès EJ, Recht B. 2009. Exact matrix completion via convex optimization. Found. Comput. Math. 9:717
3. Çetinkaya, A., Zhang, Y.Z., Hao, Y.M. and Ma, X.Y., How do you know when a stock will go up or down?(STJ Stock Forecast). AC Investment Research Journal, 101(3).
4. F. A. Oliehoek, M. T. J. Spaan, and N. A. Vlassis. Optimal and approximate q-value functions for decentralized pomdps. J. Artif. Intell. Res. (JAIR), 32:289–353, 2008
5. Burkov A. 2019. The Hundred-Page Machine Learning Book. Quebec City, Can.: Andriy Burkov
6. J. Filar, L. Kallenberg, and H. Lee. Variance-penalized Markov decision processes. Mathematics of Opera- tions Research, 14(1):147–161, 1989
7. Challen, D. W. A. J. Hagger (1983), Macroeconomic Systems: Construction, Validation and Applications. New York: St. Martin's Press.
Frequently Asked QuestionsQ: What is the prediction methodology for SEA:TSX stock?
A: SEA:TSX stock prediction methodology: We evaluate the prediction models Inductive Learning (ML) and Pearson Correlation
Q: Is SEA:TSX stock a buy or sell?
A: The dominant strategy among neural network is to Hold SEA:TSX Stock.
Q: Is Seabridge Gold Inc. stock a good investment?
A: The consensus rating for Seabridge Gold Inc. is Hold and is assigned short-term Ba1 & long-term Ba1 estimated rating.
Q: What is the consensus rating of SEA:TSX stock?
A: The consensus rating for SEA:TSX is Hold.
Q: What is the prediction period for SEA:TSX stock?
A: The prediction period for SEA:TSX is (n+8 weeks)
## People also ask
What are the top stocks to invest in right now? | 2023-03-27 10:25:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 2, "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": 0.5337420701980591, "perplexity": 9192.850865140208}, "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-2023-14/segments/1679296948620.60/warc/CC-MAIN-20230327092225-20230327122225-00045.warc.gz"} |
https://repository.asu.edu/items/53949 | ## On K-derived quartics and invariants of local fields
Abstract This dissertation will cover two topics. For the first, let $K$ be a number field. A $K$-derived polynomial $f(x) \in K[x]$ is a polynomial thatfactors into linear factors over $K$, as do all of its derivatives. Such a polynomial is said to be {\it proper} ifits roots are distinct. An unresolved question in the literature is whether or not there exists a proper $\Q$-derived polynomial of degree 4. Some examples are known of proper $K$-derived quartics for a quadratic number field $K$, although other than $\Q(\sqrt{3})$, these fields have quite large discriminant. (The second known field is $\Q(\sqrt{3441})$.) I will describe a search for quadratic fields $K$ over which there exist proper $K$-derived quartics. The search find... (more) 2019 Carrillo, Benjamin (Author) / Jones, John (Advisor) / Bremner, Andrew (Advisor) / Childress, Nancy (Committee member) / Fishel, Susanna (Committee member) / Kaliszewski, Steven (Committee member) / Arizona State University (Publisher) Mathematics Doctoral Dissertation 2610 pages English Doctoral Dissertation Mathematics 2019 Graduate College / ASU Library MODS / OAI Dublin Core / RIS
This content is under embargo until May 01, 2021
Full Text
6.8 MB application/pdf | 2019-06-20 15:55:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6741056442260742, "perplexity": 3808.810319973294}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627999261.43/warc/CC-MAIN-20190620145650-20190620171650-00153.warc.gz"} |
https://physics.stackexchange.com/questions/352495/why-is-the-rate-of-change-of-velocity-taken-using-time-and-not-distance | # Why is the rate of change of velocity taken using time and not distance? [closed]
Why is the change in velocity taken with respect to time, why not distance? I am confused, because it makes sense - as the velocity increases, the distance increases or as the velocity decreases, distance covered is decreased. Please don't use calculus to prove, I'd much rather like an explanation.
## closed as unclear what you're asking by lemon, Jon Custer, honeste_vivere, Wolpertinger, M. EnnsAug 18 '17 at 18:22
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
• "as the velocity increases, the distance increases" - only if the force applied is constant and co-linear with velocity, which in most cases it isn't. – OrangeDog Aug 18 '17 at 13:24
• And the "as the velocity decreases, distance covered is decreased" is not true. When you slow down you are still moving forward and the distance traveled increases as well. – nasu Aug 18 '17 at 13:34
Simply because, when you're considering $F=ma$ with a constant force $F$, then the rate of change is >>constant<< when taken with respect to time, but not with repsect to distance. If you've got another kind of law in mind, then maybe with respect to distance might make more sense. But most people are pretty happy with Newton:)
Time is not necessarily proportional to distance. When an object is accelerating more distance is travelled per unit of time after it has accelerated for a while compared to before it has accelerated.
Measuring acceleration per distance does not make sense -> we want to find out the rate of change in velocity, independent of current velocity (distance travelled is dependent on current velocity).
• I've got a feeling that I understood what you are trying to say here....................but plz also explain why my idea of - the more I cover distance the more my velocity increases; is wrong ? If you know what I mean. – Geeta Aug 18 '17 at 11:45
• It is true in general that the more distance you travel the more your velocity increases, e.g if you start at 0 velocity v^2 = 2aS where v is ending velocity, a is acceleration and S is displacement. However, since velocity is not directly proportional to displacement but only velocity squared, change in velocity per distance is still dependent on initial velocity making for inconvenient calculations. – Copper Aug 18 '17 at 11:51
This is a definition so ultimately it's a matter of convenience and uniformity. Obviously you define velocity as a rate of change over time: $v=\Delta x/\Delta t$ and so it is natural to define acceleration also as a rate of change over time: $$a=\frac{\Delta v}{\Delta t}=\frac{\Delta^2x}{\Delta t^2}$$ This way, all the information about the motion can be recovered from knowledge of a single position vs time graph.
This way rates are measured using the same quantity: the time interval. Finally, if you use $\Delta v/\Delta x$, then your unit of acceleration would be $(m/s)/m$, i.e. your acceleration would have units of "per second" without reference to any other basic quantity.
• But the derivative of $v$ wrt $x$ can also be recovered from the knowledge of a single position vs time graph. And this answer doesn't really explain why having a unit of one per second for acceleration would be a problem. – JiK Aug 18 '17 at 12:47
• $dv/dx = (dv/dt) / (dx/dt)$. If you can find $dv/dt$, you can find $dv/dx$. – JiK Aug 18 '17 at 13:37
• But I thought this answer claims you can get $dv/dt$ "from knowledge of a single position vs time graph"... What am I missing? – JiK Aug 18 '17 at 13:55
• @JiK Maybe I'm the one missing your point. It is easy enough to get $\Delta v/\Delta t$ from $(x(t+\Delta t)-2x(t)+x(t-\Delta t))/(\Delta t)^2$. – ZeroTheHero Aug 18 '17 at 14:02
• Why isn't it then easy enough to get $\Delta v / \Delta x$ just by dividing your formula for $\Delta v/\Delta t$ by your preferred formula for $\Delta x/ \Delta t$? – JiK Aug 19 '17 at 18:16
It is actually done, for example, in velocity vector fields in a fluid.
However, it is not useful outside those subjects. When you talk about velocity, it is usually referred to one single particle, which is placed anywhere. You compute the rate of change of the velocity following the particle. It's understood that it happens along its movement, but either if it's at rest or not, time is the only quantity that unavoidably runs along.
So, basically, if you consider that velocity "belongs" to the particle, it's kind of "useless" to get $\nabla v$. However, if you consider that the particle "gets" the velocity associated to that point (velocity is a vector field in space, like in fluid mechs.) then it is useful.
Velocity is the change in distance with respect to time. Acceleration is the change in velocity with respect to time. By definition, it doesn't make sense to say 'change in velocity with respect to distance.'
For example, if I increase my velocity from 5ms^-1 to 10ms^-1, my acceleration is 2.5ms^-2. If I increase my velocity from 10ms^-1 to 15ms^-1 in 2 seconds, this acceleration is also 2.5ms^-2. However in the latter case, i will travel a greater distance because the velocity is greater.
• This is just repeating the definition. – JiK Aug 18 '17 at 12:49 | 2019-10-16 17:55:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7963010668754578, "perplexity": 385.4955142548314}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1570986669057.0/warc/CC-MAIN-20191016163146-20191016190646-00061.warc.gz"} |